Ein Repository für den Kurs Lineare Algebra 1, WiSe 2020–21.
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linalg2020/docs/zusatz.tex

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%% ********************************************************************************
%% AUTHOR: Raj Dahya
%% CREATED: November 2020
%% EDITED: —
%% TYPE: Notizen
%% TITLE: Zusatzaufgaben
%% DOI: —
%% DEPARTMENT: Fakultät for Mathematik und Informatik
%% INSTITUTE: Universität Leipzig
%% ********************************************************************************
%% ********************************************************************************
%% DOCUMENT STRUCTURE:
%% ~~~~~~~~~~~~~~~~~~~
%%
%% - root.tex;
%% |
%% ---- parameters.tex;
%% |
%% ---- srclocal/index.tex;
%% |
%% ---- src/setup-type.tex;
%% |
%% ---- src/setup-packages.tex;
%% |
%% ---- src/setup-parameters.tex;
%% |
%% ---- src/setup-macros.tex;
%% |
%% ---- src/setup-environments.tex;
%% |
%% ---- src/setup-layout.tex;
%% |
%% ---- srclocal/setup-localmacros.tex;
%% |
%% ---- front/index.tex;
%% |
%% ---- front/title.tex;
%% |
%% ---- front/foreword.tex;
%% |
%% ---- front/contents.tex;
%% |
%% ---- body/index.tex;
%% |
%% ---- body/linear-extensions.tex;
%% |
%% ---- body/linear-systems.tex;
%% |
%% ---- back/index.tex;
%% |
%% ---- ########;
%%
%% DOCUMENT-RANDOM-SEED: 5637845
%% ********************************************************************************
%% ********************************************************************************
%% FILE: root.tex
%% ********************************************************************************
%% ********************************************************************************
%% FILE: parameters.tex
%% ********************************************************************************
%% ********************************************************************************
%% FILE: srclocal/index.tex
%% ********************************************************************************
\makeatletter
%% ********************************************************************************
%% FILE: src/setup-type.tex
%% ********************************************************************************
\documentclass[
12pt,
a4paper,
oneside,
openright,
center,
chapterbib,
crosshair,
fleqn,
headcount,
headline,
indent,
indentfirst=false,
portrait,
phonetic,
oldernstyle,
onecolumn,
sfbold,
upper,
]{scrbook}
%% ********************************************************************************
%% FILE: src/setup-packages.tex
%% ********************************************************************************
\PassOptionsToPackage{T2A,OT1}{fontenc} % T1,OT1,T2A,OT2
\PassOptionsToPackage{utf8}{inputenc} % utf8
\PassOptionsToPackage{british,english,ngerman,russian}{babel}
\PassOptionsToPackage{
english,
ngerman,
russian,
capitalise,
}{cleveref}
\PassOptionsToPackage{
bookmarks=true,
bookmarksopen=false,
bookmarksopenlevel=0,
bookmarkstype=toc,
colorlinks=false,
raiselinks=true,
hyperfigures=true,
}{hyperref}
\PassOptionsToPackage{
reset,
left=1in,
right=1in,
top=20mm,
bottom=20mm,
heightrounded,
}{geometry}
\PassOptionsToPackage{
framemethod=TikZ,
}{mdframed}
\PassOptionsToPackage{normalem}{ulem}
\PassOptionsToPackage{
amsmath,
thmmarks,
}{ntheorem}
\PassOptionsToPackage{table}{xcolor}
\PassOptionsToPackage{
all,
color,
curve,
frame,
import,
knot,
line,
movie,
rotate,
textures,
tile,
tips,
web,
xdvi,
}{xy}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{ntheorem} % <— muss nach den ams* Packages vorkommen!!
\usepackage{array}
\usepackage{babel}
\usepackage{bbding}
\usepackage{bbm}
\usepackage{calc}
\usepackage{sectsty}
\usepackage{titlesec}
\usepackage{fancyhdr}
\usepackage{footmisc}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{ifpdf}
\usepackage{ifthen}
\usepackage{ifnextok}
\usepackage{longtable}
\usepackage{multicol}
\usepackage{multirow}
\usepackage{nameref}
\usepackage{nowtoaux}
\usepackage{paralist}
\usepackage{enumerate} %% nach [paralist]
\usepackage{pgf}
\usepackage{pgfplots}
\usepackage{proof}
\usepackage{refcount}
\usepackage{relsize}
\usepackage{savesym}
\usepackage{stmaryrd}
\usepackage{subfigure}
\usepackage{yfonts} %% <— Altgotische Fonts
\usepackage{tikz}
\usepackage{xy}
\usepackage{undertilde}
\usepackage{ulem} %% <– f\"ur besseren \underline-Befehl (\ul)
\usepackage{xcolor}
\usepackage{xspace}
\usepackage{xstring}
\usepackage{hyperref}
\usepackage{cleveref} % must vor hyperref geladen werden.
\pgfplotsset{compat=newest}
\usetikzlibrary{
angles,
arrows,
automata,
calc,
decorations,
decorations.pathmorphing,
decorations.pathreplacing,
math,
positioning,
patterns,
quotes,
snakes,
}
%% \var ≈ alter Befehl
%% \xvar ≈ wie das neue Package \var interpretieren soll.
