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TITLE: Lebesgue integration calculation problem?
QUESTION [1 upvotes]: Let $f:[0,1]\to \Bbb R$ be a bounded, Lebesgue measurable function with satisfies $$\int_{[0,1]}f(x)x^kdx=\frac{1}{(k+2)(k+3)}=\frac{1}{k+2}-\frac{1}{k+3} $$ for each $k\in \Bbb N \cup{0}$. Show that $f(x)=x-x^2$ almost everywhere(with respect to Lebesgue measure) on $[0,1]$.
This problem looks obvious by calculus. I don't know where to start. I cannot thinks a theorem in Lebesgue integration I can use. Could someone kindly provide a hint? Thanks!
REPLY [2 votes]: We can relax the boundedness assumption. Suppose $f\in L^1(0,1)$ and the integral condition holds for all $k=0,1,\dots $ Let $g(x) = f(x)-(x-x^2).$ Then $\int_0^1g(x)x^k\,dx = 0$ for all $k.$ Hence $\int_0^1g\cdot p = 0$ for all polynomials $p.$
If $h$ is continuous on $[0,1],$ then by Weierstrass there is a sequence of polynomials $p_n \to h$ uniformly. A simple DCT argument then shows $\int_0^1g\cdot h = 0.$
Define $s(x) = x/|x|, x \ne 0,s(0) = 0.$ Then $s(g(x))$ is bounded and measurable on $[0,1].$ Hence by a well-known corollary of Lusin's theorem, there is a sequence $h_n\in C([0,1]), |h_n(x)|\le 1$ for $x\in [0,1],$ such that $h_n(x) \to s(g(x))$ a.e. By the DCT, we get
$$\int_0^1|g| = \int_0^1 g\cdot s(g) = \lim_{n\to \infty}\int_0^1g\cdot h_n =0.$$
Thus $g=0$ a.e.
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\begin{document}
\maketitle
\begin{abstract}
In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors (e.g., signal processing, statistics) and low-rank matrices (e.g., control, statistics), as well as several others including sums of a few permutations matrices (e.g., ranked elections, multiobject tracking), low-rank tensors (e.g., computer vision, neuroscience), orthogonal matrices (e.g., machine learning), and atomic measures (e.g., system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the \emph{atomic norm}. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.
{\bf Keywords}: Convex optimization; semidefinite programming; atomic norms; real algebraic geometry; Gaussian width; symmetry.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and
engineering. A commonly encountered difficulty that arises in such inverse problems is the limited availability of data relative to the ambient dimension of the signal to be estimated. However many interesting signals or models in practice contain few degrees of freedom relative to their ambient dimension. For instance a small number of genes may constitute a signature for disease, very few parameters may be required to specify the correlation structure in a time series, or a sparse collection of geometric constraints might completely specify a molecular configuration. Such low-dimensional structure plays an important role in making inverse problems well-posed. In this paper we propose a unified approach to transform notions of simplicity into convex penalty functions, thus obtaining convex optimization formulations for inverse problems.
We describe a model as simple if it can be written as a nonnegative combination of a few elements from an atomic set. Concretely let $\bx \in \R^p$ be formed as follows:
\begin{equation}
\bx = \sum_{i=1}^k c_i \ba_i, ~~~ \ba_i \in \A, c_i \geq 0, \label{eq:simp1}
\end{equation}
where $\A$ is a set of atoms that constitute simple building blocks of general signals. Here we assume that $\bx$ is \emph{simple} so that $k$ is relatively small. For example $\A$ could be the finite set of unit-norm one-sparse vectors in which case $\bx$ is a sparse vector, or $\A$ could be the infinite set of unit-norm rank-one matrices in which case $\bx$ is a low-rank matrix. These two cases arise in many applications, and have received a tremendous amount of attention recently as several authors have shown that sparse vectors and low-rank matrices can be recovered from highly incomplete information \cite{CanRT2006,Don2006a,Don2006b,RecFP2010,CanR2009}. However a number of other structured mathematical objects also fit the notion of simplicity described in \eqref{eq:simp1}. The set $\A$ could be the collection of unit-norm rank-one tensors, in which case $\bx$ is a low-rank tensor and we are faced with the familiar challenge of low-rank tensor decomposition. Such problems arise in numerous applications in computer vision and image processing \cite{AjaGTL2009}, and in neuroscience \cite{BecS2005}. Alternatively $\A$ could be the set of permutation matrices; sums of a few permutation matrices are objects of interest in ranking \cite{JagS2010} and multi-object tracking. As yet another example, $\A$ could consist of measures supported at a single point so that $\bx$ is an atomic measure supported at just a few points. This notion of simplicity arises in problems in system identification and statistics.
In each of these examples as well as several others, a fundamental problem of interest is to recover $\bx$ given limited \emph{linear} measurements. For instance the question of recovering a sparse function over the group of permutations (i.e., the sum of a few permutation matrices) given linear measurements in the form of partial Fourier information was investigated in the context of ranked election problems \cite{JagS2010}. Similar linear inverse problems arise with atomic measures in system identification, with orthogonal matrices in machine learning, and with simple models formed from several other atomic sets (see Section~\ref{subsec:ex} for more examples). Hence we seek tractable computational tools to solve such problems. When $\A$ is the collection of one-sparse vectors, a method of choice is to use the $\ell_1$ norm to induce sparse solutions. This method has seen a surge in interest in the last few years as it provides a tractable convex optimization formulation to exactly recover sparse vectors under various conditions \cite{CanRT2006,Don2006a,Don2006b}. More recently the nuclear norm has been proposed as an effective convex surrogate for solving rank minimization problems subject to various affine constraints \cite{RecFP2010,CanR2009}.
Motivated by the success of these methods we propose a general convex optimization framework in Section~\ref{sec:def} in order to recover objects with structure of the form \eqref{eq:simp1} from limited linear measurements. The guiding question behind our framework is: how do we take a concept of simplicity such as sparsity and derive the $\ell_1$ norm as a convex heuristic? In other words what is the natural procedure to go from the set of one-sparse vectors $\A$ to the $\ell_1$ norm? We observe that the convex hull of (unit-Euclidean-norm) one-sparse vectors is the unit ball of the $\ell_1$ norm, or the cross-polytope. Similarly the convex hull of the (unit-Euclidean-norm) rank-one matrices is the nuclear norm ball; see Figure~\ref{fig:fig1} for illustrations. These constructions suggest a natural generalization to other settings. Under suitable conditions the convex hull $\ch(\A)$ defines the unit ball of a norm, which is called the \emph{atomic norm} induced by the atomic set $\A$. We can then minimize the atomic norm subject to measurement constraints, which results in a convex programming heuristic for recovering simple models given linear measurements. As an example suppose we wish to recover the sum of a few permutation matrices given linear measurements. The convex hull of the set of permutation matrices is the \emph{Birkhoff polytope} of doubly stochastic matrices \cite{Zie1995}, and our proposal is to solve a convex program that minimizes the norm induced by this polytope. Similarly if we wish to recover an orthogonal matrix from linear measurements we would solve a \emph{spectral norm} minimization problem, as the spectral norm ball is the convex hull of all orthogonal matrices. As discussed in Section~\ref{subsec:why} the atomic norm minimization problem is, in some sense, the best convex heuristic for recovering simple models with respect to a given atomic set.
\begin{figure}
\begin{center}
\subfigure[]{\epsfig{file=l1.eps,width=4.5cm,height=4cm}}
\hspace{0.3in}
\subfigure[]{\epsfig{file=trace.eps,width=4.5cm,height=4cm}}
\hspace{0.3in}
\subfigure[]{\epsfig{file=linf.eps,width=4.5cm,height=4cm}}
\caption{Unit balls of some atomic norms: In each figure, the set of atoms is graphed in red and the unit ball of the associated atomic norm is graphed in blue. In (a), the atoms are the unit-Euclidean-norm one-sparse vectors, and the atomic norm is the $\ell_1$ norm. In (b), the atoms are the $2 \times 2$ symmetric unit-Euclidean-norm rank-one matrices, and the atomic norm is the nuclear norm. In (c), the atoms are the vectors $\{-1,+1\}^2$, and the atomic norm is the $\ell_\infty$ norm.} \label{fig:fig1}
\end{center}
\end{figure}
We give general conditions for exact and robust recovery using the atomic norm heuristic. In Section~\ref{sec:gaussian} we provide concrete bounds on the number of generic linear measurements required for the atomic norm heuristic to succeed. This analysis is based on computing certain \emph{Gaussian widths} of tangent cones with respect to the unit balls of the atomic norm \cite{Gor1988}. Arguments based on Gaussian width have been fruitfully applied to obtain bounds on the number of Gaussian measurements for the special case of recovering sparse vectors via $\ell_1$ norm minimization \cite{RudV2006,Sto2009}, but computing Gaussian widths of general cones is not easy. Therefore it is important to exploit the special structure in atomic norms, while still obtaining sufficiently general results that are broadly applicable. An important theme in this paper is the connection between Gaussian widths and various notions of \emph{symmetry}. Specifically by exploiting symmetry structure in certain atomic norms as well as convex duality properties, we give bounds on the number of measurements required for recovery using very general atomic norm heuristics. For example we provide precise estimates of the number of generic measurements required for exact recovery of an orthogonal matrix via spectral norm minimization, and the number of generic measurements required for exact recovery of a permutation matrix by minimizing the norm induced by the Birkhoff polytope. While these results correspond to the recovery of individual atoms from random measurements, our techniques are more generally applicable to the recovery of models formed as sums of a few atoms as well. We also give tighter bounds than those previously obtained on the number of measurements required to robustly recover sparse vectors and low-rank matrices via $\ell_1$ norm and nuclear norm minimization. In all of the cases we investigate, we find that the number of measurements required to reconstruct an object is proportional to its intrinsic dimension rather than the ambient dimension, thus confirming prior folklore. See Table~\ref{tab:sum} for a summary of these results.
\begin{table}[t]
\centering
\begin{tabular}{||c|c|c||}\hline
Underlying model & Convex heuristic & $\#$ Gaussian measurements \\ \hline\hline
$s$-sparse vector in $\R^p$ & $\ell_1$ norm & $2s \log(p/s)+ 5s/4 $ \\ \hline
$m \times m$ rank-$r$ matrix & nuclear norm & $3r(2m-r)$ \\
\hline
sign-vector $\{-1,+1\}^p$ & $\ell_\infty$ norm & $p / 2$\\ \hline
$m \times m$ permutation matrix & norm induced by Birkhoff polytope & $9 \,m \log (m)$\\
\hline
$m \times m$ orthogonal matrix & spectral norm & $(3 m^2 - m)/4$\\
\hline
\end{tabular}
\caption{A summary of the recovery bounds obtained using Gaussian width arguments.} \label{tab:sum}
\end{table}
Although our conditions for recovery and bounds on the number of measurements hold generally, we note that it may not be possible to obtain a computable representation for the convex hull $\ch(\A)$ of an arbitrary set of atoms $\A$. This leads us to another important theme of this paper, which we discuss in Section~\ref{sec:rep}, on the connection between algebraic structure in $\A$ and the semidefinite representability of the convex hull $\ch(\A)$. In particular when $\A$ is an algebraic variety the convex hull $\ch(\A)$ can be approximated as (the projection of) a set defined by linear matrix inequalities. Thus the resulting atomic norm minimization heuristic can be solved via semidefinite programming. A second issue that arises in practice is that even with algebraic structure in $\A$ the semidefinite representation of $\ch(\A)$ may not be computable in polynomial time, which makes the atomic norm minimization problem intractable to solve. A prominent example here is the tensor nuclear norm ball, obtained by taking the convex hull of the rank-one tensors. In order to address this problem we give a hierarchy of semidefinite relaxations using \emph{theta bodies} that approximate the original (intractable) atomic norm minimization problem \cite{GouPT2010}. We also highlight that while these semidefinite relaxations are more tractable to solve, we require more measurements for exact recovery of the underlying model than if we solve the original intractable atomic norm minimization problem. Hence there is a tradeoff between the complexity of the recovery algorithm and the number of measurements required for recovery. We illustrate this tradeoff with the cut polytope and its relaxations.
\textbf{Outline} Section~\ref{sec:def} describes the construction of the atomic norm, gives several examples of applications in which these norms may be useful to recover simple models, and provides general conditions for recovery by minimizing the atomic norm. In Section~\ref{sec:gaussian} we investigate the number of generic measurements for exact or robust recovery using atomic norm minimization, and give estimates in a number of settings by analyzing the Gaussian width of certain tangent cones. We address the problem of semidefinite representability and tractable relaxations of the atomic norm in Section~\ref{sec:rep}. Section~\ref{sec:comp} describes some algorithmic issues as well as a few simulation results, and we conclude with a discussion and open questions in Section~\ref{sec:conc}.
\section{Atomic Norms and Convex Geometry}
\label{sec:def}
In this section we describe the construction of an atomic norm from a collection of simple atoms. In addition we give several examples of atomic norms, and discuss their properties in the context of solving ill-posed linear inverse problems. We denote the Euclidean norm by $\|\cdot\|$.
\subsection{Definition}
\label{subsec:def}
Let $\A$ be a collection of atoms that is a compact subset of $\R^p$. We will assume throughout this paper that no element $\ba \in \A$ lies in the convex hull of the other elements $\ch(\A \backslash \ba)$, i.e., the elements of $\A$ are the extreme points of $\ch(\A)$. Let $\|\bx\|_\A$ denote the gauge of $\A$ \cite{Roc1996}:
\begin{equation}
\|\bx\|_{\A} = \inf\{t>0 ~:~ x \in t~ \ch(\A)\}. \label{eq:atnorm}
\end{equation}
Note that the gauge is always a convex, extended-real valued function for any set $\A$. By convention this function evaluates to $+\infty$ if $\bx$ does not lie in the affine hull of $\ch(\A)$. We will assume without loss of generality that the centroid of $\ch(\A)$ is at the origin, as this can be achieved by appropriate recentering. With this assumption the gauge function can be rewritten as \cite{Bon1991}:
\begin{equation*}
\|\bx\|_{\A} = \inf\left\{ \sum_{\ba \in\A} c_\ba ~:~ \bx = \sum_{\ba\in \A} c_\ba \ba, ~~ c_\ba \geq 0 ~\forall \ba \in \A\right\}.
\end{equation*}
If $\A$ is centrally symmetric about the origin (i.e., $\ba \in \A$ if and only if $-\ba \in \A$) we have that $\|\cdot\|_\A$ is a norm, which we call the \emph{atomic norm} induced by $\A$. The support function of $\A$ is given as:
\begin{equation}
\|\bx\|_\A^\ast = \sup \left\{\langle \bx, \ba \rangle ~ : ~ \ba \in \A \right\}. \label{eq:atnormd}
\end{equation}
If $\|\cdot\|_\A$ is a norm the support function $\|\cdot\|^\ast_\A$ is the dual norm of this atomic norm. From this definition we see that the unit ball of $\|\cdot\|_\A$ is equal to $\ch(\A)$. In many examples of interest the set $\A$ is not centrally symmetric, so that the gauge function does not define a norm. However our analysis is based on the underlying convex geometry of $\ch(\A)$, and our results are applicable even if $\|\cdot\|_\A$ does not define a norm. Therefore, with an abuse of terminology we generally refer to $\|\cdot\|_\A$ as the atomic norm of the set $\A$ even if $\|\cdot\|_\A$ is not a norm. We note that the duality characterization between \eqref{eq:atnorm} and \eqref{eq:atnormd} when $\|\cdot\|_\A$ is a norm is in fact applicable even in infinite-dimensional Banach spaces by Bonsall's atomic decomposition theorem \cite{Bon1991}, but our focus is on the finite-dimensional case in this work. We investigate in greater detail the issues of representability and efficient approximation of these atomic norms in Section~\ref{sec:rep}.
Equipped with a convex penalty function given a set of atoms, we propose a convex optimization method to recover a ``simple'' model given limited linear measurements. Specifically suppose that $\bxs$ is formed according to \eqref{eq:simp1} from a set of atoms $\A$. Further suppose that we have a known linear map $\Phi: \R^p \rightarrow \R^n$, and we have linear information about $\bxs$ as follows:
\begin{equation}
\by = \Phi \bxs.
\end{equation}
The goal is to reconstruct $\bxs$ given $\by$. We consider the following convex formulation to accomplish this task:
\begin{equation}
\begin{aligned}
\hat{\bx} = \arg \min_{\bx} & ~~~ \|\bx\|_\A \\ \mbox{s.t.} & ~~~ \by = \Phi \bx.
\end{aligned}
\label{eq:atomic-norm-primal}
\end{equation}
When $\A$ is the set of one-sparse atoms this problem reduces to standard $\ell_1$ norm minimization. Similarly when $\A$ is the set of rank-one matrices this problem reduces to nuclear norm minimization. More generally if the atomic norm $\|\cdot\|_\A$ is tractable to evaluate, then \eqref{eq:atomic-norm-primal} potentially offers an efficient convex programming formulation for reconstructing $\bxs$ from the limited information $\by$. The \emph{dual problem} of \eqref{eq:atomic-norm-primal} is given as follows:
\begin{equation}
\begin{aligned}
\max_{\bz} & ~~~ \by^T \bz \\ \mbox{s.t.} & ~~~ \|\Phi^\dag \bz\|^\ast_\A \leq 1.
\end{aligned}
\label{eq:atomic-norm-dual}
\end{equation}
Here $\Phi^\dag$ denotes the adjoint (or transpose) of the linear measurement map $\Phi$.
The convex formulation \eqref{eq:atomic-norm-primal} can be suitably modified in case we only have access to inaccurate, noisy information. Specifically suppose that we have noisy measurements $\by = \Phi \bxs + \omega$ where $\omega$ represents the noise term. A natural convex formulation is one in which the constraint $\by = \Phi \bx$ of \eqref{eq:atomic-norm-primal} is replaced by the relaxed constraint $\|\by - \Phi \bx\| \leq \delta$, where $\delta$ is an upper bound on the size of the noise $\omega$:
\begin{equation}
\begin{aligned}
\hat{\bx} = \arg \min_{\bx} & ~~~ \|\bx\|_\A \\ \mbox{s.t.} & ~~~ \|\by - \Phi \bx\| \leq \delta.
\end{aligned}
\label{eq:noisy-atomic-norm-primal}
\end{equation}
We say that we have \emph{exact recovery} in the noise-free case if $\hat{\bx} = \bxs$ in \eqref{eq:atomic-norm-primal}, and \emph{robust recovery} in the noisy case if the error $\|\hat{\bx}-\bxs\|$ is small in \eqref{eq:noisy-atomic-norm-primal}. In Section~\ref{subsec:reccond} and Section~\ref{sec:gaussian} we give conditions under which the atomic norm heuristics \eqref{eq:atomic-norm-primal} and \eqref{eq:noisy-atomic-norm-primal} recover $\bxs$ exactly or approximately. Atomic norms have found fruitful applications in problems in approximation theory of various function classes \cite{Pis1981,Jon1992,Bar1993,DeVT1996}. However this prior body of work was concerned with infinite-dimensional Banach spaces, and none of these references consider nor provide recovery guarantees that are applicable in our setting.
\subsection{Examples}
\label{subsec:ex}
Next we provide several examples of atomic norms that can be viewed as special cases of the construction above. These norms are obtained by convexifying atomic sets that are of interest in various applications.
\textbf{Sparse vectors.} The problem of recovering sparse vectors from limited measurements has received a great deal of attention, with applications in many problem domains. In this case the atomic set $\A \subset \mathbb{R}^p$ can be viewed as the set of unit-norm one-sparse vectors $\{\pm \mathbf{e}_i\}_{i=1}^p$, and $k$-sparse vectors in $\mathbb{R}^p$ can be constructed using a linear combination of $k$ elements of the atomic set. In this case it is easily seen that the convex hull $\ch(\A)$ is given by the \emph{cross-polytope} (i.e., the unit ball of the $\ell_1$ norm), and the atomic norm $\|\cdot\|_\A$ corresponds to the $\ell_1$ norm in $\mathbb{R}^p$.
\textbf{Low-rank matrices.} Recovering low-rank matrices from limited information is also a problem that has received considerable attention as it finds applications in problems in statistics, control, and machine learning. The atomic set $\A$ here can be viewed as the set of rank-one matrices of unit-Euclidean-norm. The convex hull $\ch(\A)$ is the \emph{nuclear norm ball} of matrices in which the sum of the singular values is less than or equal to one.
\textbf{Sparse and low-rank matrices.} The problem of recovering a sparse matrix and a low-rank matrix given information about their sum arises in a number of model selection and system identification settings. The corresponding atomic norm is constructed by taking the convex hull of an atomic set obtained via the union of rank-one matrices and (suitably scaled) one-sparse matrices. This norm can also be viewed as the \emph{infimal convolution} of the $\ell_1$ norm and the nuclear norm, and its properties have been explored in \cite{ChaSPW2011,CanLMW2011}.
\textbf{Permutation matrices.} A problem of interest in a ranking context \cite{JagS2010} or an object tracking context is that of recovering permutation matrices from partial information. Suppose that a small number $k$ of rankings of $m$ candidates is preferred by a population. Such preferences can be modeled as the sum of a few $m \times m$ permutation matrices, with each permutation corresponding to a particular ranking. By conducting surveys of the population one can obtain partial linear information of these preferred rankings. The set $\A$ here is the collection of permutation matrices (consisting of $m!$ elements), and the convex hull $\ch(\A)$ is the \emph{Birkhoff polytope} or the set of doubly stochastic matrices \cite{Zie1995}. The centroid of the Birkhoff polytope is the matrix $\ones \ones^T / m$, so it needs to be recentered appropriately. We mention here recent work by Jagabathula and Shah \cite{JagS2010} on recovering a sparse function over the symmetric group (i.e., the sum of a few permutation matrices) given partial Fourier information; although the algorithm proposed in \cite{JagS2010} is tractable it is not based on convex optimization.
\textbf{Binary vectors.} In integer programming one is often interested in recovering vectors in which the entries take on values of $\pm 1$. Suppose that there exists such a sign-vector, and we wish to recover this vector given linear measurements. This corresponds to a version of the multi-knapsack problem \cite{ManR2009}. In this case $\A$ is the set of all sign-vectors, and the convex hull $\ch(\A)$ is the \emph{hypercube} or the unit ball of the $\ell_\infty$ norm. The image of this hypercube under a linear map is also referred to as a zonotope \cite{Zie1995}.
\textbf{Vectors from lists.} Suppose there is an unknown vector $\bx \in \mathbb{R}^p$, and that we are given the entries of this vector without any information about the locations of these entries. For example if $\bx = [3 ~ 1 ~ 2 ~ 2 ~ 4]'$, then we are only given the list of numbers $\{1, 2, 2, 3, 4\}$ without their positions in $\bx$. Further suppose that we have access to a few linear measurements of $\bx$. Can we recover $\bx$ by solving a convex program? Such a problem is of interest in recovering partial rankings of elements of a set. An extreme case is one in which we only have two preferences for rankings, i.e., a vector in $\{1,2\}^p$ composed only of one's and two's, which reduces to a special case of the problem above of recovering binary vectors (in which the number of entries of each sign is fixed). For this problem the set $\A$ is the set of all permutations of $\bx$ (which we know since we have the list of numbers that compose $\bx$), and the convex hull $\ch(\A)$ is the \emph{permutahedron} \cite{Zie1995,SanSS2009}. As with the Birkhoff polytope, the permutahedron also needs to be recentered about the point $\ones^T \bx / p$.
\textbf{Matrices constrained by eigenvalues.} This problem is in a sense the non-commutative analog of the one above. Suppose that we are given the eigenvalues $\lambda$ of a symmetric matrix, but no information about the eigenvectors. Can we recover such a matrix given some additional linear measurements? In this case the set $\A$ is the set of all symmetric matrices with eigenvalues $\lambda$, and the convex hull $\ch(\A)$ is given by the \emph{Schur-Horn orbitope} \cite{SanSS2009}.
\textbf{Orthogonal matrices.} In many applications matrix variables are constrained to be orthogonal, which is a non-convex constraint and may lead to computational difficulties. We consider one such simple setting in which we wish to recover an orthogonal matrix given limited information in the form of linear measurements. In this example the set $\A$ is the set of $m \times m$ orthogonal matrices, and $\ch(\A)$ is the \emph{spectral norm ball}.
\textbf{Measures.} Recovering a measure given its moments is another question of interest that arises in system identification and statistics. Suppose one is given access to a linear combination of moments of an atomically supported measure. How can we reconstruct the support of the measure? The set $\A$ here is the moment curve, and its convex hull $\ch(\A)$ goes by several names including the \emph{Caratheodory orbitope} \cite{SanSS2009}. Discretized versions of this problem correspond to the set $\A$ being a finite number of points on the moment curve; the convex hull $\ch(\A)$ is then a \emph{cyclic polytope} \cite{Zie1995}.
\textbf{Cut matrices.} In some problems one may wish to recover low-rank matrices in which the entries are constrained to take on values of $\pm 1$. Such matrices can be used to model basic user preferences, and are of interest in problems such as collaborative filtering \cite{SreS2005}. The set of atoms $\A$ could be the set of rank-one signed matrices, i.e., matrices of the form $\bz \bz^T$ with the entries of $\bz$ being $\pm 1$. The convex hull $\ch(\A)$ of such matrices is the \emph{cut polytope} \cite{DezL1997}. An interesting issue that arises here is that the cut polytope is in general intractable to characterize. However there exist several well-known tractable semidefinite relaxations to this polytope \cite{DezL1997,GoeW1995}, and one can employ these in constructing efficient convex programs for recovering cut matrices. We discuss this point in greater detail in Section~\ref{subsec:tradeoff}.
\textbf{Low-rank tensors.} Low-rank tensor decompositions play an important role in numerous applications throughout signal processing and machine learning \cite{KolB2009}. Developing computational tools to recover low-rank tensors is therefore of great interest. In principle we could solve a tensor nuclear norm minimization problem, in which the tensor nuclear norm ball is obtained by taking the convex hull of rank-one tensors. A computational challenge here is that the tensor nuclear norm is in general intractable to compute; in order to address this problem we discuss further convex relaxations to the tensor nuclear norm using theta bodies in Section~\ref{sec:rep}. A number of additional technical issues also arise with low-rank tensors including the non-existence in general of a singular value decomposition analogous to that for matrices \cite{Kol2001}, and the difference between the rank of a tensor and its border rank \cite{deSL2008}.
\textbf{Nonorthogonal factor analysis.} Suppose that a data matrix admits a factorization $X=AB$. The matrix nuclear norm heuristic will find a factorization into \emph{orthogonal} factors in which the columns of $A$ and rows of $B$ are mutually orthogonal. However if \emph{a priori} information is available about the factors, precision and recall could be improved by enforcing such priors. These priors may sacrifice orthogonality, but the factors might better conform with assumptions about how the data are generated. For instance in some applications one might know in advance that the factors should only take on a discrete set of values~\cite{SreS2005}. In this case, we might try to fit a sum of rank-one matrices that are bounded in $\ell_\infty$ norm rather than in $\ell_2$ norm. Another prior that commonly arises in practice is that the factors are non-negative (i.e., in non-negative matrix factorization). These and other priors on the basic rank-one summands induce different norms on low-rank models than the standard nuclear norm \cite{Faz2002}, and may be better suited to specific applications.
\subsection{Background on Tangent and Normal Cones}
\label{subsec:bgtn}
In order to properly state our results, we recall some basic concepts from convex analysis. A convex set $\mathcal{C}$ is a \emph{cone} if it is closed under positive linear combinations. The polar $\mathcal{C}^\ast$ of a cone $\mathcal{C}$ is the cone
\begin{equation*}
\mathcal{C}^\ast = \{ x \in \R^p ~:~ \langle x,z\rangle \leq 0~\forall z\in \mathcal{C}\}.
\end{equation*}
Given some nonzero $\bx \in \R^p$ we define the \emph{tangent cone} at $\bx$ with respect to the scaled unit ball $\|\bx\|_\A \ch(\A)$ as
\begin{equation}
T_\A(\bx) = \mathrm{cone}\{\mathbf{z}-\bx~:~ \|\mathbf{z}\|_\A \leq \|\bx\|_\A\}. \label{eq:tcone}
\end{equation}
The cone $T_\A(\bx)$ is equal to the set of \emph{descent directions} of the atomic norm $\|\cdot\|_\A$ at the point $\bx$, i.e., the set of all directions $\mathbf{d}$ such that the directional derivative is negative.
The \emph{normal cone} $N_{\A}(\bx)$ at $\bx$ with respect to the scaled unit ball $\|\bx\|_\A \ch(\A)$ is defined to be the set of all directions $\mathbf{s}$ that form obtuse angles with every descent direction of the atomic norm $\|\cdot\|_\A$ at the point $\bx$:
\begin{equation}
N_\A(\bx) = \{\mathbf{s} : \langle \mathbf{s},\mathbf{z}-\bx \rangle \leq 0 ~ \forall \mathbf{z} ~ \mathrm{s.t.~} \|\mathbf{z}\|_\A\leq \|\bx\|_\A\}. \label{eq:ncone}
\end{equation}
The normal cone is equal to the set of all normals of hyperplanes given by normal vectors $\mathbf{s}$ that support the scaled unit ball $\|\bx\|_\A \ch(\A)$ at $\bx$. Observe that the polar cone of the tangent cone $T_\A(\bx)$ is the normal cone $N_\A(\bx)$ and vice-versa. Moreover we have the basic characterization that the normal cone $N_{\A}(\bx)$ is the conic hull of the subdifferential of the atomic norm at $\bx$.
\subsection{Recovery Condition}
\label{subsec:reccond}
The following result gives a characterization of the favorable underlying geometry required for exact recovery. Let $\n(\Phi)$ denote the nullspace of the operator $\Phi$.
\begin{proposition}\label{prop:null-intersection}
We have that $\hat{\bx} = \bxs$ is the unique optimal solution of \eqref{eq:atomic-norm-primal} if and only if $\n(\Phi) \cap T_{\A}(\bxs) = \{0\}$.
\end{proposition}
\begin{proof}
Eliminating the equality constraints in \eqref{eq:atomic-norm-primal} we have the equivalent optimization problem
\begin{equation*}
\min_{\mathbf{d}} ~\|\bxs + \mathbf{d}\|_\A \quad \mathrm{s.t.}~ \mathbf{d} \in \n(\Phi).
\end{equation*}
Suppose $\n(\Phi)\cap T_{\A}(\bxs)=\{0\}$. Since $\|\bxs + \mathbf{d}\|_\A\leq \|\bxs\|$ implies $\mathbf{d} \in T_{\A}(\bxs)$, we have that $\|\bxs + \mathbf{d}\|_\A > \|\bxs\|_\A$ for all $\mathbf{d} \in \n(\Phi)\setminus\{0\}$. Conversely $\bxs$ is the unique optimal solution of \eqref{eq:atomic-norm-primal} if $\|\bxs + \mathbf{d}\|_\A > \|\bxs\|_\A$ for all $\mathbf{d} \in \n(\Phi)\setminus\{0\}$, which implies that $\mathbf{d} \not\in T_{\A}(\bxs)$.
\end{proof}
Proposition~\ref{prop:null-intersection} asserts that the atomic norm heuristic succeeds if the nullspace of the sampling operator does not intersect the tangent cone $T_\A(\bxs)$ at $\bxs$. In Section~\ref{sec:gaussian} we provide a characterization of tangent cones that determines the number of Gaussian measurements required to guarantee such an empty intersection.
A tightening of this empty intersection condition can also be used to address the noisy approximation problem. The following proposition characterizes when $\bxs$ can be \emph{well-approximated} using the convex program \eqref{eq:noisy-atomic-norm-primal}.
\begin{proposition}\label{prop:noisy-recovery}
Suppose that we are given $n$ noisy measurements $\by = \Phi \bxs + \omega$ where $\| \omega\| \leq \delta$, and $\Phi: \R^p \rightarrow \R^n$. Let $\hat{\bx}$ denote an optimal solution of \eqref{eq:noisy-atomic-norm-primal}. Further suppose for all $\bz \in T_{\A}(\bxs)$ that we have $\|\Phi \bz\| \geq \epsilon\|\bz\|$. Then $\|\hat{\bx}-\bxs\| \leq \frac{2\delta}{\epsilon}$.
\end{proposition}
\begin{proof}
The set of descent directions at $\bxs$ with respect to the atomic norm ball is given by the tangent cone $T_{\A}(\bxs)$. The error vector $\hat{\bx} - \bxs$ lies in $T_{\A}(\bxs)$ because $\hat{\bx}$ is a minimal atomic norm solution, and hence $\|\hat{\bx}\|_\A \leq \|\bxs\|_\A$. It follows by the triangle inequality that
\begin{equation}
\|\Phi(\hat{\bx}-\bxs)\| \leq \|\Phi\hat{\bx} - \by\| + \|\Phi \bxs - \by\| \leq 2 \delta.
\end{equation}
By assumption we have that
\begin{equation}
\|\Phi(\hat{\bx}-\bxs)\| \geq \epsilon \|\hat{\bx}-\bxs\|,
\end{equation}
which allows us to conclude that $\|\hat{\bx}-\bxs\|\leq \frac{2\delta}{\epsilon}$.
\end{proof}
Therefore, we need only concern ourselves with estimating the minimum value of $\frac{\|\Phi \bz\|}{\|\bz\|}$ for non-zero $\bz\in T_{\A}(\bxs)$. We denote this quantity as the \emph{minimum gain} of the measurement operator $\Phi$ restricted to the cone $T_{\A}(\bxs)$. In particular if this minimum gain is bounded away from zero, then the atomic norm heuristic also provides robust recovery when we have access to noisy linear measurements of $\bxs$. Minimum gain conditions have been employed in recent recovery results via $\ell_1$-norm minimization, block-sparse vector recovery, and low-rank matrix reconstruction \cite{CanT2005,BicRT2009,VanB2009,NegRWY2010}. All of these results rely heavily on strong decomposability conditions of the $\ell_1$ norm and the matrix nuclear norm. However, there are several examples of atomic norms (for instance, the $\ell_\infty$ norm and the norm induced by the Birkhoff polytope) as specified in Section~\ref{subsec:ex} that do not satisfy such decomposability conditions. As well see in the sequel the more geometric viewpoint adopted in this paper provides a fruitful framework in which to analyze the recovery properties of general atomic norms.
\subsection{Why Atomic Norm?}
\label{subsec:why}
The atomic norm induced by a set $\A$ possesses a number of favorable properties that are useful for recovering ``simple'' models from limited linear measurements. The key point to note from Section~\ref{subsec:reccond} is that the smaller the tangent cone at a point $\bxs$ with respect to $\ch(\A)$, the easier it is to satisfy the empty-intersection condition of Proposition~\ref{prop:null-intersection}.
Based on this observation it is desirable that points in $\ch(\A)$ with smaller tangent cones correspond to simpler models, while points in $\ch(\A)$ with larger tangent cones generally correspond to more complicated models. The construction of $\ch(\A)$ by taking the convex hull of $\A$ ensures that this is the case. The extreme points of $\ch(\A)$ correspond to the simplest models, i.e., those models formed from a single element of $\A$. Further the low-dimensional faces of $\ch(\A)$ consist of those elements that are obtained by taking linear combinations of a few basic atoms from $\A$. These are precisely the properties desired as points lying in these low-dimensional faces of $\ch(\A)$ have smaller tangent cones than those lying on larger faces.
We also note that the atomic norm is, in some sense, the best possible convex heuristic for recovering simple models. Any reasonable heuristic penalty function should be constant on the set of atoms $\A$. This ensures that no atom is preferred over any other. Under this assumption, we must have that for any $\ba \in \A$, $\ba'-\ba$ must be a descent direction for all $\ba'\in\A$. The best convex penalty function is one in which the cones of descent directions at $\ba\in\A$ are as small as possible. This is because, as described above, smaller cones are more likely to satisfy the empty intersection condition required for exact recovery. Since the tangent cone at $\ba \in \A$ with respect to $\ch(\A)$ is precisely the conic hull of $\ba'-\ba$ for $\ba'\in \A$, the atomic norm is the best convex heuristic for recovering models where simplicity is dictated by the set $\A$.
Our reasons for proposing the atomic norm as a useful convex heuristic are quite different from previous justifications of the $\ell_1$ norm and the nuclear norm. In particular let $f: \R^p \rightarrow \R$ denote the cardinality function that counts the number of nonzero entries of a vector. Then the $\ell_1$ norm is the \emph{convex envelope} of $f$ restricted to the unit ball of the $\ell_\infty$ norm, i.e., the best convex underestimator of $f$ restricted to vectors in the $\ell_\infty$-norm ball. This view of the $\ell_1$ norm in relation to the function $f$ is often given as a justification for its effectiveness in recovering sparse vectors. However if we consider the convex envelope of $f$ restricted to the Euclidean norm ball, then we obtain a very different convex function than the $\ell_1$ norm! With more general atomic sets, it may not be clear \emph{a priori} what the bounding set should be in deriving the convex envelope. In contrast the viewpoint adopted in this paper leads to a natural, unambiguous construction of the $\ell_1$ norm and other general atomic norms. Further as explained above it is the favorable \emph{facial structure} of the atomic norm ball that makes the atomic norm a suitable convex heuristic to recover simple models, and this connection is transparent in the definition of the atomic norm.
\section{Recovery from Generic Measurements}
\label{sec:gaussian}
We consider the question of using the convex program \eqref{eq:atomic-norm-primal} to recover ``simple'' models formed according to \eqref{eq:simp1} from a \emph{generic} measurement operator or map $\Phi: \R^p \rightarrow \R^n$. Specifically, we wish to compute estimates on the number of measurements $n$ so that we have exact recovery using \eqref{eq:atomic-norm-primal} for \emph{most} operators comprising of $n$ measurements. That is, the measure of $n$-measurement operators for which recovery fails using \eqref{eq:atomic-norm-primal} must be exponentially small. In order to conduct such an analysis we study random \emph{Gaussian} maps whose entries are independent and identically distributed Gaussians. These measurement operators have a nullspace that is uniformly distributed among the set of all $(p-n)$-dimensional subspaces in $\R^p$. In particular we analyze when such operators satisfy the conditions of Proposition~\ref{prop:null-intersection} and Proposition~\ref{prop:noisy-recovery} for exact recovery.
\subsection{Recovery Conditions based on Gaussian Width}
\label{subsec:width}
Proposition~\ref{prop:null-intersection} requires that the nullspace of the measurement operator $\Phi$ must miss the tangent cone $T_{\A}(\bxs)$. Gordon \cite{Gor1988} gave a solution to the problem of characterizing the probability that a random subspace (of some fixed dimension) distributed uniformly misses a cone. We begin by defining the Gaussian width of a set, which plays a key role in Gordon's analysis.
\begin{definition}
The \emph{Gaussian width} of a set $S \subset \R^p$ is defined as:
\begin{equation*}
w(S) := \E_\bg\left[\sup_{\bz \in S} ~ \bg^T \bz \right],
\end{equation*}
where $\bg \sim \mathcal{N}(0,I)$ is a vector of independent zero-mean unit-variance Gaussians.
\end{definition}
Gordon characterized the likelihood that a random subspace misses a cone $\mathcal{C}$ purely in terms of the dimension of the subspace and the Gaussian width $w(\mathcal{C} \cap \Sp^{p-1})$, where $\Sp^{p-1} \subset \R^p$ is the unit sphere. Before describing Gordon's result formally, we introduce some notation. Let $\lambda_k$ denote the expected length of a $k$-dimensional Gaussian random vector. By elementary integration, we have that $\lambda_k = \sqrt{2}\Gamma(\tfrac{k+1}{2})/\Gamma(\tfrac{k}{2})$. Further by induction one can show that $\lambda_k$ is tightly bounded as $\frac{k}{\sqrt{k+1}}\leq \lambda_k \leq \sqrt{k}$.
The main idea underlying Gordon's theorem is a bound on the minimum gain of an operator restricted to a set. Specifically, recall that $\n(\Phi) \cap T_\A(\bxs) = \{0\}$ is the condition required for recovery by Proposition~\ref{prop:null-intersection}. Thus if we have that the minimum gain of $\Phi$ restricted to vectors in the set $T_\A(\bxs) \cap \Sp^{p-1}$ is bounded away from zero, then it is clear that $\n(\Phi) \cap T_\A(\bxs) = \emptyset$. We refer to such minimum gains restricted to a subset of the sphere as \emph{restricted minimum singular values}, and the following theorem of Gordon gives a bound these quantities:
\begin{theorem}[Cor.~1.2,~\cite{Gor1988}] \label{theo:escape}
Let $\Omega$ be a closed subset of $\Sp^{p-1}$. Let $\Phi: \R^p \rightarrow \R^n$ be a random map with i.i.d. zero-mean Gaussian entries having variance one. Then
\begin{equation}
\E\left[ \min_{\bz\in \Omega} \|\Phi \bz\|_2\right] \geq \lambda_n - w(\Omega)\,.
\end{equation}
\end{theorem}
Theorem~\ref{theo:escape} allows us to characterize exact recovery in the noise-free case using the convex program \eqref{eq:atomic-norm-primal}, and robust recovery in the noisy case using the convex program \eqref{eq:noisy-atomic-norm-primal}. Specifically, we consider the number of measurements required for exact or robust recovery when the measurement map $\Phi: \R^p \rightarrow \R^n$ consists of i.i.d. zero-mean Gaussian entries having variance $1/n$. The normalization of the variance ensures that the columns of $\Phi$ are approximately unit-norm, and is necessary in order to properly define a signal-to-noise ratio. The following corollary summarizes the main results of interest in our setting:
\begin{corollary} \label{corl:width}
Let $\Phi: \R^p \rightarrow \R^n$ be a random map with i.i.d. zero-mean Gaussian entries having variance $1/n$. Further let $\Omega = T_\A(\bxs) \cap \Sp^{p-1}$ denote the spherical part of the tangent cone $T_\A(\bxs)$.
\begin{enumerate}
\item Suppose that we have measurements $\by = \Phi \bxs$ and solve the convex program \eqref{eq:atomic-norm-primal}. Then $\bxs$ is the unique optimum of \eqref{eq:atomic-norm-primal} with probability at least $1-\exp\left(-\tfrac{1}{2}\left[\lambda_n - w(\Omega)\right]^2\right)$ provided \begin{equation*}
n \geq w(\Omega)^2+1\,.
\end{equation*}
\item Suppose that we have noisy measurements $\by = \Phi \bxs + \omega$, with the noise $\omega$ bounded as $\|\omega\| \leq \delta$, and that we solve the convex program \eqref{eq:noisy-atomic-norm-primal}. Letting $\hat{\bx}$ denote the optimal solution of \eqref{eq:noisy-atomic-norm-primal}, we have that $\|\bxs - \hat{\bx}\| \leq \frac{2 \delta}{\epsilon}$ with probability at least $1-\exp\left(-\tfrac{1}{2}\left[\lambda_n - w(\Omega) -\sqrt{n}\epsilon\right]^2\right)$ provided
\begin{equation*}
n \geq \frac{w(\Omega)^2+3/2}{(1-\epsilon)^2} \,.
\end{equation*}
\end{enumerate}
\end{corollary}
\begin{proof}
The two results are simple consequences of Theorem~\ref{theo:escape} and a concentration of measure argument. Recall that for an function $f:\R^d \rightarrow \R$ with Lipschitz constant $L$ and a random Gaussian vector, $\bg\in\R^d$, with mean zero and identity variance
\begin{equation}\label{eq:conc-of-meas}
\P\left[ f(\bg) \geq \E[f]-t \right] \geq 1 - \exp\left(-\frac{t^2}{2L^2} \right)
\end{equation}
(see, for example,~\cite{LedT1991,Pis1986}). For any set $\Omega \subset \mathbb{S}^{p-1}$, the function
\[
\Phi \mapsto \min_{\bz\in \Omega} \|\Phi \bz\|_2
\]
is Lipschitz with respect to the Frobenius norm with constant $1$. Thus, applying~\ref{theo:escape} and (\ref{eq:conc-of-meas}), we find that
\begin{equation}\label{eq:main-conc-ineq}
\P\left[ \min_{\bz\in \Omega} \|\Phi \bz\|_2 \geq \epsilon \right] \geq 1 - \exp\left(-\frac{1}{2}(\lambda_n - w(\Omega) - \sqrt{n}\epsilon)^2\right)
\end{equation}
provided that $\lambda_n - w(\Omega) - \sqrt{n}\epsilon\geq 0$.
The first part now follows by setting $\epsilon = 0$ in (\ref{eq:main-conc-ineq}). The concentration inequality is valid provided that $\lambda_n \geq w(\Omega)$. To verify this, note
\[
\lambda_n \geq \frac{n}{\sqrt{n+1}} \geq \sqrt{\frac{w(\Omega)^2+1}{1+1/n} }
\geq \sqrt{\frac{w(\Omega)^2+w(\Omega)^2/n}{1+1/n} } = w(\Omega)\,.
\]
Here, both inequalities use the fact that $n\geq w(\Omega)^2+1$.
For the second part, we have from (\ref{eq:main-conc-ineq}) that
\[
\|\Phi(\bz)\| = \|\bz\| \left\|\Phi\left(\frac{\bz}{\|\bz\|}\right)\right\| \geq \epsilon \|\bz\|
\]
for all $\bz\in \cT_\A(\bxs)$ with high probability if $\lambda_n \geq w(\Omega)+\sqrt{n}\epsilon$. In this case, we can apply Proposition~\ref{prop:noisy-recovery} to conclude that $\|\hat{\bx}-\bxs\|\leq \frac{2\delta}{\epsilon}$. To verify that concentration of measure can be applied is more or less the same as in the proof of Part 1. First, note that under the assumptions of the theorem
\[
w(\Omega)^2 +1 \leq n(1-\epsilon)^2 -1/2 \leq n(1-\epsilon)^2 -2\epsilon(1-\epsilon) +\frac{\epsilon^2}{n} = \left(\sqrt{n}(1-\epsilon) - \frac{\epsilon}{\sqrt{n}} \right)^2
\]
as $\epsilon(1-\epsilon)\leq 1/4$ for $\epsilon\in (0,1)$. Using this fact, we then have
\[
\lambda_n - \sqrt{n} \epsilon \geq \frac{n- (n+1)\epsilon}{\sqrt{n+1}} \geq \sqrt{\frac{w(\Omega)^2+1}{1+1/n}} \geq w(\Omega)
\]
as desired.
\end{proof}
Gordon's theorem thus provides a simple characterization of the number of measurements required for reconstruction with the atomic norm. Indeed the Gaussian width of $\Omega = T_\A(\bxs) \cap \Sp^{p-1}$ is the only quantity that we need to compute in order to obtain bounds for both exact and robust recovery. Unfortunately it is in general not easy to compute Gaussian widths. Rudelson and Vershynin \cite{RudV2006} have worked out Gaussian widths for the special case of tangent cones at sparse vectors on the boundary of the $\ell_1$ ball, and derived results for sparse vector recovery using $\ell_1$ minimization that improve upon previous results. In the next section we give various well-known properties of the Gaussian width that are useful in computations. In Section~\ref{subsec:newwidthprop} we discuss a new approach to width computations that gives near-optimal recovery bounds in a variety of settings.
\subsection{Properties of Gaussian Width}
\label{subsec:widthprop}
The Gaussian width has deep connections to convex geometry. Since the length and direction of a Gaussian random vector are independent, one can verify that for $S \subset \R^p$
\[
w(S) = \frac{\lambda_p}{2} \int_{\Sp^{p-1}} \left(\max_{\bz\in S} \bu^T \bz -\min_{\bz\in S} \bu^T \bz\right)\,d\bu = \frac{\lambda_p}{2} \, b(S)
\]
where the integral is with respect to Haar measure on $\Sp^{p-1}$ and $b(S)$ is known as the \emph{mean width} of $S$. The mean width measures the average length of $S$ along unit directions in $\R^p$ and is one of the fundamental \emph{intrinsic volumes} of a body studied in combinatorial geometry~\cite{KlaR1997}. Any continuous valuation that is invariant under rigid motions and homogeneous of degree 1 is a multiple of the mean width and hence a multiple of the Gaussian width. We can use this connection with convex geometry to underscore several properties of the Gaussian width that are useful for computation.
The Gaussian width of a body is invariant under translations and unitary transformations. Moreover, it is homogeneous in the sense that $w(tK)$ = $tw(K)$ for $t>0$. The width is also monotonic. If $S_1 \subseteq S_2 \subseteq \R^p$, then it is clear from the definition of the Gaussian width that
\begin{equation*}
w(S_1) \leq w(S_2).
\end{equation*}
Less obvious, the width is modular in the sense that if $S_1$ and $S_2$ are convex bodies with $S_1\cup S_2$ convex, we also have
\[
w(S_1 \cup S_2) + w(S_1 \cap S_2) = w(S_1)+w(S_2)\,.
\]
This equality follows from the fact that $w$ is a valuation~\cite{Bar2002}. Also note that if we have a set $S \subseteq \R^p$, then the Gaussian width of $S$ is equal to the Gaussian width of the convex hull of $S$:
\begin{equation*}
w(S) = w(\mathrm{conv}(S)).
\end{equation*}
This result follows from the basic fact in convex analysis that the maximum of a convex function over a convex set is achieved at an extreme point of the convex set.
If $V \subset \R^p$ is a subspace in $\R^p$, then we have that
\begin{equation*}
w(V \cap \Sp^{p-1}) = \sqrt{\mathrm{dim}(V)},
\end{equation*}
which follows from standard results on random Gaussians. This result also agrees with the intuition that a random Gaussian map $\Phi$ misses a $k$-dimensional subspace with high probability as long as $\mathrm{dim}(\n(\Phi)) \geq k + 1$. Finally, if a cone $S \subset \R^p$ is such that $S = S_1 \oplus S_2$, where $S_1 \subset \R^p$ is a $k$-dimensional cone, $S_2 \subset \R^p$ is a $(p-k)$-dimensional cone that is orthogonal to $S_1$, and $\oplus$ denotes the direct sum operation, then the width can be decomposed as follows:
\begin{equation*}
w(S \cap \Sp^{p-1})^2 \leq w(S_1 \cap \Sp^{p-1})^2 + w(S_2 \cap \Sp^{p-1})^2.
\end{equation*}
These observations are useful in a variety of situations. For example a width computation that frequently arises is one in which $S = S_1 \oplus S_2$ as described above, with $S_1$ being a $k$-dimensional subspace. It follows that the width of $S \cap \Sp^{p-1}$ is bounded as
\begin{equation}
w(S \cap \Sp^{p-1})^2 \leq k + w(S_2 \cap \Sp^{p-1})^2. \label{eq:width-sub-cone}
\end{equation}
Another tool for computing Gaussian widths is based on Dudley's inequality \cite{Dud1967,LedT1991}, which bounds the width of a set in terms of the covering number of the set at all scales.
\begin{definition}
Let $S$ be an arbitrary compact subset of $\mathbb{R}^p$. The \emph{covering number} of $S$ in the Euclidean norm at resolution $\epsilon$ is the smallest number, $\mathfrak{N}(S,\epsilon)$, such that $\mathfrak{N}(S,\epsilon)$ Euclidean balls of radius $\epsilon$ cover $S$.
\end{definition}
\begin{theorem}[Dudley's Inequality]
\label{theo:dudley}
Let $S$ be an arbitrary compact subset of $\R^p$, and let $\bg$ be a random vector with i.i.d. zero-mean, unit-variance Gaussian entries. Then
\begin{equation}
w(S) \leq 24 \int_0^\infty \sqrt{ \log(\mathfrak{N}(S,\epsilon))} d\epsilon.
\end{equation}
\end{theorem}
We note here that a weak converse to Dudley's inequality can be obtained via Sudakov's Minoration \cite{LedT1991} by using the covering number for just a single scale. Specifically, we have the following \emph{lower bound} on the Gaussian width of a compact subset $S \subset \R^p$ for any $\epsilon > 0$:
\begin{equation*}
w(S) \geq c \epsilon \sqrt{ \log(\mathfrak{N}(S,\epsilon))}.
\end{equation*}
Here $c > 0$ is some universal constant.
Although Dudley's inequality can be applied quite generally, estimating covering numbers is difficult in most instances. There are a few simple characterizations available for spheres and Sobolev spaces, and some tractable arguments based on Maurey's empirical method \cite{LedT1991}. However it is not evident how to compute these numbers for general convex cones. Also, in order to apply Dudley's inequality we need to estimate the covering number at all scales. Further Dudley's inequality can be quite loose in its estimates, and it often introduces extraneous polylogarithmic factors. In the next section we describe a new mechanism for estimating Gaussian widths, which provides near-optimal guarantees for recovery of sparse vectors and low-rank matrices, as well as for several of the recovery problems discussed in Section~\ref{subsec:newrec}.
\subsection{New Results on Gaussian Width}
\label{subsec:newwidthprop}
We now present a framework for computing Gaussian widths by bounding the Gaussian width of a cone via the distance to the dual cone. To be fully general let $\mathcal{C}$ be a non-empty convex cone in $\R^p$, and let $\mathcal{C}^\ast$ denote the polar of $\mathcal{C}$. We can then upper bound the Gaussian width of any cone $\mathcal{C}$ in terms of the polar cone $\mathcal{C}^\ast$:
\begin{proposition} \label{prop:dual-width}
Let $\mathcal{C}$ be any non-empty convex cone in $\R^p$, and let $\bg \sim \mathcal{N}(0,I)$ be a random Gaussian vector. Then we have the following bound:
\begin{equation*}
w(\mathcal{C} \cap \Sp^{p-1}) \leq \E_\bg \left[\mathrm{dist}(\bg, \mathcal{C}^\ast) \right],
\end{equation*}
where $\mathrm{dist}$ here denotes the Euclidean distance between a point and a set.
\end{proposition}
The proof is given in Appendix~\ref{app:dual-width}, and it follows from an appeal to convex duality. Proposition~\ref{prop:dual-width} is more or less a restatement of the fact that the support function of a convex cone is equal to the distance to its polar cone. As it is the square of the Gaussian width that is of interest to us (see Corollary~\ref{corl:width}), it is often useful to apply Jensen's inequality to make the following approximation:
\begin{equation}\label{eq:square-jensen-bound}
\E_\bg[\mathrm{dist}(\bg,\mathcal{C}^\ast)]^2 \leq \E_\bg[\mathrm{dist}(\bg,\mathcal{C}^\ast)^2].
\end{equation}
The inspiration for our characterization in Proposition~\ref{prop:dual-width} of the width of a cone in terms of the expected distance to its dual came from the work of Stojnic \cite{Sto2009}, who used linear programming duality to construct Gaussian-width-based estimates for analyzing recovery in sparse reconstruction problems. Specifically, Stojnic's relatively simple approach recovered well-known phase transitions in sparse signal recovery \cite{DonT2005}, and also generalized to block sparse signals and other forms of structured sparsity.
This new dual characterization yields a number of useful bounds on the Gaussian width, which we describe here. In the following section we use these bounds to derive new recovery results. The first result is a bound on the Gaussian width of a cone in terms of the Gaussian width of its polar.
\begin{lemma} \label{lemm:dual-width}
Let $\mathcal{C} \subseteq \R^p$ be a non-empty closed, convex cone. Then we have that
\begin{equation*}
w(\mathcal{C} \cap \Sp^{p-1})^2 + w(\mathcal{C}^\ast \cap \Sp^{p-1})^2 \leq p.
\end{equation*}
\end{lemma}
\begin{proof}
Combining Proposition~\ref{prop:dual-width} and \eqref{eq:square-jensen-bound}, we have that
\begin{equation*}
w(\mathcal{C} \cap \Sp^{p-1})^2 \leq \E_\bg\left[\mathrm{dist}(\bg,\mathcal{C}^\ast)^2 \right],
\end{equation*}
where as before $\bg \sim \mathcal{N}(0,I)$. For any $\bz \in \R^p$ we let $\Pi_\mathcal{C}(\bz) = \arg \inf_{\bu \in \mathcal{C}} \|\bz - \bu \|$ denote the projection of $\bz$ onto $\mathcal{C}$. From standard results in convex analysis \cite{Roc1996}, we note that one can decompose any $\bz \in \R^p$ into orthogonal components as follows:
\begin{equation*}
\bz = \Pi_\mathcal{C}(\bz) + \Pi_{\mathcal{C}^\ast}(\bz), ~~~ \langle \Pi_\mathcal{C}(\bz), \Pi_{\mathcal{C}^\ast}(\bz) \rangle = 0.
\end{equation*}
Therefore we have the following sequence of bounds:
\begin{eqnarray*}
w(\mathcal{C} \cap \Sp^{p-1})^2 &\leq& \E_\bg\left[\mathrm{dist}(\bg,\mathcal{C}^\ast)^2 \right] \\ &=& \E_\bg\left[ \|\Pi_{\mathcal{C}}(\bg)\|^2\right] \\ &=& \E_\bg\left[ \|\bg\|^2 - \|\Pi_{\mathcal{C}^\ast}(\bg)\|^2\right] \\ &=& p - \E_\bg \left[\|\Pi_{\mathcal{C}^\ast}(\bg)\|^2 \right] \\ &=& p - \E_\bg \left[ \mathrm{dist}(\bg, \mathcal{C})^2 \right] \\ &\leq& p - w(\mathcal{C}^\ast \cap \Sp^{p-1})^2.
\end{eqnarray*}
\end{proof}
In many recovery problems one is interested in computing the width of a self-dual cone. For such cones the following corollary to Lemma~\ref{lemm:dual-width} gives a simple solution:
\begin{corollary} \label{corl:selfdual}
Let $\mathcal{C} \subset \R^p$ be a self-dual cone, i.e., $\mathcal{C} = -\mathcal{C}^\ast$. Then we have that
\begin{equation*}
w(\mathcal{C} \cap \Sp^{p-1})^2 \leq \frac{p}{2}.
\end{equation*}
\end{corollary}
\begin{proof}
The proof follows directly from Lemma~\ref{lemm:dual-width} as $w(\mathcal{C} \cap \Sp^{p-1})^2 = w(\mathcal{C}^\ast \cap \Sp^{p-1})^2$.
\end{proof}
Our next bound for the width of a cone $\mathcal{C}$ is based on the volume of its polar $\mathcal{C}^\ast \cap \Sp^{p-1}$. The \emph{volume} of a measurable subset of the sphere is the fraction of the sphere $\Sp^{p-1}$ covered by the subset. Thus it is a quantity between zero and one.
\begin{theorem} [Gaussian width from volume of the polar]
\label{theo:angle}
Let $\mathcal{C} \subseteq \R^p$ be any closed, convex, solid cone, and suppose that its polar $\mathcal{C}^\ast$ is such that $\mathcal{C}^\ast \cap \Sp^{p-1}$ has a volume of $\Theta \in [0, 1]$. Then for $p \geq 9$ we have that
\begin{equation*}
w(\mathcal{C} \cap \Sp^{p-1}) \leq 3 \sqrt{\log\left(\frac{4}{\Theta}\right)}.
\end{equation*}
\end{theorem}
The proof of this theorem is given in Appendix~\ref{app:width-angle}. The main property that we appeal to in the proof is \emph{Gaussian isoperimetry}. In particular there is a formal sense in which a spherical cap\footnote{A \emph{spherical cap} is a subset of the sphere obtained by intersecting the sphere $\Sp^{p-1}$ with a halfspace.} is the ``extremal case'' among all subsets of the sphere with a given volume $\Theta$. Other than this observation the proof mainly involves a sequence of integral calculations.
Note that if we are given a specification of a cone $\mathcal{C} \subset \R^p$ in terms of a membership oracle, it is possible to efficiently obtain good numerical estimates of the volume of $\mathcal{C} \cap \Sp^{p-1}$ \cite{DyeFK1991}. Moreover, simple symmetry arguments often give relatively accurate estimates of these volumes. Such estimates can then be plugged into Theorem~\ref{theo:angle} to yield bounds on the width.
\subsection{New Recovery Bounds}
\label{subsec:newrec}
We use the bounds derived in the last section to obtain new recovery results. First using the dual characterization of the Gaussian width in Proposition~\ref{prop:dual-width}, we are able to obtain sharp bounds on the number of measurements required for recovering sparse vectors and low-rank matrices from random Gaussian measurements using convex optimization (i.e., $\ell_1$-norm and nuclear norm minimization).
\begin{proposition}\label{prop:l1}
Let $\bxs \in \R^p$ be an $s$-sparse vector. Letting $\A$ denote the set of unit-Euclidean-norm one-sparse vectors, we have that
\begin{equation*}
w(T_{\A}(\bxs) \cap \Sp^{p-1})^2 \leq 2s \log\left(\tfrac{p}{s}\right) + \tfrac{5}{4}s\,.
\end{equation*}
Thus, $2s \log\left(\tfrac{p}{s}\right) + \tfrac{5}{4}s+1$ random Gaussian measurements suffice to recover $\bxs$ via $\ell_1$ norm minimization with high probability.
\end{proposition}
\begin{proposition}\label{prop:nuclear}
Let $\bxs$ be an $m_1 \times m_2$ rank-$r$ matrix with $m_1 \leq m_2$. Letting $\A$ denote the set of unit-Euclidean-norm rank-one matrices, we have that
\begin{equation*}
w\left(T_{\A}(\bxs) \cap \Sp^{m_1 m_2-1}\right)^2 \leq 3 r(m_1+m_2-r).
\end{equation*}
Thus $3 r(m_1+m_2-r)+1$ random Gaussian measurements suffice to recover $\bxs$ via nuclear norm minimization with high probability.
\end{proposition}
The proofs of these propositions are given in Appendix~\ref{app:direct}. The number of measurements required by these bounds is on the same order as previously known results. In the case of sparse vectors, previous results getting $2s\log (p/s)$ were asymptotic \cite{DonT2009}. Our bounds, in contrast, hold with high probability in finite dimensions. In the case of low-rank matrices, our bound provides considerably sharper constants than those previously derived (as in, for example~\cite{CanP2009}). We also note that we have robust recovery at these thresholds. Further these results do not require explicit recourse to any type of restricted isometry property \cite{CanP2009}, and the proofs are simple and based on elementary integrals.
Next we obtain a set of recovery results by appealing to Corollary~\ref{corl:selfdual} on the width of a self-dual cone. These examples correspond to the recovery of individual atoms (i.e., the extreme points of the set $\ch(\A)$), although the same machinery is applicable in principle to estimate the number of measurements required to recover models formed as sums of a few atoms (i.e., points lying on low-dimensional faces of $\ch(\A)$). We first obtain a well-known result on the number of measurements required for recovering sign-vectors via $\ell_\infty$ norm minimization.
\begin{proposition} \label{prop:sign}
Let $\A \in \{-1,+1\}^p$ be the set of sign-vectors in $\R^p$. Suppose $\bxs \in \R^p$ is a vector formed as a convex combination of $k$ sign-vectors in $\A$ such that $\bxs$ lies on a $k$-face of the $\ell_\infty$-norm unit ball. Then we have that
\begin{equation*}
w(T_{\A}(\bxs) \cap \Sp^{p-1})^2 \leq \frac{p+k}{2}.
\end{equation*}
Thus $\tfrac{p+k}{2}$ random Gaussian measurements suffice to recover $\bxs$ via $\ell_\infty$-norm minimization with high probability.
\end{proposition}
\begin{proof}
The tangent cone at $\bxs$ with respect to the $\ell_\infty$-norm ball is the direct sum of a $k$-dimensional subspace and a (rotated) $(p-k)$-dimensional nonnegative orthant. As the orthant is self-dual, we obtain the required bound by combining Corollary~\ref{corl:selfdual} and \eqref{eq:width-sub-cone}.
\end{proof}
This result agrees with previously computed bounds in \cite{ManR2009,DonT2010}, which relied on a more complicated combinatorial argument. Next we compute the number of measurements required to recover orthogonal matrices via spectral-norm minimization (see Section~\ref{subsec:ex}). Let $\mathbb{O}(m)$ denote the group of $m \times m$ orthogonal matrices, viewed as a subgroup of the set of nonsingular matrices in $\R^{m \times m}$.
\begin{proposition} \label{prop:ortho}
Let $\bxs \in \R^{m \times m}$ be an orthogonal matrix, and let $\A$ be the set of all orthogonal matrices. Then we have that
\begin{equation*}
w(T_{\A}(\bxs) \cap \Sp^{m^2-1})^2 \leq \frac{3 m^2 - m}{4}.
\end{equation*}
Thus $\tfrac{3 m^2 - m}{4}$ random Gaussian measurements suffice to recover $\bxs$ via spectral-norm minimization with high probability.
\end{proposition}
\begin{proof}
Due to the symmetry of the orthogonal group, it suffices to consider the tangent cone at the identity matrix $I$ with respect to the spectral norm ball. Recall that the spectral norm ball is the convex hull of the orthogonal matrices. Therefore the \emph{tangent space} at the identity matrix with respect to the orthogonal group $\mathbb{O}(m)$ is a subset of the tangent cone $T_\A(I)$. It is well-known that this tangent space is the Lie Algebra of all $m \times m$ skew-symmetric matrices. Thus we only need to compute the component $S$ of $T_\A(I)$ that lies in the subspace of symmetric matrices:
\begin{eqnarray*}
S &=& \mathrm{cone}\{M - I : \|M\|_\A \leq 1, ~ M \mathrm{~ symmetric} \} \\ &=& \mathrm{cone}\{U D U^T - U U^T : \|D\|_\A \leq 1, ~ D \mathrm{~ diagonal}, ~ U \in \mathbb{O}(m) \} \\ &=& \mathrm{cone}\{U (D-I) U^T: \|D\|_\A \leq 1, ~ D \mathrm{~ diagonal}, ~ U \in \mathbb{O}(m) \} \\ &=& -\mathrm{PSD}_m.
\end{eqnarray*}
Here $\mathrm{PSD}_m$ denotes the set of $m \times m$ symmetric positive-semidefinite matrices. As this cone is self-dual, we can apply Corollary~\ref{corl:selfdual} in conjunction with the observations in Section~\ref{subsec:widthprop} to conclude that
\begin{equation*}
w(T_\A(I) \cap \Sp^{m^2-1})^2 \leq {m \choose 2} + \frac{1}{2}{m+1 \choose 2} = \frac{3m^2 - m}{4}.
\end{equation*}
\end{proof}
We note that the number of degrees of freedom in an $m \times m$ orthogonal matrix (i.e., the dimension of the manifold of orthogonal matrices) is $\tfrac{m(m-1)}{2}$. Proposition~\ref{prop:sign} and Proposition~\ref{prop:ortho} point to the importance of obtaining recovery bounds with sharp constants. Larger constants in either result would imply that the number of measurements required exceeds the ambient dimension of the underlying $\bxs$. In these and many other cases of interest Gaussian width arguments not only give order-optimal recovery results, but also provide precise constants that result in sharp recovery thresholds.
Finally we give a third set of recovery results that appeal to the Gaussian width bound of Theorem~\ref{theo:angle}. The following measurement bound applies to cases when $\ch(\A)$ is a \emph{symmetric polytope} (roughly speaking, all the vertices are ``equivalent''), and is a simple corollary of Theorem~\ref{theo:angle}.
\begin{corollary} \label{corl:symm}
Suppose that the set $\A$ is a finite collection of $m$ points, with the convex hull $\ch(\A)$ being a vertex-transitive polytope \cite{Zie1995} whose vertices are the points in $\A$. Using the convex program \eqref{eq:atomic-norm-primal} we have that $9 \log(m)$ random Gaussian measurements suffice, with high probability, for exact recovery of a point in $\A$, i.e., a vertex of $\ch(\A)$.
\end{corollary}
\begin{proof}
We recall the basic fact from convex analysis that the normal cones at the vertices of a convex polytope in $\R^p$ provide a partitioning of $\R^p$. As $\ch(\A)$ is a vertex-transitive polytope, the normal cone at a vertex covers $\tfrac{1}{m}$ fraction of $\R^p$. Applying Theorem~\ref{theo:angle}, we have the desired result.
\end{proof}
Clearly we require the number of vertices to be bounded as $m \leq \exp\{\tfrac{p}{9}\}$, so that the estimate of the number of measurements is not vacuously true. This result has useful consequences in settings in which $\ch(\A)$ is a \emph{combinatorial polytope}, as such polytopes are often vertex-transitive. We have the following example on the number of measurements required to recover permutation matrices\footnote{While Proposition~\ref{prop:birkhoff} follows as a consequence of the general result in Corollary~\ref{corl:symm}, one can remove the constant factor $9$ in the statement of Proposition~\ref{prop:birkhoff} by carrying out a more refined analysis of the Birkhoff polytope.}:
\begin{proposition} \label{prop:birkhoff}
Let $\bxs \in \R^{m \times m}$ be a permutation matrix, and let $\A$ be the set of all $m \times m$ permutation matrices. Then $9 \, m \log(m)$ random Gaussian measurements suffice, with high probability, to recover $\bxs$ by solving the optimization problem \eqref{eq:atomic-norm-primal}, which minimizes the norm induced by the Birkhoff polytope of doubly stochastic matrices.
\end{proposition}
\begin{proof}
This result follows from Corollary~\ref{corl:symm} by noting that there are $m!$ permutation matrices of size $m \times m$.
\end{proof}
\section{Representability and Algebraic Geometry of Atomic Norms}
\label{sec:rep}
All of our discussion thus far has focussed on arbitrary atomic sets $\A$. As seen in Section~\ref{sec:def} the geometry of the convex hull $\ch(\A)$ completely determines conditions under which exact recovery is possible using the convex program \eqref{eq:atomic-norm-primal}. In this section we address the question of computing atomic norms for general sets of atoms. These issues are critical in order to be able to solve the convex optimization problem \eqref{eq:atomic-norm-primal}. Although the convex hull $\ch(\A)$ is always a mathematically well-defined object, testing membership in this set is in general undecidable (for example, if $\A$ is a fractal). Further, even if these convex hulls are computable they may not admit efficient representations. For example if $\A$ is the set of rank-one signed matrices (see Section~\ref{subsec:ex}), the corresponding convex hull $\ch(\A)$ is the cut polytope for which there is no known tractable characterization. Consequently, one may have to resort to efficiently computable approximations of $\ch(\A)$. The tradeoff in using such approximations in our atomic norm minimization framework is that we require more measurements for robust recovery. This section is devoted to providing a better understanding of these issues.
\subsection{Role of Algebraic Structure}
\label{subsec:algst}
In order to obtain exact or approximate representations (analogous to the cases of the $\ell_1$ norm and the nuclear norm) it is important to identify properties of the atomic set $\A$ that can be exploited computationally. We focus on cases in which the set $\A$ has algebraic structure. Specifically let the ring of multivariate polynomials in $p$ variables be denoted by $\R[\bx] = \R[\bx_1,\dots,\bx_p]$. We then consider real algebraic varieties \cite{BocCR1998}:
\begin{definition}
A \emph{real algebraic variety} $S \subseteq \R^p$ is the set of real solutions of a system of polynomial equations:
\begin{equation*}
S = \{\bx : g_j(\bx) = 0, ~ \forall j\},
\end{equation*}
where $\{g_j\}$ is a finite collection of polynomials in $\R[\bx]$.
\end{definition}
Indeed all of the atomic sets $\A$ considered in this paper are examples of algebraic varieties. Algebraic varieties have the remarkable property that (the closure of) their convex hull can be arbitrarily well-approximated in a constructive manner as (the projection of) a set defined by linear matrix inequality constraints \cite{GouPT2010,Par2003}. A potential complication may arise, however, if these semidefinite representations are intractable to compute in polynomial time. In such cases it is possible to approximate the convex hulls via a hierarchy of tractable semidefinite relaxations. We describe these results in more detail in Section~\ref{subsec:psatz}. Therefore the atomic norm minimization problems such as \eqref{eq:noisy-atomic-norm-primal} arising in such situations can be solved exactly or approximately via semidefinite programming.
Algebraic structure also plays a second important role in atomic norm minimization problems. If an atomic norm $\| \cdot \|_\A$ is intractable to compute, we may approximate it via a more tractable norm $\|\cdot\|_{app}$. However not every approximation of the atomic norm is equally good for solving inverse problems. As illustrated in Figure~\ref{fig:fig2} we can construct approximations of the $\ell_1$ ball that are tight in a \emph{metric} sense, with $(1-\epsilon) \| \cdot \|_{app} \leq \|\cdot\|_{\ell_1} \leq (1+\epsilon) \|\cdot\|_{app}$, but where the tangent cones at sparse vectors in the new norm are halfspaces. In such a case, the number of measurements required to recover the sparse vector ends up being on the same order as the ambient dimension. (Note that the $\ell_1$-norm is in fact tractable to compute; we simply use it here for illustrative purposes.) The key property that we seek in approximations to an atomic norm $\|\cdot\|_\A$ is that they \emph{preserve algebraic structure} such as the vertices/extreme points and more generally the low-dimensional faces of the $\ch(\A)$. As discussed in Section~\ref{subsec:why} points on such low-dimensional faces correspond to simple models, and algebraic-structure preserving approximations ensure that the tangent cones at simple models with respect to the approximations are not too much larger than the corresponding tangent cones with respect to the original atomic norms (see Section~\ref{subsec:tradeoff} for a concrete example).
\begin{figure}
\begin{center}
\epsfig{file=fig2.eps,width=4cm,height=4cm} \caption{The convex body given by the dotted line is a good metric approximation to the $\ell_1$ ball. However as its ``corners'' are ``smoothed out'', the tangent cone at $\bxs$ goes from being a proper cone (with respect to the $\ell_1$ ball) to a halfspace (with respect to the approximation).} \label{fig:fig2}
\end{center}
\end{figure}
\subsection{Semidefinite Relaxations using Theta Bodies}
\label{subsec:psatz}
In this section we give a family of semidefinite relaxations to the atomic norm minimization problem whenever the atomic set has algebraic structure. To begin with if we approximate the atomic norm $\|\cdot\|_\A$ by another atomic norm $\|\cdot\|_{\tilde{\A}}$ defined using a \emph{larger} collection of atoms $\A \subseteq \tilde{\A}$, it is clear that
\begin{equation*}
\| \cdot \|_{\tilde{\A}} \leq \|\cdot\|_\A.
\end{equation*}
Consequently outer approximations of the atomic set give rise to approximate norms that provide lower bounds on the optimal value of the problem \eqref{eq:atomic-norm-primal}.
In order to provide such lower bounds on the optimal value of \eqref{eq:atomic-norm-primal}, we discuss semidefinite relaxations of the convex hull $\ch(\A)$. All our discussion here is based on results described in \cite{GouPT2010} for semidefinite relaxations of convex hulls of algebraic varieties using theta bodies. We only give a brief review of the relevant constructions, and refer the reader to the vast literature on this subject for more details (see \cite{GouPT2010,Par2003} and the references therein). For subsequent reference in this section, we recall the definition of a polynomial ideal \cite{BocCR1998,Har95}:
\begin{definition}
A \emph{polynomial ideal} $I \subset \R[\bx]$ is a subset of the ring of polynomials that contains the zero polynomial (the polynomial that is identically zero), is closed under addition, and has the property that $f \in I, g \in \R[\bx]$ implies that $f \cdot g \in I$.
\end{definition}
To begin with we note that a \emph{sum-of-squares} (SOS) polynomial in $\R[\bx]$ is a polynomial that can be written as the (finite) sum of squares of other polynomials in $\R[\bx]$. Verifying the nonnegativity of a multivariate polynomial is intractable in general, and therefore SOS polynomials play an important role in real algebraic geometry as an SOS polynomial is easily seen to be nonnegative everywhere. Further checking whether a polynomial is an SOS polynomial can be accomplished efficiently via semidefinite programming \cite{Par2003}.
Turning our attention to the description of the convex hull of an algebraic variety, we will assume for the sake of simplicity that the convex hull is closed. Let $I \subseteq \R[\bx]$ be a polynomial ideal, and let $V_\R(I) \in \R^p$ be its real algebraic variety:
\begin{equation*}
V_\R(I) = \{\bx : f(\bx) = 0, ~ \forall f \in I\}.
\end{equation*}
One can then show that the convex hull $\ch(V_\R(I))$ is given as:
\begin{eqnarray*}
\ch(V_\R(I)) &=& \{\bx : f(\bx) \geq 0, ~ \forall f ~\mathrm{linear ~ and ~ nonnegative ~ on ~} V_\R(I)\} \\ &=& \{\bx : f(\bx) \geq 0, ~ \forall f ~\mathrm{linear ~ s.t. ~} f = h + g, ~\forall~ h ~\mathrm{nonnegative, ~} \forall ~ g \in I\} \\ &=& \{\bx : f(\bx) \geq 0, ~ \forall f ~\mathrm{linear ~ s.t. ~} f ~\mathrm{nonnegative ~ modulo~} I\}.
\end{eqnarray*}
A linear polynomial here is one that has a maximum degree of one, and the meaning of ``modulo an ideal'' is clear. As nonnegativity modulo an ideal may be intractable to check, we can consider a relaxation to a polynomial being SOS modulo an ideal, i.e., a polynomial that can be written as $\sum_{i=1}^q ~ h_i^2 ~ + ~ g$ for $g$ in the ideal. Since it is tractable to check via semidefinite programmming whether bounded-degree polynomials are SOS, the $k$-th theta body of an ideal $I$ is defined as follows in \cite{GouPT2010}:
\begin{equation*}
\mathrm{TH}_k(I) = \{\bx : f(\bx) \geq 0, ~ \forall f ~\mathrm{linear ~ s.t. ~} f ~\mathrm{is~}k\mathrm{\mbox{-}sos ~ modulo~} I\}.
\end{equation*}
Here $k$-sos refers to an SOS polynomial in which the components in the SOS decomposition have degree at most $k$. The $k$-th theta body $\mathrm{TH}_k(I)$ is a convex relaxation of $\ch(V_\R(I))$, and one can verify that
\begin{equation*}
\ch(V_\R(I)) \subseteq \cdots \subseteq \mathrm{TH}_{k+1}(I) \subseteq \mathrm{TH}_{k}(V_\R(I)).
\end{equation*}
By the arguments given above (see also \cite{GouPT2010}) these theta bodies can be described using semidefinite programs of size polynomial in $k$. Hence by considering theta bodies $\mathrm{TH}_k(I)$ with increasingly larger $k$, one can obtain a hierarchy of tighter semidefinite relaxations of $\ch(V_\R(I))$. We also note that in many cases of interest such semidefinite relaxations preserve low-dimensional faces of the convex hull of a variety, although these properties are not known in general. We will use some of these properties below when discussing approximations of the cut-polytope.
\textbf{Approximating tensor norms.} We conclude this section with an example application of these relaxations to the problem of approximating the tensor nuclear norm. We focus on the case of tensors of order three that lie in $\R^{m \times m \times m}$, i.e., tensors indexed by three numbers, for notational simplicity, although our discussion is applicable more generally. In particular the atomic set $\A$ is the set of unit-Euclidean-norm rank-one tensors:
\begin{eqnarray*}
\A &=& \{\bu \otimes \bv \otimes \bw : \bu, \bv, \bw \in \R^m, ~ \|\bu\| = \|\bv\| = \|\bw\| = 1\} \\ &=& \{N \in \R^{m^3} : N = \bu \otimes \bv \otimes \bw, ~ \bu, \bv, \bw \in \R^m, ~ \|\bu\| = \|\bv\| = \|\bw\| = 1\},
\end{eqnarray*}
where $\bu \otimes \bv \otimes \bw$ is the tensor product of three vectors. Note that the second description is written as the projection onto $\R^{m^3}$ of a variety defined in $\R^{m^3+3m}$. The nuclear norm is then given by \eqref{eq:atnorm}, and is intractable to compute in general. Now let $I_\A$ denote a polynomial ideal of polynomial maps from $\R^{m^3+3m}$ to $\R$:
\begin{equation*}
I_\A = \{g: g = \sum_{i,j,k=1}^m g_{ijk} (N_{ijk} - \bu_i \bv_j \bw_k)+g_u (\bu^T \bu-1) + g_v (\bv^T \bv-1) + g_w (\bw^T \bw-1), \forall g_{ijk},g_u,g_v,g_w\}.
\end{equation*}
Here $g_u,g_v,g_w,\{g_{ijk}\}_{i,j,k}$ are polynomials in the variables $N,\bu,\bv,\bw$. Following the program described above for constructing approximations, a family of semidefinite relaxations to the tensor nuclear norm ball can be prescribed in this manner via the theta bodies $\mathrm{TH}_k(I_\A)$.
\subsection{Tradeoff between Relaxation and Number of Measurements}
\label{subsec:tradeoff}
As discussed in Section~\ref{subsec:why} the atomic norm is the best convex heuristic for solving ill-posed linear inverse problems of the type considered in this paper. However we may wish to approximate the atomic norm in cases when it is intractable to compute exactly, and the discussion in the preceding section provides one approach to constructing a family of relaxations. As one might expect the tradeoff for using such approximations, i.e., a \emph{weaker} convex heuristic than the atomic norm, is an increase in the number of measurements required for exact or robust recovery. The reason for this is that the approximate norms have \emph{larger} tangent cones at their extreme points, which makes it harder to satisfy the empty intersection condition of Proposition~\ref{prop:null-intersection}. We highlight this tradeoff here with an illustrative example involving the cut polytope.
The cut polytope is defined as the convex hull of all cut matrices:
\begin{equation*}
\mathcal{P} = \mathrm{conv}\{\bz \bz^T : \bz \in \{-1,+1\}^m\}.
\end{equation*}
As described in Section~\ref{subsec:ex} low-rank matrices that are composed of $\pm 1$'s as entries are of interest in collaborative filtering \cite{SreS2005}, and the norm induced by the cut polytope is a potential convex heuristic for recovering such matrices from limited measurements. However it is well-known that the cut polytope is intractable to characterize \cite{DezL1997}, and therefore we need to use tractable relaxations instead. We consider the following two relaxations of the cut polytope. The first is the popular relaxation that is used in semidefinite approximations of the MAXCUT problem:
\begin{equation*}
\mathcal{P}_1 = \{M : M ~\mathrm{symmetric}, ~ M \succeq 0, ~ M_{ii} = 1, \forall i = 1,\cdots,p \}.
\end{equation*}
This is the well-studied elliptope \cite{DezL1997}, and can be interpreted as the second theta body relaxation (see Section~\ref{subsec:psatz}) of the cut polytope $\mathcal{P}$ \cite{GouPT2010}. We also investigate the performance of a second, weaker relaxation:
\begin{equation*}
\mathcal{P}_2 = \{M: M ~\mathrm{symmetric}, ~ M_{ii} = 1, \forall i, ~ |M_{ij}| \leq 1, \forall i \neq j \}.
\end{equation*}
This polytope is simply the convex hull of symmetric matrices with $\pm 1$'s in the off-diagonal entries, and $1$'s on the diagonal. We note that $\mathcal{P}_2$ is an extremely weak relaxation of $\mathcal{P}$, but we use it here only for illustrative purposes. It is easily seen that
\begin{equation*}
\mathcal{P} \subset \mathcal{P}_1 \subset \mathcal{P}_2,
\end{equation*}
with all the inclusions being strict. Figure~\ref{fig:fig3} gives a toy sketch that highlights all the main geometric aspects of these relaxations. In particular $\mathcal{P}_1$ has many more extreme points that $\mathcal{P}$, although the set of vertices of $\mathcal{P}_1$, i.e., points that have full-dimensional normal cones, are precisely the cut matrices (which are the vertices of $\mathcal{P}$) \cite{DezL1997}. The convex polytope $\mathcal{P}_2$ contains many more vertices compared to $\mathcal{P}$ as shown in Figure~\ref{fig:fig3}. As expected the tangent cones at vertices of $\mathcal{P}$ become increasingly larger as we use successively weaker relaxations. The following result summarizes the number of random measurements required for recovering a cut matrix, i.e., a rank-one sign matrix, using the norms induced by each of these convex bodies.
\begin{figure}
\begin{center}
\epsfig{file=fig3.eps,width=6cm,height=4cm} \caption{A toy sketch illustrating the cut polytope $\mathcal{P}$, and the two approximations $\mathcal{P}_1$ and $\mathcal{P}_2$. Note that $\mathcal{P}_1$ is a sketch of the standard semidefinite relaxation that has the \emph{same} vertices as $\mathcal{P}$. On the other hand $\mathcal{P}_2$ is a polyhedral approximation to $\mathcal{P}$ that has many more vertices as shown in this sketch.} \label{fig:fig3}
\end{center}
\end{figure}
\begin{proposition} \label{prop:cut}
Suppose $\bxs \in \R^{m \times m}$ is a rank-one sign matrix, i.e., a cut matrix, and we are given $n$ random Gaussian measurements of $\bxs$. We wish to recover $\bxs$ by solving a convex program based on the norms induced by each of $\mathcal{P}, \mathcal{P}_1, \mathcal{P}_2$. We have exact recovery of $\bxs$ in each of these cases with high probability under the following conditions on the number of measurements:
\begin{enumerate}
\item Using $\mathcal{P}$: $n = \mathcal{O}(m)$.
\item Using $\mathcal{P}_1$: $n = \mathcal{O}(m)$.
\item Using $\mathcal{P}_2$: $n = \tfrac{m^2-m}{4}$.
\end{enumerate}
\end{proposition}
\begin{proof}
For the first part, we note that $\mathcal{P}$ is a symmetric polytope with $2^{m-1}$ vertices. Therefore we can apply Corollary~\ref{corl:symm} to conclude that $n = \mathcal{O}(m)$ measurements suffices for exact recovery.
For the second part we note that the tangent cone at $\bxs$ with respect to the nuclear norm ball of $m \times m$ matrices contains within it the tangent cone at $\bxs$ with respect to the polytope $\mathcal{P}_1$. Hence we appeal to Proposition~\ref{prop:nuclear} to conclude that $n = \mathcal{O}(m)$ measurements suffices for exact recovery.
Finally, we note that $\mathcal{P}_2$ is essentially the hypercube in ${m \choose 2}$ dimensions. Appealing to Proposition~\ref{prop:sign}, we conclude that $n = \tfrac{m^2-m}{4}$ measurements suffices for exact recovery.
\end{proof}
It is not too hard to show that these bounds are order-optimal, and that they cannot be improved. Thus we have a rigorous demonstration in this particular instance of the fact that the number of measurements required for exact recovery increases as the relaxations get weaker (and as the tangent cones get larger). The principle underlying this illustration holds more generally, namely that there exists a tradeoff between the complexity of the convex heuristic and the number of measurements required for exact or robust recovery. It would be of interest to quantify this tradeoff in other settings, for example, in problems in which we use increasingly tighter relaxations of the atomic norm via theta bodies.
We also note that the tractable relaxation based on $\mathcal{P}_1$ is only off by a constant factor with respect to the optimal heuristic based on the cut polytope $\mathcal{P}$. This suggests the potential for tractable heuristics to approximate hard atomic norms with provable approximation ratios, akin to methods developed in the literature on approximation algorithms for hard combinatorial optimization problems.
\subsection{Terracini's Lemma and Lower Bounds on Recovery}
\label{subsec:terracini}
Algebraic structure in the atomic set $\A$ also provides a means for computing \emph{lower bounds} on the number of measurements required for exact recovery. The recovery condition of Proposition~\ref{prop:null-intersection} states that the nullspace $\n(\Phi)$ of the measurement operator $\Phi: \R^p \rightarrow \R^n$ must miss the tangent cone $T_\A(\bxs)$ at the point of interest $\bxs$. Suppose that this tangent cone contains a $q$-dimensional subspace. It is then clear from straightforward linear algebra arguments that the number of measurements $n$ must exceed $q$. Indeed this bound must hold for \emph{any} linear measurement scheme. Thus the dimension of the subspace contained inside the tangent cone (i.e., the dimension of the lineality space) provides a simple lower bound on the number of linear measurements.
In this section we discuss a method to obtain estimates of the dimension of a subspace component of the tangent cone. We focus again on the setting in which $\A$ is an algebraic variety. Indeed in all of the examples of Section~\ref{subsec:ex}, the atomic set $\A$ is an algebraic variety. In such cases simple models $\bxs$ formed according to \eqref{eq:simp1} can be viewed as elements of \emph{secant varieties}.
\begin{definition}
Let $\A \in \R^p$ be an algebraic variety. Then the $k$'th \emph{secant variety} $\A^k$ is defined as the union of all affine spaces passing through any $k+1$ points of $\A$.
\end{definition}
Secant varieties and their tangent spaces have been extensively studied in algebraic geometry \cite{Har95}. A particular question of interest is to characterize the dimensions of secant varieties and tangent spaces. In our context, estimates of these dimensions are useful in giving lower bounds on the number of measurements required for recovery. Specifically we have the following result, which states that certain linear spaces must lie in the tangent cone at $\bxs$ with respect to $\ch(\A)$:
\begin{proposition} \label{prop:tspace}
Let $\A \subset \R^p$ be a smooth variety, and let $\mathcal{T}(\bu,\A)$ denote the tangent space at any $\bu \in \A$ with respect to $\A$. Suppose $\bx = \sum_{i=1}^k c_i \ba_i, ~ \forall \ba_i \in \A, c_i \geq 0$, such that
\begin{equation*}
\|\bx\|_\A = \sum_{i=1}^k c_i.
\end{equation*}
Then the \emph{tangent cone} $T_\A(\bxs)$ contains the following linear space:
\begin{equation*}
\mathcal{T}(\ba_1,\A) \oplus \cdots \oplus \mathcal{T}(\ba_k,\A) \subset T_\A(\bxs),
\end{equation*}
where $\oplus$ denotes the direct sum of subspaces.
\end{proposition}
\begin{proof}
We note that if we perturb $\ba_1$ slightly to \emph{any} neighboring $\ba_1'$ so that $\ba_1' \in \A$, then the resulting $\bx' = c_1 \ba_1' + \sum_{i = 2}^k c_2 \ba_i$ is such that $\|\bx'\|_\A \leq \|\bx\|_\A$. The proposition follows directly from this observation.
\end{proof}
This result is applicable, for example, when $\A$ is the variety of rank-one matrices or the variety of rank-one tensors as these are smooth varieties. By Terracini's lemma \cite{Har95} from algebraic geometry the subspace $\mathcal{T}(\ba_1,\A) \oplus \cdots \oplus \mathcal{T}(\ba_k,\A)$ is in fact the estimate for the tangent space $\mathcal{T}(\bx,\A^{k-1})$ at $\bx$ with respect to the $(k-1)$'th secant variety $\A^{k-1}$:
\begin{proposition}[Terracini's Lemma] \label{prop:terracini}
Let $\A \subset \R^p$ be a smooth affine variety, and let $\mathcal{T}(\bu,\A)$ denote the tangent space at any $\bu \in \A$ with respect to $\A$. Suppose $\bx \in \A^{k-1}$ is a generic point such that $\bx = \sum_{i=1}^k c_i \ba_i, ~ \forall \ba_i \in \A, c_i \geq 0$. Then the tangent space $\mathcal{T}(\bx,\A^{k-1})$ at $\bx$ with respect to the secant variety $\A^{k-1}$ is given by $\mathcal{T}(\ba_1,\A) \oplus \cdots \oplus \mathcal{T}(\ba_k,\A)$. Moreover the dimension of $\mathcal{T}(\bx,\A^{k-1})$ is at most (and is expected to be) $\min\{p, (k+1)\mathrm{dim}(\A) + k\}$.
\end{proposition}
Combining these results we have that estimates of the dimension of the tangent space $\mathcal{T}(\bx,\A^{k-1})$ lead directly to lower bounds on the number of measurements required for recovery. The intuition here is clear as the number of measurements required must be bounded below by the number of ``degrees of freedom,'' which is captured by the dimension of the tangent space $\mathcal{T}(\bx,\A^{k-1})$. However Terracini's lemma provides us with general estimates of the dimension of $\mathcal{T}(\bx,\A^{k-1})$ for generic points $\bx$. Therefore we can directly obtain lower bounds on the number of measurements, purely by considering the dimension of the variety $\A$ and the number of elements from $\A$ used to construct $\bx$ (i.e., the order of the secant variety in which $\bx$ lies). As an example the dimension of the base variety of normalized order-three tensors in $\R^{m \times m \times m}$ is $3(m-1)$. Consequently if we were to in principle solve the tensor nuclear norm minimization problem, we should expect to require at least $\mathcal{O}(km)$ measurements to recover a rank-$k$ tensor.
\section{Computational Experiments}
\label{sec:comp}
\subsection{Algorithmic Considerations}
\label{subsec:algo}
While a variety of atomic norms can be represented or approximated by linear matrix inequalities, these representations do not necessarily translate into practical implementations. Semidefinite programming can be technically solved in polynomial time, but general interior point solvers typically only scale to problems with a few hundred variables. For larger scale problems, it is often preferable to exploit structure in the atomic set $\A$ to develop fast, first-order algorithms.
A starting point for first-order algorithm design lies in determining the structure of the proximity operator (or Moreau envelope) associated with the atomic norm,
\begin{equation}
\Pi_\A(\bx;\mu): = \arg \min_\bz \tfrac{1}{2} \|\bz-\bx\|^2 + \mu\|\bz\|_{\A}\,.
\end{equation}
Here $\mu$ is some positive parameter. Proximity operators have already been harnessed for fast algorithms involving the $\ell_1$ norm \cite{FigN2003,ComW2005,DauDD2004,HalYZ2008,WriNF2009} and the nuclear norm \cite{MaGC2008,CaiCS2008,TohY2009} where these maps can be quickly computed in closed form. For the $\ell_1$ norm, the $i$th component of $\Pi_{\A}(\bx;\mu)$ is given by
\begin{equation}
\Pi_{\A}(\bx;\mu)_i = \begin{cases} \bx_i+\mu & \bx_i<-\mu\\
0 & -\mu \leq \bx_i \leq \mu\\
\bx_i-\mu & \bx_i>\mu
\end{cases}\,.
\end{equation}
This is the so-called \emph{soft thresholding} operator. For the nuclear norm, $\Pi_{\A}$ soft thresholds the singular values. In either case, the only structure necessary for the cited algorithms to converge is the convexity of the norm. Indeed, essentially any algorithm developed for $\ell_1$ or nuclear norm minimization can in principle be adapted for atomic norm minimization. One simply needs to apply the operator $\Pi_{\A}$ wherever a shrinkage operation was previously applied.
For a concrete example, suppose $f$ is a smooth function, and consider the optimization problem
\begin{equation}\label{eq:smooth-reg}
\min_\bx~f(\bx)+\mu\|\bx\|_{\A}\,.
\end{equation}
The classical projected gradient method for this problem alternates between taking steps along the gradient of $f$ and then applying the proximity operator associated with the atomic norm. Explicitly, the algorithm consists of the iterative procedure
\begin{equation}
\bx_{k+1} = \Pi_{\A}( \bx_k - \alpha_k \nabla f(\bx_k); \alpha_k\lambda)
\end{equation}
where $\{\alpha_k\}$ is a sequence of positive stepsizes. Under very mild assumptions, this iteration can be shown to converge to a stationary point of~\eqref{eq:smooth-reg}~\cite{FukM1981}. When $f$ is convex, the returned stationary point is a globally optimal solution. Recently, Nesterov has described a particular variant of this algorithm that is guaranteed to converge at a rate no worse than $O(k^{-1})$, where $k$ is the iteration counter~\cite{Nes2007}. Moreover, he proposes simple enhancements of the standard iteration to achieve an $O(k^{-2})$ convergence rate for convex $f$ and a linear rate of convergence for strongly convex $f$.
If we apply the projected gradient method to the regularized inverse problem
\begin{equation}\label{eq:general-lasso}
\min_\bx ~ \|\Phi \bx - \by\|^2 + \lambda \|\bx\|_{\A}
\end{equation}
then the algorithm reduces to the straightforward iteration
\begin{equation}
\bx_{k+1} = \Pi_{\A}( \bx_k + \alpha_k \Phi^\dag(\by-\Phi \bx_k); \alpha_k\lambda)\,.
\end{equation}
Here \eqref{eq:general-lasso} is equivalent to~\eqref{eq:noisy-atomic-norm-primal} for an appropriately chosen $\lambda>0$ and is useful for estimation from noisy measurements.
The basic (noiseless) atomic norm minimization problem~\eqref{eq:atomic-norm-primal} can be solved by minimizing a sequence of instances of~\eqref{eq:general-lasso} with monotonically decreasing values of $\lambda$. Each subsequent minimization is initialized from the point returned by the previous step. Such an approach corresponds to the classic Method of Multipliers~\cite{Ber1996} and has proven effective for solving problems regularized by the $\ell_1$ norm and for total variation denoising~\cite{YinODG2007,CaiOS2008}.
This discussion demonstrates that when the proximity operator associated with some atomic set $\A$ can be easily computed, then efficient first-order algorithms are immediate. For novel atomic norm applications, one can thus focus on algorithms and techniques to compute the associated proximity operators. We note that, from a computational perspective, it may be easier to compute the proximity operator via dual atomic norm. Associated to each proximity operator is the dual operator
\begin{equation}\label{eq:dual-shrinkage}
\Lambda_{\A}(\bx;\mu) = \arg \min_\by \tfrac{1}{2} \|\by-\bx\|^2 ~ \mathrm{s.t.}~ \|\by\|_{\A}^\ast\leq \mu
\end{equation}
By an appropriate change of variables, $\Lambda_{\A}$ is nothing more than the projection of $\mu^{-1}\bx$ onto the unit ball in the dual atomic norm:
\begin{equation}
\Lambda_{\A}(\bx;\mu) = \arg \min_\by \tfrac{1}{2} \|\by-\mu^{-1} \bx\|^2 ~ \mathrm{s.t.}~ \|\by\|_{\A}^\ast\leq 1
\end{equation}
From convex programming duality, we have $\bx = \Pi_\A(\bx;\mu)+\Lambda_{\A}(\bx;\mu)$. This can be seen by observing
\begin{align}
\min_\bz \tfrac{1}{2} \|\bz-\bx\|^2 + \mu\|\bz\|_{\A} &= \min_\bz \max_{\|\by\|_{\A}^\ast\leq \mu} \tfrac{1}{2} \|\bz-\bx\|^2 + \langle \by, \bz\rangle\\
&= \max_{\|\by\|_{\A}^\ast\leq \mu} \min_\bz \tfrac{1}{2} \|\bz-\bx\|^2 + \langle \by, \bz\rangle\\
&= \max_{\|\by\|_{\A}^\ast\leq \mu} -\tfrac{1}{2} \|\by-\bx\|^2 + \tfrac{1}{2}\|\bx\|^2
\end{align}
In particular, $\Pi_{\A}(\bx;\mu)$ and $\Lambda_{\A}(\bx;\mu)$ form a complementary primal-dual pair for this optimization problem. Hence, we only need to able to efficiently compute the Euclidean projection onto the dual norm ball to compute the proximity operator associated with the atomic norm.
Finally, though the proximity operator provides an elegant framework for algorithm generation, there are many other possible algorithmic approaches that may be employed to take advantage of the particular structure of an atomic set $\A$. For instance, we can rewrite~\eqref{eq:dual-shrinkage} as
\begin{equation}\label{eq:dual-shrinkage-explicit}
\Lambda_{\A}(\bx;\mu) = \arg \min_\by \tfrac{1}{2} \|\by-\mu^{-1} \bx\|^2 ~~ \mathrm{s.t.}~~ \langle \by,\ba \rangle \leq 1~~\forall \ba\in \A
\end{equation}
Suppose we have access to a procedure that, given $\bz\in\R^n$, can decide whether $\langle \bz,\ba\rangle\leq 1$ for all $\ba\in \A$, or can find a violated constraint where $\langle \bz, \hat{\ba} \rangle > 1$. In this case, we can apply a cutting plane method or ellipsoid method to solve~\eqref{eq:dual-shrinkage} or~\eqref{eq:atomic-norm-dual}~\cite{Nes2004,Pol1997}. Similarly, if it is simpler to compute a subgradient of the atomic norm than it is to compute a proximity operator, then the standard subgradient method~\cite{BerNO2003,Nes2004} can be applied to solve problems of the form~\eqref{eq:general-lasso}. Each computational scheme will have different advantages and drawbacks for specific atomic sets, and relative effectiveness needs to be evaluated on a case-by-case basis.
\subsection{Simulation Results}
\label{subsec:sims}
We describe the results of numerical experiments in recovering orthogonal matrices, permutation matrices, and rank-one sign matrices (i.e., cut matrices) from random linear measurements by solving convex optimization problems. All the atomic norm minimization problems in these experiments are solved using a combination of the SDPT3 package \cite{TohTT} and the YALMIP parser \cite{Lof2004}.
\textbf{Orthogonal matrices.} We consider the recovery of $20 \times 20$ orthogonal matrices from random Gaussian measurements via \emph{spectral norm minimization}. Specifically we solve the convex program \eqref{eq:atomic-norm-primal}, with the atomic norm being the spectral norm. Figure~\ref{fig:res} gives a plot of the probability of exact recovery (computed over $50$ random trials) versus the number of measurements required.
\textbf{Permutation matrices.} We consider the recovery of $20 \times 20$ permutation matrices from random Gaussian measurements. We solve the convex program \eqref{eq:atomic-norm-primal}, with the atomic norm being the norm induced by the Birkhoff polytope of $20 \times 20$ doubly stochastic matrices. Figure~\ref{fig:res} gives a plot of the probability of exact recovery (computed over $50$ random trials) versus the number of measurements required.
\textbf{Cut matrices.} We consider the recovery of $20 \times 20$ cut matrices from random Gaussian measurements. As the cut polytope is intractable to characterize, we solve the convex program \eqref{eq:atomic-norm-primal} with the atomic norm being approximated by the norm induced by the semidefinite relaxation $\mathcal{P}_1$ described in Section~\ref{subsec:tradeoff}. Recall that this is the second theta body associated with the convex hull of cut matrices, and so this experiment verifies that objects can be recovered from theta-body approximations. Figure~\ref{fig:res} gives a plot of the probability of exact recovery (computed over $50$ random trials) versus the number of measurements required.
\begin{figure}
\begin{center}
\epsfig{file=results.eps,width=14cm,height=6cm} \caption{Plots of the number of measurements available versus the probability of exact recovery (computed over $50$ trials) for various models.} \label{fig:res}
\end{center}
\end{figure}
In each of these experiments we see agreement between the observed phase transitions, and the theoretical predictions (Propositions \ref{prop:ortho}, \ref{prop:birkhoff}, and \ref{prop:cut}) of the number of measurements required for exact recovery. In particular note that the phase transition in Figure~\ref{fig:res} for the number of measurements required for recovering an orthogonal matrix is very close to the prediction $n \approx \tfrac{3m^2-m}{4} = 295$ of Proposition~\ref{prop:ortho}. We refer the reader to \cite{DonT2005,RecFP2010,ManR2009} for similar phase transition plots for recovering sparse vectors, low-rank matrices, and signed vectors from random measurements via convex optimization.
\section{Conclusions and Future Directions}
\label{sec:conc}
This manuscript has illustrated that for a fixed set of base atoms, the atomic norm is the best choice of a convex regularizer for solving ill-posed inverse problems with the prescribed priors. With this in mind, our results in Section~\ref{sec:gaussian} and Section~\ref{sec:rep} outline methods for computing hard limits on the number of measurements required for recovery from \emph{any} convex heuristic. Using the calculus of Gaussian widths, such bounds can be computed in a relatively straightforward fashion, especially if one can appeal to notions of convex duality and symmetry. This computational machinery of widths and dimension counting is surprisingly powerful: near-optimal bounds on estimating sparse vectors and low-rank matrices from partial information follow from elementary integration. Thus we expect that our new bounds concerning symmetric, vertex-transitive polytopes are also nearly tight. Moreover, algebraic reasoning allowed us to explore the inherent trade-offs between computational efficiency and measurement demands. More complicated algorithms for atomic norm regularization might extract structure from less information, but approximation algorithms are often sufficient for near optimal reconstructions.
This report serves as a foundation for many new exciting directions in inverse problems, and we close our discussion with a description of several natural possibilities for future work:
\paragraph{Width calculations for more atomic sets.} The calculus of Gaussian widths described in Section~\ref{sec:gaussian} provides the building blocks for computing the Gaussian widths for the application examples discussed in Section~\ref{sec:def}. We have not yet exhaustively estimated the widths in all of these examples, and a thorough cataloging of the measurement demands associated with different prior information would provide a more complete understanding of the fundamental limits of solving underdetermined inverse problems. Moreover, our list of examples is by no means exhaustive. The framework developed in this paper provides a compact and efficient methodology for constructing regularizers from very general prior information, and new regularizers can be easily created by translating grounded expert knowledge into new atomic norms.
\paragraph{Recovery bounds for structured measurements.} Our recovery results focus on generic measurements because, for a general set $\A$, it does not make sense to delve into specific measurement ensembles. Particular structures of the measurement matrix $\Phi$ will depend on the application and the atomic set $\A$. For instance, in compressed sensing, much work focuses on randomly sampled Fourier coefficients~\cite{CanRT2006} and random Toeplitz and circulant matrices~\cite{HauBRN2008,Rau2009}. With low-rank matrices, several authors have investigated reconstruction from a small collection of entries~\cite{CanR2009}. In all of these cases, some notion of \emph{incoherence} plays a crucial role, quantifying the amount of information garnered from each row of $\Phi$. It would be interesting to explore how to appropriately generalize notions of incoherence to new applications. Is there a particular definition that is general enough to encompass most applications? Or do we need a specialized concept to match the specifics of each atomic norm?
\paragraph{Quantifying the loss due to relaxation.} Section~\ref{subsec:tradeoff} illustrates how the choice of approximation of a particular atomic norm can dramatically alter the number of measurements required for recovery. However, as was the case for vertices of the cut polytope, some relaxations incur only a very modest increase in measurement demands. Using techniques similar to those employed in the study of semidefinite relaxations of hard combinatorial problems, is it possible to provide a more systematic method to estimate the number of measurements required to recover points from polynomial-time computable norms?
\paragraph{Atomic norm decompositions.} While the techniques of Section~\ref{sec:gaussian} and Section~\ref{sec:rep} provide bounds on the estimation of points in low-dimensional secant varieties of atomic sets, they do not provide a procedure for actually constructing decompositions. That is, we have provided bounds on the number of measurements required to recover points $\bx$ of the form
\begin{equation*}
\bx=\sum_{\ba\in\A} c_\ba \ba
\end{equation*}
when the coefficient sequence $\{c_\ba\}$ is sparse, but we do not provide any methods for actually recovering $c$ itself. These decompositions are useful, for instance, in actually computing the rank-one binary vectors optimized in semidefinite relaxations of combinatorial algorithms~\cite{GoeW1995,Nes1997,AloN2006}, or in the computation of tensor decompositions from incomplete data~\cite{KolB2009}. Is it possible to use algebraic structure to generate deterministic or randomized algorithms for reconstructing the atoms that underlie a vector $\bx$, especially when approximate norms are used?
\paragraph{Large-scale algorithms.} Finally, we think that the most fruitful extensions of this work lie in a thorough exploration of the empirical performance and efficacy of atomic norms on large-scale inverse problems. The proposed algorithms in Section~\ref{sec:comp} require only the knowledge of the proximity operator of an atomic norm, or a Euclidean projection operator onto the dual norm ball. Using these design principles and the geometry of particular atomic norms should enable the scaling of atomic norm techniques to massive data sets.
\section*{Acknowledgements}
We would like to gratefully acknowledge Holger Rauhut for several suggestions on how to improve the presentation in Section 3. We would also like to thank Santosh Vempala, Joel Tropp, Bill Helton, and Jonathan Kelner for helpful discussions.
\appendix
\section{Proof of Proposition~\ref{prop:dual-width}}
\label{app:dual-width}
\begin{proof}
First note that the Gaussian width can be upper-bounded as follows:
\begin{equation}\label{eq:gwidth}
w(\mathcal{C} \cap \Sp^{p-1}) \leq \E_\bg\left[ \sup_{\bz \in \mathcal{C}\cap \mathcal{B}(0,1)} \bg^T \bz \right],
\end{equation}
where $\mathcal{B}(0,1)$ denotes the unit Euclidean ball.
The expression on the right hand side inside the expected value can be expressed as the optimal value of the following convex optimization problem for each $\bg \in \R^p$:
\begin{equation}\label{eq:primal}
\begin{array}{ll}
\max_{\bz} & \bg^T \bz\\
\mathrm{s.t.} & \bz \in \mathcal{C}\\
& \|\bz\|^2\leq 1
\end{array}
\end{equation}
We now proceed to form the dual problem of \eqref{eq:primal} by first introducing the Lagrangian
\begin{equation*}
\mathcal{L}(\bz,\bu,\gamma) = \bg^T \bz + \gamma (1-\bz^T \bz) - \bu^T \bz
\end{equation*}
where $\bu \in \mathcal{C}^\ast$ and $\gamma \geq 0$ is a scalar. To obtain the dual problem we maximize the Lagrangian with respect to $\bz$, which amounts to setting
\begin{equation*}
\bz = \frac{1}{2\gamma} (\bg-\bu).
\end{equation*}
Plugging this into the Lagrangian above gives the dual problem
\begin{equation*}
\begin{array}{ll}
\min & \gamma + \frac{1}{4\gamma} \|\bg-\bu\|^2\\
\mathrm{s.t.} & \bu \in \mathcal{C}^\ast\\
& \gamma \geq 0.
\end{array}
\end{equation*}
Solving this optimization problem with respect to $\gamma$ we find that $\gamma = \frac{1}{2} \|\bg-\bu\|$, which gives the dual problem to \eqref{eq:primal}
\begin{equation}\label{eq:dual}
\begin{array}{ll}
\min & \|\bg-\bu\|\\
\mathrm{s.t.} & \bu \in \mathcal{C}^\ast
\end{array}
\end{equation}
Under very mild assumptions about $\mathcal{C}$, the optimal value of \eqref{eq:dual} is equal to that of \eqref{eq:primal} (for example as long as $\mathcal{C}$ has a non-empty relative interior, strong duality holds). Hence we have derived
\begin{equation}\label{eq:dual-width}
\E_\bg \left[ \sup_{\bz\in \mathcal{C}\cap \mathcal{B}(0,1)} \bg^T \bz\right] =\E_\bg \left[\mathrm{dist}(\bg,\mathcal{C}^\ast)\right].
\end{equation}
This equation combined with the bound \eqref{eq:gwidth} gives us the desired result.
\end{proof}
\section{Proof of Theorem~\ref{theo:angle}}
\label{app:width-angle}
\begin{proof}
We set $\beta = \tfrac{1}{\Theta}$. First note that if $\beta \geq \tfrac{1}{4} \exp\{\tfrac{p}{9}\}$ then the width bound exceeds $\sqrt{p}$, which is the maximal possible value for the width of $\mathcal{C}$. Thus, we will assume throughout that $\beta \leq \tfrac{1}{4} \exp\{\tfrac{p}{9}\}$.
Using Proposition~\ref{prop:dual-width} we need to upper bound the expected distance to the polar cone. Let $\bg \sim \mathcal{N}(0,I)$ be a normally distributed random vector. Then the norm of $\bg$ is independent from the angle of $\bg$. That is, $\|\bg\|$ is independent from $\bg/\|\bg\|$. Moreover, $\bg/\|\bg\|$ is distributed as a uniform sample on $\Sp^{p-1}$, and $\E_\bg[\|\bg\|]\leq \sqrt{p}$. Thus we have
\begin{equation}
\E_\bg[\mathrm{dist}(\bg,\mathcal{C}^\ast)] \leq \E_\bg[\|\bg\|\cdot \mathrm{dist}(\bg/\|\bg\|, \mathcal{C}^\ast \cap \Sp^{p-1})]\leq \sqrt{p}\E_\bu[ \mathrm{dist}(\bu, \mathcal{C}^\ast \cap \Sp^{p-1})]
\end{equation}
where $\bu$ is sampled uniformly on $\Sp^{p-1}$.
To bound the latter quantity, we will use isoperimetry. Suppose $A$ is a subset of $\Sp^{p-1}$ and $B$ is a spherical cap with the same volume as $A$. Let $N(A,r)$ denote the locus of all points in the sphere of Euclidean distance at most $r$ from the set $A$. Let $\mu$ denote the Haar measure on $\Sp^{p-1}$ and $\mu(A;r)$ denote the measure of $N(A,r)$. Then spherical isoperimetry states that $\mu(A;r)\geq \mu(B;r)$ for all $r\geq 0$ (see, for example \cite{Led2000,Mat2002}).
Let $B$ now denote a spherical cap with $\mu(B)=\mu(\mathcal{C}^\ast\cap \Sp^{p-1})$. Then we have
\begin{align}
\E_\bu[ \mathrm{dist}(\bu, \mathcal{C}^\ast \cap \Sp^{p-1})] &= \int_0^\infty \P [ \mathrm{dist}(\bu, \mathcal{C}^\ast \cap \Sp^{p-1})>t]dt\\
&= \int_0^\infty (1-\mu(\mathcal{C}^\ast\cap \Sp^{p-1};t))dt\\
&\leq \int_0^\infty (1-\mu(B;t))dt
\end{align}
where the first equality is the integral form of the expected value and the last inequality follows by isoperimetry. Hence we can bound the expected distance to the polar cone intersecting the sphere using only knowledge of the volume of spherical caps on $\Sp^{p-1}$.
To proceed let $v(\varphi)$ denote the volume of a spherical cap subtending a solid angle $\varphi$. An explicit formula for $v(\varphi)$ is
\begin{equation}
v(\varphi)= z_p^{-1}\int_0^\varphi \sin^{p-1}(\vartheta)d\vartheta
\end{equation}
where $z_p = \int_0^\pi \sin^{p-1}(\vartheta)d\vartheta$ \cite{KlaR1997}. Let $\varphi(\beta)$ denote the minimal solid angle of a cap such that $\beta$ copies of that cap cover $\Sp^{p-1}$. Since the geodesic distance on the sphere is always greater than or equal to Euclidean distance, if $K$ is a spherical cap subtending $\psi$ radians, $\mu(K;t)\geq v(\psi+t)$. Therefore
\begin{equation}\label{eq:cap-bound}
\int_0^\infty (1-\mu(B;t))dt \leq \int_0^\infty (1-v(\varphi(\beta)+t))dt \,.
\end{equation}
We can proceed to simplify the right-hand-side integral:
\begin{align}
\int_0^\infty (1-v(\varphi(\beta)+t))dt &= \int_0^{\pi-\varphi(\beta)} (1-v(\varphi(\beta)+t))dt \\
&= \pi-\varphi(\beta) - \int_0^{\pi-\varphi(\beta)} v(\varphi(\beta)+t)dt \\
&= \pi-\varphi(\beta) - z_p^{-1} \int_0^{\pi-\varphi(\beta)} \int_0^{\varphi(\beta)+t} \sin^{p-1}\vartheta d\vartheta dt \\
\label{eq:int-order} &= \pi-\varphi(\beta) - z_p^{-1} \int_0^{\pi} \int_{\max(\vartheta-\varphi(\beta),0)}^{\pi-\varphi(\beta)} \sin^{p-1}\vartheta dt d\vartheta \\
&= \pi-\varphi(\beta) - z_p^{-1} \int_0^{\pi} \left\{\pi-\varphi(\beta)-\max(\vartheta-\varphi(\beta),0)\right\}\sin^{p-1}\vartheta d\vartheta \\
&= z_p^{-1} \int_0^{\pi} \max(\vartheta-\varphi(\beta),0)\sin^{p-1}\vartheta d\vartheta \\
&= z_p^{-1} \int_{\varphi(\beta)}^{\pi} (\vartheta-\varphi(\beta))\sin^{p-1}\vartheta d\vartheta
\end{align}
\eqref{eq:int-order} follows by switching the order of integration and the rest of these equalities follow by straight-forward integration and some algebra.
Using the inequalities that $z_p \geq \frac{2}{\sqrt{p-1}}$ (see \cite{Led2000}) and $\sin(x)\leq \exp(-(x-\pi/2)^2/2)$ for $x\in[0,\pi]$, we can bound the last integral as
\begin{align}
z_p^{-1} \int_{\varphi(\beta)}^{\pi} (\vartheta-\varphi(\beta))\sin^{p-1}\vartheta d\vartheta &\leq \frac{\sqrt{p-1}}{2} \int_{\varphi(\beta)}^{\pi} (\vartheta-\varphi(\beta)) \exp\left(-\frac{p-1}{2}(\vartheta-\tfrac{\pi}{2})^2\right)d\vartheta
\end{align}
Performing the change of variables $a = \sqrt{p-1}(\vartheta-\tfrac{\pi}{2})$, we are left with the integral
\begin{align}
& \frac{1}{2} \int_{\sqrt{p-1}(\varphi(\beta)-\pi/2)}^{\sqrt{p-1} \pi/2} \left\{\frac{a}{\sqrt{p-1}}+\left(\frac{\pi}{2}-\varphi(\beta)\right)\right\} \exp\left(-\frac{a^2}{2}\right)da\\
=& -\frac{1}{2\sqrt{p-1}} \exp\left(-\frac{a^2}{2}\right) \bigg|_{\sqrt{p-1}(\varphi(\beta)-\pi/2)}^{\sqrt{p-1} \pi/2}+
\frac{ \frac{\pi}{2}-\varphi(\beta)}{2} \int_{\sqrt{p-1}(\varphi(\beta)-\pi/2)}^{\sqrt{p-1} \pi/2} \exp\left(-\frac{a^2}{2}\right)da\\
\label{eq:final-distance-bound} \leq& \frac{1}{2\sqrt{p-1}} \exp\left(-\frac{p-1}{2}(\pi/2-\varphi(\beta))^2\right) + \sqrt{\frac{\pi}{2}}\left( \frac{\pi}{2}-\varphi(\beta)\right)
\end{align}
In this final bound, we bounded the first term by dropping the upper integrand, and for the second term we used the fact that
\begin{equation}
\int_{-\infty}^\infty \exp(-x^2/2) dx = \sqrt{2\pi}\,.
\end{equation}
We are now left with the task of computing a lower bound for $\varphi(\beta)$. We need to first reparameterize the problem. Let $K$ be a spherical cap. Without loss of generality, we may assume that
\begin{equation}
K = \{ x\in\Sp^{p-1}~:~x_1\geq h\}
\end{equation}
for some $h \in [0,1]$. $h$ is the \emph{height} of the cap over the equator. Via elementary trigonometry, the solid angle that $K$ subtends is given by $\pi/2-\sin^{-1}(h)$. Hence, if $h(\beta)$ is the largest number such that $\beta$ caps of height $h$ cover $\Sp^{p-1}$, then $h(\beta)=\sin(\pi/2-\varphi(\beta))$.
The quantity $h(\beta)$ may be estimated using the following estimate from~\cite{Bri1998}. For $h\in[0,1]$, let $\gamma(p,h)$ denote the volume of a spherical cap of $\Sp^{p-1}$ of height $h$.
\begin{lemma}[\cite{Bri1998}]
For $1\geq h\geq \frac{2}{\sqrt{p}}$,
\begin{equation}
\frac{1}{10 h \sqrt{p}}(1-h^2)^{\frac{p-1}{2}} \leq \gamma(p,h) \leq
\frac{1}{2 h \sqrt{p}}(1-h^2)^{\frac{p-1}{2}} \,.
\end{equation}
\end{lemma}
Note that for $h \geq \frac{2}{\sqrt{p}}$,
\begin{equation}
\frac{1}{2 h \sqrt{p}}(1-h^2)^{\frac{p-1}{2}} \leq \frac{1}{4}(1-h^2)^{\frac{p-1}{2}} \leq \frac{1}{4}\exp(-\tfrac{p-1}{2} h^2)\,.
\end{equation}
So if
\begin{equation}
h = \sqrt{\frac{2\log(4\beta)}{p-1}}
\end{equation}
then $h\leq 1$ because we have assumed $\beta \leq \tfrac{1}{4} \exp\{\tfrac{p}{9}\}$ and $p \geq 9$. Moreover,
$h\geq\frac{2}{\sqrt{p}}$ and the volume of the cap with height $h$ is less than or equal to $1/\beta$. That is
\begin{equation}
\varphi(\beta)\geq \pi/2 - \sin^{-1}\left( \sqrt{\frac{2\log(4\beta)}{p-1}}\right)\,.
\end{equation}
Combining the estimate \eqref{eq:final-distance-bound} with Proposition~\ref{prop:dual-width}, and using our estimate for $\varphi(\beta)$, we get the bound
\begin{equation}\label{eq:complicated-looking-bound}
w(\mathcal{C}) \leq \frac{1}{2}\sqrt{\frac{p}{p-1}}\exp\left(-\tfrac{p-1}{2} \sin^{-1}\left( \sqrt{\frac{2\log(4\beta)}{p-1}}\right)^2\right) + \sqrt{\frac{\pi p}{2}}\sin^{-1}\left( \sqrt{\frac{2\log(4\beta)}{p-1}}\right)
\end{equation}
This expression can be simplified by using the following bounds. First, $\sin^{-1}(x)\geq x$ lets us upper bound the first term by $\sqrt{\frac{p}{p-1}}\frac{1}{8\beta}$. For the second term, using the inequality $\sin^{-1}(x)\leq \tfrac{\pi}{2}x$ results in the upper bound
\begin{equation}
w(\mathcal{C}) \leq \sqrt{\frac{p}{p-1}}\left( \frac{1}{8\beta} + \frac{\pi^{3/2}}{2} \sqrt{\log(4\beta)}\right).
\end{equation}
For $p\geq 9$ the upper bound can be expressed simply as $w(\mathcal{C})\leq 3\sqrt{\log(4 \beta)}$. We recall that $\beta = \tfrac{1}{\Theta}$, which completes the proof of the theorem.
\end{proof}
\section{Direct Width Calculations}
\label{app:direct}
We first give the proof of Proposition~\ref{prop:l1}.
\begin{proof}
Let $\bxs$ be an $s$-sparse vector in $\R^p$ with $\ell_1$ norm equal to $1$, and let $\A$ denote the set of unit-Euclidean-norm one-sparse vectors. Let $\Delta$ denote the set of coordinates where $\bxs$ is non-zero. The normal cone at $\bxs$ with respect to the $\ell_1$ ball is given by
\begin{align}
N_\A(\bxs) &= \mathrm{cone}\left\{\bz\in\R^p~:~\bz_i= \mathrm{sgn}(\bxs_i)~\mbox{for}~i\in \Delta,\,\,\, |\bz_i|\leq1~\mbox{for}~i\in \Delta^c\right\}\\
&= \left\{\bz\in\R^p~:~\bz_i= t\mathrm{sgn}(\bxs_{i})~\mbox{for}~i\in \Delta,\,\,\, |\bz_i|\leq t~\mbox{for}~i\in \Delta^c~\mbox{for some}~t>0\right\}\,.
\end{align}
Here $\Delta^c$ represents the zero entries of $\bxs$. The minimum squared distance to the normal cone at $\bxs$ can be formulated as a one-dimensional convex optimization problem for arbitrary $\bz\in\R^p$
\begin{align}
\inf_{\bu\in N_\A(\bxs)} \|\bz-\bu\|_2^2 &=
\inf_{\stackrel{t\geq 0}{|\bu_i|<t,\,\,i\in \Delta^c}} \sum_{i\in \Delta} (\bz_i-t\mathrm{sgn}(\bxs_i))^2 + \sum_{j\in \Delta^c} (\bz_j - \bu_j)^2\\
&=\inf_{t\geq 0} \sum_{i\in \Delta} (\bz_i-t\mathrm{sgn}(\bxs_{i}))^2 + \sum_{j\in \Delta^c} \mathrm{shrink}(\bz_j,t)^2
\end{align}
where
\begin{equation}\label{eq:shrink}
\mathrm{shrink}(z,t) = \begin{cases} z+t & z<-t\\
0 &-t\leq z \leq t\\
z - t & z>t
\end{cases}
\end{equation}
is the $\ell_1$-shrinkage function. Hence, for any fixed $t\geq 0$ independent of $\bg$, we have
\begin{align}
\E\left[\inf_{\bu \in N_\A(\bxs) } \|\bg-\bu\|_2^2\right]
&\leq \E\left[\sum_{i\in \Delta} ( \bg_i-t\mathrm{sgn}(\bxs_i) )^2
+ \sum_{j\in \Delta^c} \mathrm{shrink}(\bg_j,t)^2\right]\\
\label{eq:ell-1-width-bound} &= s(1+t^2)
+ \E\left[\sum_{j\in \Delta^c} \mathrm{shrink}(\bg_j,t)^2\right]\,.
\end{align}
Now we directly integrate the second term, treating each summand individually. For a zero-mean, unit-variance normal random variable $g$,
\begin{align}
\E\left[ \mathrm{shrink}(g,t)^2\right] &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{-t} (g+t)^2\exp(-g^2/2) dg + \frac{1}{\sqrt{2\pi}}\int_t^\infty (g-t)^2 \exp(-g^2/2)dg\\
&= \frac{2}{\sqrt{2\pi}} \int_{t}^{\infty} (g-t)^2\exp(-g^2/2) dg \\
&= -\frac{2}{\sqrt{2\pi}} t\exp(-t^2/2) +
\frac{2(1+t^2)}{\sqrt{2\pi}} \int_{t}^{\infty} \exp(-g^2/2) dg\\
&\leq \frac{2}{\sqrt{2\pi}}\left(-t +\frac{1+t^2}{t} \right)\exp(-t^2/2)\\
&=\frac{2}{\sqrt{2\pi}}\frac{1}{t}\exp(-t^2/2)\,.
\end{align}
The first simplification follows because the $\mathrm{shrink}$ function and Gaussian distributions are symmetric about the origin. The second equality follows by integrating by parts. The inequality follows by a tight bound on the Gaussian $Q$-function
\begin{equation}
Q(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} \exp(-g^2/2) dg \leq \frac{1}{\sqrt{2\pi}}\frac{1}{x}\exp(-x^2/2)\quad\mbox{for }x>0\,.
\end{equation}
Using this bound, we get
\begin{equation}\label{eq:q-fun-bnd}
\E\left[\inf_{\bu\in N_\A(\bxs)} \|\bg-\bu\|_2^2\right] \leq s(1+t^2) +(p-s)\frac{2}{\sqrt{2\pi}}\frac{1}{t}\exp(-t^2/2)
\end{equation}
Setting $t= \sqrt{2\log(p/s)}$ gives
\begin{equation}\label{eq:l1-precise-bound}
\E\left[\inf_{\bz\in N_\A(\bxs)} \|\bg-\bz\|_2^2\right] \leq s\left(1+2\log\left(\frac{p}{s}\right)\right)+\frac{s(1-s/p)}{\pi \sqrt{\log(p/s)}}\leq
2s\log\left(p/s\right)+\tfrac{5}{4}s\,.
\end{equation}
The last inequality follows because
\begin{equation}
\frac{(1-s/p)}{\pi \sqrt{\log(p/s)}}\leq 0.204<1/4
\end{equation}
whenever $0 \leq s\leq p$.
\end{proof}
Next we give the proof of Proposition~\ref{prop:nuclear}.
\begin{proof}
Let $\bxs$ be an $m_1 \times m_2$ matrix of rank $r$ with singular value decomposition $U\Sigma V^*$, and let $\A$ denote the set of rank-one unit-Euclidean-norm matrices of size $m_1 \times m_2$. Without loss of generality, impose the conventions $m_1\leq m_2$, $\Sigma$ is $r\times r$, $U$ is $m_1 \times r$, $V$ is $m_2 \times r$, and assume the nuclear norm of $\bxs$ is equal to $1$.
Let $\bu_k$ (respectively $\bv_k$) denote the $k$'th column of $U$ (respectively $V$). It is convenient to introduce the orthogonal decomposition $\R^{m_1 \times m_2} = \Delta \oplus \Delta^\perp$ where $\Delta$ is the linear space spanned by elements of the form $\bu_k \bz^T$ and $\by \bv_k^T$, $1 \le k \le r$, where $\bz$ and $\by$ are arbitrary, and $\Delta^\perp$ is the orthogonal complement of $\Delta$. The space $\Delta^\perp$ is the subspace of matrices spanned by the family $(\by \bz^T)$, where $\by$ (respectively $\bz$) is any vector orthogonal to all the columns of $U$ (respectively $V$). The normal cone of the nuclear norm ball at $\bxs$ is given by the cone generated by the subdifferential at $\bxs$:
\begin{align}
N_\A(\bxs) &= \mathrm{cone}\left\{UV^T+W\in\R^{m_1\times m_2}~:~W^TU = 0,\,\,\,\,WV=0,\,\,\,\,\|W\|_\A^\ast\leq 1\right\}\\
&=\left\{tUV^*+W\in\R^{m_1\times m_2}~:~W^TU = 0,\,\,\,\,WV=0,\,\,\,\,\|W\|_\A^\ast \leq t,\,\,\,t\geq 0\right\}\,.
\end{align}
Note that here $\|Z\|_\A^\ast$ is the operator norm, equal to the maximum singular value of $Z$~\cite{RecFP2010}.
Let $G$ be a Gaussian random matrix with i.i.d. entries, each with mean zero and unit variance. Then the matrix
\begin{equation}
Z(G) = \|\PTc(G)\| UV^* + \PTc(G)
\end{equation}
is in the normal cone at $\bxs$. We can then compute
\begin{align}
\E\left[\|G-Z(G)\|_F^2\right] & = \E\left[\|\PT(G)-\PT(Z(G))\|_F^2\right] \\
\label{eq:matrix-indep} &= \E\left[\|\PT(G)\|_F^2\right]+\E\left[\|\PT(Z(G))\|_F^2\right] \\
&= r(m_1+m_2-r) + r\E[\|\PTc(G)\|^2]\,. \label{eq:nuclear-width-bound-direct}
\end{align}
Here \eqref{eq:matrix-indep} follows because $\PT(G)$ and $\PTc(G)$ are independent. The final line follows because $\mathrm{dim}(T)=r(m_1+m_2-r)$ and the Frobenius (i.e., Euclidean) norm of $UV^*$ is $\|UV^*\|_F=\sqrt{r}$. Due to the isotropy of Gaussian random matrices, $\PTc(G)$ is identically distributed as an $(m_1-r)\times(m_2-r)$ matrix with i.i.d. Gaussian entries each with mean zero and variance one. We thus know that
\begin{equation}
\P\left[ \|\PTc(G)\|\geq \sqrt{m_1-r}+\sqrt{m_2-r} +s\right] \leq \exp\left(-s^2/2\right)
\end{equation}
(see, for example,~\cite{DavS2001}).
To bound the latter expectation, we again use the integral form of the expected value. Letting $\mu_{T^\perp}$ denote the quantity $\sqrt{m_1-r}+\sqrt{m_2-r}$, we have
\begin{align}
\E\left[\|\PTc(G)\|^2\right] &= \int_0^\infty \P\left[\|\PTc(G)\|^2>h\right] dh\\
&\leq \mu_{T^\perp}^2 + \int_{ \mu_{T^\perp}^2}^\infty\P\left[\|\PTc(G)\|^2>h\right] dh\\
&\leq \mu_{T^\perp}^2 + \int_0^\infty\P\left[\|\PTc(G)\|^2> \mu_{T^\perp}^2 + t \right] dt\\
&\leq \mu_{T^\perp}^2 + \int_0^\infty\P\left[\|\PTc(G)\|> \mu_{T^\perp} +\sqrt{t} \right] dt\\
&\leq \mu_{T^\perp}^2 + \int_0^\infty \exp(-t/2) dt\\
&= \mu_{T^\perp}^2 + 2
\end{align}
Using this bound in~\eqref{eq:nuclear-width-bound-direct}, we get that
\begin{align}\label{eq:nuclear-crude-bound}
\E\left[\inf_{Z\in N_\A(\bxs)} \|G-Z\|_F^2\right] & \leq r(m_1+m_2-r) + r(\sqrt{m_1-r}+\sqrt{m_2-r})^2+ 2r\\
& \leq r(m_1+m_2-r) + 2r(m_1+m_2-2r)+ 2r\\
& \leq 3r(m_1+m_2-r)
\end{align}
where the second inequality follows from the fact that $(a+b)^2\leq 2a^2+2b^2$. We conclude that $3r(m_1+m_2-r)$ random measurements are sufficient to recover a rank $r$, $m_1\times m_2$ matrix using the nuclear norm heuristic.
\end{proof}
| 172,099
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The.
However, since urban markets contribute 60-65% of FMCG sales, recovery of urban markets is important, he added.
The sales of food, as well as non- food categories, have grown in June as compared to the previous month, non-food categories such as beauty and hygiene have grown significantly.
"With majority of people confined to their house, cosmetics and beauty categories were de-prioritised in the lockdown phase - categories like deodorants, hair colour and skincare had witnessed significant slowdown. The categories have witnessed a sharp bounce back in June," Nielsen said.
The sales of daily use items like toothpaste, shampoo, hair oil and washing powder that had witnessed rationalisation in the lockdown period have bounced back in June.
With most consumers preferring to have home-cooked meals, packaged atta, edible oil and cheese continue to be in the shoppers' basket.
The FMCG industry has been going through a slowdown since early this year first on account of muted demand followed by the outbreak of the coronavirus pandemic.
There has been a shift towards e-commerce as people remain sceptical of public spaces. "Consumers are minimising physical touchpoints and reliance on online shopping has increased," the Nielsen report said.
Around 62% of those who were surveyed by the market research firm said that they intend to increase online shopping by more than 20%.
Consumers have prioritised fundamentals such as healthy foods, home hygiene, medical, fitness, education, home entertainment, and investment as they prepare for uncertainty caused by the slowing economy, rising unemployment and salary cuts, according to Nielsen.
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| 133,866
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I have had a lung mass that has never changed size since 2009. The pulmonologist told me that it was scar tissue and would never be anything to worry about. Well, 2014 CT scan showed a slight change in it, but not in size, and a lymph node next to it mildly enlarged. Needle biopsy of main lung mass came up showing precancerous. Then an EBUS of lung lymph node showed suspicion for adenocarcinoma but couldn't confirm that. The PET scan had shown the 2 places in the lung and a place in the T12 bone in spine as hot spots. Third biopsy was of T12 and it confirmed I have stage 4 adenocarcinoma of the lung with metastasis to the bone at T12. I have MRI in the morning to see if it has gone to the brain, although they don't think so. I am beyond scared. I don't understand how this has happened when I had a pulmonologist watching it. Now they tell me my cancer is not curable and most stage 4 lung cancers only live 10 to 14 months. Please, any information or advise would be greatly appreciated. Thank you.
I can remember many years ago, the doctors telling my husband he had lung cancer and he only had a few months to live. They sent him to have a biopsy done and I remember the doctor coming out and telling me that he didn't have cancer but that it was a horrible lung infection. My husband after that always had scar tissue that showed up on his chest x-rays. He spent a week in the hospital on very strong antiobodics. Fast forward to 2010 when he was diagnosed with laryngeal cancer (SCC). He underwent radiation and chemo. We were told after a PET scan he was NED. Not true. Another biopsy and a trach and then referred to a head and neck specialist. Surgery followed by 14 months of NED and he was doing great. Then a second primary was found at the cervical of his esophagus. Surgery ruled out. Rare spot for cancer, but no spread (they didn't tell us they were watching some small spots in his lungs). Under round of radiation and chemo. NED for 8 months when after a PET/CT scan they found the 2nd primary back and larger and that the nodes in his right lung had grown. Our specialist want a needle biopsy asap, they didn't want to do it because the nodes were small and hard to get to. Well they did it, took 4 tries to get some cells, but it was cancer (spread). Only thing offered was chemo. My husband had already told me he wouldn't do anything else. Our doctors never gave him a time, but research showed that with or without treatment that this second primary had a survival rate of 24-25 months. So far my husband has beat that, so see doctors don't know. I have always said that only the man upstairs knows. He has no problems today with breathing, although after walking outside he does get short of breath but once he sits and rests he is fine with his breathing. A lot of pain but he always complains of his pain in his spine and back that moves around to the front sometimes. Says he feels like someone is standing on his back.
Today we are 16 months out from this diagnosis and he is doing fair. Get another opinion from someone or a cancer center that has plenty of experience in this field. That was something I wished we had done in the very beginning not waiting until the second biopsy showed his laryngeal cancer was still there because according to our specialist it never went away.
Wishing you the best -- Sharon
3.5 years ago I was told I had 10-15 months with treatment. I'm doing very well. Not cureable is not the same as untreatable. I would however, find a new oncologist. Preferably at an NCI affiliated comprehensive cancer care.
| 55,686
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\begin{document}
\title{Quantization of $\Gamma$-Lie bialgebras}
\begin{abstract}
We introduce the notion of $\Gamma$-Lie bialgebras, where $\Gamma$
is a group. These objects give rise to cocommutative
co-Poisson bialgebras, for which we construct quantization functors.
This enlarges the class of co-Poisson algebras for which a quantization
is known. Our result relies on our earlier work, where we showed that
twists of Lie bialgebras can be quantized; we complement this work
by studying the behavior of this quantization under compositions of
twists.
\end{abstract}
\author{Benjamin Enriquez}
\address{IRMA (CNRS), rue Ren\'e Descartes, F-67084 Strasbourg, France}
\email{enriquez@@math.u-strasbg.fr}
\author{Gilles Halbout}
\address{IRMA (CNRS), rue Ren\'e Descartes, F-67084 Strasbourg, France}
\email{halbout@@math.u-strasbg.fr}
\maketitle
We work over a field $\kk$ of characteristic $0$.
\section{Introduction}
Recall that a co-Poisson bialgebra is a quadruple $(U,m,\Delta_0,\delta)$,
where $(U,m,\Delta_0)$ is a cocommutative bialgebra and
$\delta: U \to \wedge^2(U)$ is a derivation (for $m$),
a coderivation (for $\Delta_0$) and satisfies the co-Jacobi identity.
A quantization of $(U,m,\Delta_0,\delta)$ is a bialgebra
$(U_\hbar,m_\hbar,\Delta_\hbar)$ such that $U_\hbar \simeq
U[[\hbar]]$,
$m_\hbar=m + O(\hbar)$, $\Delta_\hbar=\Delta + O(\hbar)$,
$\Delta_\hbar(a)-\Delta_\hbar^\op(a)=\hbar \delta(a)+ O(\hbar^2)$
($\Delta_\hbar^\op$ is the opposite coproduct).
If $(\a,\mu_\a)$ is a Lie algebra ($\mu_\a :
\wedge^2(\a) \to \a$ is the Lie bracket), the co-Poisson
bialgebra structures on $U(\a)$ correspond bijectively to the maps
$\delta_\a: \a\to \wedge^2(\a)$ such that $(\a,\mu_\a,\delta_\a)$
is a Lie bialgebra. The quantization of these co-Poisson
bialgebras was obtained in \cite{EK}.
To a triple $(\Gamma,\a,\theta_a)$, where $\Gamma$ is a group,
$\a$ is a Lie algebra
and $\theta_\a:\Gamma \to \on{Aut}(\a,\mu_\a)$ is an action of
$\Gamma$ on $\a$, one associates the $\Gamma$-graded cocommutative
bialgebra $U(\a)\rtimes \Gamma$. The $\Gamma$-graded co-Poisson
bialgebra structures on $U(\a)\rtimes\Gamma$ correspond bijectively
to pairs $(\delta_\a,f)$, where $\delta_\a:\a\to \wedge^2(\a)$
is such that $(\a,\mu_\a,\delta_\a)$ is a Lie bialgebra, and
$f:\Gamma\to \wedge^2(\a)$ satisfies some conditions (see Section
\ref{gammaLie});
in particular, $f(\gamma)$ is a twist of $(\a,\mu_\a,\delta_\a)$
for any $\gamma\in\Gamma$. We call the resulting 5-uple
$(\a,\mu_\a,\delta_\a,\theta_\a,f)$ a $\Gamma$-Lie bialgebra.
The main result of this paper is the quantization of the
corresponding co-Poisson bialgebra structures.
Examples of $\Gamma$-Lie bialgebras arise from the following situation:
$G$ is a Poisson-Lie group with Lie bialgebra $(\a,\mu_\a,\delta_\a)$,
and $\Gamma \subset G$ is a discrete subgroup. Another example is when $\a$
is a Kac-Moody Lie algebra $\a$, and $\Gamma$ is the extended Weyl group
of $\a$. In the latter case, a quantization in known (\cite{MS}).
\smallskip
To achieve our goal, we complement a result obtained in \cite{EH}, namely
the compatibility of Etingof-Kazhdan (EK) quantization functors with
twists of Lie bialgebras; this result is based on an alternative
construction of these quantization functors (\cite{En}). We
describe the behavior of this quantization under
composition of twists (Section \ref{comp:twists}).
To give an idea
of the result of \cite{EH}, we formulate its main consequence:
let $Q : \{$Lie bialgebras$\} \to
\{$quantized universal enveloping (QUE) algebras$\}$,
$(\a,\mu_\a,\delta_\a) = \a \mapsto Q(\a) = (Q(\a),m(\a),
\Delta(\a),1_\a,\eps_\a)$ be a
quantization
functor; to each classical twist $f_\a$ of $\a$ (i.e.,
$f_\a\in\wedge^2(\a)$ and $(\delta_\a \otimes
\on{id}_\a)(f_\a) + [f_\a^{13},f_\a^{23}] + $ cyclic permutations = 0),
one associates $\on{F}(\a,f_\a)\in Q(\a)^{\otimes 2}$, with the
following properties: (a) it is a cocycle for $Q(\a)$, i.e.,\footnote{We
denote by $*$ the product in $Q(\a)^{\otimes k}$ for $k\geq 1$}
$(\on{F}(\a,f_\a)\otimes 1_{\a}) * (\Delta(\a) \otimes
\on{id})(\on{F}(\a,f_\a)) = (1_{\a} \otimes \on{F}(\a,f_\a)) * (\on{id}
\otimes \Delta(\a))(\on{F}(\a,f_\a))$,
$(\varepsilon_{\a} \otimes \on{id})(\on{F}(\a,f_\a))
= (\on{id} \otimes \varepsilon_{\a})(\on{F}(\a,f_\a)) = 1_{\a}$;
this implies that\footnote{If $A$ is an algebra and $u\in A$
is invertible, $\on{Ad}(u) : A \to A$ is $x\mapsto uxu^{-1}$}
${}^{\on{F}(\a,\delta_\a)}Q(\a):= (Q(\a),m(\a),\on{Ad}(\on{F}(\a,f_\a))
\circ \Delta(\a),1_{\a},\varepsilon_{\a})$ is a QUE algebra;
(b) we have an isomorphism $\on{i}(\a,f_\a) :
{}^{\on{F}(\a,f_\a)}Q(\a) \to Q(\a_{f_\a})$ of QUE algebras
(here $\a_{f_{\a}} = (\a,\mu_\a,\delta_\a + \on{ad}(f_\a))$, where $\on{ad}
(f_\a) : \a\to \wedge^2(\a)$ is $x\mapsto [f_\a,x\otimes 1 + 1 \otimes x]$).
In Section \ref{comp:twists}, we study the behavior of
the assignment $(\a,f_\a) \mapsto
\on{F}(\a,f_\a)$ under the composition of twists. A composition of
twists is a pair $(f_\a,f'_\a)$ such that $f_\a$ is a twist of $\a$,
and $f'_\a$ is a twist of $\a_{f_\a}$. We formulate the main
consequence of our results:
(a) there exists an invertible $\on{v}(\a,f_\a,f'_\a)
\in Q(\a)$, such that $\on{F}(f_\a + f'_\a) =
\on{v}(\a,f_\a,f'_\a)^{\otimes 2} * \on{i}(\a,f_\a)^{-1}
(\on{F}(\a_{f_{\a}},f'_\a)) * F(\a,f_\a) *
\Delta(\a)(\on{v}(\a,f_\a,f'_\a))^{-1}$
(Theorem \ref{thm:i:v}); and (b) if $(f_\a,f'_\a,f''_\a)$ are
such that $f'_\a$ is a twist of $\a_{f_\a}$ and $f''_\a$ is a twist
of $\a_{f_\a}$, then $f_\a + f'_\a$ is a twist of $\a$, and
$\on{v}(\a,f_\a+f'_\a,f''_\a) * \on{v}(\a,f_\a,f'_\a)
= \on{v}(\a,f_\a,f'_\a+f''_\a) *
\on{i}(\a,f_\a)^{-1}(\on{v}(\a_{f_\a},f'_\a,f''_\a))$
(Theorem \ref{thm:i:v}).
\smallskip
We use these results in Section \ref{quantgamma} to construct a quantization
of the $\Gamma$-graded co-Poisson bialgebras $U(\a)\rtimes \Gamma$
as $\Gamma$-graded bialgebras. This quantization is based on
the facts that that for $\gamma\in\Gamma$, $f(\gamma)$ is a twist of $\a$,
and for any $\gamma,\gamma'\in\Gamma$,
$(f(\gamma),\wedge^2(\theta_\a(\gamma))(f(\gamma')))$ is a composition of
twists for $\a$; we then use the results of \cite{EH} on quantization of
twists and those of Section \ref{comp:twists} on their composition.
\smallskip
This paper is organized as follows.
In Section \ref{gammaLie}, we define $\Gamma$-Lie bialgebras,
the corresponding co-Poisson cocommutative bialgebras, and the
problem of their quantization.
In Section \ref{back}, we recall the formalism of
(quasi-multi-bi)props, which is the natural framework of
the approach of \cite{En} to quantization functors and of
the results of \cite{EH} on quantization of twists.
In Section \ref{comp:twists}, we describe the behavior of
composition of twists under quantization (Theorem \ref{thm:i:v} and
Theorem \ref{thm:v:v}).
In Section \ref{quantgamma}, we apply these results to
the construction of quantizations of $\Gamma$-Lie bialgebras.
\section{$\Gamma$-Lie bialgebras}\label{gammaLie}
\subsection{$\Gamma$-Lie algebras and equivalent categories}
Define a group Lie algebra as a triple $(\Gamma,\a,\theta_\a)$,
where $\Gamma$ is a group, $\a$ is a Lie algebra and
$\theta_\a : \Gamma \to \on{Aut}(\a)$ is a group morphism.
Group Lie algebras form a category, where a morphism
$(\Gamma,\a,\theta_\a) \to (\Gamma',\a',\theta_{\a'})$ is the data
of a group morphism $i_{\Gamma\Gamma'} : \Gamma \to \Gamma'$
and a Lie algebra morphism $i_{\a\a'} : \a\to \a'$, such that
$i_{\a\a'}(\theta_{\a,\gamma}(x)) = \theta_{\a,i_{\Gamma\Gamma'}(\gamma)}
(i_{\a\a'}(x))$.
If $\Gamma$ is a group, a $\Gamma$-Lie algebra is a pair
$(\a,\theta_\a)$, such that $(\Gamma,\a,\theta_\a)$ is a group
Lie algebra. $\Gamma$-Lie algebras form a subcategory of group
Lie algebras, where the morphisms are restricted by the condition
$i_{\Gamma\Gamma} = \on{id}_{\Gamma}$.
Define a group cocommutative bialgebra as a triple $(\Gamma,A,i)$, where
$\Gamma$ is a group, $A$ is a cocommutative bialgebra,
$A = \oplus_{\gamma\in\Gamma} A_\gamma$ is a decomposition of $A$, and
$i : \kk\Gamma \to A$ is a bialgebra morphism, such that
$A_\gamma A_{\gamma'}\subset A_{\gamma\gamma'}$, $\Delta_A(A_{\gamma})
\subset A_\gamma^{\otimes 2}$, and $i$ is compatible with the $\Gamma$-grading.
A morphism $(\Gamma,A,i)\to (\Gamma',A',i')$ is the data of a group morphism
$i_{\Gamma\Gamma'} : \Gamma\to \Gamma'$ and a bialgebra morphism
$i_{AA'} : A\to A'$, such that $i_{AA'}(A_\gamma) \subset
A'_{i_{\Gamma\Gamma'}(\gamma')}$, and $i_{AA'} \circ i = i' \circ
i_{\kk\Gamma,\kk\Gamma'}$ (where $i_{\kk\Gamma,\kk\Gamma'} : \kk\Gamma
\to \kk\Gamma'$ is the morphism induced by $i_{\Gamma\Gamma'}$).
We then define a $\Gamma$-cocommutative bialgebra as a pair $(A,i)$, such that
$(\Gamma,A,i)$ is a group cocommutative bialgebra. $\Gamma$-cocommutative
bialgebras form a
category, where as before $i_{\Gamma\Gamma} = \on{id}_{\Gamma}$.
The category of group (resp., $\Gamma$-) cocommutative bialgebras contains
as a full subcategory the category of group (resp., $\Gamma$-)
universal enveloping algebras, where $(A,\Gamma,i)$ satisfies the
additional requirement that $A_e$ is a universal enveloping algebra.
Define a group commutative bialgebra (in a symmetric monoidal category $\cS$)
as a triple $(\Gamma,\cO,j)$, where $\Gamma$ is a group, $\cO$ is a
commutative algebra (in $\cS$) with a decomposition
$\cO = \oplus_{\gamma\in\Gamma} \cO_\gamma$,
such that $\cO_{\gamma}\cO_{\gamma'} = 0$ for $\gamma\neq\gamma'$,
algebra morphisms $\Delta_{\gamma'\gamma''} : \cO_{\gamma'\gamma''} \to
\cO_{\gamma'} \otimes \cO_{\gamma''}$,
$\eta : \kk \to \cO_e$ and $\varepsilon :\cO_e \to \kk$, satisfying
axioms such that when $\Gamma$ is finite, these morphisms add up to a
bialgebra structure on $\cO$; and $j : \cO \to \kk^\Gamma$ is a morphism
of commutative algebras, compatible with the $\Gamma$-gradings and the
maps $\Delta_{\gamma'\gamma''}$ on both sides. We define
$\Gamma$-commutative bialgebras as above.
We define the category of group (resp., $\Gamma$-) formal series Hopf
(FSH) algebras as a full subcategory of the category of
group (resp., $\Gamma$-) commutative bialgebras in $\cS = \{$pro-vector
spaces$\}$
by the condition the $\cO_e$ (or equivalently, each $\cO_\gamma$) is a formal
series algebra.
\begin{proposition}
1) We have (anti)equivalences of categories $\{$group Lie algebras$\}
\leftrightarrow \{$group universal enveloping algebras$\}
\leftrightarrow\{$group FHS algebras$\}$ (the last map is an antiequivalence).
2) If $\Gamma$ is a group, these (anti)equivalences restrict to
$\{\Gamma$-Lie algebras$\}
\leftrightarrow \{\Gamma$-universal enveloping algebras$\}
\leftrightarrow\{\Gamma$-FHS algebras$\}$.
\end{proposition}
{\em Proof.} We denote the $\Gamma$-universal enveloping algebra
corresponding to a $\Gamma$-Lie algebra $(\Gamma,\a,\theta_\a)$ as
$U(\a) \rtimes \Gamma$. It is isomorphic to $U(\a) \otimes \kk\Gamma$
as a vector space;
if we denote by $x\mapsto [x]$, $\gamma\mapsto [\gamma]$ the natural maps
$\a\to U(\a) \rtimes \Gamma$, $\Gamma \to U(\a) \rtimes \Gamma$, then
the bialgebra structure of $U(\a) \rtimes \Gamma$ is given by
$[\gamma][x][\gamma^{-1}] = [\theta_\gamma(x)]$, $[\gamma][\gamma'] =
[\gamma\gamma']$, $[e] = 1$, $[x][x'] - [x'][x] = [[x,x']]$,
$\Delta([x]) = [x]\otimes 1 + 1 \otimes [x]$,
$\Delta([\gamma]) = [\gamma] \otimes [\gamma]$.
When $\Gamma$ is finite, the corresponding $\Gamma$-FSH algebra is then
$(U(\a) \rtimes \kk\Gamma)^*$, and in general, this is
$\oplus_{\gamma\in\Gamma} (U(\a) \otimes \kk\gamma)^*$.
One checks that these are (anti)equivalences of categories.
For example, if $A$ is a group universal enveloping algebra, then one recovers
$\Gamma$ as $\{$group-like elements of $A\}$ and $\a$ as $\{$primitive
elements of $A\}$. \hfill \qed \medskip
\subsection{$\Gamma$-Lie bialgebras and equivalent categories}
A group Lie bialgebra
is a 5-uple $(\Gamma,\a, \theta_\a, \delta_\a,f)$
where $(\Gamma,\a,\theta_\a)$ is a group Lie algebra,
$\delta_\a : \a\to\wedge^2(\a)$ is\footnote{We view
$\wedge^2(V)$ as a subspace of $V^{\otimes 2}$.} such that $(\a,\delta_\a)$
is a Lie bialgebra, and $f : \Gamma \to \wedge^2(\a)$
is a map $\gamma\mapsto f_\gamma$, such that:
(a) $\wedge^2(\theta_\gamma) \circ \delta \circ
\theta_\gamma^{-1}(x) = \delta(x) + [f_\gamma,x\otimes 1 + 1 \otimes x]$
for any $x\in \a$, (b) $f_{\gamma\gamma'} = f_\gamma +
\wedge^2(\theta_\gamma)(f_{\gamma'})$, and (c) $(\delta\otimes
\on{id})(f_\gamma)
+ [f_\gamma^{1,3},f_\gamma^{2,3}]$ + cyclic permutations $= 0$.
Group Lie bialgebras form a category, where a morphism
$(\Gamma,\a, \theta_\a, \delta_\a,f)\to
(\Gamma',\a', \theta_{\a'}, \delta_{\a}',f')$ is a group Lie
algebra morphism $(\Gamma,\a, \theta_\a)\to
(\Gamma',\a', \theta_{\a'})$, such that $i_{\a\a'} : \a\to \a'$
is a Lie bialgebra morphism and $\wedge^2(i_{\a\a'})(f_\gamma) =
f'_{i_{\Gamma\Gamma'}(\gamma)}$. When $\Gamma$ is fixed, one defines
the category of $\Gamma$-Lie bialgebras as above.
A co-Poisson structure on a group cocommutative bialgebra
$(\Gamma,A,i)$ is a co-Poisson structure $\delta_A : A \to \wedge^2(A)$,
such that $\delta_A(A_\gamma) \subset \wedge^2(A_\gamma)$.
Co-Poisson group cocommutative bialgebras form a category, where
a morphism $(\Gamma,A,i,\delta_A) \to (\Gamma',A',i',\delta_{A'})$
is a morphism $(\Gamma,A,i) \to (\Gamma',A',i')$ of group
cocommutative bialgebras, compatible with the co-Poisson structures.
Co-Poisson group universal enveloping algebras form a full subcategory of
the latter category. One defines the full subcategories of co-Poisson
$\Gamma$-cocommutative bialgebras and co-Poisson $\Gamma$-enveloping
algebras as above.
A Poisson structure on a group commutative bialgebra
$(\Gamma,\cO,j)$ is a Poisson bialgebra structure $\{-,-\}:
\wedge^2(\cO) \to \cO$,
such that $\{\cO_\gamma,\cO_\gamma\} \subset \cO_{\gamma}$ and
$\{\cO_\gamma,\cO_{\gamma'}\} = 0$ if $\gamma\neq \gamma'$.
Poisson group bialgebras form a category, and Poisson group FSH
algebras form a full subcategory when $\cS = \{$pro-vector spaces$\}$.
One defines the full subcategories of Poisson $\Gamma$-bialgebras and
Poisson $\Gamma$-FSH algebras as above.
