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diff --git a/buch/chapters/95-homologie/images/gausshomobasis.tex b/buch/chapters/95-homologie/images/gausshomobasis.tex new file mode 100644 index 0000000..ba21f54 --- /dev/null +++ b/buch/chapters/95-homologie/images/gausshomobasis.tex @@ -0,0 +1,109 @@ +% +% gaushomobasis.tex -- Bestimmung einer Basis der Homologiegruppen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\s{0.5} +\def\inset{0.05} +\def\w{8} + +\def\zeile#1#2{ + \fill[color=#2] ({0+\inset},{(12-#1)*\s+\inset}) + rectangle ({\w*\s-\inset},{(13-#1)*\s-\inset}); +} +\def\marke#1#2{ +\node at ({0.5*\w*\s},{12.5-#1)*\s}) {$#2\mathstrut$}; +} + +\def\gauss{ +\draw (0,0) rectangle ({\w*\s},{12*\s}); +\draw (0,{7*\s}) -- ({\w*\s},{7*\s}); +} + +\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-1)*\s}) + to[out=180,in=90] (-3.6,-2); +\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-2)*\s}) + to[out=180,in=90] (-2.2,-2); +\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-4)*\s}) + to[out=180,in=90] (-0.7,-2); + +\draw[->,color=darkgreen,line width=1pt] ({0.1*\s},{(12.5-7)*\s}) + to[out=180,in=90] (0.9,-2); +\draw[->,color=darkgreen,line width=1pt] ({0.1*\s},{(12.5-8)*\s}) + to[out=180,in=90] (1.6,-2); +\draw[->,color=darkgreen,line width=1pt] ({(\w-0.1)*\s},{(12.5-12)*\s}) + to[out=0,in=90] (2.6,-2); + +\draw[->,line width=2pt] ({\w*\s+0.1},{6*\s}) -- (5.4,{6*\s}); +\node at ({0.5*(\w*\s+5.5)},{6*\s}) [above] {Gauss}; + +\begin{scope} +\zeile{1}{red!30} +\zeile{2}{red!30} +\zeile{4}{red!30} +\zeile{7}{darkgreen!30} +\zeile{8}{darkgreen!30} +%\zeile{10}{darkgreen!30} +\zeile{12}{darkgreen!30} +\marke{1}{\scriptstyle\partial_{k+1}e_1^{(k+1)}} +\marke{2}{\scriptstyle\partial_{k+1}e_2^{(k+1)}} +\marke{3}{\scriptstyle\partial_{k+1}e_3^{(k+1)}} +\marke{4}{\vdots} +\marke{5}{\scriptstyle\partial_{k+1}e_{n_{k+1}}^{(k+1)}} +\marke{6}{\scriptstyle z_1^{(k)}} +\marke{7}{\scriptstyle z_2^{(k)}} +\marke{8}{\scriptstyle z_3^{(k)}} +\marke{9}{\scriptstyle z_4^{(k)}} +\marke{10}{\vdots} +\marke{11}{\scriptstyle z_{l-1}^{(k)}} +\marke{12}{\scriptstyle z_{l}^{(k)}} +\gauss +\end{scope} + +\begin{scope}[xshift=5.5cm] +\zeile{1}{black!20} +\zeile{2}{black!20} +\zeile{3}{black} +\marke{3}{\color{white}0} +\zeile{4}{black!20} +\zeile{5}{black} +\marke{5}{\color{white}0} +\zeile{6}{black} +\marke{6}{\color{white}0} +\zeile{7}{black!20} +\zeile{8}{black!20} +\zeile{9}{black} +\marke{9}{\color{white}0} +\zeile{10}{black} +\marke{10}{\color{white}0} +\zeile{11}{black} +\marke{11}{\color{white}0} +\zeile{12}{black!20} +\gauss +\end{scope} + +\node at (-4.4,-2) [below right] {$\{ +{\color{red}\partial_{k+1}e_1^{(k+1)}}, +{\color{red}\partial_{k+1}e_2^{(k+1)}}, +{\color{red}\partial_{k+1}e_{i_3}^{(k+1)}},\dots, +{\color{darkgreen}z_2^{(k)}}, +{\color{darkgreen}z_3^{(k)}}, +\dots +{\color{darkgreen}z_l^{(k)}} +\} = {\color{red}\mathcal{B}_k} \cup {\color{darkgreen}\mathcal{Z}_k'}$}; + +\end{tikzpicture} +\end{document} + |