%%% ====================================================================
%%%  @LaTeX-file{
%%%     author          = "American Mathematical Society",
%%%     version         = "1.2beta",
%%%     date            = "27-Oct-1994",
%%%     time            = "16:25:40 EDT",
%%%     filename        = "amsldoc.tex",
%%%     copyright       = "Copyright (C) 1994 American Mathematical Society,
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%%%                        (1) you make absolutely no changes to your copy,
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%%%     address         = "American Mathematical Society,
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%%%     telephone       = "401-455-4080 or (in the USA and Canada)
%%%                        800-321-4AMS (321-4267)",
%%%     FAX             = "401-331-3842",
%%%     checksum        = "40695 1968 9507 75586",
%%%     email           = "tech-support@math.ams.org (Internet)",
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%%%     keywords        = "latex, amslatex, ams-latex, amstex",
%%%     supported       = "yes",
%%%     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
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%%%  }
%%% ====================================================================
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used for this document
\documentclass[leqno,titlepage,openany%
%,draft
]{amsldoc}

\title{\amslatex/ Version 1.2\\User's Guide}
\author{American Mathematical Society}
\date{October 1994}

\usepackage[noamsfonts]{amstex}
\usepackage{amscd}

%    The amsldoc class includes a number of features useful for
%    documentation about TeX, including:
%
%    ---Commands \tex/, \amstex/, \latex/, ... for printing various
%    logos with proper ending spacefactor and convenient 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, \opt for
%    package or class options, etc., etc.

%\makeindex % generate index data
\hfuzz=55pt % ignore overlong lines for the time being [mjd,12-Oct-1994]
\vbadness=8000 % suppress `underfull' message, who cares?

%    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}}

%    An environment for presenting comprehensive address information:
\newenvironment{infoaddress}{%
  \par\topsep\medskipamount
  \trivlist\centering
  \item[]%
  \begin{minipage}{.7\columnwidth}%
  \raggedright
}{%
  \end{minipage}%
  \endtrivlist
}

\newenvironment{ctab}{%
  \par\topsep\medskipamount
  \trivlist\centering
  \item[]%
  \begin{tabular}%
}{%
  \end{tabular}%
  \endtrivlist
}

%    A command for ragged-right parbox in a tabular.
\newcommand{\rp}{\let\PBS\\\raggedright\let\\\PBS}

\newtheorem{prop}{Proposition}

\newcommand{\ix}{\operatorname{ix}}
\newcommand{\nul}{\operatorname{nul}}
\newcommand{\End}{\operatorname{End}}

\begin{document}

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

\maketitle
\tableofcontents
\newpage

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

\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}. As \amslatex/ is an
extension for \latex/, and \latex/ 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; but it doesn't add much 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
(roughly 1982--1986), the American Mathematical Society was throwing its
resources into the development of a different macro package known as
\amstex/, written by Michael Spivak. Eventually (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 just 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/. The lion's share of the programming work
was done by Frank Mittelbach and Rainer Sch\"opf 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/?}
\markright{WHY 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 how do these features sound:
\begin{itemize}

\item Easy access to extra math symbols as found in the AMSFonts font
package, including blackboard bold and fraktur letters. And
they change size properly through all the range of \latex/ font sizes,
including sub\slash superscript sizes.

\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\slash 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
over-printing 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 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 Horizontal arrows that automatically expand to encompass wide
subscripts or superscripts

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

\item A \env{cases} environment to make it easy to produce a typical
case statement with large left brace, as in
\begin{equation} 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}

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mainmatter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{How to use \amslatex/}

\section{Using an AMS package with a \latex/ document}\label{usepackage}
\markright{USING AN AMS PACKAGE}

A `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{amstex} package
is perhaps the single most noteworthy package, as it subsumes the
\pkg{amsfonts}, \pkg{amstext}, and \pkg{amsbsy} packages, and provides a
number of other enhancements for mathematical typesetting.
Table~\ref{packages} gives the current list of packages.

%    We use the [b] option to keep this table from floating above the
%    chapter title. Depending on page breaking this might need changing
%    in the future. [mjd,26-Oct-1994]
\begin{table}[b]
\caption{Packages}\label{packages}
\begin{ctab}{lp{.7\columnwidth}}
\pkg{amstex}& \rp Defines extra environments for multiline displayed
  equations, as well as a number of other enhancements for math\\
\pkg{amstext}& \rp Provides a \cn{text} command for proper
  typesetting of a fragment of text inside a display\\
\pkg{amsbsy}& \rp Defines \cn{boldsymbol} and \cn{pmb} commands\\
\pkg{amsintsm}& \rp Provides more logical command syntax for
  \cn{int}, \cn{sum}\\
\pkg{amscd}& \rp Provides a \env{CD} environment for simple commutative
diagrams (no support for diagonal arrows)\\
\pkg{amsthm}& \rp Provides a \env{proof} environment and
extensions for the \cn{newtheorem} command
\end{ctab}
\end{table}

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

\item[\opt{intlimits}] Place the subscripts and superscripts of integral
symbols above and below rather than to the side (in displayed equations
only).

\item[\opt{nointlimits}] (default) Always place the subscripts and
superscripts of integral symbols to the side.

\item[\opt{sumlimits}] (default) Like \opt{intlimits}, but for sums and
similar symbols ($\prod$, $\coprod$, $\bigotimes$, $\bigoplus$, and so
forth).

\item[\opt{nosumlimits}] Opposite of \opt{sumlimits}.

\item[\opt{namelimits}] Like \opt{intlimits}, but for `operator names'
such as $\det$, $\inf$, $\lim$, $\max$, $\min$ and so forth.

\item[\opt{nonamelimits}] Opposite of \opt{namelimits}.

\item[\opt{tbtags}] `Top-or-bottom tags': For a split equation, place
equation numbers level with the last (resp.\ first) line, if numbers are
on the right (resp.\ left).

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

The \pkg{amstex} package also recognizes the following options which
are usually given in the options list of the \cn{documentclass} command:
\begin{description}
\item[\opt{fleqn}] Position equations at a fixed indent from the left
margin rather than centered in the text column. This option is typically
inherited from the documentclass declaration, and so doesn't need to be
specified explicitly in the \cn{usepackage} command.

