% \iffalse meta-comment % % Copyright (C) 2019-2020 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} [2020/11/15 v1.8 .dtx mismath file] % %<*driver> \documentclass{ltxdoc} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[english]{babel} \usepackage{lmodern} \usepackage{mismath} \usepackage{amssymb} \usepackage{multicol} %\enumber \inumber \pinumber % for an example in this doc we substitute [ and ] by \OB and \CB \DeclareMathSymbol{\OB}\mathclose{UpSh}{093} \DeclareMathSymbol{\CB}\mathopen{UpSh}{091} \DisableCrossrefs %\CodelineIndex %\RecordChanges \usepackage{hyperref} \hypersetup{% colorlinks, linkcolor=blue, citecolor=blue, pdftitle={mismath}, pdfsubject={LaTeX package}, pdfauthor={Antoine Missier} } \begin{document} \DocInput{mismath.dtx} %\PrintChanges %\PrintIndex \end{document} % % \fi % %% \CheckSum{419} % % \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} % \changes{v1.1}{2019/04/20}{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, addition of paren macro} % \changes{v1.5}{2019/06/22}{small corrections in documentation} % \changes{v1.6}{2019/09/06}{Removing mathfixs package} % \changes{v1.7}{2019/12/27}{Adding a table of contents to the documentation} % \changes{v1.8}{2020/11/15}{Incompatibility and solution mentioned when using i % with accent in beamer titles small changes in documentation} % % \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{November 15, 2020} % % \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 $\e$, $\i$, $\pi$ % sould be typeset in upright shape and not in italic (sloping shape) like % variables (see~\cite{TYPMA}~\cite{NIST}~\cite{ICTNS}~\cite{ISO}). % 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 $\left\Vert \vect{AB} \right\Vert$. % % \smallskip % The package also provides other macros for: % \begin{itemize} % \item some standard operator names, % \item a few useful aliases, % \item improving some spacings 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 the 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 usefull macros. % % \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 \textsf{fixmath} package by Walter Schmidt~\cite{FIXM}, % but this feature is not implemented in \textsf{mismath} because this rule is conflicting % to the one used for instance 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} % 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}.}), % \DescribeMacro{\e} \DescribeMacro{\i} \DescribeMacro{\j} % 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 ``\i'' and ``\j'' without the point % so you can place accents on them, and in mathematical mode they produce % ``Latex warning: Command invalid in math mode''. % Redefining |\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 upright shape % 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; besides anyone is free to use them or not. % % \medskip % \DescribeMacro{\pinumber\oarg{font}} % Mathematical constant $\pi$ should also be typeset in upright shape % (see~\cite{ICTNS} and~\cite{ISO}), which differs from italicized $\itpi$. % This recommendation is even less observed than the one concerning $\e$ and $\i$~\cite{TYPMA}. % The \textsf{upgreek} package by Walter Schmidt~\cite{GREEK} makes it possible to typeset % greek letters in upright font by using commands such as |\upalpha|, |\upbeta|,\ldots\@ % To avoid typing a lot of |\uppi|, we provide the |\pinumber| macro, % which has to be put in the preamble. This command loads the \textsf{upgreek} package % with an optional \meta{font} argument: \texttt{Symbol} (by default), \texttt{Euler} % or \texttt{Symbolsmallscale} (see~\cite{GREEK}). It also redefines the |\pi| % command to typeset all |\pi| in the selected upright font. % By activating |\enumer|, |\inumber| and |\pinumber| % in the preamble, you can get for instance: % \begin{center} % |$e^{i\pi} = -1$| \hspace{6em} $\e^{i\pi}=-1$ % \end{center} % % \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 by Eddie Saudrais\footnote{\textsf{esvect} % provides \texttt{\bslash vv} macro used by \texttt{\bslash vect}.