short-math-guide (23dec17)

git-svn-id: svn://tug.org/texlive/trunk@46126 c570f23f-e606-0410-a88d-b1316a301751

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diff --git a/Master/texmf-dist/doc/latex/short-math-guide/README b/Master/texmf-dist/doc/latex/short-math-guide/README
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+README file for the Short Math Guide, v.2.0 [2017-12-22]
+
+Copyright 1995-1027, Americam Mathematical Society
+
+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.
+
+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) 455-4006
+email: tech-support@ams.org
+
+========================================================================
+
+The Short Math Guide is intended to be a concise introduction to the use
+of the facilities provided by amsmath and various other LaTeX packages
+for typesetting mathematical notation.
+
+The collection consists of three files:
+short-math-guide.tex, short-math-guide.pdf, and this README file.
+
+Macros needed for formatting the LaTeX source are embedded in the tex
+file using \filecontents and when the file is processed, an additional
+file, mathdoc.sty, will be written out.
+
+========================================================================
diff --git a/Master/texmf-dist/doc/latex/short-math-guide/short-math-guide.pdf b/Master/texmf-dist/doc/latex/short-math-guide/short-math-guide.pdf
new file mode 100644
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--- /dev/null
+++ b/Master/texmf-dist/doc/latex/short-math-guide/short-math-guide.pdf
diff --git a/Master/texmf-dist/doc/latex/short-math-guide/short-math-guide.tex b/Master/texmf-dist/doc/latex/short-math-guide/short-math-guide.tex
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+% This is file `short-math-guide.tex'.
+%
+% Copyright 1995-2017
+% 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 file has the LPPL maintenance status "maintained".
+% 
+% The Current Maintainer of this work is the American Mathematical
+% Society.
+% 
+% Note: When updating, don't forget to update \smgversion, below
+
+\begin{filecontents}{mathdoc.sty}
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesPackage{mathdoc}[2017/12/22 v2.0]
+
+\providecommand{\lat}[1]{\protect\LaTeX{}}
+\providecommand{\mdash}{\textemdash}
+\providecommand{\ndash}{\textendash}
+\newcommand{\ntt}{\fontseries{m}\fontshape{n}\ttfamily}
+\newcommand{\mntt}[1]{\mbox{\ntt #1}}
+\providecommand{\pkg}[1]{\mntt{#1}}
+\providecommand{\cls}[1]{\mntt{#1}}
+\providecommand{\opt}[1]{\mntt{#1}}
+\providecommand{\fn}[1]{\mntt{#1}}
+\providecommand{\env}[1]{\mntt{#1}}
+\chardef\bslash=`\\
+\let\charhack=\char
+\providecommand{\cn}[1]{\mntt{\bslash\charhack`#1}}
+\newcommand{\hycn}{\protect\cn}
+\@ifpackageloaded{hyperref}{%
+  \def\mntt#1{\texttt{\upshape #1}}%
+  \def\hycn#1{\texttt{\upshape\protect\bslash#1}}%
+  \begingroup \lccode`\.=`\\ \lowercase{\endgroup
+  \def\bslash{\texorpdfstring{.}{\textbackslash}}
+  }
+}{}
+\providecommand{\ncn}{\cn}
+\providecommand{\qq}[1]{\textquotedblleft#1\textquotedblright}
+
+\oddsidemargin=0pt \evensidemargin=0pt
+
+\let\Huge\Large \let\huge\Large \let\LARGE\large \let\Large\large
+
+\def\section{\@startsection{section}{1}%
+  \z@{9pt plus12pt}{1.5ex}%
+  {\bfseries\global\@afterindentfalse}}
+
+\def\subsection{\@startsection{subsection}{2}%
+  \z@{9pt plus12pt}{-.5em}%
+  {\scshape}}
+
+\def\@seccntformat#1{\csname the#1\endcsname.\enskip}
+
+%    Durn it, looks like a LaTeX kernel bug: consecutive "run-in" section
+%    titles should not allow a page break between them.
+\def\@startsection #1#2#3#4#5#6{%
+  \if@noskipsec \leavevmode\par\nobreak\vskip\medskipamount\@nobreaktrue\fi
+  \par
+  \@tempskipa#4\relax \@afterindenttrue
+  \ifdim\@tempskipa<\z@ \@tempskipa-\@tempskipa \@afterindentfalse \fi
+  \if@nobreak \everypar{}%
+  \else \addpenalty\@secpenalty \addvspace\@tempskipa
+  \fi
+  \@ifstar{\@ssect{#3}{#4}{#5}{#6}}%
+          {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}%
+}
+
+%    Redefine \@sect to add period after run-in headings
+\def\@sect#1#2#3#4#5#6[#7]#8{%
+  \ifnum #2>\c@secnumdepth
+    \let\@svsec\@empty
+  \else
+    \refstepcounter{#1}%
+    \protected@edef\@svsec{\@seccntformat{#1}\relax}%
+  \fi
+  \@tempskipa #5\relax
+  \ifdim \@tempskipa>\z@
+    \begingroup
+      #6{%
+        \@hangfrom{\hskip #3\relax\@svsec}%
+          \interlinepenalty \@M #8\@@par}%
+    \endgroup
+    \csname #1mark\endcsname{#7}%
+    \addcontentsline{toc}{#1}{%
+      \ifnum #2>\c@secnumdepth \else
+        \protect\numberline{\csname the#1\endcsname}%
+      \fi
+      #7}%
+  \else
+    \def\@svsechd{%
+      #6{\hskip #3\relax
+      \@svsec #8\@addpunct.}%
+      \csname #1mark\endcsname{#7}%
+      \addcontentsline{toc}{#1}{%
+        \ifnum #2>\c@secnumdepth \else
+          \protect\numberline{\csname the#1\endcsname}%
+        \fi
+        #7}}%
+  \fi
+  \@xsect{#5}}
+%    From amsart, allows \nopunct to omit unwanted periods
+\def\@addpunct#1{%
+  \relax\ifhmode
+    \ifnum\spacefactor>\@m \else#1\fi
+  \fi}
+\def\nopunct{\texorpdfstring{\spacefactor 1007 }{}}
+
+%    Workaround to avoid overprinting problem with first contents entry
+\let\zz@l@section\l@section
+\def\l@section{\if@noskipsec \mbox{}\par\fi\zz@l@section}
+
+\RequirePackage{keyval}\relax
+
+\define@key{symlist}{adjustwidth}{\advance\wdadjust#1\relax}
+\define@key{symlist}{adjustheight}{\htadjust#1\relax}
+\define@key{symlist}{adjustcols}{\coladjust#1\relax}
+
+\providecommand{\newcolumn}{\vfil\break}
+
+\newenvironment{symlist}[1][]{%
+  \if@noskipsec\ifvmode\nobreak\fi\leavevmode\fi
+  \par
+  \nobreak
+  \wdadjust=1.5em\relax
+  \gdef\containsMSABM{TF}%
+  \setkeys{symlist}{#1}%
+    \setbox\z@\vbox\bgroup
+      \advance\baselineskip \z@ plus2pt\relax
+      % work around splittopskip discrepancy
+      \null \penalty-\@M
+}{%
+  \par\egroup
+  \splitlist
+}
+
+\let\debugit\relax
+\def\boxo{%
+  \typeout{Box0: \the\wd0x\the\ht0+\the\dp0, splitting to
+    \the\dimen@\space for \number\cols\space columns}%
+}
+
+\newdimen\shiftlistright
+\shiftlistright=20pt
+\def\splitlist{%
+  \begingroup
+  \textwidth=400pt % fudgit
+  \dimen@=\wd0 \advance\dimen@\wdadjust \advance\dimen@\colsep
+  \cols\textwidth \advance\cols\colsep
+  \divide\cols\dimen@ \advance\cols\coladjust
+  \ifnum\cols<\z@ \cols=\@ne \fi
+  \dimen@\ht0 \divide\dimen@\cols
+  \advance\dimen@\baselineskip \divide\dimen@\baselineskip
+  \multiply\dimen@\baselineskip
+  \splittopskip10pt\relax \splitmaxdepth\maxdepth
+  \advance\dimen@\splittopskip \advance\dimen@-\baselineskip
+  \advance\dimen@\htadjust
+  \vbadness\@M % ignore underfull vbox messages
+\debugit
+  \def\do{%
+    \advance\curcol 1
+    \setbox2=\vsplit0 to\dimen@
+    \ifnum\curcol=\cols \vtop \else \vtop to\dimen@\fi
+      {\unvbox 2 }\hskip\colsep
+    \ifdim\ht0>\z@ \expandafter\do\fi
+  }%
+  \setbox2=\vsplit0 to\baselineskip % discard empty top box
+  \hbox to\textwidth{\hskip\shiftlistright\quad\curcol=0 \do\unskip\hfil}%
+  % cancel a large prevdepth
+  \nobreak\nointerlineskip\hbox{}%
+  \endgroup
+  \global\shiftlistright20pt
+}
+
+\def\dosymbol#1{\csname do#1\endcsname}
+\newcommand{\ttfont}{%
+  \normalfont\ttfamily
+  \global\expandafter\let\expandafter\ttfont\the\font
+}
+
+\newdimen\wdadjust \newdimen\htadjust
+\newskip\colsep \newcount\cols \newcount\curcol \newcount\coladjust
+\colsep=10pt plus1fil minus2pt
+
+\newbox\symstrut
+\AtBeginDocument{\setbox\symstrut\hbox{\vrule height8.5pt depth3.5pt width0pt}}
+
+\newcommand{\symbox}[2]{%
+  \hbox{\llap{\unhcopy\symstrut $ #1 $ }\ttfont#2}%
+}
+
+\edef\symnote#1{%
+  \noexpand\expandafter\noexpand\symnoteA
+  \noexpand\meaning#1\relax\string h?"00\noexpand\@nil
+}
+
+\begingroup\edef\x#1h{\endgroup #1\string h}\x
+\def\symnoteA#1h#2#3"#4#5#6\@nil{%
+  \if c#2\relax
+    \expandafter\ifx\csname#2#3\endcsname\char
+      \ifcase#5\relax \or\or\or\or
+        \symnoteB[to 0pt]{a}\or\symnoteB[to 0pt]{b}\else\fi
+    \fi
+  \fi
+}
+
+\newcommand{\symnoteB}[2][]{%
+  \hbox #1{%
+    \raise0.9ex\hbox{\fontfamily{cmr}\fontseries{m}\scriptsize#2}\hss
+  }%
+  \gdef\containsMSABM{TT}%
+}
+
+
+\newcommand{\printSymbol}[1]{%
+  \hbox{\llap{\unhcopy\symstrut $#1$ }\ttfont\string#1%
+    \symnote#1%
+  }%
+}
+
+\newcommand{\printBig}[1]{\hbox{\llap{$\Big#1$ }\ttfont\string#1}}
+
+% Arg 2 is something like "kernel" or "amssymb"; but
+% if we want to distinguish msam or msbm we have to look more closely.
+
+\newcommand{\doVar}[2]{\expandafter\printSymbol\csname#1\endcsname}
+
+\newcommand{\doVarc}[2]{%
+  \hbox{\llap{\unhcopy\symstrut$#1$ }\ttfont#1}%
+}
+
+\newcommand{\doDeL}[2]{\expandafter\printBig\csname#1\endcsname}%
+
+\newcommand{\doDeLc}[2]{%
+  \hbox{\llap{$\Big#1$ }\ttfont#1}%
+}
+
+\let\doOrd\doVar
+\let\doBin\doVar
+\let\doRel\doVar
+\let\doPun\doVar
+\let\doOrdc\doVarc
+\let\doBinc\doVarc
+\let\doRelc\doVarc
+\let\doPunc\doVarc
+\let\doInn\doVar
+\let\doCOi\doVar
+\let\doCOs\doVar
+\let\doDeR\doDeL
+\let\doDeB\doDeL
+\let\doDeBc\doDeLc
+\let\doDeA\doDeL
+
+\newcommand{\doDeLR}[3]{%
+  \hbox{\llap{$
+    \expandafter\Bigl\csname#1\endcsname\,
+    \expandafter\Bigr\csname#2\endcsname$ }%
+    \ttfont\bslash#1 \bslash#2}%
+}
+
+\newcommand{\doDeLRc}[3]{%
+  \hbox{\llap{$\Bigl#1\,\Bigr#2$ }\ttfont\string#1 \string#2}%
+}
+
+\newcommand{\doOrdx}[2]{%
+  \hbox{\llap{\unhcopy\symstrut$#1$ }\ttfont\string#1}%
+}
+
+\let\doOpn\doVar
+
+\newcommand{\doFsw}[2]{%
+  \hbox{\kern-\parindent $\csname#1\endcsname{R}$\space
+    \ttfont\string#1\string{R\string}}%
+}
+
+\newcommand{\doAcc}[2]{%
+  \hbox{\llap{\unhcopy\symstrut$\csname#1\endcsname{x}$ }%
+    \ttfont\bslash#1\string{x\string}}%
+}
+
+\newcommand{\doAccw}[2]{%
+  \hbox{\llap{\unhcopy\symstrut$\csname#1\endcsname{xxx}$ }%
+    \ttfont\bslash#1\string{xxx\string}}%
+}
+
+\newcommand{\alias}[1]{$\csname#1\endcsname$ \cn{#1}}
+
+\newcommand{\symrow}[1]{%
+  {#1}& \mathbf{#1}& \mathrm{#1}& \mathsf{#1}& \mathit{#1}& \mathcal{#1}&
+  \mathbb{#1}& \mathfrak{#1}%
+}
+%  \mathscr{#1}&  \mathbb{#1}& \mathfrak{#1}%
+%% rsfs has nothing in some of the example slots, gets ! Missing character
+
+\newenvironment{eqxample}{%
+  \par\addvspace\medskipamount
+  \noindent\begin{minipage}{.5\columnwidth}%
+  \def\producing{\end{minipage}\begin{minipage}{.5\columnwidth}%
+    \hbox\bgroup\kern-.2pt\vrule width.2pt\vbox\bgroup\parindent0pt\relax
+%    The 3pt is to cancel the -\lineskip from \displ@y
+    \abovedisplayskip3pt \abovedisplayshortskip\abovedisplayskip
+    \belowdisplayskip0pt \belowdisplayshortskip\belowdisplayskip
+    \noindent}
+}{%
+  \par
+%    Ensure that a lonely \[\] structure doesn't take up width less than
+%    \hsize.