\item[\opt{reqno}] Place equation numbers on the right.

\item[\opt{leqno}] Place equation numbers on the left. This option and
the \opt{reqno} option are typically inherited from the documentclass
declaration, so there is no default to speak of.

\end{description}

Example: \verb"\usepackage[intlimits]{amstex}". In compatibility mode
obsolete option names \opt{intlim}, \opt{nosumlim}, \opt{nonamelm},
\opt{righttag} are recognized as well.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Documentation about amsart/amsbook document classes, included here
%    pending reorganization plans.
\input{amscdoc.tx1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\section{Introduction}
The \pkg{amstex} package provides several environments for  multi-line
displayed equations.
They are similar in function to \latex/'s
\env{equation} and \env{eqnarray} environments. These environments
are:
\begin{verbatim}
 align      gather     alignat    xalignat   xxalignat
 multline   split
\end{verbatim}
Each environment, except for \env{split}, 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 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 in the starred versions of all the
\pkg{amstex} alignment structures.  See \fn{testmath.tex}
for examples of the use of \cn{tag}.

\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{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.
For example:
\begin{gather}
\ix(\Omega,\beta,[a,b])
  = \sum_{\lambda>0} \nul(\Omega +\lambda\Lambda,\beta,[a,b]),\\
\sum_{a<t<b} \nul(\Omega,\gamma \oplus\infty, [a,t]),\\
[\alpha(\lambda):\Gamma_\beta]_{\epsilon\leq\lambda\leq\lambda^+}
\end{gather}
%
\begin{verbatim}
\begin{gather}
\ix(\Omega,\beta,[a,b])
  = \sum_{\lambda>0} \nul(\Omega +\lambda\Lambda,\beta,[a,b]),\\
\sum_{a<t<b} \nul(\Omega,\gamma \oplus\infty, [a,t]),\\
[\alpha(\lambda):\Gamma_\beta]_{\epsilon\leq\lambda\leq\lambda^+}
\end{gather}
\end{verbatim}

\section{Equation groups with mutual alignment}

The \env{align} environment is used for two or more equations when
vertical alignment is desired (usually binary relations such as equal
signs are aligned).
For example:
\begin{align}
\Lambda(\omega) &= \operatorname{ix}(\Omega,\beta_{\omega},[0,1])\\
N(\omega) &= \operatorname{nul}(\Omega,\beta_{\omega},[0,1]).
\end{align}
%
\begin{verbatim}
\begin{align}
\Lambda(\omega) &= \operatorname{ix}(\Omega,\beta_{\omega},[0,1])\\
N(\omega) &= \operatorname{nul}(\Omega,\beta_{\omega},[0,1]).
\end{align}
\end{verbatim}

\section{Multiple alignment points in an equation group}
\markright{MULTIPLE ALIGNMENT POINTS}

The \env{align} environment takes up the whole width of a display. If
you want to have several ``align''-type structures side by side, there
is an \env{alignat} environment you can use.  There is one required
argument, to specify the number of ``align'' structures.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{For an argument of $n$, the number of \qc{\&}'s per line
is $2n-1$ (one ampersand for alignment within each align structure,
and ampersands to separate the align structures from one another).}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The \env{xalignat} and \env{xxalignat} environments are forms of the
\env{alignat} environment with expanded spacing between the component
align structures.  If we consider each align structure to be a
column, \env{xalignat} has equal spacing between columns and at the
margins; \env{xxalignat} has equal spacing between columns and zero
spacing at the margins.
Here is an example of \env{alignat}:
\begin{alignat}{3}
V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
 & \qquad U_i & = u_i,
 \qquad \text{for $i\ne j$;}\label{eq:C}\\
V_j & = v_j, & \qquad X_j & = x_j,
  & \qquad U_j & = u_j + \sum_{i\ne j} q_i u_i.
\label{eq:D}
\end{alignat}
%
\begin{verbatim}
\begin{alignat}{3}
V_i & =v_i - q_i v_j, & \qquad X_i & = x_i - q_i x_j,
 & \qquad U_i & = u_i,
 \qquad \text{for $i\ne j$;}\\
V_j & = v_j, & \qquad X_j & = x_j,
  & \qquad U_j & u_j | \sum_{i\ne j} q_i u_i.
\end{alignat}
\end{verbatim}

The \env{xalignat} is sometimes used to add annotations on each
line, as in the following example.
\begin{xalignat}{2}
x& =y && \text {by (\ref{eq:C})}\\
x'& = y' && \text {by (\ref{eq:D})}\\
x+ x' & = y+ y' && \text {by Axiom 1.}
\end{xalignat}
%
\begin{verbatim}
\begin{xalignat}{2}
x& =y && \text {by (\ref{eq:C})}\\
x'& = y' && \text {by (\ref{eq:D})}\\
x+x' & = y+y' && \text {by Axiom 1.}
\end{xalignat}
\end{verbatim}

\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}. For example:
\begin{multline*}
\textstyle
\sum_{t_0\leq t\leq t_1} \nul(\Omega,\infty\oplus\gamma_1,[t,b])
-\sum_{t_0\leq t\leq t_1} \nul(\Omega, \infty \oplus\gamma_2, [t,b])\\
\leq \dim(E^\nu) -\dim(\gamma_1 \cap \gamma_2).
\end{multline*}
%
\begin{verbatim}
\begin{multline*}
\textstyle
\sum_{t_0\leq t\leq t_1} \nul(\Omega,\infty\oplus\gamma_1,[t,b])
-\sum_{t_0\leq t\leq t_1} \nul(\Omega, \infty \oplus\gamma_2, [t,b])\\
\leq \dim(E^\nu) -\dim(\gamma_1 \cap \gamma_2).
\end{multline*}
\end{verbatim}
%
or
\begin{multline}
\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
\det\mathbf{K}(t=1,x_1,\dots,x_n;l \vert l )\\
=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\lvert I\rvert}\mathbf{A}^{(\lambda)}(I\vert I)
\det\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}\vert\overline I\cup\{l \}).
\label{sum-ali}
\end{multline}
%
\begin{verbatim}
\begin{multline}
\biggl(\sum_{\,i\in\mathbf{n}}a_{l _i}x_i\biggr)
\det\mathbf{K}(t=1,x_1,\dots,x_n;l \vertl )\\
=\biggl(\prod_{\,i\in\mathbf{n}}\hat x_i\biggr)
\sum_{I\subseteq\mathbf{n}-\{l \}}
(-1)^{\lvert I\rvert}\mathbf{A}^{(\lambda)}(I\vert I)
\det\mathbf{A}^{(\lambda)}
(\overline I\cup\{l \}\vert\overline I\cup\{l \}).
\label{sum-ali}
\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{amstex} 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
\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\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
\biggl[(n-l )^2-\sum^p_{j=1}(n_i-l _i)^2\biggr].\end{split}
\end{equation}
\end{verbatim}

