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Diffstat (limited to '')
| -rw-r--r-- | buch/buch.fdb_latexmk | 537 | ||||
| -rw-r--r-- | buch/buch.fls | 1033 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/fixpunkte.tex | 87 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/images/Makefile | 5 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/images/polyeder.pdf | bin | 0 -> 3270 bytes | |||
| -rw-r--r-- | buch/chapters/95-homologie/images/polyeder.tex | 109 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/mayervietoris.tex | 28 | ||||
| -rw-r--r-- | buch/chapters/95-homologie/simplex.tex | 129 | ||||
| -rw-r--r-- | buch/papers/erdbeben/teil0.tex | 2 | ||||
| -rw-r--r-- | buch/papers/verkehr/section1.tex | 16 |
10 files changed, 321 insertions, 1625 deletions
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c:/JB/LaTex/SeminarMatrizen/buch/chapters/60-gruppen/images/sl2.pdf -INPUT c:/JB/LaTex/SeminarMatrizen/buch/chapters/60-gruppen/images/sl2.pdf -INPUT c:/JB/LaTex/SeminarMatrizen/buch/chapters/60-gruppen/images/sl2.pdf -INPUT c:/JB/LaTex/SeminarMatrizen/buch/chapters/60-gruppen/images/scherungen.pdf -INPUT c:/JB/LaTex/SeminarMatrizen/buch/chapters/60-gruppen/images/scherungen.pdf -INPUT c:/JB/LaTex/SeminarMatrizen/buch/chapters/60-gruppen/images/scherungen.pdf diff --git a/buch/chapters/95-homologie/fixpunkte.tex b/buch/chapters/95-homologie/fixpunkte.tex index 1ed51ef..a03d4b5 100644 --- a/buch/chapters/95-homologie/fixpunkte.tex +++ b/buch/chapters/95-homologie/fixpunkte.tex @@ -11,15 +11,78 @@ selbst gehört die zugehörige lineare Abbildung $f_*\colon H_*(X)\to H_*(X)$ der Homologiegruppen. Diese linearen Abbildungen sind im Allgemeinen viel einfacher zu analysieren. -Zum Beispiel soll in Abschnitt~\ref{buch:subsection:lefshetz} -die Lefshetz-Spurformel abgeleitet werden, die eine Aussagen darüber -ermöglicht, ob eine Abbildung einen Fixpunkt haben kann. -In Abschnitt~\ref{buch:subsection:brower} wird gezeigt wie man damit -den Browerschen Fixpunktsatz beweisen kann, der besagt, dass jede -Abbildung eines Einheitsballs in sich selbst immer einen Fixpunkt hat. - -\subsection{Lefshetz-Spurformel -\label{buch:subsection:lefshetz}} - -\subsection{Brower-Fixpunktsatz -\label{buch:subsection:brower}} +%Zum Beispiel soll in Abschnitt~\ref{buch:subsection:lefshetz} +%die Lefshetz-Spurformel abgeleitet werden, die eine Aussagen darüber +%ermöglicht, ob eine Abbildung einen Fixpunkt haben kann. +%In Abschnitt~\ref{buch:subsection:brower} wird gezeigt wie man damit +%den Browerschen Fixpunktsatz beweisen kann, der besagt, dass jede +%Abbildung eines Einheitsballs in sich selbst immer einen Fixpunkt hat. + +%\subsection{Brower-Fixpunktsatz +%\label{buch:subsection:brower}} +% +%\begin{satz}[Brower] +%\end{satz} + +%\subsection{Lefshetz-Fixpunktsatz +%\label{buch:subsection:lefshetz}} +Eine Selbstabbildung $f_*\colon C_*\to C_*$ von Kettenkomplexen führt auf +eine Selbstabbiludng der Homologiegruppen $H(f)\colon H(C)\to H(C)$. +Da sowohl $H_k$ wie auch $C_k$ endlichdimensionale Vektorräume sind, +ist die Spur von $H_k(f)$ wohldefiniert. + +\begin{definition} +Die {\em Lefshetz-Zahl} einer Abbildung $f$ von Kettenkomplexen ist +\[ +\lambda(f) += +\sum_{k=0}^\infty +(-1)^k \operatorname{Spur}f_k += +\sum_{k=0}^\infty +(-1)^k \operatorname{Spur}(H_k(f)). +\] +\end{definition} + +Die zweite Darstellung der Lefshetz-Zahl auf der rechten Seite ist +meistens viel leichter zu berechnen als die erste. +Die einzelnen Vektorräume eines Kettenkomplexes können haben typischerweise +eine hohe Dimension, so hoch wie die Anzahl der Simplizes der Triangulation. +Die Homologiegruppen dagegen haben typischerweise sehr viel kleinere +Dimension, die Matrizen $H_k(F)$ sind also relativ klein. +Es ist aber nicht klar, dass beide Berechnungsmethoden für die +Lefshetz-Zahl auf das gleiche Resultat führen müssen. + +\begin{proof}[Beweis] +\end{proof} + +Die Lefshetz-Zahl ist eine Invariante einer topologischen Abbildung, +die Aussagen über Fixpunkte zu machen erlaubt. + +\begin{satz} +Ist $f\colon X\to X$ eine Selbstabbildung eines kompakten Polyeders und +ist $\lambda(f) \ne 0$, dann hat $f$ einen Fixpunkt. +\end{satz} + +Im Folgenden soll nur ein heuristisches Argument gegeben werden, warum +ein solcher Satz wahr sein könnte. + +Wenn eine Abbildung keinen Fixpunkt hat, dann ist $f(x) \ne x$ für alle +Punkte von $X$. +Da $X$ kompakt ist, gibt es einen minimalen Abstand $d$ zwischen $f(x)$ und $x$. +Wenn man also für $X$ eine Triangulation wählt, die wesentlich feiner ist +als dieser minimale Abstand, dann wird kein Simplex der Triangulation auf +Punkte im selben Simplex oder in einem Nachbarsimplex abgebildet wird. +Indem man nötigenfalls die Triangulation nochmals verfeinert, kann man auch +genügend Platz schaffen, dass man die Abbildung $f$ etwas modifizieren kann, +so dass auch die deformierte Abbildung immer noch diese Eigenschaft hat. + +Die zugehörige Abbildung des Kettenkomplexes der Triangulation hat damit +die Eigenschaft, dass kein Basisvektor auf sich selbst abgebildet wird. +Die Matrix der Abbildung hat daher keine Nullen auf der Diagonalen, und +damit ist auch die Spur dieser Abbildung Null: $\operatorname{Spur}(H_k(f))=0$ +für alle $k$. +Erst recht ist die Lefshetz-Zahl $\lambda(f)=0$. +Wenn also die Lefshetz-Zahl verschieden ist von Null, dann muss $f$ +notwendigerweise einen Fixpunkt haben. + diff --git a/buch/chapters/95-homologie/images/Makefile b/buch/chapters/95-homologie/images/Makefile index 82f1285..ac964ff 100644 --- a/buch/chapters/95-homologie/images/Makefile +++ b/buch/chapters/95-homologie/images/Makefile @@ -3,8 +3,11 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: dreieck.pdf +all: dreieck.pdf polyeder.pdf dreieck.pdf: dreieck.tex pdflatex dreieck.tex +polyeder.pdf: polyeder.tex + pdflatex polyeder.tex + diff --git a/buch/chapters/95-homologie/images/polyeder.pdf b/buch/chapters/95-homologie/images/polyeder.pdf Binary files differnew file mode 100644 index 0000000..3a8ba60 --- /dev/null +++ b/buch/chapters/95-homologie/images/polyeder.pdf diff --git a/buch/chapters/95-homologie/images/polyeder.tex b/buch/chapters/95-homologie/images/polyeder.tex new file mode 100644 index 0000000..9a900cc --- /dev/null +++ b/buch/chapters/95-homologie/images/polyeder.tex @@ -0,0 +1,109 @@ +% +% 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,calc} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +% add image content here +\begin{scope}[xshift=-3.5cm,scale=0.5] +\coordinate (A) at (0,0); +\coordinate (B) at (4,0); +\coordinate (C) at (5,-2); +\coordinate (D) at (8,-1); +\coordinate (E) at (7,1); +\coordinate (F) at (7,3); +\coordinate (G) at (1,3); +\coordinate (H) at (5,4); +\coordinate (I) at (9,5); +\coordinate (J) at (4,7); +\coordinate (K) at (-1,9); +\coordinate (L) at (7,11); +\coordinate (M) at (6,-0.5); + +\fill[color=gray,opacity=0.5] (A)--(B)--(H)--(G)--cycle; +\fill[color=gray,opacity=0.5] (G)--(I)--(K)--cycle; +\fill[color=gray,opacity=0.5] (G)--(L)--(K)--cycle; + +\draw (K)--(G)--(A)--(B)--(D); +\draw (C)--(E); +\draw (G)--(I)--(K); +\draw (G)--(L)--(K); +\draw (B)--(H); +\draw (B)--(F); + +\fill (A) circle[radius=0.1]; +\fill (B) circle[radius=0.1]; +\fill (C) circle[radius=0.1]; +\fill (D) circle[radius=0.1]; +\fill (E) circle[radius=0.1]; +\fill (F) circle[radius=0.1]; +\fill (G) circle[radius=0.1]; +\fill (H) circle[radius=0.1]; +\fill (I) circle[radius=0.1]; +%\fill (J) circle[radius=0.1]; +\fill (K) circle[radius=0.1]; +\fill (L) circle[radius=0.1]; +%\fill (M) circle[radius=0.1]; + +\draw[color=red] (H) circle[radius=0.5]; +\draw[color=red] (J) circle[radius=0.5]; +\draw[color=red] (M) circle[radius=0.5]; +\draw[color=red] ($0.25*(A)+0.25*(B)+0.25*(G)+0.25*(H)$) circle[radius=0.5]; + +\end{scope} + +\begin{scope}[xshift=3.5cm,scale=0.5] +\coordinate (A) at (0,0); +\coordinate (B) at (4,0); +\coordinate (C) at (5,-2); +\coordinate (D) at (8,-1); +\coordinate (E) at (7,1); +\coordinate (F) at (7,3); +\coordinate (G) at (1,3); +\coordinate (H) at (5,4); +\coordinate (I) at (9,5); +\coordinate (J) at (4,7); +\coordinate (K) at (-1,9); +\coordinate (L) at (7,11); +\coordinate (M) at (6,-0.5); + +\fill[color=gray!50] (A)--(B)--(H)--(I)--(J)--(L)--(K)--(G)--cycle; + +\draw (K)--(G)--(A)--(B)--(D); +\draw (C)--(E); +\draw (G)--(I)--(K); +\draw (G)--(L)--(K); +\draw (B)--(H); +\draw (B)--(F); +\draw (H)--(J); +\draw (A)--(H); + +\fill (A) circle[radius=0.1]; +\fill (B) circle[radius=0.1]; +\fill (C) circle[radius=0.1]; +\fill (D) circle[radius=0.1]; +\fill (E) circle[radius=0.1]; +\fill (F) circle[radius=0.1]; +\fill (G) circle[radius=0.1]; +\fill (H) circle[radius=0.1]; +\fill (I) circle[radius=0.1]; +\fill (J) circle[radius=0.1]; +\fill (K) circle[radius=0.1]; +\fill (L) circle[radius=0.1]; +\fill (M) circle[radius=0.1]; + +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/95-homologie/mayervietoris.tex b/buch/chapters/95-homologie/mayervietoris.tex deleted file mode 100644 index 57105f8..0000000 --- a/buch/chapters/95-homologie/mayervietoris.tex +++ /dev/null @@ -1,28 +0,0 @@ -% -% mayervietoris.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\section{Exaktheit und die Mayer-Vietoris-Folge -\label{buch:section:mayervietoris}} -\rhead{Exaktheit und die Mayer-Vietoris-Folge} -Die Berechnung der Homologie-Gruppen ist zwar im Wesentlichen ein -kombinatorisches Problem, trotzdem ist eher aufwändig. -Oft weiss man, wie sich toplogische Räume aus einfacheren Räumen -zusammensetzen lassen. -Eine Mannigkfaltigkeit zum Beispiel wird durch die Karten -definiert, also zusammenziehbare Teilmengen von $\mathbb{R}^n$, -die die Mannigkfaltigkeit überdecken. -Das Ziel dieses Abschnittes ist, Regeln zusammenzustellen, mit denen -man die Homologie eines solchen zusammengesetzten Raumes aus der -Homologie der einzelnen Teile und aus den ``Verklebungsabbildungen'', -die die Teile verbinden, zu berechnen. - -\subsection{Kurze exakte Folgen von Kettenkomplexen -\label{buch:subsection:exaktefolgen}} - -\subsection{Schlangenlemma und lange exakte Folgen -\label{buch:subsection:schlangenlemma}} - -\subsection{Mayer-Vietoris-Folge -\label{buch:subsection:mayervietoris}} diff --git a/buch/chapters/95-homologie/simplex.tex b/buch/chapters/95-homologie/simplex.tex index 397ba07..0cf4aa7 100644 --- a/buch/chapters/95-homologie/simplex.tex +++ b/buch/chapters/95-homologie/simplex.tex @@ -1,17 +1,17 @@ % -% simplex.tex -- simplizes und simpliziale Komplexe +% simplex.tex -- simplizes und Polyeder % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % -\section{Simplexe und simpliziale Komplexe +\section{Simplices \label{buch:section:simplexe}} -\rhead{Simplexe und simpliziale Komplexe} +\rhead{Simplices} Die Idee, das Dreieck und seinen Rand zu unterscheiden verlangt, dass wir zunächst Dreiecke und deren höherdimensionale Verallgemeinerungen, die sogenannten Simplizes entwickeln müssen. -\subsection{Simplexe und Rand -\label{buch:subsection:simplexe}} +\subsection{Simplices und Rand +\label{buch:subsection:simplices}} \subsubsection{Rand eines Dreiecks} Die Inzidenz-Matrix eines Graphen hat einer Kante die beiden Endpunkte @@ -231,8 +231,127 @@ Vorzeichen zu, die Matrix ist \] \end{definition} +\subsection{Polyeder} +\begin{figure} +\centering +\includegraphics{chapters/95-homologie/images/polyeder.pdf} +\caption{Aufbau eines zweidimensionalen Polyeders aus +verschiedenen Simplizes. +Die Schnittmenge zweier Simplizes muss ein Untersimplex beider Simplizes +sein. +Die roten Kreise im linken Bild weisen auf verschiedene Situationen +hin, wo das diese Bedingung nicht erfüllt ist. +In rechten Bild sind zusätzliche Simlizes hinzugefügt worden, um +die Bedingungen eines Polyeders zu erfüllen. +\label{buch:homologie:figure:polyeder}} +\end{figure} +Aus einzelnen Simplizes können jetzt kompliziertere geometrische +Objekte gebaut werden. +Ein Graph ist ein Beispiel für ein geometrisches Objekt, welches +als Vereinigung von 1-Simplizes entsteht. +Die Vereinigung ist aber nicht beliebig, vielmehr ist die Schnittmenge +zweier beliebiger 1-Simplizes immer entweder leer, eine Menge +mit nur einem Vertex oder ein ganzes 1-Simplex. + +Dies reicht aber nicht, wie Abbildung~\ref{buch:homologie:polyeder} +zeigt. +In einem Graphen dürfen sich Kanten nicht in einem inneren Punkt treffen, +sondern nur in Endpunkten. +Verallgemeinert auf höherdimensionale Simplizes kann man dies als die +Bedingung formulieren, dass die Schnittmenge zweier beliebiger +Simplizes immer Untersimplizes beider Simplizes sein müssen. +Wir fassen dies zusammen in der folgenden Definition. + +\begin{definition} +\index{Polyeder}% +\index{Dimension eines Polyeders}% +\index{Polyeder, Dimension eines}% +Ein {\em Polyeder} ist eine Vereingung von endlich vielen Simplizes derart, +dass die Schnittmenge zweier beliebiger Simplizes immer ein Untersimplex +beider Simplizes ist. +Die {\em Dimension} des Polyeders ist die grösste Dimension der darin +enthaltenen Simplizes. +\end{definition} + +Ein Graph ist nach dieser Definition ein eindimensionales Polyeder. +Die Mengen in der Abbildung~\ref{buch:homologie:figure:polyeder} +ist kein Polyeder, kann aber leicht zu einem Polyeder gemacht werden, +indem man einzelne Kanten mit zusätzlichen Punkten unterteilt. +Auch müssen die zweidimensionalen Simplizes aufgeteilt werden. + +Die Abbildung~\ref{buch:homologie:figure:polyeder} zeigt auch, dass +die Darstellung einer Punktmenge als Polyeder nicht eindeutig ist. +Man kann die Kanten und Flächen jederzeit weiter unterteilen, ohne +dass sich die Gestalt der gesamten Menge dadurch ändert. \subsection{Triangulation \label{buch:subsection:triangulation}} +Unser Ziel ist, geometrische Objekte besser verstehen zu können. +Dabei sind uns Deformationen ja sogar Knicke egal, es interessiert uns +nur die ``Gestalt'' des Objekts. +Entfernungen zwischen Punkten sind ebenfalls von untergeordneter +Bedeutung, da sie bei Deformation nicht erhalten bleiben. +Der Begriff des ``topologischen Raumes'' fasst diese Ideen mathematisch +präzise ein, eine genaue Definition würde aber an dieser Stelle zu weit +führen. +Stattdessen beschränken wir uns auf eine Klasse von Punktmengen, die man +mit Simplizes beschreiben kann. + +Ein topologischer Raum zeichnet sich durch einen Nachbarschaftsbegriff +von Punkte aus, der erlaubt zu definieren, was eine stetige Abbildung ist. +Ein stetige Abbildungen bildet nahe beeinander liegende Punkte wieder +auf nahe beeinander liegende Punkte ab. +Dass nahe liegende Punkte nicht plötzlich auf weit auseinander liegende +Punkte abgebildet werden gibt die Intuition wieder, dass Deformationen +möglich sein sollen, dass der Raum dabei aber nicht ``reissen'' darf. +Zwei topologische Räume $X$ und $Y$ können daher als ``gleichgestaltig'' +betrachtet werden, wenn es zwei stetige Abbildungen $f\colon X\to Y$ +und $g\colon Y\to X$ gibt, die zu einander invers