\newcommand{\End}{\mathop{\rm End}}
\newcommand{\rg}{\mathop{\rm rg}}
\newcommand{\ch}{{\rm ch}}
\newcommand{\sh}{{\rm sh}}
\newcommand{\intd}{\int\hspace{-.2ex}\int}
\newcommand{\intt}{\int\hspace{-.2ex}\int\hspace{-.2ex}\int}
\newcommand{\Arctg}{\mathop{\rm Arctg}}
\newcommand{\rot}{{\rm rot}}
\newcommand{\dive}{{\rm div}}
\newcommand{\grad}{{\rm grad}}
%
\newcommand{\PT}{{\mathsf{PolTrig}(\mathbb{T})}}
\newcommand{\tah}[1]{{\stackrel{\vee}{#1}}}
\newcommand{\rt}[1]{{\upharpoonright}_{#1}}
\renewcommand{\Im}{\mathop{\rm Im }}
\renewcommand{\Re}{\mathop{\rm Re }}
\renewcommand{\r}{\mathop{\rightarrow }}
\newcommand{\supp}{\mathop{\rm supp}}
\newcommand{\oc}[1]{{\stackrel{\circ}{#1}}}
\newcommand{\ol}[1]{{\overline{#1}}}
\newcommand{\bom}[1]{{\mbox{\boldmath $ #1 $}}}
%\newcommand{\sgn}{\mathop{\rm sgn}}
\newcommand{\card}{\mathop{\rm card}}
\newcommand{\Dom}{\mathop{\rm Dom}}
%\newcommand{\mod}{\mathop{\rm mod}}
\newcommand{\KM}{\mbox{\rm KM}}
\newcommand{\diam}{\mathop{\rm diam}}
\newcommand{\var}{\mathop{\rm var}}
\newcommand{\diag}{\mathop{\rm diag}}
\newcommand{\tr}{\mathop{\rm tr}}
\newcommand{\sgn}{\mathop{\rm sgn}}
\newcommand{\Tr}{\mathop{\rm Tr}}
\newcommand{\dd}{\mathop{\rm d}}
\newcommand{\sz}{\scriptsize}
\newcommand{\id}{\mbox{\rm1\hspace{-.2ex}\rule{.1ex}{1.44ex}}
   \hspace{-.82ex}\rule[-.01ex]{1.07ex}{.1ex}\hspace{.2ex}}
\renewcommand{\P}{\BbP}
\newcommand{\V}{\mathop{\rm Var}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\Cov}{\mathrm{Cov}}
\newcommand{\cps}{{\stackrel{{\rm p.s.}}{\rightarrow}}}
\newcommand{\elaw}{{\stackrel{\textrm{d}}{=}}}
\newcommand{\limm}{\mathop{\rm l.i.m.}}
\newcommand{\cloi}{{\stackrel{{\rm loi}}{\rightarrow}}}
\newcommand{\bra}{\langle\,}
\newcommand{\ket}{\,\rangle}
\newcommand{\obl}{/\!/}
\newcommand{\mapdown}[1]{\vbox{\vskip 4.25pt\hbox{\bigg\downarrow
  \rlap{$\vcenter{\hbox{$#1$}}$}}\vskip 1pt}}
\newcommand{\tab}{&\!\!\!}
\newcommand{\tabb}{&\!\!\!\!\!}
\renewcommand{\d}{\displaystyle}
%\newcommand{\eproof}{\hfill$\Box$}
\newcommand{\eproof}{\mbox{}\hfill$\square$}
\newcommand{\demo}{{\par\noindent{\em D\'emonstration~:~}}}
%\newcommand{\solu}{{\par\noindent{\em Solution~:~}}}
%\newcommand{\NB}{{\par\noindent{\bf Remarque~:~}}}
\newcommand{\Rk}{{\par\noindent{\bf Remark:~}}}
\newcommand{\const}{{\rm const}}
%------------------------------------------------------------
% Les lettres calligraphiques uniquement en mode math{\'e}matique
%------------------------------------------------------------
\newcommand{\cA}{{\cal A}}
\newcommand{\cB}{{\cal B}}
\newcommand{\cC}{{\cal C}}
\newcommand{\cD}{{\cal D}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cG}{{\cal G}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cI}{{\cal I}}
\newcommand{\cJ}{{\cal J}}
\newcommand{\cK}{{\cal K}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cN}{{\cal N}}
\newcommand{\cO}{{\cal O}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cQ}{{\cal Q}}
\newcommand{\cR}{{\cal R}}
\newcommand{\cS}{{\cal S}}
\newcommand{\cT}{{\cal T}}
\newcommand{\cU}{{\cal U}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cW}{{\cal W}}
\newcommand{\cX}{{\cal X}}
\newcommand{\cY}{{\cal Y}}
\newcommand{\cZ}{{\cal Z}}
%--------------------------------------------------------
% Lettres grasses majuscules
%--------------------------------------------------------
\newcommand{\bA}{{\bf A}}
\newcommand{\bB}{{\bf B}}
\newcommand{\bC}{{\bf C}}
\newcommand{\bD}{{\bf D}}
\newcommand{\bE}{{\bf E}}
\newcommand{\bF}{{\bf F}}
\newcommand{\bG}{{\bf G}}
\newcommand{\bH}{{\bf H}}
\newcommand{\bI}{{\bf I}}
\newcommand{\bJ}{{\bf J}}
\newcommand{\bK}{{\bf K}}
\newcommand{\bL}{{\bf L}}
\newcommand{\bM}{{\bf M}}
\newcommand{\bN}{{\bf N}}
\newcommand{\bP}{{\bf P}}
\newcommand{\bQ}{{\bf Q}}
\newcommand{\bR}{{\bf R}}
\newcommand{\bS}{{\bf S}}
\newcommand{\bT}{{\bf T}}
\newcommand{\bU}{{\bf U}}
\newcommand{\bV}{{\bf V}}
\newcommand{\bW}{{\bf W}}
\newcommand{\bX}{{\bf X}}
\newcommand{\bY}{{\bf Y}}
\newcommand{\bZ}{{\bf Z}}
%--------------------------------------------------------
% Lettres grasses miniscules
%--------------------------------------------------------
\newcommand{\ba}{{\bf }} 
\newcommand{\bb}{{\bf b}} 
\newcommand{\bc}{{\bf c}} 
\newcommand{\bd}{{\bf d}} 
\newcommand{\be}{{\bf e}} 
\newcommand{\boldf}{{\bf f}} 
\newcommand{\bg}{{\bf g}} 
\newcommand{\bh}{{\bf h}}
\newcommand{\bi}{{\bf i}}
\newcommand{\bj}{{\bf j}}
\newcommand{\bk}{{\bf l}}
\newcommand{\bl}{{\bf l}}
\newcommand{\bm}{{\bf m}}
\newcommand{\bn}{{\bf n}}
\newcommand{\bo}{{\bf o}}
\newcommand{\bp}{{\bf p}}
\newcommand{\bq}{{\bf q}}
\newcommand{\br}{{\bf r}}
\newcommand{\bs}{{\bf s}}
\newcommand{\bt}{{\bf t}}
\newcommand{\bu}{{\bf u}}
\newcommand{\bv}{{\bf v}}
\newcommand{\bw}{{\bf w}} 
\newcommand{\bx}{{\bf x}}
\newcommand{\by}{{\bf y}}
\newcommand{\bz}{{\bf z}}