\section{Alignment environments that don't constitute an entire display}
\markright{ALIGNMENT ENVIRONMENT UNITS}

In addition to the \env{split} environment, there are some other
equation alignment environments that do not constitute an entire
display.  They are self-contained units that can be used inside of
other formulas, or set side-by-side.  The environment names are
\env{aligned}, \env{gathered}, and \env{alignedat}.  These
environments take an optional argument to specify their vertical
positioning with respect to the material on either side.  The default
is |[c]|.  A \env{gathered} environment with the first line level with
the material on either side would be done like this.

\begin{verbatim}
\begin{gathered}[t]
...\\
...
\end{gathered}
\end{verbatim}

\section{Adjusting tag placement}

Placing equation numbers can be a very complex problem in multiline
displays. The environments of the \pkg{amstex} 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. But even so, the
placement can't be guaranteed to always be ideal. There is an
\cn{adjusttag} command provided to adjust the vertical position of the
next upcoming equation number. To move a particular number up by 6pt,
you would write |\adjusttag{-6pt}|.

\section{Vertical spacing and page breaks within multiline displays}
\markright{Spacing and page breaks in displays}

You can use the \cn{\\}|[|\<dimension>|]| command to get extra
vertical space between lines in all the \pkg{amstex} displayed
equation environments, as is usual in \latex/.  Unlike \env{eqnarray},
the \pkg{amstex} environments don't allow page breaks between lines,
unless \cn{displaybreak} or \cn{allowdisplaybreaks} is used.
The philosophy is that page breaks in such situations should receive
individual attention from the author.  \cn{displaybreak} must go
before the \cn{\\} where it is supposed 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.

There is also an optional argument for \cn{allowdisplaybreaks}.
\cn{allowdisplaybreaks} obeys the usual \latex/ scoping rules; the
normal way of limiting its scope would be to put
|{|\cn{allowdisplaybreaks} at the beginning and |}| at the end
of the desired range.  Within the scope of an \cn{allowdisplaybreaks}
command, the \cn{\\*} command can be used to prohibit a pagebreak, 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 in the middle of a
display alignment.  Its salient feature is preservation of the
alignment, which would not be possible 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.  An example of
its use follows.


\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}

To produce

\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}


Notice the word ``and'' would fall outside the display,
at the left margin.


\section{Equation numbering}

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 would probably redefine
\cn{theequation}:
\begin{verbatim}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\end{verbatim}
This works fine 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{amstex} package provides a command
\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}
As the name implies, \cn{numberwithin} can be applied to other
counters besides the equation counter, but the results are not
guaranteed because of potential complications.  Normal \latex/ methods
should be used where available, e.g., in \cn{newtheorem}.

To make cross-references to equations easier, an \cn{eqref} command 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}|.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Miscellaneous mathematics features (\pkg{amstex} package)}
\markboth{MISCELLANEOUS MATHEMATICS FEATURES}%
         {MISCELLANEOUS MATHEMATICS FEATURES}

\section{Multicase equations}

``Cases'' constructions like the following are common in
mathematics:
\begin{equation} P_{r-j}=
  \left\{
  \begin{array}{ll}
    0                &\mbox{if $r-j$ is odd},\\
    r!\,(-1)^{(r-j)/2} &\mbox{if $r-j$ is even}.
  \end{array}
  \right.
\end{equation}
and in the \pkg{amstex} package there is a \env{cases} environment:
\begin{verbatim}
\begin{equation} 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}
\end{verbatim}
Notice the use of \cn{text} and the embedded math.

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

The \pkg{amstex} package provides some environments for 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. And the maximum number of
columns is determined by the counter |MaxMatrixCols|, which you
can change if necessary using \latex/'s \cn{setcounter} or
\cn{addtocounter} commands.  I.e., suppose you have a big
matrix with 19 or 20 columns.  Then you'd write something like this:
\begin{verbatim}
\begin{equation}
\setcounter{MaxMatrixCols}{20}
A=\begin{pmatrix}
...&...&...&...&...&...&...&...&...&...&...&...&  ... \\
  ...  \\
  ...
\end{pmatrix}
\end{equation}
\end{verbatim}

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}.
\begin{verbatim}
\begin{math}
  \bigl( \begin{smallmatrix}
      a&b\\ c&d
    \end{smallmatrix} \bigr)
\end{math}
\end{verbatim}

\cn{hdotsfor}|{|\<number>|}| produces a row of dots in a matrix
spanning the given number of columns.
\begin{verbatim}
\begin{matrix} a&b&c&d\\
e&\hdotsfor{3} \end{matrix}
\end{verbatim}
would give dots spanning the last three columns in the second row.
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).

\section{Math spacing commands}

Both the spelled-out and abbreviated forms of these commands are
robust, and in addition they can also be used outside of math. The
primary math spacing commands are:
\begin{ctab}{ll@{\extracolsep{2.6em}}ll}
Abbrev.& Spelled out& Abbrev.& Spelled out\\
\hline
\cn{\,}& \cn{thinspace}& \cnbang& \cn{negthinspace}\\
\cn{\:}& \cn{medspace}& & \cn{negmedspace}\\
\cn{\;}& \cn{thickspace}& & \cn{negthickspace}\\
|@,|& & |@!|& \\
& \cn{quad}& & \\
& \cn{qquad}& &
\end{ctab}
|@,|\index{"@,@{\ntt\qcat,}} and |@!|\index{"@"!@{\ntt\qcat\qcbang}} give
one-tenth the space of \cn{\,} and {\cnbang} respectively, for extra
fine tuning where necessary.