\medskip
{\bf Example.} Let $G$ be a Poisson-Lie (e.g., algebraic) group, let
$\Gamma \subset G$ be a subgroup (which we view as an abstract group).
We define $\theta_\gamma := \on{Ad}(\gamma)$, where $\on{Ad} : G \to
\on{Aut}_{Lie}(\a)$ is the adjoint action. If $P : G \to \wedge^2(\a)$ is the
Poisson bivector, satisfying $P(gg') = P(g') + \wedge^2(\on{Ad}(g))(P(g'))$,
then we set $f_\gamma := -P(\gamma)$. Then $(\a,\Gamma,f)$ is a
$\Gamma$-Lie bialgebra.
\medskip
{\bf Example.} Assume that $(\a,r_\a)$ is a quasitriangular Lie
bialgebra and $\theta : \Gamma \to \on{Aut}(\a,t_\a)$ is an action of
$\Gamma$ on $\a$ by Lie algebra automorphisms preserving $t_\a := r_\a +
r_\a^{2,1}$. If
we set $f_\gamma := \theta_\gamma^{\otimes 2}(r) - r$, then $(\a,\theta,f)$
is a $\Gamma$-Lie bialgebra (we call this a quasitriangular $\Gamma$-Lie
bialgebra). For example, $\a$ is a Kac-Moody Lie algebra,
and $\Gamma = \tilde W$ is the extended Weyl group of $\a$.
\medskip
\begin{proposition}
1) We have category (anti)equivalences $\{$group bialgebras$\}
\leftrightarrow \{$co-Poisson group universal enveloping
algebras$\}\leftrightarrow \{$Poisson group FSH algebras$\}$.
2) These restrict to category (anti)equivalences
$\{\Gamma$-bialgebras$\}
\leftrightarrow \{$co-Poisson $\Gamma$-universal enveloping
algebras$\}\leftrightarrow \{$Poisson $\Gamma$-FSH algebras$\}$.
\end{proposition}
{\em Proof.} If $(\a,\theta_\a,\delta_\a)$ is a $\Gamma$-Lie bialgebra,
then the co-Poisson structure on $A:= U(\a) \rtimes \Gamma$ is given by
$\delta_A([x]) = [\delta_\a(x)]$, and $\delta_A([\gamma])
= -[f_\gamma]([\gamma]\otimes [\gamma])$. (Here we also denote by
$x\mapsto [x]$ the natural map $\wedge^2(\a) \to \wedge^2(U(\a) \rtimes
\Gamma$).) One checks that this establishes the desired (anti)equivalences.
\hfill \qed \medskip
\subsection{The problem of quantization of $\Gamma$-Lie bialgebras}
Define a $\Gamma$-graded bialgebra (in a symmetric monoidal category $\cS$)
as a bialgebra $A$ (in $\cS$), equipped with a grading
$A = \oplus_{\gamma\in \Gamma} A_\gamma$, such that $A_\gamma A_{\gamma'}
\subset A_{\gamma\gamma'}$ and $\Delta_A(A_{\gamma}) \subset
A_\gamma^{\otimes 2}$.
Assume that $A$ is a $\Gamma$-graded bialgebra in the category of
topologically free $\kk[[\hbar]]$-modules, quasicocommutative (in the sense
that $A_0 := A/\hbar A$ is cocommutative).
Then we get a co-Poisson structure on $A_0$. It is $\Gamma$-graded, in
the sense that $\delta_{A_0}((A_0)_\gamma) \subset \wedge^2((A_0)_\gamma)$.
We therefore get a classical limit functor $\on{class} :
\{\Gamma$-graded quasicocommutative
bialgebras$\} \to \{\Gamma$-graded co-Poisson bialgebras$\}$.
\begin{definition} A quantization functor for $\Gamma$-Lie bialgebras
is a functor $\{$co-Poisson $\Gamma$-universal enveloping
algebras$\} \to \{\Gamma$-graded quasicocommutative bialgebras$\}$,
right inverse to $\on{class}$.
\end{definition}
We define the category of group-graded bialgebras as follows: objects are
pairs $(\Gamma,A)$, where $\Gamma$ is a group and $A$ is a $\Gamma$-graded
bialgebra. A morphism $(\Gamma,A)\to (\Gamma',A')$ is the pairs of a group
morphism $i_{\Gamma\Gamma'} : \Gamma \to \Gamma'$ and a bialgebra morphism
$i_{AA'} : A \to A'$, compatible with the gradings.
One defines similarly the category of group-graded co-Poisson
bialgebras and quantization functors for group Lie bialgebras.
\subsection{Relation with quantization of co-Poisson bialgebras}
We have inclusions of full subcategories $\{$co-Poisson universal
enveloping algebras$\} \subset \{$co-Poisson group universal enveloping
algebras$\} \subset \{$co-Poisson bialgebras$\}$.
The classical limit functor is $\on{class} : \{$quasicocommutative
bialgebras$\} \to \{$co-Poisson bialgebras$\}$.
A quantization functor of Lie bialgebras is a functor
$\{$co-Poisson universal enveloping algebras$\} \to \{$quasicocommutative
bialgebras$\}$, left inverse fo class. A quantization functor for
group Lie bialgebras may then be viewed as a left inverse to class
with a wider domain.
\section{The formalism of props}\label{back}
We recall material from \cite{EH}. Polynomial Schur functors form a symmetric
monoidal abelian category Sch, equipped with an involution.
A prop $P$ is an additive symmetric monoidal
category, equipped with a tensor functor $\on{Sch} \to P$, which induces
a bijection $\on{Ob(Sch)} \simeq \on{Ob}(P)$ (\cite{McL}).
A prop morphism $P\to Q$ is a tensor functor, such that the composition
$\on{Sch}\to P \to Q$ coincides with $\on{Sch} \to Q$. A topological
prop is defined in the same way, with $\on{Sch}$
replaced by the category of ``formal series'' Schur functors
(i.e., infinite sums of homogenous Schur functors).
If $F$ is a (formal series) Schur functor and $P$ is a (topological)
prop, then $F(P)$ is
a prop defined by $(F(P))(F_1,F_2) = P(F_1\circ F,F_2\circ F)$.
Props may be defined by generators and relations. We will need the props
Bialg of bialgebras, LBA of Lie bialgebras, $\on{LBA}_f$ of Lie bialgebras
with a twist. Generators of Bialg are $m,\Delta,\varepsilon,\eta$ (the
universal analogues of the product, coproduct, counit, unit of a bialgebra);
generators of LBA are $\mu,\delta$ (universal analogues of the Lie bracket and
cobracket); $\on{LBA}_f$ has the additional generator $f$ (universal twist
element). LBA and $\on{LBA}_f$ are graded ($\mu$ has degree $0$ and $\delta,f$
have degree $1$) and can be completed into topological props
${\bf LBA}$, ${\bf LBA}_f$.
We define tensor categories $\on{Sch}_{(1)}$ and $\on{Sch}_{(1+1)}$
by $\on{Ob}(\on{Sch}_{(1)}) = \prod'_{n\geq 0} \on{Ob}(\on{Sch}_n)$
and $\on{Sch}_{(1+1)} = \prod'_{p,q\geq 0} \on{Ob}(\on{Sch}_{p+q})$,
where $\on{Ob}(\on{Sch}_n)$ is the set of polynomial Schur
multifunctors $\on{Vect}^n \to \on{Vect}$; the tensor product in these
categories is denoted $\boxtimes$. The bifunctor $\on{Sch}_{(1)}^2
\to \on{Sch}_{(1+1)}$ is denoted $(F,G)\mapsto F\ul\boxtimes G$.
A multi(bi)prop
is an additive symmetric monoidal category $\tilde P$ with a tensor functor
$\on{Sch}_{(1)} \to \tilde P$ (resp., $\on{Sch}_{(1+1)} \to \tilde P$),
inducing a bijection on the sets of objects. A prop $P$ give rises to a
multi-prop $\tilde P$ via $\tilde P(F,G):= P(c(F),c(G))$, where
$c:\on{Ob}(\on{Sch}_n)\to \on{Ob}(\on{Sch})$ is induced by the diagonal
embedding $\on{Vect}\to \on{Vect}^n$.
We introduce the notions of a trace on a symmetric monoidal category, of a
quasi-category, we show that a symmetric monoidal category with a
trace and an involution gives rise to a symmetric monoidal quasi-category
(i.e., the compositions are not always defined). In particular, a trace on a
multi-prop gives rise to a quasi-multi-bi-prop
(i.e., an additive symmetric monoidal quasi-category with a morphism from
$\on{Sch}_{(1+1)}$ inducing a bijection on objects).
We define traces on the multi-props arising from $\on{LBA}$ and
$\on{LBA}_f$; this gives rise to quasi-multi-bi-props $\Pi,\Pi_f$
with $\Pi(F\underline\boxtimes G,F'\underline\boxtimes G') =
\on{LBA}(c(F) \otimes c(G')^*,c(F')\otimes
c(G)^*)$; the morphisms in $\Pi_f$ are defined by a similar formula.
We also define topological completions
\boldmath$\Pi$\unboldmath, \boldmath$\Pi$\unboldmath$_f$.
When $F,...,G'$ are tensor products (in $\on{Sch}_{(1)}$)
of irreducible Schur functors,
$\Pi(F\underline\boxtimes G,F'\underline\boxtimes G')$
is graded by a set of oriented graphs; the composition of two (or several)
morphisms is
defined if the composition of their diagrams is acyclic. For
general $F\in \on{Ob}(\on{Sch}_n),...,G'\in\on{Ob}(\on{Sch}_{p'})$,
one can define the support of a given element of
$\Pi(F\underline\boxtimes G,F'\underline\boxtimes G')$
(again an oriented graph), and acyclicity is a sufficient condition for
the composition of two (or many) morphisms to be defined. Using this
criterion, one checks that the compositions involved in the future
computations all make sense.
The motivation for working with such structures is that when $F,...,G'$
are tensor products (in $\on{Sch}_{(1)}$) of tensor Schur functors
(i.e., objects of $\on{Sch}_1$ of the form $V\mapsto V^{\otimes n}$),
$\Pi(F\underline\boxtimes G,F'\underline\boxtimes G')$ may be viewed as
a space of acyclic oriented diagrams; composition is then defined by
connecting diagrams, and is of course only defined under acyclicity
assumptions. When $F,...,G'$ are tensor products (in $\on{Sch}_{(1)}$)
of simple Schur functors, the morphisms are obtained from
the case of tensor products of tensor functors by applying projectors
in the group algebras of products of symmetric groups, preserving a
partition of the vertices.
\section{Compositions of twists} \label{comp:twists}
A quantization functor is a prop morphism $Q : \on{Bialg} \to S({\bf LBA})$
with certain classical limit properties.
Let $Q$ be an Etingof-Kazhdan (EK) quantization functor.
It is constructed as follows.
We define elements $m_\Pi\in $\boldmath$\Pi$\unboldmath$((S\underline\boxtimes
S)^{\boxtimes 2},
S\underline\boxtimes S)$, $\Delta_0\in $\boldmath$\Pi$\unboldmath$
(S\underline\boxtimes S,
(S\underline\boxtimes S)^{\boxtimes 2})$,
$\on{J}\in $\boldmath$\Pi$\unboldmath$({\mathfrak 1}
\underline\boxtimes {\mathfrak 1},
(S\underline\boxtimes S)^{\boxtimes 2})$,
$\on{R}_+\in $\boldmath$\Pi$\unboldmath$(S\underline\boxtimes
{\mathfrak 1},S\underline\boxtimes S)$, $m_a\in $\boldmath$\Pi$\unboldmath$
((S\underline\boxtimes {\mathfrak 1})^{\boxtimes 2}, S\underline\boxtimes
{\mathfrak 1})$, $\Delta_a\in $\boldmath$\Pi$\unboldmath$
(S\underline\boxtimes {\mathfrak 1},
(S\underline\boxtimes {\mathfrak 1})^{\boxtimes 2})$.
We define $m_\Pi^{(i,j)}\in $\boldmath$\Pi$\unboldmath$(((S\underline
\boxtimes S
)^{\boxtimes j})^{\boxtimes i}, (S\underline\boxtimes S)^{\boxtimes j})$
as the $j$th tensor power of the $i$fold iterate of $m_\Pi$.
We have
$$
m_\Pi \circ \on{R}_+^{\boxtimes 2} = \on{R}_+ \circ m_a, \quad
m_\Pi^{(2,2)} \circ \Big( \on{J} \boxtimes (\Delta_0
\circ \on{R}_+)\Big) = m_\Pi^{(2,2)} \circ \Big( (\on{R}_+^{\boxtimes 2} \circ
\Delta_a)\boxtimes \on{J}\Big).
$$
Then $m_a,\Delta_a$ satisfy the bialgebra relations. The functor $Q$
is defined by $m\mapsto m_a$, $\Delta \mapsto \Delta_a$.
We now recall the results from \cite{EH} on the quantization of twists.
We define prop morphisms $\kappa_i : \on{LBA} \to \on{LBA}_f$ ($i = 1,2$) by
$\kappa_1 : (\mu,\delta) \mapsto (\mu,\delta)$
and $\kappa_2 : (\mu,\delta) \mapsto (\mu,\delta + \on{ad}(f))$.
Define $\Xi_f\in $\boldmath$\Pi$\unboldmath$_f(S\underline\boxtimes S,
S\underline\boxtimes
S)^\times$ as the universal version of the sequence of maps
$S(\a) \otimes S(\a^*)
\simeq U(D(\a)) \simeq U(D(\a_f)) \simeq S(\a) \otimes S(\a^*)$,
based on the Lie algebra isomorphism $D(\a) \simeq D(\a_f)$.
Then
\begin{equation} \label{Xi:f}
\kappa_2^\Pi(m_\Pi) = \Xi_f \circ \kappa_1^\Pi(m_\Pi) \circ
(\Xi_f^{-1})^{\boxtimes 2},
\quad \kappa_2^\Pi(\Delta_0) = \Xi_f^{\boxtimes 2} \circ
\kappa_1^\Pi(\Delta_0) \circ \Xi_f^{-1},
\end{equation}
\begin{proposition} (see \cite{EH})
There exists $(\on{F},v,\on{i})$ with $\on{F}\in
$\boldmath$\Pi$\unboldmath$_f({\mathfrak 1}\underline\boxtimes {\mathfrak 1},
(S\underline\boxtimes {\mathfrak 1})^{\boxtimes 2})$, $v\in
$\boldmath$\Pi$\unboldmath$_f({\mathfrak 1}\underline\boxtimes
{\mathfrak 1},S\underline\boxtimes S)$, and
$\on{i}\in $\boldmath$\Pi$\unboldmath$_f(S\underline\boxtimes {\mathfrak 1},
S\underline\boxtimes {\mathfrak 1})^\times$, such that $\on{F} = 1
+ degree >0$, $v = 1 + degree >0$, $\on{i} = \on{id}_{S\underline\boxtimes
{\mathfrak 1}} + degree >0$,
\begin{equation} \label{J:F}
\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ
\Big( \big(
(\Xi_f^{-1})^{\boxtimes 2} \circ \kappa_2^\Pi(\on{J}) \big) \boxtimes
\big( \Delta_0 \circ v\big) \Big)
= \kappa_1^\Pi(m_\Pi^{(3,2)}) \circ \Big( \big( v\boxtimes v \big)
\boxtimes \big( \kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ \on{F}\big)
\boxtimes \big( \kappa_1^\Pi(\on{J})\big) \Big)
\end{equation}
and
\begin{equation} \label{id:zetaf}
\kappa_1^\Pi(m_\Pi) \circ \Big( \big( \Xi_f^{-1}
\circ \kappa_2^\Pi(\on{R}_+) \circ \on{i} \big) \boxtimes v \Big) =
\kappa_1^\Pi(m_\Pi) \circ \Big( v \boxtimes
\kappa_1^\Pi(\on{R}_+) \Big).
\end{equation}
The set of triples $(\on{F}',v',\on{i}')$ satisfying these relations
is given by $v' = \kappa_1^\Pi(m_\Pi) \circ \Big(v \boxtimes
(\kappa_1^\Pi(\on{R}_+) \circ \on{u}) \Big)$,
$m_a^{(2,2)} \circ \Big( \on{F}' \boxtimes (\Delta \circ \on{u}) \Big)
= m_a^{(2,2)} \circ \Big( \on{u}^{\boxtimes 2} \boxtimes \on{F} \Big)$,
$v' = \kappa_1^\Pi\Big(v \boxtimes (\kappa_1^\Pi(\on{R}_+) \circ \on{u})\Big)$,
$\on{i}' = \on{i} \circ \underline{\on{Ad}}(\on{u})$,
where $\on{u}\in $\boldmath$\Pi$\unboldmath$_f({\mathfrak 1}
\underline\boxtimes {\mathfrak 1},
S\underline\boxtimes {\mathfrak 1})$ has the form $\on{u} = 1 + degree >0$,
and $\underline{\on{Ad}}(\on{u})\in $\boldmath$\Pi$\unboldmath$_f
(S\underline\boxtimes {\mathfrak 1},
S\underline\boxtimes {\mathfrak 1})^\times$ is such that
$m_a \circ (\underline{\on{Ad}}(\on{u}) \boxtimes \on{u}) =
m_a \circ (\on{u} \boxtimes \on{id}_{S\underline\boxtimes
{\mathfrak 1}})$.
\end{proposition}
In \cite{EH}, we prove that this proposition has the following
consequence:
\begin{theorem}
(Compatibility of quantization functors with twists) We have
$$
\kappa_2^\Pi(m_a) = \on{i} \circ \kappa_1^\Pi(m_a) \circ
(\on{i}^{-1})^{\boxtimes 2}, \quad
\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big( \big( (\on{i}^{-1})^{\boxtimes 2} \circ
\kappa_2^\Pi(\Delta_a) \circ
\on{i}\big) \boxtimes \on{F}\Big) =
\kappa_1^\Pi(m_a^{(2,2)}) \circ
\Big( \on{F} \boxtimes \kappa_1^\Pi(\Delta_a)\Big),
$$
$$
\kappa_1^\Pi(m_a^{(3,2)}) \circ \Big( (\on{F}\boxtimes 1)
\boxtimes \big( (\Delta_a \boxtimes \on{id}_{S\underline\boxtimes
{\mathfrak 1}}) \circ \on{F}\big) \Big) =
\kappa_1^\Pi(m_a^{(3,2)}) \circ \Big( (1\boxtimes \on{F})
\boxtimes \big( (\on{id}_{S\underline\boxtimes
{\mathfrak 1}}\boxtimes \Delta_a) \circ \on{F}\big) \Big).
$$
\end{theorem}
As before, $m_a^{(i,j)}$ is the $j$th tensor power of the $i$ fold
iterate to $m_a$.
We will now study the behavior of the composition of twists
under quantization.
Define a prop $\on{LBA}_{f,f'}$ by generators $\mu\in
\on{LBA}_{f,f'}(\wedge^2,{\bf id})$, $\delta\in \on{LBA}_{f,f'}({\bf id},
\wedge^2)$, $f,f'\in \on{LBA}_{f,f'}({\bf 1},\wedge^2)$ and relations:
$\mu,\delta,f$ satisfy the relations of $\on{LBA}_f$, and $f'$ is such that
\begin{equation} \label{rel:LBA}
((123) + (231) + (312)) \circ \Big(
(\delta \boxtimes \on{id}_{{\bf id}}) \circ f' +
(\mu\boxtimes \on{id}_{T_2})\circ ((1234) + (1324)) \circ (f\boxtimes f')
\Big) =0.
\end{equation}
Define prop morphisms
$\kappa_{ij} : \on{LBA}_f \to \on{LBA}_{f,f'}$, by
$\kappa_{12} : (\mu,\delta,f) \mapsto (\mu,\delta,f)$,
$\kappa_{23} : (\mu,\delta,f) \mapsto (\mu,\delta + \on{ad}(f),f')$,
$\kappa_{13} : (\mu,\delta,f) \mapsto (\mu,\delta,f+f')$.
Define prop morphisms $\bar\kappa_1, \bar\kappa_2,\bar\kappa_3 :
\on{LBA} \to \on{LBA}_{f,f'}$, $(\mu,\delta) \mapsto
(\mu,\delta)$, $(\mu,\delta + \on{ad}(f))$, $(\mu,\delta + \on{ad}(f+f'))$.
Then we have $\kappa_{1i} \circ \kappa_1 = \bar\kappa_1$,
$\kappa_{i3} \circ \kappa_2 = \bar\kappa_3$,
$\kappa_{23} \circ \kappa_1 = \kappa_{12} \circ \kappa_2 = \bar\kappa_2$.
\begin{lemma}
We have
\begin{equation} \label{Xi:Xi}
\kappa_{23}^\Pi(\Xi_f) \circ \kappa_{12}^\Pi(\Xi_f)
= \kappa_{13}^\Pi(\Xi_f).
\end{equation}
\end{lemma}
{\em Proof.} This follows from the fact that if $f_\a$ is a twist for
$\a$ and $f'_\a$ is a twist for $\a_{f_\a}$, then $f_\a + f'_\a$ is a
twist for $\a$, and $(\a_{f_\a})_{f'_\a} \simeq (\a_{f_\a})_{f'_\a}$.
\hfill \qed \medskip
\begin{theorem} \label{thm:i:v}
There exists $\on{v}\in $\boldmath$\Pi$\unboldmath$_{f,f'}({\mathfrak 1}
\underline\boxtimes {\mathfrak 1}, S \underline{\boxtimes} {\mathfrak 1})$,
such that $\on{v} = 1 + degree >0$,
\begin{equation} \label{stat:v}
\bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big(
\kappa_{13}^\Pi(\on{F}) \boxtimes
(\bar\kappa_1^\Pi(\Delta_a) \circ \on{v})\Big)
= \bar\kappa_1^\Pi(m_a^{(3,2)}) \circ \Big( \on{v}^{\boxtimes 2} \boxtimes
\big(
\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})
\big) \boxtimes \kappa_{12}^\Pi(\on{F}) \Big),
\end{equation}
and
\begin{equation} \label{stat:zeta}
\bar\kappa_1^\Pi(m_a)
\circ \Big( \on{v} \boxtimes \big( \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big) \Big)
= \bar\kappa_1^\Pi(m_a) \circ
\Big( \kappa_{13}^\Pi(\on{i}^{-1}) \boxtimes \on{v}\Big),
\end{equation}
\end{theorem}
{\em Proof.} Let us prove (\ref{stat:v}).
Applying $\kappa_{23}^\Pi$ to (\ref{J:F}), we get
\begin{align} \label{int:4}
& \bar\kappa_2^\Pi(m_\Pi^{(2,2)}) \circ
\Big( \big( \kappa_{23}^\Pi(\Xi_f^{-1})^{\boxtimes 2} \circ
\bar\kappa_3^\Pi(\on{J})\big) \boxtimes \big( \Delta_0 \circ
\kappa_{23}^\Pi(v)\big) \Big)
\\ & \nonumber
= \bar\kappa_2^\Pi(m_\Pi^{(3,2)}) \circ
\Big( \big( \kappa_{23}^\Pi(v)^{\boxtimes 2}\big)
\boxtimes \big( \bar\kappa_2^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{23}(\on{F})\big) \boxtimes \big( \bar\kappa_2^\Pi(\on{J})\big) \Big).
\end{align}
(\ref{Xi:f}) implies the identities
\begin{equation} \label{new:id}
\bar\kappa_2^\Pi(m_\Pi) = \kappa_{12}^\Pi(\Xi_f) \circ
\bar\kappa_1^\Pi(m_\Pi) \circ \kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2},
\quad
\bar\kappa_2^\Pi(\Delta_0) = \kappa_{12}^\Pi(\Xi_f)^{\boxtimes 2} \circ
\bar\kappa_1^\Pi(\Delta_0) \circ \kappa_{12}^\Pi(\Xi_f^{-1}).
\end{equation}
Left composing (\ref{int:4}) with $\kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2}$
and using these identities, we get
\begin{align*}
& \bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ
\Big( \big( [\kappa_{12}^\Pi(\Xi_f^{-1}) \circ
\kappa_{23}^\Pi(\Xi_f^{-1})]^{\boxtimes 2} \circ
\bar\kappa_3^\Pi(\on{J})\big) \boxtimes \big( \Delta_0 \circ
\kappa^\Pi_{12}(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big) \Big)
\\ & =
\kappa^\Pi_{12}(\Xi_f^{-1})^{\boxtimes 2} \circ
\bar\kappa_2^\Pi(m_\Pi^{(3,2)}) \circ
\Big( \big( \kappa_{23}^\Pi(v)^{\boxtimes 2}\big)
\boxtimes \big( \bar\kappa_2^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{23}^\Pi(\on{F})\big) \boxtimes \big(
\bar\kappa_2^\Pi(\on{J})\big) \Big).
\end{align*}
Applying $\kappa_{23}^\Pi$ to (\ref{id:zetaf}), we get
\begin{equation} \label{int:2}
\bar\kappa_2^\Pi(m_\Pi) \circ \Big( \kappa_{23}^\Pi(v) \boxtimes
\big( \bar\kappa_2^\Pi(\on{R}_+) \big) \Big)
=
\bar\kappa_2^\Pi(m_\Pi) \circ \Big(
\big( \kappa_{23}^\Pi(\Xi_f^{-1})
\circ \bar\kappa_3^\Pi(\on{R}_+) \circ \kappa_{23}^\Pi(\on{i})
\big) \boxtimes \kappa_{23}^\Pi(v) \Big),
\end{equation}
and applying $\kappa_{12}^\Pi$ to (\ref{id:zetaf}), we get
\begin{equation} \label{int:3}
\bar\kappa_1^\Pi(m_\Pi) \circ \Big( \kappa_{12}^\Pi(v) \boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \big) \Big)
= \bar\kappa_1^\Pi(m_\Pi) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1})
\circ \bar\kappa_2^\Pi(\on{R}_+) \circ \kappa_{12}^\Pi(\on{i})
\big) \boxtimes \kappa_{12}^\Pi(v) \Big).
\end{equation}
(\ref{int:2}) then implies
\begin{align*}
& \bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ
\Big( \big( [\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi
(\Xi_f^{-1})]^{\boxtimes 2} \circ
\bar\kappa_3^\Pi(\on{J})\big) \boxtimes \big( \Delta_0 \circ
\kappa^\Pi_{12}(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big) \Big)
\\ & =
\kappa^\Pi_{12}(\Xi_f^{-1})^{\boxtimes 2} \circ
\bar\kappa_2^\Pi(m_\Pi^{(3,2)}) \circ
\Big(
\big( \kappa_{23}^\Pi(\Xi_f^{-1})^{\boxtimes 2} \circ
\bar\kappa_3^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{23}^\Pi(\on{i})^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})\big)
\boxtimes \big( \kappa_{23}^\Pi(v)^{\boxtimes 2}\big) \boxtimes
\bar\kappa_2^\Pi(\on{J}) \Big)
\\ & =
\bar\kappa_1^\Pi(m_\Pi^{(3,2)}) \Big( \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(\Xi_f^{-1})
\circ \bar\kappa_3^\Pi(\on{R}_+) \circ \kappa_{23}^\Pi(\on{i})
\big)^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})\Big)
\\ &
\boxtimes \big(\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)
\big)^{\boxtimes 2} \boxtimes
\big( \kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2}
\circ \bar\kappa_2^\Pi(\on{J})\big) \Big) .
\end{align*}
Applying $\kappa_{12}^\Pi$ to (\ref{J:F}), we get
\begin{align} \label{interm}
& \nonumber \bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ \Big( \Big(
\kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2} \circ \bar\kappa_2^\Pi(\on{J})\Big)
\boxtimes \kappa_{12}^\Pi(\Delta_0 \circ v)\Big)
\\ & = \bar\kappa_1^\Pi(m_\Pi^{(3,2)}) \circ \Big( \kappa_{12}^\Pi(v)^{\boxtimes 2}
\boxtimes \big( \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{12}^\Pi(\on{F}) \big) \boxtimes \bar\kappa_1^\Pi(\on{J})\Big).
\end{align}
Therefore "right multiplication" (using $m_\Pi$)
of the previous identity by
$\kappa_{12}^\Pi(\Delta_0\circ v)$ yields
\begin{align*}
& \bar\kappa_1^\Pi(m_\Pi^{(3,2)}) \circ
\Big( \big( [\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi
(\Xi_f^{-1})]^{\boxtimes 2} \circ
\bar\kappa_3^\Pi(\on{J})\big) \boxtimes \big( \Delta_0 \circ
\kappa^\Pi_{12}(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big) \boxtimes
\kappa_{12}^\Pi(\Delta_0 \circ v) \Big)
\\ & =
\bar\kappa_1^\Pi(m_\Pi^{(4,2)}) \circ \Big( \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(\Xi_f^{-1})
\circ \bar\kappa_3^\Pi(\on{R}_+) \circ \kappa_{23}^\Pi(\on{i})
\big)^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})\Big)
\\ &
\boxtimes \big(\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)
\big)^{\boxtimes 2} \boxtimes
\big( \kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2}
\circ \bar\kappa_2^\Pi(\on{J})\big) \boxtimes
\kappa_{12}^\Pi(\Delta_0 \circ v) \Big)
\\ & =
\bar\kappa_1^\Pi(m_\Pi^{(5,2)}) \circ \Big( \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(\Xi_f^{-1})
\circ \bar\kappa_3^\Pi(\on{R}_+) \circ \kappa_{23}^\Pi(\on{i})
\big)^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})\Big)
\\ &
\boxtimes \big(\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)
\big)^{\boxtimes 2} \boxtimes
\kappa_{12}^\Pi(v)^{\boxtimes 2}
\boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{12}^\Pi(\on{F})\big)
\boxtimes
\bar\kappa_1^\Pi(\on{J})
\Big),
\end{align*}
where the last equality follows from (\ref{interm}).
According to (\ref{new:id}), the last term is equal to
\begin{align*}
& \kappa_{12}^\Pi(\Xi_f^{-1}) \circ
\bar\kappa_2^\Pi(m_\Pi^{(5,2)}) \circ \Big( \Big(
\big( \kappa_{23}^\Pi(\Xi_f^{-1})
\circ \bar\kappa_3^\Pi(\on{R}_+) \circ \kappa_{23}^\Pi(\on{i})
\big)^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})\Big)
\\ &
\boxtimes \big( \kappa_{23}^\Pi(v)
\big)^{\boxtimes 2} \boxtimes
\big(\kappa_{12}^\Pi(\Xi_f) \circ \kappa_{12}^\Pi(v) \big)^{\boxtimes 2}
\boxtimes
\big( \kappa_{12}^\Pi(\Xi_f)^{\boxtimes 2} \circ
\bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{12}^\Pi(\on{F})\big)
\boxtimes
\kappa_{12}^\Pi(\Xi_f)^{\boxtimes 2} \circ \bar\kappa_1^\Pi(\on{J})
\Big),
\end{align*}
which according to (\ref{int:2}) is equal to
\begin{align*}
& \kappa_{12}^\Pi(\Xi_f^{-1}) \circ
\bar\kappa_2^\Pi(m_\Pi^{(5,2)}) \circ \Big(
\kappa_{23}^\Pi(v)^{\boxtimes 2} \boxtimes
\big( \bar\kappa_2^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{23}^\Pi(\on{F})\big)
\\ &
\boxtimes
\big(\kappa_{12}^\Pi(\Xi_f) \circ \kappa_{12}^\Pi(v) \big)^{\boxtimes 2}
\boxtimes
\big( \kappa_{12}^\Pi(\Xi_f)^{\boxtimes 2} \circ
\bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{12}^\Pi(\on{F})\big)
\boxtimes
\kappa_{12}^\Pi(\Xi_f)^{\boxtimes 2} \circ \bar\kappa_1^\Pi(\on{J})
\Big),
\end{align*}
which we rewrite as
\begin{align*}
& \bar\kappa_1^\Pi(m_\Pi^{(5,2)}) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2}
\circ \kappa_{23}^\Pi(v)^{\boxtimes 2}\big)
\boxtimes
\big( \kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2}
\circ \bar\kappa_2^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{23}^\Pi(\on{F})\big)
\\ &
\boxtimes
\big( \kappa_{12}^\Pi(v) \big)^{\boxtimes 2}
\boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{12}^\Pi(\on{F})\big)
\boxtimes
\bar\kappa_1^\Pi(\on{J})
\Big).
\end{align*}
(\ref{int:3}) allows then to rewrite this as
\begin{align*}
& \bar\kappa_1^\Pi(m_\Pi^{(5,2)}) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1})^{\boxtimes 2}
\circ \kappa_{23}^\Pi(v)^{\boxtimes 2}\big)
\boxtimes
\kappa_{12}^\Pi(v)^{\boxtimes 2}
\\ &
\boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\kappa_{23}^\Pi(\on{F})\big)
\boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2}
\circ \kappa_{12}^\Pi(\on{F})\big)
\boxtimes
\bar\kappa_1^\Pi(\on{J})
\Big).
\end{align*}
We therefore get:
\begin{equation} \label{part:1}
\bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ \Big( \big(
(\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(\Xi_f^{-1}))^{\boxtimes 2}
\circ \bar\kappa_3^\Pi(\on{J})\big)
\boxtimes (\Delta_0 \circ v_1)\Big) =
\bar\kappa_1^\Pi(m_\Pi^{(3,2)}) \circ
\Big( v_1^{\boxtimes 2} \boxtimes F_1 \boxtimes \bar\kappa_1^\Pi(\on{J})\Big) ,
\end{equation}
where
$$
v_1 = \bar\kappa_1^\Pi(m_\Pi) \circ \Big( \big(
\kappa_{12}^\Pi(\Xi_f^{-1}) \circ
\kappa_{23}^\Pi(v)\big) \boxtimes
\kappa_{12}^\Pi(v) \Big)
$$
\begin{align*}
F_1 & = \bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ \Big( \big(
\bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})
\big) \boxtimes \big( \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\kappa_{12}^\Pi(\on{F})\big) \Big)
\\ & = \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ
\bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big(
\big(
\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})
\big) \boxtimes \kappa_{12}^\Pi(\on{F}) \Big)
= \bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ \on{F_1},
\end{align*}
where
$$
\on{F}_1 =
\bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big(
\big(
\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ \kappa_{23}^\Pi(\on{F})
\big) \boxtimes \kappa_{12}^\Pi(\on{F})
\Big).
$$
(\ref{Xi:Xi}) implies that (\ref{part:1}) is rewritten as
\begin{equation} \label{neweq}
\bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ \Big( \big(
\kappa_{13}^\Pi(\Xi_f^{-1})^{\boxtimes 2} \circ \bar\kappa_3^\Pi(\on{J})
\big) \boxtimes (\Delta_0 \circ v_1)\Big)
= \bar\kappa_1^\Pi(m_\Pi^{(3,2)}) \circ \Big( v_1^{\boxtimes 2}
\boxtimes \big(\bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ \on{F}_1 \big)
\boxtimes \bar\kappa_1^\Pi(\on{J}) \Big).
\end{equation}
On the other hand, applying $\kappa_{13}^\Pi$ to (\ref{J:F}), we get
\begin{equation} \label{part:2}
\bar\kappa_1^\Pi(m_\Pi^{(2,2)}) \circ \Big( \big(
\kappa_{13}^\Pi(\Xi_f^{-1})^{\boxtimes 2}
\circ \bar\kappa_3^\Pi(\on{J})\big)
\boxtimes (\Delta_0 \circ v'_1)\Big) =
\bar\kappa_1^\Pi(m_\Pi^{(3,2)}) \circ \Big( (v'_1)^{\boxtimes 2}
\boxtimes
\big(\bar\kappa_1^\Pi(\on{R}_+)^{\boxtimes 2} \circ \on{F}'_1 \big)
\boxtimes \bar\kappa_1^\Pi(\on{J})\Big) ,
\end{equation}
where
$$
v'_1 = \kappa_{13}^\Pi(v), \quad
\on{F}'_1 = \kappa_{13}^\Pi(\on{F}),
$$
where $\on{F}'_1 = \kappa_{13}^\Pi(\on{F})$.
The result of uniqueness (up to gauge) for solutions $(\on{F},v)
\in $\boldmath$\Pi$\unboldmath$_f({\mathfrak 1}\underline\boxtimes
{\mathfrak 1},S\underline\boxtimes S)^\times \times
$\boldmath$\Pi$\unboldmath$_f({\mathfrak 1}\underline\boxtimes
{\mathfrak 1},S\underline\boxtimes {\mathfrak 1})^\times$
of equation (\ref{J:F}), which was established in
\cite{EH}, Lemma 5.3, can be generalized as follows.
\begin{lemma}
The set of pairs $(\on{F}''_1,v''_1)$ satisfying (\ref{neweq}), where
$\on{F}''_1\in $\boldmath$\Pi$\unboldmath$_{f,f'}({\mathfrak 1}
\underline\boxtimes
{\mathfrak 1},S\underline\boxtimes {\mathfrak 1})$,
$v''_1\in $\boldmath$\Pi$\unboldmath$_{f,f'}({\mathfrak 1}\underline\boxtimes
{\mathfrak 1},S\underline\boxtimes S)$, $\on{F}''_1 = 1 + degree >1$,
$v''_1 = 1 + degree >1$, is given by
$v_1 = \bar\kappa_1^\Pi(m_\Pi) \circ \Big( v''_1 \boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \circ \on{v}''\big) \Big), \quad
\bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big( \on{F}''_1 \boxtimes
(\bar\kappa_1^\Pi(\Delta_a) \circ \on{v}'')\Big)
= \bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big( \on{v}^{\prime\prime\boxtimes 2}
\boxtimes
\on{F}_1\Big)$, where $\on{v}'' \in \Pi_{f,f'}({\mathfrak
1}\underline\boxtimes {\mathfrak 1}, S \underline\boxtimes {\mathfrak 1})$,
$\on{v}'' = 1 +$ degree $>0$.
\end{lemma}
The proof if parallel to that of \cite{EH}, Lemma 5.3. The computation of the
co-Hochschild cohomology of $({\bf U}_{n,f})_{n\geq 0}$ is replaced by that of
$({\bf U}_{n,f,f'})_{n\geq 0}$, where ${\bf U}_{f,f',n} =
\Pi_{f,f'}({\mathfrak 1}\underline\boxtimes
{\mathfrak 1},(S\underline\boxtimes S)^{\boxtimes n})$,
and the argument of the vanishing of $\on{LBA}_{f}({\bf id},{\bf 1})$
is replaced by the vanishing of $\on{LBA}_{f,f'}({\bf id},{\bf 1})$.
It follows that there exists $\on{v}\in $\boldmath$\Pi$\unboldmath$_{f,f'}
({\mathfrak
1}\underline\boxtimes {\mathfrak 1}, S \underline\boxtimes {\mathfrak 1})$,
$\on{v} = 1 +$ degree $>0$, such that
\begin{equation} \label{final:id}
v_1 = \bar\kappa_1^\Pi(m_\Pi) \circ \Big( v'_1 \boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \circ \on{v}\big) \Big), \quad
\bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big( \on{F}'_1 \boxtimes
(\bar\kappa_1^\Pi(\Delta_a) \circ \on{v})\Big)
= \bar\kappa_1^\Pi(m_a^{(2,2)}) \circ \Big( \on{v}^{\boxtimes 2} \boxtimes
\on{F}_1\Big).
\end{equation}
The second of these identities is (\ref{stat:v}).
Let us now prove (\ref{stat:zeta}). Right composing (\ref{id:zetaf})
with $\on{i}^{-1}$, applying $\kappa_{23}^\Pi$, left composing with
$\kappa_{12}^\Pi(\Xi_f^{-1})$, and right multiplying the resulting identity
by $\kappa_{12}^\Pi(v)$ using $\bar\kappa_1^\Pi(m_\Pi)$, we get
\begin{align} \label{al:1}
& \nonumber \bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{13}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_3^\Pi(\on{R}_+)\big)
\boxtimes
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \kappa_{12}^\Pi(v)\Big)
\\ & = \bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_2^\Pi(\on{R}_+)
\circ \kappa_{23}^\Pi(\on{i}^{-1})\big)
\boxtimes \kappa_{12}^\Pi(v)\Big).
\end{align}
Right composing (\ref{id:zetaf}) by $\on{i}^{-1}$, applying $\kappa_{12}^\Pi$,
right composing with $\kappa_{23}^\Pi(\on{i}^{-1})$, and left multiplying by
$\kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)$ using
$\bar\kappa_1^\Pi(m_\Pi)$, we get
\begin{align} \label{al:2}
& \nonumber \bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_2^\Pi(\on{R}_+)
\circ \kappa_{23}^\Pi(\on{i}^{-1})\big)
\boxtimes \kappa_{12}^\Pi(v)\Big)
\\ & =
\bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \kappa_{12}^\Pi(v) \boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \circ \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big)\Big).
\end{align}
The first identity of (\ref{final:id})
is rewritten as
\begin{equation} \label{vv:vv}
\bar\kappa_1^\Pi(m_\Pi) \circ \Big( \big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ
\kappa_{23}^\Pi(v) \big) \boxtimes \kappa_{12}^\Pi(v) \Big)
= \bar\kappa_1^\Pi(m_\Pi) \circ \Big( \kappa_{13}^\Pi(v) \boxtimes
(\bar\kappa_1^\Pi(\on{R}_+) \circ \on{v})\Big),
\end{equation}
therefore
\begin{align} \label{al:3}
& \nonumber \bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \kappa_{12}^\Pi(v) \boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \circ \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big)\Big).
\\ &
= \bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\kappa_{13}^\Pi(v) \boxtimes \big( \bar\kappa_1^\Pi(\on{R}_+)\circ \on{v} \big)
\boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \circ \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big)\Big).
\end{align}
Combining (\ref{al:1}), (\ref{al:2}) and (\ref{al:3}), we get:
\begin{align} \label{al:4}
& \nonumber
\bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{13}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_3^\Pi(\on{R}_+)\big)
\boxtimes
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \kappa_{12}^\Pi(v)\Big)
\\ & =
\bar\kappa_1^\Pi(m_\Pi) \circ \Big(
\kappa_{13}^\Pi(v) \boxtimes
\big[ \bar\kappa_1^\Pi(\on{R}_+) \circ \bar\kappa_1^\Pi(m_a)
\circ \Big( \on{v} \boxtimes \big( \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big) \Big) \big] \Big).
\end{align}
On the other hand, left multiplying (\ref{vv:vv}) by
$\kappa_{13}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_3^\Pi(\on{R}_+)$
using $\bar\kappa_1^\Pi(m_\Pi)$, we get
\begin{align} \label{all:1}
& \nonumber
\bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{13}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_3^\Pi(\on{R}_+)\big)
\boxtimes
\big( \kappa_{12}^\Pi(\Xi_f^{-1}) \circ \kappa_{23}^\Pi(v)\big)
\boxtimes \kappa_{12}^\Pi(v)\Big)
\\ &
= \bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{13}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_3^\Pi(\on{R}_+)\big)
\boxtimes
\kappa_{13}^\Pi(v) \boxtimes (\bar\kappa_1^\Pi(\on{R}_+) \circ \on{v}) \Big).
\end{align}
Right composing (\ref{id:zetaf}) by $\on{i}^{-1}$, applying $\kappa_{13}^\Pi$
and right multiplying by $\bar\kappa_1^\Pi(\on{R}_+) \circ \on{v}$
using $\bar\kappa_1^\Pi(m_\Pi)$, we get
\begin{align} \label{all:2}
& \nonumber
\bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big(
\big( \kappa_{13}^\Pi(\Xi_f^{-1}) \circ \bar\kappa_3^\Pi(\on{R}_+)\big)
\boxtimes
\kappa_{13}^\Pi(v) \boxtimes (\bar\kappa_1^\Pi(\on{R}_+) \circ \on{v}) \Big)
\\ & \nonumber =
\bar\kappa_1^\Pi(m_\Pi^{(3,1)}) \circ \Big( \kappa_{13}^\Pi(v) \boxtimes
\big( \bar\kappa_1^\Pi(\on{R}_+) \circ \kappa_{13}^\Pi(\on{i}^{-1})\big)
\boxtimes (\bar\kappa_1^\Pi(\on{R}_+) \circ \on{v})\Big)
\\ &
= \bar\kappa_1^\Pi(m_\Pi) \circ \Big( \kappa_{13}^\Pi(v) \boxtimes
\big[ \bar\kappa_1^\Pi(\on{R}_+) \circ \bar\kappa_1^\Pi(m_a) \circ
\Big( \kappa_{13}^\Pi(\on{i}^{-1}) \boxtimes \on{v}\Big) \big] \Big).
\end{align}
Combining (\ref{al:4}), (\ref{all:1}) and (\ref{all:2}), we get
\begin{align*}
& \bar\kappa_1^\Pi(m_\Pi) \circ \Big(
\kappa_{13}^\Pi(v) \boxtimes
\big[ \bar\kappa_1^\Pi(\on{R}_+) \circ \bar\kappa_1^\Pi(m_a)
\circ \Big( \on{v} \boxtimes \big( \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big) \Big) \big] \Big)
\\ & = \bar\kappa_1^\Pi(m_\Pi) \circ \Big( \kappa_{13}^\Pi(v) \boxtimes
\big[ \bar\kappa_1^\Pi(\on{R}_+) \circ \bar\kappa_1^\Pi(m_a) \circ
\Big( \kappa_{13}^\Pi(\on{i}^{-1}) \boxtimes \on{v}\Big) \big] \Big).
\end{align*}
Since $v$ is invertible for $m_\Pi$, and $\on{R}_+$
is left invertible, this implies
$$
\bar\kappa_1^\Pi(m_a)
\circ \Big( \on{v} \boxtimes \big( \kappa_{12}^\Pi(\on{i}^{-1}) \circ
\kappa_{23}^\Pi(\on{i}^{-1})\big) \Big)
= \bar\kappa_1^\Pi(m_a) \circ
\Big( \kappa_{13}^\Pi(\on{i}^{-1}) \boxtimes \on{v}\Big),
$$
i.e., (\ref{stat:zeta}).
\hfill \qed \medskip
We define the prop $\on{LBA}_{f,f',f''}$ by generators
$\mu,\delta,f,f',f''$, where $\mu\in\on{LBA}_{f,f',f''}(\wedge^2,{\bf id})$,
$\delta\in\on{LBA}_{f,f',f''}({\bf id},\wedge^2)$,
$f,f',f''\in\on{LBA}_{f,f',f''}({\bf 1},\wedge^2)$ and relations:
$\mu,\delta,f,f'$ satisfy the relations of $\on{LBA}_{f,f'}$, and
$(\mu,\delta,f+f',f'')$ satisfy the relation (\ref{rel:LBA}) satisfied
by $(\mu,\delta,f,f')$.
Define prop morphisms
$\kappa_{ijk} : \on{LBA}_{f,f'}
\to \on{LBA}_{f,f',f''}$, $\kappa_{123} : (\mu,\delta,f,f') \mapsto
(\mu,\delta,f,f')$, $\kappa_{124} : (\mu,\delta,f,f') \mapsto
(\mu,\delta,f,f'+f'')$, $\kappa_{134} : (\mu,\delta,f,f') \mapsto
(\mu,\delta,f+f',f'')$, $\kappa_{234} : (\mu,\delta,f,f') \mapsto
(\mu,\delta + \on{ad}(f),f',f'')$.
Define prop morphisms $\bar{\kappa}_{ij} : \on{LBA}_f \to \on{LBA}_{f,f',f''}$
by $\bar{\kappa}_{12} : (\mu,\delta,f) \mapsto (\mu,\delta,f)$,
$\bar{\kappa}_{13} : (\mu,\delta,f) \mapsto (\mu,\delta,f+f')$,
$\bar{\kappa}_{14} : (\mu,\delta,f) \mapsto (\mu,\delta,f+f'+f'')$,
$\bar{\kappa}_{23} : (\mu,\delta,f) \mapsto (\mu,\delta + \on{ad}(f),f')$,
$\bar{\kappa}_{24} : (\mu,\delta,f) \mapsto (\mu,\delta+\on{ad}(f),
f'+f'')$,
$\bar{\kappa}_{34} : (\mu,\delta,f) \mapsto
(\mu,\delta+\on{ad}(f+f'),f'')$.
Define prop morphisms $\bar{\bar{\kappa}}_i : \on{LBA} \to \on{LBA}_{f,f',f''}$
for $i = 1,...,4$, by $\bar{\bar{\kappa}}_1 : (\mu,\delta) \mapsto
(\mu,\delta)$, $\bar{\bar{\kappa}}_2 : (\mu,\delta) \mapsto
(\mu,\delta + \on{ad}(f))$,
$\bar{\bar{\kappa}}_3 : (\mu,\delta) \mapsto
(\mu,\delta + \on{ad}(f+f'))$,
$\bar{\bar{\kappa}}_4 : (\mu,\delta) \mapsto
(\mu,\delta + \on{ad}(f+f'+f''))$.
Then $\bar\kappa_{ij} \circ \kappa_1 = \bar{\bar{\kappa}}_i$,
$\bar\kappa_{ij} \circ \kappa_2 = \bar{\bar{\kappa}}_j$ (where
$1\leq i<j\leq 4$); we also have
$\kappa_{1ij} \circ \bar\kappa_1 = \bar{\bar{\kappa}}_1$ (where $2\leq
i<j\leq 4$),
$\kappa_{234} \circ \bar\kappa_1 = \kappa_{12i} \circ \bar\kappa_2 =
\bar{\bar{\kappa}}_2$ (where $i=3,4$),
$\kappa_{i34} \circ \bar\kappa_2 = \kappa_{123} \circ \bar\kappa_3 =
\bar{\bar{\kappa}}_3$ (where $i=1,2$),
$\kappa_{ij4} \circ \bar\kappa_3 = \bar{\bar{\kappa}}_4$ (where
$1\leq i<j\leq 3$); finally $\kappa_{12i} \circ \kappa_{12} = \bar\kappa_{12}$
($i=3,4$), $\kappa_{1i4} \circ \kappa_{13} = \bar\kappa_{14}$ ($i=2,3$),
$\kappa_{i34} \circ \kappa_{23} = \bar\kappa_{34}$ ($i=1,2$),
$\kappa_{134}\circ \kappa_{12} = \kappa_{123} \circ \kappa_{13}
= \bar\kappa_{13}$,
$\kappa_{234}\circ \kappa_{12} = \kappa_{123} \circ \kappa_{23}
= \bar\kappa_{23}$,
$\kappa_{234}\circ \kappa_{13} = \kappa_{124} \circ \kappa_{23}
= \bar\kappa_{24}$.
\begin{theorem} \label{thm:v:v} \label{thm:ids:v}
\begin{equation} \label{vv=vv}
\bar{\bar\kappa}_1(m_a) \circ
\big( \kappa_{134}^\Pi(\on{v}) \boxtimes \kappa_{123}^\Pi(\on{v})\big)
= \bar{\bar\kappa}_1(m_a) \circ \big(
\kappa_{124}^\Pi(\on{v}) \boxtimes
(\bar\kappa_{12}^\Pi(\on{i}^{-1})\circ \kappa_{234}^\Pi(\on{v}))\big).
\end{equation}
\end{theorem}
{\em Proof.} Applying $\kappa_{134}^\Pi$ to (\ref{stat:v}), we get
$$
\bar{\bar\kappa}_1^\Pi(m_a^{(2,2)}) \circ \big( \bar\kappa_{14}^\Pi(\on{F})
\boxtimes \kappa_{134}^\Pi(\Delta_a \circ \on{v})\big) =
\bar{\bar\kappa}_1^\Pi(m_a^{(2,2)}) \circ \Big(
\kappa_{134}^\Pi(\on{v})^{\boxtimes 2} \boxtimes \big(
\bar\kappa_{13}(\on{i}^{-1})^{\boxtimes 2} \circ \bar\kappa_{34}(\on{F})\big)
\boxtimes \bar\kappa_{13}^\Pi(\on{F})\Big) .
$$
Using the fact that $\bar\kappa^\Pi_{13}(\on{F}) = \kappa^\Pi_{123}
\circ \kappa^\Pi_{13}(\on{F})$ and the image of (\ref{stat:v})
by $\kappa_{123}^\Pi$, we get
\begin{align*}
& \bar{\bar{\kappa}}_1(m_a^{(3,2)}) \circ \Big( \bar\kappa_{14}^\Pi(\on{F})
\boxtimes ( \bar{\bar\kappa}_1^\Pi(\Delta_a)\circ
\kappa_{134}^\Pi(\on{v})) \boxtimes
(\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ \kappa_{123}^\Pi(\on{v}))\Big)
\\ & =
\bar{\bar{\kappa}}_1(m_a^{(5,2)}) \circ \Big(
\kappa^\Pi_{134}(\on{v})^{\boxtimes 2} \boxtimes
\big( \bar\kappa_{13}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{34}^\Pi(\on{F})\big)
\boxtimes \kappa_{123}^\Pi(\on{v}^{\boxtimes 2}) \boxtimes
\big( \bar\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}^\Pi(\on{F})\big) \boxtimes \bar\kappa_{12}^\Pi(\on{F})
\Big).
\end{align*}
Applying $\kappa_{123}^\Pi$ to (\ref{stat:zeta}), we get
$\bar{\bar\kappa}_1^\Pi(m_a) \circ ( \bar\kappa_{13}^\Pi(\on{i}^{-1})
\boxtimes \kappa_{123}^\Pi(\on{v})) = \bar{\bar\kappa}_1^\Pi(m_a)
\circ (\kappa_{123}^\Pi(\on{v}) \boxtimes (\bar\kappa_{12}^\Pi(\on{i}^{-1})
\circ \bar\kappa_{23}^\Pi(\on{i}^{-1})))$, which implies that
\begin{align} \label{partial:1}
& \bar{\bar{\kappa}}_1(m_a^{(2,2)}) \circ \Big( \bar\kappa_{14}^\Pi(\on{F})
\boxtimes (\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ \on{v}_1)\Big)
\\ & = \nonumber
\bar{\bar{\kappa}}_1(m_a^{(4,2)}) \circ \Big(
\on{v}_1^{\boxtimes 2} \boxtimes
\big( \bar\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{34}^\Pi(\on{F})\big)
\boxtimes
\big( \bar\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}^\Pi(\on{F})\big) \boxtimes \bar\kappa_{12}^\Pi(\on{F})
\Big),
\end{align}
where $\on{v}_1 = \bar{\bar\kappa}_1(m_a^{(2,1)}) \circ
\big( \kappa_{134}^\Pi(\on{v}) \boxtimes \kappa_{123}^\Pi(\on{v})\big)$.
Applying $\kappa_{124}^\Pi$ to (\ref{stat:v}), we get
\begin{equation} \label{interm:5}
\bar{\bar\kappa}_1^\Pi(m_a)^{(2,2)} \circ \Big( \bar\kappa_{14}^\Pi(\on{F})
\boxtimes (\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ \kappa_{124}^\Pi(\on{v}))
\Big) =
\bar{\bar\kappa}_1^\Pi(m_a^{(3,2)}) \circ \Big( \kappa_{124}^\Pi
(\on{v}^{\boxtimes 2})
\boxtimes \big( \bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{24}(\on{F})\big) \boxtimes \bar\kappa_{12}(\on{F})\Big),
\end{equation}
and applying $\kappa_{234}^\Pi$ to the same identity, we get
$$
\bar{\bar\kappa}_2^\Pi(m_a^{(2,2)}) \circ \Big( \bar\kappa_{24}(\on{F})
\boxtimes
(\bar{\bar\kappa}_2^\Pi(\Delta_a) \circ \kappa_{234}^\Pi(\on{v}))\Big)
= \bar{\bar\kappa}_2^\Pi(m_a^{(3,2)}) \circ \Big( \kappa_{234}
(\on{v}^{\boxtimes 2})
\boxtimes \big( \bar\kappa_{23}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{34}(\on{F})\big) \boxtimes \bar\kappa_{23}(\on{F})\Big).
$$
Since $\bar{\bar\kappa}_2(m_a) = \bar\kappa_{12}(\on{i}) \circ
\bar{\bar\kappa}_1(m_a) \circ (\bar\kappa_{12}(\on{i})^{\boxtimes 2})^{-1}$,
we get
\begin{align*}
& \bar{\bar\kappa}_1^\Pi(m_a^{(2,2)}) \circ \Big(
(\bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{24}(\on{F})) \boxtimes
(\bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar{\bar\kappa}_2^\Pi(\Delta_a)\circ \kappa_{234}^\Pi(\on{v}))\Big)
\\ & = \bar{\bar\kappa}_1^\Pi(m_a^{(3,2)}) \circ \Big(
(\bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\kappa_{234}^\Pi(\on{v}^{\boxtimes 2}))
\boxtimes \big( \bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{34}(\on{F})\big) \boxtimes
(\bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}(\on{F})) \Big).
\end{align*}
Right multiplying this identity by $\bar\kappa_{12}(\on{F})$ using
$\bar{\bar\kappa}_1^\Pi(m_a)$, and
using $\bar{\bar\kappa}_1^\Pi(m_a) \circ
\big( [\bar\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar{\bar\kappa}_2^\Pi(\Delta_a)] \boxtimes \bar\kappa_{12}^\Pi(\on{F}) \big)
= \bar{\bar\kappa}_1^\Pi(m_a) \circ
\big( \bar\kappa_{12}^\Pi(\on{F}) \boxtimes
[\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ \bar\kappa_{12}^\Pi(\on{i}^{-1})
] \big)$, we get
\begin{align*}
& \bar{\bar\kappa}_1^\Pi(m_a^{(3,2)}) \circ \Big(
(\bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{24}(\on{F})) \boxtimes
\bar\kappa_{12}^\Pi(\on{F}) \boxtimes
\big( \bar{\bar\kappa}_1^\Pi(\Delta_a) \circ
\bar\kappa_{12}^\Pi(\on{i}^{-1}) \circ \kappa_{234}^\Pi(\on{v})\big)
\Big)
\\ & = \bar{\bar\kappa}_1^\Pi(m_a^{(4,2)}) \circ \Big(
(\bar\kappa_{12}(\on{i}^{-1}) \circ
\kappa_{234}^\Pi(\on{v}))^{\boxtimes 2}
\boxtimes
\\ & \big( \bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{34}(\on{F})\big) \boxtimes
(\bar\kappa_{12}(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}(\on{F})) \boxtimes \bar\kappa_{12}^\Pi(\on{F})\Big).
\end{align*}
Right multiplying (\ref{interm:5}) by $\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ
\bar\kappa_{12}^\Pi(\on{i}^{-1}) \circ \kappa_{234}^\Pi(\on{v})$ using
$\bar{\bar\kappa}_1(m_a)$, we then get
\begin{align} \label{partial:2}
& \bar{\bar{\kappa}}_1(m_a^{(2,2)}) \circ \Big( \bar\kappa_{14}^\Pi(\on{F})
\boxtimes (\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ \on{v}_2)\Big)
\\ & = \nonumber
\bar{\bar{\kappa}}_1(m_a^{(4,2)}) \circ \Big(
\on{v}_2^{\boxtimes 2} \boxtimes
\big( \bar\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{34}^\Pi(\on{F})\big)
\boxtimes
\big( \bar\kappa_{12}^\Pi(\on{i}^{-1})^{\boxtimes 2} \circ
\bar\kappa_{23}^\Pi(\on{F})\big) \boxtimes \bar\kappa_{12}^\Pi(\on{F})
\Big),
\end{align}
where $\on{v}_2 = \bar{\bar\kappa}_1(m_a^{(2,1)}) \circ \big(
\kappa_{124}^\Pi(\on{v}) \boxtimes
(\bar\kappa_{12}^\Pi(\on{i}^{-1})\circ \kappa_{234}^\Pi(\on{v}))\big)$.
There exists a unique $\on{w}\in \Pi_{f,f',f''}({\mathfrak 1}\underline
\boxtimes {\mathfrak 1},
S \underline\boxtimes {\mathfrak 1})$, of the form $\on{w} = 1$ + degree $>0$,
such that $\on{v}_1 = \bar{\bar\kappa}_1^\Pi(m_a) \circ (\on{w} \boxtimes
\on{v}_2)$. Then (\ref{partial:1}) and (\ref{partial:2}) imply that
$$
\bar{\bar\kappa}_1^\Pi(m_a^{(2,2)}) \circ \Big(
\bar\kappa_{14}^\Pi(\on{F}) \boxtimes
(\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ \on{w}) \Big)
=
\bar{\bar\kappa}_1^\Pi(m_a^{(2,2)}) \circ \Big(
\on{w}^{\boxtimes 2} \boxtimes \bar\kappa_{14}^\Pi(\on{F})\Big),
$$
i.e., $\on{w}' := \bar\kappa_{14}(\on{i}) \circ \on{w}$ satisfies
$(\on{w}')^{\boxtimes 2} = \bar{\bar\kappa}_4(\Delta_a) \circ \on{w}'$.
Identity (\ref{vv=vv}) now follows from:
\begin{proposition} \label{prop:gp:like}
If $x\in $\boldmath$\Pi$\unboldmath$_{f,f',f''}({\mathfrak 1}\underline
\boxtimes {\mathfrak 1},
S \underline\boxtimes {\mathfrak 1})$
is of the form $X = 1 +$ degree $>0$
and if $x^{\boxtimes 2} = \bar{\bar\kappa}_4(\Delta_a) \circ x$, then
$x=1$.
\end{proposition}
{\em Proof of Proposition.} $\on{LBA}_{f,f',f''}$ is equipped with a prop
automorphism $\iota$, where $\iota^2 = \on{id}$, uniquely defined my
$(\mu,\delta,f,f',f'') \mapsto (\mu,\delta + \on{ad}(f+f'+f''), -f'',
-f',-f)$. Then $\iota \circ \bar{\bar\kappa}_4 = \bar{\bar\kappa}_1$.
Set $y:= \iota^\Pi(x)$, then $\bar{\bar\kappa}_1^\Pi(y)
= y^{\boxtimes 2}$.
The prop $\on{LBA}_{f,f',f''}$ is equipped with a degree,
such that $\on{deg}(\mu)=0$ and $\on{deg}(\delta) = \on{deg}(f) = \on{deg}(f')
= \on{deg}(f'') = 1$. We then decompose $y = 1 + y_1 + ...$ for this degree.
Assume that we showed $y_1 = ... = y_{n-1}=0$. We then get:
$y_n \boxtimes 1 + 1 \boxtimes y_n =$ the degree $n$ part of
$\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ y_n$, i.e., $=\Delta_0 \circ y_n$.
According to the computation of the co-Hochschild cohomology of the complex
$S^{\otimes 0} \to S \to S^{\otimes 2} \to...$ of Schur functors, we
get $y_n\in \Pi_{f,f',f''}({\mathfrak 1}\underline\boxtimes {\mathfrak 1},
{\bf id}\underline\boxtimes {\mathfrak 1})
\subset \Pi_{f,f',f''}({\mathfrak 1}\underline\boxtimes {\mathfrak 1},
S \underline\boxtimes {\mathfrak 1})$.
The degree $n+1$ part of the equation $\bar{\bar\kappa}_1^\Pi(\Delta_a) \circ y
= y^{\boxtimes 2}$ then yields (degree $n+1$ part of
$\Delta_0\circ y_{n+1} + \bar{\bar\kappa}_1^\Pi(\Delta_a) \circ y_n) =
y_{n+1} \boxtimes 1 + 1 \boxtimes y_{n+1}$. Antisymmetrizing, we get
$\delta \circ y_n = 0$.
We then show:
\begin{lemma}
The map $\on{LBA}_{f,f',f''}({\bf 1},{\bf id}) \to
\on{LBA}_{f,f',f''}({\bf 1},\wedge^2)$, $y\mapsto \delta \circ y$
is injective.
\end{lemma}
{\em Proof of Lemma.} As in \cite{EH}, we will construct
a retraction of this map. As in \cite{EH}, one shows that
$\on{LBA}_{f,f',f''}(F,G)$ is the cokernel of
$\on{LBA}(C\otimes D\otimes F,G) \to \on{LBA}(C\otimes F,G)$,
$x\mapsto x\circ ([(\on{id}_C \boxtimes p)\circ \Delta_C]\boxtimes
\on{id}_F)$ where $C = S(\wedge^2\oplus \wedge^2 \oplus \wedge^2)$,
$D = \wedge^3\oplus \wedge^3 \oplus \wedge^3$
$\Delta_C : C \to C^{\otimes 2}$ is induced by the coalgebra structure of
$S$, and $p \in \on{LBA}(C,D) = \oplus_{k\geq 0}\on{LBA}(S^k \circ
(\wedge^2 \oplus \wedge^2 \oplus \wedge^2),\wedge^3\oplus \wedge^3
\oplus \wedge^3)$ has nonzero components for $k=1,2$ only; the
$k=1$, this component specializes to $\wedge^3(\a)^{\oplus 3} \to
\wedge^3(\a)^{\oplus 3}$,
$$
(f_\a,f'_\a,f''_\a) \mapsto ( (\delta_\a\otimes \on{id}_\a)(f_\a)+ \on{c.p.},
(\delta_\a\otimes \on{id}_\a)(f'_\a)+ \on{c.p.},
(\delta_\a\otimes \on{id}_\a)(f''_\a)+ \on{c.p.}),
$$
where c.p. means cyclic permutation, and for $k=2$ is specializes to
$S^2(\wedge^3(\a)^{\oplus 3}) \to
\wedge^3(\a)^{\oplus 3}$,
\begin{align*}
& (f_\a,f'_\a,f''_\a)^{\otimes 2} \mapsto (
[f_\a^{12},f_\a^{13}] + \on{c.p.},
[f_\a^{12},f_\a^{\prime 13} + f_\a^{\prime 23}]
+ [f_\a^{\prime 12},f_\a^{\prime 13}]
+ \on{c.p.},
\\ & [f_\a^{12}+f_\a^{\prime 12},f_\a^{\prime\prime 13} + f_\a^{\prime\prime 23}]
+ [f_\a^{\prime\prime 12},f_\a^{\prime\prime 13}]
+ \on{c.p.}).
\end{align*}
Since left and right compositions commute, we have a commutative diagram,
$$
\begin{matrix}
\on{LBA}(C\otimes D,{\bf id}) & \stackrel{\delta\circ -}{\to}
& \on{LBA}(C\otimes D,\wedge^2)\\
\downarrow & & \downarrow\\
\on{LBA}(C,{\bf id}) & \stackrel{\delta \circ -}{\to} & \on{LBA}(C,\wedge^2)
\end{matrix}
$$
whose vertical cokernel is the map $\on{LBA}_{f,f',f''}({\bf 1},{\bf id}) \to
\on{LBA}_{f,f',f''}({\bf 1},\wedge^2)$, $y\mapsto \delta \circ y$.
For any Schur functor $A$, we will construct a retraction
$r_A : \on{LBA}(A,\wedge^2) \to \on{LBA}(A,{\bf id})$ of
the map $\on{LBA}(A,{\bf id}) \to \on{LBA}(A,\wedge^2)$,
such that the diagram
\begin{equation} \label{comm:diag}
\begin{matrix}
\on{LBA}(C\otimes D,\wedge^2) & \stackrel{r_{C\otimes D}}{\to}
& \on{LBA}(C\otimes D,{\bf id})\\
\downarrow & & \downarrow\\
\on{LBA}(C,\wedge^2) & \stackrel{r_C}{\to} & \on{LBA}(C,{\bf id})
\end{matrix}
\end{equation}
commutes. The vertical cokernel of this map is then the desired retraction.
We have $\on{LBA}(A,{\bf id}) = \oplus_{Z\in\on{Irr(Sch)}} \on{LCA}(A,Z)
\otimes \on{LA}(Z,{\bf id})$. As in \cite{EH}, one shows that
$\on{LCA}(Z,{\bf id} \otimes Z)$ is 1-dimensional, and one
constructs an element $\delta_Z \in \on{LCA}(Z,{\bf id} \otimes Z)$,
such that the component $(Z',Z'') = ({\bf id},Z)$ of the
map $\on{LA}(Z,{\bf id}) \stackrel{\delta \circ -}{\to}
\on{LBA}(Z,\wedge^2) \subset \on{LBA}(Z,{\bf id}^{\otimes 2}) \simeq
\oplus_{Z',Z''\in\on{Irr(Sch)}} \on{LCA}(Z,Z'\otimes Z'') \otimes
\on{LA}(Z',{\bf id})\otimes \on{LA}(Z'',{\bf id})$ is $\lambda \mapsto
\delta_Z \otimes \on{id}_{\bf id} \otimes \lambda$.
It follows that the component $Z\mapsto (Z',Z'') = ({\bf id},Z)$ of the map
$\oplus_{Z\in \on{Irr(Sch)}} \on{LCA}(A,Z) \otimes \on{LA}(Z,{\bf id}) \simeq
\on{LBA}(A,{\bf id}) \stackrel{\delta \circ}{\to}
\on{LBA}(A,\wedge^2) \subset \on{LBA}(A,{\bf id}^{\otimes 2})
\simeq \oplus_{Z',Z''\in\on{Irr(Sch)}} \on{LCA}(A,Z'\otimes Z'')
\otimes \on{LA}(Z',{\bf id}) \otimes \on{LA}(Z'',{\bf id})$ is
$\kappa \otimes \lambda \mapsto (\lambda \circ \delta_Z)
\otimes \on{id}_{{\bf id}} \otimes \kappa$.
Dually to \cite{EH}, we construct a retraction of the map $\lambda\mapsto
\lambda \circ \delta_Z$; it gives rise to the section $r_A$.
One then proves the commutativity of (\ref{comm:diag}) as in
\cite{EH}.
This ends the proof of the lemma, and therefore also of Proposition
\ref{prop:gp:like} and Theorem \ref{thm:ids:v}.
\hfill \qed \medskip
\bigskip
We now draw the consequences of the results of the previous Subsection
for the quantization of twists of Lie bialgebras.
Let $\a = (\a,\mu_\a,\delta_\a)$ be a Lie bialgebra.
Its quantization is $Q(\a) =
(S(\a)[[\hbar]],m(\a),\Delta(\a))$, where
$m(\a) := m_a(\mu_\a,\hbar\delta_\a)$, $\Delta(\a) :=
\Delta_a(\mu_\a,\hbar\delta_\a)$. We set $\on{F}(\a,f_\a) :=
\on{F}(\mu_\a,\hbar\delta_\a,\hbar f_\a)$, $\on{i}(\a,f_\a)
:= \on{i}(\mu_\a,\hbar\delta_\a,\hbar f_\a)$.
Then $\on{F}(\a,f_\a)\in Q(\a)^{\otimes 2}$ and $\on{i}(\a,f_\a) :
Q(\a) \to Q(\a_{f_\a})$ are such that
$$
m(\a_{f_\a}) = \on{i}(\a,f_\a)
\circ m(\a) \circ (\on{i}(\a,f_\a)^{\otimes 2})^{-1},
\quad
\Delta(\a_{f_\a})
= \on{i}(\a,f_\a)^{\otimes 2} \circ \on{Ad}(\on{F}(\a,f_\a)) \circ
\Delta(\a) \circ \on{i}(\a,f_\a)^{-1},
$$
$$
(\on{F}(\a,f_\a)\otimes 1) *
(\Delta(\a)\otimes \on{id})(\on{F}(\a,f_\a)) =
(1\otimes \on{F}(\a,f_\a)) * (\on{id} \otimes \Delta(\a))(\on{F}(\a,f_\a))
$$
(where the product $m(\a)$ is denoted $*$).
Assume that $f'_\a$ is a twist of $\a_{f_\a}$. Then
Theorem \ref{thm:i:v} implies that $\on{v}(\a,f_\a,f'_\a) :=
\on{v}(\mu_\a,\hbar\delta_\a,\hbar f_\a,\hbar f'_\a)$ satisfies
$$
\on{F}(\a,f_\a + f'_\a) = \on{v}(\a,f_\a,f'_\a)^{\otimes 2} *
(\on{i}(\a,f_\a)^{\otimes 2})^{-1}(\on{F}(\a_{f_\a},f'_\a)) *
\on{F}(\a,f_\a) * \Delta(\a)(\on{v}(\a,f_\a,f'_\a))^{-1},
$$
$$
\on{i}(\a,f_\a + f'_\a) = \on{i}(\a_{f_\a},f'_\a) \circ \on{i}(\a,f_\a)
\circ \on{Ad}(\on{v}(\a,f_\a,f'_\a)^{-1})
$$
(in both equalities, $m(\a)$ in understood; it is denoted $*$ in the first
equality).
Finally, Theorem \ref{thm:v:v} implies that if $f''_\a$ is a twist of
$\a_{f_\a + f'_\a}$, then
$$
\on{v}(\a,f_\a+f'_\a,f''_\a) * \on{v}(\a,f_\a,f'_\a) =
\on{v}(\a,f_\a,f'_\a+f''_\a) * \on{i}(\a,f_\a)^{-1}(\on{v}(\a_{f_\a},f'_\a,f''_\a)).
$$
\section{Quantization of $\Gamma$-Lie bialgebras}\label{quantgamma}
\subsection{}
Assume that $(\a,\theta,f)$ is a $\Gamma$-Lie bialgebra.
We construct its quantization as follows. Set $A =
S(\a) \otimes \kk\Gamma[[\hbar]]$.
We set $[x |\gamma] := x\otimes \gamma$, $[x\otimes x'|\gamma,\gamma']
:= (x\otimes \gamma) \otimes (x'\otimes \gamma')\in A^{\otimes 2}$.
There are unique linear maps $m : A^{\otimes 2}
\to A$ and $\Delta : A\to A^{\otimes 2}$, such that
$$
m : [x|\gamma][x'|\gamma'] \mapsto
[x * \on{i}(\a,f_\gamma)^{-1}(\theta_\gamma(x')) *
\on{v}(\a,f_\gamma,\wedge^2(\theta_\gamma)(f_{\gamma'}))^{-1}
|\gamma\gamma']
$$
$$
\Delta : [x|\gamma] \mapsto [\Delta(\a)(x) *
\on{F}(\a,f_\gamma)^{-1}| \gamma,\gamma].
$$
The unit for $A$ is $[1|e]$, and the counit is the map
$[x|\gamma] \mapsto \delta_{\gamma,e} \varepsilon(x)$ (recall that $*$ denotes
the product $m(\a)$ on $S(\a)[[\hbar]]$ or its tensor square).
\begin{proposition}
This defines a bialgebra structure on $A$,
quantizing the co-Poisson bialgebra structure induced by
$(\a,\theta,f)$.
\end{proposition}
{\em Proof.} This follows from the above relations on twists.
\hfill \qed \medskip
\subsection{Propic version}
The quantization of $\Gamma$-Lie bialgebras has a propic version, which
we now describe.
Define $\on{LA}_\Gamma$ as the prop with generators
$\mu\in \on{LA}(\wedge^2,{\bf id})$ and
$\theta_\gamma\in \on{LA}({\bf id},{\bf id})^\times$, and relations:
Jacobi identity on $\mu$, $\Gamma \to \on{LA}_{\Gamma}
({\bf id},{\bf id})^\times$, $\gamma\mapsto
\theta_\gamma$ is a group morphism, and $\wedge^2(\theta_\gamma) \circ \mu
\circ \theta_\gamma^{-1} = \mu$.
Define $\on{LBA}_\Gamma$ as the prop with generators $\mu\in
\on{LBA}_\Gamma(\wedge^2,{\bf id})$, $\delta\in
\on{LBA}_\Gamma({\bf id},\wedge^2)$, $\theta_\gamma\in
\on{LBA}_\Gamma({\bf id},{\bf id})$ and $f_\gamma\in
\on{LBA}({\bf 1},\wedge^2)$, and relations:
$(\mu,\delta)$ satisfy the relations of the prop LBA,
$(\mu,(\theta_\gamma)_\gamma,(f_\gamma)_\gamma)$
satisfy the relations of $\on{LA}_\Gamma$; for each $\gamma\in\Gamma$,
$(\mu,\delta,f_\gamma)$ satisfy the defining relations of
$\on{LBA}_f$, as well as $\wedge^2(\theta_\gamma) \circ \delta \circ
\theta_\gamma^{-1} = \delta + \on{ad}(f_\gamma)$,
and for each pair $\gamma,\gamma'\in \Gamma$,
$f_{\gamma\gamma'} = f_\gamma + \wedge^2(\theta_\gamma) \circ f_{\gamma'}$.
Define the prop $\on{Bialg}_\Gamma$ of $\Gamma$-bialgebras as follows.
When $\Gamma$ is finite, in addition to
the generators $m,\Delta,\eps,\eta$ of $\on{Bialg}$, it has generators
$e_\gamma\in \on{Bialg}_\Gamma({\bf id},{\bf id})$, and the additional \
relations are
$\sum_{\gamma\in\Gamma}e_\gamma = \on{id}_{{\bf id}}$,
$e_\gamma \circ e_{\gamma'} = \delta_{\gamma\gamma'} e_{\gamma}$,
$m \circ (e_{\gamma} \boxtimes e_{\gamma'}) = e_{\gamma\gamma'} \circ m$,
$\Delta \circ e_{\gamma} = e_{\gamma}^{\boxtimes 2} \circ \Delta$,
$e_{\gamma} \circ \eta = \delta_{\gamma e} \eta$, $\eps \circ e_{\gamma}
= \delta_{e \gamma} \eps$.
In general, $\on{Bialg}_\Gamma$ is defined as follows.
If $S$ is a set, define $\on{Sch}_S$ as the category of
polynomial Schur functors $\on{Vect}^S \to V$ of the form
$(V_s)_{s\in S} \to \oplus_{(Z_s)}
M_{(Z_s)} \otimes (\otimes_{s\in S} Z_s(V_s))$,
where $Z_s$ are almost all ${\bf 1}$ (the unit Schur functor ${\bf 1}(V)
= {\bf k}$). Then $\on{Sch}_S$ is a symmetric tensor category.
We define a $S$-prop as a symmetric tensor category $P$
together with a natural transformation $\on{Sch}_S \to P$, which
is the identity on objects. A $S$-prop may be defined by generators
and relations. Then $\on{Bialg}_{(\Gamma)}$ is the $\Gamma$-prop
defined by generators
$m_{\gamma,\gamma'}\in \on{Bialg}_{(\Gamma)}({\bf id}_\gamma\boxtimes
{\bf id}_\gamma',{\bf id}_{\gamma\gamma'})$
$\Delta_\gamma\in \on{Bialg}_{(\Gamma)}({\bf id}_\gamma, {\bf
id}_\gamma^{\boxtimes 2})$
$\varepsilon\in \on{Bialg}_{(\Gamma)}({\bf id}_e, {\bf 1})$,
$\eta\in \on{Bialg}_{(\Gamma)}({\bf 1},{\bf id}_e)$, and the relations
derived from the finite case. The diagonal embedding $\on{Vect}
\to \on{Vect}^\Gamma$ gives rise to a functor $\Delta : \on{Sch}
\to \on{Sch}_S$, and we set $\on{Bialg}_\Gamma(F,G) :=
\on{Bialg}_{(\Gamma)}(\Delta(F),\Delta(G))$.
Then any EK quantization functor gives rise to a prop morphism
$\on{Bialg}_\Gamma \to S({\bf LBA}_\Gamma)^\Gamma$ with suitable classical
limit properties. A group morphism $\Gamma \to \Gamma'$ gives rise to a
commutative diagram
$$
\begin{matrix}
\on{Bialg}_\Gamma & \to & S({\bf LBA}_\Gamma)^{\Gamma}\\
\downarrow & & \downarrow \\
\on{Bialg}_{\Gamma'} & \to & S({\bf LBA}_{\Gamma'})^{\Gamma'}
\end{matrix}
$$
We have therefore a quantization functor $\{$group Lie bialgebras$\}
\to \{$quasicocommutative group bialgebras$\}$ (where both sides are full
subcategories of $\{$co-Poisson cocomutative bialgebras$\}$ and
$\{$quasicocommutative bialgebras$\}$).
\subsection{Quantization of quasitriangular $\Gamma$-Lie bialgebras}
We defined a quasitriangular $\Gamma$-Lie bialgebra as a triple
$(\a,r_\a,\theta_\a)$, where
$(\a,r_\a)$ be a quasitriangular Lie bialgebra (i.e., (
$r_\a\in\a^{\otimes 2}$ satisfies the classical Yang-Baxter identity,
and $t_\a := r_\a + r_\a^{21}$ is $\a$-invariant), and
$\theta_\a : \Gamma\to \on{Aut}(\a,t_\a)$ be an action of $\Gamma$ by
Lie algebra automorphisms of $\a$, preserving $t_\a$.
It gives rise to a $\Gamma$-Lie bialgebra, with $\delta(x) =
[r,x^1+x^2]$ and $f_\gamma := \theta_\gamma^{\otimes 2}(r_\a) - r_\a$.
In that case a quantization can be constructed directly:
we set $A = U(\a)\rtimes\Gamma[[\hbar]]$, the product is
undeformed, and the coproduct is $\Delta(x) = \on{J}(\hbar r_\a)\Delta_0(x)
\on{J}(\hbar r_\a)^{-1}$ ($\Delta_0$ is the standard coproduct).
Denote by $\on{qt}_\Gamma$ the prop of quasitriangular $\Gamma$-Lie bialgebras,
and by ${\bf qt}_\Gamma$ its completion. We have a natural prop morphism
$\on{LBA}_\Gamma \to \on{qt}_\Gamma$. We claim that the
prop morphisms $\on{Bialg}_\Gamma \to S({\bf qt}_\Gamma)^\Gamma$
(the above direct construction) and the composed morphism $\on{Bialg}_\Gamma
\to S({\bf LBA}_\Gamma)^\Gamma \to S({\bf qt}_\Gamma)^\Gamma$
are equivalent (i.e., can be obtained from each other using an inner
automorphism of $S({\bf qt}_\Gamma)^\Gamma$).
This is a consequence of the following statement on twists.
Let $(\a,r_\a)$ be a quasitriangular Lie bialgebra and let
$f_\a\in\wedge^2(\a)$ be a twist. According to \cite{EK}, there exists
an invertible
$\on{j}(\a,r_\a) : U(\a)[[\hbar]] \to S(\a)[[\hbar]]$, such that
$m(\a) = \on{j}(\a,r_\a) \circ m_0 \circ (\on{j}(\a,r_\a)^{\otimes 2})^{-1}$,
$\Delta(\a) = \on{j}(\a,r_\a)^{\otimes 2} \circ \on{Ad}(\on{J}(\hbar r_\a))
\circ \Delta_0 \circ \on{j}(\a,r_\a)^{-1}$.
Then one proves that $\on{j}(\a,r_\a + f_\a) =
\on{i}(\a,f_\a) \circ \on{j}(\a,r_\a) \circ \on{Ad}(\on{v}(\a,r_\a)^{-1})$,
and $\on{J}(\a,r_\a + f_\a) = (\on{v}(\a,r_\a) \otimes \on{v}(\a,r_\a)) *
(\on{j}(\a,r_\a)^{\otimes 2})^{-1}(\on{F}(\a,f_\a)) * \on{J}(\a,r_\a) *
\Delta_0(\on{v}(\a,r_\a))^{-1}$ (here $*$ is the undeformed product on
$U(\a)^{\otimes 2}[[\hbar]]$).
\subsection{Open questions}
Let $\a$ be a simple Lie algebra and let $\tilde{W}$ be its extended Weyl group.
One expects that the only possible quantization of $U(\a)\times \tilde{W}$
is the Majid-Soibelman algebra of $\a$. When $\a$ is a
Mac-Moody Lie algebra, one expects that if $Q$ is any EK quantization
functor, then $Q(\a,\tilde{W})$ is the Majid-Soibelman algebra of $\a$.
Both statements are analogues of well-known results \cite{Dr,EK6}.
| 30,251
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President.
The.
Millett is a partner at Akin Gump Strauss Hauser & Feld who has served in the appellate section of the Justice Department’s Civil Division and as an assistant to the solicitor general. Pillard began her career at the American Civil Liberties Union before spending five years at the NAACP Legal Defense and Educational Fund. She also served as an assistant to the solicitor general. Wilkins was a leader in the effort to establish the Smithsonian National Museum of African American History and then from 2002 to 2011 was a partner at Venable LLP before his confirmation to the district court.
The announcement drew praise from liberal legal advocacy groups who have complained that Obama has not made judicial nominations enough of a priority.
“The announcement of President Obama’s decision today to nominate three highly qualified individuals to the D.C. Circuit is a watershed moment for this president and the future of this critical court,” said Doug Kendall, president of the Constitutional Accountability Center.
But it is all but certain to draw strong opposition from Senate Republicans who claim the D.C. Circuit is underworked.
Iowa Sen. Charles E. Grassley, the top Republican on the Judiciary Committee, has introduced legislation (S 699) that would reduce the size of the D.C. Circuit by three seats.
The bill would eliminate one of the court’s seats and distribute two others to the Courts of Appeal for the 2nd and 11th circuits, which he argues are busier. The legislation is co-sponsored by all of the other GOP senators on the Judiciary Committee, along with moderate Republican Susan Collins of Maine, spelling potential trouble for Obama’s new nominees.
Grassley on Monday evening had no initial reaction to the nominees, except to say they aren’t needed.
“The only thing that is controversial [about filling the spots] is that you don’t need any more judges there because they’re only working half as hard as the average of other circuits,” he said.
According to statistics from the Administrative Office of the U.S. Courts, the D.C. Circuit had the lowest caseload per authorized judge last year of the 12 circuit courts. But a spokesman for the Administrative Office emphasized that court caseloads tell only a portion of the story and that workload depends often on the kinds of cases that a court hears.
While Grassley’s bill would redistribute two of the court’s seats to the 2nd and 11th circuits, the Judicial Conference, the policy-setting body of the federal courts, has suggested that another court, the 9th Circuit, is the busiest in the nation.
The conference has recommended that the San Francisco-based court receive four new permanent judgeships, a recommendation based on factors including “filings per panel, the mix of cases, the proportion of oral hearings versus submission of briefs, the contributions of senior judges and the geography of the circuit,” according to a spokesman for the federal courts.
Humberto Sanchez and Steven T. Dennis contributed to this report.
| 102,769
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Green Mountain Energy Company is the nation's longest serving renewable energy retailer. Our mission is to change the way power is made, and we look for employees who want to help us work towards this important goal. Our company is headquartered in Austin, Texas, and we have regional operations in the following locations:
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| 312,187
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Culture Shock
Der Kaffee: Gemeinfaßliche Darstellung Der Gewinnung, Verwertung Und Beurteilung Des Kaffees Und Seiner Ersatzstoffe
Der Kaffee: Gemeinfaßliche Darstellung Der Gewinnung, Verwertung Und Beurteilung Des Kaffees Und Seiner Ersatzstoffe
by
Miranda
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The Process for Getting the Best Cremation Services
Death is one of the events of life that anyone would surely undergo since we do all know that all of us would die and we would not last long in this world that we do have at all. Many people fear death nowadays due to the fact that they would no longer exist in this world but of course the aftermath of death would truly cost you a lot since you may need to do some final goodbye to your loved. In the modern world that we do have nowadays, many families are indeed suffering from financial crisis which is why to prepare for the burial of their loved ones may be a huge burden for them since they need to prepare for it. The process of getting these cremation services nowadays have become quite complex which is why you need to prepare for it. The tips that would help you have an easy time hiring the best among these cremation services are given here as reference to everyone to ensure that you would not waste time and effort.
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Ford's vehicles sales in China fell for the third straight year, down 26% in 2019. As Fred Katayama reports, that's less than half of what it sold in 2016.
Credit: Rumble Duration: 00:50Published 4 days ago
The bathrooms in Ford's Garage in Fort Myers, Florida are every gearhead's dream.
Credit: Rumble Duration: 00:15Published 2 weeks ago
| 284,021
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Video Game Information
Call of Duty
Type: Video Game
Released: 2003
Credits
Soundrack Albums
Call Of DutyActivision Publishing, Inc.
Format: CD (39 min)
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| 302,666
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TITLE: Random walk and Occupation measure
QUESTION [1 upvotes]: This is homework so no answers please
I want to find for some $A\subset \mathbb{R}$ the limit $$\lim_{n\to \infty}\mu_{n}(A)=\lim_{n\to \infty}\frac{1}{n}E\left[\sum_{i=1}^{n}1_{A}(\frac{S_{i}}{\sqrt{n}})\right]=$$
$\sum_{k\in \mathbb{Z}}1_{A}(k)\lim_{n\to \infty}\frac{\sum_{i=1}^{n}P(\frac{S_{i}}{\sqrt{n}}=k)}{n}$
where $S_{n}=\sum_{i=1}^{n}X_{i}$ is a symmetric random walk starting at 0 and $P(X_{i}=1)=P(X_{i}=-1)=\frac{1}{2}$.
Any suggestions
Any mistakes:
One might guess: $\int_{0}^{\infty}\int_{A} P(B_{t}=x)dxdt$, where $B_{t}$ is Brownian motion.
But Donsker's invariance principle does not apply because we are not looking at an interpolation (i.e. we are not looking at $S_{\lfloor t\rfloor}+(t-\lfloor t\rfloor)(S_{\lfloor t\rfloor+1}-S_{\lfloor t\rfloor}))$
I am currently trying the LIL for random walk i.e. $$ (1-\varepsilon)\sqrt{2n\log(\log(n))}\leq S_{n}\leq (1+\varepsilon)\sqrt{2n\log(\log(n))}$$
REPLY [0 votes]: My guess is different than yours: I claim the result of the limit is
$
\int_0^1 \mathbf{P} ( B_t \in A ) dt,
$
instead of integrating from $(0,\infty)$. My advice to get this result is to write the sum inside the limit as
$$
n^{-1}\mathbf{E}\left[ \sum_{i=1}^n \mathbf{1}_A \left(\frac{ S_i }{ \sqrt{ n } } \right) \right] = \mathbf{E}\left[ \sum_{i=1}^n \mathbf{1}_A \left(\frac{ S_i }{ \sqrt{ n } } \right) \left(\frac{S_{i+1} - S_{i}}{\sqrt{n}} \right) \right]^2,
$$
and then use invariance principle combined with Ito's isometry.
| 123,651
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!
1. BTSTU – Jai Paul
Random Useless Fact: Not a fan of this initially. Dumb enough. Just goes to show. Don’t judge a cover by the person who first recommends it to you! Great fact!
2. Through The Night – Grum
R.U.F: A sample from Toto’s single “We Can Make Tonight” is used. (How good is this song?! 80’s + Electro… Dancing!)
3. Memories – Hard Mix
R.U.F: This song is oddly highly addictive. Labels himself as a ‘new-age Moby’. Explains nothing.
4. Love Is All – The Tallest Man On Earth
R.U.F: The 2010 edition of Guinness Book of World Records lists Sultan Kosen as the tallest living man as well as having the largest hands and largest feet. Good work Sultan!
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\begin{document}
\title[]{Nonconventional limit theorems in averaging}
\vskip 0.1cm
\author{ Yuri Kifer\\
\vskip 0.1cm
Institute of Mathematics\\
Hebrew University\\
Jerusalem, Israel}
\address{
Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel}
\email{ kifer@math.huji.ac.il}
\thanks{ }
\subjclass[2000]{Primary: 34C29 Secondary: 60F17, 37D20}
\keywords{averaging, limit theorems, martingales, hyperbolic dynamical
systems.}
\dedicatory{ }
\date{\today}
\begin{abstract}\noindent
We consider "nonconventional" averaging setup in the form
$\frac {dX^\epsilon(t)}{dt}=\epsilon B\big(X^\epsilon(t),\xi(q_1(t)),
\xi(q_2(t)),...,\xi(q_\ell(t))\big)$ where $\xi(t),t\geq 0$ is either a
stochastic process or a dynamical system (i.e. then $\xi(t)=F^tx$) with
sufficiently fast mixing while
$q_j(t)=\al_jt,\,\al_1<\al_2<...<\al_k$ and $q_j,\, j=k+1,...,\ell$ grow
faster than linearly.
We show that the properly normalized error term in the "nonconventional"
averaging principle is asymptotically Gaussian.
\end{abstract}
\maketitle
\markboth{Yu.Kifer}{Nonconventional averaging}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\pagenumbering{arabic}
\section{Introduction}\label{sec1}\setcounter{equation}{0}
Nonconventional ergodic theorems (see \cite{Fu}) known also after \cite{Be}
as polynomial ergodic theorems studied the limits of expressions having the
form $1/N\sum_{n=1}^NF^{q_1(n)}f_1\cdots F^{q_\ell (n)}f_\ell$ where $F$ is a
weakly mixing measure preserving transformation, $f_i$'s are bounded measurable
functions and $q_i$'s are polynomials taking on integer values on the integers.
Originally, these results were motivated by applications to multiple recurrence
for dynamical systems taking functions $f_i$ being indicators of some measurable
sets and only convergence in the $L^2$-sense was dealt with but later \cite{As}
provided also almost sure convergence under additional conditions. Recently
such results were extended in \cite{BLM} to the continuous time dynamical
systems, i.e. to expressions of the form
\[
\frac 1{\cT}\int_0^{\cT}F^{q_1(t)}f_1\cdots F^{q_\ell (t)}f_\ell dt
\]
where $F^s$ is now an ergodic measure preserving flow.
In this paper we consider the averaging setup
\begin{equation}\label{1.1}
X^\ve(n+1)=X^\ve(n)+\ve B(X^\ve(n),\xi(q_1(n)),...,\xi(q_\ell(n)))
\end{equation}
in the discrete time case and
\begin{equation}\label{1.2}
\frac {dX^\ve(t)}{dt}=\ve B(X^\ve(t),\xi(q_1(t)),...,\xi(q_\ell(t)))
\end{equation}
in the continuous time case with $\xi$ being either a stochastic process or
having the form $\xi(s)=F^sf$ where $F^s$ is a dynamical system and $f$ is a
function. Positive functions $q_1,...,q_\ell$ will satisfy certain conditions
which will be specified in the next section, in particular, first $k$ of them
are linear while others grow faster than preceeding ones. An example where
(\ref{1.2}) emerges is obtained when we consider a time dependent small
perturbation of the oscillator equation
\begin{equation}\label{1.3}
\ddot x+\la^2x=\ve g(x,\dot x,t)
\end{equation}
where the force term $g$ depends on time in a random way $g(x,y,t)=g(x,y,
\xi(q_1(t)),...,\xi(q_\ell(t)))$. Then passing to the polar coordinates
$(r,\phi)$ with $x=r\sin(\la(t-\phi))$ and $\dot x=\la r\cos(\la(t-\phi))$
the equation (\ref{1.3}) will be transformed into (\ref{1.2}). It seems
reasonable that a random force may depend on versions of a same process or a
dynamical system moving with different speeds which is what we have here.
As it is well known (see, for instance, \cite{SV}), if $B(x,y_1,...,y_\ell)$
is bounded and Lipschitz continuous in $x$ and the limit
\begin{equation}\label{1.4}
\bar B(x)=\lim_{\cT\to\infty}\frac 1\cT\int_0^\cT B(x,\xi(q_1(t)),...,
\xi(q_\ell(t)))dt
\end{equation}
exists then for any $S\geq 0$,
\begin{equation}\label{1.5}
\lim_{\ve\to 0}\sup_{0\leq t\leq S/\ve}|X^\ve(t)-\bar X^\ve(t)|=
\lim_{\ve\to 0}\sup_{0\leq t\leq S}|Z^\ve(t)-\bar Z(t)|=0
\end{equation}
where
\begin{equation}\label{1.6}
\frac {d\bar X^\ve(t)}{dt}=\ve\bar B(\bar X^\ve(t))\,\,\,\mbox{and}\,\,\,
Z^\ve(t)=X^\ve(t/\ve),\,\bar Z(t)=\bar X^\ve(t/\ve).
\end{equation}
In the discrete time case we have to take
\begin{equation}\label{1.7}
\bar B(x)=\lim_{N\to\infty}\frac 1N\sum_{n=0}^N B(x,\xi(q_1(n)),...,
\xi(q_\ell(n)))
\end{equation}
and (\ref{1.5}) remains true with $\bar X^\ve$ given by (\ref{1.6}) and
(\ref{1.7}). Almost everywhere limits in (\ref{1.4}) and (\ref{1.7}) can
be obtained by nonconventional
pointwise ergodic theorems from \cite{BLM} and \cite{As}, respectively, in
rather general circumstances in the dynamical systems case and under another
set of conditions existence of such limits follows from \cite{Ki4}.
After nonconventional ergodic theorems (or in the probabilistic language laws
of large numbers) are established the next natural step is to obtain central
limit theorem type results which was accomplished in \cite{KV}. The averaging
principle (\ref{1.5}) can be considered as an extension of the ergodic theorem
and the main goal of this paper is to extend also central limit theorem type
results to the above nonconventional averaging setup in the spirit of what was
done in the standard (conventional) averaging case in \cite{Kh1} and \cite{Ki1}.
Central limit theorem type results turn in the averaging setup into assertions
about Gaussian approximations of the slow motion $X^\ve$ given by (\ref{1.1})
or by (\ref{1.2}) where $\xi$ is a fast mixing stochastic process or a dynamical
system while unlike the standard (conventional) case we have the process $\xi$
taken simultaneously at different times $q_i(t)$ in the right hand side of
(\ref{1.1}) and (\ref{1.2}).
We prove, first, our limit theorems for stochastic
processes under rather general conditions resembling the definition of
mixingales (see \cite{ML1} and \cite{ML2}) and then check these
conditions for more familiar classes of stochastic processes and dynamical
systems. In \cite{KV} we imposed mixing assumptions in a standard way
relying on two parameter families of $\sig$-algebras (see \cite{Br}) while
our assumptions here use only filtrations (i.e. nondecreasing families)
of $\sig$-algebras which are easier to construct for various classes of
dynamical systems. As one of applications we check some form of
our conditions for Anosov
flows which serve as fast motions in our nonconventional averaging setup
where we rely on the notion of Markov families from \cite{Do1} and \cite{Do2}.
At the end of the paper we discuss a fully coupled averaging setup in our
nonconventional situation where already an averaging principle itself becomes a
problem.
\begin{acknowledgment}
A part of this paper was done during my visit to the Fields Institute in
Toronto in June 2011 whose support and excellent working conditions I
greatfully acknowledge.
\end{acknowledgment}
\section{Preliminaries and main results}\label{sec2}\setcounter{equation}{0}
Our setup consists of a $\wp$-dimensional stochastic process
$\{\xi(t),\, t\geq 0\,\,\mbox{or}\,\, t=0,1,...\}$ on a probability space
$(\Om,\cF,Pr)$ together with a filtration of $\sig$-algebras
$\cF_l\subset\cF,\, 0\leq l\leq\infty$
so that $\cF_l\subset\cF_{l'}$ if $l\leq l'$.
For convenience we extend the definitions of $\cF_l$ given only for $l\geq 0$
to negative $l$ by defining $\cF_l=\cF_0$ for $l<0$. In order to relax
required stationarity assumptions to some kind of weak "limiting stationarity"
our setup includes another probability measure $P$ on the space $(\Om,\cF)$.
Namely, we assume that the distribution of $\xi(t)$ with respect to $P$ does
not depend on $t$ and the joint distribution of
$\{\xi(t), \xi(t')\}$ for $t\geq t'$ depends only on $t-t'$ which can be
written in the form
\begin{equation}\label{2.1}
\xi(t)P=\mu\,\,\mbox{and}\,\,(\xi(t),\xi(t'))P=\mu_{t-t'}\,\,\mbox{for all}
\,\,t\geq t'
\end{equation}
where $\mu$ is a probability measure on $\bbR^\wp$ and $\mu_s,\, s\geq 0$ is
a probability measure on $\bbR^\wp\times\bbR^\wp$.
Our setup relies on two probability measures $Pr$ and $P$ in order to include,
for instance, Markov processes $\xi(t)$ satisfying the Doeblin condition
(see \cite{IL} or \cite{Do}) starting at a fixed point or with another
noninvariant distribution. Then $Pr$ will be a corresponding probability
in the path space while $P$ will be the stationary probability constructed
by the initial distribution being the invariant measure of $\xi(t)$.
Usual mixing conditions for stochastic processes are formulated in terms of
a double parameter family of $\sig$-algebras via a dependence coefficient
between widely separated past and future $\sig$-algebras (cf. \cite{Br}
and \cite{KV}) but this approach
often is not convenient for applications to dynamical systems where natural
future $\sig$-algebras do not seem to exist unless an appropriate symbolic
representation is available. By this reason we formulate below a different
set of mixing and approximation conditions for the process $\xi$ which seem
to be new and will enable us to treat some of dynamical systems models within a
class of stochastic processes satisfying our assumptions.
In order to avoid some of technicalities we restrict ourselves here mostly to
bounded functions though our results can be obtained for more general
classes of functions with polynomial growth supplemented by appropriate
moment boundedness conditions similarly to \cite{KV}. For any function
$g=g(\xi,\tilde\xi)$ on $\bbR^\wp\times\bbR^\wp$ introduce its H\" older norm
\begin{equation}\label{2.2}
|g|_\ka=\sup\{ |g(\xi,\tilde\xi)|+\frac {|g(\xi,\tilde\xi)-g(\xi',\tilde\xi')|}
{|\xi-\xi'|^\ka+|\tilde\xi-\tilde\xi'|^\ka}:\, \xi\ne \xi',\, \xi\ne \xi'\}.
\end{equation}
Here and in what follows $|\psi-\tilde\psi|^\ka$ for two vectors $\psi=(\psi_1,
...,\psi_\vr)$ and $\tilde\psi=(\tilde\psi_1,...,\tilde\psi_\vr)$ denotes the
sum $\sum_{i=1}^\vr |\psi_i-\tilde\psi_i|^\ka$.
Next, for $p,q\geq 1$ and $s\geq 0$ we define a sort of a mixing coefficient
\begin{eqnarray}\label{2.3}
&\eta_{p,\ka,s}(n)=\sup_{t\geq 0}\big\{\big\| E\big(g(\xi(n+t),\xi(n+t+s))
|\cF_{[t]}\big)\\
&-E_Pg(\xi(n+t),\xi(n+t+s))\big\|_p:\, g=g(\xi,\tilde\xi),\, |g|_\ka\leq 1
\big\},\,\,\,\eta_{p,\ka}(n)=\eta_{p,\ka,0}(n)
\nonumber\end{eqnarray}
where $\|\cdot\|_p$ is the $L^p$-norm on the space $(\Om,\cF,Pr)$, $[\cdot]$
denotes the integral part and throughout this paper we write $E$ for the
expectation with respect to $Pr$ and $E_P$ for the expectation with respect
to $P$. We will need also an (one-sided) approximation coefficient
\begin{equation}\label{2.4}
\zeta_q(n)=\sup_{t\geq 0}\| E(\xi(t)|\cF_{[t]+n})-\xi(t)\|_q.
\end{equation}
\begin{assumption}\label{ass2.1}
Given $\ka\in(0,1]$ there exist $p,q\geq 1$ and $m,\del>0$ satisfying
\begin{equation}\label{2.5}
\gam_m=E|\xi(0)|^m<\infty,\, \frac 12\geq\frac 1p+\frac 2m+\frac {\del}q,\,
\del<\ka-\frac {\vr}p,\, \ka q>1
\end{equation}
with $\vr=(\ell-1)\wp$ and such that
\begin{equation}\label{2.6}
\sum_{n=0}^\infty n(\eta^{1-\frac {\vr}{p\te}}_{p,\ka}(n)+\zeta_q^\del(n))
<\infty\,\,\mbox{and}\,\,\lim_{n\to\infty}\eta_{p,\ka,s}(n)=0\,\,\mbox{for all}
\,\, s\geq 0,
\end{equation}
where $\frac {\vr}p<\te <\ka$.
\end{assumption}
Next, let $B=B(x,\xi)=(B^{(1)}(x,\xi),...,B^{(d)}(x,\xi)),$ $\xi=(\xi_1,...,
\xi_\ell)\in\bbR^{\ell\wp}$ be a
$d$-vector function on $\bbR^d\times\bbR^{\ell\wp}$ such that for some
constant $K>0$ and all $x,\tilde x\in\bbR^d$, $\xi,\tilde\xi\in\bbR^{\ell\wp}$,
$i,j,l=1,...,d$,
\begin{eqnarray}\label{2.7}
&|B^{(i)}(x,\xi)|\leq K,\,|B^{(i)}(x,\xi)-B^{(i)}(\tilde x,\tilde\xi)|\leq
K(|x-\tilde x|+\sum_{j=1}^\ell|\xi_j-\tilde\xi_j|^\ka)\\
&\mbox{and}\,\,\quad\big\vert\frac {\partial B^{(i)}(x,\xi)}{\partial x_j}
\big\vert\leq K,\,\big\vert\frac {\partial^2 B^{(i)}(x,\xi)}{\partial x_j
\partial x_l}\big\vert\leq K.
\nonumber\end{eqnarray}
We will be interested in the central limit theorem type results as $\ve\to 0$
for the solution $X^\ve(t)=X^\ve_x(t)$ of the equation
\begin{equation}\label{2.8}
\frac {dX^\ve(t)}{dt}=\ve B\big( X^\ve(t),\xi(q_1(t)),\xi(q_2(t)),...,
\xi(q_\ell(t))\big),\, X^\ve_x(0)=x,\, t\in[0,\cT/\ve]
\end{equation}
where $q_1(t)<q_2(t)<\dots<q_\ell(t),\, t>0$ are increasing functions such
that $q_j(t)=\al_jt$ for $j\leq k<\ell$ with $\al_1<\al_2<\dots <\al_k$
whereas the remaining $q_j's$ grow faster in $t$. Namely, we assume
similarly to \cite{KV} that for any $\gam>0$ and $k+1\leq i\leq\ell$,
\begin{equation}\label{2.9}
\lim_{t\to\infty}(q_i(t+\gam)-q_i(t))=\infty
\end{equation}
and
\begin{equation}\label{2.10}
\lim_{t\to\infty}(q_i(\gam t)-q_{i-1}(t))=\infty.
\end{equation}
Set
\begin{equation}\label{2.11}
\bar B(x)=\int B(x,\xi_1,...,\xi_\ell)d\mu(\xi_1)\cdots d\mu(\xi_\ell).
\end{equation}
We consider also the solution $\bar X^\ve(t)=\bar X^\ve_x(t)$ of the averaged
equation
\begin{equation}\label{2.12}
\frac {d\bar X^\ve(t)}{dt}=\ve\bar B(\bar X^\ve(t)),\,\,\bar X^\ve_x(0)=x.
\end{equation}
It will be convenient to denote $Z^\ve(t)=X^\ve(t/\ve)$, $\bar Z(t)=
\bar X^\ve(t/\ve)$ and to introduce $Y^\ve(t)=Y^\ve_y(t)$ by
\begin{equation}\label{2.13}
Y_y^\ve(t)=y+\int_0^t B\big(\bar Z(s),\xi(q_1(s/\ve)),\xi(q_2(s/\ve)),...,
\xi(q_\ell(s/\ve))\big)ds.
\end{equation}
\begin{theorem}\label{thm2.2}
Suppose that (\ref{2.7}), (\ref{2.9}), (\ref{2.10}) and Assumption \ref{ass2.1}
hold true. Then the family of processes $G^\ve(t)=\ve^{-1/2}(Y^\ve_z(t) -
\bar Z_z(t)),\, t\in[0,\cT]$ converges weakly as $\ve\to 0$ to a Gaussian
process $G^0(t),\, t\in[0,\cT]$ having not necessarily independent increments
(see an example in \cite{KV}) with covariances of its components $G^0(t)=
(G^{0,1}(t),...,G^{0,d}(t))$ having the form $EG^{0,l}(s)G^{0,m}(t))=
\int_0^{\min(s,t)}A^{l,m}(u)du$ with the matrix function $\{A^{l,m}(u),\,
1\leq l,m\leq d\}$ computed in Section \ref{sec4}. Furthermore, the
family of processes $Q^\ve(t)=\ve^{-1/2}(Z^\ve(t)-\bar Z(t)),\, t\in[0,\cT]$
converges weakly as $\ve\to 0$ to a Gaussian process $Q^0(t),\, t\in[0,\cT]$
which solves the equation
\begin{equation}\label{2.14}
Q^0(t)=G^0(t)+\int_0^t\nabla\bar B(\bar Z(s))Q^0(s)ds.
\end{equation}
In the discrete time setup (\ref{1.1}) the similar results hold true assuming
that $q_i$'s take on integer values on integers, $\gam$ in (\ref{2.9}) is
replaced by 1, $\al_i$ is replaced by $i$ for $i=1,...,k$ and defining
$Z^\ve(t)=X^\ve([t/\ve])$ together with $Y^\ve=Y^\ve_y$ given by
\begin{equation}\label{2.15}
Y_y^\ve(t)=y+\int_0^t B\big(\bar Z(s),\xi(q_1([s/\ve])),\xi(q_2([s/\ve])),...,
\xi(q_\ell([s/\ve]))\big)ds
\end{equation}
while leaving all other definitions and assumptions the same as above.
\end{theorem}
Observe that we work with $\bar B$ defined by (\ref{2.11}) but in our
circumstances the central limit theorem type results from \cite{KV} imply
also (\ref{1.4}) and (\ref{1.7}), at least, in the $L^2$-sense while a
nonconventional law of large numbers from \cite{Ki4} and (under stationarity
assumptions) pointwise nonconventional ergodic theorems from \cite{As}
and \cite{BLM} yield (\ref{1.4}) and (\ref{1.7}) also for the almost
sure convergence. Note also that we need the full strength of (\ref{2.6})
only for one argument in Section \ref{sec4} borrowed from \cite{Kh1} but
for a standard limit theorem not in the averaging setup, i.e. when $B(x,\xi_1,
...,\xi_\ell)=B(\xi_1,...,\xi_\ell)$ does not depend on $x$, it suffices to
require only summability of the expression in brackets in (\ref{2.6}).
An important point in the proof of the first part of Theorem \ref{thm2.2}
is to introduce the representation
\begin{equation}\label{2.16}
B(x,\xi)=\bar B(x)+B_1(x,\xi_1)+\cdots +B_\ell(x,\xi_1,...,\xi_\ell)
\end{equation}
where $\xi=(\xi_1,...,\xi_\ell)$ and for $i<\ell$,
\begin{eqnarray}\label{2.17}
&B_i(x,\xi_1,...,\xi_\ell)=\\
&\int B(x,\xi_1,...,\xi_\ell)d\mu(\xi_{i+1})\cdots
d\mu(\xi_\ell)-\int B(x,\xi_1,...,\xi_\ell)d\mu(\xi_{i})\cdots d\mu(\xi_\ell)
\nonumber\end{eqnarray}
while
\begin{equation}\label{2.18}
B_\ell(x,\xi_1,...,\xi_\ell)=B(x,\xi_1,...,\xi_\ell)-\int B(x,\xi_1,...,
\xi_\ell)d\mu(\xi_\ell).
\end{equation}
Next, we introduce
\begin{eqnarray}\label{2.19}
&\quad\,\,\,\, Y_i^\ve(t)=\int_0^{t/\al_i}B_i\big( \bar Z(s),\xi(q_1(s/\ve)),
\xi(q_2(s/\ve)),...,\xi(q_\ell(s/\ve))\big)ds,\,\, i=1,...,k,\\
&Y_i^\ve(t)=\int_0^tB_i\big( \bar Z(s),\xi(q_1(s/\ve)),\xi(q_2(s/\ve)),
...,\xi(q_\ell(s/\ve))\big)ds,\, i=k+1,...,\ell
\nonumber\end{eqnarray}
and $Y_0^\ve(t)=Y^\ve_{0,y}(t)=y+\int_0^tB_i(\bar Z(s))ds$. Thus $Y^\ve_y$
from (\ref{2.13}) has the representation
\begin{equation}\label{2.20}
Y_y^\ve(t)=Y^\ve_0(t)+\sum_{i=1}^kY_i^\ve(\al_it)+\sum_{i=k+1}^\ell Y^\ve_i(t).
\end{equation}
We consider also $X_0^\ve(t)=\bar X^\ve(t)$, $X^\ve_i(t)=X^\ve_{i,x}(t)=x+
\ve\int_0^tB_i\big(X^\ve_i(s),\xi(q_1(s)),...,\xi(q_\ell(s))\big)ds$
and $Z^\ve_i(t)=X^\ve_i(t/\ve)$ for all $i\geq 0$.
For $i\geq 1$ set also
\begin{equation}\label{2.21}
G^\ve_i(t)=\ve^{-1/2}Y^\ve_i(t)\,\,\mbox{and}\,\, Q^\ve_i(t)=\ve^{-1/2}
Z^\ve_i(t).
\end{equation}
Relying on martingale approximations (which also can be done employing
mixingales from \cite{ML1} and \cite{ML2}) we will show that any linear
combination $\sum_{i=1}^k\la_iG_i^\ve$ converges weakly as $\ve\to 0$ to
a Gaussian process $\sum_{i=1}^k\la_iG_i^0$. It turns out that in the
continuous time case each $G^\ve_i,\, i=k+1,...,\ell$ converges weakly
as $\ve\to 0$ to zero, and so the processes $Y^\ve_i,\, i>k$ do not play
any role in the limit. It follows that $G^\ve$ converges weakly to a
Gaussian process $G^0$ such that $G(t)=\sum_{i=1}^k\la_iG_i^0(\al_it)$.
On the other hand, in the discrete time case each $G^\ve_i,\, i>k$ cannot
be disregarded, in general, and it converges weakly as $\ve\to 0$ to
a Gaussian process $G^0_i$ which is independent of any other $G^0_j$.
The above difference between discrete and continuous time cases is due
to the different natural forms of the assumption (\ref{2.9}) in these two
cases. These arguments yield the first part of Theorem \ref{thm2.2}
while its second part concerning convergence of $Q^\ve$ as $\ve\to 0$ is
proved via some Taylor expansion and approximation arguments.
In order to clarify the role of the coefficients $\eta_{p,\ka}$ and
$\zeta_q$ we compare them with the more familiar mixing and approximation
coefficients defined via a two parameter family of $\sig$-algebras
$\cG_{s,t}\in\cF,\, -\infty\leq s\leq t\leq\infty$ by
\begin{equation}\label{2.22}
\vp_p(n)=\sup_{s\geq 0,g}\big\{\| E(g|\cG_{-\infty,s})-E_Pg\|_p:\, g\,\,
\mbox{is}\,\,\cG_{s+n,\infty}-\mbox{measurable and}\,\, |g|\leq 1\big\}
\end{equation}
and
\begin{equation}\label{2.23}
\be_q(n)=\sup_{t\geq 0}\| E(\xi(t)|\cG_{t-n,t+n})-\xi(t)\|_q,
\end{equation}
respectively, where $\cG_{st}\subset\cG_{s't'}$ if $s'\leq s$ and $t'\geq t$.
Then setting $\cF_l=\cG_{-\infty,l}$ we obtain by the contraction property
of conditional expectations that
\begin{eqnarray}\label{2.24}
&\be_q(n)\geq\sup_{t\geq 0}\| E(\xi(t)|\cG_{t-n,t+n})- \xi(t)+\xi(t)-
E(\xi(t)|\cG_{-\infty,[t]+n+1})\|_q\\
&\geq\zeta_q(n+1)-\be_q(n)\,\,\mbox{i.e.}\,\,\be_q(n)\geq\frac 12\zeta_q(n+1).
\nonumber\end{eqnarray}
Furthermore,
\begin{eqnarray*}
&\big\| g(\xi(n+t),\xi(n+t+s))-g\big(E(\xi(n+t)|\cG_{n+t-[n/2],n+t+[n/2]}),\\
&E(\xi(n+t+s)|\cG_{n+t+s-[n/2],n+t+s+[n/2]})\big)\big\|_p
\leq 2|g|_\ka\be^\ka_{p\ka}([n/2]),
\end{eqnarray*}
and so
\begin{equation}\label{2.25}
\eta_{p,\ka}(n)\leq (\vp_p([n/2])+2\be_{p\ka}^\ka([n/2]))|g|_\ka.
\end{equation}
Thus, appropriate conditions on decay of coefficients $\vp_p$ and $\be_q$
as in \cite{KV} yield corresponding conditions on $\eta_{p,\ka}$ and
$\zeta_q$. The other direction does not hold true but still it turns out
that most of the technique from \cite{KV} can be employed in our
circumstances, as well.
The conditions of Theorem \ref{thm2.2} hold true for many important stochastic
processes. In the continuous time case they are satisfied when, for instance,
$\xi(t)=f(\Xi(t))$ where $\Xi(t)$ is either an irreducible continuous time
finite state Markov chain or a nondegenerate diffusion process on a compact
manifold while $f$ is a H\" older continuous vector function. In the discrete
time case we can take, for instance, $\xi(n)=f(\Xi(n))$ with $\Xi(n)$ being
a Markov chain satisfying the Doeblin condition (see, for instance, \cite{IL},
p.p. 367--368). In all these examples $\eta_{p,\ka}(n)$ and $\zeta_q(n)$
decay in $n$ exponentially fast while (\ref{2.6}) requires much less. In
fact, in both cases $\xi(t)$ may depend on whole paths of a Markov process
$\Xi$ assuming only certain weak dependence on their tails.
Important classes of processes satisfying our conditions come from
dynamical systems. In Section \ref{sec6} we take $\xi(t)=\xi(t,z)
=g(F^tz)$ where $F^t$ is a $C^2$ Anosov flow (see \cite{KH}) on a
compact manifold $M$ whose stable and unstable foliations are jointly
nonintegrable and $g$ is a H\" older continuous $\wp$-vector function
on $M$. It turns out that if we take the initial point $z$ on an
element $S$ of a Markov family (see Section \ref{sec6}) introduced in
\cite{Do1} distributed there at random according to a probability
measure equivalent to the volume on $S$ then Assumption \ref{ass2.1}
can be verified. This does not yield though a desirable limit theorem
where the initial point is taken at random on the whole manifold $M$
distributed according to the Sinai-Ruelle-Bowen (SRB) measure (or the
normalized Riemannian volume).
We observe that a suspension representation of Anosov flows employed in
\cite{Ki1} to derive limit theorems in the conventional averaging setup
does not work in our situation because $F^{q_i(t)}x,\, i=1,...,\ell$ arrive
at the ceiling of the suspension at different times for
different $i$'s.
In the discrete time case there are several important classes of dynamical
systems where our conditions can be verified. First,
for transformations where symbolic representations via Markov
partitions are available (Axiom A diffeomorphisms (see \cite{Bo}) and
expanding endomorphisms, some one-dimensional
maps e.g. the Gauss map (see \cite{He}) etc.) we can rely on standard mixing
and approximation assumptions based on two parameter families of
$\sig$-algebras as in (\ref{2.22}) and (\ref{2.23}). On the other hand,
for many transformations Markov partitions are not available but still it
is possible to construct one parameter increasing or decreasing filtration
of $\sig$-algebras so that our conditions can be verified. For some classes
of noninvertible transformations $F$ it is possible to choose an appropriate
initial $\sig$-algebra $\cF_0$ such that $F^{-1}\cF_0\subset\cF_0$ and then
to define a decreasing filtration $\cF_i=F^{-i}\cF_0$ (see \cite{Li}
and \cite{FMT0}). Passing to the natural extension as in Remark 3.12 of
\cite{FMT0} we can turn to an increasing filtration and to verify our
conditions.
On the other hand, our results can be derived under appropriate conditions
with respect to decreasing families of $\sig$-algebras. Namely, let
$\cF\supset\cF_0\supset\cF_1\supset\cF_2\supset\cdots$ and define mixing
and approximation coefficients by
\begin{eqnarray}\label{2.26}
&\eta_{p,\ka,s}(n)=\sup_{t\geq s}\big\{\big\| E\big(g(\xi(t),\xi(t-s))
|\cF_{[t]+n}\big)\\
&-E_Pg(\xi(t),\xi(t-s))\big\|_p:\, g=g(\xi,\tilde\xi),\, |g|_\ka\leq 1\big\},
\,\,\,\eta_{p,\ka}(n)=\eta_{p,\ka,0}(n)
\nonumber\end{eqnarray}
and
\begin{equation}\label{2.27}
\zeta_q(n)=\sup_{t\geq n}\| E(\xi(t)|\cF_{[t]-n})-\xi(t)\|_q.
\end{equation}
Then under Assumption \ref{ass2.1} we can rely on estimates of Section
\ref{sec3} below and in place of martingales there arrive at reverse
martingales and to use a limit theorem for the latter.
\begin{remark}\label{rem2.3}
If $\bar B\equiv 0$ then according to Theorem \ref{thm2.2} the process
$X^\ve(t)$ is very close to its initial point on the time interval of order
$1/\ve$. Thus, in order to see fluctuations of order 1 it makes sense to
consider longer time and to deal with $V^\ve(t)=X^\ve(t/\ve)$. Under the
stronger condition $\int B(x,\xi_1,...,\xi_\ell)d\mu(\xi_\ell)\equiv 0$
it is not difficult to mimic the proofs in \cite{Kh2} and \cite{Bor}
relying on the technique of Sections \ref{sec3} and \ref{sec4} below
in order to obtain that $V^\ve(t),\, t\in[0,T]$ converges weakly as
$\ve\to 0$ to a diffusion process with parameters obtained in the same way
as in \cite{Kh2} and \cite{Bor}. It is not clear whether, in general, this
result still holds true assuming only that $\bar B\equiv 0$. Though most of
the required estimates still go through in the latter case a convergence
of $V^\ve$ to a Markov process seems to be problematic in a general
nonconventional averaging setup.
\end{remark}
\section{Estimates and martingale approximation}\label{sec3}
\setcounter{equation}{0}
The proof of Theorem \ref{thm2.2} will employ a modification of the machinery
developed in \cite{KV}. First, we have to study the asymptotical behavior as
$\ve\to 0$ of
\begin{equation}\label{3.1}
G^\ve_i(t)=\sqrt\ve\int_0^{\tau_i(t)/\ve}B_i\big(\bar Z(\ve s),\xi(q_1(s)),
...,\xi(q_i(s))\big)ds
\end{equation}
which is obtained from the definition (\ref{2.21}) by the change of variables
$s\to s/\ve$ and where $\tau_i(t)=t/\al_i$ for $i=1,...,k$ and $\tau_i(t)=t$
for $i=k+1,...,\ell$. Observe that if $\frac 1{N+1}\leq\ve\leq\frac 1N$ and
$N\geq 1$ then by (\ref{2.7}),
\begin{equation}\label{3.2}
|G^\ve_i(t)-G^{1/N}_i(t)|\leq \frac {2Ktd}{\sqrt N},
\end{equation}
and so it suffices to study the asymptotical behavior of $G^{1/N}_i$ as
$N\to\infty$. Set
\begin{equation}\label{3.3}
I_{i,N}(n)=\int_n^{n+1}B_i\big(\bar Z(s/N),\xi(q_1(s)),...,\xi(q_i(s))\big)ds.
\end{equation}
In view of (\ref{2.7}) the asymptotical behavior of $G^{1/N}_i$ as
$N\to\infty$ is the same as of $N^{-1/2}S_{i,N}(t)$ where
\begin{equation}\label{3.4}
S_{i,N}(t)=\sum_{n=0}^{[N\tau_i(t)]}I_{i,N}(n).
\end{equation}
There are two obstructions for applying directly the results of \cite{KV}
to the sum (\ref{3.4}). First, unlike \cite{KV} the integrand in (\ref{3.3})
depends on the "slow time" $s/N$. Secondly, our mixing and approximation
coefficients look differently from the corresponding coefficients in
\cite{KV}. Still, it turns out that these obstructions can
be dealt with and after minor modifications the method of \cite{KV} start
working in our situation, as well. Namely, the dependence on the "slow time"
being deterministic will not prevent us from making estimates similar to
\cite{KV} while dependence of $I^N_i$ on $N$ will just require us to deal with
martingale arrays which creates no problems as long as we obtain appropriate
limits of variances and covariances. Concerning the second obstruction we
observe that one half of the approximation estimate from \cite{KV} is contained
in the coefficient $\zeta_p$ while another half is hidden in the coefficient
$\eta_{p,\ka}$ which also suffices for required mixing estimates.
We explain next more precisely why estimates similar to \cite{KV} hold true
in our circumstances, as well. Let $f(\psi,\xi,\tilde\xi)$ be a function on
$\bbR^\vr\times\bbR^\wp\times\bbR^\wp$ such that for any $\psi,\psi'
\in\bbR^\vr$ and $\xi,\tilde\xi,\xi',\tilde\xi'\in\bbR^\wp$,
\begin{equation}\label{3.5}
|f(\psi,\xi,\tilde\xi)-f(\psi',\xi',\tilde\xi')|\leq C(|\psi-\psi'|^\ka+
|\xi-\xi'|^\ka+|y-y'|^\ka)\,\,\mbox{and}\,\, |f(\psi,\xi,\tilde\xi)|\leq C.
\end{equation}
Then setting $g(\psi)=E_Pf(\psi,\xi(0),\xi(s))$ we obtain from (\ref{2.1}) and
(\ref{2.3}) that for all $u,v\geq 0$ and $n\in\bbN$,
\begin{equation}\label{3.6}
\big\| E\big( f(\psi,\xi(n+u),\xi(n+u+v))|\cF_{[u]}\big)-g(\psi)\big\|_p\leq
C\eta_{p,\ka,v}(n).
\end{equation}
Let $h(\psi,\om)=E(f(\psi,\xi(n+u),\xi(n+u+v))|\cF_{[u]})-g(\psi)$. Then by
(\ref{3.5}) we can choose a version of $h(\psi,\om)$ such that with
probability one simultaneously for all $\psi,\psi'\in\bbR^\vr$,
\begin{equation}\label{3.7}
|h(\psi,\om)-h(\psi',\om)|\leq 2C|\psi-\psi'|^\ka.
\end{equation}
Since, in addition, $\| h(\psi,\om)\|_p\leq C\eta_{p,\ka}(n)$ by (\ref{3.6})
for all $\psi\in\bbR^\vr$, we obtain by Theorem 3.4 from \cite{KV} that for
any random $\vr$-vector $\Psi=\Psi(\om)$,
\begin{equation}\label{3.8}
\| h(\Psi(\om),\om)\|_a\leq cC\big(\eta_{p,\ka,v}(n)\big)^{1-\frac {\vr}{p\te}}
(1+\| \Psi\|_m)
\end{equation}
where $\frac {\vr}p<\te <\ka$, $\frac 1a\geq \frac 1p+\frac 1m$ and
$c=c(\vr,p,\ka,\te)>0$ depends only on parameters in brackets. Since
\begin{equation}\label{3.9}
h(\tilde\Psi(\om),\om)=E\big( f(\tilde\Psi,\xi(n+u),\xi(n+u+v))|\cF_{[u]}\big)
(\om) \,\,\mbox{a.s.}
\end{equation}
provided $\tilde \Psi$ is $\cF_{[u]}$-measurable we obtain from
(\ref{3.7})--(\ref{3.9}) together with the H\" older inequality
(cf. Corollary 3.6 in \cite{KV}) that,
\begin{eqnarray}\label{3.10}
&\big\| E\big( f(\Psi,\xi(n+u),\xi(n+u+v))|\cF_{[u]}\big)-g(\Psi)\big\|_a\\
&\leq C(\eta_{p,\ka,v}(n))^{1-\frac {\vr}{p\te}}(1+\| \Psi\|_m)+2C\| \Psi-
E(\Psi|\cF_{[u]})\|^\del_q\nonumber
\end{eqnarray}
provided $\frac 1a\geq \frac 1p+\frac 2m+\frac {\del}q$.
We apply the above estimates in two cases. First, when $f(\psi,\xi,\tilde\xi)
=f(\psi,\xi)=B_i(x,\xi_1,...,\xi_i)$ with $\psi=(\xi_1,...,\xi_{i-1})\in
\bbR^{(i-1)\wp}$, $\xi=\xi_i\in\bbR^\wp$, $n=[(q_i(t)-q_{i-1}(t))/2]$,
$u=q_i(t)-n$ and $\Psi=(\xi(q_1(t)),\xi(q_2(t)),...,\xi(q_{i-1}(t))$. In
the second case $f(\psi,\xi,\tilde\xi)=B_i(x,\xi_1,...,\xi_i)B_j(y,\xi_1',
...,\xi_j')$ with $\psi=(\xi_1,...,\xi_{i-1},\xi_1',...,\xi_{j-1}')\in
\bbR^{(i+j-2)\wp}$, $\xi=\xi_i$, $\tilde\xi=\xi_j'\in\bbR^\wp$, $n=
\big[\big(\min(q_i(t),q_j(s))-\max(q_{i-1}(t),q_{j-1}(s))\big)/2\big]$ when
$n>0$, $u=\min(q_i(t),q_j(s))-n$ and $\Psi=\big(\xi(q_1(t)),...,
\xi(q_{i-1}(t)),\xi(q_1(s)),...,\xi(q_{j-1}(s))\big)$. The estimates for the
first case are used for martingale approximations while the second case
emerges when computing covariances.
Since $\int B_i(x,\xi_1,...,\xi_{i-1},\xi_i)d\mu(\xi_i)=0$ we obtain by
(\ref{3.10}) the estimate
\begin{equation}\label{3.11}
\big\| E\big(B_i(\xi(q_1(t)),...,\xi(q_i(t)))|\cF_{[q_i(t)]-n}\big)\big\|_a
\leq C\big((\eta_{p,\ka}(n))^{1-\frac {\vr}{p\te}}+(\zeta_q(n))^\del\big)
\end{equation}
for some $C>0$ independent of $t$ where $n=n_i(t)=[(q_i(t)-q_{i-1}(t))/2]$.
Next, for any $x\in\bbR^\vr$, $\xi_1,...,\xi_{i-1}\in\bbR^\vr$ and $r=1,2,...$
set
\begin{eqnarray*}
&B_{i,r}(x,\xi_1,...,\xi_{i-1},\xi(t))=E\big( B(x,\xi_1,...,\xi_{i-1},\xi(t))|
\cF_{[t]+r}\big)\\
&\mbox{and}\,\,\,\xi_r(t)=E\big(\xi(t)|\cF_{[t]+r}\big).
\end{eqnarray*}
Then by (\ref{2.4}) and (\ref{2.7}) together with the H\" older inequality,
\begin{eqnarray}\label{3.12}
&\big\| B_i(x,\xi_1,...,\xi_{i-1},\xi(t))-B_{i,r}(x,\xi_1,...,\xi_{i-1},
\xi(t))\big\|_q\\
&\leq 2\big\| B_i(x,\xi_1,...,\xi_{i-1},\xi(t))-B_i(x,\xi_1,...,\xi_{i-1},
E(\xi(t)|\cF_{[t]+r}))\big\|_q\nonumber\\
&\leq 2Kd\big\| |\xi(t)-E(\xi(t))|\cF_{[t]+r})|^\ka\big\|_q\leq 2Kd
\zeta^\del_q(r).
\nonumber\end{eqnarray}
Moreover, similarly to Lemma 3.12 in \cite{KV} we obtain that
\begin{equation}\label{3.13}
\big\| B_i(x,\xi(q_1(t)),...,\xi(q_i(t)))-B_{i,r}(x,\xi_r(q_1(t)),...,
\xi_r(q_{i-1}(t)))\big\|_a\leq c\zeta^\del_q(r)
\end{equation}
provided $\frac 1a\geq\frac 1p+\frac 2m+\frac {\del}q$ and $\del<\min(\ka,
1-\frac d{p\ka})$ where $c=c(\del,a,p,q)>0$ depends only on the parameters
in brackets. Set
\[
b^{l,m}_{ij}(x,y;s,t)=E\big( B^{(l)}_i(x,\xi(q_1(s)),...,\xi(q_i(s)))B^{(m)}_j
(y,\xi(q_1(t)),...,\xi(q_j(t)))\big)
\]
where, recall, $B_i^{(l)}$ is the $l$-th component of the $d$-vector $B_i$.
Now, by (\ref{2.7}), (\ref{3.11}) and (\ref{3.13}),
\begin{equation}\label{3.14}
|b^{l,m}_{ij}(x,y;s,t)|\leq C(\big((\eta_{p,\ka}(n))^{1-\frac {\vr}{p\te}}
+(\zeta_q(n))^\del\big)
\end{equation}
where $C>0$ does not depend on $s,t\geq 0$ and $n=n_{ij}(s,t)=\max(\hat
n_{ij}(s,t),\,\hat n_{ji}(t,s))$ with $\hat n_{ij}(s,t)=
[\frac 12\min(q_i(s)-q_j(t),\, q_i(s)-q_{i-1}(s))]$.
Now, set
\begin{eqnarray}\label{3.15}
&I_{i,N,r}(n)=\int_{n-1}^nB_{i,r}\big(\bar Z(s/N),\xi_r(q_1(s)),...,
\xi_r(q_{i-1}(s))\big)ds,\, S_{i,N,r}(t)=\\
&\sum_{n=1}^{[N\tau_i(t)]}I_{i,N,r}(n),\, R_{i,r}(m)=
\sum_{l=m+1}^\infty E(I_{i,N,r}(l)|\cF_{m+r}),\, D_{i,N,r}(m)=\nonumber\\
&I_{i,N,r}(m)+R_{i,r}(m)-R_{i,r}(m-1)\,\mbox{and}\,
M_{i,N,r}(t)=\sum_{n=1}^{[N\tau_i(t)]}D_{i,N,r}(n).
\nonumber\end{eqnarray}
In view of (\ref{3.11}) applied with $a=2$ we see that the series for
$R_{i,r}(m)$ converges in $L^2$, $D_{i,N,r}(m)$ is $\cF_{m+r}$-measurable
and since $E(D_{i,N,r}(m)|\cF_{m-1+r})=0$
we obtain that $\{ D_{i,N,r}(m),\cF_{m+r}\}_{0\leq m\leq [N\tau_i(T)]}$ is a
martingale differences array. Next, we proceed similarly to Sections 5
and 7 of \cite{KV} observing that the limiting behaviour of
$N^{-1/2}S_{i,N,r}(t)$ as $N\to\infty$ is the same as of $N^{-1/2}
M_{i,N,r}(t)$, then dealing with the latter by means of martingale limit
theorems and, finally, employing the representation
\begin{equation}\label{3.16}
S_{i,N}(t)=S_{i,N,1}(t)+\sum_{r=1}^\infty\big(S_{i,N,2^r}(t)
-S_{i,N,2^{r-1}}(t)\big).
\end{equation}
In order to complete this programm it remains only to compute limiting
covariances as in Section 4 of \cite{KV} taking care also of the slow
time $s/N$ entering (\ref{3.3}) and (\ref{3.15}).
\section{Limiting covariances}\label{sec4}
\setcounter{equation}{0}
In this section we show the existence and compute the limit as $N\to\infty$
of the expression
\begin{equation}\label{4.1}
E\big(G_{i,l}^\ve(s)G_{j,m}^\ve(t)\big)=\ve\int_0^{\tau_i(s/\ve)}
\int_0^{\tau_j(t/\ve)}b^{l,m}_{ij}(\bar Z(\ve u),\bar Z(\ve v);u,v)dudv.
\end{equation}
We start with showing that there exists a constant $C>0$ such that for
all $t\geq s>0,\, l=1,...,d$, $N\geq 1$ and $i=1,...,\ell$,
\begin{equation}\label{4.2}
\sup_{\ve>0}E|G^{1/N}_{i,l}(t)-G^{1/N}_{i,l}(s)|^2\leq C(t-s).
\end{equation}
In order to obtain (\ref{4.2}) we note that by (\ref{2.9}) and (\ref{2.10})
for $t\geq s$,
\begin{equation}\label{4.3}
q_i(t)-q_i(s)\geq\al_i(t-s)\,\,\mbox{and}\,\, q_i(t)-q_{i-1}(t)\geq
\al_{i-1}t\,\,\mbox{when}\,\, i=2,...,k
\end{equation}
and for any $\gam>0$ there exists $t_\gam$ such that for all $t\geq t_\gam$
and $i=k+1,...,\ell$,
\begin{equation}\label{4.4}
q_i(t)-q_i(s)\geq (t-s)+\gam^{-1}\,\,\mbox{and}\,\, q_i(t)-q_{i-1}(t)\geq
t+\gam^{-1}.
\end{equation}
Now (\ref{4.2}) follows from (\ref{2.6}), (\ref{3.14}), (\ref{4.1}),
(\ref{4.3}) and (\ref{4.4}). Observe, that by (\ref{3.2}) and (\ref{4.1})
if $\frac 1{N+1}\leq\ve\leq\frac 1N$ then
\[
|EG^\ve_{i,l}(s)G^\ve_{j,m}(t)-EG^{1/N}_{i,l}(s)G^{1/N}_{j,m}(t)|\leq
\frac {4KdC\sqrt \cT}{\sqrt N},
\]
and so it suffices to study (\ref{4.1}) as $\ve=\frac 1N$ and $N\to\infty$.
Next, we claim that if $i>j$ and $i>k$ then the limit in (\ref{4.1}) as
$\frac 1\ve=N\to\infty$ exists and equals zero. Indeed, in this case for
any small $\gam >0$ with $\gam T\leq s$,
\begin{equation}\label{4.5}
|EG^{1/N}_{i,l}(s)G^{1/N}_{j,m}(t)|\leq I_1+I_2
\end{equation}
where by (\ref{4.2}),
\begin{equation}\label{4.6}
I_1=|EG^{1/N}_{i,l}(\gam \cT)G^{1/N}_{j,m}(t)|\leq\big(E(G^{1/N}_{i,l}
(\gam \cT))^2\big)^{1/2}\big(E(G^{1/N}_{j,m}(t))^2\big)^{1/2}\leq C
\sqrt {\gam \cT t}
\end{equation}
and by (\ref{3.14}),
\begin{eqnarray}\label{4.7}
&I_2=|E(G^{1/N}_{i,l}(s)-G^{1/N}_{i,l}(\gam \cT))G^{1/N}_{j,m}(t)|=\frac 1N
\int_{\gam \cT N}^{sN}du\int_0^{\tau_j(tN)}\\
&b^{l,m}_{ij}(\bar Z(u/N),\bar Z(v/N);u,v)dv\leq\frac CN\int_{\gam \cT N}^{sN}
du\int_0^{\tau_j(tN)}\rho_{ij}(u,v)dv\nonumber
\end{eqnarray}
where
\begin{equation}\label{4.8}
\rho_{ij}(u,v)=(\eta_{p,\ka}(n_{ij}(u,v)))^{1-\frac {\vr}{p\te}}
+(\zeta_q(n_{ij}(u,v)))^\del
\end{equation}
with $n_{ij}(s,t)$ defined after (\ref{3.14}). It follows from (\ref{2.6}),
(\ref{2.9}), (\ref{2.10}) and (\ref{4.8}) that for any $\gam>0$ there exists
$N_\gam$ such that whenever $N\geq N_\gam$ and $v\in[0,\cT N]$ (cf.
Proposition 4.5 in \cite{KV}),
\[
\int_{\gam \cT N}^{sN}\rho_{ij}(u,v)du\leq\gam,
\]
and so $I_2\leq C\cT\gam$. Since $\gam>0$ is arbitrary this together with
(\ref{4.5}) and (\ref{4.6}) yields that for all $l,m=1,...,d$, $i>k$ and
$j<i$,
\begin{equation}\label{4.9}
\lim_{N\to\infty}EG^{1/N}_{i,l}(s)G^{1/N}_{j,m}(t)=0.
\end{equation}
Next, we claim that when $i>k$ then also for all $l,m=1,...,d$,
\begin{equation}\label{4.10}
\lim_{N\to\infty}EG^{1/N}_{i,l}(s)G^{1/N}_{i,m}(t)=0.
\end{equation}
Indeed, by (\ref{3.14}) and (\ref{4.8}) for $t\geq s$,
\begin{equation}\label{4.11}
|EG^{1/N}_{i,l}(s)G^{1/N}_{i,m}(t)|\leq\frac 1N\int_0^{sN}du\int_0^{tN}
\rho_{ii}(u,v)dv=I_3+I_4
\end{equation}
where
\[
I_3=\frac 2N\int_0^{sN}du\int_u^{sN}\rho_{ii}(u,v)dv\,\,\mbox{and}\,\,
I_4=\frac 1N\int_0^{sN}du\int_{sN}^{tN}\rho_{ii}(u,v)dv.
\]
Now
\begin{eqnarray}\label{4.12}
&I_3=\frac 2N\int_0^{sN}du\int_u^{u+\gam}\rho_{ii}(u,v)dv+\frac 2N
\int_0^{\gam N}du\int_{u+\gam}^{sN}\rho_{ii}(u,v)dv\\
&+\frac 2N\int_{\gam N}^{sN}du\int_{u+\gam}^{sN}\rho_{ii}(u,v)dv\leq C(s\gam
+\gam+s\be_\gam(\gam N))\nonumber
\end{eqnarray}
for some $C>0$ where by (\ref{2.6}) and (\ref{2.10}) for any $\gam>0$,
\begin{equation}\label{4.13}
\be_\gam(M)=\sup_{u\geq M}\int^\infty_{u+\gam}\rho_{ii}(u,v)dv<\infty\,
\,\mbox{and}\,\, \lim_{M\to\infty}\be_\gam(M)=0.
\end{equation}
Next,
\begin{eqnarray}\label{4.14}
&I_4=\frac 1N\int_0^{sN}du\int_{sN}^{sN+\gam}\rho_{ii}(u,v)dv\\
&+\frac 1N\int_0^{sN}du\int_{sN+\gam}^{tN}\rho_{ii}(u,v)dv\leq Cs\gam+
Cs\be_s(N).\nonumber
\end{eqnarray}
Finally, (\ref{4.10}) follows from (\ref{4.11})--(\ref{4.14}) letting, first,
$N\to\infty$ and then $\gam\to 0$.
In order to compute the limit as $\frac 1\ve=N\to\infty$ of (\ref{4.1})
for $i,j=1,2,...,
k$ we recall an argument of Lemma 3.1 from \cite{Kh1} which yields that if
uniformly in $\sig\geq 0$ and $x,y$ from a compact set the limit
\begin{equation}\label{4.15}
\lim_{N\to\infty}\frac 1N\int_{\sig/\al_i}^{(\sig+sN)/\al_i}du
\int_{\sig/\al_j}^{(\sig+sN)/\al_j}b_{ij}^{lm}(x,y;u,v)dudv=sD^{l,m}_{ij}(x,y)
\end{equation}
exists and has the form of the right hand side with a continuous
$D^{l,m}_{i,j}$ then the limit (\ref{4.1}) exists, as well, and it has the
form
\begin{equation}\label{4.16}
\lim_{N\to\infty}E(G^{1/N}_{i,l}(s)G^{1/N}_{j,m}(t))=\int_0^{\min(s,t)}
D^{l,m}_{ij}(\bar Z(u),\bar Z(u))du.
\end{equation}
Namely, set $M=M(N)=[N^{2/3}]$ and let $s_\iota=\frac {\iota s}M,\,
\iota=0,1,...,M-1$. Assume also that $s\leq t$. Let
\[
A_N=\cup_{\iota=0}^{M-1}A_{N,\iota}\,\,\mbox{with}\,\, A_{N,\iota}=\{ (u,v):\,
s_\iota N\leq u,v<(s_\iota+\frac sM)N\}
\]
and $B_N=\{(u,v):\, 0\leq u\leq sN,\, 0\leq v\leq tN\}\setminus A_N$. Then
\begin{equation}\label{4.17}
EG^{1/N}_{i,l}(s)G^{1/N}_{j,m}(t)=I_5+I_6
\end{equation}
where
\[
I_5=\frac 1{Nij}\int_{B_N}b^{lm}_{ij}(\bar Z(u/N),\bar Z(v/N),u/i,v/j)dudv
\]
and
\[
I_6=\frac 1{Nij}\int_{A_N}b^{lm}_{ij}(\bar Z(u/N),\bar Z(v/N),u/i,v/j)dudv.
\]
Now, by (\ref{3.14}) and (\ref{4.8}),
\begin{eqnarray}\label{4.18}
&|I_5|\leq\frac C{Nij}\big(\sum_{\iota=0}^{M-1}\int_0^{s_\iota/\ve}
\int_{s_\iota/\ve}^{(s_\iota+\frac sM)/\ve}(\rho_{ij}(u/\al_i,v/\al_j)\\
&+\rho_{ji}(u/\al_j,v/\al_i))dudv+\int_0^{s_\iota/\ve}
\int_{s_\iota/\ve}^{t_\iota/\ve}\rho_{ij}(u/\al_i,v/\al_j)dudv\big).\nonumber
\end{eqnarray}
Observe that by the definition of $n_{ij}(u,v)$ after (\ref{3.14}) we can
write for $i,j=1,...,k$,
\begin{equation}\label{4.19}
\rho_{ij}(u/\al_i,v/\al_j)=\zeta(|u-v|)
\end{equation}
where $\zeta\geq 0$ satisfies $\int_0^\infty w\zeta(w)dw<\infty$. Integrating
by parts we obtain for any $V\geq U\geq 0$,
\begin{equation}\label{4.20}
\int_0^Udu\int_U^V\zeta(v-u)dv\leq\int_0^Udu\int_{U-u}^\infty\zeta(w)dw=
\int_0^Ur\zeta(r)dr\leq\int_0^\infty r\zeta(r)dr.
\end{equation}
This together with (\ref{2.6}), (\ref{4.18}) and (\ref{4.19}) gives by the
choice of $M=M(N)$ that
\begin{equation}\label{4.21}
|I_5|\leq \tilde C\frac M{N\al_i\al_j}\to 0\,\,\mbox{as}\,\, N\to\infty
\end{equation}
for some $\tilde C>0$ independent of $M$ and $N$.
Next,
\begin{equation}\label{4.22}
I_6=\frac 1{M\al_i\al_j}\sum_{\iota=0}^{M-1}J_{M,N}(\iota)+I_7
\end{equation}
where
\[
J_{M,N}(\iota)=\frac MN\int_{s_\iota\leq u,v<(s_\iota+\frac sM)N}b_{ij}^{lm}
(\bar Z(s_\iota),\bar Z(s_\iota);\frac u{\al_i},\frac v{\al_j})dudv
\]
and by (\ref{2.7}) and the choice of $M=M(N)$,
\begin{equation}\label{4.23}
|I_7|\leq Cs^3NM^{-2}\to 0\,\,\mbox{as}\,\, N\to\infty
\end{equation}
where $C>0$ does not depend on $s,N$ and $M$. By (\ref{4.15}) we obtain that
\begin{equation}\label{4.24}
|J_{M,N}(\iota)-s\al_i\al_jD_{ij}^{l,m}(\bar Z(s_\iota),\bar Z(s_\iota))|\to
0\,\,\mbox{as}\,\, N\to\infty,
\end{equation}
and so
\begin{equation}\label{4.25}
|I_6-\int_0^sD_{ij}^{l,m}(\bar Z(u),\bar Z(u))du|\to
0\,\,\mbox{as}\,\, N\to\infty
\end{equation}
completing the proof of (\ref{4.16}).
In order to describe $D^{l,m}_{ij}(x,y),\, i,j\leq k$ consider all indices
$1\leq i'_1<i'_2<...<i'_{\iota_{ij}}=i$ and $1\leq j'_1<j'_2<...
<j'_{\iota_{ij}}=j$ such that there exist $0<\rho_1<...<\rho_{\iota_{ij}}=1$
satisfying $\al_{i'_l}\rho_l,\,\al_{j'_l}\rho_l\in\{\al_1,...,\al_k\}$ for
all $l=1,...,\iota_{ij}$.
Define
\begin{eqnarray}\label{4.26}
&a^{l,m}_{ij}(x,y;s_1,...,s_{\iota_{ij}})=\int B^{(l)}_i(x,\xi_1,...,\xi_i)
B_j(y,\tilde\xi_1,...,\tilde\xi_j)\\
&\prod_{\be=1}^{\iota_{ij}}d\mu_{s_\be}(\xi_{i'_\be},\tilde\xi_{j'_\be})
\prod_{i_\gam\not\in\{i'_1,...,i'_{\iota_{ij}}\},1\leq i_\gam<i}
d\mu(\xi_{i_\gam})
\prod_{j_\zeta\not\in\{j'_1,...,j'_{\iota_{ij}}\},1\leq j_\zeta<j}
d\mu(\xi_{j_\zeta}).\nonumber
\end{eqnarray}
Then in the same way as in the proof of Lemma 4.4 from \cite{KV} (see also
Section 7 there) we obtain relying on (\ref{2.6}), (\ref{3.10}) and
(\ref{3.14}) that
\begin{equation}\label{4.27}
\lim_{N\to\infty,\,\al_iNu_N-\al_jNv_N=w}b_{ij}^{l,m}(x,y;Nu_N,Nv_N)=
a_{ij}^{l,m}(x,y;\rho_1w,\rho_2w,...,\rho_{\iota_{ij}}w).
\end{equation}
This is the only place where we need Assumption \ref{ass2.1} for
$\eta_{p,\ka,s}$ with $s>0$.
It follows similarly to Section 7 of \cite{KV} that the limit (\ref{4.15})
exists and it can be written in the form
\begin{equation}\label{4.28}
D^{l,m}_{ij}(x,y)=\frac 1{\al_i\al_j}\int_{-\infty}^\infty
a_{ij}^{l,m}(x,y;\rho_1w,\rho_2w,...,\rho_{\iota_{ij}}w)dw.
\end{equation}
Collecting the results of Sections \ref{sec3} and \ref{sec4} together we
conclude that each $G_i^\ve,\, i=1,...,k$ converges weakly as $\ve\to 0$
to the corresponding Gaussian process $G^0_i$ having independent increments
while the process $G^\ve_i,\, i>k$ converge weakly as $\ve\to 0$ to zero
(in the continuous time case we are dealing with now). Moreover, the
processes $G^\ve$ converge weakly as $\ve\to 0$ to a Gaussian process
$G^0$ (with not necessarily independent increments as an example in
\cite{KV} shows) having the representation
\begin{equation}\label{4.29}
G^0(t)=\sum_{i=1}^kG^\ve_i(it).
\end{equation}
Furthermore, the covariances of different components $G^0_i(s)=(G^{0,1}_i,
...,G^{0,d}_i(s))$ of this processes are described in view of the above by
\begin{equation}\label{4.30}
EG^{0,l}_i(s)G^{0,m}_j(t)=\int_0^{\min(s,t)}D^{l,m}_{ij}(\bar Z(u),
\bar Z(u))du,
\end{equation}
and so by (\ref{4.29}),
\begin{equation}\label{4.31}
EG^{0,l}(s)G^{0,m}(t)=\int_0^{\min(s,t)}A^{l,m}(u)du
\end{equation}
where
\[
A^{l,m}(u)=\sum_{1\leq i,j\leq k}D_{ij}^{l,m}(\bar Z(iu),\bar Z(ju)).
\]
\section{Gaussian approximation of the slow motion and discrete time case}
\label{sec5}\setcounter{equation}{0}
In order to complete the proof of Theorem \ref{thm2.2} we proceed similarly
to \cite{Kh1}. First, we consider the process $H^\ve(t)$ which solves the
linear equation
\begin{equation}\label{5.1}
H^\ve(t)=G^\ve(t)+\int_0^t\nabla\bar B(\bar Z(s))H^\ve(s)ds.
\end{equation}
By (\ref{2.7}), for some $C>0$ independent of $t$ and $\ve$,
\[
|H^\ve(t)|\leq |G^\ve(t)|+C\int_0^t|H^\ve(s)|ds.
\]
Then
\[
\big\vert |H^\ve(t)|-|G^\ve(t)|\big\vert\leq C\int_0^t|G^\ve(s)|ds+
C\int_0^t\big\vert |H^\ve(s)|-|G^\ve(s)|\big\vert ds
\]
and by Gronwall's inequality
\begin{equation}\label{5.2}
|H^\ve(t)|\leq |G^\ve(t)|+Ce^{Ct}\int_0^t|G^\ve(s)|ds.
\end{equation}
It follows from Section \ref{sec3} that the family of processes $\{ G^\ve(t),
\, t\in[0,\cT]\}$ is tight which together with (\ref{5.2}) implies that the
family of processes $\{ H^\ve(t),\, t\in[0,\cT]\}$, as well, as the family
of pairs $V^\ve=\{ G^\ve,\, H^\ve\}$ are tight.
It follows that any weak limit $V^0=\{ G^0,\, H^0\}$ of $V^\ve$ as
$\ve\to 0$ must satisfy the equation
\begin{equation}\label{5.3}
H^0(t)=G^0(t)+\int_0^t\nabla\bar B(\bar Z(s))H^0(s)ds
\end{equation}
which has a unique solution. Moreover, its solution $H^0$ is a Gaussian
process. Indeed, the equation (\ref{5.3}) can be
solved by successive approximations starting from $G^0$ so that on each
step we will get a Gaussian process (in view of linearity) and the
limiting process will be Gaussian, as well. Moreover, $H^0$ depends linearly
on $G^0$ having an integral representation of the form
\begin{equation}\label{5.4}
H^0(t)=G^0(t)+\int_0^tK(t,s)G^0(s)ds
\end{equation}
with a differentiable kernel $K$ (Green's function). The latter follows
considering an operator $A$ given by
\[
Af(t)=\int_0^t\nabla\bar B(\bar Z(s))f(s)ds
\]
which has the supremum norm less than 1 if $t\in[0,\Del]$ for $\Del$ small
enough, and so we can write
\[
H^0=(I-A)^{-1}G^0=G^0+\sum_{n=1}^\infty A^nG^0.
\]
In view of the form of the integral operator $A$ above this representation
yields (\ref{5.4}) on the interval $[0,\Del]$ and then employing the same
argument successively to time itervals $[\Del,2\Del],\,[2\Del,3\Del],...$
we extend the representation (\ref{5.4}) for any $t$.
Observe that
\begin{eqnarray}\label{5.5}
&Q^\ve(t)=\ve^{-1/2}\int_0^t\big( B(Z^\ve_x(s),\xi(q_1(s/\ve)),...,
\xi(q_\ell(s/\ve)))-\bar B(\bar Z_x(s))\big)ds\\
&=G^\ve(t)+\int_0^t\nabla_x B(Z^\ve_x(s),\xi(q_1(s/\ve)),...,
\xi(q_\ell(s/\ve)))Q^\ve(s)ds+\int_0^tJ^\ve_1(s)ds\nonumber
\end{eqnarray}
where
\begin{eqnarray*}
&J^\ve_1(s)=\ve^{-1/2}\big( B(\bar Z_x(s)+\sqrt\ve Q^\ve(s),\xi(q_1(s/\ve))
,...,\xi(q_\ell(s/\ve)))-B(\bar Z_x(s),\\
&\xi(q_1(s/\ve)),...,\xi(q_\ell(s/\ve)))-\nabla_xB(\bar Z_x(s),
\xi(q_1(s/\ve)),...,\xi(q_\ell(s/\ve)))\sqrt\ve Q^\ve(s)\big).
\end{eqnarray*}
If $H^\ve$ solves (\ref{5.1}) then $U^\ve(t)=Q^\ve(t)-H^\ve(t)$ satisfies
by (\ref{5.4}) the equation
\begin{equation}\label{5.6}
U^\ve(t)-\int_0^t\nabla_xB(\bar Z_x(s),\xi(q_1(s/\ve)),...,\xi(q_\ell(s/\ve)))
U^\ve(s)ds=\int_0^t(J^\ve_1(s)+J^\ve_2(s))ds
\end{equation}
where
\[
J^\ve_2(s)=\big(\nabla_xB(\bar Z_x(s),\xi(q_1(s/\ve)),...,\xi(q_\ell(s/\ve)))
-\nabla_x\bar B(\bar Z_x(s))\big)H^\ve(s).
