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Mathematics Preprint (amsart + arXiv)

Journal-neutral math manuscript on amsart with hyperref, cleveref, mathtools and a full theorem setup. Compiles cleanly on arXiv and passes as a first submission at most math journals.

LPPL 1.3c (amscls)

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%% Mathematics preprint (amsart). The layout most journals accept for a first
%% submission and the one arXiv compiles without surprises: amsart + amsthm,
%% hyperref + cleveref, numbered theorems by section, amsplain bibliography.
\documentclass[11pt,reqno]{amsart}

\usepackage[margin=1.1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
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\usepackage{booktabs}
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\usepackage[capitalise,noabbrev]{cleveref}

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\title[Short running title]{A Fixed-Point Theorem for Contractions on Complete Metric Spaces}

\author{First Author}
\address{Department of Mathematics, University Name, City, Country}
\email{first@university.edu}

\author{Second Author}
\address{Institute Name, City, Country}
\email{second@institute.org}

\thanks{The first author was supported by Grant XXXX.}

\subjclass[2020]{Primary 47H10; Secondary 54H25}
\keywords{fixed point, contraction, complete metric space}

\date{\today}

\begin{document}

\begin{abstract}
We give a self-contained proof of the contraction mapping principle together
with an explicit rate of convergence for the Picard iterates, and we show by
example that completeness cannot be dropped.
\end{abstract}

\maketitle

\section{Introduction}\label{sec:intro}

Fixed-point theorems are among the most frequently used tools in analysis;
see \cite{granas2003fixed} for a survey and \cite{rudin1976principles} for
the classical background. Our main result is \cref{thm:banach}, stated with a
quantitative error bound.

\section{Preliminaries}\label{sec:prelim}

\begin{definition}\label{def:contraction}
Let $(X, d)$ be a metric space. A map $T \colon X \to X$ is a
\emph{contraction} if there is $q \in [0, 1)$ with
$d(Tx, Ty) \le q\, d(x, y)$ for all $x, y \in X$.
\end{definition}

\begin{lemma}\label{lem:cauchy}
If $T$ is a contraction with constant $q$ and $x_{n+1} = T x_n$, then
for all $m > n$,
\begin{equation}\label{eq:cauchy}
  d(x_m, x_n) \;\le\; \frac{q^n}{1 - q}\, d(x_1, x_0).
\end{equation}
\end{lemma}

\begin{proof}
By induction $d(x_{k+1}, x_k) \le q^k d(x_1, x_0)$; summing a geometric
series over $k = n, \dots, m-1$ gives \eqref{eq:cauchy}.
\end{proof}

\section{Main result}\label{sec:main}

\begin{theorem}[Contraction mapping principle]\label{thm:banach}
Let $(X, d)$ be a complete metric space and $T \colon X \to X$ a contraction
with constant $q$. Then $T$ has a unique fixed point $x^\ast$, and for every
$x_0 \in X$ the iterates $x_n = T^n x_0$ satisfy
$d(x_n, x^\ast) \le \frac{q^n}{1-q}\, d(x_1, x_0)$.
\end{theorem}

\begin{proof}
By \cref{lem:cauchy} the sequence $(x_n)$ is Cauchy, hence converges to some
$x^\ast$ by completeness. Continuity of $T$ gives $T x^\ast = x^\ast$.
If $y^\ast$ is another fixed point then
$d(x^\ast, y^\ast) = d(Tx^\ast, Ty^\ast) \le q\, d(x^\ast, y^\ast)$, forcing
$x^\ast = y^\ast$. Letting $m \to \infty$ in \eqref{eq:cauchy} gives the rate.
\end{proof}

\begin{example}\label{ex:incomplete}
On $X = (0, 1]$ with the usual metric, $Tx = x/2$ is a contraction without a
fixed point, so completeness cannot be dropped in \cref{thm:banach}.
\end{example}

\Cref{fig:iterates} illustrates the iterates for a one-dimensional
contraction and \cref{tab:rates} compares the bound with the observed error.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=3cm,y=3cm,>=Stealth]
  \draw[->] (0,0) -- (1.1,0) node[right] {$x$};
  \draw[->] (0,0) -- (0,1.1) node[above] {$y$};
  \draw[dashed] (0,0) -- (1,1) node[right] {$y = x$};
  \draw[thick] (0,0.3) -- (1,0.8) node[right] {$y = Tx$};
  \draw[->,blue] (0.9,0) -- (0.9,0.75) -- (0.75,0.75) -- (0.75,0.675) -- (0.675,0.675) -- (0.675,0.64);
  \fill (0.6,0.6) circle (1.2pt) node[below right] {$x^\ast$};
\end{tikzpicture}
\caption{Picard iterates of a contraction converging to the fixed point.}
\label{fig:iterates}
\end{figure}

\begin{table}[ht]
\centering
\begin{tabular}{@{}rcc@{}}
\toprule
$n$ & $d(x_n, x^\ast)$ & bound $\frac{q^n}{1-q} d(x_1,x_0)$ \\
\midrule
 1 & 0.150 & 0.900 \\
 5 & 0.009 & 0.056 \\
10 & $3\cdot10^{-4}$ & $1.8\cdot10^{-3}$ \\
\bottomrule
\end{tabular}
\caption{Observed error against the a priori bound for $q = 1/2$.}
\label{tab:rates}
\end{table}

\section*{Acknowledgements}
The authors thank their colleagues for helpful discussions.

\bibliographystyle{amsplain}
\bibliography{references}

\end{document}

Libraries and docs

amsart.cls60 KBVendored from mirrors.ctan.org on 2026-05-26.

