<gh_stars>1-10
%%% ====================================================================
%%%  @LaTeX-file{
%%%     filename        = "amsldoc.tex",
%%%     version         = "1.2b",
%%%     date            = "16-Feb-1995",
%%%     time            = "11:23:29 EST",
%%%     checksum        = "55866 3006 13763 113254",
%%%     author          = "American Mathematical Society",
%%%     address         = "American Mathematical Society,
%%%                        Technical Support,
%%%                        Electronic Products and Services,
%%%                        P. O. Box 6248,
%%%                        Providence, RI 02940,
%%%                        USA",
%%%     telephone       = "401-455-4080 or (in the USA and Canada)
%%%                        800-321-4AMS (321-4267)",
%%%     FAX             = "401-331-3842",
%%%     email           = "<EMAIL> (Internet)",
%%%     copyright       = "Copyright (C) 1995 American Mathematical Society,
%%%                        all rights reserved.  Copying of this file is
%%%                        authorized only if either:
%%%                        (1) you make absolutely no changes to your copy,
%%%                        including name; OR
%%%                        (2) if you do make changes, you first rename it
%%%                        to some other name.",
%%%     supported       = "yes",
%%%     keywords        = "latex, amslatex, ams-latex, amstex",
%%%     abstract        = "This file is the source file for the AMS-\LaTeX{}
%%%                        user's guide.",
%%%     docstring       = "The checksum field above contains a CRC-16
%%%                        checksum as the first value, followed by the
%%%                        equivalent of the standard UNIX wc (word
%%%                        count) utility output of lines, words, and
%%%                        characters.  This is produced by Robert
%%%                        Solovay's checksum utility.",
%%%  }
%%% ====================================================================
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1994/06/01]% LaTeX date must June 1994 or later
\documentclass[leqno,titlepage,openany]{amsldoc}

\title{\amslatex/ Version 1.2\\User's Guide}
\author{American Mathematical Society}
\date{January 1995}

%    Use the amsmath package and amscd package in order to print
%    examples.
\usepackage{amsmath}
\usepackage{amscd}
%\usepackage{syntonly}\syntaxonly

\makeindex % 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-intsm ams-la-tex
  amsl-doc ams-symb ams-texdf ams-text ams-xtra bmatrix bold-sym-bol
  cen-ter-tags curr-addr eqn-ar-ray idots-int int-lim-its latex
  make-title med-space mr-abbrev neg-med-space neg-thick-space
  neg-thin-space new-en-vi-ron-ment new-theo-rem no-ams-fonts
  no-int-lim-its no-name-lim-its over-left-arrow over-left-right-arrow
  over-right-arrow pmatrix ps-ams-fonts qed-sym-bol set-length side-set
  small-er subj-class tbinom the-equa-tion theo-rem theo-rem-style
  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
\let\vector=\vec
\renewcommand{\vec}[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}}

%    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
}

%    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]{---#1---}
\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{\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}}

\newtheorem{prop}{Proposition}

%    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}{}% empty for first unnumbered chapter
\makeatother

%    Include down to \section but not \subsection, in toc:
\setcounter{tocdepth}{1}

\DeclareMathOperator{\ix}{ix}
\DeclareMathOperator{\nul}{nul}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\xxx}{xxx}

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frontmatter

\maketitle
\tableofcontents
\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter*{What is `\amslatex/', and why would anyone want to use
it?}
\chaptermark{What is `\amslatex/'?}

\section*{What is `\amslatex/'?}

\leavevmode
\addcontentsline{toc}{chapter}{What is
  `\amslatex/', and why would anyone want to use it?}
\addcontentsline{toc}{section}{What is `\amslatex/'?}
%
The name \amslatex/ is used for convenience to describe a set of loosely
related files that are distributed together by the American Mathematical
Society. Basically they may be described as miscellaneous enhancements
to \latex/ for \emph{superior information structure of mathematical
documents} and \emph{superior printed output}. Because \amslatex/ is an
extension for \latex/, which in turn is a `macro
package'\gloss{macro package} for the \tex/ typesetting program, it
follows that in order to use any of the pieces of \amslatex/ you need to
have \tex/ and \latex/ installed first.

\latex/ by itself does a rather good job of typesetting mathematics,
compared to non-\tex/-based software; it doesn't add much, however, to
the basic set of mathematical capabilities that it adopted from the
Plain \tex/ macro package.

At the same time that \latex/ was being developed by Leslie
Lamport\index{Lamport, Leslie} (roughly 1982--1986), the American
Mathematical Society was throwing its resources into the development of
a different macro package known as \amstex/\index{amstex@\amstex/},
written by <NAME>\index{Spivak, Michael}. By 1987 or so it
became evident that \amstex/ and \latex/ had complementary feature sets:
\amstex/ focused on the typesetting of math formulas and on fine-tuning
typically done by publishers, and was relatively weak in other areas
(for example no automatic numbering or cross-reference facilities);
\latex/ focused on document structure and logical markup of text, and
had a comparatively limited set of features for dealing with math
formula contents. This situation led to dissatisfaction among both
\amstex/ and \latex/ users who saw desirable features tantalizingly out
of reach in the other macro package. So the American Mathematical
Society looked into the question of producing some sort of combination
of the two macro packages that would better serve mathematicians in
their writing tasks. The decision that was eventually taken was to graft
the mathematical capabilities of \amstex/ onto the base stock of \latex/
through an extension package: \amslatex/. Most of the programming work
was done by <NAME>\index{Mittelbach, Frank} and Rainer
Sch\"opf\index{Sch\"opf, Rainer} in 1989--1990 and version 1.0 of
\amslatex/ was released in mid-1990.

\section*{Why would a \latex/ user want to bother with \amslatex/?}

\leavevmode
\addcontentsline{toc}{section}{Why would a \latex/
user want to bother with \amslatex/?}
%
If you are just starting out as a \latex/ user, you'll probably have to
take our word for this (or the word of friends and colleagues), but:

If your writing contains a significant proportion of mathematics, and
you care about the quality of the printed results, then sooner or
later you'll find shortcomings in standard \latex/ and want to remedy
them. Chances are that at least the first few of the shortcomings you
encounter will be ones that are already addressed by an \amslatex/
package. If you want to have maximum mathematical typesetting power
ready at hand, rather than stop to cast about for a solution whenever
you run into some unusual demand in your writing, then \amslatex/ will
go a long way toward meeting your needs.

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 \amslatex/ 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.

\item A \cn{boldsymbol} command for printing bold versions of
individual symbols, including things like $\infty$ and lowercase Greek
letters.

\item An \pkg{amsthm} package that provides a useful \env{proof}
environment and some enhancements to the \cn{newtheorem} command:
support for multiple theorem styles in a single document and
for unnumbered theorem types.

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mainmatter
\renewcommand{\rhcn}{\thechapter. }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{How to use \amslatex/}

\section{Using an AMS package in a \latex/ document}\label{usepackage}

A `package'\gloss{package}
in \latex/ terminology is an extension written in such a
form that it can be used via the \cn{usepackage} command. Many of the
principal features of \amslatex/ are provided in separate packages so
that they can be used individually on demand. The \pkg{amsmath} package
is perhaps the single most noteworthy package, as it subsumes the
\pkg{amstext}, \pkg{amsbsy}, and \pkg{amsopn} packages, and provides a
number of other enhancements for mathematical typesetting.
The current list of packages is:
\begin{description}
\raggedright
\item[\pkg{amsmath}] Defines extra environments for multiline displayed
  equations, as well as a number of other enhancements for math.

\item[\pkg{amstext}] Provides a \cn{text} command for
  typesetting a fragment of text inside a display.

\item[\pkg{amsbsy}] Defines \cn{boldsymbol} and \cn{pmb} `poor man's
  bold' commands.

\item[\pkg{amsopn}] Provides \cn{DeclareMathOperator} for defining new
  `operator names' like \cn{sin} and \cn{lim}.

\item[\pkg{amsthm}] Provides a \env{proof} environment and
  extensions for the \cn{newtheorem} command.

\item[\pkg{amsintx}] Provides more descriptive command syntax for
  integrals and sums.

\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}.

\item[\pkg{upref}] Makes \cn{ref} print cross-reference numbers always
in an upright/roman font regardless of context.

\end{description}

\section{Options for the \pkg{amsmath} package}

The \pkg{amsmath} package has the following options:
\begin{description}

\item[\opt{centertags}] (default) For a split equation, place equation
numbers\index{equation numbers!vertical placement} vertically centered
on the total height of the equation.

\item[\opt{tbtags}] `Top-or-bottom tags': For a split equation, place
equation numbers\index{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---$\prod$, $\coprod$, $\bigotimes$,
$\bigoplus$, and so forth---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}

To use one of these package options, put the option name in the optional
argument of the \cn{usepackage} command---e.g.,
\verb"\usepackage[intlimits]{amsmath}".