\]
By Gronwall's inequality we obtain that
\begin{equation}\label{5.7}
|U^\ve(t)|\leq Cte^{ct}\int_0^t|J^\ve_1(s)+J^\ve_2(s)|ds
\end{equation}
for some $C>0$ independent of $\ve$ and $t\in[0,\cT]$.
Thus, in order to prove that $Q^\ve$ converges weakly as $\ve\to 0$ to
a Gaussian process $Q^0$ solving (\ref{2.14}) it suffices to show that
$\int_0^tJ^\ve_1(s)ds$ and $\int_0^tJ^\ve_2(s)ds$ converge to zero in
probability as $\ve\to 0$. By (\ref{2.7}),
\[
|Z^\ve_x(t)-Y^\ve_x(t)|\leq C\int_0^t|Z^\ve_x(s)-\bar Z_x(s)|ds=C\sqrt\ve
\int_0^t|Q^\ve_x(s)|ds
\]
with $C=Kd$, and so
\[
|Q^\ve_x(t)|\leq |G^\ve(t)|+C\int_0^t|Q^\ve_x(s)|ds.
\]
Hence, in the same way as in (\ref{5.2}),
\begin{equation}\label{5.8}
|Q^\ve_x(t)|\leq |G^\ve(t)|+Ce^{Ct}\int_0^t|G^\ve(s)|ds.
\end{equation}
By (\ref{2.7}) and the Taylor formula with a reminder we conclude that
\begin{equation}\label{5.9}
|J^\ve_1(s)|\leq C\sqrt\ve |Q^\ve(s)|^2
\end{equation}
which together with (\ref{4.2}) yields that $E|J^\ve_1(s)|\to 0$ as
$\ve\to 0$.
The proof of convergence to zero in probability of $\int_0^tJ^\ve_2(s)ds$
as $\ve\to 0$ is based on the integral representation (\ref{5.4}). Set
\[
\Phi(x,\xi_1,...,\xi_\ell)=B(x,\xi_1,...,\xi_\ell)-\bar B(x)
\]
and
\[
\Psi(x,\xi_1,...,\xi_\ell)=\nabla_xB(x,\xi_1,...,\xi_\ell)-\nabla_x\bar B(x).
\]
Relying on the representation (\ref{5.4}) we obtain that
\begin{equation}\label{5.10}
\big\vert E\int_0^tJ_2^\ve(s)ds\big\vert\leq |J^\ve_3(t)|+|J^\ve_4(t)|
\end{equation}
where
\begin{eqnarray}\label{5.11}
&J^\ve_3(t)=\ve^{3/2}\int_0^{t/\ve}ds\int_0^sduE\big(\Psi(\bar Z_x(\ve s),
\xi(q_1(s)),...,\xi(q_\ell(s)))\\
&\times \Phi(\bar Z_x(\ve u),\xi(q_1(u)),...,\xi(q_\ell(u)))\big)\nonumber
\end{eqnarray}
and
\begin{eqnarray}\label{5.12}
&J^\ve_4(t)=\ve^{3/2}\int_0^{t/\ve}ds\int_0^{\ve s}du\int_0^{u/\ve}dv
K(\ve s,\ve v)\\
&\times E\big(\Psi(\bar Z_x(\ve s),\xi(q_1(s)),...,\xi(q_\ell(s)))
\Phi(\bar Z_x(\ve u),\xi(q_1(u)),...,\xi(q_\ell(u)))\big).\nonumber
\end{eqnarray}
Estimating the expectations in (\ref{5.11}) and (\ref{5.12}) via
(\ref{3.10}) similarly to (\ref{3.14}) we obtain that both $J_3^\ve(t)$
and $J^\ve_4(t)$ are of order $\sqrt\ve$, and so the left hand side of
(\ref{5.10}) is of order $\sqrt\ve$, as well. For more details of a
similar argument we refer the reader to \cite{Kh1}. This completes the
proof of Theorem \ref{2.2} concerning the continuous time case.
In the discrete time case the proofs are similar but slightly simpler.
Namely, set
\begin{equation}\label{5.13}
R^{1/N}_i(t)=N^{-1/2}\sum_{n=0}^{[Nt/i]}B_i(\bar Z(nt/N),\xi(q_1(n)),
...,\xi(q_i(n)))
\end{equation}
where $B_i$'s are the same as in (\ref{2.16})--(\ref{2.18}). Then for
all $N\geq 1$,
\begin{equation}\label{5.14}
|G^{1/N}_i(t)-R^{1/N}_i(t)|\leq CN^{-1/2}
\end{equation}
for some $C>0$ independent of $N$. The asymptotical behavior of $R^{1/N}$
as $N\to\infty$ can be studied in the same way as in \cite{KV} taking
into account that we have here slightly different mixing conditions, and so
the corresponding estimates should be done as above via
(\ref{3.10})--(\ref{3.14}). The main difference of the discrete vis-\' a-vis
continuous time case is that now each $G^{1/N}_i(t),\, i=k+1,...,\ell$ converges
weakly as $N\to\infty$ to a nondegenerate Gaussian process $G^0_i(t)$ having
the covariances
\begin{equation}\label{5.15}
E\big(G^0_i(t)G^0_i(s)\big)=\int_0^{\min(s,t)}du\int\big(B_i(\bar Z(u),
\xi_1,...,\xi_i)\big)^2d\mu(\xi_1)...d\mu(\xi_i)
\end{equation}
which is proved combining arguments of Proposition 4.5 in \cite{KV} and
of Section \ref{sec4} above. The computation of other limiting covariances
proceeds in the same way as in the continuous time case.
It follows that in the discrete time case the processes $G^\ve$ converge
weakly as $\ve\to 0$ to a Gaussian process $G^0$ having the representation
\[
G^0(t)=\sum_{i=1}^k G^0_i(it)+\sum_{i=k+1}^\ell G^0_i(t)
\]
where each process $G^0_i,\, i>k$ is independent of each $G^0_j$ with
$j\ne i$ while the processes $G^0_i,\, i\leq k$ are correlated with
covariances described at the end of Section \ref{sec4} taken with
$\al_i=i,\, i=1,...,k$. The argument concerning the convergence of
processes $Q^\ve$ to $Q^0$ solving (\ref{2.14}) remains the same as
in the continuous time case.
\section{Some dynamical systems applications}\label{sec6}\setcounter{equation}
{0}
We start with recalling the setup from \cite{Do1} and \cite{Do2}. A
$C^2$-diffeomorphism $F$ of a compact Riemannian manifold $\Om$ is called
partially hyperbolic if there is a $F$-invariant splitting $E^u\oplus E^c
\oplus E^s$ of the tangent bundle of $\Om$ with $E^u\ne 0$ and constants
$\la_1\leq\la_2<\la_3\leq\la_4<\la_5\leq\la_6$, $\la_2<1,\,\la_5>1$ such
that $\|dF(v)\|/\|v\|$ is between $\la_1$ and $\la_2$ on $E^s$, between
$\la_3$ and $\la_4$ on $E^c$ and between $\la_5$ and $\la_6$ on $E^u$.
Denote by $W^u$ the foliation tangent to $E^u$ and call $S$ a u-set if $S$
belongs to a single leaf of $W^u$. $F$-invariant probability measures
which are absolutely continuous with respect to the volume on leafs $W^u$
are called u-Gibbs measures. It is assumed that $F$ has a unique u-Gibbs
measure $\SRB$ which is called the Sinai-Ruelle-Bowen (SRB) measure.
An important role in the construction is played by Markov families which
are collections $\cS$ of u-sets which cover $\Om$ and have certain
regularity properties (see \cite{Do1} and \cite{Do2}) but we formulate
here only their "Markov property" saying that for any $S\in\cS$ there
are $S_i\in\cS$ such that $FS=\cup_iS_i$. Now let $\cS$ be a Markov
family. Following \cite{Do1} and \cite{Do2} we construct on each $S\in\cS$
an increasing sequence of $\sig$-algebras $\cF_n$ in
the following recursive way. Let $\cF^S_0=\{\emptyset,S\}$. Suppose that
$\cF_n^S$ is generated by $\{ S_{j,n}\}$ with $F^nS_{j,n}\in\cS$. By
the "Markov property" we can decompose $F^{n+1}S_{j,n}=\cup_lS_{jl,n}$
and now let $\cF^S_{n+1}$ be generated by $F^{-n-1}S_{jl,n}$.
Next, for each $x_1$ and $x_2$ in a u-set $S$ put
\[
\rho(x_1,x_2)=\prod_{j=0}^\infty\frac {\mbox{det}(dF^{-1}|E^u)(F^{-1}x_1)}
{\mbox{det}(dF^{-1}|E^u)(F^{-1}x_2)}.
\]
Fix $x_0\in S$ and let $\rho_S(x)=\rho(x,x_0)(\int_S\rho(x,x_0)dx)^{-1}$.
For a Markov family $\cS$ and nonnegative constants $R,\al$ denote by
$E_1(\cS,R,\al)$ the set of probability measures $\sig$ defined for each
continuous function $g\in C(\Om)$ by
\begin{equation}\label{6.1}
\sig(g)=\int_Sg(x)e^{G(x)}\rho_S(x)dx
\end{equation}
where $S\in\cS$ and $G$ is H\" older continuous with the exponent $\al$
and the constant $R$. Denote also by $E=E(\cS,R,\al)$ the closure of the
convex hull of $E_1(\cS,R,\al)$. The decay of correlations
is measured in \cite{Do1} and \cite{Do2} via a sequence $a(n)\to 0$ as
$n\to\infty$ such that for any $\sig\in E$ and each H\" older continuous
$g$ on $\Om$,
\begin{equation}\label{6.2}
|\sig(g\circ F^n)-\SRB(g)|\leq a(n)\| g\|
\end{equation}
where $\|\cdot\|$ is a H\" older norm. An argument from Section 5
of \cite{DL} compares the coefficient $a(n)$ above with the more familiar
rate of decay of correlations $|\SRB(f\cdot(g\circ F^n))-\SRB(f)\SRB(g)|$
and it follows from there that the latter decays superpolynomially if
and only if $a(n)$ decays superpolynomially. According to \cite{Do0}
such decay of correlations holds true for $C^2$ Anosov flows with
jointly nonintegrable stable and unstable foliations and for their
time-one maps. By \cite{FMT}
this remains true for an open dense set of $C^2$ Axion A flows as well,
as for their time-one maps. For other partially hyperbolic dynamical
systems with fast decay of correlations see \cite{Do1}, \cite{Do2},
\cite{FMT} and references there.
In order to estimate $\eta_{p,\ka,s}(n)$ from (\ref{2.3}) we write in
the same way as in Lemma 4 from \cite{Do1} that on each element $S$
in $\cF_{[t]}$,
\begin{equation}\label{6.3}
A_{n,s,t}=E\big(g(f\circ F^{n+t},f\circ F^{n+t+s})|\cF_{[t]}\big)=
\int_S\rho_S(y)g_{s,t}(F^ny)dy
\end{equation}
where the expectation is with respect to $\sig$ on $S$ and $g_{s,t}(z)=
g(f(F^{t-[t]}z),f(F^{t-[t]+s}z))$. If $f$ and $g$ are H\" older
continuous then $g_{s,t}$
is H\" older continuous for fixed $s$ and $t$ and it is uniformly in $t$
H\" older continuous when $s=0$. Thus, by (\ref{6.2}) we have that
$|A_{n,s,t}-EA_{n,s,t}|$ decays in $n$ with the speed of at least $a(n)$
and this decay is uniform in $t$ if $s=0$. Hence, if $a(n)$ decays
superpolynomially then (\ref{2.6}) holds true. This yields Theorem
\ref{thm2.2} for $\xi(t)=\xi(t,z)=g(F^tz)$ on a probability space
$(S,\sig)$ for $\sig\in E$ and an element $S$ of a Markov family while
$g$ is a H\" older continuous function. We observe that the measure
$\sig$ here plays the role of the probability $Pr$ in the setup of
Section \ref{sec2} while $\SRB$ plays the role of $P$ there.
\section{Concluding remarks: fully coupled averaging}\label{sec7}
\setcounter{equation}{0}
In the nonconventional framework as discussed in this paper even the setup
of fully coupled averaging, i.e. when the fast motion depends on the slow
one, is not quite clear.
On the first sight we may want to deal with the equations
\begin{eqnarray}\label{7.1}
&X^\ve(n+1)=X^\ve(n)+\ve B(X^\ve(n),\xi(n),\xi(2n),...,\xi(\ell n)),\\
&\xi(n+1)=F_{X^\ve(n)}(\xi(n))\nonumber
\end{eqnarray}
in the discrete time case and
\begin{equation}\label{7.2}
\frac {dX^\ve(t)}{dt}=\ve B(X^\ve(t),\xi(t),\xi(2t),...,\xi(\ell t)),\quad
\frac {d\xi(t)}{dt}=b(X^\ve(t),\xi(t))
\end{equation}
in the continuous time case. The problem is that $\xi(kn)$ or $\xi(kt)$ are not
yet defined for $k>1$ at time $n$ or $t$ so we cannot insert them into the
first equation in (\ref{7.1}) or (\ref{7.2}) respectively, and so these
equations do not define properly $X^\ve$ and $\xi$.
A reasonable modification of this setup is to consider
\begin{eqnarray}\label{7.3}
&X^\ve(n+1)=X^\ve(n)+\ve B(X^\ve(n),\eta_1(n),\eta_2(n),...,\eta_\ell(n)),\\
&\eta_i^\ve(n+1)=F^i_{X^\ve(n)}(\eta_i^\ve (n)),\, i=1,...,\ell\nonumber
\end{eqnarray}
in the discrete time case and
\begin{eqnarray}\label{7.4}
&\frac {dX^\ve(t)}{dt}=\ve B(X^\ve(t),\eta_1(t),\eta_2(t),...,\eta_{\ell}(t)),\\
&\frac {d\eta_i^\ve(t)}{dt}=ib(X^\ve(t),\eta^\ve_i(t)),\, i=1,2,...,\ell
\nonumber\end{eqnarray}
in the continuous time case. We consider (\ref{7.3}) and (\ref{7.4}) as sets
of $\ell+1$ equations but require that $\eta^\ve_1(0)=\eta^\ve_2(0)=\cdots=
\eta^\ve_\ell(0)$. This approach seems to be reasonable if we consider
(\ref{7.3}) and (\ref{7.4}) as perturbations of equations with constants of
motion
\begin{equation}\label{7.5}
\eta^{(x)}(n+1)=F_x(\eta^{(x)}(n))\,\,\mbox{and}\,\,\frac {d\eta^{(x)}(t)}{dt}
=B(x,\eta^{(x)}(t)),
\end{equation}
i.e. when $x$ variable remains fixed in unperturbed equations but start
moving slowly in perturbed ones.
Then $\eta^{(x)}(i(n+1))=F^i_x(\eta^{(x)}(in))$ and $d\eta^{(x)} (it)/dt=
iB(x,\eta^{(x)}(it))$.
As it is well known in the fully coupled setup the averaging principle not
always holds true and when it takes place then usually only in the sense of
convergence in average or in measure. In the nonconventional situation the
problem is even more complicated. Consider, for instance,
\begin{eqnarray}\label{7.6}
&\frac {d\al^\ve_{\al,\vf}(t)}{dt}=\ve B(\al^\ve_{\al,\vf}(t),\vf_{1,\al}^\ve(t)
,...,\vf_{\ell,\al}^\ve(t)),\\
&\frac {d\vf_{i,\al,\vf}^\ve(t)}{dt}=i\al^\ve_{i,\al,\vf}(t),\quad \al^\ve_{\al,
\vf}(0)=\al,\,\vf_{1,\al,\vf}^\ve(0)=\cdots=\vf_{\ell,\al,\vf}^\ve(0)=\vf
\nonumber\end{eqnarray}
where $\vf$ denotes a point on an $n$-dimensional torus $\bbT^n$ and $\al$
denotes a constant $n$-vector (constant vector field on $\bbT^n$).
Then $\vf^\ve_{i,\al,\vf}=i\vf^\ve_{1,\al,\vf}-
(i-1)\vf$. Set $\tilde B(\psi,\vf)=B(\al,\psi,\psi-\vf...,\psi-(\ell-1)\vf)$.
Then the right
hand side of (\ref{7.6}) can be replaced by $\ve \tilde B(\al^\ve_{\al,\vf}(t),
\vf_{1,\al}^\ve(t),\vf)$.
If $\bar B(\al)=\int\tilde B(\al,\vf_1,\vf)d\vf_1d\vf$ and
$\frac {d\bar\al_{\al}(t)}{dt}=\bar B(\bar\al_{\al}(t)),\quad \bar\al_{\al}(0)
=\al$ then employing the technique from the proof of Theorem 2.1 in
\cite{Ki2} it is not difficult to see that for any compact $K$,
\begin{equation}\label{7.7}
\int_K\sup_{0\leq t\leq\cT/\ve}|\al^\ve_{\al,\vf}(t)-\bar \al_{\al}(\ve t)|
d\al d\vf\to 0\,\,\,\mbox{as}\,\,\, \ve\to 0.
\end{equation}
\bibliography{matz_nonarticles,matz_articles}
\bibliographystyle{alpha}
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Sci fi/Fantasy image byIda Östlund
I have no idea what this is. I just started drawing one day and here is the resoult. I thought it would look better coloured so I tried it. this is my first coloured picture.
Published More than a year ago
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\begin{document}
\maketitle
\begin{abstract}
We present a construction of semi-classical states for P\"oschl-Teller
potentials based on a supersymmetric quantum mechanics approach. The parameters of these ``coherent" states are points in the classical phase space of these systems. They minimize a special uncertainty relation. Like standard coherent states they resolve the identity with a uniform measure. They permit to establish the correspondence (quantization) between classical and quantum quantities. Finally, their time evolution is localized on the classical phase space trajectory.
\end{abstract}
\section{Introduction}
The search of quantum states that exhibit a semi-classical time behavior was initiated
in Schr\"odinger's pioneering work on ``packets of eigenmodes" of the harmonic oscillator \cite{Schrodinger-1926-14}.
\begin{quotation}
\emph{I show that the packet of eigenmodes with high quantum number $n$ and with relatively small difference in quantum numbers may represent the mass point that moves according to the usual classical mechanics, i.e. it oscillates with [classical] frequency $\nu_0$.}
\end{quotation}
These Schr\"odinger states have been qualified as coherent by Glauber within the context of quantum optics.
With regard to the huge amount of recent works on quantum dots and quantum wells in nanophysics
it has become challenging to construct quantum states for infinite wells which display localization properties comparable to those nicely displayed by the Schr\"odinger states. Infinite wells are often modeled by P\"oschl-Teller (also known as trigonometric Rosen-Morse) confining potentials \cite{Poschl-1933-83,Rosen-1932-42} used, e.g., in quantum optics \cite{Yildirim-2005-72, Wang-2007-75}. The infinite square well is a limit case of this family referred to in what follows as \mbox{${\mathcal{T}}$-potentials}. The question is to find a family of states: a) phase-space labelled, b) yielding a resolution of the identity with respect to the usual uniform measure, and c) exhibiting semi-classical phase space properties with respect to ${\mathcal{T}}$-Hamiltonian time evolution. We refer to these states as coherent states (CS) as they share many striking properties with Schr\"odinger's original semi-classical states.
${\mathcal{T}}$-potentials belong to the class of shape invariant potentials \cite{Gendenshtein-1983-38} intensively studied within the framework of supersymmetric quantum mechanics (SUSYQM) \cite{Cooper-2002}. Various semi-classical states adapted to ${\mathcal{T}}$-potentials have been proposed in previous works \cite{ Aleixo-2007-40,Antoine-2001-42,Crawford-1998-57,Shreecharan-2004-69} and references therein. However, they do not verify simultaneously a), b), and c). Moreover, correspondence between classical and quantum momenta requires a thorough analysis since there exists a well-known ambiguity in the definition of a quantum momentum operator \cite{Reed-1980, Antoine-2001-42}. This is due to the confinement of the system in an interval, unlike the harmonic oscillator case.
In this letter, we present a construction of coherent states for ${\mathcal{T}}$-potentials based on a general approach given by one of us in \cite{Bergeron-1995-36}. We examine in detail the classical-quantum correspondence based on these states (``CS quantization"). We eventually show that our states stand comparison with the Schr\"odinger CS in terms of semi-classical time behavior.
\section{Definition of SUSYQM \mbox{coherent states}}
Let us consider the motion of a particle confined in the interval $[0,L]$ and submitted to the repulsive symmetric \mbox{${\mathcal{T}}$-potential}
\begin{equation}
V_{\nu}(x)=\E_0 \frac{ \: \nu(\nu+1)}{\sin^2 \frac{\pi}{L}x}\, ,
\end{equation}
where $\nu \ge 0$ is a dimensionless parameter. The limit \mbox{$\nu \to 0$} corresponds to the infinite square well. The factor $\E_0 = \hbar^2 \pi^2 (2 m L^2)^{-1} \geq 0$ is chosen as the ground state energy of the infinite square well. On the quantum level, the Hamiltonian acts in the Hilbert space $\mathcal{H}=L^2([0,L],dx)$ as:
\begin{equation}
{\bf H}_{\nu}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V_{\nu}(x)\, .
\end{equation}
The eigenvalues $E_{n,\nu}$ and corresponding eigenstates $\ket{\phi_{n,\nu}}$ of ${\bf H}_{\nu}$ read
\begin{equation}
E_{n}=\E_0(n+\nu+1)^2, \ \ \ n=0,1,2...,
\end{equation}
\begin{equation}
\phi_{n}(x)= Z_{n} \sin^{\nu+1}\left(\frac{\pi}{L}x\right)\, \mathrm{C}_n^{\nu+1}\left(\cos \frac{\pi}{L}x\right)
\end{equation}
where $C_n^{\nu+1}$ is a Gegenbauer polynomial and
\begin{equation}
Z_{n}=\Gamma(\nu+1) \frac{2^{\nu+1/2}}{\sqrt{L}} \sqrt{\frac{n! (n+\nu+1)}{\Gamma(n+2\nu+2)}}
\end{equation}
is the normalization constant. Eigenfunctions $\phi_{n}$ obey the Dirichlet boundary conditions $\phi_{n}(0)=\phi_{n}(L)=0$. A detailed mathematical discussion on the boundary conditions and self-adjoint extensions for the ${\mathcal{T}}$-Hamiltonian can be found in \cite{Gesztesy-1985-362,Antoine-2001-42}.
In particular, the ground state eigenfunction $\phi_{0}$ is $Z_{0} \sin^{\nu+1} \frac{\pi}{L}x$
and the eigenfunctions for the infinite square well ($\nu=0$) reduce to $ \sqrt{\frac{2}{L}} \sin \frac{(n+1)\pi}{L} x.$
We define the superpotential $W_{\nu}(x)$ as
\begin{equation}
W_{\nu}(x)\eqdef-\hbar \frac{\phi'_{0}(x)}{\phi_{0}(x)}=-\frac{\hbar \pi}{L} (\nu+1) \cot \frac{\pi}{L}x
\end{equation}
and the lowering and raising operators ${\bf A}_{\nu}$ and ${\bf A}^\dag_{\nu}$ as
\begin{equation}
{\bf A}_{\nu} \eqdef W_{\nu}(x)+\hbar \frac{d}{dx} \text{ and } {\bf A}^\dag_{\nu} \eqdef W_{\nu}(x)-\hbar \frac{d}{dx}
\end{equation}
Thus, the ${\mathcal{T}}$-Hamiltonian ${\bf H}_{\nu}$ can be rewritten in terms of these operators:
\begin{equation}
{\bf H}_{\nu}=\frac{1}{2m}{\bf A}^\dag_{\nu} {\bf A}_{\nu} +E_0.
\end{equation}
As expected \cite{Cooper-2002}, the supersymmetric partner ${\bf H}^{(S)}_{\nu}$
\begin{equation}
{\bf H}^{(S)}_{\nu}=\frac{1}{2m} {\bf A}_{\nu} {\bf A}^\dag_{\nu} + E_0.
\end{equation}
coincides with the original Hamiltonian with increased $\nu$: ${\bf H}^{(S)}_{\nu}= {\bf H}_{\nu+1}.$
The classical phase space for the motion in a \mbox{${\mathcal{T}}$-potential} is defined as the infinite band in the plane:
\mbox{$
\mathcal{K}=\{ (q,p) | q\in [0,L] \text{ and } p\in \mathbb{R} \}\, .
$}
Let us introduce the operators
\begin{equation}
\label{QP}
{\bf Q}: \psi(x) \mapsto x\psi(x), \quad \textrm{and} \quad {\bf P}: \psi \mapsto -i \hbar \dfrac{d}{dx} \psi(x).
\end{equation}
We then build our coherent states $|\eta_{q,p}\rangle$ as normalized eigenvectors of ${\bf A}_{\nu} =W_{\nu}({\bf Q})+i {\bf P}$ with eigenvalue \mbox{$W_{\nu}( q)+i p$}, the latter being the classical counterpart of $A_\nu$ as shown below (see Table \ref{table1}).
\begin{equation}
\label{CSPT}
\ket{\eta_{q,p}}= N_{\nu}(q) \left\vert\xi_{W(q)+ip}^{[\nu]}\right\rangle\,, \ (q,p)\in \mathcal{K}\, ,
\end{equation}
where $\xi_z(x)= e^{z x/\hbar} \sin^{\nu+1} \left(\frac{\pi}{L}x \right)\text{ for } x \in [0,L]\, . $
The normalization coefficient $N_{\nu}(q)$ is given by
\begin{align}
\nonumber \frac{1}{N_{\nu}^2(q)}=&\frac{2^{\nu+1} |\Gamma(\nu+2-i (\nu+1) \cot \frac{\pi}{L}q)|}{\sqrt{L} \sqrt{ \Gamma(2\nu+3)}}
\exp\left[\frac{\pi}{2}(\nu+1)\cot \frac{\pi}{L}q \right].
\end{align}
For the sake of simplicity we drop off systematically in the sequel the $\nu$ dependence of various used symbols when no confusion is possible.
It is possible to show that the function $x \mapsto |\eta_{q,p}(x)|$ reaches its maximal value for $x=q$ and
$\langle \mathbf{P} \rangle_{p,q}=p$. Finally, the uncertainty relation $\Delta W_\nu(\mathbf{Q}) \Delta \mathbf{P} \geq \frac{\hbar}{2} \langle W_\nu'(\mathbf{Q}) \rangle$ is minimized by our CS as proved in \cite{Bergeron-1995-36}.
\section{CS Quantization and \mbox{expected values}}
As is proved in \cite{Bergeron-new}, the CS family \eqref{CSPT} resolves the unity with respect to the uniform measure on the phase space $\mathcal{K}$:
\begin{equation}
\label{equa:resol}
\int_{\mathcal{K}} \frac{dq\,dp}{2 \pi \hbar} \ket{\eta_{q,p}} \bra{\eta_{q,p}} = \mathbb{I}\, .
\end{equation}
As an immediate consequence we proceed with the CS quantization of ``classical observables" $f(q,p)$ through the correspondence \cite{Klauder-1985,Gazeau-2009}
\begin{equation}
\label{quantization}
f(q,p) \to {\bf F}=\int_{\mathcal{K}} \frac{dq\, dp}{2 \pi \hbar} f(q,p) \ket{\eta_{q,p}}\bra{\eta_{q,p}}\, .
\end{equation}
This operator-valued integral is understood as the sesquilinear form,
\begin{equation}
B_f(\psi_1,\psi_2)=\int_{\mathcal{K}} \frac{dq\, dp}{2 \pi \hbar} f(q,p) \scalar{\psi_1}{\eta_{q,p}}\scalar{\eta_{q,p}}{\psi_2}.
\end{equation}
The form $B_f$ is assumed to be defined on a dense subspace of the Hilbert space. If $f$ is real and at least semi-bounded, the Friedrich's extension \cite[Thm. X.23]{Reed-1975} of $B_f$ univocally defines a self-adjoint operator. However, if $f$ is not semi-bounded, there is no natural choice of a self-adjoint operator associated with $B_f$. In this case, we can consider directly the symmetric operator $\mathbf{F}$ given by Eq.\eqref{quantization} enabling us to obtain a self-adjoint extension (unique for particular operators). The question of what is the class of operators that may be so represented is a subtle one \cite{Klauder-1985,Gazeau-2009}.
In Table \ref{table1}, we give a list of operators obtained through the CS quantization of basic functions $f$. One can also compute the so-called ``lower'' or ``covariant'' symbols \cite{Klauder-1985,Gazeau-2009} of operators defined as the expectation values of the latter in the CS.
In Table \ref{table2} we give a list of functions of the most important quantum operators.
\begin{table}[htb!]
\renewcommand{\tabcolsep}{0.1cm}
\centering
\newcommand\T{\rule{0pt}{2.8ex}}
\newcommand\B{\rule[-1.5ex]{0pt}{0pt}}
\begin{tabular}{|c|c|c|c|c|}
\hline
Name \T & $f$ & $\mathbf{A}_f$ & Operator action & Properties \B\\
\hline \hline
Position \T & $q$ & $ F(\mathbf{Q}) $ (*)& multiplication &bounded \\
& & & &self-adjoint \B \\
\hline
Superpotential \T & $W_\nu(q)$ & $W_\nu(\mathbf{ Q})$& multiplication & unbounded \\
& & & & self-adjoint \B \\
\hline
Potential \T & $\frac{1}{\sin^2 \pi q/L}$&$\frac{(2\nu+3)(2\nu+2)^{-1}}{\sin^2 \pi \mathbf{ Q}/L}$ & multiplication &unbounded \\
& & & & self-adjoint \B \\
\hline
``Momentum" \T & p & $\mathbf{ P}$ & $\mathbf{P} \phi_n=-i \hbar \phi'_n $ & unbounded \\
& & & & symmetric \B \\
\hline
Hamiltonian \T & $ \frac{p^2}{2m}+ \frac{2\nu-1}{2\nu+3} \frac{\E_0(\nu+1)^2}{\sin^2 \pi q/L}$ & $\mathbf{ H}_\nu$ & Schr\"odinger & semi-bounded \\
& & &operator & self-adjoint\\
& & & & ($\nu \geq 1/2$) \B \\
\hline
\end{tabular}
\caption{Some quantized classical observables.
The operators $\mathbf{Q}$ and $\mathbf{P}$ are defined in eq. \eqref{QP}.
(*)$F(x)= \sin^{2\nu+2} (\pi x/L) \int_0^L dq\: q N_\nu^2(q) \exp (2 W_\nu(q) x/L)
$.}
\label{table1}
\end{table}
\begin{table}[htb!]
\newcommand\T{\rule{0pt}{2.8ex}}
\newcommand\B{\rule[-1.5ex]{0pt}{0pt}}
\centering
\begin{tabular}{|c|c|c|}
\hline
Name \T & $\mathbf{A}$ & $f$ \B \\
\hline
\hline
Position \T & $\mathbf{Q}$& $N_\nu^2(q)$ \\
& & \B $\times\int_0^L dx \:x \sin^{2\nu+2} \frac{\pi x}{L} e^{\frac{2 W_\nu(q) x}{L}} $ \\
\hline
``Momentum" (*) \T & $\mathbf{P}$& $p$ \B \\
\hline
Superpotential \T & $W_\nu(\mathbf{Q})$ & $W_\nu(q)$ \B \\
\hline
Potential \T & $\frac{1}{\sin^2 \pi \mathbf{Q}/ L}$ & $\frac{2\nu+2}{2\nu+1}\frac{1}{\sin^2 \pi q/ L}$ \B \\
\hline
Kinetic energy \T &$\frac{\mathbf{P}^2}{2m}$& $\frac{p^2}{2m}+ \frac{1}{2\nu+1} \frac{\E_0 (\nu+1)^2}{\sin^2 \pi q/L}$ \B \\
\hline
\end{tabular}
\caption{Some lower symbols. (*) The operator $\mathbf{P}$ is the one given in Table \ref{table1}.}\label{table2}
\end{table}
\section{Semi-classical behavior}
For any normalized state $\phi \in \mathcal{H}=L^2([0,L],dx)$, the resolution of unity \eqref{equa:resol} allows us to get its phase space representation \mbox{$\Phi(q,p) \eqdef \scalar{\eta_{q,p}}{\phi}/\sqrt{2\pi\hbar}$} and the resulting probability distribution on the phase space $\mathcal{K}$:
\begin{equation}
\label{eqn:phasespacerepre}
\mathcal{K} \ni (q,p)\mapsto \frac{1}{2\pi \hbar} |\scalar{\eta_{q,p}}{\phi}|^2=\rho_{\phi}(q,p)
\end{equation}
The phase-space distribution $\rho_{\eta_{q_0,p_0}}(q,p)$ for a particular state $\phi=|\eta_{q_0,p_0}\rangle$ is shown in Figure \ref{figure}$(a)$ for $\nu=0$ (infinite square well), see \href{http://gemma.ujf.cas.cz/~siegl/SUSYCS.html}{animation} for an animated time-evolution.
\begin{figure}[!ht]
\centering
\subfloat
[Phase space distribution \eqref{eqn:phasespacerepre} for $\nu=0$ of the state $\eta_{q_0,p_0}$ with $q_0=L/5$,
$p_0=4 \pi \hbar/L$ and $L=20\,$\AA. The thick curve is the expected phase trajectory in the infinite square well,
deduced from the semi-classical hamiltonian in Eq. \eqref{eqn:semiclassenergy}. The particle is an electron, its mean energy deduced from Eq. \eqref{eqn:semiclassenergy} is $E=1.6 \: \mathrm{eV}$. Increasing values of the function are encoded by the colors from blue to red. Note that for Schr\"odinger CS the corresponding distribution is a Gaussian localized on points of a circular trajectory in the complex plane. See \href{http://gemma.ujf.cas.cz/~siegl/SUSYCS.html}{animation} for an animated time-evolution.]
{\includegraphics[width =0.4\textwidth]{fig1.pdf}} \hspace{0.5cm}
\subfloat
[Time average of the phase space distribution of $\eta_{q_0,p_0}(t)$ evolving following the Hamiltonian of the infinite square well. The values of parameters and the thick curve are those of Figure \ref{figure}$(a)$. Increasing values of the function are encoded by the colors from blue to red. Note that for Schr\"odinger CS, the corresponding time-averaged distribution would be uniformly localized on a circular trajectory in the complex plane.]
{\includegraphics[width =0.4\textwidth]{fig2.pdf}}
\caption{Phase space distribution of CS and its time average }
\label{figure}
\end{figure}
Let us now examine the time behavior $t \mapsto \rho_{\phi(t)}(q,p)$ for a state $\phi(t)$ evolving under the action of the infinite square well Hamiltonian ${\bf H}_0$ (there is no significant difference from a generic $\nu \neq 0$ case):
\begin{equation}
\ket{\phi(t)}=e^{-i {\bf H_0} t/\hbar} \ket{\phi}=\sum_{n=0}^\infty e^{-i \E_0 (n+1)^2 t/ \hbar} \scalar{\phi_{n,0}}{\phi} \ket{\phi_{n,0}}
\end{equation}
where $\phi_{n,0}\equiv \sqrt{\frac{2}{L}} \sin \frac{(n+1)\pi}{L} x$.
With $\phi=\eta_{q_0,p_0}$ as an initial state, we have for a given $\nu$ (see Table \ref{table2})
\begin{equation}
\label{eqn:semiclassenergy}
\scalar{\eta_{q_0,p_0}(t)}{{\bf H}_0 |\eta_{q_0,p_0}(t)} =\frac{p_0^2}{2m}+ \frac{1}{2\nu+1} \frac{\E_0(\nu+1)^2}{\sin^2 \frac{\pi}{L} q_0}.
\end{equation}
Since the lower symbols of $W_\nu(\mathbf{Q})$ and $\mathbf{P}$ correspond to their classical original functions $W_\nu(q)$ and $p$ (see Table \ref{table2}), one can expect that the time average of the probability law $\rho_{\eta_{q_0,p_0}(t)}(p,q)$ corresponds to some fuzzy extension in phase space of the classical trajectory corresponding to the time-independent Hamiltonian in the r.h.s. of \eqref{eqn:semiclassenergy}.
This key result is illustrated in Figure \ref{figure}(b) where we have represented the time average distribution $\bar{\rho}$ defined as
\begin{equation}
\label{rhobar}
\bar{\rho}(q,p)=\lim_{T \to \infty} \frac{1}{T} \int_0^T \rho_{\eta_{q_0,p_0}(t)}(q,p) dt,
\end{equation}
for the same values of the parameters as in Figure \ref{figure}$(a)$.
The time average distribution $\bar{\rho}$ allows us to compare the quantum behavior with the classical trajectory, but the expression \eqref{rhobar} hides the complex details of the wave-packet dynamics. The latter exhibits a splitting of the initial wave-packet into secondary ones during the sharp reflection phase, each of them following the classical trajectory, before they amalgamate to reconstitute a unique packet (revival time). This important point makes the difference with the time behavior of the Schr\"odinger states for the harmonic oscillator.
\section{Conclusion}
We have presented a family of CS for the ${\mathcal{T}}$-potentials that sets a sort of natural bridge between the phase space and its quantum counterpart. These CS share with the Schr\"odinger ones some of their most striking properties, e.g. resolution of identity with uniform measure and saturation of uncertainty inequalities. They also possess remarkable evolution stability features (not to be confused with CS temporal stability in the sense of \cite{Antoine-2001-42} corresponding to the time parametric evolution): their time evolution generated by $\mathbf{H}_\nu$ is localized on the classical phase space trajectory.
The approach developed in this paper can be easily extended to higher dimensional bounded domains, provided that the latter be symmetric enough (e.g. square, equilateral triangle, etc) to allow shape invariance integrability.
\section*{Acknowledgments}
P.S. appreciates the support by the Grant Agency of the Czech Republic
project No. 202/08/H072 and the Czech Ministry of Education, Youth and
Sports within the project LC06002. We wish to thank A. Comtet, J. Dittrich, P. Exner, J. R. Klauder, T. Paul and J. Tolar for fruitful discussions and comments.
{\footnotesize
\bibliographystyle{unsrt}
\bibliography{C:/Data/00Synchronized/GlobalReferences}
}
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TITLE: Rotation of object on another object under rotation
QUESTION [2 upvotes]: First, I would like to know the rotational velocity of disk2 if disk1 turns at $\omega1$. The axis "x" is fixed to the ground and disk1 is allowed to turn around it. Axis "y" is fixed to disk1 and disk2 is allowed to turn around it. For me, disk2 turns around axis "x" and around axis "y" but not in the same manner. I can understand its rotational velocity is $\omega2$ and for me even disk2 turns around axis "x" and because disk1 is turning, this doesn't change its rotational velocity $\omega2$. I'm not sure. Can you explain please ? In fact, what's a rotational velocity in this case for disk2 ?
Second, with $\omega1=-\omega2$. A motor is fixed between disk1 and disk2 (in axis "y"), stator on disk1 and rotor on disk2 (idea of Floris, thanks). If a torque from rotor increase $\omega2$ this would say another torque from stator increase $\omega1$. The motor need energy for give a torque to disk2, I'm agree with that. But I don't understand why the motor need energy for apply its torque from stator to disk1. For me, like stator and rotor turn at the same rotational velocity around axis "x" it's not difficult to add torque to disk2 even the stator+rotor turn around "x" in the same time. The motor turn around the axis "x", when stator turn $d\theta$ degree in one direction, rotor turn $d\theta$ degree in the same time. So, it's not "difficult" (need energy) to apply torque on disk1 for me. If rotor moves far away from stator, in this case I'm agree, I will be more difficult to accelerate it, but here stator and rotor never move from each other.
If I need to give forces that stator gives to disk1, I'm agree I need energy, but these forces are only a reaction from the stator because rotor apply a torque. I'm not sure I explain very good what I think, I hope someone can clear this idea. I would like to understand why the motor need energy for apply its torque to disk1.
------------------------Added from comment of Floris:
I'm agree, if I'm on object that move back and I try to move an object in front of me, I need more energy. I drawn disks and 2 points A and B:
Imagine, disk2 don't turn around its center of gravity (axis y). Motor and disk2 turn only around axis "x". You could look at distance AB, it is always the same even the stator rotates of 45°, because the rotor rotates of 45° too. It's not like stator move in one direction and rotor in other.
Now, the rotor turn around axis "y" but why it's not the same with a linear trajectory ? Now, distance AB change but only because stator "attack" rotor, in the same time stator rotates around axis "x" but rotor rotates around axis "x" too.
Maybe my confusion come from I dissociate 2 movements: first the rotation around axis "x" and second the rotation around axis "y". For me, inertia and centrifugal forces rotate motor and disk2 around axis "x" and torque from motor add rotational velocity of disk2 around axis "y", I don't see where the stator move "back" like your example, could you explain more please ? Maybe I don't have the right to think like that. Sorry if my level of physics is low.
I added forces, maybe like that it's easier to explain the problem:
REPLY [0 votes]: To clear up your confusion about needing energy to add to the rotation of disk 1, let's take N easier linear problem for a second.
Imagine you are standing with your back against a solid wall, and you push for time $t$ with a force $F$ against an initially stationary mass $m$. You imparted momentum $Ft$ and the work you did was $(Ft)^2:(2m)$.
Imagine doing the same experiment in orbit. Now you would need to push against two masses - one in one direction and one in the opposite direction. In effect you are doing twice as much work - the two masses are separating twice as fast so you apply your force over twice as much distance and do twice the work. When you let go you have two masses with the same kinetic energy, so all is conserved.
The same is true in rotation - for force, read torque; for velocity read angular velocity, etc. it is harder to visualize but the underlying physics is the same: if just push off against something that is moving you have to do more work than of the something was fixed. The degree to which you do more work depends on the relative mass: the lighter your "thing you push against" the more work you have to do. A reason why the baseball pitcher firmly plants his feet before throwing ...?
| 117,814
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My parents never allowed us to own dogs when we were younger. I’ve always wanted one, but evidently my brother and mom “being allergic” gets in the way of my happiness. Either way, I’ve done plenty of research in preparation for when I get my own dog love and nurture. I found through my research that there are 5 different breeds of dogs that are lamest, most pathetic excuses for canines to ever walk the earth.
5. Squirrel Terriers
Squirrel Terriers are those little assholes that run out in front of your car when you’re driving literally anywhere in the world. It’s almost like a sixth sense. They hear something, see something, or smell something that frightens them, and then they jump out into oncoming traffic like a Russian teenager trying to impress his friends.
These little dogs are more rat than dog and litter college campuses nationwide. They can be seen with or without tails, although the ones without probably need to be put down because they survived some kind of accident that surely would have strengthened their gene pool.
The only kind of people that own squirrel terriers are rednecks and animal rights activists, one because it’s funny and the other because it’s not. If you’re ever walking by a squirrel terrier and get an urge to try and pet it, don’t. Chances are that this little bitch will stop what it is doing, which is probably digging in the dirt or climbing a tree, and run away. Avoid at all costs if you don’t want to be embarrassed by these pathetic creatures.
4. Turquoise Guppy
These dogs may look like fish, but don’t be fooled. The “G” in “guppy” actually makes the “P” sound, so everything checks out.
Have you ever had an uncle or a cousin or maybe even a foreign exchange student that didn’t ever do anything? Like, nothing. This is exactly how a turquoise guppy acts. They just swim around all day in their stupid little bowls and pretend like they don’t recognize you when you say something to them.
I once went to a friend’s pool party. There was this one guy that was lying face down in the water and wouldn’t get up. Like it was some kind of joke. Everyone got really tired of him after a couple hours because it wasn’t funny anymore. We just left him there overnight and he never came back to school. That guy was a piece of shit and so is the turquoise guppy.
3. Robopet
I’m very similar to Rick Santorum in that we both agree that if there’s one thing that we hate in this world, it’s things that we can’t control. Robopet is the antithesis of control. What’s the point of owning a slave if it won’t do what you tell it?
Robopet’s breeding or programming or whatever must have gone haywire when it was created because every time I tell it to lick the peanut butter, it just sits there like it doesn’t know what to do. Homie, you know exactly what I want you to do. Stop playing around like your programming isn’t sophisticated enough to understand my commands.
What pisses me off is that people try to say that Robopet is state-of-the-line and “the future of household robot dogs.” This is horse hockey because it still won’t eat peanut butter no matter how much I put on there.
2. Cats (Scientifically known as Catnines)
Remember when I asked about whether you had a really lazy uncle or cousin? Cats are like that, only 100 times worse. Dogs are supposed to be known for their bravery and their loyalty to their human masters. Cat breeds are the Danny Zuko’s of the dog breed world, because they’re only interested in themselves. What about me, cat? What about me?
To go along with this, cats are more reluctant than any other animal to have human assistance. A cat could be trapped in a burning building and I would come to rescue it and it would still scratch at me like I was some kind of monster.
People always say that cats love you but they don’t like to show affection for you. Well, I had plenty of girlfriends in Jr. High to know what that means. Best bet: scrap your cat breed and go for a nobler breed of dog, like a golden retriever or a German shepherd.
- Bruno Mars
Don’t be fooled by this crooning Hawaiian-looking guy: this breed of dog is the sappiest, crappiest kind of canine that you can own. Sometimes dogs bark all of the time and you just want them to shut up. This breed of dog sings all of the time and he definitely won’t shut up.
If you’re looking to listen to really corny music, Bruno Mars is right up your alley. He sings about the same thing every time: love and losing love. Sure, it worked for the first album, talking about how you can never get the girl that you want. But you’re famous now, Bruno! What kinds of girls are you going after that you can’t seem to attract? This is not only a sign of a lame-duck dog, but it’s also a sign of infertility in a dog breed. Avoid at all costs!
If you don’t believe me that Bruno Mars is a canine, then shut your mouth because I have proof. One of Bruno’s biggest hits is, “Locked Out of Heaven.” I once had a religious friend tell me that dogs didn’t go to Heaven because they don’t have souls. Bruno is singing about being locked out of Heaven. Checkmate, atheists.
One thought on “The 5 Lamest Dog Breeds”
Last one is classic! I do like some of his songs, but the others, oh god. It’s just a sappy wail fest each time I have the misfortune of listening.
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It’s time for another round of Books I Should Have Read! This time, I’ll highlight some nonfiction books that I’ve had on my TBR pile but not (yet) gotten around to reading.
These three all came out over the past year and are all works of nonfiction.
1. Paris to the Pyrenees by David Downee
Description:
Part adventure story, part cultural history the author of Paris, Paris: Journey into the City of Light explores the phenomenon of pilgrimage along the age-old Way of Saint James in France..
My 2 cents:
This book seemed like it would be perfect for me–France, hiking, etc.–and I actually started it but just couldn’t get into it, The narrator put me off a bit and the beginning that I read just wasn’t compelling enough to make me keep reading. However, I’d be interested to hear from someone else who read it in its entirety to hear what you thought.
2. My Life in Middlemarch by Rebecca Mead
Description:.
My 2 cents:
I started this book and, while interesting, it’s pretty slow going. The author has a very literary style (lots of big words!) which make it a book to be savored rather than rushed, but I’m not sure that I have the patience to finish it right now. What I have read of it is good, though, and I definitely think fans of Middlemarch will appreciate this behind-the-scenes look at Eliot’s novel as well as Mead’s personal response to it.
3. Paris Letters by Janice MacLeod
Description:.
My 2 cents:
I can’t say much about this one because I haven’t even cracked the (figurative) cover yet. I think it’s one of those books that I just need to be in the mood to read. I have a love/hate relationship with expat memoirs because while they sometimes have me nodding my head in agreement, they more often cause me to want to bang my head against the wall at the “romanticizing” of the expat experience. Hopefully I’ll find the time to read this one in the near future.
Paris to the Pyrenees sounds interesting. I’m sorry to hear that you couldn’t get into it, though.
Paris to the Pyrenees sounds very good actually, and so does Life in Middlemarch. Sad they don’t live up to the premise though.
| 20,374
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Can’t Beat This Deal!
We’d thought we’d like to offer you something really groovy in these tough times and Tuesday is the day!
$10 Tuesdays!
- Valid up to 18 holes
- Cart not included
- Must mention ad when booking
You’re free to walk 9 or 18 holes for the price of $10 but please note, a cart is recommended for the back nine.
Did you know we now offer single seat pricing for your golf cart rentals.
- $12.50 for 9 holes
- $17.50 for 18 holes
Please call us at 403-342-2830 to book your tee time.
| 88,943
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About Me...
(--update : go see--) My name is Norm Tumlinson and I am the student pastor at Grace Church in Orlando, FL. I have a wonderful wife and a pretty crazy wiener dog named Pancho Villa. We spend alot of time out in the sun, we love the beach, we enjoy grilling out and spending time with friends and family.
We are both transplants to the state of Florida. But after almost 5 years it seems like home. We love being in our new house and are looking forward to what God has in store for us. But back to the blog... This blog is really a heterogeneous mixture of life, faith, culture, student ministry, bad food, funny videos and pretty much everything else inbetween.
I know Guy Kawasaki says to "niche thyself", but my intent of this blog is not to evangelize my new book or product, or to make me look cooler than I (already) am, it's just written for entertainment and amusement, with the occasional shot of wisdom from the road in ministry. I hope you enjoy the posts within. And please feel free to comment. Peace!
| 6,366
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Great ideas!
by Eugene Brennan 2 years ago
Is there any way of doing this, similar to the way tweets can be retweeted?
by Tessa Schlesinger 8 months ago
I truly don't get it. It's a whole lot of pretty pictures. How does posting our articles to pinterest get them read?
by Susana Smith 4 weeks Melanie Shebel 5 years ago like there are no hubs on Pinterest (I noticed no Pinterest traffic today and had to investigate.)Hubs are no longer pinnable.I wish hubbers wouldn't spam social networks. :\
by Susannah Birch
| 345,187
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Multipurpose Business Flyer Template provided in 11 x 17″ size four panels roll fold brochure type, with simple design and layout and focussing on text content, this layout is suited for your marketing campaign purposes.
Image and logo placeholders are using smart object layer, so you can easily change it with your own images. The format for the design is Photoshop PSD in CMYK format with 0.125″ bleed.
Only logged in customers who have purchased this product may leave a review.
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| 114,537
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\begin{definition}[Definition:Abel Summation Method]
{{Help|It is difficult finding a concise and complete definition of exactly what the Abel Summation Method actually is. All and any advice as to how to implement this adequately is requested of anyone. This is what is said in the Spring encyclopedia on the page "Abel summation method":}}
The [[Definition:Real Series|series]]:
:$\displaystyle \sum a_n$
can be summed by the Abel method ($A$-method) to the number $S$ if, for any [[Definition:Real Number|real]] $x$ such that $0 < x < 1$, the [[Definition:Real Series|series]]:
:$\displaystyle \sum_{k \mathop = 0}^\infty a_k x^k$
is [[Definition:Convergent Real Series|convergent]] and:
:$\displaystyle \lim_{x \mathop \to 1^-} \sum_{k \mathop =0}^\infty a_k x^k = S$
{{help|This is what we have on Wikipedia page {{WP|Divergent_series|Divergent series}}: }}
:$\displaystyle \map f x = \sum_{n \mathop = 0}^\infty a_n e^{-n x} = \sum_{n \mathop = 0}^\infty a_n z^n$
where $z = \map \exp {−x}$.
Then the [[Definition:Limit of Real Function|limit]] of $\map f x$ as $x$ approaches $0$ through [[Definition:Positive Real Number|positive reals]] is the [[Definition:Limit of Power Series|limit]] of the [[Definition:Power Series|power series]] for $\map f z$ as $z$ approaches $1$ from below through [[Definition:Positive Real Number|positive reals]].
The '''Abel sum''' $\map A s$ is defined as:
:$\displaystyle \map A s = \lim_{z \mathop \to 1^-} \sum_{n \mathop = 0}^\infty a_n z^n$
{{NamedforDef|Niels Henrik Abel|cat = Abel}}
\end{definition}
| 50,985
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TITLE: Is the set $A=\{n+\frac{1}{n}:n\in\mathbb{N}\}$ closed in $\mathbb{R}$?
QUESTION [2 upvotes]: Let $A=\{n+\frac{1}{n}:n\in\mathbb{N}\}$. Is the set $A$ closed in $\mathbb{R}$?
I think that if $n$ goes to infinity, every deleted neighborhoods of $n$ would contain the points of $A.$ So, infinity is the only limit point of $A$ and infinity is contained in the set $A.$ Hence, the set $A$ is closed? Is that right?
REPLY [0 votes]: Note: Let $a_n \in A$ be denoted as $n + \frac 1n$. Then $a_1 < a_2 < ... < a_n < a_{n+1} < ....$. This can be proven via induction as $\frac 1n \le 1$ so $a_n = n + \frac 1n \le n + 1 < (n+1) + \frac {1}{n+1}= a_{n+1}$.
Let $r \in \mathbb R$.
So there is a unique $a_n \in A$ so that $a_n \le r < a_{n+1}$.
If $r > a_n$ then there exist an $\epsilon > 0$ so that $a_n < r-\epsilon < r < r+ \epsilon < a_{n+1}$ so $(r-\epsilon, r + \epsilon)$ contain no points of $A$.
If $r = a_n$ then there exists an $\epsilon > 0$ so that $a_{n-1} < r-\epsilon < a_n < r + \epsilon < a_{n+1}$ so that $(r-\epsilon, r + \epsilon)$ contain no points of $A$ other than $r = a_n$.
So $r$ is no a limit point of $A$.
So $A$ has no limit points. So vacuously $A$ contains all its limit points. So $A$ is closed.
====
Note: $\infty \not \in \mathbb R$ and even if we were using the extended real numbers to allow $\infty$ then $\infty + \frac 1{\infty} \not \in \mathbb N$ and $\infty \not \in A$. For any finite $\epsilon > 0$ then $B_\epsilon(\infty) = \{x \in \mathbb R \cup \{\infty\}| |\infty - x| < \epsilon\} = \{\infty\}$. And $B_\epsilon(\infty) \cap A = \{\infty\}\cap A = \emptyset$. So $\infty$ is not a limit point of $A$.
| 17,772
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Notts/Derbyshire-based metal band Plague of Ares will be launching their debut EP at The New Inn in Ilkeston on Friday, November 2.
Bassist Scott Moody is from Ilkeston and the band wanted to launch the four track EP, called The Plague, locally as a thanks to fans and friends for their valued support.
The female-fronted outfit will then embark on a UK promotional tour. CDs will be available on the night at a special price.
Support comes from female-fronted rock band Angel of Solace with rock disco by Wolfman & It.
Admission to the gig is free and the first band will be onstage at 9pm.
| 25,309
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Shown here are our current and upcoming sound meditation events. Please learn more about the different programs and experiences we offer.
- This event has passed.
Mindfulness for You – 2nd stage – EXPANSION – in Glen Ellyn, IL
Wednesday, April 26 @ 7:30 pm – 9:00 pm
Event Navigation
- « Mindfulness for You – 1st stage – INTEGRATION – in Glen Ellyn, IL
- Guided meditation with gongs and singing bowls at Magura in Mount Prospect, IL »
Experience a trans-formative “Mindfulness for You” workshop series over a seven week period, designed for anyone who wants to create a new segment in their lives. EXPANSION is the second stage of our 3 phase program where Psychotherapy, Life Coaching and Gongs create a powerfully and unique combination for personal transformation.
Group sessions will be held on Wednesdays 4/19; 4/26; 5/3; 5/24; 5/31; 6/7 and 6/14 from 7:30 – 9:00 pm. The “Mindfulness for You” workshop will be held at DG Counseling’s Glen Ellyn, IL location. Please contact DG Counseling or us to find out if your health insurance plan covers all of the fee, register and save your spot.
Experience a personal reset through a specifically applied practice; heal your inner child; manifest your aspirations – free of interferences; learn how to recall and apply an ideal state of being; understand how to improve your relationships, and create your “Life Plan”. Please contact us and we will gladly provide you with more specific information.
| 73,908
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TITLE: Optimization Calculus Problem- Flight
QUESTION [0 upvotes]: If exactly 230 people sign up for a charter flight, the operators of a charter airline charge Dollars 330 for a round-trip ticket. However, if more than 230 people sign up for the flight, then fare is reduced by Dollar 1 for each additional person. Assuming that more than 230 people sign up, determine how many passengers will result in a maximum revenue for the travel agency.
I have this for homework and all I could come up with was that maybe i should be using some sort of revenue equation but I am not sure how to set it up.
R=Price* Number of things sold
I know that I need another equation to get here. What I had before was R=230? and then 330?
After this I know I need to derive to find the equation and then set it equal to zero.
REPLY [1 votes]: in general the revenue is $R=p\cdot x$. With 230 People: $R=230 \cdot 330$. Now you can raise the number of people by y (persons) and reduce the price by y (Dollars): $R=(230+y)\cdot (330-y)$
You can maximize the revenue by derivating $R(y)$ in respect to y and set it 0.
greetings,
calculus
| 203,719
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History
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Allergic reactions
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Poor nutrition
Lack of periodic dental examinations
- Healthy mouth and gingiva. Note the healthy light pink color of the gingiva. The interdental papillae are sharp and fill the interdental space. No local edema is present. Image courtesy of Robert J. Lindberg, DMD.
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| 75,193
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> No, that's not what I'm talking about at all. What I'm talking about > is having the X server tell the kernel what the frame buffer geometry > is, so that the kernel doesn't *have* to whack any frame buffer > registers. I don't think that's very practical, but I'd love to hear that I'm mistaken... ;-) If I understand you correctly, that would mean having a bitmapped console driver (basically) in all kernels that would be running X--ideally this would be the same console used for ports that only have bitmapped displays. Perhaps not a bad trade-off, anyway. But on the PC (or anyone using ISA display cards and maybe even some EISA/PCI adapters), this would also require some interface for manipulating the video pages, would it not? And that would depend on more information than could be kept reasonably in the console driver... On the PC video cards, is it possible to get back to text mode in a standard fashion? (At least for (S)VGA cards?) If not, I don't know if there is a good solution. If so, X has to tell the console that it's taking over, right? So it should be possible--at least for ddb--to get the display into text mode before it starts writing to the console. -allen -- Allen Briggs - end killing - briggs@macbsd.com
| 300,778
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! Shabook 21:22, May.
| 8,255
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WWE Spoilers: Update on CM Punk’s “Indefinite Suspension”By Cassidy| June 28, 2011 Wrestling News WWE.com has posted an update to the “indefinite suspension” of CM Punk, stemming from last night’s worked shoot at the end of RAW. The site posted the following: UPDATE: As of Tuesday, June 28, WWE and CM Punk have reached an agreement that Punk will fulfill his non-televised live event obligations for the remainder of his contract, through July 17. Furthermore, both sides have mutually agreed not to disparage one another. Basically what this means is that CM Punk will only appear and participate at WWE Live Event shows for the remainder of his contract and will not be appearing on any WWE television shows (RAW, SmackDown, or the Money-in-the-Bank pay-per-view). What do you think of this update to CM Punk’s “suspension”? Post your comments in the box below.
| 243,483
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\begin{document}
\title{Asymptotic properties of MUSIC-type imaging in two-dimensional inverse scattering from thin electromagnetic inclusions}
\author{Won-Kwang Park\thanks{e-mail: parkwk@kookmin.ac.kr}}
\affil{Department of Mathematics, Kookmin University, Seoul, 136-702, Korea.}
\date{}
\maketitle
\begin{abstract}
The main purpose of this paper is to study the structure of the well-known non-iterative MUltiple SIgnal Classification (MUSIC) algorithm for identifying the shape of extended electromagnetic inclusions of small thickness located in a two-dimensional homogeneous space. We construct a relationship between the MUSIC-type imaging functional for thin inclusions and the Bessel function of integer order of the first kind. Our construction is based on the structure of the left singular vectors of the collected multistatic response matrix whose elements are the measured far-field pattern and the asymptotic expansion formula in the presence of thin inclusions. Some numerical examples are shown to support the constructed MUSIC structure.
\end{abstract}
\section{Introduction}
One of the goals of the inverse scattering problem is to identify unknown properties (e.g., the shapes, material properties, locations, and constitutions) of electromagnetic targets from measured scattered field data. This problem is generally solved by Newton-type iteration schemes or level-set method involving minimization of the difference between the measured scattered data and the computed data by generating an admissible cost functional. Related works can be found in \cite{AGJKLY,BHL,B,CR,DL,GH2,K,M2,PL4,VXB,SZ} and references therein. To execute these schemes, the Fr{\'e}chet derivative must be evaluated at each iteration step, so the computational costs are large. Unfortunately, non-convergence or the appearance of several minima arises in the iteration procedure owing to the non-convex nature of the cost functional. Furthermore, \textit{a priori} information on unknown targets is essential to guarantee a successful reconstruction, and even when the above conditions are fulfilled, reconstruction will fail if the iteration process is begun with a bad initial guess. Hence, generating a good initial guess close to the expected conditions is a priority. For this purpose, alternative non-iterative imaging algorithms such as the linear sampling method, single- and multi-frequency based Kirchhoff and subspace migrations, and the topological derivative strategy have been developed.
In pioneering research \cite{D}, the MUltiple SIgnal Classification (MUSIC) algorithm has been investigated to find the locations of point-like scatterers. It was recently applied to various problems, for example, detection of antipersonnel mines buried in the ground \cite{AIL}, searching for the locations of small inclusions \cite{AILP,CZ,GH1,HM,SCC,ZC}, identifying internal corrosion in a pipeline \cite{AKKLV}, and shape reconstruction of arbitrarily shaped thin inclusions, cracks, and extended targets \cite{AGKPS,AKLP,HSZ,JP,PL1,PL3}. On the basis of these results, the locations of small inclusions can be accurately identified using MUSIC, but owing to the intrinsic resolution limit, the complete shape of extended targets cannot be imaged. Hence, the obtained results were based on good initial guesses, and iteration-based algorithms such as the level-set method were successfully executed (see
\cite{AGJKLY,BHL,PL4}).
Although the MUSIC algorithm offers good results for small and extended targets, a detailed structural analysis must be attempted because some phenomena cannot be explained using the traditional approach, for example, the unexpected appearance of artifacts or of two curves along the boundary of targets instead of the true shape (see \cite[Figure 9(b)]{HSZ}), or an image with poor resolution (see \cite[Section 4.4]{PL3}). Numerical results in existing works motivate us to explore some properties of the MUSIC-type algorithm for imaging the arbitrarily shaped thin penetrable electromagnetic inclusions and perfectly conducting cracks considered in \cite{PL1,PL3}. Our exploration is based on the rigorously derived asymptotic expansion formula in the presence of a thin inclusion \cite{BF} and physical factorization of the so-called multistatic response (MSR) matrix \cite{HSZ}. With this, we will establish a relationship between the MUSIC-type imaging functional and the Bessel function of integer order of the first kind, and identify its properties.
This paper is organized as follows. In section \ref{sec:2}, we introduce two-dimensional direct scattering problems, the asymptotic expansion formula in the presence of thin inclusions, and the MUSIC-type algorithm for imaging thin electromagnetic inclusions. In section \ref{sec:3}, we identify the structure of a MUSIC-type functional focused on imaging of thin penetrable electromagnetic inclusions by constructing a relationship with the Bessel function of integer order of the first kind, and discuss its properties. In section \ref{sec:4}, numerical simulation results are presented to support the identified structure. This paper ends with a short conclusion in section \ref{sec:5}.
\section{Direct scattering problems and MUSIC algorithm}\label{sec:2}
\subsection{Two-dimensional direct scattering problems and asymptotic expansion formula}
Suppose that an extended electromagnetic inclusion $\Gamma$ with a small (with respect to the given wavelength) thickness $2h$ is located in the two-dimensional homogeneous space $\mathbb{R}^2$. We assume that the shape of $\Gamma$ is characterized by the supporting smooth curve $\gamma$ such that (see FIG. \ref{ThinInclusion})
\[\Gamma=\{\mx+\eta\mn(\mx):\mx\in\gamma,~\eta\in(-h,h)\}.\]
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.35\textwidth]{ThinInclusion.eps}
\caption{\label{ThinInclusion}Sketch of two-dimensional thin electromagnetic inclusion $\Gamma$.}
\end{center}
\end{figure}
Throughout this paper, we assume that $\Gamma$ and $\mathbb{R}^2$ are classified by their dielectric permittivity and magnetic permeability at a given frequency $\omega=2\pi/\lambda$, where $\lambda$ denotes the wavelength. Let $0<\eps_0<+\infty$ and $0<\mu_0<+\infty$ denote the permittivity and permeability of $\mathbb{R}^2$, respectively; analogously, $0<\eps<+\infty$ and $0<\mu<+\infty$ represent those of $\Gamma$. Then, we can define the piecewise constant dielectric permittivity $\varepsilon(\mx)$ and magnetic permeability $\mu(\mx)$,
\begin{equation}\label{EPSMU}
\varepsilon(\mx)=\left\{\begin{array}{ccl}
\varepsilon_0&\mbox{for}&\mx\in\mathbb{R}^2\backslash\overline{\Gamma}\\
\varepsilon&\mbox{for}&\mx\in\Gamma
\end{array}\right.
\quad\mbox{and}\quad
\mu(\mx)=\left\{\begin{array}{ccl}
\mu_0&\mbox{for}&\mx\in\mathbb{R}^2\backslash\overline{\Gamma}\\
\mu&\mbox{for}&\mx\in\Gamma,
\end{array}\right.
\end{equation}
respectively. For convenience, we set $\eps_0=\mu_0=1$, $\eps>\eps_0$, and $\mu>\mu_0$.
For a given fixed frequency $\omega$ (we assume that the wave number $k=\omega\sqrt{\eps_0\mu_0}=\omega$), we let
\begin{equation}\label{IncidentField}
u_0(\mx,\vt;\omega):=e^{i\omega\vt\cdot\mx},\quad\mx\in\mathbb{R}^2
\end{equation}
be a plane-wave incident field with the incident direction $\vt\in\mathbb{S}^1$, where $\mathbb{S}^1$ denotes the unit circle. Let $u(\mx,\vt;\omega)=u_0(\mx,\vt;\omega)+u_s(\mx,\vt;\omega)$ denote the time-harmonic total field that satisfies the following Helmholtz equation:
\begin{equation}\label{TotalField}
\nabla\cdot\bigg(\frac{1}{\mu(\mx)}\nabla u(\mx,\vt;\omega)\bigg) +\omega^2\eps(\mx)u(\mx,\vt;\omega)=0,
\end{equation}
with transmission conditions on the boundary of $\Gamma$. Here, $u_s(\mx,\vt;\omega)$ denotes the unknown scattered field that satisfies the Sommerfeld radiation condition
\[\lim_{r\to0}\sqrt{r}\bigg(\frac{\p u_s(\mx,\vt;\omega)}{\p r}-i\omega u_s(\mx,\vt;\omega)\bigg)=0,\quad\vv=\frac{\mx}{|\mx|}\in\mathbb{S}^1\]
uniformly in all directions $\vv$. Notice that the above radiation condition implies the asymptotic behavior
\[u_s(\mx,\vt;\omega)=\frac{e^{i\omega|\mx|}}{\sqrt{|\mx|}}u_{\infty}(\vv,\vt;\omega)+o\left(\frac{1}{\sqrt{|\mx|}}\right)\quad\mbox{for all}\quad|\mx|\longrightarrow+\infty.\]
Then, according to \cite{BF}, the far-field pattern $u_{\infty}(\vv,\vt;\omega)$ can be written using the following asymptotic expansion formula:
\[u_{\infty}(\vv,\vt;\omega)=h\frac{\omega^2(1+i)}{4\sqrt{\omega\pi}} \int_\gamma\bigg((-\vv)\cdot\mathbb{M}(\mx)\cdot\vt+(\eps-1)\bigg)e^{i\omega(\vt-\vv)\cdot\mx}d\gamma(\mx)+o(h).\]
Here, the $2\times2$ symmetric matrix $\mathbb{M}(\mx)$ is defined as follows: for $\mx\in\gamma$, let $\mt(\mx)$ and $\mn(\mx)$ denote the unit tangent and normal vectors to $\gamma$ at $\mx$, respectively. Then
\begin{itemize}
\item $\mathbb{M}(\mx)$ has eigenvectors $\mt(\mx)$ and $\mn(\mx)$.
\item The eigenvalue corresponding to $\mt(\mx)$ is $2\left(\frac{1}{\mu}-1\right)$.
\item The eigenvalue corresponding to $\mn(\mx)$ is $2\left(1-\mu\right)$.
\end{itemize}
\subsection{MUSIC-type imaging algorithm}
Next, we introduce the MUSIC algorithm for imaging $\Gamma$. For simplicity, suppose that we have $N$ incident and observation directions $\vt_l$ and $\vv_j$, respectively, for $j,l=1,2,\cdots,N$, and the incident and observation directions are the same, i.e., $\vv_j=-\vt_j$. In this paper, we consider the full-view inverse problem. Hence, we assume that $\set{\vt_n:n=1,2,\cdots,N}$ spans unit circle $\mathbb{S}^1$. Moreover, we assume that the supporting curve $\gamma$ is divided into $M$ different segments with sizes on the order of half the wavelength, $\lambda/2$. Then, keeping in mind the Rayleigh resolution limit from the far-field data, any detail less than one-half of the wavelength in size cannot be seen, and only one point at each segment is expected to contribute to the image space of the response matrix $\mathbb{K}$ (see \cite{AKLP,HSZ,PL1,PL3}, for instance). Each of these points, say $\mx_j$ for $j=1,2,\cdots,M$, can be imaged by MUSIC-type imaging. With this in mind, we consider the collected MSR matrix such that
\begin{equation}\label{MSR}
\mathbb{K}=\left[K_{jl}\right]_{j,l=1}^{N}=\left[\begin{array}{cccc}
u_{\infty}(\vv_1,\vt_1;\omega) & u_{\infty}(\vv_1,\vt_2;\omega) & \cdots & u_{\infty}(\vv_1,\vt_N;\omega)\\
u_{\infty}(\vv_2,\vt_1;\omega) & u_{\infty}(\vv_2,\vt_2;\omega) & \cdots & u_{\infty}(\vv_2,\vt_N;\omega)\\
\vdots&\vdots&\ddots&\vdots\\
u_{\infty}(\vv_N,\vt_1;\omega) & u_{\infty}(\vv_N,\vt_2;\omega) & \cdots & u_{\infty}(\vv_N,\vt_N;\omega)\\
\end{array}\right].
\end{equation}
Because the incident and observation directions are the same, $K_{jl}$ becomes
\begin{align}
\begin{aligned}\label{StructureofMSRmatrix}
K_{jl}=&u_{\infty}(-\vt_j,\vt_l;\omega)\\
&\approx h\frac{\omega^2(1+i)}{4\sqrt{\omega\pi}} \int_\gamma\bigg(\vt_j\cdot\mathbb{M}(\mx)\cdot\vt_l+(\eps-1)\bigg)e^{i\omega(\vt_j+\vt_l)\cdot\mx}d\gamma(\mx)\\
=&h\frac{\omega^2(1+i)}{4\sqrt{\omega\pi}}\frac{\mbox{length}(\gamma)}{M} \sum_{m=1}^{M}\bigg[2\left(\frac{1}{\mu}-1\right)\vt_j\cdot\mt(\mx_m)\vt_l\cdot\mt(\mx_m)\\
&+2(1-\mu)\vt_j\cdot\mn(\mx_m)\vt_l\cdot\mn(\mx_m)+(\eps-1)\bigg]e^{i\omega(\vt_j+\vt_l)\cdot\mx}d\gamma(\mx),
\end{aligned}
\end{align}
where $\mbox{length}(\gamma)$ denotes the length of $\gamma$ (refer to \cite{PL3}).
With this background, the MUSIC algorithm can be introduced as follows. Let us perform the singular value decomposition of $\mathbb{K}$:
\[\mathbb{K}=\mathbb{USV}^*\approx\sum_{m=1}^M\sigma_m\mU_m\mV_m^*,\]
where $\mU_m$ and $\mV_m$ are the left and right singular vectors of $\mathbb{K}$, respectively, and $\sigma_m$ denotes the non-zero singular values. Then, on the basis of the structure of the MSR matrix (\ref{StructureofMSRmatrix}), we define a vector $\mf(\mz)\in\mathbb{C}^{N\times1}$ as
\begin{equation}\label{Vecf}
\mf(\mz)=\bigg[\mathbf{c}_1\cdot[1,\vt_1]^Te^{i\omega\vt_1\cdot\mz}, \mathbf{c}_2\cdot[1,\vt_2]^Te^{i\omega\vt_2\cdot\mz},\cdots, \mathbf{c}_N\cdot[1,\vt_N]^Te^{i\omega\vt_N\cdot\mz}\bigg]^T,
\end{equation}
where the selection of $\mathbf{c}_n\in\mathbb{R}^3\backslash\set{\mathbf{0}}$, $n=1,2,\cdots,N$, depends on the shape of the supporting curve $\gamma(\mx)$. In fact, this is a linear combination of $\mt(\mx_m)$ and $\mn(\mx_m)$.
Let us define a projection operator onto the noise subspace:
\[P_{\mathrm{noise}}(\mf(\mz)):=\left(\mathbb{I}_{N}-\sum_{m=1}^{M}\mU_m\mU_m^*\right)\mf(\mz).\]
Then, the MUSIC-type imaging functional can be introduced:
\begin{equation}\label{MUSICfunction}
\mathbb{E}(\mz):=\frac{1}{|P_{\mathrm{noise}}(\mf(\mz))|}.
\end{equation}
Note that $\mathbb{K}$ is symmetric, but it is not self-adjoint. Therefore, we form a self-adjoint matrix
\[\mathbb{A}=\mathbb{K^*K}=\mathbb{\overline{K}K},\]
where $*$ denotes the adjoint, and the bar denotes the complex conjugate. Then, with a careful choice of $\mathbf{c}_n$, the range of $\mathbb{A}$ is spanned by the $M$ vectors $\set{\mf(\mx_1),\mf(\mx_2),\cdots,\mf(\mx_M)}$ (see \cite{AK,C} for instance). Therefore,
\[\mf(\mz)\in\mbox{Range}(\mathbb{\overline{K}K})\quad\mbox{if and only if}\quad\mz\in\set{\mx_1,\mx_2,\cdots,\mx_M};\]
i.e., equivalently $|P_{\mathrm{noise}}(\mf(\mz))|=0$. Thus, the map of (\ref{MUSICfunction}) will show a large magnitude (theoretically, $+\infty$) at $\mx_m\in\gamma$.
\begin{rem}
The results of the numerical simulations in \cite{P1,PL1,PL3} indicate that the selection of $\mathbf{c}_n$ is a strong prerequisite. The selection depends on the shape of the supporting curve $\gamma$. For purely dielectric contrast, $\mathbf{c}_n=[1,0,0]^T$ is a good choice. However, for purely magnetic contrast, $\mathbf{c}_n$ must be in the form $\mathbf{c}_n=[0,\mathbf{b}]^T$, where $\mathbf{b}$ is a linear combination of $\mt(\mx_m)$ and $\mn(\mx_m)$ for $m=1,2,\cdots,M$. Unfortunately, we have no \textit{a priori} information on the shape of $\gamma$. Therefore, in \cite{HSZ,PL1}, a large number of directions are applied to find an optimal vector $\mathbf{b}$. Applying this $\mathbf{b}$ yields a good result, but this process incurs large computational costs. Hence, motivated by recent work \cite{HSZ}, we assume that $\mathbf{c}_n$ satisfies $\mathbf{c}_n\cdot[1,\vt_n]^T=1$ for all $n$, i.e.,
\begin{equation}\label{VecF}
\mf(\mz)=\bigg[e^{i\omega\vt_1\cdot\mz},e^{i\omega\vt_2\cdot\mz},\cdots,e^{i\omega\vt_N\cdot\mz}\bigg]^T,
\end{equation}
and explore some properties of the MUSIC-type imaging algorithm.
\end{rem}
\section{Structure of certain properties of MUSIC-type imaging function}\label{sec:3}
In this section, we identify the structure of the MUSIC-type imaging function. Before starting, we recall a useful result derived in \cite{G}. This plays a key role in our identification of the structure.
\begin{lem}\label{TheoremBessel}
Assume that $\set{\vt_n:n=1,2,\cdots,N}$ spans unit circle $\mathbb{S}^1$. Then, the following identities hold for sufficiently large $N$ and $\vx,\mx\in\mathbb{R}^2$.
\begin{align*}
&\frac{1}{N}\sum_{n=1}^{N}e^{i\omega\vt_n\cdot\mx}=\frac{1}{2\pi}\int_{\mathbb{S}^1}e^{i\omega\vt\cdot\mx}dS(\vt)=J_0(\omega|\mx|),\\
&\frac{1}{N}\sum_{n=1}^{N}(\vt_n\cdot\vx)e^{i\omega\vt_n\cdot\mx}=\frac{1}{2\pi}\int_{\mathbb{S}^1}(\vt\cdot\vx)e^{i\omega\vt\cdot\mx}dS(\vt)=i\left(\frac{\mx}{|\mx|}\cdot\vx\right)J_1(\omega|\mx|),
\end{align*}
where $J_p$ denotes the Bessel function of integer order $p$ of the first kind.
\end{lem}
\subsection{Pure dielectric permittivity contrast case: $\eps\ne\eps_0$ and $\mu=\mu_0$}
First, we consider the dielectric permittivity contrast case; i.e., we assume that $\eps\ne\eps_0$ and $\mu=\mu_0$. The proof is similar to the result in \cite[Theorem 3.3]{JP}.
\begin{thm}[Pure dielectric permittivity contrast case]\label{Theorem1}
For sufficiently large $N$ ($>M$) and $\omega$, (\ref{MUSICfunction}) can be written as follows:
\begin{equation}\label{MUSIC1}
\mathbb{E}_{\eps}(\mz):=\frac{1}{|P_{\mathrm{ noise}}(\mf(\mz))|}\approx\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2+o(h)\right)^{-1/2}.
\end{equation}
\end{thm}
\begin{proof}
In this case, if we let $\mu=\mu_0$ in (\ref{StructureofMSRmatrix}), the left singular vectors are of the form
\[\mU_m\approx\frac{1}{\sqrt{N}}\bigg[e^{i\omega\vt_1\cdot \mx_m},e^{i\omega\vt_2\cdot \mx_m},\cdots,e^{i\omega\vt_N\cdot \mx_m}\bigg]^T+\mO(h),\]
where $\mO(h)$ denotes a $N\times1$ vector whose elements are $o(h)$. Then, for sufficiently large $\omega$, we can observe that
\begin{equation}\label{orthogonal1}
\mU_m\cdot\overline{\mU}_{m'}\approx\frac{1}{N}\sum_{n=1}^{N}e^{i\omega\vt_n\cdot(\mx_m-\mx_{m'})}\approx J_0(\omega|\mx_m-\mx_{m'}|)+o(h)\approx\left\{\begin{array}{ccl}
1&\mbox{if}&m=m'\\
\noalign{\medskip}0&\mbox{if}&m\ne m'.
\end{array}\right.
\end{equation}
Following elementary calculus, $P_{\mathrm{noise}}$ can be written as
\begin{align*}
P_{\mathrm{noise}}(\mf(\mz))&=\left(\mathbb{I}_{N}-\sum_{m=1}^{M}\mU_m\overline{\mU}_m^T\right)\mf(\mz)\\
&=\left[\begin{array}{c}
e^{i\omega\vt_1\cdot\mz} \\
e^{i\omega\vt_2\cdot\mz} \\
\vdots \\
e^{i\omega\vt_N\cdot\mz} \\
\end{array}\right]
-\frac{1}{N}\sum_{m=1}^{M}\left[\begin{array}{c}
\displaystyle e^{i\omega\vt_1\cdot\mz}+\sum_{n\in\mathbb{N}_1}e^{i\omega\vt_1\cdot \mx_m}e^{i\omega\vt_n\cdot(\mz-\mx_m)}+o(h)\\
\displaystyle e^{i\omega\vt_2\cdot\mz}+\sum_{n\in\mathbb{N}_2}e^{i\omega\vt_1\cdot \mx_m}e^{i\omega\vt_n\cdot(\mz-\mx_m)}+o(h)\\
\vdots \\
\displaystyle e^{i\omega\vt_N\cdot\mz}+\sum_{n\in\mathbb{N}_N}e^{i\omega\vt_1\cdot \mx_m}e^{i\omega\vt_n\cdot(\mz-\mx_m)}+o(h)
\end{array}\right],
\end{align*}
where $\mathbb{N}_n=\{1,2,\cdots,N\}\backslash\{n\}$. Because
\[e^{i\omega\vt_n\cdot\mz}=e^{i\omega\vt_n\cdot \mx_m}e^{i\omega\vt_n\cdot (\mz-\mx_m)},\] $P_{\mathrm{noise}}$ can be expressed as
\[P_{\mathrm{noise}}(\mf(\mz))=\left[
\begin{array}{c}
\displaystyle e^{i\omega\vt_1\cdot\mz}-\sum_{m=1}^{M}e^{i\omega\vt_1\cdot \mx_m}J_0(\omega|\mz-\mx_m|)+o(h) \\
\displaystyle e^{i\omega\vt_2\cdot\mz}-\sum_{m=1}^{M}e^{i\omega\vt_2\cdot \mx_m}J_0(\omega|\mz-\mx_m|)+o(h) \\
\vdots \\
\displaystyle e^{i\omega\vt_N\cdot\mz}-\sum_{m=1}^{M}e^{i\omega\vt_N\cdot \mx_m}J_0(\omega|\mz-\mx_m|)+o(h) \\
\end{array}
\right].\]
Therefore, we can obtain
\[|P_{\mathrm{noise}}(\mf(\mz))|=\bigg(P_{\mathrm{noise}}(\mf(\mz))\cdot\overline{P_{\mathrm{noise}}(\mf(\mz))}\bigg)^{1/2} =\left(\sum_{n=1}^{N}\bigg(1-\Phi_1+\Phi_2+o(h)\bigg)\right)^{1/2},\]
where
\begin{align*}
\Phi_1&=\sum_{m=1}^{M}\bigg(e^{i\omega\vt_n\cdot(\mz-\mx_m)}+e^{-i\omega\vt_n\cdot(\mz-\mx_m)}\bigg)J_0(\omega|\mz-\mx_m|)\\
\Phi_2&=\left(\sum_{m=1}^{M}e^{i\omega\vt_n\cdot\mx_m}J_0(\omega|\mz-\mx_m|)\right) \left(\sum_{m=1}^{M}e^{-i\omega\vt_n\cdot\mx_m}J_0(\omega|\mz-\mx_m|)\right).
\end{align*}
Because
\begin{align}
\begin{aligned}\label{term1}
\sum_{n=1}^{N}\Phi_1&=\sum_{n=1}^{N}\sum_{m=1}^{M}\bigg(e^{i\omega\vt_n\cdot(\mz-\mx_m)}+e^{-i\omega\vt_n\cdot(\mz-\mx_m)}\bigg)J_0(\omega|\mz-\mx_m|)\\ &=2N\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2,
\end{aligned}
\end{align}
and on the basis of the orthogonal property (\ref{orthogonal1}), we can evaluate
\begin{align}
\begin{aligned}\label{term2}
\sum_{n=1}^{N}\Phi_2&=\left(\sum_{m=1}^{M}e^{i\omega\vt_n\cdot\mx_m}J_0(\omega|\mz-\mx_m|)\right) \left(\sum_{m=1}^{M}e^{-i\omega\vt_n\cdot\mx_m}J_0(\omega|\mz-\mx_m|)\right)\\
&=\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{m'=1}^{M}\bigg(e^{-i\omega\vt_n\cdot\mx_m}J_0(\omega|\mz-\mx_m|)\bigg) \bigg(e^{i\omega\vt_n\cdot\mx_{m'}}J_0(\omega|\mz-\mx_{m'}|)\bigg)\\
&=\sum_{m=1}^{M}\sum_{m'=1}^{M}\sum_{n=1}^{N}\bigg(e^{i\omega\vt_n\cdot(\mx_{m'}-\mx_m)}J_0(\omega|\mz-\mx_m|)J_0(\omega|\mz-\mx_{m'}|)\bigg)\\
&=N\sum_{m=1}^{M}\sum_{m'=1}^{M}J_0(\omega|\mx_{m'}-\mx_m|)J_0(\omega|\mz-\mx_m|)J_0(\omega|\mz-\mx_{m'}|)\\
&=N\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2.
\end{aligned}
\end{align}
Therefore, using (\ref{term1}) and (\ref{term2}), we can obtain the following result:
\[|P_{\mathrm{noise}}(\mf(\mz))|=\sqrt{N}\left(1-\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2+o(h)\right)^{1/2}.\]
\end{proof}
Note that $J_0(x)$ has its maximum value $1$ at $x=0$. This means that plots of $\mathbb{E}_{\eps}(\mz)$ will show peaks of large ($+\infty$ in theory) and small magnitude at $\mx_m\in\gamma$ and at $\mx\notin\gamma$, respectively (see FIG. \ref{PlotMUSIC1}). This is why the MUSIC algorithm offers a good result for the pure dielectric contrast case of the full-view inverse scattering problem. We refer to FIG. \ref{MapMUSIC1} and various results in \cite{AKLP,HSZ,PL1,PL3}.
\subsection{Pure magnetic permeability contrast case: $\eps=\eps_0$ and $\mu\ne\mu_0$}
Next, we consider the magnetic permeability contrast case; i.e., we assume that $\eps=\eps_0$ and $\mu\ne\mu_0$. The result is as follows.
\begin{thm}[Pure magnetic permeability contrast case]\label{Theorem2}
For sufficiently large $N$ ($>2M$) and $\omega$, (\ref{MUSICfunction}) can be written as follows:
\begin{equation}\label{MUSIC2}
\mathbb{E}_{\mu}(\mz)=\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot(\mt(\mx_m)+\mn(\mx_m))\right)^2J_1(\omega|\mz-\mx_m|)^2+o(h)\right)^{-1/2}.
\end{equation}
\end{thm}
\begin{proof}
In this case, the left singular vectors are of the form (see \cite{AGKPS,PL3})
\[\mU_{2(m-1)+s}\approx\frac{1}{\sqrt{N}}\bigg[\vt_1\cdot\vx_s(\mx_m)e^{i\omega\vt_1\cdot\mx_m},
\cdots,\vt_N\cdot\vx_s(\mx_m)e^{i\omega\vt_N\cdot\mx_m}\bigg]^T+\mO(h),\]
where
\[\vx_s(\mx_m):=\left\{\begin{array}{rcl}
\mt(\mx_m) & \mbox{if} & s=1 \\
\noalign{\medskip}\mn(\mx_m) & \mbox{if} & s=2.
\end{array}
\right.\]
Then, based on the orthonormal property of singular vectors, we can observe the following: because $\omega$ is sufficiently large, if $m\ne m'$ or $s\ne s''$, then
\begin{multline}\label{orthogonal2}
\mU_{2(m-1)+s}\cdot\overline{\mU}_{2(m'-1)+s''}\approx\frac{1}{N} \sum_{n=1}^{N}\bigg(\vt_n\cdot(\vx_s(\mx_m)+\vx_{s''}(\mx_{m'})\bigg) e^{i\omega\vt_n\cdot(\mx_m-\mx_{m'})}\\
\approx i\frac{\mx_m-\mx_{m'}}{|\mx_m-\mx_{m'}|}\cdot\bigg(\vx_s(\mx_m)+\vx_{s''}(\mx_{m'})\bigg)J_1(\omega|\mx_m-\mx_{m'}|)+o(h)\approx0.
\end{multline}
Because we selected $\mf(\mz)$ as (\ref{VecF}), $P_{\mathrm{noise}}$ can be written as
\begin{align*}
&P_{\mathrm{noise}}(\mf(\mz)) =\left(\mathbb{I}_{N}-\sum_{m=1}^{M}\sum_{s=1}^{2}\mU_{2(m-1)+s}\overline{\mU}_{2(m-1)+s}^T\right)\mf(\mz)\\
&\approx\left[\begin{array}{c}
e^{i\omega\vt_1\cdot\mz} \\
e^{i\omega\vt_2\cdot\mz} \\
\vdots \\
e^{i\omega\vt_N\cdot\mz} \\
\end{array}\right]-
\frac{1}{N}\sum_{m=1}^{M}\sum_{s=1}^{2}\left[\begin{array}{c}
\displaystyle(\vt_1\cdot\vx_s(\mx_m))e^{i\omega\vt_1\cdot\mx_m}\sum_{n=1}^{N}(\vt_n\cdot\vx_s(\mx_m))e^{i\omega\vt_n\cdot(\mz-\mx_m)}+o(h)\\
\displaystyle(\vt_2\cdot\vx_s(\mx_m))e^{i\omega\vt_2\cdot\mx_m}\sum_{n=1}^{N}(\vt_n\cdot\vx_s(\mx_m))e^{i\omega\vt_n\cdot(\mz-\mx_m)}+o(h)\\
\vdots\\
\displaystyle(\vt_N\cdot\vx_s(\mx_m))e^{i\omega\vt_N\cdot\mx_m}\sum_{n=1}^{N}(\vt_n\cdot\vx_s(\mx_m))e^{i\omega\vt_n\cdot(\mz-\mx_m)}+o(h)
\end{array}\right]\\
&=\left[\begin{array}{c}
\displaystyle e^{i\omega\vt_1\cdot\mz}-i\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_1\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)e^{i\omega\vt_1\cdot \mx_m}J_1(\omega|\mz-\mx_m|)+o(h)\\
\displaystyle e^{i\omega\vt_2\cdot\mz}-i\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_2\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)e^{i\omega\vt_2\cdot \mx_m}J_1(\omega|\mz-\mx_m|)+o(h)\\
\vdots\\
\displaystyle e^{i\omega\vt_N\cdot\mz}-i\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_N\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)e^{i\omega\vt_N\cdot \mx_m}J_1(\omega|\mz-\mx_m|)+o(h)
\end{array}\right].
\end{align*}
Hence,
\[|P_{\mathrm{noise}}(\mf(\mz))|=\left(\sum_{n=1}^{N}\bigg(1+\Psi_1-\overline{\Psi}_1+\Psi_2\overline{\Psi}_2+o(h)\bigg)\right)^{1/2},\]
where
\begin{align*}
\Psi_1&=i\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_n\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right) e^{i\omega\vt_n\cdot(\mz-\mx_m)}J_1(\omega|\mz-\mx_m|)\\
\Psi_2&=\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_n\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)e^{i\omega\vt_n\cdot \mx_m}J_1(\omega|\mz-\mx_m|).
\end{align*}
Because
\begin{align*}
&\sum_{n=1}^{N}(\Psi_1-\overline{\Psi}_1)\\
=&i\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_n\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right) e^{i\omega\vt_n\cdot(\mz-\mx_m)}J_1(\omega|\mz-\mx_m|)\\
&+i\sum_{n=1}^{N}\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_n\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right) e^{-i\omega\vt_n\cdot(\mz-\mx_m)}J_1(\omega|\mz-\mx_m|)\\
=&\sum_{m=1}^{M}\sum_{s=1}^{2}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)\sum_{n=1}^{N}\bigg(i(\vt_n\cdot\vx_s(\mx_m))e^{i\omega\vt_n\cdot(\mz-\mx_m)}\bigg)J_1(\omega|\mz-\mx_m|)\\
&+\sum_{m=1}^{M}\sum_{s=1}^{2}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)\sum_{n=1}^{N}\bigg(i(\vt_n\cdot\vx_s(\mx_m))e^{-i\omega\vt_n\cdot(\mz-\mx_m)}\bigg)J_1(\omega|\mz-\mx_m|),
\end{align*}
applying Theorem \ref{TheoremBessel}, we can obtain
\begin{align}
\begin{aligned}\label{term3}
\sum_{n=1}^{N}(\Psi_1-\overline{\Psi}_1)&=-2N\sum_{m=1}^{M}\sum_{s=1}^{2}\bigg(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\bigg)^2J_1(\omega|\mz-\mx_m|)^2\\
&=-2N\sum_{m=1}^{M}\bigg(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\bigg(\mt(\mx_m)+\mn(\mx_m)\bigg)\bigg)^2J_1(\omega|\mz-\mx_m|)^2.
\end{aligned}
\end{align}
Next, on the basis of the orthogonal property (\ref{orthogonal2}), we can evaluate
\begin{align*}
\sum_{n=1}^{N}\Psi_2\overline{\Psi}_2=&\sum_{n=1}^{N}\left(\sum_{m=1}^{M}\sum_{s=1}^{2}(\vt_n\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)e^{i\omega\vt_n\cdot \mx_m}J_1(\omega|\mz-\mx_m|)\right)\\
&\times\left(\sum_{m'=1}^{M}\sum_{s''=1}^{2}(\vt_n\cdot\vx_{s''}(\mx_{m'}))\left(\frac{\mz-\mx_{m'}}{|\mz-\mx_{m'}|}\cdot\vx_{s''}(\mx_{m'})\right)e^{-i\omega\vt_n\cdot\mx_{m'}}J_1(\omega|\mz-\mx_{m'}|)\right)\\
=&\sum_{m=1}^{M}\sum_{n=1}^{N}\left(\sum_{s=1}^{2}(\vt_n\cdot\vx_s(\mx_m))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)e^{i\omega\vt_n\cdot \mx_m}J_1(\omega|\mz-\mx_m|)\right)\\
&\times\left(\sum_{{s''}=1}^{2}(\vt_n\cdot\vx_{s''}(\mx_{m'}))\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_{s''}(\mx_m)\right)e^{-i\omega\vt_n\cdot \mx_{m}}J_1(\omega|\mz-\mx_{m}|)\right)\\
=&2\sum_{m=1}^{M}\sum_{n=1}^{N}\left((\vt_n\cdot\vx_s(\mx_m))^2\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx_m)\right)^2J_1(\omega|\mz-\mx_m|)^2\right)\\
=&2N\sum_{m=1}^{M}\sum_{s=1}^{2}\left(\frac{1}{N}\sum_{n=1}^{N}(\vt_n\cdot\vx_s(\mx_m))^2\right)\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx)\right)^2J_1(\omega|\mz-\mx_m|)^2\\
=&\frac{N}{\pi}\sum_{m=1}^{M}\sum_{s=1}^{2}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\vx_s(\mx)\right)^2\int_{\mathbb{S}^1}(\vp\cdot\vx_s(\mx_m))^2dS(\vp)J_1(\omega|\mz-\mx_m|)^2.
\end{align*}
Now, we consider polar coordinates; because $\vp,\vx_s\in\mathbb{S}^1$, let $\vp=[\cos\phi,\sin\phi]^T$ and $\vx_s=[\cos\psi,\sin\psi]^T$; then elementary calculus yields
\[\int_{\mathbb{S}^1}(\vp\cdot\vx_s(\mx_m))^2dS(\vp)=\int_0^{2\pi}\cos^2(\phi-\psi)d\phi=\pi.\]
Hence, we can obtain
\begin{equation}\label{term4}
\sum_{n=1}^{N}\Psi_2\overline{\Psi}_2=N\sum_{m=1}^{M}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\bigg(\mt(\mx_m)+\mn(\mx_m)\bigg)\right)^2J_1(\omega|\mz-\mx_m|)^2.
\end{equation}
Therefore, from (\ref{term3}) and (\ref{term4}), we can obtain
\[|P_{\mathrm{ noise}}(\mf(\mz))|=\sqrt{N}\left(1-\sum_{m=1}^{M}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot(\mt(\mx_m)+\mn(\mx_m))\right)^2J_1(\omega|\mz-\mx_m|)^2+o(h)\right)^{1/2}.\]
\end{proof}
Unlike the permittivity contrast case, the map of (\ref{MUSIC2}) shows two curves in the neighborhood of $\gamma$ because
\[\lim_{\mz\to\mx_m}\frac{J_1(\omega|\mz-\mx_m|)^2}{|\mz-\mx_m|}=0,\]
and $J_1(x)^2$ is maximum at two points, $x_1$ and $x_2$, which are symmetric with respect to $x=0$. This is why two ghost replicas with large magnitude and many artifacts with small magnitude appear instead of the true shape of the supporting curve $\gamma$ (see FIG. \ref{PlotMUSIC2}). Some numerical simulation results can be found in FIG. \ref{MapMUSIC2} and in \cite[Section 5]{HSZ}.
Note that $J_1(x)^2\ne1$ for all $x\in\mathbb{R}$. Hence, in contrast to the permittivity contrast case, (\ref{MUSIC2}) does not blow up.
\begin{figure}[!ht]
\begin{center}
\subfigure[Graph of $|1-J_0(\omega|x|)^2|^{-1}$]{\label{PlotMUSIC1}\includegraphics[width=0.49\textwidth]{MUSIC1.eps}}
\subfigure[Graph of $|1-J_1(\omega|x|)^2|^{-1}$]{\label{PlotMUSIC2}\includegraphics[width=0.49\textwidth]{MUSIC2.eps}}
\caption{\label{PlotMUSIC}Graphs of $|1-J_p(\omega|x|)^2|^{-1}$, with $p=0$ (left) and $1$ (right), for $\omega=2\pi/0.5$.}
\end{center}
\end{figure}
Finally, by combining (\ref{MUSIC1}) and (\ref{MUSIC2}), we can immediately obtain the following result.
\begin{thm}[Both permittivity and permeability contrast case]\label{Theorem3}
Let $\eps\ne\eps_0$ and $\mu\ne\mu_0$. Then, for sufficiently large $N$ ($>3M$) and $\omega$, (\ref{MUSICfunction}) can be written as follows:
\begin{multline}\label{MUSIC3}
\mathbb{E}_{\eps,\mu}(\mz):=\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2\right.\\
\left.+\sum_{m=1}^{M}\bigg(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot(\mt(\mx_m)+\mn(\mx_m))\bigg)^2J_1(\omega|\mz-\mx_m|)^2+o(h)\bigg)\right)^{-1/2}.
\end{multline}
\end{thm}
This result shows that plots of (\ref{MUSIC3}) show a large magnitude at $\mz$ if $\mz\ne\mx_m$ and
\[\sum_{m=1}^{M}\left(J_0(\omega|\mz-\mx_m|)^2+\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot(\mt(\mx_m)+\mn(\mx_m))\right)^2J_1(\omega|\mz-\mx_m|)^2+o(h)\right)=1.\]
Thus, a result with poor resolution will appear; refer to the examples of numerical simulation in \cite[Section 4.4]{PL3}.
\begin{rem}
On the basis of recent work \cite{P1}, the structure of so-called \textbf{subspace migration} is as follows.
\begin{enumerate}
\item Permittivity contrast case:
\[\mathbb{E}_{\mathrm{SM}}(\mz)=\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2.\]
\item Permeability contrast case:
\[\mathbb{E}_{\mathrm{SM}}(\mz)=\sum_{m=1}^{M}\left\{\bigg(\frac{\mz-\mx_m}{|\mz-\mx_m|}\bigg)\cdot\bigg(\mt(\mx_m)+\mn(\mx_m)\bigg)J_1(\omega|\mz-\mx_m|)\right\}^2.\]
\end{enumerate}
Hence, we can observe the following relationship between MUSIC and subspace migration, which is derived in \cite[Formula (7.4)]{AGKLS}: Let us select the unit vector $\mf(\mz)$ of (\ref{VecF}) as
\[\mf(\mz)=\frac{1}{\sqrt{N}}\bigg[e^{i\omega\vt_1\cdot\mz},e^{i\omega\vt_2\cdot\mz},\cdots,e^{i\omega\vt_N\cdot\mz}\bigg]^T.\]
Then,
\[\mathbb{E}(\mz)=\bigg(1-\mathbb{E}_{\mathrm{SM}}(\mz)\bigg)^{-1/2}.\]
\end{rem}
\subsection{Imaging of perfectly conducting cracks}
Here, let $\Gamma$ be a smooth curve that describes the crack: for an injective piecewise smooth function $\boldsymbol{\phi}:[-1,1]\longrightarrow\mathbb{R}^2$,
\begin{equation}\label{PCrack}
\Gamma=\{\boldsymbol{\phi}(x):-1\leq x\leq1\}.
\end{equation}
Let $u(\mx,\vt;\omega)$ be the single-component electric field that satisfies the Helmholtz equation:
\[\Delta u(\mx,\vt;\omega)+\omega^2 u(\mx,\vt;\omega)=0\quad\mbox{in}\quad\mathbb{R}^2\backslash\Gamma.\]
For the sound-soft arc [transverse magnetic (TM) polarization], $u(\mx,\vt;\omega)$ satisfies the Dirichlet boundary condition on $\Gamma$ (see \cite{K}):
\[u(\mx,\vt;\omega)=0\quad\mbox{on}\quad\Gamma\]
and for the sound-hard arc [transverse electric (TE) polarization], $u(\mx,\vt;\omega)$ satisfies the Neumann boundary condition on $\Gamma$ (see \cite{M2}):
\[\frac{\p u(\mx,\vt;\omega)}{\p\mn(\mx)}=0\quad\mbox{on}\quad\Gamma\backslash\{\boldsymbol{\phi}(-1),\boldsymbol{\phi}(1)\},\]
where $\mn(\mx)$ is the unit normal vector to $\Gamma$ at $\mx$.
Then the far-field pattern $u_{\infty}(\vv,\vt;\omega)$ for the scattering of an incident field $u_0(\mx,\vt;\omega)=e^{i\omega\vt\cdot\mx}$ from $\Gamma$ is given by
\[u_{\infty}(\vv,\vt;\omega)=\left\{
\begin{array}{ll}
\displaystyle\medskip\frac{1+i}{4\sqrt{\pi\omega}}\int_\Gamma e^{-i\omega\vv\cdot\mx}\varphi(\mx,\vt;\omega)d\mx & \mbox{: sound-soft arc}\\
\displaystyle\medskip\frac{(1-i)\sqrt{\omega}}{4\sqrt{\pi}}\int_\Gamma \vv\cdot\mn(\mx)e^{-i\omega\vv\cdot\mx}\psi(\mx,\vt;\omega)d\mx & \mbox{: sound-hard arc}.
\end{array}
\right.\]
According to the physical factorization in \cite{HSZ,PL1}, if the incident and observation directions are the same, the left singular vector of the MSR matrix is of the form
\[\mU_m=\left\{
\begin{array}{ll}
\medskip\bigg[e^{i\omega\vt_1\cdot\mx_m},e^{i\omega\vt_2\cdot\mx_m},\cdots,e^{i\omega\vt_N\cdot\mx_m}\bigg]^T & \mbox{: sound-soft arc} \\
\medskip\bigg[(\vt_1\cdot\mn(\mx_m))e^{i\omega\vt_1\cdot\mx_m},\cdots,(\vt_N\cdot\mn(\mx_m))e^{i\omega\vt_N\cdot\mx_m}\bigg]^T & \mbox{: sound-hard arc}.
\end{array}
\right.\]
Thus, the structure of the left singular vectors for the sound-soft and sound-hard arcs is almost the same as in the permittivity contrast and permeability contrast cases (except for the absence of a unit tangential vector), respectively. Hence, we can obtain the following result.
\begin{thm}\label{Theorem4}
Let $N$ and $\omega$ be sufficiently large. Then (\ref{MUSICfunction}) can be written as follows.
\begin{enumerate}
\item Sound-soft arc (or TM) case:
\[\mathbb{E}_{\mathrm{TM}}(\mz)\approx\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2\right)^{-1/2}.\]
\item Sound-hard arc (or TE) case:
\[\mathbb{E}_{\mathrm{TE}}(\mz)\approx\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot\mn(\mx_m)\right)^2J_1(\omega|\mz-\mx_m|)^2\right)^{-1/2}.\]
\end{enumerate}
\end{thm}
\subsection{Imaging of small electromagnetic inclusions}
We briefly consider the use of MUSIC for imaging electromagnetic inclusions $\Sigma_m$, $m=1,2,\cdots,M$, with small diameter $r$:
\[\Sigma_m:=\mx_m+r\mathbf{B}_m,\]
where $\mathbf{B}_m$ is a simply connected smooth domain containing the origin. We assume that $\Sigma_m$ are sufficiently separate from each other, and denote the collection of such inclusions as $\Sigma$.
As in section \ref{sec:2}, we let $u(\mx,\vt;\omega)$ satisfy (\ref{TotalField}) in the presence of $\Sigma$, and $u_0(\mx,\vt;\omega)$ is given by (\ref{IncidentField}). Then, the far-field pattern can be written as the following asymptotic expansion formula (see \cite{AK}):
\begin{equation}\label{smallexpansion}
u_{\infty}(\vv,\vt;\omega)\approx r^2\frac{\omega^2(1+i)}{4\sqrt{\omega\pi}} \sum_{m=1}^{M}|\mathbf{B}_m|\bigg((-\vv)\cdot\mathbb{A}(\mx_m)\cdot\vt+(\eps-\eps_0)\bigg)e^{i\omega(\vt-\vv)\cdot\mx_m}+o(r^2).
\end{equation}
Here, $\mathbb{A}(\mx_m)$ denotes the polarization tensor corresponding to $\Sigma_m$. Then, we can obtain the following results in a similar manner:
\begin{thm}\label{Theorem5}
Let $N$ and $\omega$ be sufficiently large. Then (\ref{MUSICfunction}) can be written as follows.
\begin{enumerate}
\item Dielectric permittivity contrast case:
\[\mathbb{E}_{\eps}(\mz)\approx\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2+o(r^2)\right)^{-1/2}.\]
\item Magnetic permeability contrast case:
\[\mathbb{E}_{\mu}(\mz)\approx\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}\left\{\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot(\me_1+\me_2)\right)J_1(\omega|\mz-\mx_m|)\right\}^2+o(r^2)\right)^{-1/2},\]
where $\left\{\me_1=[1,0]^T,\me_2=[0,1]^T\right\}$ denotes an orthonormal basis in $\mathbb{R}^2$.
\item Both permittivity and permeability contrast case:
\begin{multline*}
\mathbb{E}_{\eps,\mu}(\mz)\approx\frac{1}{\sqrt{N}}\left(1-\sum_{m=1}^{M}J_0(\omega|\mz-\mx_m|)^2\right.\\
\left.+\sum_{m=1}^{M}\left(\frac{\mz-\mx_m}{|\mz-\mx_m|}\cdot(\me_1+\me_2)\right)^2J_1(\omega|\mz-\mx_m|)^2+o(r^2)\right)^{-1/2}.
\end{multline*}
\end{enumerate}
\end{thm}
\begin{rem}
If $\Sigma_m$ denotes a perfectly conducting inclusion with a small diameter, the asymptotic expansion formulas for the TM and TE cases are similar to (\ref{smallexpansion}); refer to \cite[Theorem 3.1]{G}. Hence, the structure of (\ref{MUSICfunction}) will be the same as in Theorem \ref{Theorem5}.
\end{rem}
\section{Numerical examples}\label{sec:4}
In this section, we present some numerical simulation results. For this, we choose a thin inclusion $\Gamma_1=\{\mx+\eta\mn(x):\mx\in\gamma_1,~\eta\in(-h,h)\}$ with a smooth supporting curve:
\[\gamma_1=\left\{\left[x+0.2,x^3+x^2-0.3\right]^T:-0.5\leq x\leq0.5\right\}.\]
The thickness $h$ of the thin inclusion $\Gamma_1$ is set to $0.02$, and the following parameters are chosen: $\eps_0=\mu_0=1$, and $\eps=\mu=5$. For the illumination and observation directions $\vt_n$, they are chosen as
\[\vt_n=\left[\cos\frac{2n\pi}{N},\sin\frac{2n\pi}{N}\right]^T\]
for $n=1,2,\cdots,N$. The total number of directions is $N=24$, and the applied frequency is $\omega=2\pi/\lambda$ with a wavelength of $\lambda=0.4$. The data set for the MSR matrix $\mathbb{K}$ in (\ref{MSR}) is collected by solving the forward problem introduced in \cite{NK}.
FIG. \ref{MapMUSIC} shows maps of $\mathbb{E}_{\eps}(\mz)$ and $\mathbb{E}_{\mu}(\mz)$ in the presence of $\Gamma_1$. This result demonstrates that the MUSIC algorithm offers a very accurate result for the permittivity contrast case. For the permeability contrast case, as we saw in Theorem \ref{Theorem2}, we cannot recognize the true shape of $\Gamma_1$. However, using Theorem \ref{Theorem2}, we can obtain an approximate shape of $\Gamma_1$ from the two identified curves.
\begin{figure}[!ht]
\begin{center}
\subfigure[Permittivity contrast case]{\label{MapMUSIC1}\includegraphics[width=0.49\textwidth]{Thin-Permittivity.eps}}
\subfigure[Permeability contrast case]{\label{MapMUSIC2}\includegraphics[width=0.49\textwidth]{Thin-Permeability.eps}}
\caption{\label{MapMUSIC}Shape reconstruction of thin electromagnetic inclusion $\Gamma_1$ using MUSIC algorithm.}
\end{center}
\end{figure}
For a numerical example of a perfectly conducting crack, we selected the following smooth curve in the form of (\ref{PCrack}):
\[\Gamma_2=\left\{\left[x,\frac{1}{2}\cos\frac{x\pi}{2}+\frac{1}{5}\sin\frac{x\pi}{2}-\frac{1}{10}\cos\frac{3x\pi}{2}\right]^T:-1\leq x\leq1\right\}.\]
The data set for the MSR matrix $\mathbb{K}$ in (\ref{MSR}) is collected by solving the forward problems introduced in \cite[Chapter 3]{N} and \cite[Chapter 4]{N} for the sound-soft and sound-hard arcs, respectively. FIG. \ref{MapMUSICPC} shows maps of $\mathbb{E}(\mz)$ for $N=40$ directions and $\lambda=0.4$. By comparing the results in FIG. \ref{MapMUSIC}, we can observe that Theorems \ref{Theorem1} and \ref{Theorem2} hold for the sound-soft and sound-hard arcs, respectively. Additional numerical results can be found in recent works \cite{PL1,PL3}.
\begin{figure}[!ht]
\begin{center}
\subfigure[Sound-soft case]{\label{MapMUSIC4}\includegraphics[width=0.49\textwidth]{Crack-Dirichlet.eps}}
\subfigure[Sound-hard case]{\label{MapMUSIC5}\includegraphics[width=0.49\textwidth]{Crack-Neumann.eps}}
\caption{\label{MapMUSICPC}Shape reconstruction of perfectly conducting crack $\Gamma_2$ using MUSIC algorithm.}
\end{center}
\end{figure}
\section{Concluding remarks}\label{sec:5}
On the basis of the structure of the left singular vectors of the MSR matrix, we investigated the structure of the MUSIC-type imaging function by establishing a relationship between it and the Bessel function of integer order of the first kind. Using this relationship, we examined certain properties of the MUSIC algorithm.
It is worth emphasizing that the MUSIC algorithm can be applied in limited-view inverse scattering problems. However, its structure has been identified for the sound-soft arc of small length \cite{JKP}. Hence, exploring the structure of MUSIC for the extended, sound-hard arc will be a future work.
Finally, we have been considering the imaging of two-dimensional thin electromagnetic inclusions or perfectly conducting cracks. The analysis could be extended to a three-dimensional problem; refer to \cite{AILP} for related work.
| 198,341
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Improv Everywhere strikes again. This is cute and all, kind of, but if you weren't familiar with Ghostbusters, you'd probably think KKK members were being chased by hazmat men with guns.
Watch a Ghostbusters Prank at the New York Public Library
Or are KKK members being chased by hazmat men with guns?
| 99,597
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\begin{document}
\title {\textbf{The periodic oscillation of an adiabatic piston in
two or three dimensions} \footnote {Submitted to
\emph{Communications in Mathematical Physics}}\\ preprint }
\author {Paul Wright\footnote
{Department of Mathematics, Courant Institute of Mathematical Sciences, New York University,
251 Mercer St.,
New York, NY 10012 USA. E-mail address:~\texttt{paulrite@cims.nyu.edu}}}
\date {December 2006}
\maketitle
\begin {abstract}
We study a heavy piston of mass $M$ that separates finitely many
ideal, unit mass gas particles moving in two or three dimensions.
Neishtadt and Sinai previously determined a method for finding this
system's averaged equation and showed that its solutions oscillate
periodically. Using averaging techniques, we prove that the actual
motions of the piston converge in probability to the predicted
averaged behavior on the time scale $M^ {1/2} $ when $M$ tends to
infinity while the total energy of the system is bounded and the
number of gas particles is fixed.
\end {abstract}
\textbf{Mathematics Subject Classification (2000):} 34C29, 37A60,
82C22.
\textbf{Keywords:} adiabatic piston, averaging, ergodic billiards.
\section{Introduction}\label{sct:intro}
Consider the following simple model of an adiabatic piston
separating two gas containers: A massive piston of mass $M\gg 1$
divides a container in $\mathbb{R}^d$ into two halves. The piston
has no internal degrees of freedom and can only move along one axis
of the container. On either side of the piston there are a finite
number of ideal, unit mass, point gas particles that interact with
the walls of the container and with the piston via elastic
collisions. When $M=\infty $, the piston remains fixed in place, and
each gas particle performs billiard motion at a constant energy in
its sub-container. We make an ergodicity assumption on the behavior
of the gas particles when the piston is fixed. Then we study the
motions of the piston when the number of gas particles is fixed, the
total energy of the system is bounded, but $M$ is very large.
Heuristically, after some time, one expects the system to approach a
steady state, where the energy of the system is equidistributed
amongst the particles and the piston. However, even if we could show
that the full system is ergodic, an abstract ergodic theorem says
nothing about the time scale required to reach such a steady state.
Because the piston will move much slower than a typical gas
particle, it is natural to try to determine the intermediate
behavior of the piston by averaging techniques. By averaging over
the motion of the gas particles on a time scale chosen short enough
that the piston is nearly fixed, but long enough that the ergodic
behavior of individual gas particles is observable, we will show
that the system does not approach the expected steady state on the
time scale $M^ {1/2} $. Instead, the piston oscillates periodically,
and there is no net energy transfer between the gas particles.
This paper follows earlier work by Neishtadt and Sinai~\cite{NS04,
Sin99}. They determined that for a wide variety of Hamiltonians for
the gas particles, the averaged behavior of the piston is periodic
oscillation, with the piston moving inside an effective potential
well whose shape depends on the initial position of the piston and
the gas particles' Hamiltonians. They pointed out that an averaging
theorem due to Anosov~\cite{Ano60,LM88}, proved for smooth systems,
should extend to this case. This paper proves that Anosov's theorem
extends to the particular gas particle Hamiltonian described above.
Thus, if we examine the actual motions of the piston with respect to
the slow time $\tau=t/M^ {1/2}$, then, as $M\rightarrow\infty $, in
probability (with respect to Liouville measure) most initial
conditions give rise to orbits whose actual motion is accurately
described by the averaged behavior for $0\leq\tau\leq 1$, i.e.~for
$0\leq t\leq M^ {1/2}$. Gorelyshev and
Neishtadt~\cite{GorNeishtadt06} and we~\cite{Wri06} have already
proved that when $d=1$, i.e.~when the gas particles move on a line,
the convergence of the actual motions to the averaged behavior is
uniform over all initial conditions, with the size of the deviations
being no larger than $\mathcal{O} (M^ {-1/2}) $ on the time scale
$M^ {-1/2} $.
The system under consideration in this paper is a simple model of an
adiabatic piston. The general adiabatic piston problem~\cite {Ca63},
well-known from physics, consists of the following: An insulating
piston separates two gas containers, and initially the piston is
fixed in place, and the gas in each container is in a separate
thermal equilibrium. At some time, the piston is no longer
externally constrained and is free to move. One hopes to show that
eventually the system will come to a full thermal equilibrium, where
each gas has the same pressure and temperature. Whether the system
will evolve to thermal equilibrium and the interim behavior of the
piston are mechanical problems, not adequately described by
thermodynamics~\cite{Gru99}, that have recently generated much
interest within the physics and mathematics communities. One
expects that the system will evolve in at least two stages. First,
the system relaxes toward a mechanical equilibrium, where the
pressures on either side of the piston are equal. In the second,
much longer, stage, the piston drifts stochastically in the
direction of the hotter gas, and the temperatures of the gases
equilibrate. See for example~\cite{GPL03,CL02} and the references
therein. So far, rigorous results have been limited mainly to models
where the effects of gas particles recolliding with the piston can
be neglected, either by restricting to extremely short time
scales~\cite{CLS02,CLS02b} or to infinite gas
containers~\cite{Che05}.
A recent study involving some similar ideas by Chernov and
Dolgopyat~\cite{CD06} considered the motion inside a two-dimensional
domain of a single heavy, large gas particle (a disk) of mass $M\gg
1$ and a single unit mass point particle. They assumed that for
each fixed location of the heavy particle, the light particle moves
inside a dispersing (Sinai) billiard domain. By averaging over the
strongly hyperbolic motions of the light particle, they showed that
under an appropriate scaling of space and time the limiting process
of the heavy particle's velocity is a (time-inhomogeneous) Brownian
motion on a time scale $\mathcal{O} (M^ {1/2}) $. It is not clear
whether a similar result holds for the piston problem, even for gas
containers with good hyperbolic properties, such as the Bunimovich
stadium. In such a container the motion of a gas particle when the
piston is fixed is only nonuniformly hyperbolic because it can
experience many collisions with the flat walls of the container
immediately preceding and following a collision with the piston.
The present work provides a weak law of large numbers, and it is an
open problem to describe the sizes of the deviations for the piston
problem~\cite{CD06b}. Although our result does not yield concrete
information on the sizes of the deviations, it is general in that it
imposes very few conditions on the shape of the gas container. Most
studies of billiard systems impose strict conditions on the shape of
the boundary, generally involving the sign of the curvature and how
the corners are put together. The proofs in this work require no
such restrictions. In particular, the gas container can have cusps
as corners and need satisfy no hyperbolicity conditions.
We begin in Section~\ref{sct:heuristic} by giving a physical
description of our results. Precise assumptions and our main
result, Theorem~\ref{thm:dDpiston}, are stated in
Section~\ref{sct:main_result}, and a proof is presented in the
following sections.
\section{Physical motivation for the results}\label{sct:heuristic}
\begin{figure}
\begin {center}
\setlength{\unitlength}{1 cm}
\begin{picture}(15,6)
\put(2.5,1){\line(1,0){10}}
\put(2.5,5){\line(1,0){10}}
\qbezier(2,3.5)(3.5,4)(2.5,5)
\qbezier(2,3.5)(3,2.5)(2,1)
\put(2,1){\line(1,0){1}}
\qbezier(12.5,1)(14.5,1)(14.5,3)
\qbezier(14.5,3)(14.5,5)(12.5,5)
\put(3.2,4.5){$\mathcal{D}_1(Q)$}
\put(12,4.5){$\mathcal{D}_2(Q)$}
\put(1.5,1){\line(0,1){4}}
\put(1.5,1){\line(1,0){0.15}}
\put(1.5,5){\line(1,0){0.15}}
\put(1.5,3){\line(-1,0){0.15}}
\put(1,2.9){$\ell$}
\linethickness {0.15cm}
\put(8.5,1){\line(0,1){4}}
\thinlines
\put(8.5,3){\vector(1,0){.3}}
\put(8.9,2.9){$V=\varepsilon W$}
\put(8.5,1){\line(0,-1){0.15}}
\put(8.38,0.4){$Q$}
\put(8.38,5.2){$M=\varepsilon ^ {-2}\gg 1$}
\put(4.25,1){\line(0,-1){0.15}}
\put(4.18,0.4){$0$}
\put(12,1){\line(0,-1){0.15}}
\put(11.95,0.4){$1$}
\put(5,4){\circle*{.15}}
\put(5,4){\vector(1,0){1}}
\put(7,4.5){\circle*{.15}}
\put(7,4.5){\vector(1,-2){0.7}}
\put(6,3.2){\circle*{.15}}
\put(6,3.2){\vector(-2,-1){0.6}}
\put(6.5,1.8){\circle*{.15}}
\put(6.5,1.8){\vector(3,-1){0.7}}
\put(4.5,2.1){\circle*{.15}}
\put(4.5,2.1){\vector(-2,-3){0.45}}
\put(8.9,2.5){\circle*{.15}}
\put(8.9,2.5){\vector(2,-1){1}}
\put(11.5,2.0){\circle*{.15}}
\put(11.5,2.0){\vector(4,1){0.55}}
\put(10,4.6){\circle*{.15}}
\put(10,4.6){\vector(-1,-2){0.5}}
\put(12.5,3.1){\circle*{.15}}
\put(12.5,3.1){\vector(1,1){1}}
\end{picture}
\end {center}
\caption{A gas container $\mathcal{D}\subset \mathbb{R}^2 $ separated by a piston.}
\label{fig:domain1}
\end{figure}
Before precisely stating our assumptions and results, we briefly
review the physical motivations for our results and introduce some
notation.
Consider a massive, insulating piston of mass $M$ that separates a
gas container $\mathcal{D} $ in $\mathbb{R}^d$, $ d= 2\text { or
}3$. See Figure~\ref{fig:domain1}. Denote the location of the piston
by $Q$, its velocity by $dQ/dt=V$, and its cross-sectional length
(when $ d=2$, or area, when $ d=3$) by $\ell$. If $Q$ is fixed, then
the piston divides $\mathcal{D} $ into two subdomains,
$\mathcal{D}_1(Q) =\mathcal{D}_1 $ on the left and $\mathcal{D}_2(Q)
=\mathcal{D}_2 $ on the right. By $E_i$ we denote the total energy
of the gas inside $\mathcal{D}_i$, and by $\abs{\mathcal{D}_i} $ we
denote the area (when $ d=2$, or volume, when $ d=3$) of
$\mathcal{D}_i$.
We are interested in the dynamics of the piston when the system's
total energy is bounded and $M\rightarrow \infty $. When
$M=\infty$, the piston remains fixed in place, and each energy $E_i$
remains constant. When $M$ is large but finite, $MV^2/2$ is bounded,
and so $V=\mathcal{O} (M^ {-1/2}) $. It is natural to define
\[
\varepsilon=M^ {-1/2},\quad W=\frac {V} {\varepsilon},
\]
so that $W$ is of order $1$ as $\varepsilon\rightarrow 0$. This is
equivalent to scaling time by $\varepsilon$.
If we let $P_i$ denote the pressure of the gas inside
$\mathcal{D}_i$, then heuristically the dynamics of the piston
should be governed by the following differential equation:
\[
\frac {dQ} {dt}=V,\quad M\frac {dV} {dt}=P_1\ell-P_2\ell,
\]
i.e.
\begin {equation}\label{eq:piston_force}
\frac {dQ} {dt}=\varepsilon W,\quad
\frac {dW} {dt}=\varepsilon P_1\ell-\varepsilon P_2\ell.
\end {equation}
To find differential equations for the energies of the gases,
note that in a short amount of time $dt$, the change in energy
should come entirely from the work done on a gas, i.e.~the force
applied to the gas times the distance the piston has moved, because
the piston is adiabatic. Thus, one expects that
\begin {equation}\label{eq:work}
\frac {dE_1} {dt}=-\varepsilon WP_1\ell,
\quad
\frac {dE_2} {dt} = +\varepsilon W P_2\ell.
\end {equation}
To obtain a closed system of differential equations, it is necessary
to insert an expression for the pressures. Because the pressure of
an ideal gas in $d$ dimensions is proportional to the energy
density, with the constant of proportionality $2/d$, we choose to
insert
\[
P_i=\frac {2E_i}{d\abs{\mathcal{D}_i}} .
\]
Later, we will make assumptions to justify this substitution.
However, if we accept this definition of the pressure, and define
the slow time
\[
\tau=\varepsilon t,
\]
we obtain the following ordinary differential equations for the
four macroscopic variables of the system:
\begin {equation}\label{eq:heuristic_avg_eq}
\frac{d}{d\tau}
\begin {bmatrix}
Q\\
W\\
E_1\\
E_2\\
\end {bmatrix}
=
\begin {bmatrix}
W\\
\frac{2E_1\ell}{d\abs{\mathcal{D}_1(Q)}}
-\frac{2E_2\ell}{d\abs{\mathcal{D}_2(Q)}}\\
-\frac{2WE_1\ell}{d\abs{\mathcal{D}_1(Q)}}\\
+\frac{2WE_2\ell}{d\abs{\mathcal{D}_2(Q)}}\\
\end {bmatrix}.
\end {equation}
Neishtadt and Sinai~\cite{Sin99, NS04} pointed out that the
solutions of Equation~\eqref{eq:heuristic_avg_eq} have the piston
moving according to an effective Hamiltonian. This can be seen as
follows. Since
\[
\frac {\partial\abs{\mathcal{D}_1(Q)}} {\partial Q}=\ell
=-\frac {\partial\abs{\mathcal{D}_2(Q)}} {\partial Q},
\]
$d\ln(E_i)/d\tau=-(2/d)d\ln(\abs{\mathcal{D}_i(Q)})/d\tau$, and so
\[
E_i(\tau)=E_i(0)\left (\frac{\abs{\mathcal{D}_i(Q(0))}}
{\abs{\mathcal{D}_i(Q(\tau))}}\right) ^ {2/d}.
\]
Hence
\[
\frac {d^2Q} {d\tau^2}=
\frac {2\ell} {d}
\frac{E_1(0)\abs{\mathcal{D}_1(Q(0))}^{2/d}}{\abs{\mathcal{D}_1(Q(\tau))}^{1+2/d}}
-\frac {2\ell} {d}
\frac{E_2(0)\abs{\mathcal{D}_2(Q(0))}^{2/d}}{\abs{\mathcal{D}_2(Q(\tau))}^{1+2/d}},
\]
and so $(Q, W) $ behave as if they were the coordinates of a
Hamiltonian system describing a particle undergoing motion inside a
potential well. The effective Hamiltonian may be expressed as
\begin {equation}\label{eq:d_dpot}
\frac {1} {2}W^2+
\frac{E_1(0)\abs{\mathcal{D}_1(Q(0))}^{2/d}}
{\abs{\mathcal{D}_1(Q)}^{2/d}}+
\frac{E_2(0)\abs{\mathcal{D}_2(Q(0))}^{2/d}}
{\abs{\mathcal{D}_2(Q)}^{2/d}}.
\end {equation}
The question is, do the solutions of
Equation~\eqref{eq:heuristic_avg_eq} give an accurate description of
the actual motions of the macroscopic variables when $M$ tends to
infinity? The main result of this paper is that, for an
appropriately defined system, the answer to this question is
affirmative for $0\leq t\leq M^ {1/2}$, at least for most initial
conditions of the microscopic variables. Observe that one should not
expect the description to be accurate on time scales much longer
than $\mathcal{O} (M^ {1/2}) =\mathcal{O} (\varepsilon^ {-1})$. The
reason for this is that, presumably, there are corrections of size
$\mathcal{O} (\varepsilon^ {2})$ in
Equations~\eqref{eq:piston_force} and \eqref{eq:work} that we are
neglecting. On the time scale $\varepsilon^ {-1} $, these errors
roughly add up to no more than size $\mathcal{O} (\varepsilon^ {-1}
\cdot \varepsilon^ {2}=\varepsilon) $, but on a longer time scale
they should become significant. Such higher order corrections for
the adiabatic piston were studied by Crosignani \emph{et
al.}~\cite{CD96}.
\section{Statement of the main result}\label{sct:main_result}
\subsection{Description of the model}\label{sct:model}
We begin by describing the gas container. It is a compact,
connected billiard domain $\mathcal{D} \subset\mathbb{R}^d$ with a
piecewise $\mathcal{C} ^3$ boundary, i.e.~$\partial\mathcal{D} $
consists of a finite number of $\mathcal{C} ^3$ embedded
hypersurfaces, possibly with boundary and a finite number of corner
points. The container consists of a ``tube,'' whose perpendicular
cross-section $\mathcal{P} $ is the shape of the piston, connecting
two disjoint regions. $\mathcal{P} \subset \mathbb{R} ^ {d-1} $ is
a compact, connected domain whose boundary is piecewise $\mathcal{C}
^3$. Then the ``tube'' is the region $[0,1]\times
\mathcal{P}\subset\mathcal{D} $ swept out by the piston for $ 0\leq
Q\leq 1$, and $[0,1]\times
\partial\mathcal{P}\subset\partial\mathcal{D} $. If $d=2$, $\mathcal{P} $
is just a closed line segment, and the ``tube'' is a rectangle. If
$ d=3$, $\mathcal{P} $ could be a circle, a square, a pentagon, etc.
Our fundamental assumption is as follows:
\begin{main_assu}
For almost every $Q\in [0,1]$ the billiard flow of a single particle
on an energy surface in either of the two subdomains
$\mathcal{D}_i(Q)$ is ergodic (with respect to the invariant
Liouville measure).
\end {main_assu}
\noindent If $ d=2$, the domain could be the Bunimovich
stadium~\cite{Bun79}. Another possible domain is indicated in Figure
\ref{fig:domain1}. Polygonal domains satisfying our assumptions can
also be constructed~\cite{Vorobets_1997}. Suitable domains in $d=3$
dimensions can be constructed using a rectangular box with shallow
spherical caps adjoined~\cite{BunimovichRehacek1998}. Note that we
make no assumptions regarding the hyperbolicity of the billiard flow
in the domain.
The Hamiltonian system we consider consists of the massive piston of
mass $M$ located at position $Q$, as well as $ n_1+ n_2 $ gas
particles, $ n_1$ in $\mathcal{D}_1$ and $n_2$ in $\mathcal{D}_2$.
Here $n_1$ and $n_2$ are fixed positive integers. For convenience,
the gas particles all have unit mass, though all that is important
is that each gas particle has a fixed mass. We denote the positions
of the gas particles in $\mathcal{D}_i$ by $q_{ i,j}$, $1\leq j\leq
n_i$. The gas particles are ideal point particles that interact
with $\partial\mathcal{D} $ and the piston by hard core, elastic
collisions. Although it has no effect on the dynamics we consider,
for convenience we complete our description of the Hamiltonian
dynamics by specifying that the piston makes elastic collisions with
walls located at $Q=0,\: 1$ that are only visible to the piston. We
denote velocities by $dQ/dt=V=\varepsilon W$ and $dq_{ i,j}/dt=v_{
i,j}$, and we set
\[
E_{ i,j}=v_{ i,j}^2/2,\qquad E_i=\sum_{ j=1} ^
{n_i} E_{ i,j}.
\]
Our system has $d(n_1+n_2)+1$ degrees of
freedom, and so its phase space is $(2d(n_1+n_2)+2)$-dimensional.
We let
\[
h(z)=h=(Q,W,E_{1,1},E_{1,2},\cdots,E_{1,n_1},E_{2,1},E_{2,2},\cdots,E_{2,n_2}),
\]
so that $h$ is a function from our phase space to
$\mathbb{R}^{n_1+n_2+2}$. We often abbreviate
$h=(Q,W,E_{1,j},E_{2,j})$, and we refer to $h$ as consisting of the
slow variables because these quantities are conserved when
$\varepsilon=0$. We let $h_\varepsilon(t,z)=h_\varepsilon(t) $
denote the actual motions of these variables in time for a fixed
value of $\varepsilon$. Here $z$ represents the initial condition in
phase space, which we usually suppress in our notation. One should
think of $h_\varepsilon(\cdot) $ as being a random variable that
takes initial conditions in phase space to paths (depending on the
parameter t) in $\mathbb{R}^{n_1+n_2+2}$.
\subsection {The averaged equation}
From the work of Neishtadt and Sinai~\cite{NS04}, one can derive
\begin {equation}\label{eq:d_davg}
\frac{d}{d\tau}
\begin {bmatrix}
Q\\
W\\
E_{1,j}\\
E_{2,j}\\
\end {bmatrix}
=\bar H(h):=
\begin {bmatrix}
W\\
\frac{2E_1\ell}{d\abs{\mathcal{D}_1(Q)}}
-\frac{2E_2\ell}{d\abs{\mathcal{D}_2(Q)}}\\
-\frac{2WE_{1,j}\ell}{d\abs{\mathcal{D}_1(Q)}}\\
+\frac{2WE_{2,j}\ell}{d\abs{\mathcal{D}_2(Q)}}\\
\end {bmatrix}
\end {equation}
as the averaged equation (with respect to the slow time
$\tau=\varepsilon t$) for the slow variables. Later, in Section
\ref{sct:heuristic2}, we will give another heuristic derivation of
the averaged equation that is more suggestive of our proof. As in
Section~\ref{sct:heuristic}, the solutions of
Equation~\eqref{eq:d_davg} have $(Q, W) $ behaving as if they were
the coordinates of a Hamiltonian system describing a particle
undergoing motion inside a potential well. The effective Hamiltonian
is given by Equation~\eqref{eq:d_dpot}.
Let $\bar{h} (\tau,z)=\bar{h} (\tau) $ be the solution of
\[
\frac {d\bar{h}}{d\tau} =\bar {H} (\bar {h}),\qquad \bar {h} (0)
=h_\varepsilon(0).
\]
Again, think of $\bar h(\cdot) $ as being a random variable.
\subsection{The main result}
The solutions of the averaged equation approximate the motions of
the slow variables, $h_\varepsilon(t) $, on a time scale
$\mathcal{O} (1/\varepsilon) $ as $\varepsilon\rightarrow 0$.
Precisely, fix a compact set $\mathcal{V}\subset \mathbb
R^{n_1+n_2+2}$ such that $h\in \mathcal{V} \Rightarrow
Q\subset\subset (0,1),W\subset\subset \mathbb R$, and $E_{i,j}
\subset\subset (0,\infty)$ for each $i$ and $j$.\footnote { We have
introduced this notation for convenience. For example, $h\in
\mathcal{V} \Rightarrow Q\subset\subset (0,1) $ means that there
exists a compact set $A \subset (0,1) $ such that $h\in \mathcal{V}
\Rightarrow Q\in A $, and similarly for the other variables.} We
will be mostly concerned with the dynamics when $h\in\mathcal{V} $.
Define
\[
\begin {split}
Q_{min}&=\inf_{h\in\mathcal{V}}Q,\qquad
Q_{max}=\sup_{h\in\mathcal{V}}Q,
\\
E_{min}&=\inf_{h\in\mathcal{V}}\frac{1}{2}W^2+E_1+E_2,\qquad
E_{max}=\sup_{h\in\mathcal{V}}\frac{1}{2}W^2+E_1+E_2.
\end {split}
\]
For a fixed value of $\varepsilon >0$, we only consider the dynamics
on the invariant subset of phase space defined by
\[
\begin {split}
\mathcal{M}_\varepsilon =
\{(Q,V,q_{i,j},v_{i,j})\in\mathbb{R}^ {2d(n_1+n_2)+2}:
Q\in [0,1],\;q_{i,j}\in\mathcal{D}_{i}(Q),
&
\\
E_{min}\leq \frac{M}{2}V^2+E_1+E_2\leq E_{ max}\}
&.
\end {split}
\]
Let $P_\varepsilon$ denote the probability measure obtained by
restricting the invariant Liouville measure to
$\mathcal{M}_\varepsilon $. Define the stopping time
\[
T_\varepsilon(z) =T_\varepsilon =\inf \{\tau\geq 0: \bar {h}
(\tau)\notin \mathcal{V} \text { or } h_\varepsilon(\tau
/\varepsilon) \notin \mathcal{V} \}.
\]
\begin {thm}\label{thm:dDpiston}
If $\mathcal{D} $ is a gas container in $d=2$ or $3$ dimensions
satisfying the assumptions in Subsection~\ref{sct:model} above, then
for each $T>0$,
\[
\sup_{0\leq\tau\leq T\wedge
T_\varepsilon}\abs{h_\varepsilon(\tau/\varepsilon)-\bar{h}(\tau)}
\rightarrow 0 \text { in probability as } \varepsilon=M^
{-1/2}\rightarrow 0,
\]
i.e.~for each fixed $\delta>0$,
\[
P_\varepsilon\left(\sup_{0\leq\tau\leq T\wedge
T_\varepsilon}\abs{h_\varepsilon(\tau/\varepsilon)-\bar{h}(\tau)}\geq\delta
\right)
\rightarrow 0\text { as } \varepsilon=M^
{-1/2}\rightarrow 0.
\]
\end{thm}
It should be noted that the stopping time in the above result is not
unduly restrictive. If the initial pressures of the two gasses are
not too mismatched, then the solution to the averaged equation is a
periodic orbit, with the effective potential well keeping the piston
away from the walls. Thus, if the actual motions follow the
averaged solution closely for $0\leq\tau\leq T\wedge T_\varepsilon$,
and the averaged solution stays in $\mathcal{V} $, it follows that
$T_\varepsilon
>T$.
The techniques of this paper should immediately generalize to prove
the analogue of Theorem~\ref{thm:dDpiston} above in the nonphysical
dimensions $d>3$, although we do not pursue this here.
\section{Preparatory material concerning a two-dimensional gas
container with only one gas particle on each side}
\label{sct:2dprep}
Our results and techniques of proof are essentially independent of
the dimension and the fixed number of gas particles on either side
of the piston. Thus, we focus on the case when $d=2$ and there is
only one gas particle on either side. Later, in Section
\ref{sct:generalization}, we will indicate the simple modifications
that generalize our proof to the general situation. For clarity, in
this section and next, we denote $ q_{ 1,1} $ by $ q_1$, $v_{2, 1} $
by $ v_2$, etc. We decompose the gas particle coordinates according
to whether they are perpendicular to or parallel to the piston's
face, for example $q_1= (q_1^\perp,q_1^\parallel)$. See Figure
\ref{fig:domain}.
\begin{figure}
\begin {center}
\setlength{\unitlength}{0.7 cm}
\begin{picture}(15,6)
\put(2.5,1){\line(1,0){10}}
\put(2.5,5){\line(1,0){10}}
\qbezier(2,3.5)(3.5,4)(2.5,5)
\qbezier(2,3.5)(3,2.5)(2,1)
\put(2,1){\line(1,0){1}}
\qbezier(12.5,1)(14.5,1)(14.5,3)
\qbezier(14.5,3)(14.5,5)(12.5,5)
\put(3.2,4.5){$\mathcal{D}_1$}
\put(12,4.5){$\mathcal{D}_2$}
\put(13.3,5){\vector(0,1){0.5}}
\put(13.23,5.7){$\parallel$}
\put(13.3,5){\vector(1,0){0.5}}
\put(13.95,4.9){$\perp$}
\put(1.5,1){\line(0,1){4}}
\put(1.5,1){\line(1,0){0.15}}
\put(1.5,5){\line(1,0){0.15}}
\put(1.5,3){\line(-1,0){0.15}}
\put(1,2.9){$\ell$}
\linethickness {0.1cm}
\put(8.5,1){\line(0,1){4}}
\thinlines
\put(8.5,3){\vector(1,0){.4}}
\put(9.0,2.9){$V=\varepsilon W$}
\put(8.5,1){\line(0,-1){0.15}}
\put(8.38,0.3){$Q$}
\put(8.38,5.2){$M=\varepsilon ^ {-2}\gg 1$}
\put(4.25,1){\line(0,-1){0.15}}
\put(4.15,0.3){$0$}
\put(12,1){\line(0,-1){0.15}}
\put(11.95,0.3){$1$}
\put(5,4){\circle*{.15}}
\put(5,4){\vector(1,2){.4}}
\put(5.05,3.65){$q_1$}
\put(5.45,4.5){$v_1$}
\put(11.5,2.5){\circle*{.15}}
\put(11.5,2.5){\vector(-1,-3){.3}}
\put(11.65,2.3){$q_2$}
\put(11.35,1.5){$v_2$}
\end{picture}
\end {center}
\caption{A choice of coordinates on phase space.}
\label{fig:domain}
\end{figure}
The Hamiltonian dynamics define a flow on our phase space. We denote
this flow by $z_\varepsilon(t,z) =z_\varepsilon(t)$, where
$z=z_\varepsilon(0,z) $. One should think of $z_\varepsilon(\cdot) $
as being a random variable that takes initial conditions in phase
space to paths in phase space. Then $h_\varepsilon(t)
=h(z_\varepsilon(t)) $. By the change of coordinates
$W=V/\varepsilon$, we may identify all of the
$\mathcal{M}_\varepsilon$ defined in Section~\ref{sct:main_result}
with the space
\[
\begin {split}
\mathcal{M} =
\{(Q,W,q_1,v_1,q_2,v_2)\in\mathbb{R}^ {10}:
Q\in [0,1],\;q_1\in\mathcal{D}_1(Q),\;
q_2\in\mathcal{D}_2(Q),\;
&
\\
E_{min}\leq \frac{1}{2}W^2+E_1+E_2\leq E_{ max}\}
&.
\end {split}
\]
and all of the $P_\varepsilon$ with the probability measure $P$ on
$\mathcal{M}$, which has the density
\[
dP=\C \,dQdWdq_1^\perp dq_1^\parallel dv_1^\perp dv_1^\parallel
dq_2^\perp dq_2^\parallel dv_2^\perp dv_2^\parallel.
\]
(Throughout this work we will use $\C$ to represent generic
constants that are independent of $\varepsilon$.) We will assume
that these identifications have been made, so that we may consider
$z_\varepsilon(\cdot) $ as a family of measure preserving flows on
the same space that all preserve the same probability measure. We
denote the components of $z_\varepsilon(t) $ by $Q_\varepsilon(t) $,
$q_{1,\varepsilon}^\perp(t) $, etc.
The set $\{z\in\mathcal{M}:q_1=Q=q_2\}$ has co-dimension two, and so
$\bigcup_t z_\varepsilon(t)\{q_1=Q=q_2\}$ has co-dimension one,
which shows that only a measure zero set of initial conditions will
give rise to three particle collisions. We ignore this and other
measures zero events, such as gas particles hitting singularities of
the billiard flow, in what follows.
Now we present some background material, as well as some lemmas that
will assist us in our proof of Theorem \ref{thm:dDpiston}. We begin
by studying the billiard flow of a gas particle when the piston is
infinitely massive. Next we examine collisions between the gas
particles and the piston when the piston has a large, but finite,
mass. Then we present a heuristic derivation of the averaged
equation that is suggestive of our proof. Finally we prove a lemma
that allows us to disregard the possibility that a gas particle will
move nearly parallel to the piston's face -- a situation that is
clearly bad for having the motions of the piston follow the
solutions of the averaged equation.
\subsection{Billiard flows and maps in two dimensions}\label{sct:billiard}
In this section, we study the billiard flows of the gas particles
when $M=\infty $ and the slow variables are held fixed at a specific
value $h\in\mathcal{V} $. We will only study the motions of the
left gas particle, as similar definitions and results hold for the
motions of the right gas particle. Thus we wish to study the
billiard flow of a point particle moving inside the domain
$\mathcal{D}_1$ at a constant speed $\sqrt{2E_1} $. The results of
this section that are stated without proof can be found
in~\cite{CM06}.
Let $\mathcal{TD}_1$ denote the tangent bundle to $\mathcal{D}_1$.
The billiard flow takes place in the three-dimensional space
$\mathcal{M}_h^1=\mathcal{M}^1=\{(q_1,v_1)\in\mathcal{TD}_1:q_1\in\mathcal{
D}_1,\; \abs{v_1}=\sqrt{2E_1}\}/\sim$. Here the quotient means that
when $q_1\in\partial\mathcal{ D}_1$, we identify velocity vectors
pointing outside of $\mathcal{D}_1$ with those pointing inside
$\mathcal{D}_1$ by reflecting through the tangent line to
$\partial\mathcal{D}_1$ at $ q_1$, so that the angle of incidence
with the unit normal vector to $\partial\mathcal{ D}_1$ equals the
angle of reflection. Note that most of the quantities defined in
this subsection depend on the fixed value of $h$. We will usually
suppress this dependence, although, when necessary, we will indicate
it by a subscript $h$. We denote the resulting flow by
$y(t,y)=y(t)$, where $y(0,y)=y$. As the billiard flow comes from a
Hamiltonian system, it preserves Liouville measure restricted to the
energy surface. We denote the resulting probability measure by $\mu
$. This measure has the density
$d\mu=dq_1dv_1/(2\pi\sqrt{2E_1}\abs{\mathcal{D}_1} ) $. Here $dq_1$
represents area on $\mathbb{R}^2$, and $ dv_1$ represents length on
$S^1_{\sqrt{2E_1}}=\set{v_1\in\mathbb{R}^2:\abs{v_1}=\sqrt{2E_1}}$.
There is a standard cross-section to the billiard flow, the
collision cross-section
$\Omega=\{(q_1,v_1)\in\mathcal{TD}_1:q_1\in\partial\mathcal{ D}_1,\;
\abs{v_1}=\sqrt{2E_1}\}/\sim$. It is customary to parameterize
$\Omega$ by $\{x=(r,\varphi):r\in\partial\mathcal{D}_1,\:\varphi\in
[-\pi /2,+\pi /2]\}$, where $r$ is arc length and $\varphi$
represents the angle between the outgoing velocity vector and the
inward pointing normal vector to $\partial\mathcal{D}_1$. It follows
that $\Omega$ may be realized as the disjoint union of a finite
number of rectangles and cylinders. The cylinders correspond to
fixed scatterers with smooth boundary placed inside the gas
container.
If $F:\Omega\circlearrowleft$ is the collision map, i.e.~the
return map to the collision cross-section, then $F$ preserves the
projected probability measure $\nu $, which has the density
$d\nu=\cos\varphi\, d\varphi \, dr/(2\abs{\partial\mathcal{D}_1}) $.
Here $\abs{\partial\mathcal{D}_1}$ is the length of
$\partial\mathcal{D}_1$.
We suppose that the flow is ergodic, and so $F$ is an invertible,
ergodic measure preserving transformation. Because
$\partial\mathcal{D}_1$ is piecewise $\mathcal{C} ^3$, $F$ is
piecewise $\mathcal{C} ^2$, although it does have discontinuities
and unbounded derivatives near discontinuities corresponding to
grazing collisions. Because of our assumptions on $\mathcal{D}_1$,
the free flight times and the curvature of $\partial\mathcal{D}_1$
are uniformly bounded. It follows that if $x\notin \partial\Omega
\cup F^{-1} (\partial\Omega) $, then $F$ is differentiable at $x$,
and
\begin {equation}\label{eq:2d_derivative_bound}
\norm {DF(x)}\leq\frac {\C} {\cos \varphi(Fx)},
\end{equation}
where $\varphi(Fx) $ is the value of the $\varphi$ coordinate at the
image of $x$.
Following the ideas in Appendix \ref{sct:inducing}, we induce $F$ on
the subspace $\hat\Omega$ of $\Omega$ corresponding to collisions
with the (immobile) piston. We denote the induced map by $\hat F$
and the induced measure by $\hat \nu$. We parameterize $\hat\Omega$
by $\{(r,\varphi):0\leq r\leq \ell,\:\varphi\in [-\pi/2,+\pi/2]\}$.
As $\nu \hat\Omega=\ell/\abs{\partial\mathcal{D}_1} $, it follows
that $\hat\nu $ has the density $d\hat \nu=\cos\varphi\, d\varphi \,
dr/(2\ell) $.
For $x\in\Omega $, define $\zeta x $ to be the free flight time,
i.e.~the time it takes the billiard particle traveling at speed
$\sqrt{2E_1} $ to travel from $x$ to $Fx$. If $x\notin
\partial\Omega \cup F^{-1} (\partial\Omega) $,
\begin {equation}
\label{eq:2d_time_derivative}
\norm {D\zeta (x)}\leq\frac {\C} {\cos \varphi(Fx)}.
\end {equation}
Santal\'{o}'s formula~\cite{San76,Chernov1997} tells us that
\begin {equation}
\label{eq:2d_Santalo}
E_\nu \zeta=\frac {\pi
\abs{\mathcal{D}_1}} {\abs{v_1}\abs{\partial\mathcal{D}_1}}.
\end {equation}
If $\hat\zeta:\hat\Omega\rightarrow\mathbb{R} $ is the free
flight time between collisions with the piston, then it follows from
Proposition \ref{prop:inducing} that
\begin {equation}
\label{eq:2d_flight}
E_{\hat\nu} \hat\zeta=\frac {\pi
\abs{\mathcal{D}_1}}{\abs{v_1}\ell}.
\end{equation}
The expected value of $ \abs{v_1^\perp }$ when the left gas particle
collides with the (immobile) piston is given by
\begin {equation}
\label{eq:2d_momentum}
E_{\hat\nu} \abs{v_1^\perp }=E_{\hat\nu} \sqrt{2E_1}\cos\varphi=
\frac{\sqrt{2E_1}}{2}\int_{-\pi/2}^{+\pi/2} \cos^2\varphi\,d\varphi=
\sqrt{2E_1}\frac{\pi}{4}.
\end {equation}
We wish to compute $\lim_{t\rightarrow\infty} t^{-1} \int_0^t
\abs{2v_1^\perp (s)}\delta_{q_1^\perp (s)=Q}ds$, the time average of
the change in momentum of the left gas particle when it collides
with the piston. If this limit exists and is equal for almost every
initial condition of the left gas particle, then it makes sense to
define the pressure inside $\mathcal{D}_1$ to be this quantity
divided by $\ell$. Because the collisions are hard-core, we cannot
directly apply Birkhoff's Ergodic Theorem to compute this limit.
However, we can compute this limit by using the map $\hat F$.
\begin {lem}
\label{lem:ae_convergence}
If the billiard flow $y(t) $ is ergodic, then for $\mu-a.e.$ $y\in
\mathcal{M}^1$,
\[
\lim_{t\rightarrow\infty} \frac{1}{t}
\int_0^t \abs{v_1^\perp (s)}\delta_{q_1^\perp (s)
=Q}ds=
\frac{E_1\ell}{2\abs{\mathcal{D}_1(Q)}}.
\]
\end {lem}
\begin {proof}
Because the billiard flow may be viewed as a suspension flow over
the collision cross-section with $\zeta$ as the height function, it
suffices to show that the convergence takes place for $\hat\nu-a.e.$
$x\in\hat\Omega$. For an initial condition $x\in\hat\Omega$, define
$\hat{N}_t(x)=\hat{N}_t=\#\set{s\in (0,t]:y(s,x) \in\hat\Omega}$. By
the Poincar\'e Recurrence Theorem, $\hat{N}_t\rightarrow\infty$ as
$t\rightarrow\infty$, $\hat\nu-a.e.$
But
\[
\begin {split}
\frac{\hat{N}_t}{\sum_{n=0}^{\hat{N}_t}\hat\zeta(\hat F^n x)}
\frac{1}{\hat{N}_t}
\sum_{n=1}^{\hat{N}_t}\abs{v_1^\perp }(\hat F^n x)
&\leq
\frac{1}{t}\int_0^t \abs{v_1^\perp (s)}\delta_{q_1^\perp (s)=Q}ds
\\
&\leq
\frac{\hat{N}_t}{\sum_{n=0}^{\hat{N}_t-1}\hat\zeta(\hat F^n x)}
\frac{1}{\hat{N}_t}
\sum_{n=0}^{\hat{N}_t}\abs{v_1^\perp }(\hat F^n x),
\end {split}
\]
and so the result follows from Birkhoff's Ergodic Theorem and
Equations \eqref{eq:2d_flight} and \eqref{eq:2d_momentum}.
\end {proof}
\begin {cor}
\label{cor:ae_convergence}
If the billiard flow $y(t) $ is ergodic, then for each $\delta>0$,
\[
\mu
\set{y\in \mathcal{M}^1:\abs{\frac{1}{t}
\int_0^t \abs{v_1^\perp (s)}\delta_{q_1^\perp (s)
=Q}ds-
\frac{E_1\ell}{2\abs{\mathcal{D}_1(Q)}}}\geq \delta}
\rightarrow 0\text{ as }t\rightarrow \infty.
\]
\end {cor}
\subsection{Analysis of collisions}\label{sct:collisions}
In this section, we return to studying our piston system when
$\varepsilon>0$. We will examine what happens when a particle
collides with the piston. For convenience, we will only examine in
detail collisions between the piston and the left gas particle.
Collisions with the right gas particle can be handled similarly.
When the left gas particle collides with the piston, $v_1^\perp $
and $V$ instantaneously change according to the laws of elastic
collisions:
\begin {equation*}
\begin{bmatrix}
v_1^{\perp +}\\ V^+
\end{bmatrix}
=
\frac{1}{1+M}
\begin{bmatrix}
1-M& 2M\\
2& M-1\\
\end{bmatrix}
\begin{bmatrix}
v_1^{\perp -}\\ V^-
\end{bmatrix}.
\end {equation*}
In our coordinates, this becomes
\begin {equation}\label{eq:collision_change}
\begin{bmatrix}
v_1^{\perp +}\\ W^+
\end{bmatrix}
=
\frac{1}{1+\varepsilon^2 }
\begin{bmatrix}
\varepsilon^2 -1 & 2\varepsilon\\
2\varepsilon & 1-\varepsilon^2 \\
\end{bmatrix}
\begin{bmatrix}
v_1^{\perp -}\\ W^-
\end{bmatrix}.
\end {equation}
Recalling that $ v_1, W=\mathcal{O} (1) $, we find that to first
order in $\varepsilon$,
\begin{equation}
\label{eq:v_1Wchange}
v_1^{\perp +}=-v_1^{\perp -}+\mathcal{O}(\varepsilon),\qquad
W^ +=W^ -+\mathcal{O}(\varepsilon).
\end{equation}
Observe that a collision can only take place if $v_1^{\perp
-}>\varepsilon W^ - $. In particular, $v_1^{\perp -}> -
\varepsilon\sqrt{2E_{max}}$. Thus, either $v_1^{\perp -}> 0$ or
$v_1^{\perp -}= \mathcal{O} (\varepsilon) $. By expanding
Equation~\eqref{eq:collision_change} to second order in
$\varepsilon$, it follows that
\begin{equation}
\label{eq:E_1Wchange}
\begin {split}
E_1^+ -E_1^- &=-2\varepsilon W \abs{v_1^{\perp }}
+\mathcal{O}(\varepsilon^2),\\
W^+ -W^- &=+2\varepsilon \abs{v_1^{\perp }}
+\mathcal{O}(\varepsilon^2).
\end {split}
\end{equation}
Note that it is immaterial whether we use the pre-collision or
post-collision values of $W$ and $\abs{v_1^{\perp }}$ on the right
hand side of Equation~\eqref{eq:E_1Wchange}, because any ambiguity
can be absorbed into the $\mathcal{O} (\varepsilon^2) $ term.
It is convenient for us to define a ``clean collision'' between the
piston and the left gas particle:
\begin {defn}
The left gas particle experiences a \emph{clean collision} with the
piston if and only if $v_1^{\perp -}>0$ and $v_1^{\perp +}<-\varepsilon
\sqrt{2E_{max}}$.
\end{defn}
\noindent In particular, after a clean collision, the left gas
particle will escape from the piston, i.e.~the left gas particle
will have to move into the region $q_1^{\perp }\leq 0 $ before it
can experience another collision with the piston. It follows that
there exists a constant $C_1>0$, which depends on the set
$\mathcal{V} $, such that for all $\varepsilon$ sufficiently small,
so long as $ Q\geq Q_{min} $ and $\abs{v_1^{\perp }}
>\varepsilon C_1$ when $q_1^{\perp }\in [Q_{ min},Q]$, then the left gas particle
will experience only clean collisions with the piston, and the time
between these collisions will be greater than $2Q_{min}/(\sqrt
{2E_{max}})$. (Note that when we write expressions such as
$q_1^{\perp }\in [Q_{ min},Q]$, we implicitly mean that $q_1$ is
positioned inside the ``tube'' discussed at the beginning of
Section~\ref{sct:main_result}.) One can verify that $C_1=5\sqrt
{2E_{max}}$ would work.
