% \iffalse meta-comment % % Copyright (C) 2019-2022 by Antoine Missier % % This file may be distributed and/or modified under the conditions of % the LaTeX Project Public License, either version 1.3 of this license % or (at your option) any later version. The latest version of this % license is in: % % http://www.latex-project.org/lppl.txt % % and version 1.3 or later is part of all distributions of LaTeX version % 2005/12/01 or later. % \fi % % \iffalse %<*driver> \ProvidesFile{mismath.dtx} % %<*package> \NeedsTeXFormat{LaTeX2e}[2005/12/01] \ProvidesPackage{mismath} [2023/01/06 v2.2 .dtx mismath file] % %<*driver> \documentclass{ltxdoc} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} %\usepackage{textcomp} \usepackage[english]{babel} \usepackage[charter]{mathdesign} % in case of font change, modify 2.1 (3rd §) \usepackage[ibrackets]{mismath} %\usepackage{amssymb} incompatible with mathdesign \usepackage{multicol} \usepackage[Euler]{upgreek} \usepackage{sectsty} \usepackage{xcolor} \definecolor{darkbrown}{rgb}{0.5,0.1,0.1} \allsectionsfont{\color{darkbrown}} %\enumber % commented because I need italicized e in vectors section \inumber \pinumber[piup] % For using in this doc only: \usepackage[LGR,T1]{fontenc} \DeclareSymbolFont{UpGr}{LGR}{lmr}{m}{n} \DeclareMathSymbol{\mypi}\mathalpha{UpGr}{"70} % For showing original \i and \j in text mode: \DeclareTextFontCommand{\extrafont}{\fontfamily{lmr}\selectfont} \DisableCrossrefs %\CodelineIndex %\RecordChanges \usepackage{hyperref} \hypersetup{% colorlinks, linkcolor=blue, citecolor=red, pdftitle={mismath}, pdfsubject={LaTeX package}, pdfauthor={Antoine Missier} } \begin{document} \DocInput{mismath.dtx} %\PrintChanges %\PrintIndex \end{document} % % \fi % %% \CheckSum{459} % % \CharacterTable % {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z % Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z % Digits \0\1\2\3\4\5\6\7\8\9 % Exclamation \! Double quote \" Hash (number) \# % Dollar \$ Percent \% Ampersand \& % Acute accent \' Left paren \( Right paren \) % Asterisk \* Plus \+ Comma \, % Minus \- Point \. Solidus \/ % Colon \: Semicolon \; Less than \< % Equals \= Greater than \> Question mark \? % Commercial at \@ Left bracket \[ Backslash \\ % Right bracket \] Circumflex \^ Underscore \_ % Grave accent \` Left brace \{ Vertical bar \| % Right brace \} Tilde \~} % % \changes{v0.1}{2011/12/27}{First personal version} % % \changes{v1.0}{2019/04/11}{ % - Initial published version: creating dtx and ins files} % % \changes{v1.1}{2019/04/20}{ % - Some forgotten french 'et' -> 'and', citecolor=blue, % - changing the default font for pinumber Euler -> Symbol} % % \changes{v1.2}{2019/04/26}{ % - Works fine with beamer now, % - AtBeginDocument for enumber, inumber, jnumber, % - creating general @moperator macro, % - using mathup instead of mathrm} % \changes{v1.2}{2019/04/27}{ % - Added mathtools package, % - font definition Roman -> up, % - changes in documentation, % - removing the PEroman macro} % % \changes{v1.3}{2019/05/05}{ % - Using bslash in the internal @mwarning macro, % - loading of mathfixs package} % \changes{v1.3}{2019/05/08}{ % - Many corrections in documentation} % % \changes{v1.4}{2019/05/22}{ % - Changing font definition up -> UpSh, due to incompatibility with unicode-math} % % \changes{v1.5}{2019/05/30}{ % - A solution for using mul with frac -> braces, % - addition of paren macro} % \changes{v1.5}{2019/06/22}{ % - Small corrections in documentation} % % \changes{v1.6}{2019/09/06}{ % - Removing mathfixs package, problems with fractions} % % \changes{v1.7}{2019/12/27}{ % - Adding a table of contents to the documentation} % % \changes{v1.8}{2020/11/15}{ % - Incompatibility mentioned when using `i' with accent in beamer titles -> use \^i, % - small changes in documentation} % % \changes{v1.9}{2020/10/17}{ % - UpSh replaced by "operators" font in math mode, % - PackageWarning `command exist' replaced by PackageWarningNoLine, % - medspace replaced by thickspace in the lfrac command, % - changing font in documentation, lmodern -> Palatino (mathpazo)} % % \changes{v1.10}{2020/10/25}{\ % - pinumber command code adapted to avoid incompatibility with the new frenchmath, % - default option (in upgreek) changed from Symbol to Euler.} % % \changes{v2.0}{2022/11/11}{ % - pinumber command has been improved to use other Greek letter packages, % (and by default LGR encoding), it is no longer compatible with the older, % - paren command has been removed, % - hvect and hlbar have been slighlty modified (phantom t instead of phantom h), % - several changes in documentation, % - now using the Charter font (mathdesign)} % % \changes{v2.1}{2022/12/26}{ % - Improved management of square brackets with the ibrackets package, % - new macros codim, sinc, var, eqdef*, % - removing the systematic warning for paren command % - a small change in norm command -> bars in small size % - several changes in documentation} % % \changes{v2.2}{2023/01/06}{ % - Loading of ibrackets package is now optional, due to an error % when using DeclarePairedDelimiter with square brackets.} % % \GetFileInfo{mismath.sty} % % \title{\textsf{mismath}\\ Miscellaneous mathematical macros\thanks{This document % corresponds to \textsf{mismath}~\fileversion, dated \filedate. % Thanks to François Bastouil for help in English translation.