\section{Over and under arrows}

There are some additional over and under arrow operations provided in
the \pkg{amstex} package:

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

All over and under operations, including the previously available ones
(\cn{overrightarrow}, \cn{overleftarrow}), have been modified to scale
properly in subscript  sizes. (After you have installed \amslatex/, you
can process and print the sample file \fn{testmath.tex} to see  examples
of the arrows.)

\section{Dots}

In the \pkg{amstex} package, ellipsis dots should almost
always be typed as
\cn{dots}. Placement (on the baseline or centered) is selected
according to whatever follows the \cn{dots}.  If the next thing is
a plus sign, the dots will be centered; if it's a comma, they will be on
the baseline.  These default dot placements provided by the
\pkg{amstex} package can be changed by the documentstyle if different
conventions are wanted.

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 $A_1,A_2,\ldots\,$,
the regional sum $A_1+A_2+\cdots\,$,
the orthogonal product $A_1A_2\cdots\,$,
and the infinite integral
\[\int_{A_1}\int_{A_2}\cdots\,.\]
\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 portable between places where different
conventions prevail.  The control sequences \cn{ldots} and \cn{cdots}
are still available, however, for compatibility.

\section{Accents in math}

The following accent commands automatically
give good positioning of double accents:
\begin{verbatim}
 \Hat    \Check  \Tilde  \Acute  \Grave  \Dot    \Ddot
 \Breve  \Bar    \Vec
\end{verbatim}
In ordinary \latex/ the second accent will usually be askew:
$\hat{\hat{A}}$ (\cn{hat}|{\hat{A}}|). In the \pkg{amstex} package, if
you type \cn{Hat}|{\Hat{A}}| (using the capitalized form for both
accents) the second accent will be properly positioned: $\Hat{\Hat{A}}$.

This double accent operation is complicated and tends to slow down the
processing of a \tex/ file. If your document contains many double
accents, you can use the \pkg{amsxtra} package, which provides an
\cn{accentedsymbol} command. \cn{accentedsymbol} is used like
\cn{newcommand}, in the preamble of your document, to store the result
of the double accent command in a `box register' 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{amstex} 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.

\section{Extensible arrows}

|@>>>| and |@<<<| produce arrows that extend automatically to
accommodate unusually wide subscripts or superscripts.  The text of a
superscript is typed in between the first and second |>| or
|<| symbols, and for a subscript, it's typed between the second and
third symbols. For example, |@>\xi F_k\Gamma_k\alpha>>| would have
a superscript $\xi F_k\Gamma_k\alpha$ placed above the arrow.   These
arrows were originally developed for use in commutative diagrams but can
be used elsewhere also.  (See section
\ref{s:commdiag} for more information about the \pkg{amscd} package.)

\section{Affixing symbols to other symbols}

\latex/ provides \cn{stackrel} for placing a superscript above a binary
relation. In the \pkg{amstex} 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{Operator names}\label{s:opname}

Math functions such as $\log$, $\sin$, and $\lim$ are traditionally
typeset in roman type to help avoid confusion with single math
variables, 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 \pkg{amstex} provide's a general
mechanism for producing such names: \cn{operatorname}|{xxx}|
produces {\upshape xxx} in the proper font and automatically adds proper
spacing on either side when necessary, so that you get $A\,\mathrm{xxx}\,B$
instead of $A\mathrm{xxx}B$.

Since \cn{operatorname} takes rather a lot of typing, you would usually
put some definitions in the preamble of the form
\begin{verbatim}
\newcommand{\xxx}{\operatorname{xxx}}
\newcommand{\yyy}{\operatorname{yyy}}
\end{verbatim}
for any operator names you're going to use frequently.

Some of the operator names, such as \cn{lim}, have actually been
defined using \cn{operatorname*} rather than \cn{operatorname},
because in displayed formulas if there is a subscript on \cn{lim} it
is conventionally placed underneath, like the limits on sums:
\begin{equation}
C_+ f(x)=\lim_{t\to0}C(f)(x+ it)
\end{equation}
You can use \cn{operatorname*} just like \cn{operatorname}; the only
difference is the placement of subscripts and superscripts. A few
special operator names with limits are defined for you in the
\pkg{amstex} package:  \cn{varinjlim}, \cn{varprojlim},
\cn{varliminf}, and \cn{varlimsup}; there are some examples in the
sample file \fn{testmath.tex}.

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

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

\section{Fractions and related constructions}

In addition to \cn{frac} (which was already available in \latex/),
\pkg{amstex} provides \cn{dfrac} and \cn{tfrac} as convenient
abbreviations for |{\displaystyle\frac| |...| |}| and
|{\textstyle\frac| |...| |}|. Furthermore, the
thickness of the fraction line can be varied, using an optional
argument of the \cn{frac} command.\index{dimension@{\itshape
dimension}}
\cn{frac}\5|[|\<dimension>|]|\5|{...}|\5|{...}|
makes a fraction where the thickness of the horizontal rule is
determined by the given dimension.  The sample file \fn{testmath.tex}
shows an example using a thickness of |1.5pt|.

For binomial expressions such as $\bigl({n\atop k}\bigr)$
\pkg{amstex} has \cn{binom}, \cn{dbinom} and \cn{tbinom}.

\section{Continued fractions}

The continued fraction
\begin{equation}
\def\cfrac#1#2{{\displaystyle\strut#1\over\displaystyle#2}%
 \kern-\nulldelimiterspace}
\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
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 plain \tex/ 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. In the \pkg{amstex} package \cn{smash} has optional arguments
|t| and |b|, because sometimes 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
$\sqrt{x}+\sqrt{y}$ and $\sqrt{x}+\sqrt{\smash[b]{y}}$,
where the latter was produced by
\verb"$\sqrt{x}" \verb"+" \verb"\sqrt{"\5\verb"\smash[b]{y}}$".