The \pkg{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{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}

For symmetry there should perhaps be a \verb"centereqn" option as well,
to balance with \verb"fleqn", but as things currently stand there
doesn't seem to be a genuine need for it.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Displayed equations (\pkg{amsmath} package)}

\section{Introduction}
The \pkg{amsmath} package provides a number of additional displayed
equation structures\index{displayed
equations}\index{equations|see{displayed equations}} beyond the basic
\env{equation} and \env{eqnarray} environments provided in basic
\latex/. The augmented set includes:
\begin{verbatim}
  equation    align
  gather      flalign
  multline    alignat
  split
\end{verbatim}
(Although the standard \env{eqnarray} environment remains available,
\env{align} or \env{split} are recommended instead.)

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 \cn{\\}; you can also
override\index{equation numbers!overriding} it with a tag of your own
using \cn{tag}|{|\<label>|}|, where \<label> means arbitrary text such
as |$*$| or |ii| used to ``number'' the equation. 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 \pkg{amsmath}
alignment structures. Some examples of the use of \cn{tag} may be found
in the \amslatex/ sample files \fn{testmath.tex} and \fn{subeqn.tex}.

\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}&
  a_{12}& =b_{12}\\
a_{21}& =b_{21}&
  a_{22}& =b_{22}+c_{22}
\end{flalign*}
\end{verbatim}
\producing
\begin{flalign*}
a_{11}& =b_{11}&
  a_{12}& =b_{12}\\
a_{21}& =b_{21}&
  a_{22}& =b_{22}+c_{22}
\end{flalign*}
\end{eqxample}
\end{table}

\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}.}

\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}. Intermediate lines will be centered
independently within the display width. However, it's possible to force
a line 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 using \latex/'s
\cn{setlength} and \cn{addtolength} commands. If the \env{multline}
contains more than two lines, any lines other than the first and last
will be centered individually between the margins (except when the
\opt{fleqn} option is in effect).

\section{Split equations with alignment}

Like \env{multline}, the \env{split} environment is for {\em 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, as usual. In addition, unlike the other
\pkg{amsmath} equation structures, the \env{split} environment provides
no numbering, because it is intended to be used 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}

\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}).

\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 space between equation
columns to be explicitly specified. Here the number of equation
columns must also be specified (where the number of `columns' is
calculated as $(1+\text{\&}_{\max})/2$ with $\text{\&}_{\max} =
\text{maximum}$ number of \qc{\&} markers on any line).
\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}

\section{Alignment building blocks}

Some other equation alignment environments, such as \env{aligned} and
\env{gathered}, construct self-contained units that
can be used inside of other expressions, or set side-by-side. These
environments take an optional argument to specify their vertical
positioning with respect to the material on either side. The default is
`middle' placement with the vertical midpoint of the total unit
falling on the math axis%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{The height of the cross-bar in the $+$ symbol.}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
. For example:
\begin{equation*}
\begin{aligned}
\alpha&=\alpha\alpha\\
\beta&=\beta\beta\beta\beta\beta\\
\gamma&=\gamma
\end{aligned}
\qquad\text{versus}\qquad
\begin{aligned}[t]
\delta&=\delta\delta\\
\eta&=\eta\eta\eta\eta\eta\eta\\
\varphi&=\varphi
\end{aligned}
\end{equation*}
%
\pagebreak

\begin{verbatim}
\begin{equation*}
\begin{aligned}
\alpha&=\alpha\alpha\\
\beta&=\beta\beta\beta\beta\beta\\
\gamma&=\gamma
\end{aligned}
\qquad\text{versus}\qquad
\begin{aligned}[t]
\delta&=\delta\delta\\
\eta&=\eta\eta\eta\eta\eta\eta\\
\varphi&=\varphi
\end{aligned}
\end{equation*}
\end{verbatim}

``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 \pkg{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} and the embedded math.

\section{Adjusting tag placement}

Placing equation numbers can be a rather complex problem in multiline
displays. The environments of the \pkg{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. Even so,
difficulties in accurately calculating the profile of an equation can
occasionally result in a number placement that doesn't look right. So
there is a \cn{raisetag} command provided to adjust the vertical
position of the current equation number. To move a particular number up
by six points, write |\raisetag{6pt}|. (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>|]| command to get extra
vertical space between lines in all the \pkg{amsmath} displayed
equation environments, as is usual in \latex/.  Unlike \env{eqnarray},
the \pkg{amsmath} environments don't allow \ii{page breaks} between
lines, unless \cn{displaybreak} or \env{allowdisplaybreaks} is used.
The philosophy is that page breaks in such situations should receive
individual attention from the author. \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
between 0 and 4 denoting the desirability of the pagebreak.
|\displaybreak[0]| means ``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 multi-line equation, then you might put
\cn{allowdisplaybreaks} in the preamble of your document. An optional
argument 1--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.

\section{Textual interjections within 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
display alignment. 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 ``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}

\section{Equation numbering}

\subsection{Numbering hierarchy}
In \latex/ if you wanted to have equations numbered within
sections---that is, have
equation numbers (1.1), (1.2), \dots, (2.1), (2.2),
\dots, in sections 1, 2, and so forth---you could redefine
\cn{theequation} as suggested in the \latex/ manual \cite[\S6.3, \S C.8.4]{lm}:
\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 \pkg{amsmath} package provides a command\index{equation
numbers!hierarchy} \cn{numberwithin}. To have equation numbering tied to
section numbering, with automatic reset of the equation counter, the
command would be
\begin{verbatim}
\numberwithin{equation}{section}
\end{verbatim}

\subsection{Cross references to equation numbers}

To make cross-references to equations easier, an \cn{eqref}
command\index{equation numbers!cross-references} is provided. This
automatically supplies the parentheses around the equation number, and
adds an italic correction if necessary. To refer to an equation that was
labeled with the label |e:baset|, the usage would be |\eqref{e:baset}|.

\subsection{Subordinate numbering sequences}

The \pkg{amsmath} package provides also a \env{subequations}
environment\index{equation numbers!subordinate numbering} to make it
easy to number equations in a particular 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]{lm}. For example,
redefining \cn{theequation} as follows will produce roman numerals.
\begin{verbatim}
\begin{subequations}
\renewcommand{\theequation}{\theparentequation \roman{equation}}
...
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Miscellaneous mathematics features (\pkg{amsmath} package)}
\chaptermark{Miscellaneous mathematics features}

\section{Matrices}\label{ss:matrix}

The \pkg{amsmath} package provides some environments for
matrices\index{matrices} beyond the basic \env{array} environment of
\latex/. The \env{pmatrix}, \env{bmatrix}, \env{vmatrix} and
\env{Vmatrix} have (respectively) $(\,)$, $[\,]$, $\lvert\,\rvert$, and
$\lVert\,\rVert$ 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 must resort to \env{array}.)

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 must be provided; there are no |p|,|b|,|v|,|V|
versions of \env{smallmatrix}. 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, 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 \pkg{amsmath} package slightly extends the set of math
spacing\index{horizontal space!in math mode} 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}{llllll}
Abbrev.& Spelled out& Example & Abbrev.& Spelled out& Example\\
\hline
\cn{\,}& \cn{thinspace}& \spx{\,}&
  \cnbang& \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{Over and under arrows}

Basic \latex/ provides \cn{overrightarrow} and \cn{overleftarrow}
commands. Some additional over and under arrow commands are provided
by the \pkg{amsmath} package to fill out the set:

\begin{tabbing}
\qquad\=\ncn{overleftrightarrow}\qquad\=\kill
\> \cn{overleftarrow}    \> \cn{underleftarrow} \+\\
   \cn{overrightarrow}    \> \cn{underrightarrow} \\
   \cn{overleftrightarrow}\> \cn{underleftrightarrow}
\end{tabbing}

\section{Dots}

When the \pkg{amsmath} package is used, ellipsis dots should normally be
typed as \cn{dots}. Placement (on the baseline or centered) is
determined by whatever follows the \cn{dots}. If the next thing is a
plus sign or other binary symbol, the dots will be centered; if it's
any other kind of symbol, they will be on the baseline.

If the dots fall at the end of a math formula, the next thing is
something like \cn{end} or |\)| or |$|, which does not give any
information about how to place the dots.  Then you must help by using
\cn{dotsc} for ``dots with commas,'' or \cn{dotsb} for ``dots with
binary operators/relations,'' or \cn{dotsm} for ``multiplication dots,''
or \cn{dotsi} for ``dots with integrals.'' For example, the input
\begin{verbatim}
  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{verbatim}
will produce low dots in the first instance and centered dots
in the others, with the spacing on either side of the dots
nicely adjusted:
\begin{quotation}
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{quotation}

Specifying dots this way, in terms of their meaning rather than in
terms of their visual placement, is in keeping with the general
philosophy of \latex/ and makes documents more easily adaptable to
different conventions.

\section{Nonbreaking dashes}

A command \cn{nobreakdash} is provided to suppress the possibility
of a linebreak after the following hyphen or dash. For example, if you
write `pages 1--9' as |pages 1\nobreakdash--9| then a linebreak will
never occur between the dash and the 9. You can also use
\cn{nobreakdash} to prevent undesirable hyphenations in combinations
like |$p$-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":
\newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}}
\end{verbatim}
The last example shows how to prohibit a linebreak after the hyphen but
allow normal hyphenation in the following word. (It suffices to add a
zero-width space after the hyphen.)

