
\newenvironment{demo}[1][Démonstration]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\def\enddemo{$\hfill \Box$\smallskip}


\def\quest{\vspace{0.25 cm} \noindent }
\def\subquest{\vspace{0.2cm}}

\newenvironment{Solution}[1]{{\em \vspace{0.1cm}\bf Solution #1 \newline\it \small }} { \vspace{0.25cm }}
\def\dsolution{\begin{Solution}}
\def\fsolution{\end{Solution}}
\def\real {\mathbb R}
%\def\N {\mathbb N}
%\def\Q {\mathbb Q}
\newcommand \ZZ {\mathbb Z}
\newcommand \Pro{\mathbb P}
\newcommand \EE{\mathbb E}
\newcommand \RR{\mathbb R}



\newcommand{\indic}{{1\!\!1}}
\def\R\'esum\'e{\small\noindent\bf R\'esum\'e. \rm}
\def\Abstract{\small\noindent\bf Abstract. \rm}
\def\Sum{\sum}
\def\rea {{\mathbb R}}
%\def\real{I\!\!R}
\def\natural{{\mathbb N}}
%\def\rational{{\cal Z}} Erreur : rational = Q
\def\mes{\hbox{mes}}
\def\sin{\hbox{sin}}
\def\cos{\hbox{cos}}
%rappel : \overline{cequ'onveut}

\def\Varphi{{\bf \varphi}}
\def\e {\'{e}}
\def\ee {\`{e}}
\def\eee {\^{e}}
\def\e {\'{e}}
\def\ea {\'{e} }
\def\ee {\`{e}}
\def\eee {\^{e}}
\def\a {\`{a} }
\def\u {\`{u} }
\def\skipaline{\vspace{4 mm}\noindent}












%\usepackage{amsfonts,amsthm, amscd, epsfig, amsmath, amssymb,enumerate}
% \usepackage{showkeys}
%\numberwithin{equation}{section}

\newcommand{\BB}{{\mathbb B}}
\newcommand{\CC}{{\mathbb C}}
%\newcommand{\RR}{{\mathbb R}}
\newcommand{\QQ}{{\mathbb Q}}
%\newcommand{\ZZ}{{\mathbb Z}}
\newcommand{\NN}{{\mathbb N}}
\newcommand{\PP}{{\mathbb P}}
\newcommand{\TT}{{\mathbb T}}
\newcommand{\E}{{\mathbb E}}
\newcommand{\GG}{{\mathbb G}}
\newcommand{\FF}{{\mathbb F}}
\newcommand{\KK}{{\mathbb K}}
\newcommand{\LL}{{\mathbb L}}
\def\XX{{\cal X}}
\def\YY{{\cal Y}}


\def\esp{\E}
%\def\GG{{\cal G}} CONFLIT avec CEDRIC
\def\HH{{\cal H}}
%\def\FF{{\cal F}}
%\def\LL{{\cal L}} CONFLIT avec CEDRIC
\def\TT{{\cal T}}
\def\MM{{\cal M}}
%\def\EE{{\cal E}} CONFLIT avec CEDRIC


\newcommand{\id}{{\mathbb I}}
\newcommand{\ep}{{\epsilon}}
\def\esp{\E}
\def\GG{{\cal G}}
\def\HH{{\cal H}}
\def\FF{{\cal F}}
\def\LL{{\cal L}}
\def\TT{{\cal T}}
\def\MM{{\cal M}}
\def\EE{{\cal E}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% Calligraphic letters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\cA}{\ensuremath{\mathcal A}}
\newcommand{\cB}{\ensuremath{\mathcal B}}
\newcommand{\cC}{\ensuremath{\mathcal C}}
\newcommand{\cD}{\ensuremath{\mathcal D}}
\newcommand{\cE}{\ensuremath{\mathcal E}}
\newcommand{\cF}{\ensuremath{\mathcal F}}
\newcommand{\cG}{\ensuremath{\mathcal G}}
\newcommand{\cH}{\ensuremath{\mathcal H}}
\newcommand{\cI}{\ensuremath{\mathcal I}}
\newcommand{\cJ}{\ensuremath{\mathcal J}}
\newcommand{\cK}{\ensuremath{\mathcal K}}
\newcommand{\cL}{\ensuremath{\mathcal L}}
\newcommand{\cM}{\ensuremath{\mathcal M}}
\newcommand{\cN}{\ensuremath{\mathcal N}}
\newcommand{\cO}{\ensuremath{\mathcal O}}
\newcommand{\cP}{\ensuremath{\mathcal P}}
\newcommand{\cQ}{\ensuremath{\mathcal Q}}
\newcommand{\cR}{\ensuremath{\mathcal R}}
\newcommand{\cS}{\ensuremath{\mathcal S}}
\newcommand{\cT}{\ensuremath{\mathcal T}}
\newcommand{\cU}{\ensuremath{\mathcal U}}
\newcommand{\cV}{\ensuremath{\mathcal V}}
\newcommand{\cW}{\ensuremath{\mathcal W}}
\newcommand{\cX}{\ensuremath{\mathcal X}}
\newcommand{\cY}{\ensuremath{\mathcal Y}}
\newcommand{\cZ}{\ensuremath{\mathcal Z}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% Blackboard bolds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\bbA}{{\ensuremath{\mathbb A}} }
\newcommand{\bbB}{{\ensuremath{\mathbb B}} }
\newcommand{\bbC}{{\ensuremath{\mathbb C}} }
\newcommand{\bbD}{{\ensuremath{\mathbb D}} }
\newcommand{\bbE}{{\ensuremath{\mathbb E}} }
\newcommand{\bbF}{{\ensuremath{\mathbb F}} }
\newcommand{\bbG}{{\ensuremath{\mathbb G}} }
\newcommand{\bbH}{{\ensuremath{\mathbb H}} }
\newcommand{\bbI}{{\ensuremath{\mathbb I}} }
\newcommand{\bbJ}{{\ensuremath{\mathbb J}} }
\newcommand{\bbK}{{\ensuremath{\mathbb K}} }
\newcommand{\bbL}{{\ensuremath{\mathbb L}} }
\newcommand{\bbM}{{\ensuremath{\mathbb M}} }
\newcommand{\bbN}{{\ensuremath{\mathbb N}} }
\newcommand{\bbO}{{\ensuremath{\mathbb O}} }
\newcommand{\bbP}{{\ensuremath{\mathbb P}} }
\newcommand{\bbQ}{{\ensuremath{\mathbb Q}} }
\newcommand{\bbR}{{\ensuremath{\mathbb R}} }
\newcommand{\bbS}{{\ensuremath{\mathbb S}} }
\newcommand{\bbT}{{\ensuremath{\mathbb T}} }
\newcommand{\bbU}{{\ensuremath{\mathbb U}} }
\newcommand{\bbV}{{\ensuremath{\mathbb V}} }
\newcommand{\bbW}{{\ensuremath{\mathbb W}} }
\newcommand{\bbX}{{\ensuremath{\mathbb X}} }
\newcommand{\bbY}{{\ensuremath{\mathbb Y}} }
\newcommand{\bbZ}{{\ensuremath{\mathbb Z}} }