\section{Delimiter sizes}

A subject that escapes mention in the \latex/ book is how to control
the size of large delimiters if the automatic sizing done by \cn{left}
and \cn{right} produces unpleasing results. The automatic sizing has
two limitations: First, it is applied mechanically to produce
delimiters large enough to encompass the largest contained items, 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 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}{ll}
Delimiter size& Result\\
\hline
normal size& $\displaystyle(b)(\frac{c}{d})$\rule{0pt}{3ex}\\[1.6ex]
\cn{left}, \cn{right}& $\displaystyle\left(b\right)\left(\frac{c}{d}\right)$\rule{0pt}{3ex}\\[2ex]
\cn{bigl}, \cn{bigr}& $\displaystyle\bigl(b\bigr)\bigl(\frac{c}{d}\bigr)$\rule{0pt}{2ex}\\[2ex]
\cn{Bigl}, \cn{Bigr}& $\displaystyle\Bigl(b\Bigr)\Bigl(\frac{c}{d}\Bigr)$\rule{0pt}{2ex}\\[2ex]
\cn{biggl}, \cn{biggr}& $\displaystyle\biggl(b\biggr)\biggl(\frac{c}{d}\biggr)$\rule{0pt}{2.5ex}\\[2.5ex]
\cn{Biggl}, \cn{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 \cn{biggl} and \cn{biggr} 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*}
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(\left(a_1 b_1\right) - \left(a_2 b_2\right)\right)
\left(\left(a_2 b_1\right) + \left(a_1 b_2\right)\right)
\quad\text{versus}\quad
\bigl(\left(a_1 b_1\right) - \left(a_2 b_2\right)\bigr)
\bigl(\left(a_2 b_1\right) + \left(a_1 b_2\right)\bigr)
\end{equation*}
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 \cn{bigl} and \cn{bigr} can be used
to produce delimiters 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/ \cn{big}, \cn{bigg}, \cn{Big}, and \cn{Bigg}
delimiters aren't scaled properly over the full range of \latex/ font
sizes.  With the \pkg{amstex} package they are.

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

The \cn{text} command is defined by the \pkg{amstex} 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 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.

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

The \cn{boldsymbol} and \cn{pmb} commands are defined by the
\pkg{amstex} package through a subordinate package \pkg{amsbsy} (which
can also be used independently if desired). 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. 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.
\chapter{Integrals and sums (\pkg{amstex}, \pkg{amsintsm} packages)}

\section{Multiple integral signs}

\cn{iint}, \cn{iiint}, and \cn{iiiint} give multiple
integral signs 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 \env{Sb} and \env{Sp} environments can be used to typeset several
lines as a subscript or superscript:
for example
\begin{verbatim}
\begin{equation}
  \sum\begin{Sb}
        0\le i\le m\\ 0<j<n
      \end{Sb}
    P(i,j)
\end{equation}
\end{verbatim}
produces a two-line subscript underneath the sum:
\begin{equation}
  \sum_{0\le i\le m\atop 0<j<n}
    P(i,j)
\end{equation}
\env{Sb} and \env{Sp} can be used anywhere that an ordinary subscript or
superscript can be used.

\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
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}
But if you want not only the prime but also something below or
above the sum symbol, it's not so easy.  If you have
\begin{equation}
\sum_{n<k,\;n\ \mathrm{odd}}nE_n
\end{equation}
and you want to add a prime on the sum symbol, use \cn{sideset} like
this:
\begin{verbatim}
\sideset{}{'}\sum_{...}nE_n
\end{verbatim}
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{amsintsm} package}

The \pkg{amsintsm} package, which is \emph{not} automatically included
with the \pkg{amstex} package, is an experimental package that changes
the \cn{int} and \cn{sum} commands to better mark the boundaries of
the quantity being summed or integrated. For example,
\begin{verbatim}
\int[t,a,b]{(\Omega[x] -\lambda x^2)}
\end{verbatim}
means `the integral with respect to $t$, from $a$ to $b$, over the
following quantity':
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Simulate for now to get the desired output without using the
%    amsintsm package, which is not terribly stable yet.
%    [mjd,27-Oct-1994]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\int_a^b(\Omega[x] -\lambda x^2)\mathinner{dt}$.
For more details, run \latex/ on the file \fn{amsintsm.dtx}.

\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/, but for simple diagrams without diagonal arrows the
\pkg{amscd} commands may be more convenient.

The commutative diagram
\begin{equation}\label{e:cd}
\begin{array}{ccc}
S^{{\mathcal{W}}_\Lambda}\otimes T&
   \stackrel{j}{\longrightarrow}&  T\\
\Big\downarrow&
       &\Big\downarrow\vcenter{\rlap{$\scriptstyle{\rm End}\,P$}}\\
(S\otimes T)/I& =&  (Z\otimes T)/J
\end{array}
\end{equation}
can be produced in ordinary \latex/ by
\begin{verbatim}
\begin{array}{ccc}
S^{{\mathcal{W}}_\Lambda}\otimes T&
    \stackrel{j}{\longrightarrow}&  T\\
\Big\downarrow&
        &\Big\downarrow\vcenter{%
          \rlap{$\scriptstyle\mathrm{End}\,P$}}\\
(S\otimes T)/I& =&  (Z\otimes T)/J
\end{array}
\end{verbatim}
When the \pkg{amscd} package is used you would type instead
\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}
(with \cn{End} defined as \cn{operatorname}|{End}|; see
\secref{s:opname}). This would give longer horizontal arrows than in
(\ref{e:cd}) and improved spacing between elements of the diagram:
\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*}
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 full details on font use in \latex/, see the \latex/ documentation
(\fn{fntguide.tex}) and \cite{tlc}. Math font commands in \latex/
include \cn{mathbf}, \cn{mathrm}, \cn{mathcal}, \cn{mathsf},
\cn{mathtt}, \cn{mathit}. Certain other math alphabet commands are
available through the packages \pkg{amsfonts}, \pkg{eufrak}, and
\pkg{euscript}, if the requisite extra math fonts (from a font set
called `AMSFonts') are installed on your system (see the installation
instructions, Appendix~\ref{install}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Temporarily include amsfonts documentation here until the move to
%    the AMSFonts distrib is completed. But omit it if the \fn{.fd}
%    files have not been generated.
\IfFileExists{Ueuf.fd}{%
  \input{amsfdoc.txt}%
}{%
  \IfFileExists{Ueuf57.fd}{%
    \input{amsfdoc.txt}%
  }{}}%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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 \cn{mb}
instead of \cn{mathbf}. However, for \latex/ to provide shorter names
would actually be a disservice to authors, in light of the 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 you can write |\vec{a} + \vec{b}| to produce $\vec{a} +
\vec{b}$.
\footnote{%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If you actually tried this example you would discover that the command
\cn{vec} is already defined. Looking up the command \cn{vec}, you would
find that 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.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 an instance of a
vector, if you're not absolutely sure that you used \cn{mathbf}
\emph{only} for vectors and nothing else.