Similarly, we can define clean collisions between the right gas
particle and the piston. We assume that $C_1$ was chosen
sufficiently large such that for all $\varepsilon$ sufficiently
small, so long as $ Q\leq Q_{max} $ and $\abs{v_2^{\perp }}
>\varepsilon C_1$ when $q_2^{\perp }\in [Q,Q_{max}] $, then the right gas particle
will experience only clean collisions with the piston.
Now we define three more stopping times, which are functions of the
initial conditions in phase space.
\[
\begin {split}
T_\varepsilon' =&\inf \{\tau\geq 0: Q_{min}\leq q_{1,\varepsilon}^{\perp
}(\tau/\varepsilon)\leq Q_\varepsilon(\tau/\varepsilon)\leq Q_{max}
\text { and}\abs{v_{1,\varepsilon}^{\perp }(\tau/\varepsilon)}\leq C_1\varepsilon \}
,
\\
T_\varepsilon'' =&
\inf \{\tau\geq 0:Q_{min}\leq Q_\varepsilon(\tau/\varepsilon)\leq q_{2,\varepsilon}^{\perp
}(\tau/\varepsilon)\leq Q_{max}
\text { and}\abs{v_{2,\varepsilon}^{\perp }(\tau/\varepsilon)}\leq C_1\varepsilon
\},
\\
\tilde{T}_\varepsilon =&
T\wedge T_\varepsilon\wedge T_\varepsilon'\wedge T_\varepsilon''
\end {split}
\]
Define $H(z) $ by
\[
H(z) =
\begin{bmatrix}
W\\
+2\abs{v_1^{\perp }} \delta_{q_1^{\perp }=Q}
-2\abs{v_2^{\perp }} \delta_{q_2^{\perp }=Q}\\
-2W\abs{v_1^{\perp }} \delta_{q_1^{\perp }=Q}\\
+2W\abs{v_2^{\perp }} \delta_{q_2^{\perp }=Q}\\
\end{bmatrix}.
\]
Here we make use of Dirac delta functions. All integrals involving
these delta functions may be replaced by sums.
The following lemma is an immediate consequence of Equation
\eqref{eq:E_1Wchange} and the above discussion:
\begin{lem}
\label{lem:h_int}
If $0\leq t_1\leq t_2\leq \tilde{T}_\varepsilon/\varepsilon $, the
piston experiences $\mathcal{O} ((t_2-t_1)\vee 1) $ collisions with
gas particles in the time interval $[t_1, t_2]$, all of which are
clean collisions. Furthermore,
\begin {equation*}
h_\varepsilon(t_2)-h_\varepsilon(t_1)=
\mathcal{O}(\varepsilon)+\varepsilon\int_{t_1}^{t_2}
H(z_\varepsilon(s))ds.
\end {equation*}
Here any ambiguities arising from collisions occurring at the limits
of integration can be absorbed into the $\mathcal{O} (\varepsilon) $
term.
\end {lem}
\subsection{Another heuristic derivation of the averaged
equation}\label{sct:heuristic2}
The following heuristic derivation of Equation \eqref{eq:d_davg}
when $ d=2$ was suggested in~\cite{Dol05}. Let $\Delta t $ be a
length of time long enough such that the piston experiences many
collisions with the gas particles, but short enough such that the
slow variables change very little, in this time interval. From each
collision with the left gas particle, Equation~\eqref{eq:E_1Wchange}
states that $W$ changes by an amount $+2\varepsilon
\abs{v_1^{\perp}} +\mathcal{O}(\varepsilon^2)$, and from
Equation~\eqref{eq:2d_momentum} the average change in $W$ at these
collisions should be approximately $\varepsilon\pi
\sqrt{2E_1}/2+\mathcal{O}(\varepsilon^2)$. From
Equation~\eqref{eq:2d_flight} the frequency of these collisions is
approximately $\sqrt{2E_1}\,\ell /(\pi
\abs{\mathcal{D}_1})$. Arguing
similarly for collisions with the other particle, we guess that
\[
\frac {\Delta W} {\Delta t} =
\varepsilon\frac{E_1\ell}{\abs{\mathcal{D}_1(Q)}}
-\varepsilon\frac{E_2\ell}{\abs{\mathcal{D}_2(Q)}}
+\mathcal{O}(\varepsilon^2).
\]
With $\tau=\varepsilon t$ as the slow time, a reasonable guess for
the averaged equation for $W$ is
\[
\frac {dW}{d\tau}=\frac{E_1\ell}{\abs{\mathcal{D}_1(Q)}}
-\frac{E_2\ell}{\abs{\mathcal{D}_2(Q)}}.
\]
Similar arguments for the other slow variables lead to the averaged
equation \eqref{eq:d_davg}, and this explains why we used $P_i=
E_i/\abs{\mathcal{D}_i}$ for the pressure of a $2$-dimensional gas
in Section~\ref{sct:heuristic}.
There is a similar heuristic derivation of the averaged equation in
$ d>2$ dimensions. Compare the analogues of
Equations~\eqref{eq:2d_flight} and \eqref{eq:2d_momentum} in
Subsection~\ref{sct:higher_d}.
\subsection{\textit{A priori} estimate on the size
of a set of bad initial conditions}
In this section, we give an \textit{a priori} estimate on the size
of a set of initial conditions that should not give rise to orbits
for which $\sup_{0\leq\tau\leq T\wedge
T_\varepsilon}\abs{h_\varepsilon(\tau/\varepsilon)-\bar{h}(\tau)}$
is small. In particular, when proving Theorem \ref{thm:dDpiston},
it is convenient to focus on orbits that only contain clean
collisions with the piston. Thus, we show that $P\{\tilde
{T}_\varepsilon<T\wedge T_\varepsilon \} $ vanishes as
$\varepsilon\rightarrow 0$. At first, this result may seem
surprising, since $P\{T_\varepsilon'\wedge T_\varepsilon''=0\}
=\mathcal{O}(\varepsilon)$, and one would expect $\cup_{t=0} ^
{T/\varepsilon} z_\varepsilon(-t)\{T_\varepsilon'\wedge
T_\varepsilon''=0\}$ to have a size of order $1$. However, the rate
at which orbits escape from $\{T_\varepsilon'\wedge
T_\varepsilon''=0\}$ is very small, and so we can prove the
following:
\begin {lem}
\label{lem:no_vertical}
\[
P\{\tilde {T}_\varepsilon<T\wedge T_\varepsilon \} =\mathcal{O}
(\varepsilon).
\]
\end {lem}
In some sense, this lemma states that the probability of having a
gas particle move nearly parallel to the piston's face within the
time interval $[0,T/\varepsilon ] $, when one would expect the other
gas particle to force the piston to move on a macroscopic scale,
vanishes as $\varepsilon\rightarrow 0$. Thus, one can hope to
control the occurrence of the ``nondiffusive fluctuations'' of the
piston described in~\cite{CD06} on a time scale $\mathcal{O}
(\varepsilon^ {-1}) $.
\begin {proof}
As the left and the right gas particles can be handled similarly, it
suffices to show that $P\{T_\varepsilon'<T \} =\mathcal{O}
(\varepsilon)$. Define
\[
\mathfrak{B}_\varepsilon=\{z\in\mathcal{M}:
Q_{min}\leq q_1^{\perp
}\leq Q\leq Q_{max}
\text { and}\abs{v_1^{\perp }}\leq C_1\varepsilon
\}.
\]
Then $\{T_\varepsilon'<T \}\subset \cup_{t=0} ^ {T/\varepsilon}
z_\varepsilon(-t)\mathfrak{B}_\varepsilon$, and if $\gamma=
Q_{min}/\sqrt{8 E_{max}}$,
\[
\begin {split}
P\left (\bigcup_{t=0} ^{T/\varepsilon}
z_\varepsilon(-t)\mathfrak{B}_\varepsilon\right)
&
=
P\left (\bigcup_{t=0} ^{T/\varepsilon}
z_\varepsilon(t)\mathfrak{B}_\varepsilon\right)
=
P\left(\mathfrak{B}_\varepsilon\cup\bigcup_{t=0} ^{T/\varepsilon}
((z_\varepsilon(t)\mathfrak{B}_\varepsilon) \backslash \mathfrak{B}_\varepsilon)
\right)
\\
&
\leq
P\mathfrak{B}_\varepsilon+P\left( \bigcup_{k=0}^{T/(\varepsilon\gamma
)} z_\varepsilon(k\gamma )
\Bigl[ \bigcup_{t=0}^\gamma
(z_\varepsilon(t)\mathfrak{B}_\varepsilon)\backslash
\mathfrak{B}_\varepsilon \Bigr]
\right)
\\
&
\leq
P\mathfrak{B}_\varepsilon+
\left(\frac{T}{\varepsilon\gamma }+1\right)
P\left(
\bigcup_{t=0}^\gamma
(z_\varepsilon(t)\mathfrak{B}_\varepsilon)\backslash
\mathfrak{B}_\varepsilon
\right).
\end {split}
\]
Now $P\mathfrak{B}_\varepsilon=\mathcal{O} (\varepsilon) $, so if we
can show that
$P\left(\bigcup_{t=0}^\gamma(z_\varepsilon(t)\mathfrak{B}_\varepsilon)\backslash
\mathfrak{B}_\varepsilon\right)=\mathcal{O} (\varepsilon^2)$, then
it will follow that $P\{T_\varepsilon'<T \} =\mathcal{O}
(\varepsilon)$.
If
$z\in\bigcup_{t=0}^\gamma(z_\varepsilon(t)\mathfrak{B}_\varepsilon)\backslash
\mathfrak{B}_\varepsilon$, it is still true that
$\abs{v_1^\perp}=\mathcal{O}(\varepsilon)$. This is because
$\abs{v_1^\perp}$ changes by at most $\mathcal{O} (\varepsilon) $ at
the collisions, and if a collision forces
$\abs{v_1^\perp}>C_1\varepsilon$, then the gas particle must escape
to the region $q_1^\perp\leq 0$ before $ v_1^\perp $ can change
again, and this will take time greater than $\gamma $. Furthermore,
if
$z\in\bigcup_{t=0}^\gamma(z_\varepsilon(t)\mathfrak{B}_\varepsilon)\backslash
\mathfrak{B}_\varepsilon$, then at least one of the following four
possibilities must hold:
\begin {itemize}
\item
$\abs{q_1^\perp-Q_{min}}\leq\mathcal{O}(\varepsilon)$,
\item
$\abs{Q-Q_{min}}\leq\mathcal{O}(\varepsilon)$,
\item
$\abs{Q-Q_{max}}\leq\mathcal{O}(\varepsilon)$,
\item
$\abs{Q-q_1^\perp}\leq\mathcal{O}(\varepsilon)$.
\end {itemize}
It follows that
$P\left(\bigcup_{t=0}^\gamma(z_\varepsilon(t)\mathfrak{B}_\varepsilon)\backslash
\mathfrak{B}_\varepsilon\right)=\mathcal{O} (\varepsilon^2)$. For
example,
\[
\begin {split}
\int_{\mathcal{M}}
&
1_{\{\abs{v_1^\perp}\leq
\mathcal{O}(\varepsilon),\:
\abs{q_1^\perp-Q_{min}}\leq\mathcal{O}(\varepsilon)\}}dP
\\
&
=
\C
\int_{\set {E_{min}\leq W^2/2+v_1^2/2+v_2^2/2\leq E_{max}}}
1_{\{\abs{v_1^\perp}\leq
\mathcal{O}(\varepsilon)\}}
dW dv_1^{\perp}dv_1^{\parallel} dv_2^{\perp}dv_2^{\parallel}
\\
&
\qquad\times
\int_{\set {Q\in [0,1],\, q_1\in\mathcal{D}_1,\, q_2\in\mathcal{D}_2}}
1_{\{\abs{q_1^\perp-Q_{min}}\leq\mathcal{O}(\varepsilon)\}}
dQ dq_1^{\perp}dq_1^{\parallel} dq_2^{\perp}dq_2^{\parallel}
\\
&
=\mathcal{O}(\varepsilon^2).
\end {split}
\]
\end {proof}
\section{Proof of the main result for two-dimensional gas
containers with only one gas particle on each side}
\label{sct:2dproof}
As in Section~\ref{sct:2dprep}, we continue with the case when $d=2$
and there is only one gas particle on either side of the piston.
\subsection{Main steps in the proof of convergence in probability}\label{sct:main_steps}
By Lemma \ref{lem:no_vertical}, it suffices to show that $
\sup_{0\leq\tau\leq \tilde{T}_\varepsilon}
\abs{h_\varepsilon(\tau/\varepsilon)-\bar{h}(\tau)}\rightarrow 0$ in
probability as $\varepsilon=M^{-1/2}\rightarrow 0$. Several of the
ideas in the steps below were inspired by a recent proof of Anosov's
averaging theorem for smooth systems that is due to
Dolgopyat~\cite{Dol05}.
\paragraph*{Step 1: Reduction using Gronwall's Inequality.}
Observe that $\bar {h}(\tau) $ satisfies the integral equation
\[
\bar {h}(\tau) -\bar h(0) = \int_0^{\tau}\bar H(\bar
h(\sigma))d\sigma,
\]
while from Lemma \ref{lem:h_int},
\[
\begin {split}
h_\varepsilon(\tau/\varepsilon)-h_\varepsilon(0)
&
=\mathcal{O}(\varepsilon) +\varepsilon\int_0^{\tau/\varepsilon}
H(z_\varepsilon(s))ds\\
&=\mathcal{O}(\varepsilon) +
\varepsilon\int_0^{\tau/\varepsilon}
H(z_\varepsilon(s))-
\bar H(h_\varepsilon(s))ds+
\int_0^{\tau}\bar H( h_\varepsilon(\sigma/\varepsilon))d\sigma
\end {split}
\]
for $0\leq\tau\leq \tilde{T}_\varepsilon$. Define
\[
e_\varepsilon(\tau) =\varepsilon\int_0^{\tau/\varepsilon}
H(z_\varepsilon(s))- \bar H(h_\varepsilon(s))ds.
\]
It follows from
Gronwall's Inequality that
\begin {equation}\label {eq:2d_Gronwall}
\sup_{0\leq \tau\leq \tilde{T}_\varepsilon}
\abs{h_\varepsilon(\tau/\varepsilon)-\bar h(\tau)}\leq
\left(\mathcal{O}(\varepsilon)+
\sup_{0\leq \tau\leq \tilde{T}_\varepsilon}
\abs{e_\varepsilon(\tau)}\right)e^{ \Lip{\bar
H\arrowvert_\mathcal{V}}T}.
\end {equation}
\noindent Gronwall's Inequality is usually stated for continuous
paths, but the standard proof (found in \cite{SV85}) still works for
paths that are merely integrable, and
$\abs{h_\varepsilon(\tau/\varepsilon)-\bar h(\tau)}$ is piecewise
smooth.
\paragraph*{Step 2: Introduction of a time scale for ergodization.}
Let $L(\varepsilon) $ be a real valued function such that
$L(\varepsilon)\rightarrow\infty$, but $L(\varepsilon)\ll \log
\varepsilon^ {-1} $, as $\varepsilon\rightarrow 0$. In
Section~\ref{sct:Gronwall} we will place precise restrictions on the
growth rate of $L(\varepsilon) $. Think of $L(\varepsilon) $ as
being a time scale that grows as $\varepsilon\rightarrow 0$ so that
\emph{ergodization}, i.e.~the convergence along an orbit of a
function's time average to a space average, can take place. However,
$L(\varepsilon) $ doesn't grow too fast, so that on this time scale
$z_\varepsilon(t) $ essentially stays on the submanifold
$\set{h=h_\varepsilon(0)}$, where we have our ergodicity assumption.
Set $t_{k,\varepsilon} =kL(\varepsilon) $, so that
\begin {equation}
\label{eq:2d_einfnorm}
\sup_{0\leq \tau\leq \tilde{T}_\varepsilon}\abs{e_\varepsilon(\tau)}
\leq \mathcal{O}(\varepsilon L(\varepsilon))+
\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}H(z_\varepsilon(s))-\bar
H(h_\varepsilon(s))ds}.
\end {equation}
\paragraph*{Step 3: A splitting according to particles.}
Now $H(z) -\bar H(h(z)) $ divides into two pieces, each of which
depends on only one gas particle when the piston is held fixed:
\[
H(z) -\bar H(h(z))=
\begin{bmatrix}
0\\
2\abs{v_1^{\perp }} \delta_{q_1^{\perp }=Q}
-\frac{E_1\ell}{\abs{\mathcal{D}_1(Q)}}\\
-2W\abs{v_1^{\perp }} \delta_{q_1^{\perp }=Q}
+\frac{WE_1\ell}{\abs{\mathcal{D}_1(Q)}}\\
0\\
\end{bmatrix}
+
\begin{bmatrix}
0\\
\frac{E_2\ell}{\abs{\mathcal{D}_2(Q)}}
-2\abs{v_2^{\perp }} \delta_{q_2^{\perp }=Q}\\
0\\
-\frac{WE_2\ell}{\abs{\mathcal{D}_2(Q)}}+
2W\abs{v_2^{\perp }} \delta_{q_2^{\perp }=Q}\\
\end{bmatrix}.
\]
We will only deal with the piece depending on the left gas particle,
as the right particle can be handled similarly. Define
\begin{equation}
\label{eq:G_definition}
G(z)=\abs{v_1^{\perp }} \delta_{q_1^{\perp }=Q},
\qquad
\bar G(h)=
\frac{E_1\ell}{2\abs{\mathcal{D}_1(Q)}}.
\end {equation}
Returning to Equation \eqref{eq:2d_einfnorm}, we see that in order
to prove Theorem \ref{thm:dDpiston}, it suffices to show that both
\[
\begin {split}
&\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}G(z_\varepsilon(s))-\bar
G(h_\varepsilon(s))ds} \text { and}
\\
&\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}W_\varepsilon(s)
\bigl(G(z_\varepsilon(s))-\bar
G(h_\varepsilon(s))\bigr)ds}
\end {split}
\]
converge to $0$ in probability as $\varepsilon\rightarrow 0$.
\paragraph*{Step 4: A splitting for using the triangle inequality.}
Now we let $z_{k,\varepsilon} (s) $ be the orbit of the
$\varepsilon=0$ Hamiltonian vector field satisfying
$z_{k,\varepsilon}(t_{k,\varepsilon})=z_{\varepsilon}(t_{k,\varepsilon})$.
Set $h_{k,\varepsilon} (t) =h (z_{k,\varepsilon} (t)) $. Observe
that $h_{k,\varepsilon} (t) $ is independent of $t $.
We emphasize that so long as $0\leq
t\leq\tilde{T}_\varepsilon/\varepsilon$, the times between
collisions of a specific gas particle and piston are uniformly
bounded greater than $0$, as explained before Lemma \ref{lem:h_int}.
It follows that, so long as
$t_{k+1,\varepsilon}\leq\tilde{T}_\varepsilon/\varepsilon$,
\begin{equation}
\label{eq:h_div}
\sup_{t_{k,\varepsilon}\leq t\leq t_{k+1,\varepsilon}}
\abs{h_{k,\varepsilon} (t) -h_\varepsilon (t)}
=\mathcal{O}(\varepsilon L(\varepsilon)).
\end{equation}
This is because the slow variables change by at most $\mathcal{O}
(\varepsilon) $ at collisions, and
$dQ_\varepsilon/dt=\mathcal{O}(\varepsilon)$.
Also,
\[
\begin {split}
\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}
&
W_\varepsilon(s) \bigl(G(z_\varepsilon(s))-\bar
G(h_\varepsilon(s))\bigr)ds
\\
&= \mathcal{O} (\varepsilon
L(\varepsilon)^2) +W_{k,\varepsilon}(t_{k,\varepsilon})\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}
G(z_\varepsilon(s))-\bar G(h_\varepsilon(s))ds,
\end {split}
\]
and so
\[
\begin {split}
\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}
&
\abs{\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}W_\varepsilon(s)
\bigl(G(z_\varepsilon(s))-\bar
G(h_\varepsilon(s))\bigr)ds}
\\
&\leq
\mathcal{O}(\varepsilon L(\varepsilon))+
\varepsilon\,\C\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}
G(z_\varepsilon(s))-\bar
G(h_\varepsilon(s))ds}.
\end {split}
\]
Thus, in order to prove Theorem \ref{thm:dDpiston}, it suffices to
show that
\[
\begin {split}
\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}G(z_\varepsilon(s))-\bar
G(h_\varepsilon(s))ds}
\leq
\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}
\abs{I_{k,\varepsilon}}+\abs{II_{k,\varepsilon}}+\abs{III_{k,\varepsilon}}
\end {split}
\]
converges to $0$ in probability as $\varepsilon\rightarrow 0$, where
\[
\begin {split}
I_{k,\varepsilon}
&=\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}}G(z_\varepsilon(s)) -
G(z_{k,\varepsilon}(s))ds,
\\
II_{k,\varepsilon}
&=\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}} G(z_{k,\varepsilon}(s))-
\bar G(h_{k,\varepsilon}(s))ds,
\\
III_{k,\varepsilon}
&=\int_{t_{k,\varepsilon}}^{t_{k+1,\varepsilon}} \bar G(h_{k,\varepsilon}(s))-
\bar G(h_{\varepsilon}(s))ds.
\end {split}
\]
The term $II_{k,\varepsilon}$ represents an ``ergodicity term'' that
can be controlled by our assumptions on the ergodicity of the flow
$z_0(t) $, while the terms $I_{k,\varepsilon}$ and
$III_{k,\varepsilon}$ represent ``continuity terms'' that can be
controlled by controlling the drift of $z_{\varepsilon} (t) $ from
$z_{k,\varepsilon} (t) $ for $t_{k,\varepsilon}\leq t\leq
t_{k+1,\varepsilon}$.
\paragraph*{Step 5: Control of drift from the $\varepsilon=0$ orbits.}
Now $\bar G$ is uniformly Lipschitz on the compact set $\mathcal{V}
$, and so it follows from Equation \eqref{eq:h_div} that
$III_{k,\varepsilon}=\mathcal{O}(\varepsilon L(\varepsilon)^2)$.
Thus,
$\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{III_{k,\varepsilon}} =\mathcal{O}(\varepsilon
L(\varepsilon))\rightarrow 0$ as $\varepsilon\rightarrow 0$.
Next, we show that for fixed $\delta > 0$,
$P\left(\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{I_{k,\varepsilon}}\geq
\delta\right)\rightarrow 0$ as $\varepsilon\rightarrow 0$.
For initial conditions $z\in \mathcal{M}$ and for integers $k\in
[0,T/(\varepsilon L(\varepsilon))-1]$ define
\[
\begin {split}
\mathcal{A}_{k,\varepsilon} & =\set{z:\frac{1}{L(\varepsilon)}
\abs{I_{k,\varepsilon}}
>\frac{\delta}{2T} \text { and } k\leq\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1 }
,\\
\mathcal{A}_{z,\varepsilon} & =\set{k:z\in
\mathcal{A}_{k,\varepsilon}}.
\end {split}
\]
Think of these sets as describing ``poor continuity'' between
solutions of the $\varepsilon=0$ and the $\varepsilon>0 $
Hamiltonian vector fields. For example, roughly speaking,
$z\in\mathcal{A}_{k,\varepsilon}$ if the orbit $z_{\varepsilon}(t)$
starting at $z$ does not closely follow $z_{k,\varepsilon}(t)$ for
$t_{k,\varepsilon}\leq t\leq t_{k+1,\varepsilon} $.
One can easily check that $\abs{I_{k,\varepsilon}}\leq \mathcal{O}
(L(\varepsilon))$ for $k\leq\ \tilde{T}_\varepsilon/(\varepsilon
L(\varepsilon))-1$, and so it follows that
\[
\varepsilon
\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{I_{k,\varepsilon}}\leq
\frac{\delta}{2}+\mathcal{O}(\varepsilon L(\varepsilon)
\# (\mathcal{A}_{z,\varepsilon})).
\]
Therefore it suffices to show that $P\left(\#
(\mathcal{A}_{z,\varepsilon})\geq\delta(\C\, \varepsilon
L(\varepsilon)) ^ {-1}\right)\rightarrow 0$ as
$\varepsilon\rightarrow 0$. By Chebyshev's Inequality, we need only
show that
\[
E_P (\varepsilon L(\varepsilon)\# (\mathcal{A}_{z,\varepsilon})) =
\varepsilon L(\varepsilon)\sum_{k=0}^{\frac{T}{\varepsilon
L(\varepsilon)}-1}P(\mathcal{A}_{k,\varepsilon})
\]
tends to $0$ with $\varepsilon$.
Observe that $z_\varepsilon
(t_{k,\varepsilon})\mathcal{A}_{k,\varepsilon}\subset\mathcal{A}_{0,\varepsilon}
$. In words, the initial conditions giving rise to orbits that are
``bad'' on the time interval
$[t_{k,\varepsilon},t_{k+1,\varepsilon}] $, moved forward by time
$t_{k,\varepsilon}$, are initial conditions giving rise to orbits
which are ``bad'' on the time interval
$[t_{0,\varepsilon},t_{1,\varepsilon}] $. Because the flow
$z_\varepsilon(\cdot) $ preserves the measure, we find that
\[
\varepsilon L(\varepsilon)\sum_{k=0}^{\frac{T}{\varepsilon
L(\varepsilon)}-1}P(\mathcal{A}_{k,\varepsilon})
\leq \C\, P(\mathcal{A}_{0,\varepsilon}).
\]
To estimate $P(\mathcal{A}_{0,\varepsilon})$, it is convenient to
use a different probability measure, which is uniformly equivalent
to $P$ on the set $\set{z\in\mathcal{M}:h(z)\in\mathcal{V}}
\supset\{\tilde{T}_\varepsilon\geq \varepsilon L(\varepsilon)\} $.
We denote this new probability measure by $P^f$, where the $f$
stands for ``factor.'' If we choose coordinates on $\mathcal{M} $
by using $h$ and the billiard coordinates on the two gas particles,
then $P^f$ is defined on $\mathcal{M}$ by $dP^f
=dh\,d\mu^1_h\,d\mu^2_h$, where $dh$ represents the uniform measure
on $\mathcal{V}\subset\mathbb{R}^4$, and the factor measure
$d\mu^i_h$ represents the invariant billiard measure of the $i^ {th}
$ gas particle coordinates for a fixed value of the slow variables.
One can verify that $1_{\set{h(z)\in\mathcal{V}}}dP \leq \C\, dP^f$,
but that $P^f$ is not invariant under the flow $z_\varepsilon(\cdot)
$ when $\varepsilon>0$.
We abuse notation, and consider $\mu^1_h$ to be a measure on the
left particle's initial billiard coordinates once $h$ and the
initial coordinates of the right gas particle are fixed. In this
context, $\mu^1_h$ is simply the measure $\mu$ from
Subsection~\ref{sct:billiard}. Then
\[
\begin {split}
&
P^f(\mathcal{A}_{0,\varepsilon})
\\
&\leq \int
dh\,d\mu^2_h \cdot\mu_h^1
\set{z:\abs{\frac{1}{L(\varepsilon)}\int_0^{L(\varepsilon)}
G(z_\varepsilon(s))-G(z_0(s))ds}\geq\frac{\delta}{2T}
\text { and } \varepsilon
L(\varepsilon)\leq\tilde{T}_\varepsilon },
\end {split}
\]
and we must show that the last term tends to $0$ with $\varepsilon$.
By the Bounded Convergence Theorem, it suffices to show that for
almost every $h\in\mathcal{V}$ and initial condition for the right
gas particle,
\begin {equation}\label{eq:Gronwall_probability}
\mu_h^1
\set{z:\abs{\frac{1}{L(\varepsilon)}\int_0^{L(\varepsilon)}
G(z_\varepsilon(s))-G(z_0(s))ds}\geq\frac{\delta}{2T}
\text { and } \varepsilon
L(\varepsilon)\leq\tilde{T}_\varepsilon }
\rightarrow 0\text{ as }\varepsilon\rightarrow 0.
\end {equation}
Note that if $G$ were a smooth function and $z_\varepsilon(\cdot) $
were the flow of a smooth family of vector fields $Z(z,\varepsilon)
$ that depended smoothly on $\varepsilon$, then from Gronwall's
Inequality, it would follow that $\sup_{0\leq t\leq
L(\varepsilon)}\abs{z_{\varepsilon}(t)-
z_0(t)}\leq \mathcal{O}(\varepsilon L(\varepsilon)
e^{ \Lip{Z} L(\varepsilon)}).$
If this were the case, then $\abs{L(\varepsilon)^
{-1}\int_0^{L(\varepsilon)}
G(z_\varepsilon(s))-G(z_0(s))ds}=\mathcal{O}(\varepsilon
L(\varepsilon) e^{ \Lip{Z} L(\varepsilon)})$, which would tend to
$0$ with $\varepsilon$. Thus, we need a Gronwall-type inequality
for billiard flows. We obtain the appropriate estimates in Section
\ref{sct:Gronwall}.
\paragraph*{Step 6: Use of ergodicity along fibers to
control $II_{k,\varepsilon} $.}
All that remains to be shown is that for fixed $\delta > 0$,
$P\left(\varepsilon\sum_{k=0}^{\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1}\abs{II_{k,\varepsilon}}\geq
\delta\right)\rightarrow 0$ as $\varepsilon\rightarrow 0$.
For initial conditions $z\in \mathcal{M}$ and for integers $k\in
[0,T/(\varepsilon L(\varepsilon))-1]$ define
\[
\begin {split}
\mathcal{B}_{k,\varepsilon} & =\set{z:\frac{1}{L(\varepsilon)}
\abs{II_{k,\varepsilon}}
>\frac{\delta}{2T} \text { and } k\leq\frac{\tilde{T}_\varepsilon}{\varepsilon
L(\varepsilon)}-1 }
,\\
\mathcal{B}_{z,\varepsilon} & =\set{k:z\in
\mathcal{B}_{k,\varepsilon}}.
\end {split}
\]
Think of these sets as describing ``bad ergodization.'' For
example, roughly speaking, $z\in\mathcal{B}_{k,\varepsilon}$ if the
orbit $z_{\varepsilon}(t)$ starting at $z$ spends the time between
$t_{k,\varepsilon}$ and $t_{k+1,\varepsilon} $ in a region of phase
space where the function $G(\cdot) $ is ``poorly ergodized'' on the
time scale $L(\varepsilon) $ by the flow $z_0(t) $ (as measured by
the parameter $\delta/2T$). Note that $G(z)=\abs{v_1^{\perp }}
\delta_{q_1^{\perp }=Q}$ is not really a function, but that we may
still speak of the convergence of $t^ {-1}\int_0^t G(z_0(s))ds$ as
$t\rightarrow\infty$. As we showed in
Lemma~\ref{lem:ae_convergence}, the limit is $\bar G(h_0) $ for
almost every initial condition.
Proceeding as in Step 5 above, we find that it suffices to show that
for almost every $h\in\mathcal{V}$,
\[
\mu_h^1
\set{z:\abs{\frac{1}{t}\int_0^{t}
G(z_0(s))ds-\bar G(h_0(0))}\geq\frac{\delta}{2T}}
\rightarrow 0\text{ as }t\rightarrow \infty.
\]
But this is simply a question of examining billiard flows, and it
follows immediately from Corollary \ref{cor:ae_convergence} and our
Main Assumption.
\subsection{A Gronwall-type inequality for billiards}\label{sct:Gronwall}
We begin by presenting a general version of Gronwall's Inequality
for billiard maps. Then we will show how these results imply the
convergence required in Equation~\eqref{eq:Gronwall_probability}.
\subsubsection{Some inequalities for the collision map}\label{sct:Gronwall_map}
In this section, we consider the value of the slow variables to be
fixed at $h_0\in\mathcal{V} $. We will use the notation and results
presented in Section~\ref{sct:billiard}, but because the value of
the slow variables is fixed, we will omit it in our notation.
Let $\rho $, $\gamma $, and $\lambda$ satisfy $0<\rho\ll\gamma \ll
1\ll \lambda<\infty $. Eventually, these quantities will be chosen
to depend explicitly on $\varepsilon$, but for now they are fixed.
Recall that the phase space $\Omega$ for the collision map $F$ is a
finite union of disjoint rectangles and cylinders. Let $d$ be the
Euclidean metric on connected components of $\Omega$. If $x$ and
$x'$ belong to different components, then we set $d(x,x') =\infty $.
The invariant measure $\nu$ satisfies $\nu<\C\cdot (\text {Lebesgue
measure}) $. For $A\subset\Omega$ and $a >0$, let $\mathcal{N}_a
(A) =\set {x\in\Omega:d(x,A)< a} $ be the $a$-neighborhood of $A$.
For $x\in\Omega$ let $x_k(x) = x_k =F^k x$, $ k\geq 0$, be its
forward orbit. Suppose $x\notin \mathcal{C}_{\gamma ,\lambda}$,
where
\[
\mathcal{C}_{\gamma,\lambda}=
\bigl(\cup_{k=0} ^\lambda F^
{-k}\mathcal{N}_\gamma (\partial\Omega)\bigr)\bigcup\bigl(\cup_{k=0}
^\lambda F^ {-k}\mathcal{N}_\gamma (F^ {-1}\mathcal{N}_\gamma
(\partial\Omega))\bigr).
\]
Thus for $0\leq k\leq \lambda $, $x_k$ is well defined, and from
Equation~\eqref{eq:2d_derivative_bound} it satisfies
\begin {equation}\label{eq:gron0}
d(x',x_k)\leq \gamma \;\Rightarrow\; d(Fx',x_{k+1})\leq\frac{\C}{\gamma
} d(x',x_k).
\end {equation}
Next, we consider any $\rho$-pseudo-orbit $x'_k$ obtained from $x$
by adding on an error of size $\leq\rho$ at each application of the
map, i.e.~$d(x'_0,x_0)\leq \rho$, and for $k\geq 1$, $d(x'_k,Fx'_{
k-1})\leq \rho$. Provided $d(x_j,x'_j)<\gamma $ for each $j<k$, it
follows that
\begin {equation}\label{eq:gron1}
d(x_k,x'_k)\leq
\rho\sum_{j=0}^{k}\left(\frac{\C}{\gamma}\right)^j\leq
\C\,\rho \left (\frac {\C} {\gamma}\right) ^k .
\end {equation}
In particular, if $\rho $, $\gamma $, and $\lambda$ were chosen such
that
\begin {equation}\label{eq:gron2}
\C\,\rho \left (\frac {\C} {\gamma}\right) ^\lambda<\gamma,
\end {equation}
then Equation~\eqref{eq:gron1} will hold for each $k\leq\lambda $.
We assume that Equation~\eqref{eq:gron2} is true. Then we can also
control the differences in elapsed flight times using
Equation~\eqref{eq:2d_time_derivative}:
\begin {equation}\label{eq:gron3}
\abs{\zeta x_k-\zeta x'_k}
\leq
\frac {\C\,\rho} {\gamma } \left (\frac {\C} {\gamma}\right) ^k.
\end {equation}
It remains to estimate the size $\nu \mathcal{C}_{\gamma,\lambda}$
of the set of $x$ for which the above estimates do not hold. Using
Lemma~\ref{lem:gron1} below,
\begin {equation}\label{eq:gron4}
\nu\mathcal{C}_{\gamma,\lambda}
\leq
(\lambda +1)\bigl(\nu \mathcal{N}_\gamma (\partial\Omega)+\nu \mathcal{N}_\gamma
(F^ {-1}\mathcal{N}_\gamma
(\partial\Omega)) \bigr)
\leq
\mathcal{O} (\lambda (\gamma +\gamma ^ {1/3})) =\mathcal{O} (\lambda \gamma ^
{1/3}).
\end {equation}
\begin{lem}\label{lem:gron1}
As $\gamma \rightarrow 0$,
\[
\nu \mathcal{N}_\gamma (F^ {-1}\mathcal{N}_\gamma
(\partial\Omega))=\mathcal{O}(\gamma ^ {1/3}).
\]
\end {lem}
This estimate is not necessarily the best possible. For example,
for dispersing billiard tables, where the curvature of the boundary
is positive, one can show that $\nu \mathcal{N}_\gamma (F^
{-1}\mathcal{N}_\gamma (\partial\Omega))=\mathcal{O}(\gamma )$.
However, the estimate in Lemma~\ref{lem:gron1} is general and
sufficient for our needs.
\begin {proof}
First, we note that it is equivalent to estimate $\nu
\mathcal{N}_\gamma (F\mathcal{N}_\gamma (\partial\Omega))$, as $F$
has the measure-preserving involution $\mathcal{I} (r,\varphi) = (r,
-\varphi) $, i.e.~$F^ {-1} =\mathcal{I}\circ F\circ\mathcal{I}
$~\cite{CherMark06}.
Fix $\alpha\in (0,1/2) $, and cover $\mathcal{N}_\gamma
(\partial\Omega) $ with $\mathcal{O} (\gamma ^ {-1}) $ starlike
sets, each of diameter no greater than $\mathcal{O} (\gamma) $. For
example, these sets could be squares of side length $\gamma $.
Enumerate the sets as $\set {A_i} $. Set $\mathcal{G}=\set
{i:FA_i\cap \mathcal{N}_{\gamma ^\alpha} (\partial\Omega)
=\varnothing}$.
If $i\in \mathcal{G}$, $F\arrowvert _{A_i} $ is a diffeomorphism
satisfying $\norm{DF\arrowvert_{A_i}}\leq\mathcal{O} (\gamma ^
{-\alpha}) $. See Equation~\eqref{eq:2d_derivative_bound}. Thus
$\dia{FA_i}\leq\mathcal{O} (\gamma ^ {1-\alpha}) $, and so
$\dia{\mathcal{N}_\gamma (FA_i)}\leq\mathcal{O} (\gamma ^
{1-\alpha}) $. Hence $\nu\mathcal{N}_\gamma (FA_i)\leq\mathcal{O}
(\gamma ^ {2(1-\alpha)}) $, and $\nu\mathcal{N}_\gamma
(\cup_{i\in\mathcal{G}}FA_i)\leq\mathcal{O} (\gamma ^ {1-2\alpha})
$.
If $i\notin \mathcal{G}$, $A_i\cap F^ {-1} (\mathcal{N}_{\gamma
^\alpha} (\partial\Omega)) \neq\varnothing $. Thus $A_i$ might be
cut into many pieces by $F^ {-1} (\partial\Omega) $, but each of
these pieces must be mapped near $\partial\Omega $. In fact,
$FA_i\subset\mathcal{N}_{\mathcal{O} (\gamma ^\alpha)}
(\partial\Omega) $. This is because outside $F^ {-1}
(\mathcal{N}_{\gamma ^\alpha} (\partial\Omega))$,
$\norm{DF}\leq\mathcal{O} (\gamma ^ {-\alpha}) $, and so points in
$FA_i$ are no more than a distance $\mathcal{O} (\gamma /\gamma ^
{\alpha}) $ away from $\mathcal{N}_{\gamma^\alpha} (\partial\Omega)
$, and $\gamma <\gamma ^ {1-\alpha} <\gamma ^\alpha $. It follows
that $\mathcal{N}_\gamma (FA_i) \subset\mathcal{N}_{\mathcal{O}
(\gamma ^\alpha)} (\partial\Omega)$, and
$\nu\mathcal{N}_{\mathcal{O} (\gamma ^\alpha)}
(\partial\Omega)=\mathcal{O} (\gamma ^\alpha). $
Thus $\nu \mathcal{N}_\gamma (F^ {-1}\mathcal{N}_\gamma
(\partial\Omega))=\mathcal{O}(\gamma ^ {1-2\alpha}+\gamma ^
{\alpha})$, and we obtain the lemma by taking $\alpha=1/3$.
\end {proof}
\subsubsection{Application to a perturbed billiard flow}
\label{sct:gron_ap}
Returning to the end of Step 5 in Section~\ref{sct:main_steps}, let
the initial conditions of the slow variables be fixed at
$h_0=(Q_0,W_0,E_{1,0},E_{2,0})\in\mathcal{V} $ throughout the
remainder of this section. We can assume that the billiard dynamics
of the left gas particle in $\mathcal{D}_1(Q_0) $ are ergodic. Also,
fix a particular value of the initial conditions for the right gas
particle for the remainder of this section. Then $z_\varepsilon(t)
$ and $\tilde T_\varepsilon$ may be thought of as random variables
depending on the left gas particle's initial conditions
$y\in\mathcal{M} ^1$. Now if $h_\varepsilon (t)=
(Q_\varepsilon(t),W_\varepsilon(t),E_{1,\varepsilon}(t),E_{2,\varepsilon}(t))$
denotes the actual motions of the slow variables when
$\varepsilon>0$, it follows from Equation~\eqref{eq:h_div} that,
provided $\varepsilon L(\varepsilon)\leq \tilde {T}_\varepsilon $,
\begin {equation}\label{eq:h_div2}
\sup_{0\leq t\leq
L(\varepsilon)}\abs{h_0-h_\varepsilon(t)}=\mathcal{O}(\varepsilon L(\varepsilon)).
\end {equation}
Furthermore, we only need to show that
\begin {equation}\label{eq:gron5}
\mu
\set{y\in\mathcal{M} ^1:\abs{\frac{1}{L(\varepsilon)}\int_0^{L(\varepsilon)}
G(z_\varepsilon(s))-G(z_0(s))ds}\geq\frac{\delta}{2T}
\text { and } \varepsilon
L(\varepsilon)\leq\tilde{T}_\varepsilon }
\rightarrow 0
\end {equation}
as $\varepsilon\rightarrow 0$, where $G$ is defined in
Equation~\eqref{eq:G_definition}.
For definiteness, we take the following quantities from
Subsection~\ref{sct:Gronwall_map} to depend on $\varepsilon$ as
follows:
\begin {equation}\label{eq:gron6}
\begin {split}
L(\varepsilon) &= L=\log \log\frac{1}{\varepsilon},
\\
\gamma (\varepsilon) &= \gamma =e^{-L},
\\
\lambda(\varepsilon)&=\lambda=
\frac{2}{E_{\nu}\zeta}L,
\\
\rho(\varepsilon) &=\rho=\C\frac {\varepsilon L} {\gamma }.
\end {split}
\end {equation}
The constant in the choice of $\rho$ and $\rho$'s dependence on
$\varepsilon$ will be explained in the proof of
Lemma~\ref{lem:gron3}, which is at the end of this subsection. The
other choices may be explained as follows. We wish to use continuity
estimates for the billiard map to produce continuity estimates for
the flow on the time scale $L$. As the divergence of orbits should
be exponentially fast, we choose $L$ to grow sublogarithmically in
$\varepsilon^ {-1} $. Since from Equation~\eqref{eq:2d_Santalo} the
expected flight time between collisions with
$\partial\mathcal{D}_1(Q_0)$ when $\varepsilon=0$ is
$E_{\nu}\zeta=\pi\abs{\mathcal{D}_1(Q_0)}/(\sqrt{2E_{1,0}}\abs{\partial\mathcal{D}_1(Q_0)})$,
we expect to see roughly $\lambda/2$ collisions on this time scale.
Considering $\lambda$ collisions gives us some margin for error.
Furthermore, we will want orbits to keep a certain distance, $\gamma
$, away from the billiard discontinuities. $\gamma \rightarrow 0$ as
$\varepsilon\rightarrow 0$, but $\gamma $ is very large compared to
the possible drift $\mathcal{O} (\varepsilon L) $ of the slow
variables on the time scale $L$. In fact, for each $C,m,n>0$,
\begin {equation}\label{eq:gron7}
\frac {\varepsilon L^m } {\gamma ^n}
\left (\frac {C} {\gamma}\right)^\lambda=\mathcal{O}
(\varepsilon\, e^{\C\,L^2})
\rightarrow 0\text { as }\varepsilon\rightarrow 0.
\end {equation}
Let $X:\mathcal{M} ^1\rightarrow\Omega$ be the map taking
$y\in\mathcal{M} ^1$ to $x=X(y)\in\Omega $, the location of the
billiard orbit of $y$ in the collision cross-section that
corresponds to the most recent time in the past that the orbit was
in the collision cross-section. We consider the set of initial
conditions
\[
\mathcal{E}_\varepsilon=
X^{-1}(\Omega\backslash\mathcal{C}_{\gamma ,\lambda})\bigcap
X^{-1}
\set {x\in\Omega: \sum_{k=0}^\lambda \zeta (F^k x)>
L}.
\]
Now from Equations~\eqref{eq:gron4} and~\eqref{eq:gron6},
$\nu\mathcal{C}_{\gamma ,\lambda}\rightarrow 0$ as
$\varepsilon\rightarrow 0$. Furthermore, by the ergodicity of $F$,
$\nu\set {x\in\Omega:\sum_{k=0}^\lambda \zeta (F^k x)\leq L}=\nu\set
{x\in\Omega:\lambda^ {-1}\sum_{k=0}^\lambda \zeta (F^k x)\leq E_\nu
\zeta/2}\rightarrow 0$ as $\varepsilon\rightarrow 0$. But because
the free flight time is bounded above, $\mu X^ {-1}\leq \C\cdot \nu
$, and so $\mu\mathcal{E}_\varepsilon\rightarrow 1$ as
$\varepsilon\rightarrow 0$. Hence, the convergence in
Equation~\eqref{eq:gron5} and the conclusion of the proof in
Section~\ref{sct:main_steps} follow from the lemma below and
Equation~\eqref{eq:gron7}.
\begin{lem}[Analysis of deviations along good orbits]\label{lem:gron2}
As $\varepsilon\rightarrow 0$,
\[
\sup_{y\in\mathcal{E}_\varepsilon \cap \set{\varepsilon L\leq \tilde {T}_\varepsilon
}}
\abs{\frac{1}{L}\int_0^{L}
G(z_\varepsilon(s))-G(z_0(s))ds}=
\mathcal{O} \left(\rho
\left(\frac {\C} {\gamma} \right)^ {\lambda}\right)
+\mathcal{O}(L^ {- 1})\rightarrow 0.
\]
\end {lem}
\begin {proof}
Fix a particular value of $y\in\mathcal{E}_\varepsilon \cap
\set{\varepsilon L\leq \tilde {T}_\varepsilon}$. For convenience,
suppose that $y=X(y) =x\in\Omega $. Let $y_0(t) $ denote the time
evolution of the billiard coordinates for the left gas particle when
$\varepsilon=0$. Then there is some $N\leq \lambda$ such that the
orbit $ x_k=F^k x= ( r_k,\varphi_k)$ for $0\leq k\leq N $
corresponds to all of the instances (in order) when $y_0(t) $ enters
the collision cross-section $\Omega=\Omega_{ h_0} $ corresponding to
collisions with $\partial\mathcal{D}_1(Q_0) $ for $0\leq t\leq L$.
We write $\Omega_{ h_0}$ to emphasize that in this subsection we are
only considering the collision cross-section corresponding to the
billiard dynamics in the domain $\mathcal{D}_1(Q_0) $ at the energy
level $ E_{1,0} $. In particular, $F$ will always refer to the
return map on $\Omega_{ h_0}$.
Also, define an increasing sequence of times $ t_k$ corresponding to
the actual times $ y_0(t) $ enters the collision cross-section, i.e.
\[
\begin {split}
t_0 &=0,\\
t_k & = t_{k-1} +\zeta x_{k-1}\text { for } k>0.
\end {split}
\]
Then $ x_k= y_0 ( t_k) $. Furthermore, define inductively
\[
\begin {split}
N_1&=\inf\set{k>0: t_k\text{ corresponds to a collision with the
piston}},\\
N_j&=\inf\set{k>N_{j-1}: t_k\text{ corresponds to a collision with the
piston}}.\\
\end {split}
\]
Next, let $y_\varepsilon(t) $ denote the time evolution of the
billiard coordinates for the left gas particle when $\varepsilon>0$.
We will construct a pseudo-orbit $x_{k,\varepsilon}' =
(r_{k,\varepsilon}',\varphi_{k,\varepsilon}')$ of points in
$\Omega_{ h_0}$ that essentially track the collisions (in order) of
the left gas particle with the boundary under the dynamics of
$y_\varepsilon(t) $ for $0\leq t\leq L$.
First, define an increasing sequence of times $ t_{k,\varepsilon}'$
corresponding to the actual times $ y_\varepsilon(t) $ experiences a
collision with the boundary of the gas container or the moving
piston. Define
\[
\begin {split}
N_{\varepsilon}'&=\sup\set{k\geq 0: t_{k,\varepsilon}'
\leq L},\\
N_{1,\varepsilon}'&=\inf\set{k>0: t_{k,\varepsilon}'
\text{ corresponds to a collision with the
piston}},\\
N_{j,\varepsilon}'&=\inf\set{k>N_{j-1,\varepsilon}': t_{k,\varepsilon}'
\text{ corresponds to a collision with the
piston}}.\\
\end {split}
\]
Because $L\leq \tilde {T}_\varepsilon(y)/\varepsilon$, we know that
as long as $N_{j+1,\varepsilon}'\leq N_{\varepsilon}'$, then
$N_{j+1,\varepsilon}'-N_{j,\varepsilon}'\geq 2$. See the discussion
in Subsection~\ref{sct:collisions}. Then we define
$x_{k,\varepsilon}'\in\Omega_{ h_0}$ by
\[
x_{k,\varepsilon}' =
\begin {cases}
y_\varepsilon (t_{k,\varepsilon}')
\text { if }k\notin\set {N_{j,\varepsilon}'},\\
F^ {-1}x_{k+1,\varepsilon}'
\text { if }k\in\set {N_{j,\varepsilon}'}.
\end {cases}
\]
\begin{lem}\label{lem:gron3}
Provided $\varepsilon$ is sufficiently small, the following hold for
each $k\in [0,N\wedge N_\varepsilon')$. Furthermore, the requisite
smallness of $\varepsilon$ and the sizes of the constants in these
estimates may be chosen independent of the initial condition
$y\in\mathcal{E}_\varepsilon \cap \set{\varepsilon L\leq \tilde
{T}_\varepsilon}$ and of $k$:
\begin {itemize}
\item[\emph{(a)}]
$x_{k,\varepsilon}' $ is well defined. In particular, if
$k\notin\set{N_{j,\varepsilon}'} $,
$y_\varepsilon(t_{k,\varepsilon}') $ corresponds to a
collision point on $\partial\mathcal{D}_1( Q_0)$, and not to
a collision point on a piece of $\partial\mathcal{D} $ to
the right of $ Q_0$.
\item[\emph{(b)}]
If $ k>0$ and $k\notin\set{N_{j,\varepsilon}'} $, then
$x_{k,\varepsilon}' =Fx_{k-1,\varepsilon}' $.
\item[\emph{(c)}]
If $ k>0$ and $k\in\set{N_{j,\varepsilon}'} $, then
$d(x_{k,\varepsilon}',Fx_{k-1,\varepsilon}')\leq\rho$ and
the $\varphi$ coordinate of $y_\varepsilon(t_{k,\varepsilon}') $
satisfies
$\varphi(y_\varepsilon(t_{k,\varepsilon}')) =\varphi_{k,\varepsilon}' +
\mathcal{O} (\varepsilon).$
\item[\emph{(d)}]
$d(x_k,x'_{k,\varepsilon})\leq\C\,\rho (\C/\gamma) ^k$ .
\item[\emph{(e)}]
$k=N_{j,\varepsilon}'$ if and only if $k=N_j$.
\item[\emph{(f)}]
If $ k>0$, $t_{k,\varepsilon}'-t_{k-1,\varepsilon}'
=
t_k - t_{ k-1} +
\mathcal{O}(\rho \left (\C/\gamma\right) ^k).$
\end {itemize}
\end{lem}
We defer the proof of Lemma~\ref{lem:gron3} until the end of this
subsection. Assuming that $\varepsilon$ is sufficiently small for
the conclusions of Lemma~\ref{lem:gron3} to be valid, we continue
with the proof of Lemma~\ref{lem:gron2}.
Set $M=N\wedge N_\varepsilon'-1$. Note that $M\leq\lambda\sim L $.
From (f) in Lemma~\ref{lem:gron3} and Equations~\eqref{eq:gron6} and
\eqref{eq:gron7}, we see that
\[
\begin {split}
\abs{t_M-t_{M,\varepsilon}'}
&
\leq \sum_{k=1}^M
\abs{t_{k,\varepsilon}'-t_{k-1,\varepsilon}'- (t_k - t_{ k-1})}
=
\mathcal{O}\left(\rho \frac{\C^\lambda}{\gamma^{\lambda}}\right)
\rightarrow 0\text{ as }\varepsilon\rightarrow 0.
\end {split}
\]
Because the flight times $t_{k,\varepsilon}'-t_{k-1,\varepsilon}'$
and $t_k - t_{ k-1}$ are uniformly bounded above, it follows from
the definitions of $N$ and $N_\varepsilon' $ that $t_M,\,
t_{M,\varepsilon}'\geq L-\C$. But from
Subsection~\ref{sct:collisions}, the time between the collisions of
the left gas particle with the piston are uniformly bounded away
from zero. Using (c) and Equation~\eqref{eq:h_div2}, it follows that
\[
\begin {split}
&\abs{\frac{1}{L}
\int_0^{L}G(z_\varepsilon(s))-G(z_0(s))ds}
\\
&\qquad
=\mathcal{O} (L^ {-1}) +
\sum_{k\in \set { N_j:N_j\leq M}} \abs{\sqrt{2E_{1,0}}\,\cos \varphi_k
-\sqrt{2E_{1,\varepsilon}(t_{k,\varepsilon}')}\,\cos
(\varphi_{k,\varepsilon}'+\mathcal{O} (\varepsilon))}
\\
&\qquad
=\mathcal{O} (L^ {-1}) +
\sum_{k\in \set { N_j:N_j\leq M}}
\abs{\sqrt{2E_{1,0}}\,\cos \varphi_k
-\sqrt{2E_{1,0}}\,\cos
\varphi_{k,\varepsilon}'
+\mathcal{O} (\varepsilon L)}
\\
&\qquad
=\mathcal{O} (L^ {-1}) +
\mathcal{O} (\varepsilon L^2)
+\sqrt{2E_{1,0}}\,\sum_{k\in \set { N_j:N_j\leq M}}
\abs{\cos \varphi_k
-\cos
\varphi_{k,\varepsilon}'}.
\end {split}
\]
But using (d),
\[
\begin {split}
\sum_{k\in \set { N_j:N_j\leq M}}
\abs{\cos \varphi_k-\cos
\varphi_{k,\varepsilon}'}
\leq\sum_{ k=0} ^M\mathcal{O} (\rho (\C/\gamma) ^k)
=\mathcal{O} (\rho (\C/\gamma) ^\lambda).
\end {split}
\]
Since $\varepsilon L^2=\mathcal{O}(\rho (\C/\gamma) ^\lambda) $,
this finishes the proof of Lemma~\ref{lem:gron2}.
\end {proof}
\begin {proof}[Proof of Lemma~\ref{lem:gron3}]
The proof is by induction. We take $\varepsilon$ to be so small
that Equation~\eqref{eq:gron2} is satisfied. This is possible by
Equation~\eqref{eq:gron7}.
It is trivial to verify (a)-(f) for $ k=0$. So let $0<l<N\wedge
N_\varepsilon' $, and suppose that (a)-(f) have been verified for
all $k<l$. We have three cases to consider:
\subsubsection*{Case 1: $l-1$ and $l\notin \set{N_{j,\varepsilon}'}$:}
In this case, verifying (a)-(f) for $ k=l$ is a relatively
straightforward application of the machinery developed in
Subsection~\ref{sct:Gronwall_map}, because for
$t_{l-1,\varepsilon}'\leq t\leq t_{l,\varepsilon}'$, $y_\varepsilon
(t) $ traces out the billiard orbit between $x_{l-1,\varepsilon}'$
and $x_{l,\varepsilon}'$ corresponding to free flight in the domain
$\mathcal{D}_1( Q_0) $. We make only two remarks.
First, as long as $\varepsilon$ is sufficiently small, it really is
true that $x_{l,\varepsilon}'=y_\varepsilon (t_{l,\varepsilon}')$
corresponds to a true collision point on $\partial\mathcal{D}_1(
Q_0) $. Indeed, if this were not the case, then it must be that
$Q_\varepsilon(t_{l,\varepsilon}')> Q_0 $, and $y_\varepsilon
(t_{l,\varepsilon}')$ would have to correspond to a collision with
the side of the ``tube'' to the right of $ Q_0$. But then
$x_{l,\varepsilon}'' =Fx_{l-1,\varepsilon}'\in\Omega_{ h_0}$ would
correspond to a collision with an immobile piston at $ Q_0$ and
would satisfy $d(x_k,x''_{k,\varepsilon})\leq\C\,\rho (\C/\gamma)
^k\leq \C\,\rho (\C/\gamma) ^\lambda =o(\gamma )$, using
Equations~\eqref{eq:gron1} and \eqref{eq:gron7}. But $
x_k\notin\mathcal{N}_\gamma (\partial\Omega_{ h_0}) $, and so it
follows that when the trajectory of $y_\varepsilon(t) $ crosses the
plane $\set {Q= Q_0} $, it is at least a distance $\sim \gamma $
away from the boundary of the face of the piston, and its velocity
vector is pointed no closer than $\sim \gamma $ to being parallel to
the piston's face. As $Q_\varepsilon(t_{l,\varepsilon}')-
Q_0=\mathcal{O} (\varepsilon L) =o(\gamma ) $, and it is
geometrically impossible (for small $\varepsilon$) to construct a
right triangle whose sides $ s_1,\: s_2$ satisfy $\abs{ s_1}\geq\sim
\gamma ,\:\abs{ s_2}\leq\mathcal{O} (\varepsilon L) $, with the
measure of the acute angle adjacent to $ s_1$ being greater than
$\sim \gamma$, we have a contradiction. After crossing the plane
$\set {Q= Q_0} $, $y_\varepsilon(t) $ must experience its next
collision with the face of the piston, which violates the fact that
$l\notin \set{N_{j,\varepsilon}'}$.
Second, $t_{l,\varepsilon}'-t_{l-1,\varepsilon}' =\zeta
x'_{l-1,\varepsilon}+\mathcal{O}(\varepsilon L)$, because
$v_{1,\varepsilon} =v_ {1,0} +\mathcal{O} (\varepsilon L) $. See
Equation~\eqref{eq:h_div2}. From Equation~\ref{eq:gron3},
$\abs{\zeta x_{l-1}-\zeta x'_{l-1,\varepsilon}} \leq \mathcal{O}
((\rho/\gamma) \left (\C/\gamma\right) ^{l-1})$. As $t_l - t_{
l-1}=\zeta x_{l-1}$ and $\varepsilon L=\mathcal{O} ((\rho/\gamma)
\left (\C/\gamma\right) ^{l-1})$, we obtain (f).
\subsubsection*{Case 2: There exists $i$ such that $l=N_{i,\varepsilon}'$: }
For definiteness, we suppose that
$Q_\varepsilon(t_{l,\varepsilon}')\geq Q_0$, so that the left gas
particle collides with the piston to the right of $ Q_0$. The case
when $Q_\varepsilon(t_{l,\varepsilon}')\leq Q_0$ can be handled
similarly.
We know that $ x_{l-1}, x_{l},x_{l+1}\notin \mathcal{N}_\gamma
(\partial\Omega_{ h_0})\cup\mathcal{N}_\gamma (F^
{-1}\mathcal{N}_\gamma (\partial\Omega_{ h_0}))$. Using the
inductive hypothesis and Equation~\eqref{eq:gron1}, we can define
\[
x_{l,\varepsilon}''=Fx_{l-1,\varepsilon}',\qquad
x_{l+1,\varepsilon}''=F^2x_{l-1,\varepsilon}',
\]
and $d( x_{l},x_{l,\varepsilon}'') \leq\C\,\rho (\C/\gamma) ^{l}$,
$d( x_{l+1}, x_{l+1,\varepsilon}'')\leq\C\,\rho (\C/\gamma) ^{l+1}$.
In particular, $x_{l,\varepsilon}''$ and $x_{l+1,\varepsilon}''$ are
both a distance $\sim \gamma $ away from $\partial\Omega_{ h_0} $.
Furthermore, when the left gas particle collides with the moving
piston, it follows from Equation~\eqref{eq:v_1Wchange} that the
difference between its angle of incidence and its angle of
reflection is $\mathcal{O} (\varepsilon) $. Referring to
Figure~\ref{fig:collision}, this means that
$\varphi_{l,\varepsilon}' =\varphi_{l,\varepsilon}'' +\mathcal{O}
(\varepsilon) $. Geometric arguments similar to the one given in
Case 1 above show that the $y_\varepsilon$-trajectory of the left
gas particle has precisely one collision with the piston and no
other collisions with the sides of the gas container when the gas
particle traverses the region $Q_0\leq Q\leq
Q_\varepsilon(t_{l,\varepsilon}')$. Note that $x_{l,\varepsilon}' $
was defined to be the point in the collision cross-section $\Omega_{
h_0} $ corresponding to the return of the $y_\varepsilon$-trajectory
into the region $Q\leq Q_0$. See Figure~\ref{fig:collision}. From
this figure, it is also evident that
$d(r_{l,\varepsilon}',r_{l,\varepsilon}'')\leq\mathcal{O}
(\varepsilon L/\gamma ) $. Thus $d(
x_{l,\varepsilon}'',x_{l,\varepsilon}')=\mathcal{O} (\varepsilon
L/\gamma )$, and this explains the choice of $\rho(\varepsilon) $ in
Equation~\eqref{eq:gron6}.
\begin{figure}
\begin {center}
\setlength{\unitlength}{1.0 cm}
\begin{picture}(15,10)
\thicklines
\put(1,1.5){\line(1,0){10.5}}
\put(1,8.5){\line(1,0){10.5}}
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\put(3.3,1.15){$r-$coordinate}
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\put(1,8.5){\line(1,0){11}}
\put(11.5,1.4){\line(0,1){7.1}}
\put(11.3,1.0){$Q_0$}
\put(12,1.4){\line(0,1){7.1}}
\put(11.9,1.0){$Q_\varepsilon(t_{l,\varepsilon}')$}
\put(1.5,8){$\mathcal{D}_1(Q_0) $}
\put(9.5,8.8){\line(1,0){4}}
\put(11.5,8.8){\line(0,-1){0.15}}
\put(13.5,8.8){\line(0,-1){0.15}}
\put(9.5,8.8){\line(0,-1){0.15}}
\put( 10.5,8.8){\line(0,1){0.15}}
\put(10.2,9.1){$\gamma/2 $}
\put(12.5,8.8){\line(0,1){0.15}}
\put(12.2,9.1){$\gamma/2 $}
\put(11.5,0.6){\line(1,0){0.5}}
\put(11.75,0.6){\line(0,-1){0.15}}
\put( 11.5,0.6){\line(0,1){0.15}}
\put( 12,0.6){\line(0,1){0.15}}
\put(11.4,0.1){$\mathcal{O} (\varepsilon L) $}
\put(12.5,4.5){\line(0,1){1}}
\put(12.5,4.5){\line(-1,0){0.15}}
\put(12.5,5){\line(1,0){0.15}}
\put(12.5,5.5){\line(-1,0){0.15}}
\put(12.7,4.85){$\mathcal{O} (\varepsilon L/\gamma ) $}
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\put(12.5,8.5){\line(-1,0){0.15}}
\put(12.5,7.5){\line(1,0){0.15}}
\put(12.5,6.5){\line(-1,0){0.15}}
\put(12.7,7.45){$\gamma /2 $}
\put(12.5,1.5){\line(0,1){2}}
\put(12.5,3.5){\line(-1,0){0.15}}
\put(12.5,2.5){\line(1,0){0.15}}
\put(12.5,1.5){\line(-1,0){0.15}}
\put(12.7,2.45){$\gamma /2$}
\small
\put(12,5){\line(-1,-1){3.5}}
\put(11.5,4.5){\vector(-1,1){0.8}}
\put(11.5,4.5){\line(-1,1){4}}
\put(11.65,4.4){$r_{l,\varepsilon}''$}
\put(10.35,4.7){$\varphi_{l,\varepsilon}''$}
\put(11.5,4.5){\circle*{.1}}
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\put(12,5){\vector(-1,1){1.3}}
\put(12,5){\line(-1,1){3.5}}
\put(11.65,5.4){$r_{l,\varepsilon}'$}
\put(10.35,5.7){$\varphi_{l,\varepsilon}'$}
\put(11.5,5.5){\circle*{.1}}
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\put(8.5,1.5){\circle*{.1}}
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\qbezier[24](8.5,1.5)(8.5,2.25)(8.5,3.0)
\put(8.45,1.15){$r_{l-1,\varepsilon}'$}
\put(8.57,2.6){$\varphi_{l-1,\varepsilon}'$}
\put(8.5,1.9){\vector(1,-1){0.2}}
\put(8.5,8.5){\circle*{.1}}
\put(8.3,8.7){$r_{l+1,\varepsilon}'$}
\put(7.5,8.5){\circle*{.1}}
\put(6.95,8.7){$r_{l+1,\varepsilon}''$}
\normalsize
\end{picture}
\end {center}
\caption{An analysis of the divergences of orbits when $\varepsilon>0$
and the left gas particle collides with the moving piston to the right of $Q_0$. Note that the
dimensions are distorted for visual clarity, but that $\varepsilon L$
and $\varepsilon L/\gamma $ are both $o(\gamma ) $ as $\varepsilon\rightarrow
0$.} Furthermore, $\varphi_{l,\varepsilon}''\in(-\pi/2+\gamma/2,\pi/2-\gamma/2) $
and $\varphi_{l,\varepsilon}' =\varphi_{l,\varepsilon}'' +\mathcal{O} (\varepsilon) $,
and so $r_{l,\varepsilon}' =r_{l,\varepsilon}''+\mathcal{O}
(\varepsilon L/\gamma ) $. In particular, the
$y_\varepsilon$-trajectory of the left gas particle has precisely
one collision with the piston and no other collisions with the sides
of the gas container when the gas particle traverses the region
$Q_0\leq Q\leq Q_\varepsilon(t_{l,\varepsilon}')$
\label{fig:collision}
\end{figure}
From the above discussion and the machinery of
Subsection~\ref{sct:Gronwall_map}, (a)-(e) now follow readily for
\emph{both} $ k=l$ and $ k=l+1$. Furthermore, property (f) follows
in much the same manner as it did in Case 1 above. However, one
should note that $t_{l,\varepsilon}'-t_{l-1,\varepsilon}' =\zeta
x'_{l-1,\varepsilon}+\mathcal{O}(\varepsilon L
)+\mathcal{O}(\varepsilon L/\gamma )$ and
$t_{l+1,\varepsilon}'-t_{l,\varepsilon}' =\zeta
x'_{l,\varepsilon}+\mathcal{O}(\varepsilon L
)+\mathcal{O}(\varepsilon L/\gamma )$, because of the extra distance
$\mathcal{O} (\varepsilon L/\gamma ) $ that the gas particle travels
to the right of $ Q_0$. But $\varepsilon L/\gamma =\mathcal{O}
((\rho/\gamma) \left (\C/\gamma\right) ^{l-1})$, and so property (f)
follows.
\subsubsection*{Case 3: There exists $i$ such that $l-1=N_{i,\varepsilon}'$: }
As mentioned above, the inductive step in this case follows
immediately from our analysis in Case 2.
\end {proof}
\section{Generalization to a full proof of Theorem~\ref{thm:dDpiston}}
\label{sct:generalization}
It remains to generalize the proof in Sections~\ref{sct:2dprep} and
\ref{sct:2dproof} to the cases when $ n_1, n_2\geq 1$ and $d=3$.
\subsection{Multiple gas particles on each side of the piston}\label{sct:multiple}
When $d=2$, but $n_1,n_2\geq 1$, only minor modifications are
necessary to generalize the proof above. As in
Subsection~\ref{sct:collisions}, one defines a stopping time $\tilde
{T}_\varepsilon$ satisfying $P\set {\tilde {T}_\varepsilon < T\wedge
T_\varepsilon} =\mathcal{O} (\varepsilon) $ such that for $0\leq
t\leq \tilde {T}_\varepsilon/\varepsilon$, gas particles will only
experience clean collisions with the piston.
Next, define $H(z) $ by
\[
H(z) =
\begin{bmatrix}
W\\
+2\sum_{j=1}^{n_1}\abs{v_{1,j}^{\perp }} \delta_{q_{1,j}^{\perp }=Q}
-2\sum_{j=1}^{n_2}\abs{v_{2,j}^{\perp }} \delta_{q_{2,j}^{\perp }=Q}\\
-2W\abs{v_{1,j}^{\perp }} \delta_{q_{1,j}^{\perp }=Q}\\
+2W\abs{v_{2,j}^{\perp }} \delta_{q_{2,j}^{\perp }=Q}\\
\end{bmatrix}.
\]
It follows that for $0 \leq t\leq \tilde
{T}_\varepsilon/\varepsilon$, $
h_\varepsilon(t)-h_\varepsilon(0)=
\mathcal{O}(\varepsilon)+\varepsilon\int_0^t
H(z_\varepsilon(s))ds.
$ From here, the rest of the proof follows the same steps made in
Subsection~\ref{sct:main_steps}. We note that at Step 3, we find
that $H(z) -\bar H(h(z)) $ divides into $n_1+ n_2 $ pieces, each of
which depends on only one gas particle when the piston is held
fixed.
\subsection{Three dimensions}\label{sct:higher_d}
The proof of Theorem~\ref{thm:dDpiston} in $d=3$ dimensions is
essentially the same as the proof in two dimensions given above. The
principal differences are due to differences in the geometry of
billiards. We indicate the necessary modifications.
In analogy with Section~\ref{sct:billiard}, we briefly summarize the
necessary facts for the billiard flows of the gas particles when
$M=\infty $ and the slow variables are held fixed at a specific
value $h\in\mathcal{V} $. As before, we will only consider the
motions of one gas particle moving in $\mathcal{D}_1 $. Thus we
consider the billiard flow of a point particle moving inside the
domain $\mathcal{D}_1$ at a constant speed $\sqrt{2E_1} $. Unless
otherwise noted, we use the notation from
Section~\ref{sct:billiard}.
The billiard flow takes place in the five-dimensional space
$\mathcal{M}^1=\{(q_1,v_1)\in\mathcal{TD}_1:q_1\in\mathcal{ D}_1,\;
\abs{v_1}=\sqrt{2E_1}\}/\sim$. Here the quotient means that when
$q_1\in\partial\mathcal{ D}_1$, we identify velocity vectors
pointing outside of $\mathcal{D}_1$ with those pointing inside
$\mathcal{D}_1$ by reflecting orthogonally through the tangent plane
to $\partial\mathcal{D}_1$ at $ q_1$. The billiard flow preserves
Liouville measure restricted to the energy surface. This measure
has the density $d\mu=dq_1dv_1/(8\pi E_1\abs{\mathcal{D}_1} ) $.
Here $dq_1$ represents volume on $\mathbb{R}^3$, and $ dv_1$
represents area on
$S^2_{\sqrt{2E_1}}=\set{v_1\in\mathbb{R}^3:\abs{v_1}=\sqrt{2E_1}}$.
The collision cross-section
$\Omega=\{(q_1,v_1)\in\mathcal{TD}_1:q_1\in\partial\mathcal{ D}_1,\;
\abs{v_1}=\sqrt{2E_1}\}/\sim$ is properly thought of as a fiber
bundle, whose base consists of the smooth pieces of
$\partial\mathcal{D}_1$ and whose fibers are the set of outgoing
velocity vectors at $q_1\in\partial\mathcal{ D}_1$. This and other
facts about higher-dimensional billiards, with emphasis on the
dispersing case, can be found in~\cite{BalCheSzaTot_2003}. For our
purposes, $\Omega$ can be parameterized as follows. We decompose
$\partial\mathcal{D}_1$ into a finite union $\cup_j \Gamma_j$ of
pieces, each of which is diffeomorphic via coordinates $r$ to a
compact, connected subset of $\mathbb{R}^2$ with a piecewise
$\mathcal{C} ^3$ boundary. The $\Gamma_j$ are nonoverlapping,
except possibly on their boundaries. Next, if $(q_1,v_1)\in\Omega $
and $ v_1$ is the outward going velocity vector, let $\hat v =
v_1/\abs{v_1} $. Then $\Omega$ can be parameterized by $\{x=(r,\hat
v)\}$. It follows that $\Omega$ it is diffeomorphic to $\cup_j
\Gamma_j\times S^{2 +}$, where $S^{2 +}$ is the upper unit
hemisphere, and by $\partial\Omega$ we mean the subset diffeomorphic
to $(\cup_j
\partial\Gamma_j\times S^{2 +})\bigcup (\cup_j \Gamma_j\times
\partial S^{2 +})$. If $x\in\Omega $, we let $\varphi\in [0,\pi /2]$
represent the angle between the outgoing velocity vector and the
inward pointing normal vector $n$ to $\partial\mathcal{D}_1$,
i.e.~$\cos\varphi=\langle \hat v, n\rangle$. Note that we no longer
allow $\varphi$ to take on negative values. The return map
$F:\Omega\circlearrowleft$ preserves the projected probability
measure $\nu $, which has the density $d\nu=\cos\varphi\, d\hat v \,
dr/(\pi\abs{\partial\mathcal{D}_1}) $. Here
$\abs{\partial\mathcal{D}_1}$ is the area of
$\partial\mathcal{D}_1$.
$F$ is an invertible, measure preserving transformation that is
piecewise $\mathcal{C} ^2$. Because of our assumptions on
$\mathcal{D}_1$, the free flight times and the curvature of
$\partial\mathcal{D}_1$ are uniformly bounded. The bound on $\norm
{DF(x)}$ given in Equation~\eqref{eq:2d_derivative_bound} is still
true. A proof of this fact for general three-dimensional billiard
tables with finite horizon does not seem to have made it into the
literature, although see~\cite{BalCheSzaTot_2003} for the case of
dispersing billiards. For completeness, we provide a sketch of a
proof for general billiard tables in Appendix~\ref{sct:d_bounds}.
We suppose that the billiard flow is ergodic, so that $F$ is
ergodic. Again, we induce $F$ on the subspace $\hat\Omega$ of
$\Omega$ corresponding to collisions with the (immobile) piston to
obtain the induced map $\hat F:\hat\Omega\circlearrowleft$ that
preserves the induced measure $\hat \nu$.
The free flight time $\zeta:\Omega\rightarrow \mathbb{R}$ again
satisfies the derivative bound given in
Equation~\eqref{eq:2d_time_derivative}. The generalized
Santal\'{o}'s formula\cite{Chernov1997} yields
\[
E_\nu \zeta=\frac {4
\abs{\mathcal{D}_1}} {\abs{v_1}\abs{\partial\mathcal{D}_1}}.
\]
If $\hat\zeta:\hat\Omega\rightarrow\mathbb{R} $ is the free flight
time between collisions with the piston, then it follows from
Proposition \ref{prop:inducing} that
\[
E_{\hat\nu} \hat\zeta=\frac {4
\abs{\mathcal{D}_1}}{\abs{v_1}\ell}.
\]
The expected value of $ \abs{v_1^\perp }$ when the left gas particle
collides with the (immobile) piston is given by
\[
E_{\hat\nu} \abs{v_1^\perp }=E_{\hat\nu} \sqrt{2E_1}\cos\varphi=
\frac{\sqrt{2E_1}}{\pi}\iint_{S^ {2+}} \cos^2\varphi\,d\hat v_1=
\sqrt{2E_1}\frac{2}{3}.
\]
As a consequence, we obtain
\begin {lem}
\label{lem:ae_convergence_3d}
For $\mu-a.e.$ $y\in \mathcal{M}^1$,
\[
\lim_{t\rightarrow\infty} \frac{1}{t}
\int_0^t \abs{v_1^\perp (s)}\delta_{q_1^\perp (s)
=Q}ds=
\frac{E_1\ell}{3\abs{\mathcal{D}_1(Q)}}.
\]
\end {lem}
\noindent Compare the proof of Lemma~\ref{lem:ae_convergence}.
With these differences in mind, the rest of the proof of
Theorem~\ref{thm:dDpiston} when $d=3$ proceeds in the same manner as
indicated in Sections~\ref{sct:2dprep}, \ref{sct:2dproof} and
\ref{sct:multiple} above. The only notable difference occurs in the
proof of the Gronwall-type inequality for billiards. Due to
dimensional considerations, if one follows the proof of
Lemma~\ref{lem:gron1} for a three-dimensional billiard table, one
finds that $\nu \mathcal{N}_\gamma (F^ {-1}\mathcal{N}_\gamma
(\partial\Omega))=\mathcal{O}(\gamma ^ {1-4\alpha}+\gamma ^
{\alpha})$. The optimal value of $\alpha$ is $1/5$, and so $ \nu
\mathcal{N}_\gamma (F^ {-1}\mathcal{N}_\gamma
(\partial\Omega))=\mathcal{O}(\gamma ^ {1/5})$ as $\gamma
\rightarrow 0$. Hence $\nu\mathcal{C}_{\gamma,\lambda} =\mathcal{O}
(\lambda \gamma ^ {1/5})$, which is a slightly worse estimate than
the one in Equation~\eqref{eq:gron4}. However, it is still
sufficient for all of the arguments in Section~\ref{sct:gron_ap},
and this finishes the proof.
\comment{
can also be heuristically justified by
the procedure we used to justify the averaged equation when $d=2$.
Let $B^d=\{(x_1,\dots,x_d)\in\mathbb{R}^d :\sum_{i=1}^{d}x_i^2\leq
1\}$ denote the unit ball in $\mathbb{R}^d$, and let
$S^{d-1}=\{(x_1,\dots,x_d)\in B^d :\sum_{i=1}^{d}x_i^2= 1\}$ denote
the unit $(d-1)$-sphere. Also let $(S^{d-1}) ^
+=\{(x_1,\dots,x_d)\in S^{d-1} :x_d\geq 0\}$. Then if the piston is
held fixed, the expected flight time between collisions for the left
gas particle (with respect to the invariant billiard measure for the
billiard map induced on the subspace of collisions with the piston)
is
\[
E_{\hat\nu} \hat\zeta=\frac {
\abs{\mathcal{D}_1}}{\ell\sqrt{2E_1}}\frac {\abs{S^{d-1}}}
{\abs{B^{d-1}}}.
\]
See \cite{CM06}. Furthermore,\marginal {I should say a word about
invariant measures/how to derive these equations} For future
reference, we observe that the expected value of $ \abs{v_1^\perp }$
when the left gas particle collides with the (immobile) piston is
given by
\[
E_{\hat\nu} \abs{v_1^\perp }=
\frac {\sqrt{2E_1}} {2d}\frac{\abs{S^{d-1}}}{\abs{B^{d-1}}}.
\]
}
\appendix
\section{Inducing maps on subspaces}
\label{sct:inducing}
Here we present some well-known facts on inducing measure preserving
transformations on subspaces. Let $F: (\Omega,
\mathfrak{B},\nu)\circlearrowleft$ be an invertible, ergodic,
measure preserving transformation of the probability space $\Omega$
endowed with the $\sigma$-algebra $\mathfrak{B}$ and the probability
measure $\nu$. Let $\hat\Omega\in\mathfrak{B}$ satisfy $0<\nu
\hat\Omega<1$. Define $R:\hat\Omega\rightarrow\mathbb{N}$ to be the
first return time to $\hat\Omega$, i.e.~$R\omega
=\inf\{n\in\mathbb{N}:F^n\omega \in\hat\Omega\}$. Then if
$\hat{\nu} : =\nu(\cdot\cap\hat\Omega)/\nu\hat\Omega$ and
$\hat{\mathfrak{B}}: =\{B\cap\hat\Omega:B\in \mathfrak{B}\}$,
$\hat{F}: (\hat\Omega,
\hat{\mathfrak{B}},\hat{\nu})\circlearrowleft$ defined by
$\hat{F}\omega=F^{R\omega}\omega$ is also an invertible, ergodic,
measure preserving transformation~\cite{Pet83}. Furthermore
$E_{\hat{\nu}} R=\int_{\hat\Omega}
R\,d\hat{\nu}=(\nu\hat\Omega)^{-1}$.
This last fact is a consequence of the following proposition:
\begin {prop}
\label{prop:inducing}
If $\zeta:\Omega\rightarrow\mathbb{R}_{\geq 0}$ is in $L^1(\nu)$,
then $\hat\zeta =\sum_{n=0}^{R-1}\zeta\circ F^n$ is in
$L^1(\hat{\nu})$, and
\[
E_{\hat{\nu}} \hat\zeta =\frac {1}{\nu\hat\Omega}
E_{\nu}\zeta.
\]
\end{prop}
\begin {proof}
\[
\begin {split}
\nu\hat\Omega \int_{\hat\Omega} \sum_{n=0}^{R-1}\zeta\circ F^n\,d\hat{\nu}
&=
\int_{\hat\Omega} \sum_{n=0}^{R-1}\zeta\circ F^n\,d\nu
=
\sum_{k=1}^\infty\int_{\hat\Omega\cap\{R=k\}}
\sum_{n=0}^{k-1}\zeta\circ F^n\,d\nu
\\
&=
\sum_{k=1}^\infty\sum_{n=0}^{k-1}\int_{F^n(\hat\Omega\cap\{R=k\})}
\zeta\,d\nu
=
\int_{\Omega}\zeta\,d\nu,
\end {split}
\]
because $\{F^n(\hat\Omega\cap\{R=k\}):0\leq n< k<\infty\}$ is a
partition of $\Omega$.
\end {proof}
\section{Derivative bounds for the billiard map in three dimensions}
\label{sct:d_bounds}
Returning to Section~\ref{sct:higher_d}, we need to show that for a
billiard table $\mathcal{D}_1\subset\mathbb{R}^3$ with a piecewise
$\mathcal{C} ^3$ boundary and the free flight time uniformly bounded
above, the billiard map $F$ satisfies the following: If $x_0\notin
\partial\Omega \cup F^{-1} (\partial\Omega) $, then
\begin{equation*}
\norm {DF(x_0)}\leq\frac {\C} {\cos \varphi(Fx_0)}.
\end{equation*}
Fix $x_0= (r_0,\hat v_0)\in\Omega $, and let $x_1= (r_1,\hat
v_1)=Fx_0$. Let $\Sigma$ be the plane that perpendicularly bisects
the straight line between $ r_0$ and $ r_1$, and let $r_{1/2} $
denote the point of intersection. We consider $\Sigma$ as a
``transparent'' wall, so that in a neighborhood of $ x_0$, we can
write $F=F_2\circ F_1$. Here, $F_1$ is like a billiard map in that
it takes points (i.e.~directed velocity vectors with a base) near $
x_0$ to points with a base on $\Sigma$ and a direction pointing near
$ r_1$. ($ F_1$ would be a billiard map if we reflected the image
velocity vectors orthogonally through $\Sigma$.) $ F_2$ is a
billiard map that takes points in the image of $F_1$ and maps them
near $ x_1$. Let $x_{1/2} = F_1 x_0= F_2^ {-1} x_1 $. Then $ \norm
{DF(x_0)}\leq \norm {DF_1(x_0)}\norm {DF_2(x_{1/2})}$.
It is easy to verify that $\norm {DF_1(x_0)}\leq\C$, with the
constant depending only on the curvature of $\partial\mathcal{D}_1$
at $ r_0$. In other words, the constant may be chosen independent
of $ x_0$. Similarly, $\norm {DF_2^ {-1}(x_1)}\leq\C$. Because
billiard maps preserve a probability measure with a density
proportional to $\cos\varphi $, $\text {det}DF_2^ {-1}(x_1)=\cos
\varphi_{1}/\cos\varphi_{ 1/2} =\cos\varphi_1$. As $\Omega$ is $4
$-dimensional, it follows from Cramer's Rule for the inversion of
linear transformations that
\[
\norm {DF_2(x_{1/2})}\leq \frac {\C\norm {DF_2^ {-1}(x_1)}^3}
{\text {det}DF_2^ {-1}(x_1)}\leq\frac {\C} {\cos\varphi_1},
\]
and we are done.
\vskip 1cm
\textbf{Acknowledgments.} The author is grateful to D. Dolgopyat,
who first introduced him to this problem, and who generously shared
his unpublished notes on averaging~\cite{Dol05}. The author also
thanks L.-S.~Young for useful discussions regarding this project and
P.~Balint for many helpful comments on the manuscript. This
research was partially supported by the National Science Foundation
Graduate Research Fellowship Program.
\bibliographystyle{alpha}
| 61,911
|
Eating almonds helps the function of 'good cholesterol' in the body to improve overall heart health, according to a US study.
High-density lipoprotein (HDL) is known as the "good" cholesterol because it helps remove other forms of cholesterol from your bloodstream and arteries that have been linked to a greater risk of heart disease.
Researchers at Penn Sate university recently compared the levels of HDL cholesterol in people who ate almonds every day to those who ate a muffin instead.
The study, published in the Journal of Nutrition, found participants on the almond diet had improved HDL levels and functionality.
Professor of Nutrition Penny Kris-Etherton says the findings build on previous research on the effects of almonds on cholesterol-lowering diets.
"There's a lot of research out there that shows a diet that includes almonds lowers low-density lipoprotein or LDL cholesterol, which is a major risk factor for heart disease," Prof.
In a controlled-feeding study, 48 men and women with elevated LDL cholesterol were assigned to two different six-week diets.
In both, their food intake was identical except for the daily snack. On the almond diet, participants received 43 grams - about a handful - of almonds a day. During the control period, they received a banana muffin instead.
The researchers found compared to the control diet, the almond diet increased a-1 HDL - more mature and larger HDL molecules - by 19 percent.
HDL molecules grow in size as they clean up the 'bad' cholesterol from cells and tissue.
Additionally, the almond diet improved HDL function by 6.4 percent, in participants of normal weight.
"We were able to show that there were more larger particles in response to consuming the almonds compared to not consuming almonds," Professor Kris-Etherton said.
"That would translate to the smaller particles doing what they're supposed to be doing. They're going to tissues and pulling out cholesterol, getting bigger and taking that cholesterol to the liver for removal from the body."
While almonds won't eliminate the risk of heart disease altogether they are a smart choice for a healthy snack, says Prof Kris-Etherton.
Almonds also provide a dose of good fats, vitamin E and fiber.
"If people incorporate almonds into their diet, they should expect multiple benefits, including ones that can improve heart health," Prof Kris-Etherton said.
© Nine Digital Pty Ltd 2018
| 228,958
|
TITLE: Non-abelian Order of $6$ is isomorphic to $S_3$
QUESTION [0 upvotes]: I know that it's duplicate, but , How can I prove it?
I know that must be element "$a$" of order 2, and element "$b$" of order 3.
What is the next step? In fact, from what I search it is claimed that if $ba=ab^2$, then the group is $S_3$. Can someone explain to me why?
REPLY [0 votes]: As $G$ is not Abelian $Z(G)=\{e\}$ and hence $C_G(x) = \langle x\rangle$ for all $e\ne x\in G$. So the class equation reads $6=|G|=1+2n+3m$ where $n$ and $m$ are the numbers of order $3$ and $2$ elements respectively. It follows that $n=m=1$. Pick $x,y\in G$ with $|x|=3$ and $|y|=2$. Assuming that $H=\langle y\rangle$ be a normal subgroup of $ G$. we have $y^x=y$ for all $x\in G$ ,i.e $G$ is Abelian in contradiction to the assumption. So $H$ is not normal and hence $H\cap H^x=\{e\}$ for some $x \in G$ . Then $G\to S_{G/H}\simeq S_3$
| 40,285
|
He has gone to great lengths to shake off his once dour image, but a trip to Springfield is a step too far for British Prime Minister Gordon Brown.
Brown says he has no plans to follow predecessor Tony Blair in making a cameo appearance on Fox network's The Simpsons.
Yeardley Smith, who provides the voice for the show's Lisa Simpson - the gifted sister of delinquent Bart - has said she believed Brown should be invited to appear.
| 11,722
|
TITLE: If $F_1 \cup F_2 = \mathbb{R}$ and $F_1,F_2$ are closed sets then $\mbox{Int}F_1 \neq \emptyset$ or $\mbox{Int}F_2 \neq \emptyset$
QUESTION [2 upvotes]: Prove that if $F_1 \cup F_2 = \mathbb{R}$ and $F_1,F_2$ are closed sets (in euclidean space) then $\mbox{Int}F_1 \neq \emptyset$ or $\mbox{Int}F_2 \neq \emptyset$
My idea is prove that by contradiction. So let suppose that $\mbox{Int}F_1 = \emptyset$ and $\mbox{Int}F_2 = \emptyset$. So $F_1,F_2$ are border sets. And what can I do next? I can't gain contradiction. I will grateful for yuor hints.
REPLY [1 votes]: This is Baire' Theorem for complete spaces.
Here let's suppose that $ Int F_1,IntF_2=\emptyset$. Then $\Bbb R- Int F_1=\Bbb R$ and $\Bbb R- Int F_2=\Bbb R$.
We have that $\Bbb R- Int F_1$=Cl$ (\Bbb R- F_1)$ and $\Bbb R- Int F_2$=Cl$ (\Bbb R- F_2)$.
Thus Cl$ (\Bbb R- F_1)=\Bbb R$ and Cl$ (\Bbb R- F_2)=\Bbb R$ $=>$ $G_1=(\Bbb R- F_1)$ and $G_2=(\Bbb R- F_2)$ are dense in $\Bbb R$. Moreover every $G_{i}$ is open in $\Bbb R$.
From Baire's Theorem we have that $G_1\cap G_2 \neq \emptyset=> F_1^{c}\cap F_2^{c} \neq \emptyset => (F_1\cup F_2 )^{c}\neq \emptyset=>\Bbb R ^{c}\neq \emptyset=>\emptyset \neq \emptyset$
| 149,144
|
Some are made with hydrogenated fat... BAD!
What brand?
So I have one cup of coffee during the work week in the morning. No sugar. I was using half and half but lately I've been using two little Hazlenut creamers. They have a few flavors but Hazlenut is so good. Anybody know what's in these creamers. I'm sure it's similar to half and half. I'm thinking it's ok because I only use two a day and drink water the rest of the day.
Some are made with hydrogenated fat... BAD!
What brand?
International Delight
Just looked the big bottle and there is sugar added, which I thought. Think I'm going to go back to regular half and half
You don't want to know.
Ingredients
(Non-dairy Product) Water, Sugar, Palm Oil, Corn Syrup, Contains less than 2% of Each of the Following Ingredients: Carrageenan , Mono and Diglycerides, Sodium Stearoyl Lactylate, Salt, Dipotassium Phosphate, Sodium Caseinate (A Milk Derivative), Natural and Artificial Flavors. Sodium Caseinate is not A Source of Lactose. Contains: Milk
::gag::
"You can always do more than you think you can !" Sensei Scash
Without even looking I know that it has little (or effectively no) cream and no hazelnut.
Wheat is the new tobacco. Spread the word.
edit
Last edited by Chapstick; 11-13-2011 at 08:25 AM.
Ground up polyester shirts...
| 11,068
|
These are spokes models at a Johnny Walker event at Granvill Mall. This event had something to do with sponsoring a formula one racing car and hard alcohol—not a good mix by most standards, but, whatever. A camera club showed up to photograph the models and use to all their brand new, expensive gear..”Buy This Image
| 396,605
|
Who cares? You might ask yourself. Aren't we seeing sales of new cars go off the charts?
We are, and according to Jonas, that's potentially a big problem for used-car prices:
The moment of osmosis [my emphasis] where car and money are exchanging hands is where our team spends an inordinate amount of intellectual and analytical rigor. In a US market where 90% to 95% of all units are sold on a monthly payment, we take seriously the growing signs of dealer concerns over the sustainability of practices meant to lower the monthly payment and the risk that we are taking consumers out of the normal trade cycle, pulling forward demand from the future.
I've been covering the auto industry for a decade and, before that, spent a stint on the advertising side, working fairly closely with dealers. I've never heard anyone on a car lot use a term like "moment of osmosis" to describe a new-car sale — so kudos of Jonas for pushing the envelope, as he habitually does, on how we talk/write/think about the transportation economy.
Turns of phrase aside, he's making a key point: Financing drives sales of new cars — most consumers buy with a loan — so tempting financing options means that more consumers will jump into the market, trading in their old car for a new one. Or more accurately, trading in their old car loan on a new car loan — or swapping out a paid-off vehicle for a vehicle that will have to be paid for.
This is all pushing new-car sales to a pace of 16.5 to 17.5 million for 2014 — a huge recovery from the Great Recession.
There isn't any supply of used cars if carmakers aren't selling new cars. When new-car sales cratered after the financial crisis — and when consumers couldn't afford to buy new — prices of used cars spiked. Additionally, many potential buyers of new cars slipped out of the usual trade-in cycle and hung on to their old car. The upshot is that the overall age of the US auto fleet is now 11 years — an all-time high.
(For what it's worth, a not insignificant contributing factor here is that cars are much better made than they used to be, so it often makes sense to keep an old, paid-off car and simply absorb yearly maintenance costs — as long as you don't need or want the latest technological bells and whistles.)
The used-car market has cooled off, but as Jonas notes, prices have held up. He doesn't think this can last, as all the new cars now being sold and leased must logically hit the used-car market at some point. All new cars are destined to become used cars. Or as Jonas expresses it: "[Y]ou can't unmake a car. Unless you take it out to the desert with a blowtorch or crush the units they just ... exist."
The used-car market can't evade this core existential quandary — it can only grapple with, at the level of price.
Jonas foresees a market with used cars thick on the ground, where there is no recourse but to mark these depreciating assets to market. Supply and demand kicks in and undermines pricing on the used front.
According to Jonas, this "alignment of forces" could "drive the largest decline in second-hand vehicle prices in US history."
Be afraid. Be very afraid. Of future prices of used cars.
| 30,700
|
TITLE: Non-uniform circular motion, computing the angle
QUESTION [0 upvotes]: There is an object moving in circle, with this law:
$$ \alpha = -k^2 \theta $$
With $ \alpha = \frac{ d \omega }{dt} $, and $ \omega= \frac{d \theta}{dt} $, $\theta$ = angle, $k$ positive constant.
I need to compute $ \theta $ in function of time. My attempt:
$$ \frac{d^2 \theta}{dt^2} = -k \theta $$
$$ \frac{d^2 \theta}{\theta} = -k^2 dt^2 $$
$$ \int_{\theta_0}^{\theta} {\frac{d^2 \theta}{\theta}} = - \int{k^2 dt^2} $$
$$ \ln\Big(\frac{\theta}{\theta_0}\Big) d\theta = -k^2 t \cdot dt$$
So I try to get the solution computing:
$$ \int_{\theta_0}^{\theta} \ln\Big(\frac{\theta}{\theta_0}\Big) d\theta = -\int k^2 t \cdot dt $$
But it's wrong, the solution is $\theta = \theta_0 \sin \big( kt + \phi \big) $.
REPLY [1 votes]: If you don't understand the other method, this is how you would do in the realm of real numbers :
Write $$\frac{d^2\theta}{dt^2}=\frac{d\omega}{dt}=\omega\frac{d\omega}{d\theta}$$
I have used that $\omega=\frac{d\theta}{dt}$ in the last step. Now your equation is first order and you can solve it easily.
$$\omega d \omega=-k^2 \theta d\theta$$
You can take from here I guess.
| 69,064
|
$2,500
Listed by Keller Williams Realty Assoc Partners (513) 874-3300, Denise Gifford
9062 Beacon Street, Deerfield Township, OH 45040 (MLS# 1599752) is a Condo/TownHouse property with 3 bedrooms, 2 full bathrooms and 1 partial bathroom. 9062 Beacon Street is currently listed for rent at $2,500 and was received on October 15, 2018. Want to learn more about 9062 Beacon Street? Do you have questions about finding other real estate for sale or rent in Deerfield Township? You can browse all Deerfield Township rentals or contact a Coldwell Banker agent to request more information.
| 243,823
|
Dora H. Bickwermert
June 15, 1923 - September 10, 2019 B. and Cecelia (Hopf) Schuler. She married Alva Bickwermert on May 1, 1945, in St. Mary's Catholic Church in Ireland, Indiana., IN, one grandson, three sisters, Leola Heeke, Dubois, IN, Arlene Sermersheim, Jasper, IN, and Julie (Kenneth) Persohn-Kunkler, St. Anthony, IN.:00 a.m. on Friday, September 13, 2019, at St. Joseph Catholic Church in Jasper, Indiana, with burial to follow in St. Mary Cemetery in Ireland, Indiana. A visitation will be held from 9:00 until the 11:00 a.m. service time at the church on Friday. Memorial contributions may be made to the Dubois County Humane Society or to the wishes of the family. Online condolences may be made at
| 207,844
|
TITLE: How do I prove that a free group is generated by its base?
QUESTION [1 upvotes]: Let $G$ be a free group with a base $S$.
How do I prove that $G=\langle S\rangle $ only using universal property?
Here's how I proved this. Let $F(S)$ be the free group on $S$ constructed via reduced words on $S$. This construction shows that $F(S)$ is generated by $S$. Since $S\subset F(S)$ and $S\subset G$, there is a group isomorphism between $F(S)$ and $G$ which keeps $S$ fixed. Hence $G=\langle S \rangle$.
However, I think this proof is bad. How do I prove that using only universal property?
My definition for free group:
Let $G$ be a group.
Then, $G$ is free iff there exists $S\subset G$ such that for any group $H$ and a function $f\in H^S$, there exists a unique $\phi\in Hom(G,H)$ such that $\phi\upharpoonright S=f$
REPLY [2 votes]: The map of sets $f:S \hookrightarrow G$ factors over the group $\langle S \rangle$, so by the universal property the identity homomorphism $G \to G$ obtained from $f$ factors over $\langle S \rangle \hookrightarrow G$. In particular $\langle S \rangle \hookrightarrow G$ is surjective.
| 161,087
|
Kiwifruit is an autumn fruit! It is packed with more Vitamin C than an orange. It is a delicious way to enjoy cardiovascular health as consuming a kiwifruit each day significantly lowers the risk of blood clots and reduces inflammation. If you are lost for ideas on how to make eating kiwifruit exciting try making Kiwi pops. Just add kiwifruit and yogurt in a food processor, transfer them into ice moulds and freeze overnight. Kids LOVE them!
| 164,840
|
With their stay in the convent coming to an end, the girls are set their toughest task yet – to spend a week back in their home environments and resist all temptation.
Can they do it?
A sweet, uplifting series, which has seen the girls’ confidence and respect flourish under the guidance of loving nuns.
| 28,018
|
> Interestingly as far as I see it seems there is already code in rear namely in usr/share/rear/verify/NETFS/default/05_check_NETFS_requirements.sh to automatically do appropriate MODULES_LOAD stuff for sshfs and ftpfs. But it seems either somehow that code does not work or I misunderstand something here. Kind Regards Johannes Meixner -- SUSE LINUX GmbH - GF: Felix Imendoerffer, Jane Smithard, Graham Norton - HRB 21284 (AG Nuernberg)
| 49,179
|
\begin{document}
\maketitle
\begin{abstract}
\noindent
We introduce and analyze an abstract algorithm that aims to find
the projection onto a closed convex subset of a Hilbert space.
When specialized to the fixed point set of a
quasi nonexpansive mapping, the
required sufficient condition (termed ``fixed-point closed'')
is less restrictive
than the usual conditions based on the demiclosedness principle.
A concrete example of a subgradient projector is presented which
illustrates the applicability of this generalization.
\end{abstract}
\noindent {\bf Keywords:}
convex function,
demiclosedness principle,
fixed point,
nonexpansive,
polyhedron,
projection,
quasi nonexpansive,
subgradient projection.
\noindent {\bf 2010 Mathematics Subject Classification:}
Primary 47H09; Secondary 52B55, 65K10, 90C25.
\section{Introduction}
Throughout this note, we assume that
\begin{equation}
\text{$X$ is a real Hilbert space with inner product $\scal{\cdot}{\cdot}$
and norm $\|\cdot\|$.}
\end{equation}
Suppose that
\begin{equation}
\label{e:global1}
\text{$C$ is a closed convex subset of $X$, and $x_0\in X$.}
\end{equation}
We are interested in finding the projection (nearest point mapping)
$P_Cx_0$, i.e., the unique solution to the optimization
problem
\begin{equation}
\label{e:theproblem}
d(x_0,C) := \min_{c\in C}\|x_0-c\|,
\end{equation}
especially when $C$ is the fixed point set of some operator $T\colon X\to
X$.
It will be convenient to set,
for arbitrary given vectors $x$ and $y$ in $X$,
\begin{equation}
H(x,y) := \menge{z \in X}{\|y-z\|\leq\|x-z\|} = \menge{z\in
X}{2\scal{z}{x-y}\leq\|x\|^2-\|y\|^2}.
\end{equation}
Note that $H(x,y)$ is equal to either $X$ (if $x=y$) or a halfspace;
in any case, the projection onto $H(x,y)$ is easy to compute and has a well
known closed form.
In order to solve \eqref{e:theproblem}, we shall study the following simple
abstract iteration:
\begin{algorithm}
\label{a:1}
Recall the assumption~\eqref{e:global1}, and set $C_0 = X$.
Given $\nnn$ and $x_n\in X$, pick $y_n\in X$, and set
\begin{equation}
C_{n+1} := C_n \cap H(x_n,y_n)
\;\;\text{and}\;\;
x_{n+1} = P_{C_{n+1}}x_0.
\end{equation}
\end{algorithm}
Observe that if the sequence is well defined, then
\begin{equation}
\label{e:nested}
C_0\supseteq C_1 \supseteq \cdots C_n \supseteq C_{n+1}\supseteq \cdots
\end{equation}
and so
\begin{equation}
\label{e:increases}
\|x_0-x_n\| = d(x_0,C_n) \leq d(x_0,C_{n+1})=\|x_0-x_{n+1}\|
\end{equation}
for every $\nnn$.
It then follows that
\begin{equation}
\label{e:beta}
\beta := \lim_\nnn\|x_0-x_n\| = \sup_\nnn\|x_0-x_n\| \in
\left[0,+\infty\right]
\end{equation}
is well defined.
Furthermore, if $m < n$, then
$x_n\in C_m$ which implies
\begin{equation}
\label{e:kolmo}
\scal{x_n-x_m}{x_0-x_m}\leq 0
\end{equation}
as well as
\begin{equation}
\label{e:bla}
\|y_m-x_n\|\leq\|x_m-x_n\|
\end{equation}
because
$x_n \in C_n \subseteq C_{m+1}\subseteq H(x_m,y_m)$.
\begin{lemma}
\label{l:key}
Suppose that the sequence $(x_n)_\nnn$ is generated by
Algorithm~\ref{a:1}.
Suppose also that
for every subsequence $(x_{k_n})_\nnn$ of $(x_n)$, we have
\begin{equation}
\label{e:superdemi}
\left.
\begin{array}{c}
x_{k_n}\to \bar{x}\\
x_{k_n}-y_{k_n}\to 0
\end{array}
\right\}
\;\;\Rightarrow\;\;
\bar{x}\in C.
\end{equation}
Then every bounded subsequence of $(x_n)_\nnn$ must converge to a point in
$C$.
\end{lemma}
\begin{proof}
Let $(x_{k_n})_\nnn$ be a bounded subsequence of $(x_n)_\nnn$.
It follows from \eqref{e:increases} that $\beta<+\infty$.
Let $n> m$.
Using \eqref{e:kolmo}, we obtain
\begin{subequations}
\begin{align}
\|x_{k_n}-x_{k_m}\|^2 &= \|x_{k_n}-x_0\|^2 - \|x_{k_m}-x_0\|^2 +
2\scal{x_{k_n}-x_{k_m}}{x_0-x_{k_m}}\\
&\leq \|x_{k_n}-x_0\|^2 - \|x_{k_m}-x_0\|^2\\
&\to \beta^2-\beta^2 = 0\;\;\text{as $n\geq m\to +\infty$.}
\end{align}
\end{subequations}
Hence $(x_{k_n})_\nnn$ is a Cauchy sequence.
Thus, there exists $\bar{x}\in X$ such that $x_{k_n}\to \bar{x}$.
Now, from \eqref{e:bla}, we obtain
$\|y_{k_n}-x_{k_{n+1}}\| \leq \|x_{k_n}-x_{k_{n+1}}\| \to
\|\bar{x}-\bar{x}\| = 0$ and thus $y_{k_n}-x_{k_{n+1}}\to 0$.
It follows that $x_{k_n}-y_{k_n} = (x_{k_n}-x_{k_{n+1}}) + (x_{k_{n+1}} -
y_{k_n})\to 0$.
Now apply \eqref{e:superdemi}.
\end{proof}
The previous result allows us to derive the following dichotomy result.
\begin{theorem}[dichotomy]
\label{t:main}
Suppose that $(x_n)_\nnn$ is generated by
Algorithm~\ref{a:1}, that $(\forall\nnn)$ $C\subseteq C_n$,
and that for every subsequence $(x_{k_n})_\nnn$ of $(x_n)$, we have
\begin{equation}
\left.
\begin{array}{c}
x_{k_n}\to \bar{x}\\
x_{k_n}-y_{k_n}\to 0
\end{array}
\right\}
\;\;\Rightarrow\;\;
\bar{x}\in C.
\end{equation}
Then exactly one of the following holds:
\begin{enumerate}
\item
\label{t:main1}
$C\neq\varnothing$ and $x_n\to P_Cx_0$.
\item
\label{t:main2}
$C=\varnothing$ and $\|x_n\|\to+\infty$.
\end{enumerate}
\end{theorem}
\begin{proof}
Note that
\begin{equation}
\label{e:rot}
(\forall\nnn)\;\;
\|x_0-x_{n}\| = d(x_0,C_n)\leq d(x_0,C).
\end{equation}
\ref{t:main1}:
Assume that $C\neq\varnothing$.
Then $(x_n)_\nnn$ is bounded by \eqref{e:rot}.
By Lemma~\ref{l:key}, $\bar{x} := \lim_\nnn x_n \in C$.
In turn, \eqref{e:rot} yields $\|x_0-\bar{x}\|\leq d(x_0,C)$.
Therefore, $\bar{x}=P_Cx_0$, as claimed.
\ref{t:main2}:
Suppose that $\|x_n\|\not\to+\infty$.
Then $(x_n)_\nnn$ contains a bounded subsequence which,
by Lemma~\ref{l:key}, must converge to a point in $C$.
Hence if $C=\varnothing$, then $\|x_n\|\to+\infty$.
\end{proof}
\begin{remark}
Several comments regarding Theorem~\ref{t:main} are in order.
\begin{enumerate}
\item
Algorithm~\ref{a:1} is related to a method
studied by Takahashi et al in \cite[Theorem~4.1]{TTT}.
(See also \cite[Theorem~2]{RS2,RS} for Bregman-distance based variants.)
While that method is more flexible in some ways, our method has the
advantage of requiring neither nonexpansiveness of the given operator
nor the nonemptiness of the target set.
\item
Our proofs are different because we establish strong convergence
directly via a Cauchy sequence argument. The proofs mentioned in
the previous item are based on a Kadec-Klee property or on Opial's
property.
(We expect that our proof will generalize to Bregman distances,
possibly incorporating errors and families of operators.)
\item
As we shall see in Section~\ref{s:sp} below,
our framework encompasses subgradient
projectors which are important in optimization.
\item
The computation of the sequence $(x_n)_\nnn$ requires
to compute projections of the \emph{same} initial point $x_0$
onto polyhedra (intersections of finitely many halfspaces).
While this is not necessarily an easy task, this is considered to be
a standard quadratic programming problem in convex optimization.
Moreover, since $C_{n+1}$ is constructed from $C_n$ by intersecting
with the halfspace $H(x_n,y_n)$, it seems plausible to apply
\emph{active set methods} (with a warm start) to solve these
projections. While a detailed excursion on this matter is beyond the
scope of this paper, we do refer the reader to \cite{Arioli,Huynh,Nurminski}
for references
on computing projections onto polyhedra.
\end{enumerate}
\end{remark}
\section{An application to finding nearest fixed points}
Recall that $T\colon X\to X$ is called
\emph{nonexpansive} if
\begin{equation}
(\forall x\in X)(\forall y\in X)\quad
\|Tx-Ty\|\leq\|x-y\|;
\end{equation}
moreover, $T$ is \emph{quasi nonexpansive} if
\begin{equation}
(\forall x\in X)(\forall y\in \Fix T)\quad
\|Tx-y\|\leq\|x-y\|,
\end{equation}
where $\Fix T := \menge{x\in X}{x=Tx}$.
See \cite{GK,GR,BC} for further information on the fixed point theory
of nonexpansive mappings.
The next result is readily checked.
\begin{lemma}
\label{l:easy}
Let $T\colon X\to X$ be quasi nonexpansive.
Consider the following properties:
\begin{enumerate}
\item
\label{l:easy1}
$T$ is nonexpansive.
\item
\label{l:easy2}
$T$ is continuous.
\item
\label{l:easy3}
$T$ is \emph{fixed-point closed}, i.e.,
if
$x_n\to\bar{x}$ and
$x_n-Tx_n\to 0$, then
$\bar{x}\in\Fix T$.
\end{enumerate}
Then \ref{l:easy1}$\Rightarrow$\ref{l:easy2}$\Rightarrow$\ref{l:easy3}.
\end{lemma}
\begin{remark}
It is well known that if $T\colon X\to X$ is nonexpansive,
then
\begin{equation}
\left.
\begin{array}{c}
x_n\rightharpoonup \bar{x}\\
x_n-Tx_n \to 0
\end{array}
\right\}
\;\;\Rightarrow\;\;
\bar{x}\in\Fix T;
\end{equation}
this is the famous demiclosedness principle --- to be precise,
this states that $\Id-T$ is demiclosed at $0$.
For recent results on this principle, see \cite{Demi} and the references
therein.
It is clear that demiclosedness of $\Id-T$ at $0$ implies
that $T$ is fixed-point closed; the converse, however, is false
(see Example~\ref{ex:bad} below).
\end{remark}
Our main result now yields easily the following result, which
by Lemma~\ref{l:easy} is applicable in particular when $T$ is nonexpansive.
(See also \cite[Theorem~4.1]{TTT} for extensions in the nonexpansive case.)
\begin{theorem}[trichotomy]
\label{t:fix}
Let $T\colon X\to X$ be quasi nonexpansive and fixed-point closed,
let $x_0\in X$, and set $C_0 := X$.
Given $\nnn$ and $x_n$, set
\begin{equation}
C_{n+1} := C_n \cap H(x_n,Tx_n)
\;\;\text{and}\;\;
x_{n+1} = P_{C_{n+1}}x_0.
\end{equation}
Then exactly one of the following holds:
\begin{enumerate}
\item
\label{t:fix1}
$\Fix T\neq\varnothing$ and $x_n\to P_{\Fix T}x_0$.
\item
\label{t:fix2}
$\Fix T = \varnothing$ and $\|x_n\|\to+\infty$.
\item
\label{t:fix3}
$\Fix T = \varnothing$ and the sequence is not well defined (i.e.,
$C_{n+1}$ is empty for some $n$).
\end{enumerate}
\end{theorem}
\begin{proof}
Set $C = \Fix T$, and $(y_n)_\nnn = (Tx_n)_\nnn$ provided that $(x_n)_\nnn$
is well defined.
In this case, it is clear that \eqref{e:superdemi} holds because $T$ is
fixed-point closed.
\ref{t:fix1}: Assume that $C\neq\varnothing$.
If $C_n\neq\varnothing$ and $C\subseteq C_n$, then
$(\forall c\in C)$
$\|Tx_n-c\|\leq \|x_n-c\|$ and so $c\in H(x_n,Tx_n)$.
It follows that $C\subseteq C_{n+1}$ and the sequence $(x_n)_\nnn$
is well defined.
The conclusion thus follows from Theorem~\ref{t:main}.
\ref{t:fix2}\&\ref{t:fix3}: Assume that
$C=\varnothing$.
If $(x_n)_\nnn$ is not well defined, then \ref{t:fix3} happens.
Finally, if $(x_n)_\nnn$ is well defined, then \ref{t:fix2} occurs
again by Theorem~\ref{t:main}.
\end{proof}
Let us now illustrate the three alternatives in Theorem~\ref{t:fix}.
\begin{example}
Suppose that $X=\RR$ and set $T := \alpha\Id$, where
$\alpha\in\left[0,1\right[$.
Then $T$ is nonexpansive with $\Fix T = \{0\}$.
Let $x_0\geq 0$. Then $Tx_0 = \alpha x_0$ and
$C_1 = \left]-\infty,(\alpha+1)/2x_0\right]$.
Thus, $x_1 = (\alpha+1)/2x_0$.
It follows inductively that $(x_n)_\nnn$ is well defined and
\begin{equation}
(\forall \nnn)\quad
x_n = \big((\alpha+1)/2\big)^nx_0 \to 0 = P_{\Fix T}x_0,
\end{equation}
as is also guaranteed by Theorem~\ref{t:fix}\ref{t:fix1}.
\end{example}
\begin{example}
Suppose that $X=\RR$ and set $T\colon X\to X\colon x\mapsto x+\alpha$,
where $\alpha>0$.
Clearly, $T$ is nonexpansive and $\Fix T =\varnothing$.
One checks that $x_n = x_0+n\alpha/2$;
hence, $|x_n| \to +\infty$.
\end{example}
\begin{example}
Suppose $X=\RR$, let $\sigma\colon X\to\{-1,+1\}$, and set
$T_\sigma \colon X\mapsto X\colon x\mapsto x+\sigma(x)$.
For trivial reasons, $T_\sigma$ is quasi nonexpansive (since $\Fix T_\sigma =
\varnothing$) and $T_\sigma$ is fixed-point closed (since
$\ran(\Id-T_\sigma)\subseteq \{+1,-1\}$).
We now assume that $\sigma(0)=1$ and $\sigma(1/2)=-1$.
Let $x_0=0$. Then $C_1 = \left[1/2,+\infty\right[$,
$x_1 = 1/2$ and $C_2 = C_1 \cap \left]-\infty,0\right] =
\varnothing$, which means the algorithm terminates.
\end{example}
\section{Subgradient projector}
\label{s:sp}
The astute reader will ask whether the fairly general assumptions on $T$ in
Theorem~\ref{t:fix}, i.e., that
``$T$ be quasi nonexpansive and fixed-point closed'',
are really needed in applications.
In this section, we provide an example that not only requires this
generality but that also does not satisfy the usual demiclosedness type
assumptions seen in this area.
To this end,
let
\begin{equation}
f\colon X\to\RR
\end{equation}
be convex, continuous, and G\^ateaux differentiable such that
$f\geq 0$
and
\begin{equation}
C := \menge{x\in X}{f(x)\leq 0} = \{0\}.
\end{equation}
Write $g := \nabla f$ for convenience.
The \emph{subgradient projector} in this case is defined by
\begin{equation}
T\colon X\to X\colon
x\mapsto \begin{cases}
x, &\text{if $x=0$;}\\
x - \frac{f(x)}{\|g(x)\|^2}g(x), &\text{if $x\neq 0$.}
\end{cases}
\end{equation}
Then it follows (from e.g., \cite[Proposition~2.3]{MOR}) that
$T$ is \emph{quasi firmly nonexpansive}, i.e.,
\begin{equation}
\label{e:qfne}
(\forall x\in X)(\forall y\in \Fix T)\quad
\|Tx-y\|^2 + \|x-Tx\|^2 \leq \|x-y\|^2.
\end{equation}
\begin{lemma}
\label{l:sp}
\
The following hold:
\begin{enumerate}
\item
\label{l:sp0}
$T$ is quasi nonexpansive.
\item
\label{l:sp1}
$T$ is fixed-point closed.
\item
\label{l:sp2}
$T$ is continuous at $0$.
\item
\label{l:sp3}
If $f$ is Fr\'echet differentiable, then $T$ is continuous.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{l:sp0}: This follows immediately from \eqref{e:qfne}.
\ref{l:sp1}:
Let $(x_n)_\nnn$ be a sequence in $X$ such that $x_n\to \bar{x}$
and $x_n-Tx_n\to 0$.
We assume that $\bar{x}\neq 0$ (for if $\bar{x}=0$, then the conclusion is
trivially true) and that $(x_n)_\nnn$ lies in $X\smallsetminus\{0\}$.
To reach the required contradiction, observe first
that the continuity of $f$ yields
$f(x_n)\to f(\bar{x})> 0$.
Now $x_n-Tx_n\to 0$
$\Leftrightarrow$
$\|x_n-Tx_n\|\to 0$
$\Leftrightarrow$
$f(x_n)/g(x_n)\to 0$; thus,
\begin{equation}
\label{e:blowup}
\lim_\nnn \|g(x_n)\| = +\infty.
\end{equation}
On the other hand, $g$ is strong-to-weak continuous
(see, e.g., \cite[Proposition~17.31]{BC});
therefore, the sequence
$(g(x_n))_\nnn$ converges weakly to $g(\bar{x})$.
In particular, $(g(x_n))_\nnn$ is bounded --- but this contradicts
\eqref{e:blowup}.
\ref{l:sp2}:
Convexity yields
$(\forall x\in X\smallsetminus\{0\})$
$\scal{0-x}{\nabla f(x)} \leq f(0)-f(x)$, which implies
$f(x) \leq \scal{x}{g(x)}\leq \|x\|\|g(x)\|$; thus,
$f(x)/\|g(x)\| \leq \|x\|$.
Hence $\lim_{x\to 0} Tx = 0 = T0$, as claimed.
\ref{l:sp3}:
If $f$ is Fr\'echet differentiable, then $g$ is strong-to-strong
continuous (see, e.g., \cite[Proposition~17.32]{BC}),
which in turn yields the continuity of $T$
on $\menge{x\in X}{g(x)\neq 0} = X\smallsetminus\{0\}$.
\end{proof}
Note that Lemma~\ref{l:sp} guarantees the applicability of
Theorem~\ref{t:fix} to the subgradient projector $T$.
\begin{example}
\label{ex:bad}
Suppose that $X = \ell^2 = \menge{\bx = (x_n)_{n\geq 1}}{\sum_{n\geq
1}|x_n|^2 < +\infty}$ and set
\begin{equation}
f\colon X\to\RR\colon \bx = (x_n)_{n\geq 1} \mapsto\sum_{n\geq 1}
nx_n^{2n}.
\end{equation}
Then $f$ is well defined, convex, and continuous
(see \cite[Example~7.11]{SIREV}).
Moreover, $f$ is G\^ateaux differentiable with
$g(\bx) = \nabla f(\bx) = (2n^2x_n^{2n-1})_{n\geq 1}$.
Denote the sequence of standard unit vectors by $(\be_n)_{n\geq 1}$,
and set
\begin{equation}
(\forall n\geq 1) \quad \bx_n := \be_1 + \be_n \rightharpoonup \be_1
\end{equation}
For $n\geq 2$, we have
$f(\bx_n) = 1+n$,
$g(\bx_n) = 2\be_1 + 2n^2\be_n$;
hence $\|g(\bx_n)\| = \sqrt{4+ 4n^4}$ and thus
$f(\bx_n)/\|g(\bx_n)\| \to 0$.
It follows that
$\bx_n - T(\bx_n)\to 0$.
Since
\begin{equation}
\left.
\begin{array}{c}
\bx_n\rightharpoonup{\be_1}\\
\bx_n-T(\bx_n)\to 0
\end{array}
\right\}
\;\;\not\Rightarrow\;\;
{\be_1} = 0,
\end{equation}
we see that $\Id-T$ is \emph{not} demiclosed at $0$ and
that $T$ is not weak-to-weak continuous
however,
$T$ is fixed-point closed by Lemma~\ref{l:sp}\ref{l:sp1}.
\end{example}
\begin{remark}
Some comments regarding Example~\ref{ex:bad} are in order.
\begin{enumerate}
\item
This example illustrates that some of the sufficient conditions
demi-closedness type conditions
provided in the literature (see, e.g.,
\cite[Proposition~2.2]{PLCSICON}) to guarantee convergence are actually
not applicable to the subgradient projector $T$ of the function $f$
defined in Example~\ref{ex:bad}. However, Theorem~\ref{t:fix} is applicable
with $T$ because of Lemma~\ref{l:sp}.
\item
Some additional work (which we omit here)
shows that $f$ is actually Fr\'echet differentiable
on $X$. Thus, by Lemma~\ref{l:sp}\ref{l:sp3}, $T$ is actually
strong-to-strong continuous.
\item
It also follows from the classical demiclosedness principle that $T$
is not nonexpansive.
\end{enumerate}
\end{remark}
\section*{Acknowledgments}
This research was carried out during a visit of JC in Kelowna in Fall
2012.
HHB was partially supported by the Natural Sciences and
Engineering Research Council of Canada and by the Canada Research Chair
Program.
JC was partially supported by the Academic Award for Excellent Ph.D.~
Candidates Funded by Wuhan University, the Fundamental Research
Fund for the Central Universities,
and the Ph.D.~short-time mobility program by Wuhan University.
XW was partially supported by the Natural
Sciences and Engineering Research Council of Canada.
\bibliographystyle{plain}
| 155,134
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\begin{document}
\title[Maximum orders of cyclic and abelian extendable actions]
{Maximum orders of cyclic and abelian extendable actions on surfaces}
\author{Chao Wang}
\address{School of Mathematical Sciences, University of
Science and Technology of China, Hefei 230026, CHINA}
\email{chao\_{}wang\_{}1987@126.com}
\author{Yimu Zhang}
\address{Mathematics School, Jilin University,
Changchun 130012, CHINA}
\email{zym534685421@126.com}
\subjclass[2010]{Primary 57M60, 57S17, 57S25}
\keywords{maximum order, extendable action, orbifold}
\thanks{The authors were supported by National Natural Science Foundation of China (11371034). The authors would like to thank Prof. Shicheng Wang and the referee for their helpful advices.}
\begin{abstract}
A faithful action of a group $G$ on the genus $g>1$ orientable closed surface $\Sigma_g$ is extendable (over the three dimensional sphere $S^3$), with respect to an embedding $e: \Sigma_g\hookrightarrow S^3$, if $G$ can act on $S^3$ such that $h\circ e=e\circ h$ for any $h \in G$. We show that the maximum order of extendable cyclic group actions on $\Sigma_g$ is $4g+4$ when $g$ is even, and is $4g-4$ when $g$ is odd; the maximum order of extendable abelian group actions on $\Sigma_g$ is $4g+4$. We also give the maximum orders of cyclic and abelian group actions on handlebodies.
\end{abstract}
\date{}
\maketitle
\section{Introduction}
Let $\Sigma_g$ be the genus $g>1$ orientable closed surface, $S^3$ be the three dimensional sphere, and $G$ be a finite group.
\begin{definition}\label{Def of extendable action}
A faithful $G$-action on $\Sigma_g$ is extendable, with respect to an embedding $e: \Sigma_g\hookrightarrow S^3$, if $G$ can act on $S^3$ such that $h\circ e=e\circ h$ for any $h \in G$. We also say that such an action on $\Sigma_g$ is extendable over $S^3$.
\end{definition}
In this paper we consider the following question in the smooth category (namely all manifolds and maps are smooth): what is the maximum order of extendable cyclic (abelian) group actions on $\Sigma_g$ ? And following is our main result.
\begin{theorem}\label{Thm of maximum of CandA action}
(1). The maximum order of extendable cyclic group actions on $\Sigma_g$ is $4g+4$ when $g$ is even, and is $4g-4$ when $g$ is odd.
(2). The maximum order of extendable abelian group actions on $\Sigma_g$ is $4g+4$.
\end{theorem}
Let $V_g$ be the genus $g>1$ orientable handlebody. The question below is closely related to the above one: what is the maximum order of cyclic (abelian) group actions on $V_g$ ? And we have the following answer to it.
\begin{theorem}\label{Thm of CandA on handlebody}
(1). The maximum order of cyclic group actions on $V_g$ is
$$Max\{2[m, n]\mid g=[m, n]-\frac{m+n}{(m, n)}+1, m, n\in \mathbb{Z}_+, 2\nmid m, 2\nmid n\}$$
when $g$ is even, and is
$$Max\{2kn\mid g=kn-k+1, k, n\in \mathbb{Z}_+, 2\nmid k, 2\nmid n\}$$
when $g$ is odd.
(2). The maximum order of abelian group actions on $V_g$ is $4g+4$ when $g\neq 5$, and is $32$ when $g=5$. It is equal to the maximum order of abelian group actions on $\Sigma_g$.
\end{theorem}
Here $\mathbb{Z}_+$ is the set of positive integers, $[m, n]$ is the lowest common multiple of $m$ and $n$, and $(m, n)$ is the greatest common divisor of $m$ and $n$.
If $g$ is even, then let $m=n=g+1$. And if $g$ is odd, then let $k=1, n=g$. Then by Theorem \ref{Thm of CandA on handlebody}, it is easy to see that the maximum order of cyclic group actions on $V_g$ is at least $2g+2$ when $g$ is even, and is at least $2g$ when $g$ is odd.
\begin{remark}\label{Rem of classical result}
(1). The extendable action in Definition \ref{Def of extendable action} was firstly defined in \cite{WWZZ1}. And following notations were used in \cite{WWZZ1}.
$C_g$ ($A_g$): the maximum order of orientation-preserving cyclic (abelian) group actions on $\Sigma_g$.
$CH_g$ ($AH_g$): the maximum order of orientation-preserving cyclic (abelian) group actions on $V_g$.
$CE_g$ ($AE_g$): the maximum order of extendable cyclic (abelian) group actions on $\Sigma_g$, which preserve both the orientations of $\Sigma_g$ and $S^3$.
All these numbers have been determined, see \cite{Ha, St, Wa} for $C_g$; \cite{Ma} for $A_g$; \cite{MMZ} for $CH_g$; \cite{MMZ, RZ1, WWZZ1} for $AH_g$; \cite{WWZZ1} for $CE_g$ and $AE_g$. We summarize them in Table \ref{tab of maxorder of OPcase}:
\begin{table}[h]
\caption{Maximum orders in orientation-preserving case}\label{tab of maxorder of OPcase}
\begin{tabular}{|l|l|l|l|}
\hline $C_g$ & $4g+2$ & $A_g$ & $4g+4$\\
\hline $CH_g$ & $2g+2$ ($g$ even), $2g$ ($g$ odd) & $AH_g$ & $2g+2$ ($g\neq5$), $16$ ($g=5$)\\
\hline $CE_g$ & $2g+2$ ($g$ even), $2g-2$ ($g$ odd) & $AE_g$ & $2g+2$\\\hline
\end{tabular}
\end{table}
(2). The maximum order of orientation-reversing periodic maps on $\Sigma_g$ is $4g+4$ when $g$ is even, and is $4g-4$ when $g$ is odd. Hence the maximum order of cyclic group actions on $\Sigma_g$ is $4g+4$ when $g$ is even, and is $4g+2$ when $g$ is odd, see \cite{Wa}.
\end{remark}
By comparing Theorem \ref{Thm of maximum of CandA action} and \ref{Thm of CandA on handlebody} with Remark \ref{Rem of classical result}, we see that in most cases if the maximum order is achieved, then the group will contain an element reversing the orientation of either $\Sigma_g$, $V_g$ or $S^3$.
\begin{definition}\label{Def of MO of ORaction}
For each given $g>1$, define
$C_g^-$: the maximum order of cyclic group actions on $\Sigma_g$, which contains an element reversing the orientation of $\Sigma_g$.
$A_g^-$: the maximum order of abelian group actions on $\Sigma_g$, which contains an element reversing the orientation of $\Sigma_g$.
$CH_g^-$: the maximum order of cyclic group actions on $V_g$, which contains an element reversing the orientation of $V_g$.
$AH_g^-$: the maximum order of abelian group actions on $V_g$, which contains an element reversing the orientation of $V_g$.
\end{definition}
\begin{definition}\label{Def of five type}
Define five types of extendable $G$-actions on $\Sigma_g$ as following:
$(+,+)$: $G$ preserves both the orientations of $\Sigma_g$ and $S^3$.
$(+,-)$: $G$ preserves the orientation of $\Sigma_g$, and there exists $h_1\in G$ such that $h_1$ reverses the orientation of $S^3$.
$(-,+)$: $G$ preserves the orientation of $S^3$, and there exists $h_2\in G$ such that $h_2$ reverses the orientation of $\Sigma_g$.
$(-,-)$: for any $h\in G$, either $h$ preserves both the orientations of $\Sigma_g$ and $S^3$, or $h$ reverses both the orientations of $\Sigma_g$ and $S^3$.
$(Mix)$: there exists $h_1 \in G$ such that $h_1$ preserves the orientation of $\Sigma_g$ and reverses the orientation of $S^3$, and there exists $h_2\in G$ such that $h_2$ reverses the orientation of $\Sigma_g$ and preserves the orientation of $S^3$.
\end{definition}
\begin{definition}\label{Def of MO of type}
For a given type $\mathcal{T}$ in Definition \ref{Def of five type} and a given $g>1$, define
$CE_g\mathcal{T}$: the maximum order of type $\mathcal{T}$ extendable cyclic group actions on $\Sigma_g$, if such an action exists.
$AE_g\mathcal{T}$: the maximum order of type $\mathcal{T}$ extendable abelian group actions on $\Sigma_g$, if such an action exists.
\end{definition}
Clearly we have $CE_g(+,+)=CE_g$ and $AE_g(+,+)=AE_g$. For other types we have the following result. By Remark \ref{Rem of classical result}, it will give us Theorem \ref{Thm of maximum of CandA action}.
\begin{proposition}\label{Pro of maxorder of five type}
For each type $\mathcal{T}\neq (+,+)$ and genus $g>1$, $CE_g\mathcal{T}$ and $AE_g\mathcal{T}$ are given in Table \ref{tab of maxorder of fivetype}. The notation ``---'' means that the type $\mathcal{T}$ extendable action does not exist.
\begin{table}[h]
\caption{Maximum orders of type $\mathcal{T}$ extendable actions}\label{tab of maxorder of fivetype}
\begin{tabular}{|c|l|l|}
\hline $\mathcal{T}$ & $CE_g\mathcal{T}$ & $AE_g\mathcal{T}$\\
\hline $(+,-)$ & $2g+2$ & $4g+4$\\
\hline $(-,+)$ & $4g+4$ ($g$ even), $4g-4$ ($g$ odd) & $4g+4$\\
\hline $(-,-)$ & $2g+2$ ($g$ even), $2g$ ($g$ odd) & $4g+4$\\
\hline $(Mix)$ & --- & $2g+4$ ($g$ even), --- ($g$ odd)\\\hline
\end{tabular}
\end{table}
\end{proposition}
We see that $CE_g(-,+)=C_g^-$ and $AE_g(+,-)=A_g$, see Remark \ref{Rem of classical result}. Namely some kinds of maximum symmetries on surfaces are extendable over $S^3$.
We also have the following result. By Remark \ref{Rem of classical result}, it will give us Theorem \ref{Thm of CandA on handlebody}.
\begin{proposition}\label{Pro of ORhandlebody}
For each genus $g>1$, $CH_g^-$ and $AH_g^-$ are given in Table \ref{tab of maxorder of ORhandlebody}, here $k, m, n\in \mathbb{Z}_+$. And $A_g^-$ is equal to $AH_g^-$.
\begin{table}[h]
\caption{$CH_g^-$ and $AH_g^-$}\label{tab of maxorder of ORhandlebody}
\begin{tabular}{|l|l|}
\hline $CH_g^-$ ($g$ even) & $Max\{2[m, n]\mid g=[m, n]-(m+n)/(m, n)+1, 2\nmid m, 2\nmid n\}$\\
\hline $CH_g^-$ ($g$ odd) & $Max\{2kn\mid g=kn-k+1, 2\nmid k, 2\nmid n\}$\\
\hline $AH_g^-$ & $4g+4$ ($g\neq 5$), $32$ ($g=5$)\\\hline
\end{tabular}
\end{table}
\end{proposition}
In \S2, we will give some examples which will give the lower bounds of maximum orders we have defined. In \S3, we will give some known facts about group actions on manifolds, which will be used later. In \S4, we will prove Proposition \ref{Pro of ORhandlebody}. Some results in \S4 will be used in \S5, where we will prove Proposition \ref{Pro of maxorder of five type}.
\section{Examples}\label{Sec of examples}
In this section we will give three examples about extendable actions and one example of non-extendable action. The surfaces we constructed below may be non-smooth, but using orbifold theory discussed in \S3, they can be replaced by smooth ones. The following lemma is easy to verified.
\begin{lemma}\label{Lem of change sides}
Let $\Sigma_g$ be a closed subsurface in $S^3$. Suppose $h$ is an automorphism of $S^3$ preserving $\Sigma_g$. If some of the following conditions (a), (b) and (c) happen, then exactly two of them happen.
(a). $h$ reveres the orientation of $S^3$.
(b). $h$ reveres the orientation of $\Sigma_g$.
(c). $h$ changes the two sides of $\Sigma_g$.
\end{lemma}
\begin{example}[Cage]\label{Ex of cage}
Given $g>1$, we will construct some cyclic and abelian group actions on a Heegaard splitting of $S^3$.
Let $S^3$ be the unit sphere in $\mathbb{C}^2$:
$$S^3=\{(z_1, z_2)\in\mathbb{C}^2\mid |z_1|^2+|z_2|^2=1\}.$$
Let
\begin{align*}
a_m&=(e^{\frac{m\pi}{2}i}, 0), m=0, 1, 2, 3,\\
b_n&=(0, e^{\frac{n\pi}{g+1}i}), n=0, 1, \cdots, 2g+1.
\end{align*}
Connect each $a_{2l}$ to each $b_{2k}$ with the shortest geodesic in $S^3$, and connect each $a_{2l+1}$ to each $b_{2k+1}$ with the shortest geodesic in $S^3$, here $l=0, 1$ and $k=0, 1, \cdots, g$. Then we get two graphs $\Gamma_g^c, {\Gamma_g^c}'\in S^3$. $\Gamma_g^c$ contains $a_{2l}$, $b_{2k}$, and ${\Gamma_g^c}'$ contains $a_{2l+1}$, $b_{2k+1}$. Each graph has $g+3$ vertices and $2g+2$ edges. They are in the dual positions as in Figure \ref{fig of cages} ($g=2$), here graphs have been projected into the three dimensional Euclidean space $E^3$ from $S^3-\{(-1, 0)\}$. The vertex $a_2$ is at the infinity.
\begin{figure}[h]
\centerline{\scalebox{0.48}{\includegraphics{cage.eps}}}
\caption{$\Gamma_g^c$ and ${\Gamma_g^c}'$ in $S^3$}\label{fig of cages}
\end{figure}
Let $T$ be the following torus in $S^3$:
$$T=\{(z_1, z_2)\in S^3\mid |z_1|=|z_2|=\frac{\sqrt{2}}{2}\}.$$
It splits $S^3$ into two solid tori $V_1$ and $V_2$:
\begin{align*}
V_1&=\{(z_1, z_2)\in S^3\mid |z_1|\geq \frac{\sqrt{2}}{2}\},\\
V_2&=\{(z_1, z_2)\in S^3\mid |z_2|\geq \frac{\sqrt{2}}{2}\}.
\end{align*}
Let $D_{1,m}$ and $D_{2,n}$ be following meridian disks in $V_1$ and $V_2$ respectively:
\begin{align*}
D_{1,m}&=\{(\sqrt{1-r^2}e^{\frac{\pi}{4}i+\frac{m\pi}{2}i}, re^{\theta i})\in S^3\mid 0\leq r\leq\frac{\sqrt{2}}{2}, \theta\in \mathbb{R}\},\\
D_{2,n}&=\{(re^{\theta i}, \sqrt{1-r^2}e^{\frac{\pi}{2g+2}i+\frac{n\pi}{g+1}i})\in S^3\mid 0\leq r\leq\frac{\sqrt{2}}{2}, \theta\in \mathbb{R}\}.
\end{align*}
Here $m=0, 1, 2, 3$ and $n=0, 1, \cdots, 2g+1$. All the $D_{1,m}$ cut $V_1$ into $4$ cylinders. And each cylinder contains exactly one $a_m$ for some $m$. All the $D_{2,n}$ cut $V_2$ into $2g+2$ cylinders. And each cylinder contains exactly one $b_n$ for some $n$. Hence $T$ and all these disks cut $S^3$ into $2g+6$ cylinders. Let $V_g^c$ be the union of cylinders intersecting $\Gamma_g^c$, and ${V_g^c}'$ be the union of cylinders intersecting ${\Gamma_g^c}'$. Then $V_g^c$ and ${V_g^c}'$ are two handlebodies. $\partial V_g^c=\partial {V_g^c}'$ is a Heegaard surface of $S^3$, and $\partial V_g^c\cong\Sigma_g$.
Following three isometries on $S^3$ preserve the graph $\Gamma_g^c\cup{\Gamma_g^c}'$, the torus $T$, and the union of the disks. They also preserve $\partial V_g^c$.
\begin{align*}
\tau_g: (z_1, z_2)&\mapsto(iz_1, e^{\frac{\pi}{g+1}i}z_2)\\
\rho: (z_1, z_2)&\mapsto(-z_1, z_2)\\
\sigma: (z_1, z_2)&\mapsto(\bar{z}_1, z_2)
\end{align*}
For even $g>1$, $\tau_g$ has order $4g+4$ and $\tau_g^2\rho\sigma$ has order $2g+2$.
For each $g>1$, $\tau_g\sigma$ has order $2g+2$ and $\langle\tau_g\sigma, \rho\rangle$, $\langle\tau_g, \rho\rangle$,
$\langle\tau_g^2, \rho, \sigma\rangle$ are all abelian groups of
order $4g+4$.
Notice that $\tau_g$ and $\rho$ preserve the orientation of $S^3$, and $\sigma$ reverses the orientation of $S^3$. Only $\tau_g$ changes the two sides of $\partial V_g^c$. Then combining with Lemma \ref{Lem of change sides}, we have extendable cyclic (abelian) group actions on $\Sigma_g\cong \partial V_g^c$ as following:
(1). For even $g>1$, $\langle\tau_g\rangle$ gives a type $(-,+)$ extendable cyclic group action of order $4g+4$. This also realizes the maximum order of cyclic group actions on $\Sigma_g$ when $g$ is even.
(2). For even $g>1$, $\langle\tau_g^2\rho\sigma\rangle$ gives a type $(-,-)$ extendable cyclic group action of order $2g+2$.
(2$'$). For odd $g>1$, by adding shortest geodesics $a_0a_1$ and $a_0a_3$ to ${\Gamma_{g-1}^c}'$, we get a graph $\Gamma_g$. The group $\langle\tau_{g-1}^2\rho\sigma\rangle$ preserves $\Gamma_g$. Hence it also preserves some regular neighbourhood $N(\Gamma_g)$ of $\Gamma_g$. Then $\langle\tau_{g-1}^2\rho\sigma\rangle$ gives a type $(-,-)$ extendable cyclic group action of order $2g$ on $\Sigma_g\cong\partial N(\Gamma_g)$.
(3). For each $g>1$, $\langle\tau_g\sigma\rangle$ gives a type $(+,-)$ extendable cyclic group action of order $2g+2$.
(4). For each $g>1$, $\langle\tau_g\sigma, \rho\rangle$ gives a type $(+,-)$ extendable abelian group action of order $4g+4$. This also realizes the maximum order of orientation-preserving abelian group actions on $\Sigma_g$.
(5). For each $g>1$, $\langle\tau_g, \rho\rangle$ gives a type $(-,+)$ extendable abelian group action of order $4g+4$.
(6). For each $g>1$, $\langle\tau_g^2, \rho, \sigma\rangle$ gives a type $(-,-)$ extendable abelian group action of order $4g+4$.
\end{example}
\begin{example}[Wheel]\label{Ex of wheel}
Given odd $g>1$, we will construct a cyclic group
action on a Heegaard splitting of $S^3$.
Let $S^3$ and $T$ be in $\mathbb{C}^2$ as above. Let $L=L_1\cup L_2$ be the $(2, 4)$-torus link in $T$:
\begin{align*}
L_1&=\{(\frac{\sqrt{2}}{2}e^{2\theta i}, \frac{\sqrt{2}}{2}e^{\theta i})\in S^3\mid \theta\in\mathbb{R}\},\\
L_2&=\{(\frac{\sqrt{2}}{2}e^{2\theta i}, \frac{\sqrt{2}}{2}ie^{\theta i})\in S^3\mid \theta\in\mathbb{R}\}.
\end{align*}
Let
\begin{align*}
a_m&=(\frac{\sqrt{2}}{2}e^{\frac{2m\pi}{g-1}i}, \frac{\sqrt{2}}{2}e^{\frac{2m\pi}{2g-2}i}), m=0, 1, \cdots, 2g-3,\\
b_n&=(\frac{\sqrt{2}}{2}e^{\frac{(2n+1)\pi}{g-1}i}, \frac{\sqrt{2}}{2}ie^{\frac{(2n+1)\pi}{2g-2}i}), n=0, 1, \cdots, 2g-3.
\end{align*}
Then $a_0, a_1, \cdots, a_{2g-3}$ are $2g-2$ points in $L_1$, and $b_0, b_1, \cdots, b_{2g-3}$ are $2g-2$ points in $L_2$. Connect $a_i$ to $a_{i+g-1}$ with the shortest geodesic in $S^3$, and connect $b_i$ to $b_{i+g-1}$ with the shortest geodesic in $S^3$, here $i=0, 1, \cdots, g-2$. Then we get two graphs $\Gamma_g^w, {\Gamma_g^w}'\in S^3$. $\Gamma_g^w$ contains $L_1$ and ${\Gamma_g^w}'$ contains $L_2$. Each graph has $2g-2$ vertices and $3g-3$ edges. They are in the dual positions as in Figure \ref{fig of wheels} ($g=3$), here graphs have been projected into the three dimensional Euclidean space $E^3$ from $S^3-\{(-1, 0)\}$. The middle point of $a_1a_3$ is at the infinity.
\begin{figure}[h]
\centerline{\scalebox{0.67}{\includegraphics{wheel.eps}}}
\caption{$\Gamma_g^w$ and ${\Gamma_g^w}'$ in $S^3$}\label{fig of wheels}
\end{figure}
Let $T'$ be the following torus in $S^3$:
$$T'=\{(z_1, z_2)\in S^3\mid |z_1|=\frac{\sqrt{3}}{2}, |z_2|=\frac{1}{2}\}.$$
It splits $S^3$ into two solid tori ${V_1}'$ and ${V_2}'$:
\begin{align*}
{V_1}'&=\{(z_1, z_2)\in S^3\mid |z_1|\geq\frac{\sqrt{3}}{2}\},\\
{V_2}'&=\{(z_1, z_2)\in S^3\mid |z_2|\geq\frac{1}{2}\}.
\end{align*}
Let $D_k$ be the following meridian disk in ${V_1}'$:
$$D_k=\{(\sqrt{1-r^2}e^{\frac{\pi}{2g-2}i+\frac{k\pi}{g-1}i}, re^{\theta i})\in S^3\mid 0\leq r\leq \frac{1}{2}, \theta\in \mathbb{R}\}.$$
It lies between $(e^{\frac{k\pi}{g-1}i}, 0)$ and $(e^{\frac{(k+1)\pi}{g-1}i}, 0)$, here $k=0, 1, \ldots, 2g-3$.
Then these disks cut ${V_1}'$ into $2g-2$ cylinders. And each cylinder intersects exactly one of $\Gamma_g^w$ and ${\Gamma_g^w}'$. Let $A$ be the annulus in ${V_2}'$ separating $L_1$ and $L_2$:
$$A=\{(re^{2\theta i}, \sqrt{1-r^2}e^{\frac{\pi}{4}i+\theta i})\in S^3\mid -\frac{\sqrt{3}}{2}\leq r\leq\frac{\sqrt{3}}{2}, \theta\in\mathbb{R}\}.$$
Then $A$ cuts ${V_2}'$ into $2$ solid tori. Let $V_g^w$ be the
union of the solid torus and cylinders intersecting $\Gamma_g^w$, and ${V_g^w}'$ be the union of the solid torus and cylinders intersecting ${\Gamma_g^w}'$. Then $V_g^w$ and ${V_g^w}'$ are two handlebodies. $\partial V_g^w=\partial {V_g^w}'\cong\Sigma_g$, and $\partial V_g^w$ is a Heegaard surface of $S^3$.
Following isometry on $S^3$ preserves the graph $\Gamma_g^w\cup{\Gamma_g^w}'$, the torus $T'$, the annulus $A$, and the union of the disks. It also preserves $\partial V_g^w$.
$$\varphi_g: (z_1, z_2)\mapsto(e^{\frac{\pi}{g-1}i}z_1, ie^{\frac{\pi}{2g-2}i}z_2)$$
For odd $g>1$, $\varphi_g$ has order $4g-4$. Notice that $\varphi_g$ preserves the orientation of $S^3$ and changes the two sides of $\partial V_g^w$. Combining with Lemma \ref{Lem of change sides}, for odd $g>1$, $\langle\varphi_g\rangle$ gives a type $(-,+)$ extendable cyclic group action of order $4g-4$ on $\Sigma_g\cong \partial V_g^w$. This also realizes the maximum order of orientation-reversing periodic maps on $\Sigma_g$ when $g$ is odd.
\end{example}
\begin{example}[Fork]\label{Ex of fork}
Given even $g>1$, we will construct an abelian
group action on a Heegaard splitting of $S^3$.
Let $S^3$ be the unit sphere in $\mathbb{C}^2$:
$$S^3=\{(z_1, z_2)\in\mathbb{C}^2\mid |z_1|^2+|z_2|^2=1\}.$$
Let
\begin{align*}
a_m&=(e^{\frac{(2m+1)\pi}{2}i}, 0), m=0, 1,\\
b_n&=(0, e^{\frac{2n\pi}{g+2}i}), n=0, 1, \cdots, g+1.
\end{align*}
Connect $a_0$ to each $b_{2k}$ with the shortest geodesics in $S^3$, and connect $a_1$ to each $b_{2k+1}$ with the shortest geodesics in $S^3$, here $k=0, 1, \cdots, g/2$. Then we get two graphs $\Gamma^f_g, {\Gamma^f_g}'\in S^3$. $\Gamma^f_g$ contains $a_0$, $b_{2k}$, and ${\Gamma^f_g}'$ contains $a_1$, $b_{2k+1}$. Let $S$ be the following two dimensional sphere in $S^3$:
$$S=\{(z_1, z_2)\in\mathbb{C}^2\mid z_1\in \mathbb{R}\}\cap S^3.$$
It cuts $S^3$ into $2$ three dimensional balls $B_0$ and $B_1$. $B_0$ contains $a_0$ and $B_1$ contains $a_1$. $\Gamma^f_g$, ${\Gamma^f_g}'$ and $S$ are shown in Figure \ref{fig of forks} ($g=4$), here they have been projected into the three dimensional Euclidean space $E^3$ from $S^3-\{(-1, 0)\}$.
\begin{figure}[h]
\centerline{\scalebox{0.385}{\includegraphics{fork.eps}}}
\caption{$\Gamma^f_g$, ${\Gamma^f_g}'$ and $S$ in $S^3$}\label{fig of forks}
\end{figure}
Following two isometries on $S^3$ preserve the sphere $S$ and the graph $\Gamma^f_g\cup{\Gamma^f_g}'$.
\begin{align*}
\tau_{g+1}^2: (z_1, z_2)&\mapsto(-z_1, e^{\frac{2\pi}{g+2}i}z_2)\\
\rho\sigma: (z_1, z_2)&\mapsto(-\bar{z}_1, z_2)
\end{align*}
Hence they also preserve some closed regular neighbourhood $N(\Gamma^f_g)\cup N({\Gamma^f_g}')$ of $\Gamma^f_g\cup{\Gamma^f_g}'$, and preserve the common boundary of the two handlebodies $V^f_g$ and ${V^f_g}'$:
\begin{align*}
V^f_g&=N(\Gamma^f_g)\cup \overline{B_1-N({\Gamma^f_g}')},\\
{V^f_g}'&=N({\Gamma^f_g}')\cup \overline{B_0-N(\Gamma^f_g)}.
\end{align*}
The surface $\partial V^f_g=\partial {V^f_g}'$ is a Heegaard surface of $S^3$, and $\partial V^f_g\cong\Sigma_g$.
Since $\tau_{g+1}^2\rho\sigma=\rho\sigma\tau_{g+1}^2$, $\langle\tau_{g+1}^2, \rho\sigma\rangle$ is an abelian group of order $2g+4$. Since $\tau_{g+1}^2$ preserves the orientation of $S^3$, $\rho\sigma$ reverses the orientation of $S^3$, and only $\tau_{g+1}^2$ changes the two sides of $\partial V^f_g$, combining with Lemma \ref{Lem of change sides}, one can check that $\tau_{g+1}^2\rho\sigma$ and $\tau_{g+1}^2$ satisfy the condition of type $(Mix)$ extendable group action. Hence for even $g>1$, $\langle\tau_{g+1}^2, \rho\sigma\rangle$ gives a type $(Mix)$ extendable abelian group action of order $2g+4$ on $\Sigma_g\cong \partial V^f_g$.
\end{example}
\begin{example}[Square]\label{Ex of Square}
We will construct a $\oplus^5_{i=1}\mathbb{Z}_2$-action on $V_5$.
Let $\Gamma^s$ be the boundary of the square $[0,1]^2$ in $xy$-plane in the three dimensional Euclidean space $E^3$. Let $T_r$ and ${T_r}'\subset T_r$ be following translation groups:
\begin{align*}
T_r&=\{(a, b, c)\mid a, b, c\in \mathbb{Z}\},\\
{T_r}'&=\langle(2, 0, 0), (0, 2, 0), (0, 0, 1)\rangle.
\end{align*}
An element $t=(a, b, c)\in T_r$ acts on $E^3$ as following:
$$t: (x, y, z)\mapsto(x+a, y+b, z+c).$$
Following three isometries $r_x$, $r_y$ and $R_z$ on $E^3$ preserve the graph $\bigcup_{t\in T_r}t(\Gamma^s)$.
\begin{align*}
r_x: (x, y, z)&\mapsto(x, -y, -z)\\
r_y: (x, y, z)&\mapsto(-x, y, -z)\\
R_z: (x, y, z)&\mapsto(x, y, -z)
\end{align*}
In the three dimensional torus $E^3/{T_r}'\cong T^3$ we have a graph $\Gamma^s_5=(\bigcup_{t\in T_r}t(\Gamma^s))/{T_r}'$. It has $4$ vertices and $8$ edges. Since ${T_r}'$ is a normal subgroup of $\langle T_r, r_x, r_y, R_z\rangle$, the quotient group $\langle T_r, r_x, r_y, R_z\rangle/{T_r}'\cong\oplus^5_{i=1}\mathbb{Z}_2$ acts on $E^3/{T_r}'$, and it preserves $\Gamma^s_5$. Then it also preserves some closed regular neighbourhood $N(\Gamma^s_5)\cong V_5$ of $\Gamma^s_5$.
We see this $\oplus^5_{i=1}\mathbb{Z}_2$-action on $V_5$ is ``extendable over $T^3$''. But even on $\partial V_5\cong\Sigma_5$, it is not extendable over $S^3$, by Theorem \ref{Thm of maximum of CandA action}.
\end{example}
\section{Preliminaries for the proofs}\label{Sec of preliminaries}
In this section we will give some known results about actions of discrete groups on surfaces and three manifolds, which will be used later. Most results will be presented in the language of orbifold.
\begin{remark}
For orbifold theories, one can see \cite{BMP,MMZ,Th,Zi1}. In the following, contents in \S\ref{Subsec of OT} can be found in \cite{BMP}. Lemma \ref{Lem of 2Disorb} can be found in \cite{Ei}; Lemma \ref{Lem of 3Disorb} and \ref{Lem of 3SphCyc} rely on \cite{BMP,BP,Li,Sm,Th}: \cite{Li} for the orbifold having isolated singular points, and \cite{BMP,BP,Sm,Th} for the orbifold whose singular set has dimension at least $1$; Lemma \ref{Lem of Hanorb}, \ref{Lem of Euler char equ} and \ref{Lem of handcover} can be found in \cite{MMZ,Zi1}; Lemma \ref{Lem of extend to Hand} can be found in \cite{RZ2}.
\end{remark}
\subsection{Basic concepts in orbifold theory}\label{Subsec of OT}
Let $M$ be a n-dimensional manifold with a faithful smooth action of a discrete group $G$, then $M/G$ is an n-orbifold. For any $x\in M/G$, let $x'$ be one of its pre-images in $M$, and $St(x')$ be the stable subgroup of $x'$. Then the isomorphic type of $St(x')$ does not depend on the choice of $x'$. Denote it by $G_x$. It is the local group of $M/G$ at $x$. The order of $G_x$ is the index of $x$. If $|G_x|>1$, $x$ is a singular point of $M/G$. Otherwise $x$ is a regular point of $M/G$. If we forget all local groups and indices, then we get the underlying space $|M/G|$, which is a topological space with the quotient topology.
The orbifold $M/G$ is orientable if $M$ is orientable and $G$ preserves the orientation of $M$. $M/G$ is connected if $|M/G|$ is connected. And $M/G$ is compact if $|M/G|$ is compact. $M/G$ has non-empty boundary $\partial M/G$ if $M$ has non-empty boundary $\partial M$, then $\partial M/G$ is the orbifold $(\partial M)/G$. $M/G$ is closed if it is connected, compact and has empty boundary.
Two orbifolds $M_1/G_1$ and $M_2/G_2$ are homeomorphic via $p:M_1/G_1\rightarrow M_2/G_2$ if the induced map $|M_1/G_1|\rightarrow |M_2/G_2|$ is a homeomorphism, $p$ preserves local groups, and at any point $p$ can be locally lifted to an equivariant homeomorphism between open sets of $M_1$ and $M_2$.
If $M/G\cong M'/G'$ and $M'$ is simply connected, then the fundamental group $\pi_1(M/G)\cong G'$. Covering spaces of $M/G$ can be defined, and there is a one to one correspondence between covering spaces and conjugacy classes of subgroups of $\pi_1(M/G)$. Regular covering spaces will correspond to normal subgroups. A similar Van-Kampen theorem is also valid.
\begin{definition}\label{Def of isoP}
Let $P_T$ (respectively $P_O$, $P_I$) be a regular tetrahedron (respectively octahedron, icosahedron) centered at $(0,0,0)$ in $E^3$. For a polyhedron $P$ in $E^3$, define $I^+(P)$ to be the orientation-preserving isometric group of $P$.
For $n\in \mathbb{Z}_+$, define $r_n$ to be the following isometry on $E^3$:
$$r_n: (x,y,z) \mapsto (x\cos\frac{2\pi}{n}+y\sin\frac{2\pi}{n}, -x\sin\frac{2\pi}{n}+y\cos\frac{2\pi}{n}, z).$$
\end{definition}
\begin{lemma}\label{Lem of loc2orb}
Let $x$ be a point in an orientable 2-orbifold, then $x$ has a neighbourhood which is homeomorphic to some $E^2/\langle r_n\rangle$, and $G_x\cong \langle r_n\rangle$, $n\in \mathbb{Z}_+$. Here $E^2$ is the $xy$-plane in $E^3$.
