%%%%%%%%%%%%%%
%  Version   %
%            %
% 2016.01.08 %
% 2016.01.27 %
%%%%%%%%%%%%%%






%
%
%                                                     MODULES          
%
%

%
%                                                      General 
%

\usepackage{fix-cm}
\usepackage{etex}


\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
%\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[french]{babel}


\usepackage{fourier}
%\usepackage[widespace]{fourier}

%
%                                                      AmS-Math 
%

\usepackage{amsmath,amscd,amssymb,amstext,amsgen,amsxtra,upref,amsopn,textcmds}
\usepackage{amsfonts}
%\usepackage{amsthm}

\usepackage[amsmath,thmmarks]{ntheorem}


{
\theoremstyle{plain}
\theoremheaderfont{\normalfont\upshape\bfseries}
\theorembodyfont{\normalfont\upshape}
\theoremseparator{}
\theoremsymbol{}

\newtheorem*{exercice}{Exercice}
\newtheorem{Exercice}{Exercice}
}

%
%                                                      Graphiques 
%

\usepackage[dvips]{xcolor}
\usepackage[dvips,all]{xy}
\usepackage{lastpage}
\usepackage{xspace}
\usepackage[dvips]{graphicx}
\usepackage[dvips]{graphics}
%\usepackage{gnuplottex}
\usepackage{mfpic}

\usepackage{tikz}
\usepackage{tikz-3dplot,tkz-fct,tkz-tab}
\usetikzlibrary{arrows,automata,backgrounds,shapes,calendar,er,matrix,folding,patterns,plothandlers,plotmarks,shapes.geometric,shapes.symbols,shapes.arrows,shapes.multipart,shapes.misc,topaths,trees,decorations.pathmorphing,decorations.pathreplacing,decorations.shapes,decorations.markings,decorations.fractals,chains,decorations.text,decorations.footprints,fadings,fit,through,shadows}
\usepackage{pstricks,pst-fill,pst-plot,pst-tree}

%\usepackage{extarrows}
%\usepackage{fvrb-ex}
%\usepackage{picinpar}
%\usepackage{epic,eepic,psfig,epsfig,graphpap}
%\usepackage{texlogos}
%\usepackage[dvips,final]{graphicx}
%\usepackage{qsymbols}
%\usepackage{pb-diagram,pb-xy}
%\usepackage{floatflt}
%\usepackage{multido}
%\usepackage{verbatim}
%\usepackage{emp}

%
%                                                          Tableaux 
%

%\usepackage{array}
%\usepackage{hhline}
%\usepackage{fancyvrb}
%\usepackage{tabularx,multicol}
%\usepackage{shadow}
%\usepackage{ifthen,tableau}

\usepackage{delarray}
\usepackage{caption}
\usepackage{supertabular}
\usepackage{multicol}
\usepackage{float}

%
%                                                                   Fonts 
%

\usepackage{mathrsfs,dsfont}
\usepackage{oldgerm,amsfonts}
\usepackage{yfonts}
%\usepackage{upgreek}
\usepackage{oldstyle}
\DeclareMathAlphabet{\mathpzc}{T1}{pzc}{m}{it}
\usepackage{skull}
\usepackage{eurosym}

%
%                                                                 Presentation
%

\usepackage{url}
\renewcommand{\thefootnote}{(\alph{footnote})}

\usepackage{geometry}
\usepackage{fancyhdr}
\usepackage{enumerate}

\DeclareRobustCommand{\artstyle}[4]
{\fancypagestyle{artstyle}{
\renewcommand{\headrulewidth}{1pt}
\renewcommand{\footrulewidth}{1pt}
\addtolength{\headheight}{\headsep}
\addtolength{\headwidth}{\marginparsep}
\fancyhf{}
\fancyhead[L]{\bfseries \nouppercase\leftmark}
\fancyhead[R]{\bfseries \nouppercase\rightmark}
\fancyfoot[RO,LE]{\bfseries \thepage}
\fancyfoot[LO]{\bfseries\footnotesize #1}
\fancyfoot[RE]{\bfseries\footnotesize #2}
}
\fancypagestyle{plain}{
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{1pt}
\addtolength{\headwidth}{\marginparsep}
\fancyhf{}
\fancyhead[L]{#3}
\fancyhead[R]{#4}
\fancyfoot[RO,LE]{\bfseries \thepage}
\fancyfoot[LO]{\bfseries\footnotesize #1}
\fancyfoot[RE]{\bfseries\footnotesize #2}
}
\pagestyle{artstyle}
}

\DeclareRobustCommand{\bookstyle}[2]
{\fancypagestyle{bookstyle}{
\renewcommand{\headrulewidth}{1pt}
\renewcommand{\footrulewidth}{1pt}
\addtolength{\headheight}{\headsep}
\addtolength{\headwidth}{\marginparsep}
\fancyhf{}
\fancyhead[LE]{\bfseries \nouppercase\leftmark}
\fancyhead[RO]{\bfseries \nouppercase\leftmark}
\fancyfoot[RO,LE]{\bfseries \thepage}
\fancyfoot[LO]{\bfseries\footnotesize #1}
\fancyfoot[RE]{\bfseries\footnotesize #2}
}
\fancypagestyle{plain}{
\renewcommand{\headrulewidth}{1pt}
\renewcommand{\footrulewidth}{1pt}
\addtolength{\headwidth}{\marginparsep}
\fancyhf{}
\fancyhead[LE,RO]{\bfseries \nouppercase \leftmark}
\fancyfoot[RO,LE]{\bfseries \thepage}
\fancyfoot[LO]{\bfseries\footnotesize #1}
\fancyfoot[RE]{\bfseries\footnotesize #2}
}
\pagestyle{bookstyle}
}