It might also be appropriate in some circumstances 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 complex numbers or real numbers in your document, you might
find this method more convenient than (let's say) defining a \cn{field}
command and writing |\field{C}|, |\field{R}|. But for maximum
flexibility and control, define such a \cn{field} command and then
define \cn{C} and \cn{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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Documentation about amsart/amsbook document classes, included here
%    pending reorganization plans.
\input{amscdoc.tx2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Theorems and similar structures (\pkg{amsthm} package)}
\markboth{THEOREMS AND SIMILAR STRUCTURES}{THEOREMS AND SIMILAR
STRUCTURES}
\label{amsthm}

\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. The \cls{amsart} and
\cls{amsbook} document classes automatically load the \pkg{amsthm}
package, so everything described here applies to them as well.

\section{The \cn{newtheorem} command}

In mathematical research articles and books, there are many
theorem-like elements serving authors' goal of stating theorems and
proofs: lemmas, propositions, axioms, theorems (of course),
corollaries, conjectures, definitions, remarks, proofs, 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 define environments for 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
and hierarchy of the numbering.

By default each 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 \env{lem} environment should share the \env{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}
The optional argument |[section]| means that the |prop|
counter will be reset to 0 whenever the parent counter |section|
is incremented.

\section{Changing styles for theorem-like environments}
\markright{CHANGING STYLES}

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|
[default], |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, for example,
the |plain| style, used for normally 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 \fn{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]

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

\section{Proofs}

A predefined \env{proof} environment is provided for proofs, and
produces the heading ``Proof'' with appropriate spacing and
punctuation. The proof environment is primarily intended for short
proofs, no more than a page or so in length; longer proofs should
probably be done as a separate section or subsection in your document.

A ``Q.E.D.'' symbol, \qedsymbol, is automatically appended at the end
of a proof. 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, 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 (sufficiency)'', then write
\begin{verbatim}
\begin{proof}[Proof (sufficiency)]
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Installation instructions}\label{install}
\section{Introduction}
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 (CTAN), directory
tex-archive/macros/latex, ftp addresses ftp.shsu.edu (US), ftp.dante.de
(Germany), or 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}
\markright{PUTTING FILES IN A SUITABLE PLACE}

See also the \fn{READ.ME} file for late-breaking news about 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. The basic procedure is to put all the files
in the \amslatex/ distribution into the \amslatex/ source files area,
then run \latex/ on an installation file \fn{amslatex.ins}, and then move
the resulting files to the \latex/ input area.
\begin{ctab}
{lll}
\multicolumn{3}{c}{\bfseries Typical area names}\\[3pt]
   &\amslatex/ source files area& \latex/ input area\\
\hline
DOS& \verb"c:\emtex\latex\packages\amslatex"&
  \verb"c:\emtex\latex\inputs"\\
Unix& \verb"/tex/latex/packages/amslatex"& \verb"/tex/latex/inputs"\\
Mac& \verb"OzTeX:latex:packages:amslatex"& \verb"OzTeX:latex:inputs"\\
VMS& \verb"tex_disk:[latex.packages.amslatex]"&
  \verb"tex_disk:[latex.inputs]"
\end{ctab}

In the following suggested installation process, we use \textsc{PC/DOS}
directory and file name conventions for concreteness; for other systems,
it should be obvious how to adapt the names.

The destination directory for the \amslatex/ files should be
\begin{verbatim}
  xxx\latex\packages\amslatex
\end{verbatim}
where \verb"xxx" depends on the type of computer system and \tex/ directory
structure the user has. Let's say it is em\tex/, then \verb"xxx" will normally
be \verb"c:\emtex".

So you would put the above files into the directory
\begin{verbatim}
  c:\emtex\latex\packages\amslatex
\end{verbatim}

\section{Running \latex/ on the installation script \fn{amslatex.ins}}
\markright{THE INSTALLATION SCRIPT \fn{amslatex.ins}}

Next run \latex/ on the installation file \fn{amslatex.ins}. This will
create the stripped files \fn{amstex.sty}, \fn{amssymb.sty}, \fn{amsart.cls}, etc., that
are actually used for processing documents. All these files should then
be moved to an input directory, e.g.,
\begin{verbatim}
  c:\emtex\latex\inputs
\end{verbatim}
(this directory should be added to your \tex/ search path, if it isn't
there already).

\section{Testing}

For a quick test of the installation, try printing the test file
\fn{subeqn.tex}. For more detailed and comprehensive information about
using \amslatex/, print the \amslatex/ user's guide (\fn{amsldoc.tex}).

\section{Extra math fonts}

There are \latex/ packages called \pkg{amsfonts} and \pkg{amssymb} that
are designed to exploit the extra math fonts provided by the AMSFonts
font set. If you don't have this set already on your system, and would
like to use some of the symbols in it, it can be obtained by anonymous
FTP from CTAN and from the AMS archive at \fn{e-math.ams.org}. It can
also be ordered on disk from the AMS catalog.

\section{Memory requirements}
Incomplete.

