From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 2 Sep 2019 13:46:59 +0900
Subject: Initial commit

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+%
+% Copyright 1995, 2000, 2013 American Mathematical Society.
+% Copyright 2016-2018 LaTeX3 Project and American 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
+%   https://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 LaTeX3 Project.
+%
+\documentclass[leqno,titlepage,openany]{amsldoc}[1999/12/13]
+
+\def\MaintainedByLaTeXTeam#1{%
+\begin{center}%
+\fbox{\fbox{\begin{tabular}{@{}l@{}}%
+This file is maintained by the \LaTeX{} Project team.\\%
+Bug reports can be opened (category \texttt{#1}) at\\%
+\url{https://latex-project.org/bugs/}.\end{tabular}}}\end{center}}
+
+\ifx\UndEfiNed\url
+  \ClassError{amsldoc}{%
+    This version of amsldoc.tex must be processed\MessageBreak
+    with a newer version of amsldoc.cls (2.02 or later).}{}
+\fi
+
+\renewcommand\bibname{References}
+
+%    Definitions that should be moved to amsldoc.cls
+%    when it is next updated
+\makeatletter
+\let\cleardouble@page\cleardoublepage
+\AtBeginDocument{%
+  \ifx\cleardouble@page\cleardoublepage
+    \def\cleardoublepage{\clearpage{\pagestyle{empty}\cleardouble@page}}
+  \fi
+}
+
+% #1 is empty or one of class, environment, option, package
+% could use different font in each case
+\makeatletter
+\def\@category@index#1#2{%
+    \autoindex{%
+        #2\actualchar\string\index#1font{#2}%
+        \ifx\@nil#1\@nil\else\space#1\fi
+    }%
+}
+
+\let\indexpackagefont\textsf
+\let\indexclassfont\textsf
+\let\indexoptionfont\texttt
+\let\indexenvironmentfont\texttt
+\let\indexfont\texttt
+\makeatother
+%    For a package that won't be indexed, i.e. amsmath
+\newcommand{\nipkg}{\textsf}
+\def\tikz/{Ti\textit{k}Z}
+%    End definitions for amsldoc.cls
+
+\title{User's Guide for the \nipkg{amsmath} Package (Version~2.1)}
+\author{American Mathematical Society, \LaTeX3 Project}
+\date{1999-12-13 (revised 2002-02-25, 2016-11-14, 2018-04-05)}
+\makeatletter
+\def\@thanks{\bigskip\MaintainedByLaTeXTeam{amslatex}}
+\makeatother
+
+%    Use the amsmath package and amscd package in order to print
+%    examples.
+\usepackage{amsmath}
+\usepackage{amscd}
+
+\usepackage{imakeidx}
+%    Need alternate coding for vertical bar to get it into index.
+%    Based on \qcbang and \cnbang in amsldoc.cls, refined by David Carlisle.
+%    Give it a sort field that will be at the end of the symbols;
+%    must avoid " and | which have special meanings.
+\DeclareRobustCommand{\qcvert}{\qc\|\index{{z1}@{\ntt\char'174}}}
+\DeclareRobustCommand{\cnvert}{%
+  \ncn{\|}\index{{z0}@{\ntt\bslchar\char'174}}}
+%    Need alternate index sort field for bang to allow & to sort first
+\DeclareRobustCommand{\altcnbang}{%
+  \ncn{\!}\index{"!b@{\ntt\bslchar\qcbang}}}
+\DeclareRobustCommand{\cnamp}{%
+  \index{"!a@\texttt{\&}}}
+
+\makeindex[intoc] % generate index data
+\providecommand{\see}[2]{\textit{see} #1}
+
+%    The amsldoc class includes a number of features useful for
+%    documentation about TeX, including:
+%
+%    ---Commands \tex/, \amstex/, \latex/, ... for uniform treatment
+%    of the various logos and easy handling of following spaces.
+%
+%    ---Commands for printing various common elements: \cn for command
+%    names, \fn for file names (including font-file names), \env for
+%    environments, \pkg and \cls for packages and classes, etc.
+
+%    Many of the command names used here are rather long and will
+%    contribute to poor linebreaking if we follow the \latex/ practice
+%    of not hyphenating anything set in tt font; instead we selectively
+%    allow some hyphenation.
+\allowtthyphens % defined in amsldoc.cls
+
+\hyphenation{ac-cent-ed-sym-bol add-to-counter add-to-length align-at
+  aligned-at allow-dis-play-breaks ams-art ams-cd ams-la-tex amsl-doc
+  ams-symb ams-tex ams-text ams-xtra bmatrix bold-sym-bol cen-ter-tags
+  eqn-ar-ray idots-int int-lim-its latex med-space neg-med-space
+  neg-thick-space neg-thin-space no-int-lim-its no-name-lim-its
+  over-left-arrow over-left-right-arrow over-right-arrow pmatrix
+  qed-sym-bol set-length side-set small-er tbinom the-equa-tion
+  thick-space thin-space un-der-left-arrow un-der-left-right-arrow
+  un-der-right-arrow use-pack-age var-inj-lim var-proj-lim vmatrix
+  xalign-at xx-align-at}
+
+%    Prepare for illustrating the \vec example
+\newcommand{\vect}[1]{\mathbf{#1}}
+
+\newcommand{\booktitle}[1]{\textit{#1}}
+\newcommand{\journalname}[1]{\textit{#1}}
+\newcommand{\seriesname}[1]{\textit{#1}}
+
+%    Command to insert and index a particular phrase. Doesn't work for
+%    certain kinds of special characters in the argument.
+\newcommand{\ii}[1]{#1\index{#1}}
+
+\newcommand{\vstrut}[1]{\vrule width0pt height#1\relax}
+
+%    An environment for presenting comprehensive address information:
+\newenvironment{infoaddress}{%
+  \par\topsep\medskipamount
+  \trivlist\centering
+  \item[]%
+  \begin{minipage}{.7\columnwidth}%
+  \raggedright
+}{%
+  \end{minipage}%
+  \endtrivlist
+}
+
+\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
+}
+
+%    Errors in output shouldn't be \texttt, and they may also be longer
+%    than one line, but they should be identified by the "errorbullet".
+%    (They were formerly labeled as aubsections.)
+\newenvironment{erroro}[1]{%
+  \par\addvspace\medskipamount
+  \noindent\hangindent\parindent
+  \hbox to\parindent{\errorbullet\hfil}\ignorespaces
+  #1\par\vspace{\smallskipamount}
+}{%
+  \par\addvspace\medskipamount
+}
+
+%    The chapters are so short, perhaps we shouldn't call them by the
+%    name `Chapter'. We make \chaptername read an argument in order to
+%    remove a following \space or "{} " (both possibilities are present
+%    in book.cls).
+
+\renewcommand{\chaptername}[1]{}
+\newcommand{\chapnum}[1]{\mdash #1\mdash }
+\makeatletter
+\def\@makechapterhead#1{%
+  \vspace{1.5\baselineskip}%
+  {\parindent \z@ \raggedright \reset@font
+    \ifnum \c@secnumdepth >\m@ne
+      \large\bfseries \chapnum{\thechapter}%
+      \par\nobreak
+      \vskip.5\baselineskip\relax
+    \fi
+    #1\par\nobreak
+    \vskip\baselineskip
+  }}
+\makeatother
+
+%    A command for ragged-right parbox in a tabular.
+\newcommand{\rp}{\let\PBS\\\raggedright\let\\\PBS}
+
+%    Non-indexed file name
+\newcommand{\nfn}[1]{\texttt{#1}}
+
+%    For the examples in the math spacing table.
+%%\newcommand{\lspx}{\mbox{\rule{5pt}{.6pt}\rule{.6pt}{6pt}}}
+%%\newcommand{\rspx}{\mbox{\rule[-1pt]{.6pt}{7pt}%
+%%  \rule[-1pt]{5pt}{.6pt}}}
+\newcommand{\lspx}{\mathord{\Rightarrow\mkern-1mu}}
+\newcommand{\rspx}{\mathord{\mkern-1mu\Leftarrow}}
+\newcommand{\spx}[1]{$\lspx #1\rspx$}
+
+%    For a list of characters representing document input.
+\newcommand{\clist}[1]{%
+  \mbox{\ntt\spaceskip.2em plus.1em \xspaceskip\spaceskip#1}}
+
+%    Fix weird \latex/ definition of rightmark.
+\makeatletter
+\def\rightmark{\expandafter\@rightmark\botmark{}{}}
+%    Also turn off section marks.
+\let\sectionmark\@gobble
+\renewcommand{\chaptermark}[1]{%
+  \uppercase{\markboth{\rhcn#1}{\rhcn#1}}}
+\newcommand{\rhcn}{\thechapter. }
+\makeatother
+
+%    Include down to \section but not \subsection, in toc:
+\setcounter{tocdepth}{1}
+
+%    Subheadings for the bibliography
+\newif\iffirstbibsubhead \firstbibsubheadtrue
+\newcommand{\bibsubhead}[1]{%
+  \iffirstbibsubhead \firstbibsubheadfalse
+  \else \addvspace{\medskipamount}
+  \fi
+  \item[]\hspace*{-\leftmargin}\textbf{#1}\par
+  \vspace{\smallskipamount}
+}
+
+\DeclareMathOperator{\ix}{ix}
+\DeclareMathOperator{\nul}{nul}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\xxx}{xxx}
+
+\usepackage{hyperref}
+\hypersetup{pdfborder=0 0 0}
+\makeatletter
+\let\oldcs\cs
+\def\cs#1{\texorpdfstring{\oldcs{#1}}{\@backslashchar\@backslashchar#1}}
+\let\cn\cs
+\makeatother
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\frontmatter
+
+\maketitle
+
+\pagestyle{headings}
+\tableofcontents
+\cleardoublepage % for better page number placement
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mainmatter
+\chapter{Introduction}
+
+\index{amsmath@\texttt{amsmath} package|(}
+
+The \nipkg{amsmath} package is a \LaTeX{} package that provides
+miscellaneous enhancements for improving the information structure and
+printed output of documents that contain mathematical formulas. Readers
+unfamiliar with \LaTeX{} should refer to \cite{lamport}. If you have an
+up-to-date version of \LaTeX{}, the \nipkg{amsmath} package is normally
+provided along with it. Upgrading when a newer version of the
+\nipkg{amsmath} package is released can be done via
+\url{http://mirror.ctan.org/macros/latex/required/amsmath.zip}.
+
+This documentation describes the features of the \nipkg{amsmath} package
+and discusses how they are intended to be used. It also covers some
+ancillary packages:
+\begin{ctab}{lll}
+\pkg{amsbsy}& \pkg{amsopn}& \pkg{amsxtra}\\
+\pkg{amscd}& \pkg{amstext}\\
+\end{ctab}
+These all have something to do with the contents of math formulas. For
+information on extra math symbols and math fonts, see \cite{amsfonts}
+and \url{https://www.ams.org/tex/amsfonts.html}. For documentation of the
+\pkg{amsthm} package or AMS document classes (\cls{amsart},
+\cls{amsbook}, etc.\@) see \cite{amsthdoc} or \cite{amshandbk} and
+\url{https://www.ams.org/tex/author-info.html}.
+
+If you are a long-time \latex/ user and have lots of mathematics in what
+you write, then you may recognize solutions for some familiar problems
+in this list of \nipkg{amsmath} features:
+\begin{itemize}
+
+\item A convenient way to define new `operator name' commands analogous
+to \cn{sin} and \cn{lim}, including proper side spacing and automatic
+selection of the correct font style and size (even when used in
+sub- or superscripts).
+
+\item Multiple substitutes for the \env{eqnarray} environment to make
+various kinds of equation arrangements easier to write.
+
+\item Equation numbers automatically adjust up or down to avoid
+overprinting on the equation contents (unlike \env{eqnarray}).
+
+\item Spacing around equals signs matches the normal spacing in the
+\env{equation} environment (unlike \env{eqnarray}).
+
+\item A way to produce multiline subscripts as are often used with
+summation or product symbols.
+
+\item An easy way to substitute a variant equation number for a given
+equation instead of the automatically supplied number.
+
+\item An easy way to produce subordinate equation numbers of the form
+(1.3a) (1.3b) (1.3c) for selected groups of equations.
+
+\end{itemize}
+
+The \nipkg{amsmath} package is distributed together with some small
+auxiliary packages:
+\begin{description}
+\item[\nipkg{amsmath}] Primary package, provides various features for
+  displayed equations and other mathematical constructs.
