% \iffalse meta-comment
%
% Copyright (C) 2019-2024 by Antoine Missier <antoine.missier@ac-toulouse.fr>
%
% 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}
%</driver>
%<*package> 
\NeedsTeXFormat{LaTeX2e}[2005/12/01]
\ProvidesPackage{mismath}   
    [2024/05/04 v3.0 .dtx Miscellaneous mathematical macros]
%</package>
%<*driver>
\documentclass{ltxdoc}
\usepackage[utf8]{inputenc} % run through XeLaTeX
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
%\usepackage{fontspec}
\usepackage{fourier} % in case of font change, modify 2.1 (3rd §)
\usepackage[ibrackets,decimalcomma,otherReIm]{mismath}
%\usepackage{amssymb} % unnecessary with fourier
\usepackage{multicol}
\usepackage{sectsty}
\usepackage{xcolor}
\usepackage{spacingtricks}
\usepackage{array}
\usepackage{metalogo} % for logos of XeLaTeX et LuaLaTeX

\definecolor{darkbrown}{rgb}{0.5,0.1,0.1}
\allsectionsfont{\color{darkbrown}}

% The following commands provide several shapes for pi
\DeclareFontEncoding{LGR}{}{}
\def\pilmr{\text{\usefont{LGR}{lmr}{m}{n}\symbol{112}}}
\def\piAlegreya{\text{\usefont{LGR}{Alegreya-LF}{m}{n}\symbol{112}}}
\def\piCochineal{\text{\usefont{LGR}{Cochineal-LF}{m}{n}\symbol{112}}}
\def\piLibSerif{\text{\usefont{LGR}{LibertinusSerif-LF}{m}{n}\symbol{112}}}
\def\piLibSans{\text{\usefont{LGR}{LibertinusSans-LF}{m}{n}\symbol{112}}}
\def\pilmss{\text{\usefont{LGR}{lmss}{m}{n}\symbol{112}}}
\def\pigentium{\text{\usefont{LGR}{gentium}{m}{n}\symbol{112}}}
\def\pilato{\text{\usefont{LGR}{lato-LF}{m}{n}\symbol{112}}}
\DeclareFontFamily{U}{eur}{\skewchar\font'177}
\DeclareFontShape{U}{eur}{m}{n}{%
    <-6> eurm5 <6-8> eurm7 <8-> eurm10}{}
\def\piEuler{\text{\usefont{U}{eur}{m}{n}\symbol{"19}}}
\def\piSymbol{\text{\usefont{U}{psy}{m}{n}\symbol{`p}}}
\def\piCharter{\text{\usefont{OML}{mdbch}{m}{n}\symbol{"19}}}
\def\pikp{\text{\usefont{U}{jkpmia}{m}{it}\symbol{"19}}}
\pinumber[otherpi] % fourier
\def\pipx{\text{\usefont{U}{pxmia}{m}{it}\symbol{"19}}}
\def\pitx{\text{\usefont{U}{txmia}{m}{it}\symbol{"19}}}

% Next command is used for showing original Latin Modern \i and \j in text mode:
%\DeclareTextFontCommand{\extrafont}{\fontfamily{lmr}\selectfont} % useless with fourier

