\section{Introduction}

The AdS/CFT correspondence \cite{MGW} has been explored beyond the
supergravity approximation \cite{BMN,GKP}. In the BMN limit \cite{BMN}
for the $\mathcal{N} = 4$ super Yang-Mills (SYM) theory the perturbative
scaling dimensions of gauge invariant near-BPS operators with large
R-charge can be matched with the energies of certain string states, which
has been interpreted as the semiclassical quantization of nearly
point-like string with large angular momentum along the central circle
of $S^5$ \cite{GKP}. Moreover, the energies of various semiclassical
string with several large angular momenta in $AdS_5 \times S^5$ have been
shown in \cite{FRM,FT,BMSZ,AFRT,ART,AS,AT,SS} to match with the
anomalous dimensions of the corresponding gauge invariant non-BPS
operators which can be computed by using the Bethe ansatz \cite{MZ}
for diagonalization of the dilatation operator
\cite{BKS,NB,NBS,BDS,NBP}, that is represented by
a Hamiltonian of an integrable spin chain.
From the view point of integrability the gauge/string
duality has been further confirmed by verifying the equivalence between
the classical string Bethe equation for the classical $AdS_5 \times S^5$
string sigma model and the Bethe equation for the spin chain in
the various sectors such as SU(2), SL(2), SO(6) and so \cite{KMMZ,KZ},
while it has been demonstrated at the level of effective action where
an interpolating spin chain sigma model action describing the continuum
limit of the spin chain in the coherent basis was constructed
\cite{MK,KRT,AAT}.

In \cite{LM} Lunin and Maldacena have found a supergravity background
dual to the Leigh-Strassler or $\beta$-deformation of $\mathcal{N} = 4$
SYM theory \cite{LS} by applying a sequence of T-duality transformations
and shifts of angular coordinates to the original $AdS_5 \times S^5$
background. They have taken the plane-wave limit of the deformed
$AdS_5 \times \tilde{S}^5$ background with
a deformed five-sphere $\tilde{S}^5$
and have shown that the string spectrum in the pp-wave coincides with
the spectrum of BMN-type operators in the $\beta$-deformed
$\mathcal{N} = 4$ SYM \cite{NP}.

The Lax representation for the bosonic string theory in the real
$\beta$-deformed background has been constructed \cite{SF}, which is
similar to the undeformed case \cite{AF}. The gauge/string duality
for the $\beta$-deformed case has been
investigated by comparing the energies
of semiclassical strings to the anomalous dimensions of the gauge theory
operators in the two-scalar sector \cite{FRT}. The energy of circular
string with two equal angular momenta in $\tilde{S}^5$ has been computed
from the semiclassical approach and has been also derived from the
anisotropic Landau-Lifshitz equation for the interpolating string sigma
model action which was obtained by taking
the ``fast-string" limit of the world
sheet action in $AdS_5 \times \tilde{S}^5$. This interpolating action on
the string theory side has been shown to coincide with the continuum limit
of the coherent state action of an anisotropic XXZ spin chain on the
$\beta$-deformed $\mathcal{N} = 4$ SYM theory side,
which was constructed using the
one-loop dilatation operator in the deformed gauge theory \cite{RR,BC}.
The classical string Bethe equation on the string theory side has been
derived from the Lax representation of \cite{SF} to coincide with the
thermodynamic limit of the Bethe equation for the anisotropic spin chain.
Various relevant aspects of the gauge/string duality for the
marginal deformed backgrounds have been investigated
\cite{KMSS,BDR,BR,FG,GN}.

Multi-spin configurations of strings which move in both $AdS_5$
and $\tilde{S}^5$ parts of the $\beta$-deformed background have been
constructed \cite{BDR}, where the winding numbers and the frequencies
associated with the angular momenta take the same magnitudes
respectively for each part of $AdS_5 \times \tilde{S}^5$.
This property was also seen in the circular string solution with
two equal spins \cite{FRT} which is specified by
 the same frequencies and winding numbers.
We will construct a circular string solution with two unequal spins in
$\tilde{S}^5$ which is characterized by different frequencies and
winding numbers. The energy of the string solution will be computed
and represented in terms of the unequal winding numbers, the unequal
angular momenta and the deformation parameter.
This energy spectrum on the string theory side will be compared with
that of the solution in \cite{FRT} for the anisotropic Landau-Lifshitz
equation and the $\beta$-deformed Bethe equation for the spin
configuration with the filling fraction away from one half.

\section{Rotating string solution with two unequal spins}

We consider a rotating closed string motion in the supergravity
background dual to the real $\beta$-deformation of the $\mathcal{N}=4$
SYM theory. The $\beta$-deformed background for real $\beta = \gamma$
is given by \cite{LM}
\begin{eqnarray}
ds_{str}^2 &=& R^2 \left[ ds_{AdS_5}^2 + \sum_{i=1}^{3}(d\rho_i^2 +
G\rho_i^2 d\phi_i^2 ) +  \tilde{\gamma}^2G \rho_1^2 \rho_2^2 \rho_3^2(\sum_{i=1}^{3}
d\phi_i)^2 \right], \nonumber \\
B_2 &=& R^2 \tilde{\gamma} G w_2, \hspace{1cm} w_2 \equiv
\rho_1^2\rho_2^2d\phi_1d\phi_2   + \rho_2^2\rho_3^2d\phi_2d\phi_3 +
\rho_3^2\rho_1^2d\phi_3d\phi_1,   \\
G^{-1} &=& 1 + \tilde{\gamma}^2 Q, \hspace{1cm} Q \equiv \rho_1^2\rho_2^2  +
\rho_2^2\rho_3^2 + \rho_3^2\rho_1^2,  \hspace{1cm} \sum_{i=1}^{3}\rho_i^2
 = 1, \nonumber
\end{eqnarray}
where
\begin{equation}
(\rho_1, \rho_2, \rho_3) = (\sin\alpha \cos \theta, \sin \alpha
\sin\theta, \cos\alpha), \hspace{1cm} R^4 = Ng_{YM}^2 = \lambda
\end{equation}
and the regular
deformation parameter $\tilde{\gamma}$ of the supergravity background is related with
the real deformation parameter $\gamma$ of the deformed $\mathcal{N}=4$
SYM as $\tilde{\gamma} = R^2 \gamma$. We concentrate on a configuration that a closed
string is staying at the center of $AdS_5$ and moving in the
$\tilde{S}^3$ part of the deformed five-sphere defined by
\begin{equation}
\alpha = \frac{\pi}{2}, \hspace{1cm} \mathrm{i.e.}  \hspace{1cm}
\rho_1 = \cos\theta, \; \rho_2 = \sin\theta, \; \rho_3 = 0.
\end{equation}
The relevant bosonic string action takes the form
\begin{eqnarray}
S &=& -\frac{1}{2} R^2\int d\tau \int \frac{d\sigma}{2\pi} \Bigl[
\gamma^{\alpha\beta}(-\partial_{\alpha}t \partial_{\beta}t + \partial_{\alpha}\theta
\partial_{\beta}\theta + G\cos^2\theta \partial_{\alpha}\phi_1 \partial_{\beta}\phi_1
+ G\sin^2\theta \partial_{\alpha}\phi_2 \partial_{\beta}\phi_2 ) \nonumber \\
 & & - 2\epsilon^{\alpha\beta} \tilde{\gamma} G \sin^2\theta \cos^2\theta
\partial_{\alpha}\phi_1 \partial_{\beta}\phi_2 \Bigr],
\end{eqnarray}
where $\epsilon^{01}=1, \gamma^{\alpha\beta}$ is expressed as
$\gamma^{\alpha\beta}=\sqrt{-h} h^{\alpha\beta}$ in terms of a world-sheet
metric $h^{\alpha\beta}$, and
\begin{equation}
G = \frac{1}{1 + \frac{\tilde{\gamma}^2}{4}\sin^22\theta}.
\end{equation}