}} % \author{Antoine Missier \\ \texttt{antoine.missier@ac-toulouse.fr}} % \date{January 6, 2023} % % \maketitle % \tableofcontents % % \section{Introduction} % % According to the International Standards ISO~31-0:1992 to ISO~31-13:1992, % superseded by ISO~80000-2:2009, mathematical constants $\e$, $\i$, $\pi$ % should be typeset in roman (upright shape) and not in italic (sloping shape) like % variables (see~\cite{TYPMA}~\cite{NIST}~\cite{SI}~\cite{ICTNS}). % This package provides some tools to achieve this (automatically). % % \medskip % Even if it is recommended to typeset vectors names % in bold italic style~\cite{NIST}~\cite{ICTNS}, % they are often represented with arrows % (particularly in school documents or in physics). % To draw pretty arrows above vectors, we use the \textsf{esvect} package % by Eddie Saudrais~\cite{VECT} % and we provide a few more macros related to vectors with arrows, % in particular to improve the typesetting of the norm: $\norm{\vect{AB}}$ % instead of \LaTeX\ version $\lVert\vect{AB}\rVert$ which is not vertically adjusted, % or worse $\left\Vert \vect{AB} \right\Vert$ (and even ugly with Latin Modern font family). % % \smallskip % The package also provides other macros for: % \begin{itemize} % \item some standard operator names, % \item a few useful aliases, % \item improving some spacing in mathematical formulas, % \item systems of equations and small matrices, % \item displaymath in double columns for long calculation. % \end{itemize} % % To avoid incompatibility, % a large majority of our macros will be defined only % if there is not another command with the same name in the packages loaded % before \textsf{mismath}. If a macro is already defined, % compilation will produce a warning message and \textsf{mismath} definition % will simply be ignored. To keep \textsf{mismath} command, % either load \textsf{mismath} before the other package % with which it is in conflict for the name of that command % (assuming the other package supports it), or use % |\let\|\meta{command}|\relax| before loading \textsf{mismath}. % % \medskip % \DescribeEnv{\meta{options}} % The \textsf{amsmath} package is loaded by \textsf{mismath} without option. % For using \textsf{amsmath} with options (see~\cite{AMS}), % these options can be added when calling \textsf{mismath}, or % \textsf{amsmath} has to be loaded % with the required options before \textsf{mismath}. % % Another package, % \textsf{mathtools} by Morten Høgholm and Lars Madsen~\cite{TOOL} % is also loaded. It provides many useful macros and improvements of \textsf{amsmath} package. % % \medskip % A recommendation, seldom observed, is to typeset uppercase Greek letters in italic shape % like other variables~\cite{ICTNS}. This is automatically done with the packages % \textsf{fixmath} by Walter Schmidt~\cite{FIXM}, % \textsf{isomath} by Günter Milde~\cite{ISOM} % or \textsf{pm-isomath} by Claudio Beccari~\cite{PMISO} and optionally with many others % (for instance \textsf{mathpazo} or \textsf{mathptmx} with the option \texttt{slantedGreek}), % but this feature is not implemented here because this rule is conflicting % to the one used in France where all mathematics capitals % have to be typeset in upright shape\footnote{The \textsf{frenchmath} package~\cite{FR} % takes this rule into account.}. % The choice of loading or not one of these packages remains thus to the user. % % \section{Usage} % % \subsection{Mathematical constants} % % \DescribeMacro{\mathup} % \DescribeMacro{\e} \DescribeMacro{\i} \DescribeMacro{\j} % As for classic functions identifiers, \emph{predefined} mathematical constants % should be typeset in upright shape (generally in roman family), % even if this practice is not really common and tedious % to respect. To avoid to stuff a document with |\mathrm{e}| or |\mathrm{i}| % (or better |\mathup{e}| and |\mathup{i}|\footnote{\texttt{\bslash mathup} % is based on \texttt{\bslash operatorfont} % (from \textsf{amsopn} package, automatically loaded by \textsf{amsmath}). % The \textsf{beamer} % package uses a default sans serif math font, but \texttt{\bslash mathrm} % produces a font with serif in \textsf{beamer}. This problem is solved by % using \texttt{\bslash mathup} instead % of \texttt{\bslash mathrm}.}), % the package provides |\e| command for the base of the natural logarithm % and |\i| or |\j| for imaginary numbers. % Let's notice that |\i| and |\j| already exist in \LaTeX: % using in LR mode, they produce ``\extrafont{\i,\ \j}'' without the point, % so you can place accents on them, and in mathematical mode they produce % ``\texttt{LaTeX Warning: Command \bslash i invalid in math mode on input line} \meta{line}''. % Te new definition of |\i| and |\j| concerns only mathematical mode\footnote{Due to this % \texttt{\bslash i} command redefinition, there is an incompatibility with % \textsf{beamer} when using i with accents in beamer titles. % A solution is to use the classic \texttt{\bslash \textasciicircum i} % command to produce î in beamer titles for example.}. % % \medskip % \DescribeMacro{\enumber} \DescribeMacro{\inumber} \DescribeMacro{\jnumber} % Nevertheless, it can be tiresome to type a lot of backslashes in a document % with many formulas containing $\e$ or $\i$. % So a way is proposed here to free of it by placing |\enumber|, |\inumber| or |\jnumber| % in the preamble: % $\e$, $i$ or $\j$ will then automatically be set in roman % in the whole document, no need to type |\e|, |\i| or |\j|, % let's hope that there are not many other $\mathit{e}$, $\mathit{i}$ or $j$ as variables. % However, you can still get italicized % $\mathit{e}$, $\mathit{i}$ or $\mathit{j}$ with \LaTeX\ command % |\mathit| or |\mathnormal|. Of course, this does not fully comply with \LaTeX\ philosophy: % in the document body, objects should be pointed out % by their nature rather than their typographical characteristics, % defined in the preamble. But these macros are really handy and % thanks to them it is possible to bring a document up to the standards % afterwards; moreover anyone is free to use them or not. % % \medskip % \DescribeMacro{\pinumber\oarg{command}} % The mathematical constant $\pi$ should also be typeset in upright shape % (see~\cite{TYPMA}, \cite{NIST}, \cite{ICTNS}), which differs from italicized $\itpi$. % This recommendation is even less observed than the one concerning % $\e$ and $\i$~\cite{TYPMA}. % Several packages allow to typeset mathematical Greek letters in upright shape, % let us mention \textsf{upgreek}~\cite{GREEK}, \textsf{mathdesign}~\cite{DESIGN} % (used here), \textsf{kpfonts}~\cite{KPF}, \textsf{fourier}~\cite{FOUR}, % \textsf{libertinust1math}, % \textsf{pxgreeks}, \textsf{txgreeks}, \textsf{libgreek}, etc. % A special mention to \textsf{lgrmath} of Jean-François Burnol~\cite{LGR} % which allow to use, in math mode, any Greek LGR-encoded font. % These packages provide commands like |\uppi| (\textsf{upgreek}), % |\piup| (\textsf{mathdesign}, \textsf{kpfonts}, \textsf{lgrmath}), % |\otherpi| (\textsf{fourier}), etc.