+  \hrule height0pt width\hsize
+  \egroup\vrule width.2pt\kern-.2pt\egroup
+  \end{minipage}%
+  \par\addvspace\medskipamount
+}
+
+\newcommand{\noteslabel}[1]{\hskip\labelsep\textit{#1\unskip}}
+
+\newcommand{\singlenote}{\item[\textit{Note.}]}
+\newcommand{\synonyms}{\item[\textit{Synonyms\/}:]}
+
+\newenvironment{notes}{%
+  \begin{list}{Note \arabic{enumiv}.}{%
+    \usecounter{enumiv}%
+    \footnotesize
+    \setlength{\leftmargin}{0pt}%
+    \setlength{\labelwidth}{0pt}%
+    \setlength{\topsep}{\medskipamount}%
+    \renewcommand{\makelabel}{\noteslabel}%
+  }%
+  \if\containsMSABM %
+    \item Labels \symnoteB[to 1.3em]{a,b} indicate \pkg{amssymb}
+      package, font \fn{msam} or~\fn{msbm}.%
+  \fi
+  \gdef\containsMSABM{TF}%
+}{%
+  \end{list}%
+}
+
+\newcommand{\ma}[1]{%
+  \string{{\normalfont\itshape #1}\string}\penalty9999 \ignorespaces}
+
+\newenvironment{cmdspec}[1][\linewidth]{%
+  \begin{center}\begin{minipage}{#1}%
+  \raggedright \normalfont\ttfamily \exhyphenpenalty10000
+}{%
+  \end{minipage}\end{center}%
+}
+
+%    l2h has this screwed up somehow? [mjd,1999/11/05]
+\newenvironment{tex2html_preform}{}{}
+
+\providecommand{\strong}{\textbf}
+
+\@ifundefined{ht@url}{
+  \newcommand\htlink{\href}
+  \newenvironment{makeimage}{}{}
+}{
+  \renewcommand\htlink[2]{#1\htlinkfootnote{\ht@url{#2}}}
+  \newcommand{\htlinkfootnote}[1]{}
+}
+
+\providecommand{\url}{\texttt}
+\endinput
+\end{filecontents}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\errorcontextlines=99
+
+\documentclass{article}
+
+\pagestyle{myheadings}
+%\usepackage{hthtml}
+\usepackage{amsmath}
+%\usepackage[cmex10]{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+%\usepackage[mathcal]{euscript}
+\usepackage{euscript}
+%\usepackage{mathtime}
+%\usepackage{stmaryrd}
+
+\numberwithin{equation}{section}
+
+%% at the moment, using either url or hyperref crashes
+%% get the content correct before debugging that problem
+%\usepackage{url}
+\let\url\texttt
+\usepackage[breaklinks,colorlinks]{hyperref}
+\usepackage{xcolor}
+\definecolor{hycitecolor}{rgb}{0,0.65,0}
+
+\usepackage{mathdoc}
+
+\hoffset=-1 true in
+\voffset=-1 true in
+\topmargin=0.75 true in %% \oddsidemargin\topmargin
+%\textwidth=210 true mm % A4
+\textwidth=139 true mm
+\textheight=11 true in \advance\textheight-2\topmargin
+\headheight=7pt \advance\textheight-\headheight
+\headsep=11pt \advance\textheight-\headsep
+
+\oddsidemargin=\paperwidth
+\advance\oddsidemargin-\textwidth
+\oddsidemargin=.5\oddsidemargin
+\evensidemargin=\oddsidemargin
+
+\hfuzz=14pt % suppress uninteresting warnings
+
+\providecommand{\abs}[1]{\lvert#1\rvert}
+
+\newcommand{\colhead}[1]{%
+  \textbf{\begin{tabular}[b]{@{}l@{}}#1\end{tabular}}%
+}
+
+\newcommand{\secref}[1]{Section~\ref{#1}}
+\newcommand{\tabref}[1]{Table~\ref{#1}}
+
+\newcommand{\begend}[1]{%
+  \cn{begin}\texttt{\symbol{123}#1\symbol{125}}%
+  \ \dots\ \cn{end}\texttt{\symbol{123}#1\symbol{125}}%
+}
+
+\newcommand{\dbldollars}{\texttt{\$\$} \dots\ \texttt{\$\$}}
+
+\newenvironment{lstack}[1][t]{%
+  \begin{tabular}[#1]{@{}l@{}}%
+}{%
+  \end{tabular}
+}  
+
+\newenvironment{cstack}[1][t]{%
+  \begin{tabular}[#1]{@{}c@{}}%
+}{%
+  \end{tabular}
+}  
+
+\newenvironment{llstack}[1][t]{%
+  \begin{tabular}[#1]{@{}ll@{}}%
+}{%
+  \end{tabular}
+}  
+
+%%\newcommand{\lspx}{\mbox{\rule{5pt}{.6pt}\rule{.6pt}{6pt}}}
+%%\newcommand{\rspx}{\mbox{\rule[-1pt]{.6pt}{7pt}%
+%%  \rule[-1pt]{5pt}{.6pt}}}
+%%\newcommand{\lspx}{\mathord{\otimes}}
+%%\newcommand{\rspx}{\mathord{\odot}}
+\newcommand{\lspx}{3}
+\newcommand{\rspx}{4}
+\newcommand{\spx}[1]{$\lspx #1\rspx$}
+
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\esssup}{ess\,sup}
+
+\newcommand{\dotsref}{\leavevmode\unskip\space
+  (see Section~\ref{dots})}
+\newcommand{\vertref}{\leavevmode\unskip\space
+  (see Section~\ref{verts})}
+
+\providecommand{\pdfinfo}[1]{}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{document}
+\title{Short Math Guide for \LaTeX{}}
+\newcommand{\smgversion}{\textup{2.0 (2017/12/22)}}
+\markboth{Short Math Guide for \LaTeX{}, version \protect\smgversion}
+         {Short Math Guide for \LaTeX{}, version \protect\smgversion}
+\author{Michael Downes, updated by Barbara Beeton}
+\date{American Mathematical Society}
+
+\pdfinfo{
+  /Title (Short Math Guide for LaTeX)
+  /Author (Michael Downes, updated by Barbara Beeton)
+  /Subject (Short Math Guide for LaTeX)
+  /Keywords (LaTeX,amsmath,amsfonts,amssymb,equation,math,formula)
+}
+
+\maketitle
+\begin{center}
+Version \smgversion, currently available from a link at
+\mbox{\texttt{https://www.ams.org/tex/amslatex}}
+\end{center}
+%%\begin{rawhtml}
+%%<p>
+%%<a href="../short-math-guide.pdf">Download a PDF version of this documentation.</a>
+%%</p>
+%%<hr/>
+%%\end{rawhtml}
+
+\setcounter{tocdepth}{3}
+\tableofcontents
+
+\vspace{\fill}
+\section*{Acknowledgments and plans for future work}
+Thanks to all who contributed suggestions, assistance and encouragement.
+Special thanks to David Carlisle for repairing unruly macros and to
+Jennifer Wright Sharp for applying consistent editing in AMS style.
+
+\smallskip\noindent
+Plans for a future edition include addition of an index.
+
+\smallskip\noindent
+Reports concerning errors and suggestions for improvement should be
+sent to\\[2pt]
+\hspace*{\fill}
+\href{mailto:tech-support@ams.org}{\texttt{tech-support@ams.org}}\,.
+\hspace{\fill}\null
+
+\newpage
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Introduction}
+
+This is a concise summary of recommended features in \LaTeX{} and a
+couple of extension packages for \textbf{writing math formulas}. Readers
+needing greater depth of detail are referred to the sources listed in
+the bibliography, especially \cite{lamport}, \cite{amsldoc}, and
+\cite{fntguide}. A certain amount of familiarity with standard \LaTeX{}
+terminology is assumed; if your memory needs refreshing on the \LaTeX{}
+meaning of \emph{command}, \emph{optional argument}, \emph{environment},
+\emph{package}, and so forth, see \cite{lamport}.
+
+Most of the features described here are available to you if you use
+\LaTeX{} with two extension packages published by the American Mathematical
+Society: \pkg{amssymb} and \pkg{amsmath}. Thus, the source file for this
+document begins with
+\begin{verbatim}
+\documentclass{article}
+\usepackage{amssymb,amsmath}
+\end{verbatim}
+The \pkg{amssymb} package might be omissible for documents whose math
+symbol usage is relatively modest; in \secref{mathsymbols}, the symbols
+that require \pkg{amssymb} are marked with \textsuperscript{a} or
+\textsuperscript{b} (font \fn{msam} or \fn{msbm}).  In \secref{alpha-digit},
+a few additional fonts are included; the necessary packages are identified
+there.
+
+Many noteworthy features found in other packages are not covered here;
+see \secref{other-packages}. Regarding math symbols, please note
+especially that the list given here is not intended to be comprehensive,
+but to illustrate such symbols as users will normally find already
+present in their \lat/ system and usable without installing any
+additional fonts or doing other setup work.
+
+If you have a need for a symbol not shown here, you will probably want
+to consult \emph{The Comprehensive \LaTeX{} Symbol List}~\cite{comprehensive}.
+%\[\url{http://www.ctan.org/tex-archive/info/symbols/comprehensive/}\]
+If your \lat/ installation is based on \TeX\,Live, and includes documentation,
+the list can also be accessed by typing \texttt{texdoc comprehensive} at a
+system prompt.
+
+\begin{table}[p]
+\caption[]{Multiline equations and equation groups\\
+ \phantom{Table 1:} (vertical lines indicate nominal margins).}
+\label{displays}
+\bigskip
+\begin{makeimage}
+\begin{minipage}{\textwidth}
+\begin{eqxample}
+\begin{verbatim}
+\begin{equation}\label{xx}
+\begin{split}
+a& =b+c-d\\
+ & \quad +e-f\\
+ & =g+h\\
+ & =i
+\end{split}
+\end{equation}
+\end{verbatim}
+\producing
+\begin{equation}\label{xx}
+\begin{split}
+a& =b+c-d\\
+ & \quad +e-f\\
+ & =g+h\\
+ & =i
+\end{split}
+\end{equation}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{multline}
+a+b+c+d+e+f\\
++i+j+k+l+m+n\\
++o+p+q+r+s
+\end{multline}
+\end{verbatim}
+\producing
+\begin{multline}
+a+b+c+d+e+f\\
++i+j+k+l+m+n\\
++o+p+q+r+s
+\end{multline}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{gather}
+a_1=b_1+c_1\\
+a_2=b_2+c_2-d_2+e_2
+\end{gather}
+\end{verbatim}
+\producing
+\begin{gather}
+a_1=b_1+c_1\\
+a_2=b_2+c_2-d_2+e_2
+\end{gather}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{align}
+a_1& =b_1+c_1\\
+a_2& =b_2+c_2-d_2+e_2
+\end{align}
+\end{verbatim}
+\producing
+\begin{align}
+a_1& =b_1+c_1\\
+a_2& =b_2+c_2-d_2+e_2
+\end{align}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{align}
+a_{11}& =b_{11}&
+  a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{align}
+\end{verbatim}
+\producing
+\begin{align}
+a_{11}& =b_{11}&
+  a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{align}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{alignat}{2}
+a_1& =b_1+c_1&      &+e_1-f_1\\
+a_2& =b_2+c_2&{}-d_2&+e_2
+\end{alignat}
+\end{verbatim}
+\producing
+\begin{alignat}{2}
+a_1& =b_1+c_1&      &+e_1-f_1\\
+a_2& =b_2+c_2&{}-d_2&+e_2
+\end{alignat}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{flalign}
+a_{11}& =b_{11}&
+  a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{flalign}
+\end{verbatim}
+\producing
+\begin{flalign}
+a_{11}& =b_{11}&
+  a_{12}& =b_{12}\\
+a_{21}& =b_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{flalign}
+\end{eqxample}
+\def\containsMSABM{TF}
+\begin{notes}
+\item Applying \env{*} to any primary environment will suppress the
+  assignment of equation numbers.  However, \cn{tag} may be used to
+  apply a visible label, and \cn{eqref} can be used to reference
+  such manually tagged lines.   Use of either \env{*} or a \cn{tag}
+  on a subordinate environment is an error.
+\item The \env{split} environment is something of a
+  special case. It is a subordinate environment that can be used as the
+  contents of an \env{equation} environment or the contents of one
+  \qq{line} in a multiple-equation structure such as \env{align} or
+  \env{gather}.
+\item The primary environments \env{gather}, \env{align} and \env{alignat}
+  have subordinate ``\env{-ed}'' counterparts (\env{gathered},
+  \env{aligned} and \env{alignedat}) that can be used as components of
+  more complicated displays, or within in-line math.  These ``\env{-ed}''
+  environments can be positioned vertically using an optional argument
+  \verb+[t]+, \verb+[c]+ or~\verb+[b]+.
+\item The name \env{flalign} is meant as ``full length'', not
+  ``flush left'' as often mistakenly reported.  However, since a
+  display occupying the full width will often begin at the left
+  margin, this confusion is understandable.  The indent applied to
+  \env{flalign} from both margins is set with \cn{multlinegap}.
+\end{notes}
+\end{minipage}
+\end{makeimage}
+\end{table}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Inline math formulas and displayed equations}\label{first-step}
+
+\subsection{The fundamentals}
+
+Entering and leaving math mode in \LaTeX{} is normally done with the
+following commands and environments.%
+\begin{center}
+\begin{tabular}{ccc}
+\colhead{inline formulas}&& \colhead{displayed equations}\\[3pt]
+\cline{1-1}\cline{3-3}\noalign{\medskip}
+\begin{cstack}
+  \verb'$' \dots\ \verb'$'\\
+  \verb'\(' \dots\ \verb'\)'
+\end{cstack}%
+&&
+\begin{llstack}
+\begin{lstack}\verb'\[...\]'\\[6pt]\end{lstack}&
+  unnumbered\\
+\begin{lstack}
+  \verb'\begin{equation*}'\\
+  \dots\\
+  \verb'\end{equation*}'\\[6pt]
+\end{lstack}&
+  unnumbered\\
+\begin{lstack}
+  \verb'\begin{equation}'\\
+  \dots\\
+  \verb'\end{equation}'
+\end{lstack}&
+  \begin{lstack}automatically\\numbered\end{lstack}
+\end{llstack}
+\end{tabular}
+\begin{notes}
+%  \singlenote
+\item Do not leave a blank line between text and a displayed equation.