We choose the conformal gauge $\gamma^{\alpha\beta}= \mathrm{diag}(-1,1)$
and make the following ansatz describing a closed string rotating and
wound in the $\phi_1$ and $\phi_2$ directions
\begin{equation}
t = \kappa\tau, \hspace{1cm} \phi_1 = \omega_1 \tau + m_1\sigma,
\hspace{1cm} \phi_2 = \omega_2 \tau + m_2\sigma, \hspace{1cm}
\theta = \theta_0 = \mathrm{const},
\end{equation}
where $m_1, m_2$ are the winding numbers. The string equation of motion
for $\theta$ is satisfied when the constant $\theta_0$ is specified by
\begin{eqnarray}
\left[\omega_1^2 - m_1^2 - (\omega_2^2 - m_2^2)\right] \left( 1 +
\frac{\tilde{\gamma}^2}{4}\sin^22\theta_0 \right) -2\tilde{\gamma}
(\omega_1 m_2 - \omega_2m_1)\cos 2\theta_0 \nonumber \\
+ \tilde{\gamma}^2 \cos2\theta_0 \left[\cos^2\theta_0(\omega_1^2 - m_1^2)
+ \sin^2\theta_0(\omega_2^2 - m_2^2) \right] = 0.
\label{thc}\end{eqnarray}
It has a simple string solution with two equal angular momenta, which is
described by $\theta_0 = \pi/4, \; \omega_1^2 - m_1^2 = \omega_2^2 -
m_2^2$ in ref. \cite{FRT}. Here we look for an extended solution with
two unequal angular momenta which reduces to the simple solution in
a special parameter limit. The equation (\ref{thc}) is rewritten in terms
of $x = \cos2\theta_0$ as
\begin{equation}
\frac{\tilde{\gamma}^2}{4}(\Omega_1 - \Omega_2)x^2 + \tilde{\gamma}\left( \frac{\Omega_1 +
\Omega_2}{2} \tilde{\gamma} - 2\Omega_0  \right) x +
\left(1 + \frac{\tilde{\gamma}^2}{4} \right)(\Omega_1 - \Omega_2) = 0,
\label{omx}\end{equation}
where $\Omega_i \equiv \omega_i^2 - m_i^2 \; (i=1, 2), \;
\Omega_0 \equiv \omega_1m_2 - \omega_2m_1$. This equation
determines $x$ in terms of $\omega_i, \; m_i(i=1,2)$ and $\tilde{\gamma}$.
The angular momenta $\mathcal{J}_1 = J_1/\sqrt{\lambda}$ and $\mathcal{J}_2 = J_2/\sqrt{\lambda}$
coming from the rotations in the $\phi_1$ and $\phi_2$ directions
are obtained by
\begin{eqnarray}
\mathcal{J}_1 &=& \frac{1}{1 + \frac{\tilde{\gamma}^2}{4}(1-x^2)}\left[ \frac{1+x}{2}\omega_1
+ \frac{\tilde{\gamma}}{4}( 1 - x^2 )m_2 \right], \label{tao} \\
\mathcal{J}_2 &=& \frac{1}{1 + \frac{\tilde{\gamma}^2}{4}(1-x^2)}\left[ \frac{1-x}{2}\omega_2
- \frac{\tilde{\gamma}}{4}( 1 - x^2 )m_1 \right].
\label{tat}\end{eqnarray}
From them the frequencies $\omega_1$ and $\omega_2$ are expressed as
\begin{eqnarray}
\omega_1 &=& \frac{2}{1+x}\left[ \left( 1 + \frac{\tilde{\gamma}^2}{4}(1-x^2) \right)
\mathcal{J}_1 - \frac{\tilde{\gamma}}{4}( 1 - x^2 )m_2 \right],  \label{omo} \\
\omega_2 &=& \frac{2}{1-x}\left[ \left( 1 + \frac{\tilde{\gamma}^2}{4}(1-x^2) \right)
\mathcal{J}_2 + \frac{\tilde{\gamma}}{4}( 1 - x^2 )m_1 \right].
\label{omt}\end{eqnarray}

The conformal gauge constraints imply
\begin{eqnarray}
\kappa^2 = \frac{1}{1 + \frac{\tilde{\gamma}^2}{4}(1-x^2)} \left[ \frac{1 + x}{2}
(\omega_1^2 + m_1^2 )  + \frac{1 - x}{2}(\omega_2^2 + m_2^2 )
\right], \label{kax} \\
\frac{1 + x}{2}\omega_1m_1 + \frac{1 - x}{2}\omega_2m_2 = 0.
\label{omm}\end{eqnarray}
The substitution of $\omega_1$ and $\omega_2$ in (\ref{omo}) and
(\ref{omt}) into the eq. (\ref{omm}) leads to a compact expression
\begin{equation}
\mathcal{J}_1 m_1 + \mathcal{J}_2 m_2 = 0.
\label{tam}\end{equation}
It is noted that this expression for the $\gamma$-deformed background
takes the same form as that for the undeformed background.
When $\theta_0 = \pi/4$, that is, $x = 0$, the eqs. (\ref{tao}),
(\ref{tat}) and (\ref{omm}) provide $\mathcal{J}_1 = \mathcal{J}_2 = \mathcal{J}/2, \; \omega_1
= \omega_2 = \mathcal{J} + \tilde{\gamma}(m + \tilde{\gamma}\mathcal{J}/2)/2$ with $m_1 = -m_2 \equiv m$
and the total spin $\mathcal{J} = \mathcal{J}_1 + \mathcal{J}_2$, which
is the special string solution with two equal angular momenta \cite{FRT}.
From (\ref{kax}) the energy of circular string solution is specified by
\begin{equation}
E^2 = \frac{\lambda}{1 + \frac{\tilde{\gamma}^2}{4}(1-x^2)} \left[ \frac{1 + x}{2}
(\omega_1^2 + m_1^2 )  + \frac{1 - x}{2}(\omega_2^2 + m_2^2 ) \right].
\label{exo}\end{equation}
The quadratic equation (\ref{omx}) yields a solution expressed in terms of
$\omega_1, \omega_2$, which is further inserted into (\ref{tao}),
(\ref{tat}). If we can determine  $\omega_1, \omega_2$ as functions of
$\mathcal{J}_1, \mathcal{J}_2$ from the two inserted equations, we substitute these
functions into (\ref{exo}) to obtain the energy expressed in terms of
$\mathcal{J}_1, \mathcal{J}_2$. However, it is impossible to
derive the functions so that we take
the following alternative procedure. First combining (\ref{omo}) and
(\ref{omt}) with (\ref{exo}) we have the energy expression
\begin{eqnarray}
E^2 &=& \lambda \Biggl[ 2\left( 1 + \frac{\tilde{\gamma}^2}{4}(1-x^2) \right)
\left( \frac{\mathcal{J}_1^2}{1+x} + \frac{\mathcal{J}_2^2}{1-x} \right) + \tilde{\gamma}( - m_2\mathcal{J}_1
+ m_1\mathcal{J}_2 ) \nonumber \\
&+& \tilde{\gamma}( m_2\mathcal{J}_1 + m_1\mathcal{J}_2 )x + \frac{1 + x}{2}m_1^2 +
\frac{1 - x}{2}m_2^2 \Biggr].
\label{ext}\end{eqnarray}
When $\omega_1$ and $\omega_2$ in (\ref{omo}) and (\ref{omt}) are directly
substituted into (\ref{omx}) we have an involved equation for $x$.
If a solution $x$ of the transcendental equation is obtained as a function
of $\mathcal{J}_i, m_i (i=1,2), \tilde{\gamma}$ and inserted into (\ref{ext}), then
the energy of string solution is expressed in terms of the angular
momenta, the winding numbers and the deformation parameter.

\section{Energy-spin relation}

In order to solve the transcendental equation
we consider the parameter region specified by $x \ll 1$ and take the
expansion around $x = 0$. This is the case of almost equal spins, i.e.
$\Delta J \ll J, \; \Delta J \equiv J_1 - J_2$. We use (\ref{omo})
and (\ref{omt}) to expand $\Omega_1 - \Omega_2$ in (\ref{omx})
in powers of $x$ as
\begin{equation}
\Omega_1 - \Omega_2 = A_0 + A_1x + A_2x^2 + A_3x^3 + \cdots ,
\label{oma}\end{equation}
where
\begin{eqnarray}
&A_0 = 4\left(1 + \frac{\tilde{\gamma}^2}{4}\right) \left[ \left(1 +
\frac{\tilde{\gamma}^2}{4}\right)(\mathcal{J}_1^2 - \mathcal{J}_2^2) - \frac{\tilde{\gamma}}{2}
(m_2\mathcal{J}_1 + m_1\mathcal{J}_2) - \frac{1}{4} (m_1^2 - m_2^2) \right],& \nonumber \\
&A_1 = -8 \left[\left(1 + \frac{\tilde{\gamma}^2}{4}\right)^2(\mathcal{J}_1^2 + \mathcal{J}_2^2)
+ \frac{\tilde{\gamma}}{2}\left(1 + \frac{\tilde{\gamma}^2}{4}\right)( - m_2\mathcal{J}_1
+ m_1\mathcal{J}_2 ) + \frac{\tilde{\gamma}^2}{16} (m_1^2 + m_2^2) \right],& \nonumber \\
&A_2 = 12\left(1 + \frac{\tilde{\gamma}^2}{4}\right)\left(1 + \frac{\tilde{\gamma}^2}{12}
\right) (\mathcal{J}_1^2 - \mathcal{J}_2^2) - 4\tilde{\gamma}\left(1 + \frac{\tilde{\gamma}^2}{8}\right)
(m_2\mathcal{J}_1 + m_1\mathcal{J}_2) - \frac{\tilde{\gamma}^2}{4} (m_1^2 - m_2^2),&  \nonumber \\
&A_3 = -16 \left[\left(1 + \frac{\tilde{\gamma}^2}{4}\right)(\mathcal{J}_1^2 + \mathcal{J}_2^2)
+ \frac{\tilde{\gamma}}{4}( - m_2\mathcal{J}_1 + m_1\mathcal{J}_2 ) \right]. &
\end{eqnarray}
Similarly, $(\Omega_1 + \Omega_2)\tilde{\gamma}/2 - 2\Omega_0$ in (\ref{omx})
can be expanded as
\begin{equation}
\frac{\Omega_1 + \Omega_2}{2} \tilde{\gamma} - 2\Omega_0  = B_0 + B_1x + B_2x^2 +
\cdots, \nonumber
\label{omb}\end{equation}
where
\begin{eqnarray}
&B_0 = 2\left(1 + \frac{\tilde{\gamma}^2}{4}\right)^2 \left[\tilde{\gamma}(\mathcal{J}_1^2 + \mathcal{J}_2^2)
+ 2( - m_2\mathcal{J}_1 + m_1\mathcal{J}_2 ) + \frac{\tilde{\gamma}}{4} \frac{m_1^2 + m_2^2}
{1 + \frac{\tilde{\gamma}^2}{4}} \right], &  \nonumber \\
&B_1 = -4\left(1 + \frac{\tilde{\gamma}^2}{4}\right) \left[ \tilde{\gamma}\left(1 +
\frac{\tilde{\gamma}^2}{4}\right)(\mathcal{J}_1^2 - \mathcal{J}_2^2) - \left( 1 + \frac{\tilde{\gamma}^2}{2}
\right)(m_2\mathcal{J}_1 + m_1\mathcal{J}_2) - \frac{\tilde{\gamma}}{4}
(m_1^2 - m_2^2) \right],&  \nonumber \\
&B_2 = 6\left(1 + \frac{\tilde{\gamma}^2}{4}\right) \left[\tilde{\gamma}\left(1 +
\frac{\tilde{\gamma}^2}{12}\right)(\mathcal{J}_1^2 + \mathcal{J}_2^2) + \frac{2}{3}\left(1 +
\frac{\tilde{\gamma}^2}{4}\right) ( - m_2\mathcal{J}_1 + m_1\mathcal{J}_2 )
  + \frac{\tilde{\gamma}^3}{48}
\frac{m_1^2 + m_2^2}{1 + \frac{\tilde{\gamma}^2}{4}} \right]. &
\end{eqnarray}
It is noted that the coefficients $A_k, B_k \;(k=0,1,2,\cdots)$ show the
alternate expressions for $k=$ even and $k=$ odd.
Hence by combining (\ref{oma}) and (\ref{omb}) with (\ref{omx}) and
taking account of the behaviors that $A_0$ is of order $\epsilon$, while
$A_1$ and $B_0$ are of order $\epsilon^0$, we
make the leading order estimation of $x$ as
\begin{equation}
x_1 = - \frac{\left(1 + \frac{\tilde{\gamma}^2}{4}\right)A_0}{\tilde{\gamma} B_0
+ \left(1 + \frac{\tilde{\gamma}^2}{4}\right)A_1 },
\end{equation}
which is rewritten by
\begin{equation}
x_1 = \frac{1 + \frac{\tilde{\gamma}^2}{4}}{2(\mathcal{J}_1^2 + \mathcal{J}_2^2)}
\left[ \mathcal{J}_1^2 - \mathcal{J}_2^2 - \frac{\tilde{\gamma}}{2} \frac{m_2\mathcal{J}_1 + m_1\mathcal{J}_2}
{1 + \frac{\tilde{\gamma}^2}{4}} - \frac{m_1^2 - m_2^2}{4\left(1 +
\frac{\tilde{\gamma}^2}{4}\right) } \right].
\label{xmn}\end{equation}
Using the Virasoro constraint (\ref{tam}) and
a parameter $\alpha = \mathcal{J}_1/\mathcal{J}$ with the total
spin $\mathcal{J}$ we illustrate that $x_1$ is the term of order
$\epsilon = 2\alpha - 1$
\begin{equation}
x_1 = \frac{2\alpha - 1}{1 + (2\alpha - 1)^2} \left[ 1 + \frac{\tilde{\gamma}^2}{4}
+ \frac{\tilde{\gamma}}{2} \frac{m_1}{\mathcal{J}_2} + \frac{1}{4}\left( \frac{m_1}{\mathcal{J}_2}
\right)^2 \right].
\end{equation}