\savesymbol{Diamond}
\savesymbol{emptyset}
\savesymbol{ggg}
\savesymbol{int}
\savesymbol{lll}
\savesymbol{RectangleBold}
\savesymbol{langle}
\savesymbol{rangle}
\savesymbol{hookrightarrow}
\savesymbol{hookleftarrow}
\savesymbol{Asterisk}
\usepackage{mathabx}
\usepackage{wasysym}
\let\varemptyset=\emptyset
\restoresymbol{x}{Diamond}
\restoresymbol{x}{emptyset}
\restoresymbol{x}{ggg}
\restoresymbol{x}{int}
\restoresymbol{x}{lll}
\restoresymbol{x}{RectangleBold}
\restoresymbol{x}{langle}
\restoresymbol{x}{rangle}
\restoresymbol{x}{hookrightarrow}
\restoresymbol{x}{hookleftarrow}
\restoresymbol{x}{Asterisk}
\ifpdf
\usepackage{pdfcolmk}
\fi
\usepackage{mdframed}
%% Force-Import aus MnSymbol
\DeclareFontFamily{U}{MnSymbolA}{}
\DeclareFontShape{U}{MnSymbolA}{m}{n}{
<-6> MnSymbolA5
<6-7> MnSymbolA6
<7-8> MnSymbolA7
<8-9> MnSymbolA8
<9-10> MnSymbolA9
<10-12> MnSymbolA10
<12-> MnSymbolA12
}{}
\DeclareFontShape{U}{MnSymbolA}{b}{n}{
<-6> MnSymbolA-Bold5
<6-7> MnSymbolA-Bold6
<7-8> MnSymbolA-Bold7
<8-9> MnSymbolA-Bold8
<9-10> MnSymbolA-Bold9
<10-12> MnSymbolA-Bold10
<12-> MnSymbolA-Bold12
}{}
\DeclareSymbolFont{MnSyA}{U}{MnSymbolA}{m}{n}
\DeclareMathSymbol{\lcirclearrowright}{\mathrel}{MnSyA}{252}
\DeclareMathSymbol{\lcirclearrowdown}{\mathrel}{MnSyA}{255}
\DeclareMathSymbol{\rcirclearrowleft}{\mathrel}{MnSyA}{250}
\DeclareMathSymbol{\rcirclearrowdown}{\mathrel}{MnSyA}{251}
\DeclareFontFamily{U}{MnSymbolC}{}
\DeclareSymbolFont{MnSyC}{U}{MnSymbolC}{m}{n}
\DeclareFontShape{U}{MnSymbolC}{m}{n}{
<-6> MnSymbolC5
<6-7> MnSymbolC6
<7-8> MnSymbolC7
<8-9> MnSymbolC8
<9-10> MnSymbolC9
<10-12> MnSymbolC10
<12-> MnSymbolC12%
}{}
\DeclareMathSymbol{\powerset}{\mathord}{MnSyC}{180}
%% ********************************************************************************
%% FILE: src/setup-parameters.tex
%% ********************************************************************************
\def\boolwahr{true}
\def\boolfalsch{false}
\def\boolleer{}
\let\documenttwosided\boolfalsch
\let\boolinappendix\boolfalsch
\let\boolinmdframed\boolfalsch
\let\eqtagset\boolfalsch
\let\eqtaglabel\boolleer
\let\eqtagsymb\boolleer
\newcount\bufferctr
\newcount\bufferreplace
\newcounter{columnanzahl}
\newlength\rtab
\newlength\gesamtlinkerRand
\newlength\gesamtrechterRand
\newlength\ownspaceabovethm
\newlength\ownspacebelowthm
\setlength{\rtab}{0.025\textwidth}
\setlength{\ownspaceabovethm}{0.5\baselineskip}
\setlength{\ownspacebelowthm}{0.5\baselineskip}
\setlength{\gesamtlinkerRand}{0pt}
\setlength{\gesamtrechterRand}{0pt}
\def\secnumberingpt{$\cdot$}
\def\secnumberingseppt{.}
\def\subsecnumberingseppt{}
\def\thmnumberingpt{$\cdot$}
\def\thmnumberingseppt{}
\def\thmForceSepPt{.}
\definecolor{leer}{gray}{1}
\definecolor{hellgrau}{gray}{0.85}
\definecolor{dunkelgrau}{gray}{0.5}
\definecolor{maroon}{rgb}{0.6901961,0.1882353,0.3764706}
\definecolor{dunkelgruen}{rgb}{0.015625,0.363281,0.109375}
\definecolor{dunkelrot}{rgb}{0.5450980392,0,0}
\definecolor{dunkelblau}{rgb}{0,0,0.5450980392}
\definecolor{blau}{rgb}{0,0,1}
\definecolor{newresult}{rgb}{0.6,0.6,0.6}
\definecolor{improvedresult}{rgb}{0.9,0.9,0.9}
\definecolor{hervorheben}{rgb}{0,0.9,0.7}
\definecolor{starkesblau}{rgb}{0.1019607843,0.3176470588,0.8156862745}
\definecolor{achtung}{rgb}{1,0.5,0.5}
\definecolor{frage}{rgb}{0.5,1,0.5}
\definecolor{schreibweise}{rgb}{0,0.7,0.9}
\definecolor{axiom}{rgb}{0,0.3,0.3}
%% ********************************************************************************
%% FILE: src/setup-macros.tex
%% ********************************************************************************
%% ****************************************************************
%% TEX:
%% ****************************************************************
\def\let@name#1#2{\expandafter\let\csname #1\expandafter\endcsname\csname #2\endcsname\relax}
\DeclareRobustCommand\crfamily{\fontfamily{ccr}\selectfont}
\DeclareTextFontCommand{\textcr}{\crfamily}
\def\nichtzeigen#1{\phantom{#1}}
%% ****************************************************************
%% SPACING:
%% ****************************************************************
\def\ifthenelseleer#1#2#3{\ifthenelse{\equal{#1}{}}{#2}{#1#3}}
\def\bedingtesspaceexpand#1#2#3{\ifthenelseleer{\csname #1\endcsname}{#3}{#2#3}}
\def\voritemise{\leavevmode\nvraum{1}}
\def\hraum{\null\hfill\null}
\def\vraum{\null\vfill\null}
\def\nvraum{\@ifnextchar\bgroup{\nvraum@c}{\nvraum@bes}}
\def\nvraum@c#1{\vspace*{-#1\baselineskip}}
\def\nvraum@bes{\vspace*{-\baselineskip}}
\def\erlaubeplatz{\relax\ifmmode\else\@\xspace\fi}
\def\entferneplatz{\relax\ifmmode\else\expandafter\@gobble\fi}
%% ****************************************************************
%% TAGS / BEZEICHNUNGEN / LABELLING:
%% ****************************************************************
\def\send@toaux#1{\@bsphack\protected@write\@auxout{}{\string#1}\@esphack}
%% \rlabel{LABEL}[CTR]{CREF-SHORT}{CREF-LONG}{DISPLAYTEXT}
\def\rlabel#1[#2]#3#4#5{#5\rlabel@aux{#1}[#2]{#3}{#4}{#5}}
\def\rlabel@aux#1[#2]#3#4#5{%
\send@toaux{\newlabel{#1}{{\@currentlabel}{\thepage}{{\unexpanded{#5}}}{#2.\csname the#2\endcsname}{}}}\relax%
}
%% \tag@rawscheme{CREF-SHORT}{CREF-LONG}[CTR]{LEFT-BRKT}{RIGHT-BRKT} [LABEL]{DISPLAYTEXT}
\def\tag@rawscheme#1#2[#3]#4#5{\@ifnextchar[{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}}{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}[*]}}
\def\tag@rawscheme@#1#2[#3]#4#5[#6]{\@ifnextchar\bgroup{\tag@rawscheme@@{#1}{#2}[#3]{#4}{#5}[#6]}{\tag@rawscheme@@{#1}{#2}[#3]{#4}{#5}[#6]{}}}
\def\tag@rawscheme@@#1#2[#3]#4#5[#6]#7{%
\ifthenelse{\equal{#6}{*}}{%
\ifthenelse{\equal{#7}{\boolleer}}{\refstepcounter{#3}#4\csname the#3\endcsname#5}{#4#7#5}%
}{%
\refstepcounter{#3}#4%
\ifthenelse{\equal{#7}{\boolleer}}{\rlabel{#6}[#3]{#1}{#2}{\csname the#3\endcsname}}{\rlabel{#6}[#3]{#1}{#2}{#7}}%
#5%
}%
}
%% \tag@scheme{CREF-SHORT}{CREF-LONG}[CTR] [LABEL]{DISPLAYTEXT}
\def\tag@scheme#1#2[#3]{\tag@rawscheme{#1}{#2}[#3]{\upshape(}{\upshape)}}
%% \eqtag[LABEL]{DISPLAYTEXT}
\def\eqtag@post#1{\makebox[0pt][r]{#1}}
\def\eqtag@pre{\tag@scheme{Eq}{Equation}[Xe]}
\def\eqtag{\@ifnextchar[{\eqtag@}{\eqtag@[*]}}
\def\eqtag@[#1]{\@ifnextchar\bgroup{\eqtag@@[#1]}{\eqtag@@[#1]{}}}
\def\eqtag@@[#1]#2{\eqtag@post{\eqtag@pre[#1]{#2}}}
\def\eqcref#1{\text{(\ref{#1})}}
\def\ptcref#1{\ref{#1}}
\def\punktlabel#1{\label{it:#1:\beweislabel}}
\def\punktcref#1{\eqcref{it:#1:\beweislabel}}
\def\crefit#1#2{\cref{#1}~\eqcref{it:#2:#1}}
\def\Crefit#1#2{\Cref{#1}~\eqcref{it:#2:#1}}
%% UNDER/OVERSET BEFEHLE
\def\opfromto[#1]_#2^#3{\underset{#2}{\overset{#3}{#1}}}
\def\textoverset#1#2{\overset{\text{#1}}{#2}}
\def\textunderset#1#2{\underset{#2}{\text{#1}}}
\def\crefoverset#1#2{\textoverset{\cref{#1}}{#2}}
\def\Crefoverset#1#2{\textoverset{\Cref{#1}}{#2}}
\def\crefunderset#1#2{\textunderset{#2}{\cref{#1}}}
\def\Crefunderset#1#2{\textunderset{#2}{\Cref{#1}}}
\def\eqcrefoverset#1#2{\textoverset{\eqcref{#1}}{#2}}
\def\eqcrefunderset#1#2{\textunderset{#2}{\eqcref{#1}}}
\def\mathclap#1{#1}
\def\oberunterset#1{\@ifnextchar^{\oberunterset@oben{#1}}{\oberunterset@unten{#1}}}
\def\oberunterset@oben#1^#2_#3{\underset{\mathclap{#3}}{\overset{\mathclap{#2}}{#1}}}
\def\oberunterset@unten#1_#2^#3{\underset{\mathclap{#2}}{\overset{\mathclap{#3}}{#1}}}
\def\breitunderbrace#1_#2{\underbrace{#1}_{\mathclap{#2}}}
\def\breitoverbrace#1^#2{\overbrace{#1}^{\mathclap{#2}}}
\def\breitunderbracket#1_#2{\underbracket{#1}_{\mathclap{#2}}}
\def\breitoverbracket#1^#2{\overbracket{#1}^{\mathclap{#2}}}
\def\generatenestedsecnumbering#1#2#3{%
\expandafter\gdef\csname thelong#3\endcsname{%
\expandafter\csname the#2\endcsname%
\secnumberingpt%
\expandafter\csname #1\endcsname{#3}%
}%
\expandafter\gdef\csname theshort#3\endcsname{%
\expandafter\csname #1\endcsname{#3}%
}%
}
\def\generatenestedthmnumbering#1#2#3{%
\expandafter\gdef\csname the#3\endcsname{%
\expandafter\csname the#2\endcsname%
\thmnumberingpt%
\expandafter\csname #1\endcsname{#3}%
}%
\expandafter\gdef\csname theshort#3\endcsname{%
\expandafter\csname #1\endcsname{#3}%
}%
}
%% ****************************************************************
%% ALLG. MACROS:
%% ****************************************************************
\def\+#1{\addtocounter{#1}{1}}
\def\setcounternach#1#2{\setcounter{#1}{#2}\addtocounter{#1}{-1}}
\def\textsubscript#1{${}_{\textup{#1}}$}
\def\rome#1{\overline{\underline{#1}}}
\def\textTODO{\text{[{\large\textcolor{red}{More work needed!}}]}}
\def\hlineEIGENpt{\hdashline[0.5pt/5pt]}
\def\clineEIGENpt#1{\cdashline{#1}[0.5pt/5pt]}
\def\forcepunkt#1{#1\IfEndWith{#1}{.}{}{.}}
\def\lateinabkuerzung#1#2{%
\expandafter\gdef\csname #1\endcsname{\emph{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}}
}
\def\deutscheabkuerzung#1#2{%
\expandafter\gdef\csname #1\endcsname{{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}}
}
%% ****************************************************************
%% MATHE
%% ****************************************************************
\def\matrix#1{\left(\begin{array}{#1}}
\def\endmatrix{\end{array}\right)}
\def\smatrix{\left(\begin{smallmatrix}}
\def\endsmatrix{\end{smallmatrix}\right)}
\def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{%
\expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup}
\expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
\expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
\expandafter\def\csname #1@ende\endcsname##1{#8}
}
\multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}}
\multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}}
\multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{}
\multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}}
\def\cases[#1]#2{\left\{\begin{array}[#1]{#2}}
\def\endcases{\end{array}\right.}
\def\BeweisRichtung[#1]{\@ifnextchar\bgroup{\@BeweisRichtung@c[#1]}{\@BeweisRichtung@bes[#1]}}
\def\@BeweisRichtung@bes[#1]{{\bfseries(#1).~}}
\def\@BeweisRichtung@c[#1]#2#3{{\bfseries(#2#1#3).~}}
\def\erzeugeBeweisRichtungBefehle#1#2{
\expandafter\gdef\csname #1text\endcsname##1##2{\BeweisRichtung[#2]{##1}{##2}}
\expandafter\gdef\csname #1\endcsname{%
\@ifnextchar\bgroup{\csname #1@\endcsname}{\csname #1text\endcsname{}{}}%
}
\expandafter\gdef\csname #1@\endcsname##1##2{%
\csname #1text\endcsname{\punktcref{##1}}{\punktcref{##2}}%
}
}
\erzeugeBeweisRichtungBefehle{hinRichtung}{$\Longrightarrow$}
\erzeugeBeweisRichtungBefehle{herRichtung}{$\Longleftarrow$}
\erzeugeBeweisRichtungBefehle{hinherRichtung}{$\Longleftrightarrow$}
\def\cal#1{\mathcal{#1}}
\def\brkt#1{\langle{}#1{}\rangle}
\def\mathfrak#1{\mbox{\usefont{U}{euf}{m}{n}#1}}
\def\kurs#1{\textit{#1}}
\def\rectangleblack{\text{\RectangleBold}}
\def\rectanglewhite{\text{\Rectangle}}
\def\squareblack{\blacksquare}
\def\squarewhite{\Box}
%% ********************************************************************************
%% FILE: src/setup-environments.tex
%% ********************************************************************************
%% **********************************************************************
%% CLEVEREF: ************************************************************
\def\crefname@full#1#2#3{\crefname{#1}{#2}{#3}\Crefname{#1}{#2}{#3}}
\crefname@full{chapter}{Kapitel}{Kapitel}
\crefname@full{section}{Abschnitt}{Abschnitte}
\crefname@full{figure}{Fig.}{Fig.}
\crefname@full{subfigure}{Fig.}{Fig.}
\crefname@full{proof}{Beweis}{Beweise}
\crefname@full{thm}{Theorem}{Theoreme}
\crefname@full{satz}{Satz}{Sätze}
\crefname@full{claim}{Behauptung}{Behauptungen}
\crefname@full{lemm}{Lemma}{Lemmata}
\crefname@full{cor}{Korollar}{Korollarien}
\crefname@full{folg}{Folgerung}{Folgerungen}
\crefname@full{prop}{Proposition}{Propositionen}
\crefname@full{defn}{Definition}{Definitionen}
\crefname@full{conv}{Konvention}{Konventionen}
\crefname@full{fact}{Fakt}{Fakten}
\crefname@full{rem}{Bemerkung}{Bemerkungen}
\crefname@full{qstn}{Frage}{Fragen}
\crefname@full{e.g.}{Beipsiel}{Beipsiele}
%% ****************************************************************
%% THEOREME:
%% ****************************************************************
\def\qedEIGEN#1{\@ifnextchar[{\qedEIGEN@c{#1}}{\qedEIGEN@bes{#1}}}%]
\def\qedEIGEN@bes#1{%
\parfillskip=0pt% % so \par doesnt push \square to left
\widowpenalty=10000% % so we dont break the page before \square
\displaywidowpenalty=10000% % ditto
\finalhyphendemerits=0% % TeXbook exercise 14.32
\leavevmode% % \nobreak means lines not pages
\unskip% % remove previous space or glue
\nobreak% % don’t break lines
\hfil% % ragged right if we spill over
\penalty50% % discouragement to do so
\hskip.2em% % ensure some space
\null% % anchor following \hfill
\hfill% % push \square to right
#1% % the end-of-proof mark
\par%
}
\def\qedEIGEN@c#1[#2]{%
\parfillskip=0pt% % so \par doesnt push \square to left
\widowpenalty=10000% % so we dont break the page before \square
\displaywidowpenalty=10000% % ditto
\finalhyphendemerits=0% % TeXbook exercise 14.32
\leavevmode% % \nobreak means lines not pages
\unskip% % remove previous space or glue
\nobreak% % don’t break lines
\hfil% % ragged right if we spill over
\penalty50% % discouragement to do so
\hskip.2em% % ensure some space
\null% % anchor following \hfill
\hfill% % push \square to right
{#1~{\smaller\bfseries\upshape (#2)}}%
\par%
}
\def\qedVARIANT#1#2{
\expandafter\def\csname ennde#1Sign\endcsname{#2}
\expandafter\def\csname ennde#1\endcsname{\@ifnextchar[{\qedEIGEN@c{#2}}{\qedEIGEN@bes{#2}}} %]
}
\qedVARIANT{OfProof}{$\squareblack$}
\qedVARIANT{OfWork}{\rectangleblack}
\qedVARIANT{OfSomething}{$\dashv$}
\qedVARIANT{OnNeutral}{$\lozenge$} % \lozenge \bigcirc \blacklozenge
\def\qedsymbol{\enndeOfProofSign}
\def\proofSymbol{\enndeOfProofSign}
\def\ra@pretheoremwork{
\setlength{\theorempreskipamount}{\ownspaceabovethm}
}
\def\rathmtransfer#1#2{
\expandafter\def\csname #2\endcsname{\csname #1\endcsname}
\expandafter\def\csname end#2\endcsname{\csname end#1\endcsname}
}
\def\ranewthm#1#2#3[#4]{
%% FOR \BEGIN{THM}
\theoremstyle{\current@theoremstyle}
\theoremseparator{\current@theoremseparator}
\theoremprework{\ra@pretheoremwork}
\@ifundefined{#1@basic}{\newtheorem{#1@basic}[#4]{#2}}{\renewtheorem{#1@basic}[#4]{#2}}
%% FOR \BEGIN{THM}[...]