+  This allows a page break at that location, which is bad style.  It
+  also causes the spacing between text and display to be incorrect,
+  usually larger than it should be.
+  If a visual break is desired in the input, insert a line containing
+  only a \verb+%+ at the beginning.
+  Leave a blank line between a display and following text only if a new
+  paragraph is intended.
+\item Do not group multiple display structures in the input (\verb+\[...\]+,
+  \env{equation}, etc.).  Instead, use a multiline structure with
+  substructures (\env{split}, \env{aligned}, etc.)\ as appropriate.
+\item The alternative environments \begend{math} and\\
+  \begend{displaymath} are seldom needed in practice.  Using the plain
+  \TeX{} notation \dbldollars\ for displayed equations is strongly
+  discouraged. Although it is not expressly forbidden in \LaTeX{}, it
+  is not documented anywhere in the \LaTeX{} book as being part of the
+  \LaTeX{} command set, and it interferes with the proper operation of
+  various features such as the \opt{fleqn} option.
+\item The \env{eqnarray} and \env{eqnarray*} environments described
+  in \cite{lamport} are strongly discouraged because they produce
+  inconsistent spacing of the equal signs and make no attempt to prevent
+  overprinting of the equation body by the equation number.
+\end{notes}
+\end{center}
+Environments for handling equation groups and multiline equations are
+shown in \tabref{displays}.
+
+\newpage
+
+\subsection{Automatic numbering and cross-referencing}
+
+To get an auto-numbered equation, use the \env{equation} environment; to
+assign a label for cross-referencing, use the \cn{label} command:
+\begin{verbatim}
+\begin{equation}\label{reio}
+...
+\end{equation}
+\end{verbatim}
+To get a cross-reference to an auto-numbered equation, use the
+\cn{eqref} command:
+\begin{verbatim}
+... using equations~\eqref{ax1} and~\eqref{bz2}, we
+can derive ...
+\end{verbatim}
+The above example would produce something like
+\begin{quote}
+  using equations (3.2) and (3.5), we can derive
+\end{quote}
+In other words, \verb'\eqref{ax1}' is equivalent to \verb'(\ref{ax1})',
+but the parentheses produced by \cn{eqref} are always upright.
+
+To give your equation numbers the form \textit{m.n}
+(\textit{section-number.equation-number}), use the \cn{numberwithin}
+command in the preamble of your document:
+\begin{verbatim}
+\numberwithin{equation}{section}
+\end{verbatim}
+For more details on custom numbering schemes see \cite[\S 6.3,
+\S C.8.4]{lamport}.
+
+The \env{subequations} environment provides a convenient way to number
+equations in a group with a subordinate numbering scheme. For example,
+supposing that the current equation number is \theequation, write
+\begin{verbatim}
+\begin{equation}\label{first}
+a=b+c
+\end{equation}
+some intervening text
+\begin{subequations}\label{grp}
+\begin{align}
+a&=b+c\label{second}\\
+d&=e+f+g\label{third}\\
+h&=i+j\label{fourth}
+\end{align}
+\end{subequations}
+\end{verbatim}
+to get
+\begin{equation}\label{first}
+a=b+c
+\end{equation}
+some intervening text
+\begin{subequations}\label{grp}
+\begin{align}
+a&=b+c\label{second}\\
+d&=e+f+g\label{third}\\
+h&=i+j\label{fourth}
+\end{align}
+\end{subequations}
+By putting a \cn{label} command immediately after
+\verb'\begin{subequations}' you can get a reference to the parent
+  number; \verb'\eqref{grp}' from the above example would produce \eqref{grp}
+  while \verb'\eqref{second}' would produce \eqref{second}.
+
+An example at \url{https://tex.stackexchange.com/questions/220001/} shows
+a variant of the above example, with numbering like (2.1), (2.1a),
+\dots, rather than (2.1), (2.2a), \dots.  This is accomplished by using
+\cn{tag} with a cross-reference to the principal component of the
+subequation number.
+
+\newpage
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\providecommand{\dotsref}{\leavevmode\unskip\ignorespaces}
+\providecommand{\vertref}{\leavevmode\unskip\ignorespaces}
+
+\section{Math symbols and math fonts}\label{mathsymbols}
+\subsection{Classes of math symbols}
+
+The symbols in a math formula fall into different classes that
+correspond more or less to the part of speech each symbol would have if
+the formula were expressed in words. Certain spacing and positioning
+cues are traditionally used for the different symbol classes to increase
+the readability of formulas.
+
+\begin{center}
+\begin{tabular}{clll}
+\colhead{Class\\number}& \colhead{Mnemonic}& \colhead{Description\\(part
+of speech)}& \colhead{Examples}\\\hline\noalign{\smallskip}
+0& Ord& simple/ordinary (\qq{noun})& $A\;0\;\Phi\;\infty$\\
+1& Op& prefix operator& $\sum\;\prod\;\int$\\
+2& Bin& binary operator (conjunction)& ${+}\;{\cup}\;{\wedge}$\\
+3& Rel& relation/comparison (verb)& ${=}\;{<}\;{\subset}$\\
+4& Open& left/opening delimiter& $(\;{[}\;{\lbrace}\;{\langle}$\\
+5& Close& right/closing delimiter& $)\;{]}\;{\rbrace}\;{\rangle}$\\
+6& Punct& postfix/punctuation& ${.}\;{,}\;{;}\;{!}$\\
+\end{tabular}
+\end{center}
+\begin{notes}
+\item The distinction in \TeX{} between class 0 and an additional
+class 7 has to do only with font selection issues, and it is immaterial
+here.
+\item Symbols of class 2 (Bin), notably the minus sign $-$, are
+automatically printed by \LaTeX{} as class~0 (no space) if they do not
+have a suitable left operand\mdash e.g., at the beginning of a math
+formula or after an opening delimiter.
+\end{notes}
+
+The spacing for a few symbols follows tradition instead of the general
+rule: although $/$ is (semantically speaking) of class~2, we write $k/2$
+with no space around the slash rather than $k\mathbin{/}2$. And compare
+\verb'p|q' $p\vert q$ (no space) with \verb'p\mid q' $p\mid q$
+(class-3 spacing).
+
+The proper way to define a new math symbol is discussed in
+\emph{\LaTeXe{} font selection} \cite{fntguide}. It is not really
+possible to give a useful synopsis here because one needs first to
+understand the ramifications of font specifications. But supposing one
+knows that a Cyrillic font named \fn{wncyr10} is available, here is a
+minimal example showing how to define a \LaTeX{} command to print one
+letter from that font as a math symbol:
+\begin{verbatim}
+% Declare that the combination of font attributes OT2/wncyr/m/n
+% should select the wncyr font.
+\DeclareFontShape{OT2}{wncyr}{m}{n}{<->wncyr10}{}
+% Declare that the symbolic math font name "cyr" should resolve to
+% OT2/wncyr/m/n.
+\DeclareSymbolFont{cyr}{OT2}{wncyr}{m}{n}
+% Declare that the command \Sh should print symbol 88 from the math font
+% "cyr", and that the symbol class is 0 (= alphabetic = Ord).
+\DeclareMathSymbol{\Sh}{\mathalpha}{cyr}{88}
+\end{verbatim}
+
+\subsection{Some symbols intentionally omitted here}
+
+The following math symbols that are mentioned in the \LaTeX{} book
+\cite{lamport} are intentionally omitted from this discussion because
+they are superseded by equivalent symbols when the \pkg{amssymb} package
+is loaded. If you are using the \pkg{amssymb} package anyway, the only
+thing that you are likely to gain by using the alternate name is an
+unnecessary increase in the number of fonts used by your document.
+\begin{center}
+\def\jdo#1{\cn{#1} \ $\csname #1\endcsname$}
+\begin{tabular}{r@{\,, see \ }l}
+\cn{Box}&\jdo{square}\\
+\cn{Diamond}&\jdo{lozenge}\\
+\cn{leadsto}&\jdo{rightsquigarrow}\\
+\cn{Join}&\jdo{bowtie}\\
+\cn{lhd}&\jdo{vartriangleleft}\\
+\cn{unlhd}&\jdo{trianglelefteq}\\
+\cn{rhd}&\jdo{vartriangleright}\\
+\cn{unrhd}&\jdo{trianglerighteq}
+\end{tabular}
+\end{center}
+
+Furthermore, there are \strong{many, many additional symbols} available
+for \lat/ use above and beyond the ones included here. This list is not
+intended to be comprehensive. For a much more comprehensive list of
+symbols, including nonmathematically oriented ones, such as phonetic
+alphabetic or dingbats, see \emph{The Comprehensive \LaTeX{} Symbol
+List}~\cite{comprehensive}.  (Full font tables, ordered by font name,
+for all the fonts covered by the comprehensive list are included in
+the documentation provided by \TeX~Live: \texttt{texdoc rawtables}.
+These tables do not include symbol names.)  Another source of symbol
+information is the \pkg{unicode-math} package; see~\cite{uc-math}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\newpage
+
+\subsection{Alphabets and digits\nopunct}\label{alpha-digit}
+
+\subsubsection{Latin letters and Arabic numerals}
+The Latin letters are simple symbols, class~0. The default font for them
+in math formulas is italic.
+\begin{center}
+\begin{tabular}{c}
+  $A\,B\,C\,D\,E\,F\,G\,H\,I\,J\,K\,L\,M%
+   \,N\,O\,P\,Q\,R\,S\,T\,U\,V\,W\,X\,Y\,Z$\\
+  $a\,b\,c\,d\,e\,f\,g\,h\,i\,j\,k\,l\,m%
+   \,n\,o\,p\,q\,r\,s\,t\,u\,v\,w\,x\,y\,z$
+\end{tabular}
+\end{center}
+When adding an accent to an $i$ or $j$ in math, dotless variants can be
+obtained with \cn{imath} and \cn{jmath}:
+\begin{symlist}
+\dosymbol{Var}{imath}{kernel}
+\dosymbol{Var}{jmath}{kernel}
+\symbox{\hat{\jmath}}{\string\hat\string{\string\jmath\string}}
+\end{symlist}
+
+Arabic numerals 0\ndash 9 are also of class~0. Their default font is
+upright/roman.
+\[0\,1\,2\,3\,4\,5\,6\,7\,8\,9\]
+
+\subsubsection{Greek letters}
+Like the Latin letters, the Greek letters are simple symbols, class~0.
+For obscure historical reasons, the default font for lowercase Greek
+letters in math formulas is italic while the default font for capital
+Greek letters is upright/roman. (In other fields such as physics and
+chemistry, however, the typographical traditions are somewhat
+different.) The capital Greek letters not present in this list are the
+letters that have the same appearance as some Latin letter: A for Alpha,
+B for Beta, and so on. In the list of lowercase letters there is no
+omicron because it would be identical in appearance to Latin $o$. In
+practice, the Greek letters that have Latin look-alikes are seldom used
+in math formulas, to avoid confusion.
+\begin{symlist}[adjustheight=12pt]
+\dosymbol{Var}{Gamma}{kernel}
+\dosymbol{Var}{Delta}{kernel}
+\dosymbol{Var}{Lambda}{kernel}
+\dosymbol{Var}{Phi}{kernel}
+\dosymbol{Var}{Pi}{kernel}
+\dosymbol{Var}{Psi}{kernel}
+\dosymbol{Var}{Sigma}{kernel}
+\dosymbol{Var}{Theta}{kernel}
+\dosymbol{Var}{Upsilon}{kernel}
+\dosymbol{Var}{Xi}{kernel}
+\dosymbol{Var}{Omega}{kernel}
+\newcolumn
+\dosymbol{Var}{alpha}{kernel}
+\dosymbol{Var}{beta}{kernel}
+\dosymbol{Var}{gamma}{kernel}
+\dosymbol{Var}{delta}{kernel}
+\dosymbol{Var}{epsilon}{kernel}
+\dosymbol{Var}{zeta}{kernel}
+\dosymbol{Var}{eta}{kernel}
+\dosymbol{Var}{theta}{kernel}
+\dosymbol{Var}{iota}{kernel}
+\dosymbol{Var}{kappa}{kernel}
+\dosymbol{Var}{lambda}{kernel}
+\dosymbol{Var}{mu}{kernel}
+\newcolumn
+\dosymbol{Var}{nu}{kernel}
+\dosymbol{Var}{xi}{kernel}
+%\dosymbol{Var}{omicron}{??}
+\dosymbol{Var}{pi}{kernel}
+\dosymbol{Var}{rho}{kernel}
+\dosymbol{Var}{sigma}{kernel}
+\dosymbol{Var}{tau}{kernel}
+\dosymbol{Var}{upsilon}{kernel}
+\dosymbol{Var}{phi}{kernel}
+\dosymbol{Var}{chi}{kernel}
+\dosymbol{Var}{psi}{kernel}
+\dosymbol{Var}{omega}{kernel}
+\newcolumn
+\dosymbol{Ord}{digamma}{amssymb}
+\dosymbol{Var}{varepsilon}{kernel}
+\dosymbol{Ord}{varkappa}{amssymb}
+\dosymbol{Var}{varphi}{kernel}
+\dosymbol{Var}{varpi}{kernel}
+\dosymbol{Var}{varrho}{kernel}
+\dosymbol{Var}{varsigma}{kernel}
+\dosymbol{Var}{vartheta}{kernel}
+\end{symlist}
+
+\subsubsection{Other ``basic'' alphabetic symbols}
+These are also class~0.
