%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% FONT MATHRM %%
%%%%%%%%%%%%%%%%%
%% LOWER CASE %%
%%%%%%%%%%%%%%%%
\def\a{{\mathrm{a}}}  \def\b{{\mathrm{b}}}  \def\cc{{\mathrm{c}}}
\def\d{{\mathrm{d}}}  \def\e{{\mathrm{e}}}  \def\f{{\mathrm{f}}}
\def\g{{\mathrm{g}}}  \def\h{{\mathrm{h}}}  \def\ii{{\mathrm{i}}}
\def\jj{{\mathrm{j}}} \def\k{{\mathrm{k}}}  \def\rl{{\mathrm{l}}}
\def\m{{\mathrm{m}}}  \def\n{{\mathrm{n}}}  \def\oo{{\mathrm{o}}}
\def\p{{\mathrm{p}}}  \def\q{{\mathrm{q}}}  \def\r{{\mathrm{r}}}
\def\s{{\mathrm{s}}}  \def\t{{\mathrm{t}}}  \def\uu{{\mathrm{u}}}
\def\vv{{\mathrm{v}}} \def\ww{{\mathrm{w}}} \def\x{{\mathrm{x}}}
\def\y{{\mathrm{y}}}  \def\z{{\mathrm{z}}}
%%%%%%%%%%%%%%%%
%% UPPER CASE %%
%%%%%%%%%%%%%%%%
\def\rA{{\mathrm{A}}} \def\rB{{\mathrm{B}}} \def\rC{{\mathrm{C}}}
\def\rD{{\mathrm{D}}} \def\rE{{\mathrm{E}}} \def\rF{{\mathrm{F}}}
\def\rG{{\mathrm{G}}} \def\rH{{\mathrm{H}}} \def\rI{{\mathrm{I}}}
\def\rJ{{\mathrm{J}}} \def\rK{{\mathrm{K}}} \def\rL{{\mathrm{L}}}
\def\rM{{\mathrm{M}}} \def\rN{{\mathrm{N}}} \def\rO{{\mathrm{O}}}
\def\rP{{\mathrm{P}}} \def\rQ{{\mathrm{Q}}} \def\rR{{\mathrm{R}}}
\def\rS{{\mathrm{S}}} \def\rT{{\mathrm{T}}} \def\rU{{\mathrm{U}}}
\def\rV{{\mathrm{V}}} \def\rW{{\mathrm{W}}} \def\rX{{\mathrm{X}}}
\def\rY{{\mathrm{Y}}} \def\rZ{{\mathrm{Z}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% FONT MATHBB %%
%%%%%%%%%%%%%%%%%
\def\AA{{\mathbb{A}}} \def\BB{{\mathbb{B}}} \def\CC{{\mathbb{C}}}
\def\DD{{\mathbb{D}}} \def\EE{{\mathbb{E}}} \def\FF{{\mathbb{F}}}
\def\GG{{\mathbb{G}}} \def\HH{{\mathbb{H}}} \def\II{{\mathbb{I}}}
\def\JJ{{\mathbb{J}}} \def\KK{{\mathbb{K}}} \def\LL{{\mathbb{L}}}
\def\MM{{\mathbb{M}}} \def\NN{{\mathbb{N}}} \def\OO{{\mathbb{O}}}
\def\PP{{\mathbb{P}}} \def\QQ{{\mathbb{Q}}} \def\RR{{\mathbb{R}}}
\def\SS{{\mathbb{S}}} \def\TT{{\mathbb{T}}} \def\UU{{\mathbb{U}}}
\def\VV{{\mathbb{V}}} \def\WW{{\mathbb{W}}} \def\XX{{\mathbb{X}}}
\def\YY{{\mathbb{Y}}} \def\ZZ{{\mathbb{Z}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% FONT MATHBF %%
%%%%%%%%%%%%%%%%%
\def\bA{{\mathbf{A}}} \def\bB{{\mathbf{B}}} \def\bC{{\mathbf{C}}}
\def\bD{{\mathbf{D}}} \def\bE{{\mathbf{E}}} \def\bF{{\mathbf{F}}}
\def\bG{{\mathbf{G}}} \def\bH{{\mathbf{H}}} \def\bI{{\mathbf{I}}}
\def\bJ{{\mathbf{J}}} \def\bK{{\mathbf{K}}} \def\bL{{\mathbf{L}}}
\def\bM{{\mathbf{M}}} \def\bN{{\mathbf{N}}} \def\bO{{\mathbf{O}}}
\def\bP{{\mathbf{P}}} \def\bQ{{\mathbf{Q}}} \def\bR{{\mathbf{R}}}
\def\bS{{\mathbf{S}}} \def\bT{{\mathbf{T}}} \def\bU{{\mathbf{U}}}
\def\bV{{\mathbf{V}}} \def\bW{{\mathbf{W}}} \def\bX{{\mathbf{X}}}
\def\bY{{\mathbf{Y}}} \def\bZ{{\mathbf{Z}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% FONT MATHCAL %%
%%%%%%%%%%%%%%%%%%
\def\cA{{\mathcal{A}}} \def\cB{{\mathcal{B}}} \def\cC{{\mathcal{C}}}
\def\cD{{\mathcal{D}}} \def\cE{{\mathcal{E}}} \def\cF{{\mathcal{F}}}
\def\cG{{\mathcal{G}}} \def\cH{{\mathcal{H}}} \def\cI{{\mathcal{I}}}
\def\cJ{{\mathcal{J}}} \def\cK{{\mathcal{K}}} \def\cL{{\mathcal{L}}}
\def\cM{{\mathcal{M}}} \def\cN{{\mathcal{N}}} \def\cO{{\mathcal{O}}}
\def\cP{{\mathcal{P}}} \def\cQ{{\mathcal{Q}}} \def\cR{{\mathcal{R}}}
\def\cS{{\mathcal{S}}} \def\cT{{\mathcal{T}}} \def\cU{{\mathcal{U}}}
\def\cV{{\mathcal{V}}} \def\cW{{\mathcal{W}}} \def\cX{{\mathcal{X}}}
\def\cY{{\mathcal{Y}}} \def\cZ{{\mathcal{Z}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% FONT MATHGOTH %%
%%%%%%%%%%%%%%%%%%%
\def\gA{{\mathfrak{A}}} \def\gB{{\mathfrak{B}}} \def\gC{{\mathfrak{C}}}
\def\gD{{\mathfrak{D}}} \def\gE{{\mathfrak{E}}} \def\gF{{\mathfrak{F}}}
\def\gG{{\mathfrak{G}}} \def\gH{{\mathfrak{H}}} \def\gI{{\mathfrak{I}}}