\theoremstyle{\current@theoremstyle}
\theoremseparator{\thmForceSepPt}
\theoremprework{\ra@pretheoremwork}
\@ifundefined{#1@withName}{\newtheorem{#1@withName}[#4]{#2}}{\renewtheorem{#1@withName}[#4]{#2}}
%% FOR \BEGIN{THM*}
\theoremstyle{nonumberplain}
\theoremseparator{\thmForceSepPt}
\theoremprework{\ra@pretheoremwork}
\@ifundefined{#1@star@basic}{\newtheorem{#1@star@basic}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@basic}[Xdisplaynone]{#2}}
%% FOR \BEGIN{THM*}[...]
\theoremstyle{nonumberplain}
\theoremseparator{\thmForceSepPt}
\theoremprework{\ra@pretheoremwork}
\@ifundefined{#1@star@withName}{\newtheorem{#1@star@withName}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@withName}[Xdisplaynone]{#2}}
%% GENERATE ENVIRONMENTS:
\umbauenenv{#1}{#3}[#4]
\umbauenenv{#1@star}{#3}[Xdisplaynone]
%% TRANSFER *-DEFINITION
\rathmtransfer{#1@star}{#1*}
}
\def\umbauenenv#1#2[#3]{%
%% \BEGIN{THM}...
\expandafter\def\csname #1\endcsname{\relax%
\@ifnextchar[{\csname #1@\endcsname}{\csname #1@\endcsname[*]}%
}
%% \BEGIN{THM}[ANFANG]...
\expandafter\def\csname #1@\endcsname[##1]{\relax%
\@ifnextchar[{\csname #1@@\endcsname[##1]}{\csname #1@@\endcsname[##1][*]}%
}
%% \BEGIN{THM}[ANFANG][SCHLUSS]
\expandafter\def\csname #1@@\endcsname[##1][##2]{%
\ifx*##1%
\def\enndeOfBlock{\csname end#1@basic\endcsname}
\csname #1@basic\endcsname%
\else%
\def\enndeOfBlock{\csname end#1@withName\endcsname}
\csname #1@withName\endcsname[##1]%
\fi%
\def\makelabel####1{%
\gdef\beweislabel{####1}%
\label{\beweislabel}%
}%
\ifx*##2%
\def\enndeSymbol{\qedEIGEN{#2}}
\else%
\def\enndeSymbol{\qedEIGEN{#2}[##2]}
\fi
}
%% \END{THM}
\expandafter\gdef\csname end#1\endcsname{\enndeSymbol\enndeOfBlock}
}
%% NEWTHEOREM EINSTELLUNGSOPTIONEN:
%% F\"UR \theoremstyle
%% plain Emulates original LATEX defin, except uses param \theorem...skipamount.
%% break Header followed by line break.
%% change Header, Number and Text are interchanged, without a line break.
%% changebreak =change, but with a line break after Header.
%% margin Number in left margin, without a line break.
%% marginbreak =margin, but with a line break after the header.
%% nonumberplain =plain, without number.
%% nonumberbreak =break, without number.
%% empty No number, no name. Only the optional argument is typeset.
%% \theoremclass \theoremnumbering
%% \theorempreskip \theorempostkip \theoremindent
%% \theoremprework \theorempostwork
\def\current@theoremstyle{plain}
\def\current@theoremseparator{\thmnumberingseppt}
\theoremstyle{\current@theoremstyle}
\theoremseparator{\current@theoremseparator}
\theoremsymbol{}
\newtheorem{X}{X}[chapter] % for most theorems
\newtheorem{Xe}{Xe}[chapter] % for equations
\newtheorem*{Xdisplaynone}{Xdisplaynone}[chapter] % a dummy counter, that will never be displayed.
