% \iffalse
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%%%     author          = "American Mathematical Society",
%%%     version         = "1.2beta",
%%%     date            = "11-Oct-1994",
%%%     time            = "15:19:39 EDT",
%%%     filename        = "testart.tex",
%%%     copyright       = "Copyright (C) 1994 American Mathematical Society,
%%%                        all rights reserved.  Copying of this file is
%%%                        authorized only if either:
%%%                        (1) you make absolutely no changes to your copy,
%%%                        including name; OR
%%%                        (2) if you do make changes, you first rename it
%%%                        to some other name.",
%%%     address         = "American Mathematical Society,
%%%                        Technical Support,
%%%                        Electronic Products and Services,
%%%                        P. O. Box 6248,
%%%                        Providence, RI 02940,
%%%                        USA",
%%%     telephone       = "401-455-4080 or (in the USA and Canada)
%%%                        800-321-4AMS (321-4267)",
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%%%     abstract        = "This file is part of the AMS-\LaTeX{} package.
%%%                        It is a sample document illustrating the use of
%%%                        the ``amsart'' documentclass and the ``amstex''
%%%                        package.",
%%%     docstring       = "The checksum field above contains a CRC-16
%%%                        checksum as the first value, followed by the
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% Beginning of testart.tex
%------------------------------------------------------------------------------
% This is a sample file for use with AMS-LaTeX1.2. It provides an example of
% how to set up a file to be typeset with AMS-LaTeX.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\documentclass{amsart}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\makeatletter
\def\square{\RIfM@\bgroup\else$\bgroup\aftergroup$\fi
  \vcenter{\hrule\hbox{\vrule\@height.6em\kern.6em\vrule}\hrule}\egroup}
\makeatother

\begin{document}

\title[Sample Article Paper]{Sample Article Paper}
\author[AUTHOR ONE AND AUTHOR TWO]{Author One and Author Two}
\address{Department of Mathematics, Northeastern University, Boston,
Massachusetts\linebreak 02115} % Research address for author one
\curraddr{Department of Mathematics and Statistics,
Case Western Reserve University, Cleveland, Ohio 43403}
\email{xyz@@math.ams.org}
\address{Mathematical Research Section, School of Mathematical Sciences,
Australian National University, Canberra ACT 2601, Australia} %address for
%                                                             author two

\subjclass{Primary 54C40, 14E20; Secondary 46E25, 20C20}
\date{09 October 1994 \\
Version 1.02}

%  \thanks will become a 1st page footnote.
\thanks{The first author was supported in part by NSF
Grant \#000000.}

\maketitle

\begin{abstract}
This paper is a sample prepared to illustrate for authors the use of the
\AmS-\LaTeX{} Version~1.2 package \texttt{amsart} style.
\end{abstract}


\section{This is a numbered first-level section head}
This is an example of a numbered first-level heading.

\section*{This is an unnumbered first-level section head}
This is an example of an unnumbered first-level heading.

\subsection{This is a numbered second-level section head}
This is an example of a numbered second-level heading.

\subsection*{This is an unnumbered second-level section head}
This is an example of an unnumbered second-level heading.

\subsubsection{This is a numbered third-level section head}
This is an example of a numbered third-level heading.

\subsubsection*{This is an unnumbered third-level section head}
This is an example of an unnumbered third-level heading.

\begin{lemma}
Let $f, g\in  A(X)$ and let $E$, $F$ be cozero
sets in $X$.
\begin{enumerate}
\renewcommand{\theenumi}{\alph{enumi}}
\item If $f$ is $E$-regular and $F\subseteq E$, then $f$ is $F$-regular.

\item If $f$ is $E$-regular and $F$-regular, then $f$ is $E\cup F$-%
regular.

\item If $f(x)\ge c>0$ for all $x\in E$, then $f$ is $E$-regular.

\end{enumerate}
\end{lemma}


The following is an example of a proof\footnote{Here is an
example of a footnote. Notice that this footnote text is running on so that it
can stand as an example of how a footnote with separate paragraphs should be
keyed.\endgraf And here is the beginning of the second paragraph.}.

\begin{pf} Set $j(\nu)=\max(I\backslash a(\nu))-1$. Then we have
\[
\sum_{i\notin a(\nu)}t_i\sim t_{j(\nu)+1}=\prod^{j(\nu)}_{j=0}(t_{j+1}
/t_j).
\]
Hence we have
\begin{equation}
\begin{split}
\prod_\nu\biggl(\sum_{i\notin a(\nu)}t_i\biggr)^{|a(\nu-1)|-|a(\nu)|}
&\sim\prod_\nu\prod^{j(\nu)}_{j=0}(t_{j+1}/t_j)^{|a(\nu-1)|-|a
(\nu)|}\\
&=\prod_{j\ge 0}(t_{j+1}/t_j)^{\sum_{j(\nu)\ge j}(|a(\nu-1)|-|a
(\nu)|)}.
\end{split}
\end{equation}
By definition, we have $a(\nu(j))\supset c(j)$. Hence, $|c(j)|=n-j$
implies (5.4). If $c(j)\notin a$, $a(\nu(j))c(j)$ and hence
we have (5.5).\end{pf}