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  \edef#1{\the\toks@}%
}
\AtBeginDocument{%
  \@ifundefined{MakeTextUppercase}{}{\let\uppercasenonmath\altucnm}%
}
\@ifundefined{MakeUppercase}{\let\MakeUppercase\uppercase}{}%
\newcommand{\today}{%
  \relax\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
%%\if@compatibility
%%\else
%%  \def\@obsolete@fontswitch#1#2#3{%
%%    \@latex@warning@no@line{%
%%      Command {\string#1...}\on@line\space is obsolete;\MessageBreak
%%      the LaTeX2e equivalent is \string#3{...}}%
%%    \gdef#1{\@fontswitch\relax#3}%
%%  }
%%  \DeclareRobustCommand*\cal{%
%%    \@xp\@obsolete@fontswitch\csname cal \endcsname\relax\mathcal}
%%  \DeclareRobustCommand*\mit{%
%%    \@xp\@obsolete@fontswitch\csname mit \endcsname\relax\mathnormal}
%%\fi
\renewcommand*{\title}[2][]{\gdef\shorttitle{#1}\gdef\@title{#2}}
\edef\title{\@nx\@dblarg
  \@xp\@nx\csname\string\title\endcsname}
\renewcommand{\author}[2][]{%
  \ifx\@empty\authors
    \gdef\authors{#2}%
  \else
    \g@addto@macro\authors{\and#2}%
    \g@addto@macro\addresses{\author{}}%
  \fi
  \@ifnotempty{#1}{%
    \ifx\@empty\shortauthors
      \gdef\shortauthors{#1}%
    \else
      \g@addto@macro\shortauthors{\and#1}%
    \fi
  }%
}
\edef\author{\@nx\@dblarg
  \@xp\@nx\csname\string\author\endcsname}
\let\shortauthors\@empty   \let\authors\@empty
\newif\ifresetcontrib  \resetcontribfalse
\newcommand\contrib[2][]{%
  \def\@tempa{#1}%
  \ifx\@empty\@tempa
  \else
    \ifresetcontrib \@xcontribs
    \else \global\resetcontribtrue
    \fi
  \fi
  \ifx\@empty\contribs
    \gdef\contribs{#1 #2}%
  \else
    \g@addto@macro\contribs{\and#1 #2}%
  \fi
  \@wraptoccontribs{#1}{#2}%
}
\def\wraptoccontribs#1#2{}
\def\@xcontribs{%
  \author@andify\contribs
  \ifx\@empty\xcontribs
    \xdef\xcontribs{\contribs}%
  \else
    \xdef\xcontribs{\xcontribs, \contribs}%
  \fi
  \let\contribs\@empty
}
\let\contribs\@empty  \let\xcontribs\@empty  \let\toccontribs\@empty
\let\addresses\@empty      \let\thankses\@empty
\newcommand{\address}[2][]{\g@addto@macro\addresses{\address{#1}{#2}}}
\newcommand{\curraddr}[2][]{\g@addto@macro\addresses{\curraddr{#1}{#2}}}
\newcommand{\email}[2][]{\g@addto@macro\addresses{\email{#1}{#2}}}
\newcommand{\urladdr}[2][]{\g@addto@macro\addresses{\urladdr{#1}{#2}}}
\long\def\thanks@warning#1{%
  \ClassError{\@classname}{%
    \protect\thanks\space should be given separately, not inside author name.%
  }\@ehb
}
\renewcommand{\thanks}[1]{%
  \@ifnotempty{#1}{\g@addto@macro\thankses{\thanks{#1}}}%
}
\def\enddoc@text{\ifx\@empty\@translators \else\@settranslators\fi
  \ifx\@empty\addresses \else\@setaddresses\fi}
\AtEndDocument{\enddoc@text}
\def\curraddrname{{\itshape Current address}}
\def\emailaddrname{{\itshape Email address}}
\def\urladdrname{{\itshape URL}}
\def\@setaddresses{\par
  \nobreak \begingroup
\footnotesize
  \def\author##1{\nobreak\addvspace\bigskipamount}%
  \def\\{\unskip, \ignorespaces}%
  \interlinepenalty\@M
  \def\address##1##2{\begingroup
    \par\addvspace\bigskipamount\indent
    \@ifnotempty{##1}{(\ignorespaces##1\unskip) }%
    {\scshape\ignorespaces##2}\par\endgroup}%
  \def\curraddr##1##2{\begingroup
    \@ifnotempty{##2}{\nobreak\indent\curraddrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      ##2\par}\endgroup}%
  \def\email##1##2{\begingroup
    \@ifnotempty{##2}{\nobreak\indent\emailaddrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      \ttfamily##2\par}\endgroup}%
  \def\urladdr##1##2{\begingroup
    \def~{\char`\~}%
    \@ifnotempty{##2}{\nobreak\indent\urladdrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      \ttfamily##2\par}\endgroup}%
  \addresses
  \endgroup
}
\let\@date\@empty
\def\dedicatory#1{\def\@dedicatory{#1}}
\let\@dedicatory=\@empty
\def\keywords#1{\def\@keywords{#1}}
\let\@keywords=\@empty
\newcommand*\subjclass[2][2020]{%
  \def\@subjclass{#2}%
  \@ifundefined{subjclassname@#1}{%
    \ClassWarning{\@classname}{Unknown edition (#1) of Mathematics
      Subject Classification; using '2020'.}%
  }{%
    \@xp\let\@xp\subjclassname\csname subjclassname@#1\endcsname
  }%
}
\let\@subjclass=\@empty
\def\commby#1{\def\@commby{(Communicated by #1)}}
\let\@commby=\@empty
\def\translname{Translated by}
\def\translator#1{%
  \ifx\@empty\@translators \def\@translators{#1}%
  \else\g@addto@macro\@translators{\and#1}\fi}
\let\@translators=\@empty
\def\@settranslators{\par\begingroup
  \addvspace{6\p@\@plus9\p@}%
  \hbox to\columnwidth{\hss\normalfont\normalsize
    \translname{ }%
    \andify\@translators \uppercasenonmath\@translators
    \@translators}
  \endgroup
}
\newcommand{\xandlist}[4]{\@andlista{{#1}{#2}{#3}}#4\and\and}
\def\@andlista#1#2\and#3\and{\@andlistc{#2}\@ifnotempty{#3}{%
  \@andlistb#1{#3}}}
\def\@andlistb#1#2#3#4#5\and{%
  \@ifempty{#5}{%
    \@andlistc{#2#4}%
  }{%
    \@andlistc{#1#4}\@andlistb{#1}{#3}{#3}{#5}%
  }}
\let\@andlistc\@iden
\newcommand{\nxandlist}[4]{%
  \def\@andlistc##1{\toks@\@xp{\the\toks@##1}}%
  \toks@{\toks@\@emptytoks \@andlista{{#1}{#2}{#3}}}%
  \the\@xp\toks@#4\and\and
  \edef#4{\the\toks@}%
  \let\@andlistc\@iden}
\def\@@and{and}
\newcommand{\andify}{%
  \nxandlist{\unskip, }{\unskip{} \@@and~}{\unskip, \@@and~}}
\def\and{\unskip{ }\@@and{ }\ignorespaces}
\def\maketitle{\par
  \@topnum\z@ % this prevents figures from falling at the top of page 1
  \@setcopyright
  \thispagestyle{firstpage}% this sets first page specifications
  \uppercasenonmath\shorttitle
  \ifx\@empty\shortauthors \let\shortauthors\shorttitle
  \else \andify\shortauthors
  \fi
  \@maketitle@hook
  \begingroup
  \@maketitle
  \toks@\@xp{\shortauthors}\@temptokena\@xp{\shorttitle}%
  \toks4{\def\\{ \ignorespaces}}% defend against questionable usage
  \edef\@tempa{%
    \@nx\markboth{\the\toks4
      \@nx\MakeUppercase{\the\toks@}}{\the\@temptokena}}%
  \@tempa
  \endgroup
  \c@footnote\z@
  \@cleartopmattertags
}
\def\@cleartopmattertags{%
  \def\do##1{\let##1\relax}%
  \do\maketitle \do\@maketitle \do\title \do\@xtitle \do\@title
  \do\author \do\@xauthor \do\address \do\@xaddress
  \do\contrib \do\contribs \do\xcontribs \do\toccontribs
  \do\email \do\@xemail \do\curraddr \do\@xcurraddr
  \do\commby \do\@commby
  \do\dedicatory \do\@dedicatory \do\thanks \do\thankses
  \do\keywords \do\@keywords \do\subjclass \do\@subjclass
}
\def\@maketitle@hook{\global\let\@maketitle@hook\@empty}
\def\@maketitle{%
  \normalfont\normalsize
  \@adminfootnotes
  \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}%
  \global\topskip42\p@\relax % 5.5pc   "   "   "     "     "
  \@settitle
  \ifx\@empty\authors \else \@setauthors \fi
  \ifx\@empty\@dedicatory
  \else
    \baselineskip18\p@
    \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}%
      \global\dimen@i\prevdepth}\prevdepth\dimen@i
  \fi
  \@setabstract
  \normalsize
  \if@titlepage
    \newpage
  \else
    \dimen@34\p@ \advance\dimen@-\baselineskip
    \vskip\dimen@\relax
  \fi
} % end \@maketitle
\def\@adminfootnotes{%
  \let\@makefnmark\relax  \let\@thefnmark\relax
  \ifx\@empty\@date\else \@footnotetext{\@setdate}\fi
  \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi
  \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi
  \ifx\@empty\thankses\else \@footnotetext{%
    \def\par{\let\par\@par}\@setthanks}%
  \fi
}
\AtBeginDocument{%
  \@ifundefined{publname}{%
    \let\publname\@empty
    \let\@serieslogo\@empty
  }{%
    \def\@serieslogo{\article@logo}%
  }%
}
\AtBeginDocument{%
  \@ifundefined{volinfo}{%
    \def\volinfo{%
      Volume \currentvolume, Number \number0\currentissue
      \if\@printyear , \currentmonth\ \currentyear\fi
    }%
  }{}%
}
\def\@printyear{TF}% boolean false
\def\issueinfo#1#2#3#4{\def\currentvolume{#1}\def\currentissue{#2}%
  \def\currentmonth{#3}\def\currentyear{#4}}
\issueinfo{00}% volume number
  {0}%        % issue number
  {Xxxx}%     % month
  {XXXX}%     % year
\newcommand{\copyrightinfo}[2]{%
  \def\copyrightyear{#1}%
  \@ifnotempty{#2}{\def\copyrightholder{#2}}%
}
\copyrightinfo{0000}{(copyright holder)}
\def\pagespan#1#2{\setcounter{page}{#1}%
  \ifnum\c@page<\z@ \pagenumbering{roman}\setcounter{page}{-#1}\fi
  \def\start@page{#1}\def\end@page{#2}}
\pagespan{000}{000}
\AtBeginDocument{%
  \@ifundefined{pageinfo}{%
    \def\pageinfo{%
      \ifnum\start@page=\z@
        Pages 000--000
      \else
        \ifx\start@page\end@page
          Page \start@page
        \else
          Pages \start@page--\end@page
        \fi
      \fi}%
  }{}%
}
\@ifundefined{ISSN}{\def\ISSN{0000-0000}}{}
\newcommand\PII[1]{\def\@PII{#1}}
\PII{S \ISSN(XX)0000-0}
\newinsert\copyins
\skip\copyins=1.5pc
\count\copyins=1000 % magnification factor, 1000 = 100%
\dimen\copyins=.5\textheight % maximum allowed per page
\g@addto@macro\@reinserts{%
  \ifvoid\copyins\else\insert\copyins{\unvbox\copyins}\fi
}
\def\@copyinsfontsize{\fontsize{6}{7\p@}\normalfont\upshape}
\newif\if@extracrline  \@extracrlinefalse
\let\@extracrline\@empty
\relax
\def\@setcopyright{%
  \ifx\@empty\@serieslogo
  \else\ifx\@empty\copyrightyear
  \else
    \insert\copyins{\hsize\textwidth
      \parfillskip\z@\relax
      \leftskip\z@\@plus.9\textwidth\relax \rightskip\z@\relax
      \@copyinsfontsize
      \everypar{}%
      \vskip-\skip\copyins
      \if@extracrline
        \vskip-6pt
      \fi
      \nointerlineskip
      \leavevmode\hfill\vrule\@width\z@\@height\skip\copyins
      \copyright\copyrightyear\ \copyrightholder\ignorespaces
      \if@extracrline \@extracrline \fi
      \par
      \kern\z@}%
  \fi\fi
}
\def\@combinefloats{%
  \ifx \@toplist\@empty \else \@cflt \fi
  \ifx \@botlist\@empty \else \@cflb \fi
  \ifvoid\copyins \else \@cflci \fi
}
\def\@cflci{%
  \setbox\@outputbox\vbox{%
    \unvbox\@outputbox
    \vskip\skip\copyins
    \if@twocolumn \else \vskip\z@ plus\p@ \fi
    \hbox to\columnwidth{%
      \hss\vbox to\z@{\vss
        \if@twocolumn
          \if@firstcolumn \else \unvbox\copyins \fi
        \else
          \unvbox\copyins
        \fi
        }}}%
  \if@twocolumn \if@firstcolumn
    \insert\copyins{\unvbox\copyins}%
  \fi\fi
  \global\count\copyins=999 \relax
}
\newif\if@revertcopyright  \@revertcopyrightfalse
\newcommand{\revertcopyright}{%
  \global\@revertcopyrighttrue
  \global\@extracrlinetrue}
\def\@revertcrfontsize{\fontsize{6}{7\p@}\normalfont\upshape}
\def\@extracrline{%
  \if@revertcopyright
    \unskip\\
    \@revertcrfontsize
    Reverts to public domain 28 years from publication
  \fi
}
\newcommand{\abstractname}{Abstract}
\newcommand{\keywordsname}{Key words and phrases}
\@namedef{subjclassname@1991}{%
  \textup{1991} Mathematics Subject Classification}
\@namedef{subjclassname@2000}{%
  \textup{2000} Mathematics Subject Classification}
\@namedef{subjclassname@2010}{%
  \textup{2010} Mathematics Subject Classification}
\@namedef{subjclassname@2020}{%
  \textup{2020} Mathematics Subject Classification}
\@xp\let\@xp\subjclassname\csname subjclassname@2020\endcsname
\def\@tempb{amsart}
\ifx\@classname\@tempb
  \newcommand{\datename}{\textit{Date}:}
\else
  \newcommand{\datename}{Received by the editors}
\fi
\def\@settitle{\begin{center}%
  \baselineskip14\p@\relax
    \bfseries
\uppercasenonmath\@title
  \@title
  \end{center}%
}
\def\author@andify{%
  \nxandlist {\unskip ,\penalty-1 \space\ignorespaces}%
    {\unskip {} \@@and~}%
    {\unskip ,\penalty-2 \space \@@and~}%
}
\def\@setauthors{%
  \begingroup
  \def\thanks{\protect\thanks@warning}%
  \trivlist
  \centering\footnotesize \@topsep30\p@\relax
  \advance\@topsep by -\baselineskip
  \item\relax
  \author@andify\authors
  \def\\{\protect\linebreak}%
  \MakeUppercase{\authors}%
  \ifx\@empty\contribs
  \else
    ,\penalty-3 \space \@setcontribs
    \@closetoccontribs
  \fi
  \endtrivlist
  \endgroup
}
\def\@closetoccontribs{}
\def\@setcontribs{%
  \@xcontribs
  \MakeUppercase{\xcontribs}%
}
\def\@setdate{\datename\ \@date\@addpunct.}
\def\@setsubjclass{%
  {\itshape\subjclassname.}\enspace\@subjclass\@addpunct.}
\def\@setkeywords{%
  {\itshape \keywordsname.}\enspace \@keywords\@addpunct.}
\def\@setthanks{\def\thanks##1{\par##1\@addpunct.}\thankses}
\newbox\abstractbox
\newenvironment{abstract}{%
  \ifx\maketitle\relax
    \ClassWarning{\@classname}{Abstract should precede
      \protect\maketitle\space in AMS document classes; reported}%
  \fi
  \global\setbox\abstractbox=\vtop \bgroup
    \normalfont\Small
    \list{}{\labelwidth\z@
      \leftmargin3pc \rightmargin\leftmargin
      \listparindent\normalparindent \itemindent\z@
      \parsep\z@ \@plus\p@
      \let\fullwidthdisplay\relax
    }%
    \item[\hskip\labelsep\scshape\abstractname.]%
}{%
  \endlist\egroup
  \ifx\@setabstract\relax \@setabstracta \fi
}
\def\@setabstract{\@setabstracta \global\let\@setabstract\relax}
\def\@setabstracta{%
  \ifvoid\abstractbox
  \else
    \skip@20\p@ \advance\skip@-\lastskip
    \advance\skip@-\baselineskip \vskip\skip@
    \box\abstractbox
    \prevdepth\z@ % because \abstractbox is a vtop
  \fi
}
\def\titlepage{%
  \clearpage
  \thispagestyle{empty}\setcounter{page}{0}}
\def\endtitlepage{\newpage}
\def\labelenumi{(\theenumi)}
\def\theenumi{\@arabic\c@enumi}
\def\labelenumii{(\theenumii)}
\def\theenumii{\@alph\c@enumii}
\def\p@enumii{\theenumi}
\def\labelenumiii{(\theenumiii)}
\def\theenumiii{\@roman\c@enumiii}
\def\p@enumiii{\theenumi(\theenumii)}
\def\labelenumiv{(\theenumiv)}
\def\theenumiv{\@Alph\c@enumiv}
\def\p@enumiv{\p@enumiii\theenumiii}
\def\labelitemi{$\m@th\bullet$}
\def\labelitemii{\bfseries --}% \upshape already done by \itemize
\def\labelitemiii{$\m@th\ast$}
\def\labelitemiv{$\m@th\cdot$}
\newenvironment{verse}{\let\\\@centercr
  \list{}{\itemsep\z@ \itemindent -1.5em\listparindent\itemindent
  \rightmargin\leftmargin \advance\leftmargin 1.5em}\item[]%
}{%
  \endlist
}
\let\endverse=\endlist % for efficiency
\newenvironment{quotation}{\list{}{%
    \leftmargin3pc \listparindent\normalparindent
    \itemindent\z@
    \rightmargin\leftmargin \parsep\z@ \@plus\p@}%
  \item[]%
}{%
  \endlist
}
\let\endquotation=\endlist % for efficiency
\newenvironment{quote}{%
  \list{}{\rightmargin\leftmargin}\item[]%
}{%
  \endlist
}
\let\endquote=\endlist % for efficiency
\def\trivlist{\parsep\parskip\@nmbrlistfalse
  \@trivlist \labelwidth\z@ \leftmargin\z@
  \itemindent\z@
  \let\@itemlabel\@empty
  \def\makelabel##1{\upshape##1}}
\renewenvironment{enumerate}{%
  \ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{\hss\llap{\upshape##1}}}\fi
}{%
  \endlist
}
\let\endenumerate=\endlist % for efficiency
\renewenvironment{itemize}{%
  \ifnum\@itemdepth>3 \@toodeep
  \else \advance\@itemdepth\@ne
    \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
    \list{\csname\@itemitem\endcsname}%
      {\def\makelabel##1{\hss\llap{\upshape##1}}}%
  \fi
}{%
  \endlist
}
\let\enditemize=\endlist % for efficiency
\newcommand{\descriptionlabel}[1]{\hspace\labelsep \upshape\bfseries #1:}
\newenvironment{description}{\list{}{%
  \advance\leftmargini6\p@ \itemindent-12\p@
  \labelwidth\z@ \let\makelabel\descriptionlabel}%
}{
  \endlist
}
\let\enddescription=\endlist % for efficiency
\let\upn=\textup
\AtBeginDocument{%
  \labelsep=5pt\relax
  \setcounter{enumi}{13}\setcounter{enumii}{13}%
  \setcounter{enumiii}{13}\setcounter{enumiv}{13}%
  \settowidth\leftmargini{\labelenumi\hskip\labelsep}%
  \advance\leftmargini by \normalparindent
  \settowidth\leftmarginii{\labelenumii\hskip\labelsep}%
  \settowidth\leftmarginiii{\labelenumiii\hskip\labelsep}%
  \settowidth\leftmarginiv{\labelenumiv\hskip\labelsep}%
  \setcounter{enumi}{0}\setcounter{enumii}{0}%
  \setcounter{enumiii}{0}\setcounter{enumiv}{0}%
  \leftmarginv=10pt  \leftmarginvi=\leftmarginv
  \leftmargin=\leftmargini
  \labelwidth=\leftmargini \advance\labelwidth-\labelsep
  \@listi}
\newskip\listisep
\listisep\smallskipamount
\def\@listI{\leftmargin\leftmargini \parsep\z@skip
  \topsep\listisep \itemsep\z@skip
  \listparindent\normalparindent}
\let\@listi\@listI
\def\@listii{\leftmargin\leftmarginii
  \labelwidth\leftmarginii \advance\labelwidth-\labelsep
  \topsep\z@skip \parsep\z@skip \partopsep\z@skip \itemsep\z@skip}
\def\@listiii{\leftmargin\leftmarginiii
  \labelwidth\leftmarginiii \advance\labelwidth-\labelsep}
\def\@listiv{\leftmargin\leftmarginiv
  \labelwidth\leftmarginiv \advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
  \labelwidth\leftmarginv \advance\labelwidth-\labelsep}
\def\@listvi{\leftmargin\leftmarginvi
  \labelwidth\leftmarginvi \advance\labelwidth-\labelsep}
\@ifclasswith{\@classname}{fleqn}{%
  \let\@tempa\@iden
  \AtBeginDocument{\mathindent\leftmargini}%
}{\let\@tempa\@gobble}%
\@ifpackageloaded{amsmath}{\let\@tempa\@gobble}{%
  \@ifpackageloaded{amstex}{\let\@tempa\@gobble}{}%
}
\@tempa{%
  \def\[{\relax
    \ifmmode\@badmath
    \else
      \begin{trivlist}%
      \@beginparpenalty\predisplaypenalty
      \@endparpenalty\postdisplaypenalty
      \item[]\leavevmode
      \hbox to\linewidth\bgroup$\displaystyle
      \hskip\mathindent\bgroup
    \fi}%
  \def\]{\relax
    \ifmmode
      \egroup \m@th$\hfil \egroup
      \end{trivlist}%
    \else \@badmath
    \fi}%
  \renewenvironment{equation}{%
    \@beginparpenalty\predisplaypenalty
    \@endparpenalty\postdisplaypenalty
    \refstepcounter{equation}%
    \@topsep\abovedisplayskip \trivlist
    \item[]\leavevmode
    \hbox to\linewidth\bgroup\hskip\mathindent$\displaystyle
  }{%
    \m@th$\hfil \displaywidth\linewidth \hbox{\@eqnnum}\egroup
    \endtrivlist
  }%
  \renewenvironment{eqnarray}{%
    \stepcounter{equation}\let\@currentlabel\theequation
    \global\@eqnswtrue \global\@eqcnt\z@ \tabskip\mathindent
    \let\\=\@eqncr \abovedisplayskip\topsep
    \ifvmode \advance\abovedisplayskip\partopsep \fi
    \belowdisplayskip\abovedisplayskip
    \belowdisplayshortskip\abovedisplayskip
    \abovedisplayshortskip\abovedisplayskip
    $$\everycr{}\halign to\linewidth\bgroup
      \hskip\@centering
      $\displaystyle\tabskip\z@skip####\m@th$&%
      \@eqnsel \global\@eqcnt\@ne
      \hfil${}####{}\m@th$\hfil&%
      \global\@eqcnt\tw@
      $\displaystyle ####\m@th$\hfil\tabskip\@centering&%
      \global\@eqcnt\thr@@
      \hbox to \z@\bgroup\hss####\egroup\tabskip\z@skip\cr
    }{%
      \@@eqncr \egroup \global\advance\c@equation\m@ne$$%
      \global\@ignoretrue
    }%
  \newdimen\mathindent
  \mathindent\leftmargini
}
\def\@startsection#1#2#3#4#5#6{%
 \if@noskipsec \leavevmode \fi
 \par \@tempskipa #4\relax
 \@afterindenttrue
 \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi
 \if@nobreak \everypar{}\else
     \addpenalty\@secpenalty\addvspace\@tempskipa\fi
 \@ifstar{\@dblarg{\@sect{#1}{\@m}{#3}{#4}{#5}{#6}}}%
         {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}%
}
\def\@seccntformat#1{%
  \protect\textup{\protect\@secnumfont
    \csname the#1\endcsname
    \protect\@secnumpunct
  }%
}
\def\@secnumfont{\mdseries}
\def\@sect#1#2#3#4#5#6[#7]#8{%
  \edef\@toclevel{\ifnum#2=\@m 0\else\number#2\fi}%
  \ifnum #2>\c@secnumdepth \let\@secnumber\@empty
  \else \@xp\let\@xp\@secnumber\csname the#1\endcsname\fi
  \@tempskipa #5\relax
  \ifnum #2>\c@secnumdepth
    \let\@svsec\@empty
  \else
    \refstepcounter{#1}%
    \edef\@secnumpunct{%
      \ifdim\@tempskipa>\z@ % not a run-in section heading
        \@ifnotempty{#8}{.\@nx\enspace}%
      \else
        \@ifempty{#8}{.}{.\@nx\enspace}%
      \fi
    }%
    \@ifempty{#8}{%
      \ifnum #2=\tw@ \def\@secnumfont{\bfseries}\fi}{}%
    \protected@edef\@svsec{%
      \ifnum#2<\@m
        \@ifundefined{#1name}{}{%
          \ignorespaces\csname #1name\endcsname\space
        }%
      \fi
      \@seccntformat{#1}%
    }%
  \fi
  \ifdim \@tempskipa>\z@ % then this is not a run-in section heading
    \begingroup #6\relax
    \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}%
    \endgroup
    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
  \else
  \def\@svsechd{#6\hskip #3\@svsec
    \@ifnotempty{#8}{\ignorespaces#8\unskip
       \@addpunct.}%
    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
  }%
  \fi
  \global\@nobreaktrue
  \@xsect{#5}}
\let\@ssect\relax
\newcounter{part}
\newcounter{section}
\newcounter{subsection}[section]
\newcounter{subsubsection}[subsection]
\newcounter{paragraph}[subsubsection]
\newcounter{subparagraph}[paragraph]
\renewcommand\thepart          {\arabic{part}}
\renewcommand\thesection       {\arabic{section}}
\renewcommand\thesubsection    {\thesection.\arabic{subsection}}
\renewcommand\thesubsubsection {\thesubsection .\arabic{subsubsection}}
\renewcommand\theparagraph     {\thesubsubsection.\arabic{paragraph}}
\renewcommand\thesubparagraph  {\theparagraph.\arabic{subparagraph}}
\setcounter{secnumdepth}{3}
\def\partname{Part}
\def\part{\@startsection{part}{0}%
  \z@{\linespacing\@plus\linespacing}{.5\linespacing}%
  {\normalfont\bfseries\raggedright}}
\def\specialsection{\@startsection{section}{1}%
  \z@{\linespacing\@plus\linespacing}{.5\linespacing}%
  {\normalfont\centering}}
\def\section{\@startsection{section}{1}%
  \z@{.7\linespacing\@plus\linespacing}{.5\linespacing}%
  {\normalfont\scshape\centering}}
\def\subsection{\@startsection{subsection}{2}%
  \z@{.5\linespacing\@plus.7\linespacing}{-.5em}%
  {\normalfont\bfseries}}
\def\subsubsection{\@startsection{subsubsection}{3}%
  \z@{.5\linespacing\@plus.7\linespacing}{-.5em}%
  {\normalfont\itshape}}
\def\paragraph{\@startsection{paragraph}{4}%
  \z@\z@{-\fontdimen2\font}%
  \normalfont}
\def\subparagraph{\@startsection{subparagraph}{5}%
  \z@\z@{-\fontdimen2\font}%
  \normalfont}
\def\appendix{\par\c@section\z@ \c@subsection\z@
   \let\sectionname\appendixname
   \def\thesection{\@Alph\c@section}}
\def\appendixname{Appendix}
\def\@Roman#1{\@xp\@slowromancap
  \romannumeral#1@}%
\def\@slowromancap#1{\ifx @#1% then terminate
  \else
    \if i#1I\else\if v#1V\else\if x#1X\else\if l#1L\else\if
    c#1C\else\if m#1M\else#1\fi\fi\fi\fi\fi\fi
    \@xp\@slowromancap
  \fi
}
\newcommand{\@pnumwidth}{1.6em}
\newcommand{\@tocrmarg}{2.6em}
\setcounter{tocdepth}{2}
\newswitch{toc}
\newswitch{lof}
\newswitch{lot}
\newcommand\contentsnamefont{\scshape}
\def\@starttoc#1#2{\begingroup
  \setTrue{#1}%
  \par\removelastskip\vskip\z@skip
  \@startsection{}\@M\z@{\linespacing\@plus\linespacing}%
    {.5\linespacing}{\centering\contentsnamefont}{#2}%
  \ifx\contentsname#2%
  \else \addcontentsline{toc}{section}{#2}\fi
  \makeatletter
  \@input{\jobname.#1}%
  \if@filesw
    \@xp\newwrite\csname tf@#1\endcsname
    \immediate\@xp\openout\csname tf@#1\endcsname \jobname.#1\relax
  \fi
  \global\@nobreakfalse \endgroup
  \addvspace{32\p@\@plus14\p@}%
  \let\tableofcontents\relax
}
\def\contentsname{Contents}
\def\listfigurename{List of Figures}
\def\listtablename{List of Tables}
\def\tableofcontents{%
  \@starttoc{toc}\contentsname
}
\def\listoffigures{\@starttoc{lof}\listfigurename}
\def\listoftables{\@starttoc{lot}\listtablename}
\AtBeginDocument{%
  \@for\@tempa:=-1,0,1,2,3\do{%
    \@ifundefined{r@tocindent\@tempa}{%
      \@xp\gdef\csname r@tocindent\@tempa\endcsname{0pt}}{}%
  }%
}
\def\@writetocindents{%
  \begingroup
  \@for\@tempa:=-1,0,1,2,3\do{%
    \immediate\write\@auxout{%
      \string\newlabel{tocindent\@tempa}{%
        \csname r@tocindent\@tempa\endcsname}}%
  }%
  \endgroup}
\AtEndDocument{\@writetocindents}

