Theory and Applications of Categories (tac)

\documentclass{tac}

% Prepared using tac.cls and diagxy (if you do not have diagxy, compiling this will require commenting lines 165-168)
% PLEASE READ comments in the text below
% PLEASE NOTE: source files for submission to TAC require a comment like the following
%              giving style, packages used, TeX implementation
% TAC style, 2 pp, Xy-pic ver 3.7, MikTeX version 3.1


% NOTE: packages used should be the first part of the preamble

\usepackage{xy}
% diagxy loads xy, so the line preceding is redundant

\input diagxy

\def\xypic{\hbox{\rm\Xy-pic}}

% The TAC hyperref setup should be reloaded after other packages

\usepackage[colorlinks=true]{hyperref}
\hypersetup{allcolors=[rgb]{0.1,0.1,0.4}}

% NOTE: TAC preamble macros come next...

\author{Michael Barr, Geoff Cruttwell, and Robert Rosebrugh}

% NOTE: that \thanks is outside the \author macro, unlike article style...

\thanks{We would like to thank Donald Knuth, Leslie Lamport, Kris
Rose, and all the others who make high quality math typesetting
possible.}

\address{Department of Mathematics and Statistics, McGill University\\
 805 Sherbrooke St. W, Montreal, QC, Canada H3A 2K6\\[5pt]
 Department of Mathematics and Computer Science, Mt. Allison
University\\67 York St., Sackville, NB, Canada E4L 1E6\\
}


\title{Your paper}

% NOTE: this is required...

\copyrightyear{2020}

% NOTE: the next three are optional in this style (but are required for TAC publication)

\keywords{TAC, diagrams}
\amsclass{00A00}

% NOTE: that \CR here provides a vertical listing
\eaddress{[email protected]\CR [email protected]}


% NOTE: author macros  BEGIN here
%       (they are all actually used in the article!!)
% *PLEASE* note the begin and end of author macros in the source file

\def\xypic{\hbox{\rm\Xy-pic}}

% examples of proclamation macros follow - these define environments like \begin{thm} ... \end{thm}
% by default they are italic, but roman proclamations for remarks, examples are also available
% with \newtheoremrm{}{}

\newtheorem{thm}{Theorem}
% note that \newtheorem{theorem}{Theorem} and several others are already in tac.sty
\newtheorem{theorem}{Theorem} 
\newtheorem{lem}{Lemma}

% note rm This is useful for remarks and examples
\newtheoremrm{rem}{Remark}

% TAC has a \proof environment, with abbreviations following; note also \mathrmdef{} and \mathbfdef{}

\let\pf\proof
\let\epf\endproof
\mathrmdef{Hom}
\mathbfdef{Set}

% author macros END here

\begin{document}

\maketitle
\begin{abstract}
 This is distributed as a sampler to illustrate good \textsl{TAC} style.
\end{abstract}

% NOTE: it is good practice to \label all headings (and proclamations) immediately

\section{Introduction}\label{sec-Introduction}

This note includes samples of what we consider good \textsl{TAC} style.  
There are no explicit skips nor any other explicit formatting
instructions in the \LaTeX\ code; these should be left to the journal style.  For the same
reason, the \LaTeX\ code has no explicit numbering of headings or proclamations of
Theorems and so forth. However, they \emph{are} labelled to allow
logical references in the code, such as to Theorem \ref{thm-easy-style}. 
Moreover, TAC style uses the hyperref package to create links to citations like 
\cite{LUG} and to internal references like Lemma \ref{lem-latex-2e}. 
Reference links work \emph{only} if \verb.\label.'s are used. All links are coloured a dark blue without boxing. 
Though permitted, external links are strongly deprecated because of their impermanence.

\emph{Please note that there are additional comments in the source file \texttt{sample.tex}
for this sampler that you are urged to consult. You should \textbf{also} consult
the on-line author instructions on the {\sl TAC} web site.}

\section{Main results}\label{sec-Main-results}

%NOTE: more verbosely, the next line could use \begin{lem} ... \end{lem}

\lem\label{lem-latex-2e} All papers must be in \LaTeX, version 2e.\endlem

\pf Otherwise the editors would have to do a lot of work to prepare the
paper for publication.
\epf


\thm\label{thm-easy-style} The {\sl TAC} style is easy to use.\endthm

\pf
Sectioning is the same as in \LaTeX\ article style;
proclamations such as definitions and theorems are easily specified
by macros such as \verb.\newtheorem{thm}{Theorem}.;
it is easy to use \verb.\mathrmdef{Hom}. to define a macro \verb.\Hom. that produces
roman $\Hom$ when used in math mode.  Similarly \verb.\mathbfdef{Set}. gives $\Set$
in bold.
\epf

% NOTE: If you use \cite at the beginning of a theorem-like environment,
% it must come immediately after the theorem and before any label.