+\begin{symlist}
+\dosymbol{Ord}{aleph}{kernel}
+\dosymbol{Ord}{beth}{amssymb}
+\dosymbol{Ord}{daleth}{amssymb}
+\dosymbol{Ord}{gimel}{amssymb}
+\dosymbol{Ord}{complement}{amssymb}
+\dosymbol{Ord}{ell}{kernel}
+\dosymbol{Ord}{eth}{amssymb}
+\dosymbol{Ord}{hbar}{amssymb}
+\dosymbol{Ord}{hslash}{amssymb}
+\dosymbol{Ord}{mho}{amssymb}
+\dosymbol{Ord}{partial}{kernel}
+\dosymbol{Ord}{wp}{kernel}
+\dosymbol{Ord}{circledS}{amssymb}
+\dosymbol{Ord}{Bbbk}{amssymb}
+\dosymbol{Ord}{Finv}{amssymb}
+\dosymbol{Ord}{Game}{amssymb}
+\dosymbol{Ord}{Im}{kernel}
+\dosymbol{Ord}{Re}{kernel}
+\vfil
+\end{symlist}
+\begin{notes}
+\end{notes}
+
+\subsubsection{Math font switches}\label{mathfonts}
+Not all of the fonts necessary to support comprehensive math font
+switching are commonly available in a typical \LaTeX{} setup. Here are
+the results of applying various font switches to a wide range of math
+symbols when the standard set of Computer Modern fonts is in use. It can
+be seen that the only symbols that respond correctly to all of the font
+switches are the uppercase Latin letters. In fact, \emph{nearly all}
+math symbols apart from Latin letters remain unaffected by font
+switches; and although the lowercase Latin letters, capital Greek
+letters, and numerals do respond properly to some font switches, they
+produce bizarre results for other font switches. (Use of alternative
+math font sets such as Lucida New Math may ameliorate the situation
+somewhat.)
+\[\renewcommand{\arraystretch}{1.3}
+\begin{array}{cccccccc}
+%\text{default}& \cn{mathbf}& \cn{mathsf}& \cn{mathit}& \cn{mathcal}&
+%  \cn{mathscr}&  \cn{mathbb}& \cn{mathfrak}\\
+\text{default}& \cn{mathbf}& \cn{mathrm}& \cn{mathsf}& \cn{mathit}&
+  \cn{mathcal}& \cn{mathbb}& \cn{mathfrak}\\
+\hline
+\symrow{X} \\
+\symrow{x} \\
+\symrow{0} \\
+\symrow{[\,]} \\
+\symrow{+} \\
+\symrow{-} \\
+\symrow{=} \\
+\symrow{\Xi} \\
+\symrow{\xi} \\
+\symrow{\infty} \\
+\symrow{\aleph} \\
+\symrow{\sum}\\
+\symrow{\amalg} \\
+\symrow{\Re} \\
+\end{array}\]
+A common desire is to get a bold version of a particular math symbol.
+For those symbols where \cn{mathbf} is not applicable, the
+\cn{boldsymbol} or \cn{pmb} commands can be used.
+\begin{equation}
+A_\infty + \pi A_0
+\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
+  \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
+\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
+\end{equation}
+\begin{verbatim}
+A_\infty + \pi A_0
+\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
+  \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
+\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
+\end{verbatim}
+The \cn{boldsymbol} command is obtained preferably by using
+the \pkg{bm} package, which provides a newer, more powerful version than
+the one provided by the \pkg{amsmath} package. It is usually
+ill-advised to apply \cn{boldsymbol} to more than one symbol at a time;
+if such a need seems to arise, it more likely means that there is
+another, better way of going about it.
+
+\subsubsection{Blackboard Bold letters (\fn{msbm}; no lowercase)}
+ Usage: \verb'\mathbb{R}'.  Requires \pkg{amsfonts}.
+\[
+\mathbb{A}\,\mathbb{B}\,\mathbb{C}\,\mathbb{D}\,\mathbb{E}\,\mathbb{F}
+\,\mathbb{G}\,\mathbb{H}\,\mathbb{I}\,\mathbb{J}\,\mathbb{K}\,\mathbb{L}
+\,\mathbb{M}\,\mathbb{N}\,\mathbb{O}\,\mathbb{P}\,\mathbb{Q}\,\mathbb{R}
+\,\mathbb{S}\,\mathbb{T}\,\mathbb{U}\,\mathbb{V}\,\mathbb{W}\,\mathbb{X}
+\,\mathbb{Y}\,\mathbb{Z}
+\]
+One lowercase letter is available with a distinct name:\qquad
+$\Bbbk$\quad \cn{Bbbk}
+
+%\newpage
+
+\subsubsection{Calligraphic letters (\fn{cmsy}; no lowercase)} Usage:
+\verb'\mathcal{M}'.
+\[
+\mathcal{A}\,\mathcal{B}\,\mathcal{C}\,\mathcal{D}\,\mathcal{E}
+\,\mathcal{F}\,\mathcal{G}\,\mathcal{H}\,\mathcal{I}\,\mathcal{J}
+\,\mathcal{K}\,\mathcal{L}\,\mathcal{M}\,\mathcal{N}\,\mathcal{O}
+\,\mathcal{P}\,\mathcal{Q}\,\mathcal{R}\,\mathcal{S}\,\mathcal{T}
+\,\mathcal{U}\,\mathcal{V}\,\mathcal{W}\,\mathcal{X}\,\mathcal{Y}
+\,\mathcal{Z}
+\]
+
+\subsubsection{Non-CM calligraphic and script letters}
+(\fn{rsfs}; no lowercase) Usage: \verb'\usepackage{mathrsfs}' \verb'\mathscr{B}'.
+\[
+\mathscr{A}\,\mathscr{B}\,\mathscr{C}\,\mathscr{D}\,\mathscr{E}
+\,\mathscr{F}\,\mathscr{G}\,\mathscr{H}\,\mathscr{I}\,\mathscr{J}
+\,\mathscr{K}\,\mathscr{L}\,\mathscr{M}\,\mathscr{N}\,\mathscr{O}
+\,\mathscr{P}\,\mathscr{Q}\,\mathscr{R}\,\mathscr{S}\,\mathscr{T}
+\,\mathscr{U}\,\mathscr{V}\,\mathscr{W}\,\mathscr{X}\,\mathscr{Y}
+\,\mathscr{Z}
+\]
+
+\begingroup
+\noindent
+(\fn{eusm}; no lowercase) Usage: \verb'\usepackage{euscript}' \verb'\mathscr{E}'.
+\renewcommand{\mathscr}{\EuScript}
+\[
+\mathscr{A}\,\mathscr{B}\,\mathscr{C}\,\mathscr{D}\,\mathscr{E}
+\,\mathscr{F}\,\mathscr{G}\,\mathscr{H}\,\mathscr{I}\,\mathscr{J}
+\,\mathscr{K}\,\mathscr{L}\,\mathscr{M}\,\mathscr{N}\,\mathscr{O}
+\,\mathscr{P}\,\mathscr{Q}\,\mathscr{R}\,\mathscr{S}\,\mathscr{T}
+\,\mathscr{U}\,\mathscr{V}\,\mathscr{W}\,\mathscr{X}\,\mathscr{Y}
+\,\mathscr{Z}
+\]
+\endgroup
+
+\subsubsection{Fraktur letters (\fn{eufm})}
+Usage: \verb'\mathfrak{S}'.  Requires \pkg{amsfonts}.
+\[
+\mathfrak{A}\,\mathfrak{B}\,\mathfrak{C}\,\mathfrak{D}\,\mathfrak{E}
+\,\mathfrak{F}\,\mathfrak{G}\,\mathfrak{H}\,\mathfrak{I}\,\mathfrak{J}
+\,\mathfrak{K}\,\mathfrak{L}\,\mathfrak{M}\,\mathfrak{N}\,\mathfrak{O}
+\,\mathfrak{P}\,\mathfrak{Q}\,\mathfrak{R}\,\mathfrak{S}\,\mathfrak{T}
+\,\mathfrak{U}\,\mathfrak{V}\,\mathfrak{W}\,\mathfrak{X}\,\mathfrak{Y}
+\,\mathfrak{Z}
+\]
+\[
+\mathfrak{a}\,\mathfrak{b}\,\mathfrak{c}\,\mathfrak{d}\,\mathfrak{e}
+\,\mathfrak{f}\,\mathfrak{g}\,\mathfrak{h}\,\mathfrak{i}\,\mathfrak{j}
+\,\mathfrak{k}\,\mathfrak{l}\,\mathfrak{m}\,\mathfrak{n}\,\mathfrak{o}
+\,\mathfrak{p}\,\mathfrak{q}\,\mathfrak{r}\,\mathfrak{s}\,\mathfrak{t}
+\,\mathfrak{u}\,\mathfrak{v}\,\mathfrak{w}\,\mathfrak{x}\,\mathfrak{y}
+\,\mathfrak{z}
+\]
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\subsection{Miscellaneous simple symbols}
+These symbols are also of class~0 (ordinary) which means
+they do not have any built-in spacing.%
+\begin{symlist}
+\dosymbol{Ordx}{\#}{kernel}
+\dosymbol{Ordx}{\&}{kernel}
+\dosymbol{Ord}{angle}{amssymb}
+\dosymbol{Ord}{backprime}{amssymb}
+\dosymbol{Ord}{bigstar}{amssymb}
+\dosymbol{Ord}{blacklozenge}{amssymb}
+\dosymbol{Ord}{blacksquare}{amssymb}
+\dosymbol{Ord}{blacktriangle}{amssymb}
+\dosymbol{Ord}{blacktriangledown}{amssymb}
+\dosymbol{Ord}{bot}{kernel}
+\dosymbol{Ord}{clubsuit}{kernel}
+\dosymbol{Ord}{diagdown}{amssymb}
+\dosymbol{Ord}{diagup}{amssymb}
+\dosymbol{Ord}{diamondsuit}{kernel}
+\dosymbol{Ord}{emptyset}{kernel}
+\dosymbol{Ord}{exists}{kernel}
+\dosymbol{Ord}{flat}{kernel}
+\dosymbol{Ord}{forall}{kernel}
+\dosymbol{Ord}{heartsuit}{kernel}
+\dosymbol{Ord}{infty}{kernel}
+\dosymbol{Ord}{lozenge}{amssymb}
+\dosymbol{Ord}{measuredangle}{amssymb}
+\dosymbol{Ord}{nabla}{kernel}
+\dosymbol{Ord}{natural}{kernel}
+\dosymbol{Ord}{neg}{kernel}
+\dosymbol{Ord}{nexists}{amssymb}
+\dosymbol{Ord}{prime}{kernel}
+\dosymbol{Ord}{sharp}{kernel}
+\dosymbol{Ord}{spadesuit}{kernel}
+\dosymbol{Ord}{sphericalangle}{amssymb}
+\dosymbol{Ord}{square}{amssymb}
+\dosymbol{Ord}{surd}{kernel}
+\dosymbol{Ord}{top}{kernel}
+\dosymbol{Ord}{triangle}{kernel}
+\dosymbol{Ord}{triangledown}{amssymb}
+\dosymbol{Ord}{varnothing}{amssymb}
+\end{symlist}
+\begin{notes}
+\item A common mistake in the use of the symbols $\square$ and $\#$
+  is to try to make them serve as binary operators or relation symbols
+  without using a properly defined math symbol command. If you merely
+  use the existing commands \cn{square} or \cn{\#} the intersymbol
+  spacing will be incorrect because those commands produce a class-0
+  symbol.