\section{Accents in math}

The following accent commands\index{accents!double} automatically give
good positioning of double accents:
\begin{verbatim}
  \Hat    \Check  \Tilde  \Acute  \Grave  \Dot  \Ddot
  \Breve  \Bar    \Vec
\end{verbatim}
With the usual non-capitalized math accent commands,
the second accent will sometimes be askew; for example:
$\hat{\hat{A}}$ (\cn{hat}|{\hat{A}}|). With the \pkg{amsmath} package, if
you type \cn{Hat}|{\Hat{A}}| (using the capitalized form for both
accents) the second accent will be better positioned: $\Hat{\Hat{A}}$.

This double accent operation is complicated and tends to slow down the
processing of a document. If your document contains many double
accents, you may wish to use the \pkg{amsxtra} package, which provides
an \cn{accentedsymbol} command. \cn{accentedsymbol} is a sort of
hybrid of \cn{newcommand} and \cn{savebox}; you use it in the preamble
of your document to store the result of the double accent command in a
`box' for quick retrieval.
\begin{verbatim}
  \accentedsymbol{\Ahathat}{\Hat{\Hat A}}
\end{verbatim}

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/.

\section{Roots}

In ordinary \latex/ the placement of root indices is sometimes not so
good: $\sqrt[\beta]{k}$ (|\sqrt||[\beta]{k}|).  In the
\pkg{amsmath} package \cn{leftroot} and \cn{uproot} allow you to adjust
the position of the root:
\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{Extensible arrows}

\cn{xleftarrow} and \cn{xrightarrow} produce
arrows\index{arrows!extensible} that extend automatically to accommodate
unusually wide subscripts or superscripts. These commands take one
optional argument (the subscript) and one mandatory argument (the
superscript, possibly empty):
\begin{equation}
A\xleftarrow{n+\mu-1}B \xrightarrow[T]{n\pm i-1}C
\end{equation}
\begin{verbatim}
  \xleftarrow{n+\mu-1}\quad \xrightarrow[T]{n\pm i-1}
\end{verbatim}

\section{Affixing symbols to other symbols}

\latex/ provides \cn{stackrel} for placing a
superscript\index{subscripts and superscripts} above a binary relation.
In the \pkg{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{Disallowing primitive \tex/ fraction
  commands}\label{badgfrac}

The six generalized fraction commands \cn{over}, \cn{overwithdelims},
\cn{atop}, \cn{atopwithdelims}, \cn{above}, \cn{abovewithdelims} are
expressly forbidden by the \pkg{amsmath} package, as their syntax is
decidedly out of place in \latex/; use of the forms \cn{frac},
\cn{binom}, \cn{genfrac}, and variants is required.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{%
Not only is the unusual syntax of the primitive \tex/ fraction commands
rather out of place in \latex/, but furthermore that syntax seems to be
solely responsible for one of the most significant flaws in \tex/'s
mathematical typesetting capabilities: the fact that the current
mathstyle at any given point in a math formula cannot be determined
until the end of the formula, because of the possibility that a
following generalized fraction command will change the mathstyle of the
\emph{preceding} material. As the side effects are a bit technical in
nature, they are discussed in \fn{technote.tex} rather than here.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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---numerator and
denominator---and typesets them in normal fraction form. The
\pkg{amsmath} package provides also \cn{dfrac} and \cn{tfrac} as
convenient abbreviations for |{\displaystyle\frac| |...| |}|
and\index{textstyle@\cn{textstyle}}\relax
\index{displaystyle@\cn{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 $\binom{n}{k}$
\pkg{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 (as seen in
\cn{binom}); the third is a line thickness override (\cn{binom} uses
this to set the fraction line thickness to 0---i.e., invisible); and the
fourth argument is a mathstyle override: integer values 0--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}.

\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}.

\section{Smash options}

The command \cn{smash} is used to typeset a subformula and
give it an effective height and depth of zero, which is sometimes
useful in adjusting the subformula's position with respect to adjacent
symbols. With the \pkg{amsmath} package \cn{smash} has optional arguments
|t| and |b|, because occasionally it is advantageous to be able to
``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}

\subsection{Delimiter sizes}\label{bigdel}

A subject that escapes mention in the \latex/ book is how to control the
size of large delimiters\index{delimiters!size adjustments} if the
automatic sizing done by \cn{left} and \cn{right} produces
unsatisfactory results. The automatic sizing 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.
\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&
  $\displaystyle(b)(\frac{c}{d})$&
  $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$&
  $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$&
  $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$&
  $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$&
  $\displaystyle\Biggl(b\Biggr)\Biggl(\frac{c}{d}\Biggr)$
\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 \pkg{amsmath} package they are.

\subsection{Vertical bar notations}

The \pkg{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
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, at least not without a significant cost in processing speed,
and even then not without falling somewhat short of human readers'
abilities. 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
double-bar command \cn{\|}. This immediately rules out the use of \qc{|}
and \ncn{\|}\index{"|@\cn{"\"|}} for delimiters, as in the notations for
absolute value or norm, 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}
\newcommand{\abs}[1]{\lvert#1\rvert}
\newcommand{\norm}[1]{\lVert#1\rVert}
\end{verbatim}
whereupon the document would contain |\abs{z}| to produce $\lvert
z\rvert$ and |\norm{v}| to produce $\lVert v\rVert$.

\chapter{Operator names (\pkg{amsopn}, \pkg{amsmath} packages)}

\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
\pkg{amsopn} package provides a general mechanism for defining new
`operator names'. As the \pkg{amsopn} package is loaded internally by
the \pkg{amsmath} package, the following features are available there
also. 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
space!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.

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}

A few special operator names are predefined by the
\pkg{amsopn} package: \cn{varinjlim}, \cn{varprojlim}, \cn{varliminf},
and \cn{varlimsup}:
\begin{ctab}{rl}
\cn{varlimsup}&  $\displaystyle\varlimsup_{n\rightarrow\infty}
  \mathcal{Q}(u_n,u_n-u)\le0$\\[5pt]
\cn{varliminf}&  $\displaystyle\varliminf_{n\rightarrow\infty}
  \left\lvert a_{n+1}\right\rvert/\left\lvert a_n\right\rvert=0$\\[5pt]
\cn{varinjlim}&  $\displaystyle\varinjlim (m_i^\lambda)^*\le0$\\[5pt]
\cn{varprojlim}& $\displaystyle\varprojlim_{p\in S(A)}A_p\le0$
\end{ctab}

\section{\cn{mod} and its relatives}

Commands \cn{mod}, \cn{bmod}, \cn{pmod}, \cn{pod} are provided by the
\pkg{amsopn} package to deal with the special spacing conventions of
``mod'' notation. \cn{bmod} and \cn{pmod} are available in \latex/, but
with the \pkg{amsopn} 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 ``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}

\chapter{The \cn{text} command (\pkg{amstext}, \pkg{amsmath} packages)}

The \cn{text} command is defined by the \pkg{amsmath} package through a
subordinate package \pkg{amstext} (which can also be used independently
if desired). 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 |..._{\text{word or phrase}}|, which is slightly
easier than the \cn{mbox} equivalent: |..._{\mbox{\scriptsize| |word|
|or| |phrase}}|. The other advantage is the more descriptive name.
\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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The \cn{boldsymbol} command (\pkg{amsbsy}, \pkg{amsmath}
packages)}

The \cn{boldsymbol} and \cn{pmb} commands\index{math fonts!bold symbols}
are defined by the \pkg{amsbsy} package (also loaded by \pkg{amsmath}).
The \cn{boldsymbol} command is used to obtain bold numbers and other
nonalphabetic symbols, as well as bold Greek letters, which cannot be
made bold via the \cn{mathbf} command.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{Actually, depending on which font set you use,
\cn{mathbf} may---inconsistently---work for cap Greek letters but not
for lowercase.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It can also be used to obtain bold math italic letters; compare the
results of |M|, |\mathbf{M}| and |\boldsymbol{M}|:
$M\mathbf{M}\boldsymbol{M}$.

The availability of bold symbols varies on different systems depending
on whether or not suitable fonts are installed. The \cn{boldsymbol}
command should usually work fine for the common math symbols at 10pt
size or larger, but if you find that it is not having the desired
effect for a particular symbol, you could either (a)~verify that the
necessary fonts are available and properly installed; or (b)~use
\cn{pmb}: ``poor man's bold'', which works by printing multiple copies
of the same symbol with slight offsets.
\begin{equation}
A_\infty + \pi A_0
\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
  \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
\end{equation}
\begin{verbatim}
A_\infty + \pi A_0
\sim \mathbf{A}_{\boldsymbol{\infty}} \boldsymbol{+}
  \boldsymbol{\pi} \mathbf{A}_{\boldsymbol{0}}
\sim\pmb{A}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{A}_{\pmb{0}}
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Integrals and sums (\pkg{amsmath}, \pkg{amsintx} packages)}

\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.