It is possible to estimate the non-leading term $x_2$ by substituting
$x = x_1 + x_2$ into the eq. (\ref{omx}) accompanied with (\ref{oma})
and (\ref{omb}). By considering the behaviors that $A_0, A_2$ and
$B_1$ are of order $\epsilon$, while $A_1, A_3, B_0$ and $B_2$
are of order $\epsilon^0$, we obtain
\begin{eqnarray}
x_2 = -\frac{1}{\tilde{\gamma} B_0 + \left(1 + \frac{\tilde{\gamma}^2}{4}\right)A_1 }
\Biggl[ x_1^2 \left(\frac{\tilde{\gamma}^2}{4}A_0 + \tilde{\gamma} B_1 + \left(1 +
\frac{\tilde{\gamma}^2}{4}\right)A_2 \right) \nonumber \\
+ x_1^3\left( \frac{\tilde{\gamma}^2}{4}A_1 + \tilde{\gamma} B_2 + \left(1 +
\frac{\tilde{\gamma}^2}{4}\right)A_3 \right) \Biggr],
\end{eqnarray}
which is the term of order $\epsilon^3$ as shown by
\begin{eqnarray}
x_2 &=& \frac{x_1^2}{\mathcal{J}_1^2 + \mathcal{J}_2^2} \left[ \frac{3}{2}
\left(1-\frac{\tilde{\gamma}^2}{6}\right)(\mathcal{J}_1^2 - \mathcal{J}_2^2) + \frac{\tilde{\gamma}^3}{8}
\frac{m_2\mathcal{J}_1 + m_1\mathcal{J}_2}{1 + \frac{\tilde{\gamma}^2}{4}} + \frac{\tilde{\gamma}^2}{16}
\frac{m_1^2 - m_2^2}{1 + \frac{\tilde{\gamma}^2}{4}} \right] \nonumber \\
 &-& \frac{2x_1^3}{1 + \frac{\tilde{\gamma}^2}{4}}.
\label{xtw}\end{eqnarray}
The substitution of $x = x_1 + x_2 + x_3$ where $x_3$ is the term
of order $\epsilon^5$ into the energy expression (\ref{ext})
yields the following expansion up to order $\epsilon^6$
\begin{eqnarray}
E^2 &=& \lambda \Biggl[ 2(\mathcal{J}_1^2 + \mathcal{J}_2^2) + \frac{1}{2}(m_2 - \tilde{\gamma}\mathcal{J}_1)^2
+ \frac{1}{2}(m_1 + \tilde{\gamma}\mathcal{J}_2)^2 \nonumber \\
&+& (x_1 + x_2 + x_3)\left( -2\left(1 + \frac{\tilde{\gamma}^2}{4}\right)
(\mathcal{J}_1^2 - \mathcal{J}_2^2) + \tilde{\gamma}(m_2\mathcal{J}_1 + m_1\mathcal{J}_2) +
\frac{m_1^2 - m_2^2}{2} \right) \nonumber \\
&+& 2(\mathcal{J}_1^2 + \mathcal{J}_2^2)(x_1^2 + 2x_1x_2 + 2x_1x_3 + x_2^2 + x_1^4 +
4x_1^3x_2 + x_1^6 ) \\
&-& 2(\mathcal{J}_1^2 - \mathcal{J}_2^2)( x_1^3 + 3 x_1^2x_2 + x_1^5 ) + \cdots \Biggr].
\nonumber
\end{eqnarray}
Since the fourth term can be expressed as $(x_1 + x_2 + x_3)
(-4(\mathcal{J}_1^2 + \mathcal{J}_2^2)x_1)$, we have
\begin{eqnarray}
E^2 &=& P + 2(J_1^2 + J_2^2)(x_1^4 + x_2^2 + 4x_1^3x_2 + x_1^6)
-2(J_1^2 - J_2^2)(x_1^3 + 3x_1^2x_2 + x_1^5 ) + \cdots, \\
P &\equiv& 2(J_1^2 + J_2^2)( 1- x_1^2 ) + \frac{\lambda}{2}
(m_2 - \gamma J_1)^2 + \frac{\lambda}{2}(m_1 + \gamma J_2)^2, \nonumber
\end{eqnarray}
where the $x_1x_3$ term has been canceled out. The expression $x_1$
in (\ref{xmn}) is rewritten by
\begin{equation}
x_1 = \frac{1}{1 + (2\alpha -1)^2} \left[ 2\alpha -1 + \frac{\tilde{\lambda}}{4}
(m_2 - \gamma J_1)^2 - \frac{\tilde{\lambda}}{4}(m_1 + \gamma J_2)^2 \right]
\equiv \frac{\bar{x}_1}{1+(2\alpha-1)^2},
\label{xon}\end{equation}
where $\tilde{\lambda}$ is the effective coupling constant $\tilde{\lambda} = \lambda/J^2 =
1/\mathcal{J}^2$, so that $P$ takes the form
\begin{eqnarray}
P &=& P_0 - \frac{\tilde{\lambda}^2J^2}{16}\left[(m_2 - \gamma J_1)^2 -
(m_1 + \gamma J_2)^2\right]^2 + J^2(2\alpha -1)^2(1 + (2\alpha -1)^2)
x_1^2, \\
P_0 &\equiv& J^2\left[ 1 - \frac{\tilde{\lambda}(2\alpha -1)}{2} \left(
(m_2 - \gamma J_1)^2 - (m_1 + \gamma J_2)^2 \right) \right]
+ \frac{\lambda}{2}(m_2 - \gamma J_1)^2 + \frac{\lambda}{2}
(m_1 + \gamma J_2)^2. \nonumber
\end{eqnarray}
The leading part $P_0$ which is of order $\epsilon^0$ is rearranged
into
\begin{equation}
P_0 = J^2\left[ 1 + \frac{J_1J_2}{J^2} \tilde{\lambda}
(\gamma J + m_1 - m_2)^2 + \frac{\tilde{\lambda}}{J^2}(m_1J_1 + m_2J_2 )^2 \right],
\end{equation}
which is expressed through the Virasoro constraint (\ref{tam}) as
\begin{equation}
P_0 = J^2 + J^2\alpha( 1 - \alpha )\tilde{\lambda}(\gamma J + m_1 - m_2)^2.
\end{equation}
Gathering together with the expression $x_2$ in (\ref{xtw})
described by
\begin{eqnarray}
x_2 &=& \frac{x_1^2}{1 + (2\alpha -1)^2} \bar{x}_2,  \\
\bar{x}_2 &\equiv& \left(1-\frac{\tilde{\gamma}^2}{2}
\right)(2\alpha -1) + \tilde{\lambda} \gamma ( m_2J_1 + m_1J_2 )  +
\frac{\tilde{\lambda}}{2}(m_1^2 - m_2^2) \nonumber
\end{eqnarray}
we get the following energy expression
\begin{eqnarray}
E^2 &=& P_0 - \frac{\tilde{\lambda}^2J^2}{16}\left[(m_2 - \gamma J_1)^2 -
(m_1 + \gamma J_2)^2\right]^2  \\
&+& \frac{\tilde{\lambda}^2J^2}{16}\frac{x_1^2}{1 + (2\alpha -1)^2}\left[(m_2 -
\gamma J_1)^2 - (m_1 + \gamma J_2)^2\right]^2 +
2J^2\bar{x}_1^2(2\alpha -1)^4
+ P_6 + \cdots, \nonumber
\end{eqnarray}
where
\begin{eqnarray}
P_6 &\equiv& J^2\bar{x}_1^4 \Biggl[ -2(2\alpha -1)\bar{x}_1
+ (2\alpha-1)\tilde{\lambda}(\gamma J)^2\bar{x}_2 + \bar{x}_1^2
+ \frac{\tilde{\gamma}^4}{4}(2\alpha -1)^2 \nonumber \\
&-& \left( 2\alpha -1 + \tilde{\lambda} \gamma ( m_2J_1 + m_1J_2 ) +
\frac{\tilde{\lambda}}{2}(m_1^2 - m_2^2) \right)^2 \Biggr].
\end{eqnarray}
The second term is of order $\epsilon^2$ and the third term is
of order $\epsilon^4$ for the leading part, while the fourth term and
$P_6$ are of order $\epsilon^6$.