\DisableCrossrefs         
%\CodelineIndex
%\RecordChanges

\usepackage{hyperref}
\let\C\relax % "\C already defined" with hyperref when compiling with xelatex or lualatex
\hypersetup{%
    colorlinks, 
    linkcolor=blue,
    citecolor=red,   
    pdftitle={mismath}, 
    pdfsubject={LaTeX package}, 
    pdfauthor={Antoine Missier}
}
\begin{document}
  \DocInput{mismath.dtx}
  %\PrintChanges
  %\PrintIndex
\end{document}
%</driver>
% \fi
%
% \CheckSum{845}
%
% \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}{2019/04/27}{
%    - Works fine with beamer now, 
%    - AtBeginDocument for enumber, inumber, jnumber, 
%    - creating general mm@operator macro, 
%    - using mathup instead of mathrm
%    - added mathtools package, 
%    - font definition Roman -> up, 
%    - changes in documentation, 
%    - replacing the PEroman macro by PEupright}
%
% \changes{v1.3}{2019/05/08}{
%    - Using bslash in the internal mm@warning macro, 
%    - loading of mathfixs package
%    - 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 enhanced to use other Greek letters packages,
%        (and LGR encoding pi by default), it is no longer compatible with the older, 
%    - paren command has been removed, 
%    - hvect and hlbar have been slightly modified (phantom t instead of phantom h),
%    - several changes in the documentation, 
%    - now using the Charter font in the doc (with the package 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.}
%
% \changes{v2.3}{2023/02/09}{
%    - Introducing keyval options for enumber, inumber, jnumber, PEupright
%        (corresponding old commands are maintained for now),
%    - and also for ibrackets, boldvect and arrowvect
%    - Documentation update}
%
% \changes{v2.4}{2023/02/18}{
%    - New powerful macros MathUp, MathIt and also MathNumbers, MathProba, MathNormal,
%    - keyval options are no longer useful and have been deleted,
%    - loading package ifthen has been forgotten in v2.3 (causing possible errors),
%    - no more incompatibility when using `i' with accent in beamer titles}
%
% \changes{v2.5}{2023/02/23}{
%    - Unification of the code of MathUp and MathIt,
%    - a powerful macro `apply' is used in MathNormal to act on a list,
%    - a new `tensor' command,
%    - small changes in the documentation}
%
% \changes{v2.6}{2023/03/01}{
%    - A bug correction in mm@macro
%    - incompatibility of \C macro solved, when using babel with Russian
%        (thanks to Murray Eisenberg for this bug report on TeX StackExchange)
%    - mathrm added in macro eqdef* (useful if italicized Greek uppercase Delta)
%    - Tiny changes in the documentation}
%
% \changes{v2.7}{2023/03/05}{
%    - Macros for sets of numbers (\R, \C,...) now available only in math mode
%        (following remarks by David Carlisle and Enrico Gregorio)
%    - special (not generic) warning when loading babel with russian
%        (\C will not be defined in that case)}
%
% \changes{v2.8}{2023/07/26}{
%    - Some small changes in the documentation
%    - New macro `onlymathC' designed for using '\C' when Russian language is loaded}
%
% \changes{v2.9}{2023/12/19}{
%    - New option decimalcomma
%    - Some small changes in the documentation}
%
% \changes{v2.10}{2024/02/20}{
%    - Better compatibility with unicode-math: 
%        for the options ibrackets, decimalcomma and the commands MathUp, MathIt
%    - Explicit error message when using mathcols without loading the multicol package} 
%
% \changes{v2.11}{2024/02/26}{
%    - Enhancements of the pinumber macro with keyval options:
%        * no necessity to load a Greek letters package
%        * improvements of compatibility with unicode-math
%    - Now using the Adobe Utopia font with the package fourier}
%
% \changes{v2.12}{2024/02/29}{
%    - The xparse package has been removed by mistake in v2.11,
%        this may cause some compatibility problems;
%        it is loaded again by mismath
%    - Improvements to make pinumber work perfectly with unicode-math
%    - Small corrections in the doc}
%
% \changes{v3.0}{2024/03/15}{
%    - Rewriting of the pinumber command including the pifonts macro
%    - Presenting other lgrmath values for pinumber in the doc
%    - The \C macro is now in AtBeginDocument
%    - amsmath isn't loaded anymore because mathtools loads it
%    - Bug fix with options decimalcomma and ibrackets
%    - New option 'nofunction' to lighten the package loading
%    - Addition of coker and Res in standard operator names
%    - New option 'classicReIm' to deactivate Im and Re redefinition
%    - New option 'otherReIm' to provide an alternative writing with cmsy font
%    - Removal of the PEupright command
%    - Additional space in the lfrac macro increased from \: (5mu) to 7mu 
%    - New optional parameter for adjusting the space in lfrac
%    - Changing the vphantom in hvect, hvec and lbar from `t' to `A', looks better}
%
% \GetFileInfo{mismath.sty}
%
% \title{Miscellaneous mathematical macros \\The \textsf{mismath} package\thanks{This document
% corresponds to \textsf{mismath}~\fileversion, dated \filedate. 
% Thanks to François Bastouil for initial help in English translation.}}
% \author{Antoine Missier \\ \texttt{antoine.missier@ac-toulouse.fr}}
% \date{May 04, 2024}
%
% \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 \emph{constants} $\e$, $\i$, $\pi$ 
% should be typeset in roman (upright shape) and not in italic like
% variables (see~\cite{TYPMA}~\cite{NIST}~\cite{SI}~\cite{ICTNS}).
% This package provides some tools to achieve this automatically.
%
% \medskip
% Even though it is recommended to typeset vectors names 
% in bold italic style~\cite{NIST}~\cite{ICTNS}, 
% they are often represented with arrows,
% especially in school documents or in physics. 
% To draw nice arrows above vectors, we use the \textsf{esvect} package 
% by Eddie Saudrais~\cite{VECT}.
% Additionally we provide a few more macros related to vectors with arrows, 
% particularly to improve the typesetting of the norm: $\norm{\vect{AB}}$ 
% instead of the \LaTeX\ version $\lVert\vect{AB}\rVert$, which is not vertically adjusted,
% or worse $\left\Vert \vect{AB} \right\Vert$ (when using |\left|\ldots |\right|).
% 
% \smallskip
% The package also offers other macros to typeset:
% \begin{itemize}
% \item tensors in sans serif bold italic shape 
% (ISO recommendation~\cite{TYPMA}~\cite{NIST}~\cite{SI}),
% \item some standard operator names,
% \item several commands with useful aliases,
% \item improved spacings in mathematical formulas,
% \item systems of equations and small matrices,
% \item displaymath in double columns for lengthy calculations with short expressions.
% \end{itemize}
%
% To avoid compatibility issues,
% most of our macros will only be defined 
% if there isn't already a command with the same name in the packages loaded 
% before \textsf{mismath}. If a macro is already defined,
% a warning message will be displayed and the \textsf{mismath} definition 
% will be ignored. If you wish to keep the \textsf{mismath} or the existing command,
% you can use |\let\|\meta{command}|\relax|, before loading \textsf{mismath}, or after.
%
% \medskip
% \DescribeEnv{\oarg{options}}
% The \textsf{mismath} package loads the \textsf{mathtools}  
% \footnote{The \textsf{mathtools} package offers numerous helpful macros and improvements 
% of the \textsf{amsmath} package.}
% package by Morten Høgholm and Lars Madsen~\cite{TOOL} which in turn loads
% the \textsf{amsmath} package~\cite{AMS}.
% If you want to use \mbox{\textsf{amsmath}} or \textsf{mathtools} with specific options,
% you can include these options as options of \mbox{\textsf{mismath}}, or 
% you can load \textsf{amsmath} or \textsf{mathtools}
% with the desired options before loading \mbox{\textsf{mismath}}.
% When using the package \textsf{unicode-math}~\cite{UNIC}, \textsf{mismath} 
% should be loaded before \textsf{unicode-math}, just like \textsf{amsmath}.
%
% \medskip
% An ISO recommendation, although rarely followed, is to typeset uppercase 
% Greek letters in italic shape,
% as for other variables~\cite{ICTNS}. This is automatically achieved,
% for some particular fonts, with packages 
% such as \textsf{fixmath} by Walter Schmidt~\cite{FIXM},
% \mbox{\textsf{isomath}} by Günter Milde~\cite{ISOM} 
% or \textsf{pm-isomath} by Claudio Beccari~\cite{PMISO} and optionally with many others
% (such as \textsf{mathpazo} or \textsf{mathptmx} with the option \texttt{slantedGreek}).
% When running through \LuaLaTeX\ or \XeLaTeX\ you can also get this result 
% with the option \mbox{\texttt{math-style=ISO}} provided 
% by the \textsf{unicode-math} package.
% We also have the new \mbox{\textsf{mathgreeks}} package~\cite{MGREEK} which offers 
% a wide range of fonts and different settings with Greek letters.
% However this feature is not implemented here due to a conflicting rule in France,
% where all capital letters in mathematics
% are required to be typeset in upright shape
% \footnote{The \textsf{frenchmath} package~\cite{FR}
% ensures to follow the recommended French rules.}.
% The user is free to choose loading one of these packages or not.
% 
%
% \section{Usage}
%
% \subsection{Mathematical constants} 
%
% \DescribeMacro{\mathup} 
% As for classic functions identifiers, \emph{predefined} mathematical constants
% should be typeset in upright shape (typically in roman family), 
% but this practice is not sufficiently respected, probably because it's a bit tedious. 
% A first solution is to use the |\mathup| macro, 
% which is preferable to |\mathrm|
% \footnote{The \texttt{\string \mathup} macro is based on \texttt{\string \operatorfont}, 
% which comes from the
% \textsf{amsopn} package, automatically loaded by \textsf{amsmath}. 
% In \textsf{beamer}, the default math font is sans serif, 
% but \texttt{\string \mathrm} produces a font with serifs, which might not match the 
% overall style of the presentation. Hence, using \texttt{\string \mathup} is indeed 
% a better choice in \textsf{beamer} presentations to ensure that mathematical 
% constants are typeset in upright shape and consistent 
% with the default sans serif math font.},
% for setting any group of letters in roman. For example you can use
% |\mathup{e}| to get the Euler's number.
%
% \medskip
% \DescribeMacro{\e} \DescribeMacro{\i}  \DescribeMacro{\j} 
% To avoid cluttering a document that contains many occurrences of Euler's number $\e$,
% or imaginary number $\i$, with |\mathup{e}| or |\mathup{i}|, 
% the package provides the |\e| command for Euler's number 
% and |\i| or |\j| for imaginary numbers.
% Let us notice that |\i| and |\j| already exist in LaTeX. In LR (left-to-right) mode, 
% they produce \mbox{`\i,\ \j'} without the dot,  
% allowing you to place accents on them. 
% However, in mathematical mode, they produce the warning 
% ``\texttt{LaTeX Warning: Command \string \i\space invalid in math mode on input line}
% \meta{line}''. 
% With the new definition provided by the package, |\i| and |\j| will be redefined 
% specifically for mathematical mode.
%
% \medskip
% \DescribeMacro{\MathUp} 
% Indeed, typing a lot of backslashes for constants like $\e$, $\i$, or $\j$
% in a document with numerous formulas can become tiresome. 
% To alleviate this, the package proposes another solution 
% with the macro |\MathUp|\marg{char}.
% For example, when |\MathUp{e}| is called, any subsequent occurrence of $\e$
% will automatically be set in roman (upright shape), 
% without the need to type |\e| explicitly. The effect of this macro 
% can be either global or local, depending on whether it is used outside or inside an
% environment or braces.
% Furthermore, you can also call this macro in the preamble, then the change will apply
% from the beginning of the document. This powerful command allows you 
% to bring a document up to the standards effortlessly and without changing anything
% in your mathematical formulas. 
% In fact, |\MathUp| can be applied to any valid single character, 
% offering flexibility for various use cases
% \footnote{Another use of it with probability 
% will be presented in section \ref{operators}.}.
%
% \medskip
% \DescribeMacro{\MathIt}
% When there are other occurrences of $e$, $i$ or $j$ as variables, 
% you can still obtain italicized $e$, $i$ or $j$ using \LaTeX\
% commands |\mathit| or |\mathnormal|, which are useful for a single use. 
% However, you also have the option to use the inverse switch |\MathIt|\marg{char},
% which has a global effect when used outside environments or braces, 
% or a local effect when used inside them. 
% Similar to |\MathUp|, |\MathIt| can be applied to any single character.
%
% \medskip
% \DescribeMacro{\MathNumbers} \DescribeMacro{\MathNormal}
% These macros enable you to set upright or normal (italic) typesetting 
% for multiple letters in a single command. 
% For instance, |\MathNumbers{e,i}| is equivalent to |\MathUp{e}\MathUp{i}|.
% In |\MathNumbers|, the comma separator between letters can be modified 
% or removed as needed. In fact, this macro only affects the letters e, i, or j; 
% it has no effect on other characters.
% On the other hand, |\MathNormal| accepts any comma-separated list of arguments.
% This means you can apply the normal italic math mode typesetting to various letters 
% at once using |\MathNormal|.
%
% \medskip
% \DescribeMacro{\enumber}\DescribeMacro{\inumber}\DescribeMacro{\jnumber}
% These three commands, used until version 2.2 but only functioning within the preamble, 
% serve now as aliases for the commands |\MathUp{e}|, |\MathUp{i}| or |\MathUp{j}|, 
% so they can be used anywhere 
% in the document or preamble and has an inverse switches with |\MathIt|. 
% 
% \medskip
% \DescribeMacro{\pinumber[\meta{option}]}
% The constant $\pi$  should also be typeset in upright shape 
% (see~\cite{TYPMA}, \cite{NIST}, \cite{ICTNS}), which is different from italicized $\itpi$. 
% However, this recommendation is even less commonly followed compared to the one 
% concerning  $\e$ and $\i$~\cite{TYPMA}.
% Thanks to the |\pinumber| command, the italic $\itpi$ will be replaced 
% with an upright $\pi$
% each time |\pi| is called. Thus |\pinumber| makes your document compliant with standards 
% without changing the source code of your mathematical formulas. 
% It functions in two different ways.
%
% \begin{enumerate}
% \item You can load a Greek letters package that provides the glyphs in upright shape. 
% There are many available.
% Notably, let us mention \textsf{upgreek}~\cite{GREEK}, 
% \mbox{\textsf{mathdesign}}~\cite{DESIGN},
% \mbox{\textsf{kpfonts}}~\cite{KPF}, 
% \mbox{\textsf{fourier}}~\cite{FOUR} (used in the present document), but also
% \mbox{\textsf{pxgreeks}} (using \textsf{pxfonts}~\cite{PX}), \mbox{\textsf{txgreeks}}
% (using \textsf{txfonts}~\cite{TX})
% \footnote{When using \textsf{pxgreeks} or \textsf{txgreeks},
% they should be loaded \emph{after} \textsf{mismath} to avoid an error
% due to conflict with the existing macros \texttt{\string\iint}, \texttt{\string\iiint},
% \texttt{\string\iiiint}, \texttt{\string\idotsint} in \textsf{amsmath}.}, 
% \mbox{\textsf{libertinust1math}}~\cite{LIB}, \textsf{libgreek}, etc.
% A special mention goes to \textsf{lgrmath} of Jean-François Burnol~\cite{LGR}
% which allows the use of any Greek LGR-encoded font in math mode,
% an idea taken up in \textsf{mathgreeks}~\cite{MGREEK}.
% Also note \textsf{newtxmath}~\cite{NTX} which has several font options.
% These packages provide commands like |\uppi| (\textsf{upgreek}), 
% |\piup| (\textsf{mathdesign}, \textsf{kpfonts}, \textsf{lgrmath}), 
% |\otherpi| (\textsf{fourier}), etc.
% \footnote{They have also options to typeset all the Greek lowercase 
% letters in upright shape by default, but this in not our goal here.}
%
% In this case, |\pinumber| must be called in the preamble 
% with an optional argument being the name of the command, \emph{without the backslash},
% giving access to the upright pi:
% piup, uppi, otherpi\ldots
% However, installing such a Greek letters package will modify 
% all the other Greek letter glyphs.
% ^^A even if, except for pi, they will be in italics. 
%
% By calling in the preamble |\MathNumbers{ei}\pinumber[otherpi]| 
% (assuming the \textsf{fourier} package is loaded),
% you can achieve the following result:
% \begin{center}
% |$e^{i\pi} = -1$| \quad yields \quad   $\e^{\i\pi}=-1$.
% \end{center}
%
% \item Without installing a package, it is possible to change only the glyph of pi 
% without altering the other Greek letters, 
% which are typically in italics.
%
% In this case, |\pinumber| must be called in the preamble with an optional argument 
% of the \texttt{key=value} type. The key name corresponds to a package
% providing the same glyph.
% The following table summarizes the available options.
% When a key is given without a value, |\pinumber| will choose a default value
% specified in the following text (depending on the key).
%
% \begin{center}
% \setlength{\extrarowheight}{1pt}
% \begin{tabular}{>{\ttfamily}l>{$}c<{$}}
% \hline
% {\normalfont Option} lgrmath=... & \mbox{Result} \\
% \hline
% Alegreya-LF & \piAlegreya \\
% Cochineal-LF & \piCochineal \\
% LibertinusSerif-LF & \piLibSerif \\
% LibertinusSans-LF & \piLibSans \\
% lmr & \pilmr \\
% lmss & \pilmss \\
% gentium & \pigentium\\
% lato-LF & \pilato \\
% \hline
% \end{tabular}
% \qquad
% \begin{tabular}{>{\ttfamily}l>{$}c<{$}}
% \hline
% \normalfont Other options & \mbox{Result} \\
% \hline
% fontspec=\ldots & \ldots \\
% upgreek=Euler & \piEuler \\
% upgreek=Symbol & \piSymbol \\
% mathdesign & \piCharter \\
% kpfonts & \pikp \\
% fourier & \pi\\
% pxfonts & \pipx \\
% txfonts & \pitx \\
% \hline
% \end{tabular}
% \end{center}
%
% \medskip
% \begin{itemize}
%
% \item With the \texttt{lgrmath} key, we actually have numerous possibilities 
% for values (any Greek letters math font in LGR encoding). 
% The documentation of the \textsf{lgrmath}
% package explains how to check an visualize all available fonts on your distribution.
% We have only presented seven of them. The default value is \texttt{lmr}.
% Other interesting values are \texttt{NotoSerif-LF}, \texttt{Clara-TLF},
% \texttt{droidserif}, \texttt{fct}, \texttt{llcmss}.
%
% \item When |\pinumber| is called without an argument in the preamble, 
% it corresponds to the option \texttt{lgrmath=lmr}.
% This $\pilmr$ character is well-suited
% for use with the Latin Modern font family
% \footnote{It will look the same as the one provided by \texttt{lgrmath=cmr} or by
% Günter Milde's \textsf{textalpha} package~\cite{ALPHA}.}.
%
% \item With the \texttt{fontspec} key, there are also many possible values,
% corresponding to the TrueType or OpenType fonts installed on your system
% (works with \LuaLaTeX\ or \XeLaTeX).
% See the \textsf{mathgreeks} documentation for examples.