%\usepackage{abstract}
%\usepackage{indentfirst}
%\usepackage{extramarks}



%                                                                  En-tetes

\DeclareRobustCommand{\Hline}{\rule{\linewidth}{0.5mm}\vspace{0.5cm}}
\DeclareRobustCommand{\HRule}{\rule{\linewidth}{1mm}}
\DeclareRobustCommand{\Rulesup}{\rule{\linewidth}{1mm}\\[-10pt]\rule{\linewidth}{0.5pt}}
\DeclareRobustCommand{\Ruleinf}{\rule{\linewidth}{0.5pt}\\[-8pt]\rule{\linewidth}{1mm}}
\DeclareRobustCommand{\Sep}[1]{\hrulefill\ {#1}\ \hrulefill}
\DeclareRobustCommand{\entete}[5]{\noindent\HRule\\{\bfseries  #1} \hfill {\bfseries #2} \\[0.66\baselineskip]\noindent\centerline{\LARGE{{{\bfseries   #3}}}}\\[0.66\baselineskip]\noindent {\bfseries #4} \hfill{{\bfseries #5}}\\\noindent\HRule\\}





%                                            Pages de garde
%
% Page de garde estaca : #1=date, #2=type d'éval, #3=numéro, #4=durée, #5=Algèbre ou Analyse
%
%
% 
\DeclareRobustCommand{\estacapdg}[5]
{\thispagestyle{empty}
{\Large
\noindent {\parbox[c]{10cm}{\estacalogo{48pt}}} \hfill {\bf #1}

\vspace{\stretch{1}}

\begin{center}
{\Huge\bf\textsc{#4}}
\end{center}

\vspace{\stretch{1}}

\begin{center}
{\Huge\bf\textsc{#2}}
\end{center}


\vspace{\stretch{1}}

\begin{center}{{\LARGE\bf Première année}}\end{center}

% \vspace{\stretch{1}}
% 
% \begin{center}{#4}\end{center}

\vspace{\stretch{1}}

{\bfseries
\noindent Professeurs~:~\textsc{Y.~Messoudi}~---~\textsc{S.~Moreuil}.

\vspace{\stretch{1}}

\noindent Durée : #3.\\
\noindent \emph{Aucun} document autorisé.\\
\noindent Calculatrice \emph{non} autorisée.

\vspace{\stretch{1}}

Les exercices sont indépendants les uns des autres. \emph{La propreté} des copies,  \emph{la concision} et  \emph{la précision} de l'argumentation constituent une part importante de l'évaluation des candidats.

\vspace{\stretch{2}}

\noindent Nom~:\hfill Prénom~:\hfill\ 

\vspace{\stretch{2}}

\begin{center}{\huge Ne pas dégrafer le livret}\end{center}
}

\vspace{\stretch{1}}

\begin{center}
\fbox{\parbox{5\baselineskip}{\textbf{Note~:} \vspace{3\baselineskip}}}
\end{center}
}}

%
% Page de garde estaca : #1=date, #2=type d'éval, #3=durée, #4=Algèbre ou Analyse
%

\DeclareRobustCommand{\estacapdgrattrapage}[4]
{\thispagestyle{empty}
{\Large
\noindent {\parbox[c]{10cm}{\estacalogo{48pt}}} \hfill {\bf #1}

\vspace{\stretch{1}}

\begin{center}
{\Huge\bf \textsc{#4}}
\end{center}

\vspace{\stretch{1}}

\begin{center}
{\huge\bf #2}
\end{center}


\vspace{\stretch{1}}

\begin{center}{{\LARGE\bf Première année}}\end{center}

% \vspace{\stretch{1}}
% 
% \begin{center}{#4}\end{center}

\vspace{\stretch{1}}

{\bfseries
\noindent Professeurs~:~\textsc{Y.~Messoudi}~--~\textsc{S.~Moreuil}.

\vspace{\stretch{1}}

\noindent Durée : #3.\\
\noindent Aucun document autorisé.\\
\noindent Calculatrice \emph{non} autorisée.

\vspace{\stretch{1}}

Les exercices sont indépendants les uns des autres. \emph{La propreté} des copies,  \emph{la concision} et  \emph{la précision} de l'argumentation constituent une part importante de l'évaluation des candidats.

\vspace{\stretch{2}}

\noindent Nom~:\hfill\hfill\hfill\hfill Prénom~:\hfill\hfill\hfill\hfill

\vspace{\stretch{2}}

\begin{center}{\huge Ne pas dégrafer le livret}\end{center}
}

\vspace{\stretch{1}}

\begin{center}
\fbox{\parbox{5\baselineskip}{\textbf{Note~:} \vspace{3\baselineskip}}}
\end{center}
}

\newpage
\setcounter{page}{1}
}


% %                                            Pages de garde OTC
% %
% % Page de garde « Calcul » : #1=année scolaire , #2=type d'éval , #3=numéro , #4=date , #5=durée
% %
% %
% 
% 
% \DeclareRobustCommand{\calculpdg}[5]{
% \thispagestyle{empty}
% {\Large
% \noindent {\parbox[c]{10cm}{\estacalogo{48pt}}} \hfill {Année scolaire #1}
% 
% \vspace{\stretch{1}}
% 
% \begin{center}
% {\Huge\bf Outils et techniques de calcul}
% \end{center}
% 
% \vspace{\stretch{1}}
% 
% \begin{center}
% {\huge\bf  #2~\no #3}
% \end{center}
% 
% \vspace{\stretch{1}}
% 
% \begin{center}{\Large Première année}\end{center}
% \begin{center}{ #4}\end{center}
% 
% \vspace{\stretch{2}}
% 
% {\bfseries
% \noindent Professeurs~:~\textsc{Y.~Messoudi}~--~\textsc{S.~Moreuil}.\\
% 
% \vspace{\stretch{1}}
% 
% \noindent Durée : #5.\\
% \noindent Aucun document autorisé.\\
% \noindent Calculatrice \emph{non} autorisée.
% 
% \vspace{\stretch{1}}
% 
% \noindent \emph{Seuls} les résultats doivent être reportés sur la copie. Les
% raisonnements et les calculs intermédiaires ne seront pas pris en compte.\\
% 
% \vspace{\stretch{2}}
% 
% \noindent Nom~:\hfill\hfill\hfill\hfill Prénom~:\hfill\hfill\hfill\hfill Laval~$\Box$\hfill Levallois~(A$\Box$~ou~B$\Box$)
% 
% \vspace{\stretch{2}}
% 
% \begin{center}{\huge Ne pas dégrafer le livret}\end{center}
% }
% 
% \vspace{\stretch{2}}
% 
% \begin{center}
% \fbox{\parbox{3\baselineskip}{\textbf{1.}\vspace{2\baselineskip}}}\hfill\fbox{\parbox{3\baselineskip}{\textbf{A.} \vspace{2\baselineskip}}}\hfill\fbox{\parbox{3\baselineskip}{\textbf{0.}\vspace{2\baselineskip}}}\hfill
% \end{center}
% 
% \vspace{\stretch{1}}
% 
% \begin{center}
% \fbox{\parbox{5\baselineskip}{\textbf{Note~:}\vspace{3\baselineskip}}}
% \end{center}
% }
% \newpage
% \setcounter{page}{1}
% }
% 



