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-rw-r--r--Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex69
1 files changed, 21 insertions, 48 deletions
diff --git a/Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex b/Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex
index 069944ee9d0..c0df3370153 100644
--- a/Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex
+++ b/Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex
@@ -1,41 +1,15 @@
%%% ====================================================================
-%%% @LaTeX-file{
-%%% filename = "testmath.tex",
-%%% version = "2.0",
-%%% date = "1999/11/15",
-%%% time = "15:09:17 EST",
-%%% checksum = "07762 2342 7811 82371",
-%%% author = "American Mathematical Society",
-%%% copyright = "Copyright 1995, 1999 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.",
-%%% 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 = "tech-support@ams.org (Internet)",
-%%% codetable = "ISO/ASCII",
-%%% keywords = "latex, amsmath, examples, documentation",
-%%% supported = "yes",
-%%% abstract = "This is a test file containing extensive examples of
-%%% mathematical constructs supported by the amsmath
-%%% package.",
-%%% 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.",
-%%% }
+%%% filename = testmath.tex
+%%% version = 2.0a
+%%% date = 2023/08/24
+%%% author = American Mathematical Society
+%%% copyright = Copyright 1995, 1999 American Mathematical Society
+%%% 2023 LaTeX Project.
+%%% License = https://www.latex-project.org/lppl/lppl-1-3c
+%%% keywords = latex, amsmath, examples, documentation
+%%% abstract = This is a test file containing extensive examples of
+%%% mathematical constructs supported by the amsmath
+%%% package."
%%% ====================================================================
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1994/12/01]% LaTeX date must December 1994 or later
@@ -45,7 +19,7 @@
\title{Sample Paper for the \pkg{amsmath} Package\\
File name: \fn{testmath.tex}}
\author{American Mathematical Society}
-\date{Version 2.0, 1999/11/15}
+\date{Version 2.0a, 2023/08/24}
\usepackage{amsmath,amsthm}
@@ -250,7 +224,7 @@ Then, it follows that
\end{lem}
Note that all basic properties of determinants are direct consequences
-of Lemma ~\ref{lem-det}. Write
+of Lemma~\ref{lem-det}. Write
\begin{equation}\label{sum-bij}
\sum_{j\in\mathbf{n}}b_{ij}\hat y_j=\sum_{j\in\mathbf{n}}b^{(\lambda)}
_{ij}\hat y_j+(b_{ii}-\lambda_i)\hat y_i\hat y
@@ -325,7 +299,7 @@ D(t_1,\dots,t_n)
where $\mathbf{K}(t=1,t_1,\dots,t_n; i|i)$ is the $i$th principal
submatrix of $\mathbf{K}(t=1,t_1,\dots,t_n)$.
-Theorem ~\ref{thm-main} leads to
+Theorem~\ref{thm-main} leads to
\begin{equation}\label{detK1}
\det\mathbf{K}(t_1,t_1,\dots,t_n)
=\sum_{I\in\mathbf{n}}(-1)^{\envert{I}}t^{n-\envert{I}}
@@ -339,7 +313,7 @@ Note that
^{(\lambda)}(\overline{I}|\overline{I})=0.
\end{equation}
-Let $t_i=\hat x_i,i=1,\dots,n$. Lemma ~\ref{lem-per} yields
+Let $t_i=\hat x_i,i=1,\dots,n$. Lemma~\ref{lem-per} yields
\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 |l )\\
@@ -381,8 +355,8 @@ D(t_1,\dots,t_n)}_{t_i=\left\{\begin{smallmatrix}
\section{Application}
\label{lincomp}
-We consider here the applications of Theorems~\ref{th-info-ow-ow} and
-~\ref{th-weak-ske-owf} to a complete
+We consider here the applications of Theorems~\ref{th-info-ow-ow}
+and~\ref{th-weak-ske-owf} to a complete
multipartite graph $K_{n_1\dots n_p}$. It can be shown that the
number of spanning trees of $K_{n_1\dots n_p}$
may be written
@@ -395,8 +369,7 @@ where
n=n_1+\dots+n_p.
\end{equation}
-It follows from Theorems~\ref{th-info-ow-ow} and
-~\ref{th-weak-ske-owf} that
+It follows from Theorems~\ref{th-info-ow-ow} and~\ref{th-weak-ske-owf} that
\begin{equation}\label{e:barwq}
\begin{split}
H_c&=\frac1{2n}
@@ -423,7 +396,7 @@ H_c&=\frac12\sum^{n-1}_{l =0}
\end{equation}
The enumeration of $H_c$ in a $K_{n_1\dotsm n_p}$ graph can also be
-carried out by Theorem ~\ref{thm-H-param} or ~\ref{thm-asym}
+carried out by Theorem~\ref{thm-H-param} or~\ref{thm-asym}
together with the algebraic method of \eqref{multdef}.
Some elegant representations may be obtained. For example, $H_c$ in
a $K_{n_1n_2n_3}$ graph may be written
@@ -499,7 +472,7 @@ We begin our discussion with the following definition:
A function $H\colon \Re^n \to \Re^n$ is said to be
\emph{B-differentiable} at the point $z$ if (i)~$H$ is Lipschitz
-continuous in a neighborhood of $z$, and (ii)~ there exists a positive
+continuous in a neighborhood of $z$, and (ii)~there exists a positive
homogeneous function $BH(z)\colon \Re^n \to \Re^n$, called the
\emph{B-derivative} of $H$ at $z$, such that
\[ \lim_{v \to 0} \frac{H(z+v) - H(z) - BH(z)v}{\enVert{v}} = 0. \]
@@ -1112,7 +1085,7 @@ A_l =
0,&\text{otherwise }.\end{cases}
\label{compl-bip-gr}
\end{equation}
-Theorem ~\ref{thm-H-param}
+Theorem~\ref{thm-H-param}
leads to
\begin{equation}
H_c=\frac1{n_1+n_2}n_1!n_2!\delta_{n_1n_2}.