% \item With the \texttt{upgreek} key, the default value is \texttt{Symbol}. 
% There is a third possible value, \texttt{Symbolsmallscale}, 
% which provides the same character as \texttt{Symbol} but reduced in size by 10\,\%.
%
% \item With the \texttt{mathdesign} key, there are actually 3 possible values: 
% \texttt{Utopia}, \texttt{Garamond} or \texttt{Charter} (the default value), 
% but the glyphs obtained for pi look quite similar.
%
% \item With the \texttt{kpfonts} key, we have two possible values: 
% \texttt{normal} (default) and \texttt{light}. The option \texttt{kpfonts=light}
% provides a slightly less bold character.
%
% \item The last keys, \texttt{fourier} (based on Utopia), \texttt{pxfonts} 
% (based on Palatino),
% \texttt{txfonts} (based on Times)
% are booleans whose default value is \texttt{true} (when called).
% The \texttt{txfonts} option yields the same glyph than \texttt{lgrmath=txr}.
%
% \end{itemize}
% \end{enumerate}
% 
% The \textsf{unicode-math} package~\cite{UNIC} provides |\uppi|, and you can use
% |\pinumber[uppi]| to produce automatic upright pi in the selected math font,
% but |\pinumber[uppi]| must be called \emph{after} \mbox{\textsf{unicode-math}},
% and a math font must have been explicitly chosen with |\setmathfont|.
% You can also use the \texttt{fontspec} key option to obtain pi in any font 
% that is supported by \mbox{\textsf{unicode-math}} \eg
% |\pinumber[fontspec=STIX Two Math]|.
%
% For other fonts, it can be quite complicated to make Greek letters packages 
% work with \textsf{unicode-math}. In any case, such a package must be loaded 
% after \textsf{unicode-math} and in |\AtBeginDocument|. 
% However, |\pinumber| supports \textsf{unicode-math} very well with any previous
% \texttt{key=value} option, by calling |\pinumber|\oarg{option} 
% after \textsf{unicode-math}.
%
% \medskip
% \DescribeMacro{\itpi}
% When you activate |\pinumber|, the original italic $\itpi$ 
% is still accessible using |\itpi|.
%
% \medskip
% \DescribeMacro{\pinormal}
% In fact, |\pinumber| is a toggle, with its inverse toggle being |\pinormal|. 
% The latter restores the |\pi| command to its default behavior. 
% Thus, |\pinumber| can be used anywhere in the document (like |\pinormal|), 
% but then without arguments and provided it has been initially called in the preamble,
% according to the procedures outlined above.
%
% \subsection{Vectors (and tensors)}
%
% \DescribeMacro{\vect}
% By default, the |\vect| command
% \footnote{The definition of most macros in this package, 
% will only take effect if the macro has not been previously
% defined by another package. This ensures compatibility and avoids conflicts 
% when using the \textsf{mismath} package with other \LaTeX\ packages.},
% produces vectors with arrows
% (thanks to the \textsf{esvect} package by Eddie Saudrais
% \footnote{\textsf{esvect} provides the \texttt{\string \vv} macro 
% used by \texttt{\string \vect}.})
% which are more elegant than those produced by \LaTeX's |\overrightarrow| command.
% The \textsf{esvect} package has an optional argument 
% (a single letter between \texttt{a} and \texttt{h}) to define 
% the desired type of arrow (see~\cite{VECT}).
% In \textsf{mismath}, \textsf{esvect} is loaded with the option \texttt{b}:
% |\vect{AB}| gives $\vect{AB}$.
% If you wish to use a different type of arrow, you must call \textsf{esvect}
% with the appropriate option \emph{before} loading \textsf{mismath}.  
% For example, using |\usepackage[d]{esvect}|
% will provide the same arrows that are used by default in~\cite{VECT}.
%
% \medskip 
% \DescribeMacro{\boldvect}
% The |\vect| macro allows vector names to be typeset using bold italic font,
% as recommended by ISO~\cite{NIST}~\cite{SI}, instead of using arrows. 
% By using the |\boldvect| command, you can modify the behavior of |\vect|
% locally or globally, depending on its placement in the document (inside or outside
% a group or an environment):\\[2ex]
% \begin{minipage}{8cm}
% \begin{verbatim} 
%\[ \boldvect \vect{v}
%   =\lambda\vect{e}_x+\mu\vect{e}_y \]
% \end{verbatim}
% \end{minipage} 
% \begin{minipage}{4cm}\vspace{-2ex}
% $\boldvect \vect{v}=\lambda\vect{e}_x +\mu\vect{e}_y$
% \end{minipage}
%
% \DescribeMacro{\boldvectcommand}
% By default |\boldvect| uses the |\boldsymbol| command
% \footnote{\texttt{\string \mathbf} produces upright bold font, 
% even when used in combination with \texttt{\string \mathit}.}
% from the \textsf{amsbsy} package, which is automatically loaded by \textsf{amsmath}.
% However, you may prefer other packages that produce bold italic fonts, 
% such as \textsf{fixmath} with the |\mathbold| command,
% \textsf{isomath} with |\mathbfit| 
% or \textsf{bm} with the |\bm| command;
% \textsf{unicode-math} provides the |\symbfit| command.
% To use an alternative command instead of |\boldsymbol| in \textsf{mismath}, 
% redefine |\boldvectcommand|, 
% for instance after loading \textsf{fixmath}:
%
% \medskip
% \centeredline{|\renewcommand\boldvectcommand{\mathbold}|}
%
% \medskip
% According to ISO rules, symbols for matrices are also in bold italic.
% Therefore you can use the same |\boldvect| command or create another alias.
%
% \medskip
% \DescribeMacro{\arrowvect}
% At any moment, you can revert to the default behavior using the inverse switch
% |\arrowvect|. 
% These switches can be placed anywhere, whether
% inside mathematical mode or within an environment (with a local effect) or outside 
% (with a global effect).
%
% \medskip
% \DescribeMacro{\hvect}
% When vectors with arrows are typeset side by side,
% the arrows can be set up slightly higher using |\hvect|
% (which places a vertical phantom box containing `$A$') 
% to avoid inelegant effects. For example, writing
% \begin{itemize}
% \item $\vect{AB}=\hvect{u} + \vect{AC}$, obtained with |\hvect{u}|,
% looks better than $\vect{AB}=\vect{u}+ \vect{AC}$;
% \item $\hvect{a} \cdot \vect{b}=0$, obtained with |\hvect{a}|,
% looks better than $\vect{a} \cdot \vect{b}=0$.
% \end{itemize}
% This adjustment ensures a nicer appearance when vectors 
% with arrows are combined in an equation
% \footnote{For a fine tuning you can also use the \texttt{\string\vstrut} or
% \texttt{\string\cstrut} macros from the \textsf{spacingtricks} package~\cite{SPA}.}.
% The |\boldvect| and |\arrowvect| switches have the same effect on |\hvect| 
% as they do on |\vect|.
%
% \DescribeMacro{\hvec}
% In a similar way, |\hvec| raises the little arrow produced by
% the \LaTeX\ command |\vec|, to the height of the letter `$A$'
% (but |\boldvect| have no effect on |\vec| nor |\hvec|):
% \begin{itemize}
% \item $\mathcal{P}=\vec{f}\cdot\hvec{v}$, obtained with |\hvec{v}|,
% seems better than $\mathcal{P}=\vec{f}\cdot\vec{v}$;
% ^^A (the result of |\vec| with \textsf{fourier} may be bad with high letters, \eg $\vec{h}$)
% \item $\vec{f} =m \hvec{a}$, obtained with |\hvec{a}|,
% seems better than $\vec{f} =m \vec{a}$.
% \end{itemize}
%
% \DescribeMacro{\norm}
% The norm of a vector is conventionally represented using the delimiters |\lVert| and |\rVert|
% (or |\|$\mid$ unless a plus (+) or minus (-) sign follows the opening delimiter)
% or |\left\Vert| and |\right\Vert| 
% for adaptive delimiters. Unfortunately, these delimiters 
% are always vertically centered, relatively to the mathematical center 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 $\Vert\vec{h}\Vert$ or $\left\Vert \vec{h} \right\Vert$.
% Let's notice that the height of the bars don't adjust to content, 
% but however to context: main text, subscripts or exponents, 
% \eg $X^{\norm{\vec{h}}}$. 
% This macro is useful only for arguments of special height, such as $\vec{h}$ or $\vect{AB}$
% and may give bad results in other situations.
%
% \medskip
% \DescribeMacro{\mathbfsfit}\DescribeMacro{\tensor}
% For tensors symbols, ISO rules recommend using
% sans serif bold italic, but there
% is no such math alphabet in the default \LaTeX\  mathematical style.
% However, the \mbox{\textsf{mismath}} package defines this alphabet 
% (assuming the font encoding and package you use permits it) 
% and provides the macro |\mathbfsfit| or its alias |\tensor|.
% By writing |\tensor{S}\otimes\tensor{T}|, you get $\tensor{S}\otimes\tensor{T}$.
%
% \subsection{Standard operator names}\label{operators}
%
% \DescribeMacro{\di}
% The \emph{differential} operator should be typeset in upright shape, not in
% italics, to distinguish it from variables 
% (as mentioned in \cite{TYPMA}~\cite{NIST}~\cite{ICTNS}~\cite{LSHORT}).
% To achieve this, we provide the |\di| command.
% Take a look at the following examples (notice the thin spaces before the d, 
% just like with classic function's names): \\[2ex]
% \begin{minipage}{7cm}
% \begin{verbatim} 
%\[ \iint xy\di x\di y \]
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{5cm}
% \[ \iint xy\di x\di y \]
% \end{minipage}
% \\[0.5ex]
% \begin{minipage}{7cm}
% \begin{verbatim} 
%\[ m\frac{\di^2x}{\di t^2}
%   + h\frac{\di x}{\di t} + kx = 0 \]
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{5cm}
% \[m\frac{\di^2x}{\di t^2}+h\frac{\di x}{\di t}+kx=0\]
% \end{minipage}
%
% The command |\di| can also represent the \emph{distance}, hence its name:
% \[\di(u, \cmmathcal{H}) = \frac{\abs{\langle u,v\rangle}}{\lVert v \rVert}.\]
%
% \DescribeMacro{\P} \DescribeMacro{\E}
% To refer to probability
% \footnote{\LaTeX\ provides also \texttt{\string \Pr} which gives $\Pr$.}
% and expectation the proper use is to typeset the capital letters $\P$, $\E$ 
% in roman just like any standard function identifier.
% This can be achieved with |\P| and |\E| commands.
%
% \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{\V}
% Variance is generally denoted by $\var$ or $\Var$ (see the following table),
% but some authors prefer to use $\V$, which can be produced using |\V|.
%
% \pagebreak
% \DescribeMacro{\MathProba}\DescribeMacro{\MathNormal}
% As for $\e$, $\i$ or $\j$, you can use |\MathUp{P}|, |\MathUp{E}| 
% or |\MathUp{V}| to avoid typing many |\P|, |\E| or |\V|. 
% However you can also achieve this in a single command with |\MathProba|, 
% for example |\MathProba{PE}|. We get the inverse toggle with
% |\MathIt| for any individual letter or |\MathNormal| for a list.
%
% \medskip
% \DescribeMacro{\probastyle}
% Some authors use ``outline'' font shape 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{The effect of this redefinition is global or local to the container environment 
% in which it is used.}
% brings the double-struck letters.
% The |\mathbb| command is provided by \textsf{amsfonts} package
% (which needs to be called in the preamble), but also by other complete math font
% packages such as \textsf{mathdesign}, \textsf{kpfonts}, \textsf{fourier}, 
% \textsf{unicode-math}\ldots
% \hypertarget{nofunction}{}
%
% \medskip
% The following standard operator names are defined in \textsf{mismath}:
% \begin{center}
% \begin{tabular}{rlrlrl}
% |\adj| & $\adj$ \qquad\mbox{} & |\End| & $\End$ \qquad\mbox{} & |\Res| & $\Res$ \\
% |\Aut| & $\Aut$ & |\erf| & $\erf$ & |\rot| & $\rot$ \\
% |\codim| & $\codim$ & |\grad| & $\grad$ & |\sgn| & $\sgn$ \\
% |\coker| & $\coker$ & |\id| & $\id$ & |\sinc| & $\sinc$ \\
% |\Conv| & $\Conv$ & |\Id| & $\Id$ & |\spa| & $\spa$ \\
% |\Cov| & $\Cov$ & |\im| & $\im$ & |\tr| & $\tr$ \\
% |\cov| & $\cov$ & |\lb| & $\lb$ & |\var| & $\var$ \\
% |\curl| & $\curl$ & |\lcm| & $ \lcm$ & |\Var| & $\Var$  \\
% |\divg| & $\divg$ & |\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 both are very common.
% Please note that |\div| already exists ($\div$) and |\span| is a \TeX\ primitive;
% they haven't been redefined. Therefore the provided macros are called |\divg| (divergence) 
% and |\spa| (span of a set of vectors).
% Furthermore |\Z| is used to denote the set of integers (see \ref{aliases}), 
% which is why we propose |\Zu|,
% to designate the center of a group: $\Zu(G)$ (from German Zentrum).
%
% \medskip
% The \textsf{mismath} package also provides some (inverse) circular or hyperbolic functions, 
% that are missing in \LaTeX:
% \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{[nofunction]}
% Some may find that the definition of all these operators and functions is not useful 
% for their needs. So, the above definitions (on this page)
% can be disabled with the \texttt{nofunction} option.
% \hypertarget{classicReIm}{}
%
% \medskip
% \DescribeMacro{\Re} \DescribeMacro{\Im}
% The |\Re| and |\Im| macros refer to real and imaginary part 
% of a complex number. They have been redefined to produce $\mathup{Re}$ and $\mathup{Im}$, 
% in place of outdated symbols $\oldRe$ and $\oldIm$.
% Nevertheless, it is still possible to obtain the old symbols 
% with |\oldRe| and |\oldIm|.
%
% \medskip
% \DescribeMacro{[otherReIm]}
% An alternative notation $\Re$, $\Im$ is provided by invoking the \texttt{otherReIm}
% package option,
% \DescribeMacro{[classicReIm]}
% whereas the \texttt{classicReIm} option deactivates these redefinitions.
%
% \DescribeMacro{\bigO} \DescribeMacro{\bigo} \DescribeMacro{\lito}
% Asymptotic comparison operators (in Bachmann-Landau notation) are obtained with
% |\bigO| or |\bigo| and |\lito| commands. The first one uses
% the |\cmmathcal| alphabet and the last two compose the letters `O' and `o' in 
% roman, as for any operator: 
% \[ n^2+\bigO(n\log n) \txt{or} n^2+\bigo(n\log n)\txt{and} 
% \e^x=1+x+\frac{x^2}{2}+\lito\bigl(x^2\bigr).\]
%
% \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 double-struck letters 
% ($\mathbb{R}, \mathbb{C}, \mathbb{Z}, \mathbb{N}, \mathbb{Q}$) 
% are reserved for writing at the blackboard~\cite{LSHORT}.
% Similarly, to designate a field we use $\F$ or $\K$ (Körper in German).
% We obtain these symbols with the following macros:
% \begin{center}
% |\R|, |\C|, |\Z|, |\N|, |\Q|, |\F|, |\K|.
% \end{center}
%
% \smallskip
% \DescribeMacro{\mathset}
% The |\mathset| command enables you to change the behavior 
% of all these macros in a global way. 
% By default, |\mathset| is an alias for |\mathbf|, but if you prefer outline letters, 
% you can simply use |\renewcommand\mathset{\mathbb}| (with local effect when
% inside an environment or a pair of curly braces).
%
% \smallskip
% \DescribeMacro{\onlymathC}
% The macro |\onlymathC| is designed for cases when |\C| is already defined,
% but only in text mode (usually with the Russian language). Then you get the message:
% ``\texttt{Command \string\C\space invalid in math mode}''.
% This macro preserves the original definition for text mode
% and allows you to use |\C| for the complex number set in math mode.
% Simply call |\onlymathC| once in the preamble or anywhere in the document.
%
% \medskip
% \DescribeMacro{\ds}
% The |\displaystyle| command is very common, so the |\ds| alias 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 placed under or above the symbol 
% whereas for in-line math mode, 
% they are placed on the right, as a 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 can be forced with |\displaystyle|
% or its alias |\ds|. However, when using these commands, all the rest of the current
% mathematical environment will be set in displaymath mode 
% (as shown in the previous example, where the fraction will be expanded).
% To limit the display style effect to the affected symbol only, 
% similar the the  \textsf{amsmath} command |\dfrac|, 
% we can use the following macros:
% |\dlim|, |\dsum|, |\dprod|, |\dcup|, |\dcap|.
% So 
% \begin{center}
% |$\dlim_{x\to +\infty}\frac{1}{x}$| \quad  yields \quad  $\dlim_{x \to +\infty}\frac{1}{x}$.