\let\indentlabel\@empty
\def\@tochangmeasure#1{\sbox\z@{#1}%
  \ifdim\wd\z@>\csname r@tocindent\@toclevel\endcsname\relax
    \@xp\xdef\csname r@tocindent\@toclevel\endcsname{\the\wd\z@}%
  \fi
}
\def\@toclevel{0}
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % then omit
  \else
    \par \addpenalty\@secpenalty\addvspace{#2}%
    \begingroup \hyphenpenalty\@M
    \@ifempty{#4}{%
      \@tempdima\csname r@tocindent\number#1\endcsname\relax
    }{%
      \@tempdima#4\relax
    }%
    \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax
    \rightskip\@pnumwidth plus4em \parfillskip-\@pnumwidth
    #5\leavevmode\hskip-\@tempdima #6\nobreak\relax
    \hfil\hbox to\@pnumwidth{\@tocpagenum{#7}}\par
    \nobreak
    \endgroup
  \fi}
\def\@tocpagenum#1{\hss{\mdseries #1}}
\def\@tocwrite#1{\@xp\@tocwriteb\csname toc#1\endcsname{#1}}
\def\@tocwriteb#1#2#3{%
  \begingroup
    \def\@tocline##1##2##3##4##5##6{%
      \ifnum##1>\c@tocdepth
      \else \sbox\z@{##5\let\indentlabel\@tochangmeasure##6}\fi}%
    \csname l@#2\endcsname{#1{\csname#2name\endcsname}{\@secnumber}{}}%
  \endgroup
  \addcontentsline{toc}{#2}%
    {\protect#1{\csname#2name\endcsname}{\@secnumber}{#3}}}
\def\l@section{\@tocline{1}{0pt}{1pc}{}{}}
\newcommand{\tocsection}[3]{%
  \indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3}
\def\l@subsection{\@tocline{2}{0pt}{1pc}{5pc}{}}
\let\tocsubsection\tocsection
\def\l@subsubsection{\@tocline{3}{0pt}{1pc}{7pc}{}}
\let\tocsubsubsection\tocsection
\let\l@paragraph\l@subsubsection
\let\tocparagraph\tocsection
\let\l@subparagraph\l@subsubsection
\let\tocsubparagraph\tocsection
\def\l@part{\@tocline{-1}{12pt plus2pt}{0pt}{}{\bfseries}}
\let\tocpart\tocsection
\def\l@chapter{\@tocline{0}{8pt plus1pt}{0pt}{}{}}
\let\tocchapter\tocsection
\newcommand{\tocappendix}[3]{%
  \indentlabel{#1\@ifnotem

amsbook.cls60 KBVendored from mirrors.ctan.org on 2026-05-26.

%%
%% This is file `amsbook.cls',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% amsclass.dtx  (with options: `amsbook,classes')
%% This is a generated file.
%% 
%% Copyright 1995, 1999, 2004, 2009-2020 American Mathematical Society.
%% 
%% American Mathematical Society
%% Technical Support
%% Publications Technical Group
%% 201 Charles Street
%% Providence, RI 02904
%% USA
%% tel: (401) 455-4080
%%      (800) 321-4267 (USA and Canada only)
%% fax: (401) 331-3842
%% email: tech-support@ams.org
%% 
%% This work may be distributed and/or modified under the
%% conditions of the LaTeX Project Public License, either version 1.3c
%% of this license or (at your option) any later version.
%% The latest version of this license is in
%%   http://www.latex-project.org/lppl.txt
%% and version 1.3c or later is part of all distributions of LaTeX
%% version 2005/12/01 or later.
%% 
%% This work has the LPPL maintenance status `maintained'.
%% 
%% The Current Maintainer of this work is the American Mathematical
%% Society.
%% 
%% ========================================================================
%% 
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1995/06/01]% LaTeX date must be June 1995 or later
\ProvidesClass{amsbook}[2020/05/29 v2.20.6]
\global\expandafter\let\csname ver@amsthm.sty\expandafter\endcsname
  \csname ver@\@currname.\@currext\endcsname
\let\@xp=\expandafter
\let\@nx=\noexpand
\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
\long\def\@xifempty#1#2@#3#4#5\@nil{%
  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
\long\def\@ifnotempty#1{\@ifempty{#1}{}}
\def\setboxz@h{\setbox\z@\hbox}
\def\@addpunct#1{%
  \relax\ifhmode
    \ifnum\spacefactor>\@m \else#1\fi
  \fi}
\def\nopunct{\spacefactor 1007 }
\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
\def\@tempa#1#2\@nil{\edef\@classname{#1}}
\expandafter\@tempa\@currnamestack{}{}{}\@nil
\ifx\@classname\@empty \edef\@classname{\@currname}\fi
\def\@True{00}
\def\@False{01}
\newcommand\newswitch[2][False]{%
  \expandafter\@ifdefinable\csname ?@#2\endcsname{%
    \global\expandafter\let\csname ?@#2\expandafter\endcsname
      \csname @#1\endcsname
  }%
}
\newcommand{\setFalse}[1]{%
  \expandafter\let\csname ?@#1\endcsname\@False
}
\newcommand{\setTrue}[1]{%
  \expandafter\let\csname ?@#1\endcsname\@True
}
\newswitch{}
\DeclareRobustCommand{\except}[1]{%
  \if\csname ?@#1\endcsname \expandafter\@gobble
  \else \expandafter\@firstofone
  \fi
}
\DeclareRobustCommand{\for}[1]{%
  \if\csname ?@#1\endcsname \expandafter\@firstofone
  \else \expandafter\@gobble
  \fi
}
\DeclareRobustCommand{\forany}[1]{%
  \csname for@any@01\endcsname#1,?,\@nil
}
\@namedef{for@any@\@False}#1,{%
  \csname for@any@%
    \csname ?@\zap@space#1 \@empty\endcsname
  \endcsname
}
\@namedef{?@?}{x}
\@namedef{for@any@\@True}#1\@nil#2{#2}
\def\for@any@x{\@car\@gobble}
\DeclareOption{a4paper}{\paperheight 297mm\paperwidth 210mm
  \textheight 54.5pc }
\DeclareOption{letterpaper}{\paperheight 11in\paperwidth 8.5in }
\DeclareOption{landscape}{\@tempdima\paperheight
  \paperheight\paperwidth \paperwidth\@tempdima}
\DeclareOption{portrait}{}
\DeclareOption{oneside}{\@twosidefalse \@mparswitchfalse}
\DeclareOption{twoside}{\@twosidetrue  \@mparswitchtrue}
\DeclareOption{draft}{\overfullrule5\p@
  \ClassWarningNoLine{\@classname}{%
    When the draft option is used, the
      \protect\includegraphics\MessageBreak
    command will print blank placeholder boxes\MessageBreak
    for the graphics}%
}
\DeclareOption{final}{\overfullrule\z@ }
\def\dateposted#1{\def\@dateposted{#1}}%
\let\@dateposted\@empty
\newif\if@titlepage
\DeclareOption{titlepage}{\@titlepagetrue}
\DeclareOption{notitlepage}{\@titlepagefalse}
\newif\if@openright
\DeclareOption{openright}{\@openrighttrue}
\DeclareOption{openany}{\@openrightfalse}
\@openrighttrue
\DeclareOption{onecolumn}{\@twocolumnfalse}
\DeclareOption{twocolumn}{\@twocolumntrue}
\DeclareOption{nomath}{}
\DeclareOption{noamsfonts}{}
\DeclareOption{psamsfonts}{%
  \PassOptionsToPackage{psamsfonts}{amsfonts}%
  \PassOptionsToPackage{cmex10}{amsmath}}
\newif\iftagsleft@
\DeclareOption{leqno}{%
  \tagsleft@true \PassOptionsToPackage{leqno}{amsmath}}
\DeclareOption{reqno}{%
  \tagsleft@false \PassOptionsToPackage{reqno}{amsmath}}
\newif\ifctagsplit@
\DeclareOption{centertags}{%
  \ctagsplit@true \PassOptionsToPackage{centertags}{amsmath}}
\DeclareOption{tbtags}{%
  \ctagsplit@false \PassOptionsToPackage{tbtags}{amsmath}}
\DeclareOption{fleqn}{}%
\newcommand{\@mainsize}{10}
\newcommand{\@ptsize}{0}
\newcommand{\larger}[1][1]{%
  \count@\@currsizeindex \advance\count@#1\relax
  \ifnum\count@<\z@ \count@\z@ \else\ifnum\count@>12 \count@12 \fi\fi
  \ifcase\count@
    \Tiny\or\Tiny\or\tiny\or\SMALL\or\Small\or\small
    \or\normalsize
    \or\large\or\Large\or\LARGE\or\huge\or\Huge\else\Huge
  \fi
}
\newcommand{\smaller}[1][1]{\larger[-#1]}
\def\@adjustvertspacing{%
  \bigskipamount.7\baselineskip plus.7\baselineskip
  \medskipamount\bigskipamount \divide\medskipamount\tw@
  \smallskipamount\medskipamount \divide\smallskipamount\tw@
  \abovedisplayskip\medskipamount
  \belowdisplayskip \abovedisplayskip
  \abovedisplayshortskip\abovedisplayskip
  \advance\abovedisplayshortskip-1\abovedisplayskip
  \belowdisplayshortskip\abovedisplayshortskip
  \advance\belowdisplayshortskip 1\smallskipamount
  \jot\baselineskip \divide\jot 4 \relax
}
\renewcommand\normalsize{\@xsetfontsize\normalsize 6%
  \@adjustvertspacing \let\@listi\@listI}
\DeclareRobustCommand{\Tiny}{\@xsetfontsize\Tiny 1}
\DeclareRobustCommand{\tiny}{\@xsetfontsize\tiny 2}
\DeclareRobustCommand{\SMALL}{\@xsetfontsize\SMALL 3}
\DeclareRobustCommand{\Small}{\@xsetfontsize\Small 4%
  \@adjustvertspacing
  \def\@listi{\topsep\smallskipamount \parsep\z@skip \itemsep\z@skip
    \leftmargin=\leftmargini
    \labelwidth=\leftmargini \advance\labelwidth-\labelsep
  }%
}
\DeclareRobustCommand{\small}{\@xsetfontsize\small 5\@adjustvertspacing}
\def\footnotesize{\Small}
\def\scriptsize{\SMALL}
\DeclareRobustCommand{\large}{\@xsetfontsize\large 7\@adjustvertspacing}
\DeclareRobustCommand{\Large}{\@xsetfontsize\Large 8\@adjustvertspacing}
\DeclareRobustCommand{\LARGE}{\@xsetfontsize\LARGE 9}
\DeclareRobustCommand{\huge}{\@xsetfontsize\huge{10}}
\DeclareRobustCommand{\Huge}{\@xsetfontsize\Huge{11}}
\def\@xsetfontsize#1#2{%
  \chardef\@currsizeindex#2\relax
  \edef\@tempa{\@nx\@setfontsize\@nx#1%
    \@xp\ifcase\@xp\@currsizeindex\@typesizes
      \else{99}{99}\fi}%
  \@tempa
}
\chardef\@currsizeindex=6
\widowpenalty=10000
\clubpenalty=10000
\brokenpenalty=10000
\newdimen\linespacing
\lineskip=1pt \lineskiplimit=1pt
\normallineskip=1pt \normallineskiplimit=1pt
\let\baselinestretch=\@empty
\headheight=8pt \headsep=14pt
\footskip=18pt
\textheight=50.5pc \topskip=10pt
\textwidth=30pc
\columnsep=10pt \columnseprule=0pt
\marginparwidth=90pt
\marginparsep=11pt
\marginparpush=5pt
\AtBeginDocument{\settoheight{\footnotesep}{\footnotesize M$^1$}}
\skip\footins=7pt plus11pt
\skip\@mpfootins=\skip\footins
\fboxsep=3pt \fboxrule=.4pt
\arrayrulewidth=.4pt    \doublerulesep=2pt
\labelsep=5pt           \arraycolsep=\labelsep
\tabcolsep=\labelsep    \tabbingsep=\labelsep
\floatsep=15pt plus 12pt    \dblfloatsep=15pt plus 12pt
\textfloatsep=\floatsep  \dbltextfloatsep=15pt plus 12pt
\intextsep=\floatsep
\@fptop=0pt plus1fil     \@dblfptop=0pt plus1fil
\@fpbot=0pt plus1fil     \@dblfpbot=0pt plus1fil
\@fpsep=8pt plus2fil     \@dblfpsep=8pt plus2fil\relax
\parskip=0pt \relax
\newdimen\normalparindent
\normalparindent=18pt
\parindent=\normalparindent
\partopsep=0pt \relax \parsep=0pt \relax \itemsep=0pt \relax
\@lowpenalty=51         \@medpenalty=151        \@highpenalty=301
\@beginparpenalty=-\@lowpenalty
\@endparpenalty=-\@lowpenalty
\@itempenalty=-\@lowpenalty
\DeclareOption{10pt}{\def\@mainsize{10}\def\@ptsize{0}%
  \def\@typesizes{%
    \or{5}{6}\or{6}{7}\or{7}{8}\or{8}{10}\or{9}{11}%
    \or{10}{12}% normalsize
    \or{\@xipt}{13}\or{\@xiipt}{14}\or{\@xivpt}{17}%
    \or{\@xviipt}{20}\or{\@xxpt}{24}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{11pt}{\def\@mainsize{11}\def\@ptsize{1}%
  \def\@typesizes{%
    \or{6}{7}\or{7}{8}\or{8}{10}\or{9}{11}\or{10}{12}%
    \or{\@xipt}{13}% normalsize
    \or{\@xiipt}{14}\or{\@xivpt}{17}\or{\@xviipt}{20}%
    \or{\@xxpt}{24}\or{\@xxvpt}{30}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{12pt}{\def\@mainsize{12}\def\@ptsize{2}%
  \def\@typesizes{%
    \or{7}{8}\or{8}{10}\or{9}{11}\or{10}{12}\or{\@xipt}{13}%
    \or{\@xiipt}{14}% normalsize
    \or{\@xivpt}{17}\or{\@xviipt}{20}\or{\@xxpt}{24}%
    \or{\@xxvpt}{30}\or{\@xxvpt}{30}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{8pt}{\def\@mainsize{8}\def\@ptsize{8}%
  \def\@typesizes{%
    \or{5}{6}\or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}%
    \or{8}{10}% normalsize
    \or{9}{11}\or{10}{12}\or{\@xipt}{13}%
    \or{\@xiipt}{14}\or{\@xivpt}{17}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{9pt}{\def\@mainsize{9}\def\@ptsize{9}%
  \def\@typesizes{%
    \or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}\or{8}{10}%
    \or{9}{11}% normalsize
    \or{10}{12}\or{\@xipt}{13}\or{\@xiipt}{14}%
    \or{\@xivpt}{17}\or{\@xviipt}{20}}%
  \normalsize \linespacing=\baselineskip
}
\def\ps@empty{\let\@mkboth\@gobbletwo
  \let\@oddhead\@empty \let\@evenhead\@empty
  \let\@oddfoot\@empty \let\@evenfoot\@empty
  \global\topskip\normaltopskip}
\def\ps@plain{\ps@empty
  \def\@oddfoot{\normalfont\scriptsize \hfil\thepage\hfil}%
  \let\@evenfoot\@oddfoot}
\newswitch{runhead}
\def\ps@headings{\ps@empty
  \def\@evenhead{%
    \setTrue{runhead}%
    \normalfont\scriptsize
    \rlap{\thepage}\hfil
    \def\thanks{\protect\thanks@warning}%
    \leftmark{}{}\hfil}%
  \def\@oddhead{%
    \setTrue{runhead}%
    \normalfont\scriptsize \hfil
    \def\thanks{\protect\thanks@warning}%
    \rightmark{}{}\hfil \llap{\thepage}}%
  \let\@mkboth\markboth
  \def\partmark{\@secmark\markboth\partrunhead\partname}%
  \def\chaptermark{%
    \@secmark\markboth\chapterrunhead{}}%
  \def\sectionmark{%
    \@secmark\markright\sectionrunhead\sectionname}%
}
\let\sectionname\@empty
\let\subsectionname\@empty
\let\subsubsectionname\@empty
\let\paragraphname\@empty
\let\subparagraphname\@empty
\def\leftmark{\expandafter\@firstoftwo\topmark{}{}}
\def\rightmark{\expandafter\@secondoftwo\botmark{}{}}
\long\def\@nilgobble#1\@nil{}
\def\@secmark#1#2#3#4{%
  \begingroup \let\protect\@unexpandable@protect
  \edef\@tempa{\endgroup \toks@{\protect#2{#3}{\@secnumber}}}%
  \@tempa
  \toks@\@xp{\the\toks@{#4}}%
  \afterassignment\@nilgobble\@temptokena\@themark{}\@nil
  \edef\@tempa{\@nx\@mkboth{%
    \ifx\markright#1\the\@temptokena\else\the\toks@\fi}{\the\toks@}}%
  \@tempa}
\let\@secnumber\@empty
\def\markboth#1#2{%
  \begingroup
    \@temptokena{{#1}{#2}}\xdef\@themark{\the\@temptokena}%
    \mark{\the\@temptokena}%
  \endgroup
  \if@nobreak\ifvmode\nobreak\fi\fi}
\def\ps@myheadings{\ps@headings \let\@mkboth\@gobbletwo}
\newskip\normaltopskip
\normaltopskip=10pt \relax
\let\sectionmark\@gobble
\let\subsectionmark\@gobble
\let\subsubsectionmark\@gobble
\let\paragraphmark\@gobble

