% Die von T. Keilen in seiner MSc-Arbeit verwandten Macros.


%\usepackage{graphicx}    % dient dem Einbinden von Graphikfiles
%\usepackage{epsfig}            % dient dem Einbinden von Graphikfiles
\usepackage{latexsym}
\usepackage{amssymb} 
\usepackage{theorem}
\usepackage{bbm}
\usepackage{amsmath}
\usepackage{oldgerm}
%\usepackage{euler}

% Postscript Schriftarten benutzen 
\DeclareFixedFont{\AdobeZapfChancery}{T1}{pzc}{m}{it}{10}

% Option f�r MakeIndex
\newcommand{\uu}[1]{\underline{#1}}

% Abkuerzungen 
\newcommand{\be}{\begin{enumerate}} 
\newcommand{\ee}{\end{enumerate}} 
\renewcommand{\labelenumi}{\alph{enumi}.} 
\renewcommand{\labelenumii}{(\roman{enumii})} 
\newcommand{\bi}{\begin{itemize}} 
\newcommand{\ei}{\end{itemize}} 
\newcommand{\�}{\S}

% Zu verwendende Regelsaetze 
% Alte Regels�tze mit Nummerierung nach S�tzen, Definitionen, etc.
%\theoremstyle{break} 
%\newtheorem{satz}{Satz}
%\newtheorem{thm}{Theorem} 
%\newtheorem{defn}{Definition} 
%\newtheorem{defsatz}[defn]{Definition und Satz}
%\newtheorem{defthm}[defn]{Definition and Theorem} 
%\newtheorem{konstruktion}{Konstruktion}
%\newtheorem{construction}{Construction} 
%\newtheorem{lemma}{Lemma} 
%\newtheorem{folgerung}{Folgerung}
%\newtheorem{korollar}{Korollar}
%\newtheorem{corollary}{Corollary}
%\newtheorem{beispiel}{Beispiel}
%\newtheorem{example}{Example}
%\theoremstyle{plain} 
%\newtheorem{beh}{Behauptung}
%\newtheorem{claim}{Claim} 
%\theoremstyle{break} 
%\theorembodyfont{\rm} 
%\newtheorem{bem}{Bemerkung}
%\newtheorem{remark}{Remark} 
%\theoremstyle{plain}
%\newtheorem{notation}{Notation}
% Neue Regels�tze mit Nummerierung 5.12 etc.
\theoremstyle{changebreak} 
\newtheorem{satz}{Satz}[section]
\newtheorem{thm}[satz]{Theorem}
\newtheorem{defn}[satz]{Definition}
\newtheorem{defsatz}[satz]{Definition und Satz}
\newtheorem{defthm}[satz]{Definition and Theorem}
\newtheorem{konstruktion}[satz]{Konstruktion}
\newtheorem{construction}[satz]{Construction}
\newtheorem{lemma}[satz]{Lemma}
\newtheorem{folgerung}[satz]{Folgerung}
\newtheorem{korollar}[satz]{Korollar}
\newtheorem{corollary}[satz]{Corollary}
\newtheorem{proposition}[satz]{Proposition}
\newtheorem{exercise-script}[satz]{Exercise}
\newtheorem{general-assumption}[satz]{General Assumption}
\theoremstyle{plain} 
\newtheorem{beh}{Behauptung}
\newtheorem{claim}{Claim}
\theoremstyle{changebreak} 
\theorembodyfont{\rm} 
\newtheorem{beispiel}[satz]{Beispiel}
\newtheorem{example}[satz]{Example}
\newtheorem{bem}[satz]{Bemerkung}
\newtheorem{remark}[satz]{Remark}
\newtheorem{allg}[satz]{Allgemeine Hinweise}
\newtheorem{generalvoraussetzung}[satz]{Generalvoraussetzung}
\newtheorem{notation}[satz]{Notation}
\theoremstyle{plain}
\newtheorem{aufgabe}{Aufgabe}
\newtheorem{exercise}{Exercise}
\theorembodyfont{\it}
\theoremstyle{break}
\newtheorem{exercise-lemma}{Lemma}
\newtheorem{exercise-remark}[exercise-lemma]{Remark}
\newtheorem{exercise-definition}[exercise-lemma]{Definition}
\newtheorem{exercise-theorem}[exercise-lemma]{Theorem}
\newtheorem{exercise-proposition}[exercise-lemma]{Proposition}
\theorembodyfont{\rm}
\newtheorem{exercise-break}[exercise]{Exercise}