}) % which are much more elegant than those produced by \LaTeX\ |\overrightarrow| command % (giving $\overrightarrow{AB}$). % 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}, for instance |\usepackage[d]{esvect}| % will give the arrows produced by default in~\cite{VECT}. % % \medskip % \DescribeMacro{\boldvect} % |\vect| makes also possible to typeset vector's names using bold italic % (according to ISO recommendation~\cite{ISO}) rather than arrows. % For this, calling |\boldvect| will modify the behavior of |\vect|:\\[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 |\renewcommand\boldvectcommand{\mathbold}|. % % 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 $h$) % to avoid inelegants effects: % \begin{itemize} % \item $\vect{AB}=\vect{u}+ \vect{AC}$ is less than $\vect{AB}=\hvect{u}+ \vect{AC}$, % obtained with |\hvect{u}|; % \item $\vect{a} \cdot \vect{b}=0$ is less than $\hvect{a} \cdot \vect{b}=0$, % obtained with |\hvect{a}|. % \end{itemize} % The |\boldvect| switch has no effect on the |\hvect| macro which always typesets % arrows on vectors (with the |\vv| command from the \textsf{esvect} package). % % \medskip % \DescribeMacro{\hvec} % In a similar way, |\hvec| raises the little arrow produced by % the \LaTeX\ command |\vec| (but only from height of $t$ letter): % \begin{itemize} % \item $\mathcal{P}=\vec{f}\cdot\vec{v}$ is less than % $\mathcal{P}=\vec{f}\cdot\hvec{v}$, obtained with |\hvec{v}|; % \item $\vec{f} =m \vec{a}$ is less than $\vec{f} =m \hvec{a}$, % obtained with |\hvec{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 centred, relatively to the middle of the base line, % whereas vectors with arrows are asymetrics objects, % the height above the middle of the base line % being superior to the depth under it. % The code |$\norm{\vec{h}}$| raises the double bar to produce $\norm{\vec{h}}$. % Let's notice that the height of the bars don't adjust to content, % but however to context: main text, subscripts or exponents. % % \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{\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 upright shape as for any standard function identifier. % This is obtained with |\P| and |\E|. % Variance is normally denoted by $\Var$ (see further), % but in some countries we can find $\V$ 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 operator names are also defined in \textsf{mismath}: % \begin{center} % \begin{tabular}{rlrlrl} % |\adj| & $\adj$ \qquad\mbox{} & |\erf| & $\erf$ \qquad\mbox{} & |\rank| & $\rank$ \\ % |\Aut| & $\Aut$ & |\grad| & $\grad$ & |\Re| & $\Re$ \\ % |\Conv| & $\Conv$ & |\id| & $\id$ & |\rot| & $\rot$ \\ % |\cov| & $\cov$ & |\Id| & $\Id$ & |\sgn| & $\sgn$ \\ % |\Cov| & $\Cov$ & |\im| & $\im$ & |\spa| & $\spa$ \\ % |\curl| & $\curl$ & |\Im| & $\Im$ & |\tr| & $\tr$ \\ % |\divg| & $\divg$ & |\lb| & $\lb$ & |\Var| & $\Var$ \\ % |\End| & $\End$ & |\lcm| & $ \lcm$ & |\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 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) % but ``$\Im$'' is the imaginary part of a complex number. % Notice that |\div| and |\span| already exist % and haven't been redefined, therefore the |\divg| and |\spa| macros; % |\Z| is used otherwise (see further), therefore |\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 mentionned 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} % % 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 blakboard~\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}| 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: % $\upzeta(s)=\sum_{n=1}^{\infty }\frac{1}{n^s}$ with % \[\upzeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.\] % \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 like the \textsf{amsmath} command |\dfrac| % only transforms the required fraction in display style, we can limit % 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 $\dlim_{x \to +\infty}\frac{1}{x}$. % % \medskip % \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} % $\lbar{z+z'}=\lbar{z}+\lbar{z'}$ is less than $\lbar{z+z'}=\hlbar{z}+\lbar{z'}$, % obtained with |\hlbar{z}|. % \end{center} % % \DescribeMacro{\eqdef} % The |\eqdef| macro writes equality symbol topped with ``def'' % (thanks to the \LaTeX\ command |\stackrel|):\\ % \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}{6.75cm} % \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{Improving some spacings in mathematical formulas} % % \DescribeMacro{\mul} % The multiplication symbol obtained with |\times| produces the same spacing than addition % or substraction operators, whereas division obtained with $/$ is closer to its arguments. % 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} % $\lambda+\alpha \times b-\beta \times c$ % is less than $\lambda+\alpha \mul b-\beta \mul c$, obtained with |\mul|. % \end{center} % % When using |\mul| before an operator name % or a |\left...