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diff --git a/vorlesungen/slides/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc new file mode 100644 index 0000000..a6bb320 --- /dev/null +++ b/vorlesungen/slides/0/Makefile.inc @@ -0,0 +1,17 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 20920 Prof Dr Andreas Müller, Hochschule Rapperswil +# +chapter0 = \ + ../slides/0/was.tex \ + ../slides/0/intro.tex \ + ../slides/0/resourcen.tex \ + ../slides/0/latextipps.tex \ + ../slides/0/nextsteps.tex \ + ../slides/0/themen1.tex \ + ../slides/0/themen2.tex \ + ../slides/0/themen3.tex \ + ../slides/0/chapter.tex + diff --git a/vorlesungen/slides/0/chapter.tex b/vorlesungen/slides/0/chapter.tex new file mode 100644 index 0000000..6e09557 --- /dev/null +++ b/vorlesungen/slides/0/chapter.tex @@ -0,0 +1,13 @@ +% +% chapter.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{0/intro.tex} +\folie{0/was.tex} +\folie{0/resourcen.tex} +\folie{0/latextipps.tex} +\folie{0/themen1.tex} +\folie{0/themen2.tex} +\folie{0/themen3.tex} +\folie{0/nextsteps.tex} diff --git a/vorlesungen/slides/0/intro.tex b/vorlesungen/slides/0/intro.tex new file mode 100644 index 0000000..acda6d1 --- /dev/null +++ b/vorlesungen/slides/0/intro.tex @@ -0,0 +1,98 @@ +% +% intro.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\r{4} + +\def\rad#1{ +\begin{scope}[rotate=#1] +\fill[color=blue!20] (0,0) -- (-60:\r) arc (-60:60:\r) -- cycle; +\fill[color=darkgreen!20] (0,0) -- (60:\r) arc (60:180:\r) -- cycle; +\fill[color=orange!20] (0,0) -- (180:\r) arc (180:300:\r) -- cycle; + +\node[color=darkgreen] at (120:3.7) [rotate={#1+30}] {Algebra}; +\node[color=orange] at (240:3.7) [rotate={#1+150}] {Analysis}; +\node[color=blue] at (0:3.7) [rotate={#1-90}] {Zerlegung}; +\end{scope} +} + +\begin{frame} +\frametitle{Intro --- Matrizen} + +\vspace{-25pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\only<1-8>{ + \rad{-30} + \only<2->{ \node at (90:3.0) {Rechenregeln $A^2+A+I=0$}; } + \only<3->{ \node at (90:2.5) {Polynome $\chi_A(A)=0$, $m_A(A)=0$}; } + \only<4->{ \node at (90:2.0) {Projektion: $P^2=P$}; } + \only<5->{ \node at (90:1.5) {nilpotent: $N^k=0$}; } +} + +\only<9-14>{ + \rad{90} + \only<10->{ \node at (90:2.7) {Eigenbasis: $A=\sum \lambda_k P_k$}; } + \only<11->{ \node at (90:2.2) {Invariante Räume: + $AV\subset V, AV^\perp\subset V^\perp$}; } +} + +\only<15-22>{ + \rad{210} + \only<16->{ \node at (90:3.3) {Symmetrien}; } + \only<17->{ \node at (90:2.8) {Skalarprodukt erhalten: + $\operatorname{SO}(n)$}; } + \only<18->{ \node at (90:2.3) {Konstant $\Rightarrow$ Ableitung $=0$}; } + \only<19->{ \node at (90:1.5) {$\displaystyle \exp(A) + = \sum_{k=0}^\infty \frac{A^k}{k!