\thm[\cite{LUG}]\label{thm-Lamport-1986}  The following are equivalent
\begin{enumerate}
\item Lists are best done with listing macros such as enumerate;

\item you will never have to renumber anything if you use automatic
numbering of lists and other things.
\end{enumerate}

\epf\endthm

\begin{rem}\label{rem-about-roman}
For proclaimed matter that should be set in Roman we use the {\sl TAC} macros \verb.\newtheoremrm{}{}..
\end{rem}

\section{Further comments}\label{sec-Further-comments}


\begin{list}{}{}
\item[$\bullet$] Source files should include \emph{all} and
\emph{only} author macros that are actually used. \item[$\bullet$]
Be sure to use macros for multicharacter identifiers, such as
\verb.\Hom. above. 
\item[$\bullet$] Be sure to distinguish between
$<$ and $\langle$ and similarly between $>$ and $\rangle$.  Not
only does the former character look wrong as a tuple delimiter,
but the spacing is completely wrong. 
\item[$\bullet$] For diagrams, use \xypic\ or diagxy.% , or Paul Taylor's diagram package.
They can be used together since diagxy is built on top of
\xypic. A syntax example from the diagxy manual is
\begin{verbatim}
$$\bfig
  \square/>>`>`>` >->/[A`B`C`D;e`f`g`m]
  \morphism(500,500)|m|/.>/<-500,-500>[B`C;h]
\efig$$
\end{verbatim}
which makes a familiar diagram:
% NOTE: If you do not have XY-pic and diagxy installed you will have to comment out
%       the next four lines.

$$\bfig
 \square/>>`>`>` >->/[A`B`C`D;e`f`g`m]
 \morphism(500,500)|m|/.>/<-500,-500>[B`C;h]
\efig$$
\item[$\bullet$] Although we accept most reasonable bibliographical styles, the
 following is the one we most strongly recommend.  It results in an 
 [\emph{author, year}] entry in the paper, rather than uninformative numbers in
 brackets. It allows use of e.g. \verb.\cite{LUG}. and the code for this article is:
\begin{verbatim}
\refs

\bibitem [Lamport, 1986]{LUG} L. Lamport, Latex User's Guide \&
Reference Manual. Addison-Wesley (fifth edition), 1986.

\endrefs
\end{verbatim}
\end{list}

 % Note: Although we accept most reasonable bibliographical styles, the
 % following is the one we most strongly recommend.  It results in the
 % author, year entry in the paper, rather than uninformative numbers in
 % brackets.

\refs

\bibitem [Lamport, 1986]{LUG} L. Lamport, Latex User's Guide \&
Reference Manual. Addison-Wesley (fifth edition), 1986.

\endrefs



\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tac.cls for use by TAC authors
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% Copyright 2024 by The Editors of Theory and Applications of Categories
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% 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 1.3 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 Geoff Cruttwell, Managing Editor.
%
% This work consists of the file tac.cls.


\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{tac}[2023/03/06]

\LoadClass[12pt]{article}

\DeclareOption{compromisepaper}
   {\setlength\paperheight {11.4in}%
    \setlength\paperwidth  {8.25in}}
\ProcessOptions\relax
\usepackage{xspace,mathrsfs}

\setcounter{tocdepth}{1}
\def\tableofcontents{\section*{Contents\@mkboth{CONTENTS}{CONTENTS}}
 \@starttoc{toc}}
\def\l@part#1#2{\addpenalty{\@secpenalty}
 \addvspace{2.25em plus 1\p@} \begingroup
 \@tempdima 3em \parindent \z@ \rightskip \@pnumwidth \parfillskip
-\@pnumwidth
 {\large \textbf \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}}\par
 \nobreak \endgroup}
\def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{0em}%{1.0em plus 1\p@} mod 00-4-15
\@tempdima 1.5em \begingroup
 \parindent \z@ \rightskip \@pnumwidth
 \parfillskip -\@pnumwidth
 \textbf \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip #1\nobreak\hfil
\nobreak\hbox to\@pnumwidth{\hss #2}\par
 \endgroup}

%%%%%%%%%%%% Style parameters
\lineskip 1pt
\normallineskip 1pt
\def\baselinestretch{1}
\marginparsep 10pt
\topmargin 0pt
\headheight 12pt
\headsep 15pt
\topskip = 0pt
\footskip 20pt
\textheight = 8.5in
\textwidth 6.3in
\oddsidemargin0pt
\evensidemargin0pt

%%%%%%%%%%%%%%%% Basic formatting

% Add a period after (sub)section numbers.
\def\@seccntformat#1{\csname the#1\endcsname.\kern.5em}



\def\section{\@startsection {section}{1}{\z@}{-4ex plus -1ex minus
-.2ex}{1.75ex plus .2ex}{\large}}
\def\@subsection{\@startsection{subsection}{2}{\z@}{-1.5ex plus -1ex
minus  -.2ex}{-0.3em plus -0.1em minus
-.1em}{\normalfont\normalsize\scshape}}
\def\@subsubsection{\@startsection{subsubsection}{3}{\z@}{-1.25ex plus
-1ex minus-.2ex}{-0.3em plus -.1em minus -.1em}%
{\normalfont\normalsize\scshape}}