% \end{center}
%
% \DescribeMacro{\lbar} \DescribeMacro{\hlbar}
% Large bars over expressions are obtained with |\overline|
% or its alias |\lbar|, to get for instance $\lbar{z_1z_2}$.
% Similar to vectors, you can raise the bar (from the height of `$A$') with
% the |\hlbar| command, to correct uneven bars heights.
% \begin{center}
% $\lbar{z+z'}=\hlbar{z}+\lbar{z'}$, obtained with |\hlbar{z}|,
% looks better than $\lbar{z+z'}=\lbar{z}+\lbar{z'}$.
% \end{center}
%
% \DescribeMacro{\eqdef} \DescribeMacro{\eqdef*}
% The |\eqdef| macro writes the equality symbol topped with `def',
% or with `$\scriptstyle \Delta$' for |\eqdef*|
% (thanks to the \LaTeX\ command |\stackrel|):\\[2ex] 
% \begin{minipage}{7cm}
% \begin{verbatim} 
%\[ \e^{\i\theta} \eqdef 
%   \cos\theta + \i\sin\theta \]
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{5cm}\vspace{-3ex}
% \[\e^{\i\theta}\eqdef\cos\theta + \i\sin\theta\]
% \end{minipage}
%
% \noindent\begin{minipage}{7cm}
% \begin{verbatim} 
%\[ \e^{\i\theta} \eqdef* 
%   \cos\theta + \i\sin\theta \]
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{5cm}\vspace{-3ex}
% \[\e^{\i\theta}\eqdef*\cos\theta + \i\sin\theta\]
% \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 and improved version 
% of the \texttt{\string \underbrace} command, along with many other useful macros.
% It is loaded by \textsf{mismath}.}, 
% making source code more compact.\\[2ex]
% \begin{minipage}{6.4cm}
% \begin{verbatim}
%\[ (QAP)^n = \unbr{QAP\mul QAP\mul
%   \cdots\mul QAP}_{n\text{ times}} \]
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{6cm}\vspace{-3ex}
% \[ (QAP)^n = \unbr{QAP\mul QAP\mul\cdots\mul QAP}_{n\text{ times}} \]
% \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
% just like the standard macro |\iff| does it with $\Longleftrightarrow$.
% Similarly, the |\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 as addition
% or subtraction operators, whereas division obtained with $/$ is closer to its operands.
% This actually hides the priority of multiplication over $+$ and $-$.
% That's 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|,
% looks 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 ensure the same amount of space on both sides, 
% you can use thin negative spaces |\!| 
% or enclose the function or structure with braces:
% \begin{center}
% $x\mul{\sin x}$, obtained with |x\mul{\sin x}|, is slightly better than $x\mul\sin x$;\\[1ex]
% |$\sin\!{\left( \frac{\pi}{3} \right)} \mul 2$|\\
% yields $\sin\!{\left(\frac{\pi}{3}\right)}\mul 2$, which looks better than
% $\sin\left(\frac{\pi}{3}\right)\mul 2$. 
% \end{center}
% The negative thin space after the function name is not related to |\mul|,
% but to the  ``excessive'' spaces around the |\left... \right| structure.
%
% \medskip
% \DescribeMacro{\pow}
% In the same way, when typesetting an exponent after a closing \emph{big} 
% parenthesis produced by |\right)|,
% the exponent appears to be a little to far from the parenthesis.
% To address this issue, the |\pow|\marg{expr}\marg{pow} command is provided
% \footnote{This macro gives bad results with \emph{normal-sized} parenthesis.},
% which sets the positioning of the exponent \meta{pow}
% slightly closer to the right parenthesis.
% Compare:
% \[ \e^a =\lim_{n \to +\infty}\pow{1+\frac{a}{n}}{n} 
% \txt{obtained with \texttt{\string \pow}, rather than} 
% \e^a =\lim_{n \to +\infty}\left(1+\frac{a}{n}\right)^{n}.\]
%
% \DescribeMacro{\abs}
% The correct typesetting of absolute value (or modular for a complex number) is achieved
% using |\lvert| \ldots |\rvert|, rather than $\mid$, as the latter doesn't maintain 
% proper spacing in some situations (when a sign follows the open delimiter).
% For bars whose height has to adapt to the content, 
% we can use |\left\vert| \ldots |\right\vert| or, more simply, 
% the |\abs|\{\ldots\} command, which is equivalent
% \footnote{We could also define \texttt{\string \abs} 
% using \texttt{\string \DeclarePairedDelimiter} 
% from the \mbox{\textsf{mathtools}} package~\cite{TOOL}.}.
%
% \medskip
% \DescribeMacro{\lfrac}
% The |\lfrac| macro behaves like |\frac|
% but with thicker spaces around the arguments,
% making the corresponding fraction bar slightly longer. See the following examples:
% \[ \lbar{Z} = \lfrac{\lbar{z_1-z_2}}{\lbar{z_1+z_2}} \qquad
%  u(x)= \lfrac{\frac{1-2x}{5}}{x^2+1} \qquad
% y'+xy=\lfrac[4mu]{1}{\sqrt{x}}
% \]
%
% This macro has an optional parameter |\lfrac|\oarg{space}\marg{num}\marg{denom}
% to adjust the length of the fraction bar. The optional \meta{space} argument must be given
% in \emph{math units} (\texttt{mu}); 
% the default value is \texttt{7mu} (equivalent to |\:\,|),
% \texttt{4mu} in the last example.
% \hypertarget{ibrackets}{}
%
% \medskip
% \DescribeMacro{[ibrackets]}
% Open intervals are commonly represented with parenthesis, \eg $(0, +\infty)$,
% but sometimes square brackets are used, especially in French mathematics: $]0, +\infty[$.
% In that specific case, the space around the square brackets is often inappropriate, 
% as in the expression $x \in \mathclose{]} 0, +\infty[$.
% To address this issue, we have redefined the brackets in the \textsf{ibrackets}
% package~\cite{BRACKET}.
% This one can be optionally
% loaded by \textsf{mismath} using the \texttt{ibrackets} package option.
% Thus |$x\in ]-\pi,0[ \cup ]2\pi,3\pi[$|
% \begin{align*}
% \mbox{yields \ } 
% x\in ]-\pi, 0[ \cup ]2\pi, 3\pi[ & \mbox{\ with \textsf{ibrackets}}, \\
% \mbox{instead of \ }
% x\in \mathclose{]}-\pi, 0 \mathopen{[} \cup 
% \mathclose{]} 2\pi, 3\pi \mathopen{[} 
% &\mbox{\ without \textsf{ibrackets}}.
% \end{align*}
%
% In our code, the symbols $[$ and $]$ are set as `active' characters, 
% behaving like ordinary characters and not as delimiters in most cases. 
% Therefore, a line break could occur between the two brackets, but
% it is always possible to transform them 
% into delimiters using |\left| and |\right|.
%
% However, when a bracket is \emph{immediately} followed by a + or - character,
% it becomes an open delimiter.
% Therefore, 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.
% For example if you write |$x \in ] -\infty, 0]$|, it yields $x \in  ] -\infty, 0]$
% instead of $x \in  ]-\infty, 0]$.
% Conversely, when dealing with algebraic expressions involving intervals,
% \emph{you must leave a blank space between the second bracket and the} +/- 
% \emph{operation}.
% For instance |$[a,b] +[c,d]$| yields $[a,b] +[c,d]$
% but |$[a,b]+ [c,d]$| yields $[a, b]+ [c, d]$.
%
% \medskip
% Note that there are other ways to proceed,
% for example with |\interval|,
% from the \textsf{interval} package~\cite{INT},
% or with |\DeclarePairedDelimiter|
% \footnote{You cannot use \texttt{\string\DeclarePairedDelimiter} with square brackets 
% when \textsf{ibrackets} is loaded.}
% from \mbox{\textsf{mathtools}}~\cite{TOOL}.
% \hypertarget{decimalcomma}{}
%
% \medskip
%\DescribeMacro{[decimalcomma]}
% In many countries, except notably in English-speaking countries, 
% the comma is used as a decimal separator for numbers. 
% However, in the math mode of \LaTeX, the comma is always, by default, 
% treated as a punctuation symbol and therefore is followed by a space. 
% This is appropriate in intervals: |$[a,b]$| results in $[a,b]$,
% but is not appropriate for numbers where the comma represents the decimal separator. 
% For example, |$12,5$| is displayed as $12, 5$ instead of $12{,}5$.
%
% Two very convenient packages allow handling the decimal comma in math mode:
% \textsf{icomma} by Walter Schmidt~\cite{ICOMMA} and 
% \textsf{ncccomma} by Alexander I.~Rozhenko~\cite{NCC}.
% The second package takes a more generic approach,
% however it poses several compatibility issues, 
% in particular when running through Lua\LaTeX, using \textsf{unicode-math}
% and calling |\setmathfont|.
% Therefore we propose the \textsf{decimalcomma} package~\cite{DEC},
% functionally identical to that of \textsf{ncccomma} but 
% without the aforementioned incompatibility.
% It can be loaded by \textsf{mismath} using the \texttt{decimalcomma}
% package option.
%
% \subsection{Environments for systems of equations and small matrices}
%
% \DescribeEnv{system}
% The \texttt{system} environment, defined in the \textsf{mismath} package, 
% is used to represent a system of equations:\\[2ex]
% \begin{minipage}{6.5cm}
% \begin{verbatim}
%\[ \begin{system} 
%       x=1+2t \\ y=2-t \\ z=-3-t 
%   \end{system} \]
% \end{verbatim}
% \end{minipage}
% \begin{minipage}{5cm} \vspace{-3ex}
% \[ \begin{system} x=1+2t \\ y=2-t \\z=-3-t \end{system} \]
% \end{minipage}
%
% \medskip
% \DescribeMacro{\systemsep}
% This first example could also have been achieved using the \texttt{cases} environment
% from the \textsf{amsmath} package, although \texttt{cases} places mathematical expressions 
% closer to the bracket.
% The |\systemsep| length allows you to adjust the gap between 
% the bracket and the expressions. 
% By default, the gap is set to |\medspace|. You can reduce this gap by redefining 
% the command, \eg: |\renewcommand{\systemsep}{\thinspace}|.
% Alternatively you can increase the gap using |\thickspace| 
% and the same spacing as of the \texttt{cases} environment 
% is obtained with |\renewcommand\systemsep}{}|.
% The |\systemsep| command allows for greater flexibility in adjusting the spacing 
% within the \texttt{system} environment.
%
% \medskip
% \DescribeEnv{system\oarg{coldef}}
% By default, a system is written like an \texttt{array} environment with only one column, 
% left aligned. However the \texttt{system} environment has an optional 
% argument that allows to create systems with multiple columns,
% specifying their alignment using the same syntax as the \texttt{array} environment in
% \LaTeX. For instance, using |\begin{system}[cl]| will produce a two-column system, 
% with the first column centered and
% the second column left-aligned, as shown in the following example:\\[2ex]
% \begin{minipage}{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}{5cm}\vspace{-2ex}
% \[ \begin{system}[cl] y&=\dfrac{1}{2}x-2 \\[1ex] (x,y)&\neq (0,-2) \end{system}\]
% \end{minipage}
%
% \DescribeMacro{\systemstretch}
% The default spacing between the lines of a \texttt{system} environment has been slightly 
% enlarged compared to the one used in \texttt{array} environments (using a factor of 1.2). 
% This can be adjusted by using
% |\renewcommand{\systemstretch}|\marg{stretch}, where \meta{stretch} is the desired value
% for the spacing. You can place this command 
% inside the current mathematical environment for a local change, or outside 
% for a global change. The default value for is 1.2.
% Furthermore you can also use the end of the line with a spacing option, as 
% demonstrated above with |\\[1ex]|, to control the spacing between specific lines in
% the system.
%
% 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 also mention the \textsf{systeme} package~\cite{SYST} which provides 
% a lighter syntax and automatic alignments for linear systems.
% Additionally, there is the \textsf{spalign} package~\cite{SPAL},
% which offers a convenient and easy syntax for systems and matrices with
% visually appealing alignments.
%
% \medskip
% \DescribeEnv{spmatrix}
% The \textsf{amsmath} package offers several environments to typeset matrices :
% For example, the \texttt{pmatrix} environment surrounds the matrix with parenthesis,
% and the \texttt{smallmatrix} environment creates a smaller matrix suitable for insertion 
% within a text line. We provide a combination of the these both 
% functionalities with the \texttt{spmatrix} environment:
% |$\vec{u}\begin{spmatrix}-1\\2\end{spmatrix}$| yielding
% $\vec{u}\begin{spmatrix}-1\\2\end{spmatrix}$.
%
% The \textsf{mathtools} package enhances the \textsf{amsmath} matrix environments
% and also provides a small matrix environment with parenthesis: \texttt{psmallmatrix}.
% Moreover, with the starred version |\begin{psmallmatrix*}|\oarg{col},
% you can choose the alignment inside the columns (\texttt{c}, \texttt{l} or \texttt{r}).
% However, the space before the left parenthesis is unfortunately too narrow
% compared to the space inside the parenthesis.
% To illustrate this, consider the following comparison:
% $\vec{u}\begin{spmatrix}-1\\2\end{spmatrix}$ (using \textsf{mismath}'s \texttt{spmatrix})
% vs.\@ $\vec{u}\begin{psmallmatrix}-1\\2\end{psmallmatrix}$ 
% (using \textsf{mathtools} \texttt{psmallmatrix}).
%
% \medskip
% For typesetting  various kinds of matrices, let's mention the excellent 
% \textsf{nicematrix} package by François Pantigny~\cite{NMATX}.
%
% \subsection{Displaymath in double columns}
%
% \DescribeEnv{mathcols}
% The \texttt{mathcols} environment allows you to arrange 
% lengthy calculations with short expressions across two columns
% separated by a vertical line, 
% as shown in the following example.
% However, to use this feature, the \textsf{multicol} package must be loaded in the preamble.
% The \texttt{mathcols} environment activates mathematical mode
% in display style and uses an \texttt{aligned} environment.
% \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 is used to switch to the next column,
% and alignments within the columns is done using the classic delimiters 
% |&|, to separate entries, and |\\|, to start a new row.
% 
% \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}
%
% \vspace{-4ex}
% \subsection{Summary of the package options}
%
% The following table summarizes the possible package options. 
% You can add to them any option you want to pass to \textsf{amsmath} or \textsf{mathtools}.
% The hyperlinks (in blue) redirect to the paragraphs in the documentation 
% where these options are described.
% \begin{center}
% \setlength{\extrarowheight}{1pt}
% \begin{tabular}{>{\ttfamily}ll}
% \hline
% \normalfont Option & \mbox{Effect} \\
% \hline
% \hyperlink{nofunction}{nofunction} & don't load some additional function definitions \\
% \hyperlink{classicReIm}{otherReIm} & typesets |\Re| and |\Im| as $\Re$ and $\Im$ \\
% \hyperlink{classicReIm}{classicReIm} & preserves |\Re| and |\Im| as $\oldRe$ and $\oldIm$ \\
% \hyperlink{ibrackets}{ibrackets} & loads the \textsf{ibrackets} package\\
% \hyperlink{decimalcomma}{decimalcomma} & loads the \textsf{decimalcomma} package \\
% \hline
% \end{tabular}
% \end{center}
% \StopEventually{}
% \section{Implementation}
%
% We load certain packages conditionally to avoid 'option clash' errors in cases
% where these packages have been previously loaded with other options.
% The \textsf{amsmath} package is loaded by \textsf{mathtools}.
%    \begin{macrocode}
\newif\ifmm@ibrackets % initialized to false
\DeclareOption{ibrackets}{\mm@ibracketstrue}
\newif\ifmm@decimalcomma
\DeclareOption{decimalcomma}{\mm@decimalcommatrue}
\newif\ifmm@nofunction
\DeclareOption{nofunction}{\mm@nofunctiontrue}
\newif\ifmm@otherReIm
\DeclareOption{otherReIm}{\mm@otherReImtrue}
\newif\ifmm@classicReIm
\DeclareOption{classicReIm}{\mm@classicReImtrue}
\DeclareOption*{\PassOptionsToPackage{\CurrentOption}{mathtools}}
\ProcessOptions \relax
%\@ifpackageloaded{amsmath}{}{\RequirePackage{amsmath}}
\@ifpackageloaded{mathtools}{}{\RequirePackage{mathtools}}
\@ifpackageloaded{esvect}{}{\RequirePackage[b]{esvect}}
\RequirePackage{ifthen}
\RequirePackage{xparse} % provides \NewDocumentCommand, now in LaTeX3
\RequirePackage{xspace} % for \iif command
\RequirePackage{iftex}
\RequirePackage{etoolbox} % provides \AtEndPreamble
\RequirePackage{xkeyval} % for \pinumber options