+
+\item[\pkg{amstext}] Provides a \cn{text} command for
+  typesetting a fragment of text inside a display.
+
+\item[\pkg{amsopn}] Provides \cn{DeclareMathOperator} for defining new
+  `operator names' like \cn{sin} and \cn{lim}.
+
+\item[\pkg{amsbsy}] For backward compatibility this package continues
+to exist but use of the newer \pkg{bm} package that comes with \LaTeX{}
+is recommended instead.
+
+\item[\pkg{amscd}] Provides a \env{CD} environment for simple
+  commutative diagrams (no support for diagonal arrows).
+
+\item[\pkg{amsxtra}] Provides certain odds and ends such as
+  \cn{fracwithdelims} and \cn{accentedsymbol}, for compatibility with
+  documents created using version 1.1.
+
+\end{description}
+
+The \nipkg{amsmath} package incorporates \pkg{amstext}, \pkg{amsopn}, and
+\pkg{amsbsy}. The features of \pkg{amscd} and \pkg{amsxtra}, however,
+are available only by invoking those packages separately.
+
+The independent \pkg{mathtools} package \cite{mt} provides some
+enhancements to \nipkg{amsmath}; \pkg{mathtools} loads \nipkg{amsmath}
+automatically, so there is no need to separately load \nipkg{amsmath}
+if \pkg{mathtools} is used.  Some \pkg{mathtools} facilities
+will be noted below as appropriate.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Options for the \nipkg{amsmath} package}\label{options}
+
+The \nipkg{amsmath} package has the following options:
+\index{options!amsmath@\texttt{amsmath} package options|(}
+\begin{description}
+
+\item[\opt{centertags}] (default) For an equation containing a
+\env{split} environment, place equation
+numbers\index{equationn@equation numbers!vertical placement}
+vertically centered on the total height of the equation.
+
+\item[\opt{tbtags}] `Top-or-bottom tags': For an equation containing
+a \env{split} environment, place equation
+numbers\index{equationn@equation numbers!vertical placement} level with the last
+(resp.\ first) line, if numbers are on the right (resp.\ left).
+
+\item[\opt{sumlimits}] (default) Place the subscripts and
+superscripts\index{subscripts and superscripts!placement}\relax
+\index{limits|see{subscripts and superscripts}} of summation symbols
+above and below, in displayed equations. This option also affects other
+symbols of the same type\mdash $\prod$, $\coprod$, $\bigotimes$,
+$\bigoplus$, and so forth\mdash but excluding integrals (see below).
+
+\item[\opt{nosumlimits}] Always place the subscripts and superscripts of
+summation-type symbols to the side, even in displayed equations.
+
+\item[\opt{intlimits}] Like \opt{sumlimits}, but for
+integral\index{integrals!placement of limits} symbols.
+
+\item[\opt{nointlimits}] (default) Opposite of \opt{intlimits}.
+
+\item[\opt{namelimits}] (default) Like \opt{sumlimits}, but for certain
+`operator names' such as $\det$, $\inf$, $\lim$, $\max$, $\min$, that
+traditionally have subscripts%
+\index{subscripts and superscripts!placement} placed underneath when
+they occur in a displayed equation.
+
+\item[\opt{nonamelimits}] Opposite of \opt{namelimits}.
+
+\end{description}
+
+\goodbreak
+
+\begin{description}
+\item[\opt{alignedleftspaceyes}]
+\item[\opt{alignedleftspaceno}]
+\item[\opt{alignedleftspaceyesifneg}]
+\end{description}
+
+To use one of these package options, put the option name in the optional
+argument of the \cn{usepackage} command\mdash e.g.,
+\verb"\usepackage[intlimits]{amsmath}".
+For AMS document classes and any other classes that preload \nipkg{amsmath}
+desired options must be specified with the \cn{documentclass}\mdash
+e.g.,\\
+\verb+\documentclass[intlimits,tbtags,reqno]{amsart}+.
+
+The \nipkg{amsmath} package also recognizes the following options which
+are normally selected (implicitly or explicitly) through the
+\cn{documentclass} command, and thus need not be repeated in the option
+list of the \cn{usepackage}|{amsmath}| statement.
+\begin{description}
+
+\item[\opt{leqno}] Place equation numbers on the left.%
+\index{equationn@equation numbers!left or right placement}
+
+\item[\opt{reqno}] Place equation numbers on the right.
+
+\item[\opt{fleqn}] Position equations at a fixed indent from the left
+margin rather than centered in the text column.%
+\index{displayed equations!centering}
+
+\end{description}
+
+Three options have been added to control the space to the left of
+\env{aligned} and \env{gathered} environments. Prior to the 2017 release
+a thin space was added to the left but not the right of these constructs.
+This appears to have been an accidental feature of the implementation and
+was typically corrected by prefixing the environments with \altcnbang.
+
+The new default behavior is aimed to ensure that the environments do
+not have a thin space added in most cases, and that existing documents
+using |\!\begin{aligned}| continue to work as before.
+
+\begin{description}
+\item[\opt{alignedleftspaceyes}] Always add \cn{\,} to the left of \env{aligned} and \env{gathered}.
+\item[\opt{alignedleftspaceno}] Never add \cn{\,} to the left of \env{aligned} and \env{gathered}.
+\item[\opt{alignedleftspaceyesifneg}] Only add \cn{\,} if the environment is prefixed by negative space. (New default behavior.)
+\end{description}
+
+\index{options!amsmath@\texttt{amsmath} package options|)}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Displayed equations}
+
+\section{Introduction}
+The \nipkg{amsmath} package provides a number of additional displayed
+equation structures\index{displayed equations}%
+\index{equations|see{displayed equations}} beyond the ones
+provided in basic \latex/. The augmented set includes:
+\begin{verbatim}
+  equation     equation*     align       align*
+  gather       gather*       alignat     alignat*
+  multline     multline*     flalign     flalign*
+  split
+\end{verbatim}
+(Although the standard \env{eqnarray} environment remains available,
+it is better to use \env{align} or \env{equation}+\env{split} instead.
+Within \env{eqnarray}, spacing\index{horizontal spacing} around signs
+of relation is not the preferred mathematical spacing, and is
+inconsistent with that spacing as it appears in other environments.
+Long lines in this environment may result in misplaced or overprinted
+equation numbers. This environment also does not support the use of
+\cn{qed} or \cn{qedhere} as provided by theorem packages.)
+
+Except for \env{split}, each environment has both starred and unstarred
+forms, where the unstarred forms have automatic numbering using
+\latex/'s \env{equation} counter. You can suppress the number on any
+particular line by putting \cn{notag} before the end of that line;
+\cn{notag} should not be used outside a display environment as it will
+mess up the numbering.  You can also
+override\index{equationn@equation numbers!overriding} a number with a tag
+of your own using \cn{tag}|{|\<label>|}|, where \<label> means arbitrary
+text such as |$*$| or |ii| used to \qq{number} the equation.
+A tag can reference a different tagged display by use of
+|\tag{\ref{|\<label>|}|\<modifier>|}| where \<modifier> is optional.
+If you are using \pkg{hyperref}, use \cn{ref*};
+use of the starred form of \cn{ref} prevents a reference to a modified
+tag containing a nested link from linking to the original display.
+
+There is also a
+\cn{tag*} command that causes the text you supply to be typeset
+literally, without adding parentheses around it. \cn{tag} and \cn{tag*}
+can also be used within the unnumbered versions of all the \nipkg{amsmath}
+alignment structures. Some examples of the use of \cn{tag} may be found
+in the sample files \fn{testmath.tex} and \fn{subeqn.tex} provided with
+the \nipkg{amsmath} package.
+
+The \env{split} environment is a special subordinate form that is used
+only \emph{inside} one of the others. It cannot be used inside
+\env{multline}, however. \env{split} supports only one alignment (|&|)
+column; if more are needed, \env{aligned} or \env{alignedat} should be
+used. The width of a \env{split} structure is the full line width.
+
+In the structures that do alignment (\env{split}, \env{align} and
+variants), relation symbols have an |&| before them but not
+after\mdash unlike \env{eqnarray}. Putting the |&| after the
+relation symbol will interfere with the normal spacing; it has to go
+before.
+
+In all multiline environments, lines are divided by \cn{\\}.  The \cn{\\}
+should \emph{not} be used to end the last line.  Using it there will
+result in unwanted extra vertical space following the display.
+
+In \emph{all} math environments (inline or display), blank lines
+(equivalent to \cn{par}) are not permitted, and will result in an error.
+
+\begin{table}[p]
+\caption[]{Comparison of displayed equation environments (vertical lines
+indicating nominal margins)}\label{displays}
+\renewcommand{\theequation}{\arabic{equation}}
+\begin{eqxample}
+\begin{verbatim}
+\begin{equation*}
+a=b
+\end{equation*}
+\end{verbatim}
+\producing
+\begin{equation*}
+a=b
+\end{equation*}
+\end{eqxample}
+
+\begin{eqxample}
+\begin{verbatim}
+\begin{equation}
+a=b
+\end{equation}
+\end{verbatim}
+\producing
+\begin{equation}
+a=b
+\end{equation}
+\end{eqxample}
+
+\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
+\end{multline}
+\end{verbatim}
+\producing
+\begin{multline}
+a+b+c+d+e+f\\
++i+j+k+l+m+n
+\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{flalign*}
+a_{11} + b_{11}& = c_{11}&
+  a_{12}& =b_{12}\\
+b_{21}& = c_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{flalign*}
+\end{verbatim}
+\producing
+\begin{flalign*}
+a_{11} + b_{11}& = c_{11}&
+  a_{12}& =b_{12}\\
+b_{21}& = c_{21}&
+  a_{22}& =b_{22}+c_{22}
+\end{flalign*}
+\end{eqxample}
+\end{table}
+
+%% ------------------------------------------------------------------ %%
+\enlargethispage{1\baselineskip}
+
+\section{Single equations}
+
+The \env{equation} environment is for a single equation with an
+automatically generated number. The \env{equation*} environment is the
+same except for omitting the number.%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\footnote{Basic \latex/ doesn't provide an \env{equation*} environment,
+but rather a functionally equivalent environment named
+\env{displaymath}.}
+The wrapper \verb+\[ ... \]+ is equivalent to \env{equation*}.%
+\index{[@\verb+\[ ... \]+}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Split equations without alignment}
+
+The \env{multline} environment is a variation of the \env{equation}
+environment used for equations that don't fit on a single line. The
+first line of a \env{multline} will be at the left margin and the last
+line at the right margin, except for an indention on both sides in the
+amount of \cn{multlinegap}. Any additional lines in between will be
+centered independently within the display width (unless the \opt{fleqn}
+option is in effect).%
+\index{options!behavior of particular options}
+
+Like \env{equation}, \env{multline} has only a single equation number
+(thus, none of the individual lines should be marked with \cn{notag}).
+The equation number is placed on the last line (\opt{reqno} option) or
+first line (\opt{leqno} option); vertical centering as for \env{split}
+is not supported by \env{multline}.%
+\index{options!behavior of particular options}
+
+It's possible to force one of the middle lines to the left or right with
+commands \cn{shoveleft}, \cn{shoveright}. These commands take the entire
+line as an argument, up to but not including the final \cn{\\}; for
+example
+\begin{multline}
+\framebox[.65\columnwidth]{A}\\
+\framebox[.5\columnwidth]{B}\\
+\shoveright{\framebox[.55\columnwidth]{C}}\\
+\framebox[.65\columnwidth]{D}
+\end{multline}
+\begin{verbatim}
+\begin{multline}
+\framebox[.65\columnwidth]{A}\\
+\framebox[.5\columnwidth]{B}\\
+\shoveright{\framebox[.55\columnwidth]{C}}\\
+\framebox[.65\columnwidth]{D}
+\end{multline}
+\end{verbatim}
+
+The value of \cn{multlinegap} can be changed with the usual \latex/
+commands \cn{setlength} or \cn{addtolength}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Split equations with alignment}
+
+Like \env{multline}, the \env{split} environment is for \emph{single}
+equations that are too long to fit on one line and hence must be split
+into multiple lines.  Unlike \env{multline}, however, the \env{split}
+environment provides for alignment among the split lines, using |&| to
+mark alignment points. Unlike the other \nipkg{amsmath} equation
+structures, the \env{split} environment provides no numbering, because
+it is intended to be used \emph{only inside some other displayed
+  equation structure}, usually an \env{equation}, \env{align}, or
+\env{gather} environment, which provides the numbering. For example:
+\begin{equation}\label{e:barwq}\begin{split}
+H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
+  \sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
+&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
+  \Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
+  \kern-2em % adjust equation body to the right [mjd,13-Nov-1994]
+\end{split}\end{equation}
+
+\begin{verbatim}
+\begin{equation}\label{e:barwq}\begin{split}
+H_c&=\frac{1}{2n} \sum^n_{l=0}(-1)^{l}(n-{l})^{p-2}
+  \sum_{l _1+\dots+ l _p=l}\prod^p_{i=1} \binom{n_i}{l _i}\\
+&\quad\cdot[(n-l )-(n_i-l _i)]^{n_i-l _i}\cdot
+  \Bigl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\Bigr].