}$}; + } +} + +\fill[color=red!20] (0,0) circle[radius=1.0]; +\node at (0,0.25) {Matrizen}; +\node at (0,-0.25) {$M_{m\times n}(\Bbbk)$}; + +\uncover<6->{ + \node[color=darkgreen] at (4.3,3.4) [right] {Algebra}; + \node at (4.3,2.2) [right] {\begin{minipage}{5cm} + \begin{itemize} + \item<6-> Algebraische Strukturen + \item<7-> Polynome, Teilbarkeit + \item<8-> Minimalpolynom + \end{itemize} + \end{minipage}}; +} + +\uncover<12->{ + \node[color=blue] at (4.3,0.8) [right] {Zerlegung}; + \node at (4.3,-0.4) [right] {\begin{minipage}{5cm} + \begin{itemize} + \item<12-> Eigenvektoren, -räume + \item<13-> Projektionen, Drehungen + \item<14-> Invariante Unterräume + \end{itemize} + \end{minipage}}; +} + +\uncover<20->{ + \node[color=orange] at (4.3,-1.8) [right] {Analysis}; + \node at (4.3,-3.0) [right] {\begin{minipage}{6cm} + \begin{itemize} + \item<20-> Symmetrien + \item<21-> Matrix-DGL + \item<22-> Matrix-Potenzreihen + \end{itemize} + \end{minipage}}; +} + +\end{tikzpicture} +\end{center} + +\end{frame} + +\egroup diff --git a/vorlesungen/slides/0/latextipps.tex b/vorlesungen/slides/0/latextipps.tex new file mode 100644 index 0000000..09d7c89 --- /dev/null +++ b/vorlesungen/slides/0/latextipps.tex @@ -0,0 +1,16 @@ +% +% latextipps.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\frametitle{\LaTeX-Tipps/Anforderungen} +\begin{enumerate} +\item<1-> Formeln sind Bestandteil von Sätzen, dürfen nicht alleine stehen. +\item<2-> Über die Platzierung von Abbildungen/Tabellen entscheidet das System +(mit Verweisen arbeiten). +\item<3-> Neuer Absatz: Leerzeile (nicht \texttt{\textbackslash\textbackslash}) +\item<4-> Jeden Satz auf einer neuen Zeile beginnen (GIT) +\item<5-> Bilder PDF (PNG/JPG mindestens 300 dpi) +\end{enumerate} +\end{frame} diff --git a/vorlesungen/slides/0/nextsteps.tex b/vorlesungen/slides/0/nextsteps.tex new file mode 100644 index 0000000..cb9a07e --- /dev/null +++ b/vorlesungen/slides/0/nextsteps.tex @@ -0,0 +1,21 @@ +% +% nextsteps.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\begin{frame} +\frametitle{Nächste Schritte} + +\begin{enumerate} +\item<2-> +Thema wählen, Teams bilden, Thema wird festgelegt zu Beginn der Woche 2 +\item<3-> +Grundlagen studieren (Skript, Wikipedia, Bücher) +\item<4-> +Eigenes Seminarthema vertiefen +\item <5-> +Plan für Seminararbeit und Vortrag +\end{enumerate} + +\end{frame} diff --git a/vorlesungen/slides/0/resourcen.tex b/vorlesungen/slides/0/resourcen.tex new file mode 100644 index 0000000..02a3fb8 --- /dev/null +++ b/vorlesungen/slides/0/resourcen.tex @@ -0,0 +1,37 @@ +% +% resourcen.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Resourcen} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Moodle Modul MathSem} +\begin{enumerate} +\item<2-> Skript +\begin{itemize} +\item<3-> Aktuellste Version in Github +\item<4-> regelmässige Updates in Moodle: \texttt{buch.pdf} +\end{itemize} +\item<5-> Informationen zur Planung: Kurztests, Vorträge +\item<6-> Anleitung für die Seminararbeit +\item<7-> Aufgabenstellungen +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Weitere Quellen} +\begin{enumerate} +\item<9-> Zusätzliche Literaturhinweise in der Aufgabenbeschreibung im Moodle +\item<10-> Bibliothek +\item<11-> Google +\item<12-> Google Scholar +\item<13-> Paper ist nicht öffentlich zugänglich? $\rightarrow$ kann via +Bibliothek organisiert werden +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/0/themen1.tex b/vorlesungen/slides/0/themen1.tex new file mode 100644 index 0000000..756e037 --- /dev/null +++ b/vorlesungen/slides/0/themen1.tex @@ -0,0 +1,27 @@ +% +% themen1.