%    \end{macrocode}
%
% The package \textsf{unicode-math} causes some compatibility issues
% with \textsf{ibrackets} and \textsf{decimalcomma}: theses packages
% must be loaded \emph{after} \textsf{unicode-math},
% but \textsf{mismath} (like \textsf{amsmath}) should be loaded \emph{before} 
% \textsf{unicode-math}.
% And to complicate matters, \mbox{\textsf{unicode-math}} defines all its commands 
% by |\AtBeginDocument|.
% Therefore we used the command |\AtEndPreamble|, from the \textsf{etoolbox} package,
% which makes the job (because both \textsf{ibrackets} and \textsf{decimalcomma}
% work also in |\AtBeginDocument|).
%
% Moreover the command |\mathbfsfit| (used for tensors) is already defined
% in \textsf{unicode-math} and will not be redefined if \textsf{unicode-math}
% is loaded.
%    \begin{macrocode}
\@ifpackageloaded{unicode-math}{
    \PackageWarningNoLine{mismath}{The package unicode-math\MessageBreak 
        should be loaded after mismath}
}{}
\newif\ifmm@unicodemath
\newif\ifmm@multicol
\AtEndPreamble{% necessary to work with unicode-math
    \ifmm@decimalcomma\RequirePackage{decimalcomma}\fi
    \ifmm@ibrackets\RequirePackage{ibrackets}\fi
    \@ifpackageloaded{multicol}{\mm@multicoltrue}{}
    \@ifpackageloaded{unicode-math}{\mm@unicodemathtrue}{
        \DeclareMathAlphabet{\mathbfsfit}{\encodingdefault}%
            {\sfdefault}{bx}{it}}
}