 % The following code is to add a period at the end of any sub(sub)section
 % title not otherwise punctuate.  \smartperiod looks at the spacefactor at
 % the end of the title, which is > 1000 for punctuation and = 1000 for all
 % others.  Two warnings: It doesn't work if the header is separated, which
 % is not the case in tac.cls and it doesn't work if you use \frenchspacing
 % which sets all the \sfcodes to 1000, also not used here. Start by
 % undoing the kludge Knuth did to avoid an unlikely problem.  Note that
 % the bibliographic code includes \frenchspacing.
 \sfcode`\A=1000
 \sfcode`\B=1000
 \sfcode`\C=1000
 \sfcode`\D=1000
 \sfcode`\E=1000
 \sfcode`\F=1000
 \sfcode`\G=1000
 \sfcode`\H=1000
 \sfcode`\I=1000
 \sfcode`\J=1000
 \sfcode`\K=1000
 \sfcode`\L=1000
 \sfcode`\M=1000
 \sfcode`\N=1000
 \sfcode`\O=1000
 \sfcode`\P=1000
 \sfcode`\Q=1000
 \sfcode`\R=1000
 \sfcode`\S=1000
 \sfcode`\T=1000
 \sfcode`\U=1000
 \sfcode`\V=1000
 \sfcode`\W=1000
 \sfcode`\X=1000
 \sfcode`\Y=1000
 \sfcode`\Z=1000
\def\smartperiod{\ifnum\the\spacefactor=1000 .\fi}
\def\@starsubsection#1{\@subsection*{#1\smartperiod\ }}
\def\@nostarsubsection#1{\@subsection{#1\smartperiod}}
\def\subsection{\@ifstar{\@starsubsection}{\@nostarsubsection}}
\def\@starsubsubsection#1{\@subsubsection*{#1\smartperiod\ }}
\def\@nostarsubsubsection#1{\@subsubsection{#1\smartperiod}}
\def\subsubsection{\@ifstar{\@starsubsubsection}{\@nostarsubsubsection}}

\def\paragraph{\@startsection
     {paragraph}{4}{\z@}{1ex plus1ex minus.2ex}{-.5em
     plus -.5em minus -.1em}{\reset@font \normalsize\bf}}
\def\subparagraph{\@startsection
     {subparagraph}{4}{\parindent}{.5ex plus .5ex minus
     .2ex}{-.5em plus -.5em minus -.1em}{\reset@font\normalsize\bf}}
\def\appendix{\par
  \setcounter{section}{0}
  \setcounter{subsection}{0}
  \def\thesection{\Alph{section}}}



 % The next four lines allow you to start a theorem by giving credit (or
 % naming it) by a parenthetical note that will set in upright font and
 % enclosed by square brackets.  The syntax is \theorem[...] followed by
 % the statement of the theorem.  If this parenthetical material iteslf
 % contains square brackets, then it must be enclosed in braces.  This is
 % likeliest to occur using \cite[..]{..} in which case the syntax
 % that works is \theorem[{\cite[..]{..}..}], although
 % \theorem[{\cite[..]{..}}..] will also work.  If you do
 % \theorem[\cite{..}] or \theorem[{\cite[..]{..}}] you will _not_ get two
 % pairs of brackets.  Of course, all this works with any theorem like
 % environments and with the latex \begin..\end versions.

\def\@credit[#1]{\expandafter\@@credit\@firstofone#1\@@nil}
\def\@@credit{\@ifnextchar\cite{\@@@credit}{\@@@@credit}}
\def\@@@credit#1\@@nil{{\rm#1}}
\def\@@@@credit#1\@@nil{{\rm[#1]}}

\def\state#1{\subsection{\sc#1}%
\begingroup\em\@ifnextchar[{\@credit}{}}
 \def\staterm#1{\subsection{\sc#1}
\begingroup\@ifnextchar[{\@credit}{}}
\def\endstate{\endgroup\par\addvspace{\medskipamount}}

\def\thirdleveltheorems{%
\def\state##1{\subsubsection{\sc##1}%
\begingroup\em\@ifnextchar[{\@credit}{}}
 \def\staterm##1{\subsubsection{\sc##1}
\begingroup\@ifnextchar[{\@credit}{}}}


\def\firstleveltheorems{%
\def\section{\@startsection {section}{1}{\z@}{-1.5ex plus -1ex
minus  -.2ex}{-1em plus -0.4em minus
-.4em}{\normalfont\normalsize\scshape}}
\def\state##1{\section{\sc##1}%
\begingroup\em\@ifnextchar[{\@credit}{}}
 \def\staterm##1{\subsubsection{\sc##1}
\begingroup\@ifnextchar[{\@credit}{}}}

\def\theoremstyle#1{\def\next{#1}\def\nextt{plain}
\ifx\next\nextt\relax\let\@state\state\else\let\@state\staterm\fi}
\def\newtheorem#1{\@ifnextchar[{\@newthm{#1}}{\@newthm{#1}[]}}
\def\newtheoremrm#1{\@ifnextchar[{\@newthmrm{#1}}{\@newthmrm{#1}[]}}
\def\@newthm#1[#2]#3{\@ifnextchar[{\@@newthm{#1}[#2]{#3}}{\@@newthm{#1}[#2]{#3}[ ]}}
\def\@newthmrm#1[#2]#3{\@ifnextchar[{\@@newthmrm{#1}[#2]{#3}}{\@@newthmrm{#1}[#2]{#3}[ ]}}