+\end{split}\end{equation}
+\end{verbatim}
+
+The \env{split} structure should constitute the entire body of the
+enclosing structure, apart from commands like \cn{label} that produce no
+visible material.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Equation groups without alignment}
+
+The \env{gather} environment is used for a group of consecutive
+equations when there is no alignment desired among them; each one is
+centered separately within the text width (see Table~\ref{displays}).
+Equations inside \env{gather} are separated by a \cn{\\} command.
+Any equation in a \env{gather} may consist of a \verb'\begin{split}'
+  \dots\ \verb'\end{split}' structure\mdash for example:
+\begin{verbatim}
+\begin{gather}
+  first equation\\
+  \begin{split}
+    second & equation\\
+           & on two lines
+  \end{split}
+  \\
+  third equation
+\end{gather}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Equation groups with mutual alignment}
+
+The \env{align} environment is used for two or more equations when
+vertical alignment is desired; usually binary relations such as equal
+signs are aligned (see Table~\ref{displays}).
+
+To have several equation columns side-by-side, use extra ampersands
+to separate the columns:
+\begin{align}
+x&=y       & X&=Y       & a&=b+c\\
+x'&=y'     & X'&=Y'     & a'&=b\\
+x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
+\end{align}
+%
+\begin{verbatim}
+\begin{align}
+x&=y       & X&=Y       & a&=b+c\\
+x'&=y'     & X'&=Y'     & a'&=b\\
+x+x'&=y+y' & X+X'&=Y+Y' & a'b&=c'b
+\end{align}
+\end{verbatim}
+Line-by-line annotations on an equation can be done by judicious
+application of \cn{text} inside an \env{align} environment:
+\begin{align}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+                    && \text{by \eqref{eq:C}}\\
+ & = y'\circ y^*    && \text{by \eqref{eq:D}}\\
+ & = y(0) y'        && \text {by Axiom 1.}
+\end{align}
+%
+\begin{verbatim}
+\begin{align}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+                    && \text{by \eqref{eq:C}}\\
+ & = y'\circ y^*    && \text{by \eqref{eq:D}}\\
+ & = y(0) y'        && \text {by Axiom 1.}
+\end{align}
+\end{verbatim}
+A variant environment \env{alignat} allows the horizontal space between
+equations to be explicitly specified. This environment takes one argument,
+the number of \qq{equation columns} (the number of pairs of right-left
+aligned columns; the argument is the number of pairs): count the maximum
+number of \verb'&'s in any row, add~1 and divide by~2.
+\begin{alignat}{2}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+                  &\quad& \text{by \eqref{eq:C}}\\
+ & = y'\circ y^*  && \text{by \eqref{eq:D}}\\
+ & = y(0) y'      && \text {by Axiom 1.}
+\end{alignat}
+%
+\begin{verbatim}
+\begin{alignat}{2}
+x& = y_1-y_2+y_3-y_5+y_8-\dots
+                  &\quad& \text{by \eqref{eq:C}}\\
+ & = y'\circ y^*  && \text{by \eqref{eq:D}}\\
+ & = y(0) y'      && \text {by Axiom 1.}
+\end{alignat}
+\end{verbatim}
+The environment \env{flalign} (``full length alignment'') stretches the
+space between the equation columns to the maximum possible width, leaving
+only enough space at the margin for the equation number, if present.
+\begin{flalign}
+x&=y       & X&=Y\\
+x'&=y'     & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign}
+%
+\begin{verbatim}
+\begin{flalign}
+x&=y       & X&=Y\\
+x'&=y'     & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign}
+\end{verbatim}
+%
+\begin{flalign*}
+x&=y       & X&=Y\\
+x'&=y'     & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign*}
+%
+\begin{verbatim}
+\begin{flalign*}
+x&=y       & X&=Y\\
+x'&=y'     & X'&=Y'\\
+x+x'&=y+y' & X+X'&=Y+Y'
+\end{flalign*}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Alignment building blocks}
+
+Like \env{equation}, the multi-equation environments \env{gather},
+\env{align}, and \env{alignat} are designed to produce a structure
+whose width is the full line width. This means, for example, that one
+cannot readily add parentheses around the entire structure. But
+variants\index{ed env@\texttt{-ed} environments|(}
+\env{gathered}, \env{aligned}, and \env{alignedat} are provided whose
+total width is the actual width of the contents; thus they can be used
+as a component in a containing expression. E.g.,
+\begin{equation*}
+\left.\begin{aligned}
+  B'&=-\partial\times E,\\
+  E'&=\partial\times B - 4\pi j,
+\end{aligned}
+\right\}
+\qquad \text{Maxwell's equations}
+\end{equation*}
+\begin{verbatim}
+\begin{equation*}
+\left.\begin{aligned}
+  B'&=-\partial\times E,\\
+  E'&=\partial\times B - 4\pi j,
+\end{aligned}
+\right\}
+\qquad \text{Maxwell's equations}
+\end{equation*}
+\end{verbatim}
+Like the \env{array} environment, these \texttt{-ed} variants also take
+an optional\index{options!positioning of \texttt{-ed} environments}
+|[t]|\index{t option@\texttt{t} (top) option},
+|[b]|\index{b option@\texttt{b} (bottom) option} or the default
+|[c]|\index{c option@\texttt{c} (center) option}
+argument to specify vertical
+positioning.  For maximum interoperability, do not insert a
+space\index{options!space before \texttt{[}}
+or line break\index{line break} before the option.  See
+also the note on page~\pageref{breaknote} regarding page breaking
+within the \texttt{-ed} environments.
+
+\qq{Cases} constructions like the following are common in
+mathematics:
+\begin{equation}\label{eq:C}
+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{equation}
+and in the \nipkg{amsmath} package there is a \env{cases} environment to
+make them easy to write:
+\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} (cf.~\secref{text}) and the nested math
+formulas. \env{cases} is set in \cn{textstyle}. If \cn{displaystyle}
+is wanted, it must be requested explicitly; \pkg{mathtools} provides
+a \env{dcases} environment for this purpose.
+
+The |-ed|\index{ed env@\texttt{-ed} environments|)} and \env{cases}
+environments must appear within an enclosing math environment, which can
+be either in text, between |$...$|, or in any of the display environments.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Adjusting tag placement}
+
+Placing equation numbers can be a rather complex problem in multiline
+displays. The environments of the \nipkg{amsmath} package try hard to
+avoid overprinting an equation number on the equation contents, if
+necessary moving the number down or up to a separate line. Difficulties
+in accurately calculating the profile of an equation can occasionally
+result in number movement that doesn't look right.
+A \cn{raisetag} command is provided to adjust the vertical position of the
+current equation number, if it has been shifted away from its normal
+position. To move a particular number up by six points, write
+|\raisetag{6pt}|. (At the end of a display, this also shifts up the
+text following the display.)
+This kind of adjustment is fine tuning like line
+breaks and page breaks, and should therefore be left undone until your
+document is nearly finalized, or you may end up redoing the fine tuning
+several times to keep up with changing document contents.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Vertical spacing and page breaks in multiline displays}
+
+You can use the \cn{\\}|[|\<dimension>|]|%
+\index{options!extra vertical space after \cn{\\}}
+command to get extra vertical
+space between lines in all the \nipkg{amsmath} displayed equation
+environments, as is usual in \latex/. Do not type a space between the%
+\index{options!space before \texttt{[}}
+\cn{\\} and the following |[|; \emph{only} for display environments
+defined by \nipkg{amsmath} the space is interpreted to mean that the
+bracketed material is part of the document content.
+
+When the \nipkg{amsmath} package is
+in use \ii{page breaks} between equation lines are normally disallowed;
+the philosophy is that page breaks in such material should receive
+individual attention from the author. To get an individual page break
+inside a particular displayed equation, a \cn{displaybreak} command is
+provided. \cn{displaybreak} is best placed immediately before the
+\cn{\\} where it is to take effect.  Like \latex/'s \cn{pagebreak},
+\cn{displaybreak} takes an optional argument%
+\index{options!behavior of particular options} between 0 and 4 denoting
+the desirability of the pagebreak. |\displaybreak[0]| means \qq{it is
+  permissible to break here} without encouraging a break;
+\cn{displaybreak} with no optional argument is the same as
+|\displaybreak[4]| and forces a break.
+
+If you prefer a strategy of letting page breaks fall where they may,
+even in the middle of a multiline equation, then you might put
+\cn{allowdisplaybreaks}\texttt{[1]} in the preamble of your document. An
+optional argument\index{options!behavior of particular options}
+1\ndash 4 can be used for finer control: |[1]| means
+allow page breaks, but avoid them as much as possible; values of 2,3,4
+mean increasing permissiveness. When display breaks are enabled with
+\cn{allowdisplaybreaks}, the \cn{\\*} command can be used to prohibit a
+pagebreak after a given line, as usual.
+
+\begin{bfseries}
+Note:\label{breaknote} Certain equation environments wrap their contents
+in an unbreakable box, with the consequence that neither \cn{displaybreak}
+nor \cn{allowdisplaybreaks} will have any effect on them. These include
+\env{split}, \env{aligned}, \env{gathered}, and \env{alignedat}.
+\end{bfseries}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Interrupting a display}
+
+The command \cn{intertext} is used for a short interjection of one or
+two lines of text\index{text fragments inside math} in the middle of a
+multiple-line display structure (see also the \cn{text} command in
+\secref{text}). Its salient feature is preservation of the alignment,
+which would not happen if you simply ended the display and then started
+it up again afterwards. \cn{intertext} may only appear right after a
+\cn{\\} or \cn{\\*} command. Notice the position of the word \qq{and} in
+this example.
+\begin{align}
+  A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
+  A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
+\intertext{and}
+  A_3&=\mathcal{N}(\lambda;\omega).
+\end{align}
+\begin{verbatim}
+\begin{align}
+  A_1&=N_0(\lambda;\Omega')-\phi(\lambda;\Omega'),\\
+  A_2&=\phi(\lambda;\Omega')-\phi(\lambda;\Omega),\\
+\intertext{and}
+  A_3&=\mathcal{N}(\lambda;\omega).
+\end{align}
+\end{verbatim}
+
+The \pkg{mathtools} package provides a command \cn{shortintertext} that
+is intended for use when the interjected text is only a few words; it uses
+less vertical space than \cn{intertext}. This
+is most effective when equation numbers are on the right.%
+\index{equationn@equation numbers!left or right placement}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Equation numbering}
+
+\subsection{Numbering hierarchy}
+In \latex/ if you wanted to have equations numbered within
+sections\mdash that is, have
+equation numbers (1.1), (1.2), \dots, (2.1), (2.2),
+\dots, in sections 1, 2, and so forth\mdash you could redefine
+\cn{theequation} as suggested in the \latex/ manual \cite[\S6.3, \S
+C.8.4]{lamport}:
+\begin{verbatim}
+\renewcommand{\theequation}{\thesection.\arabic{equation}}
+\end{verbatim}
+
+This works pretty well, except that the equation counter won't be reset
+to zero at the beginning of a new section or chapter, unless you do it
+yourself using \cn{setcounter}. To make this a little more convenient,
+the \nipkg{amsmath} package provides a 
+command\index{equationn@equation numbers!hierarchy} \cn{numberwithin}.
+To have equation numbering tied to section numbering, with automatic
+reset of the equation counter, write
+\begin{verbatim}
+\numberwithin{equation}{section}
+\end{verbatim}
+As its name implies, the \cn{numberwithin} command can be applied to
+any counter, not just the \texttt{equation} counter.