%
%                                                    Bibliographie & Index
%


\usepackage{appendix}
\usepackage[french]{varioref}
\usepackage{makeidx}
\usepackage{xr}
%\usepackage{index}
\makeindex

\makeatletter
\newcommand{\ldsname}{Table des symboles}
\newcommand{\listofsymboles}{
 \@mkboth{\scshape\MakeUppercase\ldsname}
           {\scshape\MakeUppercase\ldsname}
 \if@openright\cleardoublepage\else\clearpage\fi
 \phantomsection
 \addcontentsline{toc}{chapter}{\MakeUppercase{\ldsname}}
 \chapter*{\ldsname}
\setlength{\baselineskip}{1ex plus .2ex minus .2ex}
\@starttoc{lsy}
}
\newcommand{\symbcaption}[1]{\addcontentsline{lsy}{symboles}{#1}}
\newcommand{\l@symboles}{\l@figure}
\makeatother

%\usepackage{cite} % espace insécable devant \cite{...}
%\usepackage[languagenames,fixlanguage]{babelbib}
%\usepackage{amsrefs} %définit le style bibliographique par défaut comme étant amsrn.bst
%\usepackage{multind}
%\usepackage{camel}
%\usepackage{xr}
%\usepackage{prettyref}
%\usepackage{listofsymbols}
%\usepackage{hyperref}




%
%                                                             MACROS           
%


%                                                           Compteurs  



\usepackage{datetime}
\newdateformat{dotdate}{\THEYEAR\twodigit{\THEMONTH}\twodigit{\THEDAY}}
\newtimeformat{dottime}{\twodigit{\THEHOUR}\twodigit{\THEMINUTE}}
\DeclareRobustCommand{\version}{\dotdate\today/\dottime}


\newcounter{Exo}[section]
\newcounter{Pexo}[Exo]
\newcounter{Ppexo}[Pexo]
\newcounter{Numeration}[subsection]
\newcounter{CalculusNumber}[subsection]%1.4041

\newenvironment{exo}{\stepcounter{Exo}\noindent{\large\bfseries\upshape Exercice \arabic{Exo}.~}\vspace{0.7071\baselineskip}}{\vspace{2.8082\baselineskip}}
\newenvironment{Ex}{\noindent{\large\bfseries\upshape Exercice.~}\vspace{0.7071\baselineskip}}{\vspace{2.8082\baselineskip}}
\DeclareRobustCommand{\pexo}{\stepcounter{Pexo}\noindent{\bfseries\upshape\arabic{Pexo}.\quad}}
\DeclareRobustCommand{\ppexo}{\stepcounter{Ppexo}\noindent{\bfseries\upshape(\alph{Ppexo})\quad}}
\DeclareRobustCommand{\num}{\stepcounter{Numeration}\noindent{\bfseries\upshape\arabic{Numeration}.\quad}}
\DeclareRobustCommand{\calnum}{\stepcounter{CalculusNumber}\noindent{\bfseries\upshape
    Calcul~\no~\arabic{CalculusNumber}.\quad}}


\usepackage[leftbars]{changebar}
\setlength{\changebarwidth}{0.7071pt}
\setcounter{changebargrey}{0}
\setlength{\changebarsep}{1.4142em}

%\newenvironment{correction}
%{\setcounter{Pexo}{0}\vspace{0.7071\baselineskip}\cbstart\noindent{\bfseries\itshape Correction.~}}
%{\cbend\vspace{1.4041\baselineskip}}
\newenvironment{correction}
{\vspace{0.7071\baselineskip}\cbstart\noindent{\bfseries\itshape Correction.~}}
{\cbend\vspace{1.4041\baselineskip}}

\newenvironment{probleme}
{\setcounter{Pexo}{0}\noindent{\large\bfseries\upshape Problème.~}\vspace{0.7071\baselineskip}}
{\vspace{1.4041\baselineskip}}


%
%                                                               Raccourcis 
%


\DeclareRobustCommand{\estacalogo}[1]{\includegraphics[height=#1]{/home/sam/Documents/Modeles/Logos/Estaca/estacalogo.eps}}
\DeclareRobustCommand{\estaca}{\textsc{estaca}\xspace}
\DeclareRobustCommand{\Estaca}{\textsc{Estaca}\xspace}

\DeclareRobustCommand{\exer}{\textsc{Ex.}\xspace}
\DeclareRobustCommand{\td}{\textsc{t.d.}\xspace}
\DeclareRobustCommand{\TD}{\textsc{T.D.}\xspace}

\DeclareRobustCommand{\qcm}{\textsc{q.c.m.}\xspace}
\DeclareRobustCommand{\cc}{\textsc{c.c.}\xspace}

