From ada53a9c225b896c8d7608300427aac475bb7045 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Feb 2021 21:52:16 +0100 Subject: move all iamges to separate files --- buch/chapters/05-zahlen/ganz.tex | 8 ++ buch/chapters/05-zahlen/images/komplex.pdf | Bin 0 -> 18852 bytes buch/chapters/05-zahlen/images/komplex.tex | 39 ++++++++ buch/chapters/05-zahlen/komplex.tex | 22 +---- buch/chapters/05-zahlen/natuerlich.tex | 13 ++- buch/chapters/10-vektorenmatrizen/ringe.tex | 31 ------ buch/chapters/30-endlichekoerper/chapter.tex | 1 + buch/chapters/40-eigenwerte/chapter.tex | 1 + buch/chapters/40-eigenwerte/grundlagen.tex | 58 +---------- buch/chapters/40-eigenwerte/images/Makefile | 5 +- buch/chapters/40-eigenwerte/images/nilpotent.pdf | Bin 0 -> 14254 bytes buch/chapters/40-eigenwerte/images/nilpotent.tex | 76 +++++++++++++++ buch/chapters/40-eigenwerte/images/sp.pdf | Bin 24049 -> 24022 bytes buch/chapters/50-permutationen/endlich.tex | 106 +-------------------- buch/chapters/50-permutationen/images/Makefile | 20 ++++ .../50-permutationen/images/komposition.pdf | Bin 0 -> 13951 bytes .../50-permutationen/images/komposition.tex | 55 +++++++++++ .../50-permutationen/images/permutation.pdf | Bin 0 -> 13814 bytes .../50-permutationen/images/permutation.tex | 43 +++++++++ .../50-permutationen/images/transpositionen.pdf | Bin 0 -> 22548 bytes .../50-permutationen/images/transpositionen.tex | 103 ++++++++++++++++++++ .../50-permutationen/images/zyklenzerlegung.pdf | Bin 0 -> 14937 bytes .../50-permutationen/images/zyklenzerlegung.tex | 60 ++++++++++++ buch/chapters/50-permutationen/transpositionen.tex | 87 +---------------- buch/chapters/70-graphen/beschreibung.tex | 21 +--- buch/chapters/70-graphen/images/Makefile | 10 ++ buch/chapters/70-graphen/images/peterson.pdf | Bin 0 -> 10833 bytes buch/chapters/70-graphen/images/peterson.tex | 38 ++++++++ buch/chapters/80-wahrscheinlichkeit/google.tex | 40 -------- .../chapters/80-wahrscheinlichkeit/images/Makefile | 8 +- .../80-wahrscheinlichkeit/images/spielB.pdf | Bin 0 -> 9917 bytes .../80-wahrscheinlichkeit/images/spielB.tex | 59 ++++++++++++ .../80-wahrscheinlichkeit/images/spielBtilde.pdf | Bin 0 -> 14592 bytes .../80-wahrscheinlichkeit/images/spielBtilde.tex | 59 ++++++++++++ .../80-wahrscheinlichkeit/images/vergleich.pdf | Bin 120558 -> 120558 bytes buch/chapters/80-wahrscheinlichkeit/parrondo.tex | 88 +---------------- buch/chapters/95-homologie/chapter.tex | 37 +------ buch/chapters/95-homologie/images/Makefile | 10 ++ buch/chapters/95-homologie/images/dreieck.pdf | Bin 0 -> 12089 bytes buch/chapters/95-homologie/images/dreieck.tex | 54 +++++++++++ 40 files changed, 670 insertions(+), 482 deletions(-) create mode 100644 buch/chapters/05-zahlen/images/komplex.pdf create mode 100644 buch/chapters/05-zahlen/images/komplex.tex create mode 100644 buch/chapters/40-eigenwerte/images/nilpotent.pdf create mode 100644 buch/chapters/40-eigenwerte/images/nilpotent.tex create mode 100644 buch/chapters/50-permutationen/images/Makefile create mode 100644 buch/chapters/50-permutationen/images/komposition.pdf create mode 100644 buch/chapters/50-permutationen/images/komposition.tex create mode 100644 buch/chapters/50-permutationen/images/permutation.pdf create mode 100644 buch/chapters/50-permutationen/images/permutation.tex create mode 100644 buch/chapters/50-permutationen/images/transpositionen.pdf create mode 100644 buch/chapters/50-permutationen/images/transpositionen.tex create mode 100644 buch/chapters/50-permutationen/images/zyklenzerlegung.pdf create mode 100644 buch/chapters/50-permutationen/images/zyklenzerlegung.tex create mode 100644 buch/chapters/70-graphen/images/Makefile create mode 100644 buch/chapters/70-graphen/images/peterson.pdf create mode 100644 buch/chapters/70-graphen/images/peterson.tex create mode 100644 buch/chapters/80-wahrscheinlichkeit/images/spielB.pdf create mode 100644 buch/chapters/80-wahrscheinlichkeit/images/spielB.tex create mode 100644 buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.pdf create mode 100644 buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.tex create mode 100644 buch/chapters/95-homologie/images/Makefile create mode 100644 buch/chapters/95-homologie/images/dreieck.pdf create mode 100644 buch/chapters/95-homologie/images/dreieck.tex (limited to 'buch') diff --git a/buch/chapters/05-zahlen/ganz.tex b/buch/chapters/05-zahlen/ganz.tex index 4809e29..fab2dcb 100644 --- a/buch/chapters/05-zahlen/ganz.tex +++ b/buch/chapters/05-zahlen/ganz.tex @@ -28,6 +28,14 @@ Die Rechenoperationen sind wie folgt definiert: \end{aligned} \label{buch:zahlen:ganze-rechenregeln} \end{equation} +Die Darstellung ganzer Zahlen als Paare von natürlichen Zahlen +findet man auch in der Buchhaltung, wo man statt eines Vorzeichen +{\em Soll} und {\em Haben} verwendet. +Dabei kommt es nur auf die Differenz der beiden Positionen an. +Fügt man beiden Positionen den gleichen Betrag hinzu, ändert sich +nichts. +Viele der Paare $(a,b)$ müssen also als äquivalent angesehen +werden. \subsubsection{Äquivalenzrelation} Die Definition~\eqref{buch:zahlen:ganze-rechenregeln} diff --git a/buch/chapters/05-zahlen/images/komplex.pdf b/buch/chapters/05-zahlen/images/komplex.pdf new file mode 100644 index 0000000..d502e3c Binary files /dev/null and b/buch/chapters/05-zahlen/images/komplex.pdf differ diff --git a/buch/chapters/05-zahlen/images/komplex.tex b/buch/chapters/05-zahlen/images/komplex.tex new file mode 100644 index 0000000..8cda85b --- /dev/null +++ b/buch/chapters/05-zahlen/images/komplex.tex @@ -0,0 +1,39 @@ +% +% komplex.tex -- Betrag und Argument einer komplexen Zahl +% +% (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{1.5} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\pgfmathparse{atan(2/3)} +\xdef\winkel{\pgfmathresult} +\fill[color=blue!20] (0,0) -- (1.5,0) arc (0:\winkel:1.5) -- cycle; +\draw[->] (-1,0) -- (4,0) coordinate[label={$\Re z$}]; +\draw[->] (0,-1) -- (0,3) coordinate[label={right:$\Im z$}]; +\draw[line width=0.5pt] (3,0) -- (3,2); +\node at (3,1) [right] {$\Im z=b$}; +\node at (1.5,0) [below] {$\Re z=a$}; +\draw[->,color=red,line width=1.4pt] (0,0) -- (3,2); +\node at (3,2) [above right] {$z=a+bi$}; +\def\punkt#1{ + \fill[color=white] #1 