% \iffalse %%% ==================================================================== %%% @LaTeX-file{ %%% author = "American Mathematical Society", %%% version = "1.2beta", %%% date = "11-Oct-1994", %%% time = "15:08:49 EDT", %%% filename = "chap1.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)", %%% FAX = "401-331-3842", %%% checksum = "59717 246 882 7528", %%% email = "tech-support@math.ams.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "latex, amslatex, ams-latex, documentation", %%% supported = "yes", %%% abstract = "This file is part of the AMS-\LaTeX{} package. %%% It is part of the monograph sample, %%% testbook.tex (q.v.).", %%% docstring = "The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ==================================================================== % \fi %----------------------------------------------------------------------------- % Beginning of chap1.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. % % This is the data file. It is input by testbook.tex. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \part{This is a Part Title Sample} \chapter[AMS Monograph Series Sample]{AMS Monograph Series Sample\\ With a Second Line to Check Breaks} \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 first 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} \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} This is an example of a nested 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 $G_i$. \item Second subsubitem. \end{enumerate} \item Third subitem. \end{enumerate} \item Third item. \end{enumerate} \begin{theorem} Here is an example of a theorem. \end{theorem} \begin{theorem}[Marcus Theorem] Here 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} \endinput %----------------------------------------------------------------------------- % End of chap1.tex %-----------------------------------------------------------------------------