\def\@@newthm#1[#2]#3[#4]{\expandafter\def\csname #1\endcsname{%
\state{#3.}\@ifnextchar[{\@credit}{}}%
 \expandafter\let\csname
     end#1\endcsname\endstate}

\def\@@newthmrm#1[#2]#3[#4]{\expandafter\def\csname #1\endcsname{%
\staterm{#3.}\@ifnextchar[{\@credit}{}}%
 \expandafter\let\csname
     end#1\endcsname\endstate}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{definition}{Definition}
\newtheorem{scholium}{Scholium}
\newtheorem{assumption}{Assumption}

\newtheoremrm{remark}{Remark}
\newtheoremrm{example}{Example}
\newtheoremrm{remarks}{Remarks}
\newtheoremrm{examples}{Examples}
\newtheoremrm{notation}{Notation}

\def\substate#1{\subsubsection*{\textit{#1}}}


\def\Pr@@f{\subsubsection*{\textsc{Proof.}}}
\def\pr@@f[#1]{\subsubsection*{{\textsc{Proof}} #1}}
\def\proof{\@ifnextchar[{\pr@@f}{\Pr@@f}}     %

 % \def\proofof[#1]{\subsubsection*{{\textsc{Proof of #1}}}}


\def\Pr@@fsk{\subsubsection*{\textsc{Sketch of Proof.}}}
\def\pr@@fsk[#1]{\subsection*{{\textsc{Sketch of Proof}} #1}}
\def\proofsk{\@ifnextchar[{\pr@@fsk}{\Pr@@fsk}}

\def\pushright#1{{%        set up
   \parfillskip=0pt            % so \par doesnt push \square to left
   \widowpenalty=10000         % so we dont break the page before \square
   \displaywidowpenalty=10000  % ditto
   \finalhyphendemerits=0      % TeXbook exercise 14.32
   \leavevmode                 % \nobreak means lines not pages
   \unskip                     % remove previous space or glue
   \nobreak                    % don't break lines
   \hfil                       % ragged right if we spill over
   \penalty50                  % discouragement to do so
   \hskip.2em                  % ensure some space
   \null                       % anchor following \hfill
   \hfill                      % push \square to right
   {#1}                        % the end-of-proof mark (or whatever)
   \par}}                      % build paragraph

\newbox\@endproofbox
\setbox\@endproofbox\hbox{\vrule width5pt height 5pt depth 0pt}
\def\endproofbox{\copy\@endproofbox}
\def\endproof{\pushright{\endproofbox}%
\penalty-700 \par\addvspace{\medskipamount}}

\def\endproofsk{\pushright{\endproofbox}%
\penalty-700 \par\addvspace{\medskipamount}}

%%%%%%%%%%%%%% Header things
% The following are to produce the page one headers and footnotes
% Note that there must be two passes to get the page numbers correct
% AMS classification, thanks, keywords and revision dates are optional.
% The others are required.

\newif\ifthanks
\newif\ifkeywords
\newif\ifamsclass
\newif\ifdedication
\newif\ifeaddress


\def\@title{\relax}
\def\@author{\relax}
\def\@cauthor{\relax}
\def\runningauthor{\relax}
\def\runningtitle{\relax}
\def\@num{0}
\def\@copryear{\the\year}
\def\@address{\relax}
\def\@eaddress{\relax}
\def\@thanks{\relax}
\def\@keywords{\relax}
\def\@amsclass{\relax}
\def\@runningoddhead{\relax}
\def\@runningevenhead{\relax}
\def\@dedication{\relax}

 \def\@uclcnotmath#1#2#3#4{\begingroup
       #1%
       \def\({$}\let\)\(%$
       \def\NoCaseChange##1{\noexpand\NoCaseChange{\noexpand##1}}
       \@nonchangecase\label
       \@nonchangecase\ref
       \@nonchangecase\ensuremath
       \def\cite##1##{\toks@{\noexpand\cite##1}\@citex}%
       \def\@citex##1{\NoCaseChange{\the\toks@{##1}}}%
       \def\reserved@a##1##2{\let#2\reserved@a}%
       \expandafter\reserved@a\@uclclist\reserved@b{\reserved@b\@gobble}%
       \protected@edef\reserved@a{\endgroup
           \noexpand\@skipmath#3#4$\valign$}%
       \reserved@a}
 \def\@nonchangecase#1{\def#1##1{\NoCaseChange{#1{##1}}}}
 \let\NoCaseChange\@firstofone
 \def\@skipmath#1#2$#3${%
   \@skip@nonchangecase#1#2\NoCaseChange\valign
   \ifx\valign#3%
   \else
     $#3$%
     \expandafter\@skipmath\expandafter#1%
   \fi}
 \def\@skip@nonchangecase#1#2\NoCaseChange#3{%
   #1{#2}%
   \ifx\valign#3%
   \else
     #3%
     \expandafter\@skip@nonchangecase\expandafter#1%
   \fi}
 \DeclareRobustCommand\MakeTextUppercase{%
   \@uclcnotmath{\def\i{I}\def\j{J}}{##1##2}\uppercase}
 \protected@edef\MakeTextUppercase#1{\MakeTextUppercase{#1}}
 \DeclareRobustCommand\MakeTextLowercase{%
   \@uclcnotmath{}{##2##1}\lowercase}
 \protected@edef\MakeTextLowercase#1{\MakeTextLowercase{#1}}