\section{Multiline subscripts and superscripts}

The \cn{substack} command can be used to produce a multiline subscript
or superscript:\index{subscripts and superscripts!multi-line}\relax
\index{superscripts|see{subscripts and superscripts}} for example
\begin{verbatim}
\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
\end{verbatim}
produces a two-line subscript underneath the sum:
\begin{equation}
\sum_{\substack{0\le i\le m\\ 0<j<n}} P(i,j)
\end{equation}
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{equation}
\sum_{\begin{subarray}{l}
        i\in \Lambda\\ 0<j<n
      \end{subarray}}
 P(i,j)
\end{equation}
\begin{verbatim}
\sum_{\begin{subarray}{l}
        i\in\Lambda\\ 0<j<n
      \end{subarray}}
 P(i,j)
\end{verbatim}

\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 $\prod$. 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---indeed, without \cn{sideset},
it would be downright difficult. With \cn{sideset}, you can write
\begin{verbatim}
  \sideset{}{'}\sum_{n<k,\;\text{$n$ odd}} nE_n
\end{verbatim}
to get
\begin{equation}
\sideset{}{'}\sum_{n<k,\;\text{$n$ odd}} nE_n
\end{equation}
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{verbatim}
  \sideset{_*^*}{_*^*}\prod
\end{verbatim}
producing
\begin{equation}
\sideset{_*^*}{_*^*}\prod
\end{equation}

\section{The \pkg{amsintx} package}

The \pkg{amsintx} package is an experimental package that provides
variants of the \cn{int}\index{integrals} and \cn{sum}\index{sums}
commands to better mark the boundaries of the quantity being summed or
integrated. Some commands for differential notation\index{differential
notation} are also provided. If you are interested in this possibility,
run \latex/ on the documentation file \fn{amsintx.dtx} to get the most
up-to-date information on usage.

\chapter{Commutative diagrams (\pkg{amscd} package)}\label{s:commdiag}

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
\xypic/ (see \secref{xypic}), 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 ``sidescripts''.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{tlc}. Many users of \amslatex/ also obtain an auxiliary
collection of math fonts\index{AMSFonts collection} known as `AMSFonts'.
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 are available through the packages \pkg{amsfonts} and
\pkg{eucal}, if the requisite fonts are installed on your system (see
\secref{amsfonts}).

\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{\vec}[1]{\mathbf{#1}}
\end{verbatim}
whereupon%
\footnote{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If you actually tried this example you would discover that the command
\cn{vec} is already defined. It produces a different sort of notation
for vectors: a small over-arrow $\vector{x}$. The solution is to use
\cn{renewcommand} (if you expect that you will never need the
over-arrow version of the notation) or to choose a different name for
your new vector command.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
you can write |\vec{a} + \vec{b}| to produce $\vec{a} +
\vec{b}$.
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 \cn{vec}; 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 \cn{field} command and then
define \ncn{C} and \ncn{R} in terms of that command:
\begin{verbatim}
\usepackage{amsfonts}% to get the \mathbb alphabet
\newcommand{\field}[1]{\mathbb{#1}}
\newcommand{\C}{\field{C}}
\newcommand{\R}{\field{R}}
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Theorems and related structures (\pkg{amsthm} package)}
\label{amsthm}
\chaptermark{Theorems and related structures}%

\section{Introduction}

The \pkg{amsthm} package provides an enhanced version of the \latex/
command \cn{newtheorem} for defining theorem-like environments. The
\pkg{amsthm} version of the \cn{newtheorem} command recognizes a
\cn{theoremstyle} specification (as in Mittelbach's \pkg{theorem}
package) and has a |*| form for defining unnumbered environments. The
\pkg{amsthm} package also defines a \env{proof} environment that
automatically adds a Q.E.D. symbol at the end. AMS document classes
automatically load the \pkg{amsthm} package, so everything described
here applies to them as well. An example file \fn{thmtest.tex} is
provided in the \amslatex/ distribution.

\section{The \cn{newtheorem} command}

In mathematical research articles and books, theorems\index{theorems}
and proofs\index{proofs} are among the most common elements, but authors
also use many others that fall in the same general class: lemmas,
propositions, axioms, corollaries, conjectures, definitions, remarks,,
cases, steps, and so forth. As these elements form a slice of the text
stream with well-defined boundaries, they are naturally handled in
\latex/ as environments. But \latex/ document classes normally do not
provide predefined environments for theorem-like elements because
(a)~that would make it difficult for authors to exercise the necessary
control over the automatic numbering, and (b)~the variety of such
elements is so wide that it's just not possible for a document class to
provide every one that will ever be needed. Instead there is a command
\cn{newtheorem}, similar to \cn{newenvironment} in effect, that makes it
easy for authors to set up the elements required for a particular
document.

The \cn{newtheorem} command has two mandatory arguments; the first one
is the environment name that the author would like to use for this
element; the second one is the heading text. For example,
\begin{verbatim}
\newtheorem{lem}{Lemma}
\end{verbatim}
means that instances in the document of
\begin{verbatim}
\begin{lem} Text text ... \end{lem}
\end{verbatim}
will produce
\[\makebox[.8\columnwidth]{%
  \textbf{Lemma 1.} \textit{Text text \dots}\hfill}\]
where the heading consists of the specified text `Lemma' and an
automatically generated number and punctuation.

If \cn{newtheorem*} is used instead of \cn{newtheorem} in the above
example, there will not be any automatic numbers generated for
any of the lemmas in the document. This form of the command can be
useful if you have only one lemma and don't want it to be
numbered; more often, though, it is used to produce a special named
variant of one of the common theorem types. For example, if you have a
lemma whose name should be `Klein's Lemma' instead of `Lemma' +
number, then the statement
\begin{verbatim}
\newtheorem*{KL}{Klein's Lemma}
\end{verbatim}
would allow you to write
\begin{verbatim}
\begin{KL} Text text ... \end{KL}
\end{verbatim}
and get the desired output.

\section{Numbering modifications}

In addition to the two mandatory arguments, \cn{newtheorem} has two
mutually exclusive optional arguments. These affect the
sequencing\index{theorems!numbering} and hierarchy of the numbering.

By default each kind of theorem-like environment is numbered
independently. Thus if you have three lemmas and two theorems
interspersed, they will be numbered something like this: Lemma 1, Lemma
2, Theorem 1, Lemma 3, Theorem 2. If you want lemmas and theorems to
share the same numbering sequence---Lemma 1, Lemma 2, Theorem 3, Lemma
4, Theorem 5---then you should indicate the desired relationship as
follows:
\begin{verbatim}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\end{verbatim}
The optional argument |[thm]| in the second statement means that
the \texttt{lem} environment should share the \texttt{thm} numbering
sequence instead of having its own independent sequence.

To have a theorem-like environment numbered subordinately within a
sectional unit---e.g., to get propositions numbered Proposition 2.1,
Proposition 2.2, and so on in Section 2---put the name of the parent
unit in square brackets in final position:
\begin{verbatim}
\newtheorem{prop}{Proposition}[section]
\end{verbatim}
With the optional argument |[section]|, the |prop| counter will be reset
to 0 whenever the parent counter |section| is incremented.

\section{Changing styles for theorem-like environments}

\subsection{The \cn{theoremstyle} command}

The \pkg{amsthm} package supports the notion of a current theorem style,
which determines what will be produced by a given \cn{newtheorem}
command. The three theorem styles provided---|plain|,\index{plain
theo@\texttt{plain} theorem style}\relax \index{definition
theo@\texttt{definition} theorem style}\relax \index{remark
theo@\texttt{remark} theorem style} |definition|, and |remark|---receive
different typographical treatment that gives them visual emphasis
corresponding to their relative importance. The details of this
typographical treatment may vary depending on the document class, but
typically the |plain| style produces italic body text, while the other
two styles produce roman body text.

To create new theorem-like environments in the different styles,
divide your \cn{newtheorem} commands into groups and preface each
group with the appropriate \cn{theoremstyle}. If no \cn{theoremstyle}
command is given, the style used will be \texttt{plain}. Some examples:
\begin{verbatim}
\theoremstyle{plain}% default
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem*{cor}{Corollary}
\newtheorem*{KL}{Klein's Lemma}

\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]
\newtheorem{conj}{Conjecture}[section]
\newtheorem{exmp}{Example}[section]
\end{verbatim}
\pagebreak

\begin{verbatim}
\theoremstyle{remark}
\newtheorem*{rem}{Remark}
\newtheorem*{note}{Note}
\newtheorem{case}{Case}
\end{verbatim}

\subsection{Number swapping}

A not uncommon style variation for theorem heads is to have the theorem
number on the left, at the beginning of the heading, instead of on the
right. As this variation is usually applied across the board regardless
of individual \cn{theoremstyle} changes, number-swapping is done by
placing a \cn{swapnumbers} command at the beginning of the list of
\cn{newtheorem} statements that should be affected. For example:
\begin{verbatim}
\swapnumbers
\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\theoremstyle{remark}
\newtheorem{rem}{Remark}
\end{verbatim}
After the above declarations, theorem and remark heads will be printed
in the form \textbf{1.4~Theorem.}, \textit{9.1.~Remark.}

\subsection{Further customization possibilities}

More extensive customization capabilities are provided by the
\pkg{amsthm} package in the form of a \cn{newtheoremstyle} command and a
mechanism for using package options to load custom theoremstyle
definitions. As these capabilities are somewhat beyond the needs of the
average user, discussion of the details is consigned to the example file
\fn{thmtest.tex} and to the commentary in \fn{amsthm.dtx}.