circle[radius=0.04]; + \draw #1 circle[radius=0.04]; +} +\punkt{(0,0)} +\punkt{(3,2)} +\node[color=red] at (1.5,1) [rotate=\winkel,above] {$r=|z|$}; +\node[color=blue] at ({\winkel/2}:1.0) + [rotate={\winkel/2}] {$\varphi=\operatorname{arg}z$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/05-zahlen/komplex.tex b/buch/chapters/05-zahlen/komplex.tex index 3cbf473..2a9b4a9 100644 --- a/buch/chapters/05-zahlen/komplex.tex +++ b/buch/chapters/05-zahlen/komplex.tex @@ -188,27 +188,7 @@ genauer untersuchen müssen. \begin{figure} \centering -\begin{tikzpicture}[>=latex,thick,scale=1.5] -\pgfmathparse{atan(2/3)} -\xdef\winkel{\pgfmathresult} -\fill[color=blue!20] (0,0) -- (1.5,0) arc (0:\winkel:1.5) -- cycle; -\draw[->] (-1,0) -- (4,0) coordinate[label={$\Re z$}]; -\draw[->] (0,-1) -- (0,3) coordinate[label={right:$\Im z$}]; -\draw[line width=0.5pt] (3,0) -- (3,2); -\node at (3,1) [right] {$\Im z=b$}; -\node at (1.5,0) [below] {$\Re z=a$}; -\draw[->,color=red,line width=1.4pt] (0,0) -- (3,2); -\node at (3,2) [above right] {$z=a+bi$}; -\def\punkt#1{ - \fill[color=white] #1 circle[radius=0.04]; - \draw #1 circle[radius=0.04]; -} -\punkt{(0,0)} -\punkt{(3,2)} -\node[color=red] at (1.5,1) [rotate=\winkel,above] {$r=|z|$}; -\node[color=blue] at ({\winkel/2}:1.0) - [rotate={\winkel/2}] {$\varphi=\operatorname{arg}z$}; -\end{tikzpicture} +\includegraphics{chapters/05-zahlen/images/komplex.pdf} \caption{Argument und Betrag einer komplexen Zahl $z=a+ib$ in der Gaussschen Zahlenebene \label{buch:zahlen:cfig}} diff --git a/buch/chapters/05-zahlen/natuerlich.tex b/buch/chapters/05-zahlen/natuerlich.tex index 3863191..acad943 100644 --- a/buch/chapters/05-zahlen/natuerlich.tex +++ b/buch/chapters/05-zahlen/natuerlich.tex @@ -35,9 +35,20 @@ $n'\in \mathbb{N}$. \item Wenn zwei Zahlen $n,m\in\mathbb{N}$ den gleichen Nachfolger haben, $n'=m'$, dann sind sie gleich $n=m$. \item Enthält eine Menge $X$ die Zahl $0$ und mit jeder Zahl auch ihren -Nachfolger, dann ist $X\subset\mathbb{N}$. %TODO: X = N?... +Nachfolger, dann ist $\mathbb{N}\subset X$. \end{enumerate} +\subsubsection{Vollständige Induktion} +Es letzte Axiom formuliert das Prinzip der vollständigen Induktion. +Um eine Aussage $P(n)$ für alle natürlichen Zahlen $n$ +mit vollständiger Induktion zu beweisen, bezeichnet man mit +$X$ die Menge aller Zahlen, für die $P(n)$ wahr ist. +Die Induktionsverankerung beweist, dass $P(0)$ wahr ist, dass also $0\in X$. +Der Induktionsschritt beweist, dass mit einer Zahl $n\in X$ auch der +Nachfolger $n'\in X$ ist. +Nach dem letzten Axiom ist $\mathbb{N}\subset X$, oder anders ausgedrückt, +die Aussage $P(n)$ ist wahr für jede natürliche Zahl. + \subsubsection{Addition} Aus der Nachfolgereigenschaft lässt sich durch wiederholte Anwendung die vertrautere Addition konstruieren. diff --git a/buch/chapters/10-vektorenmatrizen/ringe.tex b/buch/chapters/10-vektorenmatrizen/ringe.tex index 42e2a7e..21b29c2 100644 --- a/buch/chapters/10-vektorenmatrizen/ringe.tex +++ b/buch/chapters/10-vektorenmatrizen/ringe.tex @@ -108,37 +108,6 @@ $c_0(\mathbb{Z})$ ist immer noch ein Ring, aber er hat kein Eins. \begin{figure} \centering \includegraphics{chapters/10-vektorenmatrizen/images/gausszahlen.pdf} -%\begin{tikzpicture}[>=latex,thick,scale=0.8] -%\draw[->] (-8.5,0) -- (8.5,0) coordinate[label={$\Re z$}]; -%\draw[->] (0,-4.5) -- (0,4.5) coordinate[label={right:$\Im z$}]; -%\foreach \x in {-8,...,8}{ -% \foreach \y in {-4,...,4}{ -% \fill (\x,\y) circle[radius=0.05]; -% } -%} -% -% -%\coordinate (O) at (0,0); -%\coordinate (A) at (2,2); -%\coordinate (B) at (-3,1); -%\coordinate (C) at (-8,-4); -%\coordinate (D) at (-1,3); -%\draw[line width=0.5pt] (A)--(D)--(B); -%\draw[->,color=red] (O) -- (A); -%\draw[->,color=red] (O) -- (B); -%\draw[->,color=blue] (O) -- (C); -%\draw[->,color=darkgreen] (O) -- (D); -%\fill[color=red] (A) circle[radius=0.08]; -%\fill[color=red] (B) circle[radius=0.08]; -%\fill[color=blue] (C) circle[radius=0.08]; -%\fill[color=darkgreen] (D) circle[radius=0.08]; -%\fill[color=black] (O) circle[radius=0.08]; -%\node[color=red] at (A) [above right] {$z$}; -%\node[color=red] at (B) [above left] {$w$}; -%\node[color=darkgreen] at (D) [above] {$z+w$}; -%\node[color=blue] at (C) [below right] {$z\cdot w$}; -% -%\end{tikzpicture} \caption{Der Ring der ganzen Gausschen Zahlen besteht aus den ganzahligen Gitterpunkten in der Gausschen Zahlenebene \label{buch:vektorenmatrizen:fig:ganzgauss}} diff --git a/buch/chapters/30-endlichekoerper/chapter.tex b/buch/chapters/30-endlichekoerper/chapter.tex index 2d16d80..150c719 100644 --- a/buch/chapters/30-endlichekoerper/chapter.tex +++ b/buch/chapters/30-endlichekoerper/chapter.tex @@ -37,6 +37,7 @@ lösbar werden. \section*{Übungsaufgaben} +\rhead{Übungsaufgaben} \aufgabetoplevel{chapters/30-endlichekoerper/uebungsaufgaben} \begin{uebungsaufgaben} \uebungsaufgabe{3002} diff --git a/buch/chapters/40-eigenwerte/chapter.tex b/buch/chapters/40-eigenwerte/chapter.tex index e097b8d..e769b38 100644 --- a/buch/chapters/40-eigenwerte/chapter.tex +++ b/buch/chapters/40-eigenwerte/chapter.tex @@ -38,6 +38,7 @@ Dies wird in Abschnitt~\ref{buch:section:spektraltheorie} beschrieben. \input{chapters/40-eigenwerte/spektraltheorie.tex} \section*{Übungsaufgaben} +\rhead{Übungsaufgaben} \aufgabetoplevel{chapters/40-eigenwerte/uebungsaufgaben} \begin{uebungsaufgaben} \uebungsaufgabe{4001} diff --git a/buch/chapters/40-eigenwerte/grundlagen.tex b/buch/chapters/40-eigenwerte/grundlagen.tex index 55c3344..dbc19ce 100644 --- a/buch/chapters/40-eigenwerte/grundlagen.tex +++ b/buch/chapters/40-eigenwerte/grundlagen.tex @@ -346,63 +346,7 @@ $b_{ij}=0$ für $i+k>j$ und $c_{ij}=0$ für $i+l>j$. In der folgenden graphischen Darstellung der Matrizen sind die Bereiche, wo die Matrixelemente verschwinden, weiss. \begin{center} - -\newtcbox{\myboxA}{blank,boxsep=0mm, -clip upper,minipage, -width=31.0mm,height=17.0mm,nobeforeafter, -borderline={0.0pt}{0.0pt}{white}, -} - -\begin{tikzpicture}[>=latex,thick] - -\def\cx{1.8} -\def\cy{1.2} - -\draw[line width=0.3pt] (-3,2.5) -- (6,2.5); - -\begin{scope}[xshift=-4cm] -\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; -\fill[color=red!30] (0.5,3) -- (3,0.5) -- (3,3) -- cycle; -\draw (0,0) rectangle (3,3); -\draw (0,3) -- (3,0); -\node at ({\cx+0.5*0.5},{\cy+0.5*0.5}) [rotate=-45] {$k$}; -\draw[color=blue,line width=1.4pt] (0,2.5) -- (1.0,2.5); -\draw[color=red,line width=1.4pt] (1.0,2.5) -- (3,2.5); -\node at (1,1) {$B$}; -\node at (-0.3,2.5) [left] {$i$}; -\node at (1,2.5) [above right] {$i+k$}; -\end{scope} - -\node at (-0.5,1.5) {$\mathstrut\cdot\mathstrut$}; - -\begin{scope} -\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; -\fill[color=red!30] (1.0,3) -- (3,1.0) -- (3,3) -- cycle; -\draw (0,0) rectangle (3,3); -\draw (0,3) -- (3,0); -\node at ({\cx+1.0*0.5},{\cy+1.0*0.5}) [rotate=-45] {$l$}; -\draw[color=red,line width=1.4pt] (2,3)--(2,2); -\draw[color=blue,line width=1.4pt] (2,2)--(2,0); -\node at (1,1) {$C$}; -\node at (2,3) [above] {$j$}; -\node at (2,2) [above right] {$j-l$}; -\end{scope} - -\node at (3.5,1.5) {$\mathstrut=\mathstrut$}; - -\begin{scope}[xshift=4cm] -\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; -\fill[color=red!30] (1.5,3) -- (3,1.5) -- (3,3) -- cycle; -\draw (0,0) rectangle (3,3); -\draw (0,3) -- (3,0); -\node at ({\cx+1.5*0.5},{\cy+1.5*0.5}) [rotate=-45] {$k+l$}; -\fill[color=red!50!blue] (2,2.5) circle[radius=0.1]; -\draw[line width=0.3pt] (2,3) -- (2,2.5); -\node at (2,3) [above] {$j$}; -\node at (1,1) {$D$}; -\end{scope} - -\end{tikzpicture} +\includegraphics{chapters/40-eigenwerte/images/nilpotent.pdf} \end{center} Bei der Berechnung des Elementes $d_{ij}$ wird die Zeile $i$ von $B$ mit der Spalte $j$ von $C$ multipliziert. diff --git a/buch/chapters/40-eigenwerte/images/Makefile b/buch/chapters/40-eigenwerte/images/Makefile index 5915d30..db00dac 100644 --- a/buch/chapters/40-eigenwerte/images/Makefile +++ b/buch/chapters/40-eigenwerte/images/Makefile @@ -3,7 +3,7 @@ # # (c) 2020 Prof Dr Andreas Müller, Hochschule Rappersil # -all: sp.pdf +all: sp.pdf nilpotent.pdf sp.pdf: sp.tex sppaths.tex pdflatex sp.tex @@ -11,3 +11,6 @@ sp.pdf: sp.tex sppaths.tex sppaths.tex: spbeispiel.m octave spbeispiel.m +nilpotent.pdf: nilpotent.tex + pdflatex nilpotent.tex + diff --git a/buch/chapters/40-eigenwerte/images/nilpotent.pdf b/buch/chapters/40-eigenwerte/images/nilpotent.pdf new file mode 100644 index 0000000..2106697 Binary files /dev/null and b/buch/chapters/40-eigenwerte/images/nilpotent.pdf differ diff --git a/buch/chapters/40-eigenwerte/images/nilpotent.tex b/buch/chapters/40-eigenwerte/images/nilpotent.tex new file mode 100644 index 0000000..1e6cd79 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/nilpotent.tex @@ -0,0 +1,76 @@ +% +% nilpotent.tex -- Produkt nilpotenter Matrizen +% +% (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} +\usepackage[many]{tcolorbox} + +\begin{document} +\def\skala{1} + +\newtcbox{\myboxA}{blank,boxsep=0mm, +clip upper,minipage, +width=31.0mm,height=17.0mm,nobeforeafter, +borderline={0.0pt}{0.0pt}{white}, +} + +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\cx{1.8} +\def\cy{1.2} + +\draw[line width=0.3pt] (-3,2.5) -- (6,2.5); + +\begin{scope}[xshift=-4cm] +\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; +\fill[color=red!30] (0.5,3) -- (3,0.5) -- (3,3) -- cycle; +\draw (0,0) rectangle (3,3); +\draw (0,3) -- (3,0); +\node at ({\cx+0.5*0.5},{\cy+0.5*0.5}) [rotate=-45] {$k$}; +\draw[color=blue,line width=1.4pt] (0,2.5) -- (1.0,2.5); +\draw[color=red,line width=1.4pt] (1.0,2.5) -- (3,2.5); +\node at (1,1) {$B$}; +\node at (-0.3,2.5) [left] {$i$}; +\node at (1,2.5) [above right] {$i+k$}; +\end{scope} + +\node at (-0.5,1.5) {$\mathstrut\cdot\mathstrut$}; + +\begin{scope} +\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; +\fill[color=red!30] (1.0,3) -- (3,1.0) -- (3,3) -- cycle; +\draw (0,0) rectangle (3,3); +\draw (0,3) -- (3,0); +\node at ({\cx+1.0*0.5},{\cy+1.0*0.5}) [rotate=-45] {$l$}; +\draw[color=red,line width=1.4pt] (2,3)--(2,2); +\draw[color=blue,line width=1.4pt] (2,2)--(2,0); +\node at (1,1) {$C$}; +\node at (2,3) [above] {$j$}; +\node at (2,2) [above right] {$j-l$}; +\end{scope} + +\node at (3.5,1.5) {$\mathstrut=\mathstrut$}; + +\begin{scope}[xshift=4cm] +\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; +\fill[color=red!30] (1.5,3) -- (3,1.5) -- (3,3) -- cycle; +\draw (0,0) rectangle (3,3); +\draw (0,3) -- (3,0); +\node at ({\cx+1.5*0.5},{\cy+1.5*0.5}) [rotate=-45] {$k+l$}; +\fill[color=red!50!blue] (2,2.5) circle[radius=0.1]; +\draw[line width=0.3pt] (2,3) -- (2,2.5); +\node at (2,3) [above] {$j$}; +\node at (1,1) {$D$}; +\end{scope} + +\end{tikzpicture} + +\end{document} + diff --git a/buch/chapters/40-eigenwerte/images/sp.pdf b/buch/chapters/40-eigenwerte/images/sp.pdf index 5346e06..d4de984 100644 Binary files a/buch/chapters/40-eigenwerte/images/sp.pdf and b/buch/chapters/40-eigenwerte/images/sp.pdf differ diff --git a/buch/chapters/50-permutationen/endlich.tex b/buch/chapters/50-permutationen/endlich.tex index 7669a17..700c0f2 100644 --- a/buch/chapters/50-permutationen/endlich.tex +++ b/buch/chapters/50-permutationen/endlich.tex @@ -21,31 +21,7 @@ Element} der Gruppe $S_n$ und wir auch mit $e$ bezeichnet. \subsection{Permutationen als $2\times n$-Matrizen} Eine Permutation kann als $2\times n$-Matrix geschrieben werden: \begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\sx{0.8} -\def\sy{1} -\begin{scope}[xshift=-3cm] -\foreach \x in {1,...,6}{ - \node at ({(\x-1)*\sx},\sy) [above] {$\tiny\x$}; - \fill ({(\x-1)*\sx},\sy) circle[radius=0.05]; - \fill ({(\x-1)*\sx},0) circle[radius=0.05]; -} -\draw[->] (0,\sy) to[out=-70,in=110] (\sx,0); -\draw[<-] (0,0) to[out=70,in=-110] (\sx,\sy); -\draw[->] ({2*\sx},\sy) -- ({2*\sx},0); -\draw[->] ({3*\sx},\sy) to[out=-70,in=110] ({4*\sx},0); -\draw[->] ({4*\sx},\sy) to[out=-70,in=110] ({5*\sx},0); -\draw[->] ({5*\sx},\sy) to[out=-110,in=70] ({3*\sx},0); -\end{scope} -\node at (2.4,{\sy/2}) {$\mathstrut=\mathstrut$}; -\node at (5,{\sy/2}) {$\displaystyle -\renewcommand{\arraystretch}{1.4} -\begin{pmatrix} -1&2&3&4&5&6\\ -2&1&3&5&6&4 -\end{pmatrix} -$}; -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/permutation.pdf} \end{center} Das neutrale Element hat die Matrix \[ @@ -64,43 +40,7 @@ Permutation angeordnet. Die zusammengesetzte Permutation kann dann in der zweiten Zeile der zweiten Permutation abgelesen werden: \begin{center} -\begin{tikzpicture}[>=latex,thick] -\begin{scope}[xshift=-4.5cm] -\node at (0,0) {$\displaystyle -\sigma_1=\begin{pmatrix} -1&2&3&4&5&6\\ -2&1&3&5&6&4 -\end{pmatrix}$}; -\node at (0,-1) {$\displaystyle -\sigma_2=\begin{pmatrix} -1&2&3&4&5&6\\ -3&4&5&6&1&2 -\end{pmatrix} -$}; -\end{scope} -\begin{scope} -\node at (0,0) {$\displaystyle -\begin{pmatrix} -1&2&3&4&5&6\\ -2&1&3&5&6&4 -\end{pmatrix}$}; -\node at (0,-1) {$\displaystyle -\begin{pmatrix} -2&1&3&5&6&4\\ -4&3&5&1&2&6 -\end{pmatrix} -$}; - -\end{scope} -\begin{scope}[xshift=4.5cm] -\node at (0,-0.5) {$\displaystyle -\sigma_2\sigma_1=\begin{pmatrix} -1&2&3&4&5&6\\ -4&3&5&1&2&6 -\end{pmatrix} -$}; -\end{scope} -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/komposition.pdf} \end{center} Die Inverse einer Permutation kann erhalten werden, indem die beiden Zeilen vertauscht werden und dann die Spalten wieder so angeordnet werden, @@ -130,47 +70,7 @@ Eine Permutation $\sigma\in S_n$ kann auch mit sogenanten Zyklenzerlegung analysiert werden. Zum Beispiel: \begin{center} -\begin{tikzpicture}[>=latex,thick] -\begin{scope}[xshift=-3cm] -\node at (0,0) {$\displaystyle -\sigma=\begin{pmatrix} -{\color{red}1}&{\color{red}2}&{\color{darkgreen}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}\\ -{\color{red}2}&{\color{red}1}&{\color{darkgreen}3}&{\color{blue}5}&{\color{blue}6}&{\color{blue}4} -\end{pmatrix}$}; -\end{scope} -\node at (0,0) {$\mathstrut=\mathstrut$}; -\begin{scope}[xshift=1.5cm] -\coordinate (A) at (0,0.5); -\coordinate (B) at (0,-0.5); -\draw[->,color=red] (A) to[out=-20,in=20] (0,-0.5); -\draw[->,color=red] (B) to[out=160,in=-160] (0,0.5); -\node at (A) [above] {$\tiny 1$}; -\node at (B) [below] {$\tiny 2$}; -\fill (A) circle[radius=0.05]; -\fill (B) circle[radius=0.05]; - -\coordinate (C) at (1.5,0.25); -\node at (C) [above] {$\tiny 3$}; -\draw[->,color=darkgreen] ({1.5+0.01},0.25) to[out=-10,in=-170] ({1.5-0.01},0.25); -\draw[color=darkgreen] (1.5,{0.25-0.3}) circle[radius=0.3]; -\fill (C) circle[radius=0.05]; - -\def\r{0.5} -\coordinate (D) at ({3.5+\r*cos(90)},{0+\r*sin(90)}); -\coordinate (E) at ({3.5+\r*cos(210)},{0+\r*sin(210)}); -\coordinate (F) at ({3.5+\r*cos(330)},{0+\r*sin(330)}); -\node at (D) [above] {$\tiny 4$}; -\node at (E) [below left] {$\tiny 5$}; -\node at (F) [below right] {$\tiny 6$}; -\draw[->,color=blue] (D) to[out=180,in=120] (E); -\draw[->,color=blue] (E) to[out=-60,in=-120] (F); -\draw[->,color=blue] (F) to[out=60,in=0] (D); -\fill (D) circle[radius=0.05]; -\fill (E) circle[radius=0.05]; -\fill (F) circle[radius=0.05]; - -\end{scope} -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/zyklenzerlegung.pdf} \end{center} \begin{definition} diff --git a/buch/chapters/50-permutationen/images/Makefile b/buch/chapters/50-permutationen/images/Makefile new file mode 100644 index 0000000..c2e0e9f --- /dev/null +++ b/buch/chapters/50-permutationen/images/Makefile @@ -0,0 +1,20 @@ +# +# Makefile -- Bilder +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: transpositionen.pdf zyklenzerlegung.pdf permutation.pdf \ + komposition.pdf + +permutation.pdf: permutation.tex + pdflatex permutation.tex + +komposition.pdf: komposition.tex + pdflatex komposition.tex + +transpositionen.pdf: transpositionen.tex + pdflatex transpositionen.tex + +zyklenzerlegung.pdf: zyklenzerlegung.tex + pdflatex zyklenzerlegung.tex + diff --git a/buch/chapters/50-permutationen/images/komposition.pdf b/buch/chapters/50-permutationen/images/komposition.pdf new file mode 100644 index 0000000..9e532bf Binary files /dev/null and b/buch/chapters/50-permutationen/images/komposition.pdf differ diff --git a/buch/chapters/50-permutationen/images/komposition.tex b/buch/chapters/50-permutationen/images/komposition.tex new file mode 100644 index 0000000..ef3ec65 --- /dev/null +++ b/buch/chapters/50-permutationen/images/komposition.tex @@ -0,0 +1,55 @@ +% +% komposition.tex -- Komposition zweier Permutationen +% +% (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=-4.5cm] +\node at (0,0) {$\displaystyle +\sigma_1=\begin{pmatrix} +1&2&3&4&5&6\\ +2&1&3&5&6&4 +\end{pmatrix}$}; +\node at (0,-1) {$\displaystyle +\sigma_2=\begin{pmatrix} +1&2&3&4&5&6\\ +3&4&5&6&1&2 +\end{pmatrix} +$}; +\end{scope} +\begin{scope} +\node at (0,0) {$\displaystyle +\begin{pmatrix} +1&2&3&4&5&6\\ +2&1&3&5&6&4 +\end{pmatrix}$}; +\node at (0,-1) {$\displaystyle +\begin{pmatrix} +2&1&3&5&6&4\\ +4&3&5&1&2&6 +\end{pmatrix} +$}; + +\end{scope} +\begin{scope}[xshift=4.5cm] +\node at (0,-0.5) {$\displaystyle +\sigma_2\sigma_1=\begin{pmatrix} +1&2&3&4&5&6\\ +4&3&5&1&2&6 +\end{pmatrix} +$}; +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/images/permutation.pdf b/buch/chapters/50-permutationen/images/permutation.pdf new file mode 100644 index 0000000..cdfa186 Binary files /dev/null and b/buch/chapters/50-permutationen/images/permutation.pdf differ diff --git a/buch/chapters/50-permutationen/images/permutation.tex b/buch/chapters/50-permutationen/images/permutation.tex new file mode 100644 index 0000000..ee58d4a --- /dev/null +++ b/buch/chapters/50-permutationen/images/permutation.tex @@ -0,0 +1,43 @@ +% +% permutation.tex -- Definition einer Permutation +% +% (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\sx{0.8} +\def\sy{1} +\begin{scope}[xshift=-3cm] +\foreach \x in {1,...,6}{ + \node at ({(\x-1)*\sx},\sy) [above] {$\tiny\x$}; + \fill ({(\x-1)*\sx},\sy) circle[radius=0.05]; + \fill ({(\x-1)*\sx},0) circle[radius=0.05]; +} +\draw[->] (0,\sy) to[out=-70,in=110] (\sx,0); +\draw[<-] (0,0) to[out=70,in=-110] (\sx,\sy); +\draw[->] ({2*\sx},\sy) -- ({2*\sx},0); +\draw[->] ({3*\sx},\sy) to[out=-70,in=110] ({4*\sx},0); +\draw[->] ({4*\sx},\sy) to[out=-70,in=110] ({5*\sx},0); +\draw[->] ({5*\sx},\sy) to[out=-110,in=70] ({3*\sx},0); +\end{scope} +\node at (2.4,{\sy/2}) {$\mathstrut=\mathstrut$}; +\node at (5,{\sy/2}) {$\displaystyle +\renewcommand{\arraystretch}{1.4} +\begin{pmatrix} +1&2&3&4&5&6\\ +2&1&3&5&6&4 +\end{pmatrix} +$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/images/transpositionen.pdf b/buch/chapters/50-permutationen/images/transpositionen.pdf new file mode 100644 index 0000000..ad2ea7c Binary files /dev/null and b/buch/chapters/50-permutationen/images/transpositionen.pdf differ diff --git