\usepackage{enumerate}
\usepackage{pstricks}

%%%%%%%%%%%%%%%%%%%%%%%%
% Diverses
%%%%%%%%%%%%%%%%%%%%%%%%

\let\dsp=\displaystyle
\let\ol=\overline

\newcommand{\on}{\operatorname}
\newcommand{\id}{\on{id}}
\newcommand{\grad}{\on{grad}}
\newcommand{\graph}{\on{graph}}
\newcommand{\Hom}{\on{Hom}}
\newcommand{\Div}{\on{div}}


\newcommand{\betrag}[1]{\lvert #1 \rvert}
\newcommand{\norm}[1]{\lVert #1 \rVert}
\def\res#1#2{{#1}\lower .11ex\hbox{$|$}\lower .644ex\hbox{$\scriptstyle #2$}}
\def\bres#1#2{{#1}\lower .2ex\hbox{$\big|$}\lower .8ex\hbox{$\scriptstyle #2$}}


\DeclareMathAccent{\ring}{\mathalpha}{operators}{"17}

\newdimen\nwidth
 \nwidth 1cm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fonts 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\font\dc=cmbxsl10 at 30pt
% Achtung: veraltete Deklaration !!


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Buchstabenk"urzel                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Doppelbalken
\newcommand{\mN}{\mathbb N}
\newcommand{\mZ}{\mathbb Z}
\newcommand{\mQ}{\mathbb Q}
\newcommand{\mR}{\mathbb R}
\newcommand{\mC}{\mathbb C}
\newcommand{\mH}{\mathbb H}

%%% griechische Buchstaben
\def\al{{\alpha}}
\def\be{{\beta}}
\def\de{{\delta}}
\def\De{{\Delta}}
\def\om{{\omega}}
\def\Om{{\Omega}}
\def\la{{\lambda}}
\def\ka{{\kappa}}
\def\io{{\iota}}
\def\si{{\sigma}}
\def\Si{{\Sigma}}
\def\ga{{\gamma}}
\def\ep{{\varepsilon}}
\def\Ga{{\Gamma}}
\def\Tau{\mathcal{T}}
\def\th{{\vartheta}}
\def\Th{{\Theta}}
\def\ph{{\varphi}}
\def\phi{{\varphi}}
\def\Ph{{\Phi}}
\def\rh{{\rho}}
\def\La{{\Lambda}}

%%% beinahe griechische Buchstaben
\def\na{{\nabla}}
\def\pa{{\partial}}
\def\el{{\ell}}
\def\re{{\rm {Re}}}

%%% gothische Buchstaben
\def\gothg{\mbox{\goth g}}
\def\gothz{\mbox{\goth z}}

%%% Kalligraphische Buchstaben
\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cC{\mathcal{C}}
\def\cF{\mathcal{F}}
\def\cG{\mathcal{G}}
\def\cH{\mathcal{H}}
\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\cZ{\mathcal{Z}}

\let\mo\mathopen
\let\mc\mathclose

%\newcommand{\IdMat}{\mathbf{1}}
\newcommand{\IdMat}{1\!\!1}


%%%%%%%%%%%%%%%%%%%%%%%%%
% Andere Abkuerzungen   %
%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Zeichen
\def\ohne{-}
\def\ti{\tilde}
\def\witi{\widetilde}
\def\wihat{\widehat}
\def\ol{\overline}
\def\lan{\langle}
\def\ran{\rangle}
\let\<\lan
\let\>\ran
\def\llan{\ll}
\def\rran{\gg}
\def\GaN{{\Ga^{N,}}}

%%%Pfeile 
\def\LA{\Leftarrow}
\def\RA{\Rightarrow}
\def\LRA{\Leftrightarrow}
\def\gdw{\Longleftrightarrow}
\def\einbett{\hookrightarrow}

%%% Text-Macros
\def\halbplatz{\thinspace}
\def\dh{d.\halbplatz h.\ \ignorespaces}
\def\sd{s.\halbplatz d.\ \ignorespaces}
\def\ia{i.\halbplatz a.\ \ignorespaces}
\def\Ia{I.\halbplatz a.\ \ignorespaces}
\def\zB{z.\halbplatz B.\ \ignorespaces}
\def\ZB{Z.\halbplatz B.\ \ignorespaces}
\def\oE{o.\thinspace E.\ \ignorespaces}
\def\oBdA{o.\thinspace B.\thinspace d.\thinspace A.\ \ignorespaces}
\def\OBdA{O.\thinspace B.\thinspace d.\thinspace A.\ \ignorespaces}

%%% Spezielle Metriken
\def\geukl{g_{\mathrm{eukl}}}
\def\gkan{g_{\mathrm{kan}}}
\def\ghyp{g_{\mathrm{hyp}}}
\def\gMink{g_{\mathrm{Mink}}}
\def\goH{g_{\mathcal{H}}}
\def\dhyp{d_{\mathrm{hyp}}}
\def\vol{\mathop{\mathrm{vol}}}

%%% Abbildungen
\def\Log{\mathrm{Log}}
\def\Hol{\mathrm{Hol}}
\def\O{\mathrm{O}}
\def\SL{\mathrm{SL}}
\def\PSL{\mathrm{PSL}}
\def\Isom{\mathrm{Isom}}
\def\End{\mathrm{End}}
\def\I{{{\mathrm I}}}
\def\II{{\I\!\I}}
\def\III{{\I\!\I\!\I}}
\def\Id{{{\mathrm{Id}}}}
\def\spa{{\mathop{\rm span}}}
\def\ric{{\mathop{\rm ric}}}
\def\Ric{{\mathop{\rm Ric}}}
\def\scal{{\mathop{\rm scal}}}
\def\Spur{{\mathop{\rm Spur}}}
\def\tr{{\mathop{\rm tr}}}
\def\Spin{{\mathop{\rm Spin}}}
\def\const{{\mathop{\rm const}}}
\def\clvol{{\mathop{\rm clvol}}}
\def\cl{{\mathop{\rm cl}}}
\def\mCl{{\mathop{{\mathcal C}l}}}
\def\ord{{\mathop{\rm ord}}}

\def\bmatrix#1{\left(\begin{array}{cc}#1\end{array}\right)}
\def\cmatrix#1{\begin{array}{cc}#1\end{array}}
\def\stelle#1#2{\left. {#1}\right|_{#2}}
\newcommand{\rquot}[2]{\raisebox{1ex}{$#1$}/\raisebox{-1ex}{$#2$}}

%%% Klassische Makros
\def\lipot#1#2{{#1}^{#2}}
%\def\lipot#1#2{\setbox1\hbox{$#1$}\vbox{$#1$}^{#2}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Aufgaben -Nummerierung        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcount\thea
 \thea=0
\def\a{\vskip 5mm
 \leavevmode
 \advance \thea by 1
 \lower 4pt\hbox to 0pt{\kern-\nwidth\dc\the\thea\hss}\ignorespaces
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Diverse nuetzliche Makros
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\inn#1{\stackrel{\circ}{#1}} 


\long\def\komment#1{}