% definition de macros pour les equations mathematiques

% definitions de Gabriella

\def\eq#1{(\ref{#1})}
\def\pmb#1{\setbox0=\hbox{#1}%
  \kern-.02em\copy0\kern-\wd0
  \kern-.035em\copy0\kern-\wd0
  \kern-.02em\raise.03em\box0 }
 
  
\newcommand{\matx}[1]{\mbox{\bf $#1$}}
\newcommand{\vect}[1]{\mbox{{\boldmath $#1$}}}
\newcommand{\ovec}[1]{\overline{\vect{#1}}}
\newcommand{\overmat}[1]{\overline{\matx{#1}}}
\newcommand{\hatmat}[1]{\hat{\matx{#1}}}
\newcommand{\hatvec}[1]{\hat{\vect{#1}}}

\newcommand{\mm}[1]{\mathbf{#1}}
\newcommand{\vv}[1]{\mbox{{\boldmath $#1$}}}
\newcommand{\cc}[1]{\mathcal{#1}}
 
\newcommand{\ovv}[1]{\overline{\vv{#1}}}
\newcommand{\omm}[1]{\overline{\mm{#1}}}
\newcommand{\occ}[1]{\overline{\cal{#1}}}
 
\newcommand{\tvv}[1]{\widetilde{\vv{#1}}}
\newcommand{\tmm}[1]{\widetilde{\mm{#1}}}
\newcommand{\tcc}[1]{\widetilde{\cal{#1}}}
 
 
\newcommand{\hvv}[1]{\widehat{\vv{#1}}}
\newcommand{\hmm}[1]{\widehat{\mm{#1}}}
\newcommand{\hcc}[1]{\widehat{\cal{#1}}}

%\newcommand{\dvv}[1]{\stackrel{\bullet}{\vv{#1}}}
%\newcommand{\dmm}[1]{\stackrel{\bullet}{\mm{#1}}}
%\newcommand{\dcc}[1]{\stackrel{\bullet}{\cal{#1}}}

\newcommand{\dvv}[1]{\dot{\vv{#1}}}
\newcommand{\dmm}[1]{\dot{\mm{#1}}}
\newcommand{\dcc}[1]{\dot{\cc{#1}}}

%\newcommand{\crp}{\times}
\newcommand{\tp}{^{\top}}
\newcommand{\Ht}{^{\textrm{H}}}
\newcommand{\prj}{\simeq}
\newcommand{\inverse}{^{-1}}
\newcommand{\tpinv}{^{-\top}}
\newcommand{\overstar}{^{\star}}
\setlength{\parskip}{\medskipamount}
\newcommand{\var}{\mbox{var}}

\def\proof{\noindent\hspace{2em}{\it Proof: }}
\def\endproof{\hspace*{\fill}~\par\endtrivlist\unskip}

% definitions de Radu.
\def\diag{\mbox{Diag}\:}
\def\trace{\mbox{trace}\:}
\def\rank{\mbox{rank}\:}
\def\sign{\mbox{sign}\:}
\def\spec{\mbox{spec}\:}
\def\p3{\cc{P}^3}
\def\avect{\mbox{\boldmath $a$}}
\def\bvect{\mbox{\boldmath $b$}}
\def\cvect{\mbox{\boldmath $c$}}
\def\dvect{\mbox{\boldmath $d$}}
\def\evect{\mbox{\boldmath $e$}}
\def\fvect{\mbox{\boldmath $f$}}
\def\gvect{\mbox{\boldmath $g$}}
\def\hvect{\mbox{\boldmath $h$}}
\def\ivect{\mbox{\boldmath $i$}}
\def\jvect{\mbox{\boldmath $j$}}
\def\kvect{\mbox{\boldmath $k$}}
\def\lvect{\mbox{\boldmath $l$}}
\def\mvect{\mbox{\boldmath $m$}}
\def\nvect{\mbox{\boldmath $n$}}
\def\ovect{\mbox{\boldmath $o$}}
\def\pvect{\mbox{\boldmath $p$}}
\def\qvect{\mbox{\boldmath $q$}}
\def\rvect{\mbox{\boldmath $r$}}
\def\svect{\mbox{\boldmath $s$}}
\def\tvect{\mbox{\boldmath $t$}}
\def\uvect{\mbox{\boldmath $u$}}
\def\vvect{\mbox{\boldmath $v$}}
\def\wvect{\mbox{\boldmath $w$}}
\def\xvect{\mbox{\boldmath $x$}}
\def\yvect{\mbox{\boldmath $y$}}
\def\zvect{\mbox{\boldmath $z$}}

