diff options
Diffstat (limited to 'Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex')
-rw-r--r-- | Master/texmf-dist/doc/latex-dev/amsmath/testmath.tex | 69 |
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}. |