sein. +Oder wenn sich $X$ stetig auf $Y$ abbilden lässt, so dass auch die +Umkehrabbildung stetig ist. +Eine solche Abbildung heisst ein {\em Homöomorphismus}, die beiden Räume +$X$ und $Y$ heissen {\em homomorph}. + +Eine Kugel ist natürlich kein Polyeder, aber sie kann leicht homöomorph +auf ein dreidimensionales Simplex abgebildet werden. + +\begin{beispiel} +Sei $T$ ein reguläres Tetraeder mit den Ecken auf der dreidimensionalen +Einheitskugel $B^3$. +Für jeden Richtungsvektor $x\ne 0$ sei $l(x)$ Entfernung vom Mittelpunkt des +Tetraeders bis zum Durchstosspunkt einer Geraden durch den Mittelpunkt +mit Richtungsvektor $x$ durch die Oberfläche des Tetraeders. +Dann sind die Abbildungen +\[ +f\colon +T\to B^3 +: +x \mapsto\begin{cases} +\displaystyle +\frac{x}{l(x)}&\quad\text{für $x\ne 0$}\\ +0&\quad\text{für $x=0$} +\end{cases} +\qquad\text{und}\qquad +g\colon +B^3\to T +: +x \mapsto\begin{cases} +l(x) x&\quad\text{für $x\ne 0$}\\ +0&\quad\text{für $x=0$} +\end{cases} +\] +zueinander inverse stetige Abbildungen oder Homöomorphismen. +\end{beispiel} + +Im Folgenden sollen daher nur solche topologischen Räume untersucht werden, +die homöomorph sind zu einem Polyeder. +Man nennt die homöomorphe Abbildung eines Polyeders auf so einen Raum +auch eine Triangulation. +Durch Unterteilung der Simplizes in kleiner Simplizes kann eine solche +Triangulation beliebig verfeinert werden. + + + + diff --git a/buch/papers/erdbeben/teil0.tex b/buch/papers/erdbeben/teil0.tex index c099340..c985713 100644 --- a/buch/papers/erdbeben/teil0.tex +++ b/buch/papers/erdbeben/teil0.tex @@ -74,7 +74,7 @@ Dazu wird ein Zustandsvektor definiert: \[ \left(\begin{array}{c} {s_1} \\ {s_2} \\ {f} \end{array}\right). \] -Durch Rücksubstituion ergibt sich uns folgende Systemgleichung in Matrix schreibweise, , wobei $\sot {s_1}= v$ ist: +Durch Rücksubstituion ergibt sich uns folgende Systemgleichung in Matrix schreibweise, , wobei $\dot {s_1}= v$ ist: \begin{equation} \frac{d}{dt} \left(\begin{array}{c} s(t) \\ v(t) \\ f(t) \end{array}\right) = \left( \begin{array}{ccc} diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex index 416e311..6ac86ad 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -84,15 +84,15 @@ Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseina -Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1\dot n\right\}\end{equation} +Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1\dots n\right\}\end{equation} Beim PageRank-Algorithmus wird eine abgewandelte Form der Adjazenz-Matrix verwendet. -Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt. -\[ P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \] +Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt: +\( P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \) Anschliessend multipliziert man diese Matrix $P$ mit einem Spaltenvektor $\Vec{r_0}$ mit $n$ Einträgen, für welchen gilt: -\[ \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1\dot n\right\} \] +\( \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1\dots n\right\} \) Dieser Vektor stellt ein neutrales Ranking dar. Alle Knoten werden gleich gewichtet. -Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das ``erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt. -\[ \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\] -somit -\begin{equation} \Vec{r_i} = P^i\cdot\Vec{r_0}\end{equation} +Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das ``erste'' Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt: +\( \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\) +und somit allgemein: +\( \Vec{r_i} = P^i\cdot\Vec{r_0}\) Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ der das abschliessende Ranking darstellt. |