\DeclareRobustCommand{\mq}{montrer que\xspace}
\DeclareRobustCommand{\ssi}{si et seulement si\xspace}
\DeclareRobustCommand{\ssil}{si et seulement s'il\xspace}
\DeclareRobustCommand{\coef}{coefficients\xspace}
\DeclareRobustCommand{\resp}{respectivement\xspace}
\DeclareRobustCommand{\ie}{\emph{i.e.}\xspace}
\DeclareRobustCommand{\pex}{par exemple\xspace}
\DeclareRobustCommand{\Pex}{Par exemple\xspace}
\DeclareRobustCommand{\cad}{c'est-à-dire\xspace}
\DeclareRobustCommand{\Cad}{C'est-à-dire\xspace}
\DeclareRobustCommand{\eg}{\emph{e.g.}\xspace}
\renewcommand{\etc}{~etc.\@\xspace}
\DeclareRobustCommand{\idem}{\emph{idem}\xspace}
\DeclareRobustCommand{\cf}{cf.\@\xspace}
\DeclareRobustCommand{\Cf}{Cf.\@\xspace}

\DeclareRobustCommand{\faa}{fixe algébriquement attractif\xspace}
\DeclareRobustCommand{\faas}{fixes algébriquement attractifs\xspace}
\DeclareRobustCommand{\fta}{fixe topologiquement attractif\xspace}
\DeclareRobustCommand{\ftas}{fixes topologiquement attractifs\xspace}
\DeclareRobustCommand{\far}{fixe algébriquement répulsif\xspace}
\DeclareRobustCommand{\fars}{fixes algébriquement répulsifs\xspace}
\DeclareRobustCommand{\ftr}{fixe topologiquement répulsif\xspace}
\DeclareRobustCommand{\ftrs}{fixes topologiquement répulsifs\xspace}
\DeclareRobustCommand{\fpqc}{\textsc{fpqc}\xspace}


\DeclareRobustCommand{\sev}{sous-es\-pace vectoriel\xspace}
\DeclareRobustCommand{\sevs}{sous-es\-paces vectoriels\xspace}
\DeclareRobustCommand{\Sev}{Sous-es\-pace vectoriel\xspace}
\DeclareRobustCommand{\Sevs}{Sous-es\-paces vectoriels\xspace}
\DeclareRobustCommand{\ev}{espace vectoriel\xspace}
\DeclareRobustCommand{\evs}{espaces vectoriels\xspace}
\DeclareRobustCommand{\sda}{système dynamique algébrique\xspace}
\DeclareRobustCommand{\Sda}{système dynamique algébrique\xspace}
\DeclareRobustCommand{\sdas}{systèmes dynamiques algébriques\xspace}
\DeclareRobustCommand{\Sdas}{systèmes dynamiques algébriques\xspace}
\DeclareRobustCommand{\sdl}{système différentiel linéaire\xspace}
\DeclareRobustCommand{\Sdl}{Système différentiel linéaire\xspace}
\DeclareRobustCommand{\sdls}{systèmes différentiels linéaires\xspace}
\DeclareRobustCommand{\Sdls}{Systèmes différentiels linéaires\xspace}

\DeclareRobustCommand{\flot}{--flot\xspace}
\DeclareRobustCommand{\flots}{--flots\xspace}
\DeclareRobustCommand{\periodique}{--pé\-rio\-dique\xspace}
\DeclareRobustCommand{\periodiques}{--pé\-rio\-diques\xspace}
\DeclareRobustCommand{\schema}{--sché\-ma\xspace}
\DeclareRobustCommand{\schemas}{--sché\-mas\xspace}
\DeclareRobustCommand{\sousschema}{--sous-sché\-ma\xspace}
\DeclareRobustCommand{\sousschemas}{--sous-sché\-mas\xspace}
\DeclareRobustCommand{\stable}{--sta\-ble\xspace}
\DeclareRobustCommand{\stabilite}{--sta\-bi\-li\-té\xspace}
\DeclareRobustCommand{\stables}{--sta\-bles\xspace}
\DeclareRobustCommand{\total}{--to\-ta\-le\-ment\xspace}
\DeclareRobustCommand{\invariance}{--in\-va\-riance\xspace}
\DeclareRobustCommand{\invariant}{--in\-va\-riant\xspace}
\DeclareRobustCommand{\invariants}{--in\-va\-riants\xspace}
\DeclareRobustCommand{\morphisme}{--mor\-phisme\xspace}
\DeclareRobustCommand{\morphismes}{--mor\-phismes\xspace}
\DeclareRobustCommand{\immersion}{--im\-mer\-sion\xspace}
\DeclareRobustCommand{\immersions}{--im\-mer\-sions\xspace}
\DeclareRobustCommand{\isomorphe}{--iso\-morphe\xspace}
\DeclareRobustCommand{\isomorphes}{--iso\-morphes\xspace}
\DeclareRobustCommand{\automorphisme}{--auto\-mor\-phisme\xspace}
\DeclareRobustCommand{\automorphismes}{--auto\-mor\-phismes\xspace}
\DeclareRobustCommand{\isomorphisme}{--iso\-mor\-phisme\xspace}
\DeclareRobustCommand{\isomorphismes}{--iso\-mor\-phismes\xspace}
\DeclareRobustCommand{\Endomorphisme}{--en\-do\-mor\-phisme\xspace}
\DeclareRobustCommand{\Endomorphismes}{--en\-do\-mor\-phismes\xspace}
\DeclareRobustCommand{\algebre}{--al\-gèbre\xspace}
\DeclareRobustCommand{\algebres}{--al\-gèbres\xspace}
\DeclareRobustCommand{\module}{--mo\-dule\xspace}
\DeclareRobustCommand{\modules}{--mo\-dules\xspace}
\DeclareRobustCommand{\ltf}{localement de type fini\xspace}
\DeclareRobustCommand{\tf}{de type fini\xspace}
\DeclareRobustCommand{\lpf}{localement de présentation finie\xspace}
\DeclareRobustCommand{\pf}{de présentation finie\xspace}
\DeclareRobustCommand{\reduit}{\text{réd}}
\DeclareRobustCommand{\et}{\text{ét}}
\DeclareRobustCommand{\op}{\text{op}}
\DeclareRobustCommand{\sep}{\text{sép}}
\DeclareRobustCommand{\rationnel}{--ra\-tion\-nel\xspace}
\DeclareRobustCommand{\rationnels}{--ra\-tion\-nels\xspace}