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\frametitle{Seminararbeitsthemen I} +\begin{enumerate}[<+->] +\item +Verkehrsnetze und Verkehrsfluss +\item +Mittelwert von Matrizen +\item +Pascal-Matrizen +\item +Stirling-Matrizen +\item +Vandermonde-Matrix +\item +Probabilistische Matrix-Produkt-Kontrolle +\item +Der Satz von Furrer-Hungerbühler-Jantschgi +\item +Clifford-Algebren +\end{enumerate} +\end{frame} + diff --git a/vorlesungen/slides/0/themen2.tex b/vorlesungen/slides/0/themen2.tex new file mode 100644 index 0000000..1fbdab3 --- /dev/null +++ b/vorlesungen/slides/0/themen2.tex @@ -0,0 +1,27 @@ +% +% themen2.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\begin{frame} +\frametitle{Seminararbeitsthemen II} +\begin{enumerate}[<+->] +\setcounter{enumi}{7} +\item +Schnelle Matrixmultiplikation +\item +Parkettierungen mit Dominosteinen zählen +\item +Punktgruppen und Kristallographie +\item +Symmetriegruppen und Machine Learning +\item +Floyd-Warshall-Algorithmus +\item +Laser +\item +Munkres-Algorithmus +\end{enumerate} +\end{frame} + diff --git a/vorlesungen/slides/0/themen3.tex b/vorlesungen/slides/0/themen3.tex new file mode 100644 index 0000000..5a6bdf5 --- /dev/null +++ b/vorlesungen/slides/0/themen3.tex @@ -0,0 +1,26 @@ +% +% themen3.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\begin{frame} +\frametitle{Seminararbeitsthemen III} +\begin{enumerate}[<+->] +\setcounter{enumi}{14} +\item +Iwasawa-Zerlegung +\item +Reed-Solomon Code +\item +Pauli- und Dirac-Matrizen +\item +Klassifikation der Lie-Gruppen +\item +Iterierte Funktionsschemata +\item +QR-Codes +\item +McEliece-Kryptosystem +\end{enumerate} +\end{frame} diff --git a/vorlesungen/slides/0/was.tex b/vorlesungen/slides/0/was.tex new file mode 100644 index 0000000..685ee22 --- /dev/null +++ b/vorlesungen/slides/0/was.tex @@ -0,0 +1,50 @@ +% +% was.tex -- was wird erwartet +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Was wird erwartet} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Seminararbeit} +\begin{itemize} +\item<3-> Ihr Thema: gestalten Sie es! +\item<4-> Eine spannende Story erzählen +\item<5-> Immer an den Leser denken: Ihre Kollegen +\item<6-> So lang wie nötig, so kurz wie möglich +\item<7-> Bewertet durch Seminarleiter +\end{itemize} +\end{block}} +\vspace{-5pt} +\uncover<14->{% +\begin{block}{Hilfe} +\begin{itemize} +\item Einführungsvorlesungen +\item \texttt{andreas.mueller@ost.ch} +\item \texttt{roy.seitz@ost.ch} +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Seminarvortrag} +\begin{itemize} +\item<8-> Vortrag $\ne$ Arbeit +\item<9-> So lang wie nötig, so kurz wie möglich +\item<10-> Konzentration auf das Wesentliche +\item<11-> $>30\,\text{min}$ ist fast sicher zu lang +\item<12-> Bewertet durch die Seminarteilnehmer +\end{itemize} +\end{block}} +\vspace{-5pt} +\uncover<13->{% +\begin{block}{3 Kurztests} +Ziel: Sie befassen sich auch mit den Themen +ausserhalb ihrer eigenen Seminararbeit +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/Makefile.inc b/vorlesungen/slides/1/Makefile.inc new file mode 100644 index 0000000..38b47b3 --- /dev/null +++ b/vorlesungen/slides/1/Makefile.inc @@ -0,0 +1,23 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 20920 Prof Dr Andreas Müller, Hochschule Rapperswil +# +chapter1 = \ + ../slides/1/zahlensysteme.tex \ + ../slides/1/peano.tex \ + ../slides/1/ganz.tex \ + ../slides/1/bruch.tex \ + ../slides/1/ring.tex \ + ../slides/1/schwierigkeiten.tex \ + ../slides/1/strukturen.tex \ + ../slides/1/j.tex \ + ../slides/1/vektorraum.tex \ + ../slides/1/matrixalgebra.tex \ + ../slides/1/algebrastruktur.tex \ + ../slides/1/speziell.tex \ + ../slides/1/dreieck.tex \ + ../slides/1/hadamard.tex \ + ../slides/1/chapter.tex + diff --git a/vorlesungen/slides/1/algebrastruktur.tex b/vorlesungen/slides/1/algebrastruktur.tex new file mode 100644 index 0000000..fd474eb --- /dev/null +++ b/vorlesungen/slides/1/algebrastruktur.tex @@ -0,0 +1,93 @@ +% +% algebrastruktur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\begin{frame}[t] +\frametitle{Algebra über $\Bbbk$} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\pgfmathparse{atan(7/4)} +\xdef\a{\pgfmathresult} +\uncover<2->{ + \fill[color=red!40,opacity=0.5] + ({-4-2.5},{2+1.0}) + -- + ({-2.5},{-3-1.0}) + -- + ({2.5},{-3-1.0}) + -- + ({-4+2.5},{2+1.0}) + -- cycle; +} + +\uncover<4->{ + \fill[color=blue!40,opacity=0.5] + ({4-2.5},{2+1.0}) + -- + ({-2.5},{-3-1.0}) + -- + ({2.5},{-3-1.0}) + -- + ({4+2.5},{2+1.0}) + -- cycle; +} + +\uncover<6->{ + \fill[color=darkgreen!40,opacity=0.5] + ({-4-2.5},{2+1.0}) + -- + ({-4-2.5+2*(4/7)},{2-1}) + -- + ({+4+2.5-2*(4/7)},{2-1}) + -- + ({+4+2.5},{2+1}) + -- + cycle; +} + +\node at ({-3-0.5},2) {Skalarmultiplikation}; + +\node at (3.5,2.2) {Multiplikation}; +\node at (3.5,1.8) {\tiny Monoid}; + +\node at (0,-2.8) {Addition}; +\node at (0,-3.2) {\tiny Gruppe}; + +\uncover<4->{ + \node[color=blue] at (4.8,-0.5) [rotate=\a] {Ring\strut}; +} + +\uncover<2->{ + \node[color=red] at (-4.8,-0.5) [rotate=-\a] {Vektorraum\strut}; +} + +\uncover<6->{ + \node[color=darkgreen] at (0,2.6) {$(\lambda a)b=\lambda(ab)$}; +} + +\uncover<3->{ + \node[color=red] at (-2.5,-0.5) {$\displaystyle + \begin{aligned} + \lambda(a+b)&=\lambda a + \lambda b\\ + (\lambda+\mu)a&=\lambda a +\mu a + \end{aligned}$}; +} + +\uncover<5->{ + \node[color=blue] at (2.5,-0.5) {$\displaystyle + \begin{aligned} + a(b+c)&=ab+ac\\ + (a+b)c&=ac+bc + \end{aligned}$}; +} + +\end{tikzpicture} +\end{center} +\end{frame} + +\egroup diff --git a/vorlesungen/slides/1/bruch.tex b/vorlesungen/slides/1/bruch.tex new file mode 100644 index 0000000..65521a2 --- /dev/null +++ b/vorlesungen/slides/1/bruch.tex @@ -0,0 +1,73 @@ +% +% bruch.