\section{Proofs}

A predefined \env{proof} environment provided by the \pkg{amsthm}
package produces the heading ``Proof'' with appropriate spacing and
punctuation. The proof environment is primarily intended for short
proofs, no more than a page or two in length; longer proofs are
usually better done as a separate \cn{section} or \cn{subsection} in
your document.

A `Q.E.D.' symbol, \qedsymbol, is automatically appended at the end of
a \env{proof} environment. To substitute a different end-of-proof
symbol, use \cn{renewcommand} to redefine the command \cn{qedsymbol}.
For a long proof done as a subsection or section instead of with the
\env{proof} environment, you can obtain the symbol and the usual
amount of preceding space by using \cn{qed}.

Placement of the Q.E.D. symbol can be problematic if the last part of
a \env{proof} environment is a displayed equation or list environment
or something of that nature. Adequate results can sometimes be
obtained by using \cn{qed} at the appropriate spot and then undefining
\cn{qed} just before the end of the proof.  (The effect will be
automatically localized to the current proof by normal \latex/ scoping
rules.)  For example:
\begin{verbatim}
\begin{proof}
...
\begin{equation}
G(t)=L\gamma!\,t^{-\gamma}+t^{-\delta}\eta(t) \qed
\end{equation}
\renewcommand{\qed}{}\end{proof}
\end{verbatim}

An optional argument of the proof environment allows you to substitute
a different name for the standard ``Proof''. If you want
the proof heading to be, say, ``Proof of the Main Theorem'', then write
\begin{verbatim}
\begin{proof}[Proof of the Main Theorem]
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\appendix
\renewcommand{\chapnum}[1]{---Appendix\ #1---}
\renewcommand{\rhcn}{Appendix \thechapter. }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Installation instructions}\label{install}

\section{Introduction}\label{ctan}

To use version 1.2 of \amslatex/ it is necessary for you to have a
recent version of \latex/ (June 1994 or later, `\LaTeXe'). If you're not
sure about the version, look at the startup message that is printed on
screen and in the \tex/ log when you run \latex/. It should mention the
\latex/ version number and date somewhere in the first ten lines. If
your version of \latex/ is older than June 1994, we suggest getting the
latest version from the Comprehensive \tex/ Archive
Network\index{Comprehensive \tex/ Archive Network|see{CTAN}}
(\ii{CTAN}), directory \nfn{tex-archive/macros/latex}, ftp addresses
\fn{ftp.shsu.edu} (US), \fn{ftp.dante.de} (Germany), or
\fn{ftp.tex.ac.uk} (UK). If ftp file transfer is not an option for you,
contact the source from which you originally obtained \latex/.

\section{Putting files in a suitable place on your system}

See the \fn{READ.ME} file for possible updates about \ii{installation}
procedures.

There are two `areas' (directories or folders) on your system that are
involved in installing \amslatex/: an \amslatex/ source files area,
and a \latex/ input files area. All files in the \fn{inputs}
subdirectory of the \amslatex/ distribution should be placed in the
\latex/ input directory or folder on your system. Consult your \tex/
documentation if you don't know where this is. (You could also try
looking for the file \fn{article.cls}; the place where you find it is
almost surely your \latex/ input files area.)

All other files in the \amslatex/ distribution (the ones in the
\fn{math} and \fn{classes} subdirectories) can be placed in an
\amslatex/ source files area; if you are installing \amslatex/ for the
first time, create a new folder or directory for this purpose.

\section{Testing}

For a quick test of the installation, try printing the test file
\fn{subeqn.tex}. For more extensive tests print the \amslatex/ user's
guide\index{amslatex user@\amslatex/ user's guide} (\fn{amsldoc.tex})
or \fn{testmath.tex}.

\section{Extra math fonts}

For information on the AMSFonts collection, a set of extra math
fonts that supplements the standard set of \latex/ math fonts,
see \secref{amsfonts}.

\section{Memory requirements}
\label{as:capacity}

\tex/ divides up the memory available to it into various categories.
On most systems the sizes of these categories are fixed at the
beginning of a \tex/ run and cannot dynamically grow to meet
unexpected demands. (In fact certain implementations of \tex/ have the
sizes fixed at the time a format file is created, or even when the
\tex/ program is compiled.) Use of extra packages places burdens on
certain memory categories (string pool, hash size, main memory) in
proportion to the total size of the packages. Table~\ref{t:cap}
lists the recommended capacities in various categories for successful
use of the \amslatex/ major documentstyles or the \pkg{amsmath}
package. Not all categories are listed; the ones that appear are the
ones where problems tend to occur nowadays.

Note in particular that the base value for string pool needs to
be much larger than the values typically found at the end of a \latex/
log. This is because the string pool capacity reported by \tex/ in
response to a \verb'\tracingstats' command is not the base value,
but the result of subtracting from the base value the number of
characters in \tex/'s built-in error messages, the names of primitive
control sequences, and the names of all additional control sequences
defined in the format file (in our case, the whole of \latex/), not to
mention font names and file names. Thus the reported value only measures
the amount of string capacity that remains to the user after the format
file is loaded. The reported value for number of strings is reduced in
the same way.

\begin{table}[ht]
\caption[]{Recommended values for selected \tex/ memory
categories}\label{t:cap}
\edef\twohd{\noexpand\multicolumn{2}{c\string|}}
\let~=\thinspace
\begin{ctab}{|r|r|r|l|}
\hline
&\twohd{Capacity}&\\
Category& Adequate& Generous& WEB variable\\
\hline
strings& 5~000& 30~000& \verb'max_strings'\\
string characters& 80~000& 300~000& \verb'pool_size'\\
macro string pool$^*$& 50~000& 270~000& \verb'string_vacancies'\\
main memory& 80~000& 250~000& \verb'main_mem'\\
control sequences& 5~000& 20~000& \verb'hash_size'\\
font information& 60~000& 300~000& \verb'font_mem_size'\\
number of fonts& 128& 256& \verb'font_max'\\
input buffer& 1~000& 5~000& \verb'buf_size'\\
save stack& 2~000& 10~000& \verb'save_size'\\
\hline
\end{ctab}

\smallskip

\begin{center}
\parbox{.8\columnwidth}{\noindent\llap{$^*$}The number of string
characters left for macro packages and user commands, after all
primitives and built-in error messages have been loaded---i.e., the
total number of string characters available for a format file and
individual documents using that format file.}
\end{center}
\end{table}

\section{Files included in this distribution}

As files are occasionally added or removed from the distribution,
you should check the \fn{READ.ME} file if you want the most up-to-date
possible list.

%    File name and description:
\newcommand{\filed}[2]{\>\fn{#1}%
  \>\parbox[t]{.7\columnwidth}{\raggedright
     \strut#2\ifhmode\strut\fi\par}\\}

\begin{tabbing}
\quad\=\nfn{amslatex.bug}\quad\=\kill
\>\textbf{File name} \>\textbf{Description}\\
\>\rule{.9\columnwidth}{.4pt}\\
\strut
\filed{amsldoc.tex}{user's guide}
\filed{amslatex.faq}{frequently asked questions}
\filed{amslatex.bug}{description of bug fixes and other changes}
\filed{diff12.tex}{description of differences between versions 1.1
  and 1.2}
\filed{technote.tex}{some technical notes}
\filed{amslatex.ins}{installation file}
\filed{testmath.tex}{test file for general math features}
\filed{subeqn.tex}{test file for `subequations' environment}
\filed{amsbsy.dtx}{for \cn{boldsymbol} and \cn{pmb}}
\filed{amscd.dtx}{for commutative diagrams}
\filed{amsgen.dtx}{auxiliary file}
\filed{amsintx.dtx}{alternative command syntax for integrals and
  sums}
\filed{amsmath.dtx}{equations and other math}
\filed{amsopn.dtx}{operator names}
\filed{amstext.dtx}{\cn{text} command}
\filed{amsxtra.dtx}{misc rarely used commands}
\filed{amstex.sty}{frozen version of old \pkg{amstex} package}
\filed{amsdtx.dtx}{document class for printing AMS \nfn{.dtx} files}
\\
\filed{instr-l.tex}{instructions for using AMS document classes}
\filed{amsclass.dtx}{Source for \cls{amsart}, \cls{amsbook}, and
  \cls{amsproc} document classes}
\filed{amsthm.dtx}{provides \cn{theoremstyle}, \cn{newtheorem*}}
\filed{upref.dtx}{makes \cn{ref} always produce roman/upright
  numbers}
\filed{thmtest.tex}{Test file for the \pkg{amsthm} package}
\filed{amsalpha.bst}{Bibliography style for \bibtex/}
\filed{amsplain.bst}{Bibliography style for \bibtex/}
\filed{mrabbrev.bib}{\bibtex/ abbreviations for MR journal names}
\end{tabbing}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Error messages and output problems}

\section{General remarks}

This is a supplement to Chapter~8 of the \latex/ manual \cite{lm} (first
edition: Chapter~6). For the reader's convenience, the set of error
messages discussed here overlaps somewhat with the set in that chapter,
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|.