Now we use the resulting expression up to order $\epsilon^4$
as well as up to order $\tilde{\lambda}^2$
\begin{equation}
E^2 = P_0 -  \frac{\tilde{\lambda}^2J^2}{16}\left[(m_2 - \gamma J_1)^2 -
(m_1 + \gamma J_2)^2\right]^2( 1 - (2\alpha -1)^2) + \cdots
\end{equation}
to extract the energy of the solution as
\begin{eqnarray}
E &=& J + \frac{\lambda}{2J}\alpha(1-\alpha)(\gamma J + m_1 - m_2)^2
\label{eja}   \\
&-& \frac{\lambda^2}{8J^3}\alpha(1-\alpha)\left[ \left((m_2 - \gamma J_1)^2 -
(m_1 + \gamma J_2)^2 \right)^2 + \alpha(1-\alpha)
(\gamma J + m_1 - m_2)^4\right] + \cdots. \nonumber
\end{eqnarray}
The second term of order $\tilde{\lambda}$, that is, the ``one-loop" energy
correction agrees with the one-loop result of ref. \cite{FRT}, where the
$\beta$-deformed Landau-Lifshitz action produced from
``fast-string" expansion of the string sigma model action is shown to
have a rational solution with two unequal momenta and the one-loop
anomalous dimension is computed also by solving the Bethe equation for
the corresponding anisotropic spin chain.

The undeformed limit $\gamma \rightarrow 0$ for (\ref{eja}) yields
\begin{equation}
E = J + \frac{\lambda}{2J}m(n-m) - \frac{\lambda^2}{8J^3}m(n-m)(n^2 -3nm + 3m^2)
+ \cdots,
\label{emn}\end{equation}
where the relations provided from (\ref{tam}) for the undeformed case
\begin{equation}
\frac{J_1}{J} = -\frac{m_2}{m_1 - m_2}, \hspace{1cm}  \frac{J_2}{J} =
\frac{m_1}{m_1 - m_2}
\label{jra}\end{equation}
have been used and the winding numbers $m_1, m_2$ have been replaced
by $m_1 = n - m, \; m_2 = -m$. The reduced expression (\ref{emn})
including the ``one-loop" and ``two-loop" corrections was presented
\cite{KMMZ} by analyzing the classical Bethe
equation for the classical $AdS_5 \times
S^5$ string sigma model as well as the Bethe equation for the spin
chain in the SU(2) sector. On the other hand in ref. \cite{ART}
a general class of rotating string solutions in the undeformed
background was derived and the ``one-loop" correction in (\ref{emn})
was presented by solving  the relations for the two spin sector
\begin{equation}
\mathcal{E}^2 = 2\sum_{i=1}^2\omega_i\mathcal{J}_i - \nu^2, \hspace{1cm}
\sum_{i=1}^2\frac{\mathcal{J}_i}{\omega_i} = 1, \hspace{1cm}
\sum_{i=1}^2m_i\mathcal{J}_i = 0,
\end{equation}
where $\omega_i^2 - m_i^2 = \nu^2\; (i=1,2)$ and the last equation indeed
takes the same form as (\ref{tam}). Here we demonstrate that these
relations also yield the ``two-loop" correction. Indeed the
following large $\mathcal{J}$ expansion
\begin{equation}
\mathcal{E}^2 = \mathcal{J}^2 + \sum_{i=1}^2m_i^2\frac{\mathcal{J}_i}{\mathcal{J}} +
\frac{1}{4\mathcal{J}^2}\left[ \left(\sum_{i=1}^2m_i^2\frac{\mathcal{J}_i}{\mathcal{J}}\right)^2
- \sum_{i=1}^2m_i^4\frac{\mathcal{J}_i}{\mathcal{J}} \right] + \cdots
\end{equation}
leads to
\begin{equation}
E = \sqrt{\lambda}\mathcal{E} = J + \frac{\lambda}{2J}\sum_{i=1}^2m_i^2
\frac{J_i}{J} - \frac{\lambda^2}{8J^3}\sum_{i=1}^2m_i^4\frac{J_i}{J}
+ \cdots.
\label{emj}\end{equation}
The substitution of $m_1 = n - m, \; m_2= -m$ and (\ref{jra}) into
(\ref{emj}) reproduces (\ref{emn}).

Following the argument of \cite{FRT}, the structure of the energy of
a multi-spin solution can be captured as double expansions in $\tilde{\lambda}$
and $\tilde{\gamma}$. The smoothness of the deformation gives
\begin{equation}
E = \sqrt{\lambda}\mathcal{E}(\tilde{\gamma},\mathcal{J}) = \sqrt{\lambda}\left[ \mathcal{E}_0(\mathcal{J}) +
\tilde{\gamma} f_1(\mathcal{J}) + \tilde{\gamma}^2 f_2(\mathcal{J}) + \tilde{\gamma}^3 f_3(\mathcal{J}) + \tilde{\gamma}^4 f_4(\mathcal{J})
+ \cdots \right],
\label{ex}\end{equation}
while the energy for the undeformed case has the usual regular
large $\mathcal{J}$ or small $\tilde{\lambda}$ expansion
\begin{equation}
E_0 = \sqrt{\lambda}\mathcal{E}_0(\mathcal{J}) = Jf_0(\tilde{\lambda}) =
J(1 + c_1\tilde{\lambda} + c_2\tilde{\lambda}^2 + \cdots).
\end{equation}
Through $\tilde{\gamma} = \sqrt{\tilde{\lambda}}\gamma J$ the expansion (\ref{ex})
turns out to be
\begin{equation}
E = J\left[ f_0(\tilde{\lambda}) + \tilde{\lambda} \gamma Jf_1(\mathcal{J}) + \tilde{\lambda} (\gamma J)^2
\frac{f_2(\mathcal{J})}{\mathcal{J}} + \tilde{\lambda}^2 (\gamma J)^3f_3(\mathcal{J}) +
\tilde{\lambda}^2 (\gamma J)^4\frac{f_4(\mathcal{J})}{\mathcal{J}} + \cdots \right].
\end{equation}
If $f_1(\mathcal{J}), f_2(\mathcal{J})/\mathcal{J}, f_3(\mathcal{J}), f_4(\mathcal{J})/\mathcal{J}$ have the
regular expansions in $1/\mathcal{J}^2 = \tilde{\lambda}$ as
\begin{equation}
f_1 = \sum_{k=0}^{\infty}c_k^1\tilde{\lambda}^k, \hspace{1cm} \frac{f_2}{\mathcal{J}} =
\sum_{k=0}^{\infty}c_k^2\tilde{\lambda}^k, \hspace{1cm} f_3 = \sum_{k=0}^{\infty}
c_k^3\tilde{\lambda}^k, \hspace{1cm} \frac{f_4}{\mathcal{J}} =
\sum_{k=0}^{\infty}c_k^4\tilde{\lambda}^k,
\end{equation}
then $E$ also takes the following regular expansion
\begin{eqnarray}
E &=& J\Bigl[ 1 + \tilde{\lambda}\left(c_1 + c_0^1\gamma J + c_0^2(\gamma J)^2
\right)  \nonumber \\
 &+& \tilde{\lambda}^2 \left( c_2 + c_1^1\gamma J + c_1^2(\gamma J)^2 +
c_0^3(\gamma J)^3 + c_0^4(\gamma J)^4 \right) + \cdots \Bigr].
\end{eqnarray}
Hence we can see that the ``two-loop" part of order $\tilde{\lambda}^2$ contains
the terms $(\gamma J)^n$ with $n \le 4$. This behavior is indeed
seen in our explicit ``two-loop" result
\begin{eqnarray}
E_2 &=&  -\frac{\lambda^2}{8J^3}\alpha(1-\alpha)\Bigl[ \left((1-2\alpha)\gamma
J  + m_1 + m_2\right)^2(\gamma J + m_1 - m_2)^2 \nonumber \\
&+& \alpha(1-\alpha)(\gamma J + m_1 - m_2)^4\Bigr].
\end{eqnarray}

\section{Conclusion}

Analyzing the closed string motion in the $\gamma$-deformed
$AdS_5 \times \tilde{S}^5$ background by the semiclassical
approach, we have presented a transcendental equation which determines
the energy of a solution describing a rotating and wound string
with two unequal spins in $\tilde{S}^5$. By using an expansion
procedure for the case of almost equal spins we have solved the
transcendental equation to extract the string energy in terms of the
angular momenta, the winding numbers and the deformation parameter.