 
% Umgebung fuer Voraussetzungen 
\newenvironment{hypo}{\parindent0cm\par\sl}{\par} 
%\newenvironment{claim}{\parindent0cm\par\bf Claim: \it}{\par} 
\newenvironment{claimohne}{\parindent0cm\par\it}{\par} 
%\newenvironment{beweis}{\par\rm \underline{Proof:}}{\mbox{$\hspace{0.5cm} _\Box$}\par\vspace{0.3cm}} 
\newenvironment{proof}{\parindent0cm\par\rm {\bf Proof:}}{\par\hspace*{\fill}\mbox{$ ^{\Box}$}\par} 
\newenvironment{beweis}{\parindent0cm\par\rm {\bf Beweis:}}{\par\hspace*{\fill}\mbox{$ ^{\Box}$}\par}
\newenvironment{proof-named}[1]{\parindent0cm\par\rm {\bf Proof #1:}}{\par\hspace*{\fill}\mbox{$ ^{\Box}$}\par} 
\newenvironment{beweis-bezeichnet}[1]{\parindent0cm\par\rm {\bf Beweis #1:}}{\par\hspace*{\fill}\mbox{$ ^{\Box}$}\par}

% Raumkeime 1: allgemein                 2: mit Basispunkt 0 
%           3: allgemein (verkleinert)   4: mit Basispunkt 0 (verkleinert) 
\newcommand{\germ}[2]{\mbox{$\left( \rm #1,\mbox{\scriptsize #2} \right)$}} 
\newcommand{\spacegerm}[1]{\germ{#1}{0}} 
\newcommand{\germklein}[2]{\mbox{$\mbox{$\scriptstyle ($} \rm \scriptstyle #1,\mbox{\tiny #2} )$}} 
\newcommand{\spacegermklein}[1]{\germklein{#1}{0}} 
% Abblidungen  1: Abbildung                        2: Inklusion 
%              3: Abbildung (Name �ber dem Pfeil)  4: Inklusion (Name �ber dem Pfeil) 
%              4: Einschraenkung (ohne Definitions- & Wertebereich 
%              5: Einschraenkung (mit  Definitions- & Wertebereich 
\newcommand{\abbil}[3]{\mbox{$#1 : #2 \rightarrow #3$}} 
\newcommand{\einbet}[3]{\mbox{$#1 : #2 \hookrightarrow #3$}} 
\newcommand{\abbild}[3]{\mbox{$#2 \stackrel{#1}{\longrightarrow} #3$}} 
\newcommand{\einbett}[3]{\mbox{$#2 \stackrel{#1}{\hookrightarrow} #3$}} 
\newcommand{\einschr}[2]{\mbox{$#1_{\mid #2}$}} 
\newcommand{\einschra}[4]{\abbil{\einschr{#1}{#2}}{#3}{#4}} 
% Abbildungskeime  1: Abbildung                         2: Inklusion 
%                  3: Abbildung (Name ueber dem Pfeil)  4: Inklusion (Name ueber dem Pfeil) 
%                  5: Einschraenkung 
\newcommand{\abbilgerm}[3]{\abbil{#1}{\spacegerm{#2}}{\spacegerm{#3}}} 
\newcommand{\einbetgerm}[3]{\einbet{#1}{\spacegerm{#2}}{\spacegerm{#3}}} 
\newcommand{\abbildgerm}[3]{\abbild{#1}{\spacegerm{#2}}{\spacegerm{#3}}} 
\newcommand{\einbettgerm}[3]{\einbett{#1}{\spacegerm{#2}}{\spacegerm{#3}}} 
\newcommand{\einschragerm}[4]{\einschra{#1}{#2}{\spacegerm{#3}}{\spacegerm{#4}}} 
% Zeichen fuer Zahlsysteme  N, Z, Q, R, C, C (verkleinert) und C^n x C^k 
%\newcommand{\N}{{\rm I\hspace{-3pt} N}} 
%\newcommand{\Z}{{\bf Z}} 
%\newcommand{\Q}{\mbox{\rm Q\hspace{-6pt}\rule[0.5pt]{0.5pt}{7pt} \hspace{3pt}}} 
%\newcommand{\R}{{\rm I\hspace{-3pt} R}} 
%\newcommand{\C}{\mbox{\rm C\hspace{-5.5pt}\rule[0.5pt]{0.5pt}{6.8pt} \hspace{3pt}}} 
%\newcommand{\kleinC}{\mbox{\scriptsize \rm C\hspace{-4pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{2pt}}} 
%\newcommand{\CC}[2]{\mbox{$\C^#1\times\C^#2$}} 
%\newcommand{\CCC}[1]{\mbox{$\C\times\C^#1$}} 
%\newcommand{\N}{\mbox{$\mathbbm{N}$}} 
%\newcommand{\Z}{\mbox{$\mathbbm{Z}$}} 
%\newcommand{\Q}{\mbox{$\mathbbm{Q}$}} 
%\newcommand{\R}{\mbox{$\mathbbm{R}$}} 
%\newcommand{\C}{\mbox{$\mathbbm{C}$}} 
\newcommand{\kleinC}{\mbox{$\scriptstyle \mathbbm{C}$}} 
\newcommand{\CC}[2]{\mbox{$\C^#1\times\C^#2$}} 
\newcommand{\CCC}[1]{\mbox{$\C\times\C^#1$}} 
% Stellenalgebren  1: allgemein   2: C^n   3: C^n x C^k   4: C x C^n   5: C^2 
\newcommand{\alg}[1]{\mbox{$\cal O_{\mbox{$\scriptstyle\rm #1,\mbox{\tiny 0}$}}$}} 
\newcommand{\algC}[1]{\alg{\mbox{$\kleinC^{\;\mbox{\tiny #1}}$}}} 
\newcommand{\algCC}[2]{\alg{\mbox{$\kleinC^{\;\mbox{\tiny #1}}\mbox{$\scriptstyle \times$}\kleinC^{\;\mbox{\tiny #2}}$}}} 
\newcommand{\algCCC}[1]{\alg{\mbox{$\kleinC\mbox{$\scriptstyle \times$}\kleinC^{\;\mbox{\tiny #1}}$}}} 
\newcommand{\algz}{\algC{2}} 
 
