Merge branch 'master' into master

9 files changed, 331 insertions, 1 deletions

diff --git a/buch/chapters/70-graphen/images/Makefile b/buch/chapters/70-graphen/images/Makefile
index b42cbae..5db54c8 100644
--- a/buch/chapters/70-graphen/images/Makefile
+++ b/buch/chapters/70-graphen/images/Makefile
@@ -3,11 +3,14 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	peterson.pdf adjazenzu.pdf adjazenzd.pdf kreis.pdf
+all:	peterson.pdf adjazenzu.pdf adjazenzd.pdf kreis.pdf fundamental.pdf \
+	petersonchrind.pdf nine.pdf gh.pdf
 
 peterson.pdf:	peterson.tex
 	pdflatex peterson.tex
 
+petersonchrind.pdf:	petersonchrind.tex
+	pdflatex petersonchrind.tex
 adjazenzu.pdf:	adjazenzu.tex
 	pdflatex adjazenzu.tex
 
@@ -17,3 +20,12 @@ adjazenzd.pdf:	adjazenzd.tex
 kreis.pdf:	kreis.tex
 	pdflatex kreis.tex
 
+fundamental.pdf:	fundamental.tex
+	pdflatex fundamental.tex
+
+nine.pdf:	nine.tex
+	pdflatex nine.tex
+
+gh.pdf:	gh.tex
+	pdflatex gh.tex
+
diff --git a/buch/chapters/70-graphen/images/fundamental.pdf b/buch/chapters/70-graphen/images/fundamental.pdf
new file mode 100644
index 0000000..66b82ca
--- /dev/null
+++ b/buch/chapters/70-graphen/images/fundamental.pdf
diff --git a/buch/chapters/70-graphen/images/fundamental.tex b/buch/chapters/70-graphen/images/fundamental.tex
new file mode 100644
index 0000000..b7fe9c4
--- /dev/null
+++ b/buch/chapters/70-graphen/images/fundamental.tex
@@ -0,0 +1,54 @@
+%
+% fundamental.tex -- template for standalon tikz images
+%
+% (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}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}[xshift=-4.6cm]
+	\draw[color=red,line width=2pt] (1.8,0) -- (1.8,2);
+	\draw[color=red,line width=2pt] (0,0) -- (4,0);
+	\node at (1.8,0) [below] {$i$};
+	\draw[->] (-0.1,0) -- (4.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+
+	\node at (2,-2.3) [below] {Standarbasis};
+\end{scope}
+
+\begin{scope}
+	\draw[color=red,line width=1.4pt] 
+		plot[domain=0:360,samples=100] ({\x/90},{2*sin(\x)});
+	\draw[color=blue,line width=1.4pt] 
+		plot[domain=0:360,samples=100] ({\x/90},{2*cos(\x)});
+	\node[color=blue] at (1,-1) {$\Re f_i$};
+	\node[color=red] at (2,1) {$\Im f_i$};
+	\draw[->] (-0.1,0) -- (4.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+	\node at (2,-2.3) [below] {Eigenbasis};
+\end{scope}
+
+\begin{scope}[xshift=4.6cm]
+	\foreach \t in {0.02,0.05,0.1,0.2,0.5}{
+		\draw[color=red,line width=1.0pt]
+			plot[domain=-1.8:2.2,samples=100]
+				({\x+1.8},{exp(-\x*\x/(4*\t))/(sqrt(4*3.1415*\t))});
+	}
+	\fill[color=red] (1.8,0) circle[radius=0.08];
+	\node at (1.8,0) [below] {$\xi$};
+	\draw[->] (-0.1,0) -- (4.3,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+	\node at (2,-2.3) [below] {Fundamentallösung};
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/70-graphen/images/gh.pdf b/buch/chapters/70-graphen/images/gh.pdf
new file mode 100644
index 0000000..c6e48d7
--- /dev/null
+++ b/buch/chapters/70-graphen/images/gh.pdf
diff --git a/buch/chapters/70-graphen/images/gh.tex b/buch/chapters/70-graphen/images/gh.tex
new file mode 100644
index 0000000..fcceb5f
--- /dev/null
+++ b/buch/chapters/70-graphen/images/gh.tex
@@ -0,0 +1,55 @@
+%
+% gh.tex -- Lokalsierungsfunktionen für Wavelets auf einem Graphen
+%
+% (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}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\kurve#1#2{
+	\draw[color=#2,line width=1.4pt]
+		plot[domain=0:6.3,samples=400]
+			({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1});
+}
+
+\begin{scope}
+
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\kurve{1}{red}
+\foreach \k in {0,...,4}{
+	\pgfmathparse{0.30*exp(ln(2)*\k)}
+	\xdef\l{\pgfmathresult}
+	\kurve{\l}{blue}
+}
+
+\node[color=red] at ({0.7*1},3) [above] {$g(\lambda)$};
+\node[color=blue] at ({0.7*0.3*16},3) [above] {$g_i(\lambda)$};
+
+\draw[->] (0,-0.1) -- (0,3.3);
+\end{scope}
+
+\begin{scope}[xshift=7cm]
+
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\draw[color=darkgreen,line width=1.4pt]
+	plot[domain=0:6.3,samples=100] 
+		({\x},{3*exp(-(\x/0.5)*(\x/0.5)});
+
+\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}];
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/70-graphen/images/nine.pdf b/buch/chapters/70-graphen/images/nine.pdf
new file mode 100644
index 0000000..2ae9f68
--- /dev/null
+++ b/buch/chapters/70-graphen/images/nine.pdf