+
+\subsection{Cross references to equation numbers}
+
+To make cross-references to equations easier, an \cn{eqref}
+command\index{equationn@equation numbers!cross-references} is provided. This
+automatically supplies the parentheses around the equation number. I.e.,
+if \verb'\ref{abc}' produces 3.2 then \verb'\eqref{abc}' produces
+(3.2). The parentheses around an \cn{eqref} equation number will be
+set in upright type regardless of type style of the context.
+
+\subsection{Subordinate numbering sequences}
+
+The \nipkg{amsmath} package provides also a wrapper environment,
+\env{subequations}\index{equationn@equation numbers!subordinate numbering}, to make it
+easy to number equations in a particular \cn{align} or similar group with
+a subordinate numbering scheme. For example
+\begin{verbatim}
+\begin{subequations}
+...
+\end{subequations}
+\end{verbatim}
+causes all numbered equations within that part of the document to be
+numbered (4.9a) (4.9b) (4.9c) \dots, if the preceding numbered
+equation was (4.8). A \cn{label} command immediately after
+\verb/\begin{subequations}/ will produce a \cn{ref} of the parent
+number 4.9, not 4.9a. The counters used by the subequations
+environment are \verb/parentequation/ and \verb/equation/ and
+\cn{addtocounter}, \cn{setcounter}, \cn{value}, etc., can be applied
+as usual to those counter names. To get anything other than lowercase
+letters for the subordinate numbers, use standard \latex/ methods for
+changing numbering style \cite[\S6.3, \S C.8.4]{lamport}. For example,
+redefining \cn{theequation} as follows will produce roman numerals.
+\begin{verbatim}
+\begin{subequations}
+\renewcommand{\theequation}{\theparentequation \roman{equation}}
+...
+\end{verbatim}
+
+\subsection{Numbering style}
+
+The default equation number is set in \cn{normalfont}.  This means that
+in bold section headings, bold is suppressed; a workaround for this is
+to use \verb+(\ref{+\ldots\verb+})+ rather than \verb+\eqref{+\ldots\verb+}+.
+
+If smaller type is specified for a numbered display, the size of the
+equation number will also be small.  The default size can be ensured
+throughout a document by applying this patch in the preamble:
+\begin{verbatim}
+\makeatletter
+\renewcommand{\maketag@@@}[1]{\hbox{\m@th\normalsize\normalfont#1}}%
+\makeatother
+\end{verbatim}
+(This modification may be included in a future version of
+\nipkg{amsmath}.)
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Miscellaneous mathematical features}
+
+\section{Matrices}\label{ss:matrix}
+
+The \nipkg{amsmath} package provides some environments for
+matrices\index{matrices} beyond the basic \env{array} environment of
+\latex/. The \env{pmatrix}, \env{bmatrix}, \env{Bmatrix}, \env{vmatrix}
+and \env{Vmatrix} have (respectively) $(\,)$, $[\,]$, $\lbrace\,\rbrace$,
+$\lvert\,\rvert$, and $\lVert\,\rVert$ delimiters\index{delimiters}
+built in. For naming consistency there is a
+\env{matrix} environment sans delimiters. This is not entirely redundant
+with the \env{array} environment; the matrix environments all use more
+economical horizontal spacing than the rather prodigal spacing of the
+\env{array} environment. Also, unlike the \env{array} environment, you
+don't have to give column specifications for any of the matrix
+environments; by default you can have up to 10 centered columns.%
+\footnote{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+More precisely: The maximum number of columns in a matrix is determined
+by the counter |MaxMatrixCols| (normal value = 10), which you can change
+if necessary using \latex/'s \cn{setcounter} or \cn{addtocounter}
+commands.%
+}\space%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+(If you need left or right alignment in a column or other special
+formats you may use \env{array}, or the \pkg{mathtools}
+package which provides \verb+*+ variants%
+\index{matrix@\indexenvironmentfont{matrix} environment!\verb+*+ variants}
+of these environments with
+an optional argument to specify left or right alignment.)
+
+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. Delimiters\index{delimiters} must be provided. (The
+\pkg{mathtools} package provides |p|,|b|,|B|,|v|,|V| versions of
+\env{smallmatrix},%
+\index{smallmatrix@\indexenvironmentfont{smallmatrix} environment!variant delimiters}
+as well as |*| variants%
+\index{smallmatrix@\indexenvironmentfont{smallmatrix} environment!\verb+*+ variants}
+as described above.)
+The above example was produced by
+\begin{verbatim}
+\bigl( \begin{smallmatrix}
+  a&b\\ c&d
+\end{smallmatrix} \bigr)
+\end{verbatim}
+
+\cn{hdotsfor}|{|\<number>|}| produces a row of dots in a
+matrix\index{matrices!ellipsis dots}\index{ellipsis dots!in
+matrices}\index{dots|see{ellipsis dots}} spanning the given number of
+columns. For example,
+\begin{center}
+\begin{minipage}{.3\columnwidth}
+\noindent$\begin{matrix} a&b&c&d\\
+e&\hdotsfor{3} \end{matrix}$
+\end{minipage}%
+\qquad
+\begin{minipage}{.45\columnwidth}
+\begin{verbatim}
+\begin{matrix} a&b&c&d\\
+e&\hdotsfor{3} \end{matrix}
+\end{verbatim}
+\end{minipage}%
+\end{center}
+
+The spacing of the dots can be varied through use of a square-bracket
+option,\index{options!behavior of particular options}
+for example, |\hdotsfor[1.5]{3}|.  The number in square brackets
+will be used as a multiplier (i.e., the normal value is 1.0).
+\begin{equation}\label{eq:D}
+\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
+-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
+\hdotsfor[2]{4}\\
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix},
+\end{equation}
+\begin{verbatim}
+\begin{pmatrix} D_1t&-a_{12}t_2&\dots&-a_{1n}t_n\\
+-a_{21}t_1&D_2t&\dots&-a_{2n}t_n\\
+\hdotsfor[2]{4}\\
+-a_{n1}t_1&-a_{n2}t_2&\dots&D_nt\end{pmatrix}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Math spacing commands}
+
+The \nipkg{amsmath} package slightly extends the set of math
+spacing\index{horizontal spacing} commands, as shown below.
+Both the spelled-out and abbreviated forms of these commands are robust,
+and they can also be used outside of math.
+\begin{ctab}{lll|lll}
+Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\
+\hline
+\vstrut{2.5ex}
+& no space& \spx{}& & no space & \spx{}\\
+\cn{\,}& \cn{thinspace}& \spx{\,}&
+  \altcnbang& \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{ctab}
+For the greatest possible control over math spacing, use \cn{mspace}
+and `math units'. One math unit, or |mu|, is equal to 1/18 em. Thus to
+get a negative \cn{quad} you could write |\mspace{-18.0mu}|.
+
+%% ------------------------------------------------------------------ %%
+
+\section{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. By using the semantically oriented commands
+\begin{itemize}
+\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}
+\begin{tabular}{@{}l@{}l@{}}
+\begin{minipage}[t]{.54\textwidth}
+\begin{verbatim}
+Then we have the series $A_1, A_2,
+\dotsc$, the regional sum $A_1
++A_2 +\dotsb $, the orthogonal
+product $A_1 A_2 \dotsm $, and
+the infinite integral
+\[\int_{A_1}\int_{A_2}\dotsi\].
+\end{verbatim}
+\end{minipage}
+&
+\begin{minipage}[t]{.45\textwidth}
+\noindent
+Then 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}
+For most situations, the undifferentiated \cn{dots} can be used, and
+\nipkg{amsmath} will output the most suitable form based on the
+immediate context; if an inappropriate form results, it can be corrected
+after examining the output.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Nonbreaking dashes}
+
+A command \cn{nobreakdash} is provided to suppress the possibility of a
+line break\index{line break|(} after the following hyphen or dash. For
+example, if you write `pages 1\ndash 9' as |pages 1\nobreakdash--9|
+then a line break will never occur between the dash and the 9. You can
+also use \cn{nobreakdash} to prevent undesirable hyphenations in
+combinations like |$p$-adic|. For frequent use, it's advisable to make
+abbreviations, e.g.,
+\begin{verbatim}
+\newcommand{\p}{$p$\nobreakdash}% for "\p-adic"
+\newcommand{\Ndash}{\nobreakdash--}% for "pages 1\Ndash 9"
+%    For "\n dimensional" ("n-dimensional"):
+\newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}}
+\end{verbatim}
+The last example shows how to prohibit a line break\index{line break|)}
+after the hyphen but allow normal hyphenation in the following
+word. (It suffices to add a zero-width space after the hyphen.)
+
+%% ------------------------------------------------------------------ %%
+
+\section{Accents in math}
+
+In ordinary \LaTeX{} the placement of the second accent in doubled math
+accents is often poor. With the \nipkg{amsmath} package you
+will get improved placement of the second accent:
+$\hat{\hat{A}}$ (\cn{hat}|{\hat{A}}|).
+
+The commands \cn{dddot} and \cn{ddddot} are available to produce triple
+and quadruple dot accents in addition to the \cn{dot} and \cn{ddot}
+accents already available in \latex/.
+
+To get a superscripted hat or tilde character, load the \pkg{amsxtra}
+package and use \cn{sphat} or \cn{sptilde}. Usage is \verb'A\sphat'
+(note the absence of the \verb'^' character).
+
+
+To place an arbitrary symbol in math accent position, or to get under
+accents, see the \pkg{accents} package by Javier Bezos. (\nipkg{amsmath}
+must be loaded before \pkg{accents}.)
+
+%% ------------------------------------------------------------------ %%
+
+\section{Roots}
+
+In ordinary \latex/ the placement of root indices is sometimes not so
+good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|).  In the
+\nipkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
+the position of the root:%
+\index{options!adjust positioning}
+\begin{verbatim}
+  \sqrt[\leftroot{-2}\uproot{2}\beta]{k}
+\end{verbatim}
+will move the beta up and to the right:
+$\sqrt[\leftroot{-2}\uproot{2}\beta]{k}$. The negative argument used
+with \cn{leftroot} moves the $\beta$ to the right. The units are a small
+amount that is a useful size for such adjustments.
+
+%% ------------------------------------------------------------------ %%
+
+\section{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}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Over and under arrows}
+
+Basic \latex/ provides \cn{overrightarrow} and \cn{overleftarrow}
+commands. Some additional over and under arrow commands are provided
+by the \nipkg{amsmath} package to extend the set:
+
+\begin{tabbing}
+\qquad\=\ncn{overleftrightarrow}\qquad\=\kill
+\> \cn{overleftarrow}    \> \cn{underleftarrow} \+\\
+   \cn{overrightarrow}    \> \cn{underrightarrow} \\
+   \cn{overleftrightarrow}\> \cn{underleftrightarrow}
+\end{tabbing}
+
+%% ------------------------------------------------------------------ %%
+
+\section{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\index{options!behavior of particular options}
+(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}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Affixing symbols to other symbols}
+
+\latex/ provides \cn{stackrel} for placing a
+superscript\index{subscripts and superscripts} above a binary relation.
+In the \nipkg{amsmath} package there are somewhat more general commands,
+\cn{overset} and \cn{underset}, that can be used to place one symbol
+above or below another symbol, whether it's a relation or something
+else. The input |\overset{*}{X}| will place a superscript-size $*$ above
+the $X$: $\overset{*}{X}$; \cn{underset} is the analog for adding a
+symbol underneath.
+
+See also the description of \cn{sideset} in \secref{sideset}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Fractions and related constructions}
+
+\subsection{The \cn{frac}, \cn{dfrac}, and \cn{tfrac} commands}
+
+The \cn{frac} command, which is in the basic command set of
+\latex/,\index{fractions|(} takes two arguments\mdash numerator and
+denominator\mdash and typesets them in normal fraction form. The
+\nipkg{amsmath} package provides also \cn{dfrac} and \cn{tfrac} as
+convenient abbreviations for |{\displaystyle\frac| |...| |}|
+and\indexcs{textstyle}\relax
+\indexcs{displaystyle} |{\textstyle\frac| |...| |}|.