% Bilder und Urbilder von einzelnen Punkten 
\newcommand{\bild}[2]{\mbox{$#1 \mbox{$\scriptstyle (#2)$}$}} 
\newcommand{\urbild}[2]{\mbox{$#1^{-1} \mbox{$\scriptstyle (#2)$}$}} 
 
% Verschiedenes 
\newcommand{\abl}[2]{\mbox{$\frac{\partial #1}{\partial #2}$}} 
\newcommand{\rankdf}[1]{\mbox{\bf rank\mbox{$\left( \bild{d#1}{0} \right)$}}} 
\newcommand{\abbilvorschrift}[2]{\mbox{$\left( #1 \right) \mapsto \left( #2 \right)$}} 
\newcommand{\homo}[1]{\mbox{$H_1 \left( #1,\Z \right)$}} 
\newcommand{\redhomo}[2]{\mbox{$\tilde{H}_{#1} \left( #2,\Z \right)$}} 
\newcommand{\transvers}{\mbox{$\,\cap\hspace{-9pt}\mid\;$}} 
 
% Abkuerzungen 
\newcommand{\Ideal}{\mbox{$\left(\omega_1,\abl{F}{\omega_2},\abl{F}{\omega_3}\right)\algC{3}$}} 
\newcommand{\oa}{\omega_1} 
\newcommand{\ob}{\omega_2} 
\newcommand{\oc}{\omega_3} 
\newcommand{\x}{\lambda} 
%\newcommand{\s}{\sigma} 
\newcommand{\ba}{\mbox{$\mid a \mid$}} 
\newcommand{\wba}{\mbox{$\sqrt{\ba}$}} 
\newcommand{\wdr}{\mbox{$\sqrt{3}$}} 
\newcommand{\grenze}{\mbox{$\klam{\frac{\ba}{3}}^\frac{3}{2}$}} 
\newcommand{\myinjection}{\abbilgerm{f}{\C^2}{\C^3}} 
\newcommand{\cosys}{\mbox{$\left( \omega_1,\omega_2,\omega_3 \right)$\ }} 
\newcommand{\urphi}{\urbild{\varphi}{0}\ } 
\newcommand{\urpsi}{\urbild{\psi}{0}\ } 
\newcommand{\cdim}[1]{\cdimension{\algz}{#1}} 
 