\DeclareOption{makeidx}{}
\ExecuteOptions{leqno,centertags,letterpaper,portrait,%
  10pt,twoside,onecolumn,final}
\ProcessOptions\relax
\if@compatibility
  \def\@tempa{\RequirePackage{amstex}\relax}%
\else
  \@ifclasswith{\@classname}{nomath}{%
    \let\@tempa\relax
  }{%
    \def\@tempa{\RequirePackage{amsmath}\relax}%
  }%
\fi
\@tempa % load amstex.sty or amsmath.sty
\@ifundefined{numberwithin}{%
  \newcommand{\numberwithin}[3][\arabic]{%
    \@ifundefined{c@#2}{\@nocounterr{#2}}{%
      \@ifundefined{c@#3}{\@nocnterr{#3}}{%
        \@addtoreset{#2}{#3}%
        \@xp\xdef\csname the#2\endcsname{%
          \@xp\@nx\csname the#3\endcsname .\@nx#1{#2}}}}%
  }
  \csname newtoks\endcsname\@emptytoks
}{}
\if@compatibility
\else
  \@ifclasswith{\@classname}{noamsfonts}{%
    % amsfonts package is not wanted
  }{%
    % amsfonts package IS wanted; test whether a recent enough version
    % seems to be installed
    \begingroup \fontencoding{U}\fontfamily{msa}\try@load@fontshape\endgroup
    \global\@xp\let\csname U+msa\endcsname\relax % reset
    \@ifundefined{U/msa/m/n}{%
      \ClassError{\@classname}{%
  Package `amsfonts' not installed, or version too old?\MessageBreak
  Unable to get font info for the `msam' fonts in the expected form%
      }{%
  The amsfonts package will not be loaded, to avoid probable\MessageBreak
  incompatibility problems. You can (a) use the `noamsfonts'
      documentclass\MessageBreak
  option next time, or (b) check that the amsfonts package is
      installed\MessageBreak
  correctly, and is not too old to be compatible.%
      }%
    }{%
      \RequirePackage{amsfonts}[1995/01/01]\relax
    }%
  }
\fi % end yesamsfonts branch
\let\cleardouble@page\cleardoublepage
\AtBeginDocument{%
  \ifx\cleardouble@page\cleardoublepage
    \def\cleardoublepage{\clearpage{\pagestyle{empty}\cleardouble@page}}
  \fi
}
\newcommand{\uppercasenonmath}[1]{\toks@\@emptytoks
  \@xp\@skipmath\@xp\@empty#1$$%
  \edef#1{{\@nx\protect\@nx\@upprep\the\toks@}}%
}
\newcommand{\@upprep}{%
  \spaceskip1.3\fontdimen2\font plus1.3\fontdimen3\font
  \upchars@}
\newcommand{\upchars@}{%
  \def\ss{SS}\def\i{I}\def\j{J}\def\ae{\AE}\def\oe{\OE}%
  \def\o{\O}\def\aa{\AA}\def\l{\L}\def\Mc{M{\scshape c}}}
\providecommand{\Mc}{Mc}
\newcommand{\@skipmath}{}
\long\def\@skipmath#1$#2${%
  \@xskipmath#1\(\)%
  \@ifnotempty{#2}{\toks@\@xp{\the\toks@$#2$}\@skipmath\@empty}}%
\newcommand{\@xskipmath}{}
\long\def\@xskipmath#1\(#2\){%
  \uppercase{\toks@\@xp\@xp\@xp{\@xp\the\@xp\toks@#1}}%
  \@ifnotempty{#2}{\toks@\@xp{\the\toks@\(#2\)}\@xskipmath\@empty}}%
\def\altucnm#1{%
  \MakeTextUppercase{\toks@{#1}}%
  \edef#1{\the\toks@}%
}
\AtBeginDocument{%
  \@ifundefined{MakeTextUppercase}{}{\let\uppercasenonmath\altucnm}%
}
\@ifundefined{MakeUppercase}{\let\MakeUppercase\uppercase}{}%
\newcommand{\today}{%
  \relax\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
%%\if@compatibility
%%\else
%%  \def\@obsolete@fontswitch#1#2#3{%
%%    \@latex@warning@no@line{%
%%      Command {\string#1...}\on@line\space is obsolete;\MessageBreak
%%      the LaTeX2e equivalent is \string#3{...}}%
%%    \gdef#1{\@fontswitch\relax#3}%
%%  }
%%  \DeclareRobustCommand*\cal{%
%%    \@xp\@obsolete@fontswitch\csname cal \endcsname\relax\mathcal}
%%  \DeclareRobustCommand*\mit{%
%%    \@xp\@obsolete@fontswitch\csname mit \endcsname\relax\mathnormal}
%%\fi
\renewcommand*{\title}[2][]{\gdef\shorttitle{#1}\gdef\@title{#2}}
\edef\title{\@nx\@dblarg
  \@xp\@nx\csname\string\title\endcsname}
\renewcommand{\author}[2][]{%
  \ifx\@empty\authors
    \gdef\authors{#2}%
  \else
    \g@addto@macro\authors{\and#2}%
    \g@addto@macro\addresses{\author{}}%
  \fi
  \@ifnotempty{#1}{%
    \ifx\@empty\shortauthors
      \gdef\shortauthors{#1}%
    \else
      \g@addto@macro\shortauthors{\and#1}%
    \fi
  }%
}
\edef\author{\@nx\@dblarg
  \@xp\@nx\csname\string\author\endcsname}
\let\shortauthors\@empty   \let\authors\@empty
\let\addresses\@empty      \let\thankses\@empty
\newcommand{\address}[2][]{\g@addto@macro\addresses{\address{#1}{#2}}}
\newcommand{\curraddr}[2][]{\g@addto@macro\addresses{\curraddr{#1}{#2}}}
\newcommand{\email}[2][]{\g@addto@macro\addresses{\email{#1}{#2}}}
\newcommand{\urladdr}[2][]{\g@addto@macro\addresses{\urladdr{#1}{#2}}}
\long\def\thanks@warning#1{%
  \ClassError{\@classname}{%
    \protect\thanks\space should be given separately, not inside author name.%
  }\@ehb
}
\renewcommand{\thanks}[1]{%
  \@ifnotempty{#1}{\g@addto@macro\thankses{\thanks{#1}}}%
}
\def\curraddrname{{\itshape Current address}}
\def\emailaddrname{{\itshape Email address}}
\def\urladdrname{{\itshape URL}}
\def\@setaddresses{\par
  \nobreak \begingroup
  \def\author##1{\nobreak\addvspace\bigskipamount}%
  \def\\{\unskip, \ignorespaces}%
  \interlinepenalty\@M
  \def\address##1##2{\begingroup
    \par\addvspace\bigskipamount\indent
    \@ifnotempty{##1}{(\ignorespaces##1\unskip) }%
    {\scshape\ignorespaces##2}\par\endgroup}%
  \def\curraddr##1##2{\begingroup
    \@ifnotempty{##2}{\nobreak\indent\curraddrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      ##2\par}\endgroup}%
  \def\email##1##2{\begingroup
    \@ifnotempty{##2}{\nobreak\indent\emailaddrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      \ttfamily##2\par}\endgroup}%
  \def\urladdr##1##2{\begingroup
    \def~{\char`\~}%
    \@ifnotempty{##2}{\nobreak\indent\urladdrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      \ttfamily##2\par}\endgroup}%
  \addresses
  \endgroup
}
\let\@date\@empty
\def\dedicatory#1{\def\@dedicatory{#1}}
\let\@dedicatory=\@empty
\def\keywords#1{\def\@keywords{#1}}
\let\@keywords=\@empty
\newcommand*\subjclass[2][2020]{%
  \def\@subjclass{#2}%
  \@ifundefined{subjclassname@#1}{%
    \ClassWarning{\@classname}{Unknown edition (#1) of Mathematics
      Subject Classification; using '2020'.}%
  }{%
    \@xp\let\@xp\subjclassname\csname subjclassname@#1\endcsname
  }%
}
\let\@subjclass=\@empty
\def\translname{Translated by}
\def\translator#1{%
  \ifx\@empty\@translators \def\@translators{#1}%
  \else\g@addto@macro\@translators{\and#1}\fi}
\let\@translators=\@empty
\newcommand{\xandlist}[4]{\@andlista{{#1}{#2}{#3}}#4\and\and}
\def\@andlista#1#2\and#3\and{\@andlistc{#2}\@ifnotempty{#3}{%
  \@andlistb#1{#3}}}
\def\@andlistb#1#2#3#4#5\and{%
  \@ifempty{#5}{%
    \@andlistc{#2#4}%
  }{%
    \@andlistc{#1#4}\@andlistb{#1}{#3}{#3}{#5}%
  }}
\let\@andlistc\@iden
\newcommand{\nxandlist}[4]{%
  \def\@andlistc##1{\toks@\@xp{\the\toks@##1}}%
  \toks@{\toks@\@emptytoks \@andlista{{#1}{#2}{#3}}}%
  \the\@xp\toks@#4\and\and
  \edef#4{\the\toks@}%
  \let\@andlistc\@iden}
\def\@@and{and}
\newcommand{\andify}{%
  \nxandlist{\unskip, }{\unskip{} \@@and~}{\unskip, \@@and~}}
\def\and{\unskip{ }\@@and{ }\ignorespaces}
\newcommand{\copyrightinfo}[2]{%
  \def\copyrightyear{#1}%
  \@ifnotempty{#2}{\def\copyrightholder{#2}}%
}
\copyrightinfo{0000}{(copyright holder)}
\newinsert\copyins
\skip\copyins=1.5pc
\count\copyins=1000 % magnification factor, 1000 = 100%
\dimen\copyins=.5\textheight % maximum allowed per page
\g@addto@macro\@reinserts{%
  \ifvoid\copyins\else\insert\copyins{\unvbox\copyins}\fi
}
\def\@copyinsfontsize{\fontsize{6}{7\p@}\normalfont\upshape}
\newif\if@extracrline  \@extracrlinefalse
\let\@extracrline\@empty
\relax
\def\@setcopyright{%
  \ifx\@empty\@serieslogo
  \else\ifx\@empty\copyrightyear
  \else
    \insert\copyins{\hsize\textwidth
      \parfillskip\z@\relax
      \leftskip\z@\@plus.9\textwidth\relax \rightskip\z@\relax
      \@copyinsfontsize
      \everypar{}%
      \vskip-\skip\copyins
      \if@extracrline
        \vskip-6pt
      \fi
      \nointerlineskip
      \leavevmode\hfill\vrule\@width\z@\@height\skip\copyins
      \copyright\copyrightyear\ \copyrightholder\ignorespaces
      \if@extracrline \@extracrline \fi
      \par
      \kern\z@}%
  \fi\fi
}
\def\@combinefloats{%
  \ifx \@toplist\@empty \else \@cflt \fi
  \ifx \@botlist\@empty \else \@cflb \fi
  \ifvoid\copyins \else \@cflci \fi
}
\def\@cflci{%
  \setbox\@outputbox\vbox{%
    \unvbox\@outputbox
    \vskip\skip\copyins
    \if@twocolumn \else \vskip\z@ plus\p@ \fi
    \hbox to\columnwidth{%
      \hss\vbox to\z@{\vss
        \if@twocolumn
          \if@firstcolumn \else \unvbox\copyins \fi
        \else
          \unvbox\copyins
        \fi
        }}}%
  \if@twocolumn \if@firstcolumn
    \insert\copyins{\unvbox\copyins}%
  \fi\fi
  \global\count\copyins=999 \relax
}
\newcommand{\abstractname}{Abstract}
\newcommand{\keywordsname}{Key words and phrases}
\@namedef{subjclassname@1991}{%
  \textup{1991} Mathematics Subject Classification}
\@namedef{subjclassname@2000}{%
  \textup{2000} Mathematics Subject Classification}
\@namedef{subjclassname@2010}{%
  \textup{2010} Mathematics Subject Classification}
\@namedef{subjclassname@2020}{%
  \textup{2020} Mathematics Subject Classification}
\@xp\let\@xp\subjclassname\csname subjclassname@2020\endcsname
\def\@tempb{amsbook}
\ifx\@classname\@tempb
  \newcommand{\datename}{\textit{Date}:}
\else
  \newcommand{\datename}{Received by the editors}
\fi
\def\@setthanks{\def\thanks##1{\par##1\@addpunct.}\thankses}
\newbox\abstractbox
\newenvironment{abstract}{%
  \ifx\maketitle\relax
    \ClassWarning{\@classname}{Abstract should precede
      \protect\maketitle\space in AMS document classes; reported}%
  \fi
  \global\setbox\abstractbox=\vtop \bgroup
    \normalfont\Small
    \list{}{\labelwidth\z@
      \leftmargin3pc \rightmargin\leftmargin
      \listparindent\normalparindent \itemindent\z@
      \parsep\z@ \@plus\p@
      \let\fullwidthdisplay\relax
    }%
    \item[\hskip\labelsep\scshape\abstractname.]%
}{%
  \endlist\egroup
  \ifx\@setabstract\relax \@setabstracta \fi
}
\def\@setabstract{\@setabstracta \global\let\@setabstract\relax}
\def\@setabstracta{%
  \ifvoid\abstractbox
  \else
    \skip@20\p@ \advance\skip@-\lastskip
    \advance\skip@-\baselineskip \vskip\skip@
    \box\abstractbox
    \prevdepth\z@ % because \abstractbox is a vtop
  \fi
}
\def\titlepage{%
  \cleardoublepage
  \thispagestyle{empty}\setcounter{page}{0}}
\def\endtitlepage{\newpage}
\def\labelenumi{(\theenumi)}
\def\theenumi{\@arabic\c@enumi}
\def\labelenumii{(\theenumii)}
\def\theenumii{\@alph\c@enumii}
\def\p@enumii{\theenumi}
\def\labelenumiii{(\theenumiii)}
\def\theenumiii{\@roman\c@enumiii}
\def\p@enumiii{\theenumi(\theenumii)}
\def\labelenumiv{(\theenumiv)}
\def\theenumiv{\@Alph\c@enumiv}
\def\p@enumiv{\p@enumiii\theenumiii}
\def\labelitemi{$\m@th\bullet$}
\def\labelitemii{\bfseries --}% \upshape already done by \itemize
\def\labelitemiii{$\m@th\ast$}
\def\labelitemiv{$\m@th\cdot$}
\newenvironment{verse}{\let\\\@centercr
  \list{}{\itemsep\z@ \itemindent -1.5em\listparindent\itemindent
  \rightmargin\leftmargin \advance\leftmargin 1.5em}\item[]%
}{%
  \endlist
}
\let\endverse=\endlist % for efficiency
\newenvironment{quotation}{\list{}{%
    \leftmargin3pc \listparindent\normalparindent
    \itemindent\z@
    \rightmargin\leftmargin \parsep\z@ \@plus\p@}%
  \item[]%
}{%
  \endlist
}
\let\endquotation=\endlist % for efficiency
\newenvironment{quote}{%
  \list{}{\rightmargin\leftmargin}\item[]%
}{%
  \endlist
}
\let\endquote=\endlist % for efficiency
\def\trivlist{\parsep\parskip\@nmbrlistfalse
  \@trivlist \labelwidth\z@ \leftmargin\z@
  \itemindent\z@
  \let\@itemlabel\@empty
  \def\makelabel##1{\upshape##1}}
\renewenvironment{enumerate}{%
  \ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{\hss\llap{\upshape##1}}}\fi
}{%
  \endlist
}
\let\endenumerate=\endlist % for efficiency
\renewenvironment{itemize}{%
  \ifnum\@itemdepth>3 \@toodeep
  \else \advance\@itemdepth\@ne
    \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
    \list{\csname\@itemitem\endcsname}%
      {\def\makelabel##1{\hss\llap{\upshape##1}}}%
  \fi
}{%
  \endlist
}
\let\enditemize=\endlist % for efficiency
\newcommand{\descriptionlabel}[1]{\hspace\labelsep \upshape\bfseries #1:}
\newenvironment{description}{\list{}{%
  \advance\leftmargini6\p@ \itemindent-12\p@
  \labelwidth\z@ \let\makelabel\descriptionlabel}%
}{
  \endlist
}
\let\enddescription=\endlist % for efficiency
\let\upn=\textup
\AtBeginDocument{%
  \labelsep=5pt\relax
  \setcounter{enumi}{13}\setcounter{enumii}{13}%
  \setcounter{enumiii}{13}\setcounter{enumiv}{13}%
  \settowidth\leftmargini{\labelenumi\hskip\labelsep}%
  \advance\leftmargini by \normalparindent
  \settowidth\leftmarginii{\labelenumii\hskip\labelsep}%
  \settowidth\leftmarginiii{\labelenumiii\hskip\labelsep}%
  \settowidth\leftmarginiv{\labelenumiv\hskip\labelsep}%
  \setcounter{enumi}{0}\setcounter{enumii}{0}%
  \setcounter{enumiii}{0}\setcounter{enumiv}{0}%
  \leftmarginv=10pt  \leftmarginvi=\leftmarginv
  \leftmargin=\leftmargini
  \labelwidth=\leftmargini \advance\labelwidth-\labelsep
  \@listi}
\newskip\listisep
\listisep\smallskipamount
\def\@listI{\leftmargin\leftmargini \parsep\z@skip
  \topsep\listisep \itemsep\z@skip
  \listparindent\normalparindent}
\let\@listi\@listI
\def\@listii{\leftmargin\leftmarginii
  \labelwidth\leftmarginii \advance\labelwidth-\labelsep
  \topsep\z@skip \parsep\z@skip \partopsep\z@skip \itemsep\z@skip}
\def\@listiii{\leftmargin\leftmarginiii
  \labelwidth\leftmarginiii \advance\labelwidth-\labelsep}
\def\@listiv{\leftmargin\leftmarginiv
  \labelwidth\leftmarginiv \advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
  \labelwidth\leftmarginv \advance\labelwidth-\labelsep}
\def\@listvi{\leftmargin\leftmarginvi
  \labelwidth\leftmarginvi \advance\labelwidth-\labelsep}
\@ifclasswith{\@classname}{fleqn}{%
  \let\@tempa\@iden
  \AtBeginDocument{\mathindent\leftmargini}%
}{\let\@tempa\@gobble}%
\@ifpackageloaded{amsmath}{\let\@tempa\@gobble}{%
  \@ifpackageloaded{amstex}{\let\@tempa\@gobble}{}%
}
\@tempa{%
  \def\[{\relax
    \ifmmode\@badmath
    \else
      \begin{trivlist}%
      \@beginparpenalty\predisplaypenalty
      \@endparpenalty\postdisplaypenalty
      \item[]\leavevmode
      \hbox to\linewidth\bgroup$\displaystyle
      \hskip\mathindent\bgroup
    \fi}%
  \def\]{\relax
    \ifmmode
      \egroup \m@th$\hfil \egroup
      \end{trivlist}%
    \else \@badmath
    \fi}%
  \renewenvironment{equation}{%
    \@beginparpenalty\predisplaypenalty
    \@endparpenalty\postdisplaypenalty
    \refstepcounter{equation}%
    \@topsep\abovedisplayskip \trivlist
    \item[]\leavevmode
    \hbox to\linewidth\bgroup\hskip\mathindent$\displaystyle
  }{%
    \m@th$\hfil \displaywidth\linewidth \hbox{\@eqnnum}\egroup
    \endtrivlist
  }%
  \renewenvironment{eqnarray}{%
    \stepcounter{equation}\let\@currentlabel\theequation
    \global\@eqnswtrue \global\@eqcnt\z@ \tabskip\mathindent
    \let\\=\@eqncr \abovedisplayskip\topsep
    \ifvmode \advance\abovedisplayskip\partopsep \fi
    \belowdisplayskip\abovedisplayskip
    \belowdisplayshortskip\abovedisplayskip
    \abovedisplayshortskip\abovedisplayskip
    $$\everycr{}\halign to\linewidth\bgroup
      \hskip\@centering
      $\displaystyle\tabskip\z@skip####\m@th$&%
      \@eqnsel \global\@eqcnt\@ne
      \hfil${}####{}\m@th$\hfil&%
      \global\@eqcnt\tw@
      $\displaystyle ####\m@th$\hfil\tabskip\@centering&%
      \global\@eqcnt\thr@@
      \hbox to \z@\bgroup\hss####\egroup\tabskip\z@skip\cr
    }{%
      \@@eqncr \egroup \global\advance\c@equation\m@ne$$%
      \global\@ignoretrue
    }%
  \newdimen\mathindent
  \mathindent\leftmargini
}
\def\@startsection#1#2#3#4#5#6{%
 \if@noskipsec \leavevmode \fi
 \par \@tempskipa #4\relax
 \@afterindenttrue
 \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi
 \if@nobreak \everypar{}\else
     \addpenalty\@secpenalty\addvspace\@tempskipa\fi
 \@ifstar{\@dblarg{\@sect{#1}{\@m}{#3}{#4}{#5}{#6}}}%
         {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}%
}
\def\@seccntformat#1{%
  \protect\textup{\protect\@secnumfont
    \csname the#1\endcsname
    \protect\@secnumpunct
  }%
}
\let\@secnumfont\@empty
\def\@sect#1#2#3#4#5#6[#7]#8{%
  \edef\@toclevel{\ifnum#2=\@m 0\else\number#2\fi}%
  \ifnum #2>\c@secnumdepth \let\@secnumber\@empty
  \else \@xp\let\@xp\@secnumber\csname the#1\endcsname\fi
  \@tempskipa #5\relax
  \ifnum #2>\c@secnumdepth
    \let\@svsec\@empty
  \else
    \refstepcounter{#1}%
    \edef\@secnumpunct{%
      \ifdim\@tempskipa>\z@ % not a run-in section heading
        \@ifnotempty{#8}{.\@nx\enspace}%
      \else
        \@ifempty{#8}{.}{.\@nx\enspace}%
      \fi
    }%
    \protected@edef\@svsec{%
      \ifnum#2<\@m
        \@ifundefined{#1name}{}{%
          \ignorespaces\csname #1name\endcsname\space
        }%
      \fi
      \@seccntformat{#1}%
    }%
  \fi
  \ifdim \@tempskipa>\z@ % then this is not a run-in section heading
    \begingroup #6\relax
    \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}%
    \endgroup
  \csname #1mark\endcsname{#7}%
    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
  \else
  \def\@svsechd{#6\hskip #3\@svsec
    \@ifnotempty{#8}{\ignorespaces#8\unskip
       \@addpunct.}%
    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
  }%
  \fi
  \global\@nobreaktrue
  \@xsect{#5}}
\let\@ssect\relax
\newcounter{part}
\newcounter{chapter}
\newcounter{section}[chapter]
\def\thesection{\arabic{section}}
\newcounter{subsection}[section]
\newcounter{subsubsection}[subsection]
\newcounter{paragraph}[subsubsection]
\newcounter{subparagraph}[paragraph]
\renewcommand\thepart          {\arabic{part}}
\renewcommand\thechapter       {\arabic{chapter}}
\renewcommand\thesection       {\arabic{section}}
\renewcommand\thesubsection    {\thesection.\arabic{subsection}}
\renewcommand\thesubsubsection {\thesubsection .\arabic{subsubsection}}
\renewcommand\theparagraph     {\thesubsubsection.\arabic{paragraph}}
\renewcommand\thesubparagraph  {\theparagraph.\arabic{subparagraph}}
\setcounter{secnumdepth}{3}
\def\partname{Part}
\def\specialsection{\@startsection{section}{1}%
  \z@{2\linespacing\@plus\linespacing}{.5\linespacing}%
  {\large\scshape\centering}}
\def\part{\cleardoublepage \thispagestyle{empty}%
  \null\vfil  \markboth{}{}\secdef\@part\@spart}
\def\@part[#1]#2{%
  \ifnum \c@secnumdepth >-2\relax \refstepcounter{part}%
    \addcontentsline{toc}{part}{\partname\ \thepart.
        \protect\enspace\protect\noindent#1}%
  \else
    \addcontentsline{toc}{part}{#1}\fi
  \begingroup\centering
  \ifnum \c@secnumdepth >-2\relax
       {\fontsize{\@xviipt}{22}\bfseries
         \partname\ \thepart} \vskip 20\p@ \fi
  \fontsize{\@xxpt}{25}\bfseries
      #1\vfil\vfil\endgroup \newpage\thispagestyle{empty}}