\end{lemma}
\begin{lemma}\label{Lem of loc3orb}
Let $x$ be a point in an orientable 3-orbifold, then $x$ has a neighbourhood which is homeomorphic to some $E^3/H$, and $G_x\cong H$. $H$ is one of the groups $\langle r_n\rangle, \langle r_n, r_x\rangle, I^+(P_T), I^+(P_O), I^+(P_I), n\in \mathbb{Z}_+$. Here $r_x$ is defined in Example \ref{Ex of Square}
\begin{figure}[h]
\centerline{\scalebox{0.6}{\includegraphics{locmod.eps}}}
\caption{Local models of 3-orbifolds}\label{fig of 3orbiLoc}
\end{figure}
\end{lemma}
\subsection{Results about discal, spherical and handlebody orbifolds}
\begin{definition}\label{Def of DisSphHan}
Let $B^n$ and $S^n$ be the n-ball and n-sphere respectively. An orbifold which is homeomorphic to some $B^n/G$ (respectively $S^n/G$, $V_g/G$) is called a discal n-orbifold (respectively spherical n-orbifold, handlebody orbifold).
\end{definition}
\begin{lemma}\label{Lem of 2Disorb}
An orientable discal 2-orbifold is homeomorphic to some $B^2_u/\langle r_n\rangle$. Here $B^2_u$ is the unit 2-ball centered at $(0,0,0)$ in the $xy$-plane.
\end{lemma}
\begin{lemma}\label{Lem of 3Disorb}
An orientable discal 3-orbifold is homeomorphic to some $B^3_u/H$, and an orientable spherical 2-orbifold is homeomorphic to some $S^2_u/H$. Here $B^3_u$ and $S^2_u$ are the unit 3-ball and 2-sphere centered at $(0,0,0)$ in $E^3$ respectively. $H$ is one of $\langle r_n\rangle$, $\langle r_n, r_x\rangle$, $I^+(P_T)$, $I^+(P_O)$, $I^+(P_I)$, $n\in \mathbb{Z}_+$.
\end{lemma}
\begin{lemma}\label{Lem of 3SphCyc}
Let $S^3/G$ be a spherical orbifold, here $G$ is a non-trivial cyclic group.
If $S^3/G$ is orientable, then the set of singular points of index $|G|$ is $\emptyset$ or a $S^1$.
If $S^3/G$ is not orientable, then the set of singular points of index $|G|$ is a $S^0$ or a $S^2$. Here $S^0\cong \{0,1\}$ with discrete topology.
\end{lemma}
\begin{lemma}\label{Lem of Hanorb}
An orientable handlebody orbifold is a union of finitely many orientable discal 3-orbifolds $\{B^3_i/G_i\}^n_{i=1}, n\in \mathbb{Z}_+$, such that:
(a). In each $\partial B^3_i/G_i$ finitely many disjoint orientable discal 2-orbifolds are given;
(b). For $i\neq j$, if $B^3_i/G_i\cap B^3_j/G_j\neq\emptyset$, then $B^3_i/G_i\cap B^3_j/G_j$ is a union of finitely many orientable discal 2-orbifolds in (a);
(c). Each discal 2-orbifold in (a) is in exactly two elements in $\{B^3_i/G_i\}^n_{i=1}$.
(d). There is a choice of orientations of $\{B^3_i/G_i\}^n_{i=1}$, such that on any common discal 2-orbifold, the induced orientations are opposite.
Conversely, if a connected orientable 3-orbifold is a union of finitely many orientable discal 3-orbifolds $\{B^3_i/G_i\}^n_{i=1}$ satisfying (a--d), then it is an orientable handlebody orbifold. $\bigcup^n_{i=1}B^3_i/G_i$ is called a discal orbifold decomposition of it.
\end{lemma}
\begin{definition}\label{Def of graph of groups}
A finite graph of finite groups $\mathcal{G}$ is a labelled finite graph. Each vertex and edge is labelled by a finite group. If $G_v$ is a vertex adjacent to an edge $G_e$, then there is an injective homomorphism $\phi_{ev}:G_e\rightarrow G_v$. If we forget all labels, then we get its underlying graph $|\mathcal{G}|$.
\end{definition}
\begin{definition}\label{Def of induced GofGs}
Let $\mathcal{H}$ be an orientable handlebody orbifold. $\mathcal{H}=\bigcup^n_{i=1}B^3_i/G_i$ is a discal orbifold decomposition as in Lemma \ref{Lem of Hanorb}. Then $\mathcal{H}$ has an induced finite graph of finite groups (induced by the discal orbifold decomposition) defined as following: it contains $\{G_i\}^n_{i=1}$ as vertices; there is an edge $G$ between $G_i$ and $G_j$ if $B^3_i/G_i$ and $B^3_j/G_j$ share a common discal 2-orbifold $B^2/G$.
\end{definition}
\begin{definition}\label{Def of Euler char}
Let $\mathcal{G}$ be a finite graph of finite groups. $V$ and $E$ are its vertex set and edge set respectively. Define its Euler characteristic:
$$\chi(\mathcal{G})=\sum_{G_v\in V}\frac{1}{|G_v|}-\sum_{G_e\in E}\frac{1}{|G_e|}.$$
Let $\mathcal{H}$ be an orientable handlebody orbifold, and $\mathcal{H}\cong V_g/G$. Define its Euler characteristic:
$$\chi(\mathcal{H})= \frac{1-g}{|G|}.$$
\end{definition}
\begin{definition}\label{Def of finite injsurj}
A homomorphism $\phi: H\rightarrow G$ between groups is a finitely injective surjection if it is surjective and on any finite subgroup of $H$ it is injective.
\end{definition}
\begin{lemma}\label{Lem of Euler char equ}
Let $\mathcal{H}$ be an orientable handlebody orbifold. Let $\mathcal{G}$ be an induced finite graph of finite groups of $\mathcal{H}$. Then $\chi(\mathcal{H})=\chi(\mathcal{G})$.
\end{lemma}
\begin{lemma}\label{Lem of handcover}
Let $\mathcal{H}$ be an orientable handlebody orbifold, and $G$ be a finite group. Then $G$ can orientation-preservingly act on $V_g$ such that $V_g/G\cong \mathcal{H}$ if and only if there is a finitely injective surjection $\phi:\pi_1(\mathcal{H})\rightarrow G$ and $1-g=\chi(\mathcal{H})|G|$.
\end{lemma}
\begin{lemma}\label{Lem of extend to Hand}
Let $G$ be an abelian group acting on $\Sigma_g$. Suppose $\Sigma_g/G$ is orientable, having singular points $p_1,\cdots, p_{\alpha}$ and $q_1, \cdots, q_{\beta}$. Index of $p_j (1\leq j\leq \alpha)$ is $n_j$ and $n_j>2$; index of $q_k (1\leq k\leq \beta)$ is $2$. Then for some $\gamma\in \mathbb{Z}_+\cup\{0\}$ we have
\begin{align*}
\pi_1(\Sigma/G)=&\langle a_1,b_1,\cdots,a_{\gamma},b_{\gamma}, x_1,\cdots,x_{\alpha},y_1,\cdots,y_{\beta}\mid \\ &\prod^{\gamma}_{i=1}[a_i,b_i] \prod^{\alpha}_{j=1}x_j\prod^{\beta}_{k=1}y_k=1, x_j^{n_j}=y_k^2=1,1\leq j\leq \alpha,1\leq k\leq \beta\rangle.
\end{align*}
The action corresponds to a finitely injective surjection $\phi:\pi_1(\Sigma_g)\rightarrow G$. Then the following (a), (b) and (c) are equivalent:
(a). The $G$-action on $\Sigma_g$ extends to a handlebody $V_g$, with $\partial V_g=\Sigma_g$.
(b). The $G$-action on $\Sigma_g$ extends to a compact 3-manifold $M$, with $\partial M=\Sigma_g$.
(c). The generators $x_1,\cdots,x_\alpha$ corresponding to $p_1,\cdots, p_{\alpha}$ can be partitioned into pairs $x_s$, $x_t$ such that $\phi(x_s)=\phi(x_t^{-1})$.
\end{lemma}
\section{Maximum orders of cyclic and abelian actions on handlebodies}\label{Sec of maxOrdHan}
In this section we will determine the maximum orders of cyclic and abelian group actions on orientable handlebodies. Firstly we need some lemmas.
\subsection{Some useful lemmas}
\begin{lemma}\label{Lem of hand classify}
Let $V_g/G$ be an orientable handlebody orbifold. Here $G$ is cyclic and $|G|>g-1$. Then $V_g/G$ has an induced finite graph of finite groups as one of the following, here $l,m,n>1$.
\xymatrix{(A).&\mathbb{Z}_l \ar@{-}[r] & \mathbb{Z}_m \ar@{-}[r] & \mathbb{Z}_n&(1/l+1/m+1/n>1)}
\xymatrix{(B).&\mathbb{Z}_m \ar@{-}[r] & \mathbb{Z}_n& (0<1/m+1/n<1)}
\xymatrix{(C).&\mathbb{Z}_n \ar@(ur,dr)@{-}[]}
\xymatrix{(D).&\mathbb{Z}_n \ar@{-}[r] & \mathbb{Z}_m \ar@(ur,dr)@{-}[]^{\mathbb{Z}_m}}
\end{lemma}
\begin{proof}
Let $V_g/G=\bigcup^n_{i=1}B^3_i/G_i$ as in Lemma \ref{Lem of Hanorb}, and $\mathcal{G}$ be the induced finite graph of finite groups. Since $|G|>g-1$, by Definition \ref{Def of Euler char} and Lemma \ref{Lem of Euler char equ}, we have: $$\chi(\mathcal{G})=\chi(V_g/G)=\frac{1-g}{|G|}\in (-1,0).$$
Since $G$ is cyclic, all the vertices and edges of $\mathcal{G}$ are cyclic groups. Hence if an edge $G_e$ is non-trivial, then any vertex of it must be isomorphic to it, and if a vertex $G_v$ is non-trivial, then there are at most two non-trivial edges adjacent to it. Let $V$ and $E$ be the vertex set and edge set of $\mathcal{G}$ respectively, and $V'$ and $E'$ be the non-trivial vertex set and edge set of $\mathcal{G}$ respectively. By Definition \ref{Def of Euler char}, we have:
\begin{align*}
\chi(\mathcal{G})&=\chi(|\mathcal{G}|)+\sum_{G_v\in V}(\frac{1}{|G_v|}-1)-\sum_{G_e\in E}(\frac{1}{|G_e|}-1)\\ &=\chi(|\mathcal{G}|)+\sum_{G_v\in V'}(\frac{1}{|G_v|}-1)-\sum_{G_e\in E'}(\frac{1}{|G_e|}-1)\\
&<\chi(|\mathcal{G}|).
\end{align*}
Hence $\chi(|\mathcal{G}|)\in \{0,1\}$.
If $\chi(|\mathcal{G}|)=1$, then $|\mathcal{G}|$ is a tree. If $\mathcal{G}$ contains a trivial vertex $G_i$, and $G_e$ is an edge adjacent to $G_v$, $G_j$ is the other vertex of $G_e$, then $G_e$ is trivial; if $\mathcal{G}$ contains a non-trivial edge $G_e$, and $G_i$, $G_j$ are its two vertices, then $G_e\cong G_i\cong G_j$. In each case $B^3_i/G_i\cup B^3_j/G_j\cong B^3_j/G_j$. Replacing $B^3_i/G_i$ and $B^3_j/G_j$ by their union, we get a new discal decomposition of $V_g/G$. The new induced finite graph of finite groups will have less trivial vertices or less non-trivial edges. Hence we can assume $V'=V$ and $E'=\emptyset$. Then
$$\chi(\mathcal{G})=1+\sum_{G_v\in V'}(\frac{1}{|G_v|}-1).$$
Since for $G_v\in V'$ we have $|G_v|>1$, $|V'|=|V|$ must be $2$ or $3$. Hence $\mathcal{G}$ will be some finite graph of finite groups in the class (A) or (B).
If $\chi(|\mathcal{G}|)=0$, then we can delete an edge $G_e$ of $\mathcal{G}$ to get a maximal tree $\mathcal{G}'$ of it. Similar to the above discussion, we can assume $V'=V$ and edges in $\mathcal{G}'$ are trivial. The edge $G_e$ may be trivial or non-trivial. Then
$$\chi(\mathcal{G})=\sum_{G_v\in V'}(\frac{1}{|G_v|}-1)-(\frac{1}{|G_e|}-1).$$
If $G_e$ is trivial, then $|V'|=|V|=1$. Hence $\mathcal{G}$ will be some finite graph of finite groups in the class (C). If $G_e$ is non-trivial, then $|V'|=|V|=2$. If $G_e$ is a loop, then $\mathcal{G}$ will be some finite graph of finite groups in the class (D). Otherwise, as above we can get a new $\mathcal{G}$ with less non-trivial edges. Then $\mathcal{G}$ will be some finite graph of finite groups in the class (C).
\end{proof}
Let $\mathcal{C}$ be one of the classes (A--D) in Lemma \ref{Lem of hand classify}. In the following, we say that an orientable handlebody orbifold $\mathcal{H}$ is in the class $\mathcal{C}$ if $\mathcal{H}$ has an induced finite graph of finite groups belonging to $\mathcal{C}$. Following Lemma \ref{Lem of FundG} and \ref{Lem of EulerNo} can be derived easily from the Van-Kampen theorem and Definition \ref{Def of Euler char}.
\begin{lemma}\label{Lem of FundG}
The fundamental groups of the handlebody orbifolds in classes (A--D) are given in Table \ref{tab of FundG}.
\begin{table}[h]
\caption{Fundamental groups of classes (A--D)}\label{tab of FundG}
\begin{tabular}{|l|l|}
\hline (A) & $\mathbb{Z}_l\ast\mathbb{Z}_m\ast\mathbb{Z}_n$\\
\hline (B) & $\mathbb{Z}_m\ast\mathbb{Z}_n$\\
\hline (C) & $\mathbb{Z}_n\ast\mathbb{Z}$\\
\hline (D) & $\mathbb{Z}_n\ast (\mathbb{Z}_m\oplus\mathbb{Z})$ \\\hline
\end{tabular}
\end{table}
\end{lemma}
\begin{lemma}\label{Lem of EulerNo}
The Euler characteristic of the finite graphs of finite groups in classes (A--D) are given in Table \ref{tab of EulerNo}.
\begin{table}[h]
\caption{Euler characteristic of classes (A--D)}\label{tab of EulerNo}
\begin{tabular}{|l|l|}
\hline (A) & $1/l+1/m+1/n-2$\\
\hline (B) & $1/m+1/n-1$\\
\hline (C) & $1/n-1$\\
\hline (D) & $1/n-1$\\\hline
\end{tabular}
\end{table}
\end{lemma}
\begin{lemma}\label{Lem of odd order}
Let $\langle h\rangle$ be a finite cyclic group acting faithfully on a manifold $M$. $\overline{h}$ is the image of $h$ in $\langle h\rangle/\langle h^2\rangle$. Then $\overline{h}$ acts on $M/\langle h^2\rangle$. Suppose $x$ is a fixed point of $\overline{h}$ and $x$ has index $|G_x|$ in $M/\langle h^2\rangle$, then $|\langle h^2\rangle|/|G_x|$ is odd.
\end{lemma}
\begin{proof}
Let $x'$ be a pre-image of $x$ in $M$. Since $x$ is a fixed point of $\overline{h}$, the orbit $\{h^i(x')\mid i\in\mathbb{Z}\}$ and $\{h^{2i}(x')\mid i\in\mathbb{Z}\}$ are the same. Hence the pre-image of $x$ in $M$ contains odd points. Namely $|\langle h^2\rangle|/|G_x|$ is odd.
\end{proof}
\begin{lemma}\label{Lem of fixed point}
Let $h$ be a periodic map of order $2$ on a compact manifold $M$. If the Euler characteristic of $M$ is odd, then $h$ has a fixed point.
\end{lemma}
\begin{proof}
Since $h$ has order $2$, the induced linear map $h_{*i}:H_i(M,\mathbb{R})\rightarrow H_i(M,\mathbb{R})$ can only have eigenvalues $\pm 1$. Let $L(h)$ be the Lefschetz number of $h$, and $\chi(M)$ be the Euler characteristic of $M$. Then $L(h)\equiv\chi(M) (mod\, 2)$. Since $\chi(M)$ is odd, $L(h)\neq 0$. Then by the Lefschetz fixed point theorem, $h$ has a fixed point.
\end{proof}
\begin{lemma}\label{Lem of Tocyclic}
Let $l,m,n\in\mathbb{Z}_+$, and for $1\leq i\leq l$, $m_i \in\mathbb{Z}_+$. Let $[m_1,m_2,\cdots ,m_l]$ be the lowest common multiple of $m_i$, $1\leq i\leq l$.
(1). If $\phi:\mathbb{Z}_{m_1}\ast\mathbb{Z}_{m_2}\ast\cdots \ast\mathbb{Z}_{m_l}\rightarrow \mathbb{Z}_n$ is a finitely injective surjection, then $n=[m_1,m_2,\cdots ,m_l]$.
(2). If $\phi:\mathbb{Z}_{m_1}\ast\mathbb{Z}_{m_2}\ast\cdots \ast\mathbb{Z}_{m_l}\ast(\mathbb{Z}_{m}\oplus\mathbb{Z})\rightarrow \mathbb{Z}_n$ is a finitely injective surjection, then $n=t[m_1,m_2,\cdots, m_l,m]$ for some $t\in \mathbb{Z}_+$.
\end{lemma}
\begin{proof}
(1). The homomorphism $\phi$ induces a surjective homomorphism
$$\phi':\mathbb{Z}_{m_1}\oplus\mathbb{Z}_{m_2}\oplus\ldots \oplus\mathbb{Z}_{m_l}\rightarrow \mathbb{Z}_n,$$
which is injective on $\mathbb{Z}_{m_i}$, $1\leq i\leq l$. For any prime number $p$, let $\mathbb{Z}_{p^{\alpha_i}}$ be the $p$-component of $\mathbb{Z}_{m_i}$, and $\mathbb{Z}_{p^{\beta}}$ be the $p$-component of $\mathbb{Z}_n$. Since $\phi'$ is surjective, we have $max\{\alpha_1,\alpha_2,\cdots,\alpha_l\}\geq\beta$. Otherwise generators of $\mathbb{Z}_{p^{\beta}}$ will not lie in the image of $\phi'$. Since $\phi'$ is injective on $\mathbb{Z}_{m_i}$, we have $\beta\geq max\{\alpha_1,\alpha_2,\cdots,\alpha_l\}$. Hence $\beta=max\{\alpha_1,\alpha_2,\cdots,\alpha_l\}$, namely $n=[m_1,m_2,\cdots,m_l]$.
(2). The homomorphism $\phi$ induces a surjective homomorphism
$$\phi':\mathbb{Z}_{m_1}\oplus\mathbb{Z}_{m_2}\oplus\ldots \oplus\mathbb{Z}_{m_l}\oplus (\mathbb{Z}_{m}\oplus\mathbb{Z})\rightarrow \mathbb{Z}_n,$$
which is injective on $\mathbb{Z}_{m}$ and $\mathbb{Z}_{m_i}$, $1\leq i\leq l$. Let $\mathbb{Z}_{p^{\alpha}}$ be the $p$-component of $\mathbb{Z}_{m}$, then $\beta\geq max\{\alpha_1,\cdots,\alpha_l,\alpha\}$. Hence $n=t[m_1,\cdots,m_l,m]$ for some $t\in \mathbb{Z}_+$.
\end{proof}
\subsection{Maximum orders of cyclic and abelian actions on handlebodies}
\begin{proposition}\label{Pro of chg}
Table \ref{tab of maxorder of ORhandlebody} gives an upper bound of $CH_g^-$.
\end{proposition}
\begin{proof}
Let $\langle h\rangle$ be a cyclic group acting on $V_g$. $h$ reverses the orientation of $V_g$. Let $\overline{h}$ be the image of $h$ in $\langle h\rangle/\langle h^2\rangle$. Then $\overline{h}$ acts on $V_g/\langle h^2\rangle$ and has order $2$.
By (2) and (2$'$) in Example \ref{Ex of cage}, we have $CH_g^->2g-2$. If $|\langle h\rangle|=CH_g^-$, then $|\langle h^2\rangle|>g-1$. By Lemma \ref{Lem of hand classify}, $V_g/\langle h^2\rangle$ is in classes (A--D). Let $\Theta$ be the set of singular points in $V_g/\langle h^2\rangle$, and $N(\Theta)$ be a $\overline{h}$-invariant regular neighborhood of $\Theta$. Then $\overline{h}$ acts on $\Theta$ and $\overline{(V_g/\langle h^2\rangle)\backslash N(\Theta)}$ respectively.
{\bf Case 1}: $V_g/\langle h^2\rangle$ is in the class (A).
Then $|\langle h^2\rangle|$ is even. By Lemma \ref{Lem of fixed point}, $\overline{h}$ will have a fixed point in $\Theta$. Then by Lemma \ref{Lem of odd order}, for a fixed point $x$ we have that $|\langle h^2\rangle|/|G_x|$ is odd. Hence the index of $x$ must be even. Then we have $\{l,m,n\}=\{2,3,3\}$ or $\{2,2,2k\}$, $k\in \mathbb{Z}_+$.
If $\{l,m,n\}=\{2,3,3\}$, then by Lemma \ref{Lem of handcover}, there is a finitely injective surjection $\phi:\pi_1(V_g/\langle h^2\rangle)\rightarrow \langle h^2\rangle$. By Lemma \ref{Lem of FundG} and \ref{Lem of Tocyclic}, $\pi_1(V_g/\langle h^2\rangle)\cong \mathbb{Z}_2\ast\mathbb{Z}_3\ast\mathbb{Z}_3$ and $\langle h^2\rangle\cong \mathbb{Z}_6$. Hence $|\langle h\rangle|=12$. Then by Lemma \ref{Lem of EulerNo}, we have $g=6$.
Similarly, if $\{l,m,n\}=\{2,2,2k\}$, then $\langle h^2\rangle\cong \mathbb{Z}_{2k}$, $|\langle h\rangle|=4k$ and $g=2k$.
{\bf Case 2}: $V_g/\langle h^2\rangle$ is in the classes (B--D).
By Lemma \ref{Lem of fixed point}, $\overline{h}$ will have a fixed point $x$ in $\overline{(V_g/\langle h^2\rangle)\backslash N(\Theta)}$. By Lemma \ref{Lem of odd order}, $|\langle h^2\rangle|/|G_x|=|\langle h^2\rangle|$ is odd. Then by Lemma \ref{Lem of handcover}, \ref{Lem of FundG}, \ref{Lem of Tocyclic} and \ref{Lem of EulerNo}, we have:
If $V_g/\langle h^2\rangle$ is in the class (B), then $\pi_1(V_g/\langle h^2\rangle)\cong \mathbb{Z}_m\ast\mathbb{Z}_n$, and $\langle h^2\rangle\cong \mathbb{Z}_{[m,n]}$. Then $|\langle h\rangle|=2[m,n]$, and $g=[m,n]-(m+n)/(m,n)+1$. Since $n$ and $m$ are odd, $g$ is even.
If $V_g/\langle h^2\rangle$ is in the class (C), then $\pi_1(V_g/\langle h^2\rangle)\cong \mathbb{Z}_n\ast\mathbb{Z}$, and $\langle h^2\rangle\cong \mathbb{Z}_{kn}$. Then $|\langle h\rangle|=2kn$, and $g=kn-k+1$. Since $k$ and $n$ are odd, $g$ is odd.
If $V_g/\langle h^2\rangle$ is in the class (D), then $\pi_1(V_g/\langle h^2\rangle)\cong \mathbb{Z}_n\ast(\mathbb{Z}_m\oplus\mathbb{Z}$), and we have $\langle h^2\rangle\cong \mathbb{Z}_{k[m,n]}$. Then $|\langle h\rangle|=2k[m,n]$, and $g=k[m,n]-k[m,n]/n+1$. Since $k$, $m$ and $n$ are odd, $g$ is odd. Let $m=m'(m,n)$, then $|\langle h\rangle|=2(km')n$, and we have $g=(km')n-km'+1$.
\end{proof}
\begin{proposition}\label{Pro of chgcons}
Suppose $k,m,n\in \mathbb{Z}_+$. If $g=[m,n]-(m+n)/(m,n)+1$, then there is a cyclic group $\langle h\rangle$ of order $2[m,n]$ acting on $V_g$. And $h$ reverses the orientation of $V_g$.
If $g=kn-k+1$, then there is a cyclic group $\langle h\rangle$ of order $2kn$ acting on $V_g$. And $h$ reverses the orientation of $V_g$.
\end{proposition}
\begin{proof}
We can assume $m,n>1$. For $g=[m,n]-(m+n)/(m,n)+1$, let $\mathcal{F}$ be a 2-orbifold. It has underlying space $B^2$ and contains two singular points of indices $m$ and $n$ in the interior of $B^2$. Let $\mathcal{H}=\mathcal{F}\times [-1,1]$. It is in the class (B). $\pi_1(\mathcal{H})=\mathbb{Z}_m\ast\mathbb{Z}_n$, and there is a finitely injective surjection $\phi:\pi_1(\mathcal{H})\rightarrow \mathbb{Z}_{[m,n]}$. Hence by Lemma \ref{Lem of handcover}, there is a $\mathbb{Z}_{[m,n]}$-action on $V_g$ such that $V_g/\mathbb{Z}_{[m,n]}\cong \mathcal{H}$.
The pre-image of $\mathcal{F}\times \{0\}$ in $V_g$ is a surface $F$ and $V_g=F\times [-1,1]$. Then the product $Id_F\times (-Id_{[-1,1]})$ acts of $V_g$ and has order $2$. It is commutative with the $\mathbb{Z}_{[m,n]}$-action. Hence we have a cyclic action of order $2[m,n]$ on $V_g$. Its generator reverses the orientation of $V_g$.
For $g=kn-k+1$, the construction is similar. In this case $\mathcal{F}$ has underlying space an annulus and contains one singular point of index $n$ in its interior. $\mathcal{H}=\mathcal{F}\times [-1,1]$ is in the class (C).
\end{proof}
\begin{proposition}\label{Pro of ahg}
$AH_g^-=2AH_g$.
\end{proposition}
\begin{proof}
Clearly $AH_g^-\leq 2AH_g$. By (6) in Example \ref{Ex of cage} and Example \ref{Ex of Square}, we have $AH_g^-\geq 2AH_g$.
\end{proof}
\begin{proposition}\label{Pro of ag}
$A_g^-=AH_g^-$.
\end{proposition}
\begin{proof}
Let $G$ be an abelian group acting on $\Sigma_g$. $\Sigma_g/G$ is not orientable. Let $G_o$ be the subgroup of $G$ containing all elements which preserve the orientation of $\Sigma_g$. Then we have a 2-sheet cover $\Sigma_g/G_o\rightarrow\Sigma_g/G$, and the covering transformation $\tau$ reverses the orientation of $\Sigma_g/G_o$.
Let $\{p_1,\cdots, p_{\alpha}, q_1, \cdots, q_{\beta}\}$ be the set of singular points of $\Sigma_g/G_o$. $p_j (1\leq j\leq \alpha)$ has index bigger than $2$, and $q_k (1\leq k\leq \beta)$ has index $2$. Then $p_1, \cdots, p_{\alpha}$ can be partitioned into pairs by $\tau$. Since $G$ is abelian and $\tau$ reverses the orientation of $\Sigma_g/G_o$, their corresponding generators $x_1, \cdots, x_{\alpha}$ in $\Sigma_g/G_o$ can be partitioned into pairs satisfying the condition of Lemma \ref{Lem of extend to Hand}. Then $G_o$ can extend to a handlebody. Hence $|G_o|\leq AH_g$, and $A_g^-\leq AH_g^-$. Then $A_g^-=AH_g^-$.
\end{proof}
By Proposition \ref{Pro of chg}--\ref{Pro of ag} and Remark \ref{Rem of classical result}, we finished the proof of Proposition \ref{Pro of ORhandlebody}, hence finished the proof of Theorem \ref{Thm of CandA on handlebody}.
\section{Maximum orders of extendable actions on surfaces}\label{Sec of maxOrdSur}
In this section we will determine the maximum orders of extendable cyclic and abelian group actions on orientable closed surfaces. For an extendable action on $\Sigma_g$, we will identify $\Sigma_g$ and its image $e(\Sigma_g)$ in $S^3$. Firstly we will prove two lemmas.
\begin{lemma}\label{Lem of Fix circle}
Let $\langle h\rangle$ be a non-trivial cyclic group acting faithfully on $S^3$. $h$ preserves the orientation of $S^3$ and has a fixed point. Then the set of singular points in $S^3/\langle h\rangle$ is a $S^1$, with index $|\langle h\rangle|$, and $\pi_1(|S^3/\langle h\rangle|)$ is trivial.
\end{lemma}
\begin{proof}
Let $k$ be a factor of $|\langle h\rangle|$ and $k\neq |\langle h\rangle|$. By Lemma \ref{Lem of 3SphCyc}, the set of fixed points of $\langle h^k\rangle$ is a $S^1$. Hence it belongs to the set of fixed points of $\langle h\rangle$. Namely the set of singular points in $S^3/\langle h\rangle$ is a $S^1$, with index $|\langle h\rangle|$. Since $\pi_1(S^3/\langle h\rangle)\cong\langle h\rangle$, $\pi_1(|S^3/\langle h\rangle|)$ must be trivial.
\end{proof}
\begin{lemma}\label{Lem of homology}
Let $S_H^3$ be an integer homology 3-sphere, and $S_g$ be a genus $g\geq 0$ orientable closed subsurface in $S_H^3$. Then $S_g$ splits $S_H^3$ into two compact manifolds $M$ and $N$, each has the same integer homology groups as $V_g$.
\end{lemma}
\begin{proof}
Since $S_H^3$ is orientable, $S_g$ is two sided in $S_H^3$. If it does not split $S_H^3$, then there is a surjective homomorphism from $H_1(S_H^3)$ to $\mathbb{Z}$, which is a contradiction. Then we have the following MV-sequence:
\centerline{\xymatrix{H_3(S_H^3) \ar[r] & H_2(S_g) \ar[r] & H_2(M)\oplus H_2(N) \ar[r] & H_2(S_H^3)}}
\centerline{\xymatrix{H_2(S_H^3) \ar[r] & H_1(S_g) \ar[r] & H_1(M)\oplus H_1(N) \ar[r] & H_1(S_H^3)}}
Clearly we have $H_i(M)=H_i(N)=0$ for $i>2$. By the first exact sequence, $H_2(M)=H_2(N)=0$. By the second exact sequence, $H_1(M)\oplus H_1(N)\cong \oplus_{i=1}^{2g}\mathbb{Z}$.
Then by the ``half live half die'' theorem, $H_1(M)\cong H_1(N)\cong \oplus_{i=1}^{g}\mathbb{Z}$.
\end{proof}
\subsection{Maximum orders of cyclic extendable actions on surfaces}
\begin{proposition}\label{Pro of cemp}
$CE_g(-,+)=C_g^-$.
\end{proposition}
\begin{proof}
Clearly $CE_g(-,+)\leq C_g^-$. By (1) in Example \ref{Ex of cage} and Example \ref{Ex of wheel}, we have $CE_g(-,+)=C_g^-$.
\end{proof}
\begin{proposition}\label{Pro of cepm}
$CE_g(+,-)=2g+2$.
\end{proposition}
\begin{proof}
Let $\langle h\rangle$ be a cyclic group acting on $\Sigma_g$. And the action is a type $(+,-)$ extendable action realizing the maximum order $CE_g(+,-)$. Then the action of $\langle h^2\rangle$ on $\Sigma_g$ can extend to some 3-manifold. Hence by Lemma \ref{Lem of extend to Hand}, it can extend to a handlebody $V_g$. By (3) in Example \ref{Ex of cage}, we have $|\langle h^2\rangle|>g-1$. Hence $V_g/\langle h^2\rangle$ is in the classes (A--D) in Lemma \ref{Lem of hand classify}.
Since $h$ reverses the orientation of $S^3$, it has a fixed point. Then $h^2$ has a fixed point. By Lemma \ref{Lem of Fix circle}, the set of singular points of $S^3/\langle h^2\rangle$ is a $S^1$, with index $|\langle h^2\rangle|$. Then $V_g/\langle h^2\rangle$ can only belong to classes (A--C). By Lemma \ref{Lem of EulerNo}, we have:
If $V_g/\langle h^2\rangle$ is in the class (A), then $|\langle h^2\rangle|=l=m=n=2$. Hence $|\langle h\rangle|=4$, and $g=2$.
If $V_g/\langle h^2\rangle$ is in the class (B), then $|\langle h^2\rangle|=m=n$. Hence $|\langle h\rangle|=2n$, $g=n-1$.
If $V_g/\langle h^2\rangle$ is in the class (C), then $|\langle h^2\rangle|=n$. Hence $|\langle h\rangle|=2n$, $g=n$.
Hence $|\langle h\rangle|\leq 2g+2$. Then by (3) in Example \ref{Ex of cage}, $CE_g(+,-)=2g+2$.
\end{proof}
\begin{proposition}\label{Pro of cemm}
$CE_g(-,-)=2g+1+(-1)^g$.
\end{proposition}
\begin{proof}
The proof is the same as above, except that if $V_g/\langle h^2\rangle$ is in the class (B).
In this case, $\Sigma_g/\langle h^2\rangle$ is a $S^2$ with four singular points. Let $\Theta$ be the set of singular points in $S^3/\langle h^2\rangle$. By Lemma \ref{Lem of Fix circle}, $\Theta$ is a $S^1$, and $S^3/\langle h^2\rangle$ is an integer homology 3-sphere. By Lemma \ref{Lem of homology}, $|\Sigma_g/\langle h^2\rangle|$ splits $|S^3/\langle h^2\rangle|$ into two compact manifolds $M$ and $N$. Let $\overline{h}$ be the image of $h$ in $\langle h\rangle/\langle h^2\rangle$, and $N(\Theta)$ be a $\overline{h}$-invariant regular neighbourhood of $\Theta$. Then $\overline{h}$ acts on $\overline{M-N(\Theta)}$. By Lemma \ref{Lem of homology}, $\overline{M-N(\Theta)}$ will have the same homology groups as $V_2$. Then by Lemma \ref{Lem of fixed point}, $\overline{h}$ has a fixed point in $\overline{M-N(\Theta)}$. By Lemma \ref{Lem of odd order}, we have that $|\langle h^2\rangle|$ is odd. Hence $g$ is even. Namely we have $CE_g(-,-)\leq 2g+1+(-1)^g$.
Then by (2) and (2$'$) in Example \ref{Ex of cage}, $CE_g(-,-)=2g+1+(-1)^g$.
\end{proof}
\begin{proposition}\label{Pro of cemix}
The type $(Mix)$ extendable cyclic group action does not exist.
\end{proposition}
\begin{proof}
For a type $(Mix)$ extendable $G$-action, there is a surjective homeomorphism $G\rightarrow\mathbb{Z}_2\oplus \mathbb{Z}_2$. Hence $G$ can not be cyclic.
\end{proof}
\subsection{Maximum orders of abelian extendable actions on surfaces}
\begin{proposition}\label{Pro of aealmost}
$AE_g(+,-)=AE_g(-,+)=AE_g(-,-)=2AE_g$.
\end{proposition}
\begin{proof}
Let $\mathcal{T}$ be one of $(+,-)$, $(-,+)$ and $(-,-)$. Clearly $AE_g\mathcal{T}\leq 2AE_g$. Then by (4), (5) and (6) in Example \ref{Ex of cage}, we have $AE_g\mathcal{T}=2AE_g$.
\end{proof}
\begin{proposition}\label{Pro of abelnot}
For odd $g>1$, the type $(Mix)$ extendable abelian group action does not exist.
\end{proposition}
\begin{proof}
Suppose there is a type $(Mix)$ extendable abelian $G$-action on $\Sigma_g$. Then there exist $h_1, h_2\in G$ such that $h_1$ preserves the orientation of $\Sigma_g$ and reverses the orientation of $S^3$, and $h_2\in G$ reverses the orientation of $\Sigma_g$ and preserves the orientation of $S^3$. Then both $h_1$ and $h_2$ change the two sides of $\Sigma_g$.
By Lemma \ref{Lem of homology}, $\Sigma_g$ splits $S^3$ into $M$ and $N$, and $H_1(M)\simeq H_1(N)\simeq \oplus_{i=1}^{g}\mathbb{Z}$. Let $i_{M}: \Sigma_g\rightarrow M$, $i_{N}: \Sigma_g\rightarrow N$ be the inclusions. We can choose a basis of $H_1(\Sigma_g)$,
denoted by $\{\alpha_1, \alpha_2, \cdots, \alpha_g, \beta_1, \beta_2, \cdots, \beta_g\}$, satisfying the following conditions:
(1) $\{{i_M}_*(\alpha_k)\}$ is a basis of $H_1(M)$, $k=1,2,\cdots,g$.
(2) $\{{i_N}_*(\beta_k)\}$ is a basis of $H_1(N)$, $k=1,2,\cdots,g$.
(3) ${i_N}_*(\alpha_k)=0$ and ${i_M}_*(\beta_k)=0$, $k=1, 2, \cdots,g$.
Then consider the intersection product of $H_1(\Sigma_g)$. By the
condition (3) we have $\alpha_i\bullet\alpha_j=0$ and
$\beta_i\bullet\beta_j=0$, $1\leq i, j\leq g$, hence we have:
\[ \begin{pmatrix}
\alpha_1\\ \vdots\\ \alpha_g\\ \beta_1\\ \vdots\\
\beta_g
\end{pmatrix} \bullet
\begin{pmatrix}
\alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g
\end{pmatrix} =
\begin{pmatrix}
0 & X \\ -X^t & 0
\end{pmatrix} \]
Here $X=(x_{i,j})$ is a $g\times g$ matrix, and $x_{i,j}=\alpha_i\bullet\beta_j$. $X^t$ is the transposition of $X$. There are $g\times g$ matrices $A,B,C,D$, such that:
\[ {h_1}_*\begin{pmatrix}
\alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g
\end{pmatrix} =
\begin{pmatrix}
\alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g
\end{pmatrix}
\begin{pmatrix}
0 & A \\ B & 0
\end{pmatrix} \]
\[ {h_2}_*\begin{pmatrix}
\alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g
\end{pmatrix} =
\begin{pmatrix}
\alpha_1, \cdots, \alpha_g, \beta_1, \cdots, \beta_g
\end{pmatrix}
\begin{pmatrix}
0 & C \\ D & 0
\end{pmatrix} \]
By computing the intersection product, we have the following:
\[ \begin{pmatrix}
0 & B^t \\ A^t & 0
\end{pmatrix}
\begin{pmatrix}
0 & X \\ -X^t & 0
\end{pmatrix}
\begin{pmatrix}
0 & A \\ B & 0
\end{pmatrix} =
\begin{pmatrix}
0 & X \\ -X^t & 0
\end{pmatrix} \]
\[ \begin{pmatrix}
0 & D^t \\ C^t & 0
\end{pmatrix}
\begin{pmatrix}
0 & X \\ -X^t & 0
\end{pmatrix}
\begin{pmatrix}
0 & C \\ D & 0
\end{pmatrix} =
\begin{pmatrix}
0 & -X \\ X^t & 0
\end{pmatrix} \]
Hence we have $A^tXB=-X^t$ and $C^tXD=X^t$. Since the group $G$ is abelian, $h_1h_2=h_2h_1$. Then we have:
\[ \begin{pmatrix}
0 & A \\ B & 0
\end{pmatrix}
\begin{pmatrix}
0 & C \\ D & 0
\end{pmatrix} =
\begin{pmatrix}
0 & C \\ D & 0
\end{pmatrix}
\begin{pmatrix}
0 & A \\ B & 0
\end{pmatrix} \]
This is equivalent to $AD=CB$ and $BC=DA$.
The $g\times g$ matrices $A$, $B$, $C$, $D$ and $X$ are all non-degenerate. By computing the determinants we have
$|A||X||B|=(-1)^g|X|$, $|C||X||D|=|X|$ and $|A||D|=|C||B|$. Hence
$(-1)^g=|ABCD|=|AD|^2$, and $g$ is even.
\end{proof}
\begin{proposition}\label{Pro of aeeven}
For even $g>1$, $AE_g(Mix)=2g+4$.
\end{proposition}
\begin{proof}
Let $G$ be an abelian group acting on $\Sigma_g$. The action is a type $(Mix)$ extendable action, and $|G|=AE_g(Mix)$. Let $G_o$ be the subgroup of $G$ containing all elements preserving both the orientations of $\Sigma_g$ and $S^3$, then $|G|=4|G_o|$.
Choose $h\in G$ such that it reverses the orientation of $S^3$. Let $Fix(h)$ be the set of fixed points of $h$. By Lemma \ref{Lem of 3SphCyc}, $Fix(h)$ is a $S^0$ or $S^2$. Since $G$ is abelian, every element of $G$ keeps $Fix(h)$ invariant. Let $G_o'$ be the subgroup of $G$ containing all elements preserving both the orientations of $Fix(h)$ and $S^3$. If $Fix(h)\cong\{0, 1\}$, then preserving the orientation of $Fix(h)$ means fixing $0$ and $1$.
{\bf Case 1}: $G_o'\ncong\mathbb{Z}_2\oplus \mathbb{Z}_2$ and $|G_o'|>2$.
Then $G_o'$ is a cyclic group of order $|G_o'|\geq 3$, and the $G_o'$-action on $S^3$ has a fixed point. By Lemma \ref{Lem of Fix circle}, the set of singular points of $S^3/G_o'$ is a circle $\Theta$, with index $|G_o'|$, and $\pi_1(|S^3/G_o'|)$ is trivial. $|S^3/G_o'|$ is an integer homology 3-sphere.
Let $\Theta'$ be the pre-image of $\Theta$ in $S^3$, then $\Theta'$ is also a circle. Since $|G_o'|\geq 3$, $\Sigma_g$ intersects $\Theta'$ transversely. Hence $|\Sigma_g/G_o'|$ is an orientable closed surface in $|S^3/G_o'|$. Then $\Sigma_g/G_o'$ is an orientable 2-suborbifold in $S^3/G_o'$, namely $G_o'$ preserves both the orientations of $\Sigma_g$ and $S^3$. Since $|G/G_o'|\leq 4$, we have $G_o'=G_o$. By Lemma \ref{Lem of homology}, $|\Sigma_g/G_o|$ splits $|S^3/G_o|$ into two compact manifolds $M$ and $N$.
Let $g'$ be the genus of $|\Sigma_g/G_o|$, and $k$ be the number of singular points in $\Sigma_g/G_o$. Then $k$ must be even. By the Riemann-Hurwitz formula, we have:
$$2-2g=|G_o|(2-2g'-k(1-\frac{1}{|G_o|})).$$
By Example \ref{Ex of fork}, $|G_o|\geq (2g+4)/4=g/2+1$. Then we have:
$$(g',k,|G_o|)=(2,0,g-1),(1,2,g),(0,4,g+1),(0,6,g/2+1).$$
Notice that there is a $G/G_o\cong\mathbb{Z}_2\oplus \mathbb{Z}_2$-action on $(\Sigma_g/G_o,S^3/G_o)$. It satisfies a similar condition of the type $(Mix)$ extendable action.
If $(g', k, |G_o|)=(2, 0, g-1)$, then $\Theta$ is contained in one side of
$|\Sigma_g/G_o|$ in $|S^3/G_o|$. Hence the required $\mathbb{Z}_2\oplus\mathbb{Z}_2$-action does not exist.
If $(g', k, |G_o|)=(1, 2, g)$, then $|\Sigma_g/G_o|\simeq T^2$. In the proof of Proposition \ref{Pro of abelnot}, we only used homology theories, and the proof does not depend on $g>1$. Hence by a similar proof, the required $\mathbb{Z}_2\oplus\mathbb{Z}_2$-action on $(|\Sigma_g/G_o|,|S^3/G_o|)$ does not exist.
If $(g', k, |G_o|)=(0, 4, g+1)$, then $|\Sigma_g/G_o|\simeq S^2$ and $k=4$. Suppose $|\Sigma_g/G_o|$ intersects $\Theta$ at $\{A, B, C, D\}$, see Figure \ref{fig:twoBoneC}.
\begin{figure}[h]
\centerline{\scalebox{0.5}{\includegraphics{SandL.eps}}}
\caption{$\Sigma_g/G_o$ and $\Theta$ in $S^3/G_o$}\label{fig:twoBoneC}
\end{figure}
If the required $\mathbb{Z}_2\oplus \mathbb{Z}_2$-action exists, then there exists $\eta\in \mathbb{Z}_2\oplus \mathbb{Z}_2$, it preserves the orientation of $S^3/G_o$ and reverses the orientation of $\Sigma_g/G_o$. Then on $\{A, B, C, D\}$ it has no fixed point. Otherwise, in $S^3/G$ the image of the fixed point will have non-abelian local group. Since $\eta$ changes the two sides of $S^3/G_o$, we can assume $\eta(AB)=AD$. Then $\eta$ will have order $4$, which is a contradiction.
Hence we have $(g', k, |G_o|)=(0, 6, g/2+1)$, and $|G|=2g+4$.
{\bf Case 2}: $G_o'\cong \mathbb{Z}_2\oplus \mathbb{Z}_2$ or
$|G_o'|=2$.
If $|G|=AE_g(Mix)>2g+4$, then $|G|>8$. Hence $|G_o'|\neq 2$. Then $G_o'\cong \mathbb{Z}_2\oplus \mathbb{Z}_2$, and $|G|=16$. By Proposition \ref{Pro of abelnot} and \ref{Pro of ORhandlebody}, $g$ can only be $4$.
Let $G'$ be the subgroup of $G$ containing all elements preserving the orientation of $\Sigma_4$. By the proof of Proposition \ref{Pro of ag}, the $G'$-action can extend to $V_4$. Since $|G/G_o'|\leq 4$ and $|G|=16$, we have $G/G_o'\cong \mathbb{Z}_2\oplus \mathbb{Z}_2$. Then $G'\cong \mathbb{Z}_2\oplus \mathbb{Z}_2\oplus \mathbb{Z}_2$ or $\mathbb{Z}_4\oplus \mathbb{Z}_2$. Let $\mathcal{G}$ be the induced finite graph of finite groups of $V_4/G'$. On one hand $\chi(\mathcal{G})=(1-4)/8=-3/8$. On the other hand, vertices and edges of $\mathcal{G}$ have orders $1$, $2$ or $4$. Hence $4\chi(\mathcal{G})\in \mathbb{Z}$. This is a contradiction.
\end{proof}
By Proposition \ref{Pro of cemp}--\ref{Pro of aeeven} and Remark \ref{Rem of classical result}, we finished the proof of Proposition \ref{Pro of maxorder of five type}, hence finished the proof of Theorem \ref{Thm of maximum of CandA action}.
\bibliographystyle{amsalpha}
| 63,819
|
TITLE: Isomorphism map keep smoothness?
QUESTION [1 upvotes]: Let $E$ be a smooth curve, that is, one dimensional projective variety with dimension one, and Jacobi matrix is non-singular at any point.
And suppose that the map $φ$, which sends $E$ to $E'$, is isomorphism.
Isomorphism means there are two morphism on both direction which composites identity.
Then, can we say that $E'$ is smooth?
REPLY [3 votes]: Yes.
An isomorphism $\varphi$ of varieties induces an isomorphism of local rings $\mathcal O_x\cong \mathcal O_{\varphi(x)}$ at every point, hence an isomorphism of the maximal ideals $\mathfrak m_x\cong \mathfrak m_{\varphi(x)}$ and hence an isomorphism of the cotangent spaces $\mathfrak m_x/\mathfrak m^2_x\cong \mathfrak m_{\varphi(x)}/\mathfrak m^2_{\varphi(x)}.$ It follows that the dimension of the cotangent spaces at $x$ and $\varphi(x)$ are the same, so $x$ is a smooth point of $E$ exactly when $\varphi(x)$ is a smooth point of $E'.$
| 24,596
|
The broadest spectrum of identification falls under this category. From simple room numbers to towering monoliths, simple aisle markers in parking lots or vehicle directional information monuments on mall perimeter drives, whether a practical choice, an economic choice or an environmental choice to utilize ambient or natural light a well designed image with effective contrast this sign type can enhance any company image or complete a unified design
Non-Illuminated Signs
| 240,903
|
TITLE: Is limit of $\sin x$ at infinity finite?
QUESTION [2 upvotes]: As $x$ tends to infinity, sin x oscillates rapidly between $1$ and $-1$. So we are not able to pinpoint exactly what the limit is. But whatever it is, can we say that it would be finite? Or do we always have to say that its limit doesn't exist or it's undefined?
REPLY [2 votes]: The limit does not exist in the usual sense. If $\lim\limits_{x\to\infty} \sin x = L$ and $L$ is some particular number, then it would be possible to assure that $\sin x$ is between $L\pm0.001$ by making $x$ big enough, and we would probably be able to figure out how big is big enough in this case. But $\pm1$ both occur as values of $\sin x$ no matter how big $x$ gets and $\pm1$ cannot both differ from the same number $L$ by less than $0.001$.
But I wrote "in the usual sense". I cannot rule out the possibility that in some contexts an unusual sense might be appropriate. For example
$$
\lim_{A\to\infty} \Big( \text{average value of $\sin x$ in the interval } 0\le x\le A \Big) = 0,
$$
etc.
| 123,334
|
\begin{document}
\title
{Subring subgroups in symplectic groups in characteristic 2}
\author{Anthony Bak}
\address
{Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany}
\email{bak@mathematik.uni-bielefeld.de}
\author{Alexei Stepanov}
\address
{St.Petersburg State University\newline
and\newline
St.Petersburg Electrotechnical University
}
\email{stepanov239@gmail.com}
\thanks
{The work of the second named author under this publication is supported by
Russian Science Foundation, grant N.14-11-00297}
\begin{abstract}
In 2012 the second author obtained a description of the lattice of subgroups of a Chevalley group
$G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided
$\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$.
It turns out that this lattice is a disjoint union of ``sandwiches''
parameterized by subrings $R$ such that $K\subseteq R\subseteq A$.
In the current article a similar result is proved for $\Phi=C_n$, $n\ge3$,
and $2=0$ in $K$.
In this setting one has to introduce more sandwiches, namely, sandwiches which are parameterized
by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$.
The result we get generalizes Ya.\,N.\,Nuzhin's theorem of
2013 concerning the root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description
of the subgroup lattice is obtained, but under the condition that $A$ is an algebraic extension
of a field~$K$.
\end{abstract}
\maketitle
\section*{Introduction}
Throughout this paper $K$, $R$ and $A$ will denote commutative rings.
Let $G=\G_P(\Phi,\blank)$ denote a Chevalley--Demazure group scheme
with a reduced irreducible root system $\Phi$ and weight lattice $P$.
If the weight lattice $P$ is not important, we leave it out of the notation.
Denote by $E(A)=\E_P(\Phi,A)$ the elementary subgroup
of $G(A)$, i.\,e. the subgroup generated by all elementary root
unipotent elements $x_\alpha(t)$, \ $\alpha\in\Phi$, \ $t\in A$.
Let $K$ be a subring of $A$. We study the lattice
$\mathcal L=L\bigl(E(K),G(A)\bigr)$ of subgroups of $G(A)$,
containing $E(K)$.
The standard description of $\mathcal L$ is called a \emph{sandwich classification
theorem}. It states that for each $H\in\mathcal L$ there
exists a unique subring $R$ between $K$ and $A$ such that $H$
lies between $E(R)$ and its normalizer $N_A(R)$ in $G(A)$.
The lattice $L\bigl(E(R),N_A(R)\bigr)$ of all subgroups of $G(A)$ that lie between $E(R)$ and
$N_A(R)$ is called a \emph{standard sandwich}.
Thus, the sandwich classification theorem holds iff $\mathcal L$ is the disjoint union of all standard
sandwiches. In~\cite{StepStandard} the second author proved the sandwich classification
theorem provided that $\Phi$ is doubly laced and $2$ is invertible in $K$.
\emph{In this article we consider the symplectic case of rank $n\ge3$ with $2=0$ in $A$,
in particular, we always assume that $\Phi=C_n$}.
In this situation we show that the sandwich classification theorem, as formulated above, does not hold.
Let $R$ be a subring of $A$,
containing $K$. Recall~\cite{BakBook} that an additive subgroup $\Lambda$ of $R$ is called a
(symplectic) \emph{form parameter} if it contains $2R$ and is closed under multiplication by $\xi^2$
for all $\xi\in R$.
If $2=0$, the set $R^2=\{\xi^2\mid \xi\in R\}$ is a subring of $R$,
and a form parameter $\Lambda$ in $R$ is just an $R^2$-submodule of $R$. Let $\Ep(R,\Lambda)$
denote the subgroup of $\Sp(R)$ generated by all root unipotents $x_\alpha(\mu)$ and $x_\beta(\lambda)$,
where $\alpha$ is a short root and $\mu \in R$ and $\beta$ is a long root and $\lambda\in\Lambda$. Suppose now
that $\Lambda\supseteq K$. Then clearly $\Ep(R,\Lambda)\ge\Ep(K)$. But one can check
that if $\Lambda\ne R$ then $\Ep(R,\Lambda)$ is not contained in any standard sandwich
$L\bigl(\Ep(R'),N_A(R')\bigr)$ such that $K \subseteq R'$. This shows that $\mathcal L$
cannot be a union of standard
sandwiches, unless we enlarge our standard sandwiches or introduce additional ones. It turns out that the
latter approach is the correct one. It is analogous to that followed by the first author
in~\cite{BakThesis} and by E.\,Abe and K.Suzuki in~\cite{AbeSuzuki,AbeNormal} in order
to provide enough sandwiches to classify subgroups of $\Sp(R)$, which are normalized by $\Ep(R)$.
If $A$ and $K$ are field and $A$ is algebraic over $K$, then the sandwich classification
theorem was obtained by Ya.\,N.\,Nuzhin in~\cite{Nuzhin,Nuzhin13}.
In~\cite{Nuzhin13} he considered the case of doubly laced root systems
in characteristic $2$ and $G_2$ in characteristics $2$ and $3$. In all these cases sandwiches
were parameterized by pairs; each pair consists of a subring and an additive subgroup,
satisfying certain properties.
Let $F$ be a field of characteristic $2$. Then there are injections
$\Sp(F)\rightarrowtail\operatorname{SO}_{2n+1}(F)\rightarrowtail\Sp(F)$, which turn to isomorphisms
if $F$ is perfect, see~\cite{Steinberg} Theorem~28, Example~(a) after this theorem, and Remark before Theorem~29.
Note that in this case the Chevalley groups $\operatorname{G}_P(C_n,F)$ and $\operatorname{G}_P(B_n,F)$
do not depend on the weight lattice $P$ (\cite{Steinberg}, Exercise after Corollary~5 to Theorem~$4'$).
Therefore, the lattice $L\bigl(\operatorname{E}(\Phi,K),\operatorname{G}(\Phi,F)\bigr)$ embeds to the lattice
$L\bigl(\operatorname{Ep}_{2n}(\mathbb F_2),\Sp(F)\bigr)$ for $\Phi=B_n,C_n$ and a subring $K$ of $F$,
and its description follows from the main result of the current article.
Actually, the injections above are constructed in~\cite{Steinberg} on the level of Steinberg groups.
Since over a field $K_2(\Phi,F)$ is generated by Steinberg symbols, they induce the injections of Chevalley
groups. Over a ring $A$ there is no appropriate description of $K_2$ available, therefore we can not
use the above arguments. Moreover, in the ring case the group $\operatorname{G}_P(\Phi,A)$
depends on $P$. By these reasons, the adjoint group of type $C_n$ and groups of type $B_n$ over rings will be
considered in a subsequent article.
We state now our main result. Following~\cite{BakBook}, we call a pair $(R,\Lambda)$ consisting
of a ring $R$ and a form parameter $\Lambda$ in $R$ a \emph{form ring}.
\begin{ithm}\label{MAIN}
Let $A$ denote a (commutative) ring such $2=0$ in $A$. Let $K$ be a subring of $A$. If $H$ is a subgroup
of\/ $\Sp(A)$ containing\/ $\Ep(K)$, $n\ge 3$, then there is a unique form ring $(R,\Lambda)$ such that
$K \subseteq \Lambda \subseteq R \subseteq A$ and
$$
\Ep(R,\Lambda)\le H\le N_A(R,\Lambda),
$$
\noindent
where $N_A(R,\Lambda)$ is the normalizer of\/ $\Ep(R,\Lambda)$ in\/ $\Sp(A)$.
\end{ithm}
Let $\mathcal L(R,\Lambda)=L\bigl(\Ep(R,\Lambda), N_A(R,\Lambda)\bigr)$ denote the lattice of all
subgroups $H$ of $\Sp(A)$ such that $\Ep(R,\Lambda)\le H\le N_A(R,\Lambda)$.
From now on, $\mathcal L(R,\Lambda)$ is what we shall mean by a \emph{standard sandwich}.
In view of Theorem~\ref{MAIN} it is natural to study the lattice structure of $\mathcal L(R,\Lambda)$.
By definition $\Ep(R,\Lambda)$ is normal in $N_A(R,\Lambda)$. Therefore the lattice structure of
$\mathcal L(R,\Lambda)$ is the same as that of the quotient group $N_A(R,\Lambda)/\Ep(R,\Lambda)$.
The lattice $\mathcal L(R,\Lambda)$
contains an important subgroup $\Sp(R,\Lambda)$, which is called a Bak symplectic group,
whose definition will be recalled in Section~1. A group will be called \emph{quasi-nilpotent} if it is a
direct limit of nilpotent subgroups, i.\,e. the group has a directed system of nilpotent subgroups
whose colimit is the group itself.
In the following result we use the notion of Bass--Serre
dimension of a ring introduced by the first author in~\cite[\S~4]{BakNonabelian}.
Recall that the Krull (or combinatorial) dimension of a topological space is the supremum
of the lengths of proper chains of nonempty closed irreducible subsets.
Bass--Serre dimension $d=\operatorname{BS-dim} R$ of a ring $R$ is the smallest integer such that the
maximal spectrum $\operatorname{Max}R$ is a union of a finite number of irreducible Noetherian
subspaces of Krull dimension not greater than $d$. If there is no such integer, then
$\operatorname{BS-dim} R=\infty$.
\begin{ithm}\label{nilpotent}
Let $A$ denote a (commutative) ring and let $R$ be a subring of $A$
and $(R,\Lambda)$ a form ring. Suppose $n\ge 2$.
\begin{enumerate}
\item $\Sp(R,\Lambda) = \Sp(R) \cap N_A(R,\Lambda)$.
\item $\Sp(R,\Lambda)$ is normal in $N_A(R,\Lambda)$.
\item $N_A(R,\Lambda)/\Sp(R,\Lambda)$ is abelian.
\item $\Sp(R,\Lambda)/\Ep(R,\Lambda)$ is quasi-nilpotent.
\item Let $R_0$ denotes the subring of $R$, generated by all elements $\xi^2$ such that
$\xi\in R$. If Bass--Serre dimension of $R_0$ is finite, then $\Sp(R,\Lambda)/\Ep(R,\Lambda)$ is nilpotent.
\end{enumerate}
In particular, the sandwich quotient group $N_A(R,\Lambda)/\Ep(R,\Lambda)$ is quasi-nilpotent
by abelian or nilpotent by abelian if $\operatorname{BS-dim} R_0<\infty$.
\end{ithm}
Although Theorem~\ref{MAIN} is proven under the assumptions $2=0$ and $n\ge3$, we do not invoke these
assumptions until the proof of the theorem in Section~6. In particular, Theorem~\ref{nilpotent} is
proven without these assumptions. Of course, we always assume that $n\ge2$.
\subsection*{Notation}
Let $H$ be a group.
For two elements $x,y\in H$ we write $[x,y]=xyx^{-1}y^{-1}$ for their
commutator and $x^y=y^{-1}xy$ for the $y$-conjugate of $x$.
For subgroups $X,Y\le H$ we let $X^Y$ denote the normal closure of
$X$ in the subgroup generated by $X$ and $Y$,
while $[X,Y]$ stands for the mixed commutator group generated by $X$ and $Y$.
By definition it is the group generated by all commutators $[x,y]$ such that
$x\in X$ and $y\in Y$.
The commutator subgroup $[X,X]$ of $X$ will be also denoted by $D(X)$ and we
set $D^k(X)=\bigl[D^{k-1}(X),D^{k-1}(X)\bigr]$.
Recall that a group $X$ is called \emph{perfect} if $D(X)=X$.
The identity matrix is denoted by $e$ as well as the identity element of a Chevalley group.
We denote by $e_{ij}$ the matrix with $1$ in position~$(ij)$
and zeroes elsewhere. The entries of a matrix $g$ are denoted by $g_{ij}$.
For the entries of the inverse matrix we use abbreviation $(g^{-1})_{ij}=g'_{ij}$.
The transpose of the matrix $g$ is denoted by $\T g$, thus $(\T g)_{ij}=g_{ji}$.
\section{The symplectic group}\label{Sp}
The symplectic group $\Sp(R)$, its elementary root unipotent elements, and its elementary
subgroup $\Ep(R)$ will be recalled below. The groups $\Sp(R,\Lambda)$ and their elementary
subgroups $\Ep(R,\Lambda)$ will be defined in Section~\ref{Sp(R,Lambda)}.
Since the groups $\Sp(R,\Lambda)$ are not in general algebraic
(in fact, $\Sp(R,\Lambda)$ is algebraic iff $\Lambda=R$ or $\Lambda=\{0\}$),
it is convenient to work with the standard matrix representation of $\Sp(R)$.
On the other hand, we want to use the notions of parabolic subgroup, unipotent radical, etc.\
from the theory of algebraic groups. Thus, to simplify the exposition, we define below these notions
directly in terms of the matrix representation we use.
Following Bourbaki~\cite{Bourbaki4-6} we view the root system $C_n$ and its set of fundamental
roots $\Pi$ in the following way.
Let $V_n$ denote $n$-dimensional euclidean space with the orthonormal basis $\eps_1,\dots,\eps_n$.
Let
\begin{align*}
C_n&=\{\pm\eps_i\pm\eps_j\mid 1\le i<j\le n\}\cup\{\pm2\eps_k\mid 1\le k\le n\},\\
\Pi&=\{\alpha_i=\eps_i-\eps_{i+1},\text{ where }i=1,\dots,n-1,\ \alpha_{n}=2\eps_n\}.
\end{align*}
\noindent
The elements $\pm\eps_i\pm\eps_j$ are called \emph{short roots} and the elements $\pm2\eps_k$ \emph{long roots}.
From the perspective of algebraic groups, we want the standard Borel subgroup of $\Sp(R)$ to be the
group of all upper triangular matrices in $\Sp(R)$.
Accordingly, we take the following matrix description of $\Sp(R)$.
Let $J$ denote the $n\times n$-matrix with 1 in each antidiagonal position and zeroes elsewhere. Let
$F=\left(\begin{smallmatrix}0&J\\ -J& 0\end{smallmatrix}\right)$.
Then, the group $\Sp(R)$ is the
subgroup of $\GL_{2n}(R)$ consisting of all matrices which preserve the bilinear
form whose matrix is $F$. In other words,
$$
\Sp(R)=\{g\in\GL_{2n}(R)\mid \T{g}Fg=F\}.
$$
Let $I=(1,\dots,n,-n,\dots,-1)$ denote the linearly ordered set whose linear ordering is obtained
by reading from left to right.
We enumerate the rows and columns of the matrices of
$\GL_{2n}(R)$ by indexes from $I$. Thus the position of a coordinate of a matrix in
$\GL_{2n}(R)$ is denoted by a pair $(i,j)\in I\times I$.
In the language of algebraic groups the set of all diagonal matrices in $\Sp(R)$ is a \emph{maximal torus}.
Let $e$ denote the $2n\times 2n$ identity matrix.
The following matrices are \emph{elementary root unipotent elements} of $\Sp(R)$
with respect to the torus above:
\begin{align*}
x_{\eps_i-\eps_j}(\xi)&=T_{ij}(\xi)=T_{-j,-i}(-\xi)=e+\xi e_{ij}-\xi e_{-j,-i},\\
x_{\eps_i+\eps_j}(\xi)&=T_{i,-j}(\xi)=T_{j,-i}(\xi)=e+\xi e_{i,-j}+\xi e_{j,-i},\\
x_{-\eps_i-\eps_j}(\xi)&=T_{-i,j}(\xi)=T_{-j,i}(\xi)=e+\xi e_{-i,j}+\xi e_{-j,i},\\
x_{2\eps_k}(\xi)&=T_{k,-k}(\xi)=e+\xi e_{k,-k},\\
x_{-2\eps_k}(\xi)&=T_{-k,k}(\xi)=e+\xi e_{-k,k},\\
&\text{where }\xi\in R,\ 1\le i,j,k\le n,\ i\ne j.\\
\end{align*}
Note that the subscripts $(i,j)$, $(i,j)$, $(i,-j)$, $(j,-i)$, $(-j,-i)$, $(k,-k)$, and $(-k,k)$
on the $T$ above all belong to the set
$\tilde C_n=I\times I\setminus \{(k,k)\mid k\in I\}$ of nondiagonal positions of a matrix
from $\Sp(R)$ and exhaust $\tilde C_n$.
There is a surjective map $p:\tilde C_n\to C_n$ defined by the following rule.
\begin{align*}
p(i,j)&=p(-j,-i)=\eps_i-\eps_j;\\
p(i,-j)&=p(j,-i)=\eps_i+\eps_j;\\
p(-i,j)&=p(-j,i)=-\eps_i-\eps_j;\\
p(k,-k)&=2\eps_k;\\
p(-k,k)&=-2\eps_k;\\
\text{where }&\xi\in R,\ 1\le i,j,k\le n,\ i\ne j.\\
\end{align*}
\noindent
With this notation the correspondence between elementary symplectic transvections $T_{ij}(\xi)$
and root elements $x_\alpha(\xi)$ looks as follows.
$$
T_{ij}(\xi)=x_{p(i,j)}(-\sign(ij)\xi)\qquad\text{for all } i\ne j\in I.
$$
Note that $p$ maps symmetric (with respect to the antidiagonal) positions to the same root.
Therefore, for $(ij)\in I$ and $\xi\in R$ we have $T_{i,j}(\xi) = T_{-j,-i}(-\sign(ij)\xi)$.
The root subgroup scheme $X_\alpha$ is defined by $X_\alpha(R)=\{x_\alpha(\xi)\mid\xi\in R\}$.
The scheme $X_\alpha$ is naturally isomorphic to $\mathbb G_a$, i.\,e.
$x_\alpha(\xi)x_\alpha(\mu)=x_\alpha(\xi+\mu)$ for all $\xi,\mu\in R$.
The following commutator formulas are
well known in matrix language, cf.~\cite[\S~3]{BakVavHyp1}. They are special cases of the Chevalley
commutator formula in the algebraic group theory.
\begin{align*}
[x_\alpha(\lambda),x_\beta(\mu)]&=x_{\alpha+\beta}(\pm\lambda\mu),
\text{ if }\alpha+\beta\in\Phi,\,\widehat{\alpha\beta}=2\pi/3;\\
[x_\alpha(\lambda),x_\beta(\mu)]&=x_{\alpha+\beta}(\pm2\lambda\mu),
\text{ if }\alpha+\beta\in\Phi,\,\widehat{\alpha\beta}=\pi/2;\\
[x_\alpha(\lambda),x_\beta(\mu)]&=x_{\alpha+\beta}(\pm\lambda\mu)x_{\alpha+2\beta}(\pm\lambda\mu^2),
\text{ if }\alpha+\beta,\alpha+2\beta\in\Phi,\,\widehat{\alpha\beta}=3\pi/4;\\
[x_\alpha(\lambda),x_\beta(\mu)]&=e,
\text{ if }\alpha+\beta\notin\Phi\cup\{0\}
\end{align*}
In our proofs we make frequent use of the \emph{parabolic subgroup}~$P_1$ of $\Sp$.
In the matrix language above it is defined as follows:
$$
P_1(R)=\{g\in\Sp{R}\mid g_{i1}=g_{-1-i}=0\ \forall i\ne1\}
$$
\noindent
The definition above of $\Sp(R)$ shows that $g_{11}=g^{-1}_{-1,-1}$ for
any matrix $g\in P_1(R)$. The \textit{unipotent radical} $U_1$ of $P_1$ is the subgroup
generated by all root subgroups $T_{1i}$ such that $i\ne 1$.
The \emph{Levi subgroup} $L_1(R)$ of $P_1(R)$ consists of all $g\in P_1(R)$
such that $g_{1i}=g_{-i,-1}=0$ for all $i\ne 1$. As a group scheme it is isomorphic
to $\mathbb G_m\times\operatorname{Sp}_{2n-2}$.
\section{Bak symplectic groups}\label{Sp(R,Lambda)}
The Bak symplectic group $\Sp(R,\Lambda)$ is the particular case of the Bak general
unitary group, where the involution is trivial and the symmetry $\lambda=-1$.
The main references for the definition and the structure of the general unitary group
is the book~\cite{BakBook} and the paper~\cite{BakVavHyp1} by N.\,Vavilov and the first author.
In this section we recall definitions and simple properties to be used in the sequel.
Let $R$ be a commutative ring.
An additive subgroup $\Lambda$ of $R$ is called a \emph{symplectic form parameter}
in $R$, if it contains $2R$ and is closed under multiplication by squares, i.\,e. $\mu^2\lambda\in\Lambda$
for all $\mu\in R$ and $\lambda\in\Lambda$.
Define $\Sp(R,\Lambda)$ as the subgroup of $\Sp(R)$ consisting of all matrices preserving
the quadratic form which takes values in $R/\Lambda$ and is defined by the matrix
$\left(\begin{smallmatrix}0 & J\\ 0 & 0\end{smallmatrix}\right)$.
Let $*$ denote the involution on the matrix ring $\M_n(R)$ given by the formula
$a^*=J\T aJ$. Note that this is the reflection of a matrix with respect to the antidiagonal.
Define
$$
\M_n(R,\Lambda)=\{a\in\M_n(R)\mid a=a^*,\,a_{n-k+1\,k}\in\Lambda\text{ for all }k=1,\dots,n\}.
$$
Write a matrix of degree $2n$ in the block form
$\left(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\right)$, where $a,b,c,d\in\M_n(R)$.
It follows from~\cite[Lemma~2.2]{BakVavHyp1} and its proof that under the notation above
we have the following formula.
\begin{lem}\label{BakSp}
$
\Sp(R,\Lambda)=\left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix}\mid
a^*d-c^*b=e\text{ and }c^*a,d^*b\in\M_n(R,\Lambda)\right\}.
$
\end{lem}
It is easy to check that $\M_n(R,\Lambda)$ is a form parameter in the ring $\M_n(R)$ with
involution $*$, corresponding to the symmetry $\lambda=-1$.
The minimal form parameter in $\M_n(R)$ with the same involution and symmetry is
$\M_n(R,2R)$; it is denoted by $\Min(R)$.
Let $\bar\M_n(R,\Lambda)$ denote the additive group $\M_n(R,\Lambda)/\Min(R)$.
\begin{lem}\label{barM}
The group $\bar\M_n(R,\Lambda)$
has a natural structure of a left $M_n(R)$-module under the operation
$a\circ\bar b=aba^*\mod\Min(R)$, where $a\in\M_n(R)$, \ $\bar b\in\bar\M_n(R,\Lambda)$,
and $b$ is a preimage of $\bar b$ in $\M_n(R,\Lambda)$.
\end{lem}
By abuse of notation for $a\in\M_n(R)$ and $b\in\M_n(R,\Lambda)$ we shall write $a\circ b$ instead of
$a\circ(b+\Min(R))$.
It is easy to check that an elementary root unipotent element $x_\alpha(\xi)\in\Sp(R)$ belongs to
$\Sp(R,\Lambda)$
if and only if $\alpha$ is a short root or $\xi\in\Lambda$. Denote by $\Ep(R,\Lambda)$ the group
generated by all such elements:
$$
\Ep(R,\Lambda)=\langle x_\alpha(\xi)\mid
\alpha\in C_n^{short}\,\&\,\xi\in R\,\lor\alpha\in C_n^{long}\,\&\,\xi\in\Lambda\rangle.
$$
\section{Subgroups generated by elementary root unipotents}\label{bottom}
In this section we assume that $n\ge3$.
Let $H$ be a subgroup of $\Sp(A)$, containing $\Ep(K)$.
The following lemma shows that we can uncouple a short root element from
a long one inside $H$.
\begin{lem}\label{RightSide}
Let $\alpha,\beta\in C_n$ be a short and a long root, respectively, such that
$\alpha+\beta$ is not a root.
Let $g=x_{\alpha}(\mu)x_{\beta}(\lambda)$, where $\lambda,\mu\in A$.
If\/ $\Ep(K)^g\le H$ (e.\,g. $g\in H$), then each factor of $g$ belongs to $H$.
\end{lem}
\begin{proof}
Since $n\ge3$, there exists a short root $\gamma$ such that $\gamma+\alpha$
is a short root and $\gamma+\beta$ is not a root. Then, $X_\beta(A)$ commutes with
$X_\alpha(A)$ and $X_\gamma(A)$, hence $[g,x_\gamma(1)]=x_{\alpha+\gamma}(\pm\mu)\in H$.
Conjugating this element by an appropriate element from the Weyl group over $K$
and taking the inverse if necessary, we get $x_\alpha(\mu)\in H$. It follows that
$\Ep(K)^{x_{\beta}(\lambda)}\le H$.
Now, take a short root $\delta$ such that $\beta+\delta$ is a root. Then,
$[x_\delta(1),x_\beta(\lambda)]=
x_{\delta+\beta}(\pm\lambda)x_{2\delta+\beta}(\pm\lambda)\in H$.
Notice that the root $\delta+\beta$ is short whereas $2\delta+\beta$ is long.
As in the first paragraph of the proof one concludes that
$x_{\delta+\beta}(\pm\lambda)\in H$, hence $x_{2\delta+\beta}(\pm\lambda)\in H$.
Again, using the action of the Weyl group and taking inverse if necessary, one
shows that $x_{\beta}(\lambda)\in H$.
\end{proof}
Put $P_\alpha(H)=\{t\in A\,|\,x_\alpha(t)\in H\}$.
Since the Weyl group acts transitively on the set of roots of the same length,
it is easy to see that $P_\alpha(H)=P_\beta(H)$ if $|\alpha|=|\beta|$.
Let $R=R_H=P_\alpha(H)$ for any short root $\alpha$, and let
$\Lambda=\Lambda_H=P_\beta(H)$ for any long root $\beta$.
\begin{lem}\label{FormIdeal}
With the above notation $(R,\Lambda)$ is a form ring and
$K\subseteq \Lambda\subseteq R\subseteq A$.
\end{lem}
\begin{proof}
Clearly, $P_\alpha(H)$ is an additive subgroup of $A$.
Since $n\ge3$, there are two short roots $\alpha,\alpha'$ such that
$\alpha+\alpha'$ also is short. The commutator formula
$$
[x_\alpha(\lambda),x_{\alpha'}(\mu)]=x_{\alpha+\alpha'}(\pm\lambda\mu)
$$
shows that $R$ is a ring.
Now, let $\alpha,\alpha'$ be short roots which are orthogonal in $V_n$ and such that
$\beta=\alpha+\alpha'\in\Phi$ is a long root. Then
$$
[x_\alpha(\mu),x_{\alpha'}(1)]=x_{\beta}(\pm2\mu).
$$
If $\mu\in R$, then this element belongs to $H$. This proves that
$2R\subseteq\Lambda$. Finally, we show that $\Lambda$ is closed under multiplication
by squares in $R$. Let
$$
g=[x_\beta(\lambda),x_{-\alpha}(\mu)]=
x_{\alpha'}(\pm\lambda\mu)x_{\beta-2\alpha}(\pm\lambda\mu^2)
$$
If $\lambda\in\Lambda$ and $\mu=1$, then $g\in H$, and Lemma~\ref{RightSide} shows
that $x_{\alpha'}(\pm\lambda)\in H$. Therefore, $\Lambda\subseteq R$.
On the other hand, if $\mu\in R$ and $\lambda\in\Lambda$, then $g$ also lies in $H$,
and by Lemma~\ref{RightSide} we have $x_{\beta-2\alpha}(\pm\lambda\mu^2)\in H$.
This shows that $\Lambda$ is stable under multiplication by squares of elements of $R$.
\end{proof}
If $H$ is a subgroup of $\Sp(A)$ containing $\Ep(K)$,
then $(R_H,\Lambda_H)$ is called the \textit{form ring associated with} $H$.
\section{The normalizer}\label{top}
Let $(R,\Lambda)$ be a form subring of a ring $A$ such that $1\in\Lambda$.
In this section we develop properties of the normalizer
$N_A(R,\Lambda)$ of the group $\Ep(R,\Lambda)$ in $\Sp(A)$ and prove Theorem~\ref{nilpotent}.
Here we do not assume that $n\ge3$.
By a result of Bak and Vavilov~\cite[Theorem~1.1]{BakVavNormal}
we know that $\Ep(R,\Lambda)$ is normal in
$\Sp(R,\Lambda)$, thus $\Sp(R,\Lambda)\le N_A(R,\Lambda)$.
First, we show that the quotient $N_A(R,\Lambda)/\Sp(R,\Lambda)$ is abelian.
\begin{lem}\label{NormalizerElem}
Let $g\in\Sp(A)$.
If\/ $\Ep(R,\Lambda)^g\le \GL_{2n}(R)$ then
$g_{ij}g_{kl}\in R$ for all $i,j,k,l\in I$.
\end{lem}
\begin{proof}
To begin we express a matrix unit $e_{jk}$ as a linear combination $\sum\xi_m a^{(m)}$
for some $\xi_m\in R$ and $a^{(m)}\in\Ep(R,\Lambda)$. Note that the set of such linear combinations
is closed under multiplication.
Suppose $j\ne k$ and let $i\ne\pm j,\pm k$. Then $e_{jk}=(T_{ji}(1)-e)(T_{ik}(1)-e)$ and
$e_{jj}=e_{jk}e_{kj}$. It follows that if $\Ep(R,\Lambda)^g\le \GL_{2n}(R)$, then
$g^{-1}e_{jk}g\in\M_n(R)$. Thus $g'_{ij}g_{kl}\in R$ for all $i,j,k,l\in I$
(recall that $g'_{ij}=(g^{-1})_{ij}$).
The conclusion of the lemma follows since the entries of $g^{-1}$ coincide with the entries of
$g$ up to sign and a permutation.
\end{proof}
The next proposition describes the normalizer in the case $A=R$.
\begin{prop}\label{Normalizer for R=A}
$N_R(R,\Lambda)=\Sp(R,\Lambda)$.
\end{prop}
\begin{proof}
Let $g=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in N_R(R,\Lambda)$.
Since $g\in\Sp(R)$, then $d^*a-b^*c=e$.
We have to prove that $c^*a,\,d^*b\in M_n(R,\Lambda)$.
Since
$h=\left(\begin{smallmatrix}e&J\\0&e\end{smallmatrix}\right)\in\Ep(R,\Lambda)$,
we have
$$
f=ghg^{-1}=\begin{pmatrix}e-aJc^* & aJa^* \\
-cJc^* & e+cJa^*\end{pmatrix}\in\Ep(R,\Lambda)
$$
It follows that the matrices $-(cJc^*)^*(e-aJc^*)$ and $(e+cJa^*)^*(aJa^*)$ belong
to $M_n(R,\Lambda)$. Since $cJc^*$ and $aJa^*$ are automatically in
$M_n(R,\Lambda)$, we have that $cJc^*aJc^*,\,aJc^*aJa^*\in M_n(R,\Lambda)$.
Modulo $\Min(R)$ we can write
$(cJ)\circ (c^*a),\,(aJ)\circ (c^*a)\in \bar M_n(R,\Lambda)$.
By Lemma~\ref{barM} $\bar M_n(R,\Lambda)$ is an $M_n(R)$ module, therefore
\begin{multline*}
c^*a+\Min(R)=\bigl(J(d^*a-b^*c)J\bigr)\circ (c^*a)=\\
(Jd^*)\circ\bigl((aJ)\circ(c^*a)\bigr)-
(Jb^*)\circ\bigl((cJ)\circ(c^*a)\bigr)\in \bar M_n(R,\Lambda).
\end{multline*}
The proof that $d^*b\in M_n(R,\Lambda)$ is essentially the same.
\end{proof}
\begin{cor}\label{NormComm}
$[N_A(R,\Lambda),N_A(R,\Lambda)]\le\Sp(R,\Lambda)$.
\end{cor}
\begin{proof}
If $g,h\in N_A(R,\Lambda)$ and $f=[g,h]$, then by Lemma~\ref{NormalizerElem}
for all indexes $p,q$ we have
$$
f_{pq}=\sum\limits_{i,j,k=1}^n g'_{pi}h'_{ij}g_{jk}h_{kq}=
\sum\limits_{i,j,k=1}^n (g'_{pi}g_{jk})(h'_{ij}h_{kq})\in R
$$
\noindent
Therefore,
$$
f\in\Sp(R)\cap N_A(R,\Lambda)=N_R(R,\Lambda)=\Sp(R,\Lambda).
$$
\end{proof}
The first 3 items of Theorem~\ref{nilpotent} are already proved.
Our next goal is to show that the main theorem of~\cite{HazDim} implies that
$\Sp(R,\Lambda)/\Ep(R,\Lambda)$ is nilpotent, provided that $R_0$ has finite
Bass--Serre dimension. This will imply the rest of Theorem~\ref{nilpotent}.
Recall that $R_0$ denotes the subring of $R$ generated by all elements $\xi^2$
such that $\xi\in R$.
\begin{lem}\label{semilocal}
If $R_0$ is semilocal and $1\in\Lambda$, then $\Sp(R,\Lambda)=\Ep(R,\Lambda)$.
\end{lem}
\begin{proof}
Since $R$ is integral over $R_0$, it is a direct limit of $R_0$-subalgebras $R'\subseteq R$
such that $R'$ is module finite and integral over $R_0$. By the first theorem of Cohen--Seidenberg,
each $R'$ is semilocal. Thus by~\cite[Lemma~4]{Bak75} $\Sp(R',\Lambda\cap R')=\Ep(R',\Lambda\cap R')$
for each $R'$. Since $\Sp$ and $\Ep$ commute with direct limits, it follows that
$\Sp(R,\Lambda)=\Ep(R,\Lambda)$.
\end{proof}
\begin{lem}\label{FinGen}
If $R$ is a finitely generated $\Z$-algebra then so is $R_0$.
\end{lem}
\begin{proof}
Since $2\xi=(\xi+1)^2-\xi^2-1$, we have $2R\subseteq R_0$.
Let $S$ be a finite set of generators for $R$ as a $\Z$-algebra.
We set $S_0=\{s^2\mid s\in S\}\cup\{2\prod_{s\in S'}s\mid S'\subseteq S\}$
and denote by $R_1$ the $\Z$-subalgebra generated by $S_0$.
Clearly, $2R\subseteq R_1$. The map $\xi\mapsto\xi^2$ is a ring epimorphism
$R/2R\to R_0/2R$, therefore $R_0/2R$ is generated by the images of generators of
$R$. It follows that $R_0/2R=R_1/2R$, hence $R_0=R_1$ is finitely generated.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{nilpotent}]
Since $N_R(R,\Lambda)=N_A(R,\Lambda)\cap\Sp(R)$, the first statement follows from
Proposition~\ref{Normalizer for R=A}, the second is a particular case of~\cite[Theorem~1.1]{BakVavNormal},
and the third one coincides with Corollary~\ref{NormComm}.
Recall that $\Sp^0(R,\Lambda)=\cap_\ph\operatorname{Ker}\ph$, where $\ph$ ranges over all group homomorphisms
$\Sp(R,\Lambda)\to\Sp(\tilde R,\tilde \Lambda')/\Ep(\tilde R,\tilde \Lambda)$ induced by form ring morphisms
$(R,\Lambda)\to(\tilde R,\tilde\Lambda)$ such that $\tilde R_0$ is semilocal. By Lemma~\ref{semilocal}
$\Sp^0(R,\Lambda)=\Sp(R,\Lambda)$.
Since $R$ is integral over $R_0$, it is a direct limit of $R_0$-subalgebras $R'\subseteq R$
such that $R'$ is module finite over $R_0$.
By~\cite{HazDim}[Theorem~3.10]
the group $\Sp^0(R',\Lambda\cap R')/\Ep(R',\Lambda\cap R')$ is nilpotent, provided that $R_0$ has
finite Bass--Serre dimension. Since $\Sp$ and $\Ep$ commute with direct limits, this proves (5).
Any commutative ring is a direct limit of finitely generated $\Z$-algebras. Let
$R=\injlim R^{(i)}$, where $i$ ranges over some index set $I$ and each $R^{(i)}$
is a finitely generated $\Z$-algebra. Then $R_0=\injlim R^{(i)}_0$
and $\Sp(R,\Lambda\cap R)/\Ep(R,\Lambda)=\injlim Sp(R^{(i)},\Lambda\cap R^{(i)})/\Ep(R^{(i)},\Lambda\cap R^{(i)})$.
By Lemma~\ref{FinGen} each $R^{(i)}_0$ is a finitely generated $\Z$-algebra. Therefore,
this ring has finite Krull dimension and hence finite Bass--Serre dimension. By~(5) each
group $Sp(R^{(i)},\Lambda\cap R^{(i)})/\Ep(R^{(i)},\Lambda\cap R^{(i)})$ is
nilpotent which implies item~(4).
\end{proof}
The following statement is crucial for the proof of our main theorem.
For Noetherian rings it is almost immediate consequence of Theorem~\ref{nilpotent}.
\begin{lem}\label{normalizer}
Let $g\in\Sp(A)$, $n\ge3$. If\/ $\Ep(R,\Lambda)^g\le N_A(R,\Lambda)$, then
$g\in N_A(R,\Lambda)$. Moreover, $\Ep(R,\Lambda)$ is a characteristic
subgroup of $N_A(R,\Lambda)$.
\end{lem}
\begin{proof}
Let $\theta$ be either an automorphism of $N_A(R,\Lambda)$ or an automorphism of $\Sp(A)$
such that $\Ep(R,\Lambda)^\theta\le N_A(R,\Lambda)$
(we denote by $h^\theta$ the image of an element $h\in N_A(R)$
under the action of $\theta$). Since $n\ge3$, the Chevalley commutator formula (see section~\ref{Sp})
implies that the group $\Ep(R,\Lambda)$ is perfect. By Corollary~\ref{NormComm} we have
\begin{multline*}
\Ep(R,\Lambda)^\theta=[\Ep(R,\Lambda),\Ep(R,\Lambda)]^\theta\le \\
[N_A(R,\Lambda),N_A(R,\Lambda)]\le\Sp(R,\Lambda).
\end{multline*}
We shall prove that $h^\theta\in \Ep(R,\Lambda)$ for any $h\in\Ep(R,\Lambda)$.
Write $h$ as a product of elementary root unipotents
$x_{\alpha_1}(s_1)\cdots x_{\alpha_m}(s_m)$. Let $R'$ denote the $\Z$-subalgebra of $R$ generated by
all $s_i$'s, and let $\Lambda'$ denote the form parameter of $R'$ generated by
those $s_j$ for which $\alpha_j$ is a long root.
Clearly $h\in\Ep(R',\Lambda')$ and $\Ep(R',\Lambda')$ is a finitely generated group.
Let $R''$ denote the $R'$-algebra generated by all entries of the matrices
$y^\theta$, where $y$ ranges over all generators of $\Ep(R',\Lambda')$.
Let $\Lambda''=\Lambda\cap R''$.
The inclusion $\Ep(R,\Lambda)^\theta\le\Sp(R,\Lambda)$ shows that $R''\subseteq R$.
Note that $\Ep(R',\Lambda')^\theta\le\Sp(R'',\Lambda'')$ by the choice of $R''$.
Since $R''$ is a finitely generated $R'$-algebra, it is
a finitely generated $\Z$-algebra. By Lemma~\ref{FinGen} $R''_0$ is a finitely generated
$\Z$-algebra. Therefore, it has finite Krull dimension and hence finite Bass--Serre dimension.
Thus by Theorem~\ref{nilpotent}(5), the $k$th commutator subgroup $D^k\Sp(R'',\Lambda'')$ equals to
$\Ep(R'',\Lambda'')$ for some positive integer $k$.
Now, since $\Ep(R',\Lambda')$ is perfect, it is equal to
$D^k\Ep(R',\Lambda')$. It follows that
$$
\Ep(R',\Lambda')^\theta=D^k\Ep(R',\Lambda')^\theta\le
D^k \Sp(R'',\Lambda'')=\Ep(R'',\Lambda'').
$$
In particular, $h^\theta\in \Ep(R'',\Lambda'')\le \Ep(R,\Lambda)$.
Thus, $\Ep(R,\Lambda)$ is invariant under $\theta$.
If $\theta$ is an automorphism of $N_A(R)$ this means that
$\Ep(R,\Lambda)$ is a characteristic subgroup of $N_A(R)$.
If $\theta$ is an inner automorphism defined by $g\in G(A)$, then
the statement we proved is the first assertion of the lemma.
\end{proof}
The following straightforward corollary shows that the normalizers of
all subgroups of the sandwich $L\bigl(\Ep(R,\Lambda),N_A(R,\Lambda)\bigr)$
lie in that sandwich.
\begin{cor}
For any $H\le N_A(R,\Lambda)$ containing $\Ep(R,\Lambda)$ its normalizer
is contained in $N_A(R,\Lambda)$.
In particular, the group $N_A(R,\Lambda)$ is self normalizing.
\end{cor}
\section{Inside a parabolic subgroup}\label{parabolic}
Let $H$ be a subgroup of $\Sp(A)$, normalized by $\Ep(K)$.
Denote by $(R,\Lambda)$ the form ring, associated with $H$. In the proof of the
following lemma we keep using the ordering on the index set $I$, defined in
section~\ref{Sp}. For example, the product $\prod_{j=2}^{-1}$ means that
$j=2,\dots,n,-n,\dots,-1$. We assume that the order of factors agrees with the
ordering on $I$.
\begin{lem}\label{InUniRad}
If $g\in U_1(R)$ and $\Ep(K)^g\in H$, then $g\in\Ep(R,\Lambda)$.
\end{lem}
\begin{proof}
Let $g=\prod_{j=2}^{-1}T_{1j}(\mu_j)$.
We have to prove that $\mu_j\in R$ for any $j\ne -1$ and $\mu_{-1}\in\Lambda$.
Let $h$ be the smallest element from $I$ such that $\mu_j\ne0$.
We proceed by going down induction on $h$.
If $h=-1$, then $g$ consists of a single factor and
there is nothing to prove. If $h=-2$, then the result follows from
Lemma~\ref{RightSide}.
Now, let $h\le-2$. Denote by $i$ the successor of $h$ in $I$.
Then, $[T_{h,i}(-1),g]=T_{1,i}(\mu_h)\prod\limits_{j>i}T_{1,j}(\xi_h)$.
By induction hypothesis $\mu_h\in R$. The element $T_{1h}(-\mu_h)g$ satisfies the
conditions of the lemma. Again by induction hypothesis it belongs to
$\Ep(R,\Lambda)$. Thus, $g\in\Ep(R,\Lambda)$.
\end{proof}
It is known that in a Chevalley group the unipotent radicals of two opposite
standard parabolic subgroups span the elementary group (see e.\,g.~\cite[Lemma~2.1]{StepComput}).
The next lemma shows that this holds for the parabolic subgroup $P_1$ of $\Sp(R,\Lambda)$ as well.
\begin{lem}
The set
$$
\{T_{1i}(\mu),\,T_{i1}(\mu),\,T_{\pm1\,\mp1}(\lambda)\mid i\ne\pm1,\,\mu\in R,\,\lambda\in\Lambda\}
$$
generates the elementary group $\Ep(R,\Lambda)$.
\end{lem}
\begin{proof}
If $i\ne\pm j,\pm1$ then $T_{ij}(\mu)=[T_{i1}(\mu),T_{1j}(1)]$.
On the other hand, for $\lambda\in\Lambda$ we have
$T_{-ii}(\lambda)=[T_{-11}(\mu),T_{1i}(1)]T_{-i1}(-\lambda)$.
\end{proof}
\begin{lem}\label{InParabolic}
Let $H$ be a subgroup of $\Sp(A)$, containing $\Ep(K)$ and let
$(R,\Lambda)$ be a form ring, associated with $H$.
Suppose that $g$ commutes with a long root subgroup $X_\gamma(K)$
and $\Ep(K)^g\in H$. Then $g\in N_A(R,\Lambda)$.
\end{lem}
\begin{proof}
Without loss of generality we may assume that
$\gamma=2\eps_1$ is the maximal root. Then $g$ belongs to the standard parabolic subgroup
$P_1$, corresponding to the simple root $\alpha_1=\eps_1-\eps_2$,
in other words, $g_{i1}=g_{-1i}=g_{-11}=0$ for all
$i\ne\pm1$. Moreover, $g_{11}=g_{-1-1}=g'_{11}=g'_{-1-1}=\mu$, where $\mu^2=1$.
Then $g=ab$ for some $b\in U_1(A)$ and $a\in L_1(A)$ ($a$ and $b$
are not necessarily in $H$).
For any $d\in U_1(K)$ the element $d^g$ belongs to $H\cap U_1(A)$.
By Lemma~\ref{InUniRad}, $d^g\in \Ep(R,\Lambda)$.
If $d=T_{1i}(1)$, then the above inclusion implies that $\mu g_{ij}\in R$
for all $i\ne1$ and all $j\in I$. It follows that $T_{1i}(1)^a$ and
$T_{i1}(1)^a$ belong to $\Ep(R,\Lambda)$. On the other hand,
$a$ commutes with root subgroups $X_{\pm\gamma}(K)$. By the previous lemma
we conclude that $a$ normalizes $\Ep(R,\Lambda)$.
Now, we have $\Ep(K)^b\le\Ep(R,\Lambda)^g\in H$ and by
Lemma~\ref{InUniRad} $b\in\Ep(R,\Lambda)$.
Thus, $g\in N_A(R,\Lambda)$ as required.
\end{proof}
\section{Proof of Theorem~1}\label{2=0}
In this section we assume that $2=0$ in $K$.
Let $G$ be a Chevalley group with a not simply laced root system, e.\,g. $G=\Sp$.
Recall that in this case a short root unipotent element $h$
is called a \emph{small unipotent element}, see~\cite{Gordeev}.
The terminology reflects the fact that the conjugacy class of $h$ is small.
In our settings a small unipotent element is conjugate to $T_{ij}(\mu)$,
where $i\ne\pm j$ and $\mu\in R$.
The following lemma was obtained over a field by Golubchik and Mikhalev
in~\cite{GolMikh}, Gordeev in~\cite{Gordeev} and
Nesterov and Stepanov in~\cite{NestStepF4}.
\begin{lem}\label{identity}
Let $\Phi=B_n,C_n,F_4$ and let $R$ be a ring such that $2=0$.
Let $\alpha$ be a long root and $g\in G(\Phi,R)$.
If $h\in G(R)$ is a small unipotent element, then
$X_\alpha(R)^{h^g}$ commutes with $X_\alpha(R)$.
\end{lem}
\begin{proof}
The identity with constants $[X_\alpha(R)^{h^g},X_\alpha(R)]=\{1\}$ is inherited by
subrings and quotient rings. Any commutative ring with $2=0$ is a quotient of a
polynomial ring over $\mathbb F_2$ which is a subring of a field of
characteristic~$2$.
\end{proof}
The next lemma is the last ingredient for the proof of Theorem~\ref{MAIN}.
It follows from the normal structure of the general unitary group
$\operatorname{GU}_{2n}(R,\Lambda)$ obtained by Bak and Vavilov
at the middle of 1990-s. Since this result has not been published yet,
we give a prove of a very simple special case of it.
\begin{lem}\label{NSofGU}
A normal subgroup $N$ of\/ $\Ep(R,\Lambda)$, containing a root element
$T_{ij}(1)$, coincides with\/ $\Ep(R,\Lambda)$.
\end{lem}
\begin{proof}
First, suppose that $i\ne\pm j$.
Take $k\ne\pm i,\pm j$. Then, $[T_{ij}(1),T_{jk}(\xi)]=T_{ik}(\mu)\in N$.
Since the Weyl group acts transitively on the set of all short roots,
we have $T_{lm}(\mu)\in N$ for all $l\ne\pm m$ and $\mu\in R$.
Further,
$$T_{l,-m}(-\lambda)[T_{lm}(1),T_{m,-m}(\lambda)]=T_{l,-l}(\lambda)\in N$$
for all $\lambda\in\Lambda$ and $l\in I$.
Now, let $i=m$, $j=-m$ and $l\ne\pm m$. Put $\lambda=1$ in the latter commutator identity.
Then, $T_{l,-m}(1)T_{l,-l}(1)\in N$. By transitivity of the Weyl group we know that
$T_{l,-l}(1)\in N$, therefore $T_{l,-m}(1)\in N$. By the first paragraph of the proof,
$N$ contains all the generators of $\Ep(R,\Lambda)$.
\end{proof}
Now we are ready to prove Theorem~\ref{MAIN}.
The idea of the proof is the same as for the main result
of~\cite{StepStandard}.
\begin{proof}[Proof of Theorem~\ref{MAIN}]
Let $(R,\Lambda)$ be the form subring associated with $H$.
Put $h=T_{12}(1)$ and $x=T_{1,-1}(\lambda)$, where $\lambda\in\Lambda$.
Take two arbitrary elements
$a,b$ from $\Ep(R,\Lambda)$ and consider the element $c=x^{h^{agb}}\in H$.
By Lemma~\ref{identity} this element commutes with a long root subgroup
and by Lemma~\ref{InParabolic} $c\in N_A(R)$. Rewrite $c$ in the form
$$
c=\bigl(g^{-1} (a^{-1}h^{-1}a)g
(bxb^{-1})
g^{-1} (a^{-1}h a)g\bigr)^b
$$
\noindent
Since $b\in\Ep(R,\Lambda)$, the element $bcb^{-1}$ is in $N_A(R)$.
Fix $a$ and let $b$ and $\lambda$ vary. The subgroup generated by
$bxb^{-1}$ is normal in $\Ep(R,\Lambda)$. By
Lemma~\ref{NSofGU} it must coincide with $\Ep(R,\Lambda)$. Thus,
$\Ep(R,\Lambda)^{g^{-1} (a^{-1}h^{-1}a)g}\in N_A(R)$,
and by Lemma~\ref{normalizer} $(a^{-1}h^{-1}a)^g\in N_A(R)$.
Again, elements of the form $a^{-1}h^{-1}a$, as $a$ ranges over $\Ep(R,\Lambda)$,
generate a normal subgroup in $\Ep(R,\Lambda)$.
The minimal normal
subgroup of $\Ep(R,\Lambda)$ containing $h$ must be equal to $\Ep(R,\Lambda)$
by Lemma~\ref{NSofGU}.
Therefore, $\Ep(R,\Lambda)^g\in N_A(R)$. By Lemma~\ref{normalizer} one
has $g\in N_A(R)$, which completes the proof.
\end{proof}
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Brief Company Introduction
Zoltrix has been established in the year 1989. Modem and sound card are the major products of our company at the beginning stage. Our company is one of the biggest organization in Asia in the manufacturing of modems under our own brand – ‘Zoltrix’. We have our own worldwide sales channels in USA; Canada; Iran; Spain; Germany; Poland; Israel; Pakistan; South Africa; Egypt …. and our annual sales is around US$18 – 22 million during our peak period 1995 - 2005.
In addition to modem and sound card; we also have a strong sales line of computer accessories such as speakers; earphones; headphones; mouse; keyboard.
Start from the year 2000; our company has stepped into the manufacturing of memory products such USB Drive because we have very strong support of NAND Flash from Samsung. Our company is also one of the first few companies in the designing and manufacturing of USB Drive and MP3.
Nowadays; our main products lines are Visual Doorbell; Tablet; Sport DV; Car Recorder; Speaker; Power Bank; USB Drive and various kinds of mobile phone and computer accessories.
Our manufacturing plants are located in Dongguan and Shenzhen which has been established since 1995. The total factory space is 7500 square meter. The number of workers is more than 500 in the peak manufacturing season. Our factory has a very strong team of engineers to develop our own Zoltrix brand and support the OEM/ODM business as well. She also has a team of responsible and experienced management staff and QA team to support our marketing team to supply good quality products to our customers punctually. WORLD CLASS LEADING TECHNOLOGY is our company slogan.
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TITLE: Problem on $\sigma$-algebra from Rudin
QUESTION [1 upvotes]: Does there exist an infinite $\sigma$-algebra which has only countably many members ?
Proof: Suppose that $\sigma$-algebra $\mathfrak{M}$ has countably many members, namely $\{A_i\}_{i=1}^{\infty}$. By definition of $\sigma$-algebra the set $F=\bigcap \limits_{i=1}^{\infty}A_i^c$ lies in $\mathfrak{M}$. Since it lies in $\mathfrak{M}$ then $F=A_j$ for some $j\in \mathbb{N}$, i.e. $\bigcap \limits_{i=1}^{\infty}A_i^c=A_j$ which is equivalent to $$A_j^c\cap \bigcap\limits_{i\geqslant 1 \atop{i\neq j}}A_i=A_j$$ but the last equality is false since $x\in A_j$ then $x\notin A_j^c$ then $x$ does not lie in LHS.
But one moment seems to me confusing it's when $A_j$ is empty.
Sorry if this topic appeared in MSE before.
REPLY [0 votes]: Suppose $\mathcal{B}$ is an infinite algebra of sets (we don't need $\sigma$-algebra just yet). We will show that there are infinitely many pairwise disjoint non-empty elements of $\mathcal{B}$.
There are in fact two cases to consider, related to possible atoms:
A definition: a subset $A \in \mathcal{B}$ is called an atom if it is non-empty and there is no proper subset $B$ under $A$ in the algebra, i.e. for no $B \in \mathcal{B}$ do we have $\emptyset \subsetneq B \subsetneq A$.
Basic facts: two distinct atoms $A_1$ and $A_2$ are disjoint. Also, for any set in $C \in \mathcal{B}$, and any atom $A$, $C \cap A = \emptyset$ or $C \cap A = A$ (or it would be a proper set in-between).
The first case is where $\mathcal{B}$ is atomic, which means that every non-empty member of $\mathcal{B}$ has a subset that is an atom.
In that case, there cannot be finitely many atoms. Because if there are only atoms $A_1,\ldots,A_n$, then the map that sends $A \in \mathcal{B}$ to $(A \cap A_1,\ldots, A \cap A_n)$ has $2^n$ many values (as the intersections are empty or $A_i$ for every $i$) and is a bijection (here we use that $\mathcal{B}$ is atomic), showing that $\mathcal{B}$ is finite, contradicting the assumption. So $\mathcal{B}$ has infintiely many atoms if it is atomic, and so an infinite pairwise dijsoint family (of atoms).
In the other case, $\mathcal{B}$ is not atomic, so there is some $Y \neq \emptyset$ in $\mathcal{B}$ such that $Y$ contains no atoms. So construct $Y_1,Y_2,\ldots$, strictly decreasing, by recursion: $Y_1 = Y$ and $Y_n$ constructed, let $Y_{n+1}$ be a proper non-empty subset of $Y_{n}$ (as $Y_{n}$ is not an atom). Then $Y_{n} \setminus Y_{n+1}$, $n=1,2,\ldots$ are a family of pairwise disjoint members of $\mathcal{B}$.
Having such a countable pairwise disjoint family $A_n, n \in \mathbb{N}$ of non-empty subsets, for every subset $I$ of $\mathbb{N}$ we define $A(I) = \bigcup_{n \in I} A_n$ is in $\mathcal{B}$ (here we use that we have a $\sigma$-algebra!) and for different $I$, these unions are all different (by the disjointness condition).
So $|\mathcal{B}| \ge 2^{\aleph_0}$. So any infinite $\sigma$-algebra is uncountable.
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January was a good month. My original goal was 12 finishes because I had 52 quilt tops on January 1 and that would get me down to 40 tops. I'm happy with the 10 that I did though. I purposely chose a lot of the smaller pieces to finish. The colorwashes are 19" - 28". The largest 2 quilts are in the 62" x 76" range, and one 50" x 70". Two others were baby quilt size and one large wallhanging.
In February I want to have at least 4 finishes and I know I will make close to 4 new tops. I will hopefully finish the Hunter Star top and at least 2 TAW tops.
We got 14" of snow from 9 p.m. Sat. night until 9 p.m. last night. It continued to snow after 9 and the wind was blowing and drifting the snow. This photo was taken through my kitchen window. These branches are usually up even with the gutters on my house. The wet heavy snow piled up on everything and branches were bent to the max.
This is the forsythia bush by my other kitchen window. If you look closely you will see little birds taking refuge in the branches. I'm glad I filled the bird feeders Saturday afternoon.
I shoveled the sidewalk and porch 3 times and got to try out my Yaktrax. Hopefully I'll never slip and fall again.
23 comments:
Nice list of finishes and plans for Feb!... Stay careful... outside. We didn't get any of the weather... this time. Much better than last year, but we have had some cold temps.... Mild today tho!!!
Nice job for January. They are all beautiful as always.
On the news they said my area received 15+ inches of snow. I lost power at 10AM and didn't go back on until 2PM. I was just about to leave when thankfully it went on. I was running out of oxygen so was packing up to go when it finally came back on. I did get an embroidery pattern traced to make a new pillow. Gonna be a cold day today, stay warm.
Wonderful finishes. I think you were quite successful this month. Let us know how your new traction works....stay safe. That's a huge bunch of snow!!
I hope the yaktrax works for you it does for me. I am sorry you need to do the shoveling yourself though it is exercise in just take it easy! So many pretty quilts they all look so good the snow is so pretty but I no most would rather just have a little bit if they are going to get it.
It is not nearly so cold here but the wind is howling and after a night of rain, it looks like we will be getting more.
Congrats on the finishes.
Congratulations on 10 finishes! That is super! This is from someone who has many tops hanging in the closet. Careful in the snow and cold. We are still getting it now and my husband is out snowblowing at the moment. Good day to sew.
I've been looking forward to your January recap and it is wonderful! That's an awful lot of snow outside your window.
Your YakTrax are looking good! I posted today how I lost mine and had to go hunt for them. Deep crispy snow made mine pop off even though they wrap well around my boots. Congrats on all those finishes!
Wanda, these are a great invention, and are also sold down here in NZ, wonderful finishes. Hope you keep warm, guess the shovelling was warming enough.
impressive finishes! and I love the birds taking shelter in your branches! Keep warm!
I love to see your end of month finishes. Always a treat. January's were awesome! Each one a wonderful piece of art. Thank you for sharing.
We got rain last night. Were expecting ice.
I hope the snow work for you. Stay warm, dear.
I shoveled three times yesterday, and once this morning! 20+ inches in the city, probably lake effect. Isn't it beautiful though, today?
Great finishes! Stay warm!
So good to see your collection of finishes for January, they look beautiful altogether - must have a look at Picasa for producing a collection when I have a few finishes.
Beautiful, beautiful quilts. You got an amazing amount done. Your snow looks beautiful to me, too, but I'm glad I don't live in an area with that much. We need snow robots just like Rumba vacuum cleaners to help you guys out! Stay warm and safe!
Beth
TEN Spectacular January finishes ! You can be proud of each one of them.
OMG ! ! ! that is too much snow for me. Must say it is beautiful and I love the photo of the little birds all snuggled in amongst the snowy branches.
Just a word of warning about your Yak Trax's... don't wear them on ice, they work like ice skates and you will go flying. They work great on snow packed walks
JJM.
Another great month of finishes - you are a wonder!!
Shoveling snow must be such a hard chore - especially if one lives alone and is getting older. What happens if if is just too physically demanding for anyone? Would you have to rely on a friendly neighbour, perhaps? There must be a lot of people who can't manage. We don't have this problem where I live, in New Zealand, just a little snow on the tops of hills and good coverings on the mountain ranges.
Beautiful finishes Wanda! We got 12 inches here and I almost fell a couple times when I was shoveling. I had never heard of the Yaktrax until now, how did you like them and where did you buy them?
What a great lot you got done in January! Beautiful, every one! I am so sorry about the snow. It does make you feel good, though, to know that you are helping the poor birds in the cold, doesn't it? I like your "snow tires"! Good idea.
Wow, lots of projects done in January, and an ambitious list for February. Lovely snow pictures, but I'm getting a little tired of the snow here, lol! Those things you put on the bottoms of your shoes work great! We bought them when we lived in Iceland and have been using them ever since. If you walk around indoors with them, like at the mall or something, be careful. They're slippery on the floor.
I just saw JJM's comment about not using the Yaktrax on the ice. Ours look like yours, and we use them on icy sidewalks and stuff, and they seem to work well. Guess you'll have to try it out--carefully!
Even though I knew what was coming, that January recap awed me! It's stunning! Such beautiful quilts, every one of them and such an amazing amount of work!
What beautiful quilts you continue to make and always so full of gorgeous fabrics and color!! You have accomplished quite a lot quilt-wise in January. Congratulations for meeting your goals!
Your finish just impresses the heck out of me!! Those yaktraxs are the best thing. They got me through 26 years of winter delivering mail on walking routes without a major fall. Good stuff!
| 171,761
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Tom in tune with Kagawa
Tom Cleverley has spoken glowingly of his on-pitch relationship with Shinji Kagawa.
Both players have started the season brightly and Cleverley feels their similar pass-and-move styles make for a natural understanding.
“I’ve really enjoyed playing with Shinji,” Tom reveals in the latest edition of Inside United, on sale now. “He’s really on my wavelength when it comes to playing one and two-touch football.”
Cleverely also praised the other new arrivals in midfield and attack.
“They’ve all brought so much,” he says. “We all know what Robin van Persie is about and Nick Powell is a really good footballer and a good lad. He’s definitely one for the future.”
| 132,440
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TITLE: Existence of divisors of degree one on a curve over a finite field
QUESTION [18 upvotes]: Let $C$ be a smooth, geometrically irreducible projective curve defined over a finite field $\mathbb{F}_q$.
Given a (scheme-theoretic) point $x \in C$, define the degree of $x$ to be the degree of the extension $[k(x): \mathbb{F}_q]$. The degree of a divisor on $C$ can thus be defined by linearity.
We proved today in my algebraic number theory class that there is always a divisor of degree one. The argument was to suppose that all the degrees $[k(x): \mathbb{F}_q]$ were divisible by some $m >1$, and then to apply (a very weak form of) the Cebotarev density theorem to the extension of function fields $k(C) \to k(C) \otimes_{\mathbb{F}_q} \mathbb{F}_{q^m}$. All places would have to split completely if $\mathbb{F}_{q^m}$ is contained in every residue field, which is a contradiction.
I also learned another proof from a comment of Felipe Voloch on MO: for large $n$, the Weil bound on the number of $\mathbb{F}_{q^n}$-rational points implies that there is a point of degree $n$ and a point of degree $n+1$. Taking the difference gives a divisor of degree one.
Is there an elementary geometric way of seeing this? (Related question: Are there other fields for which this is true?)
REPLY [8 votes]: This is not an elementary proof, but it is a little different to the others mentioned here:
The space of degree $1$ divisors on $C$ is a torsor over Jac$(C)$, and you are asking for a proof that it is the trivial torsor.
The space of such torsors is computed by $H^1(Gal(\overline{k}/k), \mathrm{Jac}(C))$. Now when $C$ is finite, the absolute Galois group is procyclic, generated by Frobenius, and there is an argument of Lang that shows for any connected smooth commutative group scheme $G$ over $k$ that $H^1(Gal(\overline{k}/k),G)$ vanishes in this case.
Namely, the endomorphism $\mathrm{Frob} - 1$ of $G$ is surjective,
because it is smooth (take its deritative, and use that fact that the derivative of Frob vanishes), hence has open image (which is also closed, being a subgroup) and $G$ is connected.
This allows you to trivialize any 1-cocycle.
In your particular case, we should be able to make this concrete: choose a point of $C$ defined over some extension of $\mathbb F_q$, say $P$, then consider the degree zero divisor $\mathrm{Frob}_q(P) - P$. Applying Lang's argument to $\mathrm{Jac}(C)$, we find another degree zero divisor $D$ on $C$ (defined over some extension of $\mathbb F_q$) such that $\mathrm{Frob}_q(D) - D = \mathrm{Frob}_q (P) - P.$ Then $P - D$ is a degree one divisor that is fixed
by $\mathrm{Frob}_q$, and so defined over $\mathbb F_q$.
So there is not really any cohomology involved (not that the above Galois cohomology is particular difficult), but you need to know that degree zero divisors are parameterized by a connected commutative smooth group scheme (the Jacobian).
Edit: The above argument actually shows that there is a degree one divisor whose linear equivalence class is defined over $\mathbb F_q$. An extra argument is needed to actually get a degree one divisor defined over $\mathbb F_q$. One writes downs the various obvious exacts sequences involving non-zero functions, principal divisors, all divisors, and Pic, and takes Galois cohomology. Using Hilbert Thm. 90 plus the vanishing of the Brauer group $H^2(G_{\mathbb F_q},
\overline{\mathbb F}_q^{\times})$, one finds that a Galois invariant linear equiv. class does indeed lift to an invariant divisor.
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If you are anything like me, you will need this Cooking Conversion Guide because you will look at loads of recipes in books and online. You will see that there are various cooking weights, temperatures and liquid amounts which are shown in cups, lbs, fluid oz, grams, imperial, metric etc. It gets a bit confusing, especially if you only work in one measurement style. So I thought I would make it a lot easier, it is something that I use all the time when I’m cooking. Weight Conversion Guide I’ve
| 254,822
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supper
[suhp-er]
- the evening meal, often the principal meal of the day.
- any light evening meal, especially one taken late in the evening: an after-the-theater supper.
Show More
- of or relating to supper: the supper dishes.
- for, during, or including supper: a supper party.
Show More
Origin of supper
Dictionary.com Unabridged Based on the Random House Unabridged Dictionary, © Random House, Inc. 2018
Examples from the Web for supper
Contemporary Examples
They were going to get the supper clubs and Vegas, just like Sam wanted.How Martin Luther King Jr. Influenced Sam Cooke’s ‘A Change Is Gonna Come’
Peter Guralnick
December 28, 2014
Served in lieu of morning pancakes or bread at supper, ployes are nothing if not versatile.On the Canadian Border, It's Pancakes for Every Meal
Jane & Michael Stern
July 6, 2014
Other events include a Burns Night Supper on January 24, and a willow-weaving workshop in March.Get Lessons From Kate's Florist At Charles's Home
January 10, 2014
Kane spoke to The Daily Beast about his Las Supper venture, posing semi-nude with Madonna, and his hometown of Brooklyn.Big Daddy Kane: The Hip-Hop MC on Las Supper, Madonna, Jay-Z, and What’s Next
Curtis Stephen
April 16, 2013
After one ship's supper, Keith reflects that there were worse places to spend the war than the Caine.David's Bookclub: The Caine Mutiny
David Frum
January 28, 2013
Historical Examples
He made his way to the dining-room, where supper was under way.The Spenders
Harry Leon Wilson
We did justice to the supper, as we had not had anything to eat for thirty-two hours.Explorations in Australia
John Forrest
Only, my dear, do not disgrace my report when you come to supper.
Does he believe, that the disgrace which I supper on his account, will give him a merit with me?
When Banstead took the chorus out to supper he had the ready repartee of his kind.
supper
- an evening meal, esp a light one
- an evening social event featuring a supper
- sing for one's supper to obtain something by performing a service
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- (tr) rare to give supper to
- (intr) rare to eat supper
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Word Origin
C13: from Old French soper; see sup 1
Collins English Dictionary - Complete & Unabridged 2012 Digital Edition © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012
Word Origin and History for supper
n.
late 13c., "the last meal of the day," from Old French super "supper," noun use of super "to eat the evening meal," which is of Germanic origin (see sup (v.1)).
Formerly,.
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Online Etymology Dictionary, © 2010 Douglas Harper
Idioms and Phrases with supper
supper
see sing for one's supper.
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The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.
| 235,647
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TITLE: Prove that the norm in an inner product space is $ \ge 0$
QUESTION [3 upvotes]: Macdonald Linear and Geometric Algebra defines an Inner Product Space in the following way (pg 57):
"An inner product space is a vector space with a product called an inner product. The inner product of two vectors is a scalar. Axioms $I1-I4$ must be satisfied for all scalars $a$ and all vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}.$
$$
I1.\ (a\mathbf{u})\cdot\mathbf{v} = a(\mathbf{u}\cdot\mathbf{v})
$$
$$
I2.\ (\mathbf{u}+\mathbf{v})\cdot\mathbf{w} = \mathbf{u}\cdot\mathbf{w}+\mathbf{v}\cdot\mathbf{w}
$$
$$
I3.\ \mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}
$$
$$
I4.\ If \ \mathbf{v} \neq \mathbf{0}, then \ \mathbf{v}\cdot\mathbf{v} > 0
$$"
Furthermore, the norm is defined in the following way (pg 58):
"The norm of a vector $\mathbf{v}$ in an inner product space is given by $|\mathbf{v}|^2 = \mathbf{v}\cdot\mathbf{v}$."
This is slightly unconventional (to me) in the sense that every other linear algebra text I've looked at defines the norm as $\sqrt{\mathbf{v}\cdot\mathbf{v}}$. The Macdonald definition leaves open the possibility that $|\mathbf{v}|$ could be less than zero. However, we are indeed then asked to prove (pg 59):
"$N1.\ If \ \mathbf{v} \neq \mathbf{0}, |\mathbf{v}| > 0.$"
...and so the definitions are equivalent. But I can't see how to prove that from only the definitions given. What am I missing?
REPLY [1 votes]: When we say a $|\cdot|$ is a norm, we already assume its non-negativity, by the original definition of a norm. So by saying
$$|\mathbf{v}|^2 = \mathbf{v} \cdot \mathbf{v}$$
the author actually means
$$|\mathbf{v}| = \sqrt{\mathbf{v} \cdot \mathbf{v}}$$
as you usually read, or otherwise it does not define a norm. But I think it is clearer to say "a norm is induced by $|\mathbf{v}|^2 = \mathbf{v} \cdot \mathbf{v}$" rather than "the norm is given by $|\mathbf{v}|^2 = \mathbf{v} \cdot \mathbf{v}$", since the original definition of a norm does not come from an inner product.
Added:
$|\mathbf{v}| \geq 0$ is true by definition of a norm. If the non-negativity is not clear, then calling it a norm is not legit, as it has not fulfilled the original definition of a norm. If you are showing $|\mathbf{v}| \geq 0$ for all $\mathbf{v}$, you are not showing "the norm is nonnegative" but "$|\cdot|$ is a norm". So no matter he writes $|\mathbf{v}|^2 = \mathbf{v} \cdot \mathbf{v}$ is a norm or $|\mathbf{v}| = \sqrt{\mathbf{v} \cdot \mathbf{v}}$ is a norm, it is not a new definition but some kind of a proposition. Anyway, I agree that the author should be more specific in writing the statement.
| 30,498
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SPRINGFIELD — The GOP candidate for Illinois governor broadened his criticism of state agriculture officials Wednesday, saying the department is “full of cronyism” and has “folks running things that generally don’t have much expertise.”
Winnetka businessman Bruce Rauner’s statements at a forum coordinated by the Illinois Farm Bureau featuring candidates for governor and U.S. Senate follow comments this month that the agency’s director “must have agriculture experience” — a shot at Bob Flider, a former Democratic lawmaker who was named to the post after he lost a 2010 re-election race for the Illinois House.
Flider had voted, as a lame duck in the House, for Quinn’s 67 percent income-tax increase after earlier opposing it.
“The Department of Agriculture has been decimated,” Rauner said. “A lot of the folks with deep expertise have left and been replaced by folks who have been there for other reasons.”
When pressed on specifics, the Rauner campaign pointed to the department’s employment of Shayna Cherry, a former Illinois Department of Transportation employee whose father is an attorney with ties to top Democrats, including Tony Rezko, a former adviser and fundraiser to imprisoned former Gov. Rod Blagojevich. Rezko, like Blagojevich, is serving a sentence in federal prison.
Brooke Anderson, a spokeswoman for Democratic Gov. Pat Quinn’s campaign, responded for the Department of Agriculture. She said it was “false and rather sad that Mr. Rauner would attack a young woman staffer who is doing an excellent job for the state of Illinois.”
Rauner “is the only one familiar with cronyism and corruption in this race.”
If elected, Rauner pledged to infuse the department with “farmers and farm experts.” The self-portrayed government outsider in the nationally watched race stressed the importance of rotating the state’s political “crops” and ending Democrats’ one-party rule in Springfield.
Quinn, who spoke in a separate, one-hour time slot at a Bloomington farm, used the forum as an opportunity to tout agricultural achievements under his governorship and pledge to increase funding for the agency, juxtaposing himself with Rauner’s promise to return Illinois to a 3 percent income tax rate for individuals within four years, down from 5 percent.
Once fully rolled back, the move would create an $8 billion annual hole in the budget, the Quinn campaign has said.
“The idea that you pull back and starve a particular agency, not have proper personnel there, that is a very bad formula for success,” Quinn said. “Nobody likes paying taxes, I certainly don’t. But if you have candidates running for governor that say you can cut the budget by $8 billion, don’t expect to have good schools, a well-funded department of agriculture.”
Page 2 of 2 - Quinn and Rauner are scheduled to appear at another forum Thursday in Chicago, hosted by the Metropolitan Planning Council. As on Wednesday, both are scheduled to address the crowd but won’t face each other or be on stage at the same time.
| 105,404
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- Listen Live
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| 364,246
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PCDome:
"Alan Wake is like a good series or a striking thriller: slowly evolving story, sinister events, eccentric actors, and depressive mood gradually captivate captivate the victims, pulling them deeper and deeper, until he will be unable to get rid of it. These criteria, only the best games of the features of the Alan Wake is one of them despite the flaws."
| 355,185
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TITLE: Are there infinitely many composite $a$ such that $\sum_{k=1}^{a}(k,a)\equiv1\pmod{a-1}$?
QUESTION [2 upvotes]: Denote $\gcd(a, b)$ as $(a, b)$
Let
$$f(a) =\sum_{k=1}^a (k,a)$$
Clearly $f(a)\equiv 1\pmod{a-1}$ if $a$ is prime.
Can it be shown that, there are infinitely many composite numbers satisfied $f(a)\equiv1\pmod{a-1}$?
First four composite numbers are $41124, 230867, 358267, 37539572$.
Formula
$$f(a)=\sum_{d\mid a} d\phi(a/d)$$
Thus $f$ is multiplicative.
We can compute $f(p^n)=np^{n-1}(p-1)+p^n$, when $p$ is prime. Which is a general formula for $f(a)$ in terms of the prime factorization of $a.$
Source Code PARI/GP:
forcomposite(a=2,10^8,s=0;fordiv(a,d,s+=d*eulerphi(a/d));if(s%(a-1)==1,print([a])))
Credit: viper found above four composite number through given code and Thomas Andrews, write a formula for $f(a)$.
This problem is sequel to my MSE post (link).Your suggestions, comments, the answer are very valuable to me. Apologies if the problem is just unsolvable. Thank you.
REPLY [2 votes]: I do not know answer to your question, but here is an approach how one can extend a given integer $m$ to a solution $mp$ (if one exists) with a prime $p\nmid m$.
Using the multiplicativity, we want
$$f(mp) = f(m)(2p-1) \equiv 1\pmod{pm-1}.$$
That is, there should be an integer $k$ such that
$$f(m)(2p-1) - 1 = k(pm-1),$$
which is equivalent to
$$(mp-1)(2f(m)-mk) = (m-2)f(m) + m.$$
Hence, we need to find a divisor $d$ of $(m-2)f(m) + m$ such that $d\equiv -1\pmod{m}$ and test if $p:=\frac{d+1}m$ is prime, and also if $p\nmid m$. From each such divisor, we obtain a suitable prime $p$.
Using the above approach for $m\leq 10^8$, I computed seven more solutions (although they may be not in order):
$$148025049, 235167249, 242788284, 1085464188, 142772845653, 202728626748, 62763888399737.$$
Also, it can be used to come up with an heuristic argument (how likely for such $d$ to exist, how likely $\frac{d+1}m$ is prime etc.) for infinitude of solutions.
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\begin{document}
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\title{Monopole Floer homology for codimension-3 Riemannian foliation}
\date{}
\author{Dexie Lin}
\maketitle
\begin{abstract}
In this paper, we give a systematic study of Seiberg-Witten theory on closed
oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten invariant and the monopole Floer homologies $\overline{HM}(M,F,\mfs;\Gamma),~\widehat{HM}(M,F,\mfs;\Gamma),
~\widecheck{HM}(M,F,\mfs;\Gamma)$, for each transverse \spinc structure $\mfs$, where $\Gamma$ is a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the Novikov ring on basic monopole Floer homologies.
\end{abstract}
\tableofcontents
\section{Introduction}
The interaction between geometry in dimension $4$ and equation is a theme which runs through a great deal of work by many mathematicians on gauge theory over the past decades. In particular, the Seiberg--Witten equation, is one of the main tools in the study of the differential
topology and low dimensional manifolds. Since the foundational paper \cite{W} by Witten, a lot of work has
been done to apply this theory to various aspects of $3$ and $4$-dimensional
manifolds. Seiberg-Witten theory can also be applied for orbifold(see Baldridge's work \cite{Bald} for the extension to $3$-orbifolds). This article lays the groundwork for the case in which the higher-dimensional
manifold admits a Riemannian foliation of codimension $3$. From the viewpoint of analysis, gauge theory is closely related to the study of (nonlinear)Fredholm operator and the index of its linearized operator.
A natural idea to extend the framework of gauge theory to the manifold with Riemannian foliation is to study the transverse (nonlinear)elliptic operator on Riemannian foliation. For instance, the compactness of the basic Seiberg-Witten moduli space for manifolds with codimension $4$ Riemannian foliation
is showed by Kordyukov, Lejmi and Weber \cite{KLW}. The author gives a construction of basic cohomotopy Seiberg-Witten invariant for codimension 4 Riemannian foliation \cite{Lin}. In the same paper, the author gives an application to the basic index of the basic Dirac operator. The
theme of this article is to extending well-known constructions of Seiberg--Witten theory in $3$-manifolds to the manifolds with codimension $3$ Riemannian foliation structure.
For closed oriented $3$-manifold $M$ with a \spinc structure $\mfs$, monopole Floer homologies $\overline{HM}(M,\mfs),~\widehat{HM}(M,\mfs),~\widecheck{HM}(M,\mfs)$ were constructed by Kroheimer and Mrowka in their celebrated book \cite{KM}. The main result of this paper is to construct the monopole Floer homologies for the manifold with codimension $3$ Riemannian foliation $(M,F)$ satisfying a certain condition.
The idea is to extend the arguments of the non-exact perturbation of the monopole Floer homologies to the case of Riemannian foliation. The following theorem could summarize the main part of this paper.
\begin{thm}
Let $(M,F)$ be an oriented closed manifold with codimension 3 oriented taut Riemannian foliation $F$ and admits a transverse \spinc structure $\mathfrak s$. Suppose that $H^1_b(M)\cap H^1(M,\mathbb Z)\subset H^1(M)$ is a lattice of $H^1_b(M)$. Then, using a bundle-like metric $g$, a generic perturbation $\eta$ and Novikov ring $\Gamma$, we construct the basic monopole Floer homologies $$\overline{HM}(M,F,\mathfrak s, g,\eta;\Gamma),~\widehat{HM}(M,F,\mathfrak s, g,\eta;\Gamma),~\widecheck{HM}(M,F,\mathfrak s, g,\eta;\Gamma),$$ Moreover, we have that these homologies are independent of the bundle-like metric $g$ and the generic perturbation $\eta$, which are denoted by
\[\overline{HM}(M,F,\mathfrak s ;\Gamma),~\widehat{HM}(M,F,\mathfrak s;\Gamma ),~\widecheck{HM}(M,F,\mathfrak s ;\Gamma).\]
\end{thm}
Some notations and terms will be defined in the later sections.
The reason that we need the Novikov ring to construct the homologies $\overline{HM}(M,F,\mathfrak s ),~\widehat{HM}(M,F,\mathfrak s ),~\widecheck{HM}(M,F,\mathfrak s )$ is to define the partial operator of the Floer complex. Explicitly speaking, there might be infinitely many terms in general. Note that for basic Dirac operator, there is an index theorem \cite{BKR}, which were given by Br\"uning Kamber and Richardson. We give a necessary condition to avoid the Novikov ring in Section 7.3. In this paper, we inherit the convenience in the book \cite{KM}. For simplicity, we mainly consider $\mathbb F=\mathbb Z_2$ coefficient or a $\mathbb F$ module local system to construct the monopole Floer homologies.
The structure of this paper is as follows: in Section 2, we review some notions and properties about the Riemannian foliation;
in Section 3, we give some analysis properties for some transverse equations which are necessary for the later sections; in Section 4, we construct the basic Seiberg-Witten invariant for manifold with codimension $3$ Riemannian foliation; in Section 5, we construct the basic Chern-Simons-Dirac functional and give some properties of it; in Section 6, we show the gluing theorem for the basic moduli spaces, which is essential to construct the basic monopole Floer homologies; in Section 7, we give a proof of the above theorem; in Section 8, we construct the monopole Floer homologies for some $3$ orbifold, and we give some examples and a method to construct the Riemannian foliation satisfying the assumption of the above theorem.
\vspace{3mm}
{\bf Acknowledgement:} The author warmly thanks Mikio Furuta for his long time invaluable help in both mathematics and life. The author is grateful to Kim. A. Fr\o yshov for his helpful discussion on Floer homology and Ken. Richardson for the discussion on Riemannian foliation. The research is partially sponsored by the FMSP of The University of Tokyo.
\section{Preliminary}
In this section, we review some results of the previous work.
Let $M$ be a closed oriented $n$ dimensional manifold with dimension $p$ oriented $F$. We denote codimension of this foliation by $q=n-p$. For more details of this subsection, we give a reference \cite{Ton2}.
\begin{defi}
A Riemannian metric $g_Q$ on $Q$ is said to be bundle-like, if
\[L_Xg_Q\equiv0,\]
for any $X\in \Gamma(F)$, where $Q=TM/F$. We say $(M,F)$ is a Riemannian foliation, if $Q$ admits a bundle-like metric.
\end{defi}
Given a metric $g$ on $TM$, $Q$ is identified with the orthogonal complement to $F^\perp$ by $g$. In turn, $Q$ inherits a metric $g_{F^\perp}$, where $g_{F^\perp}=g|_{F^\perp}$. We have the following equivalence,
\[\mbox{a metric }g\mbox{ of }TM\mbox{ corresponds a triple }(g_F,\pi_F, g_Q),\]
where $g_F=g|_{F}$ and $\pi_F$ is the projection $TM\to F$.
\noindent
A Riemannian metric $g$ on $TM$ is said to be \emph{bundle-like}, if the induced metric $g_{F^\perp}$ is bundle-like.
By the work of Reinhart $\cite{Reinhart}$, it is known that the bundle-like metric can be locally written as $g=\sum_{i,j}g_{ij}(x,y)\omega^i\otimes \omega^j+\sum_{k,l}g_{k,l}(y)dy^k\otimes dy^l$, where $(x,y)$ is in the foliated chart of $M$ and $\omega^i=dx^i+a^i_\alpha(x,y) dy^\alpha$.
In this paper, we always assume that $(M,F)$ is a Riemannian foliation.
Let $\pi_Q$ be the canonical projection $TM\to Q$. We define a connection $\nabla^{T}$ on $Q$, by
$$\nabla^{T}_Xs=\begin{cases}
\pi_Q([X,Z_s])& X\in \Gamma(F),\\
\pi_Q(\nabla^g_X Z_s)& X\in \Gamma(F^\perp),
\end{cases}$$
for any section $s\in\Gamma(Q)$,
where $Z_s\in \Gamma(TM)$ is a lift of $s$, i.e. $\pi_Q(Z_s)=s$ and $\nabla^g$ denotes the Levi-Civita connection of $g$. We call $\nabla^T$ transverse Levi-Civita connection. If $(M,F)$ is a Riemannian foliation, then by the Koszul-formula \cite[Theorem 5.9]{Ton2}, we have that $\nabla^T$ is uniquely determined by $g_{F^\perp}$. Moreover, one can verify that it is torsion free and metric-compatible, whose leafwise restriction coincides with the Bott-connection. We set $R^T$ as the curvature of this connection.
We define the transverse Ricci curvature and scalar curvature by
\[Ric^{T}(Y)=\sum^{q}_{i=1}R^{T}(Y,e_i)e_i,~
Scal^{T}=\sum^{q}_{i=1}g_Q(Ric^{T}(e_i),e_i),\]
where $\{e_i\}$ is a local orthonormal frame of $Q$.
We define the basic forms as follows:
\[\Omega^r_b(M)=\{\omega\in\Omega^r(M)\big|~\iota_X(\omega)=0,~L_X(\omega)=0,
\mbox{ for all } X\in\Gamma(F)\}.\]
By the work of Alvarez L\`opez \cite{AL}, we have the following $L^2$ orthogonal decomposition for the forms on $M$, i.e.
\[\Omega(M)=\Omega_b(M)\oplus\Omega^\perp_b(M),\]
with respect to the $C^\infty$-Fr\'echet topology.
Choosing a local orthonormal basis $\{e_i\}_{1\leq i\leq p}$ of $F$, we define the character form of the foliation $\chi_F$ by,
$\chi_F(Y_1,\cdots, Y_p)=\det(g_F(e_i,Y_j))_{1\leq i,j\leq p}$,
for any section $Y_1,\cdots, Y_p\in \Gamma(TM)$. By the metric $g_Q$($g_{F^\perp}$), we have the basic Hodge-star operator, $$\bar*:\bigwedge^rQ^*\to \bigwedge^{q-r}Q^*.$$
The basic Hodge-star operator is related to the usual Hodge-star operator by the formula $\bar*\alpha=(-1)^{(q-r)\dim(F)}*(\alpha\wedge\chi_F)$, moreover we have $\bar *:\Omega^r_b(M)\to \Omega^{q-r}_b(M)$ and the volume density formula,
$dvol_M=dvol_Q\wedge\chi_F$.
For a section $\alpha\in \Omega^r_b(M)$, we define its $L^2$ norm by
\[\|\alpha\|^2_{L^2}=\int_M\alpha\wedge\bar*\alpha\wedge\chi_F.\]
We set $d_b$ as the restriction of $d$ to the basic forms, the complex $d_b:\Omega^r_b(M)\to\Omega^{r+1}_b(M)$ is a subcomplex of the deRham complex, whose cohomology is called basic cohomology $H^r_b(M)$. It is known that $H^1_b(M)\subset H^1(M)$.
We define $b^r_b=\dim H^r_b(M)$.
\begin{defi}
The mean curvature vector field is defined by $\tau=\sum^{\dim F}_{i=1}\pi_Q(\nabla^g_{\xi_i}\xi_i)\in\Gamma(Q)$, where $\{\xi_i\}$ is a local orthonormal basis of $F$. Let $\kappa\in \Gamma(Q^*)$ be the dual to $\tau$ via the metric $g_Q$.
\end{defi}
\begin{prop}[Rummler \cite{Rummler}]\label{formula-Rummler}
For any metric $g$ on $TM$,
we get
\[d\chi_F=-\kappa\wedge\chi_F+\phi_0,\]
where $\phi_0$ belongs to $F^2\Omega^{p+1}=\{\omega\in\Omega^{p+1}(M)\big|\iota_{X_1}\cdots\iota_{X_p}\omega=0,\mbox{ for any }X_1,\cdots,X_p\in \Gamma(F)\}$. This implies that $w\wedge\phi_0=0$ for any $w\in \Gamma(\bigwedge^{q-1}Q^*)$.
\end{prop}
By the decomposition, we have that $\kappa=\kappa_b+\kappa_0$ for a bundle-like metric $g$, where $\kappa_b\in \Omega^1_b(M)$ and $(\kappa_0,\omega_b)_{L^2}=0$ for any basic one form $\omega_b$. Dominguez \cite{D} shows that
any Riemannian foliation $F$ carries a tense bundle-like metrics, i.e. having basic mean curvature form $\kappa=\kappa_b$.
We call $\kappa_b$ the basic mean curvature form. It is known that
$d\kappa_b=0$,
and the cohomology class $[\kappa_b]$ is independent of any bundle-like metric \cite{AL}.
\begin{defi}
We say a foliation is \emph{taut}, if there is a metric on $M$ such that $\kappa=0$, i.e. all leaves are minimal submanifolds.
\end{defi}
In this paper,
we call a bundle-like metric taut, if the induced mean curvature form vanishes.
For a fixed Riemannian foliation $F$, the taut condition has a topological obstruction.
\begin{prop}[Tondeur {\cite[Page 96]{Ton2}}]\label{prop-taut}
Let $(M,F)$ be a Riemannian foliation. Suppose that $M$ is closed oriented and each leaf is also oriented. Then, the following statements are equivalent
\begin{itemize}
\item $H^q_b(M)\neq0$, $q$ is the codimension of this foliation $F$.
\item $[\kappa_b]=0~\in H^1_b(M)$,
\item the foliation is taut.
\end{itemize}
\end{prop}
By \cite[Page 99]{Ton2}, we have that for taut Riemannian foliation the Poincare duality for basic cohomologies holds, i.e. $H^r_b(M)\cong H^{q-r}_b(M)$.
\begin{prop}[Tondeur {\cite[Theorem 7.18]{Ton2}}]
Let $d_b$ denote the restriction of $d$ on the basic forms.
Then $L^2$-formal adjoint of $d_b$ is $\delta_b=(-1)^{q(*+1)+1}\bar*(d_b-\kappa_b\wedge)\bar*$.
\end{prop}
We define the basic Laplacian operator by $\Delta_b=d_b\delta_b+\delta_bd_b$.
Before introducing the transverse elliptic operator, we review the definitions of foliated vector bundle and basic connections.
\begin{defi}
A principal bundle $P\to M$ is called foliated, if it is equipped with a lifted foliation $F_P$ invariant under the structure group action, transversal to the tangent space to the fiber and $F_P$ projects isomorphically onto $F$. We say a vector bundle $E\to M$ is foliated, if its principal bundle $P_E$ is foliated.
\end{defi}
\begin{defi}
A connection $\omega$ of the foliated principal bundle $P$ is called adapted, if the horizontal distribution associated to this connection contains the foliation $F_P$. A covariant connection on a foliated vector bundle is called adapted, if its associated connection on the principal bundle is. An adapted connection $\omega$ is called basic, it is a Lie algebra valued basic form. The similar notion for the covariant connection.
\end{defi}
Using an adapted connection, we define the basic sections by \[\Gamma_b(E)=\{s\in \Gamma(E)\big|~ \nabla_Xs\equiv0,~\mbox{for all }X\in \Gamma(F)\},\]
where $\nabla$ is an adapted connection. It is known that the space of basic sections is independent of the choice of the adapted connection.
\begin{defi}
A transverse Clifford module $E$ is a complex vector bundle over $M$ equipped with a hermitian metric satisfying the following properties:
\begin{enumerate}
\item $E$ is a bundle of $Cl(Q)$-modules, and the Clifford action $Cl(Q)$ on $E$ is skew-symmetry, i.e. \[(s\cdot\psi_1,\psi_2)+(\psi_1,s\cdot\psi_2)=0,\]
for any $s\in\Gamma(Q)$ and $\psi_1,\psi_2\in\Gamma(E)$;
\item $E$ admits a basic metric-compatible connection, and this connection is compatible with the Clifford action.
\end{enumerate}
\end{defi}
We say $(M,F)$ admits a transverse \spinc structure, if $Q$ is \spinc and associated the spinor bundle $S$ is a transverse Clifford module over $(M,F)$.
\begin{defi}\label{defi-transverse-Dirac}Let $E$ be a transverse Clifford module $E$ over $(M,F)$.
Fixing a basic connection $\nabla^E$, we define the transverse Dirac operator $\Dirac^T$ by $\Dirac^T=\sum^q_{i=1}e_i\cdot\nabla^E_{e_i}$ action on $\Gamma(E)$, where $\{e_i\}$ is a local orthonormal basis of $Q$.
\end{defi}
Note that $\Dirac^T$ is not formally self adjoint in general, whose adjoint operator is $\Dirac^{T,*}=\Dirac^T-\tau_b$. We set $\Dirac_b=\Dirac^T-\frac12\tau_b$, which is called basic Dirac operator.
By straightforward calculation, we have that the basic Dirac operator is a formally self-adjoint operator and maps the basic sections $\Gamma_b(E)=\{s\in\Gamma(M,E)\big|\nabla_Xs\equiv0,\mbox{ for any }X\in\Gamma(F)\}$ to itself.
Let $E$ be a foliated vector bundle on $M$ equipped with a basic Hermitian structure
and a compatible basic connection $\nabla^E$, we define the basic $\|\|_{L^p_k}$-norm by
\[\|u\|_{L^p_k}=\sum^k_{j=0}(\int_M|(\nabla^E)^ju|^pdvol_M)^{\frac1p},\]
for any $u\in \Gamma_b(E)$. Let $L^p_k$ be the completion of $\Gamma_b(E)$ with respect to such a norm.
One has the similar Sobolev embedding and Sobolev multiplication properties for basic sections, which are shown in \cite[Theorem 9, 10, 11]{KLW}.
\begin{defi}\label{defi-transverse-elliptic}
Let $E_1$ and $E_2$ be two foliated vector bundles over $M$ with compatible basic connections. A differential operator $L:\Gamma(E_1)\to \Gamma(E_2)$, is called basic, if in any foliated chart $(x,y)\in U\times V$ with distinguished local trivialization of $E_1$ and $E_2$, then locally one can write
\[L|_{U\times V}=\sum a_\alpha(y)\frac{\partial^{|\alpha|}}{\partial^{\alpha_1} y_1\cdots\partial^{\alpha_q}y_q}.\]
A basic differential operator $L$ defined as the above, is said \emph{transverse elliptic}, if its transverse symbol is an
isomorphism away from the $0$-section, i.e. $\sigma(p,y)$ is an isomorphism
for any $p\in M$ and non-zero $y\in Q^*_p$.
\end{defi}
For transverse elliptic operator, we have the regularity estimate \cite[Theorem 12]{KLW}.
\section{Analysis of basic forms}
In this section, let $(M,F)$ be a closed oriented manifold with taut Riemannian foliation. We give some basic tools for analysis on Sobolev space of basic sections. The goal of this section is to prepare for the analysis on the moduli space of the later sections.
We recall the Unique Continuation Property on Hilbert space(see Kroheimer and Mrowka \cite[Chpater 7, 14]{KM}).
\begin{lemma}[c.f. {\cite[Lemma 7.1.3]{KM}}]\label{lemma-7.1.3}
Let $z:[t_1,t_2]\to H$ be a solution of the equation
\[\frac{d}{dt}z(t)+L(t)z(t)=f(t),\]
where $L(t)$ is a first-order transverse elliptic formal self-adjoint operator, $H$ is a Hilbert space and $f(t)$ is an element of $ H$ with $\|f(t)\|\leq C\|z(t)\|$ for some constant $C$. If $z(t)$ is zero at one-point, then it vanishes identically.
\end{lemma}
On the finite cylinder $Z=[a,b]\times M$, we have the following trace theorem.
\begin{thm}[c.f. {\cite[Theorem B10]{Wehrheim}}]\label{trace-thm}
Let $Z=[a,b]\times M$ and $1\leq p<\infty$ and $n=codim(F)$. In the case $p<n+1$ assume $1\leq q
\leq\frac{(n+1)p-p}{1+n-p}$, and in the case $p\geq1+n$ we assume $1\leq q<\infty$. Then, we have the basic trace theorem for all basic functions of $Z$, i.e. $L^p_1(Z)\to L^q(\partial Z)$.
\end{thm}
By the same idea of the \cite[Appendix B]{Hormander}, we have the following theorem.
\begin{thm}
For $j>\frac12$, we have a continuous restriction map between the Sobolev basic sections,
\[r:L^2_j(Z,E)\to L^2_{j-\frac12}(\partial Z,E_0),\]
where $E$ is the pull-back foliated bundle of $E_0\to M$.
\end{thm}
We
consider the equation
\[\begin{cases}
\Delta_bu=f&Z\\
\frac{\partial u}{\partial \nu}=g&\partial Z,
\end{cases}\]
where $\nu$ denotes the unit normal vector field along the boundary $\partial Z$.
We assume that $$\int_Zf+\int_{\partial Z}g=0. $$
Recall that we have that $\Delta_bu=\Delta u$ for any basic function $u$(see \cite[Page 86]{Ton2}).
Similar to the standard Laplacian equation with Neumann boundary condition,
we have the following theorems.
\begin{thm}\label{Neumann-estimate-thm}
For any basic function $u$ on $Z$, we have \[\|u\|_{L^2_{k+2}}\leq C(\|\Delta_bu\|_{L^2_k}+\|u|_{\partial Z}\|_{L^2_{k+1/2}}+\|u\|_{L^2_{k+1}}).\] Furthermore, for $\frac{\partial u}{\partial \nu}=0$, we have \[\|u\|_{L^2_{k+2}}\leq C(\|\Delta_b u\|_{L^2_k}+\|u\|_{L^2_{k+1}}).\]
\end{thm}
\begin{pf} Since $\Delta_b u=\Delta u$ for the basic function $u$, the proof follows by the standard theory, c.f. {\cite[Formula 7.37 ]{Tay}} for the first formula and {\cite[Formula 7.34]{Tay}} for the second one.
\end{pf}
Similarly, we have the theorem below.
\begin{thm}\label{CN-thm}
Let $f\in L^2_k(Z)$ and $g\in L^2_{k+1/2}(\partial Z)$ be the basic functions. If the formula $\int_Zf+\int_{\partial Z}g=0$ holds, then there is a solution for the equation
\[\begin{cases}
\Delta_bu=f& Z\\
\frac{\partial u}{\partial \nu}=g&\partial Z.
\end{cases}\].
\end{thm}
\begin{pf} The idea similar to the arguments of \cite[Theorem 3.1]{Wehrheim}.
Let $v\in
L^2_{2+k}(Z)$ such that $\frac{\partial v}{\partial\nu}=g\big|_{\partial Z}$, this can be realized by letting $v=\phi(t)g$ for some smooth function $\phi(t)$ with support near the boundary and near each the boundary we have $\phi(t)=t-a$ for $t\in[a,a+\epsilon)$ and $\phi(t)=t-b$ for $t\in(b-\epsilon,b]$.
Now we have
\[\int_Z(f-\Delta_bv)=\int_Zf+\int_{\partial Z}\frac{\partial v}{\partial \nu}=
0.\]
Thus, by Theorem \ref{Neumann-estimate-thm} and Rellich embedding, there exists a solution $u_1\in L^2_{k+2}(Z)$ of the Neumann problem with $f$
replaced by $f-\Delta_b v$. The solution is given by $u=u_1+v$.
\end{pf}
\begin{thm}\label{Weitzenbock-thm}
On $Z=[a,b]\times M$, we assume that the foliation of $M$ is taut. If the basic one-form $a\in\Omega^1_b(Z)$ satisfies the condition $a(\nu)=0$ on $\partial Z$, then we have
\begin{equation}
\int_Z(\nabla_{tr}a,\nabla_{tr}a)+Ric^T(a,a)=
\int_Z(d_ba,d_ba)+(\delta_ba,\delta_ba).\label{Weitzenbock-formula}
\end{equation}
\end{thm}
The proof is similar to the classical formula, which is stated by the following proposition.
\begin{prop}[Jung {\cite{Jung}}]
Let $Z=(M,F)$ or $Z=(I\times M,F)$ for a taut bundle-like. Suppose that $\alpha\in \Omega^1_b(Z)$. Then, we have that
\begin{equation}
\Delta_b\alpha=(\nabla^T)^*\nabla^T\alpha+Ric^T(\alpha). \label{formula-ricci-laplacian}
\end{equation}
\end{prop}
We set $Z=I\times M$, where $I\subset (-\infty,\infty)$ is a compact interval, we establish the following lemma.
\begin{lemma}\label{lemma-poincare-dual}
$H^r_b(Z)$ is dual to $H^{m+1-r}_{b,c}(Z)$, where $m=co\dim(F)$ in $M$ and $H^{l}_{b,c}(Z)$ denotes the basic deRham cohomology which is vanishing at the ends.
\end{lemma}
\begin{pf}
Let $e$ be a $1$-form on $I$ with integral $1$, which is vanishing at the ends. We define
\[e_*:\Omega^{*}_b(M)\to \Omega^{*+1}_{b,c}(Z),\]
by\[\alpha\mapsto \alpha\wedge e.\] We set $\pi_*:\Omega^*_{b,c}(Z)\to \Omega^{*-1}_b(M)$ as the integration along the $I$-direction. Similar to the arguments of \cite[Proposition 4.6]{BT}, we have that the induced cohomology map $e_*: H^{*-1}_b(M)\to H^*_{b,c}(Z)$ is an isomorphism, whose inverse is the induced cohomology map of $\pi_*$. Since $H^r_b(M)$ is dual to $H^{m-r}_b(M)$, the lemma follows.
\end{pf}
\begin{lemma}
We have the isomorphism
\[H^1_b(Z)\cong H^1_b(M).\]
\end{lemma}
\begin{pf}
The idea is the same as for the usual deRham cohomology. It is not hard to see that $\pi^*_Z :H^1_b(M)\to H^1_b(Z)$ is an injective, where $\pi_Z$ is the canonical projection $Z\to M$. Let $i:M\to Z,~p\mapsto (p,t_1)$ where $t_1$ is the left endpoint. If there is an element $[\omega]\in H^1_b(Z)$ such that $i^*([\omega])=0$, i.e. $i^*\omega=d_bf$ for some basic function on $M$. Consider $\omega'=\omega-d_b\pi^*_Zf$, we rewrite it as
\[\omega'=\alpha+f'dt.\]
The condition $d\omega'=0$ implies that
\[d\alpha=0,~\dot\alpha-df'=0. \]
We have that $\alpha|_{\{t_1\}\times M}=0$, and $\alpha(p,t)=\int^t_{t_1}df'(s)ds=d\int^t_{t_1}f'(s)ds$, which implies that $[\omega']=[\omega]=0$.
\end{pf}
For any non-trivial homotopy map $u:Z\to S^1$, by Theorem \ref{CN-thm}, we can find an element $v$ of this homotopy class satisfying the equation
\begin{equation}
\begin{cases}
\delta_b(v^{-1}d_bv)=0&\mbox{ in } Z,\\
dv(\nu)=0& \mbox{ on } \partial Z.
\end{cases}\label{eqn-gauge-Neumann-condition}
\end{equation}
Notice that for basic one form $\alpha$ we have that $\delta_b\alpha=\delta\alpha$.
Let $\Gamma^1_b$ be the lattice of $H^1_b(M)\cap H^1(M,\mathbb Z)$. $\Gamma^1_b$ also corresponds to a lattice of $H^1_b(Z)\cap H^1(Z,\mathbb Z)$. Choose a basis $\{a_i\}$ of this lattice, by the pairing we have a basis $\{\beta_i\}$ of $H^3_{b,c}(Z)$, which is
dual to $\{a_i\}$.
\section{Basic Seiberg-Witten invariant on codimension $3$ foliation}
In this section, we will define a basic Seiberg-Witten invariant on manifold with codimension $3$ foliation under a certain condition.
\subsection{Basic Seiberg-Witten equations on codimension $3$ foliation}
In this subsection, we focus on the manifold with foliation satisfying the following assumption.
\begin{assump}\label{assum-main}
Let $(M,F)$ be an oriented closed manifold with codimension 3 oriented Riemannian foliation $F$ and admits a transverse \spinc structure $\mathfrak s$. Suppose that $H^1_b(M)\cap H^1(M,\mathbb Z)\subset H^1(M)$ is a lattice of $H^1_b(M)$.
\end{assump}
Let $\mathcal A_b(\mathfrak s)$ be the space of basic \spinc connections. We define basic Seiberg-Witten equations for manifold with codimension-$3$ Riemannian foliation by
\begin{equation}
\begin{cases}
\Dirac_{b,A}\Psi=0,\\
\frac12\bar*F_{A^t}-q(\Psi)=0,
\end{cases}\label{eqn-SW-equation-1}
\end{equation} for $(A,\Psi)\in \mathcal A_b(\mathfrak s)\times \Gamma_b(S)$,
where we identify the traceless endomorphism of the spinor bundle with the imaginary valued cotangent bundle(we use $q(\Psi)$ instead of $\rho^{-1}(\Psi\Psi^*)_0$ in the book \cite[Formula 4.4]{KM}), $A^t$ denotes the connection on the determinate bundle of $S$, see \cite[Notation 1.2.1]{KM} and $\Dirac_{b,A}$ denotes the twisted basic Dirac operator with the basic connection $A$. The basic gauge group \[\mathcal G_b=\{u:M\to U(1)\big|~L_Xu\equiv0,\mbox{ for all }X\in \Gamma(F)\}, \] acts on $\mathcal C_b(\mfs)= \mathcal A_b(\mathfrak s)\times \Gamma_b(S)$ as follows(\cite[Formula 4.5]{KM})
\[u:(A,\Psi)\mapsto (A-u^{-1}du, u\Psi).\]
To construct the basic Seiberg-Witten invariant, we need to consider the moduli space, which is defined as below.
\begin{defi}
The moduli space $\mathcal M(M,F,\mfs)$ of the basic Seiberg-Witten equations on $M$ is the solution of space of the above Seiberg-Witten equations modulo the gauge transformation group $\mathcal G_b$. The moduli space $\mathcal M^*(M,F,\mfs)$ is the irreducible part of $\mathcal M_b(M,F,\mfs)$, i.e. spinor field part is not identically zero.
\end{defi}
By the standard argument, we consider the following complex
\[L^2_{2+k}(\Omega^0_{b}(M,i\mathbb R))\overset{G_{(A,\Psi)}}\longrightarrow
L^2_{1+k}(\Omega^1_{b}(Y,i\mathbb R))\oplus L^2_{1+k}(\Gamma_{b}(S))\overset{L_{(A,\Psi)}}\longrightarrow L^2_{k}(\Omega^1_{b}(M,i\mathbb R))\oplus L^2_k(\Gamma_{b}(S)),\]
where $G_{(A,\Psi)}f=(-d_bf,f\Psi)$ and $L_{(A,\Psi)}(a,\Phi)=(-\frac12\bar*da+q(\Psi,\Phi),\Dirac_{b,A}\Phi+a\Psi)$.