\newtheorem{Xsp}{Xsp}[chapter] % for special theorems
\generatenestedthmnumbering{arabic}{chapter}{X}
\generatenestedthmnumbering{arabic}{chapter}{Xe}
\generatenestedthmnumbering{Roman}{chapter}{Xsp}
\let\theXsp\theshortXsp
\theoremheaderfont{\upshape\bfseries}
\theorembodyfont{\slshape}
\ranewthm{thm}{Theorem}{\enndeOnNeutralSign}[X]
\ranewthm{satz}{Satz}{\enndeOnNeutralSign}[X]
\ranewthm{claim}{Behauptung}{\enndeOnNeutralSign}[X]
\ranewthm{lemm}{Lemma}{\enndeOnNeutralSign}[X]
\ranewthm{cor}{Korollar}{\enndeOnNeutralSign}[X]
\ranewthm{folg}{Folgerung}{\enndeOnNeutralSign}[X]
\ranewthm{prop}{Proposition}{\enndeOnNeutralSign}[X]
\theorembodyfont{\upshape}
\ranewthm{defn}{Definition}{\enndeOnNeutralSign}[X]
\ranewthm{conv}{Konvention}{\enndeOnNeutralSign}[X]
\ranewthm{obs}{Beobachtung}{\enndeOnNeutralSign}[X]
\ranewthm{e.g.}{Beipsiel}{\enndeOnNeutralSign}[X]
\ranewthm{fact}{Fakt}{\enndeOnNeutralSign}[X]
\ranewthm{rem}{Bemerkung}{\enndeOnNeutralSign}[X]
\ranewthm{qstn}{Frage}{\enndeOnNeutralSign}[X]
\ranewthm{exer}{Aufgabe}{\enndeOnNeutralSign}[X]
\ranewthm{soln}{Lösung}{\enndeOnNeutralSign}[X]
\theoremheaderfont{\itshape\bfseries}
\theorembodyfont{\upshape}
\ranewthm{proof@tmp}{Beweis}{\enndeOfProofSign}[Xdisplaynone]
\rathmtransfer{proof@tmp*}{proof}
\def\behauptungbeleg@claim{%
\iflanguage{british}{Claim}{%
\iflanguage{english}{Claim}{%
\iflanguage{ngerman}{Behauptung}{%
\iflanguage{russian}{Утверждение}{%
Claim%
}}}}%
}
\def\behauptungbeleg@pf@kurz{%
\iflanguage{british}{Pf}{%
\iflanguage{english}{Pf}{%
\iflanguage{ngerman}{Bew}{%
\iflanguage{russian}{Доказательство}{%
Pf%
}}}}%
}
\def\behauptungbeleg{\@ifnextchar\bgroup{\behauptungbeleg@c}{\behauptungbeleg@bes}}
\def\behauptungbeleg@c#1{\item[{\bfseries \behauptungbeleg@claim\erlaubeplatz #1.}]}
\def\behauptungbeleg@bes{\item[{\bfseries \behauptungbeleg@claim.}]}
\def\belegbehauptung{\item[{\bfseries\itshape\behauptungbeleg@pf@kurz.}]}
%% ****************************************************************
%% ALTE UMGEBUNGEN:
%% ****************************************************************
\newcolumntype{\RECHTS}[1]{>{\raggedleft}p{#1}}
\newcolumntype{\LINKS}[1]{>{\raggedright}p{#1}}
\newcolumntype{m}{>{$}l<{$}}
\newcolumntype{C}{>{$}c<{$}}
\newcolumntype{L}{>{$}l<{$}}
\newcolumntype{R}{>{$}r<{$}}
\newcolumntype{0}{@{\hspace{0pt}}}
\newcolumntype{\LINKSRAND}{@{\hspace{\@totalleftmargin}}}
\newcolumntype{h}{@{\extracolsep{\fill}}}
\newcolumntype{i}{>{\itshape}}
\newcolumntype{t}{@{\hspace{\tabcolsep}}}
\newcolumntype{q}{@{\hspace{1em}}}
\newcolumntype{n}{@{\hspace{-\tabcolsep}}}
\newcolumntype{M}[2]{%
>{\begin{minipage}{#2}\begin{math}}%
{#1}%
<{\end{math}\end{minipage}}%
}
\newcolumntype{T}[2]{%
>{\begin{minipage}{#2}}%
{#1}%
<{\end{minipage}}%
}
\setlength{\LTpre}{\baselineskip}
\setlength{\LTpost}{0pt}
\def\center{\centering}
\def\endcenter{}
\def\punkteumgebung@genbefehl#1#2#3{
\punkteumgebung@genbefehl@{#1}{#2}{#3}{}{}
\punkteumgebung@genbefehl@{multi#1}{#2}{#3}{
\setlength{\columnsep}{10pt}%
\setlength{\columnseprule}{0pt}%
\begin{multicols}{\thecolumnanzahl}%
}{\end{multicols}\nvraum{1}}
}
\def\punkteumgebung@genbefehl@#1#2#3#4#5{
\expandafter\gdef\csname #1\endcsname{
\@ifnextchar\bgroup{\csname #1@c\endcsname}{\csname #1@bes\endcsname}
}%]
\expandafter\def\csname #1@c\endcsname##1{
\@ifnextchar[{\csname #1@c@\endcsname{##1}}{\csname #1@c@\endcsname{##1}[\z@]}
}%]
\expandafter\def\csname #1@c@\endcsname##1[##2]{
\@ifnextchar[{\csname #1@c@@\endcsname{##1}[##2]}{\csname #1@c@@\endcsname{##1}[##2][\z@]}
}%]
\expandafter\def\csname #1@c@@\endcsname##1[##2][##3]{
\let\alterlinkerRand\gesamtlinkerRand
\let\alterrechterRand\gesamtrechterRand
\addtolength{\gesamtlinkerRand}{##2}
\addtolength{\gesamtrechterRand}{##3}
\advance\linewidth -##2%
\advance\linewidth -##3%
\advance\@totalleftmargin ##2%
\parshape\@ne \@totalleftmargin\linewidth%
#4
\begin{#2}[\upshape ##1]%
\setlength{\parskip}{0.5\baselineskip}\relax%
\setlength{\topsep}{\z@}\relax%
\setlength{\partopsep}{\z@}\relax%
\setlength{\parsep}{\parskip}\relax%
\setlength{\itemsep}{#3}\relax%
\setlength{\listparindent}{\z@}\relax%
\setlength{\itemindent}{\z@}\relax%
}
\expandafter\def\csname #1@bes\endcsname{
\@ifnextchar[{\csname #1@bes@\endcsname}{\csname #1@bes@\endcsname[\z@]}
}%]