\def\@spart#1{\addcontentsline{toc}{part}{\protect\noindent#1}%
  \begingroup\centering
  \fontsize{\@xxpt}{25}\bfseries
     #1\vfil\vfil\endgroup \newpage\thispagestyle{empty}}
\def\partrunhead#1#2#3{%
  \@ifnotempty{#2}{\uppercase{\ignorespaces#1 #2\unskip}\@ifnotempty{#3}{. }}%
  \def\@tempa{#3}%
  \ifx\@empty\@tempa\else
    \begingroup \def\\{ \ignorespaces}% defend against questionable usage
    \uppercasenonmath\@tempa\@tempa
    \endgroup
  \fi
}
\let\chapterrunhead\partrunhead
\let\sectionrunhead\partrunhead
\def\section{\@startsection{section}{1}%
  \z@{.7\linespacing\@plus\linespacing}{.5\linespacing}%
  {\normalfont\bfseries\centering}}
\def\subsection{\@startsection{subsection}{2}%
  \normalparindent{.5\linespacing\@plus.7\linespacing}{-.5em}%
  {\normalfont\bfseries}}
\def\subsubsection{\@startsection{subsubsection}{3}%
  \normalparindent\z@{-.5em}%
  {\normalfont\itshape}}
\def\paragraph{\@startsection{paragraph}{4}%
  \normalparindent\z@{-\fontdimen2\font}%
  \normalfont}
\def\subparagraph{\@startsection{subparagraph}{5}%
  \z@\z@{-\fontdimen2\font}%
  \normalfont}
\def\@Roman#1{\@xp\@slowromancap
  \romannumeral#1@}%
\def\@slowromancap#1{\ifx @#1% then terminate
  \else
    \if i#1I\else\if v#1V\else\if x#1X\else\if l#1L\else\if
    c#1C\else\if m#1M\else#1\fi\fi\fi\fi\fi\fi
    \@xp\@slowromancap
  \fi
}
\def\frontmatter{\cleardoublepage\pagenumbering{roman}}
\def\mainmatter{\cleardoublepage\pagenumbering{arabic}}
\def\backmatter{%
  \if@openright\cleardoublepage\else\clearpage\fi
  \let\chaptername\relax}
\def\aufm#1{\par\vspace*{12pt}{\flushright #1\par}}
\newenvironment{xcb}{%
  \setcounter{enumi}{0}%
  \settowidth{\leftmargini}{\labelenumi\hskip\labelsep}%
  \setcounter{enumii}{4}% letter d
  \settowidth{\leftmarginii}{\labelenumii\hskip\labelsep}%
  \@startsection{section}% counter name; ignored because of the
                                % * below
  {1}% sectioning level
  {\z@}% indent to the left of the section title
  {18\p@\@plus2\p@}% vertical space above
  {1sp}% Space below of 13pt base-to-base, so none needs to be added
      % here; but \z@ would cause the following text to be run-in, so we
      % use 1sp instead.
  {\bfseries}% The font of the subsection title
  *% always unnumbered
}{%
  \par
}
\def\chapter{%
  \if@openright\cleardoublepage\else\clearpage\fi
  \thispagestyle{plain}\global\@topnum\z@
  \@afterindenttrue \secdef\@chapter\@schapter}
\def\@chapter[#1]#2{\refstepcounter{chapter}%
  \ifnum\c@secnumdepth<\z@ \let\@secnumber\@empty
  \else \let\@secnumber\thechapter \fi
  \typeout{\chaptername\space\@secnumber}%
  \def\@toclevel{0}%
  \ifx\chaptername\appendixname \@tocwriteb\tocappendix{chapter}{#2}%
  \else \@tocwriteb\tocchapter{chapter}{#2}\fi
  \chaptermark{#1}%
  \addtocontents{lof}{\protect\addvspace{10\p@}}%
  \addtocontents{lot}{\protect\addvspace{10\p@}}%
  \@makechapterhead{#2}\@afterheading}
\def\@schapter#1{\typeout{#1}%
  \let\@secnumber\@empty
  \def\@toclevel{0}%
  \ifx\chaptername\appendixname \@tocwriteb\tocappendix{chapter}{#1}%
  \else \@tocwriteb\tocchapter{chapter}{#1}\fi
  \chaptermark{#1}%
  \addtocontents{lof}{\protect\addvspace{10\p@}}%
  \addtocontents{lot}{\protect\addvspace{10\p@}}%
  \@makeschapterhead{#1}\@afterheading}
\newcommand\chaptername{Chapter}
\newcommand\appendixname{Appendix}
\def\@makechapterhead#1{\global\topskip 7.5pc\relax
  \begingroup
  \fontsize{\@xivpt}{18}\bfseries\centering
    \ifnum\c@secnumdepth>\m@ne
      \leavevmode \hskip-\leftskip
      \rlap{\vbox to\z@{\vss
          \centerline{\normalsize\mdseries
              \uppercase\@xp{\chaptername}\enspace\thechapter}
          \vskip 3pc}}\hskip\leftskip\fi
     #1\par \endgroup
  \skip@34\p@ \advance\skip@-\normalbaselineskip
  \vskip\skip@ }
\def\@makeschapterhead#1{\global\topskip 7.5pc\relax
  \begingroup
  \fontsize{\@xivpt}{18}\bfseries\centering
  #1\par \endgroup
  \skip@34\p@ \advance\skip@-\normalbaselineskip
  \vskip\skip@ }
\def\appendix{\par
  \c@chapter\z@ \c@section\z@
  \let\chaptername\appendixname
  \def\thechapter{\@Alph\c@chapter}}
\newcommand{\@pnumwidth}{1.6em}
\newcommand{\@tocrmarg}{2.6em}
\setcounter{tocdepth}{1}
\newswitch{toc}
\newswitch{lof}
\newswitch{lot}
\def\@starttoc#1#2{%
  \begingroup
  \setTrue{#1}%
  \let\secdef\@gobbletwo \chapter
  \let\@secnumber\@empty % for \@tocwrite and \chaptermark
  \ifx\contentsname#2%
  \else \@tocwrite{chapter}{#2}\fi
  \typeout{#2}\@xp\chaptermark\@xp{#2}%
  \@makeschapterhead{#2}\@afterheading
  \parskip\z@skip
  \makeatletter
  \@input{\jobname.#1}%
  \if@filesw
    \@xp\newwrite\csname tf@#1\endcsname
    \immediate\@xp\openout\csname tf@#1\endcsname \jobname.#1\relax
  \fi
  \global\@nobreakfalse \endgroup
  \newpage
}
\def\contentsname{Contents}
\def\listfigurename{List of Figures}
\def\listtablename{List of Tables}
\def\tableofcontents{%
  \@starttoc{toc}\contentsname
}
\def\listoffigures{\@starttoc{lof}\listfigurename}
\def\listoftables{\@starttoc{lot}\listtablename}
\AtBeginDocument{%
  \@for\@tempa:=-1,0,1,2,3\do{%
    \@ifundefined{r@tocindent\@tempa}{%
      \@xp\gdef\csname r@tocindent\@tempa\endcsname{0pt}}{}%
  }%
}
\def\@writetocindents{%
  \begingroup
  \@for\@tempa:=-1,0,1,2,3\do{%
    \immediate\write\@auxout{%
      \string\newlabel{tocindent\@tempa}{%
        \csname r@tocindent\@tempa\endcsname}}%
  }%
  \endgroup}
\AtEndDocument{\@writetocindents}