\section{Files included in this distribution}

[Incomplete]
See the \fn{READ.ME} file for details on the contents of the individual
files.

\begin{flushleft}
\ttfamily

\begin{tabular}[t]{l}
READ.ME\\
amslatex.ins\\
amslatex.chg\\
amslatex.bug\\
amsbsy.dtx\\
amscd.dtx\\
amsgen.dtx\\
amsintsm.dtx\\
amstex.dtx\\
amstext.dtx\\
amsxtra.dtx
\end{tabular}
%
\begin{tabular}[t]{l}
amsldoc.dtx\\
subeqn.tex\\
testmath.tex\\
dif11-12.tex
\end{tabular}
%
\end{flushleft}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Error messages}

Incomplete.

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

\chapter{Where to find other information}

\markright{Differences between versions 1.1 and 1.2}
\section{Differences between \amslatex/ version 1.1 and \amslatex/
version 1.2}
\markright{Differences between versions 1.1 and 1.2}

These are described in the document \fn{dif11-12.tex} (incomplete).

\section{Differences between \amstex/ and \amslatex/}
These are described in the document \fn{amstexdf.tex} (incomplete).

\section{Technical notes}
These are found in \fn{technote.tex} (incomplete).

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

\chapter{Getting help}\label{help}

Comments or questions regarding the \amslatex/ package should be sent
to:
\begin{infoaddress}
American Mathematical Society\\
Technical Support\\
P. O. Box 6248\\
Providence, RI 02940\\[3pt]
Phone: 800-321-4AMS (321-4267) \quad or \quad 401-455-4080\\
Internet: \mail{tech-support@@math.ams.org}
\end{infoaddress}
If you are reporting a problem you should include
the following information:
\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-math.ams.org}
can be obtained by sending a request through electronic mail to:
\[\mbox{\mail{e-math@@math.ams.org}}\]

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{cust-serv@@math.ams.org}
\end{infoaddress}

The \tex/ Users Group is a nonprofit organization that publishes
a journal (\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{tug@@tug.org}\\
\end{infoaddress}
Membership in the \tex/ Users Group is a good way to support
continued development of \tex/-related public-domain software.

There are also many \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}

\bibitem{amsfonts}\booktitle{AMSFonts version 2.1---user's guide},
American Mathematical Society, Providence, R.I., 1991; distributed
with the AMSFonts package.

\bibitem{author-guidelines}\booktitle{Instructions for preparing
electronic manuscripts---\amslatex/},
American Mathematical Society, Providence, R.I., 1990.

\bibitem{tlc} Michel Goossens, Frank Mittelbach, and Alexander Samarin,
\booktitle{The \latex/ Companion}, Addison-Wesley, 1993.

\bibitem{kn} Donald Knuth, \booktitle{The \tex/book}, Addison-Wesley,
1984.

\bibitem{lm} Leslie Lamport, \booktitle{\latex/: A document preparation
system}, Addison-Wesley, 2nd edition, 1994.

\bibitem{msf} Frank Mittelbach and Rainer Sch\"opf,
\textit{The new font family selection---user
interface to standard \latex/}, \journalname{TUGboat} \textbf{11},
no.~2 (June 1990), pp.~297--305.

\bibitem{jt} Michael Spivak, \booktitle{The joy of \tex/}, 2nd ed.,
American Mathematical Society, Providence, R.I., 1990.

\end{thebibliography}

%    This is amsldoc.ind generated on 26-Oct-1994. [mjd]
\begin{theindex}

  \item {\normalfont  \ttfamily  \bslchar \qcbang  }, 9
  \item \cn  {\,}, 9
  \item \cn  {\:}, 9
  \item \cn  {\;}, 9
  \item {\ntt\qcat\qcbang}, 9
  \item {\ntt\qcat,}, 9
  \item \cn  {\\}, 2, 6
  \item \cn  {\\*}, 6

  \indexspace

  \item \cn  {accentedsymbol}, 10, 11
  \item \cn  {addtocounter}, 8
  \item \cn  {addtolength}, 5
  \item \cn  {adjusttag}, 6
  \item \env  {align} environment, 3, 5
  \item \env  {alignat} environment, 3, 4
  \item \env  {aligned} environment, 6
  \item \env  {alignedat} environment, 6
  \item \cn  {allowdisplaybreaks}, 6
  \item \fn  {amsart.cls}, 20
  \item \pkg  {amsbsy} package, 1, 15
  \item \pkg  {amscd} package, 1, 11, 17
  \item \pkg  {amsfonts} package, 1, 20
  \item \pkg  {amsintsm} package, 1, 15, 16
  \item \fn  {amsintsm.dtx}, 16
  \item \fn  {amslatex.ins}, 19, 20
  \item \fn  {amsldoc.tex}, 20
  \item \pkg  {amssymb} package, 20
  \item \fn  {amssymb.sty}, 20
  \item \pkg  {amstex} package, 1, 2, 5--16
  \item \fn  {amstex.sty}, 20
  \item \fn  {amstexdf.tex}, 21
  \item \pkg  {amstext} package, 1, 14
  \item \pkg  {amsxtra} package, 10
  \item \env  {array} environment, 8

  \indexspace

  \item \cn  {Big}, 14
  \item \cn  {big}, 14
  \item \cn  {Bigg}, 14
  \item \cn  {bigg}, 14
  \item \cn  {Biggl}, 14
  \item \cn  {biggl}, 14
  \item \cn  {Biggr}, 14
  \item \cn  {biggr}, 14
  \item \cn  {Bigl}, 14
  \item \cn  {bigl}, 14
  \item \cn  {Bigr}, 14
  \item \cn  {bigr}, 14
  \item \cn  {binom}, 12
  \item \env  {bmatrix} environment, 8
  \item \cn  {bmod}, 12
  \item \cn  {boldsymbol}, v, 1, 15
  \item \cn  {boxed}, 11

  \indexspace

  \item \cn  {C}, 18
  \item \env  {cases} environment, v, 8
  \item \env  {CD} environment, 1, 17
  \item \cn  {cdots}, 10
  \item \opt  {centertags} option, 1
  \item \cn  {cfrac}, 13
  \item \fn  {comp.text.tex}, 22