+\item Synonyms: \alias{lnot}
+
+\end{notes}
+
+\subsection{Binary operator symbols\nopunct}
+\begin{symlist}
+\dosymbol{Binc}{*}{kernel}
+\dosymbol{Binc}{+}{kernel}
+\dosymbol{Binc}{-}{kernel}
+\dosymbol{Bin}{amalg}{kernel}
+\dosymbol{Bin}{ast}{kernel}
+\dosymbol{Bin}{barwedge}{amssymb}
+\dosymbol{Bin}{bigcirc}{kernel}
+\dosymbol{Bin}{bigtriangledown}{kernel}
+\dosymbol{Bin}{bigtriangleup}{kernel}
+\dosymbol{Bin}{boxdot}{amssymb}
+\dosymbol{Bin}{boxminus}{amssymb}
+\dosymbol{Bin}{boxplus}{amssymb}
+\dosymbol{Bin}{boxtimes}{amssymb}
+\dosymbol{Bin}{bullet}{kernel}
+\dosymbol{Bin}{cap}{kernel}
+\dosymbol{Bin}{Cap}{amssymb}
+\dosymbol{Bin}{cdot}{kernel}
+\dosymbol{Bin}{centerdot}{amssymb}
+\dosymbol{Bin}{circ}{kernel}
+\dosymbol{Bin}{circledast}{amssymb}
+\dosymbol{Bin}{circledcirc}{amssymb}
+\dosymbol{Bin}{circleddash}{amssymb}
+\dosymbol{Bin}{cup}{kernel}
+\dosymbol{Bin}{Cup}{amssymb}
+\dosymbol{Bin}{curlyvee}{amssymb}
+\dosymbol{Bin}{curlywedge}{amssymb}
+\dosymbol{Bin}{dagger}{kernel}
+\dosymbol{Bin}{ddagger}{kernel}
+\dosymbol{Bin}{diamond}{kernel}
+\dosymbol{Bin}{div}{kernel}
+\dosymbol{Bin}{divideontimes}{amssymb}
+\dosymbol{Bin}{dotplus}{amssymb}
+\dosymbol{Bin}{doublebarwedge}{amssymb}
+\dosymbol{Bin}{gtrdot}{amssymb}
+\dosymbol{Bin}{intercal}{amssymb}
+\dosymbol{Bin}{leftthreetimes}{amssymb}
+\dosymbol{Bin}{lessdot}{amssymb}
+\dosymbol{Bin}{ltimes}{amssymb}
+\dosymbol{Bin}{mp}{kernel}
+\dosymbol{Bin}{odot}{kernel}
+\dosymbol{Bin}{ominus}{kernel}
+\dosymbol{Bin}{oplus}{kernel}
+\dosymbol{Bin}{oslash}{kernel}
+\dosymbol{Bin}{otimes}{kernel}
+\dosymbol{Bin}{pm}{kernel}
+\dosymbol{Bin}{rightthreetimes}{amssymb}
+\dosymbol{Bin}{rtimes}{amssymb}
+\dosymbol{Bin}{setminus}{kernel}
+\dosymbol{Bin}{smallsetminus}{amssymb}
+\dosymbol{Bin}{sqcap}{kernel}
+\dosymbol{Bin}{sqcup}{kernel}
+\dosymbol{Bin}{star}{kernel}
+\dosymbol{Bin}{times}{kernel}
+\dosymbol{Bin}{triangleleft}{kernel}
+\dosymbol{Bin}{triangleright}{kernel}
+\dosymbol{Bin}{uplus}{kernel}
+\dosymbol{Bin}{vee}{kernel}
+\dosymbol{Bin}{veebar}{amssymb}
+\dosymbol{Bin}{wedge}{kernel}
+\dosymbol{Bin}{wr}{kernel}
+\end{symlist}
+\begin{notes}
+\synonyms \alias{land}, \alias{lor}, \alias{doublecup}, \alias{doublecap}
+\end{notes}
+
+\subsection{Relation symbols:
+    \texorpdfstring{$<$ $=$ $>$ $\succ$ $\sim$}{< + > succeed ~}
+    and variants\nopunct}
+\begin{symlist}[adjustheight=10pt]
+\dosymbol{Relc}{<}{kernel}
+\dosymbol{Relc}{=}{kernel}
+\dosymbol{Relc}{>}{kernel}
+\dosymbol{Rel}{approx}{kernel}
+\dosymbol{Rel}{approxeq}{amssymb}
+\dosymbol{Rel}{asymp}{kernel}
+\dosymbol{Rel}{backsim}{amssymb}
+\dosymbol{Rel}{backsimeq}{amssymb}
+\dosymbol{Rel}{bumpeq}{amssymb}
+\dosymbol{Rel}{Bumpeq}{amssymb}
+\dosymbol{Rel}{circeq}{amssymb}
+\dosymbol{Rel}{cong}{kernel}
+\dosymbol{Rel}{curlyeqprec}{amssymb}
+\dosymbol{Rel}{curlyeqsucc}{amssymb}
+\dosymbol{Rel}{doteq}{kernel}
+\dosymbol{Rel}{doteqdot}{amssymb}
+\dosymbol{Rel}{eqcirc}{amssymb}
+\dosymbol{Rel}{eqsim}{amssymb}
+\dosymbol{Rel}{eqslantgtr}{amssymb}
+\dosymbol{Rel}{eqslantless}{amssymb}
+\dosymbol{Rel}{equiv}{kernel}
+\dosymbol{Rel}{fallingdotseq}{amssymb}
+\dosymbol{Rel}{geq}{kernel}
+\dosymbol{Rel}{geqq}{amssymb}
+\dosymbol{Rel}{geqslant}{amssymb}
+\dosymbol{Rel}{gg}{kernel}
+\dosymbol{Rel}{ggg}{amssymb}
+\dosymbol{Rel}{gnapprox}{amssymb}
+\dosymbol{Rel}{gneq}{amssymb}
+\dosymbol{Rel}{gneqq}{amssymb}
+\dosymbol{Rel}{gnsim}{amssymb}
+\dosymbol{Rel}{gtrapprox}{amssymb}
+\dosymbol{Rel}{gtreqless}{amssymb}
+\dosymbol{Rel}{gtreqqless}{amssymb}
+\dosymbol{Rel}{gtrless}{amssymb}
+\dosymbol{Rel}{gtrsim}{amssymb}
+\dosymbol{Rel}{gvertneqq}{amssymb}
+\dosymbol{Rel}{leq}{kernel}
+\dosymbol{Rel}{leqq}{amssymb}
+\dosymbol{Rel}{leqslant}{amssymb}
+\dosymbol{Rel}{lessapprox}{amssymb}
+\dosymbol{Rel}{lesseqgtr}{amssymb}
+\dosymbol{Rel}{lesseqqgtr}{amssymb}
+\dosymbol{Rel}{lessgtr}{amssymb}
+\dosymbol{Rel}{lesssim}{amssymb}
+\dosymbol{Rel}{ll}{kernel}
+\dosymbol{Rel}{lll}{amssymb}
+\dosymbol{Rel}{lnapprox}{amssymb}
+\dosymbol{Rel}{lneq}{amssymb}
+\dosymbol{Rel}{lneqq}{amssymb}
+\dosymbol{Rel}{lnsim}{amssymb}
+\dosymbol{Rel}{lvertneqq}{amssymb}
+\dosymbol{Rel}{ncong}{amssymb}
+\dosymbol{Rel}{neq}{kernel}
+\dosymbol{Rel}{ngeq}{amssymb}
+\dosymbol{Rel}{ngeqq}{amssymb}
+\dosymbol{Rel}{ngeqslant}{amssymb}
+\dosymbol{Rel}{ngtr}{amssymb}
+\dosymbol{Rel}{nleq}{amssymb}
+\dosymbol{Rel}{nleqq}{amssymb}
+\dosymbol{Rel}{nleqslant}{amssymb}
+\dosymbol{Rel}{nless}{amssymb}
+\dosymbol{Rel}{nprec}{amssymb}
+\dosymbol{Rel}{npreceq}{amssymb}
+\dosymbol{Rel}{nsim}{amssymb}
+\dosymbol{Rel}{nsucc}{amssymb}
+\dosymbol{Rel}{nsucceq}{amssymb}
+\dosymbol{Rel}{prec}{kernel}
+\dosymbol{Rel}{precapprox}{amssymb}
+\dosymbol{Rel}{preccurlyeq}{amssymb}
+\dosymbol{Rel}{preceq}{kernel}
+\dosymbol{Rel}{precnapprox}{amssymb}
+\dosymbol{Rel}{precneqq}{amssymb}
+\dosymbol{Rel}{precnsim}{amssymb}
+\dosymbol{Rel}{precsim}{amssymb}
+\dosymbol{Rel}{risingdotseq}{amssymb}
+\dosymbol{Rel}{sim}{kernel}
+\dosymbol{Rel}{simeq}{kernel}
+\dosymbol{Rel}{succ}{kernel}
+\dosymbol{Rel}{succapprox}{amssymb}
+\dosymbol{Rel}{succcurlyeq}{amssymb}
+\dosymbol{Rel}{succeq}{kernel}
+\dosymbol{Rel}{succnapprox}{amssymb}
+\dosymbol{Rel}{succneqq}{amssymb}
+\dosymbol{Rel}{succnsim}{amssymb}
+\dosymbol{Rel}{succsim}{amssymb}
+\dosymbol{Rel}{thickapprox}{amssymb}
+\dosymbol{Rel}{thicksim}{amssymb}
+\dosymbol{Rel}{triangleq}{amssymb}
+\end{symlist}
+\begin{notes}
+  \synonyms \alias{ne}, \alias{le}, \alias{ge}, \alias{Doteq}, \alias{llless}, \alias{gggtr}
+\end{notes}
+
+\subsection{Relation symbols: arrows}
+See also \secref{notations}.
+\begin{symlist}[adjustheight=10pt]
+\dosymbol{Rel}{circlearrowleft}{amssymb}
+\dosymbol{Rel}{circlearrowright}{amssymb}
+\dosymbol{Rel}{curvearrowleft}{amssymb}
+\dosymbol{Rel}{curvearrowright}{amssymb}
+\dosymbol{Rel}{downdownarrows}{amssymb}
+\dosymbol{Rel}{downharpoonleft}{amssymb}
+\dosymbol{Rel}{downharpoonright}{amssymb}
+\dosymbol{Rel}{hookleftarrow}{kernel}
+\dosymbol{Rel}{hookrightarrow}{kernel}
+\dosymbol{Rel}{leftarrow}{kernel}
+\dosymbol{Rel}{Leftarrow}{kernel}
+\dosymbol{Rel}{leftarrowtail}{amssymb}
+\dosymbol{Rel}{leftharpoondown}{kernel}
+\dosymbol{Rel}{leftharpoonup}{kernel}
+\dosymbol{Rel}{leftleftarrows}{amssymb}
+\dosymbol{Rel}{leftrightarrow}{kernel}
+\dosymbol{Rel}{Leftrightarrow}{kernel}
+\dosymbol{Rel}{leftrightarrows}{amssymb}
+\dosymbol{Rel}{leftrightharpoons}{amssymb}
+\dosymbol{Rel}{leftrightsquigarrow}{amssymb}
+\dosymbol{Rel}{Lleftarrow}{amssymb}
+\dosymbol{Rel}{longleftarrow}{kernel}
+\dosymbol{Rel}{Longleftarrow}{kernel}
+\dosymbol{Rel}{longleftrightarrow}{kernel}
+\dosymbol{Rel}{Longleftrightarrow}{kernel}
+\dosymbol{Rel}{longmapsto}{kernel}
+\dosymbol{Rel}{longrightarrow}{kernel}
+\dosymbol{Rel}{Longrightarrow}{kernel}
+\dosymbol{Rel}{looparrowleft}{amssymb}
+\dosymbol{Rel}{looparrowright}{amssymb}
+\dosymbol{Rel}{Lsh}{amssymb}
+\dosymbol{Rel}{mapsto}{kernel}
+\dosymbol{Rel}{multimap}{amssymb}
+\dosymbol{Rel}{nLeftarrow}{amssymb}
+\dosymbol{Rel}{nLeftrightarrow}{amssymb}
+\dosymbol{Rel}{nRightarrow}{amssymb}
+\dosymbol{Rel}{nearrow}{kernel}
+\dosymbol{Rel}{nleftarrow}{amssymb}
+\dosymbol{Rel}{nleftrightarrow}{amssymb}
+\dosymbol{Rel}{nrightarrow}{amssymb}
+\dosymbol{Rel}{nwarrow}{kernel}
+\dosymbol{Rel}{rightarrow}{kernel}
+\dosymbol{Rel}{Rightarrow}{kernel}
+\dosymbol{Rel}{rightarrowtail}{amssymb}
+\dosymbol{Rel}{rightharpoondown}{kernel}
+\dosymbol{Rel}{rightharpoonup}{kernel}
+\dosymbol{Rel}{rightleftarrows}{amssymb}
+\dosymbol{Rel}{rightleftharpoons}{amssymb}
+\dosymbol{Rel}{rightrightarrows}{amssymb}
+\dosymbol{Rel}{rightsquigarrow}{amssymb}
+\dosymbol{Rel}{Rrightarrow}{amssymb}
+\dosymbol{Rel}{Rsh}{amssymb}
+\dosymbol{Rel}{searrow}{kernel}
+\dosymbol{Rel}{swarrow}{kernel}
+\dosymbol{Rel}{twoheadleftarrow}{amssymb}
+\dosymbol{Rel}{twoheadrightarrow}{amssymb}
+\dosymbol{Rel}{upharpoonleft}{amssymb}
+\dosymbol{Rel}{upharpoonright}{amssymb}
+\dosymbol{Rel}{upuparrows}{amssymb}
+\end{symlist}
+\begin{notes}
+  \synonyms \alias{gets}, \alias{to}, \alias{restriction}
+\end{notes}
+
+\subsection{Relation symbols: miscellaneous\nopunct}
+\begin{symlist}[adjustheight=10pt]
+\dosymbol{Rel}{backepsilon}{amssymb}
+\dosymbol{Rel}{because}{amssymb}
+\dosymbol{Rel}{between}{amssymb}
+\dosymbol{Rel}{blacktriangleleft}{amssymb}
+\dosymbol{Rel}{blacktriangleright}{amssymb}
+\dosymbol{Rel}{bowtie}{kernel}
+\dosymbol{Rel}{dashv}{kernel}
+\dosymbol{Rel}{frown}{kernel}
+\dosymbol{Rel}{in}{kernel}
+\dosymbol{Rel}{mid}{kernel}
+\dosymbol{Rel}{models}{kernel}
+\dosymbol{Rel}{ni}{kernel}
+\dosymbol{Rel}{nmid}{amssymb}
+\dosymbol{Rel}{notin}{kernel}
+\dosymbol{Rel}{nparallel}{amssymb}
+\dosymbol{Rel}{nshortmid}{amssymb}
+\dosymbol{Rel}{nshortparallel}{amssymb}
+\dosymbol{Rel}{nsubseteq}{amssymb}
+\dosymbol{Rel}{nsubseteqq}{amssymb}
+\dosymbol{Rel}{nsupseteq}{amssymb}
+\dosymbol{Rel}{nsupseteqq}{amssymb}
+\dosymbol{Rel}{ntriangleleft}{amssymb}
+\dosymbol{Rel}{ntrianglelefteq}{amssymb}
+\dosymbol{Rel}{ntriangleright}{amssymb}
+\dosymbol{Rel}{ntrianglerighteq}{amssymb}
+\dosymbol{Rel}{nvdash}{amssymb}
+\dosymbol{Rel}{nVdash}{amssymb}
+\dosymbol{Rel}{nvDash}{amssymb}
+\dosymbol{Rel}{nVDash}{amssymb}
+\dosymbol{Rel}{parallel}{kernel}
+\dosymbol{Rel}{perp}{kernel}
+\dosymbol{Rel}{pitchfork}{amssymb}
+\dosymbol{Rel}{propto}{kernel}
+\dosymbol{Rel}{shortmid}{amssymb}
+\dosymbol{Rel}{shortparallel}{amssymb}
+\dosymbol{Rel}{smallfrown}{amssymb}
+\dosymbol{Rel}{smallsmile}{amssymb}
+\dosymbol{Rel}{smile}{kernel}
+\dosymbol{Rel}{sqsubset}{amssymb}
+\dosymbol{Rel}{sqsubseteq}{kernel}
+\dosymbol{Rel}{sqsupset}{amssymb}
+\dosymbol{Rel}{sqsupseteq}{kernel}
+\dosymbol{Rel}{subset}{kernel}
+\dosymbol{Rel}{Subset}{amssymb}
+\dosymbol{Rel}{subseteq}{kernel}
+\dosymbol{Rel}{subseteqq}{amssymb}
+\dosymbol{Rel}{subsetneq}{amssymb}
+\dosymbol{Rel}{subsetneqq}{amssymb}
+\dosymbol{Rel}{supset}{kernel}
+\dosymbol{Rel}{Supset}{amssymb}
+\dosymbol{Rel}{supseteq}{kernel}
+\dosymbol{Rel}{supseteqq}{amssymb}
+\dosymbol{Rel}{supsetneq}{amssymb}
+\dosymbol{Rel}{supsetneqq}{amssymb}
+\dosymbol{Rel}{therefore}{amssymb}
+\dosymbol{Rel}{trianglelefteq}{amssymb}
+\dosymbol{Rel}{trianglerighteq}{amssymb}
+\dosymbol{Rel}{varpropto}{amssymb}
+\dosymbol{Rel}{varsubsetneq}{amssymb}
+\dosymbol{Rel}{varsubsetneqq}{amssymb}
+\dosymbol{Rel}{varsupsetneq}{amssymb}
+\dosymbol{Rel}{varsupsetneqq}{amssymb}
+\dosymbol{Rel}{vartriangle}{amssymb}
+\dosymbol{Rel}{vartriangleleft}{amssymb}
+\dosymbol{Rel}{vartriangleright}{amssymb}
+\dosymbol{Rel}{vdash}{kernel}
+\dosymbol{Rel}{Vdash}{amssymb}
+\dosymbol{Rel}{vDash}{amssymb}
+\dosymbol{Rel}{Vvdash}{amssymb}
+\end{symlist}
+\begin{notes}
+  \synonyms \alias{owns}
+\end{notes}
+
+\subsection{Cumulative (variable-size) operators\nopunct}
+\begin{symlist}[adjustcols=-1]
+\openup3pt
+\dosymbol{COi}{int}{kernel}
+\dosymbol{COi}{oint}{kernel}
+\dosymbol{COs}{bigcap}{kernel}
+\dosymbol{COs}{bigcup}{kernel}
+\dosymbol{COs}{bigodot}{kernel}
+\dosymbol{COs}{bigoplus}{kernel}
+\dosymbol{COs}{bigotimes}{kernel}
+\dosymbol{COs}{bigsqcup}{kernel}
+\dosymbol{COs}{biguplus}{kernel}
+\dosymbol{COs}{bigvee}{kernel}
+\dosymbol{COs}{bigwedge}{kernel}
+\dosymbol{COs}{coprod}{kernel}
+\dosymbol{COs}{prod}{kernel}
+\dosymbol{COs}{smallint}{kernel}
+\dosymbol{COs}{sum}{kernel}
+\end{symlist}
+
+\subsection{Punctuation\nopunct}
+\begin{symlist}[adjustcols=-4]
+\openup2pt
+\dosymbol{Ordc}{.}{kernel}
+\dosymbol{Ordc}{/}{kernel}
+\dosymbol{Ordc}{|}{kernel}
+\dosymbol{Punc}{,}{kernel}
+\dosymbol{Punc}{;}{kernel}
+\dosymbol{Pun}{colon}{kernel}
+\dosymbol{Relc}{:}{kernel}
+%\dosymbol{DeR}{!}{kernel}
+%\dosymbol{DeR}{?}{kernel}
+\dosymbol{Punc}{!}{kernel}
+\dosymbol{Punc}{?}{kernel}
+\dosymbol{Inn}{dotsb}{kernel}
+\dosymbol{Inn}{dotsc}{kernel}
+\dosymbol{Inn}{dotsi}{kernel}
+\dosymbol{Inn}{dotsm}{kernel}
+\dosymbol{Inn}{dotso}{kernel}
+\dosymbol{Inn}{ddots}{kernel}
+\dosymbol{Ord}{vdots}{kernel}
+\end{symlist}
+\begin{notes}
+\item The \verb':' by itself produces a colon with
+  class-3 (relation) spacing. The command \cn{colon} produces special
+  spacing for use in constructions such as \verb'f\colon A\to B'
+  $f\colon A\to B$.