diff --git a/buch/chapters/70-graphen/images/nine.tex b/buch/chapters/70-graphen/images/nine.tex
new file mode 100644
index 0000000..f214c1e
--- /dev/null
+++ b/buch/chapters/70-graphen/images/nine.tex
@@ -0,0 +1,67 @@
+%
+% nine.tex -- Nine node graph to illustrate Wilf's theorem
+%
+% (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}
+\def\kante#1#2{
+        \draw[shorten >= 0.2cm,shorten <= 0.2cm] (#1) -- (#2);
+}
+\def\knoten#1#2{
+                \fill[color=#2!30] (#1) circle[radius=0.2];
+                \draw[color=#2] (#1) circle[radius=0.2];
+                \draw (#1) circle[radius=0.2];
+}
+\def\R{1.5}
+\definecolor{rot}{rgb}{1,0,0}
+\definecolor{gruen}{rgb}{0,0.6,0}
+\definecolor{blau}{rgb}{0,0,1}
+
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\coordinate (A) at (0:\R);
+\coordinate (B) at (40:\R);
+\coordinate (C) at (80:\R);
+\coordinate (D) at (120:\R);
+\coordinate (E) at (160:\R);
+\coordinate (F) at (200:\R);
+\coordinate (G) at (240:\R);
+\coordinate (H) at (280:\R);
+\coordinate (I) at (320:\R);
+
+\knoten{A}{rot}
+\knoten{B}{blau}
+\knoten{C}{gruen}
+\knoten{D}{blau}
+\knoten{E}{rot}
+\knoten{F}{blau}
+\knoten{G}{rot}
+\knoten{H}{gruen}
+\knoten{I}{blau}
+
+\kante{A}{B}
+\kante{B}{C}
+\kante{C}{D}
+\kante{D}{E}
+\kante{E}{F}
+\kante{F}{G}
+\kante{G}{H}
+\kante{H}{I}
+\kante{I}{A}
+
+\kante{A}{C}
+\kante{A}{D}
+\kante{D}{G}
+
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/70-graphen/images/petersonchrind.pdf b/buch/chapters/70-graphen/images/petersonchrind.pdf
new file mode 100644
index 0000000..23ef6e9
--- /dev/null
+++ b/buch/chapters/70-graphen/images/petersonchrind.pdf
diff --git a/buch/chapters/70-graphen/images/petersonchrind.tex b/buch/chapters/70-graphen/images/petersonchrind.tex
new file mode 100644
index 0000000..4ae9f39
--- /dev/null
+++ b/buch/chapters/70-graphen/images/petersonchrind.tex
@@ -0,0 +1,142 @@
+%
+% tikztemplate.tex -- template for standalon tikz images
+%
+% (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}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\Ra{2}
+\def\Ri{1}
+\def\e{1.0}
+\def\r{0.2}
+
+\begin{scope}[xshift=-3.5cm]
+
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\fill[color=blau] (PA) circle[radius=\r];
+\fill[color=rot] (PB) circle[radius=\r];
+\fill[color=rot] (PC) circle[radius=\r];
+\fill[color=gruen] (PD) circle[radius=\r];
+\fill[color=gruen] (PE) circle[radius=\r];
+
+\fill[color=rot] (QA) circle[radius=\r];
+\fill[color=blau] (QB) circle[radius=\r];
+\fill[color=gruen] (QC) circle[radius=\r];
+\fill[color=rot] (QD) circle[radius=\r];
+\fill[color=blau] (QE) circle[radius=\r];
+
+\draw (PA) circle[radius=\r];
+\draw (PB) circle[radius=\r];
+\draw (PC) circle[radius=\r];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QA) circle[radius=\r];
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QD) circle[radius=\r];
+\draw (QE) circle[radius=\r];
+
+\node at (0,{-\Ra}) [below] {$\operatorname{chr}P=3\mathstrut$};
+
+\end{scope}
+
+\begin{scope}[xshift=3.5cm]
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+\definecolor{gelb}{rgb}{0,0,1}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\fill[color=rot] (QA) circle[radius={1.5*\r}];
+\fill[color=rot!40] (QB) circle[radius=\r];
+\fill[color=rot!40] (QE) circle[radius=\r];
+\fill[color=rot!40] (PA) circle[radius=\r];
+
+\fill[color=blau] (PB) circle[radius={1.5*\r}];
+\fill[color=blau!40] (PD) circle[radius=\r];
+\fill[color=blau!40] (PE) circle[radius=\r];
+\fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r];
+
+\fill[color=gruen] (PC) circle[radius={1.5*\r}];
+\fill[color=gruen!40] (QC) circle[radius=\r];
+\fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r];
+\fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r];
+
+\fill[color=gelb] (QD) circle[radius={1.5*\r}];
+\fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r];
+\fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r];
+\fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r];
+
+\draw (PA) circle[radius=\r];
+\draw (PB) circle[radius={1.5*\r}];
+\draw (PC) circle[radius={1.5*\r}];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QA) circle[radius={1.5*\r}];
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QD) circle[radius={1.5*\r}];
+\draw (QE) circle[radius=\r];
+
+\node at (0,{-\Ra}) [below] {$\operatorname{ind}P=4\mathstrut$};
+
+\end{scope}
+
+
+
+\end{tikzpicture}
+\end{document}
+