\def\dedication#1{\def\@dedication{{\let\cr=\\ \def\\{\protect\cr[1\p@]}
\large\textit{#1}}}\dedicationtrue}

% 01-12-14 for multiple emails
\def\CR{\\\phantom{\rm Email: }}

\long\def\address#1{\global\def\@address{{\par\addvspace{12\p@}
\baselineskip11.5pt\parindent0pt
\let\newline=\par
\noindent\textit{#1} \\ %\vspace{5pt}
\ifeaddress Email: \texttt \@eaddress\fi }}}
\def\eaddress#1{\def\@eaddress{#1}\eaddresstrue}

{\catcode`\/=\active\gdef/{\char47\penalty-100 }
 \catcode`\.=\active \gdef.{\char46\penalty-100 }
 \catcode`\_=\active \gdef_{\char95\penalty-100 }
}
\def\url{%
   \bgroup
       \catcode`\/=\active
       \catcode`\.=\active
       \catcode`\_=\active
       \def~{$\sim$}\@url}
\def\@url#1{\texttt{#1}\egroup}


\def\copyrightyear#1{\def\@copryear{{#1}}}
\def\keywords#1{\def\@keywords{Key words and phrases: #1.}\keywordstrue}
\def\amsclass#1{\def\@amsclass{2020 Mathematics Subject
Classification: #1.}\amsclasstrue}
\let\subjclass\amsclass
\def\@@thanks[#1]#2{\def\thempfn{}\footnotetext{\thmark{#1}#2}}
\def\@@@thanks#1{\def\@thanks{#1}\thankstrue}
\def\thanks{\@ifnextchar[{\@@thanks}{\@@@thanks}}
\def\thmark#1{${}^{#1)}$}

%%%%%%%%%%% Fancy headings
\@twosidetrue

% fancyheadings.sty version 1.1
% Fancy headers and footers.
% Piet van Oostrum, Dept of Computer Science, University of Utrecht
% Padualaan 14, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands
% Telephone: +31-30-531806. [email protected] (mcvax!hp4nl!ruuinf!piet)
% Sep 23, 1992.

\def\lhead{\@ifnextchar[{\@xlhead}{\@ylhead}}
\def\@xlhead[#1]#2{\gdef\@elhead{#1}\gdef\@olhead{#2}}
\def\@ylhead#1{\gdef\@elhead{#1}\gdef\@olhead{#1}}

\def\chead{\@ifnextchar[{\@xchead}{\@ychead}}
\def\@xchead[#1]#2{\gdef\@echead{#1}\gdef\@ochead{#2}}
\def\@ychead#1{\gdef\@echead{#1}\gdef\@ochead{#1}}

\def\rhead{\@ifnextchar[{\@xrhead}{\@yrhead}}
\def\@xrhead[#1]#2{\gdef\@erhead{#1}\gdef\@orhead{#2}}
\def\@yrhead#1{\gdef\@erhead{#1}\gdef\@orhead{#1}}

\def\lfoot{\@ifnextchar[{\@xlfoot}{\@ylfoot}}
\def\@xlfoot[#1]#2{\gdef\@elfoot{#1}\gdef\@olfoot{#2}}
\def\@ylfoot#1{\gdef\@elfoot{#1}\gdef\@olfoot{#1}}

\def\cfoot{\@ifnextchar[{\@xcfoot}{\@ycfoot}}
\def\@xcfoot[#1]#2{\gdef\@ecfoot{#1}\gdef\@ocfoot{#2}}
\def\@ycfoot#1{\gdef\@ecfoot{#1}\gdef\@ocfoot{#1}}

\def\rfoot{\@ifnextchar[{\@xrfoot}{\@yrfoot}}
\def\@xrfoot[#1]#2{\gdef\@erfoot{#1}\gdef\@orfoot{#2}}
\def\@yrfoot#1{\gdef\@erfoot{#1}\gdef\@orfoot{#1}}

\newdimen\headwidth
\newif\if@fancyplain \@fancyplainfalse
\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi}


\lhead[\fancyplain{}{\sl\rightmark}]{\fancyplain{}{\sl\leftmark}}
%  i.e. empty on ``plain'' pages \rightmark on even, \leftmark on odd pages
\chead{}
\rhead[\fancyplain{}{\sl\leftmark}]{\fancyplain{}{\sl\rightmark}}
%  i.e. empty on ``plain'' pages \leftmark on even, \rightmark on odd pages
\lfoot{}
\cfoot{\rm\thepage} % page number
\rfoot{}

% Put together a header or footer given the left, center and
% right text, fillers at left and right and a rule.
% The \lap commands put the text into an hbox of zero size,
% so overlapping text does not generate an errormessage.