%    \end{macrocode}
%
% \medskip
% \DescribeMacro{\bslash}
% The |\bslash| macro originates from Frank Mittelbach's \textsf{doc.sty} package. 
% It can be employed in other documents as an alternative to |\textbackslash|,
% especially in situations where |\textbackslash| does not work correctly, such as
% inside warning messages.
%    \begin{macrocode}
{\catcode`\|=\z@ \catcode`\\=12 |gdef|bslash{\}} % \bslash command

%    \end{macrocode}
%
% \DescribeMacro{\mm@warning} \DescribeMacro{\mm@macro} \DescribeMacro{\mm@operator} 
% The next three internal macros serve as meta commands for
% conditionally defining macros while providing a warning message if the macro already exists.
% These macros can be useful in other packages as well.
% \medskip
%    \begin{macrocode}
\newcommand\mm@warning[1]{
    \PackageWarningNoLine{mismath}{Command \bslash #1 already exist 
        \MessageBreak and will not be redefined}
}
\newcommand\mm@macro[2]{
    \@ifundefined{#1}{
        \expandafter\def\csname #1\endcsname{#2}
    }{\mm@warning{#1}}
}
\NewDocumentCommand\mm@operator{O{#3}mm}{%
    \@ifundefined{#1}{
        \DeclareMathOperator{#2}{#3}
    }{\mm@warning{#1}}
}
    
%    \end{macrocode}
%
% To produce the correct upright shape font when working 
% with the \textsf{beamer} package, you don't have to use |\mathrm|
% but rather |\mathup| (based on |\operatorfont| from the \textsf{amsopn} package).
% This command also works fine with other sans serif fonts like \textsf{cmbright}.
%
% Moreover for \textsf{beamer}, which changes the default font family (to sans serif),
% |\e|, |\i|, |\j| have no effect without |\AtBeginDocument|.
% |\AtBeginDocument| is also necessary to redefine |\i| when calling 
% the \textsf{hyperref} package which overwrites the |\i| definition.
% \medskip
%    \begin{macrocode}
\@ifundefined{mathup}{
    \providecommand*{\mathup}[1]{{\operatorfont #1}}
    }{\mm@warning{mathup} } % also in kpfonts and unicode-math
\mm@macro{e}{\mathup{e}}
\AtBeginDocument{\let\oldi\i \let\oldj\j
    \renewcommand{\i}{\TextOrMath{\oldi}{\mathup{i}}}
    \renewcommand{\j}{\TextOrMath{\oldj}{\mathup{j}}} }

%    \end{macrocode}
%
% \DescribeMacro{\MathFamily}
% The following macros |\MathUp| and |\MathIt| are switches that transform 
% any chosen letter in math mode to roman or italic style. 
% These switches can be used anywhere in the document or preamble.
% They are based on the generic macro |\MathFamily|.
% To obtain a letter in roman style instead of italic, we need to change the 
% mathcode digit that represents the font family: 1 to 0.
%
% For example, except for \LuaLaTeX, mathcode of the `e' letter is:
% `e="7165 (decimal 29029), with the second digit `1' indicating ``italic'' style.
% To get a roman `e', we need to change its mathcode to "7065.
%
% When used in the preamble, we call |\MathFamily| by |\AtBeginDocument|
% for working with the \textsf{beamer} package.
% Let's notice that |\MathFamily| has an erratic behavior when \textsf{unicode-math}
% is loaded, but fortunately, in that case, the |\DeclareMathSymbol| can be
% used instead, even outside the preamble.
% \medskip
%    \begin{macrocode}
\newcount\mm@charcode
\newcount\mm@charclass
\newcount\mm@charfam
\newcount\mm@charslot

\newcommand*\MathFamily[2]{%
    \mm@charfam=#2
    \ifluatex
        \mm@charclass=\Umathcharclass`#1
        %\mm@charfam=\Umathcharfam`#1
        \mm@charslot=\Umathcharslot`#1
        \Umathcode`#1= \mm@charclass \mm@charfam \mm@charslot
    \else
        \mm@charcode=\mathcode`#1
        % extract charclass
        \@tempcnta=\mm@charcode
        \divide\@tempcnta by "1000
        \multiply\@tempcnta by "1000 % charclass
        \mm@charclass=\@tempcnta
        % extract charslot
        \@tempcnta=\mm@charcode
        \@tempcntb=\mm@charcode
        \divide\@tempcnta by "100
        \multiply\@tempcnta by "100 % charclass + charfam
        \advance\@tempcntb by -\@tempcnta % charslot
        \mm@charslot=\@tempcntb
        % construct charcode
        \mm@charcode=\mm@charclass
        \multiply\mm@charfam by "100
        \advance\mm@charcode by \mm@charfam
        \advance\mm@charcode by \mm@charslot
        \mathcode`#1=\mm@charcode
    \fi
}

\newcommand*\MathUp[1]{%
    \ifx\@onlypreamble\@notprerr % not in preamble
        \ifmm@unicodemath
            \DeclareMathSymbol{#1}{\mathalpha}{operators}{`#1}
        \else
            \MathFamily{#1}{0}
        \fi
    \else % in preamble
      \AtBeginDocument{
        \ifmm@unicodemath
            \DeclareMathSymbol{#1}{\mathalpha}{operators}{`#1}
        \else
            \MathFamily{#1}{0}
        \fi
      }
    \fi
}

\newcommand*\MathIt[1]{%
    \ifx\@onlypreamble\@notprerr % not in preamble
        \ifmm@unicodemath
            \DeclareMathSymbol{#1}{\mathalpha}{letters}{`#1}
        \else
            \MathFamily{#1}{1}
        \fi
    \else % in preamble
      \AtBeginDocument{
        \ifmm@unicodemath
            \DeclareMathSymbol{#1}{\mathalpha}{letters}{`#1}
        \else
            \MathFamily{#1}{1}
        \fi
      }
    \fi
}

%    \end{macrocode}
% With a similar approach we could also create additional macros to set any letter
% in bold or sans serif. However, there is no default family number associated 
% with these typefaces. The family number depends on the font package being loaded and
% may vary depending on specific |\DeclareSymbolFont| used.
% Therefore, setting letters in bold or sans serif requires additional consideration 
% and may not have a straightforward solution.
%
% \medskip
% In addition to |\MathUp| and |\MathIt|, we also offer the following command
% to set a group of letters, among `e, i, j', in roman typeface.
%    \begin{macrocode}
\newcommand*\MathNumbers[1]{%
    \in@{e}{#1} \ifin@ \MathUp{e} \fi
    \in@{i}{#1} \ifin@ \MathUp{i} \fi
    \in@{j}{#1} \ifin@ \MathUp{j} \fi
}