\footnote{They also have options to % typeset all the Greek lowercase letters in upright shape by default, % but this in not our goal here.} % To preserve default sloped lowercase Greek letters except for pi, and to % avoid typing a lot of |\uppi| or |\piup|, we provide the |\pinumber| macro, % which has to be put in the preamble. This command redefines |\pi| to match % the optional command given, for instance |piup|, assuming the appropriate % package has been loaded before. % % By activating |\enumer|, |\inumber| and |\pinumber[piup]| in the preamble, % you can get for instance : % \begin{center} % |$e^{i\pi} = -1$| \enskip \hspace{3em} \enskip $\e^{i\pi}=-1$. % \end{center} % % When no argument is given, |\pinumber| defines |\pi| with an LGR encoding of Greek letters % to produce $\mypi$. It looks the same as the one supplied with % Günter Milde's \textsf{textalpha} package~\cite{ALPHA}. % This $\mypi$ is particularly suitable % for use with the default Computer Modern or Latin Modern font family\footnote{This % default $\mypi$ doesn't fit well with many text fonts, more bold than Computer Modern; % the \textsf{upgreek} package~\cite{GREEK} provides often a better $\uppi$ % and it has also a \texttt{Symbol} option (using Adobe Symbol font) % that fits well with several text fonts, for instance Times.}. % % \DescribeMacro{\itpi} % When |\pinumber| is activated, the original italic $\itpi$ is still available with |\itpi|. % % \subsection{Vectors} % % \DescribeMacro{\vect} % By default, the |\vect| command\footnote{As for many macros of this package, % the definition will take effect only if this macro is not defined before % by another package.}, % produces vectors with arrows % (thanks to the \textsf{esvect} package of Eddie Saudrais\footnote{\textsf{esvect} % provides \texttt{\bslash vv} macro used by \texttt{\bslash vect}.}) % which are more elegant than those produced by \LaTeX's |\overrightarrow| command. % The \textsf{esvect} package has an optional argument % (one letter between \texttt{a} and \texttt{h}) defining % the required type of arrow (see~\cite{VECT}). % In \textsf{mismath}, \textsf{esvect} is loaded with the option \texttt{b}: % |\vect{AB}| gives $\vect{AB}$. % To choose another type of arrow, \textsf{esvect} must be called % with the required option \emph{before} \textsf{mismath}, % e.g.\@ |\usepackage[d]{esvect}| % will give the arrows produced by default in~\cite{VECT}. % % \medskip % \DescribeMacro{\boldvect} % The |\vect| macro allow to typeset vector's names using bold italic % (according to ISO recommendation~\cite{NIST}~\cite{SI}) rather than arrows. % For this, calling |\boldvect| will modify the behavior of |\vect|, % globally or locally, depending on where |\boldvect| is placed:\\[1ex] % \begin{minipage}{8cm} % \begin{verbatim} %\[ \boldvect \vect{v} % =\lambda\vect{e}_x+\mu\vect{e}_y. \] % \end{verbatim} % \end{minipage} % \begin{minipage}{6cm} % $\boldvect \vect{v}=\lambda\vect{e}_x +\mu\vect{e}_y$. % \end{minipage} % % \DescribeMacro{\boldvectcommand} % By default |\boldvect| uses the |\boldsymbol| % command\footnote{\texttt{\bslash mathbf} gives upright bold font, % even if used in combination with \texttt{\bslash mathit}.} % from \textsf{amsbsy} package, loaded by \textsf{amsmath}. % But other packages producing bold italic can be preferred, e.g.\@ % \texttt{\bslash bm} from \textsf{bm} package or |\mathbold| from \textsf{fixmath} % package or |\mathbfit| from \textsf{isomath}. % For that, redefine |\boldvectcommand|, for instance: % \begin{center} |\renewcommand\boldvectcommand{\mathbold}|. \end{center} % % By setting |\boldvectcommand| to |\mathbf|, |\vect| produces vectors % in bold \emph{upright} shape, % which tends to be used instead of bold \emph{italic} % (but probably for bad reasons). % % \medskip % \DescribeMacro{\arrowvect} % At any moment, you can get back to the default behavior with the inverse switch % |\arrowvect|. These switches can be placed anywhere: % inside mathematical mode or inside an environment (with local effect) or outside % (with global effect). % % \medskip % \DescribeMacro{\hvect} % When vectors with arrows are typeset side by side, % arrows can be set up a bit higher (with a vertical phantom box containing $t$) % to avoid inelegant effects: % \begin{itemize} % \item $\vect{AB}=\hvect{u}+ \vect{AC}$, obtained with |\hvect{u}|, % is better than $\vect{AB}=\vect{u}+ \vect{AC}$; % \item $\hvect{a} \cdot \vect{b}=0$, obtained with |\hvect{a}|, % is better thant $\vect{a} \cdot \vect{b}=0$. % \end{itemize} % The |\boldvect| switch has the same effect on |\hvect| than on |\vect|. % % \medskip % \DescribeMacro{\hvec} % In a similar way, |\hvec| raises the little arrow produced by % the \LaTeX\ command |\vec| (from height of $t$ letter): % \begin{itemize} % \item $\mathcal{P}=\vec{f}\cdot\hvec{v}$, obtained with |\hvec{v}|, % is better than $\mathcal{P}=\vec{f}\cdot\vec{v}$. % \item $\vec{f} =m \hvec{a}$, obtained with |\hvec{a}|, % is better than $\vec{f} =m \vec{a}$. % \end{itemize} % % \DescribeMacro{\norm} % The norm of a vector is classically produced by the delimiters |\lVert| and |\rVert| % (rather than \texttt{\bslash}$\mid$) or |\left\Vert| and |\right\Vert| % for delimiters adapting to the content. Unfortunately, these delimiters % are always vertically centered, relatively to the middle of the base line, % whereas vectors with arrows are asymmetric objects. % The code |$\norm{\vec{h}}$| raises a smaller double bar to produce $\norm{\vec{h}}$ % instead of $\left\Vert \vec{h} \right\Vert$. % Let's notice that the height of the bars don't adjust to content, % but however to context: main text, subscripts or exponents, % e.g. $\e^{\norm{\vec{h}}}$. % % \subsection{Standard operator names} % % \DescribeMacro{\di} % The \emph{differential} operator should be typeset in upright shape and not in % italic, to make it different from variables % (as mentioned in \cite{TYPMA}~\cite{NIST}~\cite{ICTNS}~\cite{LSHORT}). % For this, we provide the |\di| command. % See the following examples (notice the thin spaces before the d, % as for classic function's names): \\ % \begin{minipage}[t]{7cm} % \begin{verbatim} %\[ \iint xy\di x\di y \] % \end{verbatim} % \end{minipage} % \begin{minipage}{6cm} % \[ \iint xy\di x\di y \] % \end{minipage} % \\ % \begin{minipage}[t]{7cm} % \begin{verbatim} %\[ m\frac{\di^2x}{\di t^2} % + h\frac{\di x}{\di t} + kx = 0 \] % \end{verbatim} % \end{minipage} % \begin{minipage}[t]{6cm} % \[m\frac{\di^2x}{\di t^2}+h\frac{\di x}{\di t}+kx=0\] % \end{minipage} % % This command can also stand for \emph{distance} (hence its name): % \[\lambda\di(A,\mathcal{F})+\mu\di(B,\mathcal{H}).