\right| structure, additionnal spacing occur on the right side of |\mul|. % A solution to get the same amount of space on the two sides of |\mul|, is to % enclose the operator name (or the structure) with brackets: % \begin{center} % Compare $x\mul\sin x$ with $x\mul{\sin x}$ % obtained with |x\mul{\sin x}|. % \end{center} % % \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, % automatically 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{\paren} % Spaces around parenthesis produced by |\left(...\right)| may be too large, for example % after a function name or a point name with coordinates. % A solution is to add a thin negative space |\!| before the opening (or after the closing) % parenthesis, or to enclose the % |\left(...\right)| structure by brackets, or to use the |\paren| macro: % \begin{center} % $\sin\left(\frac{\pi}{3}\right)\mul 2$ is less than $\sin\paren{\frac{\pi}{3}}\mul 2$ % obtained with \\[1ex] |\sin\paren{\frac{\pi}{3}}\mul 2|. % \end{center} % % \DescribeMacro{\pow} % 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: % \[ \e^a \sim\left(1+\frac{a}{n}\right)^n \txt{and} \e^a \sim\pow{1+\frac{a}{n}}{n}.\] % % \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}.}. % % \DescribeMacro{\lfrac} % This macro behaves like |\frac| % but with medium 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{[ ]} % Brackets symbols $[$ and $]$ have been redefined for mathematical mode because, % in standard \LaTeX, the space before them can be unsuitable\footnote{The % \textsf{interval} package~\cite{INT} gives another solution, less direct, % based on an \texttt{\bslash interval} macro.}: % \begin{center} % |$x\in ]0,\pi[ \cup ]2\pi,3\pi[$| \hspace{0.5em} % $\begin{cases} % x\in \OB 0,\pi \CB \cup \OB2\pi, 3\pi \CB &\text{ without \textsf{mismath}}\\ % x\in ]0, \pi[ \cup ]2\pi, 3\pi[ &\text{ with \textsf{mismath}} % \end{cases}$ % \end{center} % In our code, $[$ and $]$ symbols are not defined anymore as delimiters. % Thereby a line break could occur between the two, but % it is always possible to transform them % into delimiters with |\left| and |\right|\footnote{Is \LaTeX\ definition % of [ as |mathopen| really appropriate % where this symbol could almost also logically have been defined as |mathclose|?}. % And consider that these symbols are most of the time preceded or followed by relational, % binary or punctuation symbols and therefore spaces are correct without a delimiter definition. % % \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 get back to 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 centred, % 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 a carriage return 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; % aligment 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{} % \vspace{-4ex} % \section{Implementation} % % \begin{macrocode} \DeclareOption*{\PassOptionsToPackage{\CurrentOption}{amsmath}} \ProcessOptions \relax \@ifpackageloaded{amsmath}{}{\RequirePackage{amsmath}} \@ifpackageloaded{esvect}{}{\RequirePackage[b]{esvect}} \RequirePackage{ifthen} \RequirePackage{xspace} \RequirePackage{mathtools} % \end{macrocode} % The above conditional packages loading avoids ``option clash'' errors if the packages % have been previously loaded with (other) options. % % \medskip % The three following internal commands are meta commands for a % conditional macro definition with warning message if the macro already exists. % \DescribeMacro{\bslash} % The |\bslash| macro used inside |\@mwarning| comes from \textsf{doc.sty} package % by Frank Mittelbach. It can also be used in other documents instead of |\textbackslash| % (which doesn't work here). % \begin{macrocode} {\catcode`\|=\z@ \catcode`\\=12 |gdef|bslash{\}} % the \bslash command \newcommand\@mwarning[1]{ \PackageWarning{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, TODO: by default #1=#3 \ifthenelse{\equal{#1}{}}{ \@ifundefined{#3}{ \DeclareMathOperator{#2}{#3} }{\@mwarning{#3}} }{ \@ifundefined{#1}{ \DeclareMathOperator{#2}{#3} }{\@mwarning{#1}} } } % \end{macrocode} % % To work correctly with the \textsf{beamer} package, we did not use |\mathrm| % but |\mathup| (based on |\operatorfont| from the \textsf{mathopn} package) % to produce the