\DeclareRobustCommand{\tvi}{\textsc{tvi}\xspace}
\DeclareRobustCommand{\Tvi}{\textsc{Tvi}\xspace}
\DeclareRobustCommand{\iaf}{\textsc{iaf}\xspace}
\DeclareRobustCommand{\Iaf}{\textsc{Iaf}\xspace}
\DeclareRobustCommand{\iafg}{\textsc{iafg}\xspace}
\DeclareRobustCommand{\Iafg}{\textsc{Iafg}\xspace}
\DeclareRobustCommand{\taf}{\textsc{taf}\xspace}
\DeclareRobustCommand{\Taf}{\textsc{Taf}\xspace}
\DeclareRobustCommand{\tafg}{\textsc{tafg}\xspace}
\DeclareRobustCommand{\Tafg}{\textsc{Tafg}\xspace}
\DeclareRobustCommand{\speg}{\textsc{speg}\xspace}
\DeclareRobustCommand{\sgep}{\textsc{sgep}\xspace}

%
%
%


%
%                                                                      Maths 
%

\allowdisplaybreaks

\DeclareRobustCommand{\bbmC}{\mathbbm{C}}
\DeclareRobustCommand{\bbmR}{\mathbbm{R}}
\DeclareRobustCommand{\bbmN}{\mathbbm{N}}
\DeclareRobustCommand{\bbmZ}{\mathbbm{Z}}
\DeclareRobustCommand{\bbmQ}{\mathbbm{Q}}

\DeclareRobustCommand{\C}{\mathds{C}}
\DeclareRobustCommand{\R}{\mathds{R}}
\DeclareRobustCommand{\N}{\mathds{N}}
\DeclareRobustCommand{\Z}{\mathds{Z}}
\DeclareRobustCommand{\D}{\mathds{D}}
\DeclareRobustCommand{\Q}{\mathds{Q}}
\DeclareRobustCommand{\K}{\mathds{K}}
\renewcommand{\P}{\mathds{P}}
\DeclareRobustCommand{\A}{\mathds{A}}
\DeclareRobustCommand{\I}{\mathds{I}}
\DeclareRobustCommand{\J}{\mathds{J}}
\DeclareRobustCommand{\Fini}{\mathds{F}}

\DeclareRobustCommand{\ZZ}[1]{\Z\!/\! #1 \Z}

\DeclareRobustCommand{\V}{\mathcal{V}}
\DeclareRobustCommand{\U}{\mathtt{U}}

%\DeclareRobustCommand{\oldepsilon}{\epsilon}
%\renewcommand{\epsilon}{\varepsilon}
%\renewcommand{\phi}{\varphi}

\DeclareRobustCommand{\sg}[2]{\Gamma(#1,#2)}
\renewcommand{\O}{\mathcal{O}}
\DeclareRobustCommand{\F}{\mathcal{F}}
\DeclareRobustCommand{\G}{\mathcal{G}}
\DeclareRobustCommand{\B}{\mathcal{B}}

\DeclareRobustCommand{\Cn}[2]{\mathcal{C}^{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Dn}[2]{\mathcal{D}^{#1}\!\left(#2\right)}
\DeclareRobustCommand{\interieur}[1]{\mathring{#1}}

\DeclareRobustCommand{\kahler}[1]{\Omega_{#1}}
\DeclareRobustCommand{\Kahler}[2]{\Omega_{#1}^{#2}}

\DeclareRobustCommand{\tri}{\triangle}

\DeclareRobustCommand{\prolim}{\varprojlim\nolimits}
\DeclareRobustCommand{\indlim}{\varinjlim\nolimits}

\DeclareRobustCommand{\Orbite}[3]{\operatorname*{O}\nolimits^{#1}_{#2}(#3)}
\DeclareRobustCommand{\cyclo}[2]{\operatorname*{\mu}\nolimits_{{#1}}\!\left(#2\right)}
\DeclareRobustCommand{\prePer}[1]{\operatorname*{prePer}\!\left(#1\right)}
\DeclareRobustCommand{\Per}[1]{\operatorname*{Per}\!\left(#1\right)}
\DeclareRobustCommand{\ifs}[2]{\operatorname*{IFS}\!\left(#1,#2\right)}
\DeclareRobustCommand{\ifsa}[2]{\operatorname*{IFS}_{\text{a}}\!\left(#1,#2\right)}
\DeclareRobustCommand{\ifscoa}[2]{\operatorname*{IFS}^{\text{a}}\!\left(#1,#2\right)}
\DeclareRobustCommand{\Spr}[1]{\operatorname*{Spr}\!\left(#1\right)}
\DeclareRobustCommand{\Spro}[1]{\operatorname*{Spro}\!\left(#1\right)}
\DeclareRobustCommand{\Sprf}[1]{\operatorname*{Sprf}\!\left(#1\right)}
\DeclareRobustCommand{\BA}[1]{\operatorname*{BA}\!\left(#1\right)}