+\item Although the commands \cn{cdots} and \cn{ldots} are frequently
+  used, we recommend the more semantically oriented commands
+  \cn{dotsb} \cn{dotsc} \cn{dotsi} \cn{dotsm} \cn{dotso} for most
+  purposes\dotsref.
+\end{notes}
+
+\subsection{Pairing delimiters (extensible)}\label{pair-delims}
+See Section~\ref{delim} for more information.
+\begin{symlist}
+\openup7pt
+\dosymbol{DeLRc}{(}{)}{kernel}
+\dosymbol{DeLRc}{[}{]}{kernel}
+\dosymbol{DeLR}{lbrace}{rbrace}{kernel}
+\dosymbol{DeLR}{lvert}{rvert}{kernel}
+\dosymbol{DeLR}{lVert}{rVert}{kernel}
+\dosymbol{DeLR}{langle}{rangle}{kernel}
+\dosymbol{DeLR}{lceil}{rceil}{kernel}
+\dosymbol{DeLR}{lfloor}{rfloor}{kernel}
+\dosymbol{DeLR}{lgroup}{rgroup}{kernel}
+\dosymbol{DeLR}{lmoustache}{rmoustache}{kernel}
+\end{symlist}
+
+\subsection{Nonpairing extensible symbols\nopunct}
+\begin{symlist}
+\dosymbol{DeB}{vert}{kernel}
+\dosymbol{DeB}{Vert}{kernel}
+\dosymbol{DeBc}{/}{kernel}
+\dosymbol{DeB}{backslash}{kernel}
+\dosymbol{DeB}{arrowvert}{kernel}
+\dosymbol{DeB}{Arrowvert}{kernel}
+\dosymbol{DeB}{bracevert}{kernel}
+\end{symlist}
+\begin{notes}
+\item Using \cn{vert}, \verb'|', \cn{Vert}, or \cn{|} for paired
+delimiters is not recommended\vertref.  Instead, use delimiters from
+the list in Section~\ref{pair-delims}.
+  \synonyms \alias{|}
+\end{notes}
+
+\subsection{Extensible vertical arrows\nopunct}
+\begin{symlist}
+\dosymbol{DeA}{uparrow}{kernel}
+\dosymbol{DeA}{Uparrow}{kernel}
+\dosymbol{DeA}{downarrow}{kernel}
+\dosymbol{DeA}{Downarrow}{kernel}
+\dosymbol{DeA}{updownarrow}{kernel}
+\dosymbol{DeA}{Updownarrow}{kernel}
+\end{symlist}
+
+\subsection{Math accents\nopunct}\label{accents}
+\begin{symlist}[adjustcols=1]
+\dosymbol{Acc}{acute}{kernel}
+\dosymbol{Acc}{grave}{kernel}
+\dosymbol{Acc}{ddot}{kernel}
+\dosymbol{Acc}{tilde}{kernel}
+\dosymbol{Acc}{bar}{kernel}
+\dosymbol{Acc}{breve}{kernel}
+\dosymbol{Acc}{check}{kernel}
+\dosymbol{Acc}{hat}{kernel}
+\dosymbol{Acc}{vec}{kernel}
+\dosymbol{Acc}{dot}{kernel}
+\dosymbol{Acc}{ddot}{amsmath}
+\dosymbol{Acc}{dddot}{amsmath}
+\dosymbol{Acc}{mathring}{amsmath}
+\dosymbol{Accw}{widetilde}{kernel}
+\dosymbol{Accw}{widehat}{kernel}
+\end{symlist}
+
+\subsection{Named operators}
+These operators are represented by a multiletter abbreviation.
+\shiftlistright40pt
+\begin{symlist}[adjustcols=-1]
+\dosymbol{Opn}{arccos}{kernel}
+\dosymbol{Opn}{arcsin}{kernel}
+\dosymbol{Opn}{arctan}{kernel}
+\dosymbol{Opn}{arg}{kernel}
+\dosymbol{Opn}{cos}{kernel}
+\dosymbol{Opn}{cosh}{kernel}
+\dosymbol{Opn}{cot}{kernel}
+\dosymbol{Opn}{coth}{kernel}
+\dosymbol{Opn}{csc}{kernel}
+\dosymbol{Opn}{deg}{kernel}
+\dosymbol{Opn}{det}{kernel}
+\dosymbol{Opn}{dim}{kernel}
+\dosymbol{Opn}{exp}{kernel}
+\dosymbol{Opn}{gcd}{kernel}
+\dosymbol{Opn}{hom}{kernel}
+\dosymbol{Opn}{inf}{kernel}
+\dosymbol{Opn}{injlim}{kernel}
+\dosymbol{Opn}{ker}{kernel}
+\dosymbol{Opn}{lg}{kernel}
+\dosymbol{Opn}{lim}{kernel}
+\dosymbol{Opn}{liminf}{kernel}
+\dosymbol{Opn}{limsup}{kernel}
+\dosymbol{Opn}{ln}{kernel}
+\dosymbol{Opn}{log}{kernel}
+\dosymbol{Opn}{max}{kernel}
+\dosymbol{Opn}{min}{kernel}
+\dosymbol{Opn}{Pr}{kernel}
+\dosymbol{Opn}{projlim}{kernel}
+\dosymbol{Opn}{sec}{kernel}
+\dosymbol{Opn}{sin}{kernel}
+\dosymbol{Opn}{sinh}{kernel}
+\dosymbol{Opn}{sup}{kernel}
+\dosymbol{Opn}{tan}{kernel}
+\dosymbol{Opn}{tanh}{kernel}
+\dosymbol{Opn}{varinjlim}{kernel}
+\dosymbol{Opn}{varprojlim}{kernel}
+\dosymbol{Opn}{varliminf}{kernel}
+\dosymbol{Opn}{varlimsup}{kernel}
+  \end{symlist}
+
+To define additional named operators outside the above list, use the
+\cn{DeclareMathOperator} command; for example, after
+\begin{verbatim}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\esssup}{ess\,sup}
+\end{verbatim}
+one could write
+\begin{center}
+\begin{tabular}{rl}
+\verb'\rank(x)'& $\rank(x)$\\
+\verb'\esssup(y,z)'& $\esssup(y,z)$
+\end{tabular}
+\end{center}
+The star form \cn{DeclareMathOperator*} creates an operator that takes
+limits in a displayed formula, such as $\sup$ or $\max$.
+
+When predefining such a named operator is problematic (e.g., when using
+one in the title or abstract of an article), there is an alternative
+form that can be used directly:
+\[\verb'\operatorname{rank}(x)'\quad
+  \rightarrow\quad\operatorname{rank}(x)\]
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Notations}
+\label{notations}
+
+\subsection{Top and bottom embellishments}
+
+These are visually similar to accents but generally span multiple
+symbols rather than being applied to a single base symbol. For ease of
+reference, \cn{widetilde} and \cn{widehat} are redundantly included here
+and in the table of math accents.
+\begin{symlist}
+\dosymbol{Accw}{widetilde}{kernel}
+\dosymbol{Accw}{widehat}{kernel}
+\dosymbol{Accw}{overline}{kernel}
+\dosymbol{Accw}{underline}{kernel}
+\dosymbol{Accw}{overbrace}{kernel}
+\dosymbol{Accw}{underbrace}{kernel}
+\dosymbol{Accw}{overleftarrow}{kernel}
+\dosymbol{Accw}{underleftarrow}{amsmath}
+\dosymbol{Accw}{overrightarrow}{kernel}
+\dosymbol{Accw}{underrightarrow}{amsmath}
+\dosymbol{Accw}{overleftrightarrow}{amsmath}
+\dosymbol{Accw}{underleftrightarrow}{amsmath}
+\end{symlist}
+
+\subsection{Extensible arrows}
+
+\cn{xleftarrow} and \cn{xrightarrow} produce
+arrows\index{arrows!extensible} that extend automatically to accommodate
+unusually wide subscripts or superscripts. These commands take one
+optional argument (the subscript) and one mandatory argument (the
+superscript, possibly empty):
+\begin{equation}
+A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C
+\end{equation}
+\begin{verbatim}
+  \xleftarrow{n+\mu-1}\quad \xrightarrow[T]{n\pm i-1}
+\end{verbatim}
+
+\subsection{Affixing symbols to other symbols}
+
+In addition to the standard accents (Section~\ref{accents}), other
+symbols can be placed above or below a base symbol with the \cn{overset}
+and \cn{underset} commands. For example, writing \verb|\overset{*}{X}|
+will place a superscript-size $*$ above the $X$, thus: $\overset{*}{X}$.
+See also the description of \cn{sideset} in \secref{sideset}.
+
+\subsection{Matrices}\label{ss:matrix}
+
+The environments \env{pmatrix}, \env{bmatrix}, \env{Bmatrix},
+\env{vmatrix}, and \env{Vmatrix} have (respectively) $(\,)$, $[\,]$,
+$\lbrace\,\rbrace$, $\lvert\,\rvert$, and $\lVert\,\rVert$ delimiters
+built in. There is also a \env{matrix} environment without delimiters
+and an \env{array} environment that can be used to obtain left alignment
+or other variations in the column specs.
+\begin{center}
+\begin{minipage}{.4\columnwidth}
+\begin{verbatim}
+\begin{pmatrix}
+\alpha& \beta^{*}\\
+\gamma^{*}& \delta
+\end{pmatrix}
+\end{verbatim}
+\end{minipage}
+\qquad
+\begin{minipage}{.4\columnwidth}
+\[
+\begin{pmatrix}
+\alpha& \beta^{*}\\
+\gamma^{*}& \delta
+\end{pmatrix}
+\]
+\end{minipage}
+\end{center}
+To produce a small matrix suitable for use in text, there is a
+\env{smallmatrix} environment (e.g.,
+\begin{math}
+\bigl( \begin{smallmatrix}
+  a&b\\ c&d
+\end{smallmatrix} \bigr)
+\end{math})
+that comes closer to fitting within a single text line than a normal
+matrix. This example was produced by
+\begin{verbatim}
+\bigl( \begin{smallmatrix}
+  a&b\\ c&d
+\end{smallmatrix} \bigr)
+\end{verbatim}
+By default, all elements in a matrix are centered horizontally.
+The \pkg{mathtools} package provides starred versions of all the matrix
+environments that facilitate other alignments.  That package also provides
+fenced versions of \env{smallmatrix} with parallel names in both starred
+and nonstarred versions.
+
+To produce a row of dots in a matrix\index{matrices!ellipsis
+  dots}\index{ellipsis dots!in matrices}\index{dots|see{ellipsis dots}}
+spanning a given number of columns, use \cn{hdotsfor}. For example,
+\verb'\hdotsfor{3}' in the second column of a four-column matrix will
+print a row of dots across the final three columns.
+
+For piecewise function definitions there is a \env{cases} environment:
+\begin{verbatim}
+P_{r-j}=\begin{cases}
+    0&  \text{if $r-j$ is odd},\\
+    r!\,(-1)^{(r-j)/2}&  \text{if $r-j$ is even}.
+  \end{cases}
+\end{verbatim}
+Notice the use of \cn{text} and the embedded math.
+
+\begin{notes}
+  \singlenote The plain \TeX{} form \verb'\matrix{...\cr...\cr}' and the
+  related commands \cn{pmatrix}, \cn{cases} should be avoided in
+  \LaTeX{} (and when the \pkg{amsmath} package is loaded they are
+  disabled).