There is also a section discussing some output errors, i.e., instances
where the printed document has something wrong but there was no \latex/
error during typesetting.

\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}{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}

\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{AMSFonts collection}
distribution are considered part of standard \latex/ (as of June 1994):
\fn{cmex7}--\texttt{9}, \fn{cmmib5}--\texttt{9}, and
\fn{cmbsy5}--\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
\fn{cmex10} for the \pkg{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}

\begin{error}{Foreign command \over; use \frac or \genfrac instead}
\errexa
\begin{verbatim}
! Package amsmath Error: Foreign command: \over; use \frac or \genfrac instead.

See the amsmath package documentation for explanation.
Type  H <return>  for immediate help.
 ...

l.49 Fractions ${1\over
                        2}$ and
?
\end{verbatim}
\errexpl
The primitive generalized fraction commands of \tex/ are disallowed by
the \pkg{amsmath} package because their syntax is foreign to \latex/.
The substitutes \cn{frac} and \cn{genfrac} are provided instead.
See \secref{badgfrac} for further information.
\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}

\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 \env{amsmath} 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 linebreak 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]{lm}, even if it occurs on the following
line and is intended to be part of the contents. For example
\begin{verbatim}
\begin{array}
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]{lm}:
\begin{verbatim}
\begin{array}
a+b\\
{[f,g]}\\
m+n
\end{array}
\end{verbatim}

\end{error}

\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 linebreak inside a
\cn{left}-\cn{right} pair of delimiters in a \env{multline} or
\env{split} environment:
\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 of fixed size (\cn{bigl} \cn{bigr}
\cn{biggl} \cn{biggr} \dots; see \secref{bigdel}); or (2)~use null
delimiters to break up the \cn{left}-\cn{right} pair into parts for each
line:
\begin{verbatim}
AAA\left(BBB\right.\\
  \left.CCC\right)
\end{verbatim}
The latter solution may result in mismatched delimiter sizes;
ensuring that they match requires using \cn{vphantom} in the line
that has the smaller delimiter (or possibly \cn{smash} in the line that
has the larger delimiter). In the argument of \cn{vphantom} put a copy
of the tallest element that occurs in the other line, e.g.,
\begin{verbatim}
xxx \left(\int_t yyy\right.\\
  \left.\vphantom{\int_t} zzz ... \right)
\end{verbatim}
\end{error}

\begin{error}{Paragraph ended before \xxx was complete}
\errexa
\begin{verbatim}
Runaway argument?

! Paragraph ended before \equation 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 misspelling in the |\end{equation}| command,
e.g.,
\begin{verbatim}
\begin{equation}
...
\end{equatin}
\end{verbatim}
or by using abbreviations such as |\beq| and |\eeq| for
|\begin{equation}| and |\end{equation}|:
\begin{verbatim}
\beq
...
\eeq
\end{verbatim}
That kind of abbreviation works in ordinary \latex/ but for technical
reasons it does not work with any of the displayed equation environments
(\env{gather}, \env{align}, \env{split}, \env{equation}, etc.\@) when
the \pkg{amsmath} package is used. Work to re-enable the use of such
abbreviations is under way, but nontrivial technical complications are
involved (cf.\@ \fn{technote.tex}).
\end{error}

\begin{error}{Runaway argument?}
See the discussion for the error message
\texttt{Paragraph ended before \cn{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 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{Wrong output}

\subsection{Section numbers 0.1, 5.1, 8.1 instead of 1, 2, 3}
\label{numinverse}

This most likely means that you have the arguments for \cn{numberwithin}
in reverse order:
\begin{verbatim}
\numberwith{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}
\numberwith{equation}{section}
\end{verbatim}

\subsection{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
\secref{numinverse}.

\subsection{Double accent command failed to position the second accent
properly}

The capitalization of the command names is important. Check whether
all of the accent commands are capitalized. Compare
\[\hat{\hat{A}}\quad\Hat{\hat{A}}\quad\Hat{\Hat{A}}\]
\begin{verbatim}
\[\hat{\hat{A}}\quad\Hat{\hat{A}}\quad\Hat{\Hat{A}}\]
\end{verbatim}

\subsection{The \cn{boldsymbol} command didn't work}

This probably means that the font necessary to produce a bold version of
the symbol in question is not available on your system or not installed
in a way that \latex/ can use it. For example, bold versions of the
AMS extra symbol fonts \fn{msam} and \fn{msbm} do not currently exist,
so \cn{boldsymbol}|{|\cn{square}|}| will merely produce the same result
as \cn{square} alone.

Also, the weight of the fonts \fn{cmmib} and \fn{cmbsy} is `bold'
rather than `bold extended' as for \fn{cmbx}. Depending on the symbol
in question and the resolution of your printer (or previewer), a
symbol that was taken from one of those two bold symbol fonts might
be nearly indistinguishable from the non-bold version.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Other useful items for mathematical documents}

\section{AMS documentclasses (\cls{amsart}, \cls{amsbook},
\cls{amsproc})}

The American Mathematical Society provides custom \amslatex/ document
classes for AMS journals and book series. These may obtained by
anonymous FTP from the e-MATH system \fn{e-math.ams.org}, directory, in
the \fn{author-info} area. (Note that directory structures may
occasionally change; if you have trouble locating files you need, please
contact AMS Technical Support; see Appendix~\ref{help}.) Specific and
generic classes can be requested by e-mail to
\mail{<EMAIL>} or ordered on floppy disk through the AMS
Customer Services Department at \mail{<EMAIL>} or
800-321-4267.

\section{Extra math fonts (the AMSFonts collection)}
\label{amsfonts}

The `AMSFonts' font collection is a set of extra math\index{AMSFonts
collection} fonts that supplements the standard set of \latex/ math
fonts. It includes \latex/ packages called \pkg{amsfonts} and
\pkg{amssymb} that make it easy to use the extra fonts in a \latex/
document. If you don't have the AMSFonts collection already on your
system, and would like to use some of the symbols in it, it can be
obtained by anonymous FTP from CTAN\index{FTP}\index{CTAN} (see
\secref{ctan}) or from the e-MATH system at \fn{e-math.ams.org}. It can
also be ordered on disk from the AMS Customer Services Department at
\mail{<EMAIL>} or 800-321-4267.

\section{Syntax checking (the \pkg{syntonly} package)}

If you are working on a relatively slow computer system you might want
to try the \pkg{syntonly} package. This makes \latex/ skim through your
document only checking for proper syntax and usage of the commands, but
not producing any printed pages. As \latex/ runs quite a bit faster
under those conditions, for early error-checking runs of a document this
can save you valuable time. The \pkg{syntonly} package is a
standard \latex/ package so you should find it already on your system;
to use it, write
\begin{verbatim}
\usepackage{syntonly}
\syntaxonly
\end{verbatim}
When you want to produce pages, comment out the second line by adding a
percent character:
\begin{verbatim}
%\syntaxonly
\end{verbatim}

\section{Verbatim and comments (the \pkg{verbatim} package)}

The \pkg{verbatim} package is another standard \latex/ package, like
\pkg{syntonly}. In addition to some minor enhancements for the
\env{verbatim} environment, it provides a \env{comment} environment that
skips everything between |\begin{comment}| and the next
|\end{comment}|. A command \cn{verbatiminput} is also provided for
typesetting a whole file in verbatim style.

\section{Commutative diagrams and other diagrams (packages \pkg{diagram},
\pkg{xypic}, \pkg{pstricks})}\label{xypic}

The \pkg{amscd} package for \ii{commutative diagrams} that is included
in the \amslatex/ distribution is quite limited. The following packages
are worth investigation for authors who need more powerful diagram
features: \pkg{diagram}, \pkg{xypic}, \pkg{pstricks}. These are
available by anonymous FTP\index{FTP} from CTAN\index{CTAN} (see
\secref{ctan}). The \pkg{diagram} package, by <NAME>, focuses on
commutative diagrams and is thus smaller than the others; it uses
\latex/ arrows as produced by \cn{vector} for diagram construction. The
\pkg{xypic} package, by <NAME>, is a more general diagram
package; it requires special arrow fonts (included in the \pkg{xypic}
distribution) or PostScript capabilities in your system. The
\pkg{pstricks} package, by <NAME>, also provides general
diagram object commands; it is strictly for PostScript use, as the name
implies.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Where to find other information}

\section{Technical notes}
The file \fn{technote.tex} contains some remarks on miscellaneous
technical questions related to \amslatex/; they are relegated to a
separate document because they are unlikely to be of interest to most
users.