\] % % \DescribeMacro{\P} \DescribeMacro{\E} \DescribeMacro{\PEupright} \DescribeMacro{\V} % To refer to probability\footnote{\LaTeX\ provides % also \texttt{\bslash Pr} which gives $\Pr$.} % and expectation the proper use is to typeset capital letters $\P$, $\E$ % in roman as for any standard function identifier. % This is obtained with |\P| and |\E|. % In the same way as for $\i$ and $\j$, you can use |\PEupright| in the preamble % to avoid typing many |\P| or |\E|. % Variance is generally denoted by $\var$ or $\Var$ (see table below), % but some authors prefer to use $\V$, produced by |\V|. % % \medskip % \DescribeMacro{\Par} % The |\P| command already existed to refer to the end of paragraph symbol \Par\ % and has been redefined, but this symbol can still be obtained with |\Par|. % % \medskip % \DescribeMacro{\probastyle} % Some authors use ``blackboard bold'' font % to represent probability, expectation and variance: $\mathbb{P}, \mathbb{E}, \mathbb{V}$. % The |\probastyle| macro sets the appearance of |\P|, |\E| and |\V|: % for instance |\renewcommand\probastyle{\mathbb}|\footnote{As for % \texttt{\bslash boldvect} and \texttt{\bslash arrowvect}, % effect is local to the container environment.} % brings the previous ``openwork'' letters. % |\mathbb| comes from \textsf{amsfonts} package % (loaded by \textsf{amssymb} % but also available standalone) % which has to be called in the preamble. % % \medskip % The following standard operator names are defined in \textsf{mismath}: % \begin{center} % \begin{tabular}{rlrlrl} % |\adj| & $\adj$ \qquad\mbox{} & |\erf| & $\erf$ \qquad\mbox{} & |\Re| & $\Re$ \\ % |\Aut| & $\Aut$ & |\grad| & $\grad$ & |\rot| & $\rot$ \\ % |\codim| & $\codim$ & |\id| & $\id$ & |\sgn| & $\sgn$ \\ % |\Conv| & $\Conv$ & |\Id| & $\Id$ & |\sinc| & $\sinc$ \\ % |\cov| & $\cov$ & |\im| & $\im$ & |\spa| & $\spa$ \\ % |\Cov| & $\Cov$ & |\Im| & $\Im$ & |\tr| & $\tr$ \\ % |\curl| & $\curl$ & |\lb| & $\lb$ & |\var| & $\var$ \\ % |\divg| & $\divg$ & |\lcm| & $ \lcm$ & |\Var| & $\Var$ \\ % |\End| & $\End$ & |\rank| & $\rank$ & |\Zu| & $\Zu$ % \end{tabular} % \end{center} % % By default, operators returning vectors, |\grad| and |\curl| (or its synonym |\rot| % rather used in Europe), are written with an arrow on the top. % When |\boldvect| is activated, they are typeset in bold style: % $\boldvect \grad, \curl, \rot$. % For the variance, the covariance and the identity function, % two notations are proposed, with or without a first capital letter, % because they are both very common. % On the other hand, ``$\im$'' stands for the image of a linear transformation % (like ``$\ker$'' for the kernel) % whereas ``$\Im$'' is the imaginary part of a complex number. % Notice that |\div| already exist ($\div$) and |\span| is a \TeX\ primitive % (used in |\multicolumn|); % they haven't been redefined, therefore the macros |\divg| (divergence) % and |\spa| (span of a set of vectors) ; % |\Z| is used for the set of integers (see \ref{aliases}), therefore we used |\Zu|, % to designate the center of a group: $\Zu(G)$ (from German Zentrum). % %\medskip % \DescribeMacro{\oldRe} \DescribeMacro{\oldIm} % The |\Re| and |\Im| macros already existed, to refer to real and imaginary part % of a complex number, producing outdated symbols $\oldRe$ and $\oldIm$. % They have been redefined according to actual use, as mentioned in the above table, % but it's still possible to get the old symbols with |\oldRe| and |\oldIm|. % % \medskip % Some (inverse) circular or hyperbolic functions, missing % in \LaTeX, are also provided by \textsf{mismath}: % \begin{center} % \begin{tabular}{rlrlrl} % |\arccot| & $\arccot$\qquad\mbox{} & |\arsinh| & $\arsinh$\qquad\mbox{} % & |\arcoth| & $\arcoth$ \\ % |\sech| & $\sech$ & |\arcosh| & $\arcosh$ & |\arsech| & $\arsech$ \\ % |\csch| & $\csch$ & |\artanh| & $\artanh$ & |\arcsch| & $\arcsch$ % \end{tabular} % \end{center} % % \DescribeMacro{\bigO} \DescribeMacro{\bigo} \DescribeMacro{\lito} % Asymptotic comparison operators (in Landau notation) are obtained with % |\bigO| or |\bigo| and |\lito| commands: % \[ n^2+\bigO(n\log n) \txt{or} n^2+\bigo(n\log n)\txt{and} \e^x=1+x+\lito(x^2).\] % % \subsection{A few useful aliases} \label{aliases} % % In the tradition of Bourbaki and D.~Knuth, proper use requires % that classic sets of numbers are typeset in bold roman: % $\R, \C, \Z, \N, \Q$, % whereas ``openwork'' letters ($\mathbb{R}, \mathbb{Z}, \ldots$) % are reserved for writing at blackboard~\cite{LSHORT}; % and likewise to designate a field: $\F$ or $\K$ (Körper in German). % We get these symbols with the macros: % \begin{center} % |\R|, |\C|, |\Z|, |\N|, |\Q|, |\F|, |\K|. % \end{center} % % \DescribeMacro{\mathset} % The |\mathset| command enables to change the behavior of all these macros in a global way: % by default, |\mathset| is an alias for |\mathbf|, but if you prefer openwork letters, % just place |\renewcommand\mathset{\mathbb}| where you want, for instance in the preamble, % after loading \textsf{amsfonts} package (which provides the ``blackboard bold'' typeface, % also loaded by \textsf{amssymb}). % % \medskip % \DescribeMacro{\ds} % The |\displaystyle| command being very common, alias |\ds| is provided. % Not only it eases typing but also it makes source code more readable. % % \medskip % Symbols with limits behave differently for in-line formulas or for displayed equations. % In the latter case, ``limits'' are put under or above whereas for in-line math mode, % they are placed on the right, as subscript or exponent. Compare: % $\zeta(s)=\sum_{n=1}^{\infty }\frac{1}{n^s}$ with % \[\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.