\def\@fancyhead#1#2#3#4#5{#1\hbox to\headwidth{\vbox{\hbox
{\rlap{\parbox[b]{\headwidth}{\raggedright#2\strut}}\hfill
\parbox[b]{\headwidth}{\centering#3\strut}\hfill
\llap{\parbox[b]{\headwidth}{\raggedleft#4\strut}}} %\headrule}}#5}
}}#5}


\def\@fancyfoot#1#2#3#4#5{#1\hbox to\headwidth{\vbox{ %\footrule
\hbox{\rlap{\parbox[t]{\headwidth}{\raggedright#2\strut}}\hfill
\parbox[t]{\headwidth}{\centering#3\strut}\hfill
\llap{\parbox[t]{\headwidth}{\raggedleft#4\strut}}}}}#5}


\def\ps@fancy{
\let\@mkboth\markboth
\@ifundefined{chapter}{\def\sectionmark##1{\markboth
{\uppercase{\ifnum \c@secnumdepth>\z@
 \thesection\hskip 1em\relax \fi ##1}}{}}
\def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne
 \thesubsection\hskip 1em\relax \fi ##1}}}
{\def\chaptermark##1{\markboth {\uppercase{\ifnum \c@secnumdepth>\m@ne
 \@chapapp\ \thechapter. \ \fi ##1}}{}}
\def\sectionmark##1{\markright{\uppercase{\ifnum \c@secnumdepth >\z@
 \thesection. \ \fi ##1}}}}
\ps@@fancy
\headwidth\textwidth}
\def\ps@fancyplain{\ps@fancy \let\ps@plain\ps@plain@fancy}
\def\ps@plain@fancy{\@fancyplaintrue\ps@@fancy}
\def\ps@@fancy{
\def\@oddhead{\@fancyhead\relax\@olhead\@ochead\@orhead\hss}
\def\@oddfoot{\@fancyfoot\relax\@olfoot\@ocfoot\@orfoot\hss}
\def\@evenhead{\@fancyhead\hss\@elhead\@echead\@erhead\relax}
\def\@evenfoot{\@fancyfoot\hss\@elfoot\@ecfoot\@erfoot\relax}
}
\def\maketitle{\par
 \begingroup
 \def\thefootnote{\fnsymbol{footnote}}
 \def\@makefnmark{\hbox
 to\z@{$^{\@thefnmark}$\hss}}
 \global\@topnum\z@ \@maketitle \thispagestyle{plain}
 \endgroup
 \setcounter{footnote}{0}
 \let\@maketitle\relax
{\def\thempfn{}
\ifthanks\footnotetext{\@thanks}\fi
\ifamsclass\footnotetext{\@amsclass}\fi
\ifkeywords\footnotetext{\@keywords}\fi
% mod 2001-7-7
\def\relaxp[##1]{\relax}
\def\footnotemark{\@ifnextchar[{\relaxp}{\relax}}
\footnotetext{\copyright\ \@cauthor,\ \@copryear.
Permission to copy for private use granted.}
}}
\pagestyle{fancyplain}


\def\Uppercase#1{\let\i=I\let\j=J\def\ss{SS}\uppercase{#1}}
\let\MakeUppercase=\Uppercase
% The next two lines specify the font and size for the headers (and footer)
\def\headstyle#1{\let\\=\relax\footnotesize\MakeUppercase{#1}}
\def\thefnpage{{\rm\thepage}}

%\headrulewidth0pt
\headheight 14.5pt
\lhead[\fancyplain{}{\thefnpage}]{\fancyplain{}{}}
\chead[\fancyplain{}{\runningauthor}]{\fancyplain{}{\runningtitle}}
\rhead[]{\fancyplain{}{\thefnpage}}
\lfoot{}
\cfoot[\fancyplain{\thefnpage}{}]{\fancyplain{\thefnpage}{}}
\rfoot{}


\def\@@title[#1]#2{\def\@title{{\let\cr=\\ \def\\{\protect\cr[1\p@]}
\large\MakeUppercase{#2}}}\def\runningtitle{\headstyle{#1}}}
\def\@@@title#1{\def\@title{{\let\cr=\\ \def\\{\protect\cr[1\p@]}
\large\MakeUppercase{#1}}}\def\runningtitle{\headstyle{#1}}}
\def\title{\@ifnextchar[{\@@title}{\@@@title}}


\def\@@author[#1]#2{\def\@author{\MakeUppercase{#2}}\def\@cauthor
{\def\thmark####1{\relax}#2}\def\runningauthor{\headstyle{#1}}}
\def\@@@author#1{\def\@author{\MakeUppercase{#1}}\def\@cauthor
{\def\thmark####1{\relax}#1} \def\runningauthor
{\def\thmark####1{\relax}\headstyle{#1}}}
\def\author{\@ifnextchar[{\@@author}{\@@@author}}
\def\cauthor#1{\def\@cauthor{#1}}

%%% end of fancyheadings mods


\def\@maketitle{\newpage\vspace*{12\p@}
{\topsep\z@ \partopsep\z@
\begin{center}
\ifdedication
   \@title\\[12\p@]
   \@dedication\\[18\p@]
\else
   \@title\\[14\p@]
\fi
\@author
\end{center}}
\vskip24\p@}

\def\abstractname{Abstract}
\def\abstract{\bgroup\par
\footnotesize\begin{list}{}{\rightmargin=13mm \leftmargin=13mm
\parsep5\p@ \listparindent=\z@ \topsep=\z@\partopsep=\z@}
\item\leavevmode% \hbox to 1em{\hss} deleted indent 00-1-20
{\textsc{\abstractname.}}\hskip 1em}
\def\endabstract{\end{list}\par\egroup}