a/buch/chapters/50-permutationen/images/transpositionen.tex b/buch/chapters/50-permutationen/images/transpositionen.tex new file mode 100644 index 0000000..0670a2c --- /dev/null +++ b/buch/chapters/50-permutationen/images/transpositionen.tex @@ -0,0 +1,103 @@ +% +% transpositionen.tex -- Darstellung eines Zyklus aus Transpositionen +% +% (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\kreuz#1#2#3{ + \draw[->] ({(#1)-1},#2) to[out=-90,in=90] ({#1},{#2-1}); + \draw[->] ({#1},#2) to[out=-90,in=90] ({#1-1},{#2-1}); + \node at ({(#1)-0.5+0.2},{#2-0.5}) [right] {$#3$}; +} +\begin{tikzpicture}[>=latex,thick,scale=\skala] +\foreach \x in {1,2,3,6,7,8,9}{ + \fill ({\x-1},0) circle[radius=0.05]; +} +\foreach \x in {1,2,3}{ + \node at ({\x-1},0) [above] {$\tiny \x$}; +} +\node at (8,0) [above] {$\tiny k$}; +\node at (7,0) [above] {$\tiny k-1$}; +\node at (6,0) [above] {$\tiny k-2$}; +\node at (5,0) [above] {$\tiny k-3$}; +\foreach \x in {1,2,3,4,7,8,9}{ + \fill ({\x-1},-8) circle[radius=0.05]; +} +\foreach \x in {1,2,3,4}{ + \node at ({\x-1},-8) [below] {$\tiny \x$}; +} +\node at (6,-8) [below] {$k-2$}; +\node at (7,-8) [below] {$k-1$}; +\node at (8,-8) [below] {$k$}; + +\foreach \x in {3,3.2,...,5}{ + \fill (\x,{-8+\x}) circle[radius=0.02]; + \fill ({\x+0.5},-8) circle[radius=0.02]; + \fill ({\x-0.5},0) circle[radius=0.02]; +} + +\kreuz{8}{0}{\tau_{k-1,k}} +\kreuz{7}{-1}{\tau_{k-2,k-1}} +\kreuz{6}{-2}{\tau_{k-3,k-2}} +%\kreuz{5}{-3}{\tau_{56}} +%\kreuz{4}{-4}{\tau_{45}} +\kreuz{3}{-5}{\tau_{34}} +\kreuz{2}{-6}{\tau_{23}} +\kreuz{1}{-7}{\tau_{12}} + +\draw[->,color=gray] (0,0) -- (0,-7); +\draw[->,color=gray] (1,0) -- (1,-6); +\draw[->,color=gray] (2,0) -- (2,-5); +%\draw[->,color=gray] (3,0) -- (3,-4); +%\draw[->,color=gray] (4,0) -- (4,-3); +\draw[->,color=gray] (5,0) -- (5,-2); +\draw[->,color=gray] (6,0) -- (6,-1); + +\draw[->,color=gray] (8,-1) -- (8,-8); +\draw[->,color=gray] (7,-2) -- (7,-8); +\draw[->,color=gray] (6,-3) -- (6,-8); +%\draw[->,color=gray] (5,-4) -- (5,-8); +%\draw[->,color=gray] (4,-5) -- (4,-8); +\draw[->,color=gray] (3,-6) -- (3,-8); +\draw[->,color=gray] (2,-7) -- (2,-8); + +\fill (6,-1) circle[radius=0.05]; +\fill (7,-1) circle[radius=0.05]; +\fill (8,-1) circle[radius=0.05]; + +\fill (5,-2) circle[radius=0.05]; +\fill (6,-2) circle[radius=0.05]; +\fill (7,-2) circle[radius=0.05]; + +%\fill (4,-3) circle[radius=0.05]; +\fill (5,-3) circle[radius=0.05]; +\fill (6,-3) circle[radius=0.05]; + +%\fill (3,-4) circle[radius=0.05]; +%\fill (4,-4) circle[radius=0.05]; +%\fill (5,-4) circle[radius=0.05]; + +\fill (2,-5) circle[radius=0.05]; +\fill (3,-5) circle[radius=0.05]; +%\fill (4,-5) circle[radius=0.05]; + +\fill (1,-6) circle[radius=0.05]; +\fill (2,-6) circle[radius=0.05]; +\fill (3,-6) circle[radius=0.05]; + +\fill (0,-7) circle[radius=0.05]; +\fill (1,-7) circle[radius=0.05]; +\fill (2,-7) circle[radius=0.05]; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf b/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf new file mode 100644 index 0000000..72d9353 Binary files /dev/null and b/buch/chapters/50-permutationen/images/zyklenzerlegung.pdf differ diff --git a/buch/chapters/50-permutationen/images/zyklenzerlegung.tex b/buch/chapters/50-permutationen/images/zyklenzerlegung.tex new file mode 100644 index 0000000..c197579 --- /dev/null +++ b/buch/chapters/50-permutationen/images/zyklenzerlegung.tex @@ -0,0 +1,60 @@ +% +% zyklenzerlegung.tex -- Zerlegung einer Permutation in Zyklen +% +% (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} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope}[xshift=-3cm] +\node at (0,0) {$\displaystyle +\sigma=\begin{pmatrix} +{\color{red}1}&{\color{red}2}&{\color{darkgreen}3}&{\color{blue}4}&{\color{blue}5}&{\color{blue}6}\\ +{\color{red}2}&{\color{red}1}&{\color{darkgreen}3}&{\color{blue}5}&{\color{blue}6}&{\color{blue}4} +\end{pmatrix}$}; +\end{scope} +\node at (0,0) {$\mathstrut=\mathstrut$}; +\begin{scope}[xshift=1.5cm] +\coordinate (A) at (0,0.5); +\coordinate (B) at (0,-0.5); +\draw[->,color=red] (A) to[out=-20,in=20] (0,-0.5); +\draw[->,color=red] (B) to[out=160,in=-160] (0,0.5); +\node at (A) [above] {$\tiny 1$}; +\node at (B) [below] {$\tiny 2$}; +\fill (A) circle[radius=0.05]; +\fill (B) circle[radius=0.05]; + +\coordinate (C) at (1.5,0.25); +\node at (C) [above] {$\tiny 3$}; +\draw[->,color=darkgreen] ({1.5+0.01},0.25) to[out=-10,in=-170] ({1.5-0.01},0.25); +\draw[color=darkgreen] (1.5,{0.25-0.3}) circle[radius=0.3]; +\fill (C) circle[radius=0.05]; + +\def\r{0.5} +\coordinate (D) at ({3.5+\r*cos(90)},{0+\r*sin(90)}); +\coordinate (E) at ({3.5+\r*cos(210)},{0+\r*sin(210)}); +\coordinate (F) at ({3.5+\r*cos(330)},{0+\r*sin(330)}); +\node at (D) [above] {$\tiny 4$}; +\node at (E) [below left] {$\tiny 5$}; +\node at (F) [below right] {$\tiny 6$}; +\draw[->,color=blue] (D) to[out=180,in=120] (E); +\draw[->,color=blue] (E) to[out=-60,in=-120] (F); +\draw[->,color=blue] (F) to[out=60,in=0] (D); +\fill (D) circle[radius=0.05]; +\fill (E) circle[radius=0.05]; +\fill (F) circle[radius=0.05]; + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/50-permutationen/transpositionen.tex b/buch/chapters/50-permutationen/transpositionen.tex index baed2fb..604e010 100644 --- a/buch/chapters/50-permutationen/transpositionen.tex +++ b/buch/chapters/50-permutationen/transpositionen.tex @@ -35,92 +35,7 @@ also die Permutation, die $1\to2\to3\to\dots\to k-2\to k-1\to k\to 1$ abbildet. Dieser Zyklus lässt sich wie folgt aus Transpositionen zusammensetzen: \begin{center} -\def\kreuz#1#2#3{ - \draw[->] ({(#1)-1},#2) to[out=-90,in=90] ({#1},{#2-1}); - \draw[->] ({#1},#2) to[out=-90,in=90] ({#1-1},{#2-1}); - \node at ({(#1)-0.5+0.2},{#2-0.5}) [right] {$#3$}; -} -\begin{tikzpicture}[>=latex,thick] -\foreach \x in {1,2,3,6,7,8,9}{ - \fill ({\x-1},0) circle[radius=0.05]; -} -\foreach \x in {1,2,3}{ - \node at ({\x-1},0) [above] {$\tiny \x$}; -} -\node at (8,0) [above] {$\tiny k$}; -\node at (7,0) [above] {$\tiny k-1$}; -\node at (6,0) [above] {$\tiny k-2$}; -\node at (5,0) [above] {$\tiny k-3$}; -\foreach \x in {1,2,3,4,7,8,9}{ - \fill ({\x-1},-8) circle[radius=0.05]; -} -\foreach \x in {1,2,3,4}{ - \node at ({\x-1},-8) [below] {$\tiny \x$}; -} -\node at (6,-8) [below] {$k-2$}; -\node