The ``one-loop" part of the string energy expanded in $\lambda/J^2$
has reproduced the one-loop anomalous dimension of long two-scalar
operator which was found in ref. \cite{FRT} as a solution for the
anisotropic Landau-Lifshitz and Bethe equations for the
$\gamma$-deformed $\mathcal{N}=4$ SYM
for the $SU(2)_{\gamma}$ sector. This agreement
at the one-loop level confirms the gauge/string duality between the
superstring theory in the $\gamma$-deformed $AdS_5 \times \tilde{S}^5$
background and the $\gamma$-deformed $\mathcal{N}=4$ SYM theory.
We have observed that the ``two-loop" part of the string energy
consists of the terms $(\gamma J)^n$ with $n \le 4$, and recovers
the corresponding expected ``two-loop" string energy for the undeformed
case when we take the undeformed limit $\gamma \rightarrow 0$.
In order to confirm the gauge/string duality at the two-loop level
it is desirable to construct some two-loop dilatation operator for
the $\gamma$-deformed $\mathcal{N}=4$ SYM theory and compute the
two-loop anomalous dimension of the relevant gauge invariant scalar
operator.


\section{Introduction}
\qquad The uncapacitated facility location (UFL) problem has received lots of attention due to its connection to operational research and infrastructure planning.  In the UFL problem, we are given a set of potential facility locations $F$, each $i\in F$ with a facility cost $f_i$, a set of clients $C$ and a metric $d$ over $F\cup C$. We need to open a set of facilities $F'$, and assign each client to an opened facility, so that the sum of the total facility opening cost and the total assignment cost is minimized.

The UFL is NP-hard and the current best algorithm for this problem was designed and analyzed by Li \cite{ShiLi}, that reached an approximation ratio of 1.488. Another algorithm by Byrka, parametrized by $\gamma$ achieved an optimal bi-factor approximation ratio when $\gamma>1.6774$. All these algorithms are based on the algorithm of Chudak and Shmoys \cite{CS} (known as the Chudak-Shmoys algorithm) and a filtering technique of Lin and Vitter\cite{Lin}. The algorithms of Byrka and Li also utilized a (1.11,1.78) bifactor algorithm by Jain et al.\cite{Jain1}.

On the negative side, Guha and Kuller \cite{Guha} showed that there is no 1.463-approximation unless \textbf{NP}$\subseteq$\textbf{DTIME}$(n^{O(\log\log n)})$. This condition was later strengthened by Sviridenko \cite{Svir} to be "\textbf{NP}=\textbf{P}". Jain et al. \cite{Jain1} generalized these work and proved that there is no $(\gamma_f,\gamma_c)$ bifactor approximation algorithm for $\gamma_c<1+2e^{-\gamma_f}$ unless  \textbf{NP}$\subseteq$\textbf{DTIME}$(n^{O(\log\log n)})$. We therefore refer to this result as the hardness curve of the UFL problem (notice that the hardness curve pass the point (1.463,1.463)). The algorithm by Byrka \cite{Byrka1}\cite{Byrka2} was optimal for $\gamma_f>1.6774$, but for $\gamma_f<1.6774$, there is no current algorithm that can reach this hardness curve.

The main difficulty in tackling the problem is to understand the regularity and irregularity of the instance. A instance is said to "regular" if all facilities to which a client is served in the optimal fractional solution to the LP, which is shown below, is of approximate the same distance to that client. While in an irregular case, the facilities to which a client is served in the optimal fractional solution varies in distance to that client. We have to notice that irregularity is not an absolute definition. The degree to which the instance is irregular may be different for different instances. We now consider an adversary that might produce instances to fight against our algorithms. It was shown that to fight against Byrka's algorithm for $\gamma>\gamma_0$ \ref{Byrka} or Li's algorithm\ref{Li}, the adversary would like to produce an completely regular case. In the case of Byrka's algorithm, that means that the case when $d_{ave}^C(j)=d_{ave}^D(j)$ will make the bounds tight. In the case of Li's algorithm, that means that the best strategy by an adversary would be render the characteristic function a threshold function. On the other hand, however, the regular case is also the easiest case to deal with, as is indicated by section 3.1 in Byrka's paper \cite{Byrka2}. Our failing to come up with an algorithm that does better than that of Li's indicates the existence of a worst case to Li's algorithm that is a very irregular case.

This research serves as an attempt to improve over the 1.488-approximation algorithm of Li. He allowed the algorithm to make a random choice of $\gamma$, which was the parameter in Byrka's algorithm, and in combination with the $(1.11,1.78)$ JMS algorithm, obtained the 1.488-approximation algorithm. In his paper, he developed a "zero-sum game" whose value equals the approximation ratio of the algorithm and found the best strategy for the adversary that made it easy to analyze the algorithm. This best strategy turned out to be a very "regular" instance which was rather easy to handle if the parameter $\gamma$ was chosen optimally instead of randomly. We also observed that the so-called "zero-sum game" was in fact not a "zero-sum game". Instead, it was a vector game at its core and thus the Minimax Principle didn't apply. Based on these observations, we attempted to improve the algorithm by making an optimal choice of $\gamma$ instead of a random choice. Our analysis of the algorithm used Blackwell's Approachability Theorem. Unfortunately, our computer simulation showed that our algorithm did not  actually improve Li's algorithm. However, this failure might bring more insights into either the UFL problem or the theory of vector games.

\section{Byrka's Algorithm and Analysis}
\label{Byrka}
\quad In this section, we review the algorithm by Byrka\cite{Byrka2}, yet our notation follows that of Li's\cite{ShiLi}. The algorithm starts by solving the following linear programming for the UFL problem:
\begin{eqnarray*}
\min &&\quad \sum_{i\in F, j\in C} d(i,j)x_{i,j}+\sum_{i\in F}f_iy_i\quad s.t.\\
&&\quad \sum_{i\in F}=1\qquad \forall j\in C\\
&&\quad x_{i,j}\leq y_i\qquad \forall i\in F,j\in C\\
&&\quad x_{i,j},y_i\geq 0 \qquad \forall i\in F,j\in C
\end{eqnarray*}
Throughout this article, we will assume that we have solved the above LP and obtain an optimal solution $x^*$ and $y^*$. For each client $j$, the set of facilities to which it is served more than 0 is denoted as $F_j$.

The algorithm starts by a filtering process with parameter $\gamma$: we create a "filtered" solution $\overline{x}$, $\overline{y}$ as follows: $\overline{y}=\gamma y^*$ . We may split those facilities with $\overline{y}>1$ to two facilities. Then since the $\overline{y}$ are fixed, we may reassign the $\overline{x}$ facilities in a greedy manner: for each client $j$, sort the facilities by their distances to $j$ and then for each facility $i$ in order, assign $\overline{x}_{ij}=\overline{y_i}$ until the sum of all assignments of client $j$ reaches 1. W.l.o.g., we may assume that the solution obtained in this way is a complete solution, that is $\overline{x}_{ij}$ is either $\overline{y_i}$ or 0. This can be done via a facility splitting argument. A more detailed one can be found in \cite{}.

For each client $j\in C$, after the filtering process, we may distinguish between two kinds of facilities with respect to that client: close facilities and distant facilities. We say a facility $i$ is a close facility to client $j$ if $\overline{x}_{ij}>0$. We say a facility $j$ is a distant facility to client $j$ if $\overline{x}_{ij}=0$ but $x^*_{ij}>0$. We denote the set of close and distant facilities to client $j$ as $F_j^C$ and $F_j^D$ respectively.

Now we define a distance from a client $j$ to a set of facilities $F'\subseteq F$ $d(j, F')$ as the average distance to the facilities $F'$ with respect to the solution $\overline{y}$. We define $d_{ave}^C(j)=d(j,F_j^C)$, $d_{ave}^D(j)=d(j,F_j^D)$ and $d_{ave}(j)=d(j,F_j)$. We also define $d_{max}^C(j)$ as the maximum distance from $j$ to a facility in $F_j^C$.

Byrka's algorith follows the idea developed Chudak and Shmoys, which first select some facilities as the cluster centers, according to some criteria quantity. However, the criteria quantity chosen here is $K(j)=d_{max}^C(j)+d_{ave}^C(j)$. The process of choosing clustering centers is as follows. We consider each client in the order of increasing criteria quantity $K$. Suppose we are now at client $j$, if it is not assigned to any clustering center, we make it a clustering center, and for all those clients $j'$ that satisfies $F^C_{j'}\cap F^C_j\neq \emptyset$, we assign client $j'$ to the cluster center $j$. After this process, the clients are divided into several clusters. Call the set of all cluster centers $C'$.

We now round the solution $(\overline{x},\overline{y})$ randomly. For each cluster center, we open exactly one of its close facilitis randomly with probabilities $\overline{y}_i$. For each facility that is not a close facility of any cluster centers, open it independently with probability $\overline{y}_i$. Then connect each client $j$ to its closest facility. Let $C_j$ be the connection cost of client $j$.

The expected cost of the algorithm is just $\gamma$. The key to bounding the expected total connection cost is to bound $C_j$ under the condition that there is no open facility in $F_j$. Byrka was able to show that $d(j,F_{j'}^C\backslash F_j)\leq d_{ave}^D(j)+d_{max}^C(j)+d_{ave}^C(j)$ and thus bound the $\E[C_j]\leq (1-e^{-1}+e^{-\gamma})d_{ave}^C(j)+(e^{-1}+e^{-\gamma})d_{ave}^D(j)$. But the connection cost of the optimal solution is $d_{ave}(j)=\frac{1}{\gamma}d_{ave}^C(j)+\frac{\gamma-1}{\gamma}d_{ave}^D(j)$. By computing the maximum ratio between these two costs, we have an optimal bifactor algorithm for $\gamma>\gamma_0\approx 1.67736$ that achieves a bifactor approximation ratio of $(\gamma,1+2e^{-\gamma})$. Also by taking $\gamma=\gamma_0$ and in combination with the $(1.11,1.7764)$-bifactor approximation by Jain, Mahadian and Saberi, we can get a 1.50-approximation algorithm for the UFL problem.