%----------------------------------------------------------
% Standard Black board bold
% --------------------------------------------------------
%\input mssymb

\newcommand{\BbA}{\mathbb{A}}
\newcommand{\BbB}{\mathbb{B}}
\newcommand{\BbC}{\mathbb{C}}
\newcommand{\BbD}{\mathbb{D}}
\newcommand{\BbE}{\mathbb{E}}
\newcommand{\BbF}{\mathbb{F}}
\newcommand{\BbG}{\mathbb{G}}
\newcommand{\BbH}{\mathbb{H}}
\newcommand{\BbI}{\mathbb{I}}
\newcommand{\BbJ}{\mathbb{J}}
\newcommand{\BbK}{\mathbb{K}}
\newcommand{\BbL}{\mathbb{L}}
\newcommand{\BbM}{\mathbb{M}}
\newcommand{\BbN}{\mathbb{N}}
\newcommand{\BbO}{\mathbb{O}}
\newcommand{\BbP}{\mathbb{P}}
\newcommand{\BbQ}{\mathbb{Q}}
\newcommand{\BbR}{\mathbb{R}}
\newcommand{\BbS}{\mathbb{S}}
\newcommand{\BbT}{\mathbb{T}}
\newcommand{\BbU}{\mathbb{U}}
\newcommand{\BbV}{\mathbb{V}}
\newcommand{\BbW}{\mathbb{W}}
\newcommand{\BbX}{\mathbb{X}}
\newcommand{\BbY}{\mathbb{Y}}
\newcommand{\BbZ}{\mathbb{Z}}
%
\newcommand{\Bba}{\mathbb{a}} 
\newcommand{\Blackbb}{\mathbb{b}}
\newcommand{\Bbc}{\mathbb{c}}
\newcommand{\Bbd}{\mathbb{d}}
\newcommand{\Bbe}{\mathbb{e}}
\newcommand{\Bbf}{\mathbb{f}}
\newcommand{\Bbg}{\mathbb{g}}
\newcommand{\Bbh}{\mathbb{h}}
\newcommand{\Bbi}{\mathbb{i}}
\newcommand{\Bbj}{\mathbb{j}}
\newcommand{\Bbk}{\mathbb{l}}
\newcommand{\Bbl}{\mathbb{l}}
\newcommand{\Bbm}{\mathbb{m}}
\newcommand{\Bbn}{\mathbb{n}}
\newcommand{\Bbo}{\mathbb{o}}
\newcommand{\Bbp}{\mathbb{p}}
\newcommand{\Bbq}{\mathbb{q}}
\newcommand{\Bbr}{\mathbb{r}}
\newcommand{\Bbs}{\mathbb{s}}
\newcommand{\Bbt}{\mathbb{t}}
\newcommand{\Bbu}{\mathbb{u}}
\newcommand{\Bbv}{\mathbb{v}}
\newcommand{\Bbw}{\mathbb{w}}
\newcommand{\Bbx}{\mathbb{x}}
\newcommand{\Bby}{\mathbb{y}}
\newcommand{\Bbz}{\mathbb{z}}

%-----------------------------------------------------------
%       Date selon l'{\'e}criture europ{\'e}enne jj/mm/aa
%-----------------------------------------------------------
%en anglais
%
\def\eurtoday{\number\day \space\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\year}
%
%et en fran{\c c}ais
%
\def\aujourdhui{\number\day \space\ifcase\month\or
 janvier\or f{\'e}vrier\or mars\or avril\or mai\or juin\or
 juillet\or ao{\^u}t\or septembre\or octobre\or novembre\or d{\'e}cembre\fi
 \space\number\year}