at (7,-8) [below] {$k-1$}; -\node at (8,-8) [below] {$k$}; - -\foreach \x in {3,3.2,...,5}{ - \fill (\x,{-8+\x}) circle[radius=0.02]; - \fill ({\x+0.5},-8) circle[radius=0.02]; - \fill ({\x-0.5},0) circle[radius=0.02]; -} - -\kreuz{8}{0}{\tau_{k-1,k}} -\kreuz{7}{-1}{\tau_{k-2,k-1}} -\kreuz{6}{-2}{\tau_{k-3,k-2}} -%\kreuz{5}{-3}{\tau_{56}} -%\kreuz{4}{-4}{\tau_{45}} -\kreuz{3}{-5}{\tau_{34}} -\kreuz{2}{-6}{\tau_{23}} -\kreuz{1}{-7}{\tau_{12}} - -\draw[->,color=gray] (0,0) -- (0,-7); -\draw[->,color=gray] (1,0) -- (1,-6); -\draw[->,color=gray] (2,0) -- (2,-5); -%\draw[->,color=gray] (3,0) -- (3,-4); -%\draw[->,color=gray] (4,0) -- (4,-3); -\draw[->,color=gray] (5,0) -- (5,-2); -\draw[->,color=gray] (6,0) -- (6,-1); - -\draw[->,color=gray] (8,-1) -- (8,-8); -\draw[->,color=gray] (7,-2) -- (7,-8); -\draw[->,color=gray] (6,-3) -- (6,-8); -%\draw[->,color=gray] (5,-4) -- (5,-8); -%\draw[->,color=gray] (4,-5) -- (4,-8); -\draw[->,color=gray] (3,-6) -- (3,-8); -\draw[->,color=gray] (2,-7) -- (2,-8); - -\fill (6,-1) circle[radius=0.05]; -\fill (7,-1) circle[radius=0.05]; -\fill (8,-1) circle[radius=0.05]; - -\fill (5,-2) circle[radius=0.05]; -\fill (6,-2) circle[radius=0.05]; -\fill (7,-2) circle[radius=0.05]; - -%\fill (4,-3) circle[radius=0.05]; -\fill (5,-3) circle[radius=0.05]; -\fill (6,-3) circle[radius=0.05]; - -%\fill (3,-4) circle[radius=0.05]; -%\fill (4,-4) circle[radius=0.05]; -%\fill (5,-4) circle[radius=0.05]; - -\fill (2,-5) circle[radius=0.05]; -\fill (3,-5) circle[radius=0.05]; -%\fill (4,-5) circle[radius=0.05]; - -\fill (1,-6) circle[radius=0.05]; -\fill (2,-6) circle[radius=0.05]; -\fill (3,-6) circle[radius=0.05]; - -\fill (0,-7) circle[radius=0.05]; -\fill (1,-7) circle[radius=0.05]; -\fill (2,-7) circle[radius=0.05]; - -\end{tikzpicture} +\includegraphics{chapters/50-permutationen/images/transpositionen.pdf} \end{center} Es ist also \[ diff --git a/buch/chapters/70-graphen/beschreibung.tex b/buch/chapters/70-graphen/beschreibung.tex index 6e8e59b..2dcc78f 100644 --- a/buch/chapters/70-graphen/beschreibung.tex +++ b/buch/chapters/70-graphen/beschreibung.tex @@ -236,26 +236,7 @@ sind. \begin{beispiel} \begin{figure} \centering -\begin{tikzpicture}[>=latex,thick] -\def\l{0.25} -\def\r{1} -\def\punkt#1{({\r*sin(((#1)-1)*72)},{\r*cos(((#1)-1)*72)})} -\def\R{2} -\def\Punkt#1{({\R*sin(((#1)-6)*72)},{\R*cos(((#1)-6)*72)})} -\draw \Punkt{6} -- \Punkt{7} -- \Punkt{8} -- \Punkt{9} -- \Punkt{10} -- cycle; -\draw \punkt{1} -- \punkt{3} -- \punkt{5} -- \punkt{2} -- \punkt{4} -- cycle; -\foreach \k in {1,...,5}{ - \draw \punkt{\k} -- \Punkt{(\k+5)}; - \fill[color=white] \punkt{\k} circle[radius=\l]; - \node at \punkt{\k} {$\k$}; - \draw \punkt{\k} circle[radius=\l]; -} -\foreach \k in {6,...,10}{ - \fill[color=white] \Punkt{\k} circle[radius=\l]; - \node at \Punkt{\k} {$\k$}; - \draw \Punkt{\k} circle[radius=\l]; -} -\end{tikzpicture} +\includegraphics{chapters/70-graphen/images/peterson.pdf} \caption{Peterson-Graph mit zehn Knoten. \label{buch:figure:peterson}} \end{figure} diff --git a/buch/chapters/70-graphen/images/Makefile b/buch/chapters/70-graphen/images/Makefile new file mode 100644 index 0000000..2199ddc --- /dev/null +++ b/buch/chapters/70-graphen/images/Makefile @@ -0,0 +1,10 @@ +# +# Makefile -- Bilder für Kapitel Graphen +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: peterson.pdf + +peterson.pdf: peterson.tex + pdflatex peterson.tex + diff --git a/buch/chapters/70-graphen/images/peterson.pdf b/buch/chapters/70-graphen/images/peterson.pdf new file mode 100644 index 0000000..259d808 Binary files /dev/null and b/buch/chapters/70-graphen/images/peterson.pdf differ diff --git a/buch/chapters/70-graphen/images/peterson.tex b/buch/chapters/70-graphen/images/peterson.tex new file mode 100644 index 0000000..c0af98b --- /dev/null +++ b/buch/chapters/70-graphen/images/peterson.tex @@ -0,0 +1,38 @@ +% +% 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\l{0.25} +\def\r{1} +\def\punkt#1{({\r*sin(((#1)-1)*72)},{\r*cos(((#1)-1)*72)})} +\def\R{2} +\def\Punkt#1{({\R*sin(((#1)-6)*72)},{\R*cos(((#1)-6)*72)})} +\draw \Punkt{6} -- \Punkt{7} -- \Punkt{8} -- \Punkt{9} -- \Punkt{10} -- cycle; +\draw \punkt{1} -- \punkt{3} -- \punkt{5} -- \punkt{2} -- \punkt{4} -- cycle; +\foreach \k in {1,...,5}{ + \draw \punkt{\k} -- \Punkt{(\k+5)}; + \fill[color=white] \punkt{\k} circle[radius=\l]; + \node at \punkt{\k} {$\k$}; + \draw \punkt{\k} circle[radius=\l]; +} +\foreach \k in {6,...,10}{ + \fill[color=white] \Punkt{\k} circle[radius=\l]; + \node at \Punkt{\k} {$\k$}; + \draw \Punkt{\k} circle[radius=\l]; +} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/80-wahrscheinlichkeit/google.tex b/buch/chapters/80-wahrscheinlichkeit/google.tex index 3616760..ca78b3d 100644 --- a/buch/chapters/80-wahrscheinlichkeit/google.tex +++ b/buch/chapters/80-wahrscheinlichkeit/google.tex @@ -41,46 +41,6 @@ zerstört. \begin{figure} \centering \includegraphics{chapters/80-wahrscheinlichkeit/images/internet.pdf} -%\begin{tikzpicture}[>=latex,thick] -%\foreach \x in {0,3,6,9}{ -% \foreach \y in {0,3}{ -% \fill[color=white] ({\x},{\y}) circle[radius=0.3]; -% \draw ({\x},{\y}) circle[radius=0.3]; -% } -%} -%\node at (0,3) {$1$}; -%\node at (0,0) {$2$}; -%\node at (3,3) {$3$}; -%\node at (3,0) {$4$}; -%\node at (6,3) {$5$}; -%\node at (6,0) {$6$}; -%\node at (9,3) {$7$}; -%\node at (9,0) {$8$}; -%% 1 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (0,3) -- (3,3); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (0,3) -- (0,0); -%% 2 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (0,0) to[out=-20,in=-160] (3,0); -%% 3 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (3,3) -- (6,3); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (3,3) -- (0,0); -%% 4 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (3,0) -- (6,3); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (3,0) -- (6,0); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (3,0) to[out=160,in=20] (0,0); -%% 5 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (6,3) -- (9,3); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (6,3) -- (9,0); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (6,3) -- (6,0); -%% 6 