\def\gJ{{\mathfrak{J}}} \def\gK{{\mathfrak{K}}} \def\gL{{\mathfrak{L}}}
\def\gM{{\mathfrak{M}}} \def\gN{{\mathfrak{N}}} \def\gO{{\mathfrak{O}}}
\def\gP{{\mathfrak{P}}} \def\gQ{{\mathfrak{Q}}} \def\gR{{\mathfrak{R}}}
\def\gS{{\mathfrak{S}}} \def\gT{{\mathfrak{T}}} \def\gU{{\mathfrak{U}}}
\def\gV{{\mathfrak{V}}} \def\gW{{\mathfrak{W}}} \def\gX{{\mathfrak{X}}}
\def\gY{{\mathfrak{Y}}} \def\gZ{{\mathfrak{Z}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\le=\leqslant \let\leq=\leqslant
\let\ge=\geqslant \let\geq=\geqslant
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% GUILLEMETS (FRENCH QUOTES) %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\og{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle\langle\!\langle$~}}
\def\fg{\leavevmode\raise.3ex\hbox{~$\!\scriptscriptstyle\,\rangle\!\rangle$}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator{\id}{id}       %identity
\DeclareMathOperator{\Aut}{Aut}     %automorphisms
\DeclareMathOperator{\supp}{supp}   %support
\DeclareMathOperator{\clos}{clos}   %closure
\DeclareMathOperator{\im}{im}       %image
\DeclareMathOperator{\vol}{vol}     %Volume
\DeclareMathOperator{\diam}{diam}   %diameter
\DeclareMathOperator{\Div}{div}     %divergence
\DeclareMathOperator{\grad}{grad}   %gradient
\DeclareMathOperator{\rot}{rot}     %rotationel
\DeclareMathOperator{\curl}{curl}   %
\DeclareMathOperator{\dist}{dist}   %distance
\DeclareMathOperator{\ord}{ord}     %order
\DeclareMathOperator{\End}{End}     %endomorphisms
\DeclareMathOperator{\Hom}{Hom}     %homomorphisms
\DeclareMathOperator{\Span}{span}   %spanned vector space
\DeclareMathOperator{\diag}{diag}   %diagonal
\DeclareMathOperator{\spec}{Spec}   %spectra
\DeclareMathOperator{\trace}{trace} %trace
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\pr}{pr}
\DeclareMathOperator{\res}{res}     %restriction or residue
\DeclareMathOperator{\coker}{coker} %cokernel
\DeclareMathOperator{\Char}{Char}   %characteristic
\DeclareMathOperator{\ind}{ind}     %indices
\DeclareMathOperator{\sgn}{sgn}     %sign
\DeclareMathOperator{\Ric}{Ric}     %Ricci curvature
\DeclareMathOperator{\scal}{scal}   %scalar curvature
\DeclareMathOperator{\meas}{meas}   %measure, in English
\DeclareMathOperator{\mes}{mes}     %mesure, in French
\DeclareMathOperator{\dirlim}{dirlim}
\DeclareMathOperator{\invlim}{invlim}
%{\catcode`@=11\global\let\c@equation=\c@theorem}

%\allowdisplaybreaks[2]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% LIMITES INDUCTIVE ET MILITE PROJECTIVE  %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\limind{\setbox1=\hbox{\oalign{\vadjust{\vskip -2pt}%
     \rm lim\cr \vadjust{\vskip -2pt}
      \hidewidth$-\mkern -12mu\rightarrow$\hidewidth\cr}}
       \mathop{\box1}}
\def\limproj{\setbox1=\hbox{\oalign{\vadjust{\vskip -2pt}%
     \rm lim\cr \vadjust{\vskip -2pt}
      \hidewidth$\leftarrow\mkern -12mu-$\hidewidth\cr}}
       \mathop{\box1}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% INF ESS, SUP ESS,LIM INF, LIM SUP  %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\infess{\mathop{\rm inf\,ess}\limits}
\def\supess{\mathop{\rm sup\,ess}\limits}
\def\essinf{\mathop{\rm ess\,inf}\limits}
\def\esssup{\mathop{\rm ess\,sup}\limits}
\def\liminf{\mathop{\rm lim\,inf}\limits}
\def\limsup{\mathop{\rm lim\,sup}\limits}
%\def\limproj{\mathop{\lim}_{\raise2pt
%     \hbox to 5mm{\leftarrowfill}}\limits}
%\def\limind{\mathop{\lim}_{\raise2pt
%     \hbox to 5mm{\rightarrowfill}}\limits}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TO BE COMPLETED
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