correct upright shape font. % This command works also fine with other sans serif fonts like \textsf{cmbright}. % Moreover for \textsf{beamer}, |\enumber| must use % the family default font defined by the \textsf{beamer} package (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}} \@mmacro{e}{\mathup{e}} \AtBeginDocument{\let\oldi\i \let\oldj\j \renewcommand{\i}{\TextOrMath{\oldi}{\mathup{i}}} \renewcommand{\j}{\TextOrMath{\oldj}{\mathup{j}}} } \DeclareSymbolFont{UpSh}{\encodingdefault}{\familydefault}{m}{n} \newcommand{\enumber}{ \AtBeginDocument{\DeclareMathSymbol{e}\mathalpha{UpSh}{`e}} } \newcommand{\inumber}{ \AtBeginDocument{\DeclareMathSymbol{i}\mathalpha{UpSh}{`i}} } \newcommand{\jnumber}{ \AtBeginDocument{\DeclareMathSymbol{j}\mathalpha{UpSh}{`j}} } \newcommand*{\pinumber}[1][Symbol]{ \@ifpackageloaded{upgreek}{}{\usepackage[#1]{upgreek}} \let\itpi\pi \renewcommand{\pi}{\uppi} } \newboolean{arrowvect} \setboolean{arrowvect}{true} \newcommand{\arrowvect}{\setboolean{arrowvect}{true}} \newcommand{\boldvect}{\setboolean{arrowvect}{false}} \newcommand{\boldvectcommand}{\boldsymbol} % needs bm package \@mmacro{vect}{\ifthenelse{\boolean{arrowvect}}{\vv}{\boldvectcommand}} \newcommand*{\hvect}[1]{\vv{\vphantom{h}#1}} \newcommand*{\hvec}[1]{\vec{\vphantom{t}#1}} \newcommand*{\@norm}[1]{ \mbox{\raisebox{1.75pt}{$\bigl\Vert$}} #1 \mbox{\raisebox{1.75pt}{$\bigr\Vert$}} } % 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}{$\Vert$}} } \providecommand*{\norm}[1]{ \mathchoice{\@norm{#1}}{\@norm{#1}}{\@@norm{#1}}{\@@@norm{#1}} } \newcommand{\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{UpSh}{`P} \DeclareMathSymbol{E}\mathalpha{UpSh}{`E} } } \@moperator{\adj}{adj} \@moperator{\Aut}{Aut} \@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[spa]{\spa}{span} \@moperator{\tr}{tr} \@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} \newcommand{\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} \providecommand*{\hlbar}[1]{\overline{\vphantom{h}#1}} \@mmacro{eqdef}{\stackrel{\mathup{def}}{=}} \@mmacro{unbr}{\underbrace} \@mmacro{iif}{if and only if\xspace} \@mmacro{mul}{\mathord{\times}} \@mmacro{then}{\ \Longrightarrow \ \mbox{} } % \end{macrocode} % Without |\mbox{}|, the space produced by |\| would be suppressed in tables. % \medskip % \begin{macrocode} \providecommand*{\txt}[1]{\quad\text{#1}\quad} \providecommand*{\paren}[1]{\mathopen{\left(#1\right)}} \providecommand*{\pow}[2]{\left( #1 \right)^{\!#2}} \providecommand*{\abs}[1]{\left\vert#1\right\vert} \providecommand*{\lfrac}[2]{\frac{\:#1\:}{\:#2\:}} \DeclareMathSymbol{]}\mathord{UpSh}{093} % originally \mathclose \DeclareMathSymbol{[}\mathord{UpSh}{091} % originally \mathopen \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} % % \begin{thebibliography}{17} % \bibitem{TYPMA} \emph{Typesetting mathematics for science and technology according % to ISO 31/XI}, Claudio Beccari, TUGboat Volume 18 (1997), No.~1. % \bibitem{NIST} \emph{Typefaces for Symbols in Scientific Manuscripts}.\\ % https://www.physics.nist.gov/cuu/pdf/typefaces.pdf. % \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 Nomenclature and Symbols), % dec 1999. % \bibitem{ISO} \emph{ISO 80000-2}. https://en.wikipedia.org/wiki/ISO\_80000-2 % \bibitem{AMS} \emph{The \textsf{amsmath} package}. Frank Mittelbach, Rainer Schöpf, % Michael Downes, Davis M.~Jones, David Carlisle, CTAN, v2.17b 2018/12/01. % \bibitem{TOOL} \emph{The \textsf{mathtool} package}. Morten Høgholm, Lars Madsen, CTAN, % v1.21 2018/01/08. % \bibitem{VECT} \emph{Typesetting vectors with beautiful arrow with \LaTeXe}. % \textsf{esvect} package by Eddie Saudrais, CTAN, v1.3 2013/07/11. % \bibitem{GREEK} \emph{The \textsf{upgreek} package for \LaTeXe}, Walter Schmidt, % CTAN, v2.0 2003/02/12. % \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 04/06/2012. % \bibitem{INT} \emph{The \textsf{interval} package}. Lars Madsen, CTAN, % v0.3 2014/08/04. % \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, v1.5 2020/11/02. % \bibitem{LSHORT} \emph{The Not So Short Introduction to \LaTeXe}. \textsf{lshort} package by % Tobias Oetiker, Hubert Partl, Irene Hyna and Elisabeth Schlegl, CTAN, v6.2 2018/02/28. % \bibitem{COMP} \emph{The \LaTeX\ Companion}. Frank Mittelbach, Michel Goossens, % Johannes Braams, David Carlisle, Chris Rowley, 2nd edition, Pearson Education, 2004. % \end{thebibliography} % \Finale \endinput