\let\indentlabel\@empty
\def\@tochangmeasure#1{\sbox\z@{#1}%
  \ifdim\wd\z@>\csname r@tocindent\@toclevel\endcsname\relax
    \@xp\xdef\csname r@tocindent\@toclevel\endcsname{\the\wd\z@}%
  \fi
}
\def\@toclevel{0}
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % then omit
  \else
    \par \addpenalty\@secpenalty\addvspace{#2}%
    \begingroup \hyphenpenalty\@M
    \@ifempty{#4}{%
      \@tempdima\csname r@tocindent\number#1\endcsname\relax
    }{%
      \@tempdima#4\relax
    }%
    \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax
    \rightskip\@pnumwidth plus4em \parfillskip-\@pnumwidth
    #5\leavevmode\hskip-\@tempdima #6\nobreak\relax
    \hfil\hbox to\@pnumwidth{\@tocpagenum{#7}}\par
    \nobreak
    \endgroup
  \fi}
\def\@tocpagenum#1{\hss{\mdseries #1}}
\def\@tocwrite#1{\@xp\@tocwriteb\csname toc#1\endcsname{#1}}
\def\@tocwriteb#1#2#3{%
  \begingroup
    \def\@tocline##1##2##3##4##5##6{%
      \ifnum##1>\c@tocdepth
      \else \sbox\z@{##5\let\indentlabel\@tochangmeasure##6}\fi}%
    \csname l@#2\endcsname{#1{\csname#2name\endcsname}{\@secnumber}{}}%
  \endgroup
  \addcontentsline{toc}{#2}%
    {\protect#1{\csname#2name\endcsname}{\@secnumber}{#3}}}
\def\l@section{\@tocline{1}{0pt}{1pc}{}{}}
\newcommand{\tocsection}[3]{%
  \indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3}
\def\l@subsection{\@tocline{2}{0pt}{1pc}{5pc}{}}
\let\tocsubsection\tocsection
\def\l@subsubsection{\@tocline{3}{0pt}{1pc}{7pc}{}}
\let\tocsubsubsection\tocsection
\let\l@paragraph\l@subsubsection
\let\tocparagraph\tocsection
\let\l@subparagraph\l@subsubsection
\let\tocsubparagraph\tocsection
\def\l@part{\@tocline{-1}{12pt plus2pt}{0pt}{}{\bfseries}}
\let\tocpart\tocsection
\def\l@chapter{\@tocline{0}{8pt plus1pt}{0pt}{}{}}
\let\tocchapter\tocsection
\newcommand{\tocappendix}[3]{%
  \indentlabel{#1\@ifnotempty{#2}{ #2}.\quad}#3}
\def\l@figure{\@tocline{0}{3pt plus2pt}{0pt}{1.5pc}{}}
\let\l@table=\l@figure
\def\refname{References}
\def\bibname{Bibliography}
\def\@defaultbiblabelstyle#1{#1.}
\def\bibliographystyle#1{%
    \if@filesw\immediate\write\@auxout{\string\bibstyle{#1}}\fi
    \def\@tempa{#1}%
    \def\@tempb{amsplain}%
    \def\@tempc{}%
    \ifx\@tempa\@tempb
        \def\@biblabel##1{\@defaultbiblabelstyle{##1}}%
        \def\bibsetup{}%
    \else
        \def\bibsetup{\labelsep6\p@}%
        \ifx\@tempa\@tempc
            \def\@biblabel##1{}%
            \def\bibsetup{\labelwidth\z@ \leftmargin24\p@
                \itemindent-\leftmargin
                \labelsep\z@ }%
        \fi
    \fi}
\newcommand{\bibliofont}{\footnotesize}
\newcommand{\@bibtitlestyle}{%
  \@xp\chapter\@xp*\@xp{\bibname}%
}
\newenvironment{thebibliography}[1]{%
  \@bibtitlestyle
  \normalfont\bibliofont\labelsep .5em\relax
  \renewcommand\theenumiv{\arabic{enumiv}}\let\p@enumiv\@empty
  \list{\@biblabel{\theenumiv}}{\settowidth\labelwidth{\@biblabel{#1}}%
    \leftmargin\labelwidth \advance\leftmargin\labelsep
    \usecounter{enumiv}}%
  \sloppy \clubpenalty\@M \widowpenalty\clubpenalty
  \sfcode`\.=\@m
}{%
  \def\@noitemerr{\@latex@warning{Empty `thebibliography' environment}}%
  \endlist
}
\def\bysame{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\def\newblock{}
\newcommand\MR[1]{\relax\ifhmode\unskip\spacefactor3000 \space\fi
  MR~\MRhref{#1}{#1}}
\let\MRhref\@gobble
\newcommand\URL{\begingroup
  \def\@sverb##1{%
    \def\@tempa####1##1{\@URL{####1}\egroup\endgroup}%
    \@tempa}%
  \verb}
\let\URLhref\@gobble
\def\@URL#1{\URLhref{#1}#1}
\newif\if@restonecol
\def\indexchap#1{\global\topskip 7.5pc\relax
  \twocolumn[{\fontsize{\@xivpt}{18}\bfseries\centering
    \vskip\topskip\hbox{}\vskip-\baselineskip% adjust top space
    #1\par
   \global\topskip 34\p@\relax
   \ifx\@empty\indexintro
   \else
     \begingroup \normalsize
       \skip@\topskip \advance\skip@ -\baselineskip
       \vskip\skip@
       \parbox[t]{24pc}{\normalfont\indexintro\par}%
     \endgroup
     \global\topskip 24\p@\relax
   \fi
  }]%
}
\newcommand{\indexintro}{}
\newcommand{\@indextitlestyle}{%
  \let\@makeschapterhead\indexchap
  \@xp\chapter\@xp*\@xp{\indexname}%
}
\def\theindex{\@restonecoltrue\if@twocolumn

amsproc.cls59 KBVendored from mirrors.ctan.org on 2026-05-26.

%%
%% This is file `amsproc.cls',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% amsclass.dtx  (with options: `amsproc,classes')
%% This is a generated file.
%% 
%% Copyright 1995, 1999, 2004, 2009-2020 American Mathematical Society.
%% 
%% American Mathematical Society
%% Technical Support
%% Publications Technical Group
%% 201 Charles Street
%% Providence, RI 02904
%% USA
%% tel: (401) 455-4080
%%      (800) 321-4267 (USA and Canada only)
%% fax: (401) 331-3842
%% email: tech-support@ams.org
%% 
%% This work may be distributed and/or modified under the
%% conditions of the LaTeX Project Public License, either version 1.3c
%% of this license or (at your option) any later version.
%% The latest version of this license is in
%%   http://www.latex-project.org/lppl.txt
%% and version 1.3c or later is part of all distributions of LaTeX
%% version 2005/12/01 or later.
%% 
%% This work has the LPPL maintenance status `maintained'.
%% 
%% The Current Maintainer of this work is the American Mathematical
%% Society.
%% 
%% ========================================================================
%% 
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1995/06/01]% LaTeX date must be June 1995 or later
\ProvidesClass{amsproc}[2020/05/29 v2.20.6]
\global\expandafter\let\csname ver@amsthm.sty\expandafter\endcsname
  \csname ver@\@currname.\@currext\endcsname
\let\@xp=\expandafter
\let\@nx=\noexpand
\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
\long\def\@xifempty#1#2@#3#4#5\@nil{%
  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
\long\def\@ifnotempty#1{\@ifempty{#1}{}}
\def\setboxz@h{\setbox\z@\hbox}
\def\@addpunct#1{%
  \relax\ifhmode
    \ifnum\spacefactor>\@m \else#1\fi
  \fi}
\def\nopunct{\spacefactor 1007 }
\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
  \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
\def\@tempa#1#2\@nil{\edef\@classname{#1}}
\expandafter\@tempa\@currnamestack{}{}{}\@nil
\ifx\@classname\@empty \edef\@classname{\@currname}\fi
\def\@True{00}
\def\@False{01}
\newcommand\newswitch[2][False]{%
  \expandafter\@ifdefinable\csname ?@#2\endcsname{%
    \global\expandafter\let\csname ?@#2\expandafter\endcsname
      \csname @#1\endcsname
  }%
}
\newcommand{\setFalse}[1]{%
  \expandafter\let\csname ?@#1\endcsname\@False
}
\newcommand{\setTrue}[1]{%
  \expandafter\let\csname ?@#1\endcsname\@True
}
\newswitch{}
\DeclareRobustCommand{\except}[1]{%
  \if\csname ?@#1\endcsname \expandafter\@gobble
  \else \expandafter\@firstofone
  \fi
}
\DeclareRobustCommand{\for}[1]{%
  \if\csname ?@#1\endcsname \expandafter\@firstofone
  \else \expandafter\@gobble
  \fi
}
\DeclareRobustCommand{\forany}[1]{%
  \csname for@any@01\endcsname#1,?,\@nil
}
\@namedef{for@any@\@False}#1,{%
  \csname for@any@%
    \csname ?@\zap@space#1 \@empty\endcsname
  \endcsname
}
\@namedef{?@?}{x}
\@namedef{for@any@\@True}#1\@nil#2{#2}
\def\for@any@x{\@car\@gobble}
\DeclareOption{a4paper}{\paperheight 297mm\paperwidth 210mm
  \textheight 54.5pc }
\DeclareOption{letterpaper}{\paperheight 11in\paperwidth 8.5in }
\DeclareOption{landscape}{\@tempdima\paperheight
  \paperheight\paperwidth \paperwidth\@tempdima}
\DeclareOption{portrait}{}
\DeclareOption{oneside}{\@twosidefalse \@mparswitchfalse}
\DeclareOption{twoside}{\@twosidetrue  \@mparswitchtrue}
\DeclareOption{draft}{\overfullrule5\p@
  \ClassWarningNoLine{\@classname}{%
    When the draft option is used, the
      \protect\includegraphics\MessageBreak
    command will print blank placeholder boxes\MessageBreak
    for the graphics}%
}
\DeclareOption{final}{\overfullrule\z@ }
\def\dateposted#1{\def\@dateposted{#1}}%
\let\@dateposted\@empty
\def\@logofont{\fontsize{6}{7\p@}\selectfont}
\long\def\set@logo#1{%
  \vbox to\headheight{%
    \@parboxrestore \@logofont
    \noindent#1\par\vss
  }%
}
\def\procart@logo{%
  \set@logo{\publname
    \ifx\@empty\volinfo \else\newline\volinfo\fi}%
}
\newif\if@titlepage
\DeclareOption{titlepage}{\@titlepagetrue}
\DeclareOption{notitlepage}{\@titlepagefalse}
\DeclareOption{onecolumn}{\@twocolumnfalse}
\DeclareOption{twocolumn}{\@twocolumntrue}
\DeclareOption{nomath}{}
\DeclareOption{noamsfonts}{}
\DeclareOption{psamsfonts}{%
  \PassOptionsToPackage{psamsfonts}{amsfonts}%
  \PassOptionsToPackage{cmex10}{amsmath}}
\newif\iftagsleft@
\DeclareOption{leqno}{%
  \tagsleft@true \PassOptionsToPackage{leqno}{amsmath}}
\DeclareOption{reqno}{%
  \tagsleft@false \PassOptionsToPackage{reqno}{amsmath}}
\newif\ifctagsplit@
\DeclareOption{centertags}{%
  \ctagsplit@true \PassOptionsToPackage{centertags}{amsmath}}
\DeclareOption{tbtags}{%
  \ctagsplit@false \PassOptionsToPackage{tbtags}{amsmath}}
\DeclareOption{fleqn}{}%
\newcommand{\@mainsize}{10}
\newcommand{\@ptsize}{0}
\newcommand{\larger}[1][1]{%
  \count@\@currsizeindex \advance\count@#1\relax
  \ifnum\count@<\z@ \count@\z@ \else\ifnum\count@>12 \count@12 \fi\fi
  \ifcase\count@
    \Tiny\or\Tiny\or\tiny\or\SMALL\or\Small\or\small
    \or\normalsize
    \or\large\or\Large\or\LARGE\or\huge\or\Huge\else\Huge
  \fi
}
\newcommand{\smaller}[1][1]{\larger[-#1]}
\def\@adjustvertspacing{%
  \bigskipamount.7\baselineskip plus.7\baselineskip
  \medskipamount\bigskipamount \divide\medskipamount\tw@
  \smallskipamount\medskipamount \divide\smallskipamount\tw@
  \abovedisplayskip\medskipamount
  \belowdisplayskip \abovedisplayskip
  \abovedisplayshortskip\abovedisplayskip
  \advance\abovedisplayshortskip-1\abovedisplayskip
  \belowdisplayshortskip\abovedisplayshortskip
  \advance\belowdisplayshortskip 1\smallskipamount
  \jot\baselineskip \divide\jot 4 \relax
}
\renewcommand\normalsize{\@xsetfontsize\normalsize 6%
  \@adjustvertspacing \let\@listi\@listI}
\DeclareRobustCommand{\Tiny}{\@xsetfontsize\Tiny 1}
\DeclareRobustCommand{\tiny}{\@xsetfontsize\tiny 2}
\DeclareRobustCommand{\SMALL}{\@xsetfontsize\SMALL 3}
\DeclareRobustCommand{\Small}{\@xsetfontsize\Small 4%
  \@adjustvertspacing
  \def\@listi{\topsep\smallskipamount \parsep\z@skip \itemsep\z@skip
    \leftmargin=\leftmargini
    \labelwidth=\leftmargini \advance\labelwidth-\labelsep
  }%
}
\DeclareRobustCommand{\small}{\@xsetfontsize\small 5\@adjustvertspacing}
\def\footnotesize{\Small}
\def\scriptsize{\SMALL}
\DeclareRobustCommand{\large}{\@xsetfontsize\large 7\@adjustvertspacing}
\DeclareRobustCommand{\Large}{\@xsetfontsize\Large 8\@adjustvertspacing}
\DeclareRobustCommand{\LARGE}{\@xsetfontsize\LARGE 9}
\DeclareRobustCommand{\huge}{\@xsetfontsize\huge{10}}
\DeclareRobustCommand{\Huge}{\@xsetfontsize\Huge{11}}
\def\@xsetfontsize#1#2{%
  \chardef\@currsizeindex#2\relax
  \edef\@tempa{\@nx\@setfontsize\@nx#1%
    \@xp\ifcase\@xp\@currsizeindex\@typesizes
      \else{99}{99}\fi}%
  \@tempa
}
\chardef\@currsizeindex=6
\widowpenalty=10000
\clubpenalty=10000
\brokenpenalty=10000
\newdimen\linespacing
\lineskip=1pt \lineskiplimit=1pt
\normallineskip=1pt \normallineskiplimit=1pt
\let\baselinestretch=\@empty
\headheight=8pt \headsep=14pt
\footskip=12pt
\textheight=50.5pc \topskip=10pt
\textwidth=30pc
\columnsep=10pt \columnseprule=0pt
\marginparwidth=90pt
\marginparsep=11pt
\marginparpush=5pt
\AtBeginDocument{\settoheight{\footnotesep}{\footnotesize M$^1$}}
\skip\footins=7pt plus11pt
\skip\@mpfootins=\skip\footins
\fboxsep=3pt \fboxrule=.4pt
\arrayrulewidth=.4pt    \doublerulesep=2pt
\labelsep=5pt           \arraycolsep=\labelsep
\tabcolsep=\labelsep    \tabbingsep=\labelsep
\floatsep=15pt plus 12pt    \dblfloatsep=15pt plus 12pt
\textfloatsep=\floatsep  \dbltextfloatsep=15pt plus 12pt
\intextsep=\floatsep
\@fptop=0pt plus1fil     \@dblfptop=0pt plus1fil
\@fpbot=0pt plus1fil     \@dblfpbot=0pt plus1fil
\@fpsep=8pt plus2fil     \@dblfpsep=8pt plus2fil\relax
\parskip=0pt \relax
\newdimen\normalparindent
\normalparindent=18pt
\parindent=\normalparindent
\partopsep=0pt \relax \parsep=0pt \relax \itemsep=0pt \relax
\@lowpenalty=51         \@medpenalty=151        \@highpenalty=301
\@beginparpenalty=-\@lowpenalty
\@endparpenalty=-\@lowpenalty
\@itempenalty=-\@lowpenalty
\DeclareOption{10pt}{\def\@mainsize{10}\def\@ptsize{0}%
  \def\@typesizes{%
    \or{5}{6}\or{6}{7}\or{7}{8}\or{8}{10}\or{9}{11}%
    \or{10}{12}% normalsize
    \or{\@xipt}{13}\or{\@xiipt}{14}\or{\@xivpt}{17}%
    \or{\@xviipt}{20}\or{\@xxpt}{24}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{11pt}{\def\@mainsize{11}\def\@ptsize{1}%
  \def\@typesizes{%
    \or{6}{7}\or{7}{8}\or{8}{10}\or{9}{11}\or{10}{12}%
    \or{\@xipt}{13}% normalsize
    \or{\@xiipt}{14}\or{\@xivpt}{17}\or{\@xviipt}{20}%
    \or{\@xxpt}{24}\or{\@xxvpt}{30}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{12pt}{\def\@mainsize{12}\def\@ptsize{2}%
  \def\@typesizes{%
    \or{7}{8}\or{8}{10}\or{9}{11}\or{10}{12}\or{\@xipt}{13}%
    \or{\@xiipt}{14}% normalsize
    \or{\@xivpt}{17}\or{\@xviipt}{20}\or{\@xxpt}{24}%
    \or{\@xxvpt}{30}\or{\@xxvpt}{30}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{8pt}{\def\@mainsize{8}\def\@ptsize{8}%
  \def\@typesizes{%
    \or{5}{6}\or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}%
    \or{8}{10}% normalsize
    \or{9}{11}\or{10}{12}\or{\@xipt}{13}%
    \or{\@xiipt}{14}\or{\@xivpt}{17}}%
  \normalsize \linespacing=\baselineskip
}
\DeclareOption{9pt}{\def\@mainsize{9}\def\@ptsize{9}%
  \def\@typesizes{%
    \or{5}{6}\or{5}{6}\or{6}{7}\or{7}{8}\or{8}{10}%
    \or{9}{11}% normalsize
    \or{10}{12}\or{\@xipt}{13}\or{\@xiipt}{14}%
    \or{\@xivpt}{17}\or{\@xviipt}{20}}%
  \normalsize \linespacing=\baselineskip
}
\def\ps@empty{\let\@mkboth\@gobbletwo
  \let\@oddhead\@empty \let\@evenhead\@empty
  \let\@oddfoot\@empty \let\@evenfoot\@empty
  \global\topskip\normaltopskip}
\def\ps@plain{\ps@empty
  \def\@oddfoot{\normalfont\scriptsize \hfil\thepage\hfil}%
  \let\@evenfoot\@oddfoot}
\newswitch{runhead}
\def\ps@headings{\ps@empty
  \def\@evenhead{%
    \setTrue{runhead}%
    \normalfont\scriptsize
    \rlap{\thepage}\hfil
    \def\thanks{\protect\thanks@warning}%
    \leftmark{}{}\hfil}%
  \def\@oddhead{%
    \setTrue{runhead}%
    \normalfont\scriptsize \hfil
    \def\thanks{\protect\thanks@warning}%
    \rightmark{}{}\hfil \llap{\thepage}}%
  \let\@mkboth\markboth
}
\let\sectionname\@empty
\let\subsectionname\@empty
\let\subsubsectionname\@empty
\let\paragraphname\@empty
\let\subparagraphname\@empty
\def\leftmark{\expandafter\@firstoftwo\topmark{}{}}
\def\rightmark{\expandafter\@secondoftwo\botmark{}{}}
\def\ps@firstpage{\ps@plain
  \def\@oddfoot{\normalfont\scriptsize \hfil\thepage\hfil
     \global\topskip\normaltopskip}%
  \let\@evenfoot\@oddfoot
  \def\@oddhead{\@serieslogo\hss}%
  \let\@evenhead\@oddhead % in case an article starts on a left-hand page
}
\long\def\@nilgobble#1\@nil{}
\def\markboth#1#2{%
  \begingroup
    \@temptokena{{#1}{#2}}\xdef\@themark{\the\@temptokena}%
    \mark{\the\@temptokena}%
  \endgroup
  \if@nobreak\ifvmode\nobreak\fi\fi}
\def\ps@myheadings{\ps@headings \let\@mkboth\@gobbletwo}
\newskip\normaltopskip
\normaltopskip=10pt \relax
\let\sectionmark\@gobble
\let\subsectionmark\@gobble
\let\subsubsectionmark\@gobble
\let\paragraphmark\@gobble