+\begin{equation}
+\frac{1}{k}\log_2 c(f)\quad\tfrac{1}{k}\log_2 c(f)\quad
+\sqrt{\frac{1}{k}\log_2 c(f)}\quad\sqrt{\dfrac{1}{k}\log_2 c(f)}
+\end{equation}
+\begin{verbatim}
+\begin{equation}
+\frac{1}{k}\log_2 c(f)\;\tfrac{1}{k}\log_2 c(f)\;
+\sqrt{\frac{1}{k}\log_2 c(f)}\;\sqrt{\dfrac{1}{k}\log_2 c(f)}
+\end{equation}
+\end{verbatim}
+
+\subsection{The \cn{binom}, \cn{dbinom}, and \cn{tbinom} commands}
+
+For binomial expressions\index{binomials} such as $\binom{n}{k}$
+\nipkg{amsmath} has \cn{binom}, \cn{dbinom} and \cn{tbinom}:
+\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 \cn{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\index{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\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{center}\begin{minipage}{.85\columnwidth}
+\raggedright \normalfont\ttfamily \exhyphenpenalty10000
+\newcommand{\ma}[1]{%
+  \string{{\normalfont\itshape#1}\string}\penalty9999 \ignorespaces}
+\string\genfrac \ma{left-delim} \ma{right-delim} \ma{thickness}
+\ma{mathstyle} \ma{numerator} \ma{denominator}
+\end{minipage}\end{center}
+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}
+If you find yourself repeatedly using \cn{genfrac} throughout a document
+for a particular notation, you will do yourself a favor (and your
+publisher) if you define a meaningfully-named abbreviation for that
+notation, along the lines of \cn{frac} and \cn{binom}.
+
+The primitive generalized fraction commands \cs{over}, \cs{overwithdelims},
+\cs{atop}, \cs{atopwithdelims}, \cs{above}, \cs{abovewithdelims} produce
+warning  messages if used with the \nipkg{amsmath} package, for reasons
+discussed in \fn{technote.tex}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{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}|[l]| or \cn{cfrac}|[r]| instead of
+\cn{cfrac}.\index{options!adjust positioning}%
+\index{fractions|)}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Smash options}
+
+The command \cn{smash} is used to typeset a subformula with an
+effective height and depth of zero, which is sometimes
+useful in adjusting the subformula's position with respect to adjacent
+symbols. With the \nipkg{amsmath} package \cn{smash} has optional
+arguments\index{options!behavior of particular options}
+|[t]|\index{t option@\texttt{t} (top) option} and
+|[b]|\index{b option@\texttt{b} (bottom) option} because occasionally it is
+advantageous to be able to \qq{smash} only the top or only the bottom
+of something while retaining the natural depth or height. For example,
+when adjacent radical symbols are unevenly sized or positioned because
+of differences in the height and depth of their contents, \cn{smash}
+can be employed to make them more consistent. Compare
+$\sqrt{x}+\sqrt{y}+\sqrt{z}$ and $\sqrt{x}+\sqrt{\smash[b]{y}}+\sqrt{z}$,
+where the latter was produced by
+\verb"$\sqrt{x}" \verb"+"
+\verb"\sqrt{"\5\verb"\smash[b]{y}}" \verb"+" \verb"\sqrt{z}$".
+
+%% ------------------------------------------------------------------ %%
+
+\section{Delimiters}
+
+\index{delimiters|(}
+
+\subsection{Delimiter sizes}\label{bigdel}
+
+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 is not even approximately continuous but has fairly large
+quantum jumps. This means that a math fragment that is infinitesimally
+too large for a given delimiter size will get the next larger size, a
+jump of 3pt or so in normal-sized text. There are two or three
+situations where the delimiter size is commonly adjusted, using a set of
+commands that have `big' in their names.%
+\index{delimiters!fixed size|(}
+\begin{ctab}{l|llllll}
+Delimiter&
+  text& \ncn{left}& \ncn{bigl}& \ncn{Bigl}& \ncn{biggl}& \ncn{Biggl}\\
+size&
+  size& \ncn{right}& \ncn{bigr}& \ncn{Bigr}& \ncn{biggr}& \ncn{Biggr}\\
+\hline
+Result\vstrut{5ex}&
+  $\displaystyle(b)(\frac{c}{d})$&
+  $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$&
+  $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$&
+  $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$&
+  $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$&
+  $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$
+\end{ctab}
+The first kind of situation is a cumulative operator with limits above
+and below. With \cn{left} and \cn{right} the delimiters usually turn out
+larger than necessary, and using the |Big| or |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 slightly larger than the base size but still able to
+fit within the normal line spacing:
+$\bigl\lvert\frac{b'}{d'}\bigr\rvert$.
+
+In ordinary \latex/ \ncn{big}, \ncn{bigg}, \ncn{Big}, and \ncn{Bigg}
+delimiters aren't scaled properly over the full range of \latex/ font
+sizes.  With the \nipkg{amsmath} package they are.%
+\index{delimiters!fixed size|)}
+
+\subsection{Vertical bar notations}
+
+The \nipkg{amsmath} package provides commands \cn{lvert}, \cn{rvert},
+\cn{lVert}, \cn{rVert} (compare \cn{langle}, \cn{rangle}) to address the
+%problem of overloading for the vert bar character \qc{\|}. This
+problem of overloading for the vert bar character \qcvert. This
+character is currently used in \latex/ documents to represent a wide
+variety of mathematical objects: the `divides' relation in a
+number-theory expression like $p\vert q$, or the absolute-value
+operation $\lvert z\rvert$, or the `such that' condition in set
+notation, or the `evaluated at' notation $f_\zeta(t)\bigr\rvert_{t=0}$.
+The multiplicity of uses in itself is not so bad; what is bad, however,
+is that fact that not all of the uses take the same typographical
+treatment, and that the complex discriminatory powers of a knowledgeable
+reader cannot be replicated in computer processing of mathematical
+documents. It is recommended therefore that there should be a one-to-one
+correspondence in any given document between the vert bar character
+%\qc{\|} and a selected mathematical notation, and similarly for the
+\qcvert\ and a selected mathematical notation, and similarly for the
+%double-bar command \ncn{\|}\index{"|@\verb"*+"\"|+}. This immediately
+double-bar command \cnvert. This immediately
+rules out the use of \qc{|}
+%and \ncn{\|}\index{"|@\verb"*+"\"|+} for delimiters, because left and right
+and \cnvert\ for delimiters, because left and right
+delimiters are distinct usages that do not relate in the same way to
+adjacent symbols; recommended practice is therefore 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 the document would contain |\abs{z}| to produce $\lvert
+z\rvert$ and |\norm{v}| to produce $\lVert v\rVert$.
+The \pkg{mathools} provides the command \cn{DeclarePairedDelimiter}
+for defining |\abs|-like macros with scaling delimiters.
+\index{delimiters|)}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Operator names}
+
+\section{Defining new operator names}\label{s:opname}
+
+Math functions\index{operator names}\relax \index{function
+names|see{operator names}} such as $\log$, $\sin$, and $\lim$ are
+traditionally typeset in roman type to make them visually more distinct
+from one-letter math variables, which are set in math italic. The more
+common ones have predefined names, \cn{log}, \cn{sin}, \cn{lim}, and so
+forth, but new ones come up all the time in mathematical papers, so the
+\nipkg{amsmath} package provides a general mechanism for defining new
+`operator names'. To define a math function \ncn{xxx} to work like
+\cn{sin}, you write
+\begin{verbatim}
+\DeclareMathOperator{\xxx}{xxx}
+\end{verbatim}
+whereupon ensuing uses of \ncn{xxx} will produce {\upshape xxx} in the
+proper font and automatically add proper spacing\index{horizontal
+spacing!around operator names} on either side when necessary, so that you
+get $A\xxx B$ instead of $A\mathrm{xxx}B$. In the second argument of
+\cn{DeclareMathOperator} (the name text), a pseudo-text mode prevails:
+the hyphen character \qc{\-} will print as a text hyphen rather than a
+minus sign and an asterisk \qc{\*} will print as a raised text asterisk
+instead of a centered math star. (Compare
+\textit{a}-\textit{b}*\textit{c} and $a-b*c$.) But otherwise the name
+text is printed in math mode, so that you can use, e.g., subscripts and
+superscripts there.
+
+%%%%xxx%%%\enlargethispage{1\baselineskip}
+If the new operator should have subscripts and superscripts placed in
+`limits' position above and below as with $\lim$, $\sup$, or $\max$, use
+the \qc{\*} form of the \cn{DeclareMathOperator} command:
+\begin{verbatim}
+\DeclareMathOperator*{\Lim}{Lim}
+\end{verbatim}
+See also the discussion of subscript placement in
+Section~\ref{subplace}.
+
+\goodbreak
+
+The following operator names are predefined:
+
+\vspace{-1\baselineskip}
+\begin{ctab}{@{}rlrlrlrl}
+\cn{arccos}& $\arccos$ &\cn{deg}& $\deg$ &      \cn{lg}& $\lg$ &        \cn{projlim}& $\projlim$\\
+\cn{arcsin}& $\arcsin$ &\cn{det}& $\det$ &      \cn{lim}& $\lim$ &      \cn{sec}& $\sec$\\
+\cn{arctan}& $\arctan$ &\cn{dim}& $\dim$ &      \cn{liminf}& $\liminf$ &\cn{sin}& $\sin$\\
+\cn{arg}& $\arg$ &      \cn{exp}& $\exp$ &      \cn{limsup}& $\limsup$ &\cn{sinh}& $\sinh$\\
+\cn{cos}& $\cos$ &      \cn{gcd}& $\gcd$ &      \cn{ln}& $\ln$ &        \cn{sup}& $\sup$\\
+\cn{cosh}& $\cosh$ &    \cn{hom}& $\hom$ &      \cn{log}& $\log$ &      \cn{tan}& $\tan$\\
+\cn{cot}& $\cot$ &      \cn{inf}& $\inf$ &      \cn{max}& $\max$ &      \cn{tanh}& $\tanh$\\
+\cn{coth}& $\coth$ &    \cn{injlim}& $\injlim$ &\cn{min}& $\min$\\
+\cn{csc}& $\csc$ &      \cn{ker}& $\ker$ &      \cn{Pr}& $\Pr$
+\end{ctab}
+\par\nobreak
+\vspace{-1.2\baselineskip}
+\begin{ctab}{rlrl}
+\cn{varinjlim}&  $\displaystyle\varinjlim$&
+\cn{varliminf}&  $\displaystyle\varliminf$\\
+\cn{varprojlim}& $\displaystyle\varprojlim$&
+\cn{varlimsup}&  $\displaystyle\varlimsup$
+\end{ctab}
+
+There is also a command \cn{operatorname} such that using
+\begin{verbatim}
+\operatorname{abc}
+\end{verbatim}
+in a math formula is equivalent to a use of \ncn{abc} defined by
+\cn{DeclareMathOperator}. This may be occasionally useful for
+constructing more complex notation or other purposes. (Use the variant
+\cn{operatorname*} to get limits.)
+
+%% ------------------------------------------------------------------ %%
+
+\section{\cn{mod} and its relatives}
+
+Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} are provided to deal
+with the special spacing conventions of \qq{mod} notation. \cn{bmod} and
+\cn{pmod} are available in \latex/, but with the \nipkg{amsmath} package
+the spacing of \cn{pmod} will adjust to a smaller value if it's used in
+a non-display-mode formula. \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}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\enlargethispage{1\baselineskip}
+
+\chapter{The \cn{text} command}\label{text}
+
+The main use of the command \cn{text} is for words or
+phrases\index{text fragments inside math} in a display. It is very
+similar to the \latex/ command \cn{mbox} in its effects, but has a
+couple of advantages. If you want a word or phrase of text in a
+subscript, you can type |..._{\textrm{word or phrase}}|, which is slightly
+easier than the \cn{mbox} equivalent: |..._{\mbox{\rmfamily\scriptsize| |word|
+|or| |phrase}}|. Note that the standard \cn{textrm} command will use the
+\nipkg{amsmath} \cn{text} definition, but ensure the \verb+\rmfamily+
+font is used.
+\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}
+
+The font used for \cn{text} is the same as that of the surrounding
+environment; i.e., within a theorem, the contents of \cn{text} will
+be set in italic.
+
+If a math expression is included in a \cn{text} string, it must be
+explicitly marked as math (|$...$|).