\expandafter\def\csname #1@bes@\endcsname[##1]{
\@ifnextchar[{\csname #1@bes@@\endcsname[##1]}{\csname #1@bes@@\endcsname[##1][\z@]}
}%]
\expandafter\def\csname #1@bes@@\endcsname[##1][##2]{
\let\alterlinkerRand\gesamtlinkerRand
\let\alterrechterRand\gesamtrechterRand
\addtolength{\gesamtlinkerRand}{##1}
\addtolength{\gesamtrechterRand}{##2}
\advance\linewidth -##1%
\advance\linewidth -##2%
\advance\@totalleftmargin ##1%
\parshape\@ne \@totalleftmargin\linewidth%
#4
\begin{#2}%
\setlength{\parskip}{0.5\baselineskip}\relax%
\setlength{\topsep}{\z@}\relax%
\setlength{\partopsep}{\z@}\relax%
\setlength{\parsep}{\parskip}\relax%
\setlength{\itemsep}{#3}\relax%
\setlength{\listparindent}{\z@}\relax%
\setlength{\itemindent}{\z@}\relax%
}
\expandafter\gdef\csname end#1\endcsname{%
\end{#2}#5
\setlength{\gesamtlinkerRand}{\alterlinkerRand}
\setlength{\gesamtlinkerRand}{\alterrechterRand}
}
}
\def\ritempunkt{{\Large\textbullet}} % \textbullet, $\sqbullet$, $\blacktriangleright$
\setdefaultitem{\ritempunkt}{\ritempunkt}{\ritempunkt}{\ritempunkt}
\punkteumgebung@genbefehl{itemise}{compactitem}{\parskip}{}{}
\punkteumgebung@genbefehl{kompaktitem}{compactitem}{\z@}{}{}
\punkteumgebung@genbefehl{enumerate}{compactenum}{\parskip}{}{}
\punkteumgebung@genbefehl{kompaktenum}{compactenum}{\z@}{}{}
\let\ALTthebibliography\thebibliography
\renewenvironment{thebibliography}[1]{%
\begin{ALTthebibliography}{#1}
\addcontentsline{toc}{part}{\bibname}
}{%
\end{ALTthebibliography}
}
%% ****************************************************************
%% NEUE UMGEBUNGEN:
%% ****************************************************************
\def\matrix#1{\left(\begin{array}[mc]{#1}}
\def\endmatrix{\end{array}\right)}
\def\smatrix{\left(\begin{smallmatrix}}
\def\endsmatrix{\end{smallmatrix}\right)}
\def\vector{\begin{matrix}{c}}
\def\endvector{\end{matrix}}
\def\svector{\begin{smatrix}}
\def\endsvector{\end{smatrix}}
\def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{%
\expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup}
\expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
\expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
\expandafter\def\csname #1@ende\endcsname##1{#8}
}
\multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}}
\multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}}
\multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{}
\multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}}
\def\underbracenodisplay#1{%
\mathop{\vtop{\m@th\ialign{##\crcr
$\hfil\displaystyle{#1}\hfil$\crcr
\noalign{\kern3\p@\nointerlineskip}%
\upbracefill\crcr\noalign{\kern3\p@}}}}\limits%
}
\def\mathe[#1]#2{%
\ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\begin{escapeeinzug}}
\noindent%
\let\eqtagset\boolfalsch
\let\eqtaglabel\boolleer
\let\eqtagsymb\boolleer
\let\alteqtag\eqtag
\def\eqtag{\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}}%
\def\eqtag@loc@[##1]{\@ifnextchar\bgroup{\eqtag@loc@@[##1]}{\eqtag@loc@@[##1]{}}}%
\def\eqtag@loc@@[##1]##2{%
\gdef\eqtagset{\boolwahr}
\gdef\eqtaglabel{##1}
\gdef\eqtagsymb{##2}
}%
\def\verticalalign{}%
\IfBeginWith{#1}{t}{\def\verticalalign{t}}{}%
\IfBeginWith{#1}{m}{\def\verticalalign{c}}{}%
\IfBeginWith{#1}{b}{\def\verticalalign{b}}{}%
\def\horizontalalign{\null\hfill\null}%
\IfEndWith{#1}{l}{}{\null\hfill\null}%
\IfEndWith{#1}{r}{\def\horizontalalign{}}{}%
\begin{math}
\begin{array}[\verticalalign]{0#2}%
}
\def\endmathe{%
\end{array}
\end{math}\horizontalalign%
\let\eqtag\alteqtag
\ifthenelse{\equal{\eqtagset}{\boolwahr}}{\eqtag[\eqtaglabel]{\eqtagsymb}}{}
\ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\end{escapeeinzug}}%
}
\def\longmathe[#1]#2{\relax
\let\altarraystretch\arraystretch
\renewcommand\arraystretch{1.2}\relax
\begin{longtable}[#1]{\LINKSRAND #2}
}
\def\endlongmathe{
\end{longtable}
\renewcommand\arraystretch{\altarraystretch}
}
\def\einzug{\@ifnextchar[{\indents@}{\indents@[\z@]}}%]
\def\indents@[#1]{\@ifnextchar[{\indents@@[#1]}{\indents@@[#1][\z@]}}%]
\def\indents@@[#1][#2]{%
\begin{list}{}{\relax
\setlength{\topsep}{\z@}\relax
\setlength{\partopsep}{\z@}\relax
\setlength{\parsep}{\parskip}\relax
\setlength{\listparindent}{\z@}\relax
\setlength{\itemindent}{\z@}\relax
\setlength{\leftmargin}{#1}\relax
\setlength{\rightmargin}{#2}\relax