By the straightforward calculation, we have that \[L\comp G=0.\]
To read off the virtual dimension of the moduli space,
it is convenient to consider the form operator(see \cite[Formula 2.7]{B.L. Wang}),
\[Q_{(A,\Psi)}:L^2_{2+k}(\Omega^1_{b}(M,i\mathbb R))\oplus L^2_{2+k}(\Omega^0_{b}(M,i\mathbb R))\oplus L^2_{2+k}(\Gamma_{b}(S))\]\[\to
L^2_{1+k}(\Omega^1_{b}(M,i\mathbb R))\oplus L^2_{1+k}(\Omega^0_{b}(M,i\mathbb R))\oplus L^2_{1+k}(\Gamma_{b}(S)),\]
\begin{equation}
Q_{(A,\Psi)}=\left(\begin{array}{cc}
L_{(A,\Psi)}&G_{(A,\Psi)}\\
G^*_{(A,\Psi)}&0\\
\end{array}\right), \label{Hessian-operator}
\end{equation}
where $G^*_{(A,\Psi)}$ is the formal self-adjoint of $G_{(A,\Psi)}$.
To show the smoothness of the moduli space, we need perturb the above equation.
Fixing a basic perturbation $\eta\in i\Omega^1_b(M)$, we denote the moduli space of the perturbed basic Seiberg-Witten equations by $\mathcal M_{g,\eta}(M,F,\mfs)$, i.e. consider the solutions of equation
\[\begin{cases}
\Dirac_{b,A}\Psi=0,\\
\frac12\bar*F_{A^t}-q(\Psi)=\eta.
\end{cases}\]
modulo the gauge action.
\begin{defi} We say $[A,\Psi]\in \mathcal M_{b,\eta}(M,F,\mfs)$ is \emph{non-degenerate} if \[\ker(L_{(A,\Psi)})/ImG_{(A,\Psi)}=0.\]\end{defi}
\begin{prop}
If $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, then for a generic perturbation
the irreducible moduli space of basic Seiberg-Witten equations is a compact manifold with formal dimension zero.
\end{prop}
\begin{pf}
To show that moduli space is a compact and smooth manifold, we repeat the similar arguments of \cite[Lemma 2.2.3, Lemma 2.2.6, Theorem 2.2.8]{B.L. Wang}.
To prove the formal dimension of the moduli space is zero, we can calculate the index the operator \eqref{Hessian-operator}.
We recall that the operator \eqref{Hessian-operator} is a compact perturbation of
\[\left(\begin{array}{ccc}
\frac12\bar*d_b&-d_b&0\\
-\delta_b&0&0\\
0&0&\Dirac_{b,A}\\
\end{array}\right),\]
which is a first-order transverse elliptic operator, however it is not formal self-adjoint in general.
It is easy to see that the operator
\[\left(\begin{array}{ccc}
\frac12(\bar*d_b-\frac12\kappa)&-d_b&0\\
-\delta_b&0&0\\
0&0&\Dirac_{b,A}\\
\end{array}\right)\]
is a formal self-adjoint operator and the difference
\[\left(\begin{array}{ccc}
\frac12\bar*d_b&-d_b&0\\
-\delta_b&0&0\\
0&0&\Dirac_{b,A}\\
\end{array}\right)-\left(\begin{array}{ccc}
\frac12(\bar*d_b-\frac12\kappa)&-d_b&0\\
-\delta_b&0&0\\
0&0&\Dirac_{b,A}\\
\end{array}\right)\]
is a compact operator, hence they have the same index zero. To equip an orientation of the moduli space, we just need to equip an orientation of the determinant line bundle of \eqref{Hessian-operator}.
\end{pf}
The above proposition implies that the determine line bundle of the operator \eqref{Hessian-operator} is trivial over the moduli space $\mathcal M^*_{g,\eta}(M,F,\mfs)$. Hence we have a natural orientation for the moduli space. The basic Seiberg-Witten invariant $SW_{g,\eta}(M,F,\mfs)$ on manifold with codimension $3$ Riemannian foliation is defined by the signed counting of the moduli space $\mathcal M^*_{g,\eta}(M,F,\mfs)$.
\subsection{Basic Seiberg Witten invariant on codimension $3$ foliation}
In this subsection, we show how the Seiberg-Witten invariant depends on the bundle-like metric and basic perturbation.
We review the notion of the reducible solution.
Let $(A,\Psi)$ be a solution of the basic Seiberg-Witten equations \eqref{eqn-SW-equation-1}. When $\Psi=0$, the basic Seiberg-Witten equations reduce to a single equation
for the basic connection $\frac12\bar *F_{A^t}=\eta$. If $\eta=0$, then the reducible class is identified with the moduli space of the flat basic $U(1)$-connection of $\det(\mathfrak s)$. We denote its first Chern class by $c_1(\mathfrak s)$.
\begin{lemma}
The equation $\frac12\bar *F_{A^t}=\eta$ has a solution if and only if we have that $d_b\bar*\eta=0$ and $\pi i[c_1(\mathfrak s)]=[\bar *\eta]$ in $iH^2_b(M)$. In particular, if $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, then the set of reducible solutions modulo gauge action is isomorphic to $H^1_b(M)/(H^1_b(M)\cap H^1(M,\mathbb Z))$.
\end{lemma}
\begin{pf}
Obviously, $[\frac12F_{A^t}]=[\bar *\eta]$ is a necessary condition to solving the equation.
Conversely, suppose that $\pi i[c_1(\mathfrak s)]=[\bar *\eta]$. We fix a basic connection $A_0$ such that $[\frac12 F_{A^t_0}]=[\bar *\eta]$. It suffices to get a basic one-form $a$ such that $da=\bar *\eta-\frac12 F_{A^t_0}$. Since $\bar *\eta-\frac12 F_{A^t_0}$ is an exact basic form, such a one-form $a$ always exists. By choosing one solution $a_0$ of the above equation, one can represent all the others as
\[a_0+\mbox{ closed one form}.\]
Any two solutions $a_1$ and $a_2$ are equivalent, if and only if $a_1=a_2+u^{-1}du$ for some $u\in\mathcal G_b$.
\end{pf}
\begin{thm}
Let $(M,F)$ be manifold with foliation satisfying Assumption \ref{assum-main}.
When $b^2_b>1$, we have that the basic Seiberg-Witten invariant is well-defined, i.e. it is independent of the generic choice of the basic perturbation and bundle-like metric.
\end{thm}
\begin{pf}
To show that the basic Seiberg-Witten invariant is independent of bundle-like metric, we apply the similar proof in codimension $4$ case \cite[Chapter 5]{Morgan}. Here we get a sketch of the proof.
Denote by $\mathcal N=\{a\in i\Omega^1_b|~d_b\bar*a=0\}$, which can be identified with the closed basic two-forms. Set
\[\mathcal W_{\mathfrak s}=\mathcal W_{\mathfrak s}(g)=\{\eta\in \mathcal N|~[\bar*\eta]=\pi ic_1(\mathfrak s)\}.\]
$\mathcal W_{\mathfrak s}$ is a codimension $b^2_b$ affine subspace of $\mathcal N$.
When $b^2_b>1$, $\mcWs$ is of codimension two or more. For two perturbations $\eta_{i}\in \mathcal N\setminus \mcWs$ for $i=1,~2$, there are no reducible solutions. Since, $b^2_b>1$, one can choose a generic path $\eta_s$ connection these two perturbations $\eta_1$ and $\eta_2$, such that for each $s$, $\eta_{s}\in \mathcal N\setminus \mcWs$. This completes the proof.
\end{pf}
At the end of this subsection, we show the dependence on the bundle-like metric for basic Seiberg-Witten invariant, when $b^2_b=1$.
We choose an orientation of the one dimensional space $iH^2_b(M)$. There exists an unique unit $g$-harmonic basic two form $\omega$, i.e. $\|\omega\|_{L^2}=1$.
The wall $\mcWs$ can be regarded by the linear equation
\[(\bar*\eta-\pi ic_1(\mfs),\omega)=0.\]
Set $\mathcal N^\pm=\{\eta\in\mathcal N|~\pm(\bar*\eta-\pi i c_1(\mfs),\omega)>0\}$.
We consider a family of $(g_t,\eta_t)_{t\in[-1,1]}$ crossing the wall transversely once at $t=0$, such that $g_t$ is locally constant near $-1,~0$ and $1$. The one-parameter family moduli space
\[\mathcal M_{[-1,1]}=\bigcup_{t\in[-1,1]}\mathcal M_{g_t,\eta_t}(\mfs)\]
has reducible solution at $t=0$.
Under Assumption \ref{assum-main}, the reducible space $\mathcal M^0$,
\[\mathcal M^0=\{(A,0)\big|~\frac12F_A=\bar*\eta^0+d_b\alpha\}/\mathcal G_b\]
can be identified as torus $T^{b^1_b}$, where $b^1_b\geq1$.
We decompose the connection $A$ as $A=A_0+\alpha+a$, where $A_0$ is a connection such that $\frac12F_{A^t_0}=\bar*\eta^0$ and $a$ is a harmonic one-form.
\begin{lemma}
Suppose that $\mathcal M^0$ is indexed by $a\in H^1_b(M)/H^1_b(M)\cap H^1(M,\mathbb Z)$ with the decomposition $A=A_0+\alpha+a$. If $\ker(\Dirac_{b,A})=0$, then there is no irreducible solution connection $A$ in
$\mathcal M_{[-1,1]}$.
\end{lemma}
\begin{pf} The proof is similar to \cite[Lemma2.3.7]{B.L. Wang}.
Let $(A_t,\Psi_t)$ be a family of solution to the perturbed basic Seiberg-Witten equations with $(A_t,\Psi_t)|_{t=0}=(A,0)$. Near $(A,0)$ we write $(A_t,\Psi_t)=(A+a_t,\Psi_t)$. We let $(a_t,\Psi_t)$ satisfies
\[\begin{cases}
\delta_ba_t=0\\
\bar*d_ba_t=q(\Psi(t))\\
\Dirac_{b,A}(\Psi_t)+a_t\Psi_t=0\\
\end{cases}\]
near $t=0$. Locally, we write $\Psi_t=\sum_{i\geq1}t^i\Psi_i$ and $a_t=\sum_{i\geq1}t^ia_i$. Differentiating the third equation with respect to $t$, we get the result.
\end{pf}
\begin{defi}
A reducible solution is called regular, if $\ker\Dirac_{b,A}=0$.
\end{defi}
Therefore, we can perturb the equation so that regular part $\mcM^0_{reg}$ of the reducible solutions is isolated from the irreducible solution in $\mathcal M_{[-1,1]}$.
In particular, when the foliation is taut($H^1_b(M)\cong H^2_b(M)$), we can find a generic perturbation
so that there are only finitely many points in $\mcM^{0}$ meet the irreducible solutions in $\mcM_{[-1,1]}$ and whose kernel of the associated twisted Dirac operator $\Dirac_{b,\omega+a_\theta}$ is of 1 dimension. Following the same arguments as in \cite[Proposition 2.3.8]{B.L. Wang}, we have the following proposition.
\begin{prop}
Let $(M,F)$ be a manifold with foliation satisfying Assumption \ref{assum-main}, and foliation is taut.
Suppose that $(g_{-1},\eta_{-1})$ and $(g_{1},\eta_{1})$ belong to the different parts of separated by $\mathcal W$, choosing an orientation of $H^1_b(M)$, then $$SW_{g_1,\eta_1} (M,F,\mathfrak s )-SW_{g_{-1},\eta_{-1}}(M,F,\mathfrak s)=
SF(\Dirac_{A(\theta)}),$$
where $A(\theta)$ defines a connection joining $A_{-1}, ~A_1$ and the $SF(\Dirac_{A(\theta)})$ denotes the spectral flow of the corresponding basic Dirac operator.
\end{prop}
\begin{pf}
We choose a family of metrics and perturbations $(g_t,\eta_t),t\in[-1,1]$ such that it crosses $\mathfrak R=\{(g_Q,\eta)| \eta=\eta^0+\bar*_{g_Q}d\alpha,~[\eta^0]=\pi i[c_1(\mfs)],~\eta\mbox{ is harmonic }\}$ at $t=0$ with finite singular points on $\mathcal M^0$. Similar to \cite[Proposition 2.3.8]{B.L. Wang}, we have that $SW_{g_1,\eta_1} (M,F,\mathfrak s )-SW_{g_2,\eta_2}(M,F,\mathfrak s)$ is the same as the contribution of the spectral flow of the twisted basic Dirac operator along $\mathcal M^0$, which proves the proposition.
\end{pf}
Summarizing the arguments, we have the following results.
\begin{thm}
Suppose that $(M,F)$ is a manifold with foliation $F$ satisfying Assumption \ref{assum-main}. Then for each transverse \spinc structure $\mathfrak s$ and for generic perturbation $\eta$ and a bundle-like metric $g$, we define the basic Seiberg-Witten invariant $SW_{\eta,g}(M,F,\mathfrak s)$. Moreover, we have the properties:
\begin{itemize}
\item If $b^2_b>1$, then for generic bundle-like metric and perturbation the basic Seiberg-Witten moduli space is a smooth compact manifold, and $SW_{g,\eta} (M,F,\mathfrak s )$ is independent of the generic choice of the general bundle-like metric and the perturbation.
\item If $b^2_b=1$, $W_{\eta,g} (M,F,\mathfrak s )$ depends only on the component of $H^2_b(M)\setminus \pi c_1(\mfs)$.
\end{itemize}
\end{thm}
\section{Chern-Simons-Dirac functional and moduli space for foliation}
The purpose of this section is to give the preparation to show the compactness for the moduli space.
\subsection{Chern-Simons-Dirac functional for foliation}
From this subsection, we let $(M,F)$ satisfy the assumption below.
\begin{assump}\label{assum-main-1}
Let $(M,F)$ be a oriented closed manifold with codimension 3 oriented Riemannian foliation $F$ and admits a transverse \spinc structure $\mathfrak s$. Suppose that $H^1_b(M)\cap H^1(M,\mathbb Z)\subset H^1(M)$ is a lattice of $H^1_b(M)$ and $F$ is taut.
\end{assump}
Fixing a bundle-like metric and a transverse \spinc structure $\mathfrak s$,
we define the basic Chern-Simons-Dirac functional over $M$ by
\[\L(A,\Psi)=-\frac18\int_M(A^t-A^t_0)\wedge(F_{A^t}+F_{A^t_0})\wedge\chi_F
+ \frac12\int_M(\Psi,\Dirac_A\Psi)dvol_M,\]
for any $(A,\Psi)\in \mathcal A_b(\mathfrak s)\times\Gamma_b(S)$.
The formal gradient of the Chern-Simons-Dirac functional is given by the following lemma.
\begin{lemma}
We have the following identity
\[grad\L(A,\Psi)=(\frac12 \bar*(F_{A^t}+\frac12(A^t-A^t_0)\wedge\kappa_b)-
q(\Psi)),\Dirac_{b,A}\Psi). \]
\end{lemma}
Note that the gradient is not gauge-invariant in general.
\begin{pf}
Let $(A+ta,\Psi+t\Phi)\in \mathcal A_b(\mathfrak s)\times\Gamma_b(S)$, by the standard variation one deduces that
\begin{eqnarray*}
&&\partial_t(\L(A+ta,\Psi+t\Phi))\big|_{t=0}\\
&=&-\frac18\int_M(2a\wedge(F_{A^t}+F_{A^t_0})+(A^t-A^t_0)\wedge 2da)\wedge
\chi_F
\\
&&+ \frac12\int_M(\langle\Phi,\Dirac_A\Psi\rangle+
\langle\Dirac_A\Psi,\Phi\rangle+\langle\Psi,a\cdot\Psi\rangle)dvol_Q\wedge\chi_F\\
&=&\int_M a\wedge(-\frac12F_{A^t}-\frac14(A^t-A^t_0)\wedge\kappa_b+
\bar*q(\Psi))\wedge\chi_F
+\int_M Re\langle\Dirac_A\Psi,\Phi\rangle dvol_Q\wedge \chi_F\\
&=&\int_M(a,\frac12\bar*(F_{A^t}+\frac12(A^t-A^t_0)\wedge\kappa_b)-q(\Psi))dvol_M+
\int_M Re\langle\Dirac_A\Psi,\Phi\rangle dvol_M,
\end{eqnarray*}
where we used Rummler formula \eqref{formula-Rummler} to deduce the second identity.
\end{pf}
We say $(A,\Psi)$ is a critical point of $\L$, if its gradient vanishes at $(A,\Psi)$.
\noindent
For any gauge action $u\in\mathcal G_b(M,S^1)$, we have
\[\L(u(A,\Psi))-\L(A,\Psi)=\frac12\int_Mu^{-1}du\wedge (F_{A^t})\wedge \chi_F,\]
in general it however does depend on the choice of the representation of the cohomology class $[u]=[\frac{-i}{2\pi}u^{-1}du]$.
When $F$ is taut, the critical points of the basic Chern-Simons-Dirac-functional coincide with the solutions of the basic Seiberg-Witten equations \eqref{eqn-SW-equation-1}.
We consider the gradient flow equations for the
Chern-Simons-Dirac functional
\[\frac{d}{dt}(A(t),\Psi(t))=-grad\L(A(t),\Psi(t))\]
for a path $(A(t),\Psi(t))$ of configuration space $\mathcal C_b(M,F,\mfs)=\mathcal A_b(M)\times\Gamma_b(S)$.
Let $\mathcal M([\alpha],[\beta])$ denote the moduli space of trajectories connecting the critical points up to gauge, i.e. the solution of the basic Seiberg-Witten equations
\[\begin{cases}
\frac12F^+_{A^t}=q(\Psi),\\
\Dirac^+_A\Psi=0,
\end{cases}\]
on $\mathbb R\times M$, such that $[A(t),\Psi(t)]\to[\alpha]$ as $t\to-\infty$ and $[A(t),\Psi(t)]\to[\beta]$ as $t\to\infty$,
where $[\alpha],~[\beta]$ denote the gauge equivalence classes of the critical points. The components of this space have different dimensions corresponding to the different lifts of $[\alpha],~[\beta]$. This is a manifestation of the fact that the quotient space $\mathcal B_b(M,F,\mfs)=\mathcal C_b(M,F,\mfs)/\mathcal G_b$ may have non-trivial fundamental group.
We have a decomposition
\[\mathcal M([\alpha],[\beta])=\bigcup_{z\in\pi_1([\alpha],[\beta])}
\mathcal M_{z}([\alpha],[\beta])\]
as the union over the moduli spaces in a given relative homotopy class, where $\pi_1([\alpha],[\beta])$ denotes all the homotopy classes of paths joining $[\alpha]$ and $[\beta]$.
Given two critical points $\alpha,~\beta\in\mathcal C(M,F,\mfs)$, we define the quantity
\[gr(\alpha,\beta)\in\mathbb Z\]
by the spectral flow of the Hessian operator \eqref{Hessian-operator} of a path connecting them. This is well defined because the spectral flow is invariant under homotopy and the configuration space $\mathcal C(M,F,\mfs)$ is simply connected. Moreover, such a number computes the formal dimension of the moduli space of trajectories connecting $\alpha$ and $\beta$ by the path $z$, i.e. $\dim\mathcal M_{z}([\alpha],[\beta])$.
\subsection{Compactification of the moduli space}
In this subsection, we give a compactification of the moduli space on the cylinder. The original idea to using the energy functional, which was introduced by Kroheimer and Mrowka \cite[Chapter 5]{KM}. Here we gave a foliated version of their work.
Let $Z=I\times M$, $I\subset (-\infty,\infty)$, and $\alpha,~\beta$ be the critical points of the $\L $. Denoting by $\mathcal M([\alpha],[\beta])$ the moduli spaces of trajectories in $\mathcal B_b(M,F,\mfs)$ and $\check{\mathcal M} ([\alpha],[\beta])$ the unparameterized moduli space. We show that by adding broken trajectories it can be compactified, which is denoted by $\check{\mathcal M}^+ ([\alpha],[\beta])$.
\begin{lemma}
Let $(Z=I\times M,F)$ be compact taut Riemannian foliation with a taut bundle-like product metric $g$. For a basic one-form $\alpha $ on $Z$, satisfying the boundary condition $(\alpha,\nu)=0$, where $\nu$ denotes the outward unit vector field. If there exists a constant $C_0$ such that $\int_Z\beta_i\wedge \alpha\wedge\chi_F\in[ -C_0, C_0]$ for each $\{\beta_i\}$, where $\{\beta_i\}$ is a basis of $H^3_{b,c}(Z)$. Then, there are constants $C_1$ and $C_2$ such that
$$\|\alpha\|^2_{L^2_1(Z)}\leq C_1\int_Z(|\delta_b\alpha|^2+|d_b\alpha|^2)+C_2.$$
\end{lemma}
\begin{pf}
Recall that by Theorem \ref{Weitzenbock-thm}, we have
\begin{equation}
\int_Z|\nabla^T\alpha|+Ric^T(\alpha,\alpha)=\int_Z|\delta_b\alpha|^2+|
d_b\alpha|^2. \label{formula-basic-ric-laplacian-1}
\end{equation}
Since we use the product metric on this finite cylinder, the $Ric^T$ has a uniform bound. Moreover, by the $L^2_1$-bound of $\alpha$, the estimate follows from the proof of \cite[Lemma 5.1.2]{KM}.
\end{pf}
Recall that $(M,F)$ is a closed oriented taut Riemannian foliation with codimension 3, and it admits transverse \spinc structure. For the spinor bundle $S_Z=S^+\oplus S^-$ on $Z=I\times M$, we take $S\oplus S$. For the Clifford multiplication $\rho_Z:TZ\to Hom(S_Z,S_Z)$, we take
\[\rho_Z(\partial_t)=\left(\begin{array}{cc}
0&-1\\
1&0\\
\end{array}\right),~ \rho_Z(v)=\left(\begin{array}{cc}
0&-\rho(v)^*\\
\rho(v)&0\\
\end{array}\right),\]
for $v\in Q$. A time-dependent \spinc connection $B$ on $S$ gives a \spinc connection $A$ on $S_Z$,
whose $t$ component is ordinary differentiation, i.e.
\[\nabla_A=\frac{d}{dt}+\nabla_B.\]
We call such a connection $A$ in temporal gauge.
With a basic connection $A$ as above, we have the Dirac operator
\[\Dirac^+_{b,A}:\Gamma_b(S^+)\to \Gamma_b(S^-),~\Dirac^+_{b,A}=\frac{d}{dt}+\Dirac_B.\]
For a general basic connection $A$, we can write
\[A=B+(cdt),\]
where $c$ is a basic function. The corresponding basic Seiberg-Witten equations for $(A,\Psi)$ can written as
\begin{equation}
\begin{cases}
\frac{d}{dt}B-dc=-(\frac12\bar*F_{B^t}-q(\Psi)),\\
\frac{d}{dt}\Psi+c\Psi=-\Dirac_B\Psi.
\end{cases}
\end{equation}
We define the analytic energy by
\begin{equation}
\mathcal E^a(A,\Psi)=2(\L(t_1)-\L(t_2))+\int_Z|\dot\gamma(t)+
grad(\L)(A,\Psi)|^2,\label{formula-analytic energy}
\end{equation}
where $\gamma=(A,\Psi)$.
By the transverse Weitzenb\"ock formula \cite{GK} and similar arguments of \cite[Section 4.5]{KM}, we have that
\begin{eqnarray*}
\mathcal E^a(A,\Psi)
&=&\int_Z(|\nabla_A\Psi|^2+ \frac{Scal^T}{4}|\Psi^2|+\frac14|\Psi|^4)+\frac14\int_Z|F_{A^t}|^2.
\end{eqnarray*}
The topological energy $\mathcal E^{top}$ is defined by the twice of drop of the basic Chern-Simons-Dirac functional on the cylinder, i.e.
\[\mathcal E^{top}(A,\Psi)=2(\L(t_1)-\L(t_2)).\]
For convenience, we denote the Seiberg-Witten map by $\mathfrak F$.
\begin{thm}\label{thm-coro5.1.8}
Let $\gamma_n\in\mcC(Z)$ be a sequence of solutions basic Seiberg-Witten equations on manifold with the codimension $4$ foliation, $\mathfrak F(\gamma)=0$, on $Z=[t_1,t_2]\times M$. Suppose that $$\L(\gamma_n(t_1))-\L(\gamma_n(t_2))\leq C.$$
Then, there is a sequence of gauge transformations $u_n\in \mcG(Z)$, such that, after passing to a subsequence, the transformed solutions $u_n\gamma_n$ converge in the $C^\infty$ topology on $[t'_1,t'_2]\times M$ for a smaller interval $[t'_1,t'_2]$ in the interior of $[t_1,t_2]$.
\end{thm}
\begin{pf}
The formula \eqref{formula-analytic energy} implies that there is a uniform bound for the analytic energy. We take a gauge-fixing action by Theorem \ref{Neumann-estimate-thm}. The arguments of \cite[Theorem 5.1.1]{KM} shows that we have a uniform bounds for the following terms
\[\|\Phi_n\|_{L^4}\leq C,~ \|F_{A_n}
\|_{L^2}\leq C.\]
For each $a_n=A_n-A_0$ we can choose a gauge action $u_n$, to make $a^1_n=u_n(a_n)$ satisfy the Coulomb-Neumann condition(see \cite[Page102]{KM}). For each homotopy class we can choose an element satisfying the equation \eqref{eqn-gauge-Neumann-condition}, for each $a_n$ we can find a gauge action $v_n$ satisfying the equation \eqref{eqn-gauge-Neumann-condition} such that there is a uniform constant $C$(independent of $n$) to make the following estimate holds:
\begin{equation}
\int_Z\beta_i\wedge a^1_n\wedge\chi_F\in [-C,C]
\end{equation}
where $\beta_i$ is dual basic to the lattice $H^1_b(Z)\cap H^1(Z;\mathbb Z)$.
We have a uniform $L^2_1$-bound on $(A_n-A_0,\Psi_n)$ up to gauge. Let $(A,\Psi)$ be the weak limit of $(A_n,\Psi_n)$, we have that
\[\sup\mathcal E^a(A_n,\Psi_n)\geq \mathcal E^a(A,\Psi).\]
The drop of the Chern-Simons-Dirac functional is bounded above implies that
up to subsequence we have that
\[\mathcal E^a(A_n,\Psi_n)\to \mathcal E^a(A,\Psi).\]
Thus, we have that $(A_n-A_0,\Psi_n)$ converges in $L^2_1$ up to a subsequence. The rest argument is the same as in \cite[Page 107-108]{KM}.
\end{pf}
We also need to define the perturbation, which is similar to \cite[Section 10, 11]{KM}.
Let $\mathcal V_k(Z)$ be the $L^2_k$-completion of $\Omega^+(Z)\oplus \Gamma(Z,S^-)$.
\begin{defi}[c.f. {\cite[Definition 10.5.1]{KM}}]\label{defin-perturbation}
A perturbation $\mathfrak q$ is called $k$-\emph{tame}, if the it is a formal gradient of a continuous $\mcG_b(M)$-invariant function on $\mcC(M)$ and satisfies the following properties:
\begin{enumerate}
\item the corresponding codimension $4$ perturbation $\hat{\mathfrak q}$ defines a smooth section $\hat{\mathfrak q}: \mcC_k(Z)\to \mathcal V_k(Z)$(see \cite[Formula 10.2]{KM}), where $Z=I\times M$;
\item for each $i\in[1,k]$, the codimension $4$ perturbation $\hat{\mathfrak q}$ defines a continuous section
$\hat{\mathfrak q}: \mcC_j(Z)\to \mathcal V_j(Z)$;
\item the derivative
$$D\hat{\mathfrak q}: \mcC_k(Z)\to
Hom(T\mcC_k(Z),\mathcal V_k(Z))$$
extends to a map
$$D\hat{\mathfrak q}: \mcC_k(Z)\to
Hom(T\mcC_j(Z),\mathcal V_j(Z)),$$
for $j\in[-k,k]$;
\item we have the estimate
\[\|\mathfrak q(A,\Psi)\|_{L^2}\leq C(\|\Psi\|_{L^2}+1),\]
for some constant $C$ and each $(A,\Psi)\in\mcC_k(M)$;
\item for any reference connection $A_0$, we have
\[\|\hat{\mathfrak q}(A,\Psi)\|_{L^2_{1,A}}\leq f_1(\|A-A_0,\Psi\|_{L^2_{1,A_0}}),\]
where $f_1$ is a real function and $(A,\Psi)\in\mcC_k(Z)$;
\item $\mathfrak q$ defines a $C^1$-section
$\mcC_1(M)\to\mathcal T_0$.
\end{enumerate}
We say $\mathfrak q$ is \emph{tame}, if it is $k$-\emph{tame} for all $k\geq2$.
\end{defi}
Using Theorem \ref{thm-coro5.1.8} and the same idea of \cite[Proposition 16.2.1]{KM}, we establish the following proposition.
\begin{prop}[c.f. {\cite[Proposition-16.2.1]{KM}}]\label{cmpt-prop}
For any $C>0$, there are only finitely many $[\alpha],~[\beta]$ and pathes $z$ with
$\mathcal E^{top}_q(z)\leq C$ and such that the space $\check{\mathcal M}^+_z([\alpha],[\beta])$ is non-empty. Furthermore, each $\check{\mathcal M}^+_z([\alpha],[\beta])$ is compact.
\end{prop}
\section{Compact of moduli space }
The purpose of this section is to give some preparation to construct the monopole Floer homologies in the next section.
\subsection{Gluing trajectories}
In this subsection we show the gluing theorem in gauge version.
We define the blow-up configuration space of $(M,F)$ as the space of triples, see \cite[Chapter 9]{KM}
\[\mathcal C^\sigma_k(M,F,\mfs)=\{(A,s,\phi)\big| (A,\phi)\in \mathcal C_k(X),~ s\in\mathbb R^{\geq0},~\|\phi\|_{L^2}=1\}.\]
We define the quotient space $\mathcal B^\sigma_k(M,F,\mfs)=\mathcal C^\sigma_k(M,F,\mfs,F,\mfs)/\mathcal G_{k+\frac12}$, which is a Hilbert manifold with boundary.
The basic Seiberg-Witten equations naturally extend to the equations
\[\begin{cases}
\Dirac_{b,A}\phi=0\\
\frac12\bar*F_{A^t}=s^2q(\phi).
\end{cases}\]
We define \[\tilde{\mathcal C}^\tau_k(Z,F,\mfs)\subset \mathcal A_k(Z,\mfs)\times L^2_k(I,\mathbb R)\times L^2_k(S^+) \]
to the subset consisting of triples $(A,s,\phi)$ with $\|\phi(t)\|_{L^2(M)}=1$ for each $t\in I$. We have an involution map $\mathfrak i: \tilde{\mathcal C}^\tau_k(Z,F,\mfs)\to \tilde{\mathcal C}^\tau_k(Z,F,\mfs)$ defined by $(A,s,\phi)\mapsto (A,-s,\phi)$.
Similarly, we define the $\tau$-module for the configuration space for $(Z,F)$, where $Z=I\times M$
\[{\mathcal C}^\tau_k(Z,F,\mfs)\subset \mathcal A_k(Z,\mfs)\times L^2_k(I,\mathbb R)\times L^2_k(S^+) \]
to the subset consisting of triples $(A,s,\phi)$ with
\[s(t)\geq0,~\|\phi(t)\|_{L^2(M)}=1,\]
for each $t\in I$.
There is a well-defined map
$$\pi: \mathcal C^\sigma_k(M,F,\mfs)\to \mathcal C_k(M,F,\mfs),~(A,s,\phi)\mapsto (A,s\phi).$$
From this map, we know that any vector field on $\mathcal C_k(M,F,\mfs)$ lifts to a vector field on $\mathcal C^\sigma_k(M,F,\mfs)$.
In order to get the
transversality condition, we need to add a perturbation $\mathfrak p$ as defined as \eqref{defin-perturbation} on $grad(\L)$. The sum $grad(\L)+\mathfrak p$ is a gauge invariant and gives rise to a vector field $v^\sigma_q$,
\[v^\sigma_q:\mathcal B^\sigma_{k+1/2}(M,F,\mfs)\to \mathcal T_{k-1/2}(M),\]
where $\mathcal T_{k-1/2}(M)$ denotes the $L^2_{k-/2}$-completion of the tangent bundle of $\mathcal B^\sigma_{k-1/2}(M,F,\mfs)$. $grad^\sigma(\L)$ is defined as follows
\[grad^\sigma(\L)(A,r,\psi)=(\frac12\bar*F_{A^t}-r^2q(\psi),\Lambda(A,r,\psi)r,\Dirac_A\psi-
\Lambda(A,r,\psi)\psi),\]
where $\Lambda(A,r,\psi)=\langle\psi,\Dirac_A\psi\rangle_{L^2}$.
A trajectory $\gamma(t)=(A(t),r(t),\psi(t))$ is a solution of the equation
\begin{equation}
\begin{cases}
\frac12\frac{d}{dt}A=-(\frac12\bar*F_A-r^2q(\psi)),\\
\frac{d}{dt}r=-\Lambda(A,r,\psi)r,\\
\frac{d}{dt}\psi=-(\Dirac_A\psi-
\Lambda(A,r,\psi)\psi)).
\end{cases}
\end{equation}
We call the perturbation $\mathfrak q$ admissible if all critical points of $v^\sigma_q$ are nondegenerate and moduli spaces of the flow lines connecting them are regular(See \cite[Defition 22.1.1]{KM}).
We categorize the set $C$ of critical points in $\mathcal B^\sigma_k(M,F,\mfs)$ into the disjoint union of three subsets:
\begin{itemize}
\item $C^o$, the set of irreducible points;
\item $C^s$, the set of reducible boundary stable(where the spinor part locates on the positive eigenspace part) critical points;
\item $C^u$, the set of reducible boundary unstable(where the spinor part locates on the negative eigenspace part) critical points.
\end{itemize}
We set
\[M([\mfa],[\mfb])=\bigcup_{z\in\pi_1([\mfa],[\mfb])}
M_z([\mfa],[\mfb])\]
as the union over the moduli spaces in a given relative homotopy class, where $\pi_1([\mfa],[\mfb])$ denotes the homotopy class of path connecting the two critical points in the quotient space. We recall two notions which are given in \cite[Page 261]{KM}.
\begin{defi}
We say a moduli space $M([\mfa],[\mfb])$ is \emph{boundary-obstructed}, if $[\mfa],~[\mfb]$ are reducible, $[\mfa]\in C^s$ and $[\mfb]\in C^u$.
\end{defi}
\begin{defi}
Let $[\gamma]\in M_z([\mfa],[\mfb])$. When the moduli space is not boundary-obstructed, we say that $\gamma$ is \emph{regular} if the linearized Seiberg-Witten map $Q_\gamma$ along $\gamma$ is surjective. In the boundary-obstructed case, we say $\gamma$ is \emph{regular}, if the linearized Seiberg-Witten map restriction along the boundary $Q^\partial_\gamma$ is surjective. We say that $ M_z([\mfa],[\mfb])$ is regular if all its elements are regular.
\end{defi}
We topologize the space of unparameterized broken trajectories as follows\cite[Page 276]{KM}.
We choose $[\breve\gamma]=([\breve\gamma_1],\cdots,[\breve\gamma_n])\in \breve M^+_z([\mfa],[\mfb])$, where $[\breve\gamma_i]\in \breve M_{z_i}([\mfa_{i-1}],[\mfa_i])$ is represented by a trajectory
\[[\gamma_i]\in M_{z_i}([\mfa_{i-1}],[\mfa_i]).\]
Let $U_i\in \mcB^\tau_{k,loc}(Z,F,\mfs)=\mathcal C^\tau_{k,loc}(Z,F,\mfs)/\mcG_{k+1,loc}$ be an open neighborhood of $[\gamma_i]$ and $T\in \mathbb R^+$. We define $\Omega=\Omega(U_1,\cdots,U_n,T)$ to be the subset of $\breve M_z([\mfa][\mfb])$ consisting of unparameterized broken trajectories
$\breve\delta=([\breve\delta_1],\cdots,[\breve\delta_m])$ satisfying the following condition: there exists a map
$$(\iota,s):\{1,\cdots, n\}\to \{1,\cdots,m\}\times \mathbb R$$
such that
\begin{itemize}
\item $[\tau_{s(i)}\delta_{\iota(i)}]\in U_i$,
\item if $1\leq i_1<i_2\leq n$, then either $\iota(i_1)<\iota(i_2)$ or $\iota(i_1)=\iota(i_2)$ and $s(i_1)+T\leq s(i_2)$.
\end{itemize}
To prove the compactness theorem for the blow-up model, we need to get a bound of $\Lambda$ of the moduli space.
For any $C>0$ and any $[\mfa],~[\mfb]$ with energy $\mathcal E^a_q\leq C$ for which $M^+([\mfa],[\mfb])$ is non-empty. Moreover, the space of broken trajectories $[\gamma]\in M^+([\mfa],[\mfb])$ with energy $\mathcal E_q\leq C$ is compact.
For a trajectory $\gamma^\tau\in M([\mfa],[\mfb])$, we define $K(\gamma^\tau)$ as the total variation of $\Lambda_{\mathfrak q}$ by(see \cite[Section 16.3]{KM})
\[K(\gamma^\tau)=\int_{\mathbb R}|\frac{d\Lambda_{\mathfrak q}(\gamma^\tau)}{dt}|dt.\]
We set
\[K_+(\gamma^\tau)=\int_{\mathbb R}\left(\frac{d\Lambda_{\mathfrak q}(\gamma^\tau)}{dt}\right)^+dt.\]
Proposition \ref{cmpt-prop} gives a bound on the number of components for which the blow-down is non-constant.
To get the energy bound, we need the proposition below.
\begin{prop}[c.f. {\cite[Proposition 16.1.4]{KM}}]\label{prop-16.1.4}
The space of broken trajectories $[\breve\gamma]\in\breve M^+([\mfa],[\mfb])$ with topology energy $\mathcal E_{\mathfrak q}(\breve\gamma)\leq C$ is compact.
\end{prop}
We set $Z^T=[-T,T]\times M$ and $Z^\infty=(\mathbb R^{\leq}\times M)\coprod(\mathbb R^{\geq}\times M)$. Let $\mfa$ be a critical point on $\mathcal C_{k}(M,F,\mfs)$, we write $\gamma_\mfa$ as a translation-invariant solution on $Z^T$ or $Z^\infty$ in temporal gauge.
We suppose that $\mfa=(A_0,\Phi_0)$ is non-degenerate by choosing a generic perturbation. We describe the quotient space
\[\mathcal B_k(Z^\infty,[\mfa])=\mathcal C_k(Z^\infty,[\mfa])/\mathcal G_{k+1}(Z^\infty),\]
where $C_k(Z^\infty,[\mfa])=\{\gamma\in\mathcal C_{k,loc}(Z^\infty,F,\mfs)\big|
\gamma-\gamma_\mfa\in L^2_{k,A_0}\}$ and $\mathcal G_{k+1}=\{u\in \Gamma_b(Z^\infty,S^1)\big| u\in L^2_{k+1,loc}~,1-u\in L^2_{k+1}\}$.
Let \[\mathcal K_{s,\mfa}(M)\]
be the $L^2_{s}$-completion of the complement $\mathcal K_{\mfa}$ to the gauge orbit, where $\mathcal K_{\mfa}$ is the orthogonal complement to the gauge-orbit. Similarly, we denote $\mathcal K^\sigma$
by the blow-up model of $\mathcal K$. Similar to \cite[Proposition 9.3.4]{KM}, we have the proposition below.
\begin{prop}\label{prop-slice}
Let $\mathcal J_{k,\gamma}$ be the image of $d_\gamma: L^2_{k+1,b}(M,i\mathbb R)\to \mathcal T_{k,\gamma}$, via
$\xi\mapsto (-d\xi,\xi \Phi_0)$, where $\mathcal T_{k,\gamma}$ denotes the tangent space at $\gamma$. As $\gamma$ varies over $\mathcal C^*_k(M)$, we define $\mathcal K_{k,\gamma}$ to be the subspace of $\mathcal T_{k,\gamma}$ which is orthogonal to $\mathcal J_{k,\gamma}$ with respect to the $L^2$-inner product. Then, we have the decomposition
\[\mathcal T_{k,\gamma}=\mathcal J_{k,\gamma}\oplus\mathcal K_{k,\gamma}.\]
\end{prop}
\begin{pf}
We denote by $\gamma=(A_0,\Phi_0)$. By doing integral by part, we define $\mathcal K_{k,\gamma}=\{(a,\phi)\big|
-\delta_ba+iRe\langle i\Phi_0,\phi\rangle=0\}$. It is clear that $\mathcal K_{k,\gamma}$ is orthogonal to $\mathcal J_{k,\gamma}$. We need to show that
\[\mathcal T_{k,\gamma}=\mathcal J_{k,\gamma}\oplus\mathcal K_{k,\gamma}.\]
This is sufficient to show that for any $(a,\phi)$, we have a unique solution for the equation
\[
\delta_b((a,\phi)+d_\gamma(\xi))=0,
\]
which is equivalent to
\[\Delta_b\xi+|\Phi_0|^2\xi=c,\]
where $c=G^*_\gamma(a,\phi)$.
Since $\Phi_0$ is non-zero, this equation has a unique solution by Theorem \ref{CN-thm}.
\end{pf}
Following the arguments of \cite[Proposition 9.3.5, 9.4.1]{KM}, we can show the similar decompositions for $\sigma$-model and $\tau$-model.
By doing the integral part and the taut condition, one has that the slice\[S_{k,\mfa}(Z^T)\subset \mathcal C_{k,b}(Z^T)\] can be represented by \[S_{k,\mfa}(Z^T)=\{(A_0+a,\Phi)\big| -\delta_ba+iRe\langle i\Phi_0,\Phi\rangle=0,~(a,\overrightarrow{n})\big|_{\partial Z^T}=0.\},\]where $\gamma_\mfa=(A_0,\Phi_0)$.
Similarly, we define the slice $\mathcal S^\sigma_{k,\mfa}$ and $\mathcal S^\tau_{k,\mfa}$(see \cite[Page 144, Page 147]{KM}).
We can run the arguments in \cite[Section 18.4]{KM} in foliation case.
Here we give a sketch.
The boundary of $Z^T$ is $\bar{M}\amalg M$, we have the restriction map\[r:\mathcal{\tilde C^\tau}_{k,b}(Z^T)\to \mathcal{\tilde C^\tau}_{k-\frac12,b}(\bar{M}\amalg M)\times L^2_{k-\frac12,b}(\bar{M}\amalg M,i\mathbb R),\]where the second component is defined as the normal component of the basic connection $A$ at the boundary and
$\bar M$ is a copy of $M$ with the reversing orientation by reversing the orientation of $Q$.
For the non-degeneracy of the Hessian operator \eqref{Hessian-operator} at $\mfa$, we have the decomposition $\mathcal K^\sigma_{k-\frac12}\big|_{\mfa}= \mathcal K^+\oplus \mathcal K^-$. Let $H^-_M$ and $H^-_{\bar M}$ be defined by\[ H^-_M=\{0\}\oplus \mathcal K^-\oplus L^2_{k-\frac12}(M,i\mathbb R),~H^-_{\bar M}=\{0\}\oplus \mathcal K^+\oplus L^2_{k-\frac12}(M,i\mathbb R).\]
We set $H=H^-_M\oplus H^-_{\bar M}$ and define $\Pi_M= \Pi^-_M\oplus \Pi^-_{\bar M}$ by the projection to the space $\mathcal K^-\oplus \mathcal K^+$, i.e.
\[\Pi_M:\mathcal T_{k-\frac12}\big|_{\mfa}(M\amalg \bar M)
\oplus L^2_{k-\frac12}(M\amalg\bar M,i\mathbb R)\to \mathcal K^-\oplus \mathcal K^+.\]
For $\gamma $ in a small neighborhood of $\gamma_\mfa\in \mathcal C^\tau_{k,b}(Z,F,\mfs)$ on $Z=Z^T$ or $Z^\infty$, we consider the equation
\[\begin{cases}
\mathfrak F_q(\gamma)=0,\\
\gamma\in S^\tau_{k,\mfa}(Z),\\
(\Pi_M\comp r)(\gamma)=h,
\end{cases}\]
where $h\in H$. We can write
the equations as
\[\begin{cases}
(Q_{\gamma_\mfa}+\alpha)\gamma=0,\\
(\Pi_M\comp r)\gamma=h,
\end{cases}\]
where $Q_{\gamma_\mfa}$ is defined as $D_{\gamma_\mfa}\mathfrak F_q$
and $\alpha$ denotes the remainder terms.
We write $Q_{\gamma_\mfa}=\frac{d}{dt}+L_b$, let $H^\pm_L$ be the spectral subspaces of $L_b$ in $L^2_{\frac12}(M)$.
By Proposition \ref{prop-17.2.7}, we get that the linear map
\begin{equation}
(Q_{\gamma_\mfa},\Pi^-_L\comp r) \label{Fredholm-formula}
\end{equation}
is an isomorphism, where $\Pi^-_L$ is the spectral projection with kernel $H^+_L$.
Let $K$ denote the kernel of $Q_{\gamma_\mfa}$, the domain can be decomposed as $C\oplus K$, we rewrite the above operator as
\[\left(\begin{array}{cc}
Q_{\gamma_\mfa}\big|_C&0\\
*&(\Pi^-_L\comp r)\big|_K\\
\end{array}\right).\]
The isomorphism of two components on diagonal implies that the matrix is an invertible operator. Thus, we have verified abstract hypothesis \cite[Hypothesis 18.3.1]{KM}.
By the definition of tame perturbation, the abstract hypothesis \cite[Hypothesis 18.3.3]{KM} follows. Setting $\mathcal K=\mathcal K^+\oplus \mathcal K^-$,
we get that there is an $\eta_1>0$ and the maps from the $B_{\eta_1}(\mathcal K)$ to the slices parameterizing the subsets of the set of solutions, i.e.
\[u(T,\cdot): B_{\eta_1}(\mathcal K)\to S^\tau_{k,\gamma_\mfa}(Z^T)\cap SW^{-1}_q(0),~
u(\infty,\cdot): B_{\eta_1}(\mathcal K)\to S^\tau_{k,\gamma_\mfa}(Z^\infty)\cap \mathfrak F^{-1}_q(0).\]
By the parallel arguments of the proof of \cite[Theorem 18.2.1]{KM}, one has the proposition below.
\begin{prop}
There is an $\eta_0$, such that all $\eta<\eta_0$, there is $\eta'$, independent of $T$, such that: \begin{enumerate}
\item the map \[\mu:\{\gamma\in S^\tau_{k,\gamma_\mfa}(Z^T)\big |\|\gamma-\gamma_\mfa\|_{L^2_k}\leq\eta\}\to \tilde{\mathcal B^\tau_k}(Z^T) \]is a diffeomorphism onto its image, where $\tilde{\mathcal B^\tau_k}(Z^T) $ denotes the quotient of $\tilde{\mathcal C^\tau_{k}}(Z^T)$ by the gauge action;
\item the image of the above map contains all gauge-equivalent classes $[\gamma]\in \tilde{\mathcal B^\tau_k}(Z^T) $ represented by the elements $\gamma$ satisfying \[\|\gamma-\gamma_\mfa\|_{L^2_k}\leq \eta'.\] \end{enumerate}\end{prop}
By the similar arguments, we have the following propositions for the foliated case.
\begin{prop}[c.f. {\cite[Proposition 17.2.7]{KM}}]\label{prop-17.2.7} Let $Z=(-\infty,0]\times M$ and $D_0:C^\infty(Z,E)\to L^2(Z,E)$ be a transverse elliptic operator of the form \[D_0=\frac{d}{dt}+L_0,\] where $L_0:C^\infty(M,E_0)\to C^\infty(M,E_0)$ is a transverse self-adjoint elliptic operator on $M$. Suppose that the spectrum of $L_0$ does not contain zero. Then, the operator \[D_0\oplus(\Pi_0\comp r):L^2_j(Z,E)\to L^2_{j-1}(Z,E_0)\oplus (H^-_0\cap L^2_{j-\frac12}(M,E_0))\] is an isomorphism for all $j\geq1$, where $\Pi_0$ denotes the projection to the negative eigen-vector part of $L_0$. Moreover, we have that $H^-_0\cap L^2_{j-\frac12}(M,E_0)=Im(r\big|_{\ker(D_0)})$. \end{prop}
The proof only needs the parametrix patching and regularity, we omit it here. Let $Z=I\times M$ be a closed finite cylinder, suppose that $I=I_1\cup I_2$ with $I_1\cap I_2=\{0\}$. We denote by $Z=Z_1\cup Z_2$, where $Z_1=I_1\times M$ and $Z_2=I_2\times M$. Let $D:C^\infty(Z,E)\to L^2(Z,E)$ be an elliptic operator of the form\[D=\frac{d}{dt}+L_0+h(t),\]where $L_0$ is a self-adjoint operator on $M$, $h: L^2_j(Z,E)\to L^2_{j}(Z,E)$ is a bounded operator. We set $D_1,~D_2$ being the restriction of these operators to the two subcylinders respectively, and\[H^i_{j-\frac12}\subset L^2_{j-\frac12}(\{0\}\times M,E_0)\]being the image of the $\ker(D_i)$ under the restriction map\[r_i: L^2_{j}(Z_i,E)\to L^2_{j-\frac12}(\{0\}\times M,E_0). \]Denoting by $D_0=-\frac{d}{dt}+L_0$.
We have the following lemma.
\begin{prop}[c.f. {\cite[Proposition 17.2.8]{KM}} ]\label{prop-surjective-on-subinterval}
Suppose that $D:L^2_j(Z)\to L^2_{j-1}(Z)$ is surjective for $2\leq j$. Then, we get the decomposition
\[L^2_{j-\frac12}(\{0\}\times M,E_0) = H^1_{j-\frac12}+H^2_{j-\frac12}. \]
Conversely, if the above formula holds and $D_1,~D_2$ are surjective, then $D$ is surjective.
\end{prop}
Let $Z$ be a finite cylinder.
We set $\tilde M(Z)=\{[\gamma]\in\tilde{\mathcal B^\tau_k}(Z)|~\mathfrak F^\tau_{\mathfrak q}(\gamma)=0 \}$.
We have the following theorem.
\begin{thm}[c.f. {\cite[Theorem 17.3.1]{KM}}]\label{thm-17.3.1}
The subspace $\tilde M(Z)\subset \tilde{\mathcal B^\tau_k}(Z)$ is a closed Hilbert submanifold.
The subset $M(Z)$ is a Hilbert submanifold with boundary: it can be
identified as the quotient of $\tilde M(Z)$ by the involution $\mathfrak i$.
\end{thm}
Let $[\gamma]\in\tilde M(Z)$ and let $\bar\mfa$ and $\mfa$ be the restrictions of $\gamma$ to the two boundary components. We have the restriction maps
\[R_+:\tilde M(Z)\to\mcB^\sigma_{k-1/2}(M),~R_{-}:\tilde M(Z)\to \mcB^\sigma_{k-1/2}(\bar M).\]
\begin{thm}[c.f. {\cite[Theorem 17.3.2]{KM}}]\label{thm-17.3.2}
Let $\gamma,~\mfa$ and $\bar\mfa$ be as above, and let $\Pi:\mathcal K^\sigma_{\bar\mfa}(\bar M)\oplus \mathcal K^\sigma_{\mfa}( M)\to \mathcal K^-_{\bar \mfa}(\bar M)\oplus \mathcal K^-_{\mfa}( M)$ be the projection with kernel $\mathcal K^+_{\bar \mfa,\mfa}(\bar M\amalg M)$. Then, the two composition maps
$\Pi\comp (DR_-,R_+)$ and $(1-\Pi)\comp (DR_-,R_+)$ are respectively Fredholm and compact, where $DR_-$ and $DR_+$ denote the derivatives of $R_-$ and $R_+$ respectively.
\end{thm}
We can prove the foliated version of \cite[Lemma 16.5.3, Proposition 16.5.2, Proposition 16.5.5]{KM}. By Lemma \ref{lemma-7.1.3}, Proposition \ref{prop-surjective-on-subinterval}, Theorem \ref{thm-17.3.1} and Theorem \ref{thm-17.3.2},
there is no difficulty to apply similar same arguments of \cite[Section 19.2,Section 19.3]{KM}.
Repeat the similar arguments of \cite[Section 19.1, Section 19.2, Section 19.3 and Section 19.4]{KM}, one establishes the following theorems.
\begin{thm}[c.f. {\cite[Theorem 19.5.4]{KM}}]\label{thm-19.5.4} Suppose that the moduli space $M_z([\mfa],[\mfb])$ is $d$-dimensional and contains irreducible trajectories, so that the moduli space $\breve M^+([\mfa],[\mfb])$ is a $(d-1)$-dimensional space stratified by manifolds(see \cite[Definition 16.5.1]{KM}). Let $M'\subset \breve M^+([\mfa],[\mfb])$ be any component of the codimension-1 stratum. Then along $M'$, the moduli space is either a $C^0$-manifold with boundary, or has a codimension-1 $\delta$-structure in the sense of \cite[Definition 19.5.3]{KM}. The latter
occurs only when $M'$ consists of 3-component broken trajectories, with the
middle component boundary-obstructed .
\end{thm}
\subsection{Finite result on moduli space}
In this subsection, we will give some properties for the compactified moduli space, which are necessary to construct the basic monopole Floer homologies without using Novikov ring.
We recall that a reducible critical point $\mfa$ corresponds to a pair $(\alpha,\lambda)$, where $\alpha=(B,0)=\pi(\mfa)$ is a critical point in
$\mathcal C_k(M,F,\mfs)$, and $\lambda$ is an element of the spectrum of
$\Dirac_{B,\mathfrak q}$. For such $\mfa$, we define(see \cite[Fromula 16.2]{KM})
\[\iota(\mfa)=\begin{cases}
|Spec(\Dirac_{B,\mathfrak q})\cap [0,\lambda)|,&\lambda>0,\\
1/2-|Spec(\Dirac_{B,\mathfrak q})\cap[0,\lambda]|,&\lambda<0.
\end{cases}\]
For the moduli space of reducible trajectories, we have simple structure. We denote by $M^{red}([\mfa],[\mfb])\subset M([\mfa],[\mfb])$ consisting of the
reducible trajectories. It is always a manifold without boundary.
For its dimension, we have the formula(see \cite[Formula 16.9]{KM})
\[\dim(M^{red}_z([\mfa],[\mfb]))=\bar{gr}_z([\mfa],[\mfb])=gr_z([\mfa],[\mfb])-
o[\mfa]+o[\mfb],\]
where $o[\mfa]=0$ when $[\mfa]\in C^s$ and $o[\mfa]=1$ when $[\mfa]\in C^u$.
For an irreducible critical point $\mfa$, we set $\iota(\mfa)=0$. If $[\mfa]$ and $[\mfa']$ are two critical points whose images under $\pi$ equals to the same critical point $[\alpha]\in\mathcal B_k(M)$, then
\[gr_{z_0}([\mfa],[\mfa'])=2(\iota(\mfa)-\iota(\mfa'))\]
for a trivial homotopy class $z_0$(see \cite[Formula 16.3]{KM}).
\begin{lemma}\label{lemma-16.4.4}
If all moduli spaces are regular and there is positive number $C_0>0$ such that \begin{equation}
\mathcal E^{top}_{\mathfrak q} (z_u)+C_0gr_{z_u}([\mfa],[\mfa])=
0,\label{qunatity-zero}
\end{equation}
where $z_u$ is the closed loop joins $\mfa$ to $u\mfa$ for any $u\in\mathcal G_b(M)$.
then there exists a constant $C$ such that for every $[\mfa],[\mfb]$ an each broken trajectory $[\breve\gamma]\in \breve M^+([\mfa],[\mfb])$, we have the energy bound
\[\mathcal E^{top}_{\mathfrak q}(\gamma)\leq C+C(\iota([\mfa])-\iota([\mfb])).\]
\end{lemma}
\begin{pf}
The idea is similar to \cite[Lemma 16.4.4]{KM}.
Let $[\breve \gamma]=([\breve\gamma_1],\cdots,[\breve \gamma_l])$ be a broken trajectory in $\breve M^+_z([\mfa],[\mfb])$ with $[\breve\gamma_i]\in \breve M_{z_i}([\mfa_{i-1}],[\mfa_i])$. The space $\breve M_{z_i}([\mfa_{i-1}],[\mfa_i])$ is non-empty, and it is manifold of dimension $1$, possibly with boundary.
Hence $\dim (\breve M_{z_i}([\mfa_{i-1}],[\mfa_i]))$ is either $gr_{z_i}([\mfa_{i-1}],[\mfa_i])-1$ or $gr_{z_i}([\mfa_{i-1}],[\mfa_i])$. In either case, $gr_{z_i}([\mfa_{i-1}],[\mfa_i])\geq0$. By adding the grading, we have $$gr_z([\mfa_0],[\mfa_l])\geq0.$$
The energy $\mathcal E^a_{\mathfrak q}(\gamma)$ is equal to $\mathcal E^{top}_{\mathfrak q}(z)$, defined as twice the change in $\L$ along any path $\tilde\zeta$ in $\mcC^\sigma(M,F,\mfs)$ whose image $\zeta$ in $\mcB^\sigma(M,F,\mfs)$ belongs to the class $z\in \pi_1(\mcB^\sigma(M,F,\mfs),[\mfa],[\mfb])$.
By the condition, we have that the quantity
$\mathcal E^{top}_{\mathfrak q} (w)+C_0gr_{w}([\mfa],[\mfa])$ depends only on $[\mfa]$ and $[\mfb]$ not on the homotopy class.
The quantity \begin{equation}
\mathcal E^{top}_{\mathfrak q} (w)+C_0(gr_{w}([\mfa],[\mfa])-2\iota(\mfa)+2\iota(\mfb))
\label{quantity-bound}
\end{equation}depends only on the critical points $[\alpha]=[\pi\mfa]$ and $[\beta]=[\pi\mfb]$. Since there are only finitely many critical points in $\mcB(M,F,\mfs)$, there is a constant $C$ such that this quantity is at most $C$, which proves the lemma.
\end{pf}
\noindent{\bf Remark}:
In particular when $b^1_b=0$, the above lemma automatically holds.
In the $3$-manifold case, i.e. $F=0$, Kroheimer and Mrowka consider the quantity
\[\mathcal E_{\mathfrak q}(z)+4\pi^2 gr_z([\mfa],[\mfb]),\]
where $z$ is a homotopy class connecting $[\mfa],[\mfb]$. When $b^1 >0$, the difference of the above quantity for two different homotopy classes is the class of a closed loop whose lift to the configuration space joins $\mfa$ to $u\mfa$, for some gauge action $u$. By Atiyah-Singer theory on closed oriented manfiold, the difference is zero. For the basic Dirac operator $\Dirac_A$, Br\"uning, F. W. Kamber, K. Richardson gave an expression for its index \cite{BKR}. They showed that
\[Ind(\Dirac)=\int_{\bar M_0/\bar F}A_{0,b}|\tilde{dx}|+\sum^r_{j=1}\beta(M_j),\]
\[\beta(M_j)=\frac12\sum_\tau\frac1{n_\tau rank(W^\tau)}(-\eta(D^{S^+,\tau}_j)+h(D^{S^+,\tau}_j))\int_{\bar M_j/\bar F}A^{\tau}_{j,b}(x)|\tilde{dx}|,\]
where the integrands $A_{0,b},~A^{\tau}_{j,b}(x)$ are similar to Atiyah-Singer integrands and notations are explained in their paper. Here we pose a question.
\noindent{\bf Question}: For $b^1_b>1$, under what topological condition, we have a constant $C_0$, such that for any non-degenerate critical point $\mfs$, we have $$C_0gr(\mfa,u\mfa)-\int_M[F_A]\wedge[u]\wedge\chi_F=0,$$ where $A$ is the connection component of $\mfa$ and $u\in \mathcal G_b(M)$.
By Lemma \ref{lemma-16.4.4},
we have the following proposition, which is analog to \cite[Proposition 16.4.1]{KM}.
\begin{prop}\label{prop-16.4.1}
Suppose all the moduli spaces $M_z([\mfa],[\mfb])$ are regular and \eqref{qunatity-zero} holds. Then, there are finitely many homotopy classes $z$ for which space $\check M^+([\mfa],[\mfb])$ is non-empty.
\end{prop}
\begin{prop}\label{prop-16.4.3}
Suppose \eqref{qunatity-zero} holds and the moduli space $M_z([\mfa],[\mfb])$ is regular. If $c_1(\mfs)=0\in H^2_b(M)$, then for a given $[\mfa]$ and $d\geq0$, there are finitely many pairs $([\mfb,]z$ for which the moduli space $\check M^+([\mfa],[\mfb])$ is non-empty and of dimension $d$. If $c_1(\mfs)\neq0\in H^2_b(M)$, then for a generic perturbation there are only finitely many
triples $([\mfa],[\mfb],z)$ for which the moduli space $\check M^+([\mfa],[\mfb])$ is non-empty.
\end{prop}
\begin{pf} The idea is no different to \cite[Proposition 16.4.3]{KM}, here we just show the
case $c_1(\mfs)=0$. The functional $\L$ descends to a well-defined function on
$\mcB^\sigma(M,F,\mfs)$, which is pulled back from $\mcB(M,F,\mfs)$.
Since the image of critical points in $\mcB(M,F,\mfs)$ is finite, $\L$ takes finitely many values, the energy $\mathcal E^{top}_q$ of a trajectory is the twice of the drop of $\L$, so there is
a uniform bound on the energy of all solutions. The expression \eqref{quantity-bound} depends on $[\pi\mfa],~[\pi\mfb]$, which takes only finitely many values. Assumed that the dimension is bounded and $[\mfa]$ is fixed, hence we have that $\iota([\mfb])$ is uniform bounded, which leaves finitely many choices.
\end{pf}
\section{Basic monopole Floer homologies on manifold with codimension $3$ foliation }
In this section, we show the main result of this paper, i.e. to construct the basic monopole Floer homologies.
\subsection{Basic Seiberg-Witten Floer homology for $b^1_b>1$}
In this subsection, we assume that $(M,F)$ is an oriented closed taut Riemannian foliation admitting a transverse \spinc structure and whose basic first deRham cohomology is nontrivial. $\mathcal B(M,F,\mfs)$ is not simply connected in general, which implies that the index of critical points
$$gr([\mfa],[\mfb])\in\mathbb Z$$
might not be
well defined. However we can still define relative gradings.
On the other hand, the components of the moduli space $\mathcal M([\mfa],[\mfb])$, trajectories connecting the
critical points mod the gauge action, might have different dimensions corresponding
to the different lifts of $[\mfa]$ and $[\mfb]$, where $[\mfa]$ and $[\mfb]$ are the gauge equivalence classes
of the critical points. Recall that
we decompose the space of trajectories
$\mcM([\mfa],[\mfb])=\bigcup_{z\in\pi_1([\mfa],[\mfb])}
\mcM_z([\mfa],[\mfb])$
as the union over the moduli spaces in a given relative homotopy class, where $\pi_1([\mfa],[\mfb])$ denotes the homotopy class of path connecting the two critical points in the quotient space.
For one critical point $[\mfa]\in\mathcal B(M,F,\mfs)$, we might have different lifts in $\mathcal C(M,F,\mfs)$, say $\mfa$ and $u\mfa$, we can measure their spectral by the following index,
\[gr(\mfa,u\mfa)=Ind(\Dirac^+_{u}),\]
where $Ind(\Dirac^+_{u})$ denotes the index of the basic Dirac operator on the product space $(M\times S^1,F)$.
\begin{prop}
The index $Ind(\Dirac^+_{u})$ defined above lifts to a homomorphism
\[Ind:\pi_0(\mathcal G)\to \mathbb Z.\]
\end{prop}
\begin{pf}
We need to show that for different critical points the index is unchanged, since it is clear to see that for the same homotopy class, the index is well-defined. For another critical point $\mfb$, the connection difference is a one-form, which is a compact operator. Hence the index lifts to a homomorphism.
\end{pf}
We define
\[d(\mathfrak s)=gcd(Ind:\pi_0(\mathcal G)\to \mathbb Z).\]
For two distinct irreducible critical points $\mfa$ and $\mfb$, we denote by $ \mcM^i([\mfa],[\mfb])$ the dimension $i$ component of ${\mathcal M}([\mfa],[\mfb])$. Let $\check{\mathcal M}([\mfa],[\mfb])$ be the unparameterized space of ${\mathcal M}([\mfa],[\mfb])$, i.e. $\check{\mathcal M}([\mfa],[\mfb])={\mathcal M}([\mfa],[\mfb])/\mathbb R$. At the irreducible critical points, the slice decomposition, i.e. Proposition \ref{prop-slice} holds.
By Theorem \ref{thm-19.5.4}, we have the following proposition.
\begin{prop}[c.f. {\cite[Corollary 3.1.24]{Wang}}]
Suppose that $[\mfa_0],~[\mfa_2]$ are two irreducible critical points with the relative index $gr([\mfa_2],[\mfa_0])=2\bmod d(\mfs)$. Then the boundary of $\breve\mcM^2([\mfa_0],[\mfa_2])$ consists of union
\[\bigcup_{[\mfa_1]\in Crit}\breve\mcM^1([\mfa_0],[\mfa_1])\times \breve\mcM^1([\mfa_1],[\mfa_2]),\]
where $\mfa_1$ runs over critical points with $gr([\mfa_1],[\mfa_0])=1\bmod d(\mfs)$ and $Crit$ denotes the set of irreducible critical points in $\mcB(M,F,\mfs)$.
\end{prop}
We define the relative Floer complex is generated by the irreducible critical points of Chern-Simon-Dirac functional with grading given by the relative indices $\mathbb Z_{d(\mfs)}$ or $\mathbb Z$
\[C(M)=\bigoplus_{\mfa\in Crit}\mathbb Z_2\mfa.\]
The boundary operator of the complex is defined by
\[\partial :C(M)\to C(M),~\partial([\mfa])=\sum_{[\mfb]}\sharp\breve\mcM^1([\mfa],[\mfb]),\]
where $\sharp\breve\mcM^1([\mfa],[\mfb])\in \mathbb Z_2$ denotes the signed number of points in $\ \breve\mcM^1([\mfa],[\mfb])$ mod $2$.
\begin{lemma} \label{lemma-partial^2}
$\partial^2=0$.
\end{lemma}
\begin{pf}
By definition we have that
$\partial^2 ([\mfa])=\sum_{[\mfb],} \sharp\breve\mcM^1([\mfa],[\mfb])\sharp\breve\mcM^1([\mfb],[\mfc])([\mfc])$, where $[\mfb]$ runs over the irreducible critical points with relative index $1$. By the above proposition, it is known that each term $\sharp\breve\mcM^1([\mfa],[\mfb])\sharp\breve\mcM^1([\mfb],[\mfc])([\mfc])$ is the sum of the number of oriented boundary points of a compact 1-dimensional manifold, which is zero.
\end{pf}
We define basic Seiberg-Witten Floer homology as $HF(M,F,\mfs,\eta,g)=\ker(\partial)/Im(\partial)$, which is $\mathbb Z_{d(\mfs)}$-relative grading(or $\mathbb Z$-grading).
\begin{prop}\label{prop-independent}
Suppose that $(M,F)$ satisfies Assumption \ref{assum-main-1}. Then, for $b^1_b>1$, we have that the relative Floer homology is independent of the taut bundle-like metric and perturbation. We denote the basic Seiberg-Witten Floer homology group by $HF(M,F,\mfs)$.
\end{prop}
\begin{pf}
For a bundle-like metric $g$, it corresponds a triple
\[g~\leftrightarrow~(g_F,g_Q,s),\]
where $g_F$ is the leafwise restriction, $s$ corresponds to the decomposition
\[s: Q\to TM,~\pi_Q\comp s=Id_Q\]
and $g_Q$ is the transverse restriction.
It is clear to see that the domain $\mathcal A_b\times \Gamma_b(M,S)$ and the Seiberg-Witten equations \eqref{eqn-SW-equation-1} are independent of the leafwise metric $g_F$ and the decomposition $s$. For two distinct leafwise metrics $g_F$ and $g'_F$ with the same $(p,l)$, the two Sobolev spaces $L^p_l$ and $L'^p_l$ are mutually equivalent to each other. Therefore, we have that the basic Seiberg-Witten Floer homology group is invariant under the leafwise metric. For two distinct decomposition $s$ and $s'$, we can apply the same argument, as the character form $\chi_F$ only depends on the leafwise metric $g_F$ and the decomposition $s$.
The remaining part is to verify that the Floer homology group is independent of the generic basic perturbation and metric $g_Q$. The idea is exactly the same, which was originally posed by Floer \cite{Floer}.
\end{pf}
\subsection{Basic monopole Floer homologies for $b^1_b=0$}
The purpose of this subsection is to construct the basic monopole Floer homologies and show that they are independent of the perturbation and taut bundle-like metric in a special case $b^1_b=0$ .
We define
the basic monopole Floer homologies $\overline{HM}(M,F,\mfs;\mathbb F)$, $\widecheck{HM}(M,F,\mfs;\mathbb F)$ and $\widehat{HM}(M,F,\mfs;\mathbb F)$ as the homologies of the chain complexes freely generated by $\bar C=C^s\cup C^u,~\check{C}=C^o\cup C^s,~\hat{C}=C^o\cup C^u$ respectively(see \cite[Section 22]{KM}), where $\mathbb F=\mathbb Z_2$. . The differentials on them are given in components as
\[\bar\partial =\left(\begin{array}{cc}
\bar\partial^s_s&\bar\partial^u_s\\
\bar\partial^s_u& \bar\partial^u_u\\
\end{array}\right),~
\breve\partial =\left(\begin{array}{cc}
\partial^o_o&\partial^u_o\bar\partial^s_u\\
\partial^o_s& \bar\partial^s_s+\partial^u_s\bar\partial^s_u\\
\end{array}\right),~
\hat\partial =\left(\begin{array}{cc}
\partial^o_o&\partial^u_o \\
\bar\partial^s_u\partial^o_s& \bar\partial^u_u+\bar\partial^s_u\partial^u_s\\
\end{array}\right).\]
The linear maps
\[\partial^o_o:C^o\to C^o,~\partial^o_s:C^o\to C^s,\]
\[\partial^u_o:C^u\to C^o,~\partial^u_s:C^u\to C^s\]
are defined by the formula
\[\partial^o_o[\mfa]=\sum_{[\mfb]\in C^o}\sharp \breve\mcM([\mfa],[\mfb])[\mfb],~[\mfa]\in C^o,\]
where $\sharp \breve\mcM([\mfa],[\mfb])\in \mathbb F$ is the signed counting number, the other three are defined similarly. By considering the number $\sharp \breve\mcM^{red}([\mfa],[\mfb])$, we similarly define the linear maps
\[\bar\partial^s_s:C^s\to C^s,~\bar\partial^s_u:C^s\to C^u,\]
\[\bar\partial^u_s:C^u\to C^s,~\bar\partial^u_u:C^u\to C^u.\]
When $b^1_b=0$, it is clear that
for given $[\mfa]$, there are finitely many pairs $([\mfb],z)$ such that the moduli space $ M_z([\mfa],[\mfb])$ is non-empty and of dimension $1$.
\begin{prop}[c.f. {\cite[Proposition 22.1.4]{KM}}]
\[\bar\partial^2=0,~\breve\partial^2=0,~\hat\partial^2=0.\]
\end{prop}
\begin{pf}
The proof is by showing that $\bar\partial^2=0$, which is the same as the blow-down case(Lemma \ref{lemma-partial^2}), and following identities
\begin{enumerate}
\item $\partial^o_o\partial^o_o+\partial^u_o\bar\partial^s_u \partial^o_s=0$;
\item $\partial^o_s\partial^o_o+\bar\partial^s_s\partial^0_s+
\partial^u_s\bar\partial^s_u\partial^o_u=0$;
\item $\partial^o_o\partial^u_o+\partial^u_o\bar\partial^u_u+
\partial^u_o\bar\partial^s_u\partial^u_s=0$;
\item $\bar\partial^u_s+\partial^o_s\partial^u_o+\bar\partial^s_s\partial^u_s
+\partial^u_s\bar\partial^u_u+\partial^u_s\bar\partial^s_u\partial^u_s=0$.
\end{enumerate}
Each of the four formulas is proved by considering a moduli space $\breve M_z([\mfa],[\mfb])$ of dimension $1$. By Theorem \ref{thm-19.5.4}, we can run the similar arguments as the proof of \cite[Proposition 22.1.4]{KM}.
\end{pf}
We give a grading for these homologies.
Let $\mathcal P$ be the space of the perturbations. We define $\mathbb J$ by the quotient of $\mathcal B^\sigma(M,F,\mfs)\times\mathcal P\times \mathbb Z/\sim$, where the equivalent relation $\sim$ is defined as follows(see \cite[Section 22.3]{KM}):
for any two elements $([\mfa],\mathfrak q_1,m),~([\mfb],\mathfrak q_2,n)\in \mathcal B^\sigma(M,F,\mfs)\times\mathcal P\times \mathbb Z$, let $\zeta$ be a path joining $[\mfa]$ and $[\mfb]$ and $\mathfrak p$ be a path of perturbation joining $\mathfrak q_1$ and $\mathfrak q_2$. We have a Fredholm operator $P_{\zeta,\mathfrak p}$ as defined on \eqref{Fredholm-formula}, we say that $([\mfa],\mathfrak q_1,m)\sim ([\mfb],\mathfrak q_2,n)$, if there is a path $\zeta$ such that
\[Ind(P_{\zeta,\mathfrak p})=n-m.\]
The map $([\mfa],\mathfrak q,m)\mapsto ([\mfa],\mathfrak q,m+1)$ descends to $\mathbb J$, and raises to an action of $\mathbb Z$.
Note that the above construction of the index set $\mathbb J$ is also available when $b^1_b>0$. Let $\mathfrak q$ be a fixed admissible perturbation, for a critical point $[\mfa]$, we define
\[gr([\mfa])=([\mfa],\mathfrak q,0)/\sim\in \mathbb J.\]
For reducible critical points, we define the modified grading by
\[\bar{gr}([\mfa])=\begin{cases}
gr([\mfa])&[\mfa]\in C^s\\
gr([\mfb])-1&[\mfa]\in C^u.
\end{cases}\]
We show that the invariance of the basic monopole Floer homologies under the perturbation and the bundle-like metric.
Let $W=[0,1]\times M$ and $W^*=(-\infty,0]\times M\cup W\cup [1,\infty)\times M$. To tell the distinguish, we denote $Y_-$ to be the left boundary
$\{0\}\times M$ with metric and perturbation and $Y_+$ to be the right boundary $\{1\}\times M$ with another metric and perturbation. We consider the moduli space $M([\mfa],W^*,[\mfb])$, as defined in \cite[Section 25]{KM}. Using broken trajectories, we denote its compactification by $M^+([\mfa],W^*,[\mfb])$.
Fix a positive integer $d_0$, we consider a pair $([\mfa],[\mfb])$ for which the moduli space $M([\mfa],W^*,[\mfb])$ or $M^{red}([\mfa],W^*,[\mfb])$ has dimension $d_0$ at most. To prove the independence of the metric, we need to define the maps induced by the trivial cobordism, which are given by
counting the number of solutions in zero dimensional moduli spaces.
We define linear operators,
\[m^o_o:
C^o_*(Y_-)\to C^o_*(Y_+),~m^o_s:
C^o_*(Y_-)\to C^s_*(Y_+)\]
\[m^u_o:
C^u_*(Y_-)\to C^o_*(Y_+),~m^u_s:
C^u_*(Y_-)\to C^s_*(Y_+)\]
by
\[m^o_o(-)=\sum_{[\mfa]\in C^o(Y_-)}\sum_{[\mfb]\in C^o(Y_+)}
\sharp M([\mfa],W^*,[\mfb]),\]
for the first one and similarly for the others. We similarly define operators on the reducible part of the Floer complexes: we have an operator
\[\bar m:
\bar C_*(Y_-)\to \bar C_*(Y_+)\]
\[\bar m=\left(\begin{array}{cc}
\bar m^s_s&\bar m^u_s\\
\bar m^s_u&\bar m^u_u\\
\end{array}\right)\]
where $\bar m^s_s(-)=\sum_{[\mfa]\in C^s(Y_-)}\sum_{[\mfb]\in C^s(Y_+)}
\sharp M([\mfa],W^*,[\mfb]) $, and the others are defined similarly. On $\check{C}_*$, we define
\[\check m:
\check{C}_*(Y_-)
\to \check{C}_*(Y_+)\]
by the formula
\[\check m=\left(\begin{array}{cc}
m^o_o&m^u_o\bar\partial^s_u(Y_-)+\partial^u_o(Y_+)\bar m^s_u\\
m^o_s&\bar m^s_s+m^u_s\bar\partial^s_u(Y_-)+\partial^u_s(Y_+)\bar m^s_u\\
\end{array}\right),\]
where $\partial^u_o(Y_\pm)$, for example, denotes the operator $\partial^u_o$ on $Y_\pm$. On $\hat C_*$, we define
\[\hat m:
\hat C_*(Y_-)\to \hat C_*(Y_+)\]
by the formula
\[\hat m=\left(\begin{array}{cc}
m^o_o&m^u_o\\
\bar m^s_u\partial^o_s(Y_-)+\bar\partial^s_u(Y_+)m^o_s&
\bar m^u_s+\bar m^s_u\partial^u_s(Y_-)
\end{array}\right).\]
By considering the zero-dimension moduli space, we have the following proposition.
\begin{prop}\label{prop-cobordism}
The operators $\breve m,~\hat m$ and $\bar m$ satisfy the identities:
\[\begin{cases}
\breve\partial(Y_+)\breve m_{-+}=\breve m_{-+}( \partial(Y_-)),\\
\hat\partial(Y_+)\hat m_{-+}=\hat m_{-+}( \bar\partial(Y_-)),\\
\bar\partial(Y_+)\bar m_{-+}=\bar m_{-+}( \bar \partial(Y_-)).
\end{cases}\]
In particular, we give rise to the operators
\[\begin{cases}
\breve m_{-+} : \widecheck{HM}(Y_-)\to \widecheck{HM}(Y_+)\\
\hat m_{-+} : \widehat{HM}(Y_-)\to \widehat{HM}(Y_+)\\
\bar m_{-+}:\overline{HM}(Y_-)\to \overline{HM}(Y_+).
\end{cases}\]
Moreover, the above operators only depend on the data of $Y_-$ and $Y_+$.
\end{prop}
Note that since we focus on the zero dimension part, the proof is much easier than {\cite[Proposition 25.3.8]{KM}}.
The last step is to prove the composition law. Let $Y_-$, $Y_0$ and $Y_+$ be the same $(M,F)$ with three metrics and basic perturbations, let $W_1$ be the cobordism from $Y_-$ to $Y_0$ such that near each collar, the metric of $W_1$ is the product metric and $W_2$ be the cobordism from $Y_0$ to $Y_+$ with the same condition on the metric.
Repeat the same argument as in \cite[Section 26.1]{KM}. We have the composition law below for the cobordisms.
\begin{prop}[c.f. { \cite[Proposition 26.1.2]{KM}}]
Let $(M,F)$ be the manifold with foliation satisfying Assumption \ref{assum-main-1}. Fix a transverse \spinc structure.
Let $Y_-,~Y_0,~Y_+$ be three data of bundle-like metrics and basic perturbations, and let $W_{-0}$ be the cobordism from $Y_-$ to $Y_0$, $W_{0+}$ be the cobordism from $Y_0$ to $Y_+$ and $W_{-+}$ be the composition of the $W_{-0}$ and $W_{0+}$. Suppose that $m_{-0}$, $m_{0+}$ and $m_{-+}$
are the operators in Proposition \ref{prop-cobordism}. Then we have that
\[m_{0+}\comp m_{-0}=m_{-+}.\]
\end{prop}
The above proposition implies the corollary below.
\begin{cor}
The monopole Floer homologies are independent of the generic choice of the perturbation and the bundle-like metric, which are denoted by $\widecheck{HM}_*(M,F,\mfs;\mathbb F)$, $\widehat{HM}_*(M,F,\mfs;\mathbb F)$ and $\overline{HM}_*(M,F,\mfs;\mathbb F)$.
\end{cor}
\begin{prop}[c.f. {\cite[Proposition 22.2.1]{KM}}]
Let $(M,F)$ satisfy Assumption \ref{assum-main-1}, there is an exact sequence
\[...\overline{HM}_*(M,F,\mfs;\mathbb F)\overset{i_*}\to
\widecheck{HM}_*(M,F,\mfs;\mathbb F)\overset{j_*}\to \widehat{HM}_*(M,F,\mfs;\mathbb F)\overset{p_*}\to \overline{HM}_*(M,F,\mfs;\mathbb F)
\overset{i_*}\to...\]
in which the maps $i_*,~j_*$ and $p_*$ arise from the chain-maps
\[i:\bar C\to \breve{C},~j:\breve{C}\to \hat{C},~
p:\hat{C}\to \breve{C},\]
which are defined by
\[i=\left(\begin{array}{cc}
0&-\partial^u_o\\
1&-\partial^u_s\\
\end{array}\right),~
j=\left(\begin{array}{cc}
1&0\\
0&-\bar\partial^s_u\\
\end{array}\right),~
p=\left(\begin{array}{cc}
\partial^o_s&\partial^u_s\\
0&1\\
\end{array}\right).\]
Here $i$ and $j$ are genuine chain maps, however $p$ is a anti chain map, i.e. $p\hat\partial+\hat\partial p=0$.
\end{prop}
We review the completion of graded group, c.f. \cite[Defintion 3.1.3]{KM}.
Let $G_*$ be an abelian group graded by the set $\mathbb J$ equipped with a $\mathbb Z$-action. Let $O_a(a\in A)$ be the set of free $\mathbb Z$-orbits in $\mathbb J$ and fix an element $j_a\in O_a$ for each $a$. Consider the subgroups
\[G_*[n]=\bigoplus_a\bigoplus_{m\geq n} G_{j_a-m},\]
which form a decreasing filtration of $G_*$. We define the negative completion of $G_*$ as the topological group $G_\bullet\supset G_*$ obtained by completing with respect to this filtration. We define the negative completions
\[\widecheck{HM}_\bullet(M,F,\mfs;\mathbb F),~\widehat{HM}_\bullet(M,F,\mfs;\mathbb F),~\overline{HM}_\bullet(M,F,\mfs;\mathbb F),\]
of the basic monopole Floer homologies defined as the above. If we want to consider all transverse \spinc
structures at the same time, we need to consider the completed basic monopole Floer homology
\[\widecheck{HM}_\bullet(M,F;\mathbb F)=\bigoplus_\mfs\widecheck{HM}_\bullet(M,F,\mfs;\mathbb F).\]
We make similar definitions for $\widehat{HM}_\bullet(M,F;\mathbb F)$ and $ \overline{HM}_\bullet(M,F;\mathbb F)$.
By the previous argument, we construct the basic monopole Floer homology groups and prove that these homology groups are independent of the basic perturbation and bundle-like metric.
\subsection{Basic monopole Floer homologies for $b^1_b>0$}
In this subsection, we construct the basic monopole Floer homologies
for $b^1_b>0$ with Novikov ring.
We recall the notion of local system $\Gamma$ over a topological space $X$.
\begin{defi}
A local system on a topological space $X$, is a system to distribute abelian groups $\{\Gamma_a\}$ for each point $a\in X$, such that for each relative homotopy class of paths $z$ from $a$ to $b$, we have an isomorphism
\[\Gamma(z):\Gamma_a\to \Gamma_b\] satisfying the composition law for composite paths.
\end{defi}
We review the classical result of \cite[Section 22, 29, 30]{KM},
we can take $\Gamma$ to be a local system of abelian groups on $\mcB^\sigma_b(M,F,\mfs)$, such that to each point $[\mfa]\in \mcB^\sigma_b(M,F,\mfs)$ there is an associated group $\Gamma[\mfa]$ and to each homotopy class $z$ of the paths from $[\mfa]$ to $[\mfb]$, there is an associated isomorphism $\Gamma(z):\Gamma[\mfa]\to\Gamma[\mfb]$.
By using the above local system $\Gamma$, the boundary maps
are well-defined. For instance, we consider
\[C^o(Y,\mfs,c,\Gamma)=\bigoplus_{[\mfa]}
\Gamma[\mfa], \] where $[\mfa]$ denotes the irreducible critical point.
We define the partial
\[\partial^o_o =\sum_{[\mfa]}\sum_{[\mfb]}\sum_z\sum_{[\gamma]}\in \check M_z([\mfa],[\mfb])\otimes \Gamma(z),\]
where the sum is over all the moduli space $M_z([\mfa],[\mfb])$ with dimension $1$ and $[\mfb]$ denotes the irreducible critical point. The contribution for a given pair of critical points takes the form
\[\sum_zn_z\Gamma(z).\]
Before proceeding, we review some definitions and notions which are given in \cite[Section 30]{KM}.
\begin{defi}[c.f. {\cite[Definition 30.2.1]{KM}}]
Let $\mathcal E^{top}_{\mathfrak q}$ be a corresponding perturbation of the topological energy. A subset $S\subset \pi_1([\mfa],[\mfb])$ is called $c$-finite, where $\pi_1([\mfa],[\mfb])$ denotes the homotopy classes of paths joining $[\mfa]$ and $[\mfb]$ in $\mcB^\sigma(M,F,\mfs)$, if the following conditions are satisfied:
\begin{itemize}
\item for all $C$, $S \cap \{z|\mathcal E^{top}_{\mathfrak q}(z)\leq C\}$ is finite;
\item there exists $d\geq0$ such that $|gr_z([\mfa],[\mfb])|\leq d$ for all $z\in S$.
\end{itemize}
\end{defi}
We consider a local system of complete topological abelian groups $\Gamma$ on $\mcB^\sigma(M,F,\mfs)$, i.e. each $\Gamma[\mfa]$ is a complete topological group and the homomorphism $\Gamma(z):\Gamma[\mfa]\to\Gamma[\mfb]$ is continuous. Assume that $0\in\Gamma[\mfa]$ has a neighborhood basis consisting of subgroups, so that $\Gamma[\mfa]$ is a complete filtered group, filtered by the open subgroups. Let $Hom(\Gamma[\mfa],\Gamma[\mfb])$ be the group of continuous homomorphisms, equipped with the compact-open topology. A neighborhood basis for $0$ in $Hom(\Gamma[\mfa],\Gamma[\mfb])$ consists of subgroups
\[\Omega(N,V)=\{k:\Gamma[\mfa]\to\Gamma[\mfb]|k(N)\subset V\}, \]
where $N$ runs over all precompact subsets of $\Gamma[\mfa]$ and $V$ runs all open subgroups of $\Gamma[\mfb]$. Note that a subset $N\subset \Gamma[\mfa]$ is precompact if and only if $(N+U)/U$ is finite for all open subgroups $U$ of $\Gamma[\mfa]$.
\begin{defi}
A countable series $\sum_{k\in K}k$ of $Hom(\Gamma[\mfa],\Gamma[\mfb])$ is said to be equicontinuous, if for each open subgroup $U$ $\subset\Gamma[\mfb]$, there exists an open subgroup $V$ such that $k(V)\subset U$ for each $k\in K$.
\end{defi}
\begin{defi}[c.f. {\cite[Definition 30.2.2]{KM}}]
A local system of complete filtered abelian groups $\Gamma$ is called $c$-complete, if it satisfies the following properties for each $[\mfa],~[\mfb]$:
\begin{itemize}
\item for any $c$-finite set $S\subset \pi_1(\mcB^\sigma,[\mfa],[\mfb])$, the set $\{\Gamma(z)|z\in S\}\subset Hom(\Gamma[\mfa],\Gamma[\mfb])$ is equicontinuous;
\item for any $c$-finite set $S\subset \pi_1(\mcB^\sigma,[\mfa],[\mfb])$, $\Gamma(z)$ converges to zero as $z$ runs through $S$ in the compact-open topology.
\end{itemize}
\end{defi}
We set the support as
\[supp(n)=\{z| ~n_z\neq0\}.\]
By the definition of c-complete, we have that
\[supp(n)\cap \{z|\mathcal E^{top}_{\omega,\mathfrak q}(z)\leq C\}\] is finite. Using the completeness of the local system, we have that the form
\[\sum_zn_z\Gamma(z)\] is convergent. Similarly, one can verify that the maps $\check\partial$, $\hat\partial$ and $\bar\partial$ are well-defined.
Combining with the equicontinuous property of the local system $\Gamma$, the proofs of $\check\partial^2$, $\hat\partial^2$ and $\bar\partial^2$ go through as the non-exact perturbation of 3 manifold case(see \cite[Section 30.2]{KM}).
\noindent We give an example of such a local system, e.g. Novikov ring \cite{Novikov}. We have a homomorphism
\[\mathcal E^{top}:\pi_1(\mcB (M,F,\mfs))\to \mathbb R,~z\mapsto \mathcal E^{top}(z),\] where $\mathcal E^{top}(z)$ denotes the difference of the Chern-Simons-Dirac functional between the different representatives of the quotient point. Since $\pi_1(\mcB^\sigma(M,F,\mfs))\cong \mathbb Z^{b^1_b}$, we can choose a basis $\{z_i\}_{1\leq i \leq b^1_b}$, such that each element $z$ can be written as
$z=k_1z_1+\cdots k_{b^1_b}z_{b^1_b},$
where $k_i\in\mathbb Z$ for $i=1,\cdots, b^1_b$ and $\mathcal E^{top}(z_i)\geq 0$. Moreover, we may assume that for $1\leq i\leq l$, we have that $\mathcal E^{top}(z_i)>0$. It is not hard to see that such a basis $\{z_i\}_{1\leq i \leq l}$ is independent of the metric $g$.
Choosing a commutative ring $R$(e.g. $\mathbb Z_2$), we define $\mathbb F[t,,t^{-1}]$ by
\[\mathbb F[t,t^{-1}]=\{\sum _{-k\leq i\leq K} r_it^i|\mbox{ only for finitely many }i,~r_i\ne0\}.\]For $k\in\mathbb Z$, let $U_{-k}$ be the $\mathbb F$-module spanned by the generators $t^i$, satisfying $i\leq -k$. Using these as open neighborhoods of $0$, we form the completion $\bar R[I]$, i.e. each element is of the form $$\sum^C_{i= -\infty} r_it^i.$$ We define a local system by taking at $[\mfa_0]$ to be $\bar {\mathbb F}[t,t^{-1}]$, and specifying that for each closed loop $z$ based at $[\mfa_0]$, the automorphism $\Gamma(z)$ be the multiplication by $t^{-(k_1+\cdots k_{l})}$ for $z=k_1z_1+\cdots k_lz_l+\cdots k_{b^1_b}z_{b^1_b}$.
Repeat the parallel arguments of previous subsection, together with Proposition \ref{prop-16.4.1} and Proposition \ref{prop-16.4.3}, we have that:
\begin{thm}
Let $(M,F)$ satisfy Assumption \ref{assum-main-1}. For a complete local system $\Gamma$, e.g. Novikov ring, we can construct the basic monopole Floer homologies. Moreover,
The monopole Floer homologies are independent of the generic choice of the perturbation and the bundle-like metric, which are denoted by $\widecheck{HM}_*(M,F,\mfs;\Gamma)$, $\widehat{HM}_*(M,F,\mfs;\Gamma)$ and $\overline{HM}_*(M,F,\mfs;\Gamma)$. Moreover, if \eqref{qunatity-zero} holds, then for any local system, we have the well-defined basic monopole Floer homologies.
\end{thm}
In general, we consider the (non-exact)perturbed basic Chern-Simons-Dirac functional defined as below:
given a class $c\in H^2_b(M)$, we write
\[\L_\omega(A,\Psi)=\L(A,\Psi)-\frac12\int_M(A^t-A^t_0)\wedge\omega\wedge\chi_F,\]
where $\omega\in \frac{2\pi}ic$.
It is known that a critical point $(A,s,\psi)$ in the blow-up model $\mcC^\sigma(M,F,\mfs)$ is defined by
\begin{equation}
\begin{cases}
\frac12\bar*(F_{A^t}-\omega)=s^2q(\psi),\\
\Dirac_A\psi=0.
\end{cases}\label{eqn-non-exact-SW-29.2}
\end{equation}
The corresponding perturbed equations for $(A,s,\phi)\in\mcC^\tau(\mathbb R\times M)$ are defined by
\begin{equation}
\begin{cases}
\frac12(F^+_{A^t}-\omega^+)=s^2q(\phi),\\
\frac{d}{dt}s+\Lambda(A,s,\phi)s=0,\\
\Dirac^+_A\phi-\Lambda(A,s,\phi)\phi=0.
\end{cases}\label{eqn-non-exact-SW-29.3}
\end{equation}
Applying the classical argument of the manifold case, we have the following lemma.
With the non-exact perturbation, the space of broken trajectories space $\check M^+_z([\mfa],[\mfb])$ can be defined as the manner of exact perturbation.
Following the same strategy of the construction in Section 6(or see \cite[Section 20-Section 26]{KM}),
we have the following theorem.
\begin{thm}
Let $\Gamma$ be a complete local system, e.g. a Novikov ring, and $\L_\omega$ be the non-exact perturbation for the Chern-Simons-Dirac functional defined as above. Then we have the basic monopole Floer homologies
\[\widecheck{HM}_*(M,F,\mfs,c;\Gamma), ~ \widehat{HM}_*(M,F,\mfs ,c;\Gamma),~ \overline{HM}_*(M,F,\mfs,c;\Gamma),\]
where $c\in H^2_b(M)$. These homologies depend only on the isomorphism class of the \spinc structure $\mfs$, $c$ and $(M,F)$, however they are independent of the metric or the perturbation.
\end{thm}
At the end of this subsection,
we give a necessary condition to avoid the complete local system or Novikov ring.
\begin{thm}
Let $(M,F,\mfs,c)$ be as above. Let $g$ be a bundle like metric and $\chi_F$ be the character form of the foliation. Suppose that there is constant $t$ such that the identity holds
\[-\int_M(c_1(\mfs)-c)\wedge[u]\wedge\chi_F+t\cdot gr(\mfa,u\mfa)=0,\]
for a non-degenerate critical point. Then, with any local coefficient $\Gamma$ we have the basic monopole Floer homologies
\[\overline{HM}_*(M,F,\mfs,c;\Gamma),~\widehat{HM}_*(M,F,\mfs,c;\Gamma),~
\widecheck{HM}_*(M,F,\mfs,c;\Gamma).\]
\end{thm}
\begin{pf}
Here we give a sketch of the proof. The idea is to show that $\sum_zn_z\Gamma(z)$ is of finitely many sum, for each $z\in M_z([\mfa],[\mfa])$, where $M_z([\mfa],[\mfb])$ is a moduli space of dimension $1$ and $[\mfa],~[\mfb]$ are two regular critical points. It is sufficient to show a foliated version of Kronheimer Mrowka's proposition \cite[Proposition 29.2.1]{KM}, which is stated as below.
\end{pf}
\begin{prop}\label{prop-29.2.1}
Let $(M,F,\mfs,c)$ be as above, let $g$ be a bundle like metric and $\chi_F$ be the character form of the foliation. Suppose that there is constant $t$ such that the identity holds
\[-\int_M(c_1(\mfs)-c)\wedge[u]\wedge\chi_F+t\cdot gr(\mfa,u\mfa)=0,\]
for a non-degenerate critical point. Then, we have the following:
\begin{enumerate}
\item When $t\leq0$, then for a given $[\mfa]$ and a non-negative integer $d_0$, there are only finitely many pairs $([\mfb],z)$ for which the moduli space $\check{M}^+_z([\mfa],[\mfb])$ is non-empty and of dimension at most $d_0$.
\item When $t>0$, then for a given $[\mfa]$, there are only finitely many pairs $([\mfb],z)$ for which the moduli space $\check{M}^+_z([\mfa],[\mfb])$ is non-empty.
\end{enumerate}
\end{prop}
\begin{pf}
\begin{itemize}
\item When $t>0$, we repeat the same argument of Proposition \ref{prop-16.4.3} to get the conclusion.
\item
When $t=0$, and the moduli space $M_{z_i}([\mfa],[\mfb])$ are non-empty. It is known that the image of the critical set under the blow-down map $\pi:\mcB^\sigma(M,F,\mfs)\to \mcB(M,F,\mfs)$ is a set of finite points. We may assume that $\pi[b_i]=[\beta]$ for all $i$. $\L$ descends to a single-valued function on $\mcB(M,F,\mfs)$, hence the energy of the trajectories in all these moduli spaces has an up-bound. For the blow-down case, Proposition \ref{cmpt-prop} implies that there are only finitely many choices for the homotopy class of the path $\pi(z_i)$ in $\mcB(M,F,\mfs)$. In addition, $d_0$ gives a lower-bound and up-bound for $\iota([b_i])$, we have that there are only finitely many $[b_i]$.
\item When $t<0$, there is a negative number $t$ such that
$\mathcal E^{top}_{\mathfrak q}(z)+tgr_z([\mfa],[\mfb])$ is independent of $z$. To give a bound for the dimension, it suffices to give a bound for $gr_z([\mfa],[\mfb])$. Since $t<0$, we have an above bound for $\mathcal E^{top}_{\omega,\mathfrak q}(z)$. Assume that the dimension is bounded by $d_0$ and $[\mfa]$ is fixed, so we have that there are finitely many $([\mfb],z)$ such that $\iota([\mfb])$ is bounded above and below, and $gr_z([\mfa],[\mfb])\geq0$. The energy bound implies that only finitely many
of these moduli spaces can be non-empty, and there are only finitely many critical points in the absence of reducibles, so the conclusion also holds.
\end{itemize}
\end{pf}
It is known that $gr(\mfa,u\mfa)$ equals to the index of basic Dirac operator on $M\times S^1$, by \cite{APS}. We rewrite the above formula as
\[\int_M(c_1(\mfs)-c)\wedge[u]\wedge\chi_F+t(\cdot\int_{\bar M_0\times S^1/\bar F}A_{0,b}|\tilde{dx}|+\sum^r_{j=1}\beta(M_j\times S^1))=0,\]
where
\[\beta(M_j\times S^1)=\frac12\sum_\tau\frac1{n_\tau rank(W^\tau)}(-\eta(D^{S^+,\tau}_j)+h(D^{S^+,\tau}_j))\int_{\bar M_j\times S^1/\bar F}A^{\tau}_{j,b}(x)|\tilde{dx}|,\]
the integrands $A_{0,b},~A^{\tau}_{j,b}(x)$ are similar to Atiyah-Singer integrands, $\bar M_0\times S^1$ is the principal domain of $M\times S^1$ and $\bar M_j\times S^1$ are the finite desingularities of $M\times S^1$, more detail are explained in the paper \cite{BKR}.
\section{Examples}
In this section, we will give a family of manifold with foliation satisfying Assumption \ref{assum-main-1}.
\subsection{Fibration and orbifold}
The easiest model is to consider $M=Y\times F$, where $Y$ is a closed oriented $3$ manifold and $F$ is a closed oriented manifold. Given a metric $g_Y$ and a \spinc structure $\mathfrak s$ of $Y$, by pulling back, one has a data $(M,F,\pi^*g_Y\oplus g_F, \pi^*\mathfrak s)$, where $\pi:M\to Y$. Such a manifold with foliation $(M,F)$ satisfies Assumption \ref{assum-main-1}. We can generalize the global product model to the local product model, i.e. the fibration over $Y$.
\noindent Let $Y$ be a closed oriented $3$ manifold, and $M\to Y$ be a fibration over $Y$, such that $M$ is closed and oriented. Fix a metric $g_Y$ and a \spinc structure $\mathfrak s$ of $Y$, via pulling back, we have a bundle like metric and a transverse \spinc structure, still denoted by $\mathfrak s$. Since the volume form of $Y$ is closed, by pulling back, one has that $H^3_b(M)\neq0$. We have that $(M,F)$ satisfies Assumption \ref{assum-main-1}, by Proposition \ref{prop-taut}. By the identification between the basic forms(sections) of $M$ and the forms(sections) of $Y$, one establishes the proposition below.
\begin{prop}\label{prop-fibration}
Let $(M,F,\mathfrak s)$ be defined as above. Then,
the basic monopole Floer homology groups $\overline{HM}(M,F,\mathfrak s)$, $\widehat{HM}(M,F, \mathfrak s)$, $\widecheck{HM}(M,F, \mathfrak s)$ are isomorphic to the basic monopole Floer homology groups $\overline{HM}(Y,\mathfrak s)$, $\widehat{HM}(Y, \mathfrak s)$, $\widecheck{HM}(Y,\mathfrak s)$ respectively.
\end{prop}
One can generalize the model of fibration over manifold to the model of fibration over orbifold. First, we recall the notion of orbifold, which was first introduced by Satake \cite{Satake}.
\begin{defi}[c.f. \cite{Bald}]
An $n$-dimensional orbifold $Y$ is a Hausdorff space $|Y|$ together with an atlas $(\{U_i\},\{\phi_i\},\{\tilde U_i\},\{\Gamma_i\}$, with transition maps $\{\phi_{ij}\}$, which satisfies
\begin{itemize}
\item $\{U_i\}$ is locally finite;
\item $\{U_i\}$ is closed under finite intersections;
\item For each $U_i$, the finite group $\Gamma_i$ actions smoothly and effectively on a connected open subset $\tilde U_i\subset \mathbb R^n$, and there is a homeomorphism $\phi_i*\tilde U_i/\Gamma_i\to U_i$;
\item If $U_i\subset U_j$, then there exists a monomorphism $f_{ij}:\Gamma_i\to \Gamma_j$ and a smooth embedding $\phi_{ij}: \tilde U_i\to \tilde U_j$ such that for any $g\in\Gamma_i,~x\in\tilde U_i$, we have $\phi_{ij}(g\cdot x)=f_{ij}(g)\cdot \phi_{ij}(x)$ and the following diagram commute: \[\xymatrix{ \tilde U_i \ar[d]\ar[r]^{\phi_{ij}} &\tilde U_j\ar[d] \\ \tilde U_i/\Gamma_i\ar[r]^{f_{ij}}\ar[d]_{\phi_i} & \tilde U_j/\Gamma_j\ar[d]_{\phi_j}\\
U_i\ar[r]& U_j}\]
where $f_{ij}$ is induced by the monomorphism and the canonical projection.
\end{itemize}
\end{defi}
An $n$-dimensional orbifold bundle over $Y$ is defined in the similar manner.
\begin{defi}[c.f. \cite{Bald}]
An orbifold $E$ is called an orbifold bundle over $Y$, if there exists a smooth orbifold map $p:E\to Y$, such that
\begin{itemize}
\item there is an atlas $(\{V_i\},\{\tilde V_i\},G_i)$ of $E$, satisfying $V_i=p^{-1}(U_i)$ and $\tilde V_i=\tilde U_i\times E_0$, where $(\{U_i\},\{\phi_i\},\{\tilde U_i\},\{\Gamma_i\} $ is an atlas of $Y$ and $E_0$ is a standard fiber;
\item the following diagram commutes
\[\xymatrix{ \tilde U_i\times E_0 \ar[d]\ar[r]^{\tilde p} &\tilde U_i\ar[d] \\ \tilde V_i/G_i\ar[d] & \tilde U_i/\Gamma_i\ar[d] \\
V_i\ar[r]^p& U_i}\]
\end{itemize}
where $\tilde p$ is a $(G_i,\Gamma_i)$-equivariant map.
\end{defi}
When $G_i$ acts freely, $E$ becomes a manifold, e.g. the frame bundle of an oriented orbifold(see \cite[Theorem 1.3]{ALR}).
Let $Y$ be an oriented closed $3$-orbifold. Suppose the singular set $\Sigma Y=\{x\in Y| ~G_x\neq{1}\}$ is a set of disjoint union of finite circles, where $G_x$ denotes the isotropy group at $x$. We rewrite $$\Sigma Y=\cup_{1\leq i\leq n}l_i$$ and each circle $l_i$ is assigned a positive integer $\alpha_i$ given by its isotropy group $\mathbb Z_{\alpha_i}$. Let $D$ be the unit disk and $\mathbb Z_{\alpha_i}$ acts on it by rotation. Near each $l_i$, we have an atlas,
\[\phi_i:(S^1\times D,S^1\times\{0\})\to (U_i,l_i),\] where $\phi_i$ induces a homeomorphism from $((S^1\times D)/\mathbb Z_{\alpha_i}, S^1\times\{0\})$ to $(U_i,l_i)$. It is known that $TY$ always lifts to an orbifold $spin^c$-bundle for such a $3$-orbifold. The definition of the Seiberg-Witten invariant can be generalized to 3-orbifold, see Baldridge \cite{Bald} and Chen \cite{Chen}. For Seiberg-Witten invariant, we have the following proposition, which is similar to the manifold case.
\begin{prop}
Let $Y$ be a closed oriented $3$-orbifold and $M\to Y$ be a fibration over $Y$. Suppose that $\mathfrak s$ is a transverse \spinc structure which comes from the pull-back \spinc structure of $Y$ and $M$ is a closed oriented manifold. Then, we have that basic Seiberg-Witten invariant of $M$ is equal to the Seiberg-Witten invariant of $Y$, for $b^1(Y)>1$.
\end{prop}
\noindent Under tensor product, the topological isomorphism classes of orbifold line bundles form a group. We give a local description for each class of such a group. We define on orbifold line bundle over $Y$, which is a trivial line bundle over $Y\setminus \Sigma Y$, and over each $U_i$, it is given by $(S^1\times D\times \mathbb C)/\mathbb Z_{\alpha_i}$, where $ \mathbb Z_{\alpha_i}$ action is defined by,
\[a\cdot(t,w,z)\mapsto (t,e^{\frac{2\pi ia}{\alpha}}w,e^{\frac{2\pi ia}{\alpha}}z),\] for each element $a\in \mathbb Z_{\alpha_i}$. This bundle is glued together by a transition function $\varphi(t,w)=w$ on the overlap $\partial(S^1\times D)$. Each $l_i$ generates a line bundle $E_i$. Let $L$ be a line bundle over $Y$. There is a collection of integers $\{\beta_1,\cdots,\beta_n\}$ such that
\begin{itemize}
\item $0\leq\beta_i<\alpha_i$, for each $i=1,\cdots,n$;
\item the bundle $L\otimes E^{-\beta_1}_1\cdots\otimes E^{-\beta_n}_n$ is trivial over each neighborhood of $l_i$.
\end{itemize}
By forgetting the orbifold structure, it can be naturally identified with a smooth line bundle (denoted by $|L|$) over the smooth manifold $|Y |$. We will list some necessary results of such orbifolds.
\begin{thm}[Baldridge \cite{Bald}]
The tangent bundle $TY$ lifts to an orbifold \spinc bundle.
\end{thm}
\begin{lemma}[Chen \cite{Chen}]
Let $Y$ be defined as above. Then we have that
\[\pi_0(C^\infty(Y,S^1))\cong H^1(|Y|,\mathbb Z).\]
\end{lemma}
\begin{prop}\label{prop-foliated-deRham}
Let $Y$ be the orbifold as before. Then, we have the following isomorphism
\[H^*(|Y|,\mathbb R)\cong H^*_{dR}(Y,\mathbb R). \]
\end{prop}
\begin{pf}
We have the fine resolution below for orbifold $Y$,
\[0\to\mathbb R\to \mathcal A^0\overset{d}\to\mathcal A^1\cdots\]
of the constant sheaf $\mathbb R$. By the double complex argument, we have the isomorphism $$\check H^*(Y,\mathbb R)\cong H^*_{dR}(Y,\mathbb R), $$
where the first cohomology group is the \v{C}eck-cohomology group. Since we can find a finite covering $\{\mathcal U_i\}$, such that for each finitely many intersection $U_i\cap U_j\cdot U_n$ is contractible, \v{C}ech cohomology is isomorphic to the singular cohomology of the of the underlying space $|Y|$. Thus, we have that
\[H^*(|Y|,\mathbb R)\cong \check H^*(Y,\mathbb R)\cong H^*_{dR}(M,\mathbb R). \]
\end{pf}
For an oriented closed $3$-orbifold $Y$ with a metric $g$ and \spinc structure $\mathfrak s$ whose determinant line bundle has the Seifert data $(b,\beta_1,\cdots, \beta_n)$, one can define the Chern-Simons-Dirac functional
\[\L(A,\Psi)=-\frac18\int_Y(A^t-A^t_0)\wedge(F_{A^t}+F_{A^t_0})
+\frac12\int_Y(\Psi,\Dirac_A\Psi)dvol_Y,\]
for any $(A,\Psi)\in \mathcal C(Y,\mfs)$. Let $u\in\mathcal G(Y)$, we have that
\[\L(A,\Psi)-\L(u(A,\Psi))=-\frac12\int_Yu^{-1}du\wedge F_{A^t_0}=-2\pi^2\langle c_1(\mathfrak s),[u]\rangle,\] where $c_1(\mathfrak s)=[\frac{i}{2\pi}F_{A^t_0}]$ and $[u]=[\frac{-i}{2\pi}u^{-1}du]$. Similar to the manifold case, we define the critical points of the Chern-Simons-Dirac functional and the blow-up configuration space.
By Proposition \ref{prop-foliated-deRham} and Poincar\'e duality, it is known that there is a unique second cohomology class $c(\mfs,Y)\in H^2(Y,\mathbb R)$ such that
\[gr(\mfa,u\mfa)=\langle c_1(\mfs)-c(\mfs,Y),[u]\rangle, \]
where $gr(\mfa,u\mfa)$ denotes the grading between $\mfa$ and $u\mfa$ for a non-degenerate critical point $\mfa\in Crit^\sigma(\L)$. Using complete local system $\Gamma$, we can construct the monopole Floer homologies for $(Y,\mfs)$.
When $c(\mfs,Y)$ is propositional to $c_1(\mfs)$, i.e. there is a real constant $k$ such that
\[c(\mfs,Y)=kc_1(\mfs).\] Suppose that $k\neq1$, then we can find a real constant $t$, such that
\[\L_\omega(A,\Psi)-L_\omega(u(A,\Psi)+t gr(\mfa,u\mfa)=0,\]
which is equivalent to the formula
\begin{equation}
-c_1(\mfs)+t(c_1(\mfs)-c(\mfs,Y))=0.\label{formula-orbifold}
\end{equation}
\begin{prop}\label{prop-29.2.1-energy-bound}
Let $(Y,\mfs)$ be a closed oriented $3$-orbifold as above.
Suppose that and all the moduli spaces $M_z([\mfa],[\mfb])$ for the perturbation $\mathfrak q$ are regular and the formula \eqref{formula-orbifold} holds for each non-degenerate critical $\mfa$ and $(A,\Psi)=\pi(\mfa)$. Then, the following holds:
\begin{enumerate}
\item When $t\leq0$, then for a given $[\mfa]$ and a non-negative integer $d_0$, there are only finitely many pairs $([\mfb],z)$ for which the moduli space $\check{M}^+_z([\mfa],[\mfb])$ is non-empty and of dimension at most $d_0$.
\item When $t>0$, then for a given $[\mfa]$, there are only finitely many pairs $([\mfb],z)$ for which the moduli space $\check{M}^+_z([\mfa],[\mfb])$ is non-empty.$([\mfb],z)$ for which the moduli space $\check{M}^+_z([\mfa],[\mfb])$ is non-empty and has dimension no more than $d_0$.
\end{enumerate}
\end{prop}
The proof is no different to Proposition \ref{prop-29.2.1}, here we omit it.
The space of broken trajectories $\check M^+_z([\mfa],[\mfb])$ can be identified by manifold model. This space is still compact for fixed $[\mfa]$, $[\mfb]$ and $z$ as in \cite[Theorem 16.1.3]{KM}. We apply the same arguments of \cite[Section 20-Section 25]{KM} or of the previous section to establish the following theorem.
\begin{thm}
Let $\Gamma$ be any local system of abelian groups on $\mcB^\sigma(Y,\mfs)$ and let $(Y,\mfs)$ be a closed oriented $3$-orbifold as above. Suppose that the the formula \eqref{formula-orbifold} holds for each non-degenerate critical point. Then we construct the basic monopole Floer homologies
\[\widecheck{HM}_*(Y,\mfs; \Gamma), ~ \widehat{HM}_*(Y,\mfs;\Gamma),~ \overline{HM}_*(Y,\mfs;\Gamma).\]
\end{thm}
\noindent We give an example of such a complete local system, i.e. Novikov ring \cite{Novikov}. Let $I\subset\mathbb R$ be the set of the image of the homomorphism
\[\mathcal E^{top}:\pi_1(\mcB^\sigma(Y,\mfs))\to \mathbb R,~z\mapsto \mathcal E^{top}(z),\] where $\mathcal E(z)$ denotes the difference of the Chern-Simons-Dirac functional between the different representatives of the quotient point. We define $\mathbb F[I]$ by
\[\mathbb F[I]=\{\sum _{i\in I} r_it^i|\mbox{ only for finitely many }i,~r_i\ne0\}.\]For $k\in\mathbb R$, let $U_{-k}$ be the $\mathbb F$-module spanned by the generators $t^i$, $i\in I$ satisfying $i\leq -k$. Using these as open neighborhoods of $0$, we form the completion $\bar{\mathbb F}[I]$, i.e. each element is of the form $$\sum^C_{i= -\infty} r_it^i.$$ We define a local system by taking at $[\mfa_0]$ to be $\bar R[I]$, and specifying that for each closed loop $z$ based at $[\mfa_0]$, the automorphism $\Gamma(z)$ be the multiplication by $t^{-\mathcal E^{top}(z)}$. This is a $c$-complete local system.
Similar to the foliation case of the previous section or to the non-exact perturbation on manifold case, we have the following theorem.
\begin{thm}
Let $(Y,\mfs)$ be a $3$ orbifold defined as above with a \spinc structure $\mfs$. Then we have the monopole Floer homologies
\[\widecheck{HM}_*(Y,\mfs,c;\Gamma), ~ \widehat{HM}_*(Y,\mfs ,c;\Gamma),~ \overline{HM}_*(Y,\mfs,c;\Gamma).\]
where $\Gamma$ is a complete local system. Moreover, these homologies depend only on the isomorphism class of the \spinc structure $\mfs$, $c$ and $Y$,and are independent of the metric or the perturbation.
\end{thm}
We give a necessary condition to avoid the complete local system or Novikov ring.
\begin{thm}
Let $Y$ be a closed oriented $3$-orbifold defined as above, and $(\mfs,c)$ as above. Suppose that there is a some constant $t$ such that $$-(c_1(\mfs)-c)+t(c_1(\mfs)-c(\mfs,Y))=0.$$ Then, for any local system the monopole Floer homologies are well-defined.
\end{thm}
\subsection{Suspension}
Another way to construct the foliation is by suspension, here we give two references of this subsection, see \cite[Chapter 3.8]{Molino} and \cite{Richard}. Let $(Y,g)$ be a closed oriented $3$ Riemannian manifold. Suppose that a compact Lie group $G$ actions on $(Y,g)$ isometrically and preserving the orientation of $Y$, and we have a representation
\[f: \pi_1(X)\to G\]
such that the closure of $Im(f)$ is $G$, where
$X$ is a closed oriented manifold with fundamental group $\pi_1(X)$.
We set $M=\tilde X\times Y/f$, where $\tilde X$ denotes the universal covering of $X$ and $(x,y)\sim (x[\gamma]^{-1},f([\gamma])y)$ for $[\gamma]\in \pi_1(X)$. Fixing a point $p= [y_0,x_0]\in M$, its leaf is defined by the set of the form\[\mathcal F_{p}=\{[x,y_0]\big| ~x\in \tilde X\}. \]
Before preceding, we have the following lemma(see \cite{Lin}).
Since one can find a $G$-invariant volume form over $Y$, lifting back on $M$ we have that $H^3_b(M, F)\neq0$, which implies that the foliation is taut by Proposition \ref{prop-taut}.
\begin{lemma}
Let $(M,F)$ be defined as above. Then, we have an identification
\[\pi_0(Map^G(Y,S^1))\cong H^1(M,\mathbb Z)\cap H^1_b(M),\]
where $Map^G(Y,S^1)$ denotes the space of $G$-invariant $S^1$-valued functions.
\end{lemma}
Suppose there is a $G$-equivariant \spinc structure. Given a $G$-equivariant spinor bundle $$S'\to Y,$$ we construct a foliated spinor bundle $S=\tilde X\times S'/f$ where the action of $[\gamma]\in \pi_1(X)$ is defined by $[\gamma](x,s_p)=(x[\gamma]^{-1},f[\gamma]s_p)$. . By \cite{Richard}, it is known that there is an identification\[\Gamma^G(Y,S')\cong\Gamma_b(M,S).\]
Summarizing the above arguments, we have the following proposition(see \cite{Lin}).
\begin{prop}
Let $(M,F)$ be a manifold with foliation constructed as above and $Y$ admits $G$-equivariant spinor bundle. Suppose $rank(\pi_0(Map^G(Y,S^1)))=b^G_1(Y)$, where $Map^G(Y,S^1)$ denotes the set of $G$-invariant $S^1$-valued functions and $b^G_1$ denotes the dimension of the first cohomology for the $G$-invariant deRham complex. Then $(M,F)$ satisfies the Assumption \ref{assum-main-1}.
\end{prop}
{\bf Remark}:
\begin{enumerate}
\item The condition that $rank(\pi_0(Map^G(Y,S^1)))=b^G_1(Y)$ is necessary. For example, let $Y=T^3=(S^1)^3$, $G=S^1$ action canonically on one of first slot of $Y$, and $X=S^1$ with $f:\pi_1(X)\to S^1$ by sending the generator element of $\pi_1(X)$ to a dense element of $S^1$, e.g. $1\mapsto e^{i2\pi\theta}$ for $\theta\notin\mathbb Q$. We have that $\dim H^1_b(M)=\dim H^{1,G}_{dR}(Y)=3$, and $H^1_b(M)\cap H^1(M,\mathbb Z)\cong\pi_0(\{u:M\to S^1|\mbox{ such that} L_\xi u\equiv0,\mbox{ for any }\xi\in\Gamma(F)\}))\cong\mathbb Z^2$.
\item When $G$ is connected, we have that $b^{G}_1=b_1$, since any homology cycle $\sigma$ is homotopic to $g_*\sigma$, for any $g\in G$. When $G$ actions freely, we have that $\pi_0(Map^G(Y,S^1))\cong H^1(Y/G,\mathbb Z)$, and $b_1(Y/G)=b^G_1(Y)$.
\end{enumerate}
At the end of this section, we give an explicit example. Let $Y=SO(3)$, and $T_1,~T_2$ and $T_3$ be three maximal tori(circles), such that their Lie algebras span the Lie algebra of $Y$, i.e. $so(3)$. We choose a closed oriented manifold $X$ whose fundamental groups is isomorphic to $\mathbb Z*\mathbb Z*\mathbb Z$, e.g. $X=\sharp_3 S^1\times S^k$ with $k\geq2$. We can consider a family of representations
\[f_t: \pi_1(X)\to T_1, T_2, T_3\]
such that the first component $(1,0,0)$ sends to an element $\left(\begin{array}{ccc}
\cos 2\pi t& -\sin 2\pi t &\\
\sin2\pi t&\cos 2\pi t &\\
&&1\\
\end{array}\right)$ of $T_1$, the second component $(0,1,0)$ sends to an element $\left(\begin{array}{ccc}
\cos 2\pi t& & \sin 2\pi t\\
& 1&\\
-\sin 2\pi t&&\cos2\pi t\\
\end{array}\right)$ of $T_2$ and the third component $(0,0,1)$ sends to an element $\left(\begin{array}{ccc}
1&&\\
&\cos2\pi t& -\sin2\pi t \\
&\sin2\pi t&\cos2\pi t \\
\end{array}\right)$ of $T_3$. We set $M_t=Y\times \tilde X/f_t$, the codimension $3$ foliation $F_t$ on $M_t$ is definite by letting the leaves be sets of the form
\[\mathcal F_{t,y}=\{[x,y]\big| ~x\in \tilde X\}. \] We choose a trivial $SO(3)$-equivariant spin structure of $Y$.
\begin{itemize}
\item When the group $G_t$ is a finite group of $Y$. Since $Y$ admits a metric of positive scalar curvature, by Proposition \ref{prop-fibration} and \cite[Proposition 36.1.3]{KM}, one can deduce that \[\overline{HM}(M_t,F_t)\cong \mathbb F[U,U^{-1}],~ \widecheck{HM}(M_t,F_t)\cong \mathbb F[U,U^{-1}]/\mathbb F[U],~ \widehat{HM}(M_t,F_t)\cong \mathbb F[U].\]
\item When $G_t$ is dense in $M_t$. Since the transverse \spinc structure $\mathfrak s$ is trivial, then $\Gamma_b(S)=\mathbb C^2$. By the argument at the beginning of this subsection, we have that $\Omega^1_b(M_t)\cong \Omega^{1,G}(Y)\cong\mathbb R^3$. Let $g_Y$ be a bi-invariant metric of $Y$ with positive scalar curvature, then we have that the associated $\Dirac^{g_Y}$ is an $SO(3)$ equivariant, which corresponds to a basic Dirac operator $\Dirac_0$ with spin connection $A_0$, their spectrums have a one-to-one corresponding. Therefore, the solutions of the basic Seiberg-Witten equations \eqref{eqn-SW-equation-1} corresponds to the solution of $SO(3)$-invariant Seiberg-Witten equations, i.e.
\[\begin{cases}
\Dirac^A_0\Psi=0\\
\frac12*_{g_Y}F_A=q(\Psi),
\end{cases}\] where $(A,\Psi)\in\mathcal C^{SO(3)}(Y)=\{A_0+i\Omega^{1,SO(3)}(Y)\times \mathbb C^2\}$. Since $g_Y$ has a positive scalar curvature, it is known that $\Psi\equiv0$, i.e. there is no irreducible solution for the basic Seiberg-Witten solution. Consider the reducible solutions, we have that $dA=0$ and $\Psi$ is an eigenvector of $\Dirac^A_0$. Since $A=A_0+a$ and $d A_0=0$, this implies that $a$ is closed. Combining with $H^{1,G}(Y)=0$, we have that $a=df$ for a $SO(3)$-invariant function $f$, which implies that $f$ is constant and $a=0$. Thus, all reducible solutions are eigenvector of $\Dirac_0$. Recall that $\Dirac_0=\sum_i e^i\nabla'_{e_i}$, where $\{e_i\}$ is an orthonormal frame of $TY$; and
\[\nabla'=d+\frac12\sum_{i<j}\omega_{ij}e^ie^j\]
where $d$ is the flat connection of $S'$ and $\omega_{ij}$ is the Levi-Civita connection associated to $g_Y$. Since $\Dirac_0$ is independent of the choice of the orthonormal frame, we choose $\{e_1,e_2,e_3\}$ that are generated by the left action of a frame $\{L_x,L_y,L_z\}$ of $T_1Y$, where $L_x=\left(\begin{array}{ccc}
&&\\
&&-1\\
&1&\\
\end{array}\right)$, $L_y=\left(\begin{array}{ccc}
&&1\\
&&\\
-1&&\\
\end{array}\right)$ and $L_z=\left(\begin{array}{ccc}
&-1&\\
1&&\\
&&\\
\end{array}\right)$. Since $g_Y$ is bi-invariant $\{e_1,e_2,e_3\}$ are left-invariant, we have that
\[\omega_{12}(e_3)=\frac12g_Y([e_3,e_1],e_2)=\frac12g_Y(e_2,e_2)=\frac12,\]
\[\omega_{13}(e_2)=\frac12g_Y([e_2,e_1],e_3)=-\frac12g_Y(e_3,e_3)=-\frac12,\]
\[\omega_{23}(e_1)=\frac12g_Y([e_1,e_2],e_3)=\frac12g_Y(e_2,e_2)=\frac12.\]
We inherit the convection from the book \cite{KM} that $e^1\cdot e^2\cdot e^3=Id$, by assigning
\[e^1\mapsto\left(\begin{array}{cc}
i&\\
&-i\\
\end{array}\right),~e^2\mapsto\left(\begin{array}{cc}
&-1\\
1&\\
\end{array}\right),~ e^3\mapsto\left(\begin{array}{cc}
&i\\
i&\\
\end{array}\right).\]
We have that for any $\Psi\in\Gamma^{SO(3)}(Y,S')\cong\mathbb C^2$,
\[\Dirac_0\Psi=\sum_ke^kd\Psi+\sum_{i<j,k}e^k\frac12\omega_{ij}(e_k)e^ie^j\Psi=
\frac34\Psi.\]
Hence, $\Dirac_0$ acts as a diagonal matrix with eigenvalues $(\frac34,\frac34)$. Summarizing the above arguments, we have that
\[\overline{HM}(M_t,F_t,\mfs)\cong \mathbb F\oplus \mathbb F ,~ \widecheck{HM}(M_t,F_t,\mfs)\cong \mathbb F\oplus \mathbb F, \widehat{HM}(M_t,F_t,\mfs)\cong 0.\]
\end{itemize}
\vspace{3mm}
{\bf Remark:}
{ Note} that the closure of the image of $f_t$ can not be of one dimensional. Otherwise, let $H$ be this one-dimensional closed subgroup in $SO(3)$. It is well known that $H$ preserves a vector in $S^2\subset \mathbb R^3$, say $(x,y,z)^t\in S^2$. We have that the group generated by $\left(\begin{array}{ccc}
1&&\\
&\cos2\pi t& -\sin2\pi t \\
&\sin2\pi t&\cos2\pi t \\
\end{array}\right)$ preserves $(x,y,z)^t$, which implies that either $y=z=0$ or $t=0,1$. We can apply the same arguments for the other two subgroups. In conclusion we have that either $x=y=z=0$ or $t=0,1$, which contradicts to our assumption.
\vspace{3mm}
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| 330,659
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You all remember when Trump banned travel from China due to COVID and Joe Biden called the decision xenophobic, don’t you? Of course you do, because it really happened.
This weekend, when a reporter asked Biden press secretary Jen Psaki about Biden’s new travel bans and what they should be called, and she tried to claim that this never happened.
She tried to spin it by bringing up the ‘Muslim ban’ that was not actually a Muslim ban.
The mainstream media isn’t going to call her out. Hell, they are saying outright they won’t report Biden’s lies. But we surely will.:
Again, Biden did call the China travel ban xenophobic and we all know it.
| 249,481
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Consejo Area Food and Art Festival
at Casa Blanca By The Sea
The second annual Consejo Area Food and Art Festival was held on Sunday July 8th at the Casa Blanca By The Sea hotel. This was well organized, heavily advertised and much anticipated event. There was lots of great food to be consumed and live music scheduled for the late afternoon and evening. Unfortunately, Mother Nature didn't cooperate. We had a lot of rain the night before with more forecast, so with Corozal-Consejo Road a muddy mess, many of the families expected from Corozal and other points South didn't make the trip.
The disappointing turnout didn't stop those who came from having a great time. Several local artist displayed their wares while a several local families set up booths to provide food and drink. The Consejo Yacht Club offered rides on one of their Catamarans, as well as running a couple of fund raising games like "Chicken Drop" and "Darts".
The band was setting up as we left to avoid the next rain shower. The festivities went on until around 7PM.
We would welcome any photos or comments from the late afternoon and evening doings to complete the coverage of this event
| 126,915
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TITLE: Solving a 3rd order differential equation with a non constant coefficient
QUESTION [0 upvotes]: I am trying to solve this non linear differential equation with 3 boundary conditions :
$$ R\dfrac{d^3 h}{d R^3}+2\dfrac{d^2 h}{d R^2} + k\dfrac{d h}{d R} = 0$$
With : $\frac{d h}{d R}(R=0)=0$, $h(R=L) = \frac{d h}{d R}(R=L)=0$
Unfortunately, I am stuck because I haven't been able to find a reliable way to solve it correctly.
How do you think it would be possible to find an analytical solution to this equation ?
Thank you,
Cheers.
REPLY [2 votes]: Since you have derivatives of $h$ of orders $1$, $2$, and $3$ (but not $0$, namely $h$ itself), we can make the substitution $y = \frac{dh}{dR}$, resulting in the second-order linear equation in $y$:
$$
R\, \frac{d^2 y}{dR^2} + 2\, \frac{dy}{dR} + k\, y = 0.
$$
We can multiply through by $R$, yielding:
$$
R^2\, \frac{d^2 y}{dR^2} + 2R\, \frac{dy}{dR} + kR\, y = 0.
$$
Now the first two terms are a derivative of a product:
$$
\frac{d}{dR} \biggl( R^2\, \frac{d y}{dR} \biggr) + kR\, y = 0.
$$
This is the standard form for a (homogeneous) Sturm–Liouville equation. There is a robust theory for studying solutions to equations of this form, however you're probably not going to see an analytic solution in a closed form unless you're willing to use Bessel functions and other special functions.
You can, of course, find a series solution, starting with the assumption that
$$
y = \sum_{n=0}^\infty a_n R^n,
$$
and using the differential equation in its original form to determine recurrence relations among the coefficients (and use the boundary values to get initial values).
| 66,373
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Saturday, September 9, 2017.
Century ride begins at 7:00 a.m. with staggered starts to follow...
Post Event Barbeque at Stage Fort Park, Gloucester, MA
12:00 to 4:00 – free for registered riders, guests welcome for additional fee.
We thank our sponsors:
| 411,300
|
TITLE: $P(z)=0$ iff $Q(z)=0$, $P(z)=1$ iff $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x$
QUESTION [13 upvotes]: Assume $P(x)$ and $Q(x)$ are polynomials with complex coefficients with degree greater than or equal to $1$ such that $P(z)=0$ if and only if $Q(z)=0$, $P(z)=1$ if and only if $Q(z)=1$. Prove that $P(x)=Q(x)$ for all $x\in \mathbb{C}$.
I have no idea for this problem. The only thing I can say is $P(x)$ and $Q(x)$ share the same roots.
Thanks so much for any help.
REPLY [8 votes]: Let $deg(P)=n>0$ and assume without loss of generality that $deg(Q) \leq deg(P)$. Consider the polynomial $R=(P-Q)P'$ ($P'$ denote the derivative of $P$). We have :
$$deg(R)\leq 2n-1 $$
Now if $r$ is a root of multiplicity $k$ of $P$ then $r$ is a root of $P'$ of multiplicity $k-1$ and because $Q(r)=0$, $r$ is a root of $P-Q$. hence $r$ is a root of $R$ of multiplicity at least $k$. So the n zeros of $P$ produces at least $n$ zeros of $R$, when multiplicities are counted.
The same pattern can be applied to $P-1$, every root $r$ of multiplicity $k$ of $P-1$ is a root of $P-Q$ and a root of multiplicity $k-1$ of $(P-1)'=P'$. So the n zeros of $P-1$ produces at least another $n$ zeros of $R$ when multiplicities are counted.
(the zeros of $P$ are all different with the zeros of $P-1$ it's obvious).
This means that $R$ has at least $2n$ roots and beacuse $deg(R)\leq 2n-1 $ we conclude that $R=0$ which implies $P=Q$.
| 101,494
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Some Really Bad Movies that got Decent Reviews
Critics no matter how high we hold them, they do make mistakes, and that sometimes cost us dearly than we imagine. Sometimes really bad movies end up getting decent reviews, and money so the producers keep making more them. Following are a few good examples!
You Don’t Mess with the Zohan
This may be the start of Adam Sandler’s fall. His career has been on a down path ever since. The movie is a guilty pleasure for whoever watched it . It follows the lead character Zohan played by Adam sandler who is tired of being a superhero, trying to save the world, and leaves that to pursue his dream, to become a barber. It have some laugh but wasn’t worth the decent reviews it managed.
Twilight
Stephanie Meyer blossomed the romance between an 18 year old, and 200 years old vampire. I don’t know about the novel, but Twilight really kick started the whole frenzy and the world went crazy for it. I am speaking as someone who hates the whole franchise, and didn’t had the time to waste on the whole series. But one thing is for sure, we never had to bear the weight of five movies if the first one didn’t went well with the critics, and reviews.
Spiderman 3
The movie that killed Sam Raimi Spiderman Series, Spiderman was never meant to be a conclusion for the previous 2 films but death of Harry Obsborn, and the whole goth think with Tobey Maguire was too much for the audience. The movie got decent reviews but lets face it, it killed a possible 4th installment and that makes it bad enough. I guess it was Venom’s curse, he was such a bad charge that we haven’t seen his live action adaption even after 10 year, and 3 movies.
| 275,673
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| 57,153
|
TITLE: Can sufficiency of James's Theorem be derived from Hahn-Banach Theorem?
QUESTION [3 upvotes]: Notation: Let $V$ be a normed vector space and $V^*$ to be its continuous dual space.
One of the consequences of the Hahn-Banach Theorem is the existence of non-trivial dual space.
Corollary 1: Let $z$ be any nonzero element in $V.$
Then there exists a continuous linear functional $\psi$ on $V$ such that $\|\psi\|\leq 1$ and $\psi(z)=\|z\|.$
Since $V^*$ is always a Banach space, we can apply corollary $1$ to obtain the following.
Corollary 2: For any nonzero element $\psi$ in $V^*,$ there exists a continuous linear functional $\mu$ on $V^*$ such that $\|\mu\|\leq 1$ and $\mu(\psi) = \|\psi\|.$
Now, recall James's Theorem.
James's Theorem: Let $V$ be a Banach space. Then $V$ is reflexive if and only if every continuous linear functional on $V$ attains its supremum on the closed unit ball $B_V,$ that is, for every $\psi\in V^*,$ there exists $z\in B_V$ such that $\|\psi\| = \psi(z).$
So here comes my question.
Question: Can we use Corollary $2$ to prove sufficiency of reflexivity in James's Theorem, that is, if a Banach space $V$ is reflexive, then every continuous linear functional on $V$ attains its supremum on $B_V?$
It seems to me that this is possible.
Assume that $V$ is a reflexive Banach space.
Pick any nonzero $\psi\in V^*.$
By Corollary $2$ and reflexivity of $V,$ there exists $z\in V$ such that $\|z\|\leq 1$ and $\psi(z) = \|\psi\|.$
(Since $V$ s reflexive, there exists some $z\in V$ such that $\mu = e_z$ where $e_z\in V^{**}$ is an evaluation functional). So sufficiency is proven.
Is there any mistake above?
REPLY [1 votes]: Your reasoning is correct. This direction of the theorem is easy; what is actually due to James is its converse.
| 32,778
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i’d be panicked, too
In the article below, Rob Port details the recent acts of desperation by Heidi Heitkamp, including her most recent immigration flip-flop and the excuses she’s already making for losing this election.
It’s no surprise her campaign is in full-panic mode. If I had nothing more than Heitkamp’s liberal, out-of-touch record and excuses the voters aren’t buying, I’d be panicked too.
In case you missed it…
It Kinda Seems Like the Heitkamp Campaign Is Panicking
Rob Port | Say Anything Blog
8.10.18
A couple of things have happened which make me think that Senator Heidi Heitkamp’s campaign is in full-on panic mode.
For one thing the Democratic Senator gave an exclusive interviewto Breitbart– yes, that Breitbart, a news outlet derided by Heitkamp’s political base as a digital den for white nationalists – in which she touted her supposedly long-time support for a border wall.
“I’ve always supported increased and enhanced border security along our southwest border with Mexico – including physical barriers, sensors, drones, and more resources at our ports of entry – and yes, wall funding as well. My votes clearly reflect a commitment to robust border security funding,” Heitkamp said.
I’m not sure that squares with the Senator’s voting record. “I just wish we could get beyond it,” she said of the border wall issue last year. She voted against a border fence in 2013. She voted against cracking down on sanctuary cities in 2015, 2016, and 2018.
This year she also voted for legislation which would have provided permanent residence for illegal immigrants without funding a border wall.
But it’s not just that Heitkamp, despite the aforementioned voting record, is suddenly pitching herself as a border hawk to an “alt-right” website (to use the parlance of the Senator’s political base). She’s also playing the Russia card:
Heitkamp, responding to a question during a meeting with The Forum Editorial Board, said she believes the Russians could try to intercede on Cramer’s behalf in the race.
“I would be a fool if I didn’t think that was true,” Heitkamp said.
To be fair to Heitkamp she was responding to a question from theFargo Forumeditorial board, who in turn was working off a batty columnby left wing commentator Lloyd Omdahl, but if I had to guess I’d say the editorial board was probably surprised that Heitkamp took a bite at that particular apple.
Cramer, for his part, laughed it off. “It sounds like a really pathetic excuse for poor performance in the election in advance,” he said.
That’s what I thought too when I read the initial story about Heitkamp’s comments (which, at the time, didn’t yet include Cramer’s response).
Is Heitkamp laying the ground work to excuse what would be a devastating loss on the statewide ballot for North Dakota Democrats this cycle?
It kinda seems like it. Blaming the Russians would mean Democrats wouldn’t have to acknowledge that they stand for things most voters here just don’t want.
###
| 30,685
|
\begin{document}
\title[Polynomial Approximation]
{Regular polynomial interpolation and approximation of global solutions of linear partial differential equations}
\author{ J\"org Kampen}
\address{
Weierstrass Institute
for Applied Analsis and
Stochastics
Mohrenstrasse 39\\
10117 Berlin\\
Germany}
\email{kampen@wias-berlin.de}
\thanks{This work was completed with the support of DFG (Matheon)}
\subjclass{65D05; 35G05}
\keywords{extended Newtonian interpolation, linear systems of partial differential equations, error estimates}
\date{Mai 31, 2007}
\begin{abstract}
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations.
Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations.
Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.
\end{abstract}
\maketitle
\section{Introduction}
This work shows how multivariate interpolation techniques can be combined with analytic information of linear partial differential equations (i.e. a priori estimates and/or WKB representations of solutions) in order to design efficient and accurate numerical schemes for solving (systems) of linear partial differential equations. These schemes are nothing but sequences of multivariate polynomials which are constructed recursively such that they solve a given linear system of partial differential equations on a finite discrete set of interpolation points. However, additional information is needed in order to ensure that the sequence of interpolation polynomials converges to a (or, if uniqueness is proved, the) global solution of a given linear system of partial differential equations. As we shall see, this information can be provided by a priori estimates which in turn lead us to error estimates in regular norms dependent on the regularity of the solution. We examine the situation in the case of linear elliptic equations with variable coefficients. Another possibility is that (more or less) explicit representations of solutions are known which lead to problems which are easier to solve. A prominent example is the WKB-expansion which was investigated in \cite{Ka}. The recursive structure of WKB coefficient functions and the error analysis lead us to the problem of regular polynomial approximation. In this introductionary Section we our method on an abstract level.
\subsection{Regular polynomial interpolation}
Since we are interested in the relationship between multivariate polynomial interpolation and approximation of solutions of partial differential equations, our focus will be on multivariate polynomial interpolation. However, in order to make basic ideas more accessible we shall describe algorithms in the univariate case first and then generalize to the multivariate case. It is well known that polynomial interpolation in the multivariate case is quite different from the univariate case in general. However, in our approach which aims at solving linear systems of partial differential equations or aims at supplementing certain strategies of solving partial differential equations many features are already present in the univariate framework. In order to avoid misunderstandings, we dwell a little on this point.
Classically, the problem of multivariate interpolation can be stated as follows (cf. \cite{Sau2}):
{\it
Given a set of interpolation points $\Theta=\left\lbrace x_1,\cdots ,x_N \right\rbrace $ and an N-dimensional
space $P_{\Theta}$ of polynomials find, for given values $y_1,\cdots,y_N$, a unique polynomial $f\in P$ such that
\begin{equation}
f(x_j)=y_j,~j\in 1, \cdots ,N.
\end{equation}\it}
In this form it turns out that there is an intricate relation between sets of interpolation points and interpolation spaces that must be satisfied in order that the problem can be considered to be well-posed. Either we have to make some restrictions concerning the set of interpolation points $\Theta$ (cf. \cite{Sau2}) or we consider $\Theta$ to be fixed and consider the problem of constructing the polynomial space $P_{\theta}$ (cf.\cite{boor}). This amounts to a construction of the map
\begin{equation}
\Theta \rightarrow P_{\Theta}
\end{equation}
with additional constraints such as minimality of degree (cf. \cite{Sau2, boor}) or monotonicity (cf. \cite{boor}). In this paper we are interested in interpolation algorithms with the following features
\begin{itemize}
\item there are no essential restriction on the discrete set $\Theta$ of interpolation points except that $\Theta \subset D$, where $D$ is the domain of the function to be interpolated.
\item the map $\Theta \rightarrow P_{\Theta}$ is monoton (indeed our basic algorithm is an extension of multivariate versions of Newton's interpolation algorithm).
\item the algorithm can be extended to vector valued interpolation functions $g: D\subseteq {\mathbb R}^n\rightarrow {\mathbb R}^k$ and if $g$ satisfies a system of linear partial differential equations, then the interpolation polynomial $p$ solves the same system of linear partial differential equations on the given set $\Theta$ of interpolation points.
\item the algorithm is numerically stable and practical with respect to the problem that the interpolation function $f$ and arbitrary set of partial derivatives of $f$ are to be interpolated simultaneously. For the application of higher order approximation of the fundamental solution of linear parabolic equations we comute accurate approximations of derivatives of smooth functions up to order $10$ in order to obtain an approxmation of order $5$ of the WKB-expansion of the fundamental solution.
\item the algorithm can be refined in order to solve well-posed linear systems of partial differential equations directly.
\item the algorithm can be combined with collocation methods in an efficient way; it can be partially parallelized.
\item the algorithm allows for error estimates which depend on the regularity of the solution such that the algorithm is compatible with methods for higher dimensional problems of linear systems of partial differential equations such as sparse grids, adaptive sparse grids, and weighted Monte-Carlo.
\end{itemize}
First we consider the problem of polynomial approximation $p$ of a regular (i.e smooth or finitely many times differentiable) function
\begin{equation}
f:D\subseteq {\mathbb R}^n\rightarrow {\mathbb R}
\end{equation}
defined on discrete subset of $\Theta\subset D$ where for $m$ given linear partial differential operators
\begin{equation}
L_i=\sum_{|\alpha| \leq \beta_i} a^i_{\alpha}(x)\partial_{\alpha},
\end{equation}
we require that
\begin{equation}
L_i f(x_j)=L_i p(x_j) \mbox{ for } 1\leq i \leq m
\end{equation}
for some finite set of points $x_j\in \Theta\subset D$. As indicated above we shall allow that the interpolation set $\Theta$ can be constructed recursively (and, hence, extended arbitrarily within the domain of the interpolation function).
Investigations of specific instances of this problem can be found in the literature on polynomial interpolation (cf. the survey paper of \cite{Sau} for the development up to the year 2001). Note that other algorithms of natural interpolation of $C^k$-functions have been proposed (cf.\cite{Gasca} for hints at the history and further references).
The paper is organized as follows. In Section 1.2 we introduce the partial differential equations for which we seek global regular interpolation polynomials of their global solutions. All basic types of partial differential equations, i.e. elliptic equations, parabolic equations, and hyperbolic equations are considered. While the basic algorithm is quite similar for each type of partial differential equation, we shall see, however, that the convergence of the scheme of recursively defined interpolation polynomials depends on very different a priori estimates for different type of equations. In case of second order elliptic equations classical Schauder boundary estimates can be used, while in the case of hyperbolic equations energy estimates are considered. In the case of parabolic equations we refer back to Safanov-Krylov estimates considered in the context of the truncation error analysis of WKB-expansions. In Section 2.1 we introduce first an extension of Newton's polynomial algorithm which interpolates a given function and its derivatives up to some given order $k$ simultaneously. Section 2.2. describes a variation of this algorithm which interpolates a given function such that a given set of partial differential equations is preserved. Section 3 discusses the extension to the multvariate case. In Section 4 we refine the algorithm and construct polynomials which satisfy a given linear (i.g. partial) differential equation on a given set of interpolation points, i.e. there is no given function to be interpolated. In Section 5 we consider refinements which show how polynomials constructed on disjoint sets of interpolation points can be synthesized in order to get one polynomial which interpolates on the union of sets of interpolation points. Naturally, parallelization is consideredin this context. In Section 6 we show how a priori estimates of elliptic equations (standard Schauder boundary estimates) and hyperbolic equations (energy estimates) lead to convergent schemes implied by error estimates. Section 7 discusses a special use of regular polynomial interpolation for parabolic equations where the global solution is given in the form of a WKB- expansion. Section 8 provides a numerical example of global regular polynomial interpolation of a locally analytic function up to the third derivative. In Section 9 we provide a summary and give an outlook on current research and research in the near future. Before we start with the description of the algorithm, we state the typical linear partial differential equations and indicate the different types of approximations and error estimates which we aim at.
\subsection{Regular interpolation and partial differential equations}
We consider the three standard types of linear partial differential equations, namely elliptic equations, parabolic equations, and hyperbolic equations, and exemplify different types of application and extension.
\begin{itemize}
\item The most popular examples of elliptic partial differential equations are of the second order form, i.e.
\begin{equation}
\sum_{j, k}^na_{jk}(x)\frac{\partial^{2}u}{\partial x_j \partial x_k}+\sum_{l}b_l(x)\frac{\partial u}{\partial x_l }+c(x)u=f(x),
\end{equation}
to be solved on a domain $\Omega\subseteq {\mathbb R}^n$ with the
boundary condition
\begin{equation}
u\mbox{\Big |}_{\partial \Omega}=g
\end{equation}
for some function
$f:{\partial \Omega}\rightarrow {\mathbb R}$ which is usually assumed to be Lipschitz continuous at least.
Here, $a_{jk}$ are (at least) measurable coefficient functions satisfying for some constant $c$, and ellipticity means that
\begin{equation}
\sum_{jk}a_{jk}(x)\xi_i\xi_j\geq c>0 \mbox{ (uniformly in $x$)}.
\end{equation}
We construct an extension of the polynomial interpolation algorithm which produces a multivariate polynomial solving this elliptic equation on an arbitrary grid of interpolation points. In order to obtain error estimates b standard boundary Schauder estimates in this paper we shall make some regularity assumptions. We derive convergence of the family of multivariate polynomials constructed by our our interpolation scheme to the global solution of the linear elliptic equation on a bounded domain and we derive error estimates from a priori estimates.
\item Parabolic equations of the form
\begin{equation}\label{parab}
\frac{\partial u}{\partial t}-Lu=0,
\end{equation}
on $D:=\Omega\times (0,T)$, ($\Omega\subseteq {\mathbb R}^n$, with
\begin{equation}
u(0,x)=\delta_y(x):=\delta(x-y),~y\in{\mathbb R}^n,
\end{equation}
where $\delta$ is the Dirac delta distribution, and where
\begin{equation}
Lu\equiv \frac{1}{2}\sum_{ij}a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum_i b_i(x)\frac{\partial u}{\partial x_i}
\end{equation}
is an elliptic operator. The solution of this equation is called fundamental solution, because solutions of standard parabolic initial-value boundary problems can be represented by convolution integrals of data functions with the fundamental solution. The standard assumptions for such a fundamental solution to exist are
\begin{itemize}
\item[(A)] The operator $L$ is uniformly parabolic in ${\mathbb R}^n$, i.e. there exists $0<\lambda <\Lambda <\infty$
such that for all $\xi\in {\mathbb R}^n\setminus \{0\}$
$$
0<\lambda |\xi|^2\leq \sum_{i,j=1}^n a_{ij}(x)\xi_i\xi_j\leq \Lambda |\xi|^2.
$$
\item[(B)] The coefficients of $L$ are bounded functions in ${\mathbb R}^n$ which are uniformly
H\"older continuous of exponent $\alpha$ ($\alpha \in (0,1)$).
\end{itemize}
If some regularity assumptions on the coefficients hold in addition, then it can be shown that the fundamental solution $p$ is of the form
\begin{equation}\label{frep}
p(t,x,y)=\frac{1}{\sqrt{2\pi t}^n}\exp\left( -\frac{d^2(x,y)}{2 t}+\sum_{k\geq 0}c_k(x,y)t^k\right),
\end{equation}
with some regular coefficient functions $d^2$ and $c_k$. We shall show how our regular polynomial interpolation algorithm can be used to compute the fundamental solution in terms of this representation.
\begin{rem}
The algorithm designed in the case of elliptic equations can be applied to the parabolic case directly, of course. However, it turns out that the convergence is better if the special representation \eqref{frep} is used.
\end{rem}
\item
As an example of a hyperbolic equation we consider an equation of the form
\begin{equation}
Lu=f \mbox{ in } \Omega,
\end{equation}
where
\begin{equation}
Lu\equiv \sum_{ij} h_{ij}\frac{\partial u}{\partial x_i\partial x_j}+\sum_i\frac{\partial}{\partial x_j}+c(x)u
\end{equation}
and $(h_{ij})$ is a symmetric matrix of signature $(n,1)$, if $\mbox{dim}\Omega =n+1$. We assume that some $O\subset \Omega$ is bounded by two spacelike surfaces $\Sigma_i$ and $\Sigma_e$ and swept out by a family of spacelike surfaces $\Sigma_e(s)$. We assume the initial conditions
\begin{equation}
u=g \mbox{ and } du=\omega
\end{equation}
where $g$ is a function on $\Omega$ and $\Omega$ is a 1-form.
\end{itemize}
\section{Interpolation algorithm (univariate case)}
We start with the description of the algorithm which produces polynomials which satisfy some given requirements on interpolation points. Our starting point is an extension of Newton's polynomial interpolation method such that the interpolation polynomial and its derivatives up to a given order $k$ (an integer) equal a given function and its derivatives up to order $k$ at the interpolation points. For simplicity of representation and since the essential features of the algorithm can be demonstrated for one dimensional functions, we describe our ideas first in the univariate case and then generalize to the multivariate case in the next section.
\subsection{Extension of Newton's method}
Let us recall the Newtonian interpolation for an univariate function
\begin{equation}
\begin{array}{ll}
f:[a,b]\subset {\mathbb R}\rightarrow {\mathbb R}.
\end{array}
\end{equation}
Given a discrete set of interpolation points $D=\{x_0,x_1\cdots ,x_N\}\subset [a,b]$ we want to construct a polynomial
\begin{equation}
\begin{array}{ll}
p:[a,b]\subset {\mathbb R}\rightarrow {\mathbb R} \mbox{ such that}\\
\\
f(x_i)=p(x_i) \mbox{ for all } x_i \in D.
\end{array}
\end{equation}
The idea of the basic Newton interpolation algorithm is that instead of looking for some polynomial of form $\sum_{i=1}^N b_i x^i$ for some constants $b_i$ we may write
\begin{equation}
\sum_{l=0}^N a_l \Phi_l(x)
\end{equation}
with
\begin{equation}
\Phi_0(x)=1 \mbox{ and } \Phi_l(x)=
\Pi_{i=0}^{l}(x-x_i) \mbox{ for $l\geq 1$.}
\end{equation}
In order to determine $a_0,\cdots a_{N}$ we then may solve the system
\begin{equation}
R_0a:=\left[ \begin{array}{ccccccc}
&1 &0 & 0 &\cdots &0\\
&1 &\phi_1(x_1) & 0&\cdots &0\\
&1 & \phi_1(x_2) &\phi_2(x_2) &\cdots & 0\\
&\vdots & \vdots &\vdots &\vdots\\
&1 & \phi_1(x_{N}) &\phi_2(x_{N}) &\cdots & \phi_N(x_{N})
\end{array}
\right] \left[ \begin{array}{ccccccc}
a_0\\
a_1\\
a_2\\
\vdots\\
a_{N}
\end{array}
\right] =\left[ \begin{array}{ccccccc}
f(x_0)\\
f(x_1)\\
f(x_2)\\
\vdots\\
f(x_{N})
\end{array}
\right]
\end{equation}
This leads to an $L^2$-approximation of the function $f$ similar to the Gaussian algorithm. Note however, that the matrix $R_0$ is a lower diagonal. Hence the linear system can be solved easily. Moreover the matrix condition number is much better than that of the Vandermonde matrix used in the classical Gaussian interpolation. We extend this idea to a $C^k$-norm interpolation, i.e. we design an algorithm that approximates $f$ up to the $k$-th derivative, i.e. we construct a polynomial
\begin{equation}
\begin{array}{ll}
q:[a,b]\subset {\mathbb R}\rightarrow {\mathbb R} \mbox{ such that}\\
\\
f^{(l)}(x_i)=q^{(l)}(x_i) \mbox{ for all } x_i \in D \mbox{ and all } l\leq k,
\end{array}
\end{equation}
where for a function $g:[a,b]\subset {\mathbb R}\rightarrow {\mathbb R}$ $g^{(l)}$ denotes the derivative of order $l$ while $g=g^{0}$. We consider the polynomial
\begin{equation}
\sum_{m=0}^{(N+1)(k+1)-1}a_m\Phi_{m,k}(x)
\end{equation}
where
\begin{equation}
\Phi_{m,k}(x)=(x-x_{m \mbox{ div}(k+1)})^{m \mbox{ mod}(k+1)}\Pi_{l=0}^{m \mbox{div}(k+1)-1}(x-x_l)^{k+1},
\end{equation}
where, by convention, we understand
\begin{equation}
\Pi_{l=0}^{-1}(x-x_l)^{k+1}:=1.
\end{equation}
For simplicity of notation we sometimes use the abbreviations
\begin{equation}
p(m)= m \mbox{div} (k+1) \mbox{ and } q(m)=m\mbox{mod}(k+1).
\end{equation}
Next we define
\begin{equation}
\Phi^{(l)}_{m,k}(x):= \frac{d}{dx^l}\Phi_{m,k}(x),
\end{equation}
and for each $k\geq 1$ the linear system
\begin{equation}
R_k\left[ \begin{array}{ccccccc}
a_0\\
a_1\\
a_2\\
\vdots\\
a_{(k+1)(N+1)-1}
\end{array}
\right] =\left[ \begin{array}{ccccccc}
f(x_0)\\
f'(x_0)\\
\vdots\\
f^{(k)}(x_0)\\
f(x_1)\\
\vdots\\
f^{(k)}(x_{(k+1)(N+1)-1})
\end{array}
\right]
\end{equation}
where $R_k$ is a $(N+1)(k+1)\times (N+1)(k+1)$-matrix determined by $(k+1)\times (k+1)$ matrices $A_k^{lm}$ as follows:
\begin{equation}
R_k:=\left[ \begin{array}{ccccccc}
&A^{00}_k &Z_k & Z_k &Z_k &\cdots &Z_k\\
&A^{10}_k &A^{11}_k & Z_k & Z_k &\cdots &Z_k\\
&A^{20}_k &A^{21}_k & A^{31}_k & Z_k &\cdots &Z_k\\
&\vdots &\vdots & \vdots &\vdots &\vdots &\vdots\\
&A^{N0}_k & A^{N1}_k & A^{N2}_k &A^{N3}_k &\cdots &A^{NN}_k\\
\end{array}
\right],
\end{equation}
where $Z_k$ is the $(k+1)\times (k+1)$ matrix with $0$ entries, and
\begin{equation}
A^{ij}_k=A^i_k(x_j)
\end{equation}
with
\begin{equation}
A^{ij}_k:=\left[ \begin{array}{ccccccc}
&\Phi_{(k+1)p(i),k}(x_j) &\Phi_{(k+1)p(i)+1,k}(x_j) & \Phi_{(k+1)p(i)+2,k}(x_j) &\cdots &\Phi_{(k+1)p(i)+k,k}(x_j)\\
&\Phi^{(1)}_{(k+1)p(i),k}(x_j) &\Phi^{(1)}_{(k+1)p(i)+1,k}(x_j) & \Phi^{(1)}_{(k+1)p(i)+2,k}(x_j) &\cdots &\Phi^{(1)}_{(k+1)p(i)+k,k}(x_j)\\
&\Phi^{(2)}_{(k+1)p(i),k}(x_j) &\Phi^{(2)}_{(k+1)p(i)+1,k}(x_j) & \Phi^{(2)}_{(k+1)p(i)+2,k}(x_j) &\cdots &\Phi^{(2)}_{(k+1)p(i)+k,k}(x_j)\\
&\vdots &\vdots & \vdots &\vdots &\vdots\\
&\Phi^{(k)}_{(k+1)p(i),k}(x_j) &\Phi^{(k)}_{(k+1)p(i)+1,k}(x_j) & \Phi^{(k)}_{(k+1)p(i)+2,k}(x_j) &\cdots &\Phi^{(k)}_{(k+1)p(i)+k,k}(x_j)
\end{array}
\right].