\] % % \DescribeMacro{\dlim} \DescribeMacro{\dsum} \DescribeMacro{\dprod} % \DescribeMacro{\dcup} \DescribeMacro{\dcap} % With in-line math mode, displaymath behavior can be forced with |\displaystyle| % or its alias |\ds|, but then, all the rest of the current mathematical % environment will be set in displaymath mode too (in the previous example, % the fraction will be expanded). % Just as the \textsf{amsmath} command |\dfrac| % only transforms the required fraction in display style, we can limit % the display style effect to the affected symbol, by using the following macros: % |\dlim|, |\dsum|, |\dprod|, |\dcup|, |\dcap|. % So |$\dlim_{x\to +\infty}\frac{1}{x}$| gives $\dlim_{x \to +\infty}\frac{1}{x}$. % % \bigskip % \DescribeMacro{\lbar} \DescribeMacro{\hlbar} % Large bars over expressions are obtained with |\overline| % or, shorter, its alias |\lbar|, to get for instance $\lbar{z_1z_2}$. % Such as for vectors, you can raise the bar (from the height of $h$) with % the |\hlbar| command, in order to correct uneven bars heights. % \begin{center} % $\lbar{z+z'}=\hlbar{z}+\lbar{z'}$, obtained with |\hlbar{z}|, % is better than $\lbar{z+z'}=\lbar{z}+\lbar{z'}$. % \end{center} % % \DescribeMacro{\eqdef} \DescribeMacro{\eqdef*} % The |\eqdef| macro writes equality symbol topped with ``def'' % or with ``$\scriptstyle \Delta$'' for |\eqdef*| % (thanks to the \LaTeX\ command |\stackrel|): % % \noindent\begin{minipage}[t]{8cm} % \begin{verbatim} %$ \e^{\i\theta} \eqdef % \cos\theta + \i\sin\theta $ % \end{verbatim} % \end{minipage} % \begin{minipage}{6cm} % $\e^{\i\theta}\eqdef\cos\theta + \i\sin\theta$ % \end{minipage} % % \DescribeMacro{\unbr} % |\unbr| is an alias for |\underbrace|\footnote{The \textsf{mathtools} % package by Morten Høgholm and Lars Madsen~\cite{TOOL} % provides a new improved version of \texttt{\bslash underbrace} command % (as many other usefull macros); % it is loaded by \textsf{mismath}.}, making source code more compact.\\[2ex] % \begin{minipage}{7.5cm} % \begin{verbatim} %$ (QAP)^n = \unbr{QAP\mul QAP\mul % \cdots\mul QAP}_{n\text{ times}} $ % \end{verbatim} % \end{minipage} % \begin{minipage}{6.5cm} % $ (QAP)^n = \unbr{QAP\mul QAP\mul\cdots\mul QAP}_{n\text{ times}} $ % \end{minipage} % % \DescribeMacro{\iif} % |\iif| is an alias for ``\iif'', to be used in text mode. % % \subsection{Improved spacing in mathematical formulas} % % \DescribeMacro{\then} % The |\then| macro produces the symbol $\Longrightarrow$ surrounded by large spaces % as the standard macro |\iff| does it with $\Longleftrightarrow$. % In a similar way, |\txt| \DescribeMacro{\txt} % based on the |\text| macro (from the \textsf{amstext} package, % loaded by \textsf{amsmath}), % leaves em quad spaces (|\quad|) around the text. See the following example: % \begin{center} % |\[ \ln x=a \then x=\e^a \txt{rather than}| \\ % | \ln x=a \Longrightarrow x=\e^a \]| \\[1ex] % $ \ln x=a \then x=\e^a \txt{rather than} \ln x=a \Longrightarrow x=\e^a$ % \end{center} % % \DescribeMacro{\mul} % The multiplication symbol obtained with |\times| produces the same spacing than addition % or subtraction operators, whereas division obtained with $/$ is closer to its operands. % This actually hides the priority of the multiplication on $+$ and $-$. % This is why we provide the |\mul| macro, behaving like $/$ % (ordinary symbol) and leaving less space around than |\times|: % \begin{center} % $\lambda+\alpha \mul b-\beta \mul c$, obtained with |\mul|, % is better than $\lambda+\alpha \times b-\beta \times c$. % \end{center} % % When using |\mul| before a function name or around a |\left...\right| structure, % the space may be too large on one side of |\mul|. % To get the same amount of space on the two sides of |\mul|, % you can use thin negative spaces |\!| % or enclose the function or the structure with braces: % \begin{center} % $x\mul{\sin x}$, obtained with |x\mul{\sin x}|, is slightly better than $x\mul\sin x$.\\[1ex] % |$\sin\!{\left( \frac{\pi}{3} \right)} \mul 2$| gives\\ % $\sin\!{\left(\frac{\pi}{3}\right)}\mul 2$ which is better than % $\sin\left(\frac{\pi}{3}\right)\mul 2$. % \end{center} % The thin negative space after the function name is not relative to |\mul|, % but is due to the fact that spaces around a |\left...\right| structure % are bigger than those produced by single parenthesis |(...)|. % % \medskip % \DescribeMacro{\pow} % In the same way, when typesetting an exponent after a closing \emph{big} % parenthesis produced by |\right)|, % the exponent is little to far from the parenthesis. % The command |\pow|\marg{expr}\marg{pow} sets % \meta{expr} between parentheses and puts the exponent \meta{pow} % slightly closer to the right parenthesis\footnote{This macro gives bad % results with normal sized parenthesis.}. Compare: % \[ \e^a \sim\pow{1+\frac{a}{n}}{n} \txt{may be better than} % \e^a \sim\left(1+\frac{a}{n}\right)^{n}.\] % % \DescribeMacro{\abs} % Absolute value (or modular for a complex number) should be typeset with % |\lvert| \ldots |\rvert| rather than $\mid$ which doesn't respect correct % spaces for delimiters; for bars whose height has to adapt to content, % we use |\left\vert| \ldots |\right\vert| or, more simply, % the |\abs|\{\ldots\} command which is equivalent\footnote{Another % solution is to define \texttt{\bslash abs} with the % \texttt{\bslash DeclarePairedDelimiter} command % from the \textsf{mathtool} package~\cite{TOOL}.}. % % \medskip % \DescribeMacro{\lfrac} % This macro behaves like |\frac| % but with thick spaces around the arguments, % so the corresponding fraction bar is perceptibly a little bit longer:\\ % \begin{minipage}[t]{8cm} % \begin{verbatim} %\[ \lbar{Z} = % \lfrac{\lbar{z_1-z_2}}{\lbar{z_1+z_2}} \] % \end{verbatim} % \end{minipage} % \begin{minipage}[t]{4cm} % \[ \lbar{Z} = \lfrac{\lbar{z_1-z_2}}{\lbar{z_1+z_2}} \] % \end{minipage} % % \DescribeMacro{[ibrackets]} % Open intervals are usually represented with parenthesis, e.g.\@ $(0, +\infty)$, % but sometimes we find also % square brackets, for example in French mathematics. % In that case the space around them is often unsuitable, % e.g.\@ $x \in \mathclose{]} 0, +\infty[$. % % We have redefine brackets in the \textsf{ibrackets} package~\cite{BRACKET} % which can be optionally\footnote{This functionality is optional because it causes % error when using a command defined by \texttt{\bslash DeclarePairedDelimiter}~\cite{TOOL} % with square brackets.