\let\alterlinkerRand\gesamtlinkerRand
\let\alterrechterRand\gesamtrechterRand
\addtolength{\gesamtlinkerRand}{#1}
\addtolength{\gesamtrechterRand}{#2}
}\relax
\item[]\relax
}
\def\endeinzug{%
\setlength{\gesamtlinkerRand}{\alterlinkerRand}
\setlength{\gesamtlinkerRand}{\alterrechterRand}
\end{list}%
}
\def\escapeeinzug{\begin{einzug}[-\gesamtlinkerRand][-\gesamtrechterRand]}
\def\endescapeeinzug{\end{einzug}}
\def\programmiercode{
\modulolinenumbers[1]
\begin{einzug}[\rtab][\rtab]%
\begin{linenumbers}%
\fontfamily{cmtt}\fontseries{m}\fontshape{u}\selectfont%
\setlength{\parskip}{1\baselineskip}%
\setlength{\parindent}{0pt}%
}
\def\endprogrammiercode{
\end{linenumbers}
\end{einzug}
}
\def\schattiertebox@genbefehl#1#2#3{
\expandafter\gdef\csname #1\endcsname{%
\@ifnextchar[{\csname #1@args\endcsname}{\csname #1@args\endcsname[#3]}%]%
}
\expandafter\def\csname #1@args\endcsname[##1]{%
\@ifnextchar[{\csname #1@args@l\endcsname[##1]}{\csname #1@args@n\endcsname[##1]}%]%
}
\expandafter\def\csname #1@args@l\endcsname[##1][##2]{%
\@ifnextchar[{\csname #1@args@l@r\endcsname[##1][##2]}{\csname #1@args@l@n\endcsname[##1][##2]}%]%
}
\expandafter\def\csname #1@args@n\endcsname[##1]{%
\let\boolinmdframed\boolwahr
\begin{mdframed}[#2leftmargin=0,rightmargin=0,outermargin=0,innermargin=0,##1]
}
\expandafter\def\csname #1@args@l@n\endcsname[##1][##2]{%
\let\boolinmdframed\boolwahr
\begin{mdframed}[#2leftmargin=##2/2,rightmargin=##2/2,outermargin=##2/2,innermargin=##2/2,##1]
}
\expandafter\def\csname #1@args@l@r\endcsname[##1][##2][##3]{%
\let\boolinmdframed\boolwahr
\begin{mdframed}[#2leftmargin=##2,rightmargin=##3,outermargin=##2,innermargin=##3,##1]
}
\expandafter\gdef\csname end#1\endcsname{%
\end{mdframed}
\let\boolinmdframed\boolfalsch
}
}
\schattiertebox@genbefehl{schattiertebox}{
splittopskip=0,%
splitbottomskip=0,%
frametitleaboveskip=0,%
frametitlebelowskip=0,%
skipabove=1\baselineskip,%
skipbelow=1\baselineskip,%
linewidth=2pt,%
linecolor=black,%
roundcorner=4pt,%
}{
backgroundcolor=leer,%
nobreak=true,%
}
\schattiertebox@genbefehl{schattierteboxdunn}{
splittopskip=0,%
splitbottomskip=0,%
frametitleaboveskip=0,%
frametitlebelowskip=0,%
skipabove=1\baselineskip,%
skipbelow=1\baselineskip,%
linewidth=1pt,%
linecolor=black,%
roundcorner=2pt,%
}{
backgroundcolor=leer,%
nobreak=true,%
}
\def\algorithm{\schattiertebox[backgroundcolor=hellgrau,nobreak=false]}
\def\endalgorithm{\endschattiertebox}
\def\tikzsetzenode#1{%
\tikz[remember picture,baseline,overlay]{\node #1;}%
}
\def\tikzsetzepfeil#1{%
\begin{tikzpicture}[remember picture,overlay,>=latex]%
\draw #1;%
\end{tikzpicture}%
}
\def\tikzsetzeoverlay#1{%
\begin{tikzpicture}[remember picture,overlay,>=latex]%
#1%
\end{tikzpicture}%
}
\def\tikzsetzekreise[#1]#2#3{%
\tikzsetzepfeil{%
[rounded corners,#1]%
([shift={(-\tabcolsep,0.75\baselineskip)}]#2)%
rectangle%
([shift={(\tabcolsep,-0.5\baselineskip)}]#3)
}%
}
\tikzset{
>=stealth,
auto,
thick,
main node/.style={
circle,draw,font=\sffamily\Large\bfseries,minimum size=0pt
},
}
%% ********************************************************************************
%% FILE: src/setup-layout.tex
%% ********************************************************************************
\pagestyle{fancyplain}
\@ifundefined{setcitestyle}{%
%% do nothing
}{%
\setcitestyle{numeric-comp,open={[},close={]}}
}
\def\crefpairconjunction{ und }
\def\crefmiddleconjunction{, }
\def\creflastconjunction{, und }
\raggedbottom %% <- pushes footers up
\sloppy
\def\headrulewidth{0pt}
\def\footrulewidth{0pt}
\setlength{\columnsep}{20pt}
\setlength{\columnseprule}{1pt}
\setlength{\headheight}{11pt}
\setlength{\partopsep}{0pt}
\setlength{\topsep}{\baselineskip}
\setlength{\topskip}{0.5\baselineskip}
\setlength{\footskip}{-1\baselineskip}
\setlength{\maxdepth}{0pt}
\renewcommand{\baselinestretch}{1}
\renewcommand{\arraystretch}{1}
\setcounter{LTchunksize}{\infty}
\setlength{\abovedisplayskip}{0pt}
\setlength{\parskip}{1\baselineskip}
\def\firstparagraph{\noindent}
\def\continueparagraph{\noindent}
\hypersetup{
hidelinks=true,
}
\@addtoreset{chapter}{part} %% nötig für Hyperref.
\def\partfont{\documentfont\fontseries{bx}\Huge\selectfont}
\def\chapterfont{\documentfont\fontseries{bx}\huge\selectfont}
\def\sectionfont{\documentfont\fontseries{bx}\Large\selectfont}