\section{Differences between \amslatex/ version 1.1 and \amslatex/
version 1.2}
\markright{\thesection. DIFFERENCES BETWEEN VERSIONS 1.1 AND 1.2}

Version 1.2 of \amslatex/ fills in some gaps and corrects some mistakes
in the \amslatex/ 1.1 feature set. The significant differences, from a
user's perspective, are described in the document \fn{diff12.tex}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Getting help}\label{help}

Questions or comments regarding the \amslatex/ package 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{<EMAIL>}
\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}

\section{Further information}\label{a:furtherinfo}
Information about obtaining AMSFonts or other \tex/-related
software from the AMS Internet archive \fn{e-<EMAIL>}
can be obtained by sending a request through electronic mail to:
\mail{<EMAIL>}.\pagebreak

Information about obtaining \amslatex/ on diskette from the AMS is
available from:
\begin{infoaddress}
American Mathematical Society\\
Customer Services\\
P. O. Box 6248\\
Providence, RI 02940\\[3pt]
Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4000\\
Internet: \mail{<EMAIL>}
\end{infoaddress}

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 clearing-house of general information about
\tex/ and \tex/-related software.
\begin{infoaddress}
\tex/ Users Group\\
P. O. Box 869\\
Santa Barbara, CA 93102-0869\\[3pt]
Phone: (805) 963-1338\\
Internet: \mail{<EMAIL>}\\
\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.

There is a Usenet newsgroup called \fn{comp.text.tex} that is a fairly
good source of information about \latex/ and \tex/ in general. If you
don't know about reading newsgroups, check with your local system
administrator to see if newsgroup service is available at your site.

\begin{thebibliography}{9}
\addcontentsline{toc}{chapter}{Bibliography}

\bibitem{amsfonts}\booktitle{AMSFonts version \textup{2.2}---user's guide},
Amer. Math. Soc., Providence, RI, 1994; distributed
with the AMSFonts package.

\bibitem{author-guidelines}\booktitle{Instructions for preparation of
papers and monographs---\amslatex/},
Amer. Math. Soc., Providence, RI, 1994.

\bibitem{tlc} <NAME>, <NAME>, and <NAME>,
\booktitle{The \latex/ companion}, Addison-Wesley, Reading, MA, 1994.

\bibitem{kn} <NAME>, \booktitle{The \tex/book},
Addison-Wesley, Reading, MA, 1984.

\bibitem{lm} <NAME>, \booktitle{\latex/: A document preparation
system}, 2nd revised ed., Addison-Wesley, Reading, MA, 1994.

\bibitem{msf} <NAME> and <NAME>,
\textit{The new font family selection---user
interface to standard \latex/}, \journalname{TUGboat} \textbf{11},
no.~2 (June 1990), pp.~297--305.

\bibitem{jt} <NAME>, \booktitle{The joy of \tex/}, 2nd revised ed.,
Amer. Math. Soc., Providence, RI, 1990.

\end{thebibliography}

\newcommand{\toss}[2]{#1{#2}\addcontentsline{toc}{chapter}{Index}}
\toss % toss the desired addcontentsline command past the following
      % \begin{index} in order to ensure the right page number
%    This is amsldoc.ind generated on 16-Feb-1995. [mjd]
\begin{theindex}

  \item {\normalfont  \ttfamily  \bslchar \qcbang  }, 10
  \item \cn{\,}, 10
  \item \cn{\:}, 10
  \item \cn{\;}, 10
  \item \cn{\\}, 3, 7, 8, 30, 32
  \item \cn{\\*}, 8
  \item \cn{\|}, 16

  \indexspace

  \item \cn{above}, 13
  \item \cn{abovewithdelims}, 13
  \item \cn{accentedsymbol}, 1, 12
  \item accents
    \subitem double, 12
  \item \cn{addtocounter}, 9, 10
  \item \cn{addtolength}, 3
  \item \env{align} environment, 2, 5, 6, 34
  \item \env{alignat} environment, 6, 30, 32
  \item \env{aligned} environment, 6
  \item \cn{allowdisplaybreaks}, 8
  \item \env{allowdisplaybreaks} environment, 7
  \item \fn{amsalpha.bst}, 29
  \item \cls{amsart} class, 29, 35
  \item \cls{amsbook} class, 29, 35
  \item \pkg{amsbsy} package, 1, 18
  \item \fn{amsbsy.dtx}, 29
  \item \pkg{amscd} package, 1, 20, 21, 36
  \item \fn{amscd.dtx}, 29
  \item \fn{amsclass.dtx}, 29
  \item \fn{amsdtx.dtx}, 29
  \item \pkg{amsfonts} package, 21, 36
  \item AMSFonts collection, 21, 31, 36
  \item \fn{amsgen.dtx}, 29
  \item \pkg{amsintx} package, 1, 19, 20
  \item \fn{amsintx.dtx}, 20, 29
  \item \amslatex/ user's guide, 27
  \item \fn{amslatex.bug}, 28
  \item \fn{amslatex.faq}, 28
  \item \fn{amslatex.ins}, 29
  \item \fn{amsldoc.tex}, 27, 28
  \item \env{amsmath} environment, 32
  \item \pkg{amsmath} package, 1--3, 5, 7, 9--19, 28, 31, 34
  \item \fn{amsmath.dtx}, 29
  \item \pkg{amsopn} package, 1, 17, 18
  \item \fn{amsopn.dtx}, 29
  \item \fn{amsplain.bst}, 29
  \item \cls{amsproc} class, 29, 35
  \item \pkg{amssymb} package, 36
  \item \amstex/, v
  \item \pkg{amstex} package, 29
  \item \fn{amstex.sty}, 29
  \item \pkg{amstext} package, 1, 18
  \item \fn{amstext.dtx}, 29
  \item \pkg{amsthm} package, vi, 1, 22, 24, 25, 29
  \item \fn{amsthm.dtx}, 25, 29
  \item \pkg{amsxtra} package, 1, 12
  \item \fn{amsxtra.dtx}, 29
  \item \env{array} environment, 9, 10, 32
  \item arrows
    \subitem extensible, 13
    \subitem in commutative diagrams, 21
  \item \fn{article.cls}, 27
  \item \cn{atop}, 13
  \item \cn{atopwithdelims}, 13
  \item \fn{author-info}, 35

  \indexspace

  \item BaKoMa fonts, 31
  \item \cn{big}, \cn{Big}, \cn{bigg}, \dots\ delimiters, 16
  \item \cn{biggl}, 33
  \item \cn{biggr}, 33
  \item \cn{bigl}, 33
  \item \cn{bigr}, 33
  \item \cn{binom}, 13, 14
  \item binomials, 14
  \item \env{bmatrix} environment, 9
  \item \cn{bmod}, 18
  \item \cn{boldsymbol}, vi, 1, 18, 19, 29, 35
  \item \cn{boxed}, 13

  \indexspace

  \item \cn{C}, 22
  \item \env{cases} environment, 7
  \item \env{CD} environment, 1, 21
  \item \opt{centertags} option, 1
  \item \cn{cfrac}, 15
  \item \fn{classes}, 27
  \item \fn{cmbsy}, 35
  \item \fn{cmbsy5}, 31
  \item \fn{cmbx}, 35
  \item \fn{cmex}, 31
  \item \fn{cmex10}, 31
  \item \fn{cmex7}, 31
  \item \fn{cmmib}, 35
  \item \fn{cmmib5}, 31
  \item \env{comment} environment, 36
  \item commutative diagrams, 36
  \item \fn{comp.text.tex}, 38
  \item Comprehensive \tex/ Archive Network, \see{CTAN}{27}
  \item continued fractions, 14
  \item CTAN, 27, 36

  \indexspace

  \item \cn{dbinom}, 14
  \item \cn{ddddot}, 12
  \item \cn{dddot}, 12
  \item \cn{ddot}, 12
  \item \cn{DeclareMathOperator}, 1, 17
  \item \texttt{definition} theorem style, 24
  \item delimiters
    \subitem size adjustments, 15
  \item \cn{dfrac}, 13, 14
  \item \pkg{diagram} package, 36
  \item \fn{diff12.tex}, 29, 37
  \item differential notation, 20
  \item \cn{displaybreak}, 7, 8
  \item displayed equations, 2
    \subitem centering, 2
  \item \env{displaymath} environment, 3
  \item \cn{displaystyle}, 14
  \item \cn{documentclass}, 2
  \item \cn{dot}, 12
  \item dots, \see{ellipsis dots}{10}
  \item \cn{dots}, 11
  \item \cn{dotsb}, 11
  \item \cn{dotsc}, 11
  \item \cn{dotsi}, 11
  \item \cn{dotsm}, 11