+\[
+\partial_s f(x) = \frac{\partial}{\partial x_0} f(x)\quad
+  \text{for $x= x_0 + I x_1$.}
+\]
+\begin{verbatim}
+\partial_s f(x) = \frac{\partial}{\partial x_0} f(x)\quad
+  \text{for $x= x_0 + I x_1$.}
+\end{verbatim}
+
+Function names should \emph{not} be entered as \cn{text}.  Instead,
+use \cn{mathrm} or \cn{DeclareMathOperator} as appropriate.  These are
+fixed entities that should not change depending on outside content
+(such as appearing within a theorem that is set in italic), and in the
+case of declared operators, proper spacing\index{horizontal
+spacing!around operator names} is applied automatically.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Integrals and sums}
+
+\section{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{ctab}{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{ctab}
+A slightly more generalized form is the \env{subarray} environment which
+allows you to specify that each line should be left-aligned instead of
+centered, as here:
+\begin{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sum_{\begin{subarray}{l}
+        i\in\Lambda\\ 0<j<n
+      \end{subarray}}
+ P(i,j)
+\end{verbatim}
+\end{minipage}
+&
+$\displaystyle
+  \sum_{\begin{subarray}{l}
+        i\in \Lambda\\ 0<j<n
+      \end{subarray}}
+ P(i,j)$
+\end{ctab}
+
+%% ------------------------------------------------------------------ %%
+
+\section{The \cn{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
+large operator symbol such as $\sum$ or $\prod$. \emph{Note: this
+command is not designed to be applied to anything other than sum-class
+symbols.} The prime
+example is the case when you want to put a prime on a sum symbol. If
+there are no limits above or below the sum, you could just use
+\cn{nolimits}: here's
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+|\sum\nolimits' E_n| in display mode:
+\begin{equation}
+\sum\nolimits' E_n
+\end{equation}
+If, however, you want not only the prime but also something below or
+above the sum symbol, it's not so easy\mdash indeed, without
+\cn{sideset}, it would be downright difficult. With \cn{sideset}, you
+can write
+\begin{ctab}{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{ctab}
+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{ctab}{ll}
+\begin{minipage}[t]{.6\columnwidth}
+\begin{verbatim}
+\sideset{_*^*}{_*^*}\prod
+\end{verbatim}
+\end{minipage}
+&$\displaystyle
+\sideset{_*^*}{_*^*}\prod
+$
+\end{ctab}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Placement of subscripts and limits}\label{subplace}
+
+The default positioning for subscripts depends on the
+base symbol involved. The default for sum-class symbols is
+`displaylimits' positioning: When a sum-class symbol appears
+in a displayed formula, subscript and superscript are placed in `limits'
+position above and below, but in an inline formula, they are placed to
+the side, to avoid unsightly and wasteful spreading of the
+surrounding text lines.
+The default for integral-class symbols is to have sub- and
+superscripts always to the side, even in displayed formulas.
+(See the discussion of the \opt{intlimits} and related
+options\index{options!behavior of particular options} in
+Section~\ref{options}.)
+
+Operator names such as $\sin$ or $\lim$ may have either `displaylimits'
+or `limits' positioning depending on how they were defined. The standard
+operator names are defined according to normal mathematical usage.
+
+The commands \cn{limits} and \cn{nolimits} can be used to override the
+normal behavior of a base symbol:
+\begin{equation*}
+\sum\nolimits_X,\qquad \iint\limits_{A},
+\qquad\varliminf\nolimits_{n\to \infty}
+\end{equation*}
+To define a command whose subscripts follow the
+same `displaylimits' behavior as \cn{sum}, put
+\cn{displaylimits} at the tail end of the definition. When multiple
+instances of \cn{limits}, \cn{nolimits}, or \cn{displaylimits} occur
+consecutively, the last one takes precedence.
+
+%% ------------------------------------------------------------------ %%
+
+\section{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.
+\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}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Commutative diagrams}\label{s:commdiag}
+\index{commutative diagrams}
+
+Some commutative diagram commands like the ones in \amstex/ are
+available as a separate package, \pkg{amscd}. For complex commutative
+diagrams authors will need to turn to more comprehensive packages like
+\tikz/\index{TikZ@\tikz/ package} (in particular, \pkg{tikz-cd})
+or \xypic/\index{XY-pic@\xypic/ package},
+but for simple diagrams without diagonal
+arrows\index{arrows!in commutative diagrams} the \pkg{amscd} commands
+may be more convenient. Here is one example.
+\begin{equation*}
+\begin{CD}
+S^{\mathcal{W}_\Lambda}\otimes T   @>j>>   T\\
+@VVV                                    @VV{\End P}V\\
+(S\otimes T)/I                  @=      (Z\otimes T)/J
+\end{CD}
+\end{equation*}
+\begin{verbatim}
+\begin{CD}
+S^{\mathcal{W}_\Lambda}\otimes T   @>j>>   T\\
+@VVV                                    @VV{\End P}V\\
+(S\otimes T)/I                  @=      (Z\otimes T)/J
+\end{CD}
+\end{verbatim}
+In the \env{CD} environment the commands |@>>>|,
+|@<<<|, |@VVV|, and |@AAA| give respectively right, left, down, and up
+arrows. For the horizontal arrows, material between the first and second
+|>| or |<| symbols will be typeset as a superscript, and material
+between the second and third will be typeset as a subscript. Similarly,
+material between the first and second or second and third |A|s or |V|s
+of vertical arrows will be typeset as left or right \qq{sidescripts}.
+The commands |@=| and \verb'@|' give horizontal and vertical double lines.
+A \qq{null arrow} command |@.| can be used instead of a visible arrow
+to fill out an array where needed.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Using math fonts}
+
+\section{Introduction}
+
+For more comprehensive information on font use in \latex/, see the
+\latex/ font guide (\fn{fntguide.tex}) or \booktitle{The \latex/
+Companion} \cite{tlc2}. The basic set of math font commands\index{math
+fonts}\relax \index{math symbols|see{math fonts}} in \latex/ includes
+\cn{mathbf}, \cn{mathrm}, \cn{mathcal}, \cn{mathsf}, \cn{mathtt},
+\cn{mathit}. Additional math alphabet commands such as
+\cn{mathbb} for blackboard bold, \cn{mathfrak} for Fraktur, and
+\cn{mathscr} for Euler script are available through the packages
+\pkg{amsfonts} and \pkg{euscript} (distributed separately).
+
+%% ------------------------------------------------------------------ %%
+\enlargethispage{1\baselineskip}
+
+\section{Recommended use of math font commands}
+
+If you find yourself employing math font commands frequently in your
+document, you might wish that they had shorter names, such as \ncn{mb}
+instead of \cn{mathbf}. Of course, there is nothing to keep you from
+providing such abbreviations for yourself by suitable \cn{newcommand}
+statements. But for \latex/ to provide shorter names would actually be a
+disservice to authors, as that would obscure a much better alternative:
+defining custom command names derived from the names of the underlying
+mathematical objects, rather than from the names of the fonts used to
+distinguish the objects. For example, if you are using bold to indicate
+vectors, then you will be better served in the long run if you define a
+`vector' command instead of a `math-bold' command:
+\begin{verbatim}
+  \newcommand{\vect}[1]{\mathbf{#1}}
+\end{verbatim}
+you can write |\vect{a} + \vect{b}| to produce $\vect{a} +
+\vect{b}$.
+If you decide several months down the road that you want to use the bold
+font for some other purpose, and mark vectors by a small over-arrow
+instead, then you can put the change into effect merely by changing the
+definition of \ncn{vect}; otherwise you would have to replace all
+occurrences of \cn{mathbf} throughout your document, perhaps even
+needing to inspect each one to see whether it is indeed
+an instance of a vector.
+
+It can also be useful to assign distinct
+command names for different letters of a particular font:
+\begin{verbatim}
+\DeclareSymbolFont{AMSb}{U}{msb}{m}{n}% or use amsfonts package
+\DeclareMathSymbol{\C}{\mathalpha}{AMSb}{"43}
+\DeclareMathSymbol{\R}{\mathalpha}{AMSb}{"52}
+\end{verbatim}
+These statements would define the commands \cn{C} and \cn{R} to produce
+blackboard-bold letters from the `AMSb' math symbols font. If you refer
+often to the\break complex numbers or real numbers in your document, you
+might find this method more convenient than (let's say) defining a
+\ncn{field} command and writing\break |\field{C}|, |\field{R}|. But for
+maximum flexibility and control, define such a \ncn{field} command and
+then define \ncn{C} and \ncn{R} in terms of that
+command:\indexcs{mathbb}
+\begin{verbatim}
+\usepackage{amsfonts}% to get the \mathbb alphabet
+\newcommand{\field}[1]{\mathbb{#1}}
+\newcommand{\C}{\field{C}}
+\newcommand{\R}{\field{R}}
+\end{verbatim}
+
+%% ------------------------------------------------------------------ %%
+\enlargethispage{3\baselineskip}
+
+\section{Bold math symbols}
+
+The \cn{mathbf} command is commonly used to obtain bold Latin letters in
+math, but for most other kinds of math symbols it has no effect, or its
+effects depend unreliably on the set of math fonts that are in use. For
+example, writing
+\begin{verbatim}
+\Delta \mathbf{\Delta}\mathbf{+}\delta \mathbf{\delta}
+\end{verbatim}
+produces $\Delta \mathbf{\Delta}\mathbf{+}\delta
+\mathbf{\delta}$; the \cn{mathbf} has no effect on the plus
+sign or the small delta.
+% should be paragraph break; avoid overfull box on next line
+The \nipkg{amsmath} package therefore provides two additional commands,
+\cn{boldsymbol} and \cn{pmb}, that can be applied to other kinds of math
+symbols. \cn{boldsymbol} can be used for a math symbol that remains
+unaffected by \cn{mathbf} if (and only if) your current math font set
+includes a bold version of that symbol. \cn{pmb} can be used as a last
+resort for any math symbols that do not have a true bold version
+provided by your set of math fonts; \qq{pmb} stands for \qq{poor man's
+bold} and the command works by typesetting multiple copies of the symbol
+with slight offsets. The quality of the output is inferior, especially
+for symbols that contain any hairline strokes. When the standard default set of
+\latex/ math fonts are in use (Computer Modern), the only symbols that
+are likely to require \cn{pmb} are large operator symbols like \cn{sum},
+extended delimiter\index{delimiters} symbols, or the extra math symbols
+provided by the \pkg{amssymb} package \cite{amsfonts}.
+
+The following formula shows some of the results that are possible:
+\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}
+\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*}
+If you want to use only the \cn{boldsymbol} command without loading the
+whole \nipkg{amsmath} package, the \pkg{bm} package is recommended (this
+is a standard \latex/ package, not an AMS package; you probably have it
+already if you have a 1997 or newer version of \latex/).
+
+%% ------------------------------------------------------------------ %%
+
+\section{Italic Greek letters}
+
+For italic versions of the capital Greek letters, the following commands
+are provided:
+\begin{ctab}{rlrl}
+\cn{varGamma}& $\varGamma$& \cn{varSigma}& $\varSigma$\\
+\cn{varDelta}& $\varDelta$& \cn{varUpsilon}& $\varUpsilon$\\
+\cn{varTheta}& $\varTheta$& \cn{varPhi}& $\varPhi$\\
+\cn{varLambda}& $\varLambda$& \cn{varPsi}& $\varPsi$\\
+\cn{varXi}& $\varXi$& \cn{varOmega}& $\varOmega$\\
+\cn{varPi}& $\varPi$
+\end{ctab}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Error messages and output problems}
+
+\section{General remarks}
+
+This is a supplement to Chapter~8 of the \latex/ manual \cite{lamport}
+(first edition: Chapter~6). Appendix~B of the
+\textit{Companion}~\cite{tlc2} lists all \latex/ errors including those
+for \nipkg{amsmath} with likely causes for each.  For the reader's
+convenience, the set of error messages discussed here overlaps
+somewhat with the set in those references,
+but please be aware that we don't provide exhaustive coverage here.
+The error messages are arranged in alphabetical order, disregarding
+unimportant text such as |! LaTeX Error:| at the beginning, and
+nonalphabetical characters such as \qc{\\}. Where examples are given, we
+show also the help messages that appear on screen when you respond to an
+error message prompt by entering |h|.
+
+Remember that the important error message is the first for a given line.
+When running in nonstop mode, errors accumulate, and the first error
+may create a further error condition from which it is impossible to
+recover.  In such a situation processing will stop after 100~errors,
+so the reporting of genuine errors may not be complete, and it may be
+impossible to determine which reported errors are genuine and which
+are not.