+\end{notes}
+
+\subsection{Math spacing commands}
+
+When the \pkg{amsmath} package is used, all of these math spacing
+commands can be used both in and out of math mode.
+\begin{center}\begin{tabular}{llllll}
+Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\
+\hline
+\strut & no space& \spx{}& & no space& \spx{}\\
+\cn{\,}& \cn{thinspace}& \spx{\,}&
+  \cn{!}& \cn{negthinspace}& \spx{\!}\\
+\cn{\:}& \cn{medspace}& \spx{\:}&
+  & \cn{negmedspace}& \spx{\negmedspace}\\
+\cn{\;}& \cn{thickspace}& \spx{\;}&
+  & \cn{negthickspace}& \spx{\negthickspace}\\
+& \cn{quad}& \spx{\quad}\\
+& \cn{qquad}& \spx{\qquad}
+\end{tabular}\end{center}
+For finer control over math spacing, use \cn{mspace}
+and `math units'. One math unit, or \verb|mu|, is equal to 1/18 em. Thus to
+get a negative half \cn{quad} write \verb|\mspace{-9.0mu}|.
+
+There are also three commands that leave a space equal to the height
+and/or width of a given fragment of \lat/ material:
+\begin{center}\begin{tabular}{ll}
+\colhead{Example}& \colhead{Result}\\
+\hline
+\verb'\phantom{XXX}'& space as wide and high as three X's\strut \\
+\verb'\hphantom{XXX}'& space as wide as three X's; height 0\\
+\verb'\vphantom{X}'& space of width 0, height = height of X
+\end{tabular}\end{center}
+
+\subsection{Dots}\label{dots}
+
+For preferred placement of ellipsis dots (raised or on-line) in various
+contexts there is no general consensus. It may therefore be considered a
+matter of taste.  In most situations, the generic \cn{dots} can be used,
+and \pkg{amsmath} will interpret it in the manner preferred by the AMS,
+namely low dots (\cn{ldots}) between commas or raised dots (\cn{cdots})
+between binary operators and relations, etc.  If what follows the dots is
+ambiguous as to the choice, the specific form of the command can be used.
+However, by using the semantically oriented commands
+\begin{itemize}
+\setlength{\itemsep}{0pt}
+\item \cn{dotsc} for \qq{dots with commas}
+\item \cn{dotsb} for \qq{dots with binary operators/relations}
+\item \cn{dotsm} for \qq{multiplication dots}
+\item \cn{dotsi} for \qq{dots with integrals}
+\item \cn{dotso} for \qq{other dots} (none of the above)
+\end{itemize}
+instead of \cn{ldots} and \cn{cdots}, you make it possible for your
+document to be adapted to different conventions on the fly, in case (for
+example) you have to submit it to a publisher who insists on following
+house tradition in this respect. The default treatment for the various
+kinds follows American Mathematical Society conventions:
+\begin{center}
+\vspace{-\topsep}
+\begin{tabular}{@{}ll@{}}
+\begin{minipage}[t]{.50\textwidth}
+\small
+\begin{verbatim}
+We have the series $A_1,A_2,\dotsc$,
+the regional sum $A_1+A_2+\dotsb$,
+the orthogonal product $A_1A_2\dotsm$,
+and the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi\].
+\end{verbatim}
+\end{minipage}
+&
+\begin{minipage}[t]{.48\textwidth}
+\noindent
+We have the series $A_1,A_2,\dotsc$,
+the regional sum $A_1+A_2+\dotsb$,
+the orthogonal product $A_1A_2\dotsm$,
+and the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi.\]
+\end{minipage}
+\end{tabular}
+\end{center}
+
+\subsection{Nonbreaking dashes}
+
+The command \cn{nobreakdash} suppresses the possibility
+of a linebreak after the following hyphen or dash. For example, if you
+write `pages 1\ndash 9' as \verb|pages 1\nobreakdash--9| then a linebreak will
+never occur between the dash and the 9. You can also use
+\cn{nobreakdash} to prevent undesirable hyphenations in combinations
+like \verb|$p$-adic|. For frequent use, it's advisable to make abbreviations,
+e.g.,
+\begin{verbatim}
+\newcommand{\p}{$p$\nobreakdash}% for "\p adic" ("p-adic")
+\newcommand{\Ndash}{\nobreakdash\textendash}% for "pages 1\Ndash 9"
+%    For "\n dimensional" ("n-dimensional"):
+\newcommand{\n}{$n$\nobreakdash-\hspace{0pt}}
+\end{verbatim}
+The last example shows how to prohibit a linebreak after the hyphen but
+allow normal hyphenation in the following word. (Add a zero-width space
+after the hyphen.)
+
+\subsection{Roots}
+The command \cn{sqrt} produces a square root. To specify an explicit
+radix, give it as an optional argument.
+\[
+\verb'\sqrt{\frac{n}{n-1} S}'\quad\sqrt{\frac{n}{n-1} S}, \qquad
+\verb'\sqrt[3]{2}'\quad
+\sqrt[3]{2}
+\]
+
+\subsection{Boxed formulas}
+
+The command \cn{boxed} puts a box around its
+argument, like \cn{fbox} except that the contents are in math mode:
+\begin{equation}
+\boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
+\end{equation}
+\begin{verbatim}
+  \boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))}
+\end{verbatim}
+If you need to box an equation including the equation number, it may be
+difficult, depending on the context; there are some suggestions in the
+AMS author FAQ; see the entry outlined in red on the page
+\url{https://www.ams.org/faq?faq_id=290}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Fractions and related constructions}
+
+\subsection{The \hycn{frac}, \hycn{dfrac}, and
+  \hycn{tfrac} commands}
+
+The \cn{frac} command\index{fractions} takes two arguments\mdash
+numerator and denominator\mdash and typesets them in normal fraction
+form. Use \cn{dfrac} or \cn{tfrac} to overrule \LaTeX{}'s guess about
+the proper size to use for the fraction's contents (t = text style, d =
+display style).
+\begin{equation}
+\frac{1}{k}\log_2 c(f),\quad\dfrac{1}{k}\log_2 c(f),\quad\tfrac{1}{k}\log_2 c(f)
+\end{equation}
+\begin{verbatim}
+\begin{equation}
+\frac{1}{k}\log_2 c(f),\quad\dfrac{1}{k}\log_2 c(f),
+    \quad\tfrac{1}{k}\log_2 c(f)
+\end{equation}
+\end{verbatim}
+\begin{equation}
+\Re{z} =\frac{n\pi \dfrac{\theta +\psi}{2}}{
+        \left(\dfrac{\theta +\psi}{2}\right)^2 + \left( \dfrac{1}{2}
+        \log \left\lvert\dfrac{B}{A}\right\rvert\right)^2}.
+\end{equation}
+\begin{verbatim}
+\begin{equation}
+\Re{z} =\frac{n\pi \dfrac{\theta +\psi}{2}}{
+        \left(\dfrac{\theta +\psi}{2}\right)^2 + \left( \dfrac{1}{2}
+        \log \left\lvert\dfrac{B}{A}\right\rvert\right)^2}.
+\end{equation}
+\end{verbatim}
+
+\subsection{The \hycn{binom}, \hycn{dbinom}, and
+        \hycn{tbinom} commands}
+
+For binomial expressions\index{binomials} such as $\binom{n}{k}$
+there are \cn{binom}, \cn{dbinom} and \cn{tbinom} commands:
+\begin{equation}
+2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
+\end{equation}
+\begin{verbatim}
+2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}
+\end{verbatim}
+
+\subsection{The \hycn{genfrac} command}
+
+The capabilities of \cn{frac}, \cn{binom}, and their variants are
+subsumed by a generalized fraction command \cn{genfrac} with six
+arguments. The last two correspond to \cn{frac}'s numerator and
+denominator; the first two are optional delimiters (as seen in
+\cn{binom}); the third is a line thickness override (\cn{binom} uses
+this to set the fraction line thickness to 0 pt\mdash i.e., invisible);
+and the fourth argument is a mathstyle override: integer values
+0\ndash 3 select, respectively, \cn{displaystyle}, \cn{textstyle},
+\cn{scriptstyle}, and \cn{scriptscriptstyle}. If the third argument is
+left empty, the line thickness defaults to ``normal''.
+
+\begin{cmdspec}[25em]
+\string\genfrac \ma{left-delim} \ma{right-delim} \ma{thickness}
+\ma{mathstyle} \ma{numerator} \ma{denominator}
+\end{cmdspec}
+To illustrate, here is how \cn{frac}, \cn{tfrac}, and
+\cn{binom} might be defined.
+\begin{verbatim}
+\newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}}
+\newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}}
+\newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}}
+\end{verbatim}
+
+\begin{notes}
+  \singlenote For technical reasons, using the primitive fraction
+  commands \cn{over}, \cn{atop}, \cn{above} in a \LaTeX{} document is
+  not recommended (see, e.g., \url{https://www.ams.org/faq?faq\_id=288},
+  the entry outlined in red).
+\end{notes}
+
+\subsection{Continued fractions}
+
+The continued fraction\index{continued fractions}
+\begin{equation}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+  \cfrac{1}{\sqrt{2}+\cdots
+}}}
+\end{equation}
+can be obtained by typing
+{\samepage
+\begin{verbatim}
+\cfrac{1}{\sqrt{2}+
+ \cfrac{1}{\sqrt{2}+
+  \cfrac{1}{\sqrt{2}+\dotsb
+}}}
+\end{verbatim}
+}% End of \samepage
+This produces better-looking results than straightforward use of
+\cn{frac}. Left or right placement of any of the numerators is
+accomplished by using \cn{cfrac}\verb|[l]| or \cn{cfrac}\verb|[r]| instead of
+\cn{cfrac}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Delimiters}\label{delim}
+
+\subsection{Delimiter sizes}\label{bigdel}
+
+Unless you indicate otherwise, delimiters in math formulas will remain
+at the standard size regardless of the height of the enclosed material.
+To get larger sizes, you can either select a particular size using a
+\cn{big...} prefix (see below), or you can use \cn{left} and \cn{right}
+prefixes for autosizing.
+
+The automatic delimiter sizing done by \cn{left} and \cn{right} has two
+limitations: first, it is applied mechanically to produce delimiters
+large enough to encompass the largest contained item, and second, the
+range of sizes has fairly large quantum jumps. This means that an
+expression that is infinitesimally too large for a given delimiter size
+will get the next larger size, a jump of 6pt or so (3pt top and bottom)
+in normal-sized text. There are two or three situations where the
+delimiter size is commonly adjusted. These adjustments are
+done using the following commands:
+\begin{center}\begin{tabular}{l|llllll}
+Delimiter&
+  no size& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\
+size&
+  specified& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\[4pt]
+%\hline\omit\rule{0pt}{1ex}\\
+\hline\omit\rule{0pt}{1ex}\\[-1ex]
+Result $\vphantom{\Bigg|^{\frac{1}{2}}}$ & % force height to avoid gap in vertical
+  $\displaystyle(b)(\frac{c}{d})$&
+  $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$&
+  $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$&
+  $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$&
+  $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$&
+  $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$
+\end{tabular}\end{center}
+The first kind of adjustment is done for cumulative operators with
+limits, such as summation signs. With \cn{left} and \cn{right} the
+delimiters usually turn out larger than necessary, and using the
+\verb|Big| or \verb|bigg| sizes\index{big@\cn{big}, \cn{Big}, \cn{bigg},
+  \dots\ delimiters} instead gives better results:
+\begin{equation*}
+\left[\sum_i a_i\left\lvert\sum_j x_{ij}\right\rvert^p\right]^{1/p}
+\quad\text{versus}\quad
+\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}
+\end{equation*}
+\begin{verbatim}
+\biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p}
+\end{verbatim}
+The second kind of situation is clustered pairs of delimiters, where
+\cn{left} and \cn{right} make them all the same size (because that is
+adequate to cover the encompassed material), but what you really want
+is to make some of the delimiters slightly larger to make the nesting
+easier to see.
+\begin{equation*}
+\left((a_1 b_1) - (a_2 b_2)\right)
+\left((a_2 b_1) + (a_1 b_2)\right)
+\quad\text{versus}\quad
+\bigl((a_1 b_1) - (a_2 b_2)\bigr)
+\bigl((a_2 b_1) + (a_1 b_2)\bigr)
+\end{equation*}
+\begin{verbatim}
+\left((a_1 b_1) - (a_2 b_2)\right)
+\left((a_2 b_1) + (a_1 b_2)\right)
+\quad\text{versus}\quad
+\bigl((a_1 b_1) - (a_2 b_2)\bigr)
+\bigl((a_2 b_1) + (a_1 b_2)\bigr)
+\end{verbatim}
+The third kind of situation is a slightly oversize object in running
+text, such as $\left\lvert\frac{b'}{d'}\right\rvert$ where the
+delimiters produced by \cn{left} and \cn{right} cause too much line
+spreading. In that case \ncn{bigl} and \ncn{bigr}\index{big@\cn{big},
+\cn{Big}, \cn{bigg}, \dots\ delimiters} can be used to produce
+delimiters that are larger than the base size but still able to
+fit within the normal line spacing:
+$\bigl\lvert\frac{b'}{d'}\bigr\rvert$.
+
+The \pkg{mathtools} package provides a feature \cn{DeclarePairedDelimiter}
+that can simplify sizing; see the package documentation for details.
+
+\subsection{Vertical bar notations}\label{verts}
+
+The use of the \verb'|' character to produce paired delimiters is not
+recommended. There is an ambiguity about the directionality of the
+symbol that will in rare cases produce incorrect spacing\mdash e.g.,
+\verb'|k|=|-k|' produces $|k|=|-k|$, and \verb'|\sin x|' produces $|\sin x|$
+instead of the correct $\lvert\sin x\rvert$. Using \cn{lvert} for a \qq{left
+vert bar} and \cn{rvert} for a \qq{right vert bar} whenever they are
+used in pairs will prevent this problem; compare $\lvert -k\rvert$,
+produced by \verb'\lvert -k\rvert'. For double bars there are analogous
+\cn{lVert}, \cn{rVert} commands. Recommended practice is to define
+suitable commands in the document preamble for any paired-delimiter use
+of vert bar symbols:
+\begin{verbatim}
+\providecommand{\abs}[1]{\lvert#1\rvert}
+\providecommand{\norm}[1]{\lVert#1\rVert}
+\end{verbatim}
+whereupon \verb|\abs{z}| would produce $\lvert z\rvert$ and
+\verb|\norm{v}| would produce $\lVert v\rVert$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{The \hycn{text} command}
+
+The main use of the command \cn{text} is for words or phrases\index{text
+  fragments inside math} in a display. It is similar to \cn{mbox} in its
+effects but, unlike \cn{mbox}, automatically produces subscript-size
+text if used in a subscript.