\end{equation}
Note that
\begin{equation}
A^{00}_k:=\left[ \begin{array}{ccccccc}
&1 &0 & 0 &0 &\cdots &0\\
&0 &1 & 0 &0 &\cdots &0\\
&0 &0 & 2 &0 &\cdots &0\\
&\vdots &\vdots & \vdots &\vdots &\vdots &\vdots\\
&0 &0 & 0 &0 &\cdots &k!
\end{array}
\right].
\end{equation}
This leads to a system which can be solved row by row. It is therefore very easy to implement and numerically well-conditioned.
\begin{rem}
In order to avoid large entries in the matrices $A^{lm}_k$ one may consider basis functions of form $\frac{1}{l!}\Phi^{(l)}_{(k+1)p(i),k}$, but we do not deal with the peculiar niceties of computation here.
\end{rem}
\subsection{Interpolation preserving linear systems of differential equations}
The preceding algorithm can be adapted it in order to construct a polynomial approximation $p$ of $f$ where the $k$ differential operators
\begin{equation}
L_if(x)=\sum_{j \leq q_i} a^i_{j}(x)\frac{d}{d x^j}f(x), i=1,\cdots,k
\end{equation}
are preserved on a discrete set of points $\Theta=\{x_0,\cdots ,x_N\}$ in the sense that
\begin{equation}
L_if(x_j)=L_ip(x_j) \mbox{ for } x_j\in \Theta.
\end{equation}
At this point the linear system of the operators $\left\lbrace L_i|1\leq i\leq k\right\rbrace $ is quite arbitrary; we just assume that the operators are defined pointwise, i.e. $x\rightarrow a^i_j(x)$ are classical functions which can be evaluated pointwise (at least on the set of interpolation points). Note that we do not ask about convergence of a family of interpolation polynomials to at this point. There are several possibilities to extend our preceding algorithm. One is the following.
Let
\begin{equation}
Q_i:=\left\lbrace j|a^i_j\neq 0\right\rbrace
\end{equation}
and define
\begin{equation}
L_i^m=\sum_{j\in Q_i, j\leq m}a^1_{i_j}(x)\frac{d^{i_j}}{d x^{i_j}}.
\end{equation}
We start with
\begin{equation}
Q_1=\left\lbrace i_{11},\cdots ,i_{1r_1}\right\rbrace ,
\end{equation}
and assume that
\begin{equation}
i_{11}< \cdots < i_{1r_1}
\end{equation}
We consider first the interpolation point $x_0$ and start with the following ansatz for the interpolation polynomial
\begin{equation}
p_{10}(x)=\sum_{i_{1j}\in Q_1}b^{10}_{i_j}(x-x_{0})^{i_{1j}}.
\end{equation}
We assume $f(x_0)=p_{10}(x_0)=0$ w.l.o.g. ; we shall see later how we interpolate values of $f$ different from zero at the other interpolation points $x_1,\cdots, x_N$. First we apply the operator
\begin{equation}
L_1^{i_1}\equiv a^1_{i_1}(x)\frac{d^{i_1}}{d x^{i_1}}
\end{equation}
to $f$ and $p_{10}$ at $x_0$. This leads to
\begin{equation}
i_1!b^{10}_{i_1}=a^1_{i_1}(x_0)\frac{d^{i_1} f}{d x^{i_1}}(x_0)~~\Rightarrow~~b^{10}_{i_1}=\frac{1}{i_1!}a^1_{i_1}\frac{d^{i_j} f}{d x^{i_1}}(x_0 )
\end{equation}
Inductively we assume that the coefficients $b^{10}_{i_j}$ have been defined up to the index $i_m$ for some $m< r_1$ and that the operator $L_1^{i_m}$ has been defined accordingly. We apply the operator
\begin{equation}
L^{i_{m+1}}_1\equiv L^{i_{m}}_1 + a^1_{i_{m+1}}(x)\frac{d^{i_{m+1}}}{d x^{i_{m+1}}}
\end{equation}
to $f$ and $p_{10}$ at $x_0$. For an integer $s$ with $m+1\leq s\leq r_1$ define
\begin{equation}
p_{10}^s(x)=\sum_{j=1}^{s}b^{10}_{i_j}(x-x_0)^{i_j}.
\end{equation}
Then we have
\begin{equation}
\begin{array}{ll}
L^{i_{m+1}}_1p_{10}(x_0)= L^{i_m}_1p_{10}(x_0) +a^1_{i_{m+1}}(x_0)\frac{d^{i_{m+1}}}{d x^{i_{m+1}}}p_{10}(x_0)=\\
\\
L^{i_m}_1p_{10}^{i_m}(x_0)+i_{m+1}! a^1_{i_{m+1}}(x_0)b^{10}_{i_{m+1}}=L^{i_{m+1}}_1f(x_0).
\end{array}
\end{equation}
This gives $b^{10}_{i_{m+1}}$.
Next inductively assume that an interpolation polynomial $p_{1k}$ has been constructed which interpolates $f$ on the set of interpolation points $\left\lbrace x_0,\cdots ,x_k\right\rbrace $ for some positive integer $k$ with $k< N $ subject to the condition
\begin{equation}
L_1 f(x_i)=p_{1k}(x_i) \mbox{ for } 1\leq i\leq k.
\end{equation}
First we extend that polynomial in order to interpolate $f$ at the point $x_{k+1}$. We consider the ansatz
\begin{equation}
p^0_{1(k+1)}(x)=p_{1k}(x)+b^{1(k+1)}_0\Pi_{l=0}^k(x-x_l)^{q_1}.
\end{equation}
We then get $b^{1(k+1)}_0$ from the equation
\begin{equation}
p^0_{1(k+1)}(x_{k+1})=f(x_{k+1}).
\end{equation}
The ansatz for $p_{1(k+1)}$ (i.e. the interpolation polynomial which preserves $L_1 f$ on the set of interpolation points $\left\lbrace x_1,\cdots ,x_{k+1}\right\rbrace$) is
\begin{equation}
p_{1(k+1)}(x)=p^0_{1(k+1)}(x)+\sum_{i_j\in Q_1} b^{1(k+1)}_{i_j}(x-x_{k+1})^{i_j}\Pi_{l=0}^k(x-x_l)^{q_1+1}
\end{equation}
and the determination of coefficient constants $b^{1(k+1)}_{i_j}$ is similar to the procedure for the interpolation point $x_0$ described above.
Proceeding inductively, we are lead to the polynomial $p_{1}$ which interpolates $f$ at the interpolation points of $\Theta =\left\lbrace x_0,\cdots ,x_N\right\rbrace $ such that
\begin{equation}
L_1 p_1(x_j)= L_1f(x_j) \mbox{ for all } x_j\in \Theta.
\end{equation}
Finally assuming that for some integer $s<k$ the polynomial $p_s$ satisfies the condition that
\begin{equation}
\begin{array}{ll}
p_s(x_j)=f(x_j) \mbox{ for } x_j\in \Theta\\
\\
L_ip_s(x_j)=L_if(x_j) \mbox{ for } x_j\in \Theta \mbox{ and } i\leq s,
\end{array}
\end{equation}
it is clear that we only need to consider the reduced operator
\begin{equation}
L_{s+1}\equiv \sum_{i_jj\in Q_{s+1}\setminus \cup_{i=1}^{s} Q_i}a^{s+1}_{i_j}(x)\frac{d^{i_j}}{d x^{i_j}}.
\end{equation}
and proceed analogously.
\section{Extension to the multivariate case}
Next we consider generalizations to the multivariate case. There are several possibilities but the most simple seems to be the following. First we formulate the problem in a way that will turn out to be useful in the context of polynomial interpolation of global solutions of linear systems of partial differential equations. In its most simple form it is a form of multivariate Newton interpolation: given a function
\begin{equation}
\begin{array}{ll}
f: S\subset {\mathbb R}^n\rightarrow {\mathbb R}
\end{array}
\end{equation}
we want to construct a polynomial
\begin{equation}
\begin{array}{ll}
p:S\subset {\mathbb R}^n\rightarrow {\mathbb R} \mbox{ such that}\\
\\
f(x_i)=p(x_i) \mbox{ for all } x_i \in D \subseteq S,
\end{array}
\end{equation}
where $D=\left\lbrace x_0,x_1,\cdots ,x_n\right\rbrace $ is some discrete sets of points in ${\mathbb R}^n$ whose coordinates will be denoted by superscript indices as $x_i^j,~j=1,\cdots ,n$. This is done then by recursive definition of polynomials $p_0, p_1,\cdots $. First, define
\begin{equation}
\begin{array}{ll}
p_0(x)\equiv f(x_0).
\end{array}
\end{equation}
Next, ansatz and equation
\begin{equation}
\begin{array}{ll}
p_1(x)\equiv f(x_0)+a_1\Pi_{i=1}^n(x^i-x_0^i)=f(x_1)
\end{array}
\end{equation}
leads to the determination of $p_1$ by
\begin{equation}
\begin{array}{ll}
a_1=\frac{f(x_1)-f(x_0)}{\Pi_{i=1}^n(x^i-x_0^i)}
\end{array}
\end{equation}
Next assume that $p_0,p_1,\cdots, p_q$ have been defined. Then ansatz and equation
\begin{equation}
\begin{array}{ll}
p_{q+1}(x_{q+1})\equiv p(x_{q+1})+a_{q+1}\Pi_{k=0}^{q}\Pi_{i=1}^n(x^i-x_k^i)=f(x_{q+1})
\end{array}
\end{equation}
leads to the determination of $p_{q+1}$ by
\begin{equation}
\begin{array}{ll}
a_{q+1}=\frac{f(x_{q+1})-p_q(x_{q+1})}{\Pi_{k=0}^q\Pi_{i=1}^n(x^i-x_k^i)}
\end{array}
\end{equation}
\subsection{Extension of Newton's method}
Next we extend a multivariate version of Newton's method, i.e. we design an algorithm that approximates $f$ up to the $\beta$-th derivative ($\beta=(\beta_1,\cdots ,\beta_n)$ being some multiindex) where we construct a polynomial
\begin{equation}
\begin{array}{ll}
q: S\subset {\mathbb R}\rightarrow {\mathbb R} \mbox{ such that}\\
\\
\frac{\partial f}{\partial x^{\gamma}}(x_i)
=\frac{\partial q}{\partial x^{\gamma}}(x_i)
\mbox{ for all }
x_i \in D\subseteq S \mbox{ and all } \gamma \leq \beta.
\end{array}
\end{equation}
where $\beta$ is given (i.e. the multivariate substitute for $k$ in the univariate case described above), and ordering is in the following sense:
\begin{defn}
Let $x^{\alpha}$ and $x^{\beta}$ be monomials in ${\mathbb R}\left[x_1,\cdots,x_n\right]$. We say that $x^{\alpha}>x^{\beta}$ ( lexicographical order) if $\sum_{i}\alpha^i > \sum_i \beta^i$ or $\sum_{i}\alpha^i=\sum_i \beta^i$, and in the difference $\alpha -\beta\in {\mathbb Z}^n$ the left-most non zero entity is positive.
\end{defn}
Now, let $\alpha_0,\alpha_1,\cdots,\alpha_m,\cdots $ an enumeration of multiindices with respect to this ordering. We define a sequence of polynomials $p_{\alpha_0}, p_{\alpha_1},\cdots ,p_{\alpha_m},\cdots $ recursively. First, let
\begin{equation}
p_{\alpha_0}(x)=a_{\alpha_0}+\sum_{\gamma \leq \beta}a_{\alpha_0\gamma}\Pi_{i=1}^n(x^i-x^i_{\alpha_0})^{\gamma_i}.
\end{equation}
If $p_{\alpha_0},\cdots ,p_{\alpha_{m-1}}$ have been defined, then we define
\begin{equation}
\begin{array}{ll}
p_{\alpha_m}(x)=p_{\alpha_{m-1}}(x)+\\
\\
\sum_{\gamma \leq \beta}a_{\alpha_{m-1}\gamma}\Pi_{i=1}^n(x^i-x^i_{\alpha_{m-1}})^{\gamma^i}\Pi_{j=0}^{m-1}\Pi_{i=1}^n(x^i-x^i_{\alpha_j})^{\beta^i+1}.
\end{array}
\end{equation}
This leads to a linear system to be solved for a vector $\left(a_{\alpha_0},\cdots , a_{\alpha_N\beta} \right)$ of length $(N+1)\left(\sum_i \beta^i+1\right)$
\begin{equation}
R_{\beta}\left[ \begin{array}{ccccccc}
a_{\alpha_0}\\
\vdots\\
a_{\alpha_0\beta}\\
a_{\alpha_1}\\
\vdots\\
a_{\alpha_N\beta}
\end{array}
\right] =\left[ \begin{array}{ccccccc}
f(x_{\alpha_0})\\
\vdots\\
f^{(\beta)}(x_{\alpha_0})\\
f(x_{\alpha_1})\\
\vdots\\
f^{(\beta)}(x_{\alpha_N})
\end{array}
\right]
\end{equation}
with
\begin{equation}
R_{\beta}:=\left[ \begin{array}{ccccccc}
&A^{00}_{\beta} &Z_{\beta} & Z_{\beta} &Z_{\beta} &\cdots &Z_{\beta}\\
&A^{10}_{\beta} &A^{11}_{\beta} & Z_{\beta} & Z_{\beta} &\cdots &Z_{\beta}\\
&A^{20}_{\beta} &A^{21}_{\beta} & A^{31}_{\beta} & Z_{\beta} &\cdots &Z_{\beta}\\
&\vdots &\vdots & \vdots &\vdots &\vdots &\vdots\\
&A^{N0}_{\beta} & A^{N1}_{\beta} & A^{N2}_{\beta} &A^{N3}_{\beta} &\cdots &A^{NN}_{\beta}\\
\end{array}
\right]
\end{equation}
We abbreviate $\sum \beta = \sum_i (\beta^i+1)$ and defining $p(m)=m\div \sum \beta$
we have
\begin{equation}
A^{ij}_k:=\left[ \begin{array}{ccccccc}
&\Phi_{{\tiny \sum} \beta p(i),\beta}(x_j) &\Phi_{{\tiny \sum} \beta p(i)+\beta_1,\beta}(x_j) & \Phi_{{\tiny \sum} \beta p(i)+\beta_2,\beta}(x_j) &\cdots &\Phi_{{\tiny \sum} \beta p(i)+\beta,\beta}(x_j)\\
&\Phi^{(\beta_1)}_{{\tiny \sum} \beta p(i),\beta}(x_j) &\Phi^{(1)}_{{\tiny \sum} \beta p(i)+\beta_1,\beta}(x_j) & \Phi^{(\beta_1)}_{(k+1)p(i)+\beta_2,\beta}(x_j) &\cdots &\Phi^{(\beta_1)}_{{\tiny \sum} \beta p(i)+\beta,\beta}(x_j)\\
&\Phi^{(\beta_2)}_{{\tiny \sum} \beta p(i),\beta}(x_j) &\Phi^{(\beta_2)}_{{\tiny \sum} \beta p(i)+\beta_1,\beta}(x_j) & \Phi^{(\beta_2)}_{\sum \beta p(i)+\beta_2,k}(x_j) &\cdots &\Phi^{(\beta_2)}_{\sum \beta p(i)+\beta ,\beta }(x_j)\\
&\vdots &\vdots & \vdots &\vdots &\vdots\\
&\Phi^{(\beta)}_{\sum\beta p(i),\beta}(x_j) &\Phi^{(\beta )}_{\sum\beta p(i)+\beta_1,\beta}(x_j) & \Phi^{(\beta)}_{\sum\beta p(i)+\beta_2,\beta}(x_j) &\cdots &\Phi^{(\beta )}_{\sum\beta p(i)+\beta,\beta}(x_j)
\end{array}
\right].
\end{equation}
\subsection{Multivariate Interpolation preserving linear systems of PDEs }
Similar to the univariate case one can adapt the preceding algorithm to the interpolation of multivariate functions, i.e. interpolate $f$ by a polynomial $p$ such that $f=p$, and
\begin{equation}
L_if(x)=L_ip(x) \mbox{ for } x\in \Theta.
\end{equation}
where $\Theta=\{x_0,\cdots ,x_N\}$ is the set of interpolation points, and the partial differential operators are defined by
\begin{equation}
L_if(x)=\sum_{|\alpha|\leq q_i} a^i_{\alpha}(x)\partial^{\alpha}f(x), ĩ=1,\cdots,k.
\end{equation}
The procedure is analogue to that described in Section 2.2. (cf.also \cite{Ka2}).
\section{Approximation of global solutions of linear partial differential equations}
We refine the algorithm further in order to solve linear partial differential equations globally. In this case the function $u$ to be approximated is not known. In this section we shall simply describe an algorithm which constructs a polynomial which satifies a linear system of partial differential equations
on an arbitrary set of interpolation points. It is not clear, however, if this polynomial approximation converges to the solution of the system. To ensure that and in order to estimate the rate of convergence we shall need the a priori estimates and regularity results. Note however, that the regularity constraints on the solution maybe low for problems on compact domains as any continuous solution functions $u$ can be approximated by a families of polynomial functions approximating $u$. Therefore, principally, the families of polynomial functions constructed here may approximate continuous global solutions in viscosity sense. An investigation of this problem will be considered elsewhere in a more general framework where we include some class of nonlinear problems. In order to make the basic ideas transparent we consider first scalar linear problems. We exemplify our algorithm first in the case of dimension $n=1$ and then generalize to the case $n>2$. What we have in mind here are elliptic equations but we need the ellipticity condition only when we wan to prove that the family of polynomials construxted converges to the global solutions. Then we exemplify our method in the case of a typical linear first order system. It is then clear how to generalize to systems of linear equations of any order.
\subsection{The case scalar second order equations of dimension $n=1$}
We consider the simple boundary value problem
\begin{equation}\label{bound1dim}
L_1u\equiv a(x)\frac{d^2 u}{dx^2}+b(x)\frac{d u}{dx}+c(x)u=f(x) \mbox{ on } (d,e)\subset {\mathbb R},
\end{equation}
with the boundary condition $u(d)=c_d$ and $u(e)=c_e$ (actually an ordinary differential equation). If $a(x)\geq \lambda >0$ for all $x\in {\mathbb R}$, then we have an elliptic operator, but this is not an assumption which we need to construct an univariate polynomial which satisfies the boundary problem on the interpolation points.
We start with the point $d$. We construct a list of polynomial $q_m, m\geq 0$. We define the $q_m$ in substeps. Let $p_0=a_0$. In order that $p_0$ satisfies the boundary condition at $x=d$ we impose
\begin{equation}
p_0=a_0=c_d
\end{equation}
Next we define
\begin{equation}
p_1(x)=a_0+a_1(x-d)
\end{equation}
In order to satisfy the second boundary condition we get
\begin{equation}
p_1(e)=a_0+a_1(e-d)=c_d+a_1(e-d)=c_e \Rightarrow a_1=\frac{c_e-c_d}{e-d}.
\end{equation}
It is clear that $p_1$ preserves the boundary conditions, i.e. $p(d)=u(d)=c_d$ and $p(e)=u(e)=c_e$.
Next let $x_0$ be the first interpolation point (any point in the interval $\left(d,e\right)$. We want to ensure that
\begin{equation}
a(x_0)\frac{d^2 p}{dx^2}(x_0)+b(x_0)\frac{d p}{dx}(x_0)+c(x_0)p(x_0)=f(x_0).
\end{equation}
In order to ensure this, we define a polynomial which is an extension of $p_0$ in three steps. First, define
\begin{equation}
p_2(x)=a_0+a_1(x-d)+a_4(x-x_0)^2(x-d)(x-e)
\end{equation}
Plugging in and evaluating at $x=x_0$ we get
\begin{equation}
a(x_0)2a_4(x_0-d)(x_0-e)+b(x_0)a_1+c(x_0)(a_0+a_1(x_0-d))=f(x_0)
\end{equation}
Since $a_0,a_1$ are known we get (recall that $x_0\neq d$ and $x_0\neq e$)
\begin{equation}
a_4=\frac{f(x_0)-c(x_0)(a_0+a_1(x_0-d))-b(x_0)a_1}{2a(x_0)(x_0-d)(x_0-e)}.
\end{equation}
Next define
\begin{equation}
p_3(x)=p_2(x)+a_3(x-x_0)(x-x_d)(x-x_e).
\end{equation}
Plugging in and evaluating at $x=x_0$ we get (assuming that )
\begin{equation}
\begin{array}{ll}
L_1p_3(x_0)=&
L_1p_2(x_0)+a(x_0)a_3(2(x_0-d)\\
\\
&+2(x_0-x_e))+b(x_0)a_3(x_0-d)(x_0-e) =f(x_0).
\end{array}
\end{equation}
Hence, (provided that $x_0\neq d$ and $x_0\neq e$),
\begin{equation}
a_3=\frac{f(x_0)-L_1p_2(x_0)}{a(x_0)(2(x_0-d)+2(x_0-e))+b(x_0)(x_0-d)(x_0-e)}
\end{equation}
Finally, finishing the first inductive step of recursive definition of the polynomial family $(q_m)_{m\in {\mathbb N}}$
\begin{equation}
p_4(x)=p_3(x)+a_2(x-d)(x-e).
\end{equation}
Plugging in and evaluating at $x=x_0$ we get (assuming that )
\begin{equation}
L_1p_4(x_0)=L_1p_3(x_0)+2a(x_0)a_2+b(x_0)((x_0-d)+(x_0-e))=f(x_0).
\end{equation}
Hence, (recall again that $x_0\neq d$ and $x_0\neq e$),
\begin{equation}
a_2=\frac{f(x_0)-L_1p_3(x_0)-b(x_0)((x_0-d)+(x_0-e))}{2a(x_0)((x_0-d)+2(x_0-e))}
\end{equation}
Now we can define
\begin{equation}
q_1(x)=p_4(x)
\end{equation}
Next assume that the polynomials $q_1, \cdots, q_k$ have been defined. This means that we have computed the polynomial coefficients $a_0,a_1,\cdots, a_{2+3k}$. Then $q_{k+1}$ is defined via
\begin{equation}
q_{k+1}(x)=q_k(x)+(x-d)^3(x-e)^3\Pi_{l=0}^{k}(x-x_l)^3z_k(x),
\end{equation}
where $z_k$ is a polynomial function which will be defined in three substeps.
First, let
\begin{equation}
q_{k+1,1}(x)=q_k(x)+a_{2+3(k+1)}(x-x_{k+1})^2(x-d)^3(x-e)^3\Pi_{l=0}^{k}(x-x_l)^3
\end{equation}
Plugging in leads to
\begin{equation}
\begin{array}{ll}
L_1q_{k+1,1}(x_{k+1})=&L_1q_k(x_{k+1})+a(x_{k+1})2a_{2+3(k+1)}(x_{k+1}-d)^3\times\\
\\
&(x_{k+1}-e)^3\Pi_{l=0}^{k}(x_{k+1}-x_l)^3=f(x_{k+1}).
\end{array}
\end{equation}
Hence,
\begin{equation}
a_{2+3(k+1)}=\frac{f(x_{k+1})-L_1q_k(x_{k+1})}{a(x_{k+1})2(x_{k+1}-d)^3(x_{k+1}-e)^3\Pi_{l=0}^{k}(x_{k+1}-x_l)^3}
\end{equation}
Next, let
\begin{equation}
q_{k+1,2}(x)=q_{k+1,1}(x)+a_{2+3k+2}(x-x_{k+1})(x-d)^3(x-e)^3\Pi_{l=0}^{k}(x-x_l)^3
\end{equation}
We define
\begin{equation}
R(x)=(x-d)^3(x-e)^3\Pi_{l=0}^{k}(x-x_l)^3.
\end{equation}
Plugging in leads to
\begin{equation}
\begin{array}{ll}
L_1q_{k+1,2}(x_{k+1})=L_1q_{k+1,1}(x_{k+1})+\\
\\
a(x_{k+1})2a_{2+3k+2}\frac{d^2}{dx^2}R(x_{k+1})+b(x_{k+1})a_{2+3k+2}\frac{d}{dx}R (x_{k+1})=f(x_{k+1}).
\end{array}
\end{equation}
Hence,
\begin{equation}
a_{2+3k+2}=\frac{f(x_{k+1})-L_1q_{k+1,1}(x_{k+1})}{a(x_{k+1})\frac{d^2}{dx^2}R (x_{k+1})+b(x_{k+1})\frac{d}{dx}R (x_{k+1})}
\end{equation}
Finally, let
\begin{equation}
\begin{array}{ll}
q_{k+1,3}(x)&=q_{k+1,2}(x)+a_{2+3k+1}(x-d)^3(x-e)^3\Pi_{l=0}^{k}(x-x_l)^3\\
\\
&=a_{2+3k+1}R(x)
\end{array}
\end{equation}
Plugging in leads to
\begin{equation}
\begin{array}{ll}
L_1q_{k+1,3}(x_{k+1})=L_1q_{k+1,2}(x_{k+1})+
a(x_{k+1})a_{2+3k+1}\frac{d^2}{dx^2}R(x_{k+1})\\
\\
+b(x_{k+1})a_{2+3k+1}\frac{d}{dx}R (x_{k+1})+c(x_{k+1})a_{2+3k+1}R(x_{k+1})=f(x_{k+1}).
\end{array}
\end{equation}
Hence,
\begin{equation}
a_{2+3k+2}=\frac{f(x_{k+1})-L_1q_{k+1,2}(x_{k+1})}{a(x_{k+1})\frac{d^2}{dx^2}R(x_{k+1})
+b(x_{k+1})\frac{d}{dx}R (x_{k+1})+c(x_{k+1})R(x_{k+1})}.
\end{equation}
It is clear how to proceed inductively in order to get a family of interpolation polynomials which satisfy the differential equation on an increasing set of interpolation points. Note,however,that we have not used any structural information about the coefficients at this point. This means that the equation may be ill-posed,and convergence cannot be guaranteed.
\subsection{The case of scalar linear partial differential equations}
For a positive integer $k$ consider an equation of form
\begin{equation}
L_ku\equiv\sum_{|\alpha|\leq k}a_{\alpha}(x)\frac{\partial^{\alpha}u}{\partial x^{\alpha}}=g(x),
\end{equation}
to be solved on the domain $\Omega$ where
\begin{equation}
u\mbox{\Big |}_{\partial \Omega}=f
\end{equation}
What we have in mind is an elliptic equation f order $k$, but ellipticity is not required in order to describe the algorithm which produces a family of multivariate polynomials which satisfy the equation on a set of interpolation points in $\Omega$. Ellipticity becomes important when we want to show that the family of polynomial converges to the solution of the equation (assuming that there is an unique global solution). For simplicity of notation we consider the case $k=2$, i.e. the situation of \eqref{parab}. Assume that $f\in C^k$ and choose a discrete interpolation set $\Theta_b\subset \partial\Omega$. Then we can apply the extended Newton algorithm of Section 3 in order to produce a polynomial $p_b:{\mathbb R}^n\rightarrow {\mathbb R}$ such that
\begin{equation}
\begin{array}{ll}
p_b(x)=f(x) \mbox{ for all } x\in \Theta_b\\
\\
\frac{\partial p_b}{\partial x^{\alpha}}=\frac{\partial p_b}{\partial x^{\alpha}}\mbox{ for all } \alpha \mbox{ with } |\alpha|\leq l \mbox{ and }x\in \Theta_b
\end{array}
\end{equation}
We assume that $\Theta_b=\left\lbrace x_{0b},\cdots ,x_{Mb}\right\rbrace $ with $x_{ib}=(x^1_{ib},\cdots ,x^n_{ib})$ and define
\begin{equation}
\Phi_b(x)=\Pi_{i=0b}^{Mb}\Pi_{j=1}^n(x^j-x^j_{i})^{l+1}.
\end{equation}
Next let $\theta_{int}\subset \Omega \setminus \partial \Omega$ be a set of interpolation points in the interior of $\Omega$. Let
\begin{equation}
\Theta_{int}=\left\lbrace x_0,\cdots ,x_N\right\rbrace .
\end{equation}
We enumerate (case $k=2$) the $q:=\frac{(n+1)n}{2}$ diffusion coefficients $a_{\alpha_1},\cdots ,a_{\alpha_q}$ (arbitrary order), where we assume $\alpha_l=(\alpha_{l1},\alpha_{l2})$ and define first $q$ polynomials $p^{\mbox{{\tiny diff}},l}_0(x), l=1,\cdots,q$. Let
\begin{equation}
p^{\mbox{{\tiny diff}},1}_0(x)=p_b(x)+\Phi_b(x)a_{\alpha_1}(x^{\alpha_{11}}-x_0^{\alpha_{11}})(x^{\alpha_{12}}-x_0^{\alpha_{12}}).
\end{equation}
Then we have
\begin{equation}
L_2p^{\mbox{{\tiny diff}},1}_0(x_0)=L_2p_b(x_0)+\Phi_b(x_0)(1+\delta_{\alpha_{11}\alpha_{12}})a_{\alpha_1}=f(x_0),
\end{equation}
which leads to
\begin{equation}
a_{\alpha_1}=\frac{f(x_0)-L_2p_b(x_0)}{(1+\delta_{\alpha_{11}\alpha_{12}})\Phi_b(x_0)}
\end{equation}
Having defined $p^{\mbox{{\tiny diff}},1}_0(x),\cdots , p^{\mbox{{\tiny diff}},l}_0(x)$ (and therefore computed $a_{\alpha_1},\cdots ,a_{\alpha_l}$) we
define
\begin{equation}
p^{\mbox{{\tiny diff}},l+1}_0(x)=p^{\mbox{{\tiny diff}},l}_0(x)+\Phi_b(x)a_{\alpha_{l+1}}(x^{\alpha_{(l+1)1}}-x_0^{\alpha_{(l+1)1}})(x^{\alpha_{(l+1)2}}-x_0^{\alpha_{(l+1)2}}),
\end{equation}
and evaluation leads to
\begin{equation}
a_{\alpha_{l+1}}=\frac{f(x_0)-L_2p^{\mbox{{\tiny diff}},l}_0(p_(x_0)}{(1+\delta_{\alpha_{(l+1)1}\alpha_{(l+1)2}})\Phi_b(x_0)}.
\end{equation}
Proceeding inductively we get a $p^{\mbox{{\tiny diff}},q}_0(x)$ which equals together with its derivatives up to order $l$ the function $f$ and such that the diffusion part of the operator applied to $p^{\mbox{{\tiny diff}},q}_0(x)$ equals $g$ at $x_0$. It is now clear how this procedure can be extended such that an extended polynomial $p_0(x)$ equals together with its derivatives up to order $l$ the function $f$ and such that the total operator applied to $p_0(x)$ equals $g$ at $x_0$. As in Section 3 the ansatz for the interpolation polynomial $p_{\Theta}$ which satisfies the linear equation on the set of interpolation points $\Theta=\left\lbrace x_0,\cdots ,x_N \right\rbrace $ then is
\begin{equation}
p_{\Theta}(x)=\sum_{i=0}^N \Pi_{j=1}^i\Pi_{k=1}^n(x^k-x^k_{j-1})^3p_i(x),
\end{equation}
where $p_i$ for $i\geq 2$ are then constructed as $p_0$ above.
\subsection{The case of a linear hyperbolic equation}
We consider the hyperbolic equation mentioned above of the form
\begin{equation}
Lu=f \mbox{ in } \Omega,
\end{equation}
where
\begin{equation}
Lu\equiv \sum_{ij} h_{ij}\frac{\partial u}{\partial x_ii\partial x_j}+\sum_i\frac{\partial}{\partial x_j}+c(x)u
\end{equation}
and $(h_{ij})$ is a symmetric matrix of signature $(n,1)$, if $\mbox{dim}\Omega =n+1$. Note that the operator $L$ can be transformed into the form
\begin{equation}
Lu\equiv \square u + L_1u,
\end{equation}
where $L_1u$ is some first order differential operator on $\Omega$. We assume the initial conditions
\begin{equation}
u=g \mbox{ and } du=\omega ,
\end{equation}
where $g$ and $\omega$ ($1-form$) are initial data.
It is clear that the algorithm described in the preceding section can be used in the present situation. Later we shall see that energy estimates imply convergence of the scheme.
\section{Further refinements: collocation and parallelization}
Numerical experiments show that the coefficients of the recursively computed polynomials have to be computed with increasing accuracy in order to control effects of the truncation error of the coefficients of the polynomials. In the numerical example below, where we computed a polynomial approximation of degree $74$ of the locally analytic function
\begin{equation}
x\rightarrow \frac{1}{1+x}
\end{equation}
and its derivatives up to order $3$ on the interval $[0,5.4]$ such effects are not observed. However, if we increase the number of derivatives to be approximated up to order $k=10$ and increase the number of interpolation points, effects of truncation errors can be observed for polynomials of degrees larger than 200. The error increases as $|x|$ becomes large and truncation errors increase. This error can be reduced by a more precise representation of the computational approximation of the real numbers involved in the computation. However, as we point out in this section, we can compute $m$ polynomials $p_1^{\Theta_1},\cdots,p_m^{\Theta_m}$ of degree $N_1,N_2\cdots N_m$ parallel which interpolate a given linear system of partial differential equations on some interpolation sets $\Theta_1,\cdots, \Theta_m$ using our basic algorithm, and then compute one polynomial $p_{\sum\Theta}$ which interpolates the same linear system of partial differential equations on the set
$\sum\Theta =\Theta_1,\cup\cdots, \cup\Theta_m$. It turns out that this can be in such a way that the truncation error of the resulting polynomial $p_{\sum\Theta}$ is much smaller than in case of a direct extension of one polynomial $p_{\Theta_i}$ using the basic algorithm. We call this method the collocation extension of our basic algorithm. We shall assume that the sets of interpolation points are mutually disjunct, i.e.
\begin{equation}
\Theta_i\cap \Theta_j=\oslash \mbox{ iff } i\neq j.
\end{equation}
It is clear that the computation of the polynomials $p_1^{\Theta_1},\cdots,p_m^{\Theta_m}$ can be done parallel and only the step of synthesizing has to be done non-parallel. Next we describe that step in case of two polynomials for simplicit of notation. Extension to $m>2$ polynomials will be clear from that description. So let $\Theta_1,\Theta_2\subset \Omega \subset {\mathbb R}^n$ be two discrete finite sets of interpolation points of a linear system of partial differential equations $Lu=f$ to be solved on a domain $\Omega$ and such that $\Theta_1\cap\Theta_2=\oslash$. We write down the polynomial in the univariate case because this simplifies the notation, and the multivariate case is quite similar.
Then we define a regular polynomial interpolation formula on $\Theta_1\cup\Theta_2$ by
\begin{equation}
\begin{array}{ll}
\sum_{j=1}^N\Pi_{k\neq j, }\frac{(x-x^{\Theta_1}_k)^{k+1}}{(x_j^{\Theta_1}-x_k^{\Theta_1})^{k+1}}\Pi_{i=1}^{M}
\frac{(x-x^{\Theta_2}_i)^{k+1}}
{(x^{\Theta_1}_j-x_i^{\Theta_2})^{k+1}}
p_{\Theta_1}(x)\\
\\
-\sum_j\Pi_{p\in\{1,2\}~l\neq j}
(x-x_l^{\Theta_p})^{k+1}
a^j_{i,1}(x-x_j^{\Theta_1})^i\\
\\
+\sum_{j=1}^M\Pi_{k\neq j}
\frac{(x-x^{\Theta_1}_k)^{k+1}}{(x_j^{\Theta_1}
-x_k^{\Theta_1})^{k+1}}\Pi_{i=1}^{N}
\frac{(x-x^{\Theta_2}_i)^{k+1}}{(x^{\Theta_1 }_j-x_i^{\Theta_2})^{k+1}}p_{\Theta_2}(x)\\
\\
-\sum_j\Pi_{p\in\{1,2\}~l\neq j}
(x-x_l^{\Theta_p})^{k+1}
a^j_{i,2}(x-x_j^{\Theta_2})^i\\
\\
=:\sum_j q^{1j}_{\Theta_2,\Theta_1}(x)p_{\Theta_1}(x)+h_1^a(x)\\
\\
+\sum_j q^{2j}_{\Theta_1,\Theta_2}(x)p_{\Theta_2}(x)+h_2^a(x),
\end{array}
\end{equation}
where the constants $a^j_{i,p},p\in\{1,2\}$ are computed recursively as follows:
For each $j$ we can define $a^j_{0,p}=0$. If $a_{1,p}^j,\cdots, a_{l-1,p}^j$ are determined, then compute $a_{1,p}^j$ via
\begin{equation}
\sum_{1\leq r\leq l}\left(\begin{array}{cc}l\\r\end{array} \right)
D_x^rq^{pj}_{\Theta_1\Theta_2}(x_j)D^{l-r}_x p_{\Theta_p}(x_j)=D^l_x h^a_p(x_j)
\end{equation}
for each $j$.
Note that this 'synthesis of polynomials' improves the computational power of our method dramatically. In the example below, where we approximate a simple locally analytic function
\begin{equation}
x\rightarrow \frac{1}{1+x}
\end{equation}
(with convergence radius $1$) and its derivatives up to the third derivative on the interval $[0,5.4]$ with $19$ interpolation points $\Theta_1=\{k0.3|k=0,\cdots 18\}$ we compute a polynomial of degree $74$ in half a minute on a modest laptop machine. If we want to compute a polynomial which gives the same kind of approximation on the interval $[0,5836,8]$ it will take several weeks. However, using parallelization and synthesis, and using the rough estimate that synthesis takes in average the same time as building the $1024$ basis polynomials of degree $74$ on the intervals $[0,5.4]$ and $[k5.7,(k+1)5.7], k=1,\cdots 1023$ we need $10$ steps of parallel synthesis of pairs of polynomials of cost of a less than a minute to get a regular approximation polynomial which is at least of degree $75776$!
It is clear fromthe preceding remarks how to extend this to the multivariate case (cf. also \cite{Ka2}).
\section{Convergence of polynomial approximations of global solutions of linear elliptic PDE and error estimates by a priori estimates}
Up to now we just considered (regular) polynomial interpolation on given sets of interpolation points.
In this section we consider standard problems in the theory of linear partial differential equations and derive the convergence of our algorithm and error estimates (as the mesh size of the sets of interpolation points converges to zero).
We start with elliptic equations and then consider hyperbolic problems. Similar results can be obtained for initial-value boundary problems for parabolic equations (since analogous error estimates can be obtained). In this case, however, it turns out that (at least for regular data) a WKB-expansion of the fundamental solution has better convergence properties and error estimates can be obtained by Safanov a priori estimates (cf. \cite{KKS} and \cite{Ka}). We shall consider application of our algorithm to this case in the next section. Note that Since to get an error from simple Taylor expansion in genera, because the interpolated function is unknown.
\subsection{Convergence for elliptic equations with regular data}
We consider the Dirichlet problem for elliptic equations, i.e. an equation of the form
\begin{equation}\label{elliptsc}
Lu=\sum_{|\alpha|\leq k}a_{\alpha}(x)\frac{\partial u}{\partial x^{\alpha}}=f(x)
\end{equation}
on a domain $\Omega \subseteq {\mathbb R}^n$. coefficient functions
\begin{equation}
x\rightarrow a_{\alpha}(x),
\end{equation}
and where $u$ is given on the boundary, i.e.
\begin{equation}
u{\big|}_{\partial \Omega}=g.
\end{equation}
We consider the classical case where $k=2$ and $\Omega$ is bounded. We assume uniform ellipticity, i.e. there exists a constant $K>0$ such that for all $x\in \Omega$
\begin{equation}\label{ell}
\sum_{ij=1}^n a_{ij}(x)\xi_i\xi_j\geq K|\xi|^2.
\end{equation}
In the classical case Schauder boundary estimates are available. We cite them in the context of a standard existence result. for a scalar function $h$ in $\Omega$ we introduce the norms
\begin{equation}
\|h\|_{k}^{bd}=\sum_{j=0}^k\sum_{|\delta|=j}\|D^{\delta}h\|_0
\end{equation}
where
\begin{equation}
\|h\|_0:=\sup_{x\in \Omega}|h(x)|,
\end{equation}
and
\begin{equation}
\|h\|_{k+\alpha}^{bd}=\|h\|_{k}^{bd}+\sum_{j=0}^k\sum_{|\delta|=j}H^{bd}_{\alpha}\left( D^{\delta}h\right),
\end{equation}
where $H_{\alpha}^{bd}(f)$ is the H\"older coefficient of a given function $f$ in $\Omega$.
We assume that the coefficient functions $x\rightarrow a_{ij}(x)$ (diffusion terms), $x\rightarrow b_i(x)$ (drift terms), the potential term ($x\rightarrow c(x)$), and the right side $x\rightarrow f(x)$ are uniformly H\"older continuous (exponent $\alpha$) such that
\begin{equation}\label{coco}
\|a_{ij}\|
^{bd}_{\alpha}
\leq C,
\|b_{i}\|^{bd}_{\alpha}\leq C, \|c\|^{bd}_{\alpha}\leq C, \|f\|^{bd}_{\alpha}\leq C
\end{equation}
for some generic constant $C$.
\begin{thm}
Assume that conditions \eqref{ell} and \eqref{coco} hold, and assume that $c\leq 0$. Furthermore, assume that $\partial \Omega$ belongs to $C^{2+\alpha}$ and that $g$ belongs to $C_{2\alpha}^bd$. Then the inequalities
\begin{equation}
\begin{array}{ll}
\|u\|_{2+\alpha}^{bd}&\leq C\left(\|g\|+\|u\|_0 +\|f\|_{\alpha}^{bd} \right)\\
\\
&\leq C\left(\|g\|+\sup_{\partial \Omega}|g|+C\sup_{\Omega}|f| +\|f\|_{\alpha}^{bd} \right)
\end{array}
\end{equation}
hold. Furthermore there exist a unique solution $u\in C_{2+\alpha}^{bd}$ to the Dirichlet problem.
\end{thm}
The interpolation polynomial $p_{\Theta}$ described in the preceding section is by construction such that
\begin{equation}\label{elliptsc}
L(u-p_{\Theta})=\sum_{|\alpha|\leq k}a_{\alpha}(x)\frac{\partial (u-p_{\theta})}{\partial x^{\alpha}}=\Delta f(x),
\end{equation}
and
\begin{equation}
u-p_{\Theta}{\big|}_{\partial \Omega}=\Delta g.
\end{equation}
It follows that
\begin{thm} Assume the same conditions as in theorem 6.1.. Then
\begin{equation}
\begin{array}{ll}
\|u-p_{\Theta}\|_{2+\alpha}^{bd}
\leq C\left(\|\Delta g\|+\sup_{\partial \Omega}|\Delta g|+C\sup_{\Omega}|\Delta f| +\|\Delta f\|_{\alpha}^{bd} \right)
\end{array}
\end{equation}
\end{thm}
Note that this implies an $L^2$-error even for the second derivatives of the global solution function, hence essentially an estimate in $H^2(\Omega)$. Even stronger results can be obtained if additional equations for the derivatives of $u$ are considered (cf. \cite{Ka2}).
\subsection{Convergence for a hyperbolic linear partial differential equations equation}
We consider again the hyperbolic equation mentioned above of the form
\begin{equation}\label{hyperb}
Lu=f \mbox{ on } O\subset \Omega,
\end{equation}
where
\begin{equation}
Lu\equiv \sum_{ij} h_{ij}\frac{\partial u}{\partial x_ii\partial x_j}+\sum_i\frac{\partial}{\partial x_j}+c(x)u
\end{equation}
and $(h_{ij})$ is a symmetric matrix of signature $(n,1)$, if $\mbox{dim}\Omega =n+1$. We assume that some $O\subset \Omega$ is bounded by two spacelike surfaces$\Sigma_i$ and $\Sigma_e$ and swept out by a family of spacelike surfaces $\Sigma_e(s)$. Recall that the initial conditions
\begin{equation}\label{boundary}
u=g \mbox{ and } du=\omega .
\end{equation}
Let $p$ be the interpolation polynom described above such that
\begin{equation}
L(u-p)=\Delta f \mbox{ on } O\subset \Omega.
\end{equation}
\begin{equation}\label{boundary}
u-p=\Delta g \mbox{ and } du=\Delta \omega .
\end{equation}
Then we use the following energy estimate
\begin{prop}
Let $u$ solve the intial value problem \eqref{hyperb}, \eqref{boundary}. Let
\begin{equation}
O(s)=\overline{O}\cap \left\lbrace t\leq s \right\rbrace
\end{equation}
(swept out by the spacelike surfaces $\Sigma_e(s)$). Then
\begin{equation}
\begin{array}{ll}
\int_{O(s)}|u|^2dV\leq \\
\\
\int_{\Sigma^b_i(s)} |g|^2dS+C(s-s_0)\int_{\Sigma_i}\left( |g|^2+|\omega|^2 \right) dS+C\int_{O(s)}|f|^2dV
\end{array}
\end{equation}
for $s\in \left[s_0,s_1\right]$.
\end{prop}
This implies
\begin{thm} With the same assumptions as in propostion 6.2. we have
\begin{equation}
\begin{array}{ll}
\int_{O(s)}|u-p|^2dV\leq \\
\\
\int_{\Sigma^b_i(s)} |\Delta g|^2dS+C(s-s_0)\int_{\Sigma_i}\left( |\Delta g|^2+|\omega|^2 \right) dS+C\int_{O(s)}|\Delta f|^2dV
\end{array}
\end{equation}
for $s\in \left[s_0,s_1\right]$.
\end{thm}
Hence the polynomial interpolation scheme described in Section 4 leads to $L^2$-convergence. One can improve this scheme assuming regularity of solutions and considering systems of equations including equations for derivatives of the solution $u$ (cf. \cite{Ka2}).
\section{Applications to parabolic equations (connection to WKB-expansions)}
We summarize some results concerning WKB-expansions of parabolic equations (cf. \cite{Ka} for details).
Let us consider the parabolic diffusion operator
\begin{equation}\label{PPDE}
\begin{array}{l}
\frac{\partial u}{\partial t}-Lu\equiv \frac{\partial u}{\partial t}-\frac{1}{2}\sum_{i,j}a_{ij}\frac{\partial^2 u}{\partial x_i\partial x_j}-
\sum_i b_i\frac{\partial u}{\partial x_i},
\end{array}
\end{equation}
where the diffusion coefficients $a_{ij}
$ and the first order coefficients $b_i$ in \eqref{PPDE}
depend on the spatial variable $x$ only. In the following let $\delta t=T-t$, and let
\begin{equation}
(x,y)\rightarrow d(x,y)\ge0,~~(x,y)\rightarrow c_k(x,y),~k\geq 0
\end{equation}
denote some smooth functions on the domain ${\mathbb R}^n\times {\mathbb R}^n$.
Then a set of (simplified) conditions sufficient for pointwise valid WKB-representations
of the form
\begin{equation}\label{WKBrep}
p(\delta t,x,y)=\frac{1}{\sqrt{2\pi \delta t}^n}\exp\left(-\frac{d^2(x,y)}{2\delta t}+\sum_{k= 0}^{\infty}c_k(x,y)\delta t^k\right),
\end{equation}
for the solution $(t,x)\rightarrow p(\delta t,x,y)$.
\begin{equation}
\begin{array}{ll}
\frac{\partial u}{\partial \delta t}- Lu =0, \mbox{with final value}\\
\\
u(0,x,y)=\delta(x-y),
\end{array}
\end{equation}
is given by
\begin{itemize}
\item[(A)] The operator $L$ is uniformly elliptic in ${\mathbb R}^n$, i.e. the matrix norm of $(a_{ij}(x))$ is bounded below and above by $0<\lambda <\Lambda <\infty$ uniformly in~$x$,
\item[(B)] the smooth functions $x\rightarrow a_{ij}(x)$ and $x\rightarrow b_i(x)$ and all their derivatives are bounded.
\end{itemize}
For more subtle (and partially weaker conditions) we refer to \cite{Ka}. We consider the case where there exists a global transformation to the Laplace operator.
If we add the uniform boundedness condition
\begin{itemize}
\item[(C)] there exists a constant $c$ such that for each multiindex $\alpha$ and for all $1\leq i,j,k\leq n,$
\begin{equation}\label{unibd}
{\Big |}\frac{\partial
a_{jk}}{\partial x^{\alpha}}{\Big |},~{\Big |}\frac{\partial
b_{i}}{\partial x^{\alpha}}{\Big |}\leq c\exp\left(c|x|^2 \right),
\end{equation}
\end{itemize}
then the function $d^2=(x-y)^2$ (in the transformed coordinates and $c_k$ equals its Taylor expansion around $y\in {\mathbb R}^n$, i.e $c_k,k\geq 0$ have the power series representations
\begin{equation}
\begin{array}{ll}
c_k(x,y)&=\sum_{\alpha}c_{k,\alpha}(y)\delta x^{\alpha}, k\geq 0.
\end{array}
\end{equation}
Moreover $c_k, k\geq 0$ are determined by the recursive equations
\begin{equation}\label{c01e}
-\frac{n}{2}+\frac{1}{2}Ld^2+\frac{1}{2}\sum_{i} \left( \sum_j\left( a_{ij}(x)+a_{ji}(x)\right) \frac{d^2_{x_j}}{2}\right) \frac{\partial c_{0}}{\partial x_i}(x,y)=0,
\end{equation}
where the boundary condition
\begin{equation}\label{c01b}
c_0(y,y)=-\frac{1}{2}\ln \sqrt{\mbox{det}\left(a^{ij}(y) \right) }
\end{equation}
determines $c_0$ uniquely for each $y\in {\mathbb R}^n$, and
for $k+1\geq 1$ we have
\begin{equation}\label{1gaa}
\begin{array}{ll}
(k+1)c_{k+1}(x,y)+\frac{1}{2}\sum_{ij} a_{ij}(x)\Big(
\frac{d^2_{x_i}}{2}\frac{\partial c_{k+1}}{\partial x_j}
+\frac{d^2_{x_j}}{2} \frac{\partial c_{k+1}}{\partial x_i}\Big)\\
\\
=\frac{1}{2}\sum_{ij}a_{ij}(x)\sum_{l=0}^{k}\frac{\partial c_l}{\partial x_i} \frac{\partial c_{k-l}}{\partial x_j}
+\frac{1}{2}\sum_{ij}a_{ij}(x)\frac{\partial^2 c_k}{\partial x_i\partial x_j}
+\sum_i b_i(x)\frac{\partial c_{k}}{\partial x_i},
\end{array}
\end{equation}
with boundary conditions
\begin{equation}\label{Rk}
c_{k+1}(x,y)=R_k(y,y) \mbox{ if }~~x=y,
\end{equation}
$R_k$ being the right side of \eqref{1gaa}. In case $a_{ij}=\delta_{ij}$
we have the representations
\begin{equation}\label{eqd}
d^2(x,y)=\sum_i (x_i-y_i)^2,
\end{equation}
\begin{equation}\label{solc0}
c_0(x,y)=\sum_i(y_i-x_i)\int_0^1 b_i(y+s(x-y))ds,
\end{equation}
and
\begin{equation}\label{solck}
c_{k+1}(x,y)=\int_0^1 R_k(y+s(x-y),y)s^{k}ds,
\end{equation}
$R_k$ being again the right-hand-side of \eqref{1gaa}. The integrals can be taken out if the functions $x\rightarrow b_i(x)$ are given by multivariate power series and error estimates for the truncation error in space and time are obtained (cf. \cite{Ka, KKS}). However, even if the coefficient functions are analytic, i.e. equal locally a power series, it is not possible to approximate such a function globally by their Taylor polynomial. As an example consider the equation
\begin{equation}
\frac{\partial u}{\partial t}-\frac{1}{2}\Delta u -\sum_i^n \frac{1}{1+x_i}\frac{\partial u}{\partial x_i}=0
\end{equation}
Here, the coefficient functions
\begin{equation}\label{coeff}
x_i\rightarrow \frac{1}{1+x_i} =b_i(x)
\end{equation}
are univariate locally analytic function with convergence radius $1$. Such type of equations occur in praxis of finance (cf. \cite{FrKa, KKS}). In order to obtain an approximation of the WKB-expansion say up to order $5$, i.e. compute the coefficient functions
\begin{equation}
x\rightarrow c_k(x,y), k=0,\cdots, 5 ,
\end{equation}
we need a global approximation of the functions \eqref{coeff} and their derivatives up to order 10! This is due to the recursion equations for the $c_k, k\geq 1$ which involve second derivatives of $c_{k-1}$. If we have $20$ interpolation points on the $x$-axis this implies that our regular interpolation algorithm computes a polynomial of order $231$. We do the computation in a more modest example in order to keep the resulting polynomial representable on one page in the following section.
\section{A numerical example}
The following polynomial is a similtaneous approximation of the function
\begin{equation}
\begin{array}{ll}
f: [0,5,4]\subseteq {\mathbb R}\rightarrow {\mathbb R}\\
\\
f(x)=\frac{1}{1+x}
\end{array}
\end{equation}
and its first, second, and third derivative on the domain $[0,5,4]$ with $19$ interpolation points. Hence the degree of this univariate polynomial is $74$. Note that the convergence radius of $f$ is $1$.
\begin{equation}
p_{76}(x)=\sum_{m=0}^{75}a_m(x-x_{m \mbox{{\tiny div}}4})^{m \mbox{{\tiny mod}}4}\Pi_{l=0}^{{ m \mbox{{\tiny div}} 4 -1}}(x-x_l)^{4}
\end{equation}
Note that
\begin{equation}
\frac{d^n}{d x^n}\left(\frac{1}{1+x}\right)|_{x=0}=\frac{(-1)^n n!}{(1+x)^{n+1}}|_{x=0}=(-1)^n n!
\end{equation}
This leads to the values $a_0=1$, $a_1=-1$, $a_2=1$, and $a_3=-1$ for the coefficients of our interpolation polynomial at $x_0=0$. Note that the coefficients $a_i$ of the interpolation polynomial tend to become smaller for large indexes $i$ as you would expect.
\begin{equation}
\begin{array}{ll}
a_0=1.0\\
\\
a_1=-1.000000000000,\hspace{0.1cm}
a_2=1.000000000000,\hspace{0.1cm}
a_3=-1.000000000000\\
\\
a_4=0.769230765432, \hspace{0.1cm}
a_5=-0.591715921811,\hspace{0.1cm}
a_6=0.455165657066\\
\\
a_7=-0.350124490177,\hspace{0.1cm}
a_8=0.218822618520,\hspace{0.1cm}
a_9=-0.136753442090\\
\\
a_{10}=0.085452696109,\hspace{0.1cm}
a_{11}=-0.053404312398,\hspace{0.1cm}
a_{12}=0.028144617397\\
\\
a_{13}=-0.014953338935,\hspace{0.1cm}
a_{14}=0.008262188243,\hspace{0.1cm}
a_{15}=-0.005370784216\\
\\
a_{16}=0.003734873988,\hspace{0.1cm}
a_{17}=-0.003633027430,\hspace{0.1cm}
a_{18}=0.004645502211\\
\\
a_{19}=-0.006813570816,\hspace{0.1cm}
a_{20}=0.007086610952,\hspace{0.1cm}
a_{21}=-0.007144432312\\
\\
a_{22}=0.006238342564,\hspace{0.1cm}
a_{23}=-0.002646146059,\hspace{0.1cm}
a_{24}=-0.002374282360\\
\\
a_{25}=0.008387675067,\hspace{0.1cm}
a_{26}=-0.015766978592,\hspace{0.1cm}
a_{27}=0.024857498610\\
\\
a_{28}=-0.025373687351,\hspace{0.1cm}
a_{29}=0.025025735340,\hspace{0.1cm}
a_{30}=-0.023974098174\\
\\
a_{31}=0.022321168853,\hspace{0.1cm}
a_{32}=-0.015945627926,\hspace{0.1cm}
a_{33}=0.011155207224\\
\\
a_{34}=-0.007619506803,\hspace{0.1cm}
a_{35}=0.005069120726,\hspace{0.1cm}
a_{36}=-0.002759684498\\
\\
a_{37}=0.001479716734,\hspace{0.1cm}
a_{38}=-0.000790172686,\hspace{0.1cm}
a_{39}=0.000430223475\\
\\
a_{40}=-0.000208511304,\hspace{0.1cm}
a_{41}=0.000106279314,\hspace{0.1cm}
a_{42}=-0.000056281013\\
\\
a_{43}=0.000028862889,\hspace{0.1cm}
a_{44}=-0.000011153733,\hspace{0.1cm}
a_{45}=0.000002201139\\
\\
a_{46}=0.000002233629,\hspace{0.1cm}
a_{47}=-0.000004202246,\hspace{0.1cm}
a_{48}=0.000003699371\\
\\
a_{49}=-0.000002870941,\hspace{0.1cm}
a_{50}=0.000002068390,\hspace{0.1cm}
a_{51}=-0.000001402599\\
\\
a_{52}=0.000000753699,\hspace{0.1cm}
a_{53}=-0.000000375935,\hspace{0.1cm}
a_{54}=0.000000159621\\
\\
a_{55}=-0.000000037499,\hspace{0.1cm}
a_{56}=-0.000000015690,\hspace{0.1cm}
a_{57}=0.000000032004\\
\\
a_{58}=-0.000000031616,\hspace{0.1cm}
a_{59}=0.000000023406,\hspace{0.1cm}
a_{60}=-0.000000010968\\
\\
a_{61}=0.000000001564,\hspace{0.1cm}
a_{62}=0.000000005590,\hspace{0.1cm}
a_{63}=-0.000000011521\\
\\
a_{64}=0.000000012190,\hspace{0.1cm}
a_{65}=-0.000000012095,\hspace{0.1cm}
a_{66}=0.000000011769\\
\\
a_{67}=-0.000000011467,\hspace{0.1cm}
a_{68}=0.000000008698,\hspace{0.1cm}
a_{69}=-0.000000006652\\
\\
a_{70}=0.000000005132,\hspace{0.1cm}
a_{71}=-0.000000003988,\hspace{0.1cm}
a_{72}=0.000000002523\\
\\
a_{73}=-0.000000001600,\hspace{0.1cm}
a_{74}=0.000000001015,\hspace{0.1cm}
a_{75}=-0.000000000643
\end{array}
\end{equation}
\section{Conclusion}
We have designed regular polynomial interpolation algorithms and variations which produce families of multivariate polynomials which solve linear systems of partial differential equations on arbitrary sets of interpolation points. In our basic algorithm the members of the family of polynomials are defined recursively each being an extension of the preceding member in the sense that the preceding member agrees with a given member on the set of interpolation points on which the preceding member satisfies the linear system of partial differential equations. We have shown that the family of multivariate polynomials has the global solution as its natural limit if some a priori information on the system of partial differential equations is available. The information needed can variate from case to case. In any case a solution should exist. We have shown how to use a priori estimates of elliptic equations and of hyperbolic systems of equations in order to obtain error estimates adapted to the regularity of the solution. Similar is true for parabolic equations. All this makes our approach compatible with new techniques like sparse grids or weighted Monte-Carlo algorithms developed in order to treat systems of higher dimension. In case of parabolic equations we showed how regular polynomial interpolation of known functions can be used in order to compute higher order approximations of WKB-expansions of fundamental solutions. We also constructed extensions where the algorithm is parallelized on different set of interpolation points an showed how these partial polynomial approximations can be patched together to one multivariate polynom which fits the given system of linear partial differential equations on the union of sets of interpolation points.
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How to Spend $1K Marketing Your Nonprofit Online.
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Ar.
Literary Supplement to Al Maghrib Journal, Thursday, June 2, 1938
We know of no nation, amongst all who populate the globes surface, which finds itself in the 20th century while continuing to live to rhythms of centuries long gone, centuries of darkness from the point of view of organization, tradition, myth and all that these entail to such an (exaggerated) extent that in the eyes of strangers, the (whole) nation passes off as a museum. A museum where tourists can go to observe the flow of time, the evolution of successive generations and that of different stages of backward living. They are amused, deriding the expositions of this immense museum which they traverse up and down. It was as if they were in a cinema viewing strange and bizarre scenes. We hardly recognize (our) Morocco as the nation bearing the title of this film, a country which until recently could count itself amongst the great states along the shores of the Mediterranean. After having been one of the most important cradles of civilization and culture, it has become these days a center of amusement for other people. It finds itself at the mercy of world travel agencies who advertise in their bulletin boards that this is a land where the vestiges of times past are still intact, that its inhabitants live in the 20th century despite themselves and that nothing binds them to the progress of modern life.
These travel agencies compete with (much) ingenuity in the choice of promotional campaigns designed to persuade their European clientele to visit our country. Sometimes they even publish imagery that have nothing to do with Morocco such as these posters which depict Moroccans as black Africans dressed in skirts made with plants bearing exotic ornaments. Sometimes they attribute to our country customs and traditions whose origins are unknown to us. All this to arouse in the potential tourist a desire to have an experience unlike any other so he can describe upon his return home the extraordinary one thousand and one night adventures suggested to him by so called tourist guides who (invariably) add fictional color to the sights he would have seen during his Moroccan voyage.
We believe that these travel agencies bring harm to our country by exploiting this colonial bias which they spread about Morocco exclusively for their commercial gain. Meanwhile one is forced to acknowledge that they have had a huge success with the advertising slogan they chose to brand the Moroccan touristy product. This slogan has touched the heart of the matter even though (for us) therein lies nothing to be proud of. The agencies owe this success to the diffusion of the slogan through the use of huge mural billboards bearing print with vibrant colors to attract attention with the slogan according to which a tourist who visits Morocco can see parading before his eyes, 20 Centuries in 20 Days.
In fact when we examine scenes from our daily lives, we realize there is little relation between (our current) Moroccan society and the 20th century. There is an enormous gap between our lives and those of modern times. The latter are based on true understanding of all aspects of living while we lead lives driven by pervasive laziness and ignorance. Modern living relies on power and a fighting spirit while we lend proof to weakness each time we are confronted by any difficulty, no matter how small. Modern civilization teaches man to live for the community. Meanwhile we only look after our personal interests and declare at each opportunity that we dont care what happens after we are gone.
All these abstruse dispositions have led to a lack of (proper) upbringing and to a degenerate environment at a time when the roots of our past civilization have begun to wither and die. While centuries (helped) brew our languor, we turned our backs against basic principles paying attention only to means of making fortunes, to the possession of property and to show off some superficial ornament, object of our pride.
And so we stumbled and perished against adversity in the journey of life until the moment when we were ambushed by modern times. We were left stunned and we knelt before it without knowing how to reach for it. It inspired us with great admiration.
But the secrets of its superiority always eluded us. This is how we became the objects of ridicule and wonder to foreigners for whom we represented the centuries past and for whom we reassembled all that was chased away by progress and left behind by the development of science. And that is how 20 centuries managed to squeeze themselves into our lives.
It would be commendable to preserve traditions and customs that are neither shocking nor harmful to the reputation of our nation but we must first of all live in our (current) century and we must adapt to the living requirements of our era. Only in this manner can we distance ourselves from the charge of apathy that is leveled so well by the travel agency slogan 20 Centuries in 20 Days.
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Deepika Ranveer Wedding
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Featuring 52 Scriptures and messages on joy, this 'Joy' inspirational wooden message box is wonderful for encouraging women of all ages with a new, uplifting message every day, or one for every week of the year, or as needed. This wooden message box is perfect to display on desk or tabletop and makes a thoughtful gift for any occasion.
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Centric L3 Smartphone Launched Alongside A 256 Gb Expandable Retentiveness Together With Cost Is Nether 7,000₹ Or 120$
Centric L3 is launched inward Republic of Republic of India alongside a toll of 6,794₹. This is a novel entry-level smartphone. Recently Centric A1 was launched alongside a three GB RAM, which was launched at the toll of 10,999₹. The telephone is a Dual-Sim phone, which comes alongside a back upwardly of 4G VoLTE.
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TITLE: Zero Cohomology means zero homology?
QUESTION [6 upvotes]: Suppose we have a space $X$, which has zero cohomology (except in degree zero). Does he neccesarly have zero homology (except in degree zero)?
If not, what if $X$ is a manifold?
Universal Coefficient theorem gives me, that $Hom(H_*(X),\mathbb{Z})$ vanishes, but this isn't enough to conclude, that $H_*(X)$ vanishes, right?
REPLY [7 votes]: $\newcommand{\Z}{\mathbb{Z}}$We can assume $X$ path-connected, by restricting to each path-component.
Then yes, vanishing cohomology does imply vanishing homology. In Hatcher's Algebraic topology, proposition 3F.12, it is proven that $\tilde{H}^*(X) = 0$ implies that $\tilde{H}_n(X)$ is finitely generated for all $n$, because otherwise $H^*(X)$ would be uncountable.
Now the universal coefficient theorem tells you that for every $n > 0$, $$\begin{align}
\hom(H_n(X), \Z) & = 0
& \mathrm{Ext}(H_n(X), \Z) & = 0
\end{align}$$
But since $H_n(X)$ is a finitely generated abelian group, you can write it as
$$H_n(X) = \Z^r \oplus \Z/p_1^{\alpha_1} \oplus \dots \oplus \Z/p_s^{\alpha_s}$$
The first equation gives you $\hom(H_n(X), \Z) \simeq \Z^r = 0 \Rightarrow r = 0$;
The second equation gives you $\mathrm{Ext}(H_n(X), \Z) \simeq \Z/p_1^{\alpha_1} \oplus \dots \oplus \Z/p_s^{\alpha_s} = 0 \Rightarrow s = 0$.
Therefore $H_n(X) = 0$.
| 145,181
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High followed a one-out hit-batter in the first inning with his second home run of the series and the season just inside the right-field foul pole off Adam Morgan. Two innings later, he led off the third with a single and scored on Felix Perez's double to right center. … Over his last seven games against Lehigh Valley dating to the 2012 season, Soto is batting 406 (13-for-32) with two doubles, four home runs and 13 RBI. … Morgan (1-2) allowed four runs on six hits in six innings, with one walk and four strikeouts. … Corcino allowed just five hits, including back-to-back doubles in the fourth that accounted for the IronPigs' lone run off him. Lehigh Valley also loaded the bases in the fifth, but the rally ended when shortstop Kristopher Negron made a leaping grab of a Delmon Young line drive and doubled off Morgan at third. … Corcino (1-4), who came into the game with a 10.29 ERA, allowed six hits for his first Triple-A victory. The Puerto Rico native is ranked among baseball's top 100 prospects by Baseball America. … Cesar Hernandez's RBI single in the seventh inning cut the deficit to 4-2, but J.C. Ramirez gave up two runs in the bottom of the inning. They were the first runs the right-hander has allowed in 12 innings this year between Reading and the IronPigs. … Hernandez had three more hits to hike his average to .380 and extend his hitting streak to 11, the longest in one season since Domonic Brown hit in 13 straight in 2011 as part of his team-record 21-game hitting streak. … Darin Ruf, who had four hits, including two home runs Sunday, to break out of his slump, added a fifth-inning single to his double. … Leandro Castro went 1-for-4 and has hit in six of his last seven games (10-for-27, .370). … Young went 1-for-4 and is hitting .294 (5-for-17) in four games.
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\begin{document}
\begin{abstract}
G\"ottsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the cohomology spaces. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the cohomology of Hilbert schemes of points to the cohomology of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.
\end{abstract}
\maketitle
\section{Introduction}
Let $S$ be a smooth projective K3 surface over $\mathbb{C}$. In \cite{YZ96} and \cite{Bea99}, the number of rational curves in an integral linear system on $S$ is calculated using the relative compactified Jacobian. The idea is that the Euler characteristic of the relative compactified jacobian equals the number of maximally degenerate fibers if all rational fibers are nodal, and these are the raional curves we want. But the relative compactified jacobian is birational to the Hilbert scheme of points of $S$, the Euler characteristic of which is computed in \cite{Go90}. Hence we get the number of rational curves and the generating series:
\[
\sum_{n=0}^{\infty}N(n) t^{n}=\sum_{n=0}^{\infty}e({\overline{J^{n}(\mathcal{C}_{n})}})t^{n}=\sum_{n=0}^{\infty}e(S^{[n]}) t^{n}=\prod_{n=1}^{\infty}(1-t^{n})^{-24}=\frac{t}{\Delta(t)}
\]
where $N(n)$ is the number of rational curves contained in an $n$-dimensional linear system $|\mathcal{L}|$, $\mathcal{C}_{n}$ is the tautological family of curves over $|\mathcal{L}|$ with fibers being integral, $S^{[n]}$ is the Hilbert scheme of $n$ points of $S$ and $\Delta(t)=t\prod_{n\geq 1}(1-t^n)^{24}$ is the unique cusp form of weight 12 for $SL_{2}(\mathbb{Z})$.
In this paper $G$ will always be a finite group. We will consider a smooth projective K3 surface over $\mathbb{C}$ with a $G$-action, and ask whether we can prove a similar equality for $G$-representations. Since the main idea is to count the number of rational points in finite fields and then use comparison theorems between singular cohomology and $l$-adic cohomology, we will also consider the situation in characteristic $p$. $H^{i}(\cdot,\mathbb{C})$ denotes the singular cohomology and $H^{i}(\cdot,\mathbb{Q}_{l})$ denotes the $l$-adic cohomology. We will consider power series with coefficients lying in the ring of virtual graded $G$-representations $R_{k}(G)$, of which the elements are the formal differences of isomorphism classes of finite dimensional graded $k$-representations of $G$. The addition is given by direct sum and the multiplication is given by tensor product.
The main results are as follows.
{\bf Theorem 1.1.} {\it Let $S$ be a smooth projective surface over $\mathbb{F}_{q}$ with a $G$-action, where $q$ is a power of $p$. Suppose $p\nmid|G|$. Let $S^{[n]}$ be the Hilbert scheme of n points of $S$, and let $S^{(n)}$ be the $n$-th symmetric power of $S$. Then we have the following equality as virtual graded $G$-representations.}
\[
\sum_{n=0}^{\infty}[H^{*}({S^{[n]}_{\overline{\mathbb{F}_{p}}}}, \mathbb{Q}_l)]t^{n}=\prod_{m=1}^{\infty}\left(\sum_{n=0}^{\infty}[H^{*}({S^{(n)}_{\overline{\mathbb{F}_{p}}}},\mathbb{Q}_l)][-2n(m-1)]t^{mn}\right)
\]
\[
=\prod_{m=1}^{\infty}\frac{\left(\sum_{i=0}^{b_{1}}[(\wedge^{i}H^{1})\otimes \mathbb{Q}_{l,q^{i(m-1)}}]t^{mi}\right)\left(\sum_{i=0}^{b_{3}}[(\wedge^{i}H^{3})\otimes \mathbb{Q}_{l,q^{i(m-1)}}]t^{mi}\right)}{\Bigl(1-[\mathbb{Q}_{l,q^{m-1}}]t^m\Bigr)\left(\sum_{i=0}^{b_{2}}(-1)^{i}[(\wedge^{i}H^{2})\otimes \mathbb{Q}_{l,q^{i(m-1)}}]t^{mi}\right)\Bigl(1-[\mathbb{Q}_{l,q^{m+1}}]t^m\Bigr)}
\]
{\it where the coefficients lie in $R_{\mathbb{Q}_{l}}(G)$, $H^{*}:=\oplus H^{i}$ and $[-2n(m-1)]$ indicates shift in degrees. $\wedge^{i}H^{j}$ means $\wedge^{i}H^{j}(S_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})$ of degree $ij$, and $\mathbb{Q}_{l,q^n}$ denotes the one-dimensional trivial $G$-representation $\mathbb{Q}_{l}[-2n]$. In particular, the above equalities hold in $R_{\mathbb{C}}(G)[[t]]$ for the singular cohomology if $S$ is a smooth projective surface over $\mathbb{C}$ with a $G$-action by comparison theorems.}
\[
\sum_{n=0}^{\infty}[H^{*}(S^{[n]}, \mathbb{C})]t^{n}=\prod_{m=1}^{\infty}\left(\sum_{n=0}^{\infty}[H^{*}(S^{(n)},\mathbb{C})][-2n(m-1)]t^{mn}\right)
\]
\[
=\prod_{m=1}^{\infty}\frac{\left(\sum_{i=0}^{b_{1}}[(\wedge^{i}H^{1})\otimes \mathbb{C}_{q^{i(m-1)}}]t^{mi}\right)\left(\sum_{i=0}^{b_{3}}[(\wedge^{i}H^{3})\otimes \mathbb{C}_{q^{i(m-1)}}]t^{mi}\right)}{\Bigl(1-[\mathbb{C}_{q^{m-1}}]t^m\Bigr)\left(\sum_{i=0}^{b_{2}}(-1)^{i}[(\wedge^{i}H^{2})\otimes \mathbb{C}_{q^{i(m-1)}}]t^{mi}\right)\Bigl(1-[\mathbb{C}_{q^{m+1}}]t^m\Bigr)}.
\]
{\bf Corollary 1.2.} {\it Let $S$ be a smooth projective surface over $\mathbb{C}$ with a $G$-action. If we fix $i\geq 0$, then $H^{i}(S^{[n]},\mathbb{C})$ become stable for $n\geq i$ as $G$-representations.}
{\bf Theorem 1.3.} {\it Let $X$ and $Y$ be smooth projective algebraic varieties over $\mathbb{C}$ with a $G$-action. If $X$ and $Y$ have trivial canonical bundles and there is a birational map $f:X\rightarrow Y$ which commutes with the $G$-action, then}
\[
H^{*}(X,\mathbb{C})\cong H^{*}(Y,\mathbb{C})
\]
{\it as graded G-representations.}
Note that Theorem 1.3 is the finite group actions version of Batyrev's result \cite{Ba99}. As explained in \cite[Prop 3.1]{Ba99}, there exists $U\subset X$ (resp. $V\subset Y$) which is the maximal open subset where $f$ (resp. $f^{-1}$) is defined, and we have $f:U\rightarrow V$ is an isomorphism. The requirement of commuting with the $G$-action in Theorem 1.3 means that $U, V$ are $G$-stable, and $f:U\rightarrow V$ commutes with the $G$-action.
Recall that a linear system $|\mathcal{L}|$ is called an integral linear system if every effective divisor in it is integral. $|\mathcal{L}|$ is called $G$-stable if $G$ induces an action on the projective space $|\mathcal{L}|$, which means $G$ maps an effective divisor in $|\mathcal{L}|$ again to an effective divisor in $|\mathcal{L}|$.
{\bf Corollary 1.4.} {\it Let $S$ be a smooth projective K3 surface over $\mathbb{C}$ with a $G$-action, and let $\mathcal{C}_n$ be the tautological family of curves over any $n$-dimensional integral $G$-stable linear system. Then we have the following equalities as virtual graded $G$-representations.}
\[
\sum_{n=0}^{\infty}[e({\overline{J^{n}(\mathcal{C}_n)}})]t^{n}=\sum_{n=0}^{\infty}[e(S^{[n]})]t^{n}=\prod_{m=1}^{\infty}\left(\sum_{n=0}^{\infty}[e({S^{(n)}})][-2n(m-1)]t^{mn}\right)\tag{$*$}
\]
\[
=\prod_{m=1}^{\infty}\frac{\left(\sum_{i=0}^{b_{1}}(-1)^{i}[\wedge^{i}H^{1}][-2i(m-1)]t^{mi}\right)\left(\sum_{i=0}^{b_{3}}(-1)^{i}[\wedge^{i}H^{3}][-2i(m-1)]t^{mi}\right)}{\Bigl(1-\mathbb{C}[-2(m-1)]t^m\Bigr)\left(\sum_{i=0}^{b_{2}}(-1)^{i}[\wedge^{i}H^{2}][-2i(m-1)]t^{mi}\right)\Bigl(1-\mathbb{C}[-2(m+1)]t^m\Bigr)}
\]
{\it where the coefficients lie in $R_{\mathbb{C}}(G)$, $e(X)$ means $\sum_{i}(-1)^{i}H^{i}(X,\mathbb{C})$, and $\wedge^{i}H^{j}$ means $\wedge^{i}H^{j}(S,\mathbb{C})$ of degree $ij$.}
{\bf Remark 1.5.} Note that we are fixing the surface $S$ here, so the equality above should be understood as: if $S$ admits an $n$-dimensional integral $G$-stable linear system, then $[e(\overline{J^{n}(\mathcal{C}_n)})]$ equals the coefficient of $t^n$ on the right hand side.
Recall that for a complex K3 surface $S$ with an automorphism $g$ of finite order $n$, $H^{0}(S,K_{S})=\mathbb{C}\omega_{S}$ has dimension 1, and we say $g$ acts symplectically on $S$ if it acts trivially on $\omega_{S}$, and $g$ acts non-symplectically otherwise, namely, $g$ sends $\omega_{S}$ to $\zeta_{n}^{k}\omega_{S}$, $0<k<n$, where $\zeta_{n}$ is a primitive $n$-th root of unity.
For the right hand side of $(*)$, we have
{\bf Theorem 1.6.} {\it Let $G$ be a finite group which acts faithfully and symplectically on a complex K3 surface $S$. Then}
\[
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(S^{[n]})]}t^{n}=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{\epsilon(\text{ord}(g^{k}))t^{mk}}{k}\right)
\]
{\it for all $g\in G$, where $\epsilon(n)=24\left(n\prod_{p|n}\left(1+\frac{1}{p}\right)\right)^{-1}$. In particular, if $G$ is generated by a single element $g$ of order $N\leq 8$, then we have}
\begin{center}
\begin{tabular}{c||c}
$N$ & $\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(S^{[n]})]}t^{n}$ \\
\hline\hline
$1$ & $t/\eta(t)^{24}$ \\
$2$ & $t/\eta(t)^{8}\eta(t^{2})^{8}$ \\
$3$ & $t/\eta(t)^{6}\eta(t^{3})^{6}$ \\
$4$ & $t/\eta(t)^{4}\eta(t^{2})^{2}\eta(t^{4})^{4}$ \\
$5$ & $t/\eta(t)^{4}\eta(t^{5})^{4}$ \\
$6$ & $t/\eta(t)^{2}\eta(t^{2})^{2}\eta(t^{3})^{2}\eta(t^{6})^{2}$ \\
$7$ & $t/\eta(t)^{3}\eta(t^{7})^{3}$ \\
$8$ & $t/\eta(t)^{2}\eta(t^{2})\eta(t^{4})\eta(t^{8})^{2}$
\end{tabular}
\end{center}
{\it where $\eta(t)=t^{1/24}\prod_{n=1}^{\infty}(1-t^{n})$.}
{\bf Remark 1.7.} Notice that if $g$ acts symplectically on $S$, then $g$ has order $\leq 8$ by \cite[Corollary 15.1.8]{H16}. We know the generating series for topological Euler characteristic $\sum_{n=0}^{\infty}e(S^{[n]})t^{n}=\sum_{n=0}^{\infty}\text{Tr}(1)|_{[e(S^{[n]})]}t^{n}$ equals $t/\Delta(t)$, where $\Delta(t)=\eta(t)^{24}$ is a level 1 cusp form of weight 12. But by Theorem 1.6, we deduce that when an element $g$ of order $N$ acts faithfully and symplectically on $S$, we have $\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(S^{[n]})]}t^{n}=t/F(t)$, where $F(t)$ is also a cusp form for $\Gamma_{0}(N)$ of weight $\lceil\frac{24}{N+1}\rceil$ (\cite[Proposition 5.9.2]{CS17}). This coincides with the results by Jim Bryan and \'Ad\'am Gyenge in \cite{BG19} when $G$ is a cyclic group. See also \cite[Lemma 3.1]{BO18}.
{\bf Theorem 1.8.} {\it Let $G=\left<g\right>$ be a finite group generated by an automorphism $g$ of order $p$, which acts non-symplectically on a complex K3 surface $S$. Then we have}
\[
\sum_{n=0}^{\infty}\text{Tr}(1)|_{[e(S^{[n]})]}t^{n}=\left(\prod_{m=1}^{\infty}(1-t^{m})\right)^{-24}
\]
\[
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(S^{[n]})]}t^{n}=\left(\prod_{m=1}^{\infty}(1-t^{m})\right)^{dp-24}\left(\prod_{m=1}^{\infty}(1-t^{mp})\right)^{-d}
\]
{\it for all $g\in G$, $g\neq 1$, where $d=\frac{{\rm{rank}}\,T(g)}{p-1}$, and $T(g):=(H^{2}(S,\mathbb{Z})^{g})^{\bot}$ is the orthogonal complement of the $g$-invariant sublattice.}
For the left hand side of $(*)$, recall that in \cite{Bea99}, for the curve $C$ in $\mathcal{C}_n$, we have $e(\overline{J^{n}(C)})=0$ if the normalization $\tilde{C}$ has genus $\geq$ 1, and $e(\overline{J^{n}(C)})=1$ if $C$ is a nodal rational curve. Hence intuitively that is why $e(\overline{J^{n}(\mathcal{C}_{n})})$ counts the number of rational curves in $\mathcal{C}_{n}$ if we assume all rational curves in $\mathcal{C}_{n}$ are nodal. But in our situation, $e(\overline{J^{n}(C)})=0$ does not mean $[e(\overline{J^{n}(C)})]=0$ as $G$-representations. Hence non-rational curves may also contribute to $[e({\overline{J^{n}(\mathcal{C}_n)}})]$, and certain $G$-orbits of curves contribute certain representations (See Example 1, 2 and 3). Nevertheless, we show that a $G$-orbit of curves with nodal singularities will contribute nothing if the normalization of the curve quotient by its stablizer is not rational. By this method, we are able to understand certain $G$-orbits in the linear system. We denote by $[e(X)]$ the alternating sum of the compactly supported $l$-adic cohomology $\sum_{n=0}(-1)^{n}[H^{n}_{c}(X_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})]$ when we are in the situation of characteristic $p$.
{\bf Theorem 1.9.} {\it Let $C$ be an integral curve over $\overline{\mathbb{F}_{p}}$ with nodal singularities and a $G$-action. Suppose $p\nmid|G|$. Denote by $\tilde{C}$ its normalization. If $\tilde{C}/\left<g\right>$ is not a rational curve {\rm(}i.e. $\mathbb{P}^{1}${\rm)} over $\overline{\mathbb{F}_{p}}$ for every $g\in G$, then $[e(\overline{J^{n}C})]=0$ as $G$-representations.}
{\bf Corollary 1.10.} {\it Let $C$ be an integral curve over $\overline{\mathbb{F}_{p}}$ with nodal singularities and a $G$-action. Suppose $p\nmid|G|$. Denote by $\tilde{C}$ its normalization. If $\tilde{C}/G$ is not a rational curve {\rm(}i.e. $\mathbb{P}^{1}${\rm)} over $\overline{\mathbb{F}_{p}}$, then $[e(\overline{J^{n}C})]=0$ as $G$-representations.}
By the above discussions, we show that the representation $[e(\overline{J^{n}(\mathcal{C}_{n})})]$ actually `counts' the curves in $\mathcal{C}_{n}$ whose normalization quotient by its stablizer is rational (See Example 1, 2, and 3).
This paper is organized as follows. Section 2 explains how we can compare cohomology groups using the Weil conjectures. In Section 3, we work with Hilbert schemes of points and prove Theorem 1.1 and Corollary 1.2. In Section 4, we review the method of $p$-adic integrals in \cite{Ba99} and prove Theorem 1.3 by applying it in our case. In Section 5, we deal with compactified jacobians and prove Theorem 1.9 and Corollary 1.10. In Section 6, we prove Corollary 1.4, Theorem 1.6 and Theorem 1.8 by using the results in previous sections. Then we give three explicit examples when $G$ equals $\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/3\mathbb{Z}$. Finally, when $G=PSL(2,7), A_{6}, A_{5}\ {\rm or}\ S_{5}$, we determine the smooth projective curve $C$ over $\mathbb{C}$ with a faithful $G$-action if there exists $g\in G$ such that $C/{\left<g\right>}=\mathbb{P}^{1}$.
\begin{center}
\sc{Acknowledgements}
\end{center}
\vspace{0.1 in}
I thank my advisor Professor Michael Larsen for his guidance and valuable discussions throughout this work, and in particular, for suggesting the problem. I also thank Professor Jim Bryan for several helpful suggestions and comments.
\section{Preliminaries}
Let $X$ be a smooth projective variety over $\mathbb{C}$. Then we can choose a finitely generated $\mathbb{Z}$-subalgebra $\mathcal{R}\subset\mathbb{C}$ such that $X\cong\mathcal X\times_{\mathcal S}\text{Spec}\mathbb{C}$ for a regular projective scheme $\mathcal X$ over $\mathcal{S}=\text{Spec}\mathcal{R}$, and we can choose a maximal ideal $\mathfrak q$ of $\mathcal R$ such that $\mathcal X$ has good reduction modulo $\mathfrak q$. We denote by $\bar{X}$ the smooth projective variety over ${\mathbb{F}_{q}}$ after reduction, where $q$ is some power of a prime number $p$.
Denote $\bar{X}\otimes_{\mathbb{F}_{q}}\overline{\mathbb{F}_{p}}$ by $\bar{X}_{\overline{\mathbb{F}_{p}}}$. By comparison theorems between \'etale cohomology and singular cohomology for smooth projective varieties, we have $H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})\otimes\mathbb{C}\cong H^{i}({X},\mathbb{Q}_{l})\otimes\mathbb{C}\cong H^{i}(X,\mathbb{C})$, and this isomorphism is compatible with the $G$-action by functoriality if there is a finite group $G$ acting on $X$. Also by the fact that $H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}}/G,\mathbb{Q}_{l})\cong H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})^{G}$ for a projective variety $X$ with a finite group $G$-action, we have
\[
H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}}/G,\mathbb{Q}_{l})\otimes\mathbb{C}\cong H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})^{G}\otimes\mathbb{C}\cong H^{i}({X},\mathbb{C})^{G}\cong H^{i}(X/G,\mathbb{C})
\]
for a smooth projective $X$.
Now we first recall the proof of the following well-known fact, the idea of which will be used throughout the paper.
{\bf Proposition 2.1.} {\it Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$. By considering good reductions of $\mathcal X$ and $\mathcal Y$ modulo some $\mathfrak q$, if $|\bar{X}(\mathbb{F}_{q^n})|=|\bar{Y}(\mathbb{F}_{q^n})|$ for every $n\geq 1$, then $H^{n}(X,\mathbb{C})\cong H^{n}(Y,\mathbb{C})$ as vector spaces over $\mathbb{C}$.}
{\bf Proof.} By the Lefschetz fixed point formula for Frobenius, we have $|\bar{X}(\mathbb{F}_{q^n})|=\sum_{i=0}^{\infty}(-1)^{i}\text{Tr}(F_{q^n}, H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l))$, where $F_{q^n}$ denotes the geometric Frobenius. Let $\alpha_{n,i},i=1,2,...,a_n$ (resp. $\beta_{n,i},i=1,2,...,b_n$) denote the eigenvalues of $F_q$ acting on $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$ (resp. $H^{n}(\bar{Y}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$), where $a_n$ (resp. $b_n$) is the $n$-th betti number. Then since $|\bar{X}(\mathbb{F}_{q^n})|=|\bar{Y}(\mathbb{F}_{q^n})|$, we have $\sum_{i=0}^{\infty}(-1)^{i}\sum_{j=1}^{a_i}\alpha_{i,j}^{n}=\sum_{i=0}^{\infty}(-1)^{i}\sum_{j=1}^{b_i}\beta_{i,j}^{n}$ for every $n\geq 1$. Then by linear independence of the characters $\chi_{\alpha}:\mathbb{Z}^{+}\rightarrow\mathbb{C}, n\mapsto \alpha^n$ and taking into consideration that $\alpha_{i,j},\beta_{i,j},j=1,2,...$ all have absolute value $q^{i/2}$ by Weil's conjecture, we deduce that for each $i$, the list of eigenvalues ${\alpha_{i,1},...,\alpha_{i,a_i}}$ must be the same as the list of eigenvalues ${\beta_{i,1},...,\beta_{i,b_i}}$ up to reordering, which implies, in particular, $a_{i}=b_{i}$ for each $i$. Hence the proposition follows. $\square$
Since we only know that the multiplicities of the eigenvalues of two sides are the same and we do not know whether the Frobenius action is semisimple, we cannot get $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})\cong H^{n}(\bar{Y}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})$ as Galois representations. Actually what we proved above is that $[H^{*}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})]=[H^{*}(\bar{Y}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})]$ in the graded Grothendieck group $G_{0}(\mathbb{Q}_{l}[Gal_{\mathbb{F}_{q}}])$, which is the abelian group generated by the set $\{[X]|X\in {\rm graded}\ \mathbb{Q}_{l}[Gal_{\mathbb{F}_{q}}]-\rm{Mod}\}$ of isomorphism classes of finitely generated graded Galois modules modulo the relations $[A]-[B]+[C]=0$ if there is a graded short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. Using the same idea, we can prove Macdonald's formula \cite{M62} by the Weil conjectures.
{\bf Proposition 2.2.} {\it Let $X$ be a smooth projective variety of dimension $N$ over ${\mathbb{C}}$. Then by choosing some good reduction of $\mathcal{X}$ over $\mathfrak q$, we have}
\[
\sum_{k=0}^{\infty} [H^{*}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})]t^{k}=\prod_{j=0}^{2N}\left(\prod_{i=1}^{b_j}(1+(-1)^{j+1}[\mathbb{Q}_{l,\alpha_{j,i}}]t)\right)^{(-1)^{j+1}}
\]
{\it where the coefficients lie in $G_{0}(\mathbb{Q}_{l}[Gal_{\mathbb{F}_{q}}])$, $\alpha_{j,i}$ denote the eigenvalues of $F_q$ acting on $H^{j}(\bar{X}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$, and $\mathbb{Q}_{l,\alpha_{j,i}}$ denotes the one-dimensional Galois representation of degree $j$ with eigenvalue $\alpha_{j,i}$ by the geometric Frobenius $F_{q}$. }
{\bf Proof.} First we note that for a smooth projective variety $\bar{X}$ over $\mathbb{F}_{q}$, $\bar{X}^{(k)}$ does have the purity property. Namely, the absolute value of the eigenvalues of Frobenius $F_{q}$ on $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})$ is $q^{n/2}$. This is because $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})\cong H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{k},\mathbb{Q}_{l})^{S_{k}}\subset H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{k},\mathbb{Q}_{l})$, and $\bar{X}^{k}$ is smooth projective, hence pure. Now by the Weil conjectures, we have
\[
\text{exp}(\sum_{r=1}^{\infty}|\bar{X}(\mathbb{F}_{q^r})|\frac{t^r}{r})=\sum_{k=0}^{\infty}|\bar{X}^{(k)}(\mathbb{F}_{q})|t^{k}=\prod_{j=0}^{2N}\left(\prod_{i=1}^{b_j}(1-\alpha_{j,i}t)\right)^{(-1)^{j+1}}.
\]
The first equality is a combinatorial fact \cite[Remark 1.2.4]{Go94}. Let $\{\beta_{k,n,i}\}$ denote the eigenvalues of $F_q$ on $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})$. Then we have
\[
\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}(-1)^{m}\sum_{i}\beta_{k,m,i}t^{k}=\prod_{j=0}^{2N}\left(\prod_{i=1}^{b_j}(1-\alpha_{j,i}t)\right)^{(-1)^{j+1}}.
\]
Replacing $q$ by $q^n$, we get the following equality for every $n\geq 1$
\[
\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}(-1)^{m}\sum_{i}\beta_{k,m,i}^{n}t^{k}=\prod_{j=0}^{2N}\left(\prod_{i=1}^{b_j}(1-\alpha_{j,i}^{n}t)\right)^{(-1)^{j+1}}.
\]
Note that if we define the weight of $\alpha$ as $2\log_{q}|\alpha|$, then $\alpha_{j,i}$ has an odd weight if $j$ is odd and there is a minus sign before it. On the other hand, $\alpha_{j,i}$ has an even weight if $j$ is even and there is a plus sign before it since $1/(1-\alpha t)=\sum_{m=0}^{\infty}(\alpha t)^{m}$. On the left hand side, the same situation holds, namely if the eigenvalues have odd (resp. even) weights, then there are minus (resp. plus) signs before them. Hence by linear independence of the characters, we deduce the desired equality in $G_{0}(\mathbb{Q}_{l}[F_{q}])$. $\square$
Recall the definition of the Poincar\'e polynomial $P(X,z):=\sum_{n=0}^{\infty}\text{dim}H^{n}(X,\mathbb{C})z^{n}$.
{\bf Corollary 2.3.} {\it Let $X$ be a smooth projective variety of dimension $N$ over $\mathbb{C}$. Then the generating series of Poincar$\acute e$ polynomials of $X^{(n)}$ is given by}
\[
\sum_{n=0}^{\infty}P(X^{(n)},z)t^{n}=\prod_{j=0}^{2N}\Bigl(1+(-1)^{j+1}z^{j}t\Bigr)^{(-1)^{j+1}b_{j}}.
\]
{\bf Proof.} Take the Poincar\'e polynomial of both sides of the equality. $\square$
{\bf Corollary 2.4.} {\it Let $X$ be a smooth projective curve, an abelian variety, or a smooth projective K3 surface over ${\mathbb{C}}$. Denote the dimension of $X$ by $N$. Then by choosing some good reduction of $\mathcal{X}$ over $\mathfrak q$, we have}
\[
\sum_{k=0}^{\infty} [H^{*}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})]t^{k}=\prod_{j=0}^{2N}\left(\sum_{i=0}^{b_j}(-1)^{(j+1)i}[\wedge^{i}H^{j}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})]t^{i}\right)^{(-1)^{j+1}}
\]
{\it where the coefficients lie in $R_{\mathbb{Q}_{l}}(Gal_{\mathbb{F}_{q}})$, and $\wedge^{i}H^{j}(\bar{X}_{\overline{\mathbb{F}_{p}}})$ has degree $ij$.}
{\bf Proof.} This follows from the fact that the action of the Frobenius on $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})$ is semisimple if $X$ is a smooth projective curve, an abelian variety or a K3 surface \cite{De72}. Hence the action of the Frobenius on $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})$ is also semisimple since
\[
H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})\cong H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{k},\mathbb{Q}_{l})^{S_{k}}\cong \bigoplus_{n_{1}+...+n_{k}=n}\left(\bigotimes_{i=1}^{k}H^{n_{i}}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})\right)^{S_{k}}.
\]
So the equality in Proposition 2.2 is actually an equality as graded Galois modules. $\square$
Now if we have a smooth projective variety with finite group actions, we then need a lemma corresponding to Proposition 2.1. The idea is that if $X$ is a quasi-projective variety over $\overline{\mathbb{F}_{p}}$ with an automorphism $\sigma$ of finite order, then $X$ and $\sigma$ can be defined over some $\mathbb{F}_q$. Let $F_q$ be the corresponding geometric Frobenius. Then for $n \geq 1$, the composite $F_{q}^{n}\circ\sigma$ is the Frobenius map relative to some new way of lowering the field of definition of $X$ from $\overline{\mathbb{F}_{p}}$ to $\mathbb{F}_{q^n}$ (\cite[Prop.3.3]{DL76} and \cite[Appendix(h)]{Ca85}). Then the Grothendieck trace formula implies that $\sum_{k=0}^{\infty}(-1)^{k}\text{Tr}((F_{q}^{n}\sigma)^{*},H_{c}^{k}(X,\mathbb{Q}_{l}))$ is the number of fixed points of $F_{q}^{n}\sigma$.
{\bf Proposition 2.5.} {\it Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$ with finite group $G$-actions. By considering good reductions of $\mathcal{X}$ and $\mathcal{Y}$ modulo some $\mathfrak{q}$ such that $G$-actions can be defined over $\mathbb{F}_{q}$, if $|\bar{X}(\overline{\mathbb{F}_{p}})^{gF_{q^n}}|=|\bar{Y}(\overline{\mathbb{F}_{p}})^{gF_{q^n}}|$ for every $n\geq 1$ and $g\in G$, then $H^{n}(X,\mathbb{C})\cong H^{n}(Y,\mathbb{C})$ as $G$-representations for every $n$.}
{\bf Proof.} Fix $g\in G$. Since $g$ commutes with $F_{q}$ and the action of $g$ on the cohomology group is semisimple, there exists a basis of the cohomology group such that the actions of $g$ and $F_{q}$ are in Jordan normal forms simultaneously. Let $\alpha_{n,i},i=1,2,...,a_n$ (resp. $\beta_{n,i},i=1,2,...,b_n$) denote the eigenvalues of $F_q$ acting on $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$ (resp. $H^{n}(\bar{Y}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$) in such a basis, where $a_n$ (resp. $b_n$) is the $n$-th betti number. Let $c_{n,i},i=1,2,...,a_n$ (resp. $d_{n,i},i=1,2,...,b_n$) denote the eigenvalues of $g$ acting on the same basis of $H^{n}(\bar{X}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$ (resp. $H^{n}(\bar{Y}_{\overline{\mathbb{F}_{p}}}, \mathbb{Q}_l)$). Then since
\[
|\bar{X}(\overline{\mathbb{F}_{p}})^{gF_{q^n}}|=\sum_{i=0}^{\infty}(-1)^{i}\text{Tr}((gF_{q^{n}})^{*},H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l}))
\]
and $|\bar{X}(\overline{\mathbb{F}_{p}})^{gF_{q^n}}|=|\bar{Y}(\overline{\mathbb{F}_{p}})^{gF_{q^n}}|$ for every $n\geq 1$, we have
\[
\sum_{i=0}^{\infty}(-1)^{i}\sum_{j=1}^{a_i}c_{i,j}\alpha_{i,j}^{n}=\sum_{i=0}^{\infty}(-1)^{i}\sum_{j=1}^{b_i}d_{i,j}\beta_{i,j}^{n}
\]
for every $n\geq 1$. By the linear independence and eigenvalue discussions as before, we have $a_{i}=b_{i}$ and $\sum_{j=1}^{a_i}c_{i,j}=\sum_{j=1}^{b_i}d_{i,j}$ for each $i$. But since $g$ is arbitrary, this means that the characters for the $G$-representations $H^{i}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})$ and $H^{i}(\bar{Y}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})$ are the same. Hence $H^{n}(X,\mathbb{C})\cong H^{n}(Y,\mathbb{C})$ as $G$-representations for every $n$ by comparison theorems. $\square$
By a similar argument as in the proof of Proposition 2.2, we obtain
{\bf Corollary 2.6.} {\it Let $X$ be a smooth projective curve, an abelian variety, or a smooth projective K3 surface over ${\mathbb{C}}$. Denote the dimension of $X$ by $N$. Then by choosing some good reduction of $\mathcal{X}$ over $\mathfrak q$ such that the $G$-action can be defined over $\mathbb{F}_{q}$, we have}
\[
\sum_{k=0}^{\infty} [H^{*}({X}^{(k)},\mathbb{C})]t^{k}=\prod_{j=0}^{2N}\left(\sum_{i=0}^{b_j}(-1)^{(j+1)i}[\wedge^{i}H^{j}({X},\mathbb{C})]t^{i}\right)^{(-1)^{j+1}}
\]
{\it where the coefficients lie in $R_{\mathbb{C}}(G)$. }
{\bf Proof.} Similar as before, except that now we have
\[
\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}(-1)^{m}\sum_{i}h_{k,m,i}\beta_{k,m,i}^{n}t^{k}=\prod_{j=0}^{2N}\left(\prod_{i=1}^{b_j}(1-g_{j,i}\alpha_{j,i}^{n}t)\right)^{(-1)^{j+1}}
\]
where $h_{k,m,i}$ are the eigenvalues of $g$ on $H^{m}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(k)},\mathbb{Q}_{l})$, and $g_{j,i}$ are the eigenvalues of $g$ on $H^{j}(\bar{X}_{\overline{\mathbb{F}_{p}}},\mathbb{Q}_{l})$. Hence we deduce that the trace of $g$ on the left hand side equals the trace of $g$ on the right hand side for the equality in Corollary 2.6. $\square$
\section{Hilbert scheme of points}
Let $X^{[n]}$ denote the component of the Hilbert scheme of $X$ parametrizing subschemes of length $n$ of $X$. For properties of Hilbert scheme of points, see references \cite{I77}, \cite{Go94} and \cite{N99}. The following theorem is proved for smooth projective surfaces over $\mathbb{C}$ in \cite{Go90}, and for quasi-projective surfaces over $\mathbb{C}$ in \cite{GS93}.
{\bf Theorem 3.1.} {\it The generating function of the Poincar\'e polynomials of the Hilbert scheme $X^{[n]}$ is given by }
\[
\sum_{n=0}^{\infty} p(X^{[n]},z)t^n=\prod_{m=1}^{\infty} \frac{(1+z^{2m-1}t^m)^{b_{1}(X)}(1+z^{2m+1}t^{m})^{b_{3}(X)}}{(1-z^{2m-2}t^m)^{b_{0}(X)}(1-z^{2m}t^m)^{b_{2}(X)}(1-z^{2m+2}t^m)^{b_{4}(X)}}.
\]
By analyzing the structure of $X^{[n]}$ \cite[Lemma 2.9]{Go90}, we know that
\[
\sum_{n=0}^{\infty}|\bar{X}^{[n]}(\mathbb{F}_{q^{k}})|t^{n}=\prod_{m=1}^{\infty}\left(\sum_{n=0}^{\infty}|\bar{X}^{(n)}(\mathbb{F}_{q^{k}})|q^{kn(m-1)}t^{nm}\right)
\]
for some $q$ and every $k\geq 1$, which implies
\[
\sum_{n=0}^{\infty}[H^{*}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{[n]},\mathbb{Q}_{l})]t^{n}=\prod_{m=1}^{\infty}\left(\sum_{n=0}^{\infty}[H^{*}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{(n)},\mathbb{Q}_{l})](-n(m-1))[-2n(m-1)]t^{nm}\right).
\]
Now using Proposition 2.2 and replacing $t$ by $[\mathbb{Q}_{l,q^{m-1}}]t^{m}$, we have
\[
\sum_{n=0}^{\infty}[H^{*}(\bar{X}_{\overline{\mathbb{F}_{p}}}^{[n]},\mathbb{Q}_{l})]t^{n}=\prod_{m=1}^{\infty}\left(\frac{\prod_{i=1}^{b_1}(1+[\mathbb{Q}_{l,\alpha_{1,i}}\otimes\mathbb{Q}_{l,q^{m-1}}]t^{m})\prod_{i=1}^{b_3}(1+[\mathbb{Q}_{l,\alpha_{3,i}}\otimes\mathbb{Q}_{l,q^{m-1}}]t^{m})}{(1-[\mathbb{Q}_{l,q^{m-1}}]t^{m})\prod_{i=1}^{b_2}(1-[\mathbb{Q}_{l,\alpha_{2,i}}\otimes\mathbb{Q}_{l,q^{m-1}}]t^{m})(1-[\mathbb{Q}_{l,q^{m+1}}]t^{m})}\right).
\]
Taking the Poincar\'e polynomials of both sides, we obtain Theorem 3.1 for smooth projective surfaces. We notice that the first term in the product on the right hand side of Theorem 3.1 is the generating series of the Poincar\'e polynomial for $X^{(n)}$, which seems like a coincidence at first glance. But actually each term is some twisted generating series of the Poincar\'e polynomial for $X^{(n)}$.
Now suppose we have a smooth projective surface over $\mathbb{C}$ with a $G$-action, and we want a similar equality as above. By considering some good reduction of the surface, we can restrict ourselves to the characteristic $p$ and $p\nmid|G|$.
The following discussion will be used to deduce the key Lemma 3.3.
Let $S$ be a smooth projective surface over $\mathbb{F}_{q}$ with an automorphism $g$ over $\mathbb{F}_{q}$ of finite order. If $x\in S(\overline{\mathbb{F}_{q}})^{gF_{q}}$ where $F_{q}$ is the geometric Frobenius, then $x$ lies over a closed point $y\in S$. Denote the residue degree of $y$ by $N$. Hence $x\in S(\mathbb{F}_{q^{N}})$ and there are $N$ geometric points $x, F_{q}(x),..., F_{q}^{N-1}(x)$ lying over $y$.
Let us study the relative Hilbert scheme of $n$ points at a closed point.
\[
{\rm Hilb}^{n}({\rm Spec}(\widehat{\mathcal{O}_{S,y}})/{\rm Spec}\mathbb{F}_{q})\cong{\rm Hilb}^{n}({\rm Spec}(\mathbb{F}_{q^{N}}[[s,t]])/{\rm Spec}\mathbb{F}_{q}).
\]
Since $g$ and $\mathbb{F}_{q}$ fix $y$, they act on this Hilbert scheme. Over $\overline{\mathbb{F}_{q}}$, we have
\[
{\rm Hilb}^{n}({\rm Spec}(\widehat{\mathcal{O}_{S,y}})/{\rm Spec}\mathbb{F}_{q})\otimes_{\mathbb{F}_{q}}\overline{\mathbb{F}_{q}}\cong{\rm Hilb}^{n}({\rm Spec}(\overline{\mathbb{F}_{q}}\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^{N}}[[s,t]])/{\rm Spec}\overline{\mathbb{F}_{q}})
\]
by the base change property of the Hilbert scheme. Denote by $u$ a primitive element of the field extension $\mathbb{F}_{q^{N}}/\mathbb{F}_{q}$ and denote by $f(x)$ the irreducible polynomial of $u$ over $\mathbb{F}_{q}$. Since we have an $\overline{\mathbb{F}_{q}}$-algebra isomorphism
\[
\overline{\mathbb{F}_{q}}\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^{N}}\cong\overline{\mathbb{F}_{q}}\otimes_{\mathbb{F}_{q}}(\mathbb{F}_{q}[x]/(f(x)))\cong \overline{\mathbb{F}_{q}}(x)/(x-u)\times...\times\overline{\mathbb{F}_{q}}(x)/(x-u^{q^{N-1}})
\]
by the Chinese Remainder Theorem, we deduce that
\[
\begin{aligned}
{\rm Hilb}^{n}({\rm Spec}(\widehat{\mathcal{O}_{S,y}})/{\rm Spec}\mathbb{F}_{q})\otimes_{\mathbb{F}_{q}}\overline{\mathbb{F}_{q}}&\cong{\rm Hilb}^{n}({\rm Spec}((\overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}})[[s,t]])/{\rm Spec}\overline{\mathbb{F}_{q}})\\
&\cong{\rm Hilb}^{n}(\coprod {\rm Spec}\overline{\mathbb{F}_{q}}[[s,t]]/{\rm Spec}\overline{\mathbb{F}_{q}}).
\end{aligned}
\]
Hence the $\overline{\mathbb{F}_{q}}$-valued points of ${\rm Hilb}^{n}({\rm Spec}(\widehat{\mathcal{O}_{S,y}})/{\rm Spec}\mathbb{F}_{q})$ correspond to the closed subschemes of degree $n$ of $\coprod {\rm Spec}\overline{\mathbb{F}_{q}}[[s,t]]$, i.e. the closed subschemes of degree $n$ of $S$ whose underlying space is a subset of the points $x, F_{q}(x),..., F_{q}^{N-1}(x)$.
Since $F_{q}$ acts on $\mathbb{F}_{q^{N}}[[s,t]]$ by sending $s$ to $s^{q}$, $t$ to $t^{q}$ and $c\in\mathbb{F}_{q^{N}}$ to $c^{q}$, we deduce from the above discussion that $F_{q}$ acts on $(\overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}})[[s,t]]$ by sending $s$ to $s^{q}$, $t$ to $t^{q}$ and $(\alpha_0,\alpha_1,...,\alpha_{N-2},\alpha_{N-1})\in \overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}}$ to $(\alpha_1,\alpha_{2},...,\alpha_{N-1},\alpha_{0})$.
Let $\sigma$ be an element of ${\rm Gal}(\mathbb{F}_{q^{N}}/\mathbb{F}_{q})$. Recall that for an $\mathbb{F}_{q^{N}}$-vector space $V$, a $\sigma$-linear map $f:V\rightarrow V$ is an additive map on $V$ such that $f(\alpha v)=\sigma(\alpha)f(v)$ for all $\alpha\in\mathbb{F}_{q^{N}}$ and $v\in V$.
{\bf Lemma 3.2.} {\it Let $H=\left<g\right>$. Suppose $p\nmid|H|$, then we can choose $s$ and $t$ such that $g$ acts on $\mathbb{F}_{q^{N}}[[s,t]]$ $\sigma$-linearly, where $\sigma$ is the inverse of the Frobenius automorphism of ${\rm Gal}(\mathbb{F}_{q^{N}}/\mathbb{F}_{q})$.}
{\bf Proof.} $g$ acts as an $\mathbb{F}_{q}$-automorphism on $\mathbb{F}_{q^{N}}[[s,t]]$ sending $(s,t)$ to $(s,t)$ and $\mathbb{F}_{q^{N}}$ to $\mathbb{F}_{q^{N}}$. Since we know $F_{q}$ sends $(\alpha_0,\alpha_1,...,\alpha_{N-2},\alpha_{N-1})\in \overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}}$ to $(\alpha_1,\alpha_{2},...,\alpha_{N-1},\alpha_{0})$ and $gF_{q}$ fixes the geometric points $x, F_{q}(x),..., F_{q}^{N-1}(x)$, we deduce that $g$ sends $(\alpha_0,\alpha_1,...,\alpha_{N-2},\alpha_{N-1})\in \overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}}$ to $(\alpha_{N-1},\alpha_{0},...,\alpha_{N-3},\alpha_{N-2})$. Hence $g(\alpha)=\sigma(\alpha)$ for all $\alpha\in\mathbb{F}_{q^{N}}$ where $\sigma$ is the inverse of the Frobenius automorphism.
Now we write $g(s)=as+bt+...$ and $g(t)=cs+dt+...$ where $a,b,c,d\in\mathbb{F}_{q}$ since $g$ commutes with $F_{q}$. Define an automorphism $\rho(g)$ of $\mathbb{F}_{q^{N}}[[s,t]]$ by $\rho(g)(s)=as+bt$, $\rho(g)(t)=cs+dt$ and the action of $\rho(g)$ on $\mathbb{F}_{q^{N}}$ is the same as the action of $g$. Then we denote the $\mathbb{F}_{q^{N}}$-automorphism $\frac{1}{|H|}\sum_{h\in H}h\rho(h)^{-1}$ by $\theta$. Notice that $\theta$ is an automorphism because the linear term of $\theta$ is an invertible matrix. We deduce that $g\theta=\theta\rho(g)$, which implies $\theta^{-1}g\theta=\rho(g)$. Hence we are done. $\square$
Notice that Lemma 3.2 is the only place we use the assuption $p\nmid|G|$ for Theorem 1.1.
From Lemma 3.2, we now know that the $g$-action on $(\overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}})[[s,t]]$ is given by sending $s$ to $(a,...,a)s+(b,...,b)t$, $t$ to $(c,...,c)s+(d,...,d)t$ and $(\alpha_0,\alpha_1,...,\alpha_{N-2},\alpha_{N-1})\in \overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}}$ to $(\alpha_{N-1},\alpha_{0},...,\alpha_{N-3},\alpha_{N-2})$.
Hence the action of $gF_{q}$ on $(\overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}})[[s,t]]$ is given by sending $s$ to $(a,...,a)s^{q}+(b,...,b)t^{q}$, $t$ to $(c,...,c)s^{q}+(d,...,d)t^{q}$ and $(\alpha_0,\alpha_1,...,\alpha_{N-2},\alpha_{N-1})\in \overline{\mathbb{F}_{q}}\times...\times\overline{\mathbb{F}_{q}}$ to itself. Hence $gF_{q}$ acts on each complete local ring, which is what we expected since $gF_{q}$ fixes each geometric point over $y$. In particular, it acts on $\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}}\cong(\overline{\mathbb{F}_{q}}\times\{0\}\times...\times\{0\})[[s,t]]\cong\overline{\mathbb{F}_{q}}[[s,t]]$.
Recall that ${\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})(\overline{\mathbb{F}_{q}})$ parametrizes closed subschemes of degree $n$ of $S_{\overline{\mathbb{F}_{q}}}$ supported on $x$.
{\bf Lemma 3.3.}
\[
|{\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})(\overline{\mathbb{F}_{q}})^{gF_{q}}|=|{\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])(\overline{\mathbb{F}_{q}})^{F_{q}}|
\]
{\bf Proof.} First we define an $\mathbb{F}_{q}$-automorphism $\tilde{g}$ on $\mathbb{F}_{q}[[s,t]]$ by
\[
\tilde{g}(s)=as+bt\ \ {\rm and}\ \ \tilde{g}(t)=cs+dt
\]
Recall that the action of $F_{q}$ on $\mathbb{F}_{q}[[s,t]]$ is an $\mathbb{F}_{q}$-endomorphism sending $s$ to $s^{q}$ and $t$ to $t^{q}$. By Lemma 3.2 and the above discussion, we observe that the action of $gF_{q}$ on $\overline{\mathbb{F}_{q}}[[s,t]]$ on the left is the same as the action of $\tilde{g}{F_{q}}$ on $\overline{\mathbb{F}_{q}}[[s,t]]$ on the right. Hence we have
\[
|{\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})(\overline{\mathbb{F}_{q}})^{gF_{q}}|=|{\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])(\overline{\mathbb{F}_{q}})^{\tilde{g}{F_{q}}}|.
\]
Now for the right hand side, $\tilde{g}$ is an automorphism of finite order and $F_{q}$ is the geometric Frobenius. Then by the Grothendieck trace formula, we have
\[
|{\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])(\overline{\mathbb{F}_{q}})^{\tilde{g}{F_{q}}}|=\sum_{k=0}^{\infty}(-1)^{k}{\rm Tr}((\tilde{g}{F_{q}})^{*}, H^{k}_{c}({\rm Hilb}^{n}(\overline{\mathbb{F}_{q}}[[s,t]]),\mathbb{Q}_{l})).
\]
But the action of $\tilde{g}$ factors through $GL_{2}(\overline{\mathbb{F}_{q}})$. Now we use the fact that if $G$ is a connected algebraic group acting on a separated and finite type scheme $X$, then the action of $g\in G$ on $H^{*}_{c}(X,\mathbb{Q}_{l})$ is trivial [DL76, Corollary 6.5]. Hence we have
\[
\begin{aligned}
|{\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])(\overline{\mathbb{F}_{q}})^{\tilde{g}{F_{q}}}|&=\sum_{k=0}^{\infty}(-1)^{k}{\rm Tr}(({F_{q}})^{*}, H^{k}_{c}({\rm Hilb}^{n}(\overline{\mathbb{F}_{q}}[[s,t]]),\mathbb{Q}_{l}))\\
&=|{\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])(\overline{\mathbb{F}_{q}})^{F_{q}}|.\ \ \square
\end{aligned}
\]
From Lemma 3.3 we observe that $|{\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})(\overline{\mathbb{F}_{q}})^{gF_{q}}|$ is a number independent of the choice of the $gF_{q}$-fixed point $x$.
{\bf Lemma 3.4.}
\[
\sum_{n=0}^{\infty}|S^{[n]}(\overline{\mathbb{F}_q})^{gF_{q}}|t^{n}=\prod_{r=1}^{\infty}\left(\sum_{n=0}^{\infty}|{\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})(\overline{\mathbb{F}_{q}})^{g^{r}F_{q}^{r}}|t^{nr}\right)^{|P_{r}(S,gF_{q})|}
\]
{\it where ${\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})$ is the punctual Hilbert scheme of $n$ points at some $g^{r}F_{q}^{r}$-fixed point $x\in S(\overline{\mathbb{F}_{q}})$, and $P_{r}(S,gF_{q})$ denotes the set of primitive 0-cycles of degree $r$ of $gF_{q}$ on $S$, whose elements are of the form $\sum_{i=0}^{r-1}g^{i}F_{q}^{i}(x)$ with $x\in S(\overline{\mathbb{F}_q})^{g^{r}F_{q}^{r}}\backslash(\cup_{j<r}S(\overline{\mathbb{F}_q})^{g^{j}F_{q}^{j}})$.}
{\bf Proof.} Let $Z\in S^{[n]}(\overline{\mathbb{F}_q})^{gF_{q}}$. Suppose $(n_{1},...,n_{r})$ is a partition of $n$ and $Z=(Z_{1},...,Z_{r})$ with $Z_{i}$ being the closed subscheme of $Z$ supported at a single point with length $n_{i}$. Then Supp$Z$ decomposes into $gF_{q}$ orbits. We can choose an ordering $\leq$ on $S(\overline{\mathbb{F}_q})$. In each orbit, we can find the smallest $x_{j}\in S(\overline{\mathbb{F}_q})$. Suppose $Z_{j}$ with length $l$ is supported on $x_{j}$ and $x_{j}$ has order $k$. Then the component of $Z$ which is supported on the orbit of $x_{j}$ is determined by $Z_{j}$, namely, it is $\cup_{i=0}^{k-1}g^{i}F_{q}^{i}(Z_{j})$ with length $kl$. Also notice that $Z_{j}$ is fixed by $g^{k}F_{q}^{k}$. Hence, to give an element of $S^{[n]}(\overline{\mathbb{F}_q})^{gF_{q}}$ is the same as choosing some $gF_{q}$ orbits and for each orbit choosing some element in ${\rm Hilb}^{n}(\widehat{\mathcal{O}_{S_{\overline{\mathbb{F}_{q}}},x}})(\overline{\mathbb{F}_{q}})^{g^{k}F_{q}^{k}}$ for some $g^{k}F_{q}^{k}$-fixed point $x$ in this orbit such that the final length altogether is $n$. Combining all of these into power series, we get the desired equality. $\square$
The idea we used above is explained in detail in \cite[lemma 2.7]{Go90}. We implicitly used the fact that $\pi:(S^{[n]}_{(n)})_{red}\rightarrow S$ is a locally trivial fiber bundle in the Zariski topology with fiber ${\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])_{red}$ \cite[Lemma 2.1.4]{Go94}, where $S^{[n]}_{(n)}$ parametrizes closed subschemes of length $n$ that are supported on a single point. Denote ${\rm Hilb}^{n}(\mathbb{F}_{q}[[s,t]])$ by $V_{n}$.
Now combining Lemma 3.3 and Lemma 3.4, we have
\[
\sum_{n=0}^{\infty}|S^{[n]}(\overline{\mathbb{F}_q})^{gF_{q}}|t^{n}=\prod_{r=1}^{\infty}\left(\sum_{n=0}^{\infty}|V_{n}(\overline{\mathbb{F}_{q}})^{F_{q}^{r}}|t^{nr}\right)^{|P_{r}(S,gF_{q})|}.
\]
Recall the following structure theorem for the punctual Hilbert scheme of points \cite[Prop 4.2]{ES87}.
{\bf Proposition 3.5.} {\it Let $k$ be an algebraically closed field. Then ${\rm Hilb}^{n}(k[[s,t]])$ over $k$ has a cell decomposition, and the number of $d$-cells is $P(d,n-d)$, where $P(x,y):=\#$$\{\text{partition of x into parts}\leq y\}$.}
Denote by $p(n,d)$ the number of partitions of $n$ into $d$ parts. Then $p(n,d)=P(n-d,d)$. Now we can proceed similarly as in the proof of \cite[Lemma 2.9]{Go90}.
{\bf Proof of Theorem 1.1} Since we have
\[
\prod_{i=1}^{\infty}\left(\frac{1}{1-z^{i-1}t^{i}}\right)=\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}p(n,n-i)t^{n}z^{i},
\]
by Proposition 3.5 we get
\[
\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\#\{\text{m-dim cells of }{\rm Hilb}^{n}(k[[s,t]])\}t^{n}z^{m}=\prod_{i=1}^{\infty}\frac{1}{1-z^{i-1}t^{i}}.
\]
Let $l\in\mathbb{N}$. Then by choosing sufficiently large $q$ powers $Q$ such that the cell decomposition of $V_{n,\overline{\mathbb{F}_{q}}}$ is defined over $\mathbb{F}_{Q}$, we have
\[
\sum_{n=0}^{\infty}|V_{n,\overline{\mathbb{F}_{q}}}(\mathbb{F}_{Q^{r}})|t^{nr}\overset{t^l}\equiv\prod_{i=1}^{\infty}\frac{1}{1-Q^{r(i-1)}t^{ri}}.
\]
Hence
\[
\begin{aligned}
\sum_{n=0}^{\infty}|S^{[n]}(\overline{\mathbb{F}_q})^{gF_{Q}}|t^{n}&\overset{t^l}\equiv\prod_{r=1}^{\infty}\prod_{i=1}^{\infty}\left(\frac{1}{1-Q^{r(i-1)}t^{ri}}\right)^{|P_{r}(S,gF_{Q})|}\\
&=\text{exp}\left(\sum_{i=1}^{\infty}\sum_{r=1}^{\infty}\sum_{h=1}^{\infty}|P_{r}(S,gF_{Q})|Q^{hr(i-1)}t^{hri}/h\right)\\
&=\text{exp}\left(\sum_{i=1}^{\infty}\sum_{m=1}^{\infty}(\sum_{r|m}r|P_{r}(S,gF_{Q})|)Q^{m(i-1)}t^{mi}/m\right)\\
&=\prod_{i=1}^{\infty}\text{exp}\left(\sum_{m=1}^{\infty}|S(\overline{\mathbb{F}_{Q}})^{g^{m}F_{Q}^{m}}|Q^{m(i-1)}t^{mi}/m\right)\\
&=\prod_{i=1}^{\infty}\sum_{n=0}^{\infty}|S^{(n)}(\overline{\mathbb{F}_{Q}})^{gF_{Q}}|Q^{n(i-1)}t^{ni}.
\end{aligned}
\]
By replacing $Q$ by $Q$-powers and using the Grothendieck trace formula as in the proof of Proposition 2.5, we get
\[
\sum_{n=0}^{\infty}[H^{*}({S_{\overline{\mathbb{F}_{p}}}^{[n]}}, \mathbb{Q}_l)]t^n=\prod_{m=1}^{\infty}\left(\sum_{n=0}^{\infty}[H^{*}({S_{\overline{\mathbb{F}_{p}}}^{(n)}},\mathbb{Q}_l)][-2n(m-1)]t^{mn}\right)
\]
as graded $G$-representations. Now by Corollary 2.6 and replacing $t$ by $\mathbb{Q}_{l,q^{m-1}}t^{m}$, we have
\[
\sum_{n=0}^{\infty}[H^{*}(S_{\overline{\mathbb{F}_{p}}}^{[n]},\mathbb{Q}_{l})]t^{n}
\]
\[=\prod_{m=1}^{\infty}\frac{\left(\sum_{i=0}^{b_{1}}[(\wedge^{i}H^{1})\otimes \mathbb{Q}_{l,q^{i(m-1)}}]t^{mi}\right)\left(\sum_{i=0}^{b_{3}}[(\wedge^{i}H^{3})\otimes \mathbb{Q}_{l,q^{i(m-1)}}]t^{mi}\right)}{\Bigl(1-[\mathbb{Q}_{l,q^{m-1}}]t^m\Bigr)\left(\sum_{i=0}^{b_{2}}(-1)^{i}[(\wedge^{i}H^{2})\otimes \mathbb{Q}_{l,q^{i(m-1)}}]t^{mi}\right)\Bigl(1-[\mathbb{Q}_{l,q^{m+1}}]t^m\Bigr)}.
\]
$\square$
{\bf Corollary 3.6.} {\it For a smooth projective surface $S$ over $\mathbb{C}$, we have}
\[
\sum_{n=0}^{\infty}[e(S^{[n]})]t^{n}
\]
\[=\prod_{m=1}^{\infty}\frac{\left(\sum_{i=0}^{b_{1}}(-1)^{i}[\wedge^{i}H^{1}][-2i(m-1)]t^{mi}\right)\left(\sum_{i=0}^{b_{3}}(-1)^{i}[\wedge^{i}H^{3}][-2i(m-1)]t^{mi}\right)}{\Bigl(1-\mathbb{C}[-2(m-1)]t^m\Bigr)\left(\sum_{i=0}^{b_{2}}(-1)^{i}[\wedge^{i}H^{2}][-2i(m-1)]t^{mi}\right)\Bigl(1-\mathbb{C}[-2(m+1)]t^m\Bigr)}
\]
where the coefficients lie in $R_{\mathbb{C}}(G)$.
{\bf Remark 3.7.} Notice that the generating series of Euler numbers of $S^{[n]}$ is $\sum_{n=0}^{\infty}e(S^{[n]})t^{n}=\prod_{m=1}^{\infty}(1-t^{m})^{-e(S)}$. But this is not the case if we consider $G$-representations and regard $\prod_{m=1}^{\infty}(1-t^{m})^{-[e(S)]}$ as $\text{exp}(\sum_{m=1}^{\infty}[e(S)](-\log(1-t^{m})))=\text{exp}(\sum_{m=1}^{\infty}[e(S)](\sum_{k=1}^{\infty}t^{mk}/k)))$. What we have is actually
\[
\sum_{n=0}^{\infty}{\rm Tr}(g)|_{[e(S^{[n]})]}t^{n}=\prod_{m=1}^{\infty}\left(\frac{(\prod_{i=1}^{b_1}(1-{g_{1,i}}t^{m}))(\prod_{i=1}^{b_3}(1-{g_{3,i}}t^{m}))}{(1-t^{m})(\prod_{i=1}^{b_2}(1-{g_{2,i}}t^{m}))(1-t^{m})}\right)
\]
\[
=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{t^{mk}}{k}\left(1-\sum_{i=1}^{b_{1}}{g_{1,i}}^{k}+\sum_{i=1}^{b_{2}}g_{2,i}^{k}-\sum_{i=1}^{b_{3}}{g_{3,i}}^{k}+1\right)\right).
\]
We will use this expression to determine the $G$-representation $[e(S^{[n]})]$ later when $S$ is a K3 surface.
{\bf Proof of Corollary 1.2} We let
\[
G(t):=(1-t)\sum_{n=0}^{\infty}[H^{*}(S^{[n]},\mathbb{C})]t^{n}
\]
\[
=\prod_{m=1}^{\infty}\frac{(1-t)\left(\sum_{i=0}^{b_{1}}[\wedge^{i}H^{1}][-2i(m-1)]t^{mi}\right)\left(\sum_{i=0}^{b_{3}}[\wedge^{i}H^{3}][-2i(m-1)]t^{mi}\right)}{\Bigl(1-\mathbb{C}[-2(m-1)]t^m\Bigr)\left(\sum_{i=0}^{b_{2}}(-1)^{i}[\wedge^{i}H^{2}][-2i(m-1)]t^{mi}\right)\Bigl(1-\mathbb{C}[-2(m+1)]t^m\Bigr)}
\]
We denote by $a_{i,j}$ the degree $i$ part of the coefficient of $t^{j}$. If $i<j$, then $a_{i,j}(G(t))=0$ as $G$-representation. Now fix $i$, take $n\geq i$, and then we have
\[
\begin{aligned}
H^{i}(S^{[n]},\mathbb{C})&=a_{i,n}\Bigl(\Bigl(\sum_{k=0}^{\infty}t^{k}\Bigr)G(t)\Bigr)\\
&=\sum_{k=0}^{n}a_{i,k}(G(t))\\
&=\sum_{k=0}^{\infty}a_{i,k}(G(t))\\
&=a_{i,0}(G(1)).
\end{aligned}
\]
Notice that $a_{i,0}(G(1))$ is a representation independent of $n$. Hence $H^{i}(S^{[n]},\mathbb{C})$ become stable as $G$-representations for $n\geq i$. $\square$
\section{Birational varieties with trivial canonical bundles}
We first review the idea in the proof of the following theorem.
{\bf Theorem 4.1. \cite{Ba99}} {\it Let $X$ and $Y$ be smooth n-dimensional projective varieties over $\mathbb{C}$. Assume that the canonical line bundles $\omega_{X}^{n}$ and $\omega_{Y}^{n}$ are trivial and that $X$ and $Y$ are birational. Then $X$ and $Y$ have the same Betti numbers, that is,}
\[
H^{i}(X,\mathbb{C})\cong H^{i}(Y,\mathbb{C})\quad {\rm for}\ {\rm all}\ i\geq 0 .
\]
Let $F$ be a finite extension of the $p$-adic field $\mathbb{Q}_{p}$, $R\subset F$ be the maximal compact subring, $\mathfrak q\subset R$ the maximal ideal, and $\mathbb{F}_{q}=R/\mathfrak{q}$ the residue field with $|\mathbb{F}_{q}|=q=p^r$.
Now let $X$ be a smooth $n$-dimensional algebraic variety over $F$. Assume that $X$ admits an extension $\mathcal X$ to a regular $S$-scheme, where $S=\text{Spec} R$. Then if the relative dualizing sheaf $\Omega_{\mathcal{X}/S}^{n}$ is trivial, there exists a global section $\omega\in\Gamma(\mathcal{X},\Omega_{\mathcal{X}/S}^{n})$ which has no zeros in $\mathcal{X}$. This $\omega$ is called a gauge form, and it determines a canonical $p$-adic measure $d\mu_{\omega}$ on the locally compact $p$-adic topological space $\mathcal{X}(F)$ of $F$-rational points. This $d\mu_{\omega}$ is called the Weil $p$-adic measure associated with the gauge form $\omega$, and it is defined as follows:
Let $x\in\mathcal{X}(F)$ be an $F$-point and let $t_1,...,t_n$ be local $p$-adic analytic parameters at $x$. Then $t_1,...,t_n$ define a $p$-adic homeomorphism $\theta:U\rightarrow \mathbb{A}^{n}(F)$ of an open subset $U\subset\mathcal{X}(F)$ and $\theta(U)\subset\mathbb{A}^{n}(F)$ in the $p$-adic topology. Let $\omega=\theta^{*}(g\text{d}t_{1}\wedge...\wedge \text{d}t_{n})$, where $g$ is a $p$-adic analytic function on $\theta(U)$ having no zeros. Then $d\mu_{\omega}$ on $U$ is defined to be the pullback with respect to $\theta$ of the $p$-adic measure $||g(t)||d\bf{t}$ on $\theta(U)$, where $||\cdot||$ can be taken as the absolute value on $F$ extended from $||\cdot||_{p}$ on $\mathbb{Q}_{p}$, and $d\bf{t}$ is the standard $p$-adic Haar measure on $\mathbb{A}^{n}(F)$ normalized by the condition $\int_{\mathbb{A}^{n}(R)}d{\bf t}=1$.
Notice that if we choose another global section $\omega'$, then $d\mu_\omega$ and $d\mu_{\omega'}$ may not agree on $\mathcal{X}(F)$, but they agree on $\mathcal{X}(R)$ since $\omega'=\omega h$ for an invertible function $h$ and $h$ has $p$-adic norm 1 on $\mathcal{X}(R)$.
Hence in general even if we do not assume the canonical bundle is trivial, since it is locally trivial, we can still define a $p$-adic measure $d\mu$ at least on the compact $\mathcal{X}(R)$.
Now we list some properties of these measures \cite{Ba99}:
{\bf Theorem 4.2.} {\it Let $\mathcal{X}$ be a regular $S$-scheme with a gauge form $\omega$, and let $d\mu_{\omega}$ be the corresponding Weil $p$-adic measure on $\mathcal{X}(F)$. Then}
\[
\int_{\mathcal{X}(R)}d\mu_{\omega}=\frac{|\mathcal{X}(\mathbb{F}_{q})|}{q^n}
\]
{\bf Theorem 4.3.} {\it Let $\mathcal{X}$ be a regular $S$-scheme, and let $d\mu$ be the $p$-adic measure on $\mathcal{X}(R)$. Then}
\[
\int_{\mathcal{X}(R)}d\mu=\frac{|\mathcal{X}(\mathbb{F}_{q})|}{q^n}
\]
{\bf Theorem 4.4.} {\it Let $\mathcal{X}$ be a regular $S$-scheme and let $\mathcal{Z}\subset\mathcal{X}$ be a closed reduced subscheme of codimension $\geq$ 1. Then the subset $\mathcal{Z}(R)\subset\mathcal{X}(R)$ has zero measure with respect to the canonical $p$-adic measure $d\mu$ on $\mathcal{X}(R)$.}
Combining all of these, let $X$ and $Y$ be smooth projective birational varieties of dimension $n$ over $\mathbb{C}$ with trivial canonical bundles. Then there exist Zariski open dense subsets $U\subset X$ and $V\subset Y$ with ${\rm codim}_{X}(X\backslash U)\geq 2$ and ${\rm codim}_{Y}(Y\backslash V)\geq 2$ and an isomorphism $\phi:U\rightarrow V$. By a standard argument, we can choose a finitely generated $\mathbb{Z}$-subalgebra ${R}\subset\mathbb{C}$ such that `everything' (e.g. $X, Y, U, V$) can be defined over Spec${R}$ and have good reductions over some maximal ideal $\mathfrak{q}$. Let $\omega_{X}$ and $\omega_{Y}$ be gauge forms on $X$ and $Y$ respectively and $\omega_{U}$ and $\omega_{V}$ their restriction to $U$ and $V$. Since $\phi^{*}\omega_{V}$ is another gauge form on $U$, $\phi^{*}\omega_{V}=h\omega_{U}$ for some nowhere vanishing regular function $h\in\Gamma(U,\mathcal{O}_{X}^{*})$. The property ${\rm codim}_{X}(X\backslash U)\geq 2$ implies that $h\in\Gamma(X,\mathcal{O}_{X}^{*})$. Hence $||h(x)||=1$ for all $x\in X(F)$ (notice that $X(F)=X(R)$), which means the Weil $p$-adic measures on $U(F)$ associated with $\phi^{*}{\omega_{V}}$ and $\omega_{U}$ are the same. Hence
\[
\int_{U(F)}d\mu_{X}=\int_{V(F)}d\mu_{Y}.
\]
By Theorem 4.4, we have
\[
\int_{X(R)}d\mu_{X}=\int_{X(F)}d\mu_{X}=\int_{U(F)}d\mu_{X}=\int_{V(F)}d\mu_{Y}=\int_{Y(F)}d\mu_{Y}=\int_{Y(R)}d\mu_{Y}.
\]
Applying Theorem 4.2, we get
\[
\begin{aligned}
\frac{|{X}(\mathbb{F}_{q})|}{q^n}&=\frac{|{Y}(\mathbb{F}_{q})|}{q^n}\\
|{X}(\mathbb{F}_{q})|&=|{Y}(\mathbb{F}_{q})|.
\end{aligned}
\]
Replacing $R$ by its cyclotomic extension obtained by adjoining all $(q^{r}-1)$-th roots of unity, the residue field $\mathbb{F}_{q}$ will become $\mathbb{F}_{q^r}$ and we get the same equality regarding $\mathbb{F}_{q^r}$-rational points for $r\geq 1$. Hence we have
\[
H^{i}(X,\mathbb{C})\cong H^{i}(Y,\mathbb{C})\quad {\rm for}\ {\rm all}\ i\geq 0.
\]
Now let $X$ and $Y$ be smooth projective algebraic varieties over $\mathbb{C}$ with finite group $G$-actions. If $X$ and $Y$ have trivial canonical bundles and there is a birational map $f:X\rightarrow Y$ which commutes with $G$-actions, where the meaning of commuting with $G$-actions is explained after Theorem 1.3, then as above there exists ${R}$, which is the maximal compact subring in a local $p$-adic field, such that `everything' (e.g. $X, Y, U, V$) can be defined over Spec${R}$ and have good reductions over some maximal ideal $\mathfrak{q}$. For a fixed $\sigma\in G$, we have $X(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}\subset X(\mathbb{F}_{q^N})$ and $Y(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}\subset Y(\mathbb{F}_{q^N})$ for some positive integer $N$. Let $R'$ be the integral closure of $R$ in the unramified extension of $F$ such that the residue field is $\mathbb{F}_{q^N}$. Then we have
{\bf Proposition 4.5.}
\[
\int_{\tilde{{X}}(R')}d\mu_{\omega}=\frac{|X(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}|}{q^{Nn}}
\]
{\it where $\tilde{X}(R')=\{x\in X(R')|\bar{x}\in X(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}\}$.}
{\bf Proof.} Let $\phi:X(R')\rightarrow X(\mathbb{F}_{q^N})$, $x\rightarrow\bar{x}$ be the reduction map. If $\bar{x}\in X(\mathbb{F}_{q^N})$ is a closed $\mathbb{F}_{q^N}$-point and $g_1,...,g_n$ are generators of the maximal ideal of $\mathcal{O}_{X,\bar{x}}$, then they define a $p$-adic analytic homeomorphism $\gamma:\phi^{-1}(\bar{x})\rightarrow\mathbb{A}^{n}(\mathfrak q')$, where $\phi^{-1}(\bar{x})$ is the fiber of $\phi$ over $\bar{x}$ and $\mathbb{A}^{n}(\mathfrak q')$ is the set of $R'$-rational points of $\mathbb{A}^{n}$ whose coordinates lie in $\mathfrak q'$. Now as in the proof of \cite[Theorem 2.5]{Ba99}, we know that
\[
\int_{\phi^{-1}(\bar{x})}d\mu_{\omega}=\int_{\mathbb{A}^{n}(\mathfrak q')}d{\bf t}=\frac{1}{q^{Nn}}.
\]
Hence we have
\[
\int_{\tilde{X}(R')}d\mu_{\omega}=\frac{|X(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}|}{q^{Nn}}.
\]
$\square$
{\bf Proof of Theorem 1.3.} The Weil $p$-adic measures on $U(F')$ associated with $f^{*}\omega_{V}$ and $\omega_{U}$ are the same, which implies the following equality
\[
\int_{\tilde{U}(F')}d\mu_{X}=\int_{\tilde{V}(F')}d\mu_{Y}
\]
where $\tilde{U}(F')=\{x\in U(F')|\bar{x}\in X(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}\}$, $\tilde{V}(F')=\{x\in V(F')|\bar{x}\in Y(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}\}$ and $f$ induces a bijection between $\tilde{U}(F')$ and $\tilde{V}(F')$.
Hence
\[
\int_{\tilde{X}(R')}d\mu_{X}=\int_{\tilde{U}(F')}d\mu_{X}=\int_{\tilde{V}(F')}d\mu_{Y}=\int_{\tilde{Y}(R')}d\mu_{Y}.
\]
Then by Proposition 4.5, we have
\[
|X(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}|=|Y(\overline{\mathbb{F}_{q}})^{\sigma F_{q}}|.
\]
Replacing $R$ by its cyclotomic extension and then by Proposition 2.5, we get
\[
H^{*}(X,\mathbb{C})\cong H^{*}(Y,\mathbb{C})
\]
as graded G-representations. $\square$
\section{Compactified Jacobians }
Recall some facts from \cite{AK76}, \cite{A04} and \cite{EGK00}. Let $C/S$ be a flat projective family of integral curves. By a torsion-free rank-1 sheaf $\mathcal I$ on $C/S$, we mean an $S$-flat coherent $\mathcal{O}_C$-module $\mathcal I$ such that, for each point $s$ of $S$, the fiber $\mathcal{I}_s$ is a torsion-free rank-1 sheaf on the fiber $C_s$. We say that $\mathcal I$ is of degree $n$ if $\chi(\mathcal{I}_{s})-\chi(\mathcal{O}_{C_{s}})=n$ for each $s$.
Given $n$, consider the \'etale sheaf associated to the presheaf that assigns to each locally Noetherian $S$-scheme $T$ the set of isomorphism classes of torsion-free rank-1 sheaves of degree $n$ on $C_{T}/T$. This sheaf is representable by a projective $S$-scheme, denoted $\bar{J}_{C/S}^{n}$. It contains $J^{n}:=\rm{Pic}^{n}_{C/S}$ as an open subscheme. For every $S$-scheme $T$, we have a natural isomorphism $\bar{J}_{C_{T}/T}^{n}=\bar{J}_{C/S}^{n}\times T$. If $S=\rm{Spec}$$k$ for an algebraically closed field $k$, we denote $\bar{J}_{C/S}^{n}$ by $\overline{J^{n}C}$.
Recall that at the beginning we are considering $\mathcal{C}$, which is the tautological family of curves over an $n$-dimensional integral $G$-stable linear system. Since $\mathcal{C}$ has a stratification according to the geometric genus of the fibers and the $G$-action (see $\S 6$), we can temporarily focus our attention on $\overline{J^{n}C}$ for a single singular curve $C$ with a $G$-action (note that our $G$-action on $\overline{J^{n}C}$ is given by pushing forward the torsion-free rank-1 sheaves). This is reasonable since we have
{\bf Lemma 5.1.} {\it Let $X$ be an algebraic variety over $\overline{\mathbb{F}_{p}}$ with a $G$-action, $U$ an open subvariety of $X$, $Z:=X\backslash U$ the closed subvariety. If both $U$ and $Z$ are $G$-stable, then }
\[
[e(X)]=[e(U)]+ [e(Z)]
\]
{\it as $G$-representations, i.e. in $R_{\mathbb{Q}_{l}}(G)$, and $[e(X)]:=\sum_{n=0}^{\infty}(-1)^{n}[H_{c}^{n}(X,\mathbb{Q}_{l})]$.}
{\bf Proof.} One way to see this is to consider the bounded long exact sequence $0\rightarrow H^{0}_{c}(U)\rightarrow H^{0}_{c}(X)\rightarrow H^{0}_{c}(Z)\rightarrow ...$ and check that Tr$(g)=0$ on $-[e(U)]+[e(X)]-[e(Z)]$ for every $g$. Another way is to use the description of Tr$(g)$ on $e(X)$ without involving cohomology \cite[Appendix(h)]{Ca85}. $\square$
Now we have an integral curve $C$ over $\overline{\mathbb{F}_{p}}$. Recall that $\overline{J^{n}C}$ parametrizes the isomorphism classes of torsion-free rank-1 sheaves of degree $n$ on $C$, and we have the following facts \cite{Bea99}.
{\bf Proposition 5.2.} {\it Let $C$ be an integral curve over an algebraically closed field $k$.
(1) If $L\in \overline{J^{n}C}$ is a non-invertible torsion-free rank 1 sheaf, then $L=f_{*}L'$, where $L'$ is some invertible sheaf on some partial normalization $f:C'\rightarrow C$.
(2) If $f:C'\rightarrow C$ is a partial normalization of $C$, then the morphism $f_{*}:\overline{J^{n}C'}\rightarrow\overline{J^{n}C}$ is a closed embedding.}
Using these two facts, we obtain the following corollary.
{\bf Corollary 5.3.} {\it Let all the singularities of an integral curve $C$ be nodal singularities. Then $\overline{J^{n}C}$ has the following stratification}
\[
\overline{J^{n}C}=\coprod_{C'\rightarrow C} {J^{n}C'}
\]
{\it where $\overline{J^{n}C}$ parametrizes rank-1 torsion-free sheaves of degree n, and $C'$ goes through all partial normalizations of $C$ {\rm(}including $C$ itself{\rm)}.}
Now let $J^{n}C'$ be some stratum which is preserved by $G$. We want to calculate the $G$-representation $[e(J^{n}C')]$. Here we need to make use of the short exact sequence of algebraic groups
\[
0\rightarrow L\rightarrow J^{n}C'\rightarrow J^{n}\tilde{C'}\rightarrow 0
\]
where $L$ is a smooth connected linear algebraic group \cite[$\S$9 Corollary 11]{BLR90}, and $\widetilde{C'}$ is the normalization of $C'$. Since $L$ is linear, we have that $J^{n}C'$ is a principal Zariski fiber bundle over $J^{n}\tilde{C'}$ \cite[Chapter VII, Proposition 6]{Se88}. Now we need to prove the following lemma.
{\bf Lemma 5.4.} {\it Let $B$ and $F$ be separated schemes of finite type over $\overline{\mathbb{F}_{p}}$, and let $G$ be a finite group acting on $B\times F$ and $B$ such that the projection $B\times F\rightarrow B$ is $G$-equivariant. Suppose $B$ is connected. Then we have}
\[
[e(B\times F)]=[e(B)][e(F)]
\]
{\it as virtual $G$-representations, i.e. in $R_{\mathbb{Q}_{l}}(G)$, where the action of $g\in G$ on $H_{c}^{*}(F,\mathbb{Q}_{l})$ is given by $H_{c}^{*}(\{g(b)\}\times F,\mathbb{Q}_{l})\rightarrow H_{c}^{*}(\{b\}\times F,\mathbb{Q}_{l})$ by choosing any closed point $b\in B$.}
{\bf Proof.} We begin with a homotopy argument. Fix $g\in G$. By assumption, we have a commutative diagram
\[
\begin{CD}
B\times F @>g>> B\times F\\
@V\pi VV @VV\pi V\\
B @>g>> B
\end{CD}
\]
Hence we have a map $\phi=(g,\pi)$ from $B\times F$ to the fiber product $B\times F$, which maps $(b,f)$ to $(b,g_{b}(f))$, where $g_{b}:F\cong \{b\}\times F\rightarrow \{g(b)\}\times F\cong F$, and the diagram
\[
\begin{CD}
B\times F @>\phi>> B\times F\\
@V\pi VV @VV\pi V\\
B @>id>> B
\end{CD}
\]
commutes. On the other hand, we have
\[
\begin{CD}
B\times F @>>> F\\
@V\pi VV @VVV\\
B @>>> \overline{\mathbb{F}_{p}}
\end{CD}
\]
Hence $R^{i}\pi_{!}(\mathbb{Z}/n\mathbb{Z})$ is the constant sheaf $H_{c}^{i}(F,\mathbb{Z}/n\mathbb{Z})$ on $B$. The automorphism $\phi$ acts on it and, at $b\in B$ it acts the way $g_{b}$ acts on $H_{c}^{i}(F,\mathbb{Z}/n\mathbb{Z})$. Since an endomorphism of a constant sheaf over a connected base is constant, the action of $\phi$ is the same everywhere. Passing to limit, we deduce that the actions of $g_{b}$ on $H_{c}^{*}(F,\mathbb{Q}_{l})$ are the same for every $b\in B$.
By the same idea we used before, now it suffices to prove the following:
\[
|(B\times F)(\overline{\mathbb{F}_{p}})^{gF_{Q}}|=|B(\overline{\mathbb{F}_{p}})^{gF_{Q}}||F(\overline{\mathbb{F}_{p}})^{gF_{Q}}|
\]
But by what we just proved and the Lefschetz trace formula, we have $|(\{b_1\}\times F)(\overline{\mathbb{F}_{p}})^{gF_{Q}}|=|(\{b_{2}\}\times F)(\overline{\mathbb{F}_{p}})^{gF_{Q}}|$, where $b_{1}, b_{2}$ are fixed points of $gF_{Q}$. Hence the equality follows. $\square$
{\bf Corollary 5.5.} {\it Let $B$, $E$ and $F$ be separated schemes of finite type over $\overline{\mathbb{F}_{p}}$. Suppose $E$ is a Zariski-locally trivial fiber bundle over $B$ with fiber $F$ and let $G$ be a finite group acting on $E$ and $B$, the action of which is compatible with the projection $\pi:E\rightarrow B$. If $B$ is irreducible, then we have}
\[
[e(E)]=[e(B)][e(F)]
\]
{\it as virtual $G$-representations, i.e. in $R_{\mathbb{Q}_{l}}(G)$.}
{\bf Proof.} Fixing $g\in G$, it suffices to prove that the action of $g_{b_1}$ on $H_{c}^{*}(F,\mathbb{Q}_{l})$ is the same as the action of $g_{b_{2}}$ for any $b_{1}, b_{2}\in B$ fixed by $g$. Take open neighborhoods $U_{1}, U_{2}$ of $b_{1}, b_{2}$ which trivialize the bundle. Replacing $U$ by $\cap_{n=0}^{\infty} g^{n}(U)$, we can assume $U_{1}, U_{2}$ are $g$-stable and connected since $B$ is irreducible. Now let $V=U_{1}\cap U_{2}$ and take any closed point $b_{0}\in V$. By Lemma 5.4, we deduce that the action of $g_{b_1}$ is the same as the action of $g_{b_{0}}$, which is the same as the action of $g_{b_{2}}$. Hence we have
\[
|E(\overline{\mathbb{F}_{p}})^{gF_{Q}}|=|B(\overline{\mathbb{F}_{p}})^{gF_{Q}}||F(\overline{\mathbb{F}_{p}})^{gF_{Q}}|
\]
and we are done. $\square$
{\bf Corollary 5.6.} {\it Let $C$ be an integral projective curve over $\overline{\mathbb{F}_{p}}$ with finite group $G$-actions. Then }
\[
[e(J^{n}C)]=[e(L)][e(J^{n}\tilde{C})]
\]
{\it as virtual $G$-representations, i.e. in $R_{\mathbb{Q}_{l}}(G)$, where $L$ is a linear algebraic group and $\tilde{C}$ is the normalization of $C$.}
{\bf Proof.} Let $f^{*}:J^{n}C\rightarrow J^{n}\tilde{C}$ be the pullback map. Since $g$ is an automorphism on $C$ and $\tilde{C}$, we have $g_{*}f^{*}=f^{*}g_{*}$. Now we use Corollary 5.5. $\square$
Now to prove Theorem 1.9, we first prove the following statement about $e(J^{n}C)$.
{\bf Lemma 5.7.} {\it Let $C$ be an integral curve over $\overline{\mathbb{F}_{p}}$ with nodal singularities and a $G$-action. Suppose $p\nmid|G|$. If $\tilde{C}/\left<g\right>$ is not a rational curve for every $g\in G$, then $[e({J^{n}C})]=0$ as $G$-representations.}
{\bf Proof.} By Corollary 5.6, it suffices to prove $[e(J^{n}\tilde{C})]=0$, which is equivalent to ${\rm Tr}(g)|_{[e(J^{n}\tilde{C})]}=0$ for any $g\in G$. But $J^{n}\tilde{C}$ is an abelian variety, which means $H^{i}(J^{n}\tilde{C},\mathbb{Q}_{l})\cong\wedge^{i}H^{1}(J^{n}\tilde{C},\mathbb{Q}_{l})$. Since $p\nmid|G|$, $\tilde{C}/\left<g\right>$ is smooth. Then since $\tilde{C}/\left<g\right>$ is not rational, we have $H^{1}(\tilde{C}/\left<g\right>,\mathbb{Q}_{l})\neq 0$. Hence $H^{1}(J^{n}\tilde{C},\mathbb{Q}_{l})^{\left<g\right>}\neq 0$, which implies ${\rm Tr}(g)|_{[e(J^{n}\tilde{C})]}=0$. This is because $H^{1}(J^{n}\tilde{C},\mathbb{Q}_{l})=V_{0}\oplus V_{1}$, where $V_{0}$ is the non-empty eigenspace of $g$ with eigenvalue 1, and $V_{1}$ is its complement. We have $\sum(-1)^{i}\wedge^{i}H^{1}(J^{n}\tilde{C},\mathbb{Q}_{l})=(\sum(-1)^{i}\wedge^{i}V_{0})\otimes(\sum(-1)^{i}\wedge^{i}V_{1}$) and $\sum(-1)^{i}\wedge^{i}V_{0}$ has trace 0. $\square$
Now with the help of Corollary 5.3, we have
{\bf Proof of Theorem 1.9.} Fix $g\in G$. Recall that we have $\overline{J^{n}C}=\coprod_{C'\rightarrow C} {J^{n}C'}$ by Corollary 5.3. Depending on the action of $g$ on the nodes of $C$, $g_{*}$ permutes or acts on the strata $J^{n}C'$. For any union of two or more strata permuted by $g_{*}$ cyclically, the trace of $g$ on any $H^{i}_{c}$ equals 0 since $g$ acts by cyclically permuting the $H^{i}_{c}$ of the components. For the stratum which is stable under $g$, the trace of $g$ is also 0 by Lemma 5.7. Hence $[e(\overline{J^{n}C})]=0$ by Lemma 5.1. $\square$
{\bf Proof of Corollary 1.10.} If $\tilde{C}/G$ is not a rational curve, then $\tilde{C}/\left<g\right>$ is not rational for any $g\in G$. $\square$
{\bf Remark 5.8.} We are dealing with characteristic $p$ in this section because we do not know whether Corollary 5.5 is true for singular cohomology in characteristic 0. Since $B$, $E$, and $F$ are not assumed to be smooth projective, we cannot use the proper smooth base change theorem.
\section{Rational curves on surfaces}
Let $S$ be a smooth projective $K3$ surface over $\mathbb{C}$ with a $G$-action, and let $\mathcal{C}$ be the tautological family of curves over an $n$-dimensional integral linear system $|\mathcal{L}|$ acted on by $G$. Then $\overline{J^{n}\mathcal{C}}$ is a smooth projective variety over $|\mathcal{L}|$ whose fiber over a point $t\in\mathcal{L}$ is the compactified jacobian $\overline{J^{n}C_{t}}$. Choose some good reduction over $q$ such that `everything' ($\overline{J^{n}\mathcal{C}}$, $S$, $G$-action etc.) is defined over $\mathbb{F}_{q}$, and we assume $|\mathcal{L}|$ is still integral. Then $|\mathcal{L}|$ has a stratification where each stratum $B$ satisfies ${\rm Stab}_{G}(t)=H$ for every $t\in B$ and some subgroup $H$, and the fibers $\mathcal{C}_{t}$ of the stratum have the same geometric genus. This is because for any subgroup $H$ in $G$, $|\mathcal{L}|^{H}\backslash\cup_{H'\supsetneqq H}|\mathcal{L}|^{H'}$ is a locally closed subspace. The reason for the stratification by the geometric genus is that the geometric genus gives a lower semicontinuous function in our case \cite{Sh12}. Now notice that $gB=B$ if $g\in N_{G}(H)$ and $gB\cap B=\emptyset$ if $g\notin N_{G}(H)$. Hence we have a new stratification of $|\mathcal{L}|$ where each stratum $\cup_{g\in G}gB$ is $G$-invariant. Let $\pi:\overline{J^{n}\mathcal{C}}\rightarrow|\mathcal{L}|$ be the compactified jacobian of the family of curves. Then by considering $\pi^{-1}(\cup_{g\in G}gB)$ for all $B$, we obtain a stratification of $\overline{J^{n}\mathcal{C}}$, where each stratum is $G$-stable and is a Zariski-locally trivial fiber bundle over $\cup_{g\in G}gB$ for the corresponding $B$. Then since
\[
H_{c}^{*}(\pi^{-1}(\cup_{g\in G}gB),\mathbb{Q}_{l})={\rm Ind}_{N_{G}(H)}^{G}H_{c}^{*}(\pi^{-1}B,\mathbb{Q}_{l}),
\]
we know that $[e(\overline{J^{n}\mathcal{C}})]$ equals
\[
\sum [e(\pi^{-1}(\cup_{g\in G}gB))]=\sum{\rm Ind}_{N_{G}(H)}^{G}[e(\pi^{-1}B)]=\sum {\rm Ind}_{N_{G}(H)}^{G} [e(\overline{J^{n}C_{0}})][e(B)]
\]
by Lemma 5.1 and Corollary 5.5, where $C_{0}$ is the fiber $\mathcal{C}_{t}$ for some $t\in B$, ${\rm Ind}_{N_{G}(H)}^{G}$ is the induced representation, and $e(\overline{J^{n}C_{0}})$ can be expressed using Corollary 5.3, 5.6 or Theorem 1.9.