\DeclareRobustCommand{\Top}{\mathfrak{Top}}
\DeclareRobustCommand{\TopLC}{\mathfrak{TopLC}}
\DeclareRobustCommand{\Set}{\mathfrak{Set}}
\DeclareRobustCommand{\Sch}{\mathfrak{Sch}}
\DeclareRobustCommand{\Ab}{\mathfrak{Ab}}
\DeclareRobustCommand{\Alg}{\mathfrak{Alg}}
\DeclareRobustCommand{\AlgQC}{\mathfrak{AlgQC}}
\DeclareRobustCommand{\Noeth}{\mathfrak{Noeth}}
\DeclareRobustCommand{\SchAffNoeth}{\mathfrak{SchAffNoeth}}
\DeclareRobustCommand{\Et}{\mathfrak{\acute{E}t}}
\DeclareRobustCommand{\LPF}{\mathfrak{LPF}}

\DeclareRobustCommand{\p}{\mathfrak{p}}
\renewcommand{\q}{\mathfrak{q}}
\renewcommand{\a}{\mathfrak{a}}
\DeclareRobustCommand{\m}{\mathfrak{m}}


\DeclareRobustCommand{\nul}[1]{0_{{ #1}}}

\DeclareRobustCommand{\res}[1]{\operatorname*{Res}\!\left(#1\right)}
\DeclareRobustCommand{\Hom}[3]{\operatorname*{Hom}\nolimits_{#1}\!\left(#2,#3\right)}
\renewcommand{\hom}[1]{\operatorname*{Hom}\!\left(#1\right)}
\DeclareRobustCommand{\End}[1]{\operatorname*{End}\!\left(#1\right)}
\DeclareRobustCommand{\Endo}[2]{\operatorname*{End}\nolimits_{{#1}}\!\left(#2\right)}
\DeclareRobustCommand{\L}[1]{\mathcal{L}\!\left(#1\right)}
\DeclareRobustCommand{\LK}[2]{\mathcal{L}_{#1}\!\left(#2\right)}
\DeclareRobustCommand{\aut}[1]{\operatorname*{Aut}\!\left(#1\right)}
\DeclareRobustCommand{\Aut}[2]{\operatorname*{Aut}\nolimits_{{#1}}\!\left(#2\right)}

\DeclareRobustCommand{\dom}[1]{\operatorname*{dom}\!\left(#1\right)}
\DeclareRobustCommand{\codom}[1]{\operatorname*{codom}\!\left(#1\right)}
\DeclareRobustCommand{\ob}[1]{\operatorname*{\mathtt{Ob}}\!\left(#1\right)}
\DeclareRobustCommand{\fl}[1]{\operatorname*{\mathtt{Fl}}\!\left(#1\right)}
\DeclareRobustCommand{\Sheaf}[1]{\operatorname*{\mathbf{{S}}}\!\left(#1\right)}
\DeclareRobustCommand{\Presheaf}[1]{\operatorname*{\mathbf{P}}\!\left(#1\right)}
\DeclareRobustCommand{\Spec}[1]{\operatorname*{Spec}\!\left(#1\right)}


%


\DeclareRobustCommand{\pgcd}[1]{\operatorname*{pgcd}\!\left(#1\right)}
\DeclareRobustCommand{\ppcm}[1]{\operatorname*{ppcm}\!\left(#1\right)}
\DeclareRobustCommand{\conj}[1]{\overline{#1}}
\DeclareRobustCommand{\der}[1]{\operatorname*{d}\!\left(#1\right)}
\renewcommand{\mod}[1]{\ \operatorname*{mod}\!\left(#1\right)}


\newcommand{\Max}[2]{\operatorname*{Max}{#1}\!\left(#2\right)}
\newcommand{\Min}[2]{\operatorname*{Min}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Sup}[2]{\operatorname*{Sup}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Inf}[2]{\operatorname*{Inf}{#1}\!\left(#2\right)}

\renewcommand{\max}[1]{\operatorname*{Max}\!\left(#1\right)}
\renewcommand{\min}[1]{\operatorname*{Min}\!\left(#1\right)}
\renewcommand{\sup}[1]{\operatorname*{Sup}\!\left(#1\right)}
\renewcommand{\inf}[1]{\operatorname*{Inf}\!\left(#1\right)}
\renewcommand{\deg}[1]{\operatorname*{deg}\!\left(#1\right)}
\newcommand{\val}[1]{\operatorname*{val}\!\left(#1\right)}


\DeclareRobustCommand{\sgn}[1]{\operatorname*{signe}\!\left(#1\right)}
\DeclareRobustCommand{\Def}[1]{\operatorname*{D\acute{e}f}\!\left(#1\right)}

%

\renewcommand{\sin}[1]{\operatorname*{sin}\!\left(#1\right)}
\DeclareRobustCommand{\sinus}{\operatorname*{sin}}
\DeclareRobustCommand{\Sin}[2]{\operatorname*{sin}{#1}\!\left(#2\right)}
\renewcommand{\cos}[1]{\operatorname*{cos}\!\left(#1\right)}
\DeclareRobustCommand{\cosinus}{\operatorname*{cos}}
\DeclareRobustCommand{\Cos}[2]{\operatorname*{cos}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\cotan}[1]{\operatorname*{cotan}\!\left(#1\right)}
\DeclareRobustCommand{\cotangente}{\operatorname*{cotan}}
\DeclareRobustCommand{\Cotan}[2]{\operatorname*{cotan}{#1}\!\left(#2\right)}
\renewcommand{\tan}[1]{\operatorname*{tan}\!\left(#1\right)}
\DeclareRobustCommand{\tangente}{\operatorname*{tan}}
\DeclareRobustCommand{\Tan}[2]{\operatorname*{tan}{#1}\!\left(#2\right)}

\renewcommand{\arccos}[1]{\operatorname*{Arccos}\!\left(#1\right)}
\DeclareRobustCommand{\Arccos}[2]{\operatorname*{Arccos}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\arccosinus}{\operatorname*{Arccos}}
\renewcommand{\arcsin}[1]{\operatorname*{Arcsin}\!\left(#1\right)}
\DeclareRobustCommand{\Arcsin}[2]{\operatorname*{Arcsin}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\arcsinus}{\operatorname*{Arcsin}}
\renewcommand{\arctan}[1]{\operatorname*{Arctan}\!\left(#1\right)}
\DeclareRobustCommand{\Arctan}[2]{\operatorname*{Arctan}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\arctangente}{\operatorname*{Arctan}}
\DeclareRobustCommand{\arccotan}[1]{\operatorname*{Arccotan}\!\left(#1\right)}
\DeclareRobustCommand{\Arccotan}[2]{\operatorname*{Arccotan}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\arccotangente}{\operatorname*{Arccotan}}