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (6,0) to[out=20,in=160] (9,0); -%% 7 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (9,3) .. controls (7.5,4) .. (6,4) -- (3,4) .. controls (1.5,4) .. (0,3); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (9,3) to[out=-110,in=110] (9,0); -%% 8 -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (9,0) to[out=-160,in=-20] (6,0); -%\draw[->,shorten >= 0.3cm, shorten <= 0.3cm] (9,0) to[out=70,in=-70] (9,3); -%\end{tikzpicture} \caption{Modell-Internet als Beispiel für die Link-Matrix und die Google-Matrix. \label{buch:figure:modellinternet}} \end{figure} diff --git a/buch/chapters/80-wahrscheinlichkeit/images/Makefile b/buch/chapters/80-wahrscheinlichkeit/images/Makefile index b04f6ff..8d34217 100644 --- a/buch/chapters/80-wahrscheinlichkeit/images/Makefile +++ b/buch/chapters/80-wahrscheinlichkeit/images/Makefile @@ -5,7 +5,8 @@ # all: dreieck.pdf trenn.pdf vergleich.pdf vergleich.jpg \ positiv.pdf positiv.jpg diffusion.png diffusion.pdf \ - konvex.pdf internet.pdf markov.pdf markov2.pdf markov3.pdf + konvex.pdf internet.pdf markov.pdf markov2.pdf markov3.pdf \ + spielB.pdf spielBtilde.pdf # Visualisierung diffusion in einer primitiven Matrix diffusion.pdf: diffusion.tex diffusion.jpg @@ -71,3 +72,8 @@ markov2.pdf: markov2.tex markov3.pdf: markov3.tex pdflatex markov3.tex +spielB.pdf: spielB.tex + pdflatex spielB.tex + +spielBtilde.pdf: spielBtilde.tex + pdflatex spielBtilde.tex diff --git a/buch/chapters/80-wahrscheinlichkeit/images/spielB.pdf b/buch/chapters/80-wahrscheinlichkeit/images/spielB.pdf new file mode 100644 index 0000000..466974d Binary files /dev/null and b/buch/chapters/80-wahrscheinlichkeit/images/spielB.pdf differ diff --git a/buch/chapters/80-wahrscheinlichkeit/images/spielB.tex b/buch/chapters/80-wahrscheinlichkeit/images/spielB.tex new file mode 100644 index 0000000..92989ed --- /dev/null +++ b/buch/chapters/80-wahrscheinlichkeit/images/spielB.tex @@ -0,0 +1,59 @@ +% +% spielB.tex -- Zutandsdiagramm für Spiel B +% +% (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\R{2} +\def\r{0.5} +\coordinate (A) at (0,\R); +\coordinate (B) at ({\R*sqrt(3)/2},{-0.5*\R}); +\coordinate (C) at ({-\R*sqrt(3)/2},{-0.5*\R}); + +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (B); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (C); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) -- (B); + +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=90,in=-30] (A); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) to[out=90,in=-150] (A); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=-150,in=-30] (C); + +\pgfmathparse{0.93*\R} +\xdef\Rgross{\pgfmathresult} + +\node at (30:\Rgross) {$\frac34$}; +\node at (150:\Rgross) {$\frac14$}; +\node at (-90:\Rgross) {$\frac14$}; + +\pgfmathparse{0.33*\R} +\xdef\Rklein{\pgfmathresult} + +\node at (-90:\Rklein) {$\frac34$}; +\node at (30:\Rklein) {$\frac9{10}$}; +\node at (150:\Rklein) {$\frac1{10}$}; + +\fill[color=white] (A) circle[radius=\r]; +\draw (A) circle[radius=\r]; +\node at (A) {$0$}; + +\fill[color=white] (B) circle[radius=\r]; +\draw (B) circle[radius=\r]; +\node at (B) {$2$}; + +\fill[color=white] (C) circle[radius=\r]; +\draw (C) circle[radius=\r]; +\node at (C) {$1$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.pdf b/buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.pdf new file mode 100644 index 0000000..7812c9c Binary files /dev/null and b/buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.pdf differ diff --git a/buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.tex b/buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.tex new file mode 100644 index 0000000..b2d4b01 --- /dev/null +++ b/buch/chapters/80-wahrscheinlichkeit/images/spielBtilde.tex @@ -0,0 +1,59 @@ +% +% spielBtilde.tex -- Zustandsdiagramm des modifzierten Spiels +% +% (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\R{2.5} +\def\r{0.5} +\coordinate (A) at (0,\R); +\coordinate (B) at ({\R*sqrt(3)/2},{-0.5*\R}); +\coordinate (C) at ({-\R*sqrt(3)/2},{-0.5*\R}); + +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (B); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (C); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) -- (B); + +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=90,in=-30] (A); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) to[out=90,in=-150] (A); +\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=-150,in=-30] (C); + +\pgfmathparse{0.93*\R} +\xdef\Rgross{\pgfmathresult} + +\node at (30:\Rgross) {$\frac34-\varepsilon$}; +\node at (150:\Rgross) {$\frac14+\varepsilon$}; +\node at (-90:\Rgross) {$\frac14+\varepsilon$}; + +\pgfmathparse{0.32*\R} +\xdef\Rklein{\pgfmathresult} + +\node at (-90:\Rklein) {$\frac34-\varepsilon$}; +\node at (30:\Rklein) {$\frac9{10}+\varepsilon$}; +\node at (150:\Rklein) {$\frac1{10}-\varepsilon$}; + +\fill[color=white] (A) circle[radius=\r]; +\draw (A) circle[radius=\r]; +\node at (A) {$0$}; + +\fill[color=white] (B) circle[radius=\r]; +\draw (B) circle[radius=\r]; +\node at (B) {$2$}; + +\fill[color=white] (C) circle[radius=\r]; +\draw (C) circle[radius=\r]; +\node at (C) {$1$}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf index c6173ce..b7215b4 100644 Binary files a/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf and b/buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf differ diff --git a/buch/chapters/80-wahrscheinlichkeit/parrondo.tex b/buch/chapters/80-wahrscheinlichkeit/parrondo.tex index 3bdba9a..a62d813 100644 --- a/buch/chapters/80-wahrscheinlichkeit/parrondo.tex +++ b/buch/chapters/80-wahrscheinlichkeit/parrondo.tex @@ -61,48 +61,7 @@ Fälle recht gross, in einem Fall aber sehr klein. \subsubsection{Übergangsmatrix im Spiel $B$} \begin{figure} \centering -\begin{tikzpicture}[>=latex,thick] -\def\R{2} -\def\r{0.5} -\coordinate (A) at (0,\R); -\coordinate (B) at ({\R*sqrt(3)/2},{-0.5*\R}); -\coordinate (C) at ({-\R*sqrt(3)/2},{-0.5*\R}); - -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (B); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (C); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) -- (B); - -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=90,in=-30] (A); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) to[out=90,in=-150] (A); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=-150,in=-30] (C); - -\pgfmathparse{0.93*\R} -\xdef\Rgross{\pgfmathresult} - -\node