\DeclareOption{makeidx}{}
\ExecuteOptions{leqno,centertags,letterpaper,portrait,%
  10pt,twoside,onecolumn,final}
\ProcessOptions\relax
\if@compatibility
  \def\@tempa{\RequirePackage{amstex}\relax}%
\else
  \@ifclasswith{\@classname}{nomath}{%
    \let\@tempa\relax
  }{%
    \def\@tempa{\RequirePackage{amsmath}\relax}%
  }%
\fi
\@tempa % load amstex.sty or amsmath.sty
\@ifundefined{numberwithin}{%
  \newcommand{\numberwithin}[3][\arabic]{%
    \@ifundefined{c@#2}{\@nocounterr{#2}}{%
      \@ifundefined{c@#3}{\@nocnterr{#3}}{%
        \@addtoreset{#2}{#3}%
        \@xp\xdef\csname the#2\endcsname{%
          \@xp\@nx\csname the#3\endcsname .\@nx#1{#2}}}}%
  }
  \csname newtoks\endcsname\@emptytoks
}{}
\if@compatibility
\else
  \@ifclasswith{\@classname}{noamsfonts}{%
    % amsfonts package is not wanted
  }{%
    % amsfonts package IS wanted; test whether a recent enough version
    % seems to be installed
    \begingroup \fontencoding{U}\fontfamily{msa}\try@load@fontshape\endgroup
    \global\@xp\let\csname U+msa\endcsname\relax % reset
    \@ifundefined{U/msa/m/n}{%
      \ClassError{\@classname}{%
  Package `amsfonts' not installed, or version too old?\MessageBreak
  Unable to get font info for the `msam' fonts in the expected form%
      }{%
  The amsfonts package will not be loaded, to avoid probable\MessageBreak
  incompatibility problems. You can (a) use the `noamsfonts'
      documentclass\MessageBreak
  option next time, or (b) check that the amsfonts package is
      installed\MessageBreak
  correctly, and is not too old to be compatible.%
      }%
    }{%
      \RequirePackage{amsfonts}[1995/01/01]\relax
    }%
  }
\fi % end yesamsfonts branch
\let\cleardouble@page\cleardoublepage
\AtBeginDocument{%
  \ifx\cleardouble@page\cleardoublepage
    \def\cleardoublepage{\clearpage{\pagestyle{empty}\cleardouble@page}}
  \fi
}
\newcommand{\uppercasenonmath}[1]{\toks@\@emptytoks
  \@xp\@skipmath\@xp\@empty#1$$%
  \edef#1{{\@nx\protect\@nx\@upprep\the\toks@}}%
}
\newcommand{\@upprep}{%
  \spaceskip1.3\fontdimen2\font plus1.3\fontdimen3\font
  \upchars@}
\newcommand{\upchars@}{%
  \def\ss{SS}\def\i{I}\def\j{J}\def\ae{\AE}\def\oe{\OE}%
  \def\o{\O}\def\aa{\AA}\def\l{\L}\def\Mc{M{\scshape c}}}
\providecommand{\Mc}{Mc}
\newcommand{\@skipmath}{}
\long\def\@skipmath#1$#2${%
  \@xskipmath#1\(\)%
  \@ifnotempty{#2}{\toks@\@xp{\the\toks@$#2$}\@skipmath\@empty}}%
\newcommand{\@xskipmath}{}
\long\def\@xskipmath#1\(#2\){%
  \uppercase{\toks@\@xp\@xp\@xp{\@xp\the\@xp\toks@#1}}%
  \@ifnotempty{#2}{\toks@\@xp{\the\toks@\(#2\)}\@xskipmath\@empty}}%
\def\altucnm#1{%
  \MakeTextUppercase{\toks@{#1}}%
  \edef#1{\the\toks@}%
}
\AtBeginDocument{%
  \@ifundefined{MakeTextUppercase}{}{\let\uppercasenonmath\altucnm}%
}
\@ifundefined{MakeUppercase}{\let\MakeUppercase\uppercase}{}%
\newcommand{\today}{%
  \relax\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
%%\if@compatibility
%%\else
%%  \def\@obsolete@fontswitch#1#2#3{%
%%    \@latex@warning@no@line{%
%%      Command {\string#1...}\on@line\space is obsolete;\MessageBreak
%%      the LaTeX2e equivalent is \string#3{...}}%
%%    \gdef#1{\@fontswitch\relax#3}%
%%  }
%%  \DeclareRobustCommand*\cal{%
%%    \@xp\@obsolete@fontswitch\csname cal \endcsname\relax\mathcal}
%%  \DeclareRobustCommand*\mit{%
%%    \@xp\@obsolete@fontswitch\csname mit \endcsname\relax\mathnormal}
%%\fi
\renewcommand*{\title}[2][]{\gdef\shorttitle{#1}\gdef\@title{#2}}
\edef\title{\@nx\@dblarg
  \@xp\@nx\csname\string\title\endcsname}
\renewcommand{\author}[2][]{%
  \ifx\@empty\authors
    \gdef\authors{#2}%
  \else
    \g@addto@macro\authors{\and#2}%
    \g@addto@macro\addresses{\author{}}%
  \fi
  \@ifnotempty{#1}{%
    \ifx\@empty\shortauthors
      \gdef\shortauthors{#1}%
    \else
      \g@addto@macro\shortauthors{\and#1}%
    \fi
  }%
}
\edef\author{\@nx\@dblarg
  \@xp\@nx\csname\string\author\endcsname}
\let\shortauthors\@empty   \let\authors\@empty
\newif\ifresetcontrib  \resetcontribfalse
\newcommand\contrib[2][]{%
  \def\@tempa{#1}%
  \ifx\@empty\@tempa
  \else
    \ifresetcontrib \@xcontribs
    \else \global\resetcontribtrue
    \fi
  \fi
  \ifx\@empty\contribs
    \gdef\contribs{#1 #2}%
  \else
    \g@addto@macro\contribs{\and#1 #2}%
  \fi
  \@wraptoccontribs{#1}{#2}%
}
\def\wraptoccontribs#1#2{}
\def\@xcontribs{%
  \author@andify\contribs
  \ifx\@empty\xcontribs
    \xdef\xcontribs{\contribs}%
  \else
    \xdef\xcontribs{\xcontribs, \contribs}%
  \fi
  \let\contribs\@empty
}
\let\contribs\@empty  \let\xcontribs\@empty  \let\toccontribs\@empty
\let\addresses\@empty      \let\thankses\@empty
\newcommand{\address}[2][]{\g@addto@macro\addresses{\address{#1}{#2}}}
\newcommand{\curraddr}[2][]{\g@addto@macro\addresses{\curraddr{#1}{#2}}}
\newcommand{\email}[2][]{\g@addto@macro\addresses{\email{#1}{#2}}}
\newcommand{\urladdr}[2][]{\g@addto@macro\addresses{\urladdr{#1}{#2}}}
\long\def\thanks@warning#1{%
  \ClassError{\@classname}{%
    \protect\thanks\space should be given separately, not inside author name.%
  }\@ehb
}
\renewcommand{\thanks}[1]{%
  \@ifnotempty{#1}{\g@addto@macro\thankses{\thanks{#1}}}%
}
\def\enddoc@text{\ifx\@empty\@translators \else\@settranslators\fi
  \ifx\@empty\addresses \else\@setaddresses\fi}
\AtEndDocument{\enddoc@text}
\def\curraddrname{{\itshape Current address}}
\def\emailaddrname{{\itshape Email address}}
\def\urladdrname{{\itshape URL}}
\def\@setaddresses{\par
  \nobreak \begingroup
\footnotesize
  \def\author##1{\nobreak\addvspace\bigskipamount}%
  \def\\{\unskip, \ignorespaces}%
  \interlinepenalty\@M
  \def\address##1##2{\begingroup
    \par\addvspace\bigskipamount\indent
    \@ifnotempty{##1}{(\ignorespaces##1\unskip) }%
    {\scshape\ignorespaces##2}\par\endgroup}%
  \def\curraddr##1##2{\begingroup
    \@ifnotempty{##2}{\nobreak\indent\curraddrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      ##2\par}\endgroup}%
  \def\email##1##2{\begingroup
    \@ifnotempty{##2}{\nobreak\indent\emailaddrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      \ttfamily##2\par}\endgroup}%
  \def\urladdr##1##2{\begingroup
    \def~{\char`\~}%
    \@ifnotempty{##2}{\nobreak\indent\urladdrname
      \@ifnotempty{##1}{, \ignorespaces##1\unskip}\/:\space
      \ttfamily##2\par}\endgroup}%
  \addresses
  \endgroup
}
\let\@date\@empty
\def\dedicatory#1{\def\@dedicatory{#1}}
\let\@dedicatory=\@empty
\def\keywords#1{\def\@keywords{#1}}
\let\@keywords=\@empty
\newcommand*\subjclass[2][2020]{%
  \def\@subjclass{#2}%
  \@ifundefined{subjclassname@#1}{%
    \ClassWarning{\@classname}{Unknown edition (#1) of Mathematics
      Subject Classification; using '2020'.}%
  }{%
    \@xp\let\@xp\subjclassname\csname subjclassname@#1\endcsname
  }%
}
\let\@subjclass=\@empty
\def\translname{Translated by}
\def\translator#1{%
  \ifx\@empty\@translators \def\@translators{#1}%
  \else\g@addto@macro\@translators{\and#1}\fi}
\let\@translators=\@empty
\def\@settranslators{\par\begingroup
  \addvspace{6\p@\@plus9\p@}%
  \hbox to\columnwidth{\hss\normalfont\normalsize
    \translname{ }%
    \andify\@translators \uppercasenonmath\@translators
    \@translators}
  \endgroup
}
\newcommand{\xandlist}[4]{\@andlista{{#1}{#2}{#3}}#4\and\and}
\def\@andlista#1#2\and#3\and{\@andlistc{#2}\@ifnotempty{#3}{%
  \@andlistb#1{#3}}}
\def\@andlistb#1#2#3#4#5\and{%
  \@ifempty{#5}{%
    \@andlistc{#2#4}%
  }{%
    \@andlistc{#1#4}\@andlistb{#1}{#3}{#3}{#5}%
  }}
\let\@andlistc\@iden
\newcommand{\nxandlist}[4]{%
  \def\@andlistc##1{\toks@\@xp{\the\toks@##1}}%
  \toks@{\toks@\@emptytoks \@andlista{{#1}{#2}{#3}}}%
  \the\@xp\toks@#4\and\and
  \edef#4{\the\toks@}%
  \let\@andlistc\@iden}
\def\@@and{and}
\newcommand{\andify}{%
  \nxandlist{\unskip, }{\unskip{} \@@and~}{\unskip, \@@and~}}
\def\and{\unskip{ }\@@and{ }\ignorespaces}
\def\maketitle{\par
  \@topnum\z@ % this prevents figures from falling at the top of page 1
  \@setcopyright
  \thispagestyle{firstpage}% this sets first page specifications
  \uppercasenonmath\shorttitle
  \ifx\@empty\shortauthors \let\shortauthors\shorttitle
  \else \andify\shortauthors
  \fi
  \@maketitle@hook
  \begingroup
  \@maketitle
  \toks@\@xp{\shortauthors}\@temptokena\@xp{\shorttitle}%
  \toks4{\def\\{ \ignorespaces}}% defend against questionable usage
  \edef\@tempa{%
    \@nx\markboth{\the\toks4
      \@nx\MakeUppercase{\the\toks@}}{\the\@temptokena}}%
  \@tempa
  \endgroup
  \c@footnote\z@
  \@cleartopmattertags
}
\def\@cleartopmattertags{%
  \def\do##1{\let##1\relax}%
  \do\maketitle \do\@maketitle \do\title \do\@xtitle \do\@title
  \do\author \do\@xauthor \do\address \do\@xaddress
  \do\contrib \do\contribs \do\xcontribs \do\toccontribs
  \do\email \do\@xemail \do\curraddr \do\@xcurraddr
  \do\dedicatory \do\@dedicatory \do\thanks \do\thankses
  \do\keywords \do\@keywords \do\subjclass \do\@subjclass
}
\def\@maketitle@hook{\global\let\@maketitle@hook\@empty}
\def\@maketitle{%
  \normalfont\normalsize
  \@adminfootnotes
  \@mkboth{\@nx\shortauthors}{\@nx\shorttitle}%
  \global\topskip8pc\relax % 10pc to base of first title line
  \@settitle
  \ifx\@empty\authors \else \@setauthors \fi
  \ifx\@empty\@dedicatory
  \else
    \baselineskip26\p@
    \vtop{\centering{\footnotesize\itshape\@dedicatory\@@par}%
      \global\dimen@i\prevdepth}\prevdepth\dimen@i
  \fi
  \@setabstract
  \normalsize
  \if@titlepage
    \newpage
  \else
    \dimen@34\p@ \advance\dimen@-\baselineskip
    \vskip\dimen@\relax
  \fi
} % end \@maketitle
\def\@adminfootnotes{%
  \let\@makefnmark\relax  \let\@thefnmark\relax
  \ifx\@empty\@subjclass\else \@footnotetext{\@setsubjclass}\fi
  \ifx\@empty\@keywords\else \@footnotetext{\@setkeywords}\fi
  \ifx\@empty\thankses\else \@footnotetext{%
    \def\par{\let\par\@par}\@setthanks}%
  \fi
}
\AtBeginDocument{%
  \@ifundefined{publname}{%
    \let\publname\@empty
    \let\@serieslogo\@empty
  }{%
    \def\@serieslogo{\procart@logo}%
  }%
}
\AtBeginDocument{%
  \@ifundefined{volinfo}{\let\volinfo\@empty}{}
}
\def\issueinfo#1#2#3#4{\def\currentvolume{#1}\def\currentissue{#2}%
  \def\currentmonth{#3}\def\currentyear{#4}}
\issueinfo{00}% volume number
  {0}%        % issue number
  {Xxxx}%     % month
  {XXXX}%     % year
\newcommand{\copyrightinfo}[2]{%
  \def\copyrightyear{#1}%
  \@ifnotempty{#2}{\def\copyrightholder{#2}}%
}
\copyrightinfo{0000}{(copyright holder)}
\def\pagespan#1#2{\setcounter{page}{#1}%
  \ifnum\c@page<\z@ \pagenumbering{roman}\setcounter{page}{-#1}\fi
  \def\start@page{#1}\def\end@page{#2}}
\pagespan{000}{000}
\newinsert\copyins
\skip\copyins=1.5pc
\count\copyins=1000 % magnification factor, 1000 = 100%
\dimen\copyins=.5\textheight % maximum allowed per page
\g@addto@macro\@reinserts{%
  \ifvoid\copyins\else\insert\copyins{\unvbox\copyins}\fi
}
\def\@copyinsfontsize{\fontsize{6}{7\p@}\normalfont\upshape}
\newif\if@extracrline  \@extracrlinefalse
\let\@extracrline\@empty
\relax
\def\@setcopyright{%
  \ifx\@empty\@serieslogo
  \else\ifx\@empty\copyrightyear
  \else
    \insert\copyins{\hsize\textwidth
      \parfillskip\z@\relax
      \leftskip\z@\@plus.9\textwidth\relax \rightskip\z@\relax
      \@copyinsfontsize
      \everypar{}%
      \vskip-\skip\copyins
      \if@extracrline
        \vskip-6pt
      \fi
      \nointerlineskip
      \leavevmode\hfill\vrule\@width\z@\@height\skip\copyins
      \copyright\copyrightyear\ \copyrightholder\ignorespaces
      \if@extracrline \@extracrline \fi
      \par
      \kern\z@}%
  \fi\fi
}
\def\@combinefloats{%
  \ifx \@toplist\@empty \else \@cflt \fi
  \ifx \@botlist\@empty \else \@cflb \fi
  \ifvoid\copyins \else \@cflci \fi
}
\def\@cflci{%
  \setbox\@outputbox\vbox{%
    \unvbox\@outputbox
    \vskip\skip\copyins
    \if@twocolumn \else \vskip\z@ plus\p@ \fi
    \hbox to\columnwidth{%
      \hss\vbox to\z@{\vss
        \if@twocolumn
          \if@firstcolumn \else \unvbox\copyins \fi
        \else
          \unvbox\copyins
        \fi
        }}}%
  \if@twocolumn \if@firstcolumn
    \insert\copyins{\unvbox\copyins}%
  \fi\fi
  \global\count\copyins=999 \relax
}
\newcommand{\abstractname}{Abstract}
\newcommand{\keywordsname}{Key words and phrases}
\@namedef{subjclassname@1991}{%
  \textup{1991} Mathematics Subject Classification}
\@namedef{subjclassname@2000}{%
  \textup{2000} Mathematics Subject Classification}
\@namedef{subjclassname@2010}{%
  \textup{2010} Mathematics Subject Classification}
\@namedef{subjclassname@2020}{%
  \textup{2020} Mathematics Subject Classification}
\@xp\let\@xp\subjclassname\csname subjclassname@2020\endcsname
\def\@tempb{amsproc}
\ifx\@classname\@tempb
  \newcommand{\datename}{\textit{Date}:}
\else
  \newcommand{\datename}{Received by the editors}
\fi
\def\@settitle{\begin{center}%
 \Large
    \bfseries
  \@title
  \end{center}%
}
\def\author@andify{%
  \nxandlist {\unskip ,\penalty-1 \space\ignorespaces}%
    {\unskip {} \@@and~}%
    {\unskip ,\penalty-2 \space \@@and~}%
}
\def\@setauthors{%
  \begingroup
  \def\thanks{\protect\thanks@warning}%
  \trivlist
  \centering \fontsize{11}{13\p@}\selectfont\@topsep13\p@\relax
  \item\relax
  \author@andify\authors
  \def\\{\protect\linebreak}%
  \authors
  \ifx\@empty\contribs
  \else
    ,\penalty-3 \space \@setcontribs
    \@closetoccontribs
  \fi
  \endtrivlist
  \endgroup
}
\def\@closetoccontribs{}
\def\@setcontribs{%
  \@xcontribs
  \xcontribs
}
\def\@setdate{\datename\ \@date\@addpunct.}
\def\@setsubjclass{%
  {\itshape\subjclassname.}\enspace\@subjclass\@addpunct.}
\def\@setkeywords{%
  {\itshape \keywordsname.}\enspace \@keywords\@addpunct.}
\def\@setthanks{\def\thanks##1{\par##1\@addpunct.}\thankses}
\newbox\abstractbox
\newenvironment{abstract}{%
  \ifx\maketitle\relax
    \ClassWarning{\@classname}{Abstract should precede
      \protect\maketitle\space in AMS document classes; reported}%
  \fi
  \global\setbox\abstractbox=\vtop \bgroup
    \normalfont\Small
    \list{}{\labelwidth\z@
      \leftmargin3pc \rightmargin\leftmargin
      \listparindent\normalparindent \itemindent\z@
      \parsep\z@ \@plus\p@
      \let\fullwidthdisplay\relax
    }%
    \item[\hskip\labelsep\scshape\abstractname.]%
}{%
  \endlist\egroup
  \ifx\@setabstract\relax \@setabstracta \fi
}
\def\@setabstract{\@setabstracta \global\let\@setabstract\relax}
\def\@setabstracta{%
  \ifvoid\abstractbox
  \else
    \skip@20\p@ \advance\skip@-\lastskip
    \advance\skip@-\baselineskip \vskip\skip@
    \box\abstractbox
    \prevdepth\z@ % because \abstractbox is a vtop
  \fi
}
\def\titlepage{%
  \clearpage
  \thispagestyle{empty}\setcounter{page}{0}}
\def\endtitlepage{\newpage}
\def\labelenumi{(\theenumi)}
\def\theenumi{\@arabic\c@enumi}
\def\labelenumii{(\theenumii)}
\def\theenumii{\@alph\c@enumii}
\def\p@enumii{\theenumi}
\def\labelenumiii{(\theenumiii)}
\def\theenumiii{\@roman\c@enumiii}
\def\p@enumiii{\theenumi(\theenumii)}
\def\labelenumiv{(\theenumiv)}
\def\theenumiv{\@Alph\c@enumiv}
\def\p@enumiv{\p@enumiii\theenumiii}
\def\labelitemi{$\m@th\bullet$}
\def\labelitemii{\bfseries --}% \upshape already done by \itemize
\def\labelitemiii{$\m@th\ast$}
\def\labelitemiv{$\m@th\cdot$}
\newenvironment{verse}{\let\\\@centercr
  \list{}{\itemsep\z@ \itemindent -1.5em\listparindent\itemindent
  \rightmargin\leftmargin \advance\leftmargin 1.5em}\item[]%
}{%
  \endlist
}
\let\endverse=\endlist % for efficiency
\newenvironment{quotation}{\list{}{%
    \leftmargin3pc \listparindent\normalparindent
    \itemindent\z@
    \rightmargin\leftmargin \parsep\z@ \@plus\p@}%
  \item[]%
}{%
  \endlist
}
\let\endquotation=\endlist % for efficiency
\newenvironment{quote}{%
  \list{}{\rightmargin\leftmargin}\item[]%
}{%
  \endlist
}
\let\endquote=\endlist % for efficiency
\def\trivlist{\parsep\parskip\@nmbrlistfalse
  \@trivlist \labelwidth\z@ \leftmargin\z@
  \itemindent\z@
  \let\@itemlabel\@empty
  \def\makelabel##1{\upshape##1}}
\renewenvironment{enumerate}{%
  \ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{\hss\llap{\upshape##1}}}\fi
}{%
  \endlist
}
\let\endenumerate=\endlist % for efficiency
\renewenvironment{itemize}{%
  \ifnum\@itemdepth>3 \@toodeep
  \else \advance\@itemdepth\@ne
    \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
    \list{\csname\@itemitem\endcsname}%
      {\def\makelabel##1{\hss\llap{\upshape##1}}}%
  \fi
}{%
  \endlist
}
\let\enditemize=\endlist % for efficiency
\newcommand{\descriptionlabel}[1]{\hspace\labelsep \upshape\bfseries #1:}
\newenvironment{description}{\list{}{%
  \advance\leftmargini6\p@ \itemindent-12\p@
  \labelwidth\z@ \let\makelabel\descriptionlabel}%
}{
  \endlist
}
\let\enddescription=\endlist % for efficiency
\let\upn=\textup
\AtBeginDocument{%
  \labelsep=5pt\relax
  \setcounter{enumi}{13}\setcounter{enumii}{13}%
  \setcounter{enumiii}{13}\setcounter{enumiv}{13}%
  \settowidth\leftmargini{\labelenumi\hskip\labelsep}%
  \advance\leftmargini by \normalparindent
  \settowidth\leftmarginii{\labelenumii\hskip\labelsep}%
  \settowidth\leftmarginiii{\labelenumiii\hskip\labelsep}%
  \settowidth\leftmarginiv{\labelenumiv\hskip\labelsep}%
  \setcounter{enumi}{0}\setcounter{enumii}{0}%
  \setcounter{enumiii}{0}\setcounter{enumiv}{0}%
  \leftmarginv=10pt  \leftmarginvi=\leftmarginv
  \leftmargin=\leftmargini
  \labelwidth=\leftmargini \advance\labelwidth-\labelsep
  \@listi}
\newskip\listisep
\listisep\smallskipamount
\def\@listI{\leftmargin\leftmargini \parsep\z@skip
  \topsep\listisep \itemsep\z@skip
  \listparindent\normalparindent}
\let\@listi\@listI
\def\@listii{\leftmargin\leftmarginii
  \labelwidth\leftmarginii \advance\labelwidth-\labelsep
  \topsep\z@skip \parsep\z@skip \partopsep\z@skip \itemsep\z@skip}
\def\@listiii{\leftmargin\leftmarginiii
  \labelwidth\leftmarginiii \advance\labelwidth-\labelsep}
\def\@listiv{\leftmargin\leftmarginiv
  \labelwidth\leftmarginiv \advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
  \labelwidth\leftmarginv \advance\labelwidth-\labelsep}
\def\@listvi{\leftmargin\leftmarginvi
  \labelwidth\leftmarginvi \advance\labelwidth-\labelsep}
\@ifclasswith{\@classname}{fleqn}{%
  \let\@tempa\@iden
  \AtBeginDocument{\mathindent\leftmargini}%
}{\let\@tempa\@gobble}%
\@ifpackageloaded{amsmath}{\let\@tempa\@gobble}{%
  \@ifpackageloaded{amstex}{\let\@tempa\@gobble}{}%
}
\@tempa{%
  \def\[{\relax
    \ifmmode\@badmath
    \else
      \begin{trivlist}%
      \@beginparpenalty\predisplaypenalty
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    \item[]\leavevmode
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}
\def\@seccntformat#1{%
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\let\@secnumfont\@empty
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  \fi
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\let\@ssect\relax
\newcounter{part}
\newcounter{section}
\newcounter{subsection}[section]
\newcounter{subsubsection}[subsection]
\newcounter{paragraph}[subsubsection]
\newcounter{subparagraph}[paragraph]
\renewcommand\thepart          {\arabic{part}}
\renewcommand\thesection       {\arabic{section}}
\renewcommand\thesubsection    {\thesection.\arabic{subsection}}
\renewcommand\thesubsubsection {\thesubsection .\arabic{subsubsection}}
\renewcommand\theparagraph     {\thesubsubsection.\arabic{paragraph}}
\renewcommand\thesubparagraph  {\theparagraph.\arabic{subparagraph}}
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\def\partname{Part}
\def\part{\@startsection{part}{0}%
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\def\specialsection{\@startsection{section}{1}%
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\def\section{\@startsection{section}{1}%
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\def\subsection{\@startsection{subsection}{2}%
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  {\normalfont\bfseries}}
\def\subsubsection{\@startsection{subsubsection}{3}%
  \normalparindent\z@{-.5em}%
  {\normalfont\itshape}}
\def\paragraph{\@startsection{paragraph}{4}%
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  \normalfont}
\def\subparagraph{\@startsection{subparagraph}{5}%
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  \normalfont}
\def\appendix{\par\c@section\z@ \c@subsection\z@
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   \def\thesection{\@Alph\c@section}}
\def\appendixname{Appendix}
\def\@Roman#1{\@xp\@slowromancap
  \romannumeral#1@}%
\def\@slowromancap#1{\ifx @#1% then terminate
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    \if i#1I\else\if v#1V\else\if x#1X\else\if l#1L\else\if
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}
\def\aufm#1{\par\vspace*{12pt}{\flushright #1\par}}
\newcommand{\@pnumwidth}{1.6em}
\newcommand{\@tocrmarg}{2.6em}
\setcounter{tocdepth}{1}
\newswitch{toc}
\newswitch{lof}
\newswitch{lot}
\newcommand\contentsnamefont{\scshape}
\def\@starttoc#1#2{\begingroup
  \setTrue{#1}%
  \par\removelastskip\vskip\z@skip
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    {.5\linespacing}{\centering\contentsnamefont}{#2}%
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  \global\@nobreakfalse \endgroup
  \addvspace{32\p@\@plus14\p@}%
  \let\tableofcontents\relax
}
\def\contentsname{Contents}
\def\listfigurename{List of Figures}
\def\listtablename{List of Tables}
\def\tableofcontents{%
  \@starttoc{toc}\contentsname
}
\def\listoffigures{\@starttoc{lof}\listfigurename}
\def\listoftables{\@starttoc{lot}\listtablename}
\AtBeginDocument{%
  \@for\@tempa:=-1,0,1,2,3\do{%
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}
\def\@writetocindents{%
  \begingroup
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    \immediate\write\@auxout{%
      \string\newlabel{tocindent\@tempa}{%
        \csname r@tocindent\@tempa\endcsname}}%
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  \endgroup}
\AtEndDocument{\@writetocindents}