  \indexspace

  \item \cn  {dbinom}, 12
  \item \cn  {ddddot}, 11
  \item \cn  {dddot}, 11
  \item \cn  {ddot}, 11
  \item \cn  {dfrac}, 12
  \item \fn  {dif11-12.tex}, 21
  \item {\itshape dimension}, 12
  \item \cn  {displaybreak}, 6
  \item \env  {displaymath} environment, 2
  \item \cn  {documentclass}, 2
  \item \cn  {dot}, 11
  \item \cn  {dots}, 10
  \item \cn  {dotsb}, 10
  \item \cn  {dotsc}, 10
  \item \cn  {dotsi}, 10
  \item \cn  {dotsm}, 10

  \indexspace

  \item \fn  {e-math.ams.org}, 20, 21
  \item \cn  {End}, 17
  \item \cn  {end}, 10
  \item \env  {eqnarray} environment, v, 2, 6
  \item \cn  {eqref}, 7
  \item \env  {equation} environment, v, 2, 4, 5
  \item \env  {equation*} environment, 2

  \indexspace

  \item \cn  {fbox}, 11
  \item \cn  {field}, 18
  \item \opt  {fleqn} option, 2, 5
  \item \fn  {fntguide.tex}, 18
  \item \cn  {frac}, 12

  \indexspace

  \item \env  {gather} environment, 3, 5
  \item \env  {gathered} environment, 6

  \indexspace

  \item \cn  {Hat}, 10
  \item \cn  {hat}, 10
  \item \cn  {hdotsfor}, 9

  \indexspace

  \item \cn  {idotsint}, 15
  \item \cn  {iiiint}, 15
  \item \cn  {iiint}, 15
  \item \cn  {iint}, 15
  \item \cn  {int}, 1, 16
  \item \cn  {intertext}, 6
  \item \opt  {intlim} option, 2
  \item \opt  {intlimits} option, 1

  \indexspace

  \item \cn  {ldots}, 10
  \item \cn  {left}, 13, 14
  \item \cn  {leftroot}, 11
  \item \opt  {leqno} option, 2
  \item \cn  {lim}, v, 11, 12
  \item \cn  {log}, 11

  \indexspace

  \item \cn  {mathbb}, 18
  \item \cn  {mathbf}, 15, 18
  \item \cn  {mathcal}, 18
  \item \cn  {mathfrak}, 18
  \item \cn  {mathit}, 18
  \item \cn  {mathrm}, 18
  \item \cn  {mathsf}, 18
  \item \cn  {mathtt}, 18
  \item \env  {matrix} environment, 8
  \item \cn  {mb}, 18
  \item \cn  {mbox}, 15
  \item \cn  {medspace}, 9
  \item \cn  {mod}, 12
  \item \env  {multline} environment, 4, 5
  \item \cn  {multlinegap}, 4, 5

  \indexspace

  \item \opt  {namelimits} option, 1
  \item \cn  {negmedspace}, 9
  \item \cn  {negthickspace}, 9
  \item \cn  {negthinspace}, 9
  \item \cn  {newcommand}, 11
  \item \cn  {newtheorem}, 7
  \item \opt  {nointlimits} option, 1
  \item \cn  {nolimits}, 16
  \item \opt  {nonamelimits} option, 1
  \item \opt  {nonamelm} option, 2
  \item \opt  {nosumlim} option, 2
  \item \opt  {nosumlimits} option, 1
  \item \cn  {notag}, 2
  \item \cn  {numberwithin}, 7

  \indexspace

  \item \cn  {operatorname}, 12, 17
  \item \cn  {operatorname*}, 12
  \item \cn  {overleftarrow}, 9
  \item \cn  {overleftrightarrow}, 9
  \item \cn  {overrightarrow}, 9
  \item \cn  {overset}, 11

  \indexspace

  \item \cn  {pagebreak}, 6
  \item \env  {pmatrix} environment, 8
  \item \cn  {pmb}, 1, 15
  \item \cn  {pmod}, 12
  \item \cn  {pod}, 12

  \indexspace

  \item \cn  {qquad}, 9
  \item \cn  {quad}, 9

  \indexspace

  \item \cn  {R}, 18
  \item \fn  {READ.ME}, 20
  \item \cn  {renewcommand}, 18
  \item \opt  {reqno} option, 2
  \item \cn  {right}, 13, 14
  \item \opt  {righttag} option, 2

  \indexspace

  \item \env  {Sb} environment, 15, 16
  \item \cn  {setcounter}, 7, 8
  \item \cn  {setlength}, 5
  \item \cn  {sideset}, 11, 16
  \item \cn  {sin}, v, 11
  \item \env  {smallmatrix} environment, 8
  \item \cn  {smash}, 13
  \item \env  {Sp} environment, 15, 16
  \item \env  {split} environment, 2, 5
  \item \cn  {stackrel}, 11
  \item \fn  {subeqn.tex}, 20
  \item \cn  {sum}, 1, 16
  \item \opt  {sumlimits} option, 1

  \indexspace

  \item \cn  {tag}, 2
  \item \cn  {tag*}, 2
  \item \cn  {tbinom}, 12
  \item \opt  {tbtags} option, 1
  \item \fn  {technote.tex}, 21
  \item \fn  {testmath.tex}, 2, 9, 12
  \item \cn  {text}, 1, 8, 14
  \item \cn  {tfrac}, 12
  \item \cn  {theequation}, 7
  \item \cn  {thickspace}, 9
  \item \cn  {thinspace}, 9

  \indexspace

  \item \cn  {underleftarrow}, 9
  \item \cn  {underleftrightarrow}, 9
  \item \cn  {underrightarrow}, 9
  \item \cn  {underset}, 11
  \item \cn  {uproot}, 11
  \item \cn  {usepackage}, 1, 2

  \indexspace

  \item \cn  {varinjlim}, 12
  \item \cn  {varliminf}, 12
  \item \cn  {varlimsup}, 12
  \item \cn  {varprojlim}, 12
  \item \cn  {vec}, 18
  \item \env  {Vmatrix} environment, 8
  \item \env  {vmatrix} environment, 8

  \indexspace

  \item \env  {xalignat} environment, 3, 4
  \item \env  {xxalignat} environment, 3

\end{theindex}

\end{document}