\def\Avect{\mbox{\boldmath $A$}}
\def\Bvect{\mbox{\boldmath $B$}}
\def\Cvect{\mbox{\boldmath $C$}}
\def\Dvect{\mbox{\boldmath $D$}}
\def\Evect{\mbox{\boldmath $E$}}
\def\Fvect{\mbox{\boldmath $F$}}
\def\Gvect{\mbox{\boldmath $G$}}
\def\Hvect{\mbox{\boldmath $H$}}
\def\Ivect{\mbox{\boldmath $I$}}
\def\Jvect{\mbox{\boldmath $J$}}
\def\Kvect{\mbox{\boldmath $K$}}
\def\Lvect{\mbox{\boldmath $L$}}
\def\Mvect{\mbox{\boldmath $M$}}
\def\Nvect{\mbox{\boldmath $N$}}
\def\Ovect{\mbox{\boldmath $O$}}
\def\Pvect{\mbox{\boldmath $P$}}
\def\Qvect{\mbox{\boldmath $Q$}}
\def\Rvect{\mbox{\boldmath $R$}}
\def\Svect{\mbox{\boldmath $S$}}
\def\Tvect{\mbox{\boldmath $T$}}
\def\Uvect{\mbox{\boldmath $U$}}
\def\Vvect{\mbox{\boldmath $V$}}
\def\Wvect{\mbox{\boldmath $W$}}
\def\Xvect{\mbox{\boldmath $X$}}
\def\Yvect{\mbox{\boldmath $Y$}}
\def\Zvect{\mbox{\boldmath $Z$}}

\def\tauvect{\mbox{\boldmath $\tau$}}
\def\alphavect{\mbox{\boldmath $\alpha$}}
\def\muvect{\mbox{\boldmath $\mu$}}
\def\xivect{\mbox{\boldmath $\xi$}}
\def\chivect{\mbox{\boldmath $\chi$}}
\def\epsvect{\mbox{\boldmath $\epsilon$}}
\def\phivect{\mbox{\boldmath $\phi$}}
\def\gammavect{\mbox{\boldmath $\gamma$}}

\def\amat{\mbox{\bf a}}
\def\bmat{\mbox{\bf b}}
\def\cmat{\mbox{\bf c}}
\def\dmat{\mbox{\bf d}}
\def\emat{\mbox{\bf e}}
\def\fmat{\mbox{\bf f}}
\def\gmat{\mbox{\bf g}}
\def\hmat{\mbox{\bf h}}
\def\imat{\mbox{\bf i}}
\def\jmat{\mbox{\bf j}}
\def\kmat{\mbox{\bf k}}
\def\lmat{\mbox{\bf l}}
\def\mmat{\mbox{\bf m}}
\def\nmat{\mbox{\bf n}}
\def\omat{\mbox{\bf o}}
\def\pmat{\mbox{\bf p}}
\def\qmat{\mbox{\bf q}}
\def\rmat{\mbox{\bf r}}
\def\smat{\mbox{\bf s}}
\def\tmat{\mbox{\bf t}}
\def\umat{\mbox{\bf u}}
\def\vmat{\mbox{\bf v}}
\def\wmat{\mbox{\bf w}}
\def\xmat{\mbox{\bf x}}
\def\ymat{\mbox{\bf y}}
\def\zmat{\mbox{\bf z}}