} % loaded by \textsf{mismath}. % Simply type |$x\in ]-\pi,0[ \cup ]2\pi,3\pi[$| to get % \begin{align*} % x\in ]-\pi, 0[ \cup ]2\pi, 3\pi[ &\mbox{\quad with \textsf{ibrackets}}, \\ % \mbox{instead of \quad} % x\in \mathclose{]}-\pi, 0 \mathopen{[} \cup \mathclose{]} 2\pi, 3\pi \mathopen{[} % &\mbox{\quad without \textsf{ibrackets}}. % \end{align*} % In our code, $[$ and $]$ symbols become ``active'' % and are not defined by default as delimiters. % Thereby a line break could occur between the two brackets, but % it is always possible to transform them % into delimiters with |\left| and |\right|. % % With \textsf{ibrackets}: a bracket becomes an ordinary character % but an open delimiter when it is immediately followed by a + or - character. % Thus, when the left bound contains an operator sign, % \emph{you don't have to leave a space between the first bracket and the sign}, % otherwise, the spaces surrounding the operator will be too large: % e.g.\@ |$x \in ] -\infty, 0]$| yields $x \in ] -\infty, 0]$. % Contrariwise, when you want to write algebra on intervals then % \emph{you must leave a blank space between the second bracket and the} +/- % \emph{operations}, % e.g.\@ |$[a, b] + [c, d]$| yields $[a, b] + [c, d]$ % but |$[a, b]+ [c, d]$| yields $[a, b]+ [c, d]$. % % Let us also mention other approaches with the |\interval| macro % from the \textsf{interval} package~\cite{INT}, % or |\DeclarePairedDelimiters| from the \textsf{mathtool} package~\cite{TOOL} % (but the latter is incompatible with \textsf{ibrackets} % for brackets management). % % \subsection{Environments for systems of equations and small matrices} % % \DescribeEnv{system} % The \texttt{system} environment produces a system of equations:\\ % \begin{minipage}[t]{6.5cm} % \begin{verbatim} %$\begin{system} % x=1+2t \\ y=2-t \\ z=-3-t %\end{system}$ % \end{verbatim} % \end{minipage} % \begin{minipage}[t]{5cm} % \[ \begin{system} x=1+2t \\ y=2-t \\z=-3-t \end{system} \] % \end{minipage} % % \medskip % \DescribeMacro{\systemsep} % This first example could also have been produced with \texttt{cases} environment % from \textsf{amsmath} package, although \texttt{cases} places mathematical expressions % closer to the bracket (which makes sense considering it's use). % |\systemsep| enables to set the gap between the bracket and the expressions, % set by default to |\medspace|. This gap may be reduced, for instance: % |\renewcommand{\systemsep}{\thinspace}|, % or enlarged with |\thickspace| (and with |\renewcommand\systemsep}{}| % we obtain what \texttt{cases} do). % % \medskip % \DescribeEnv{system\oarg{coldef}} % By default, a system is written like an \texttt{array} environment with only one column, % left aligned. The environment has an optional argument to create several columns, % specifying their alignment, with the same syntax than the \texttt{array} environment of % \LaTeX : |\begin{system}[cl]| produces a two-column system, the first one being centered, % the second being left aligned, such as in the following example:\\ % \begin{minipage}[t]{7cm} % \begin{verbatim} %$\begin{system}[cl] % y & =\dfrac{1}{2}x-2 \\[1ex] % (x,y) & \neq (0,-2) %\end{system}$ % \end{verbatim} % \end{minipage} % \begin{minipage}[t]{5cm} % \[ \begin{system}[cl] y&=\dfrac{1}{2}x-2 \\[1ex] (x,y)&\neq (0,-2) \end{system}\] % \end{minipage} % % \DescribeMacro{\systemstretch} % Default spacing between the lines of a \texttt{system} environment has been slightly % enlarged compared to the one from \texttt{array} environments (from 1.2 factor). % This spacing may be changed by typing |\renewcommand{\systemstretch}|\marg{stretch}, % inside the current mathematical environment (for a local change) or outside % (for a global change). By default, stretch's value is 1.2. % In addition we can use the end of line with a spacing option such % as it has been done above with |\\[1ex]|. % % Another example with |\begin{system}[rl@{\quad}l]|\footnote{\texttt{@\{\ldots\}} % sets inter-column space.}: % \begin{equation*} % \begin{system}[rl@{\quad}l] % x+3y+5z&=0 & R_1\\ 2x+2y-z&=3 & R_2\\ 3x-y+z&=2 & R_3 % \end{system} % \iff % \begin{system}[rl@{\quad}l] % x+3y+5z&=0 & R_1\\ % 4y+11z&=3 & R_2 \gets 2R_1-R_2 \\ % 5y+7z&=-1 & R_3 \gets \frac{1}{2}\left(3R_1-R_3\right) % \end{system} % \end{equation*} % % Let's mention the \textsf{systeme} package~\cite{SYST} which deals with linear systems % with a lighter syntax and automatic alignments on $+$, $-$, $=$, % and also the \textsf{spalign} package~\cite{SPAL} which moreover produces nice alignments % for matrices (with spaces and semi­colons as de­lim­iters). % % \medskip % \DescribeEnv{spmatrix} % The \textsf{amsmath} package provides various environments to typeset matrices: % for instance \texttt{pmatrix} surrounds the matrix with parenthesis % or \texttt{smallmatrix} typesets a small matrix that can even be inserted % in a text line. We provide a combination of the two with \texttt{spmatrix}:\\ % |$\vec{u}\begin{spmatrix}-1\\2\end{spmatrix}$| yielding % $\vec{u}\begin{spmatrix}-1\\2\end{spmatrix}$. % % The \textsf{mathtools} package enhance \textsf{amsmath} matrices environments % and provides also a small matrix environment with parenthesis. % Furthermore, with starred version |\begin{psmallmatrix*}|\oarg{col}, % you can choose the alignment inside the columns (\texttt{c}, \texttt{l} or \texttt{r}). % But sadly, the space before the left parenthesis is too narrow % regarding to the space inside the parenthesis. % Compare previous $\vec{u}\begin{spmatrix}-1\\2\end{spmatrix}$ % with $\vec{u}\begin{psmallmatrix}-1\\2\end{psmallmatrix}$. % % \subsection{Displaymath in double columns} % % \DescribeEnv{mathcols} % The \texttt{mathcols} environment activates mathematical mode and enables to arrange % ``long''calculation in double columns, separated with a central rule, % as shown in the following example. % But you have to load the \textsf{multicol} package in the preamble. % \begin{mathcols} % & \frac{1}{2 \mul {\pow{\frac{1}{4}}{n}} + 1} \geq 0.999 \\ % \iff\ & 1 \geq 1.998 \pow{\frac{1}{4}}{n} + 0.999 \\ % \iff\ & 0.001 \geq \frac{1.998}{4^n} \\ % \changecol % & \iff 4^n \geq 1998 \\ % & \iff n \ln 4 \geq \ln(1998) \\ % & \iff n \geq \frac{\ln(1998)}{\ln 4} \approx 5.4 \\ % & \iff n \geq 6 % \end{mathcols} % % \DescribeMacro{\changecol} % The |\changecol| macro causes a change of column; % alignment is produced using the classic delimiters |&| and |\\|. % % \begin{verbatim} %\begin{mathcols} % & \frac{1}{2 \mul {\pow{\frac{1}{4}}{n}} + 1} \geq 0.999 \\ % \iff\ & 1 \geq 1.998 \pow{\frac{1}{4}}{n} + 0.999 \\ % \iff\ & 0.001 \geq \frac{1.998}{4^n} \\ %\changecol % & \iff 4^n \geq 1998 \\ % & \iff n \ln 4 \geq \ln(1998) \\ % & \iff n \geq \frac{\ln(1998)}{\ln 4} \approx 5.4 \\ % & \iff n \geq 6 %\end{mathcols} % \end{verbatim} % % \StopEventually{} % \section{Implementation} % % \begin{macrocode} \RequirePackage{ifthen} \newboolean{ibrackets} \DeclareOption{ibrackets}{\setboolean{ibrackets}{true}} \DeclareOption*{\PassOptionsToPackage{\CurrentOption}{amsmath}} \ProcessOptions \relax \@ifpackageloaded{amsmath}{}{\RequirePackage{amsmath}} \@ifpackageloaded{esvect}{}{\RequirePackage[b]{esvect}} \RequirePackage{xspace} \RequirePackage{mathtools} \ifthenelse{\boolean{ibrackets}}{\RequirePackage{ibrackets}}{} % \end{macrocode} % The above conditional packages loading avoids ``option clash'' errors if the packages % have been previously loaded with other options. % % \medskip % \DescribeMacro{\bslash} % The |\bslash| macro comes from Frank Mittelbach's \textsf{doc.sty} package. % It can also be used in other documents instead of |\textbackslash| % (which doesn't work inside warnings). % \begin{macrocode} {\catcode`\|=\z@ \catcode`\\=12 |gdef|bslash{\}} % \bslash command % \end{macrocode} % \medskip % \DescribeMacro{\@mwarning} \DescribeMacro{\@mmacro} \DescribeMacro{\@moperator} % The three following internal macros are meta commands for a % conditional macro definition with a warning message if the macro already exists. % They could be useful in other packages. % \begin{macrocode} \newcommand\@mwarning[1]{ \PackageWarningNoLine{mismath}{ Command \bslash #1 already exist and will not be redefined} } \newcommand\@mmacro[2]{ \@ifundefined{#1}{ \expandafter\def\csname #1\endcsname{#2} }{\@mwarning{#1}} } \newcommand\@moperator[3][]{% this macro is ugly, by default #1=#3 \ifthenelse{\equal{#1}{}}{ \@ifundefined{#3}{ \DeclareMathOperator{#2}{#3} }{\@mwarning{#3}} }{ \@ifundefined{#1}{ \DeclareMathOperator{#2}{#3} }{\@mwarning{#1}} } } % \end{macrocode} % % To produce the correct upright shape font even when working % with the \textsf{beamer} package, we did not use |\mathrm| % but |\mathup| (based on |\operatorfont| from the \textsf{amsopn} package). % This command works also fine with other sans serif fonts like \textsf{cmbright}. % Moreover for \textsf{beamer}, |\enumber| must be typeset in % the family default font (sans serif), % therefore the |\AtBeginDocument| inside the macro (otherwise it has no effect). % The same holds for |\inumber| and |\jnumber|. % % |\AtBeginDocument| is also necessary to redefine |\i| when calling % the \textsf{hyperref} package which overwrites the |\i| definition. % % \medskip % \begin{macrocode} \providecommand{\mathup}[1]{{\operatorfont #1}} % also in kpfonts \@mmacro{e}{\mathup{e}} \AtBeginDocument{\let\oldi\i \let\oldj\j \renewcommand{\i}{\TextOrMath{\oldi}{\mathup{i}}} \renewcommand{\j}{\TextOrMath{\oldj}{\mathup{j}}} } \newcommand{\enumber}{ \AtBeginDocument{\DeclareMathSymbol{e}\mathalpha{operators}{`e}}} \newcommand{\inumber}{ \AtBeginDocument{\DeclareMathSymbol{i}\mathalpha{operators}{`i}}} \newcommand{\jnumber}{ \AtBeginDocument{\DeclareMathSymbol{j}\mathalpha{operators}{`j}}} \newcommand*{\pinumber}[1][defaultpi]{ \@ifundefined{itpi}{\let\itpi\pi}{\@mwarning{itpi}} \ifthenelse{\equal{#1}{defaultpi}}{ \usepackage[LGR,T1]{fontenc} \DeclareSymbolFont{UpGr}{LGR}{lmr}{m}{n} \DeclareMathSymbol{\pi}\mathalpha{UpGr}{"70} }{ \@ifundefined{#1}{ \PackageWarningNoLine{mismath}{ \bslash pinumber command has changed since v2.0, \MessageBreak option #1 must be a valid command name \MessageBreak (look at the documentation), but command \bslash #1 is undefined, \MessageBreak I cannot use it for replacement to \bslash pi. \MessageBreak Perhaps a missing package} }{\renewcommand{\pi}{\csname #1\endcsname}} } } \newboolean{arrowvect} \setboolean{arrowvect}{true} \newcommand{\arrowvect}{\setboolean{arrowvect}{true}} \newcommand{\boldvect}{\setboolean{arrowvect}{false}} \newcommand{\boldvectcommand}{\boldsymbol} % from amsbsy package \@mmacro{vect}{\ifthenelse{\boolean{arrowvect}}{ \vv}{\boldvectcommand}} \newcommand*{\hvect}[1]{\vect{\vphantom{t}#1}} \newcommand*{\hvec}[1]{\vec{\vphantom{t}#1}} \newcommand*{\@norm}[1]{ \mbox{\raisebox{1.75pt}{\small$\bigl\Vert$}} #1 \mbox{\raisebox{1.75pt}{\small$\bigr\Vert$}} } % works better than with relative length \newcommand*{\@@norm}[1]{ \mbox{\footnotesize\raisebox{1pt}{$\Vert$}} #1 \mbox{\footnotesize\raisebox{1pt}{$\Vert$}} } \newcommand*{\@@@norm}[1]{ \mbox{\tiny\raisebox{1pt}{$\Vert$}} #1 \mbox{\tiny\raisebox{1pt}{$\Vert$}} } \@ifundefined{norm}{\providecommand*{\norm}[1]{ \mathchoice{\@norm{#1}}{\@norm{#1}}{\@@norm{#1}}{\@@@norm{#1}} } }{\@mwarning{norm} } % bad result with libertinust1math \@mmacro{di}{\mathop{}\!\mathup{d}} \newcommand\probastyle{} \let\Par\P % end of paragraph symbol \renewcommand{\P}{\operatorname{\probastyle{P}}} \@mmacro{E}{\operatorname{\probastyle{E}}} \@mmacro{V}{\operatorname{\probastyle{V}}} \newcommand{\PEupright}{ \AtBeginDocument{% necessary for working with beamer \DeclareMathSymbol{P}\mathalpha{operators}{`P} \DeclareMathSymbol{E}\mathalpha{operators}{`E} } } \@moperator{\adj}{adj} \@moperator{\Aut}{Aut} \@moperator{\codim}{codim} \@moperator{\Conv}{Conv} \@moperator{\cov}{cov} \@moperator{\Cov}{Cov} \@mmacro{curl}{\operatorname{\vect{\mathup{curl}}}} \@moperator[divg]{\divg}{div} \@moperator{\End}{End} \@moperator{\erf}{erf} \@mmacro{grad}{\operatorname{\vect{\mathup{grad}}}} \@moperator{\id}{id} % mathop or mathord ? \@moperator{\Id}{Id} \@moperator{\im}{im} \let\oldIm\Im \renewcommand{\Im}{\operatorname{Im}} \@moperator{\lb}{lb} \@moperator{\lcm}{lcm} \@moperator{\rank}{rank} \let\oldRe\Re \renewcommand{\Re}{\operatorname{Re}} \@mmacro{rot}{\operatorname{\vect{\mathup{rot}}}} \@moperator{\sgn}{sgn} \@moperator{\sinc}{sinc} \@moperator[spa]{\spa}{span} \@moperator{\tr}{tr} \@moperator{\var}{var} \@moperator{\Var}{Var} \@moperator[Zu]{\Zu}{Z} \@moperator{\arccot}{arccot} \@moperator{\sech}{sech} \@moperator{\csch}{csch} \@moperator{\arsinh}{arsinh} \@moperator{\arcosh}{arcosh} \@moperator{\artanh}{artanh} \@moperator{\arcoth}{arcoth} \@moperator{\arsech}{arsech} \@moperator{\arcsch}{arcsch} \@moperator[bigO]{\bigO}{\mathcal{O}} \@moperator[bigo]{\bigo}{O} \@moperator[lito]{\lito}{o} \@mmacro{mathset}{\mathbf} \@mmacro{R}{\ensuremath{\mathset{R}}\xspace} \@mmacro{C}{\ensuremath{\mathset{C}}\xspace} \@mmacro{N}{\ensuremath{\mathset{N}}\xspace} \@mmacro{Z}{\ensuremath{\mathset{Z}}\xspace} \@mmacro{Q}{\ensuremath{\mathset{Q}}\xspace} \@mmacro{F}{\ensuremath{\mathset{F}}\xspace} \@mmacro{K}{\ensuremath{\mathset{K}}\xspace} \@mmacro{ds}{\displaystyle} \@mmacro{dlim}{\lim\limits} \@mmacro{dsum}{\sum\limits} \@mmacro{dprod}{\prod\limits} \@mmacro{dcup}{\bigcup\limits} \@mmacro{dcap}{\bigcap\limits} \@mmacro{lbar}{\overline} \@ifundefined{hlbar}{ \providecommand*{\hlbar}[1]{\overline{\vphantom{t}#1}}}{ \@mwarning{hlbar} } \newcommand\@eqdef{\stackrel{\mathup{def}}{=}} \newcommand\@@eqdef{\stackrel{\Delta}{=}} \@mmacro{eqdef}{\@ifstar{\@@eqdef}{\@eqdef}} \@mmacro{unbr}{\underbrace} \@mmacro{iif}{if and only if\xspace} \@mmacro{then}{\ \Longrightarrow \ \mbox{} } % \end{macrocode} % Without |\mbox{}|, the space produced by |\| would be suppressed in tables. % \medskip % \begin{macrocode} \@ifundefined{txt}{ \providecommand*{\txt}[1]{\quad\text{#1}\quad} }{ \@mwarning{txt} } \@mmacro{mul}{\mathord{\times}} \providecommand\paren{\PackageWarning{mismath}{Command \bslash paren is no longer supported}} \@ifundefined{pow}{ \providecommand*{\pow}[2]{\left( #1 \right)^{\!#2}} }{ \@mwarning{pow} } \@ifundefined{abs}{ \providecommand*{\abs}[1]{\left\vert#1\right\vert} }{ \@mwarning{abs} } \@ifundefined{lfrac}{ \providecommand*{\lfrac}[2]{\frac{\;#1\;}{\;#2\;}} }{ \@mwarning{lfrac} } \newcommand{\systemstretch}{1.2} \newcommand{\systemsep}{\medspace} \newenvironment{system}[1][l]{ \renewcommand{\arraystretch}{\systemstretch} \setlength{\arraycolsep}{0.15em} \left\{\begin{array}{@{\systemsep}#1@{}} % }{\end{array}\right.} \newenvironment{spmatrix}{ \left(\begin{smallmatrix} }{\end{smallmatrix}\right)} \newenvironment{mathcols}{% needs multicol package \renewcommand{\columnseprule}{0.1pt} \begin{multicols}{2} \par\noindent\hfill \begin{math}\begin{aligned}\displaystyle }{% \end{aligned}\end{math} \hfill\mbox{} \end{multicols} } \newcommand{\changecol}{% \end{aligned}\end{math} \hfill\mbox{} \par\noindent\hfill \begin{math}\begin{aligned}\displaystyle } % \end{macrocode} % \bigskip % \begin{thebibliography}{23} % \begin{raggedright} % \bibitem{TYPMA} \emph{Typesetting mathematics for science and technology according % to ISO 31/XI}, Claudio Beccari, TUGboat Volume 18 (1997), No.~1. % \url{http://www.tug.org/TUGboat/tb18-1/tb54becc.pdf}. % \bibitem{NIST} \emph{Typefaces for Symbols in Scientific Manuscripts}, % \url{https://www.physics.nist.gov/cuu/pdf/typefaces.pdf}. % \bibitem{SI} \emph{Guide for the Use of the International System of Units (SI)}, % NIST (National Institute of Standards and Technology), updated March 4, 2020 % \url{https://www.nist.gov/pml/special-publication-811}. % \bibitem{ICTNS} \emph{On the Use of Italic and up Fonts for Symbols in Scientific Text}, % I.M.~Mills and W.V.~Metanomski, ICTNS (Interdivisional Committee % on Terminology, Nomenclature and Symbols), dec 1999, % \url{https://old.iupac.org/standing/idcns/italic-roman_dec99.pdf}. % \bibitem{VECT} \emph{\textsf{esvect} -- Typesetting vectors with beautiful % arrow with \LaTeXe}, Eddie Saudrais, CTAN, v1.3 2013/07/11. % \bibitem{AMS} \emph{\textsf{amsmath} -- \AmS\ mathmatical facilities for \LaTeX}, % Frank Mittelbach, Rainer Schöpf, Michael Downes, Davis M.~Jones, David Carlisle, % CTAN, v2.17n 2022/04/08. % \bibitem{TOOL} \emph{The \textsf{mathtool} package}, Morten Høgholm, Lars Madsen, CTAN, % v1.29 2022/06/29. % \bibitem{FIXM} \emph{The \textsf{fixmath} package for \LaTeXe}, Walter Schmidt, % CTAN, v0.9 2000/04/11. % \bibitem{ISOM} \emph{\textsf{isomath} -- Mathematical style for science and technology}. % Günter Milde, CTAN, v0.6.1 2012/09/04. % \bibitem{PMISO} \emph{\textsf{PM-ISOmath}, The Poor Man ISO math bundle}, % the \textsf{pm-isomath} package by Claudio Beccari, CTAN, v1.2.00 2021/08/04. % \bibitem{GREEK} \emph{The \textsf{upgreek} package for \LaTeXe}, Walter Schmidt, % CTAN, v2.0 2003/02/12. % \bibitem{DESIGN} \emph{The \textsf{mathdesign} package}, % Paul Pichaureau, CTAN, v2.31 2013/08/29. % \bibitem{ALPHA} \emph{The \textsf{textalpha} package} % (part of the \textsf{greek-fontenc} bundle), Günter Milde, CTAN, v2.1 14/06/2022. % \bibitem{KPF} \emph{\textsf{Kp-Fonts} -- The Johannes Kepler project}, % Christophe Caignaert, CTAN, v3.34 20/09/2022. % \bibitem{FOUR} \textsf{Fourier-GUT\hspace{-0.1em}\emph{enberg}}, % Michel Bovani, CTAN, v1.3 30/01/2005. % \bibitem{LGR} \emph{The \textsf{lgrmath} package}, Jean-François B., CTAN, v1.0 2022/11/16. % \bibitem{INT} \emph{The \textsf{interval} package}. Lars Madsen, CTAN, % v0.4 2019/03/06. % \bibitem{SYST} \emph{L'extension pour \TeX\ et \LaTeX\ \textsf{systeme}}, % Christian Tellechea, CTAN v0.32 2019/01/13. % \bibitem{SPAL} \emph{The \textsf{spalign} package}, Joseph Rabinoff, CTAN, 2016/10/05. % \bibitem{FR} \emph{L'extension \textsf{frenchmath}}, Antoine Missier, CTAN, v2.3 2022/12/26. % \bibitem{BRACKET} \emph{Intelligent brackets -- The \textsf{ibrackets} package} % Antoine Missier, CTAN, v1.0, 2022/12/19. % \bibitem{LSHORT} \emph{The Not So Short Introduction to \LaTeXe}, the \textsf{lshort} package by % Tobias Oetiker, Hubert Partl, Irene Hyna and Elisabeth Schlegl, CTAN, v6.4 2021/04/09. % \url{http://tug.ctan.org/info/lshort/english/lshort.pdf}. % \bibitem{COMP} \emph{The \LaTeX\ Companion}, Frank Mittelbach, Michel Goossens, % Johannes Braams, David Carlisle, Chris Rowley, 2nd edition, Pearson Education, 2004. % \end{raggedright} % \end{thebibliography} % \Finale \endinput