+
+A final section discusses some output errors, i.e., instances where
+the printed document has something wrong but there was no \latex/
+error during typesetting.
+
+%% ------------------------------------------------------------------ %%
+%% **********
+\enlargethispage{-2\baselineskip}
+
+\section{Error messages}
+
+\begin{error}{\begin{split} won't work here.}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: \begin{split} won't work here.
+ ...
+
+l.8 \begin{split}
+
+? h
+\Did you forget a preceding \begin{equation}?
+If not, perhaps the `aligned' environment is what you want.
+?
+\end{verbatim}
+\errexpl
+The \env{split} environment does not construct a stand-alone displayed
+equation; it needs to be used within some other environment such as
+\env{equation} or \env{gather}.
+
+\end{error}
+
+\begin{error}{Erroneous nesting of equation structures}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Erroneous nesting of equation structures;
+(amsmath)                trying to recover with `aligned'.
+
+See the amsmath package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...
+
+l.260 \end{alignat*}
+                    \end{equation*}
+\end{verbatim}
+\errexpl
+The structures \env{align}, \env{alignat}, etc., are designed
+for top-level use and for the most part cannot be nested inside some
+other displayed equation structure. The chief exception is that
+\env{align} and most of its variants can be used inside the
+\env{gather} environment.
+\end{error}
+
+\begin{error}{Extra & on this line}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Extra & on this line.
+
+See the amsmath package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...
+
+l.9 \end{alignat}
+
+? h
+\An extra & here is so disastrous that you should probably exit
+ and fix things up.
+?
+\end{verbatim}
+\errexpl
+In an \env{alignat} structure the number of alignment points per line
+is dictated by the numeric argument given after |\begin{alignat}|.
+If you use more alignment points in a line it is assumed that you
+accidentally left out a newline command \cn{\\} and the above error is
+issued.
+\end{error}
+
+%% **********
+\enlargethispage{1.5\baselineskip}
+
+\begin{error}{Font OMX/cmex/m/n/7=cmex7 not loadable ...}
+\errexa
+\begin{verbatim}
+! Font OMX/cmex/m/n/7=cmex7 not loadable: Metric (TFM) file not found.
+<to be read again>
+                   relax
+l.8 $a
+      b+b^2$
+? h
+I wasn't able to read the size data for this font,
+so I will ignore the font specification.
+[Wizards can fix TFM files using TFtoPL/PLtoTF.]
+You might try inserting a different font spec;
+e.g., type `I\font<same font id>=<substitute font name>'.
+?
+\end{verbatim}
+\errexpl
+Certain extra sizes of some Computer Modern fonts that were formerly
+available mainly through the AMSFonts\index{amsfont@AMSFonts collection}
+distribution are considered part of standard \latex/ (as of June 1994):
+\fn{cmex7}\ndash \texttt{9}, \fn{cmmib5}\ndash \texttt{9}, and
+\fn{cmbsy5}\ndash \texttt{9}. If these extra sizes are missing on your
+system, you should try first to get them from the source where you
+obtained \latex/. If that fails, you could try getting the fonts from
+CTAN (e.g., in the form of Metafont\index{Metafont source files} source
+files, directory \nfn{/tex-archive/fonts/latex/mf}, or in PostScript
+Type 1 format, directory
+\nfn{/tex-archive/fonts/cm/ps-type1/bakoma}\index{BaKoMa fonts}\relax
+\index{PostScript fonts}).
+
+If the font name begins with \fn{cmex}, there is a special option%
+\index{options!behavior of particular options}
+\opt{cmex10} for the \nipkg{amsmath} package that provides a temporary
+workaround. I.e., change the \cn{usepackage} to
+\begin{verbatim}
+\usepackage[cmex10]{amsmath}
+\end{verbatim}
+This will force the use of the 10-point size of the \fn{cmex} font in
+all cases. Depending on the contents of your document this may be
+adequate.
+\end{error}
+
+\enlargethispage{1\baselineskip}
+\begin{error}{Improper argument for math accent}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Improper argument for math accent:
+(amsmath)                Extra braces must be added to
+(amsmath)                prevent wrong output.
+
+See the amsmath package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...
+
+l.415 \tilde k_{\lambda_j} = P_{\tilde \mathcal
+                                               {M}}
+?
+\end{verbatim}
+\errexpl
+Non-simple arguments for any \LaTeX{} command should be enclosed in
+braces. In this example extra braces are needed as follows:
+\begin{verbatim}
+... P_{\tilde{\mathcal{M}}}
+\end{verbatim}
+\end{error}
+
+\begin{error}{Math formula deleted: Insufficient extension fonts}
+\errexa
+\begin{verbatim}
+! Math formula deleted: Insufficient extension fonts.
+l.8 $ab+b^2$
+
+?
+\end{verbatim}
+\errexpl
+This usually follows a previous error |Font ... not loadable|; see the
+discussion of that error (above) for solutions.
+\end{error}
+
+\enlargethispage{1\baselineskip}
+\begin{error}{Missing number, treated as zero}
+\errexa
+\begin{verbatim}
+! Missing number, treated as zero.
+<to be read again>
+                   a
+l.100 \end{alignat}
+
+? h
+A number should have been here; I inserted `0'.
+(If you can't figure out why I needed to see a number,
+look up `weird error' in the index to The TeXbook.)
+
+?
+\end{verbatim}
+\errexpl
+There are many possibilities that can lead to this error. However, one
+possibility that is relevant for the \nipkg{amsmath} package is that you
+forgot to give the number argument of an \env{alignat} environment, as
+in:
+\begin{verbatim}
+\begin{alignat}
+ a&  =b&    c& =d\\
+a'& =b'&   c'& =d'
+\end{alignat}
+\end{verbatim}
+where the first line should read instead
+\begin{verbatim}
+\begin{alignat}{2}
+\end{verbatim}
+
+Another possibility is that you have a left bracket character |[|
+following a line break\index{line break} command \cn{\\} in a multiline
+construction such as \env{array}, \env{tabular}, or \env{eqnarray}.
+This will be interpreted by \latex/ as the beginning of an `additional
+vertical space request \cite[\S C.1.6]{lamport}, even if it occurs on the
+following line and is intended to be part of the contents. For example,
+\begin{verbatim}
+\begin{array}{c}
+a+b\\
+[f,g]\\
+m+n
+\end{array}
+\end{verbatim}
+To prevent the error message in such a case, you can
+add braces as discussed in the \latex/ manual \cite[\S C.1.1]{lamport}:
+\begin{verbatim}
+\begin{array}{c}
+a+b\\
+{[f,g]}\\
+m+n
+\end{array}
+\end{verbatim}
+or precede the bracketed expression by \cn{relax}.
+\end{error}
+
+\goodbreak
+
+\begin{error}{Missing \right. inserted}
+\errexa
+\begin{verbatim}
+! Missing \right. inserted.
+<inserted text>
+                \right .
+l.10 \end{multline}
+
+? h
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+\end{verbatim}
+\errexpl
+This error typically arises when you try to insert a
+line break\index{line break} or |&|\cnamp\ between a
+\cn{left}-\cn{right} pair of delimiters\index{delimiters} in any
+multiline environment, including \env{split}.
+\begin{verbatim}
+\begin{multline}
+AAA\left(BBB\\
+  CCC\right)
+\end{multline}
+\end{verbatim}
+There are two possible solutions: (1)~instead of using \cn{left} and
+\cn{right}, use `big' delimiters\index{delimiters!fixed size} of fixed
+size (\cn{bigl} \cn{bigr} \cn{biggl} \cn{biggr} \dots; see
+\secref{bigdel}); or (2)~use null delimiters\index{delimiters!null} to
+break up the \cn{left}-\cn{right} pair into parts for each line (or cell):
+\begin{verbatim}
+AAA\left(BBB\right.\\
+  \left.CCC\right)
+\end{verbatim}
+The latter solution may result in mismatched
+delimiter\index{delimiters!mismatched sizes} sizes;
+ensuring that they match requires using \cn{vphantom} in the segment
+that has the smaller delimiter (or possibly \cn{smash} in the segment
+that has the larger delimiter). In the argument of \cn{vphantom} put a
+copy of the tallest element that occurs in the other segment, e.g.,
+\begin{verbatim}
+xxx \left(\int_t yyy\right.\\
+  \left.\vphantom{\int_t} zzz ... \right)
+\end{verbatim}
+\end{error}
+
+\goodbreak
+
+\begingroup
+\lccode`M=`M
+\lccode`@=`\}
+\lowercase{\endgroup
+\begin{error}{Missing @ inserted}}
+
+\errexa
+\begin{verbatim}
+! Missing } inserted.
+<inserted text>
+                \right .
+l.10 \end{multline}
+
+? h
+I've inserted something that you may have forgotten.
+(See the <inserted text> above.)
+With luck, this will get me unwedged. But if you
+really didn't forget anything, try typing `2' now; then
+my insertion and my current dilemma will both disappear.
+\end{verbatim}
+\errexpl
+This error is often the result of using \verb+$+ within a multiline
+display environment.  Remove such \verb+$+ signs (except when they
+appear within \verb+\text{...}+).
+\end{error}
+
+\begin{error}{Old form `\pmatrix' should be \begin{pmatrix}.}
+\errexa
+\begin{verbatim}
+! Package amsmath Error: Old form `\pmatrix' should be
+                         \begin{pmatrix}.
+
+See the amsmath package documentation for explanation.
+Type  H <return>  for immediate help.
+ ...
+
+\pmatrix ->\left (\matrix@check \pmatrix
+                                         \env@matrix
+l.16 \pmatrix
+             {a&b\cr c&d\cr}
+? h
+`\pmatrix{...}' is old Plain-TeX syntax whose use is
+ill-advised in LaTeX.
+?
+\end{verbatim}
+\errexpl
+When the \nipkg{amsmath} package is used, the old forms of \cn{pmatrix},
+\cn{matrix}, and \cn{cases} cannot be used any longer because of naming
+conflicts. Their syntax did not conform with standard \LaTeX{} syntax
+in any case.
+\end{error}
+
+\enlargethispage{2\baselineskip}
+\begin{error}{Paragraph ended before \xxx was complete}
+\errexa
+\begin{verbatim}
+Runaway argument?
+
+! Paragraph ended before \multline was complete.
+<to be read again>
+                   \par
+l.100
+
+? h
+I suspect you've forgotten a `}', causing me to apply this
+control sequence to too much text. How can we recover?
+My plan is to forget the whole thing and hope for the best.
+?
+\end{verbatim}
+\errexpl
+This might be produced by a blank line between the |\begin| and |\end|.
+Another possibility is a misspelling in the |\end{multline}| command,
+e.g.,
+\begin{verbatim}
+\begin{multline}
+...
+\end{multiline}
+\end{verbatim}
+or by using abbreviations for certain environments, such as |\bal| and
+|\eal| for |\begin{align}| and |\end{align}|:
+\begin{verbatim}
+\bal
+...
+\eal
+\end{verbatim}
+For technical reasons that kind of abbreviation does not work with
+the more complex displayed equation environments of the \nipkg{amsmath} package
+(\env{gather}, \env{align}, \env{split}, etc.; cf.\ \fn{technote.tex}).
+\end{error}
+
+\begin{error}{Runaway argument?}
+See the discussion for the error message
+\texttt{Paragraph ended before \ncn{xxx} was complete}.
+\end{error}
+
+\begin{error}{Unknown option `xxx' for package `yyy'}
+\errexa
+\begin{verbatim}
+! LaTeX Error: Unknown option `intlim' for package `amsmath'.
+...
+? h
+The option `intlim' was not declared in package `amsmath', perhaps you
+misspelled its name. Try typing  <return>  to proceed.
+?
+\end{verbatim}
+\errexpl
+This means that you misspelled the option\index{options!unknown}
+name, or the package simply does not have an option that you expected
+it to have. Consult the documentation for the given package.
+\end{error}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Warning messages}
+
+\begin{error}{Cannot use `split' here}
+\errexa
+\begin{verbatim}
+Package amsmath Warning: Cannot use `split' here;
+(amsmath)                trying to recover with `aligned'
+\end{verbatim}
+\errexpl The \env{split} environment is designed to serve as the entire
+body of an equation, or an entire line of an \env{align} or \env{gather}
+environment. There cannot be any printed material before or
+after it within the same enclosing structure:
+\begin{verbatim}
+\begin{equation}
+\left\{ % <-- Not allowed
+\begin{split}
+...