%Note that this is a redefinition.  Its purpose is to compile the
%label lastpage so that we can get the page numbers right.
%It also adds the authors' address

\AtEndDocument{%
% mod 99/12/4 - suppress breaks, label inside
% mod 2015-02-09
\nopagebreak[4]%
\@address
\label{lastpage}%
\nopagebreak[4]%
}

%%%%%%%%%%% Bibliographical styles
\def\thebibliography#1{\section*{References\@mkboth
 {REFERENCES}{REFERENCES}}\list
 {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
 \advance\leftmargin\labelsep
 \usecounter{enumi}}
 \def\newblock{\hskip .11em plus .33em minus .07em}
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\relax}
\let\endthebibliography=\endlist


% Below are the codes for the two reference styles.  They can be called by
% \begin{references} ... \end{references} and \begin{references*} ...
% \end{references*} .  In addition, the first can be called by \references
% ... \endreferences .

\newdimen\twlrm
\newdimen\refindent \refindent 8mm
\def\references{%
%\fontdimen2\twlrm=4.3pt% space instead of 3.91663
%\fontdimen3\twlrm=4.2pt%stretch instead of 1.95831
%\fontdimen4\twlrm=1.7pt%shrink instead of 1.30554
\section*{References}
\sloppy % RR deleted 2008-01-26 \footnotesize
\begin{list}{}
{\leftmargin\refindent \itemindent-\refindent \itemsep2pt \topsep0pt}}
\def\endreferences{\end{list}}

\newcounter {reflister}
\expandafter\def\csname references*\endcsname {%
%\fontdimen2\twlrm=4.3pt% space instead of 3.91663
%\fontdimen3\twlrm=4.2pt%stretch instead of 1.95831
%\fontdimen4\twlrm=1.7pt%shrink instead of 1.30554
\section*{References}
\sloppy\footnotesize
\begin{list}{{\textrm[\arabic{reflister}]}}{\usecounter{reflister}}
{\leftmargin\refindent \itemindent-\refindent \itemsep2pt \topsep0pt}}
\expandafter\let\csname endreferences*\endcsname=\endreferences


 % Here is a third reference style.  An alternative to \thebibliography, it
 % is intended to be used as follows.
 %
 % and then \cite{A} will give [Author (year)]

\def\refs{\bgroup\section*{References\@mkboth{REFERENCES}
{REFERENCES}}\def\@biblabel##1{\relax}\frenchspacing\list{}{
 \leftmargin2em \itemindent-2em}
 \def\newblock{\hskip .11em plus .33em minus .07em}
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\relax}
\def\endrefs{\endlist\egroup}

\def\aref{\bgroup\section*{Reference\@mkboth{REFERENCE}
{REFERENCE}}\def\@biblabel##1{\relax}\frenchspacing\list{}{
 \leftmargin2em \itemindent-2em}
 \def\newblock{\hskip .11em plus .33em minus .07em}
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\relax}
\def\endaref{\endlist\egroup}

 % The \tcite below allows one to put a citation within the optional
 % argument to a theorem or like, including a page citation.  If there is a
 % page citation (inside []), then the entire citation must be enclosed in
 % braces.

\def\tcite{\def\@cite##1##2{{##1\if@tempswa , ##2\fi}}\cite}

% The following is the content of breakcite.sty to allow breaking of
% citations.  This originally caused an obscure error that the latex
 % team has fixed, along with the problem of breaking.  The following is
 % exactly what is found in the current (but not older) latex.ltx


\def\@citex[#1]#2{\leavevmode
  \let\@citea\@empty
  \@cite{\@for\@citeb:=#2\do
    {\@citea\def\@citea{,\penalty\@m\ }%
     \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
     \if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
     \@ifundefined{b@\@citeb}{\hbox{\reset@font\bfseries ?}%
       \G@refundefinedtrue
       \@latex@warning
         {Citation `\@citeb' on page \thepage \space undefined}}%
       {{\csname b@\@citeb\endcsname}}}}{#1}}

% Modification of the `plain' definition, taken over by latex.  Note the
% failure to parametrize the definition, one of the many such that testify
% to Knuth's conception of plain as a provisional package.

\def\footnoterule{%\kern-3\p@
  \hrule width 1truein \kern 2.6\p@} % the \hrule is .4\p@ high

% old font commands from article.cls

\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}
\DeclareRobustCommand*\cal{\@fontswitch\relax\mathcal}
\DeclareRobustCommand*\mit{\@fontswitch\relax\mathnormal}


% Follows are a small number of procedures I have personally found very
% useful.  The first five define new math operations like Hom.  In the
% first, the word is spaced like a mathop and is used if there are to be
% arguments.  It is better not used if, like Hom, it is followed by an
% opening paranthesis.  The next two are without the spacing and are used
% for boldface, roman face, resp.  The usage is: \mathrm{Hom} that defines
% a new macro \Hom that puts the word Hom in ordinary type.

% replaced 2004-03-24 to allow an optional alternate name.  For example,
% \mathopdef[hom]{Hom} creates a macro called \hom that produces
% \mathop{\rm Hom}\nolimits.