 
% Vektoren und endliche Punktmengen 
\newcommand{\klam}[1]{\mbox{$\left(#1\right)$}} 
\newcommand{\punkt}[2]{\mbox{$\left(#1,#2\right)$}} 
\newcommand{\punktmengeA}[3]{\mbox{$\left\{#1_#2,\ldots,#1_#3\right\}$}} 
\newcommand{\punktmengeB}[4]{\mbox{$\left\{\punkt{#1_#3}{#2},\ldots,\punkt{#1_#4}{#2}\right\}$}} 
\newcommand{\punktmengeC}[4]{\mbox{$\left\{\punkt{#1}{#2_#3},\ldots,\punkt{#1}{#2_#4}\right\}$}} 
\newcommand{\vect}[3]{\mbox{$\left(#1_#2,\ldots,#1_#3\right)$}} 
\newcommand{\menge}[1]{\mbox{$\left\{#1\right\}$}} 
\newcommand{\mymatrix}[2]{\mbox{\footnotesize $\left(\begin{array}{#1} #2\end{array}\right)$}}

 
% Berechnung von Milnor und Tjurina Zahlen 
\newcommand{\partdiff}[2]{\mbox{$\frac{\partial #1}{\partial #2}$}} 
\newcommand{\mfvec}[4]{\mbox{$\left(\partdiff{#1}{#2_{#3}},\ldots,\partdiff{#1}{#2_{#4}}\right)$}} 
\newcommand{\tfvec}[4]{\mbox{$\left(#1,\partdiff{#1}{#2_{#3}},\ldots,\partdiff{#1}{#2_{#4}}\right)$}} 
\newcommand{\cdimension}[2]{\mbox{$\dim_{\kleinC} #1/#2$}} 
 
% Deformationen 
\newcommand{\Rmvu}{\mbox{$\cal R$}--miniversal unfolding\ } 
\newcommand{\Kmvu}{\mbox{$\cal K$}--miniversal unfolding\ } 
% Beispiel: 1=phi 2=z 3=1 4=tau 5=lambda 6=e 
\newcommand{\deform}[6]{\mbox{$\bild{#1}{#2} + \sum_{i=#3}^{#4} #5_i \bild{#6_i}{#2}$}} 
 


% Funktionen
\newcommand{\sqrtn}[2]{\hspace*{2pt}^#1\hspace{-4pt}\sqrt{#2}}

% maximale Untergruppe / Normalteiler / Ideal
\newcommand{\lemax}{\mbox{$<\hspace{-4pt}\cdot\hspace{2pt}$}}
\newcommand{\lhdmax}{\mbox{$\lhd\cdot$}}


% Die Macros von Pauline.

\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Baselocus}{Baselocus}
\DeclareMathOperator{\Bl}{Bl}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\Cok}{Cok}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Def}{Def}
\DeclareMathOperator{\Der}{Der}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Exp}{Exp}
\DeclareMathOperator{\Ext}{Ext}
%\DeclareMathOperator{\gcd}{gcd}
\DeclareMathOperator{\GF}{GF}
\DeclareMathOperator{\Gl}{Gl}
\DeclareMathOperator{\height}{height}
\DeclareMathOperator{\Hom}{Hom}
% Hom-Garbe - die Variable gibt die Schriftgroesse an
\newcommand{\HOM}[1]{\mbox{\DeclareFixedFont{\AZC}{T1}{pzc}{m}{it}{#1}\AZC Hom\,}}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\Mat}{Mat}
\renewcommand{\mod}{\mbox{\rm mod }}
%\DeclareMathOperator{\mod}{mod}
\DeclareMathOperator{\Mor}{Mor}
\DeclareMathOperator{\mSpec}{\textswab{m}-Spec}
\DeclareMathOperator{\mult}{mult}
\DeclareMathOperator{\ord}{ord}
\DeclareMathOperator{\pr}{pr}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\PSL}{PSL}
\DeclareMathOperator{\Quot}{Quot}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\Sec}{Sec}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\Sl}{Sl}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Syl}{Syl}
\DeclareMathOperator{\supp}{supp}