{\bf Proof of Corollary 1.4.} Recall that we have a birational map from $\overline{J^{n}\mathcal{C}}$ to $S^{[n]}$ \cite{Bea99}, which maps a pair $(C_{t},L)$ to a unique effective divisor $D$ on $C_{t}$ which is linearly equivalent to $L$, and can be viewed as a length $d$ subscheme of $S$. Note that the $G$-actions commutes with this map in the sense of Theorem 1.3. Hence by Theorem 1.3 $[e(\overline{J^{n}\mathcal{C}})]=[e(S^{[n]})]$ as $G$-representations. Now by using Corollary 3.6 we are done. $\square$
Now we want to explicitly describe $\sum_{n=0}^{\infty}[e(S^{[n]})]t^{n}$ as a $G$-representation. For that purpose, we cite the following theorems (\cite[Proposition 1.2]{Mu88} and \cite[Lemma 2.3]{AST11}).
{\bf Theorem 6.1. \cite{Mu88}} {\it Let $g$ be a symplectic automorphism of a complex K3 surface $S$ of order $n<\infty$. Then the number of fixed points of $g$ is equal to $\epsilon(n)=24\left(n\prod_{p|n}\left(1+\frac{1}{p}\right)\right)^{-1}$.}
{\bf Theorem 6.2. \cite{AST11}} {\it Let $g$ be a non-symplectic automorphism of a complex K3 surface $S$ of prime order $p$. Then the Euler characteristic of $S^{g}$ is $24-dp$, where $S^{g}$ denotes the fixed locus of $g$, $d=\frac{{\rm rank}\,T(g)}{p-1}$, and $T(g):=(H^{2}(S,\mathbb{Z})^{g})^{\bot}$.}
{\bf Proof of Theorem 1.6.} By remark 3.7, we deduce that
\[
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{\text{Tr}(g^{k})|_{[e(\bar{S})]}t^{mk}}{k}\right).
\]
Then using the Lefschetz fixed point formula and Theorem 6.1, we get the equality we want.
When $G$ is a cyclic group of order $N$, we have $N\leq 8$ by \cite[Corollary 15.1.8]{H16}. Recall the definition of the Dedekind eta function $\eta(t)=t^{1/24}\prod_{n=1}^{\infty}(1-t^{n})$, where $t=e^{2\pi iz}$. Fix a generator $g$ of $G$.
If $N$ is a prime number $p$, we notice that ord$(g^{k})=1$ if $p|k$, and ord$(g^{k})=p$ otherwise. Hence
\[
\begin{aligned}
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}&=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{\epsilon(p)t^{mk}}{k}\right)\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{(24-\epsilon(p))t^{mpk}}{pk}\right)\\
&=\left(\prod_{m=1}^{\infty}(1-t^{m})\right)^{-\epsilon(p)}\left(\prod_{m=1}^{\infty}(1-t^{mp})\right)^{\frac{\epsilon(p)-24}{p}}\\
&=\left(\prod_{m=1}^{\infty}(1-t^{m})(1-t^{mp})\right)^{-\frac{24}{p+1}}\\
&=t/\eta(t)^{\frac{24}{p+1}}\eta(t^{p})^{\frac{24}{p+1}}.
\end{aligned}
\]
If $N=4$, we have
\[
\text{Tr}(g^{k})|_{[e(\bar{S})]}=
\begin{cases}
4, & {\rm if}\ k\equiv 1,3\ ({\rm mod}\ 4)\\
8, & {\rm if}\ k\equiv 2\ ({\rm mod}\ 4)\\
24, & {\rm if}\ k\equiv 0\ ({\rm mod}\ 4).
\end{cases}
\]
Hence
\[
\begin{aligned}
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}&=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k\equiv 1,3}\frac{4t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 2}\frac{8t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 0}\frac{24t^{mk}}{k}\right)\\
&=\prod_{m=1}^{\infty}(1-t^{m})^{-4}(1-t^{2m})^{-2}(1-t^{4m})^{-4}\\
&=t/\eta(t)^{4}\eta(t^{2})^{2}\eta(t^{4})^{4}.
\end{aligned}
\]
If $N=6$, we have
\[
\text{Tr}(g^{k})|_{[e(\bar{S})]}=
\begin{cases}
2, & {\rm if}\ k\equiv 1,5\ ({\rm mod}\ 6)\\
6, & {\rm if}\ k\equiv 2,4\ ({\rm mod}\ 6)\\
8, & {\rm if}\ k\equiv 3\ ({\rm mod}\ 6)\\
24, & {\rm if}\ k\equiv 0\ ({\rm mod}\ 6).
\end{cases}
\]
Hence
\[
\begin{aligned}
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}&=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k\equiv 1,5}\frac{2t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 2,4}\frac{6t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 3}\frac{8t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 0}\frac{24t^{mk}}{k}\right)\\
&=\prod_{m=1}^{\infty}(1-t^{m})^{-2}(1-t^{2m})^{-2}(1-t^{3m})^{-2}(1-t^{6m})^{-2}\\
&=t/\eta(t)^{2}\eta(t^{2})^{2}\eta(t^{3})^{2}\eta(t^{6})^{2}.
\end{aligned}
\]
If $N=8$, we have
\[
\text{Tr}(g^{k})|_{[e(\bar{S})]}=
\begin{cases}
2, & {\rm if}\ k\equiv 1,3,5,7\ ({\rm mod}\ 8)\\
4, & {\rm if}\ k\equiv 2,6\ ({\rm mod}\ 8)\\
8, & {\rm if}\ k\equiv 4\ ({\rm mod}\ 8)\\
24, & {\rm if}\ k\equiv 0\ ({\rm mod}\ 8).
\end{cases}
\]
Hence
\[
\begin{aligned}
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}&=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k\equiv 1,3,5,7}\frac{2t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 2,6}\frac{4t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 4}\frac{8t^{mk}}{k}+\sum_{m=1}^{\infty}\sum_{k\equiv 0}\frac{24t^{mk}}{k}\right)\\
&=\prod_{m=1}^{\infty}(1-t^{m})^{-2}(1-t^{2m})^{-1}(1-t^{4m})^{-1}(1-t^{8m})^{-2}\\
&=t/\eta(t)^{2}\eta(t^{2})\eta(t^{4})\eta(t^{8})^{2}.
\end{aligned}
\]
$\square$
{\bf Proof of Theorem 1.8.} By remark 3.7, we have
\[
\sum_{n=0}^{\infty}{\rm Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{\text{Tr}(g^{k})|_{[e(\bar{S})]}t^{mk}}{k}\right).
\]
By the topological Lefschetz formula, we have
\[
e_{top}(S^{g})=\sum_{i=0}^{4}(-1)^{i}{\rm Tr}(g^{*}|H^{i}(S,\mathbb{C}))=\sum_{i=0}^{4}(-1)^{i}{\rm Tr}(g^{*}|H^{i}(\bar{S},\mathbb{Q}_{l}))={\rm Tr}(g)|_{[e(\bar{S})]}.
\]
Fix $g\neq 1$ and notice that $S^{g}$ is the same as $S^{g^{k}}$ for $p\nmid k$. We have ${\rm Tr}(g)|_{[e(\bar{S})]}={\rm Tr}(g^{k})|_{[e(\bar{S})]}=24-dp$ by Theorem 6.2. Hence
\[
\begin{aligned}
\sum_{n=0}^{\infty}\text{Tr}(g)|_{[e(\bar{S}^{[n]})]}t^{n}&=\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{(24-dp)t^{mk}}{k}\right)\text{exp}\left(\sum_{m=1}^{\infty}\sum_{k=1}^{\infty}\frac{(dp)t^{mpk}}{pk}\right)\\
&=\left(\prod_{m=1}^{\infty}(1-t^{m})\right)^{dp-24}\left(\prod_{m=1}^{\infty}(1-t^{mp})\right)^{-d}.
\end{aligned}
\]
$\square$
{\bf Example 1 ($\mathbb{Z}/2\mathbb{Z}$).} Here we look at an explicit K3 surface with a symplectic $\mathbb{Z}/2\mathbb{Z}$-action. Consider the elliptic K3 surface $S$ defined by the Weierstrass equation
\[
y^{2}=x^{3}+(t^{4}+a_{1}t^{2}+a_{2})x+(t^{12}+b_{1}t^{10}+b_{2}t^{8}+b_{3}t^{6}+b_{4}t^{4}+b_{5}t^{2}+b_{6})
\]
where $(a_{1},a_{2})\in\mathbb{C}^{2}$, $(b_{1},...,b_{6})\in\mathbb{C}^{6}$ are generic. The fibration has 24 nodal fibers (Kodaira type $I_{1}$) over the zeros of its discriminant polynomial and those zeros do not contain $0$ and $\infty$. The automorphsim of order 2
\[
\sigma(x,y,t)=(x,-y,-t)
\]
acts non-trivially on the basis of the fibration and preserves the smooth elliptic curves over $t=0$ and $t=\infty$. Now denote one of the fibers by $L$, then $|L|$ is a $\left<\sigma\right>$-invariant integral linear system and all of the singular curves in $|L|$ are nodal rational curves. We want to understand the $\sigma$-orbits in $|L|$.
Since we know explicitly the action of $\sigma$, by calculation we know that there are 4 $\sigma$-fixed points on the fiber over $t=0$ and 4 $\sigma$-fixed points on the fiber over $t=\infty$. So $\sigma$ has 8 isolated fixed points, hence it is a symplectic involution.
Now we know from Theorem 1.6 that
\[
\sum_{n=0}^{\infty}\text{Tr}(\sigma)|_{[e(\bar{S}^{[n]})]}t^{n}=\left(\prod_{k=1}^{\infty}(1-t^{k})\right)^{-8}\left(\prod_{k=1}^{\infty}(1-t^{2k})\right)^{-8}.
\]
This implies ${\rm Tr}(\sigma)|_{[e(S)]}=8$ by looking at the coefficient of $t$. Hence we have ${\rm Tr}(\sigma)|_{[e(\overline{J\mathcal{C}})]}=8$ by Corollary 1.4. But in this case, there are only two strata contributing to $[e(\overline{J\mathcal{C}})]$ as $\left<\sigma\right>$-representations (see Remark 6.3). One consists of nodal rational curves which are not $\sigma$-stable. The other consists of elliptic curves whose quotient by $\sigma$ is rational. The first stratum contributes $n_{1}[\mathbb{Z}/2\mathbb{Z}]$ if there are $n_{1}$ such $\sigma$-orbits in $|L|$, where $[\mathbb{Z}/2\mathbb{Z}]$ is the regular representation. The second stratum contributes $n_{2}(2V_{1}-2V_{-1})$ if there are $n_{2}$ such elliptic curves in $|L|$, where $V_{s}$ is the 1-dim representation on which $\sigma$ has eigenvalue $s$. Hence
\[
[e(\overline{J\mathcal{C}})]=n_{1}[\mathbb{Z}/2\mathbb{Z}]+n_{2}(2V_{1}-2V_{-1})
\]
But since we already know the representation $[e(\overline{J\mathcal{C}})]$, by calculation we have $n_{1}=12$ and $n_{2}=2$.
On the other hand, this coincides with the geometric picture. From the definition of $\sigma$ we observe that there are indeed 12 $\sigma$-orbits of nodal rational curves. Denote by $C_{0}, C_{\infty}$ the fibers over $t=0, \infty$. Since $\sigma$ preserve $C_{0}$ and there are 4 $\sigma$-fixed points, we deduce that the degree 2 morphism $C_{0}\rightarrow C_{0}/\left<\sigma\right>$ has 4 ramification points. Hence by Riemann-Hurwitz formula $C_{0}/\left<\sigma\right>$ is smooth rational. By the same argument, $C_{\infty}/\left<\sigma\right>$ is also smooth rational. This is what we have expected since there should be two such curves from the calculation of the representations.
{\bf Example 2 ($\mathbb{Z}/3\mathbb{Z}$).} Here we look at an explicit K3 surface with a non-symplectic $\mathbb{Z}/3\mathbb{Z}$-action \cite[Remark 4.2]{AST11}. Consider the elliptic K3 surface $S$ defined by the Weierstrass equation
\[
y^{2}=x^{3}+(t^6+a_{1}t^{3}+a_{2})x+(t^{12}+b_{1}t^{9}+b_{2}t^{6}+b_{3}t^{3}+b_{4})
\]
where $(a_{1},a_{2})\in\mathbb{C}^{2}$, $(b_{1},...,b_{4})\in\mathbb{C}^{4}$ are generic. The fibration has 24 nodal fibers (Kodaira type $I_{1}$) over the zeros of its discriminant polynomial and those zeros do not contain $0$ and $\infty$. The automorphsim of order 3
\[
\sigma(x,y,t)=(x,y,\zeta_{3}t)
\]
acts non-trivially on the basis of the fibration and preserves the smooth elliptic curves over $t=0$ and $t=\infty$. Now denote one of the fibers by $L$, then $|L|$ is a $\left<\sigma\right>$-invariant integral linear system and all of the singular curves in $|L|$ are nodal rational curves. We want to understand the $\sigma$-orbits in $|L|$.
First we observe that $\sigma$ fixs the fiber over $t=0$. Hence $\left<\sigma\right>$ acts non-symplectically on $S$. We know from \cite[Theorem 4.1]{AST11} that rank $T(\sigma)=14$. Then using Theorem 1.8 we have
\[
\sum_{n=0}^{\infty}\text{Tr}(\sigma)|_{[e(\bar{S}^{[n]})]}t^{n}=\left(\prod_{k=1}^{\infty}(1-t^{k})\right)^{-3}\left(\prod_{k=1}^{\infty}(1-t^{3k})\right)^{-7}.
\]
This implies ${\rm Tr}(\sigma)|_{[e(S)]}=3$ by looking at the coefficient of $t$. Hence we have ${\rm Tr}(\sigma)|_{[e(\overline{J\mathcal{C}})]}=3$ by Corollary 1.4. But in this case, there are only two strata contributing to $[e(\overline{J\mathcal{C}})]$ as $\left<\sigma\right>$-representations (see Remark 6.3). One consists of nodal rational curves which are not $\sigma$-stable. The other consists of elliptic curves whose quotient by $\sigma$ is rational. In particular, the fiber over $t=0$ will not contribute to $[e(\overline{J\mathcal{C}})]$. The first stratum contributes $n_{1}[\mathbb{Z}/3\mathbb{Z}]$ if there are $n_{1}$ such $\sigma$-orbits in $|L|$, where $[\mathbb{Z}/3\mathbb{Z}]$ is the regular representation. The second stratum contributes $n_{2}(2V_{1}-V_{\zeta_{3}}-V_{\zeta^{-1}_{3}})$ if there are $n_{2}$ such elliptic curves in $|L|$ , where $V_{s}$ is the 1-dim representation on which $\sigma$ has eigenvalue $s$. Hence
\[
[e(\overline{J\mathcal{C}})]=n_{1}[\mathbb{Z}/3\mathbb{Z}]+n_{2}(2V_{1}-V_{\zeta_{3}}-V_{\zeta^{-1}_{3}})
\]
But since we already know the representation $[e(\overline{J\mathcal{C}})]$, by calculation we have $n_{1}=8$ and $n_{2}=1$.
On the other hand, this coincides with the geometric picture. From the definition of $\sigma$ we observe that there are indeed 8 $\sigma$-orbits of nodal rational curves. Since the action of $\sigma$ is explicit, by calculation we know that there are 3 $\sigma$-fixed points on the fiber over $t=\infty$. Denote by $C_{\infty}$ the fiber over $t=\infty$. Then this implies that the degree 3 morphism $C_{\infty}\rightarrow C_{\infty}/\left<\sigma\right>$ has 3 ramification points. Hence by Riemann-Hurwitz formula $C_{\infty}/\left<\sigma\right>$ is smooth rational, which is what we have expected.
{\bf Remark 6.3 ($\mathbb{Z}/2\mathbb{Z}$ in general)}. Let us consider a complex K3 surface $S$ with a $\mathbb{Z}/2\mathbb{Z}$-action (i.e. an involution $\sigma$). Take a $\sigma$-invariant integral linear system $\mathcal{L}$ of dimension $d$. Assume all the rational curves in $\mathcal{L}$ have nodal singularities.
For the stratum of $\mathcal{L}$ which consists of curves that are not $\sigma$-stable and of geometric genus $>0$, denote by $M$ the corresponding stratum of $\overline{J^{d}\mathcal{C}}$. Then $[e(M)]=e(M/\left<\sigma\right>)[\mathbb{Z}/2\mathbb{Z}]=e(\overline{J^{d}C_{0}})e(B/\left<\sigma\right>)[\mathbb{Z}/2\mathbb{Z}]$, where $[\mathbb{Z}/2\mathbb{Z}]$ is the regular representation. But $e(\overline{J^{d}C_{0}})=\sum_{n=0}^{\infty} (-1)^{n}{\rm dim}H^{n}(\overline{J^{d}C_{0}},\mathbb{Q}_{l})=0$ by Theorem 1.9. Hence $[e(M)]=0$ as representation.
For the stratum of $\mathcal{L}$ which consists of curves that are not $\sigma$-stable and are nodal rational curves, we have $[e(M)]=n_{0}[\mathbb{Z}/2\mathbb{Z}]$ where $n_{0}$ is the number of $\sigma$-orbits of nodal rational curves, since $e(\overline{J^{d}C_{0}})=1$ by Corollary 5.3, Corollary 5.6 and $e(\mathbb{G}_{m})=0$.
For those curves that are $\sigma$-stable, we first notice that $\sigma$ fixes some curve only if $\sigma$ acts non-symplectically on $S$, and in that case the fixed curves are always smooth (\cite[$\S 2$]{AST11}). Let $C_{0}$ be a smooth curve of genus $d\geq 1$ fixed by $\sigma$. If $d=1$, then the fixed locus of $\sigma$ consists of two disjoint elliptic curves and they are linearly equivalent. If $d>1$, then it is the only curve of genus $d$ fixed by $\sigma$ (\cite[Theorem 3.1]{AST11}). In either case, the stratum consisting of $\sigma$-fixed curves contributes 0 to the representation since the Euler characteristic of an abelian variety is $0$.
For the stratum of $\mathcal{L}$ which consists of curves that are $\sigma$-stable, if the normalization of the curves quotient by $\left<\sigma\right>$ is not rational, then by Theorem 1.9, we have $[e(M)]=0$.
Hence there are only two strata contributing to $[e(\overline{J^{d}\mathcal{C}})]$. One consists of $\sigma$-orbits of nodal rational curves. The other consists of the curves whose normalization quotient by $\left<\sigma\right>$ are smooth rational.
In particular, for the stratum which consists of $\sigma$-stable smooth curves whose quotient by $\left<\sigma\right>$ are rational, by a similar argument as in the proof of Lemma 5.7, we have $[e(J^{d}(C_{0}))]=2^{2d-1}V_{1}-2^{2d-1}V_{-1}$, where $V_{s}$ is the 1-dim representation on which $\sigma$ has eigenvalue $s$.
Now let us give an example of a 2-dim $\mathbb{Z}/2\mathbb{Z}$-invariant linear system. This example is suggested by Jim Bryan.
{\bf Example 3 (2-dim $\mathbb{Z}/2\mathbb{Z}$).} Let $S$ be a K3 surface given by the double cover of $\mathbb{P}_{2}$ branched over a smooth sextic curve $C$ in $\mathbb{P}_{2}$. Let $\tau$ be the involution on $\mathbb{P}_{2}$ sending $(x: y: z)$ to $(-x: y: z)$. Denote the covering involution by $i: S\rightarrow S$. Then if we suppose $C$ is $\tau$-invariant, the `composition' of $\tau$ and $i$ will give a symplectic involution $\sigma$ on $S$ (\cite[Section 3.2]{GS07}).
The fixed locus of $\tau$ on $\mathbb{P}_{2}$ consists of a point $x_{0}=(1: 0: 0)$ and a line $l_{0}=\{(x: y: z)| x=0\}$. Denote the six intersection points of $l_{0}$ and $C$ by $x_{3}, x_{4},..., x_{8}$. Let $\pi: S\rightarrow\mathbb{P}_{2}$ be the double cover map. Denote the two points in $\pi^{-1}(x_{0})$ by $x_{1}, x_{2}$. Then the fixed locus of $\sigma$ is the eight points $x_{1}, x_{2},..., x_{8}$. Notice that $\sigma$ commutes with $i$ and the induced action of $\sigma$ on $\mathbb{P}_{2}$ is just $\tau$.
Now let $\mathcal{L}=\pi^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)$. Then the linear system $|\mathcal{L}|$ consists of the curves which are the preimages of the lines in $\mathbb{P}^{2}$ under $\pi$. For a generic choice of $C$, $|\mathcal{L}|$ is a $\sigma$-invariant integral linear system. A generic line will intersects $C$ in six points, and its preimage is a smooth genus 2 curve. Some lines will intersect $C$ in a tangent point and 4 other distinct points, and their preimages are curves with one node. The other lines are the 324 bitangents of $C$, which can be seen from the Pl\"ucker formula or the coefficient of $t^{2}$ in $\prod_{n=1}^{\infty}(1-t^{n})^{-24}$.
Let $\mathcal{C}\rightarrow|\mathcal{L}|$ be the tautological family of curves over $|\mathcal{L}|$. Now we know from Theorem 1.6 that
\[
\sum_{n=0}^{\infty}\text{Tr}(\sigma)|_{[e({S}^{[n]})]}t^{n}=\left(\prod_{k=1}^{\infty}(1-t^{k})\right)^{-8}\left(\prod_{k=1}^{\infty}(1-t^{2k})\right)^{-8}.
\]
This implies ${\rm Tr}(\sigma)|_{[e({S}^{[2]})]}=52$ by looking at the coefficient of $t^{2}$. Hence we have ${\rm Tr}(\sigma)|_{[e(\overline{J\mathcal{C}})]}=52$ by Corollary 1.4. Since we know ${\rm Tr}(1)|_{[e(S^{[2]})]}=324$, we have
\[
[e(\overline{J\mathcal{C}})]=188V_{1}+136V_{-1}
\]
where $V_{1}$ is the 1-dimensional trivial representation and $V_{-1}$ is the 1-dimensional representation on which $\sigma$ has eigenvalue $-1$.
To give a geometric interpretation of this representation, we first concentrate on the $\sigma$-invariant curves in $|\mathcal{L}|$ since non-invariant curves with zero or one node will contribute nothing to the representation, and non-invariant nodal rational curves will contribute some multiple of the regular representation. The $\sigma$-invariant curves in $|\mathcal{L}|$ are given by the preimages of the $\tau$-invariant lines, which consists of the line $l_{0}=\{(x: y: z)| x=0\}$ and all the lines passing through $x_{0}=(1: 0: 0)$, i.e., $\{(x: y: z)| by+cz=0\}$, $(b: c)\in\mathbb{P}^{1}$.
The preimage of $l_{0}$ is a smooth genus 2 curve, and it has 6 ramification points $x_{3}, x_{4},..., x_{8}$ under $\sigma$. Hence its preimage quotient by $\sigma$ is a smooth rational curve, and it will contribute $8V_{1}-8V_{-1}$ to the representation by a similar argument as in the proof of Lemma 5.7.
The preimages of $\{(x: y: z)| by+cz=0\}$ are more complicated. A generic line will intersect $C$ in six points, and its preimage is a smooth curve of genus 2. It has 2 ramification points $x_{1}, x_{2}$ under $\sigma$. Hence its preimage quotient by $\sigma$ is an elliptic curve, and it will contribute nothing to the representation.
Now let us consider those tangent lines. We first observe that if a line passes through one of the six points $x_{3}, x_{4},..., x_{8}$, then by looking at the $\sigma$-action on the preimage of the line, this point must be a tangent point. So its preimage is a curve with one node, and the normalization of it has 4 ramification points $x_{1}, x_{2}, x_{i}^{1}, x_{i}^{2}$ under $\sigma$, where $x_{i}^{1}, x_{i}^{2}$ are the two points on the normalization over the point $x_{i}$ if our line passes through $x_{i}$, $i\in\{3,4,...,8\}$. Hence the normalization of its preimage quotient by $\sigma$ is a rational curve, and we denote its contribution to the representation by $[e(\overline{J{C}_{1}})]$.
Now if there is a tangent point $y$ of the tangent line which is not one of the six points $x_{3}, x_{4},..., x_{8}$, then since both of the line and the curve $C$ are $\tau$-invariant, this line must have another tangent point $\tau(y)$. Hence this line must be a bitangent. In order to calculate the number of the $\tau$-invariant bitangents, we notice that the degree of the dual curve of $C$ is 30. We already have 6 tangent lines with one tangent point, and all the other $\tau$-invariant tangent lines are bitangents. Hence there should be 12 $\tau$-invariant bitangents. The preimage of the $\tau$-invariant bitangent is a rational curve with two nodes, and $\tau$ permutes the nodes. The normalization of the preimage has 2 ramification points $x_{1}, x_{2}$ under $\sigma$. Hence the normalization of its preimage quotient by $\sigma$ is a rational curve, and we denote its contribution to the representation by $[e(\overline{J{C}_{2}})]$.
Finally, non-invariant curves with two nodes will also contribute to the representation. We know there are 324 curves with two nodes, and 12 of them are $\sigma$-invariant by the discussion above. So there are 312 non-invariant nodal rational curves, which will contribute $156[\mathbb{Z}/2\mathbb{Z}]=156V_{1}+156V_{-1}$ to the representation.
Combining all of the above, we have
\[
\begin{aligned}
188V_{1}+136V_{-1}&=[e(\overline{J\mathcal{C}})]\\
&=8V_{1}-8V_{-1}+6[e(\overline{J{C}_{1}})]+12[e(\overline{J{C}_{2}})]+156V_{1}+156V_{-1}.
\end{aligned}
\]
Hence we only need to check
\[
[e(\overline{J{C}_{1}})]+2[e(\overline{J{C}_{2}})]=4V_{1}-2V_{-1}
\]
and this is true by the following two lemmas.
{\bf Lemma 6.4.} {\it Let $C_{1}$ be an integral curve of arithmetic genus 2 with one node over $\overline{\mathbb{F}_{p}}$. If there is a involution $\sigma$ acting on it, and the action $\sigma$ on its normalization $\tilde{C_{1}}$ has 4 fixed points, two of which are the points over the node, then we have }
\[
[e(\overline{J{C}_{1}})]=2V_{1}-2V_{-1}
\]
as $\mathbb{Z}/2\mathbb{Z}$-representations.
{\bf Lemma 6.5.} {\it Let $C_{2}$ be an integral curve of arithmetic genus 2 with two nodes over $\overline{\mathbb{F}_{p}}$. If there is a involution $\sigma$ acting on it, and $\sigma$ permutes the nodes, then we have }
\[
[e(\overline{J{C}_{2}})]=V_{1}
\]
as $\mathbb{Z}/2\mathbb{Z}$-representations.
{\bf Proof of Lemma 6.4.} Since $C_{1}$ is an integral curve of arithmetic genus 2 with one node, its normalization $\pi:\tilde{C_{1}}\rightarrow C_{1}$ is an elliptic curve, and we denote it by $E$. By Corollary 5.3 and Corollary 5.6, we have
\[
[e(\overline{J{C}_{1}})]=[e({J{C}_{1}})]+[e({J{E}})]=[e(\mathbb{G}_{m})][e(JE)]+[e(JE)].
\]
Now we notice that $\sigma$ has 4 fixed points on $E$, so $E\rightarrow E/\left<\sigma\right>$ realizes $E$ as a double cover of $\mathbb{P}^{1}$. Hence
\[
[e(JE)]=[H^{0}(E,\mathbb{Q}_{l})]-[H^{1}(E,\mathbb{Q}_{l})]+[H^{2}(E,\mathbb{Q}_{l})]=2V_{1}-2V_{-1}.
\]
On the other hand, since $\sigma$ fixes the points over the node, we deduce from the short exact sequence
\[
0\rightarrow\mathcal{O}_{C_{1}}^{*}\rightarrow\pi_{*}\mathcal{O}_{E}^{*}\rightarrow\delta\rightarrow 0
\]
that $\sigma$ acts trivially on the skyscraper sheaf $\delta$. Hence $\sigma$ acts trivially on $H^{0}(C_{1}, \delta)=\mathbb{G}_{m}$. So $[e(\mathbb{G}_{m})]=e(\mathbb{G}_{m})V_{1}=0$ since the topological Euler characteristic of $\mathbb{G}_{m}$ is 0.
Combining the above discussion, we have
\[
[e(\overline{J{C}_{1}})]=2V_{1}-2V_{-1}.
\]
$\square$
{\bf Proof of Lemma 6.5.} Since $C_{2}$ is an integral curve of arithmetic genus 2 with two nodes, its normalization $\pi:\tilde{C_{2}}\rightarrow C_{2}$ is a rational curve. It also has two partial normalizations by resolving one of the nodes $\pi_{1}:C_{2}^{'}\rightarrow C_{2}$ and $\pi_{2}:C_{2}^{''}\rightarrow C_{2}$. By Corollary 5.3 and Corollary 5.6, we have
\[
[e(\overline{J{C}_{2}})]=[e({J{C}_{2}})]+[e({J{C}^{'}_{2}})]+[e({J{C}^{''}_{2}})]+[e({J\mathbb{P}^{1}})]=[e(\mathbb{G}_{m}\times\mathbb{G}_{m})]+e(\mathbb{G}_{m})[\mathbb{Z}/2\mathbb{Z}]+V_{1}
\]
since $\sigma$ permutes two nodes.
On the other hand, $\mathbb{G}_{m}$ is an affine curve. So dim $H^{2}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})=1$ and dim $H^{0}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})=0$. Since the topological Euler characteristic of $\mathbb{G}_{m}$ is 0, we also have dim $H^{1}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})=1$. Notice that $\sigma$ permutes two $\mathbb{G}_{m}$'s, and hence by the K\"unneth formula we have
\[
\begin{aligned}{}
[e(\mathbb{G}_{m}\times\mathbb{G}_{m})]&=[H^{1}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})\otimes H^{1}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})]-[H^{1}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})\otimes H^{2}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})]\\
&-[H^{2}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})\otimes H^{1}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})]+[H^{2}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})\otimes H^{2}_{c}(\mathbb{G}_{m},\mathbb{Q}_{l})]\\
&=V_{-1}-[\mathbb{Z}/2\mathbb{Z}]+V_{1}\\
&=0.
\end{aligned}
\]
Combining the above discussion, we have
\[
[e(\overline{J{C}_{2}})]=V_{1}.
\]
$\square$
Finally, let us give some discussions when $G$ equals a certain finite simple group.
{\bf Example 4 (PSL(2,7)).} Let $S$ be a complex K3 surface acting faithfully by $G=PSL_{2}(\mathbb{F}_{7})$. Such a K3 surface exists. For example, $PSL(2,7)$ acts faithfully and symplectically on the surface $X^{3}Y+Y^{3}Z+Z^{3}X+T^{4}=0$ in $\mathbb{P}^{3}$ by means of a linear action on $\mathbb{P}^{3}$ \cite{Mu88}. We know from Theorem 1.9 that a $G$-stable curve $C$ with nodal singularities in an integral linear system does contribute to the representation $[e(\overline{J^{d}\mathcal{C}})]$ only if there exists some $g\in G$ such that $\tilde{C}/\left<g\right>=\mathbb{P}^{1}$. It turns out that if this happens, then $\tilde{C}$ must be the Klein quartic, which is the Hurwitz surface of the lowest possible genus. Notice that $G$ acts on $C$ faithfully since any non-trivial element of $G$ acts symplectically on $S$ and cannot fix curves.
{\bf Proposition 6.6.} {\it Let $C$ be a smooth projective curve over $\mathbb{C}$ with a faithful $G=PSL(2,7)$-action. If there exists $g\in PSL(2,7)$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$, then the genus of $C$ is $3$ and $g$ has order $7 $. In particular, the automorphism group of $C$ reaches its Hurwitz bound, and hence $C$ is the Klein quartic.}
{\bf Proof.} The idea is to use the equivariant Riemann-Hurwitz formula \cite[Chapter VI $\S 4$]{Se79} for $\pi:C\rightarrow C/G=\mathbb{P}^{1}$. We have
\[
[e(C)]=e(\mathbb{P}^{1})I_{\left<1\right>}-\sum_{p\in\mathbb{P}^{1}}(I_{\left<1\right>}-I_{\left<g\right>})
\]
as $G$-representations, where $\left<g\right>$ is the stablizer of some point over $p$, $I_{\left<g\right>}$ denotes the induced representation ${\rm Ind}^{G}_{\left<g\right>}\mathbbm{1}$, and $\mathbbm{1}$ is the 1-dim trivial representation. Notice that $I_{\left<g\right>}$ is independent of the point we choose over $p$.
Since $G$ acts trivially on $\mathbb{P}^{1}$, we have
\[
H^{1}(C,\mathbb{C})=\sum_{p\in\mathbb{P}^{1}}(I_{\left<1\right>}-I_{\left<g\right>})-2I_{\left<1\right>}+2\mathbbm{1}
\]
and what we are going to do is to compare the representations on both sides. For this purpose, we need the character table of $PSL(2,7)$.
\begin{center}
\begin{tabular}{c||cccccc}
& $1A_{1}$ & $2A_{21}$ & $3A_{56}$ & $4A_{42}$ & $7A_{24}$ & $7B_{24}$\\
\hline\hline
$\chi_{1}$ & 1&1&1&1&1&1\\
$\chi_{2}$ & $1^{(3)}$&$1(-1)^{(2)}$&$1\omega\bar{\omega}$&1i(-i)&$\zeta_{7}\zeta_{7}^{2}\zeta_{7}^{4}$&$\zeta_{7}^{3}\zeta_{7}^{5}\zeta_{7}^{6}$\\
$\chi_{3}$&$1^{(3)}$& $1(-1)^{(2)}$&$1\omega\bar{\omega}$&1i(-i)&$\zeta_{7}^{3}\zeta_{7}^{5}\zeta_{7}^{6}$&$\zeta_{7}\zeta_{7}^{2}\zeta_{7}^{4}$\\
$\chi_{4}$&$1^{(6)}$& $1^{(4)}(-1)^{(2)}$&$1^{(2)}\omega^{(2)}\bar{\omega}^{(2)}$&$1^{(2)}(-1)^{(2)}i(-i)$&$\zeta_{7}...\zeta_{7}^{6}$&$\zeta_{7}...\zeta_{7}^{6}$\\
$\chi_{5}$&$1^{(7)}$& $1^{(3)}(-1)^{(4)}$&$1^{(3)}\omega^{(2)}\bar{\omega}^{(2)}$&$1(-1)^{(2)}i^{(2)}(-i)^{(2)}$&$1\zeta_{7}...\zeta_{7}^{6}$&$1\zeta_{7}...\zeta_{7}^{6}$\\
$\chi_{6}$&$1^{(8)}$& $1^{(4)}(-1)^{(4)}$&$1^{(2)}\omega^{(3)}\bar{\omega}^{(3)}$&$1^{(2)}(-1)^{(2)}i^{(2)}(-i)^{(2)}$&$1^{(2)}\zeta_{7}...\zeta_{7}^{6}$&$1^{(2)}\zeta_{7}...\zeta_{7}^{6}$
\end{tabular}
\end{center}
This is a refined character table which can be deduced from the usual character table. Each entry denotes the eigenvalues of the element in given conjuacy classes acting on given irreducible representations, $nA_{m}$ denotes the conjugacy class of size $m$ in which each element has order $n$, $a^{(i)}b^{(j)}$ denotes eigenvalue $a$ with multiplicity $i$ and eigenvalue $b$ with multiplicity $j$, and $\zeta_{7}...\zeta_{7}^{6}$ means $\zeta_{7}\zeta_{7}^{2}\zeta_{7}^{3}\zeta_{7}^{4}\zeta_{7}^{5}\zeta_{7}^{6}$.
For induced representations, we have $I_{\left<g\right>}(x)=\frac{1}{|\left<g\right>|}\sum_{h\in G}\chi(hxh^{-1})$, where $\chi(x)=1$ if $x\in\left<g\right>$, and $\chi(x)=0$ otherwise. Hence using our character table and calculating by Schur orthogonality relations, we have
\[
\begin{aligned}
I_{2}&=\chi_{1}+\chi_{2}+\chi_{3}+4\chi_{4}+3\chi_{5}+4\chi_{6}\\
I_{3}&=\chi_{1}+\chi_{2}+\chi_{3}+2\chi_{4}+3\chi_{5}+2\chi_{6}\\
I_{4}&=\chi_{1}+\chi_{2}+\chi_{3}+2\chi_{4}+\chi_{5}+2\chi_{6}\\
I_{7}&=\chi_{1}+\chi_{5}+2\chi_{6}
\end{aligned}
\]
where $I_{n}$ denotes $I_{\left<g\right>}$ for the element $g$ of order $n$.
Now since there exists $g\in G$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$, we have $H^{1}(C,\mathbb{C})^{g}=H^{1}(C/\left<g\right>,\mathbb{C})=0$. But for $\chi_{1}$, $\chi_{5}$ and $\chi_{6}$, whatever $g$ is, there are always non-trivial $g$-fixed vectors. This implies that $H^{1}(C,\mathbb{C})$ does not contain $\chi_{1}$, $\chi_{5}$ and $\chi_{6}$ at all. Hence the coefficients of $\chi_{1}$, $\chi_{5}$ and $\chi_{6}$ in $\sum_{p\in\mathbb{P}^{1}}(I_{\left<1\right>}-I_{\left<g\right>})-2I_{\left<1\right>}+2\mathbbm{1}$ must be 0. This gives us only two possibilities: $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{3}-I_{4}+2\mathbbm{1}$ or $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{3}-I_{7}+2\mathbbm{1}$. If we look at the dimensions of the right hand sides, the first one gives dimension -12 and the second gives dimension 6. Hence the only possibility is $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{3}-I_{7}+2\mathbbm{1}=\chi_{2}+\chi_{3}$, which shows that the genus of $C$ is $\frac{1}{2}\text{dim}H^{1}(C,\mathbb{C})=3$. We also deduce from this argument that $g$ must has order 7 since the element of order not equal to 7 does have fixed vectors in $\chi_{2}$ and $\chi_{3}$. $\square$
Following this observation, we do the calculations for some other groups in Mukai's list \cite{Mu88}.
{\bf Example 5 ($A_6$).} $G=A_{6}$ acts faithfully and symplectically on the K3 surface $\sum_{1}^{6}X_{i}=\sum_{1}^{6}X^{2}_{i}=\sum_{1}^{6}X^{3}_{i}=0$ in $\mathbb{P}^{5}$ via permutation action of coordinates on $\mathbb{P}^{5}$. Then by Theorem 1.9, a $G$-stable curve $C$ with nodal singularities in an integral linear system will not contribute to the representation $[e(\overline{J^{d}\mathcal{C}})]$.
{\bf Proposition 6.7.} {\it Let $C$ be a smooth projective curve over $\mathbb{C}$ with a faithful $G=A_{6}$-action. Then for any $g\in A_{6}$, we have ${C}/\left<g\right>\neq\mathbb{P}^{1}$.}
{\bf Proof.} We have the following character table for $A_{6}$.
\begin{center}
\begin{tabular}{c||cccc}
& $1A_{1}$ & $2A_{45}$ & $3A_{40}$ & $3B_{40}$\\
\hline\hline
$\chi_{1}$ & 1&1&1&1\\
$\chi_{2}$ & $1^{(5)}$&$1^{(3)}(-1)^{(2)}$&$1^{(3)}\omega\bar{\omega}$&$1\omega^{(2)}\bar{\omega}^{(2)}$\\
$\chi_{3}$&$1^{(5)}$& $1^{(3)}(-1)^{(2)}$&$1\omega^{(2)}\bar{\omega}^{(2)}$&$1^{(3)}\omega\bar{\omega}$\\
$\chi_{4}$&$1^{(8)}$& $1^{(4)}(-1)^{(4)}$&$1^{(2)}\omega^{(3)}\bar{\omega}^{(3)}$&$1^{(2)}\omega^{(3)}\bar{\omega}^{(3)}$\\
$\chi_{5}$&$1^{(8)}$& $1^{(4)}(-1)^{(4)}$&$1^{(2)}\omega^{(3)}\bar{\omega}^{(3)}$&$1^{(2)}\omega^{(3)}\bar{\omega}^{(3)}$\\
$\chi_{6}$&$1^{(9)}$& $1^{(5)}(-1)^{(4)}$&$1^{(3)}\omega^{(3)}\bar{\omega}^{(3)}$&$1^{(3)}\omega^{(3)}\bar{\omega}^{(3)}$\\
$\chi_{7}$&$1^{(10)}$& $1^{(4)}(-1)^{(6)}$&$1^{(4)}\omega^{(3)}\bar{\omega}^{(3)}$&$1^{(4)}\omega^{(3)}\bar{\omega}^{(3)}$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c||ccc}
& $4A_{90}$ & $5A_{72}$ & $5B_{72}$\\
\hline\hline
$\chi_{1}$ & 1&1&1\\
$\chi_{2}$ & $1{(-1)}^{(2)}i(-i)$&$1\zeta_{5}\zeta_{5}^{2}\zeta_{5}^{3}\zeta_{5}^{4}$&$1\zeta_{5}\zeta_{5}^{2}\zeta_{5}^{3}\zeta_{5}^{4}$\\
$\chi_{3}$ & $1{(-1)}^{(2)}i(-i)$&$1\zeta_{5}\zeta_{5}^{2}\zeta_{5}^{3}\zeta_{5}^{4}$&$1\zeta_{5}\zeta_{5}^{2}\zeta_{5}^{3}\zeta_{5}^{4}$\\
$\chi_{4}$ & $1^{(2)}(-1)^{(2)}i^{(2)}(-i)^{(2)}$&$1^{(2)}\zeta_{5}\zeta_{5}^{2(2)}\zeta_{5}^{3(2)}\zeta_{5}^{4}$&$1^{(2)}\zeta_{5}^{(2)}\zeta_{5}^{2}\zeta_{5}^{3}\zeta_{5}^{4(2)}$\\
$\chi_{5}$ & $1^{(2)}(-1)^{(2)}i^{(2)}(-i)^{(2)}$&$1^{(2)}\zeta_{5}^{(2)}\zeta_{5}^{2}\zeta_{5}^{3}\zeta_{5}^{4(2)}$&$1^{(2)}\zeta_{5}\zeta_{5}^{2(2)}\zeta_{5}^{3(2)}\zeta_{5}^{4}$\\
$\chi_{6}$ & $1^{(3)}(-1)^{(2)}i^{(2)}(-i)^{(2)}$&$1\zeta_{5}^{(2)}\zeta_{5}^{2(2)}\zeta_{5}^{3(2)}\zeta_{5}^{4(2)}$&$1\zeta_{5}^{(2)}\zeta_{5}^{2(2)}\zeta_{5}^{3(2)}\zeta_{5}^{4(2)}$\\
$\chi_{7}$ & $1^{(2)}(-1)^{(2)}i^{(3)}(-i)^{(3)}$&$1^{(2)}\zeta_{5}^{(2)}\zeta_{5}^{2(2)}\zeta_{5}^{3(2)}\zeta_{5}^{4(2)}$&$1^{(2)}\zeta_{5}^{(2)}\zeta_{5}^{2(2)}\zeta_{5}^{3(2)}\zeta_{5}^{4(2)}$
\end{tabular}
\end{center}
For induced representations, we have
\[
\begin{aligned}
I_{2}&=\chi_{1}+3\chi_{2}+3\chi_{3}+4\chi_{4}+4\chi_{5}+5\chi_{6}+4\chi_{7}\\
I_{3A}&=\chi_{1}+3\chi_{2}+\chi_{3}+2\chi_{4}+2\chi_{5}+3\chi_{6}+4\chi_{7}\\
I_{3B}&=\chi_{1}+\chi_{2}+3\chi_{3}+2\chi_{4}+2\chi_{5}+3\chi_{6}+4\chi_{7}\\
I_{4}&=\chi_{1}+\chi_{2}+\chi_{3}+2\chi_{4}+2\chi_{5}+3\chi_{6}+2\chi_{7}\\
I_{5}&=\chi_{1}+\chi_{2}+\chi_{3}+2\chi_{4}+2\chi_{5}+\chi_{6}+2\chi_{7}
\end{aligned}
\]
Now suppose there exists $g\in G$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$. Then we have $H^{1}(C,\mathbb{C})^{g}=H^{1}(C/\left<g\right>,\mathbb{C})=0$. But for all the irreducible representations of $G$, whatever $g$ is, there are always non-trivial $g$-fixed vectors. This implies that $H^{1}(C,\mathbb{C})=0$. Hence $\sum_{p\in\mathbb{P}^{1}}(I_{\left<1\right>}-I_{\left<g\right>})-2I_{\left<1\right>}+2\mathbbm{1}=0$. But no combination will give this equality. Hence such $g$ does not exist. $\square$
{\bf Example 6 ($A_{5}$).} $G=A_{5}$ acts faithfully and symplectically on the K3 surface $\sum_{1}^{5}X_{i}=\sum_{1}^{6}X^{2}_{i}=\sum_{1}^{5}X^{3}_{i}=0$ in $\mathbb{P}^{5}$ via permutation action of the first 5 coordinates on $\mathbb{P}^{5}$. Then by theorem 1.9, a $G$-stable curve $C$ with nodal singularities in an integral linear system can contribute to the representation $[e(\overline{J^{d}\mathcal{C}})]$ only if $\tilde{C}$ is rational.
{\bf Proposition 6.8.} {\it Let $C$ be a smooth projective curve over $\mathbb{C}$ with a faithful $G=A_{5}$-action. If there exists $g\in A_{5}$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$, then $C$ must be a smooth rational curve.}
{\bf Proof.} We have the following character table for $A_{5}$.
\begin{center}
\begin{tabular}{c||ccccc}
& $1A_{1}$ & $2B_{15}$ & $3A_{20}$ & $5A_{12}$ & $5B_{12}$ \\
\hline\hline
$\chi_{1}$ & 1&1&1&1&1\\
$\chi_{2}$ & $1^{(3)}$&$1(-1)^{(2)}$&$1\omega\bar{\omega}$ & $1\zeta_{5}\zeta_{5}^{4}$&$1\zeta_{5}^{2}\zeta_{5}^{3}$\\
$\chi_{3}$&$1^{(3)}$& $1(-1)^{(2)}$&$1\omega\bar{\omega}$ & $1\zeta_{5}^{2}\zeta_{5}^{3}$&$1\zeta_{5}\zeta_{5}^{4}$\\
$\chi_{4}$&$1^{(4)}$& $1^{(2)}(-1)^{(2)}$&$1^{(2)}\omega\bar{\omega}$&$\zeta_{5}...\zeta_{5}^{4}$&$\zeta_{5}...\zeta_{5}^{4}$\\
$\chi_{5}$&$1^{(5)}$& $1^{(3)}(-1)^{(2)}$&$1\omega^{(2)}\bar{\omega}^{(2)}$&$1\zeta_{5}...\zeta_{5}^{4}$&$1\zeta_{5}...\zeta_{5}^{4}$
\end{tabular}
\end{center}
For induced representations, we have
\[
\begin{aligned}
I_{2}&=\chi_{1}+\chi_{2}+\chi_{3}+2\chi_{4}+3\chi_{5}\\
I_{3}&=\chi_{1}+\chi_{2}+\chi_{3}+2\chi_{4}+\chi_{5}\\
I_{5}&=\chi_{1}+\chi_{2}+\chi_{3}+\chi_{5}
\end{aligned}
\]
Now suppose there exists $g\in G$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$. Then we have $H^{1}(C,\mathbb{C})^{g}=H^{1}(C/\left<g\right>,\mathbb{C})=0$. But for $\chi_{1}$, $\chi_{2}$, $\chi_{3}$ and $\chi_{5}$, whatever $g$ is, there are always non-trivial $g$-fixed vectors. This implies that $H^{1}(C,\mathbb{C})$ does not contain $\chi_{1}$, $\chi_{2}$, $\chi_{3}$ and $\chi_{5}$ at all. Hence the coefficients of $\chi_{1}$, $\chi_{2}$, $\chi_{3}$ and $\chi_{5}$ in $\sum_{p\in\mathbb{P}^{1}}(I_{\left<1\right>}-I_{\left<g\right>})-2I_{\left<1\right>}+2\mathbbm{1}$ must be 0. This gives us only one possibility: $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{3}-I_{5}+2\mathbbm{1}=0$, which implies $C$ has genus 0. $\square$
{\bf Example 7 ($S_{5}$).} $G=S_{5}$ acts faithfully and symplectically on the K3 surface $\sum_{1}^{5}X_{i}=\sum_{1}^{6}X^{2}_{i}=\sum_{1}^{5}X^{3}_{i}=0$ in $\mathbb{P}^{5}$ via permutation action of the first 5 coordinates on $\mathbb{P}^{5}$. Then by theorem 1.9, a $G$-stable curve $C$ with nodal singularities in an integral linear system can contribute to the representation $[e(\overline{J^{d}\mathcal{C}})]$ only if $\tilde{C}$ has genus 4. In this case $\tilde{C}$ has the largest possible automorphism group for a genus $4$ curve and $\tilde{C}$ is Bring's curve.
{\bf Proposition 6.9.} {\it Let $C$ be a smooth projective curve over $\mathbb{C}$ with a faithful $G=S_{5}$-action. If there exists $g\in S_{5}$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$, then $C$ has genus $4$ and $g$ has order $5$. In particular, $C$ is Bring's curve.}
{\bf Proof.} We have the following character table for $S_{5}$.
\begin{center}
\begin{tabular}{c||cccc}
& $1A_{1}$ & $2A_{10}$ & $2A_{15}$ & $3A_{20}$\\
\hline\hline
$\chi_{1}$ & 1&1&1&1\\
$\chi_{2}$ & 1&-1&1&1\\
$\chi_{3}$&$1^{(4)}$& $1^{(3)}(-1)$&$1^{(2)}(-1)^{(2)}$&$1^{(2)}\omega\bar{\omega}$\\
$\chi_{4}$&$1^{(4)}$& $1(-1)^{(3)}$&$1^{(2)}(-1)^{(2)}$&$1^{(2)}\omega\bar{\omega}$\\
$\chi_{5}$&$1^{(5)}$& $1^{(3)}(-1)^{(2)}$&$1^{(3)}(-1)^{(2)}$&$1\omega^{(2)}\bar{\omega}^{(2)}$\\
$\chi_{6}$&$1^{(5)}$& $1^{(2)}(-1)^{(3)}$&$1^{(3)}(-1)^{(2)}$&$1\omega^{(2)}\bar{\omega}^{(2)}$\\
$\chi_{7}$&$1^{(6)}$& $1^{(3)}(-1)^{(3)}$&$1^{(2)}(-1)^{(4)}$&$1^{(2)}\omega^{(2)}\bar{\omega}^{(2)}$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c||ccc}
& $4A_{30}$ & $5A_{24}$ & $6A_{20}$\\
\hline\hline
$\chi_{1}$ & 1&1&1\\
$\chi_{2}$ &-1&1&-1\\
$\chi_{3}$ & $1{(-1)}i(-i)$&$\zeta_{5}...\zeta_{5}^{4}$&$1(-1)\omega\bar{\omega}$\\
$\chi_{4}$ & $1{(-1)}i(-i)$&$\zeta_{5}...\zeta_{5}^{4}$&$1(-1)(-\omega)(-\bar{\omega})$\\
$\chi_{5}$ & $1(-1)^{(2)}i(-i)$&$1\zeta_{5}...\zeta_{5}^{4}$&$1\omega\bar{\omega}(-\omega)(-\bar{\omega})$\\
$\chi_{6}$ & $1^{(2)}(-1)i(-i)$&$1\zeta_{5}...\zeta_{5}^{4}$&$-1\omega\bar{\omega}(-\omega)(-\bar{\omega})$\\
$\chi_{7}$ & $1(-1)i^{(2)}(-i)^{(2)}$&$1^{(2)}\zeta_{5}...\zeta_{5}^{4}$&$1(-1)\omega\bar{\omega}(-\omega)(-\bar{\omega})$
\end{tabular}
\end{center}
For induced representations, we have
\[
\begin{aligned}
I_{2A}&=\chi_{1}+3\chi_{3}+\chi_{4}+3\chi_{5}+2\chi_{6}+3\chi_{7}\\
I_{2B}&=\chi_{1}+\chi_{2}+2\chi_{3}+2\chi_{4}+3\chi_{5}+3\chi_{6}+2\chi_{7}\\
I_{3}&=\chi_{1}+\chi_{2}+2\chi_{3}+2\chi_{4}+\chi_{5}+\chi_{6}+2\chi_{7}\\
I_{4}&=\chi_{1}+\chi_{3}+\chi_{4}+\chi_{5}+2\chi_{6}+\chi_{7}\\
I_{5}&=\chi_{1}+\chi_{2}+\chi_{5}+\chi_{6}+2\chi_{7}\\
I_{6}&=\chi_{1}+\chi_{3}+\chi_{4}+\chi_{5}+\chi_{7}
\end{aligned}
\]
Now since there exists $g\in G$ such that ${C}/\left<g\right>=\mathbb{P}^{1}$, we have $H^{1}(C,\mathbb{C})^{g}=H^{1}(C/\left<g\right>,\mathbb{C})=0$. But for $\chi_{1}$, $\chi_{5}$ and $\chi_{7}$, whatever $g$ is, there are always non-trivial $g$-fixed vectors. This implies that $H^{1}(C,\mathbb{C})$ does not contain $\chi_{1}$, $\chi_{5}$ and $\chi_{7}$ at all. Hence the coefficients of $\chi_{1}$, $\chi_{5}$ and $\chi_{7}$ in $\sum_{p\in\mathbb{P}^{1}}(I_{\left<1\right>}-I_{\left<g\right>})-2I_{\left<1\right>}+2\mathbbm{1}$ must be 0. This gives us only two possibilities: $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{4}-I_{5}+2\mathbbm{1}=2\chi_{4}$ or $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{5}-I_{6}+2\mathbbm{1}=2\chi_{4}+2\chi_{6}$. For the second case, we notice that whatever conjugacy class $g$ belongs to, there always exists $g$-fixed vectors in $2\chi_{4}+2\chi_{6}$. Hence $H^{1}(C,\mathbb{C})=I_{1}-I_{2}-I_{4}-I_{5}+2\mathbbm{1}=2\chi_{4}$. It follows that $C$ has genus $4$ and $g$ has order $5$. $\square$
| 137,094
|
TITLE: A semisimple commutative Banach algebra with a non-semisimple quotient
QUESTION [2 upvotes]: I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient.
Attempt from the comments:
"I take $A$ to be the algebra of all continuously differentiable functions. Let $I$ to be ideal of all functions from $A$ such that $f(0)=f'(0)=0$. I want to prove that $A/I$ is a two-dimensional algebra which has a one-dimensional radical."
REPLY [2 votes]: Take $A$ to be the Wiener algebra $A(\mathbb T)$, the algebra of all continuous functions on $\mathbb T$ with absolutely convergent Fourier series; in other words, $A(\mathbb T)$ is the image of $\ell_1(\mathbb Z)$ under the Fourier transform. The product of $A(\mathbb T)$ is the pointwise product and the norm is induced by $\ell_1(\mathbb Z)$,
$$\Vert f\Vert_A=\sum_{n\in\mathbb Z} \vert\widehat f(n)\vert$$
It is well known that the characters of $A(\mathbb T)$ are the evaluations at points of $\mathbb T$. In particular, $A$ is semi-simple. Moreover, if $I$ is a closed ideal in $A$ with hull $h(I)=E\subset\mathbb T$, then the characters of $A/I$ are exactly the evaluations $\delta_s$ with $s\in E$. In particular, if $f\in A$ is identically $0$ on $E=h(I)$, then the spectral radius of $[f]_{A/I}$ is $0$.
For any closed set $E\subset \mathbb T$, denote by $I(E)$ the ideal of all $f\in A$ such that $f\equiv 0$ on $E$, and by $J(E)$ the ideal of all $f\in A$ such that $f\equiv 0$ on a neighbourhood of $E$. Since $A$ is a regular Banach algebra we have $h(\overline{J(E)})=E$. Moreover, by a famous result of P. Malliavin there exist $non$-$spectral$ $sets$ for $A(\mathbb T)$, i.e. closed sets $E\subset \mathbb T$ such that $\overline{J(E)}\neq I(E)$.
Now, if $E$ is a non-spectral set for $A$, then $A/\overline{J(E)}$ is not semi-simple just by definition: if you take any $f\in I(E)$ which is not in $\overline{J(E)}$, then $[f]$ is not $0$ in $A/\overline{J(E)}$ but has spectral radius $0$.
| 84,687
|
Fraser Suites Glasgow
1-19 Albion Street, Glasgow, United Kingdom
The deluxe 4-star Fraser Suites Glasgow is set in a vibrant area minutes away from Gallery of Modern Art, Merchant City and Buchanan Street. Featuring Victorian architecture, the hotel was renovated in 2010.
The property is notable for its excellent location 20 minutes’ walk from Glasgow city center, close to museums, galleries and the palace as well as parks. The hotel is approximately 10 minutes' walk from Citizens Theatre and Glasgow Cathedral.
The venue is a short walk to a shopping street, markets and boutiques.
This property offers 99 rooms including Spacious Two-Bedroom Apartment, Spacious One-Bedroom Apartment and Luxury Studio Apartment. The cozy rooms feature the city view. They also comprise family bathrooms appointed with a shower, complimentary toiletries and a hairdryer.
A fine selection of local dishes is served at the 24-hour restaurant.
A 10-minute walk will take you to Buchanan Street SPT subway station. The hotel is located within 15 km distance of Glasgow International airport.
The hotel offers a baggage storage, a beauty salon and an elevator for general needs and a meeting room, a photocopier and a copier for corporate travelers.
Guests will enjoy using a health club and a solarium available on site. On-site fitness amenities include fitness classes, a gym and fitness center.
Important information
- Children and extra beds
- All children under the age of 3 may stay free of charge when using existing bedding.
- Maximum capacity of extra beds in a room is 2.
Facilities
General
- Non-smoking property
- Wi-Fi
- Paid parking
- Safe deposit box
- 24-hour check-in
- 24-hour reception
- Late check-in
- Express check-in/ -out
- No pets allowed
- Rooms/ Facilities for disabled
- Wheelchair access
- 24-hour security
- Children are welcome
- Luggage storage
- Elevator
- Drugstore/Pharmacy
- Currency exchange
- Gift/Newsstand
- Barber shop
- Boutiques
- Florist/Flower shop
- Multilingual staff
- Smoke detectors
Dining
- Kitchen
- Refrigerator
- Electric kettle
- Microwave
- Cooking hob
- Cookware/ Kitchen utensils
- Paid breakfast
- Restaurant
- Bar/ Lounge area
- Catering service
Leisure & Sports
- Indoor swimming pool
- Casino
- Night club
- Health club
- Solarium
- Fitness center
- Golf course
Services
- Paid airport shuttle
- Room service
- Housekeeping
- Car rental
- Self-service laundry
- Dry cleaning
- Doctor on call
- Wake up service
- Shops/Commercial services
- Newspaper service
- Business center
- Business services
Room Amenities
- Individual air conditioning
- Individual heating
- In-room safe
- Living room
- Separate sitting area
- Dressing area
- Balcony
- Tea and coffee facilities
- Bottled water
- In-room desk
- Ironing facilities
- Rollaway beds
- Private bathroom
- Bath/ Shower
- Hair dryer on request
- Room toiletries
- Flat-screen TV
- Hi-Fi
- Cable/ Satellite television
- In-room movies
- DVD player
- International calls
- Direct dial telephone
- Radio
- AM/FM alarm clock
- Cribs/ Baby cots
- Babysitting/Child services
- Playground
- Carpeted floor
Location
1-19 Albion Street, Glasgow, United Kingdom
Rooms
This property offers 99 rooms including Spacious Two-Bedroom Apartment, Spacious One-Bedroom Apartment and Luxury Studio Apartment. The cozy rooms feature the city view. They also comprise family Suites Glasgow
1-19 Albion Street, Glasgow, United Kingdom
What's nearby
- Local attractions
- Tron Theatre50 m
- Emirates Arena100 m
- Britannia Panopticon Music Hall100 m
- Sharmanka Kinetic Theatre150 m
- Merchant City250 m
- Trades Hall of Glasgow350 m
- Glasgow Police Museum450 m
| 121,196
|
Synopsis
Only a year on from the coalition negotiations and the “Rose Garden” press conference, the minds of the political parties are already beginning to shift towards the next election. At an event this week, Policy Exchange considered how the battleground for the next election might look and what will come top of the policy agenda in 2015.
A further clue about how politicians should look to appeal to voters at the next election was contained in research conducted for Policy Exchange by YouGov recently. The research confirmed the importance of meritocracy and aspiration in the British psyche and how aspiration provides the basis for the British public’s approach to “fairness”. The findings of the survey suggest that, for the vast bulk of the British people, fairness and meritocracy are the same thing.
The polling shows that the British people consistently expect “something for something” across a wide range of issues and their outlook is fundamentally aspirational and meritocratic. Politicians from all parties should take note of this resounding finding.
We gave respondents three options of what fairness means to them. Some 85% supported the idea that in a fair society “people’s incomes should depend on how hard they work and how talented they are.” 63% agreed with a free market concept of fairness (fairness based on what the market will pay) and only 41% believed in an egalitarian concept (equal rewards regardless of effort or ability). Despite the success of books such as The Spirit Level in recent years, meritocracy has more than twice the appeal of egalitarianism. Throughout the polling, there is a clear emphasis that voters believe in hard work, effort, ability and ambition being rewarded. Whilst compassionate about poverty, they also believe that “something for something”, rather than “something for nothing” should be the ongoing motto.
The results confirm that the key to electoral success in British politics is appealing to aspiration and emphasising opportunity, particularly amongst the C1s and C2s of market research jargon. At the last election, the Conservatives claimed 37% of C2 voters and the Labour vote collapsed from a high of 50% in 1997 to 29% last year. The most successful electoral politicians of the past 50 years, Margaret Thatcher and Tony Blair, both gained their considerable electoral success through an appeal to meritocracy and aspiration. It is clear that both political parties have more to do to appeal to those “strivers” who can be the decisive factor in British elections.
The Conservatives failed to persuade enough voters that they understood aspiration to make crucial breakthroughs in key marginals, whilst Labour seemed to increasingly lose touch with aspirational voters. Both parties have work to do to appeal to enough of the crucial aspiration working and lower middle class to gain an overall majority.
The findings of our poll suggest that belief in meritocracy is stronger than ever and that the political party that successfully taps in to the well of aspiration will be rewarded with electoral success. As all parties set out to shape the policy agenda leading up to the next election, they should have the meritocracy and aspiration at the forefront of their minds. Connecting with aspirational voters will be the key to electoral success in 2015, just as it has been key in every election in the past 50 years.
| 412,781
|
\begin{document}
\maketitle
\begin{abstract}
\noindent
We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every
irreducible component of the moduli spaces of hyperelliptic and trielliptic curves.
In particular, we provide a proof that the $\integ/\ell$ monodromy of
the moduli space of hyperelliptic curves of genus
$g$ is the symplectic group $\sp_{2g}(\integ/\ell)$.
We prove that the $\integ/\ell$ monodromy of the
moduli space of trielliptic curves with signature $(r,s)$ is the
special unitary group
$\su_{(r,s)}(\integ/\ell\tensor\integ[\zeta_3])$.
\smallskip
\\
{\bf MSC} 11G18, 14D05, 14H40\\
{\bf keywords} monodromy, hyperelliptic, trigonal, moduli, Jacobian
\end{abstract}
\section{Introduction}
If $C \ra S$ is a relative smooth proper curve of genus $g \ge 1$ over an irreducible
base, then the $\ell$-torsion of the relative Jacobian of $C$ encodes
important information about the family. Suppose $\ell$ is invertible
on $S$, and let $s \in S$ be a
geometric point. The fundamental group $\pi_1(S,s)$ acts
linearly on the fiber $\pic^0(C)[\ell]_{s} \iso (\integ/\ell)^{2g}$, and
one can consider the mod-$\ell$ monodromy representation associated to $C$:
\begin{diagram}
\rho_{C \ra S, \ell}:\pi_1(S,s) & \rto & \aut(\pic^0(C)[\ell]_{s}) \iso
\gl_{2g}(\integ/\ell).
\end{diagram}
Let $\mono_\ell(C \ra S)$, or simply $\mono_\ell(S)$, be the image
of this representation.
If a primitive $\ell\th$ root of unity is defined globally on $S$, then $\pic^0(C)[\ell]_{s}$ is equipped
with a skew-symmetric form $\ang{\cdot,\cdot}$ and $\mono_\ell(C \ra S) \subseteq
\sp(\pic^0(C)[\ell]_s,\ang{\cdot,\cdot}) \iso
\sp_{2g}(\integ/\ell)$.
If $C \ra S$ is a sufficiently general family of curves, then
$\mono_\ell(C \ra S) \iso \sp_{2g}(\integ/\ell)$ \cite{delignemumford}.
In this paper, we compute $\mono_\ell(S)$ when $S$ is an
irreducible component of the moduli space of hyperelliptic or
trielliptic curves and $C \ra S$ is the tautological curve.
The first result implies that there is no restriction on the monodromy group
in the hyperelliptic case other than that it preserve the symplectic pairing.
As a trielliptic curve is a $\integ/3$-cover of a genus zero curve,
the $\integ/3$-action
constrains the monodromy group to lie in a unitary group associated to $\integ[\zeta_3]$.
The second result implies that this is the only additional restriction in the trielliptic case.
\paragraph{Theorem \ref{thhe}}
{\it
Let $\ell$ be an odd prime, and let $k$ be an algebraically closed field in which $2\ell$ is invertible.
For $g\ge 1$, $\mono_\ell(\calh_g\bc k)\iso
\sp_{2g}(\integ/\ell)$.}
\paragraph{Theorem \ref{thtri}}
{\it
Let $\ell\ge 5$ be prime, and let $k$ be an algebraically closed field in which $3\ell$ is invertible. Let
$\calt^{\bar\gamma}$ be any component of the moduli space of
trielliptic curves of genus $g\ge 3$. Then
$\mono_\ell(\calt^{\bar\gamma}\bc k) \iso
\sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$ (where the latter is a unitary group defined
in \eqref{eqdefsg}).}
\medskip
We also prove that the $\ell$-adic monodromy group is
$\sp_{2g}(\integ_\ell)$ in the situation of Theorem \ref{thhe} and is $\sg_{(r_\gamma,s_\gamma)}(\integ_\ell)$
in the situation of Theorem \ref{thtri}.
Theorem \ref{thhe} is an unpublished result of J.K. Yu and has already been used multiple times in the literature.
In \cite{chavdarov}, Chavdarov assumes this result to show that the numerator of the zeta function of
the typical hyperelliptic curve over a finite field is irreducible.
Kowalski also uses this result in a similar fashion \cite{kowalskisieve}.
The first author used Theorem \ref{thhe} to prove a conjecture of Friedman and
Washington on class groups of quadratic function fields \cite{achtercl}.
There are other results in the literature which are similar to Theorem \ref{thhe}
but which are not quite strong enough for the applications above.
A'Campo \cite[Th.\ 1]{acampo} computes the topological monodromy of $\calh_g \bc \CC$.
On the arithmetic side, the $\rat_\ell$,
as opposed to $\integ_\ell$, monodromy of $\calh_g$
is computed in \cite[10.1.16]{katzsarnak}. Combined with a theorem of
Larsen on compatible families of representations \cite[3.17]{larsenmax},
this shows that the mod-$\ell$ monodromy group
of $\calh_g$ is maximal for a set of
primes $\ell$ of density one (as opposed to for all $\ell \ge 3$).
There are results on $\rat_\ell$-monodromy of cyclic covers of the projective
line of arbitrary degree, e.g., \cite[Sec. 7.9]{katztwisted}. Also,
in \cite[5.5]{fkv}, the authors prove that the projective representation
$\proj \rho_{C \ra S,\ell}$ is surjective for many
families of cyclic covers of the projective line.
Due to a combinatorial hypothesis, their theorem does not apply to $\calh_g$
and applies to at most one component of the moduli space of
trielliptic curves for each genus, see Remark \ref{Rfkv}.
See also work of Zarhin, e.g., \cite{zarhincyclic}.
As an application, for all $p \geq 5$, we show using \cite{chaioort01}
that there exist hyperelliptic and trielliptic curves
of every genus (and signature) defined over $\bar \ff_p$ whose Jacobians are absolutely simple.
In contrast with the applications above,
these corollaries do not use the full strength of our results.
Related work can be found in \cite{HZhu} where the authors produce curves with absolutely simple
Jacobians over $\ff_p$ under the restriction $g \leq 3$.
\paragraph{Corollary \ref{Chypabsirr}}
{\it Let $p \not = 2$ and let $g\in\nat$. Then there exists a
smooth hyperelliptic curve of genus $g$ defined over $\bar \ff_p$ whose Jacobian is
absolutely simple.}
\paragraph{Corollary \ref{Ctriabsirr}}
{\it Let $p \not = 3$. Let $g \geq 3$ and let $(r,s)$ be a trielliptic signature for $g$
(Definition \ref{Dtrisig}).
Then there exists a smooth trielliptic curve defined over $\bar \ff_p$ with genus $g$ and signature $(r,s)$
whose Jacobian is absolutely simple.}
\medskip
Our proofs proceed by induction on the genus.
The base cases for the hyperelliptic family
rely on the fact that every curve of genus $g=1,2$ is hyperelliptic;
the claim on monodromy follows from the analogous assertion about the monodromy of $\calm_g$.
The base case $g=3$ for the trielliptic family involves a comparison with
a Shimura variety of PEL type, namely, the Picard modular variety.
An important step is to show that the monodromy group does not change in the base cases when
one adds a labeling of the ramification points to the moduli problem.
The inductive step is similar to the method used in \cite{ekedahlmono}
and uses the fact that families of smooth hyperelliptic (trielliptic)
curves degenerate to trees of hyperelliptic (trielliptic) curves of lower genus.
The combinatorics of admissible degenerations require us
to compute the monodromy exactly for the inductive step rather than up to isomorphism.
The inductive strategy using admissible degeneration developed here
should work for other families of curves, especially for more general
cyclic covers of the projective line. The difficulty is in the direct
calculation of monodromy for the necessary base cases.
We thank C.-L.\ Chai, R.\ Hain, A.J.\ de Jong, E. Kani, and J. Kass.
\section{Moduli spaces of curves with $\integ/d$-action}
\subsection{Stable $\integ/d$-covers of a genus zero curve}
Let $G=\ZZ/d$ be a cyclic group of prime order $d$. Let $G\units=G-\id_G$.
Let $S$ be an irreducible scheme over $\Spec \ZZ[1/d, \zeta_d]$. Let
$k$ be an algebraically closed field equipped with a map
$\integ[1/d,\zeta_d] \ra k$.
Let $\psi:C \to S$ be a semi-stable curve. In other words, $\psi$ is
flat and proper and the geometric fibers of $C$ are connected, reduced
curves whose only singularities are ordinary double points. If $s \in
S$, let $C_s$ denote the fiber of $C$ over $s$. Let $\sing_S(C)$
be the set of $z \in C$ for which $z$ is a singular point of the fiber
$C_{\psi(z)}$.
A {\it mark} $\Xi$ on $C/S$ is a closed subscheme of
$C-\sing_S(C)$ which is finite and \'etale over $S$. The {\em
degree} of $\Xi$ is the number of points in any geometric fiber of
$\Xi \ra S$.
A marked semi-stable curve $(C/S, \Xi)$ is {\it stably marked} if
every geometric fiber of $C$ satisfies the following condition: every
irreducible component of genus zero has at least three points which
are either in $\sing_S(C)$ or on the mark $\Xi$.
Consider a $G$-action $\iota_0: G \inject \aut_S(C)$ on $C$.
Denote the ramification locus of $C \ra C/\iota_0(G)$ by $R$, and the
smooth ramification locus by $R_\smooth = R - (R\cap \sing_S(C))$. We say
that $(C/S,\iota_0)$ is a {\em stable $G$-curve} if $C/S$ is a
semi-stable curve; if $\iota_0: G \inject \aut_S(C)$ is an action of $G$;
if $R_\smooth$ is a mark on $C/S$; and
if $(C/S,R_\smooth)$ is stably marked.
We note that the definition implies that the {\it dihedral
nodes} of \cite[Def.\ 1.3]{ekedahlhurwitz} do not occur for $(C/S, \iota_0)$.
We say that a stable $G$-curve $(C/S, \iota_0)$ is {\it admissible} if
the following conditions are satisfied for every geometric point $z \in R\cap{\rm Sing}_S(C)$.
Let $C_{z,1}$ and $C_{z,2}$ denote the two components of the formal
completion of $C_{\psi(z)}$ at $z$.
First, $\iota_0(1)$ stabilizes each branch $C_{z,i}$;
second, the characters of the action of $\iota_0$ on the tangent spaces
of $C_{z,1}$ and $C_{z,2}$ at $z$ are inverses.
Throughout the paper, we suppose that $(C/S, \iota_0)$ is an admissible stable $G$-curve.
We further assume throughout that $C/\iota_0(G)$ has arithmetic genus $0$.
Then $C/\iota_0(G)$ is also a stably marked curve \cite[Prop.\ 1.4]{ekedahlhurwitz}.
The mark on $C/\iota_0(G)$ is the smooth branch locus
$B_\smooth$, which is the (reduced subscheme of) the image of $R_\smooth$ under the morphism $C
\ra C/\iota_0(G)$.
Let $r$ be the degree of $R_\smooth$. By the Riemann-Hurwitz formula, the arithmetic
genus of each fiber of $C$ is $g=1-d+r(d-1)/2$.
Let $s$ be a geometric point of $S$ with residue field $k$
and let $a$ be a point of the fiber $R_{\smooth,s}$. Then $G$ acts on the tangent space of $C_s$ at
$a$ via a character $\chi_a: G \ra k\units$. In particular, there is
a unique choice of $\gamma_a \in (\integ/d)\units$ so that $\chi_a(1)
= \zeta_d^{\gamma_a}$.
We say that $\gamma_a$ is the {\em canonical generator of inertia} at
$a$.
The {\em inertia type} of $(C/S,\iota_0)$ is the multiset
$\st{\gamma_{a} : a \in R_{\smooth,s}}$. It is independent of the choice of $s$.
By Riemann's existence theorem,
$\sum_{a \in R_{\smooth,s}} \gamma_a = 0 \in \integ/d$.
We say that a mark $\Xi$ has a labeling if $\Xi$ is an ordered disjoint
union of sections $S \ra C$. If $\Xi$ has degree $r$, we denote the
labeling by $\eta:\st{1, \ldots, r} \ra \Xi$.
A labeling of an admissible stable $G$-curve $(C/S,\iota_0)$ is a
labeling $\eta$ of $R_\smooth$.
There is an induced labeling $\eta_0:\{1, \ldots, r\} \ra B_\smooth$.
If $(C/S,\iota_0,\eta)$ is a labeled $G$-curve, the {\em class vector}
is the map of sets $\gamma:\st{1, \ldots, r} \ra G\units$ such that
$\gamma(i) = \gamma_{\eta(i)}$. We frequently write $\gamma = (\gamma(1),
\ldots, \gamma(r))$. If $\gamma$ is a class vector, we denote
its inertia type by $\bar\gamma: G\units \ra \integ_{\ge 0}$
where $\bar\gamma(h) = \#\gamma\inv(h)$ for all $h\in G$.
\subsection{Moduli spaces}
We define moduli functors on the category of schemes
over $\spec\integ[1/d, \zeta_d]$ by describing their $S$-points:
\begin{description}
\item{$\overline{\calm}_G$} parametrizes admissible stable $G$-curves $(C/S, \iota_0)$.
\item{$\til\calm_G$} parametrizes labeled admissible stable $G$-curves $(C/S, \iota_0, \eta)$.
\item{$\til\calm_{g,r}$} parametrizes triples $(C/S, \Xi,\eta)$ where
$C/S$ is a semi-stable curve of genus $g$, $\Xi$ is a mark of degree
$r$ on $C$ such that $(C/S,\Xi)$ is stably marked, and $\eta$ is a
labeling of $\Xi$.
\end{description}
Each functor is represented by an algebraic stack, and we use the
same letter to denote both a moduli functor and its representing
stack.
For each of these moduli spaces $\calm$, we let $\calm^{\circ}$ denote the open substack
whose objects parametrize smooth curves of the appropriate type.
To work with fibers
of the structural map $\calm \ra \spec
\integ[1/d,\zeta_d]$ we
write $\calm\bc k$ for $\calm\cross_{\spec \integ[1/d,\zeta_d]}
\spec k$.
\begin{lemma} \label{Lmoduli}
The moduli spaces $\til\calm_G$ and $\bar\calm_G$ are smooth, proper
Deligne-Mumford stacks over $\spec\integ[1/d,\zeta_d]$. The subspaces
$\til\calm_G^\circ$ and $\bar{\calm}^\circ_G$ are open and dense in
$\til\calm_G$ and $\bar\calm_G$, respectively.
\end{lemma}
\begin{proof}
The moduli spaces $\til\calm_G$
and $\bar\calm_G$ are algebraic stacks. Since the
automorphism group scheme of a (labeled) admissible stable $G$-curve
over an algebraically closed field is \'etale, $\bar\calm_G$ and
$\til\calm_G$ are Deligne-Mumford stacks \cite[8.1]{lmbstacks}.
The local deformation problem
for a semi-stable curve with $G$-action is formally smooth
\cite[2.1]{ekedahlhurwitz}, so $\bar\calm_G$ is smooth. By
\cite[2.3]{ekedahlhurwitz}, the degree of the smooth
ramification divisor of a stable admissible $G$-curve is locally constant, so
$\til\calm_G$ is smooth, too.
We use the valuative criterion to show that these stacks are proper \cite[7.3]{lmbstacks}.
Let $\calo_K$ be a
complete discrete valuation ring with field of fractions $K$, and suppose
$(C,\iota_0) \in \bar\calm_G(K)$. By the stable reduction theorem,
there exists a finite extension $K'/K$ so that the curve $C$ extends
as a stable curve to $\calo_{K'}$, as does the admissible $G$-action $\iota_0$. Possibly after a
further extension of the base, one may blow up the special fiber of
$C/\calo_{K'}$ to remove any dihedral nodes
\cite[p.195]{ekedahlhurwitz}, so that no singular point of $C$ is in
the closure of $R_\smooth$ and the resulting curve is an
admissible stable $G$-curve. This shows that $\bar\calm_G$ is
proper. After a finite base change, a labeling of $R_\smooth$
also extends uniquely. Thus,
$\til\calm_G$ is also proper.
The openness and density of
$\til\calm_G^\circ$ in $\til\calm_G$ and of $\bar\calm_G^\circ$ in
$\bar\calm_G$ follow from the fact that an admissible stable
$G$-curve is equivariantly smoothable \cite[2.2]{ekedahlhurwitz}.
\end{proof}
Consequently, every connected component of $\bar\calm_G$ or $\til\calm_G$ is irreducible.
Let $\gamma: \st{1, \ldots, r} \ra (\integ/d)\units$ be a class vector
of length $r=r(\gamma)$ for $G$. Let $g(\gamma) :=1 - d+r(\gamma)(d-1)/2$.
Let $\til\calm_G^\gamma$ be the substack of $\til\calm_G$
for which $(C/S,\iota_0,\eta)$ has class vector $\gamma$.
Let $\bar\calm_G^{\bar\gamma}$ be the substack of $\bar\calm_G$ for
which $(C/S,\iota_0)$ has inertia type $\bar\gamma$.
\begin{lemma} \label{Lirreducible}
The moduli space $\til\calm_G^\gamma$ is irreducible.
\end{lemma}
\begin{proof}
Since $\til\calm_G$ is proper and smooth over $\spec \integ[1/d,
\zeta_d]$, it is sufficient to prove that $\til\calm_{G}^\gamma \bc \cx$
is irreducible \cite[IV.5.10]{faltingschai}.
Since $\til\calm_G^{\gamma,\circ}\bc \cx$ is open and dense in $\til
\calm_{G}^\gamma\bc \cx$, it suffices to prove that
$\til\calm_{G}^{\gamma,\circ}\bc \cx$ is irreducible.
Consider the functor $\beta_\gamma: \til \calm_G^\gamma \ra \til\calm_{0,r(\gamma)}$.
On $S$-points, this functor takes the isomorphism class of
$(C,\iota_0, \eta)$ to the isomorphism class of $(C/\iota_0(G),
B_\smooth,\eta_0)$, where $\eta_0$ is
the induced labeling of $B_\smooth$.
By Riemann's existence theorem \cite[Section 2.2]{Vbook},
$\beta^{\circ}_{\gamma}\bc \cx: \til \calm^{\gamma, \circ}_G\bc\cx \ra \til\calm^{\circ}_{0,r}\bc\cx$ is an isomorphism.
The statement follows since $\til\calm^{\circ}_{0,r}\bc\cx$ is irreducible.
\end{proof}
If two class vectors $\gamma$ and $\gamma'$ yield the same inertia type, so that
$\bar{\gamma} = \bar{\gamma}'$, then there is a permutation $\varpi$ of
$\st{1, \ldots, r}$ such that $\gamma' = \gamma \comp \varpi$. This
relabeling of the branch locus yields an isomorphism
\begin{diagram}[LaTeXeqno] \label{Erelabel}
\label{diagrelabellocus}
\til\calm^{\gamma}_G & \rto^\twiddle & \til \calm^{\gamma \comp \varpi}_G.
\end{diagram}
Suppose $\gamma$ and $\gamma'$ differ by an
automorphism of $G$, so that there exists $\sigma\in \aut(G)$ such
that $\gamma'= \sigma \comp \gamma$. This relabeling of
the $G$-action yields an isomorphism
\begin{diagram}[LaTeXeqno]\label{Eaut}
\label{diagrelabelG}
\til\calm^{\gamma} &\rto^\twiddle & \til \calm^{\sigma\comp \gamma}_G.
\end{diagram}
\begin{lemma}\label{lemforgetgalois}
The forgetful functor $\til\calm_G^\gamma \ra
\bar\calm_G^{\bar\gamma}$ is \'etale and Galois.
\end{lemma}
\begin{proof}
The map is \'etale since any admissible stable
$G$-curve admits a labeling of $R_\smooth$ \'etale-locally on the base.
Moreover, the set of labelings of a
fixed stable $G$-curve with rational smooth ramification locus is a
torsor under the subgroup of $\sym(r)$ consisting
of those $\varpi$ for which $\gamma = \gamma\comp \varpi$. Therefore,
$\til\calm_G^\gamma \ra \bar\calm_G^{\bar\gamma}$ is Galois.
\end{proof}
\subsection{Degeneration}
For $i=1,2,3$, let $\gamma_i$ denote a class vector with length $r_i$
and let $g_i=g(\gamma_i)$.
There is a clutching map
\begin{diagram}
\label{diagrelabelG2}
\til\calm_{g_1,r_1} \cross \til \calm_{g_2,r_2} & \rto &
\til\calm_{g_1+g_2, r_1+r_2-2}
\end{diagram}
which is a closed immersion \cite[3.9]{knudsen2}. On $S$-points, this map
corresponds to gluing $C_1$ and $C_2$ together over $S$ by
identifying the last section of $C_1$ with the first section of $C_2$.
There is a forgetful functor $\til \calm_G^\gamma \ra \til \calm_{g(\gamma), r(\gamma)}$ taking the
isomorphism class of $(C/S, \iota_0, \eta)$ to the isomorphism class of $(C/S, R_\smooth,\eta)$.
This functor is finite-to-one since ${\rm Aut}_S(C, \iota_0)$ is finite \cite[1.11]{delignemumford}.
The composition
\begin{diagram}
\til \calm^{\gamma_1}_G
\cross \til \calm^{\gamma_2}_G &\rto & \til \calm_{g_1,r_1} \cross \til
\calm_{g_2, r_2} & \rto& \til \calm_{g_1+g_2, r_1+r_2-2}
\end{diagram}
allows us to glue two labeled $G$-curves $(C_i/S, \iota_{0,i}, \eta_i)$ together to obtain a labeled
$G$-curve $C/S$ with genus $g_1+g_2$ and class vector
$$\gamma=(\gamma_1(1), \ldots, \gamma_1(r_1-1), \gamma_2(2), \ldots, \gamma_2(r_2)).$$
Moreover, $C/S$ is
equivariantly smoothable if and only if the $G$-action is admissible,
i.e., if and only if $\gamma_1(r)$ and $\gamma_2(1)$ are inverses \cite[2.2]{ekedahlhurwitz}.
In this situation, we say that
{\it $(\gamma_1,\gamma_2)$ deforms to $\gamma$} or that {\it $\gamma$
degenerates to $(\gamma_1,\gamma_2)$}, and write
\begin{diagram}
\til\calm^{\gamma_1}_G \times \til\calm^{\gamma_2}_G & \rto &\til\calm^{\gamma}_G.
\end{diagram}
The clutching maps admit a generalization to maps
\begin{diagram}
\til \calm_{g_1,r_1} \cross \til \calm_{g_2,r_2} \cross \til
\calm_{g_3,r_3} &\rto & \til \calm_{g_1+g_2+g_3, r_1+r_2 +r_3- 4}.
\end{diagram}
Assume that $(\gamma_1, \gamma_2)$ deforms to $\gamma_L$,
that $(\gamma_2, \gamma_3)$ deforms to $\gamma_R$,
and that both $(\gamma_L, \gamma_3)$ and $(\gamma_1, \gamma_R)$ deform to $\gamma$.
Then we have a commutative diagram of maps:
\begin{diagram}[LaTeXeqno]
\label{diagclutch}
\til\calm_G^{\gamma_1} \cross \til\calm_G^{\gamma_2} \cross \til\calm_G^{\gamma_3} &
\rto & \til\calm_G^{\gamma_1} \cross \til\calm_G^{\gamma_R} \\
\dto & \rdto & \dto \\
\til\calm_G^{\gamma_L} \cross \til \calm_G^{\gamma_3} & \rto &
\til\calm_G^{\gamma}.
\end{diagram}
In the situation above, suppose $g_1=g_3=1$. The image $\Delta_{1,1}$ of
$\til\calm_G^{\gamma_1} \cross \til\calm_G^{\gamma_2} \cross \til\calm_G^{\gamma_3}$
in $\til \calm_G^\gamma$
is part of the boundary of $\til\calm_G^\gamma$.
We say that a class vector $\gamma$ {\it degenerates to $\Delta_{1,1}$}
if there are $\gamma_1$, $\gamma_2$ and $\gamma_3$ as in
\eqref{diagclutch} so that $g_1=g_3=1$.
Furthermore, we say that an inertia type $\overline{\gamma}$ {\it degenerates to $\Delta_{1,1}$}
if there exists some class vector $\gamma'$ with inertia type $\overline{\gamma}' =\overline{\gamma}$
so that $\gamma'$ degenerates to $\Delta_{1,1}$.
Using Equations \eqref{diagrelabellocus} and
\eqref{diagrelabelG}, if $\overline{\gamma}$ degenerates to $\Delta_{1,1}$,
then every component of $\til \calm_G$ lying over
$\bar\calm_G^{\overline{\gamma}}$ degenerates to $\Delta_{1,1}$
(although this may require identifying different ramification points
of $C_1$, $C_2$ and $C_3$ in the clutching maps).
\subsection{The case of hyperelliptic curves: $d=2$} \label{subsechecomp}
Let $C/S$ be a $\integ/2$-curve of genus $g$. Then $C/S$ is
hyperelliptic, and $\iota_0(1)$ is the hyperelliptic involution. Over
every geometric point $s$ of
$S$, $C_s \ra C_s/\iota_0(\integ/2)$ is ramified at $2g+2$ smooth points.
There is a unique class vector $(1, \ldots, 1)$ for $\integ/2$ of
length $2g+2$ for each $g\in\nat$. By Lemma \ref{Lirreducible}, there
is a unique component of $\til\calm_{\integ/2}$
parametrizing labeled hyperelliptic curves of genus $g$. We denote
this component by $\til\calh_g$. Similarly, $\bar\calm_{\integ/2}$
has a unique component $\bar\calh_g$ which parametrizes
hyperelliptic curves of genus $g$. Let $\calh_g=\bar\calh_g^\circ$ denote
the moduli space of smooth hyperelliptic curves of genus $g$.
\begin{lemma}
\label{lemdegenhe}
If $g \geq 3$, then $\til{\calh}_g$ degenerates to $\Delta_{1,1}$.
\end{lemma}
\begin{proof}
The class vector for $\til\calh_g$ is
$\gamma=(1, \ldots, 1)$ with length $2g+2$.
Let $\gamma_1=\gamma_3=(1,1,1,1)$ and let $\gamma_2=(1, \ldots, 1)$ with length $2g-2$.
\end{proof}
\begin{remark}
When the boundary components $\Delta_1$ and $\Delta_2$ are distinct,
the inductive argument in Theorem \ref{thhe} can be revised
to rely on monodromy groups only up to
isomorphism. Unfortunately, when $g = 3$ one has $\Delta_1 =
\Delta_2$, so that we must calculate monodromy groups exactly.
\end{remark}
\subsection{The case of trielliptic curves: $d=3$} \label{subsectricomp}
A {\it trielliptic} curve is a $\ZZ/3$-curve $(C/S, \iota_0)$ so that
$C/\iota_0(\ZZ/3)$ has genus zero.
The curve $C/S$ is sometimes called a cyclic trigonal curve.
Over every geometric point
$s$ of $S$, the $\integ/3$-cover $C_s \ra C_s / \iota_0(\integ/3)$ is
ramified at $g+2$ smooth points.
\subsubsection{Components of the trielliptic locus}
The moduli space of trielliptic curves $(C/S, \iota_0)$ with $C$ of genus $g$
is not connected even for fixed $g$.
This reflects the different possibilities for the inertia type,
or equivalently for the signature, which we now describe.
By Lemma \ref{Lirreducible}, the components of $\til
\calm_{\integ/3}$ parametrizing labeled trielliptic curves of genus $g$ are in
bijection with the maps $\gamma: \st{1, \ldots, g+2} \ra
(\integ/3)\units$ such that $\sum \gamma(i) = 0 \in \integ/3$. We
denote these components by $\til\calt^\gamma$, and their projections
to $\bar\calm_{\integ/3}$ by $\bar\calt^{\overline{\gamma}}$. Let
$\calt^{\bar\gamma}={\bar\calt^{\bar\gamma\circ}}$ denote the
moduli space of smooth trielliptic curves with inertia type $\bar\gamma$.
\begin{lemma} \label{Lgamma}
The set of possible inertia types $\overline{\gamma}$ for a trielliptic curve $(C/S, \iota_0)$ of genus $g$
is in bijection with the set of pairs of integers $(d_1,d_2)$ so that
$d_1,d_2 \geq 0$, $d_1+d_2=g+2$,
and $d_1+2d_2 \equiv 0 \bmod 3$.
\end{lemma}
\begin{proof}
Let $d_1=\bar \gamma(1)$ and $d_2=\bar \gamma(2)$. The claim follows from earlier remarks.
\end{proof}
\begin{remark} \label{Rfkv}
The result \cite[5.5]{fkv} applies to
trielliptic curves only when $d_1d_2 = 0$. These inertia types occur
for curves of genus
$3g_1+1$ having an equation of the form $y^3=f(x)$ for some
separable polynomial $f(x)$ of degree $3g_1+2$.
\end{remark}
Let $S$ be an irreducible scheme over $\spec\integ[1/3, \zeta_3]$. Then
$\calo_S\tensor \integ[1/3,\zeta_3] \iso \calo_S \oplus \calo_S$. We
choose the isomorphism so that the first component has the
given structure of $\calo_S$ as a $\integ[1/3,\zeta_3]$-module.
Consider a trielliptic curve $(\psi: C \ra S, \iota_0)$.
The sheaf of relative
one-forms $\psi_*\Omega^1_{C/S}$ is a locally free
$\integ[1/3,\zeta_3]\tensor \calo_S$-module of some rank $(r,s)$, where $r+s = g$. We
call $(r,s)$ the {\em signature} of $(C/S, \iota_0)$. The signature is locally
constant on $S$, so we may calculate it at any geometric point of $S$.
If $(C/k,\iota_0)$ is a trielliptic curve,
then $H^0(C,\Omega^1_C)$ decomposes as a direct sum $W_1 \oplus W_2$ where
$\omega \in W_j$ if $\zeta_3 \circ \omega = \zeta_3^j \omega$.
The signature of $(C/k, \iota_0)$ is $(r,s) = (\dime(W_1),\dime(W_2))$.
\begin{lemma} \label{Lsig}
The signature and inertia type of a trielliptic curve of genus $g$ are related as follows:
$\overline{\gamma}(1)=2r-s+1$ and $\overline{\gamma}(2)=2s-r+1$.
There exists a trielliptic curve $(C/k, \iota_0)$ of genus $g$ with signature $(r,s)$ if and only if
$r,s \in \ZZ$, $r+s=g$, and $(g-1)/3 \leq r,s \leq (2g+1)/3$.
\end{lemma}
\begin{proof}
Let $(C/k, \iota_0)$ be a trielliptic curve with inertia type $\overline{\gamma}$.
For simplicity, let $d_1=\overline{\gamma}(1)$ and let $d_2=\overline{\gamma}(2)$
and let $N=(d_1+2d_2)/3$. There is an equation for $C$ of the form
$$y^3=\prod_{i=1}^{d_1}(x-a_i)\prod_{j=1}^{d_2}(x-b_j)^2.$$
Consider the differential
$\omega=\prod_{i=1}^{d_1}(x-a_i)^{n_i}\prod_{j=1}^{d_2}(x-b_j)^{n_j'}dx/y^m$
with $n_i, n_j' \geq 0$ and $m \geq 1$.
By \cite[Thm.\ 3]{koo91}, $\omega$ is holomorphic if and only if
$3n_i \geq m-2$, $3n_j' \geq 2m-2$, and $mN \geq \sum_{i=1}^{d_1} n_i +\sum_{j=1}^{d_2} n_j' +2$.
Let $g(x)=\prod_{j=1}^{d_2}(x-b_j)$.
Thus we have the following set of linearly independent holomorphic differentials:
$$\{dx/y, \ldots, (x-a_1)^{N-2}dx/y, g(x)dx/y^2, \ldots, (x-a_1)^{2N-2-d_2}g(x)dx/y^2\}.$$
This set is a basis of $H^0(C,\Omega^1_C)$ since its cardinality is $3N-d_2-2=g$.
Also, for any $h(x) \in k[x]$, $\zeta_3 \circ h(x)dx/y^m = \zeta_3^{-m} h(x) dx/y^m$.
It follows that the signature of $(C/k, \iota_0)$ is $(r,s)=(g-N+1,N-1)$.
It follows that $d_1=2r-s+1$ and $d_2=2s-r+1$.
Now $r,s \in \ZZ$ if and only if $d_1+2d_2 \equiv 0 \bmod 3$.
Also, $d_1+d_2=g+2$ if and only if $r+s=g$.
The conditions $d_1,d_2 \geq 0$ and $(g-1)/3 \leq r,s \leq (2g+1)/3$ are equivalent.
The second claim then follows from Lemma \ref{Lgamma}.
\end{proof}
\begin{definition} \label{Dtrisig}
Let $g \in \NN$. A {\it trielliptic signature for $g$} is a pair $(r,s)$ with
$r,s \in \ZZ$, $r+s=g$, and $(g-1)/3 \leq r,s \leq (2g+1)/3$.
\end{definition}
As in \eqref{diagrelabelG}, if $(C/S, \iota_0)$ is a trielliptic curve then so is $(C/S, \iota'_0)$
where $\iota'_0(1)=\iota_0(2)$.
Replacing $\iota_0$ with $\iota'_0$ exchanges the values of $d_1$ and $d_2$
and the values of $r$ and $s$.
\subsubsection{Degeneration of the trielliptic locus}
We show that every component $\til\calt^\gamma$ with $g(\gamma) \geq 4$
degenerates to $\Delta_{1,1}$.
Recall that there is a unique elliptic curve $E$ which admits a $\ZZ/3$-action $\iota_0$.
The class vector of $(E, \iota_0)$ is either $(1,1,1)$ or $(2,2,2)$.
\begin{proposition} \label{Pdeg311}\label{propdegentri}
If $g(\gamma) \geq 4$, then $\til \calt^\gamma$ degenerates to $\Delta_{1,1}$.
\end{proposition}
\begin{proof}
Let $g=g(\gamma) \geq 4$ and
let $\gamma:\{1, \ldots,g+2\} \ra (\ZZ/3)\units$ be a class vector for $G=\ZZ/3$.
It suffices to show that there exist class vectors $\gamma_i$ for $i=1,2,3$
with $g(\gamma_1)=g(\gamma_3)=1$ as in \eqref{diagclutch}.
In particular, we require that $\gamma_1$ and $\gamma_3$ are either $(1,1,1)$ or $(2,2,2)$.
By Equations \eqref{Erelabel} and \eqref{Eaut}, we can reorder the
values $\gamma(i)$ or replace each $\gamma(i)$ with $-\gamma(i)$.
Thus it is sufficient to restrict attention to the inertia type $\overline{\gamma}$,
or to the corresponding signature $(r,s)$ by Lemma \ref{Lsig}.
Suppose $\gamma_1=\gamma_3=(1,1,1)$. Then we need $\gamma_2:\{1,
\ldots, g\} \to (\ZZ/3) \units$ with $\gamma_2(1)=\gamma_2(g)=2$ and
$\overline{\gamma}_2(1)=\overline{\gamma}(1)-4$ and
$\overline{\gamma}_2(2)=\overline{\gamma}(2)+2$. In other words, we
need $\gamma_2$ to have four fewer points with inertia generator 1 and
two more points with inertia generator 2 than $\gamma$ does. This can
be achieved by decreasing $r$ by $2$. Similarly, the case
$\gamma_1=\gamma_3=(2,2,2)$ can be achieved by decreasing $s$ by $2$.
Suppose $\gamma_1=(1,1,1)$ and $\gamma_2=(2,2,2)$.
Then we need $\gamma_2:\{1, \ldots, g\} \to (\ZZ/3) \units$ with $\gamma_2(1)=2$ and $\gamma_2(g)=1$ and
$\overline{\gamma}_2(1)=\overline{\gamma}(1)-1$ and $\overline{\gamma}_2(2)=\overline{\gamma}(2)-1$.
In other words, we need $\gamma_2$ to have one fewer point with inertia generator 1 and
one fewer point with inertia generator 2 than $\gamma$ does.
This can be done by decreasing both $r$ and $s$ by $1$.
The next table shows that, for each inertia type $\overline{\gamma}$, there is a choice of
$\gamma_1, \gamma_2, \gamma_3$ satisfying the numerical constraints.
\[\begin{tabular}{|l|l|l|l|l|}
\hline
Component $\gamma$ & degenerates so that & and $\gamma_1$ is & and $\gamma_3$ is & under this \\
with signature & $\gamma_2$ has signature & & & condition\\
\hline
\hline
$(r,s)$ & $(r-2,s)$& $(1,1,1)$ & $(1,1,1)$ & $r \geq 2$ \\
\hline
$(r,s)$ & $(r,s-2)$ & $(2,2,2)$ & $(2,2,2)$ & $s \geq 2$ \\
\hline
$(r,s)$ & $(r-1,s-1)$ & $(1,1,1)$ & $(2,2,2)$ & $r \geq 1$, $s \geq 1$ \\
\hline
\end{tabular}
\]
\end{proof}
\begin{remark}
When $\til \calt^\gamma$ degenerates to both $\Delta_1$ and
$\Delta_2$, the inductive argument in Theorem \ref{thtri} can be
revised to rely on monodromy groups only up to isomorphism.
Unfortunately, $\til{\calt}^\gamma$ does not degenerate to $\Delta_2$
when $\bar\gamma(1)\bar\gamma(2)=0$. For $g(\gamma) \geq
4$, all other components $\til{\calt}^{\gamma}$ degenerate to
$\Delta_2$.
\end{remark}
\section{Monodromy groups}
\subsection{Definition of monodromy}
Let $(X/S,\phi)$ be a principally polarized abelian scheme of
relative dimension $g$ over an irreducible base. If $\ell$ is a
rational prime invertible on $S$, then the $\ell$-torsion $X[\ell]$ of
$\ell$ is an \'etale cover of $S$ with geometric fiber isomorphic to
$(\integ/\ell)^{2g}$.
Let $s$ be a geometric point of $S$. The fundamental group $\pi_1(S,s)$ acts
linearly on the $\ell$-torsion of $X$.
This yields a representation
\begin{diagram}
\rho_{X \ra S, s,\ell}: \pi_1(S,s) & \rto & \aut(X[\ell]_s) \iso \gl_{2g}(\integ/\ell).
\end{diagram}
The cover $X[\ell] \ra S$ both determines and is determined by the representation
$\rho_{X \ra S, s,\ell}$. The image of $\rho_{X \ra S, s,
\ell}$ is the {\it mod-$\ell$ monodromy} of $X \ra S$ and we denote it by
$\mono_\ell(X \ra S, s)$, or by $\mono_\ell(S,s)$ if the choice of
abelian scheme is clear. The isomorphism class of the
$\mono_\ell(S,s)$ is independent of the choice of base point $s$, and we
denote it by $\mono_\ell(S)$.
Let $X\dual$ be the dual abelian scheme. There
is a canonical pairing $X[\ell] \cross X\dual[\ell] \ra
\mmu_{\ell,S}$, where $\mmu_{\ell,S} := \mmu_\ell \cross S$ is
the group scheme of $\ell\th$ roots of unity.
The polarization $\phi$ induces an isomorphism $X \ra X\dual$, and
thus a skew-symmetric pairing $X[\ell] \cross X[\ell] \ra \mmu_{\ell,S}$.
Because the polarization is defined globally, the image of monodromy
$\mono_\ell(X \ra S, s)$ is contained in the group of symplectic
similitudes of $(X[\ell]_s,
\ang{\cdot,\cdot}_\phi)$, which is isomorphic to
$\gsp_{2g}(\integ/\ell)$. Moreover, if a primitive $\ell^{{\rm th}}$ root of
unity exists globally on $S$, then $\pi_1(S,s)$ acts trivially on
$\mmu_{\ell,S}$ and $\mono_\ell(X \ra S,s) \subseteq
\sp(X[\ell]_s,\ang{\cdot,\cdot}_\phi) \iso \sp_{2g}(\integ/\ell)$.
Similarly, the cover $X[\ell^n] \ra S$ defines a monodromy representation
with values in $\aut(X[\ell^n]_s) \iso\gl_{2g}(\integ/\ell^n)$. Taking
the inverse limit over all $n$, we obtain a continuous
representation on the Tate module of $X$,
\begin{diagram}
\rho_{X \ra S, \integ_\ell, s}: \pi_1(S,s) & \rto & \invlim n
\aut(X[\ell^n]_s) \iso \gl_{2g}(\integ_\ell).
\end{diagram}
We denote the image of this representation by $\mono_{\integ_\ell}(X
\ra S, s)$, and its isomorphism class by $\mono_{\integ_\ell}(X \ra
S)$ or $\mono_{\integ_\ell}(S)$. Again, there is an inclusion
$\mono_{\integ_\ell}(X \ra S) \subseteq \gsp_{2g}(\integ_\ell)$. If
$F$ is a field, let $F_{\ell^\infty} = F(\mmu_{\ell^\infty}(\bar F))$.
If
$S$ is an $F$-scheme, then $\mono_{\integ_\ell}(X \ra S,
s)/ \mono_{\integ_\ell}(X\bc{\bar F} \ra S \bc{\bar F}, s) \iso
\gal(F_{\ell^\infty}/F)$. Finally, let $\mono_{\rat_\ell}(X\ra S, s)$
be the Zariski closure of $\mono_{\integ_\ell}(X \ra S, s)$ in
$\gl_{2g}(\rat_\ell)$.
Now suppose that $\psi:C \ra S$ is a relative proper semi-stable curve.
Let $\pic^0(C) := \pic^0_{C/S}$ be the neutral component of the
relative Picard functor of $C$ over $S$. Since $C/S$ is semi-stable,
$\pic^0(C)$ is a semiabelian scheme \cite[9.4.1]{blr}. Suppose that
there is at least one geometric point $s$ such that the fiber
$\pic^0(C_s)$ is an abelian variety. (This is true if some $C_s$ is a tree
of smooth curves.) Then there is a nonempty open
subscheme $S^*$ of $S$ such that $\pic^0(C\rest{S^*})$ is an abelian scheme
over $S^*$. We define the mod-$\ell$ and $\integ_\ell$ monodromy
representations of $C$ to be those of $\pic^0(C\rest{S^*}) \ra S^*$.
(Alternatively, these may be constructed as the restrictions of
$R^1\psi_*\mmu_{\ell,S}$ and $R^1\psi_*\mmu_{\ell^\infty,S}$
to the largest subscheme of $S$ on which these sheaves are unramified.)
Thus, $\mono_\ell(C \ra S, s) = \mono_\ell(\pic^0(C\rest{S^*}) \ra S^*, s)$,
and we denote this again by $\mono_\ell(S,s)$ if the curve is clear
and by $\mono_\ell(S)$ if the base point is suppressed.
The moduli spaces $\bar\calm_G$ and $\til\calm_G$ are Deligne-Mumford
stacks, and we employ a similar formalism for \'etale covers of stacks
\cite{noohi}. Let $\cals$ be a connected Deligne-Mumford stack. The
category of Galois \'etale covers of $\cals$ is a Galois category in
the sense of Grothendieck, and thus there is an \'etale fundamental
group of $\cals$. More precisely, let $s\in \cals$ be a geometric
point. Then there is a group $\pi_1(\cals,s)$ and an equivalence of
categories between finite $\pi_1(\cals,s)$-sets and finite \'etale
Galois covers of $\cals$. If $\cals$ has a coarse moduli space
$\smod$, then $\pi_1(\cals,s)$ is the extension of $\pi_1(\smod,s)$ by
a group which encodes the extra automorphism structure on the moduli
space $\smod$ \cite[7.11]{noohi}. If $X \ra \cals$ is a family of
abelian varieties, we again let $\mono_\ell(X\ra \cals,s)$ be the
image of $\pi_1(\cals, s)$ in $\aut(X[\ell]_s)$.
Let
$\calc^\gamma$ be the tautological labeled curve over
$\til\calm^\gamma_G$. By the mod-$\ell$ or $\integ_\ell$ monodromy of
$\til\calm_G^\gamma$ we mean that of $C^\gamma \ra
\til\calm_G^\gamma$.
\subsection{Degeneration and monodromy} \label{Sinduct}
In this section, we deduce information on monodromy groups from the
degeneration of moduli spaces to the boundary.
We will use this to prove inductively that the mod-$\ell$
monodromy groups of each $\til\calh_g$ and each $\til\calt^\gamma$
are as large as possible.
\begin{lemma}
\label{lemclutchmono}
Let $k$ be an algebraically closed field in which $d\ell$ is invertible.
Suppose that the pair $(\gamma_1,\gamma_2)$ deforms to $\gamma$, so that there is a clutching map
\begin{diagram}
\kappa:& \til \calm^{\gamma_1}_G\cross \til \calm^{\gamma_2}_G & \rto & \til \calm^\gamma_G.
\end{diagram}
\begin{alphabetize}
\item There is a canonical isomorphism of sheaves on $(\til
\calm^{\gamma_1}_G \cross \til \calm^{\gamma_2}_G)\bc k$,
\begin{equation}
\label{eqclutchsum}
(\kappa\bc k)^* \pic^0(\calc^\gamma)[\ell] \iso
\pic^0(\calc^{\gamma_1})[\ell] \cross \pic^0(\calc^{\gamma_2})[\ell].
\end{equation}
\item Suppose $s_i \in \til \calm^{\gamma_i}_G(k)$ for $i= 1,
2$, and
let $s = \kappa(s_1,s_2)$. After base change to $k$, there is a commutative diagram:
\begin{diagram}
\mono_\ell(\til\calm^{\gamma_1}_G, s_1) \cross
\mono_\ell(\til\calm^{\gamma_2}_G, s_2) & \rto^\alpha_{\sim} &
\mono_\ell(\kappa(\til \calm^{\gamma_1}_G \cross \til
\calm^{\gamma_2}_G), s) & \rinject &
\mono_\ell(\til\calm^\gamma_G,s) \\
\dinject &&& \rdinject &\dinject\\
\aut(\pic^0(\calc^{\gamma_1})[\ell]_{s_1}) \cross
\aut(\pic^0(\calc^{\gamma_2})[\ell]_{s_2})
&& \rinject^\delta &&
\aut(\pic^0(\calc^{\gamma})[\ell]_{s})
\end{diagram}
\end{alphabetize}
\end{lemma}
\begin{proof}
Let $C \ra S$ be a stable curve. Suppose that $C$
is the union of two (not necessarily irreducible) proper,
connected $S$-curves $C_1$ and $C_2$ which intersect along a unique section. If $\call$ is a line bundle on
$C$ of degree zero, then for $i = 1, 2$ the restriction
$\call\rest{C_i}$ is a line bundle of degree zero on $C_i$ \cite[9.1.2
and 9.2.13]{blr}. Thus, there is a morphism of group functors
$(\kappa\bc k)^*\pic^0(C^\gamma) \ra \pic^0(C^{\gamma_1}) \cross
\pic^0(C^{\gamma_2})$, which restricts to a morphism on
$\ell$-torsion. This is an isomorphism on stalks \cite[9.2.8]{blr},
and thus an isomorphism of sheaves.
For part (b), the homomorphism $\delta$ is induced by the isomorphism
in Equation \eqref{eqclutchsum}.
For $i = 1, 2$, let $C_i \ra S_i$ denote the cover $\calc^{\gamma_i} \bc k \ra \til \calm_G^{\gamma_i} \bc k$.
To define $\alpha$, consider the kernel of the representation $\rho_{C_i \ra S_i,s_i, \ell}$.
Since it is an open normal subgroup of the fundamental group, this kernel
defines an irreducible Galois \'etale
cover $Y_i \ra S_i$.
Let $H_i$ be its Galois group.
Then $Y_1
\cross_k Y_2$ is an irreducible $H_1\cross H_2$ cover of $S_1 \cross
S_2$.
Using this and part (a), the cover
$\pic^0(\calc^\gamma)[\ell]\rest{\kappa(S_1\cross S_2)}$
is trivialized by a Galois extension with group $H_1 \cross H_2$.
Therefore, $\alpha$ is an isomorphism.
\end{proof}
\begin{lemma}
\label{lemmaxgp}
Let $\ff$ be a finite field of odd characteristic, and let $V/\ff$ be
a $g$-dimensional vector space. Suppose that
\begin{equation}
\label{eqsumdecomp}
V = V_1 \oplus V_2 \oplus V_3,
\end{equation}
and that $g_i := \dim V_i$ is positive for $i=1,2,3$. Let $H$ be
a subgroup of $\gl(V)$.
\begin{alphabetize}
\item\label{lemmaxsl} If $H\subseteq \sl(V)$ contains both $\sl(V_1\oplus V_2)$ and
$\sl(V_2\oplus V_3)$, then $H = \sl(V)$.
\item Suppose that $V$ is equipped with a nondegenerate pairing $\ang{\cdot,\cdot}$ and that for $i\not = j$, $\ang{V_i, V_j} = (0)$.
\begin{romanize}
\item\label{lemmaxsp} If the pairing is symplectic, and if $H\subseteq \sp(V)$
contains both $\sp(V_1\oplus V_2)$ and $\sp(V_2\oplus V_3)$, then $H =
\sp(V)$.
\item\label{lemmaxsu} If the pairing is Hermitian, and if $H\subseteq \su(V)$ contains both
$\su(V_1\oplus V_2)$ and $\su(V_2\oplus V_3)$, then $H =\su(V)$.
\end{romanize}
\end{alphabetize}
\end{lemma}
\begin{proof}
We use \eqref{eqsumdecomp} to identify $\gl(V_i)$
with a subgroup of $\gl(V)$.
Our proof of (a) is Lie-theoretic. Since $\sl(V)$ is split, the roots
and Weyl group of $\sl(V)$ are the same as those of
$\sl(V\tensor\bar\ff)$ \cite[1.18]{carterfglt}. Moreover,
$\sl(V)$ has a split BN-pair, and we show $H = \sl(V)$ by successively
showing that $H$ contains a maximal torus $T$ of $\sl(V)$, the
associated Weyl group, and all root subgroups.
Choose coordinates $e_1, \ldots, e_g$ so that $V_1$ is the span of
$\st{e_1, \ldots, e_{g_1}}$; $V_2$ is the span of $\st{e_{g_1+1},
\ldots, e_{g_1+g_2}}$; and $V_3$ is the span of $\st{e_{g_1+g_2+1},
\ldots, e_g}$.
First, $H$ contains the maximal split torus $T$ of $\sl(V)$
consisting of all diagonal matrices $\diag(\nu_1, \ldots, \nu_g)$ such that
$\prod_j \nu_j = 1$. To see this, choose $j$ such that $e_j\in
V_2$. Given $\nu = \diag(\nu_1, \ldots, \nu_g) \in T$, we use the fact
that $\nu_j = (\prod_{i\not= j}\nu)\inv$ to write
\begin{equation*}
\nu =\diag(\nu_1, \ldots, \nu_{j-1}, (\prod_{i=1}^{j-1}\nu_i)\inv, 1,
\ldots, 1) \cdot \diag( 1, \ldots, 1,(\prod_{i=j+1}^g \nu_i)\inv,
\nu_{j+1}, \ldots, \nu_g).
\end{equation*}
Second, $H$ contains $N_{\sl(V)}(T)$, the normalizer of $T$ in
$\sl(V)$. It suffices to show $N_H(T)/T = N_{\sl(V)}(T)/T$. The
normalizer $N_{\sl(V)}(T)$ is the set of matrices of determinant one with exactly one
nonzero entry in each row and in each column, and the quotient
$N_{\sl(V)}(T)/T$ is isomorphic to $\sym(\st{1, \ldots, g})$. Under
this identification,
$$N_{\sl(V_1 \oplus V_2)}(T\cap \sl(V_1\oplus V_2))/(T\cap \sl(V_1\oplus V_2)) =
\sym(\st{1, \ldots, g_1+g_2});$$
$$N_{\sl(V_2\oplus V_3)}(T\cap \sl(V_2\oplus V_3))/(T\cap \sl(V_2\oplus V_3)) = \sym(\st{g_1+1,
\ldots, g}).$$
Since $H$ contains $\sl(V_1\oplus V_2)$ and
$\sl(V_2\oplus V_3)$, it contains $N_{\sl(V)}(T)$.
Third, let $\Delta$ be the canonical set of simple roots associated with
$\End(V) \iso \mat_n(V)$. For each $ i \in \st{1,
\ldots, g-1}$ there is a root $\alpha_i\in \Delta$. The associated root group
is the unipotent group $U_{\alpha_i}$; the nontrivial elements of $U_{\alpha_i}$ are unipotent
matrices whose only nonzero offdiagonal entry is at $(i,i+1)$. Clearly, $H$
contains each $U_{\alpha_i}$.
Finally, the Weyl group $N_{\sl(V)}(T)/T$, acting on the set of simple roots
$\Delta$, generates the set of all roots $\Phi$. Since $H$ contains
$N_{\sl(V)}(T)$, it therefore contains all root
groups $U_\alpha$ where $\alpha\in \Phi$. By the Bruhat
decomposition, any element of $\sl(V)$ is a product of a
diagonal matrix and members of the various root groups. Therefore, $H
= \sl(V)$.
For (b) part (i), after relabeling if necessary, we may assume that
$V_1\oplus V_2\not = V_3$. Moreover, the hypothesis
ensures that $H$ contains $\sp(V_3)$, and thus it contains $\sp(V_1\oplus V_2)
\oplus \sp(V_3)$. It is known that $\sp(V_1\oplus V_2) \oplus
\sp(V_3)$ is a maximal subgroup of $\sp(V)$ \cite[Thm.\ 3.2]{king81}.
Since $\sp(V_2\oplus V_3)\not\subset
\sp(V_1\oplus V_2) \oplus \sp(V_3)$, the subgroup $H$ must equal $\sp(V)$.
The same argument proves (b) part (ii); as group-theoretic input, we use
the fact \cite[p.373]{king81} that $\su(V_1\oplus V_2)\oplus \su(V_3)$ is a
maximal subgroup of $\su(V)$.
\end{proof}
\subsection{Monodromy of hyperelliptic curves}
We show that the mod-$\ell$ monodromy of the tautological family of
hyperelliptic curves of genus $g$ is the full symplectic group
$\sp_{2g}(\integ/\ell)$. Recall (Section \ref{subsechecomp}) that for
$g\in\nat$, the moduli space $\til\calh_g$ of labeled hyperelliptic
curves of genus $g$ is irreducible. We will use Lemma
\ref{lemdisjointhe} to relate $\mono_\ell(\til\calh_g)$ to
$\mono_\ell(\bar\calh_g)$.
\begin{lemma}
\label{lemdisjointhe}
Let $\ell$ be an odd prime. Let $S$ be an irreducible scheme with a
primitive $2\ell\th$ root of unity, and let $(X \ra S, \phi)$ be a principally
polarized abelian scheme. Let $Y \ra S$ be a Galois \'etale cover.
Suppose that $\mono_\ell(X \ra S) \iso
\sp_{2g}(\integ/\ell)$ and that the groups $\gal(Y/S)$ and
$\sp_{2g}(\integ/\ell)$ have no common nontrivial quotients. Then $\mono_\ell(
X \cross_S Y \ra Y) \iso \sp_{2g}(\integ/\ell)$.
\end{lemma}
\begin{proof}
We equip $(\integ/\ell)^{2g}$ with the standard symplectic pairing
$\ang{\cdot,\cdot}_\std$. Let
\begin{equation*}
I_\ell := \Isom( (X[\ell], \ang{\cdot,\cdot}_\phi),
((\integ/\ell)^{2g},\ang{\cdot,\cdot}_{\std})).
\end{equation*}
The hypothesis on $S$ implies that $I_\ell \ra S$ is an \'etale Galois cover with group
$\sp_{2g}(\integ/\ell)$.
The hypothesis on $\mono_\ell(X \ra S)$ implies that $I_\ell$ is irreducible.
To prove the lemma, we must show that $I_\ell \cross_SY$ is
irreducible. Equivalently, we must show that $I_\ell \ra S$ and $Y
\ra S$ are disjoint. Now, $I_\ell \cross_S Y \to S$ is a (possibly
reducible) \'etale Galois cover with group $\sp_{2g}(\integ/\ell)
\cross \gal(Y/S)$. If $Z \ra S$ is any common quotient of $I_\ell \ra
S$ and $Y \ra S$, then so is each conjugate $Z^\tau \to S$ for $\tau
\in \gal(I_\ell \cross_SY/S)$. Therefore, the compositum $\til Z$ of
all such conjugates $Z^\tau$ is also a common quotient of $I_\ell \ra
S$ and $Y \ra S$. It thus suffices to show that there is no
nontrivial Galois cover $\til Z \ra S$ which is a quotient of $I_\ell
\ra S$ and $Y \ra S$. This last claim is guaranteed by the
group-theoretic hypothesis. \end{proof}
We compute the mod-$\ell$ monodromy of the moduli
space of hyperelliptic curves.
\begin{theorem}
\label{thhe}
Let $\ell$ be an odd prime, and let $k$ be an algebraically closed field in which $2\ell$ is invertible.
For $g\ge 1$, $\mono_\ell(\til\calh_g\bc k)\iso \mono_\ell(\calh_g\bc k) \simeq
\sp_{2g}(\integ/\ell)$.
\end{theorem}
\begin{proof}
Since $\calh_g\bc k$ is open and dense in $\bar\calh_g\bc k$, which is dominated
by $\til\calh_g\bc k$, it suffices to show that $\mono_\ell(\til\calh_g\bc
k) \iso \sp_{2g}(\integ/\ell)$.
Our proof is by induction on $g$.
If $g=1$ or $g= 2$, then every curve of genus $g$ is hyperelliptic,
so that $\bar\calm_g\bc k$ and $\bar\calh_g\bc k$ coincide. Therefore,
$\mono_\ell(\bar\calh_g\bc k)\iso \mono_\ell(\bar\calm_g\bc k)$. By
\cite[5.15-5.16]{delignemumford}, $\mono_\ell(\bar\calm_g\bc k) \iso
\sp_{2g}(\integ/\ell)$.
We conclude the base case $g=1$ and $g=2$ by applying Lemma \ref{lemdisjointhe} to the
cover $\til \calh_g \ra \bar \calh_g$. This cover is \'etale and Galois, with Galois
group $\sym(2g+2)$.
This group and $\sp_{2g}(\integ/\ell)$ have no common nontrivial
quotient. (To see this, recall that if $\ell$ is odd then the projective
symplectic group $\psp_{2g}(\integ/\ell)$ is simple, except that
$\psp_{2}(\integ/3) \iso A_4$. Neither $\psp_{2g}(\integ/\ell)$ nor
any quotient of $A_4$ is a nontrivial quotient of $\sym(4)$ or $\sym(6)$.)
By Lemma \ref{lemdisjointhe}, $\mono_\ell(\til\calh_g\bc
k) \iso \sp_{2g}(\integ/\ell)$.
We now assume that $g \ge 3$ and that $\mono_\ell(\til\calh_{g'}\bc k) \iso
\sp_{2g'}(\integ/\ell)$ for $1 \le g' < g$. By Lemma
\ref{lemdegenhe}, $\til\calh_g \bc k$
degenerates to $\Delta_{1,1}$, so that there is a
diagram \eqref{diagclutch}:
\begin{diagram}
(\til\calh_1 \cross \til \calh_{g-2} \cross \til \calh_1)\bc k &
\rto^{\kappa_R} &
(\til \calh_1 \cross \til \calh_{g-1})\bc k \\
\dto<{\kappa_L} & \rdto>\kappa & \dto \\
(\til \calh_{g-1} \cross \til \calh_1) \bc k& \rto & \til \calh_g \bc k.
\end{diagram}
Fix a base point $(s_1,s_2,s_3) \in (\til \calh_1 \cross \til
\calh_{g-2} \cross \til \calh_1)(k)$, and let $s =
\kappa(s_1,s_2,s_3)$. For $h \in \NN$, let $C^{h} \to \til\calh_{h}$
be the tautological labeled $\ZZ/2$-curve of genus $h$. Let $V =
\pic^0(\calc^g)[\ell]_s$, and for $i=1,2,3$ let $V_i =
\pic^0(\calc^{g_i})[\ell]_{s_i}$. Each of these is a
$\integ/\ell$-vector space equipped with a symplectic form. By Lemma
\ref{lemclutchmono}(a), there is an isomorphism of symplectic
$\integ/\ell$-vector spaces $V \iso V_1 \oplus V_2 \oplus V_3$.
Let $\til\calb_R$ (resp.\ $\til\calb_L$) be the image of $\kappa_R$ (resp.\ $\kappa_L$).
Using the decomposition in Equation \eqref{eqclutchsum},
we have inclusions:
\begin{equation}
\label{eqhecontain}
\begin{split}
\mono_\ell(\til\calb_R\bc k) & \subseteq \sp(V_1) \oplus \sp(V_2 \oplus
V_3); \\
\mono_\ell(\til\calb_L\bc k) & \subseteq \sp(V_1\oplus V_2) \oplus \sp(V_3).
\end{split}
\end{equation}
By the induction hypothesis, the inclusions in \eqref{eqhecontain} are
equalities. By Lemma \ref{lemclutchmono}(b),
$\mono_\ell(\til \calh_g\bc k)$ contains $\sp(V_1)\oplus
\sp(V_2\oplus V_3)$ and $\sp(V_1\oplus V_2) \oplus \sp(V_3)$.
By Lemma \ref{lemmaxgp}b(i),
$\mono_\ell(\til\calh_g) \iso \sp_{2g}(\integ/\ell)$.
\end{proof}
\begin{corollary}
\label{corladiche}
Let $\ell$ be an odd prime, and let $k$ be an algebraically closed
field in which $2\ell$ is invertible. For $g \ge 1$,
$\mono_{\integ_\ell}(\til \calh_g\bc k) \iso
\mono_{\integ_\ell}(\calh_g \bc k) \iso\sp_{2g}(\integ_\ell)$.
\end{corollary}
\begin{proof}
By Theorem \ref{thhe}, $\mono_\ell(\til\calh_g\bc k) \iso
\sp_{2g}(\integ/\ell)$. By construction, there is a surjection
$\mono_{\integ_\ell}(\til\calh_g\bc k) \ra \mono_\ell(\til\calh_g \bc
k)$.
Since the composition $\mono_{\integ_\ell}(\til \calh_g\bc k)
\inject \sp_{2g}(\integ_\ell) \ra \sp_{2g}(\integ/\ell)$ is
surjective, a standard group theory argument (e.g., \cite[1.3]{vasiusurj})
shows that $\mono_{\integ_\ell}(\til\calh_g \bc k) \iso
\sp_{2g}(\integ_\ell)$.
As in the proof of Theorem \ref{thhe}, $\mono_{\integ_\ell}(\calh_g \bc k) \iso \sp_{2g}(\integ_\ell)$
as well.
\end{proof}
\begin{corollary} \label{Chypabsirr}
Let $p \not = 2$ and let $g\in\nat$. Then there exists a
smooth hyperelliptic curve of genus $g$ defined over $\bar\ff_p$ whose Jacobian is
absolutely simple.
\end{corollary}
\begin{proof}
Let $\ell$ be an odd prime distinct from $p$. By Corollary
\ref{corladiche},
\begin{equation*}
\sp_{2g}(\integ_\ell) \iso \mono_{\integ_\ell}(\calh_g\bc {\bar\ff_p})
\subseteq \mono_{\integ_\ell}(\calh_g\bc\ff_p) \subseteq
\gsp_{2g}(\integ_\ell).
\end{equation*}
Moreover,
$\mono_{\integ_\ell}(\calh_g\bc\ff_p)/\mono_{\integ_\ell}(\calh_g\bc{\bar\ff_p})
\iso \gal(\ff_{p,\ell^\infty}/\ff_p)$, which has finite index in
$\aut(\mmu_{\ell^\infty}(\bar\ff_p)) \iso
\gsp_{2g}(\integ_\ell)/\sp_{2g}(\integ_\ell)$. Therefore,
$\mono_{\integ_\ell}(\calh_g\bc \ff_p)$ is an open subgroup of
$\gsp_{2g}(\integ_\ell)$; by Borel's density theorem \cite[Th. 4.10]{platonovrapinchuk}, $\mono_{\rat_\ell}(\calh_g\bc \ff_p)
\iso \gsp_{2g}(\rat_\ell)$. The claim now follows from
\cite[Prop. 4]{chaioort01}.
\end{proof}
\subsection{Monodromy of trielliptic curves}
We now compute the monodromy groups of tautological families
$\calc^\gamma \ra \til\calt^\gamma$ of labeled trielliptic curves.
The Jacobians of trielliptic curves admit an action by
$\integ[\zeta_3]$. This places a constraint on
$\mono_\ell(\til\calt^\gamma)$; in Theorem \ref{thtri}, we show that
this is the only constraint.
We need more notation concerning $\integ[\zeta_3$]-actions and unitary groups
to describe the monodromy group precisely.
\subsubsection{Unitary groups}
Let $S$ be an irreducible scheme over $\spec\integ[1/3,\zeta_3]$, and
let $X \ra S$ be an abelian scheme of relative dimension $g$ equipped
with an action $\iota:\integ[\zeta_3] \ra \End_S(X)$. Then $\Lie(X)$
is a locally free $\integ[\zeta_3]\tensor \calo_S$-module of some rank
$(r,s)$, where $r+s = g$. We call $(r,s)$ the signature of the action
of $\integ[\zeta_3]$ on $X$. If $(C/S,\iota_0)$ is a trielliptic curve,
then $\Lie(\pic^0(C))$ is the $\calo_S$-linear dual of
$\psi_*\Omega^1_{C/S}$ and the signature of $\pic^0(C)$ is the
same as that of $C$.
Let $V_{(r,s)}$ be a free $\integ[\zeta_3]$-module of rank
$g$, equipped with a $\integ[\zeta_3]$-linear pairing
$\ang{\cdot,\cdot}$ of signature $(r,s)$. Let $\gu_{(r,s)}$ be the
$\integ[1/3, \zeta_3]$-group scheme of similitudes of the pair
$(V_{(r,s)}, \ang{\cdot,\cdot})$, and let $\g_{(r,s)}$ be the
restriction of scalars of $\gu_{(r,s)}$ to $\integ[1/3]$. Let
$\su_{(r,s)} \subset \gu_{(r,s)}$ be the sub-group scheme of elements
of determinant one, and let $\sg_{(r,s)}$ be the restriction of
scalars of $\su_{(r,s)}$ to $\integ[1/3]$. Concretely, for any
$\integ[1/3]$-algebra $R$,
\begin{equation}
\label{eqdefsg}
\sg_{(r,s)}(R) = \st{ \tau \in \aut( V_{(r,s)}\tensor_{\integ[1/3]}R,
\ang{\cdot,\cdot}): \det(\tau) = 1 }.
\end{equation}
In the abstract, we used $\su_{(r,s)}(\integ/\ell\tensor\integ[\zeta_3])$ to denote
$\sg_{(r,s)}(\integ/\ell)$. The signature
condition implies that $\sg_{(r,s)}(\real)$ is isomorphic to the
complex special unitary group $\su(r,s)$.
The behavior of $\sg_{(r,s)}$ at finite
primes $\ell \geq 5$ depends on their splitting in $\integ[\zeta_3]$.
Specifically, if $\ell$ is inert in $\integ[\zeta_3]$,
then $\sg_{(r,s)}(\integ/\ell) \iso \su_g(\ff_{\ell^2})$.
Alternatively, suppose that $\ell$ splits in $\integ[\zeta_3]$, and
let $\lambda$ and $\bar\lambda$ be the two primes of $\integ[\zeta_3]$
lying over $\ell$. The factorization $\ell = \lambda \cdot
\bar\lambda$ yields a factorization $\integ[\zeta_3]\tensor\integ/\ell
\iso \integ/\ell \oplus \integ/\ell$. This induces a decomposition
$V_{(r,s)}\tensor \integ/\ell \iso V_{(r,s)}(\lambda)\oplus V_{(r,s)}(\bar\lambda)$, where
$V_{(r,s)}(\lambda)$ and $V_{(r,s)}(\bar\lambda)$ are $g$-dimensional
$\integ/\ell$-vector spaces. Moreover, the inner product $\ang{\cdot,\cdot}$ restricts
to a perfect pairing between $V_{(r,s)}(\lambda)$ and $V_{(r,s)}(\bar\lambda)$. Then
$\sg_{(r,s)}(\integ/\ell)\subset \aut(V_{(r,s)}(\lambda)\cross
V_{(r,s)}(\bar\lambda))$ is the image of $\sl_g(\integ/\ell)$ embedded
as $\tau \mapsto \tau \cross (^t\tau)\inv$ (where $^t \tau$ denotes the transpose of $\tau$).
In either case, the
isomorphism class of $\sg_{(r,s)}(\integ/\ell)$ depends only on $g$
and the splitting of $\ell$ in $\integ[\zeta_3]$, and not on the
signature $(r,s)$.
\subsubsection{Calculation of trielliptic monodromy}
Let $\til\calt^\gamma$ be any component of the moduli space of
labeled trielliptic curves. The signature of the action of $\integ[\zeta_3]$
on the Lie algebra of the relative Jacobian is locally constant, and we denote
it by $(r_\gamma, s_\gamma)$ (see Lemma \ref{Lsig}).
The monodromy group $\mono_\ell(\til
\calt^\gamma\bc k)$ must preserve both the symplectic pairing and the
$\integ[\zeta_3]$-action on the $\ell$-torsion of
$\pic^0(\calc^\gamma\bc k)$. This means that there is an
inclusion $\mono_\ell(\til \calt^\gamma\bc k) \subseteq
\sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$. We show that this is actually an
isomorphism.
As a basis for induction, we compare the moduli space of
trielliptic curves of genus three to the Picard modular variety, which
is a
component of a Shimura variety of PEL type. Let $\calsh_{(2,1)}$ be the moduli stack
parametrizing data $(X/S, \iota, \phi)$ where $(X/S, \phi)$ is a principally polarized
abelian scheme of relative dimension $3$ and $\iota:\integ[\zeta_3]\ra
\End(X)$
satisfies signature and involution constraints. Specifically, we
require that $\Lie(X)$ be a locally free $\integ[\zeta_3]\tensor
\calo_S$-module of signature $(2,1)$ and that $\iota$ take complex
conjugation on $\integ[\zeta_3]$ to the Rosati involution on
$\End(X)$. Then $\calsh_{(2,1)}$ is the Shimura variety associated to
the reductive group $\g_{(2,1)}$.
We also
consider the moduli stack $\calsh_{(2,1),\ell}$ parametrizing data $(X/S, \iota,
\phi,\xi)$ where $(X/S,\iota,\phi)$ is as above and $\xi$ is a
principal level $\ell$ structure. More precisely, $\xi$ is a
$\integ[\zeta_3]$-linear isomorphism $\xi: X[\ell] \ra (V_{(2,1)}\tensor
\integ/\ell)\tensor \calo_S$ compatible with the given pairings. The forgetful
functor $\calsh_{(2,1),\ell} \ra \calsh_{(2,1)}$ induces a Galois
cover of stacks, with covering group $\sg_{(2,1)}(\integ/\ell)$
\cite[1.4]{gordon92}.
Note
that, as in \cite[IV.6.1]{faltingschai} but in contrast to \cite{bellaiche,larsenthesis}, we have implicitly chosen an
isomorphism $\mu_\ell \ra \integ/\ell$. If
we had not chosen this isomorphism,
the covering group would be $\g_{(2,1)}(\integ/\ell)$; compare \cite[IV.6.12]{faltingschai}.
\begin{lemma}
\label{lemdisjointtri}
Let $\ell\ge 5$ be prime, and let $S$ be an irreducible scheme with a primitive $3\ell\th$ root of unity.
Let $(X/S, \iota,\phi)\in \calsh_{(2,1)}(S)$ be a principally
polarized abelian scheme with $\integ[\zeta_3]$-action.
Let $Y \ra S$ be an \'etale Galois cover. Suppose that $\mono_\ell(X \ra S)
\iso \sg_{(2,1)}(\integ/\ell)$ and that the groups $\gal(Y/S)$ and
$\sg_{(2,1)}(\integ/\ell)$ have no common nontrivial quotients. Then
$\mono_\ell(X \cross_S Y \ra Y) \iso \sg_{(2,1)}(\integ/\ell)$.
\end{lemma}
\begin{proof}
The proof is exactly the same as that of Lemma \ref{lemdisjointhe},
except that it involves
$$I_{\integ[\zeta_3],\ell} := \Isom_{\integ[\zeta_3]}((X[\ell],
\ang{\cdot,\cdot}_\phi), (V_{2,1}\tensor\integ/\ell,
\ang{\cdot,\cdot})).$$
Then $I_{\integ[\zeta_3],\ell} \to S$ is \'etale and Galois with group
$\su(V_{(2,1)}\tensor\integ/\ell) \simeq \sg_{(2,1)}(\integ/\ell)$.
The hypothesis on $\mono_\ell(X \ra S)$ is equivalent to the irreducibility of $I_{\integ[\zeta_3,],\ell}$.
\end{proof}
\begin{lemma}
\label{lemtribase}
Let $\ell\ge 5$ be prime, and let $k$ be an algebraically closed field
in which $3\ell$ is invertible.
Let
$\til\calt^\gamma$ be any component of the moduli space of
trielliptic curves. If $g(\gamma) = 3$, then
$\mono_\ell(\til\calt^\gamma\bc k) \iso \sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$.
\end{lemma}
\begin{proof}
We start by computing the monodromy group of the Picard modular
variety $\calsh_{(2,1)}$.
Both $\calsh_{(2,1)}\bc\cx$ and
$\calsh_{(2,1),\ell}\bc\cx$ are arithmetic quotients of the complex
ball, thus irreducible \cite[1.4]{gordon92}. Moreover, the existence of smooth
arithmetic compactifications \cite[Section
3]{larsenthesis} \cite[1.3.13]{bellaiche} and Zariski's
connectedness theorem (see \cite[IV.5.10]{faltingschai}) imply that
$\calsh_{(2,1)}\bc k$ and $\calsh_{(2,1),\ell}\bc k$ are also irreducible. The irreducibility of the
cover $\calsh_{(2,1),\ell}\bc k \ra \calsh_{(2,1)}\bc k$
implies that the fundamental group of
$\calsh_{(2,1)}\bc k$ acts transitively on the set of Hermitian
$\integ[\zeta_3]\tensor\integ/\ell$-bases for the $\ell$-torsion of
the tautological abelian scheme over $\calsh_{(2,1)}\bc k$.
Therefore, $\mono_\ell(\calsh_{(2,1)}\bc k) \iso
\sg_{(2,1)}(\integ/\ell)$.
We now consider $\til\calt^\gamma\bc k$. Possibly after relabeling
\eqref{diagrelabelG}, by Lemma \ref{Lsig} we may assume that
$(r_\gamma,s_\gamma) = (2,1)$ and that the inertia type $\bar\gamma$
is $\st{1,1,1,1,2}$. Moreover, since $\dim \calsh_{(2,1)}
\bc k= 2$, the Torelli map gives an inclusion of
$\calt^{\bar\gamma}$ onto an open subset of $\calsh_{(2,1)}$.
Thus $\mono_\ell(\bar\calt^{\bar\gamma}\tensor k) \iso
\mono_\ell(\calsh_{(2,1)}\tensor k)\iso \sg_{(2,1)}(\integ/\ell)$.
By Lemma \ref{lemforgetgalois}, $\til\calt^\gamma \ra
\bar\calt^{\bar\gamma}$ is Galois with group $\sym(4)$.
Since $\ell \ge 5$, the quotient of
$\sg_{(2,1)}(\integ/\ell)$ by its center is simple. Thus,
$\sym(4)$ and $\sg_{(2,1)}(\integ/\ell)$ have no common nontrivial
quotient, and $\mono_\ell(\til\calt^\gamma\bc k) \iso \sg_{(2,1)}(\integ/\ell)$
by Lemma \ref{lemdisjointtri}.
\end{proof}
\begin{theorem}
\label{thtri}
Let $\ell\ge 5$ be prime, and let $k$ be an algebraically closed
field in which $3\ell$ is invertible.
Let
$\til\calt^\gamma$ be any component of the moduli space of labeled
trielliptic curves. If $g(\gamma) \ge 3$, then
$\mono_\ell(\til\calt^\gamma\bc k) \iso \mono_\ell(\calt^{\bar \gamma}\bc k) \iso \sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$.
\end{theorem}
\begin{proof}
Since $\calt^{\bar \gamma}\bc k$ is open and dense in $\bar\calt^{\bar \gamma}\bc k$, which is dominated by $\til\calt^{\gamma}\bc k$,
it is sufficient to show $\mono_\ell(\til\calt^\gamma\bc k) \iso \sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$.
We proceed by induction on $g(\gamma)$.
The case $g(\gamma) = 3$ is supplied by Lemma \ref{lemtribase}.
Now let $\til\calt^\gamma$ be any component of $\til\calt$ with
$g = g(\gamma) \ge 4$, and
suppose the result is true for all components
$\til\calt^{\gamma'}$
with $3 \le g(\gamma') < g$.
By Proposition \ref{propdegentri}, $\til \calt^\gamma$ degenerates to $\Delta_{1,1}$.
This means that there are class vectors
$\gamma_1$, $\gamma_{2}$, $\gamma_3$, $\gamma_L$, and $\gamma_R$
such that $g(\gamma_1) = g(\gamma_3)
= 1$; $g(\gamma_2)=g-2$; and there is a diagram \eqref{diagclutch}:
\begin{diagram}
(\til \calt^{\gamma_1} \cross \til \calt^{\gamma_{2}} \cross \til
\calt^{\gamma_3})\bc k & \rto^{\kappa_R} & (\til \calt^{\gamma_1} \cross \til
\calt^{\gamma_R})\bc k \\
\dto<{\kappa_L} &\rdto^\kappa& \dto \\
(\til \calt^{\gamma_L} \cross \til \calt^{\gamma_3})\bc k & \rto &
\til \calt^\gamma\bc k.
\end{diagram}
Let $(s_1,s_2,s_3) \in (\til \calt^{\gamma_1} \cross \til \calt^{\gamma_{2}} \cross \til
\calt^{\gamma_3})(k)$, and let $s = \kappa(s_1,s_2,s_3)$.
Let $V =\pic^0(\calc^\gamma)[\ell]_s$, and for $i=1,2,3$ let $V_i=
\pic^0(\calc^{\gamma_i})[\ell]_{s_i}$. Then $V \iso V_1 \oplus V_2 \oplus V_3$
is a decomposition of $V$ as a Hermitian $(\integ[\zeta_3]\tensor
k)$-module (Lemma \ref{lemclutchmono}(a)). Let $\til \calb_R$ be the
image of $\kappa_R$, and let $\til \calb_L$ be the image of
$\kappa_L$. Lemma \ref{lemclutchmono}(b) shows that $\mono_\ell(\til
\calt^\gamma\bc k, s)$ contains $\mono_\ell(\til \calb_L\bc k)$ and $\mono_\ell(\til \calb_R\bc k) $;
the inductive hypothesis shows
that these are $\su(V_1\oplus V_2)$ and $\su(V_2\oplus V_3)$,
respectively.
If $\ell$ is inert in $\integ[\zeta_3]$, Lemma
\ref{lemmaxgp}b(ii) implies that $\mono_\ell(\til\calt^\gamma\bc k) \iso
\su_g(\FF_{\ell^2}) \iso \sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$.
Otherwise, if $\ell = \lambda \cdot \bar \lambda$ is split in
$\integ[\zeta_3]$, let $V(\lambda)$ be the eigenspace of $V$
corresponding to $\lambda$, and define $V_i(\lambda)$ analogously for
$i = 1,2,3$.
In this case, $\sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$, $\su(V)$ and $\sl(V(\lambda))$ are isomorphic.
By the inductive hypothesis,
the projection of $\mono_\ell(\til\calt^\gamma\bc k)$ to
$\sl(V(\lambda))$ contains $\sl(V_1(\lambda)\oplus V_2(\lambda))$ and
$\sl(V_2(\lambda)\oplus V_3(\lambda))$. By Lemma \ref{lemmaxgp}(a),
we see that $\mono_\ell(\til\calt^\gamma\bc k) \iso \sl(V(\lambda)) \iso
\sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$.
\end{proof}
\begin{corollary}
\label{corladictri}
Let $\ell$ be an odd prime, and let $k$ be an algebraically closed
field in which $3\ell$ is invertible. Let $\til\calt^\gamma$ be any
component of the moduli space of trielliptic curves. If $g(\gamma)
\ge 3$, then $\mono_{\integ_\ell}(\til\calt^\gamma\bc k) \iso
\mono_{\integ_\ell}(\calt^{\bar\gamma}\bc k) \iso
\sg_{(r_\gamma,s_\gamma)}(\integ_\ell)$.
\end{corollary}
\begin{proof}
The proof is parallel to that of Corollary \ref{corladiche}; any
subgroup of $\sg_{(r_\gamma,s_\gamma)}(\integ_\ell)$ which surjects
onto $\sg_{(r_\gamma,s_\gamma)}(\integ/\ell)$ is all of
$\sg_{(r_\gamma,s_\gamma)}(\integ_\ell)$.
\end{proof}
\begin{corollary} \label{Ctriabsirr}
Let $p \not = 3$. Let $g \geq 3$ and let $(r,s)$ be a trielliptic signature for $g$ (Definition \ref{Dtrisig}).
Then there exists a smooth trielliptic curve defined over $\bar \ff_p$ with genus $g$ and signature $(r,s)$
whose Jacobian is absolutely simple.
\end{corollary}
\begin{proof}
Let $\ff = \ff_{p^2}$. Let $\ell$ be an odd prime distinct from $p$
which is inert in $\integ[\zeta_3]$, and let $K_\ell =\rat_\ell(\zeta_3)$.
By Lemma \ref{Lsig}, there is a
class vector $\gamma$ whose inertia type has signature $(r,s)$. As
in the proof of Corollary \ref{Chypabsirr}, using Corollary
\ref{corladictri} one sees that
$\mono_{\rat_\ell}(\til\calt^\gamma\bc \ff)\iso\gu_{(r,s)}(K_\ell)$.
Let $F$ be a CM field of degree $[F:\rat] = 2g$ which contains
$\rat(\zeta_3)$ and is inert at $\ell$, and let $F_\ell = F
\tensor\rat_\ell$. There is a torus $H\subset
\gu_{(r,s)}(K_\ell)$ isomorphic to $F_\ell\units$.
This torus is maximal since $[F:\rat] = 2g$. The quotient of $H$ by
the center of $\gu_{(r,s)}(K_\ell)$ is isogenous to the kernel
$J$ of the norm map $F_\ell\units \ra \rat_\ell\units$.
Since $F_\ell$ is a field, $J$ is anisotropic and $H$ is
elliptic. Finally, $H$ acts irreducibly on the Tate module of the
Jacobian of the tautological curve $C^{\gamma}$, since
$T_\ell(\pic^0(\calc^\gamma))\tensor\rat_\ell$ is a one-dimensional $F_\ell$-vector space. The result then follows from
\cite[Remark 5(i)]{chaioort01}.
\end{proof}
\bibliographystyle{abbrv}
\bibliography{jda}
\end{document}
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Have I ever told you about the time I slept with a Russian model while travelling by train to one of the most romantic cities in the world? I haven’t? Right, well sit back, relax, and enjoy the greatest love story the world has ever known.
Before Sunrise is one of my favourite movies of all time. It’s about two attractive young people who have a chance meeting on a train between Budapest and Vienna, and end up falling in love during the course of one impossibly long evening. It’s a brilliant film that explores life, loss and love, so when I made my own trip to Europe, I was determined to have a similar experience while riding the rails.
Unfortunately, the closest I came to finding my soul mate was when a sour-faced Romanian told me to fuck off somewhere in the middle of Belgium, so by the time I stepped onto an overnight train between Munich and Venice, I didn’t hold out much hope for it happening. Then again, I didn’t count on Valerie being onboard.
I noticed her as soon as she stepped onto the train. Shit, it would be impossible not to, because even under a full-length coat it was obvious that her body was incredible. I stepped onto the train behind her and she turned to face me, her stunning face framed by a pair of reading glasses and cascading brown hair. She looked a little lost, maybe even awkward. I was instantly smitten, and almost gave myself a self-high five when she carried her bags into the same compartment as me.
I lay my bags on the top bunk, she lay hers on the bottom, and I lay back for a few minutes wondering what the hell I was going to say to her. I didn’t even know if I’d be able to talk to her – not because I was too nervous, but because we were in the heart of continental Europe, where most people struggle to string a sentence together unless their jabbering away in some foreign tongue.
As the train eased out of Munich and slid through the dark suburbs, the girl disappeared from the room, so I decided it was as good a time as any to find the train’s restaurant and get stuck into the beers. After all, it was a long trip, i was thirsty, and I had a pretty lady to talk to. Turns out there wasn’t a restaurant, so a little bar, but I was able to buy a beer and start drinking it as I headed back to my room.
And there she was, standing in the hallway, looking out the window towards the full moon. Her coat was gone, revealing a body that could (and almost certainly has) brought grown men to tears. I took a pull of my beer, composed myself, and came out with a line I hoped would win the heart of the fairest maiden of them all:
“Not gonna see much out there tonight, love.” Fucking hell, Mr Smooth! Thankfully, she didn’t understand a word I said, and simply looked back at me with a confused look on her face. Bloody hell, she even made being confused look good. I took my second chance with both hands and struck up a conversation with her, and discovered her name was Veleriya, she was from Siberia (and doesn’t know the song Jukebox in Siberia) and it was her birthday. Just talking to her made me feel like it was my birthday and that I was the one getting the presents. I bought her a drink to celebrate and thought I was getting somewhere, when Manny showed up.
Oh, Manny. This slice Americana was about five foot tall and eight foot round, wearing a garish yellow button-up shirt and a dazzling smile. He’d apparently been drinking all day in Munich’s Englischer Garten (as had I) and was on a 15-day train tour through 15 European cities, spending about 10 hours in each. He was looking forward to going to Milan, where he planned to drop a few grand on new clothes (from the look of him, he needed it), but right now he was more intent on getting between me and and the lovely Valerie.
Manny seemed like a nice enough bloke, but I was desperate to get rid of him so that I could give all my attention back to Valerie. She looked bored, but she even made that look good. I was on my fourth beer and trying to get Manny to stop talking about his fucking Fedora when a very strange couple fell through the door and started staggering towards us.
“Hey, we found ze party!” said the bloke, who was bald and skinny, with his eyes sunken deep within his skull. He downed what was left of his beer and dumped it out the window. His girlfriend was fat and English, scoffing a chocolate cake, barely able to stand by herself. Next thing I know I was involved in a group hug, being squished delightfully close to Valerie’s ample chest, an act that would’ve been far more romantic if not for the stench of beer and vomit from the pissheads. I decided to roll with it, and bought myself and my lady friend another drink. I might not have been enjoying the party for two I was after, but it was turning into a party anyway.
We were making so much noise that we were moved along to the nearest vestibule, where a stick-thin and solemn-faced middle-aged woman was standing, clutching a box. As I did my best to slur sweet nothings in the direction of Valerie and Manny did his best to pull down his pants and show me his silk boxer shorts, the drunk couple kept asking the thin woman what was in the box. She kept shaking her head and they kept asking her and I kept drinking and Manny kept being a fuckwit and Valerie kept being the most beautiful woman on the planet, when finally the thin woman let out an exasperated groan and opened the fucking box.
Inside was a dildo. A big one, too, purple, battery operated, flashing lights, the whole shebang. The thin woman’s composition changed and she started waving the purple pussy eater around, bonking Manny on the head with it, and making loud ‘Woop woop woop’ noises. So here I am, eight beers in, trying to chat up a gorgeous chicky babe while dildos are being waved around, drunks are pissing in the corner, and Manny is trying to show off his Hello Kitty socks.
It was all too much for Valerie, who said she was tired and needed to go to bed. When I told her it was my gentlemanly duty to escort her back to her room, Manny looked at me in open-mouthed horror and said, “But dude! We were gonna do shots!”
“Mate,” I said, “get in with the chick with the dildo. I’ve got a feeling that a good-looking, fashionable bloke like yourself is just what she’s after.” And with that, the lovely Valerie and myself headed back to our room. I won’t go into what happened after that, but I will tell you about what happened the next day. After seeing next to nothing during the night trip (not that I was focusing on what was outside the windows) it was great to be able to watch the Italian towns and villages and waterways pass by underneath the early morning sun. I was heading all the way in to Venice’s Santa Lucia, while Valerie was getting off at the earlier Mestre, on the mainland. When the train stopped, she stepped off without saying goodbye. I was probably never going to see her again, and I’d fallen for her, so I leapt off and raced after her, like something out of a romantic comedy. I’d like to say there was a happy ending, but there wasn’t.
As soon as I got close to her I was surrounded by half-a-dozen huge, bearded dudes in suits who yelled at me in some sort of foreign language. I couldn’t understand it, but I knew it wasn’t good, and when I looked at Valerie she just stared at the ground. I started to say something and one of the heavies punched me in the guts, then pushed me back towards the train, which I stepped onto just before it left. The last I saw of Valerie was her being led away by the thugs.
Venice was beautiful, but crowded, and I was in a daze as I walked to my hotel, so I didn’t take in much of it. When I finally found WiFi, I was able to look at Valerie’s profile, which told me two things. Firstly, she wasn’t just a pretty girl, she was a professional model who had worked all over the world (it came as no surprise). Secondly, she was in a relationship with a short, mean-lookin’ dude who owned a 60-foot yacht, a Ferrari 458, and a selection of shiny guns that he didn’t mind showing off all over the internet. Basically, I’d spent the night putting the moves on some Mafia heavy’s missus. So much for romance, eh?
| 364,171
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EP.
They begin with the shortest timescales and fastest dynamics, for example burstiness of call sequences between individuals, and then "zoom out" towards longer temporal and larger structural scales, from temporal motifs formed by correlated calls between multiple individuals to long-term dynamics of social groups. The authors conclude with a future outlook for CDR-based research, which faces a major challenge in the great diversity of new communication channels available and the propensity of the younger generations to adopt these new channels, even for voice. Future research will call for approaches that do not rely on a single source of data.
The authors identify emerging themes such as (i) the move from large data sets to smaller samples which compensate size with improvements in data quality and coverage of multiple channels, (ii) the shift from large aggregates to individuals, (iii) and the difficulty of sharing datasets which precludes the possibility to check the reproducibility of results.
Jari Saramäki and Esteban Moro (2015),
From seconds to months: an overview of multi-scale dynamics of mobile telephone calls,
European Physical Journal B, DOI: 10.1140/epjb/e2015-60106-6
| 94,431
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> The Watersons > Records > Lal Waterson & Oliver Knight: A Bed of Roses
Lal Waterson & Oliver Knight: A Bed of Roses
Lal Waterson died while recording this album with her son Oliver Knight; it was finished by him alone.
Recorded by Panda Sound at Robin Hood's Bay;
Engineered by Oliver Knight;
Produced by Oliver Knight & Lal Waterson;
Edited by Ray Williams;
Original rose photographs by Máté Gibb Photography;
Digital design / photo manipulation by John Haxby, Art Surgery.
Special thanks to George, Marry, Joe & Nigel, Norma, Martin & Mike,
Tony Engle, Ray Williams, No Masters Co-operative and all the musicians.
Musicians
Lal Waterson, vocals;
Oliver Knight, guitar;
Charles O'Connor, strings [2];
Maria Gilhooley (Marry Waterson), vocals [3];
Jody Stecher, mandolin [4];
Chopper, cello [6];
Alice Kinloch, trombone [9];
Jo Freya, tenor saxophone [9]
Tracks
- Memories (2.43)
- Foolish One (4.00)
- Just a Note (1.42)
- Columbine (4.06)
- At First She Starts (2.45)
- Bath Time (4.52)
- Train to Bay (4.03)
- Long Vacation (3.21)
- Party Games (3.28)
- Together (4.50)
- Migrating Bird (1.17)
- Lullaby (3.30)
All tracks Lal Waterson & Oliver Knight pub Topic Records Ltd except
Track 3 Ewan MacColl
Track 5 Lal Waterson pub Topic Records Ltd except
Tracks 7 and 12 Oliver Knight pub Topic Records Ltd except
Acknowledgements
Thanks to John Haxby for permission to use the images of the CD cover.
| 392,036
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Japanese Yarn bag, Japanese Wool bag, Yarn organiser, Wool organiser, Crochet bag, Knitting bag, Craft bag, Japanese bag, Mothers Day Gift
Sold by ChainBaySewcraft
$30.91+ $12.26 shipping
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Made to order
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| 78,939
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