\DeclareRobustCommand{\sh}[1]{\operatorname*{sh}\!\left(#1\right)}
\DeclareRobustCommand{\Sh}[2]{\operatorname*{sh}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\sinush}{\operatorname*{sh}}
\DeclareRobustCommand{\ch}[1]{\operatorname*{ch}\!\left(#1\right)}
\DeclareRobustCommand{\Ch}[2]{\operatorname*{ch}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\cosinush}{\operatorname*{ch}}
\renewcommand{\th}[1]{\operatorname*{th}\!\left(#1\right)}
\DeclareRobustCommand{\Th}[2]{\operatorname*{th}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\tangenteh}{\operatorname*{th}}
\renewcommand{\coth}[1]{\operatorname*{coth}\!\left(#1\right)}
\DeclareRobustCommand{\Coth}[2]{\operatorname*{coth}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\cotangenteh}{\operatorname*{coth}}

\DeclareRobustCommand{\argsh}[1]{\operatorname*{Argsh}\!\left(#1\right)}
\DeclareRobustCommand{\Argsh}[2]{\operatorname*{Argsh}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\argsinush}{\operatorname*{Argsh}}
\DeclareRobustCommand{\argch}[1]{\operatorname*{Argch}\!\left(#1\right)}
\DeclareRobustCommand{\Argch}[2]{\operatorname*{Argch}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\argcosinush}{\operatorname*{Argch}}
\DeclareRobustCommand{\argth}[1]{\operatorname*{Argth}\!\left(#1\right)}
\DeclareRobustCommand{\Argth}[2]{\operatorname*{Argth}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\argtangenteh}{\operatorname*{Argth}}
\DeclareRobustCommand{\argcoth}[1]{\operatorname*{Argcoth}\!\left(#1\right)}
\DeclareRobustCommand{\Argcoth}[2]{\operatorname*{Argcoth}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\argcotangenteh}{\operatorname*{Argcoth}}

\renewcommand{\ln}[1]{\operatorname*{ln}\!\left(#1\right)}
\DeclareRobustCommand{\Ln}[2]{\operatorname*{ln}{#1}\!\left(#2\right)}
\renewcommand{\log}[2]{\operatorname*{log}_{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Log}[3]{\operatorname*{log}{#1}_{#2}\!\left(#3\right)}
\DeclareRobustCommand{\LOG}[1]{\operatorname*{log}\!\left(#1\right)}

\DeclareRobustCommand{\id}[1]{\operatorname*{id}\nolimits_{#1}}
\DeclareRobustCommand{\e}[1]{\operatorname*{e}\nolimits^{#1}}
\DeclareRobustCommand{\Exp}[2]{\operatorname*{exp}{#1}\!\left(#2\right)}
\DeclareRobustCommand{\expa}[2]{\operatorname*{exp}_{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Expa}[3]{\operatorname*{exp}{#1}_{#2}\!\left(#3\right)}
\renewcommand{\exp}[1]{\operatorname*{exp}\!\left(#1\right)}
\DeclareRobustCommand{\E}[1]{\operatorname*{E}\!\left(#1\right)}

\DeclareRobustCommand{\sign}[1]{\operatorname*{sign}\!\left(#1\right)}

%

\DeclareRobustCommand{\Det}{\operatorname*{det}\!}
\renewcommand{\det}[1]{\operatorname*{det}\!\left(#1\right)}
\DeclareRobustCommand{\tr}[1]{\operatorname*{tr}\!\left(#1\right)}
\renewcommand{\t}[1]{\sideset{^{\operatorname*{t}}}{}{#1}}
\DeclareRobustCommand{\rg}[1]{\operatorname*{rg}\!\left(#1\right)}
\DeclareRobustCommand{\cof}[1]{\operatorname*{cofact}\!\left(#1\right)}
\DeclareRobustCommand{\Norm}[1]{\operatorname*{Norm}\!\left(#1\right)}
\DeclareRobustCommand{\norme}[1]{\left|\!\left|#1\right|\!\right|}
\DeclareRobustCommand{\va}[1]{\left|#1\right|}
\DeclareRobustCommand{\scal}[2]{\left\langle#1\left|#2\right\rangle\right.}
\DeclareRobustCommand{\im}[1]{\operatorname*{Im}\!\left(#1\right)}
\DeclareRobustCommand{\re}[1]{\operatorname*{Re}\!\left(#1\right)}
\renewcommand{\arg}[1]{\operatorname*{arg}\!\left(#1\right)}
\renewcommand{\ker}[1]{\operatorname*{Ker}\!\left(#1\right)}
\DeclareRobustCommand{\coker}[1]{\operatorname*{Coker}\!\left(#1\right)}

\DeclareRobustCommand{\Card}[1]{\operatorname*{Card}\!\left(#1\right)}
\DeclareRobustCommand{\Diff}[1]{\operatorname*{D}\!\left(#1\right)}
\DeclareRobustCommand{\dist}[1]{\operatorname*{d}\!\left(#1\right)}
\DeclareRobustCommand{\partie}[1]{\mathcal{P}\!\left(#1\right)}