\def\Amat{\mbox{\bf A}}
\def\Bmat{\mbox{\bf B}}
\def\Cmat{\mbox{\bf C}}
\def\Dmat{\mbox{\bf D}}
\def\Emat{\mbox{\bf E}}
\def\Fmat{\mbox{\bf F}}
\def\Gmat{\mbox{\bf G}}
\def\Hmat{\mbox{\bf H}}
\def\Imat{\mbox{\bf I}}
\def\Jmat{\mbox{\bf J}}
\def\Kmat{\mbox{\bf K}}
\def\Lmat{\mbox{\bf L}}
\def\Mmat{\mbox{\bf M}}
\def\Nmat{\mbox{\bf N}}
\def\Omat{\mbox{\bf O}}
\def\Pmat{\mbox{\bf P}}
\def\Qmat{\mbox{\bf Q}}
\def\Rmat{\mbox{\bf R}}
\def\Smat{\mbox{\bf S}}
\def\Tmat{\mbox{\bf T}}
\def\Umat{\mbox{\bf U}}
\def\Vmat{\mbox{\bf V}}
\def\Wmat{\mbox{\bf W}}
\def\Xmat{\mbox{\bf X}}
\def\Ymat{\mbox{\bf Y}}
\def\Zmat{\mbox{\bf Z}}

\def\Phimat{\mbox{\boldmath $\Phi$}}
\def\Psimat{\mbox{\boldmath $\Psi$}}
\def\Gammamat{\mbox{\boldmath $\Gamma$}}
\def\Sigmamat{\mbox{\boldmath $\Sigma$}}
\def\Lambdamat{\mbox{\boldmath $\Lambda$}}

\def\phimat{\mbox{\boldmath $\upphi$}}
\def\alphamat{\mbox{\boldmath $\upalpha$}}


\def\Abarre{\overline{\Amat}}
\def\Cbarre{\overline{\Cmat}}
\def\Mbarre{\overline{\Mvect}}
\def\Nbarre{\overline{\Nvect}}
\def\Pbarre{\overline{\Pmat}}
\def\Qbarre{\overline{\Qmat}}

\def\Phivect{\mbox{\boldmath $\Phi$}}
\def\psivect{\mbox{\boldmath $\psi$}}
\def\lambdavect{\mbox{\boldmath $\lambda$}}
\def\muvect{\mbox{\boldmath $\mu$}}
\def\thetavect{\mbox{\boldmath $\theta$}}
\def\zeromat{\mbox{\bf 0}}
\def\zerovect{\mbox{\boldmath $0$}}
\def\unmat{\mbox{\bf 1}}
\def\unvect{\mbox{\bf 1}}

\def\qquat{\mbox{\boldmath $q$}}

\def\itilde{\mbox{\bf \~\i}}
\def\jtilde{\mbox{\bf \~\j}}
\def\ihat{\mbox{\bf \^\i}}
\def\jhat{\mbox{\bf \^\j}}

\def\ahat{\hat{\avect}}
\def\uhat{\hat{u}}
\def\vhat{\hat{v}}
\def\what{\hat{w}}

\def\Rcal{\cal R}
\def\Pcal{\cal P}

\newcommand{\Reel}{I\!\!R}

\def\Ogvect{\mbox{\boldmath $\Omega$}}
\def\sigvect{\mbox{\boldmath $\sigma$}}
\def\Sigvect{\mbox{\boldmath $\Sigma$}}

\def\spc{\ \ \ \ }
\def\And{\mbox{\ \ \ and\ \ \ }}

% vector: \vector{O}{M} puts an arrow on OM. 
\newcommand{\vecteur}[2]{{\boxmaxdepth=1000pt\relax\overrightarrow{#1#2}}}

\newcommand{\diagvect}[1]{\mbox{\boldmath $\operatorname{diag}$}\{#1\}}
\newcommand{\Diagmat}[1]{\mbox{\boldmath $\operatorname{Diag}$}\{#1\}}

%\newtheorem{theorem}{Theorem}
%\newtheorem{proposition}{Proposition}
%\newtheorem{definition}{Definition}
%\newtheorem{corollary}{Corollary}[proposition]