+\end{split}
+\right. % <-- Not allowed
+\end{equation}
+\end{verbatim}
+\end{error}
+
+\begin{error}{Foreign command \over [or \atop or \above]}
+\errexa
+\begin{verbatim}
+Package amsmath Warning: Foreign command \over; \frac or \genfrac
+(amsmath)                should be used instead.
+\end{verbatim}
+\errexpl The primitive generalized fraction commands of \tex/\mdash
+\cs{over}, \cs{atop}, \cs{above}\mdash are deprecated when the
+\nipkg{amsmath} package is used because their syntax is foreign to \latex/
+and \nipkg{amsmath} provides native \latex/ equivalents. See
+\fn{technote.tex} for further information.
+\end{error}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Wrong output}
+
+\begin{erroro}{Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3}
+%
+This most likely means that you have the arguments for \cn{numberwithin}
+in reverse order:
+\begin{verbatim}
+\numberwithin{section}{equation}
+\end{verbatim}
+That means `print the section number as \textit{equation
+number}.\textit{section number} and reset to 1 every time an equation
+occurs' when what you probably wanted was the inverse
+\begin{verbatim}
+\numberwithin{equation}{section}
+\end{verbatim}
+\end{erroro}
+
+\begin{erroro}{The \cn{numberwithin} command had no effect on equation
+  numbers}
+%
+Are you looking at the first section in your document? Check the section
+numbers elsewhere to see if the problem is the one described in the
+previous item.
+\end{erroro}
+
+\begin{erroro}{Bracketed material disappears at the beginning of
+  \env{aligned} or \env{gathered}}
+%
+For most multiline equation environments, \nipkg{amsmath} disables the
+\latex/ convention to recognize a [bracketed] expression as an option%
+\index{ed env@\texttt{-ed} environments}%
+\index{options!positioning of \texttt{-ed} environments}%
+\index{options!space before \texttt{[}}
+when it occurs at the beginning of an environment or immediately
+following \cn{\\}, provided a space intervenes.  This fails at the
+beginning of \env{aligned} and \env{gathered}, and is a bug.  Insert
+\cn{relax} before the opening brace to restore the desired result.
+\end{erroro}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Additional information}
+
+\section{Compatibility with other packages}
+
+Several packages have already been mentioned as providing facilities
+lacking in \nipkg{amsmath}.
+
+\begin{itemize}
+\item \pkg{bm} is recommended for specifying bold math symbols.
+\item \pkg{mathtools} provides enhancements to some \nipkg{amsmath}
+  environments as well as additional compatible features, such as
+  \env{rcases} (similar to \env{cases} but with the brace on the
+  right-hand side), \env{multlined} (a subsidiary environment
+  comparable to \env{multline}), and \cn{shortintertext} (more
+  compact spacing than \cn{intertext}).  If \pkg{mathtools} is used,
+  \nipkg{amsmath} is loaded automatically, so it is not necessary
+  to load it separately.
+\item \pkg{unicode-math} provides commands for (most?)\ math symbols
+  in Unicode.
+%\item \pkg{empheq}
+\end{itemize}
+
+These packages have known problems when used with \nipkg{amsmath}.
+\begin{itemize}
+\item \pkg{lineno} may omit numbers before display environments.
+\item \pkg{breqn} redefines a number of commands from other packages,
+  so is best loaded after \nipkg{amsmath} and other packages with math
+  facilities.\\ \pkg{breqn} is not yet totally stable, and a number of
+  bugs have been reported via \texttt{tex.stackexchange} \cite{tex-sx}.
+\item \pkg{wasysym}, \pkg{mathabx} and perhaps other font packages also
+  define multiple integrals\index{integrals!multiple} using the same
+  names as \nipkg{amsmath}.
+\end{itemize}
+
+%% ------------------------------------------------------------------ %%
+
+\section{Converting existing documents}
+
+\subsection{Converting from plain \LaTeX{}}
+
+A \LaTeX{} document will typically continue to work the same in most
+respects if \verb'\usepackage{amsmath}' is added in the document
+preamble. By default, however, the \nipkg{amsmath} package suppresses page
+breaks inside multiple-line displayed equation structures such as
+\env{eqnarray}, \env{align}, and \env{gather}. To continue allowing page
+breaks inside \env{eqnarray} after switching to \nipkg{amsmath}, you will
+need to add the following line in your document preamble:
+\begin{verbatim}
+\allowdisplaybreaks[1]
+\end{verbatim}
+To ensure normal spacing around relation symbols, you might also want to
+change \env{eqnarray} to \env{align}, \env{multline}, or
+\env{equation}\slash\env{split} as appropriate.
+
+Most of the other differences in \nipkg{amsmath} usage can be considered
+optional refinements, e.g., using
+\begin{verbatim}
+\DeclareMathOperator{\Hom}{Hom}
+\end{verbatim}
+instead of \verb'\newcommand{\Hom}{\mbox{Hom}}'.
+
+\subsection{Converting from \texorpdfstring{\amslatex/}{amslatex} 1.1}
+See \fn{diffs-m.txt}.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Technical notes}
+The file \fn{technote.tex} contains some remarks on miscellaneous
+technical questions that are less likely to be of general interest.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Getting help}
+
+The \nipkg{amsmath} collection of packages is maintained by the \latex/
+team \cite{ltx-team}.  Bugs should be reported using the instructions
+at \url{https://www.latex-project.org/bugs/}.
+Please be careful not to confuse \nipkg{amsmath} with the AMS document
+classes (\cls{amsart}, \cls{amsbook}, and \cls{amsproc}), which
+automatically incorporate \nipkg{amsmath}; bugs in the classes should
+be reported directly to AMS, by email to \mail{tech-support@ams.org}.
+
+%Questions or comments regarding \nipkg{amsmath} and related packages
+%should be sent to:
+%\begin{infoaddress}
+%American Mathematical Society\\
+%Technical Support\\
+%Electronic Products and Services\\
+%P. O. Box 6248\\
+%Providence, RI 02940\\[3pt]
+%Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4080\\
+%Internet: \mail{tech-support@ams.org}
+%\end{infoaddress}
+%If you are reporting a problem you should include
+%the following information to make proper investigation possible:
+%\begin{enumerate}
+%\item The source file where the problem occurred, preferably reduced
+%  to minimum size by removing any material that can be removed without
+%  affecting the observed problem.
+%\item A \latex/ log file showing the error message (if applicable) and
+%  the version numbers of the document class and option files being used.
+%\end{enumerate}
+
+The online forum\index{online Q\,\&\,A forum} \texttt{tex.stackexchange}
+\cite{tex-sx} is a good source of answers to common questions on usage.
+Look first in the archive to see if your question has already been asked
+and answered; if it has not, post your own question.
+
+There is an online discussion group called \fn{comp.text.tex} \cite{ctt}
+that is a fairly good source of information about \latex/ and \tex/ in
+general. This group, formerly a Unix newsgroup, predates
+\texttt{tex.stackexchange} by decades, and the archives can be a useful
+resource for historical investigation.
+
+%% ------------------------------------------------------------------ %%
+
+\section{Of possible interest}\label{a:possible-interest}
+Information about obtaining AMSFonts or other \tex/-related
+software from the AMS web server at \fn{www.ams.org}
+can be obtained by sending an email request to
+\mail{tech-support@ams.org}.  This software is also posted on
+CTAN\index{CTAN (Comprehensive \TeX\ Archive Network)} and included
+in distributions based on \TeX\,Live\index{TEX Live@\TeX\,Live}.
+
+The \tex/ Users Group\index{TeX Users@\tex/ Users Group} is a nonprofit
+organization that publishes a journal
+(\journalname{TUGboat}\index{TUGboat@\journalname{TUGboat}}), holds
+meetings, and serves as a clearinghouse for general information about
+\tex/ and \tex/-related software.
+\begin{infoaddress}
+\tex/ Users Group\\
+PO Box 2311\\
+Portland, OR 97208-2311\\
+USA\\[3pt]
+Phone: +1-503-223-9994\\
+Email: \mail{office@tug.org}
+\end{infoaddress}
+Membership in the \tex/ Users Group is a good way to support continued
+development of free \tex/-related software. There are also many local
+\tex/ user groups in other countries; information about contacting a
+local user group can be gotten from the \tex/ Users Group.
+
+\index{amsmath@\texttt{amsmath} package|)}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newpage
+
+\begin{thebibliography}{10}
+\addcontentsline{toc}{chapter}{\bibname}
+
+\raggedright
+
+\bibsubhead{References in print}
+
+\bibitem{mil} George Gr\"atzer, \booktitle{More Math into \latex/}, fifth ed.,
+Springer, New York, 2016.
+% Deal with a line breaking problem
+%\begin{raggedright}
+%\bibitem{mil} G. Gr\"{a}tzer,
+%\emph{Math into \LaTeX{}: An Introduction to \LaTeX{} and AMS-\LaTeX{}}
+%  \url{https://www.ams.org/cgi-bin/bookstore/bookpromo?fn=91&arg1=bookvideo&itmc=MLTEX},
+%Birkh\"{a}user, Boston, 1995.\par
+%\end{raggedright}
+
+\bibitem{kn} Donald E. Knuth, \booktitle{The \tex/book},
+Addison-Wesley, Reading, MA, 1984.
+
+\bibitem{lamport} Leslie Lamport, \booktitle{\latex/: A document preparation
+system}, 2nd revised ed., Addison-Wesley, Reading, MA, 1994.
+
+\bibitem{tlc2} Frank Mittelbach, Michel Goossens, et al.,
+\booktitle{The \latex/ companion}, second ed., Addison-Wesley, Reading,
+MA, 2004.  This is now also available as an ebook, in both English and
+German; see \url{https://www.latex-project.org/help/books/}.
+The front matter, including the full Table of Contents, can be viewed
+online, from a link on the same page.
+%Michel Goossens, Frank Mittelbach, and Alexander Samarin,
+%\booktitle{The \latex/ companion}, Addison-Wesley, Reading, MA, 1994.
+%  [\emph{Note: The 1994 edition is not a reliable guide for the
+%    \nipkg{amsmath} package unless you refer to the errata for Chapter
+%    8\mdash file \fn{compan.err}, distributed with \LaTeX{}.}]
+
+\bibitem{msf} Frank Mittelbach and Rainer Sch\"opf,
+\textit{The new font family selection\mdash user
+interface to standard \latex/}, \journalname{TUGboat} \textbf{11},
+no.~2 (June 1990), pp.~297\ndash 305.
+
+\bibitem{jt} Michael Spivak, \booktitle{The joy of \tex/}, 2nd revised ed.,
+Amer. Math. Soc., Providence, RI, 1990.
+
+\bibsubhead{Package documentation}
+
+\bibitem{amshandbk}\booktitle{AMS author handbook}, separate versions
+for journal articles, monographs and proceedings articles, Amer. Math.
+Soc., Providence, RI, 2017;
+\url{https://www.ams.org/publications/authors/tex/author-handbook}.
+
+\bibitem{amsfonts}\booktitle{AMSFonts version \textup{2.2d}\mdash user's guide},
+Amer. Math. Soc., Providence, RI, 2002; distributed
+with the AMSFonts package
+\url{http://mirror.ctan.org/tex-archive/fonts/amsfonts/doc/amsfndoc.pdf}.
+
+\bibitem{amsthdoc}\booktitle{Using the \pkg{amsthm} Package}, version 2.20.3,
+Amer. Math. Soc., Providence, RI, 2017;
+\url{http://mirror.ctan.org/tex-archive/macros/latex/required/amscls/doc/amsthdoc.pdf}.
+
+\bibitem{mt} Morten H{\o}gholm, Lars Madsen, \booktitle{The \pkg{mathtools}
+package}, 2018;
+\url{http://mirror.ctan.org/tex-archive/macros/latex/contrib/mathtools/mathtools.pdf}.
+
+\bibsubhead{Online resources}
+
+\bibitem{ltx-team} The \latex/ Project, \url{https://www.latex-project.org}.
+
+\bibitem{ctt} Online discussion group \texttt{comp.text.tex},
+\url{https://groups.google.com/forum/\#!forum/comp.text.tex}.
+
+\bibitem{tex-sx} Online question and answer forum \texttt{tex.stackexchange},
+\url{https://tex.stackexchange.com}.
+
+\end{thebibliography}
+
+\printindex
+
+\end{document}
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