at (30:\Rgross) {$\frac34$}; -\node at (150:\Rgross) {$\frac14$}; -\node at (-90:\Rgross) {$\frac14$}; - -\pgfmathparse{0.33*\R} -\xdef\Rklein{\pgfmathresult} - -\node at (-90:\Rklein) {$\frac34$}; -\node at (30:\Rklein) {$\frac9{10}$}; -\node at (150:\Rklein) {$\frac1{10}$}; - -\fill[color=white] (A) circle[radius=\r]; -\draw (A) circle[radius=\r]; -\node at (A) {$0$}; - -\fill[color=white] (B) circle[radius=\r]; -\draw (B) circle[radius=\r]; -\node at (B) {$2$}; - -\fill[color=white] (C) circle[radius=\r]; -\draw (C) circle[radius=\r]; -\node at (C) {$1$}; - -\end{tikzpicture} +\includegraphics{chapters/80-wahrscheinlichkeit/images/spielB.pdf} \caption{Zustandsdiagramm für das Spiel $B$, Zustände sind die Dreierreste des Kapitals. \label{buch:wahrscheinlichkeit:fig:spielB}} @@ -454,51 +413,10 @@ U^t (G\odot B) p \] wie erwartet. -\subsubsection{Das modifizierte Spiel $B$} +\subsubsection{Das modifizierte Spiel $\tilde{B}$} \begin{figure} \centering -\begin{tikzpicture}[>=latex,thick] -\def\R{2.5} -\def\r{0.5} -\coordinate (A) at (0,\R); -\coordinate (B) at ({\R*sqrt(3)/2},{-0.5*\R}); -\coordinate (C) at ({-\R*sqrt(3)/2},{-0.5*\R}); - -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (B); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (A) -- (C); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) -- (B); - -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=90,in=-30] (A); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (C) to[out=90,in=-150] (A); -\draw[->,shorten >= 0.5cm,shorten <= 0.5cm] (B) to[out=-150,in=-30] (C); - -\pgfmathparse{0.93*\R} -\xdef\Rgross{\pgfmathresult} - -\node at (30:\Rgross) {$\frac34-\varepsilon$}; -\node at (150:\Rgross) {$\frac14+\varepsilon$}; -\node at (-90:\Rgross) {$\frac14+\varepsilon$}; - -\pgfmathparse{0.32*\R} -\xdef\Rklein{\pgfmathresult} - -\node at (-90:\Rklein) {$\frac34-\varepsilon$}; -\node at (30:\Rklein) {$\frac9{10}+\varepsilon$}; -\node at (150:\Rklein) {$\frac1{10}-\varepsilon$}; - -\fill[color=white] (A) circle[radius=\r]; -\draw (A) circle[radius=\r]; -\node at (A) {$0$}; - -\fill[color=white] (B) circle[radius=\r]; -\draw (B) circle[radius=\r]; -\node at (B) {$2$}; - -\fill[color=white] (C) circle[radius=\r]; -\draw (C) circle[radius=\r]; -\node at (C) {$1$}; - -\end{tikzpicture} +\includegraphics{chapters/80-wahrscheinlichkeit/images/spielBtilde.pdf} \caption{Zustandsdiagramm für das modifizerte Spiel $\tilde{B}$, Zustände sind die Dreierreste des Kapitals. Gegenüber dem Spiel $B$ diff --git a/buch/chapters/95-homologie/chapter.tex b/buch/chapters/95-homologie/chapter.tex index 95ecb79..eaa56c4 100644 --- a/buch/chapters/95-homologie/chapter.tex +++ b/buch/chapters/95-homologie/chapter.tex @@ -14,42 +14,7 @@ Es ist zum Beispiel nicht möglich, ein Dreieck vom Rand eines Dreiecks zu unterscheiden~\ref{buch:homologie:figure:zusammenziehbar}. \begin{figure} \centering -\begin{tikzpicture}[>=latex,thick] -\def\punkt#1{ - \fill[color=white] #1 circle[radius=0.07]; - \draw #1 circle[radius=0.07]; -} -\begin{scope}[xshift=3cm] -\draw[->] (0,0) -- (3,3); -\draw[->] (0,0) -- (4,1); -\draw[->] (4,1) -- (3,3); -\node at (0,0) [below left] {$P_0$}; -\node at (4,1) [below right] {$P_1$}; -\node at (3,3) [above] {$P_2$}; -\punkt{(0,0)} -\punkt{(4,1)} -\punkt{(3,3)} -\node at (2,0.5) [below] {$k_{01}$}; -\node at (1.5,1.5) [above left] {$k_{02}$}; -\node at (3.5,2) [right] {$k_{12}$}; -\end{scope} -\begin{scope}[xshift=-3cm] -\fill[color=gray!40] (0,0) -- (4,1) -- (3,3) -- cycle; -\draw[->] (0,0) -- (3,3); -\draw[->] (0,0) -- (4,1); -\draw[->] (4,1) -- (3,3); -\node at (0,0) [below left] {$P_0$}; -\node at (4,1) [below right] {$P_1$}; -\node at (3,3) [above] {$P_2$}; -\node at (2,0.5) [below] {$k_{01}$}; -\node at (1.5,1.5) [above left] {$k_{02}$}; -\node at (3.5,2) [right] {$k_{12}$}; -\node at (2.333,1.333) {$\triangle$}; -\punkt{(0,0)} -\punkt{(4,1)} -\punkt{(3,3)} -\end{scope} -\end{tikzpicture} +\includegraphics{chapters/95-homologie/images/dreieck.pdf} \caption{Ein Dreieck $\triangle$ (rechts) und der Rand des Dreicks (links) sind mit den Methoden der Graphentheorie nicht unterschiedbar. diff --git a/buch/chapters/95-homologie/images/Makefile b/buch/chapters/95-homologie/images/Makefile new file mode 100644 index 0000000..82f1285 --- /dev/null +++ b/buch/chapters/95-homologie/images/Makefile @@ -0,0 +1,10 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: dreieck.pdf + +dreieck.pdf: dreieck.tex + pdflatex dreieck.tex + diff --git a/buch/chapters/95-homologie/images/dreieck.pdf b/buch/chapters/95-homologie/images/dreieck.pdf new file mode 100644 index 0000000..e5d360a Binary files /dev/null and b/buch/chapters/95-homologie/images/dreieck.pdf differ diff --git a/buch/chapters/95-homologie/images/dreieck.tex b/buch/chapters/95-homologie/images/dreieck.tex new file mode 100644 index 0000000..1c28bc7 --- /dev/null +++ b/buch/chapters/95-homologie/images/dreieck.tex @@ -0,0 +1,54 @@ +% +% dreieck.tex -- Dreieck und Simplex +% +% (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\punkt#1{ + \fill[color=white] #1 circle[radius=0.07]; + \draw #1 circle[radius=0.07]; +} +\begin{scope}[xshift=3cm] +\draw[->] (0,0) -- (3,3); +\draw[->] (0,0) -- (4,1); +\draw[->] (4,1) -- (3,3); +\node at (0,0) [below left] {$P_0$}; +\node at (4,1) [below right] {$P_1$}; +\node at (3,3) [above] {$P_2$}; +\punkt{(0,0)} +\punkt{(4,1)} +\punkt{(3,3)} +\node at (2,0.5) [below] {$k_{01}$}; +\node at (1.5,1.5) [above left] {$k_{02}$}; +\node at (3.5,2) [right] {$k_{12}$}; +\end{scope} +\begin{scope}[xshift=-3cm] +\fill[color=gray!40] (0,0) -- (4,1) -- (3,3) -- cycle; +\draw[->] (0,0) -- (3,3); +\draw[->] (0,0) -- (4,1); +\draw[->] (4,1) -- (3,3); +\node at (0,0) [below left] {$P_0$}; +\node at (4,1) [below right] {$P_1$}; +\node at (3,3) [above] {$P_2$}; +\node at (2,0.5) [below] {$k_{01}$}; +\node at (1.5,1.5) [above left] {$k_{02}$}; +\node at (3.5,2) [right] {$k_{12}$}; +\node at (2.333,1.333) {$\triangle$}; +\punkt{(0,0)} +\punkt{(4,1)} +\punkt{(3,3)} +\end{scope} + +\end{tikzpicture} +\end{document} + -- cgit v1.2.1