  \indexspace

  \item \fn{e-math.ams.org}, 35--37
  \item ellipsis dots
    \subitem in matrices, 10
  \item \cn{end}, 11
  \item \env{eqnarray} environment, vi, 2, 7, 32
  \item \cn{eqref}, 9
  \item \env{equation} environment, vi, 2, 3, 5, 30, 34
  \item equation numbers
    \subitem cross-references, 9
    \subitem hierarchy, 9
    \subitem left or right placement, 2
    \subitem overriding, 3
    \subitem subordinate numbering, 9
    \subitem vertical placement, 1
  \item \env{equation*} environment, 3
  \item equations, \see{displayed equations}{2}
  \item \pkg{eucal} package, 21

  \indexspace

  \item \cn{fbox}, 13
  \item \cn{field}, 22
  \item \opt{fleqn} option, 2, 3
  \item \fn{fntguide.tex}, 21
  \item \cn{frac}, 13--15, 31
  \item fractions, 13
  \item \cn{fracwithdelims}, 1
  \item FTP, 36
  \item \fn{ftp.dante.de}, 27
  \item \fn{ftp.shsu.edu}, 27
  \item \fn{ftp.tex.ac.uk}, 27
  \item function names, \see{operator names}{17}

  \indexspace

  \item \env{gather} environment, 5, 30, 34
  \item \env{gathered} environment, 6
  \item \cn{genfrac}, 13, 14, 31

  \indexspace

  \item \cn{Hat}, 12
  \item \cn{hat}, 12
  \item \cn{hdotsfor}, 10
  \item horizontal space
    \subitem around operator names, 17
    \subitem in math mode, 10

  \indexspace

  \item \cn{idotsint}, 19
  \item \cn{iiiint}, 19
  \item \cn{iiint}, 19
  \item \cn{iint}, 19
  \item \fn{inputs}, 27
  \item installation, 27
  \item \fn{instr-l.tex}, 29
  \item \cn{int}, 20
  \item integrals, 20
    \subitem multiple, 19
    \subitem placement of limits, 2
  \item \cn{intertext}, 8
  \item \opt{intlimits} option, 2

  \indexspace

  \item \cn{label}, 9
  \item Lamport, Leslie, v
  \item \cn{langle}, 16
  \item \cn{left}, 15, 16, 33
  \item \cn{leftroot}, 12
  \item \opt{leqno} option, 2
  \item \cn{lim}, vi, 1, 17
  \item limits, \see{subscripts and superscripts}{1}
  \item \cn{log}, 17
  \item \cn{lVert}, 16
  \item \cn{lvert}, 16

  \indexspace

  \item \fn{math}, 27
  \item math fonts, 21
    \subitem bold symbols, 18
  \item math symbols, \see{math fonts}{21}
  \item \cn{mathbf}, 18, 21, 22
  \item \cn{mathcal}, 21
  \item \cn{mathit}, 21
  \item \cn{mathrm}, 21
  \item \cn{mathsf}, 21
  \item \cn{mathtt}, 21
  \item matrices, 9
    \subitem ellipsis dots, 10
  \item \env{matrix} environment, 9
  \item \cn{mbox}, 18
  \item \cn{medspace}, 10
  \item Metafont source files, 31
  \item Mittelbach, Frank, v
  \item \cn{mod}, 18
  \item \fn{mrabbrev.bib}, 29
  \item \fn{msam}, 35
  \item \fn{msbm}, 35
  \item \cn{mspace}, 11
  \item \env{multline} environment, 3, 5, 33
  \item \cn{multlinegap}, 3

  \indexspace

  \item \opt{namelimits} option, 2
  \item \cn{negmedspace}, 10
  \item \cn{negthickspace}, 10
  \item \cn{negthinspace}, 10
  \item \cn{newcommand}, 12, 21
  \item \cn{newenvironment}, 23
  \item \cn{newtheorem}, vi, 1, 22--25
  \item \cn{newtheorem*}, 23, 29
  \item \cn{newtheoremstyle}, 25
  \item \cn{nobreakdash}, 11, 12
  \item \opt{nointlimits} option, 2
  \item \cn{nolimits}, 20
  \item \opt{nonamelimits} option, 2
  \item \opt{nosumlimits} option, 2
  \item \cn{notag}, 3
  \item \cn{numberwithin}, 9, 34, 35

  \indexspace

  \item operator names, 17
  \item \cn{over}, 13
  \item \cn{overleftarrow}, 11
  \item \cn{overleftrightarrow}, 11
  \item \cn{overrightarrow}, 11
  \item \cn{overset}, 13
  \item \cn{overwithdelims}, 13

  \indexspace

  \item page breaks, 7
  \item \cn{pagebreak}, 8
  \item \texttt{plain} theorem style, 24
  \item \env{pmatrix} environment, 9
  \item \cn{pmb}, 1, 18, 19, 29
  \item \cn{pmod}, 18
  \item \cn{pod}, 18
  \item PostScript fonts, 31
  \item \env{proof} environment, vi, 1, 22, 25
  \item proofs, 23
  \item \pkg{pstricks} package, 36

  \indexspace

  \item \cn{qed}, 25
  \item \cn{qedsymbol}, 25
  \item \cn{qquad}, 10
  \item \cn{quad}, 10, 11

  \indexspace

  \item \cn{R}, 22
  \item \cn{raisetag}, 7
  \item \cn{rangle}, 16
  \item \fn{READ.ME}, 27, 28
  \item \cn{ref}, 1, 9, 29
  \item \texttt{remark} theorem style, 24
  \item \cn{renewcommand}, 22, 25
  \item \opt{reqno} option, 2
  \item \cn{right}, 15, 16, 33
  \item \cn{rVert}, 16
  \item \cn{rvert}, 16

  \indexspace

  \item \cn{savebox}, 12
  \item Sch\"opf, Rainer, v
  \item \cn{scriptscriptstyle}, 14
  \item \cn{scriptstyle}, 14
  \item \cn{section}, 25
  \item \cn{setcounter}, 9, 10
  \item \cn{setlength}, 3
  \item \cn{shoveleft}, 3
  \item \cn{shoveright}, 3
  \item \cn{sideset}, 13, 20
  \item \cn{sin}, vi, 1, 17
  \item \env{smallmatrix} environment, 10
  \item \cn{smash}, 15, 33
  \item Spivak, Michael, v
  \item \env{split} environment, 2, 5, 30, 33, 34
  \item \cn{square}, 35
  \item \cn{stackrel}, 13
  \item \env{subarray} environment, 19
  \item \fn{subeqn.tex}, 3, 27, 29
  \item \env{subequations} environment, 9
  \item subscripts and superscripts, 13
    \subitem multi-line, 19
    \subitem on sums, 20
    \subitem placement, 1, 2
  \item \cn{subsection}, 25
  \item \cn{substack}, 19
  \item \cn{sum}, 20
  \item \opt{sumlimits} option, 1, 2
  \item sums, 20
  \item superscripts, \see{subscripts and superscripts}{19}
  \item \cn{swapnumbers}, 25
  \item \pkg{syntonly} package, 36

  \indexspace

  \item \env{tabular} environment, 32
  \item \cn{tag}, 3
  \item \cn{tag*}, 3
  \item \cn{tbinom}, 14
  \item \opt{tbtags} option, 1
  \item \fn{technote.tex}, 13, 29, 34, 37
  \item \fn{testmath.tex}, 3, 27, 29
  \item \tex/ Users Group, 38
  \item \cn{text}, 1, 6, 7, 18, 29
  \item text fragments inside math, 8, 18
  \item \cn{textstyle}, 14
  \item \cn{tfrac}, 13, 14
  \item \cn{theequation}, 8, 9
  \item \pkg{theorem} package, 22
  \item theorems, 23
    \subitem numbering, 23
  \item \cn{theoremstyle}, 22, 24, 25, 29
  \item \cn{thickspace}, 10
  \item \cn{thinspace}, 10
  \item \fn{thmtest.tex}, 22, 25, 29
  \item \journalname{TUGboat}, 38

  \indexspace

  \item \cn{underleftarrow}, 11
  \item \cn{underleftrightarrow}, 11
  \item \cn{underrightarrow}, 11
  \item \cn{underset}, 13
  \item \pkg{upref} package, 1
  \item \fn{upref.dtx}, 29
  \item \cn{uproot}, 12
  \item \cn{usepackage}, 1, 2, 31

  \indexspace

  \item \cn{value}, 9
  \item \cn{varinjlim}, 17
  \item \cn{varliminf}, 17
  \item \cn{varlimsup}, 17
  \item \cn{varprojlim}, 17
  \item \cn{vec}, 22
  \item \cn{vector}, 36
  \item \env{verbatim} environment, 36
  \item \pkg{verbatim} package, 36
  \item \cn{verbatiminput}, 36
  \item \env{Vmatrix} environment, 9
  \item \env{vmatrix} environment, 9
  \item \cn{vphantom}, 33

  \indexspace

  \item \cn{xleftarrow}, 13
  \item \cn{xrightarrow}, 13
  \item \cn{xxx}, 34
  \item \pkg{xypic} package, 36

\end{theindex}

\end{document}