\DeclareFontFamily{OT1}{pzc}{}
\DeclareFontShape{OT1}{pzc}{m}{it}%
     {<-> s * [1.15] pzcmi7t}{}
\DeclareMathAlphabet{\mathzc}{OT1}{pzc}{m}{it}
\DeclareSymbolFont{script}{OT1}{pzc}{m}{it}
\DeclareMathAlphabet{\mathcmssbx}{OT1}{cmss}{bx}{n}

\def\@mathopdef[#1]#2{\expandafter\def\csname#1\endcsname
{\mathop{\rm{}#2}\nolimits\xspace}}
\def\@mathoplsdef[#1]#2{\expandafter\def\csname#1\endcsname
 {\mathop{{}#2}\xspace}}
\def\@mathbfdef[#1]#2{\expandafter\def\csname#1\endcsname
 {{\rm\bf#2}\xspace}}
\def\@mathrmdef[#1]#2{\expandafter\def\csname#1\endcsname
{{\ensuremath{\rm#2}}\xspace}}
\def\@mathsfdef[#1]#2{\expandafter\def\csname#1\endcsname
{{\ensuremath{\sf#2}}\xspace}}
\def\@mathfkdef[#1]#2{\expandafter\def\csname#1\endcsname
{\ensuremath{\frak{#2}}\xspace}}
\def\@mathzcdef[#1]#2{\expandafter\def\csname#1\endcsname
{{\ensuremath{\mathzc{#2}}}\xspace}}
\def\@mathcaldef[#1]#2{\expandafter\def\csname#1\endcsname
{{\ensuremath{\mathcal{#2}}}\xspace}}
\def\@mathssbxdef[#1]#2{\expandafter\def\csname#1\endcsname
{{\ensuremath{\mathcmssbx{#2}}}\xspace}}
\def\@mathrsfsdef[#1]#2{\expandafter\def\csname#1\endcsname
{{\ensuremath{\mathscr{#2}}}\xspace}}

\def\mathopdef{\@dblarg\@mathopdef}
\def\mathopsldef{\@dblarg\@mathopsldef}
\def\mathbfdef{\@dblarg\@mathbfdef}
\def\mathrmdef{\@dblarg\@mathrmdef}
\def\mathsfdef{\@dblarg\@mathsfdef}
\def\mathfkdef{\@dblarg\@mathfkdef}
\def\mathzcdef{\@dblarg\@mathzcdef}
\def\mathssbxdef{\@dblarg\@mathssbxdef}
% next lines added 2009-10-25
\def\mathcaldef{\@dblarg\@mathcaldef}
% replaced 2015-06-29
% put back 2015-08-02
\def\mathrsfsdef{\@dblarg\@mathrsfsdef}
%\DeclareSymbolFont{rsfscript}{OMS}{rsfs}{m}{n}
%\DeclareSymbolFontAlphabet{\mathrsfs}{rsfscript}
%\def\@mathrsfsdef[#1]#2{\expandafter\def\csname#1\endcsname
%{{\ensuremath{\mathrsfs{#2}}}\xspace}}


\def\mld#1$${\null\,\vcenter\bgroup\def\\{\cr&}\openup9pt\m@th
\ialign\bgroup\strut\hfil$\displaystyle{##}$%
&$\displaystyle{{}##}$\hfil\crcr #1
\crcr\egroup\egroup\,$$}
\def\mldepf#1$${\null\,\vbox\bgroup\def\\{\cr&}\openup9pt\m@th
\ialign\bgroup\strut\hfil$\displaystyle{##}$%
&$\displaystyle{{}##}$\hfil\crcr #1
\crcr\egroup\egroup\,\eqno{\endproofbox}$$}
\let\emld\relax % for compatibility with old code

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Here is a typical usage of \mld (multi-line display)          %%
%%                                                               %%
%% The pattern is                                                %%
%% $$\mld formula 1 & formula 2 \\                               %%
%%                    formula 3 \\                               %%
%%                    ..........\\                               %%
%%                    formula n $$                               %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\DeclareFontShape{OMX}{cmex}{m}{n}{%
      <-8>sfixed * cmex7
      <8> cmex8
      <9> cmex9
      <10> <10.95> <12> <14.4> <17.28> <20.74> <24.88>cmex10
       }{}
\DeclareSymbolFont{largesymbols}{OMX}{cmex}{m}{n}
\newdimen\big@size
\addto@hook\every@math@size{\setbox\z@\vbox{\hbox{$($}\kern\z@}%
   \global\big@size 1.2\ht\z@}
\def\bBigg@#1#2{%
   {\hbox{$\left#2\vcenter to#1\big@size{}\right.\n@space$}}}
\def\big{\bBigg@\@ne}
\def\Big{\bBigg@{1.5}}
\def\bigg{\bBigg@\tw@}
\def\Bigg{\bBigg@{2.5}}
\def\edot{\vrule height .5pt width .5pt depth 0pt\kern1.5pt}
\def\eat{\raise1.4pt\hbox{\emph{\footnotesize a}}\llap 0\kern.5pt}

\usepackage[pdfusetitle,linktoc=all,colorlinks=true]{hyperref}
\hypersetup{allcolors=[rgb]{0.1,0.1,0.4}}

\endinput