\DeclareRobustCommand{\ps}[2]{\left<#1\left|\right.#2\right>}
\renewcommand{\dim}[2]{\operatorname*{dim}\nolimits_{{ #1}}\!\left(#2\right)}
\DeclareRobustCommand{\Mat}[2]{\operatorname*{mat}\nolimits_{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Sp}[1]{\operatorname*{Sp}\!\left(#1\right)}
\DeclareRobustCommand{\GL}[2]{\operatorname*{GL}_{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Vect}[2]{\operatorname*{Vect}\nolimits_{#1}\!\left(#2\right)}
\renewcommand{\vec}[1]{\overrightarrow{#1}}
\DeclareRobustCommand{\vecteur}[1]{\begin{pmatrix}#1\end{pmatrix}}
\DeclareRobustCommand{\tvecteur}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
\DeclareRobustCommand{\matrice}[1]{\begin{pmatrix}#1\end{pmatrix}}
\DeclareRobustCommand{\vmatrice}[1]{\begin{vmatrix}#1\end{vmatrix}}
\DeclareRobustCommand{\tmatrice}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
\DeclareRobustCommand{\Com}[1]{\operatorname*{Com}\!\left(#1\right)}
\DeclareRobustCommand{\transpose}[1]{{\vphantom{#1}}^{\operatorname*{t}}\!{#1}}
\DeclareRobustCommand{\Det}[2]{\operatorname*{det}\nolimits_{#1}\!\left(#2\right)}
\renewcommand{\det}[1]{\operatorname*{det}\!\left(#1\right)}


\DeclareRobustCommand{\M}[2]{\mathcal{M}_{#1}\!\left(#2\right)}
\DeclareRobustCommand{\Orth}[1]{#1^{\perp}}
\DeclareRobustCommand{\orth}[2]{\operatorname*{O}^{#2}\!\left(#1\right)}
\DeclareRobustCommand{\SO}[1]{\operatorname*{SO}\!\left(#1\right)}
\DeclareRobustCommand{\Gal}[1]{\operatorname*{Gal}\!\left(#1\right)}

%

\DeclareRobustCommand{\Lim}[2]{\underset{{\scriptscriptstyle #1 \smallto #2}}{\longrightarrow}}
\DeclareRobustCommand{\limite}[3]{\operatorname*{lim}\limits_{{\scriptscriptstyle #1 \smallto #2}}{#3}}
%\renewcommand{\lim}[2]{\lim\limits_{{\scriptscriptstyle #1}}{#2}}
\DeclareRobustCommand{\Sim}[1]{\,\underset{{\scriptstyle #1}}{\sim}\,}
\DeclareRobustCommand{\Notsim}[1]{\,\underset{{\scriptstyle #1}}{\not\sim}\,}
\renewcommand{\o}[1]{\operatorname*{o}\!\left(#1\right)}

\renewcommand{\d}{\operatorname*{d}\!}
\DeclareRobustCommand{\derive}[3]{\frac{\d #1}{\d #2}\!\left(#3\right)}
\DeclareRobustCommand{\Derive}[3]{\dfrac{\d #1}{\d #2}\!\left(#3\right)}
\DeclareRobustCommand{\open}[1]{\mathring{#1}}
\DeclareRobustCommand{\Int}{{\displaystyle \int}}


\DeclareRobustCommand{\fonction}[5]{
\begin{eqnarray*}
  #1: #2  & \longrightarrow & #3\\[-0.2\baselineskip]
      #4  & \longmapsto     & #5 
\end{eqnarray*}}

\DeclareRobustCommand{\function}[4]{
\begin{array}{rcl}
#1  & \longrightarrow & #2\\[-0.2\baselineskip]
#3  & \longmapsto     & #4 
\end{array}}


\DeclareRobustCommand{\diagram}{\text{{\large $\circlearrowleft$}}}

\DeclareRobustCommand{\cartesien}{\ar@{-}[] +R+<0pt,0pt>;[]+RD+<0pt,0pt>\ar@{-}[]+D+<0pt,0pt>;[]+RD+<0pt,0pt>}
%\xymatrix{
%A \ar[r] \ar[d] \cartesien & B \ar[d] \\
%C \ar[r] & D }
%
\DeclareRobustCommand{\cartesienr}{\ar@{-}[]+L+<0pt,0pt>;[]+LD+<0pt,0pt>\ar@{-}[]+D+<0pt,0pt>;[]+LD+<0pt,0pt>}
%\xymatrix{ 
%A \ar[r] \ar[d] \cartesienr & B \ar[d] \\
%C \ar[r] & D }
%
\DeclareRobustCommand{\cocartesien}{\ar@{-}[]+L+<0pt,0pt>;[]+LU+<0pt,0pt>\ar@{-}[]+U+<0pt,0pt>;[]+LU+<0pt,0pt>}
%\xymatrix{ 
%A \ar[r] \ar[d] & B \ar[d] \\
%C \ar[r] & D \cocartesien }
\DeclareRobustCommand{\commutatif}{\ar@{}[rd]|{\circlearrowleft}}
%\xymatrix{ 
%A \ar[r] \ar[d] \commutatif & B \ar[d] \\
%C \ar[r] & D }


\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\subset}{\subseteq}
\renewcommand{\subsetneq}{\varsubsetneq}
\renewcommand{\emptyset}{\varnothing}

\renewcommand{\i}{\imath}
\renewcommand{\j}{\jmath}

\DeclareRobustCommand{\veci}{\vec{\imath}}
\DeclareRobustCommand{\vecj}{\vec{\jmath}}
\DeclareRobustCommand{\veck}{\vec{k}}

\DeclareRobustCommand{\xbar}{\conj{x}}
\DeclareRobustCommand{\ybar}{\conj{y}}
\DeclareRobustCommand{\zbar}{\conj{z}}
\DeclareRobustCommand{\ibar}{\conj{\i}}
\DeclareRobustCommand{\jbar}{\conj{\j}}

\renewcommand{\setminus}{\smallsetminus}
\renewcommand{\to}{\longrightarrow}
\DeclareRobustCommand{\smallmapto}{\mapsto}
\DeclareRobustCommand{\mapto}{\longmapsto}
\DeclareRobustCommand{\smallto}{\rightarrow}