In his paper \cite{Byrka2}, he also pointed out that by making $\gamma$ random instead of a fixed value might improve the approximation ratio and that's the basis of Li's algorithm and analysis.
\section{Li's Algorithm and Analysis}
\label{Li}
Li's algorithm is based on Byrka's algorithm by making $\gamma$ random. Yet the analysis is very involved.

Li's analysis utilized an important concept called characteristic function (see Definition 15 in Li's paper \cite{ShiLi}.
\begin{defn}[Definition 15 in Li's paper] Given a UFL instance and its optimal fractional solution $(x,y)$, the characteristic function $h_j:[0,1]\rightarrow \mathbb{R}$ of some client $j\in C$ is defined as follows. Let $i_1,i_2,\cdots,i_m$ be the facilities in $F_j$, in the non-decreasing order of distances to $j$. Then $h_j(p)=d(i_t,j)$, where $t$ is the minimum number such that $\sum_{s=1}^ty_{i_s}\geq p$. The characteristic function of the instance is defined as $h=\sum_{j\in C}h_j$.
\end{defn}
It is not hard to see that the characteristic function is a non-decreasing piece-wise constant function. Li also came up with a better bound on the connection cost, which is Lemma 12 in his paper \cite{ShiLi}.
\begin{lemma}[Lemma 12 in Li's paper]
For some facility $j\notin C'$, let $j'$ be the cluster center of $j$. We have,
$$
d(j,F_{j'}^C\backslash F_j)\leq (2-\gamma)d_{max}^C(j)+(\gamma-1)d_{ave}^D(j)+d_{max}^C(j')+d_{ave}^C(j').
$$
\end{lemma}
With this bound, he was able to show the following bound on the expected connection cost.
\begin{lemma}[Lemma 20 in Li's paper]
The expected cost of the integral solution is
$$
\E[C]\leq \int_0^1h(p)e^{-\gamma p}\gamma \dd p+e^{-\gamma}(\gamma\int_0^1h(p)\dd p+(3-\gamma)h(\frac{1}{\gamma})).
$$
\end{lemma}
To come up with an explicit distribution for $\gamma$, Li introduced a zero-sum game. We can scale the instance so that $\int_0^1h(p)\dd p=1$. Let
$$\alpha(\gamma,h)=\int_0^1h(p)e^{-\gamma p}\gamma \dd p+e^{-\gamma}(\gamma+(3-\gamma)h(\frac{1}{\gamma}))$$.
The game is between an algorithm designer $A$ and an adversary $B$. The strategy of $A$ is a pair $(\mu,\theta)$, where $\theta$ is the probability of playing the JMS algorithm and $\mu$ is $1-\theta$ times a probability density function for $\gamma$. The strategy for $B$ is a non-decreasing piece-wise constant function $h:[0,1]\rightarrow \textbf{R}*$ such that $\int_0^1h(p)\dd p=1$. The value of the game, when the strategy of $A$ is $(\mu,\theta)$ and the strategy of $B$ is $h$, is defined as:
$$
\nu(\mu,\theta,h)=\max\{\int_1^\infty \gamma\mu(\gamma)\dd \gamma+1.11\theta,\int_1^\infty \alpha(\gamma,h)\mu(\gamma)\dd \gamma+1.78\theta\}
$$
A threshold function $h_q:[0,1]\rightarrow \mathbb{R},0\leq q < 1$ is defined as:
$$
h_q(p)=
\begin{cases}
0 &\quad p\leq q\\
\frac{1}{1-q} & \quad p>q
\end{cases}
$$

And Li was able to show that there is a best strategy for $B$ that is a threshold function. From this fact he was able to find the optimal strategy for $A$: with probability $\theta\approx 0.2$, run the JMS algorithm; with probability about 0.5, run Byrka's algorithm witth parameter $\gamma_1\approx 1.5$; with the remaining probability, run Byrka's algorithm with parameter $\gamma$ selected uniformly between $\gamma_1$ and $\gamma_2\approx 2$. A very refined analysis by Li proved that this distribution was able to achieve an approximation ratio of 1.4879.
\section{An Attempt to Improve Li's Algorithm}

 \quad In this section, we present our attempt to improve over Li's algorithm. Although computer simulation illustrates that this approach might not actually work, it does provide some insights. It also provides a framework to design better approximation algorithms.

Li's analysis showed that there is a best strategy for $B$ which is a threshold function, which corresponds to a very regular case. But if the instance is regular, we might not need to randomly pick $\gamma$. It might be better if we can examine the instance carefully and find the best $\gamma$ for the instance. This idea makes use of the fact that there is an order in the game: the adversary first pick an instance and then the algorithm designer runs his algorithm. That means the algorithm designer is able to first examine the instance and then make his dicision in choosing the best parameter $\gamma$, which might help to improve the approximation ratio.

Our method of choosing the best parameter $\gamma$ is as follows: first discretize the region $[1,M]$, where $M$ is a constant large enough, to many possible choices for the parameter $\gamma$. Then for each choice of $\gamma$, find the correspondingly best $\theta$ so that the balanced approximation ratio is the smallest. Then choose the $\gamma$ that corresponds to the smallest approximation ratio, and the corresponding $\theta$ that achieves it.

It is obvious that this approach achieves a performance at least as good as Li's algorithm, since choosing the parameter randomly cannot do better than picking it optimally. Another very important reason why this approach might does better is due to the fact that the game defined by Li is not strictly a zero-sum game. According to Von Neumann's Minimax Principle, it will not be helpful if it is a zero-sum game, since a best strategy by $B$ will not be influenced by the order of the players. But the game is not a zero-sum game, since the value of the game is a maximum between two quatities. We shall now restate this game as a vector game.

To make it easier to state and more importantly, easy to simulate with a computer, we shall apply discretization to the game. Notice the game is in fact continuous and its definition is almost the same with what we are going to present here. We shall first discretize the region in which $\gamma$ might take value to very tiny pieces and call these discretized value $\gamma_1,\gamma_2,\cdots$. These indicates the pure strategy of $A$ if he is going to run Byrka's algorithm. Another pure strategy of $A$ is the JMS algorithm. So the strategy set of $A$ can be written as $\{JMS,\gamma_1,\gamma_2,\cdots\}$, and $A$ is allowed to play a mixed strategy. The pure strategy of $A$ is denoted by $(\theta,\gamma)$, where $\theta$ is either 1 (indicating the JMS algorithm), or 0 (where in this case, $A$ runs Byrka's algorithm with parameter $\gamma$). Now we define the strategy of $B$. Since every characteristic function can be written as a linear combination of a finite number of threshold function, we might think of the strategy of $B$ as a mixed strategy over all threshold functions. We can therefore discretize the region $[0,1]$ to very tiny pieces $p_1,p_2\cdots$, and the strategy set of $B$ is thus $\{p_1,p_2,\cdots\}$. The value of the game is in fact a vector $\alpha$, which is defined as $\alpha(\theta, \gamma, p)=(C_f,C_c)(\theta,\gamma,p)=((1-\theta)\gamma+1.11\theta,(1-\theta)\alpha(\gamma,h_p)+1.78\theta)$. We should stress here that this value is defined when both players play pure strategy, thus $\theta\in \{0,1\}$. If players play mixed strategies, then we shall define the value of the game in an intuitive way, that is the expected vector value. The approximation ratio of this algorithm is determined by the region which is approachable by $A$. That is, we seek to find the smallest $\beta$ so that the 2-D region defined by the half-plains $x\leq \beta,y\leq \beta$ is approachable. In this case, $\beta$ will be the approximation ratio of this algorithm.

To find such a $\beta$, we may first guess how small can it be and we suppose our guess is $\beta_0$. Our task at this moment is to show that the region defined by $\beta_0$ is approachable. According to Blackwell's Approachability Theorem, it suffices to prove the approachability of all the supporting hyperplanes, which are just all the lines that pass through the point $(\beta_0,\beta_0)$ with non-positive slope in this case. We may assume that each of the line is defined by the equation with parameter $0\leq \phi\leq 1$: $\phi x+(1-\phi)y=\beta_0$, then if we only consider the game with respect to this line, we can get a zero-sum game with value $\phi C_f+(1-\phi)C_c-\beta_0$. We call this quantity the value of the game with respect to the line parametrized by $\phi$. When $\phi$ is determined, the remaining zero-sum game can be solved by any standard method, and in this research, we use linear programming to handle it. We seek to find a mixed strategy for $B$ such that for any pure strategy of $A$, the value of the game with respect to the line parametrized by $\phi$ is larger than 0. If we can find such a strategy for $B$, then the line parametrized by $\phi$ is not approachable. In the language of LP, the line parametrized by $\phi$ is not approachable if and only if the corresponding LP stated above has at least a solution that corresponds to a mixed strategy for $B$. We may therefore write out the LP explicitly and test whether or not a solution can be found.

If our guess for $\beta_0$ is indeed correct (notice it might not be the smallest of such correct guesses), Blackwell's Approachability Theorem tells us that for any $\phi\geq 0$, the line parametrized by $\phi$ is approachable. On the other hand, if we can check through all possible $0\leq\phi\leq 1$ and show that every one of them is approachable, we might conclude that the region defined by $\beta_0$ is approachable. To make the problem tractable to our computer, we should discretize for $\phi$ as well. Then we enumerate each possible value for $\phi$ and see if it is approachable.