\begin{quotation}
This is an example of an extract.  The magnetization $M_0$ of the Ising model
is related to the local state probability $P(a):M_0=P(1)-P(-1)$.
The equivalences are shown in Table~\ref{eqtable}.
\end{quotation}


\begin{table}[ht]
\caption{}\label{eqtable}
\renewcommand\arraystretch{1.5}
\noindent\[
\begin{array}{|c|c|c|}
\hline
&{-\infty}&{+\infty}\\
\hline
{f_+(x,k)}&e^{\sqrt{-1}kx}+s_{12}(k)e^{-\sqrt{-1}kx}&s_{11}(k)e^
{\sqrt{-1}kx}\\
\hline
{f_-(x,k)}&s_{22}(k)e^{-\sqrt{-1}kx}&e^{-\sqrt{-1}kx}+s_{21}(k)e^{\sqrt
{-1}kx}\\
\hline
\end{array}
\]
\end{table}


\begin{definition}
This is an example of the definition style.
For $f\in A(X)$, we define
\begin{equation}
\mathcal{Z} (f)=\{E\in Z[X]: \text{$f$ is $E^c$-regular}\}.
\end{equation}
\end{definition}


\begin{remark}
This is an example of the remark style.
For $f\in A(X)$, we define
\begin{equation}
\mathcal{Z} (f)=\{E\in Z[X]: \text{$f$ is $E^c$-regular}\}.
\end{equation}
\end{remark}

\begin{example}
This is an example of the example style.
For $f\in A(X)$, we define
\begin{equation}
\mathcal{Z} (f)=\{E\in Z[X]: \text{$f$ is $E^c$-regular}\}.
\end{equation}
\end{example}

The following is an example of a numbered list.
\begin{enumerate}

\item First item.
 In the case where in $G$
there is a sequence of subgroups
$$
G = G_0, G_1, G_2, \ldots, G_k = e
$$
such that each is an invariant subgroup of $G_i$.


\item Second item.
Its action on an arbitrary element $X = \lambda^\alpha X_\alpha$
has the form
\begin{equation}\label{eq:2.15}
[e^\alpha X_\alpha, X] = e^\alpha \lambda^\beta
[X_\alpha X_\beta] = e^\alpha c^\gamma_{\alpha \beta}
 \lambda^\beta X_\gamma,
\end{equation}

\begin{enumerate}
\item First subitem.
$$
- 2\psi_2(e) =  c_{\alpha \gamma}^\delta c_{\beta \delta}^\gamma
e^\alpha e^\beta.
$$
When the form $\psi_1(e)$ is not zero,
the expression on the right-hand side of this equation can be
written in the form:

\item Second subitem.
\begin{enumerate}
\item First subsubitem.
 In the case where in $G$
there is a sequence of subgroups
$$
G = G_0, G_1, G_2, \ldots, G_k = e
$$
such that each subgroup $G_{i+1}$ is an invariant subgroup
of $G_i$ and each quotient group $G_{i+1}/G_{i}$ is abelian,
 the group $G$ is called {\it solvable}.

\item Second subsubitem.
\end{enumerate}
\item Third subitem.
\end{enumerate}
\item Third item.
\end{enumerate}



\begin{theorem}
This is an example of a theorem.
\end{theorem}

\begin{theorem}[Marcus Theorem]
This is an example of a theorem with the theorem name printed also.
\end{theorem}

%  art work measures 11.5pc for figure 1, 7pc for figure 2
\begin{figure}[tb]
\vskip 5pc
\caption{This is an example of a figure caption.\label{firstfig}}
\end{figure}

\begin{figure}[tb]
\vskip 3pc
\caption{}\label{otherfig}
\end{figure}


\bibliographystyle{amsalpha}
\begin{thebibliography}{BFMO}

\bibitem [A1]{A1} T. Aoki, \textit{Calcul exponentiel des op\'erateurs
microdifferentiels d'ordre infini.} I, Ann. Inst. Fourier (Grenoble)
\textbf{33} (1983), 227--250.

\bibitem [A2]{A2} \bysame, \textit{Calcul exponential des op\'erateurs
microdifferentiels d'ordre infini.} II, Ann. Inst. Fourier
(Grenoble) \textbf{36} (1986), 143--165.

\bibitem [AKK]{AKK}
T. Aoki, M. Kashiwara, and T. Kawai, \textit{On a class of
linear differential operators of infinite order with finite index},
Adv. in Math. \textbf{62} (1986), 155--168.

\end{thebibliography}

\end{document}

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% End of testart.tex
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