%%%mit Befehl \begin{bfitemize}\end{bfitemize}erzeugt nachstehende newenvironme%%%nt itemize Umgebungen in Fettdruck. 
\newenvironment{bfitemize}
{\begin{itemize}\normalfont\bfseries}{\end{itemize}}

%%%%%
%%%new commands for  AMS blackboard characters with amstex package
%%%%%%
%\newcommand{\N}{{\Bbb N}}
%\newcommand{\A}{{\Bbb A}}
%\newcommand{\Z}{{\Bbb Z}}
%\newcommand{\R}{{\Bbb R}}
%\newcommand{\C}{{\Bbb C}}
%\newcommand{\Q}{{\Bbb Q}}
%\renewcommand{\P}{{\Bbb P}}
%\newcommand{\D}{{\Bbb D}}
%\newcommand{\F}{{\Bbb F}}
%\newcommand{\E}{{\Bbb E}}
%\newcommand{\G}{{\Bbb G}}
%\newcommand{\K}{{\Bbb K}}
%\renewcommand{\H}{{\Bbb H}}

%%%with amsmath package

\newcommand{\N}{{\mathbbm N}}
\newcommand{\A}{{\mathbbm A}}
\newcommand{\Z}{{\mathbbm Z}}
\newcommand{\R}{{\mathbbm R}}
\newcommand{\C}{{\mathbbm C}}
\newcommand{\Q}{{\mathbbm Q}}
\renewcommand{\P}{{\mathbbm P}}
\newcommand{\D}{{\mathbbm D}}
\newcommand{\F}{{\mathbbm F}}
\newcommand{\E}{{\mathbbm E}}
\newcommand{\G}{{\mathbbm G}}
\newcommand{\K}{{\mathbbm K}}
\newcommand{\V}{{\mathbbm V}}
\renewcommand{\H}{{\mathbbm H}}

%%%%%
%%% new commands for fraktur charakters with oldgerm
%%%%%
\newcommand{\m}{\textswab{m}}

%%%%%
%%% new commands for calligraphic characters with amsmath 
%%%%%
\newcommand{\ka}{{\mathcal A}}
\newcommand{\kb}{{\mathcal B}}
\newcommand{\kc}{{\mathcal C}}
\newcommand{\kd}{{\mathcal D}}
\newcommand{\ke}{{\mathcal E}}
\newcommand{\kf}{{\mathcal F}}
\newcommand{\kg}{{\mathcal G}}
\newcommand{\kh}{{\mathcal H}}
\newcommand{\ki}{{\mathcal I}}
\newcommand{\kj}{{\mathcal J}}
\newcommand{\kk}{{\mathcal K}}
\newcommand{\kl}{{\mathcal L}}
\newcommand{\km}{{\mathcal M}}
\newcommand{\kn}{{\mathcal N}}
\newcommand{\ko}{{\mathcal O}}
\newcommand{\kp}{{\mathcal P}}
\newcommand{\kq}{{\mathcal Q}}
\newcommand{\kr}{{\mathcal R}}
\newcommand{\ks}{{\mathcal S}}
\newcommand{\kt}{{\mathcal T}}
\newcommand{\ku}{{\mathcal U}}
\newcommand{\kv}{{\mathcal V}}
\newcommand{\kw}{{\mathcal W}}
\newcommand{\kx}{{\mathcal X}}
\newcommand{\ky}{{\mathcal Y}}
\newcommand{\kz}{{\mathcal Z}}
%%%%%

\newcommand{\lra}{\longrightarrow}
\newcommand{\lla}{\longleftarrow}


%nachstehend sind die .sty files zu finden
%/usr/local/tetex/texmf/tex/latex/amslatex/amsthm.sty

%%%Eigendefinition f\"ur Proof-Umgebung (amsthm-package muss geladen sein)
%%%%%%%%%
\newenvironment{Proof}[1][\proofname]
{\par   \normalfont \trivlist
  \item[\hskip\labelsep\bfseries%\itshape
    #1\,: ]\ignorespaces}
{  \qed\endtrivlist}
%%%%%%%%%
%%optional argument could be as follows
%%\begin{Proof}[Proof of the Main Theorem]
%%%%%%%%






