In our simulation, we discretized each of $\gamma$,$p$ and $\phi$ to 1000 possible values and tested for approachability with different $\beta$. Unfortunately, the smallest such $\beta$ found was $1.4880$, which showed that our attempt to improve over Li's algorithm didn't work. However, the result found by this simulation is interesting, the startegy for $B$ turned out to be very complicated: it had a value of around 0.6 for $p=0$, and very tiny values ($\leq 0.02$) for all other values of $p$. But we have to be careful here in understanding this result because the coefficients of each $h_p$ is determined by the following equation:
$$
h=\sum_{i=0}^{m-1}(c_{i+1}-c_i)(1-p_i)h_{p_i}
$$
Therefore, what the result really says is that there is an average distance of 0.6 where there are no facilities within it, but outside this region, facilities seem to be distributed very uniformly (in respect of distance) around the client.
\section{Implication of Our Results}
Although our attempt failed to improve over Li's algorithm, it had interesting implications to both the UFL problem and vector games.
\subsection{Implication to the UFL problem}
The result of our linear programming showed the existence of a highly irregular instance which could defeat the filtering process and our attempt to take the best $\gamma$ value. In other words, the filtering process alone cannot capture every aspect of the irregularity of the instance. If we want to design a better algorithm, we might need a better method to capture the irregularity of the problem.
\subsection{Implication to vector games}
As a byproduct, another interesting aspect of the implication of our result lies in the theory of vector games. At present, no corresponding result to the Minimax theorem of zero-sum games is known to vector games. However, our failure tends to show that the order of playing for vector games might not be important, under some condition. Admittedly, the vector game in our research is special in that (1) the first coordinate has nothing to do with the strategy of $B$ and (2) we only care about the maximum of the two coordinates. At this point, we don't know what aspects of this game leads to the fact that the order of playing is not important. Neither do we know whether a general conclusion exists for vector games.
\section{Acknowledgements}
I thank my instructor Li for helpful discussions.

\bibliographystyle{plain}


\section{Introduction}
Coronal mass ejections (CMEs) are energetic expulsions
of plasma from the solar corona that are driven by the
release of magnetic energy typically in the range of $10^{32-33}$ ergs.
The majority of CMEs originate from the eruption of pre-existing
large-scale helmet streamers \citep[]{WM_hundhausen93}.
Less common fast CMEs typically come from smaller, more
concentrated locations of magnetic flux referred to as
active regions.  In this case, the CMEs often occur shortly
after the flux has emerged at the photosphere, but can
also happen even as the active region is decaying.
While CMEs occur in a wide range of circumstances,
all CMEs originate above photospheric magnetic
polarity inversion lines (neutral lines), which exhibit strong
magnetic shear.  Shear implies that the magnetic field has a strong
component parallel to the photospheric line that separates magnetic
flux of opposite sign, and in this configuration, the field
possesses significant free energy.  In contrast, a potential field
runs perpendicular to the inversion line and has no free energy.

There is enormous evidence for the existence of highly sheared
magnetic fields associated with CMEs and large flares.
At the photosphere, magnetic shear is measured directly
with vector magnetographs \citep[e.g.][]{WM_hagyard84, WM_zirin93,
WM_falconer2002, WM_yang2004, WM_liu2005}.
Higher in the atmosphere, the magnetic field is difficult to
measure directly, but its direction may be inferred from
plasma structures formed within the field.
Seen in chromospheric H$\alpha$ absorption, filaments form only
over photospheric inversion lines \citep[]{WM_zirin83} along which
the magnetic field is nearly parallel \citep[]{WM_leroy89}.
Fibrils and H$\alpha$ loops that overlay photospheric bipolar
active regions are also indicative of magnetic shear \citep[]{WM_foukal71}.
Comparisons between vector magnetograms and H$\alpha$ images
show that the direction of the sheared photospheric magnetic field
coincides with the orientation of such fibril structures
\citep[]{WM_zhang95}.  Higher in the corona, evidence of magnetic
shear is found in loops visible in the extreme ultraviolet
\citep[]{WM_liu2005} and X-ray sigmoids \citep[]{WM_moore2001}.  These
structures run nearly parallel to the photospheric
magnetic inversion line prior to CMEs, and are followed by
the reformation of closed bright loops that are much more
potential in structure.  Finally, observations
by the Transition Region and Coronal Explorer (TRACE)
show that 86 percent of two-ribbon flares show a strong-to-weak
shear change of the ribbon footpoints that indicates the eruption
of a sheared core of flux \citep{WM_su2007}.

Sheared magnetic fields are at the epicenter of solar eruptive behavior.
Large flares are preferentially found to occur along the most highly
sheared portions of magnetic inversion lines \citep[]{WM_hagyard84, WM_zhang95}.
More recent analysis by \citet{WM_schrijver2005} found that shear
flows associated with flux emergence drove enhanced flaring.
Similarly, active region CME productivity is also strongly
correlated with magnetic shear as shown by
\citet{WM_falconer2001, WM_falconer2002, WM_falconer2006}.
It is not coincidental that large flares and CMEs are strongly
associated with filaments, which are known to form only along
sheared magnetic inversion lines \citep[]{WM_zirin83}.
The buildup of magnetic shear is essential for
energetic eruptions, and for this reason, it is of fundamental
importance to understanding solar activity.

Currently the majority of CME initiation models rely on the
application of artificially imposed shear flows.  Examples of
such models include
\citet{WM_antiochos99, WM_mikic94, WM_guo98, WM_amari2003}.
Until recently, there was no theory
to account for these large scale shear flows.
In this paper, we discuss a series of simulations that illustrate a
physical process by which these shear flows are self-organized
in emerging magnetic fields.  We will give a close comparison of these
simulations in the context of new observations that make a more complete
and compelling picture of a fundamental cause of eruptive solar
magnetic activity.

\section{Simulations}
\begin{figure*}[ht!]
\begin{center}
\includegraphics[angle=0,scale=0.85]{f01.ps}
\end{center}
\caption{Magnetic flux emerging from a horizontal layer into the
corona in the form of an axisymmetric arcade.
The top row illustrates the time evolution of the horizontal shear
velocity, $U_x$, shown in color with field lines confined to the
plane drawn black.  The bottom row shows the evolution of
fully three-dimensional magnetic field lines viewed from above projected
onto the horizontal $x-y$ plane.  The footpoints of the field lines at
the photosphere are shown as black dots.  The shear flow is
clearly manifest in the footpoint motion.  The magnetic field near the
center of the arcade evolves to be nearly parallel to the inversion
line, while field lines near the periphery of the system are more
nearly orthogonal.}
\label{layer}
\end{figure*}

Shearing motions and magnetic field alignment with the polarity inversion
line, so frequently observed in active regions, are readily explained
as a response to the Lorentz force that arises when magnetic flux
emerges in a gravitationally stratified atmosphere.   The motions
take the form of large-amplitude shear Alfv\'en waves in which the
magnetic tension force drives horizontal flows in opposite directions
across the polarity inversion line.  The physical process was first shown by
\citet{WM_manchester2000a, WM_manchester2000b, WM_manchester2001} and found
in later simulations by \citet{WM_fan2001, WM_magara2003, WM_archontis2004}.
As will be discussed in greater detail, what ultimately produces the
shearing Lorentz force is the nonuniform expansion of the emerging
magnetic field.

The shearing process was first simulated by \citet{WM_manchester2001},
with a two-and-a-half-dimensional (2.5D) simulation of flux emergence
from a horizontal magnetic layer placed 2-3 pressure scale heights
below the photosphere.  The crucial aspect of this simulation is that the
magnetic field is initially in a {\it sheared} configuration oriented at a 45
degree angle to the plane of variation.  In this case, for spatial
and temporal variations of the instability described by
$e^{i(\bf{k} \cdot r - \omega t)}$,  $\bf{k}$ is oblique to
the magnetic field ($\bf{B}$).  This distinguishes the mode of
instability from a Parker mode \citep[]{WM_parker66} in which
$\bf{k}$ is parallel to $\bf{B}$ or an interchange mode in which $\bf{k}$
is perpendicular to $\bf{B}$.  With $\bf{k}$ oblique
to $\bf{B}$, the instability can be thought of as a mixed mode,
\citep[e.g.][]{WM_cat90, WM_matsumoto93, WM_kusano98}.  In this
case, there will be a component of the Lorentz force out of
the plane of variation any where the magnetic component out of the plane
($B_x$ in the chosen coordinates) is not constant along field lines.
In the simulation by \citet{WM_manchester2001}, it was shown
that the departure of $B_x$ from constant values along field
lines produced strong shear flows in the emerging flux.