\def\bZ{{\Bbb Z}}
\def\bF{{\Bbb F}}
\def\bR{{\Bbb R}}
\def\bC{{\Bbb C}}
\def\bE{{\Bbb E}}
\def\bP{{\Bbb P}}
\def\bI{{\Bbb I}}
\def\bN{{\Bbb N}}
\def\bL{{\Bbb L}}
\def\bV{{\Bbb V}}






%
%%%%%%%%%%%%%%%%%%%%%%%%% GRECO
%
\let\a=\alpha \let\b=\beta
%\let\c=\chi
\let\d=\delta  \let\e=\varepsilon
\let\f=\varphi \let\g=\gamma \let\h=\eta    \let\k=\kappa  \let\l=\lambda
\let\m=\mu   \let\n=\nu   \let\o=\omega    \let\p=\pi  \let\ph=\varphi
\let\r=\rho  \let\s=\sigma \let\t=\tau   \let\th=\vartheta
\let\y=\upsilon \let\x=\xi \let\z=\zeta
\let\D=\Delta \let\F=\Phi  \let\G=\Gamma  \let\L=\Lambda \let\The=\Theta
\let\O=\Omega \let\P=\Pi   \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi
\let\Y=\Upsilon
%


%%%%%%%%%%% cedric's command


\newcommand{\reff}[1]{(\ref{#1})}
%\newcommand{\text}{\hbox}
\newcommand\of[1]{\bar{\mathfrak{#1}}}
\newcommand\der[1]{\overline{\partial_{t}\mathfrak{#1}}}
\newcommand\mf[1]{\mathfrak{#1}}
\newcommand\Psia[1]{\Psi^{\alpha}_{#1}}
\newcommand\Psiu[1]{\Psi_{#1}} %% when \alpha is understood

\newtheorem{exem}{Exemple}

\newtheorem{thf}{Théorème}[section]
\newtheorem{propf}{Proposition}[section]
\newtheorem{corf}{Corollaire}[section]
\newtheorem{remf}{Remarque}[section]
\newtheorem{lemf}{Lemme}[section]
\newtheorem{deff}{Définition}[section]
\newtheorem{Exer}{Exercice}[section]
\def\dexer{\begin{Exer}\rm}
\def\fexer{\end{Exer}}



\def\fb&{\begin{equation}}
\def\fe&{\end{equation}}


\def\rational{\mathbb Q}
\newcommand \relative {\mathbb Z}


\def\Varphi{{\bf \varphi}}

\def\ee {\`{e}}
\def\eee {\^{e}}
\def\ea {\'{e} }
\def\ee {\`{e}}
\def\eee {\^{e}}
\def\u {\`{u} }
\def\skipaline{\vspace{4 mm}\noindent}
\def\Sum{\sum}

\def\skipaline{\vspace{4 mm}\noindent}


\def\Var{\mathop {Var}}
\def\Card{\mathop {Card}}

 \newcommand \Binom[2]{\binom{#1}{#2}}
 \def\complex{\mathbb{C}}


\numberwithin{equation}{section}
\def\pal{\medskip\noindent}
