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\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

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