\let\indentlabel\@empty
\def\@tochangmeasure#1{\sbox\z@{#1}%
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    \@xp\xdef\csname r@tocindent\@toclevel\endcsname{\the\wd\z@}%
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}
\def\@toclevel{0}
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % then omit
  \else
    \par \addpenalty\@secpenalty\addvspace{#2}%
    \begingroup \hyphenpenalty\@M
    \@ifempty{#4}{%
      \@tempdima\csname r@tocindent\number#1\endcsname\relax
    }{%
      \@tempdima#4\relax
    }%
    \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax
    \rightskip\@pnumwidth plus4em \parfillskip-\@pnumwidth
    #5\leavevmode\hskip-\@tempdima #6\nobreak\relax
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    \endgroup
  \fi}
\def\@tocpagenum#1{\hss{\mdseries #1}}
\def\@tocwrite#1{\@xp\@tocwriteb\csname toc#1\endcsname{#1}}
\def\@tocwriteb#1#2#3{%
  \begingroup
    \def\@tocline##1##2##3##4##5##6{%
      \ifnum##1>\c@tocdepth
      \else \sbox\z@{##5\let\indentlabel\@tochangmeasure##6}\fi}%
    \csname l@#2\endcsname{#1{\csname#2name\endcsname}{\@secnumber}{}}%
  \endgroup
  \addcontentsline{toc}{#2}%
    {\protect#1{\csname#2name\endcsname}{\@secnumber}{#3}}}
\def\l@section{\@tocline{1}{0pt}{1pc}{}{}}
\newcommand{\tocsection}[3]{%
  \indentlabel{\@ifnotempty{#2}{\ignorespaces#1 #2.\quad}}#3}
\def\l@subsection{\@tocline{2}{0pt}{1pc}{5pc}{}}
\let\tocsubsection\tocsection
\def\l@subsubsection{\@tocline{3}{0pt}{1pc}{7pc}{}}
\let\tocsubsubsection\tocsection
\let\l@paragraph\l@subsubsection
\let\tocparagraph\tocsection
\let\l@subparagraph\l@subsubsection
\let\tocsubparagraph\tocsection
\def\l@part{\@tocline{-1}{12pt plus2pt}{0pt}{}{\bfseries}}
\let\tocpart\tocsection
\def\l@chapter{\@tocline{0}{8pt plus1pt}{0pt}{}{}}
\let\tocchapter\tocsection
\newcommand{\tocappendix}[3]{%
  \indentlabel{#1\@ifnotempty{#2}{ #2}.\quad}#3}
\def\l@figure{\@tocline{0}{3pt plus2pt}{0pt}{1.5pc}{}}
\let\l@table=\l@figure
\def\refname{References}
\def\bibname{Bibliography}
\def\@defaultbiblabelstyle#1{#1.}
\def\bibliographystyle#1{%
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    \def\@tempa{#1}%
    \def\@tempb{amsplain}%
    \def\@tempc{}%
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        \def\@biblabel##1{\@defaultbiblabelstyle{##1}}%
        \def\bibsetup{}%
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        \def\bibsetup{\labelsep6\p@}%
        \ifx\@tempa\@tempc
            \def\@biblabel##1{}%
            \def\bibsetup{\labelwidth\z@ \leftmargin24\p@
                \itemindent-\leftmargin
                \labelsep\z@ }%
        \fi
    \fi}
\newcommand{\bibliofont}{\footnotesize}
\newcommand{\@bibtitlestyle}{%
  \@xp\section\@xp*\@xp{\refname}%
}
\newenvironment{thebibliography}[1]{%
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  \normalfont\bibliofont\labelsep .5em\relax
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    \usecounter{enumiv}}%
  \sloppy \clubpenalty\@M \widowpenalty\clubpenalty
  \sfcode`\.=\@m
}{%
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  \endlist
}
\def\bysame{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\def\newblock{}
\newcommand\MR[1]{\relax\ifhmode\unskip\spacefactor3000 \space\fi
  MR~\MRhref{#1}{#1}}
\let\MRhref\@gobble
\newcommand\URL{\begingroup
  \def\@sverb##1{%
    \def\@tempa####1##1{\@URL{####1}\egroup\endgroup}%
    \@tempa}%
  \verb}
\let\URLhref\@gobble
\def\@URL#1{\URLhref{#1}#1}
\newif\i

references.bib707 B

@book{rudin1976principles,
  author    = {Rudin, Walter},
  title     = {Principles of Mathematical Analysis},
  edition   = {Third},
  publisher = {McGraw-Hill},
  address   = {New York},
  year      = {1976}
}

@book{granas2003fixed,
  author    = {Granas, Andrzej and Dugundji, James},
  title     = {Fixed Point Theory},
  series    = {Springer Monographs in Mathematics},
  publisher = {Springer},
  address   = {New York},
  year      = {2003}
}

@article{banach1922,
  author  = {Banach, Stefan},
  title   = {Sur les op\'erations dans les ensembles abstraits et leur application aux \'equations int\'egrales},
  journal = {Fund. Math.},
  volume  = {3},
  pages   = {133--181},
  year    = {1922}
}

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