+\begin{equation}
+f_{[x_{i-1},x_i]} \text{ is monotonic,}
+\quad i = 1,\dots,c+1
+\end{equation}
+\begin{verbatim}
+f_{[x_{i-1},x_i]} \text{ is monotonic,}
+\quad i = 1,\dots,c+1
+\end{verbatim}
+
+\subsection{\hycn{mod} and its relatives}
+
+Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} deal with the special
+spacing conventions of \qq{mod} notation. \cn{mod} and \cn{pod} are
+variants of \cn{pmod} preferred by some authors; \cn{mod} omits the
+parentheses, whereas \cn{pod} omits the \qq{mod} and retains the
+parentheses.
+\begin{equation}
+\gcd(n,m\bmod n) ;\quad x\equiv y\pmod b
+;\quad x\equiv y\mod c ;\quad x\equiv y\pod d
+\end{equation}
+\begin{verbatim}
+\gcd(n,m\bmod n) ;\quad x\equiv y\pmod b
+;\quad x\equiv y\mod c ;\quad x\equiv y\pod d
+\end{verbatim}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Integrals and sums}
+
+\subsection{Altering the placement of limits}
+
+The limits on integrals, sums, and similar symbols are placed either to
+the side of or above and below the base symbol, depending on convention
+and context. \lat/ has rules for automatically choosing one or the
+other, and most of the time the results are satisfactory. In the event
+they are not, there are three \lat/ commands that can be used to
+influence the placement of the limits: \cn{limits}, \cn{nolimits},
+\cn{displaylimits}. Compare
+\begin{center}
+\begin{minipage}{.4\columnwidth}
+\[\int_{\abs{x-x_z(t)}<X_0} z^6(t)\phi(x)\]
+\begin{verbatim}
+\int_{\abs{x-x_z(t)}<X_0} ...
+\end{verbatim}
+\end{minipage}\quad
+and\quad
+\begin{minipage}{.5\columnwidth}
+\[\int\limits_{\abs{x-x_z(t)}<X_0} z^6(t)\phi(x)\]
+\begin{verbatim}
+\int\limits_{\abs{x-x_z(t)}<X_0} ...
+\end{verbatim}
+\end{minipage}
+\end{center}
+The \cn{limits} command should follow immediately after the base symbol
+to which it applies, and its meaning is: shift the following subscript
+and/or superscript to the limits position, regardless of the usual
+convention for this symbol. \cn{nolimits} means to shift them to the
+side instead, and \cn{displaylimits}, which might be used in defining
+a new kind of base symbol, means to use standard positioning as for the
+\cn{sum} command.
+
+See also the description of the options \opt{intlimits} and
+\opt{nosumlimits} in \cite{amsldoc}.
+
+\subsection{Multiple integral signs}
+
+\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple integral
+signs\index{integrals!multiple} with the spacing between them nicely
+adjusted, in both text and display style. \cn{idotsint} is an extension
+of the same idea that gives two integral signs with dots between them.
+Notice the use of thin space (\cn{,})\index{thin space (\protect\cn{,})}
+before $dx$ and friends to clarify the meaning.
+\begin{gather}
+\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
+f(x,y,z)\,dx\,dy\,dz\\
+\iiiint\limits_A
+f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
+\end{gather}
+\begin{verbatim}
+\iint\limits_A f(x,y)\,dx\,dy\qquad\iiint\limits_A
+f(x,y,z)\,dx\,dy\,dz\\
+\iiiint\limits_A
+f(w,x,y,z)\,dw\,dx\,dy\,dz\qquad\idotsint\limits_A f(x_1,\dots,x_k)
+\end{verbatim}
+
+\subsection{Multiline subscripts and superscripts}
+
+The \cn{substack} command can be used to produce a multiline subscript
+or superscript:\index{subscripts and superscripts!multiline}\relax
+\index{superscripts|see{subscripts and superscripts}} for example
+\begin{center}
+\begin{tabular}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sum_{\substack{
+         0\le i\le m\\
+         0<j<n}}
+  P(i,j)
+\end{verbatim}
+\end{minipage}
+&
+$\displaystyle
+\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)$
+\end{tabular}
+\end{center}
+
+\subsection{The \hycn{sideset} command}\label{sideset}
+
+There's also a command called \cn{sideset}, for a rather special
+purpose: putting symbols at the subscript and
+superscript\index{subscripts and superscripts!on sums} corners of a
+symbol like $\sum$ or $\prod$. \emph{Note: The
+  \cn{sideset} command is only designed for use with large operator
+  symbols; with ordinary symbols the results are unreliable.}
+With \cn{sideset}, you can write
+\begin{center}
+\begin{tabular}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sideset{}{'}
+  \sum_{n<k,\;\text{$n$ odd}} nE_n
+\end{verbatim}
+\end{minipage}
+&$\displaystyle
+\sideset{}{'}\sum_{n<k,\;\text{$n$ odd}} nE_n
+$
+\end{tabular}
+\end{center}
+The extra pair of empty braces is explained by the fact that
+\cn{sideset} has the capability of putting an extra symbol or symbols at
+each corner of a large operator; to put an asterisk at each corner of a
+product symbol, you would type
+\begin{center}
+\begin{tabular}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sideset{_*^*}{_*^*}\prod
+\end{verbatim}
+\end{minipage}
+&$\displaystyle
+\sideset{_*^*}{_*^*}\prod
+$
+\end{tabular}
+\end{center}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Changing the size of elements in a formula}
+
+The \lat/ mechanisms for changing font size inside a math formula are
+completely different from the ones used outside math formulas. If you
+try to make something larger in a formula with one of the text commands
+such as \cn{large} or \cn{huge}:
+\[\text{\large \#}\qquad\verb'{\large \#}'\]
+you will get a warning message
+\begin{verbatim}
+Command \large invalid in math mode
+\end{verbatim}
+Such an attempt, however, often indicates a misunderstanding of how
+\lat/ math symbols work. If you want a \# symbol analogous to a
+summation sign in its typographical properties, then in principle the
+best way to achieve that is to define it as a symbol of type ``mathop''
+with the standard \lat/ \cn{DeclareMathSymbol} command (see
+\cite{fntguide}). (This entails, however, getting hold of a math font
+with a suitable text-size\slash display-size pair, which may not be so
+easy.)
+
+Consider the expression:
+\[\frac{\sum_{n > 0} z^n}{\prod_{1\leq k\leq n} (1-q^k)}
+\qquad\begin{minipage}{.5\columnwidth}
+\begin{verbatim}
+\frac{\sum_{n > 0} z^n}
+     {\prod_{1\leq k\leq n} (1-q^k)}
+\end{verbatim}
+\end{minipage}
+\]
+Using \cn{dfrac} instead of \cn{frac} wouldn't change anything in this
+case; if you want the sum and product symbols to appear full size, you
+need the \cn{displaystyle} command:
+\[
+\frac{{\displaystyle\sum_{n > 0} z^n}}
+     {{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}}
+\qquad\begin{minipage}{.7\columnwidth}
+\begin{verbatim}
+\frac{{\displaystyle\sum_{n > 0} z^n}}
+     {{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}}
+\end{verbatim}
+\end{minipage}
+\]
+And if you want full-size symbols but with limits on the side, use
+the \cn{nolimits} command also:
+\[
+\frac{{\displaystyle\sum\nolimits_{n> 0} z^n}}
+  {{\displaystyle\prod\nolimits_{1\leq k\leq n} (1-q^k)}}
+\qquad\begin{minipage}{.76\columnwidth}
+\begin{verbatim}
+\frac{{\displaystyle\sum\nolimits_{n> 0} z^n}}
+  {{\displaystyle\prod\nolimits_{1\leq k\leq n} (1-q^k)}}
+\end{verbatim}
+\end{minipage}
+\]
+There are similar commands \cn{textstyle}, \cn{scriptstyle}, and
+\cn{scriptscriptstyle}, to force \lat/ to use the symbol size and
+spacing that would be applied in (respectively) inline math, first-order
+subscript, or second-order subscript, even when the current context
+would normally yield some other size.
+
+\textbf{Note:} These commands belong to a special class of
+commands referred to in the \lat/ book as ``declarations''. In
+particular, notice where the braces fall that delimit the effect of the
+command:
+\begin{center}
+\textbf{Right: } \verb'{\displaystyle ...}'
+\qquad\qquad\textbf{Wrong: } \verb'\displaystyle{...}'
+\end{center}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Other packages of interest}
+\label{other-packages}
+
+Many other \LaTeX{} packages that address some aspect of mathematical
+formulas are available from CTAN (the Comprehensive \TeX{} Archive
+Network). To recommend a few examples:
+\begin{description}
+\raggedright
+\item[mathtools] Additional features extending \pkg{amsmath}; loads
+  \pkg{amsmath}.
+\item[amsthm] General theorem and proof setup.
+\item[amsfonts] Defines \cn{mathbb} and \cn{mathfrak}, and provides access
+  to many additional symbols (without names; \pkg{amssymb} provides the
+  names).
+\item[accents] Under accents and accents using arbitrary symbols.
+\item[bm] Bold math package, provides a more general and more robust
+  implementation of \cn{boldsymbol}.
+\item[mathrsfs] Ralph Smith's Formal Script, font setup.
+\item[cases] Apply a large brace to two or more equations without
+  losing the individual equation numbers.
+\item[delarray] Delimiters spanning multiple rows of an array.
+%%\item[kuvio] Commutative diagrams and other diagrams. % not in TeX Live
+\item[xypic] Commutative diagrams and other diagrams.
+\item[TikZ] Comprehensive graphical facilities, including features for
+  drawing diagrams.
+\end{description}
+
+The \TeX{} Catalogue,\\
+\null\hspace{2\parindent}
+\url{http://mirror.ctan.org/help/Catalogue/alpha.html},\\
+is a good place to look if you know a package's name.
+
+\medskip
+Questions and answers on specific \TeX-related topics are the
+\emph{raison d'\^etre} of this forum:\\
+\null\hspace{2\parindent}
+\url{https://tex.stackexchange.com/questions}\\
+Check the archives for existing answers; pointers to selected topics
+may expedite your search:\\
+\null\hspace{2\parindent}
+\url{https://tex.meta.stackexchange.com/a/2425#2425}\\
+If nothing useful turns up, ask your own question.
+
+%\newpage
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\section{Other documentation of interest}
+
+\begin{thebibliography}{AMUG}
+\raggedright
+
+\bibitem[AMUG]{amsldoc} American Mathematical Society and the \LaTeX3 Project:
+  \emph{User's Guide for the \textnormal{\ttfamily amsmath} package},
+  Version~2.$+$,
+  \url{http://mirror.ctan.org/macros/latex/required/amsmath/amsldoc.tex} and
+  \url{http://mirror.ctan.org/macros/latex/required/amsmath/amsldoc.pdf},
+  2017.
+
+\bibitem[AFUG]{amsfndoc} American Mathematical Society:
+  \emph{User's Guide, AMSFonts}, 
+  \url{http://mirror.ctan.org/fonts/amsfonts/amsfndoc.pdf}, 2002.
+
+\bibitem[CLSL]{comprehensive} Scott Pakin:
+  \emph{The Comprehensive \LaTeX{} Symbol List},
+  \url{http://mirror.ctan.org/tex-archive/info/symbols/comprehensive/},
+  January 2017.  Raw font tables, without symbol names, are shown
+  alphabetically by font name in the \fn{rawtables*.pdf} files in the
+  same area of CTAN and from \TeX\,Live with \texttt{texdoc rawtables}.
+
+\bibitem[Lam]{lamport} Leslie Lamport: \emph{\LaTeX{}: A document
+    preparation system}, 2nd edition, Addison-Wesley, 1994.
+
+\bibitem[LC]{companion} Frank Mittelbach and Michel Goossens,
+  with Johannes Braams, David Carlisle, and Chris Rowley:
+  \emph{The \LaTeX{} Companion}, 2nd edition, Addison-Wesley, 2004.
+
+\bibitem[LFG]{fntguide} \LaTeX3 Project Team: \emph{\LaTeXe{} font
+  selection}, % \fn{fntguide.tex}, November 2005.
+  \url{http://mirror.ctan.org/macros/latex/doc/fntguide.pdf}, 2005.
+
+\bibitem[LGC]{graphics-companion} Michel Goossens, Frank Mittelbach,
+  Sebastian Rahtz, Denis Roegel, and Herbert~Vo\ss:
+  \emph{The \LaTeX{} Graphics Companion}, 2nd edition, Addison-Wesley, 2008.
+
+\bibitem[LGG]{grfguide} D.~P. Carlisle, \LaTeX3 Project:
+  \emph{Packages in the `graphics' bundle}, %\fn{grfguide.tex}, 2017.
+  \url{http://mirror.ctan.org/macros/latex/required/graphics/grfguide.pdf},
+  2017.
+
+\bibitem[LUG]{usrguide} \LaTeX3 Project Team: \emph{\LaTeXe~for
+  authors}, % \fn{usrguide.tex}, 2015.
+  \url{http://mirror.ctan.org/macros/latex/doc/usrguide.pdf}, 2015.
+
+\bibitem[MML]{gratzer} George Gr\"atzer: \textit{More Math into \LaTeX},
+   5th edition, Springer, New York, 2016.
+
+\bibitem[UCM]{uc-math} Will Robertson: \emph{Every symbol
+  \textup{(}most symbols\textup{)} defined by \pkg{unicode-math}},
+  \url{http://mirror.ctan.org/macros/latex/contrib/unicode-math/unimath-symbols.pdf}, 2017; and\\
+  Will Robertson, Philipp Stephani, Joseph Wright, and Khaled Hosny:
+  \emph{Experimental Unicode mathematical typesetting: The \pkg{unicode-math}
+  package},
+  \url{http://mirror.ctan.org/macros/latex/contrib/unicode-math/unicode-math.pdf}, 2017.
+
+\end{thebibliography}
+
+\end{document}