%    \end{macrocode}
% \DescribeMacro{\apply}
% With the inverse switch |\MathNormal|, you can apply the normal (italic) style 
% on any comma-separated list of characters. This is achieved using
% the |\apply| macro, \eg |\apply\macro{arg1,arg2}|
% expands to |\macro{arg1}\macro{arg2}|.
% Thus |\apply\MathUp{e,i,j}| is equivalent to |\MathUp{e}\MathUp{i}\MathUp{j}|.
% I discovered this powerfull macro on \texttt{iterate190.rssing.com} by searching for 
% ``TeX How to iterate over a comma separated list''.
% The answer was posted under the pseudonym `wipet' on 2021/02/26.
% Let its author, Petr Ol\v{s}ák, be thanked.
% This macro allows to accomplish tasks that
% usual loop instructions like |\@for| or |\foreach| cannot achieve due to
% errors like ``!~Improper alphabetic constant''.
% For instance, if you try |\def\letter{A} \MathUp{\letter}| it will fail
% because the control sequence |\letter| is not strictly
% equivalent here to the single character `A'.
%
% \medskip
%    \begin{macrocode}
\def\apply#1#2{\apply@#1#2,\apply@,}
\def\apply@#1#2,{\ifx\apply@#2\empty
    \else #1{#2}\afterfi@{\apply@#1}\fi}
\def\afterfi@#1#2\fi{\fi#1}

\newcommand*\MathNormal[1]{% list argument
    \apply\MathIt{#1} 
}

%    \end{macrocode}
%
% The following commands were used until version 2.2 but still work.
% They were intended to set some letters in upright shape
% in math mode, but only worked in the preamble.
% This is now managed by the more powerful |\MathUp| command, and
% the old commands are maintained but as alias for |\MathUp|.
% \medskip
%
%    \begin{macrocode}
\newcommand{\enumber}{\MathUp{e}}
\newcommand{\inumber}{\MathUp{i}}
\newcommand{\jnumber}{\MathUp{j}}

%    \end{macrocode}
%
% Obtaining an upright Greek letter $\pi$ must be handled differently. 
% The switches are called |\pinumber| and |\pinormal| 
% and can be used anywhere in the document.
%
% But |\pinumber| must be called first in the preamble with an optional argument.
% This argument can be a valid command name that produces an upright pi letter
% (after having loading an appropriate package).
% Compatibility with \textsf{unicode-math} is a bit tricky!
% When given without an argument in the preamble, 
% |\pinumber| uses an LGR font encoding called \texttt{lmr}.
% A new feature (v2.11) is to use |\pinumber| with a keyval option to 
% use many other Greek pi letters without loading
% a whole package, thus without altering the other (italic) Greek letters. 
% We achieve this with |\DeclareSymbolFont| and |\DeclareMathSymbol|.
% We just have to know the ``name'' of the desired symbol font.
% 
% \medskip
%    \begin{macrocode}
\newif\ifmm@lgr
\define@cmdkey{pifonts}[mm@]{lgrmath}[lmr]{\mm@lgrtrue}
\newif\ifmm@upgreek
\define@choicekey{pifonts}{upgreek}[\mm@upgreek@option]%
    {Euler,Symbol,Symbolsmallscale}[Symbol]{\mm@upgreektrue}
\newif\ifmm@mathdesign
\define@choicekey{pifonts}{mathdesign}[\mm@mathdesign@option]%
    {Utopia,Garamond,Charter}[Charter]{\mm@mathdesigntrue}
\newif\ifmm@kpfonts
\define@choicekey{pifonts}{kpfonts}[\mm@kp@option]%
    {normal,light}[normal]{\mm@kpfontstrue}
\define@boolkeys{pifonts}[mm@]{fourier,pxfonts,txfonts}[true]
\newif\ifmm@fontspec
\define@cmdkey{pifonts}[mm@]{fontspec}[GFS Didot]{\mm@fontspectrue}
  
\newcommand*\pifonts[1]{%
    \setkeys{pifonts}{#1}
    \let\pi\relax
       
    \ifmm@lgr
        \DeclareFontEncoding{LGR}{}{}
        \DeclareSymbolFont{mmupgr}{LGR}{\mm@lgrmath}{m}{n}
        % may work with bold (b) instead of m
        \DeclareMathSymbol{\pi}{\mathord}{mmupgr}{112}

    \else\ifmm@fontspec
        \@ifpackageloaded{fontspec}{}{
            \PackageError{mismath}{\string\pinumber\space with
                the `fontspec' option\MessageBreak
                needs the fontspec package,\MessageBreak
                which must be run with LuaLaTeX or XeLaTeX}{}
        }
        \newfontfamily\mismathgreekfont{\mm@fontspec}[NFSSFamily=mgr]
        \DeclareSymbolFont{mmupgr}{TU}{mgr}{m}{n}
        \Umathchardef\pi="7 \symmmupgr "03C0
         
    \else\ifmm@upgreek
        \ifdefstring{\mm@upgreek@option}{Euler}{
            \DeclareFontFamily{U}{eur}{\skewchar\font'177}
            \DeclareFontShape{U}{eur}{m}{n}{%
              <-6> eurm5 <6-8> eurm7 <8-> eurm10}{}
            \DeclareSymbolFont{mmupgr}{U}{eur}{m}{n}
            \DeclareMathSymbol{\pi}{\mathord}{mmupgr}{"19}
        }{
        \ifdefstring{\mm@upgreek@option}{Symbol}{
            \DeclareSymbolFont{mmupgr}{U}{psy}{m}{n}
            \DeclareMathSymbol{\pi}{\mathord}{mmupgr}{`p}
        }{
        \ifdefstring{\mm@upgreek@option}{Symbolsmallscale}{
            \DeclareFontFamily{U}{fsy}{}
            \DeclareFontShape{U}{fsy}{m}{n}{<->s*[.9]psyr}{}
            \DeclareSymbolFont{mmupgr}{U}{fsy}{m}{n}
            \DeclareMathSymbol{\pi}{\mathord}{mmupgr}{`p}          
        }{}}}
          
    \else\ifmm@mathdesign
        \ifdefstring{\mm@mathdesign@option}{Utopia}{
            \DeclareSymbolFont{mmupgr}{OML}{mdput}{m}{n}
        }{
        \ifdefstring{\mm@mathdesign@option}{Garamond}{
            \DeclareSymbolFont{mmupgr}{OML}{mdugm}{m}{n}
        }{
        \ifdefstring{\mm@mathdesign@option}{Charter}{
            \DeclareSymbolFont{mmupgr}{OML}{mdbch}{m}{n}
        }{}}}

    \else\ifmm@fourier
        \DeclareFontEncoding{FML}{}{}
        \DeclareSymbolFont{mmupgr}{FML}{futm}{m}{it}

    \else\ifmm@kpfonts
        \ifdefstring{\mm@kp@option}{normal}{
            \DeclareSymbolFont{mmupgr}{U}{jkpmia}{m}{it}
        }{
        \ifdefstring{\mm@kp@option}{light}{
            \DeclareSymbolFont{mmupgr}{U}{jkplmia}{m}{it}
        }{}}

    \else\ifmm@pxfonts
        \DeclareSymbolFont{mmupgr}{U}{pxmia}{m}{it}

    \else\ifmm@txfonts
        \DeclareSymbolFont{mmupgr}{U}{txmia}{m}{it}
            
    \fi\fi\fi\fi\fi
        \DeclareMathSymbol{\pi}{\mathord}{mmupgr}{"19}    
    \fi\fi\fi
}

\newcommand*\pinumber[1][]{%
    \ifthenelse{\equal{#1}{}}{% no argument given
        \ifx\@onlypreamble\@notprerr % not in preamble
            \@ifundefined{savedpi}{
                \PackageWarning{mismath}{%
                    \string\pinumber\space
                    must be used in the preamble first}
            }{\let\pi\savedpi}
        \else % in the preamble
            \AtBeginDocument{
                \let\itpi\pi
                \pifonts{lgrmath}
                \let\savedpi\pi
            }
        \fi
    }{% command name or keyval options, necessarily in the preamble
        \AtBeginDocument{% must be here with unicode-math
            \let\itpi\pi
            \@ifundefined{#1}{%
                \pifonts{#1}
            }{
                \ifmm@unicodemath
                    \ifthenelse{\equal{#1}{uppi}}{% or "1D70B
                        \renewcommand\pi{\symup{\symbol{"03C0}}}
                        \renewcommand\itpi{\symit{\symbol{"03C0}}}
                    }{\renewcommand{\pi}{\csname #1\endcsname}}
                \else
                    \renewcommand{\pi}{\csname #1\endcsname}
                \fi
            }
            \let\savedpi\pi}
      }
}

\newcommand{\pinormal}{%
  \@ifundefined{itpi}{
    \PackageWarning{mismath}{Command \string\itpi\space undefined,
    \MessageBreak
    use \string\pinumber\space in the preamble first}
  }{
    \ifmm@unicodemath
      \@ifundefined{savedpi}{
        \PackageError{mismath}{Before using \string \pinormal,
        \MessageBreak
        you must call \string\pinumber\space in the preamble}{}}
    \fi
    \let\pi\itpi
  }
}

%    \end{macrocode}
% When |\pinumber|\oarg{keyval} has been called, you can also get 
% some other mathematical constants using Greek letters, 
% \eg $\othergamma$, the Euler-Mascheroni constant:
% \begin{verbatim}
% \let\gamma\relax
% \DeclareMathSymbol{\gamma}{\mathord}{mmupgr}{"0D}
% \end{verbatim}
% \vspace{-2ex}
% If \textsf{unicode-math} is used, you must put these commands in
% |\AtBeginDocument|.
% The hexadecimal code |"0D| depends on the option passed to |\pinumber|
% in the preamble (see the command |\pifonts| above and search in package docs). \\
% And to get the golden ratio $\othervarphi$:
% \begin{verbatim}
% \let\phi\relax
% \DeclareMathSymbol{\varphi}{\mathord}{mmupgr}{"27}
% \end{verbatim}
% \vspace{-2ex}
% To preserve the original $\gamma$ or $\varphi$, 
% you can define |\upgamma| or |\upvarphi| instead.
%
% \medskip
% And now the commands for vectors and tensors.
% \medskip
%    \begin{macrocode}
\newboolean{arrowvect}
\setboolean{arrowvect}{true}
\newcommand{\arrowvect}{\setboolean{arrowvect}{true}} 
\newcommand{\boldvect}{\setboolean{arrowvect}{false}}
\newcommand{\boldvectcommand}{\boldsymbol} % from amsbsy package
\mm@macro{vect}{\ifthenelse{\boolean{arrowvect}}{
    \vv}{\boldvectcommand}} % doesn't work well with \if... \fi
\newcommand*{\hvect}[1]{\vect{\vphantom{A}#1}}
\newcommand*{\hvec}[1]{\vec{\vphantom{A}#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}} 
    }
}{\mm@warning{norm}} % bad result with libertinust1math