When the magnetic flux rises forming an arcade as shown in Figure
\ref{layer}, expansion causes the magnitude of $B_x$ to decrease in
the arcade.  The gradient in $B_x$ along field lines results
in the Lorentz force that drives the shear flows.
Examining the expression for the $x$ component of the Lorentz
(tension) force, $F_{x} = \frac{1}{4 \pi} \nabla B_{x} \cdot ({\bf B_y + B_z})$,
we see the reason for the shear.  The gradient in $B_x$ is
negative moving up the arcade in the direction of ${\bf B}$ on
one side of the arcade, while on the opposite side, the gradient
of $B_x$ along ${\bf B}$ is positive.

\begin{figure*}[ht!]
\begin{center}
\includegraphics[angle=0,scale=0.60]{f02.ps}
\end{center}
\caption{
Partial eruption of a three-dimensional emerging magnetic flux rope.
Panel (a) illustrates the Lorentz force with magnetic stream lines
(confined to the $y-z$ plane at the central cross section of the rope)
shown with white lines while black lines show the direction of the
current density.  The existence of the Lorentz force $(\bf{j \times B})$
out of the plane is clearly seen
where the field and current density are oblique.
The magnetic field crosses $\bf{j}$ in opposite directions on
opposite sides of the rope producing the shear flow.
The large vertical gradient in the axial component
(shown in color) produces the horizontal cross-field current.
Panel (b) shows the horizontal shear velocity, which clearly occurs
where $\bf{j}$ and $\bf{B}$ are non-parallel.
The bottom panels show the resulting eruption of the flux rope.
At the photosphere, the vertical magnetic field strength is shown in color
and the horizontal direction is shown with vectors in Panel (c).
Here, the current sheet (where the magnetic field is reconnecting) is shown
with a gray isosurface, along with magnetic field lines entering the
sheet.  Coronal field lines and the current sheet
form a sigmoid structure that runs nearly parallel to the highly sheared
photospheric magnetic field.  Panel (d) shows the flux rope splitting
apart at the current sheet.  The upper part of the rope erupts into the
corona (blue and red lines) while the lower part forms deep dips that
remain just above the photosphere filled with dense plasma.}
\label{rope}
\end{figure*}

These shear flows are illustrated in color in the top row
of Figure \ref{layer}.  Here, we see the persistence of shear flows
as the magnetic field (drawn with black lines) expands in the
atmosphere.  The magnitude of the shear flow velocity reaches a peak of 15 km/s
high in the arcade, 3-4 km/s at the photosphere and approximately
1 km/s below the photosphere.  The maximum shear speed typically
reaches half the value of the local Alfv\'en speed.  The bottom
row of Figure \ref{layer} shows the time evolution of magnetic field lines
integrated in three-dimensional space, and shown from above
projected onto the $x-y$ plane.  Footpoints of the field lines
are shown with black dots.  These panels show the shear displacement
of the footpoints and how the magnetic field evolves to be parallel
to the polarity inversion line.  Note that the
field is most highly sheared in close proximity to the polarity
inversion line, and grows more nearly perpendicular to the line
with greater distance from it.

This same shearing process is found in more complex simulations such as
three-dimensional emerging flux ropes \citep[]{WM_fan2001, WM_magara2003,
WM_manchester2004, WM_archontis2004}.  Panel (a) of Figure \ref{rope} shows
in greater detail than before the structure of the magnetic field and current
systems in the emerging flux rope of \citet{WM_manchester2004} that leads
to the Lorentz force, which drives the shear flows.  Here, the axial field
strength of the rope is shown in color, while the direction of the magnetic
field and current density (confined to the plane) are shown with white
and black lines respectively on the vertical plane that is at the central cross
section of the flux rope.  The large expansion of the
upper part of the flux rope produces a vertical gradient in the
axial component of the magnetic field $(B_x)$ that results in a horizontal
current $(j_y)$.  The magnetic field crosses the current in opposite
direction on opposite sides of the flux rope.  This produces the
Lorentz force out of the plane that reverses direction across the
flux rope, and drives the horizontal shear flow.  Panel (b) shows
the magnitude of these shear flows on the same central cross section
of the flux rope.  The shear flow is greatest precisely where the
field is most nearly perpendicular to the current reaching a magnitude
of approximately 30 km/s.  At this time, the flux rope has risen twice
as high as the flux that emerged from the magnetic layer and similarly
the shear flow in the rope is also twice as fast.

\begin{figure*}[ht!]\begin{center}\includegraphics[angle=0,scale=0.80]{f03.ps}
\end{center}
\caption{Results of the magnetic arcade simulation.  The initial state
is shown on the left with the three-dimensional arcade field lines
drawn in black, and the density and numerical grid are shown on the outer
boundary of the computational domain.  The middle and right panels,
respectively, show the shear and vertical velocities in color at the
central vertical plane of the simulation.  Field lines (confined
to the plane) are drawn white. Note the much larger scale of the
arcade compared to the flux emergence simulations along with the much
higher velocities.}
\label{arcade}
\end{figure*}

In the case of \citet{WM_manchester2004} the flux rope erupted, which
was not found in earlier simulations such as \citep[]{WM_fan2001}.  This
eruption resulted from high speed shear flows close to the neutral
line that formed a highly sheared core that lifted off, and was
followed by magnetic reconnection.  The bottom panels show the magnetic
and current structure at the time of the eruption.  Panel (c) gives a
view of the system that shows the vertical magnetic field
in color and the horizontal field with arrows at the photosphere.  A
current sheet is illustrated with an isosurface of current density.
The sigmoid shape of the sheet reflects the shape of both the coronal
field lines passing close to the sheet as well as the highly sheared
field at the photosphere.  This model strongly suggests that X-ray sigmoids
seen prior to and during CMEs are indicative of a highly sheared magnetic
field geometry.  Panel (d) shows that at this current sheet,
reconnection is taking place that separates the upper part of the
rope that is erupting from the lower part that has V-shaped field lines
full of dense plasma that remains attached to the photosphere.
Such flux separations in CMEs have been suggested by \citet{WM_gilbert2000}
based on observations of filament eruptions.

Finally we discuss a numerical simulation of an arcade eruption.  This
simulation is essentially identical to the simulation of
\citet{WM_manchester2003}, with the exception that it is carried out
in fully three-dimensional space with the BATS-R-US MHD code developed
at the University of Michigan.  The left panel of Figure \ref{arcade}
shows the initial state of the simulation with field lines drawn in
black showing the sheared magnetic arcade.  On the boundaries of the
computational domain, the plasma mass density is shown in color along
with the computational grid shown with white lines.  The center and
right panels show the shear and vertical velocities respectively in
color on the central plane of the simulation.  The magnetic field
confined to the plane (ignoring the component out of the plane) is
drawn with white lines in both panels.  In these panels, we find that
the arcade erupts very violently with a shear flow that reach
a magnitude of 200 km/s.  The arcade rises at a peak velocity of
150 km/s, and drives a shock ahead of it in the corona.

In this simulation, the eruption is caused by shear flows driven by
the Lorentz force.  A shearing catastrophe occurs when the $B_x$
component of the magnetic field can not be equilibrated along field lines.
This simulation is different
from the pervious two discussed in that it does not treat flux emergence
through the photosphere, but only models the coronal plasma at a temperature
of one million degrees.  Not needing to resolve the photospheric pressure
scale hight allows much larger cells and a computational domain that is
25 times larger than that of the previous two simulations.  The result
of this coronal arcade simulation is an eruption that is 5 times faster
than the flux rope eruption, and extends out to half a solar radius
above the surface.  This progression of eruption
velocity with the size of the flux system being treating offers
compelling evidence that shear flows driven by the Lorentz are
capable of producing fast CMEs from large active regions.

\section{Discussion}
The build up of magnetic energy in active regions is essential
to the onset of CMEs and flares.  The magnetic stress must pass
from the convection zone into the corona in the form of non-potential
fields, and effectively couple layers of the atmosphere.
Observations show that such non-potential fields occurs
along magnetic polarity inversion lines where the magnetic
field is highly sheared.  The evidence that magnetic shear is
essential to CMEs and flares is provided by the very strong correlation
between photospheric shear flows, flux emergence and the onset of CMEs and
large flares \citep[]{WM_meunier2003, WM_yang2004, WM_schrijver2005}.  These
shear flows are found to be strongest along the magnetic inversion line
precisely where flares are found \citep[]{WM_yang2004, WM_deng2006}.
This complements earlier evidence that in the case of two-ribbon flares
\citep[]{WM_zirin84}, prior to the eruption, H$\alpha$ arches
over the inversion line are highly sheared, afterward, the
arches are nearly perpendicular to the inversion line.
\citet{WM_su2006} found a similar pattern of magnetic shear loss
in the apparent motion of footpoints in two-ribbon flares.
There are even now observations of subphotospheric shear flows
with a magnitude of 1-2 km/s 4-6 Mm below the photosphere that
occur during flux emergence \citep[]{WM_kosovichev2006}, which
simulations show is fully consistent with the Lorentz force driving
mechanism \citep[]{WM_manchester2007}.

These ubiquitous shear flows and sheared magnetic fields so strongly
associated with CMEs are readily explained by the Lorentz force
that occurs when flux emerges and expands in a gravitationally
stratified atmosphere.  This physical process explains and synthesizes
many observations of active regions and gives them meaning in a larger
context.  This shearing mechanism explains (1) the coincidence of the
magnetic neutral line with the velocity neutral line, (2) the impulsive
nature of shearing in newly emerged flux, (3) the magnitude of the
shear velocity in different layers of the atmosphere, (4) the large
scale pattern of magnetic shear in active regions, (5) the
transport of magnetic flux, and energy from the convection
zone into the corona, (6) eruptions such as CMEs and flares.
With so much explained, it still remains a numerical challenge to
model an active region with sufficient resolution to produce a large
scale CME by this shearing mechanism.  The rope emergence simulation
discussed here only produces a flux concentration that is one tenth
the size of an active region, which at this scale simply can not produce
an eruption the size of a CME.  However, this simulation illustrates the
basic process by which CMEs and flares
must occur, and current simulations already show a very
favorable scaling of eruption size.  With increases in computer power,
simulations of flux emergence should soon be producing CME size eruptions
by shear flows driven by the Lorentz force.

\acknowledgements{
Ward Manchester is supported in part at the University of Michigan
by NASA SR\&T grant NNG06GD62G.  The simulations shown here
were performed on NCAR and NASA super computers.}