\newcommand{\tensor}{\mathbfsfit} % isomath uses \mathsfbfit

%    \end{macrocode}

%    \begin{macrocode}
\mm@macro{di}{\mathop{}\!\mathup{d}}
\newcommand\probastyle{}
\let\Par\P % end of paragraph symbol
\renewcommand{\P}{\operatorname{\probastyle{P}}}
\mm@macro{E}{\operatorname{\probastyle{E}}}
\mm@macro{V}{\operatorname{\probastyle{V}}}

\newcommand*\MathProba[1]{%
    \in@{P}{#1} \ifin@ \MathUp{P} \fi
    \in@{E}{#1} \ifin@ \MathUp{E} \fi
    \in@{V}{#1} \ifin@ \MathUp{V} \fi
}

%    \end{macrocode}
% Classic identifiers are presented below. They will be defined only
% if the option \texttt{nofunction} has not been activated.
% \medskip
%    \begin{macrocode}
\ifmm@nofunction\else
    \mm@operator{\adj}{adj}
    \mm@operator{\Aut}{Aut}
    \mm@operator{\codim}{codim}
    \mm@operator{\coker}{coker}
    \mm@operator{\Conv}{Conv}
    \mm@operator{\cov}{cov}
    \mm@operator{\Cov}{Cov}
    \mm@macro{curl}{\operatorname{\vect{\mathup{curl}}}}
    \mm@operator[divg]{\divg}{div}
    
    \mm@operator{\End}{End}    
    \mm@operator{\erf}{erf}
    \mm@macro{grad}{\operatorname{\vect{\mathup{grad}}}}
    \mm@operator{\id}{id} % mathop or mathord?
    \mm@operator{\Id}{Id}
    \mm@operator{\im}{im}
    \mm@operator{\lb}{lb}
    \mm@operator{\lcm}{lcm}    
    \mm@operator{\rank}{rank}
    
    \mm@operator{\Res}{Res}
    \mm@macro{rot}{\operatorname{\vect{\mathup{rot}}}}
    \mm@operator{\sgn}{sgn}
    \mm@operator{\sinc}{sinc}
    \mm@operator[spa]{\spa}{span}
    \mm@operator{\tr}{tr}
    \mm@operator{\var}{var}
    \mm@operator{\Var}{Var}
    \mm@operator[Zu]{\Zu}{Z}
    
    \mm@operator{\arccot}{arccot}
    \mm@operator{\sech}{sech}
    \mm@operator{\csch}{csch}
    \mm@operator{\arsinh}{arsinh}
    \mm@operator{\arcosh}{arcosh}
    \mm@operator{\artanh}{artanh}
    \mm@operator{\arcoth}{arcoth}
    \mm@operator{\arsech}{arsech}
    \mm@operator{\arcsch}{arcsch}
\fi

%    \end{macrocode}
%
% The |\mathcal| alphabet, from the original Computer Modern font family,
% is used here to produce $\Re$, $\Im$ and $\bigO$.
% Several font packages redefines this alphabet producing glyphs that may seem less 
% suitable for the commands below.
% We have therefore retained the original |\cmmathcal| math alphabet, 
% which can be used for other letters.
% If \textsf{unicode-math} is called, it will redefine the commands |\Re| and |\Im|
% in |\AtBeginDocument|, hence the use of `|\AtEndPreamble{\AtBeginDocument{|'
% to ensure that the \textsf{mismath} redefinition occur after the actions 
% of \textsf{unicode-math}.
%
%    \begin{macrocode}
\DeclareFontFamily{U}{cmsy}{\skewchar\font48 }
\DeclareFontShape{U}{cmsy}{m}{n}{% from mathalpha
    <-5.5> cmsy5%
    <5.5-6.5> cmsy6%
    <6.5-7.5> cmsy7%
    <7.5-8.5> cmsy8%
    <8.5-9.5> cmsy9%
    <9.5-> cmsy10}{}
\DeclareMathAlphabet{\cmmathcal}{U}{cmsy}{m}{n}

\AtEndPreamble{\AtBeginDocument{
    \ifmm@classicReIm\else
        \let\oldRe\Re
        \let\oldIm\Im
    \ifmm@otherReIm
        \renewcommand{\Re}{\cmmathcal{R}\mathit{e}}
        \renewcommand{\Im}{\cmmathcal{I}\mathit{m}}    
    \else      
        \renewcommand{\Re}{\operatorname{Re}}
        \renewcommand{\Im}{\operatorname{Im}}
    \fi\fi 
}}

\mm@operator[bigO]{\bigO}{\cmmathcal{O}} 
\mm@operator[bigo]{\bigo}{O}
\mm@operator[lito]{\lito}{o}

%    \end{macrocode}
% And finally we present the remaining macros.
%
% With Cyrillic languages, the command |\C| may already be defined but only for text mode.
% Thus, it will not be redefined by \textsf{mismath}. However, when activating
% |\onlymathC|, you can to use our |\C| macro only for math mode,
% without interfering the definition of the text |\C| that is already defined.
%
% When using \XeTeX\ or \LuaTeX\ engines with the \textsf{hyperref} package,
% |\C| will be already defined and
% you get the message "\texttt{Command \string\C\space unavailable in encoding TU}".
% Therefore |\onlymathC| is automatically called in that case.
% \medskip
%    \begin{macrocode}
\mm@macro{mathset}{\mathbf}
\mm@macro{R}{\mathset{R}}
\AtBeginDocument{
    \@ifpackageloaded{hyperref}%
        {\iftutex\onlymathC\fi}% LuaTex or XeTeX engines
        {\mm@macro{C}{\mathset{C}}}
}
\providecommand\onlymathC{\let\oldC\C 
    \renewcommand{\C}{\TextOrMath{\oldC}{\mathset{C}}} }
\mm@macro{N}{\mathset{N}}
\mm@macro{Z}{\mathset{Z}}
\mm@macro{Q}{\mathset{Q}}
\mm@macro{F}{\mathset{F}}
\mm@macro{K}{\mathset{K}}

\mm@macro{ds}{\displaystyle}
\mm@macro{dlim}{\lim\limits}
\mm@macro{dsum}{\sum\limits}
\mm@macro{dprod}{\prod\limits}
\mm@macro{dcup}{\bigcup\limits}
\mm@macro{dcap}{\bigcap\limits}

\mm@macro{lbar}{\overline}
\@ifundefined{hlbar}{
    \providecommand*{\hlbar}[1]{\overline{\vphantom{A}#1}}}{
    \mm@warning{hlbar} }
\newcommand\@eqdef{\stackrel{\mathup{def}}{=}}
\newcommand\@@eqdef{\stackrel{\mathrm{\Delta}}{=}}
\mm@macro{eqdef}{\@ifstar{\@@eqdef}{\@eqdef}}
\mm@macro{unbr}{\underbrace}
\mm@macro{iif}{if and only if\xspace}

%    \end{macrocode}
% Above, we have used |\mathrm| before |\Delta| in case of defining capital Greek letters
% in italics (for example with the \textsf{fixmath} package).
%
% The use of |\mbox{}| ensures that the space produced by |\| in the |\then| macro 
% is not suppressed in tables.
% \medskip
%    \begin{macrocode}
\mm@macro{then}{\ \Longrightarrow \ \mbox{} } 
\@ifundefined{txt}{
    \providecommand*{\txt}[1]{\quad\text{#1}\quad} }{
    \mm@warning{txt} }
\mm@macro{mul}{\mathord{\times}}
\@ifundefined{pow}{
    \providecommand*{\pow}[2]{\left( #1 \right)^{\!#2}} }{
    \mm@warning{pow} }
\@ifundefined{abs}{
    \providecommand*{\abs}[1]{\left\vert#1\right\vert} }{
    \mm@warning{abs} }
\@ifundefined{lfrac}{
    \providecommand*{\lfrac}[3][7mu]{%
        \frac{\mkern#1#2\mkern#1}{\mkern#1#3\mkern#1}} }{
    \mm@warning{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
  \ifmm@multicol
    \renewcommand{\columnseprule}{0.1pt}
    \begin{multicols}{2}
        \par\noindent\hfill
        \begin{math}\begin{aligned}\displaystyle
  \else
    \PackageError{mismath}{The mathcols environment
        needs the multicol package}{Call the package multicol
        in your preamble.}
  \fi
}{%
        \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}{34}
% \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{TOOL} \emph{The \textsf{mathtools} package}, Morten Høgholm, Lars Madsen, CTAN,
% v1.29 2022/06/29.
% \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{UNIC} \emph{Experimental Unicode mathematical typesetting: 
% The \textsf{unicode-math} package}, Will Robertson, Philipp Stephani, Joseph Wright, 
% Khaled Hosny, and others, CTAN, v0.8r 2023/08/13.
% \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{MGREEK} \emph{The \textsf{mathgreeks} package}, Antoine Missier, CTAN,
% v1.1 2024/05/04.
% \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{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 2005/01/30.
% \bibitem{PX} \emph{\textsf{PX Fonts} -- Palatino-like fonts in support of mathematics}, 
% Young Ryu, CTAN, 2000/12/14.
% \bibitem{TX} \emph{\textsf{TX Fonts} -- Times-like fonts in support of mathematics}, 
% Young Ryu, CTAN, 2000/12/15.
% \bibitem{LIB} \emph{The LibertinusT1 Math Package}, Michael Sharpe, CTAN, v2.0.4 2024/01/14.
% \bibitem{LGR} \emph{The \textsf{lgrmath} package}, Jean-François B., CTAN, v1.0 2022/11/16.
% \bibitem{NTX} \emph{New TX font package}, Micahel Sharpe, CTAN, v1.735 2024/03/01.
% \bibitem{ALPHA} \emph{The \textsf{textalpha} package}
% (part of the \textsf{greek-fontenc} bundle), Günter Milde, CTAN, v2.1 2022/06/14.
% \bibitem{SPA} \emph{The \textsf{spacingtricks} package}, Antoine Missier, CTAN,
% v1.8 2023/12/06.
% \bibitem{BRACKET} \emph{Intelligent brackets -- The \textsf{ibrackets} package},
% Antoine Missier, CTAN, v1.2, 2023/07/26.
% \bibitem{INT} \emph{The \textsf{interval} package}, Lars Madsen, CTAN,
% v0.4 2019/03/06.
% \bibitem{ICOMMA} \emph{The \textsf{icomma} package for \LaTeXe},
% Walter Schmidt, CTAN, v2.0 2002/03/10.
% \bibitem{NCC} \emph{The \textsf{ncccomma} package}, Alexander I.~Rozhenko, 
% CTAN, v1.0 2005/02/10.
% \bibitem{DEC} \emph{The \textsf{decimalcomma} package}, Antoine Missier,
% CTAN, v1.4 2023/12/30.
% \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{NMATX} \emph{The package \textsf{nicematrix}}, François Pantigny, CTAN,
% v6.14 2023/02/18.
% \bibitem{FR} \emph{L'extension \textsf{frenchmath}}, Antoine Missier, CTAN, v3.0 2024/05/04.
% \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