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Brüche} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-8pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Division} +Nicht für alle $a,b\in\mathbb{Z}$ hat die Gleichung +\[ +ax=b +\uncover<2->{ +\;\Rightarrow\; +x=\frac{b}{a}} +\] +eine Lösung in $\mathbb{Z}$\uncover<2->{, nämlich wenn $b\nmid a$} +\end{block} +\uncover<3->{% +\begin{block}{Brüche} +Idee: $\displaystyle\frac{b}{a} = (b,a)$ +\begin{enumerate} +\item<4-> $(b,a)\in\mathbb{Z}\times\mathbb{Z}$ +\item<5-> Äquivalenzrelation +\[ +(b,a)\sim (d,c) +\ifthenelse{\boolean{presentation}}{ +\only<5>{ +\Leftrightarrow +\text{`` +$\displaystyle +\frac{b}{a}=\frac{d}{c} +$ +''} +}}{} +\only<6->{ +\Leftrightarrow +bc=ad +} +\] +\end{enumerate} +\vspace{-15pt} +\uncover<7->{% +$\Rightarrow$ alle Quotienten +} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<9->{% +\begin{block}{Gruppe} +$\mathbb{Q}^* = \mathbb{Q}\setminus\{0\}$ ist eine multiplikative Gruppe: +\begin{enumerate} +\item<10-> Neutrales Element: $1\in \mathbb{Q}^*$ +\item<11-> Inverses Element $q=\frac{b}{a}\in\mathbb{Q} +\Rightarrow +q^{-1}=\frac{a}{b}\in\mathbb{Q}$ +\end{enumerate} +\end{block} +} +\uncover<8->{% +\begin{block}{Rationale Zahlen} +Alle Brüche, gleiche Werte zusammengefasst: +\[ +\mathbb{Q} = \mathbb{Z}\times\mathbb{Z}/\sim +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/chapter.tex b/vorlesungen/slides/1/chapter.tex new file mode 100644 index 0000000..7bdda34 --- /dev/null +++ b/vorlesungen/slides/1/chapter.tex @@ -0,0 +1,19 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{1/zahlensysteme.tex} +\folie{1/peano.tex} +\folie{1/ganz.tex} +\folie{1/bruch.tex} +\folie{1/ring.tex} +\folie{1/schwierigkeiten.tex} +\folie{1/strukturen.tex} +\folie{1/j.tex} +\folie{1/vektorraum.tex} +\folie{1/matrixalgebra.tex} +\folie{1/algebrastruktur.tex} +\folie{1/speziell.tex} +\folie{1/dreieck.tex} +\folie{1/hadamard.tex} diff --git a/vorlesungen/slides/1/dreieck.tex b/vorlesungen/slides/1/dreieck.tex new file mode 100644 index 0000000..3797e4b --- /dev/null +++ b/vorlesungen/slides/1/dreieck.tex @@ -0,0 +1,69 @@ +% +% dreieck.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Dreiecksmatrizen} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.31\textwidth} +\begin{block}{Dreiecksmatrix} +\begin{align*} +R&= +\begin{pmatrix} +*&*&*&\dots&*\\ +0&*&*&\dots&*\\ +0&0&*&\dots&*\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\dots&* +\end{pmatrix} +\\ +U&= +\begin{pmatrix} +1&*&*&\dots&*\\ +0&1&*&\dots&*\\ +0&0&1&\dots&*\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\dots&1 +\end{pmatrix} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.31\textwidth} +\uncover<2->{% +\begin{block}{Nilpotente Matrix} +\[ +N= +\begin{pmatrix} +0&*&*&\dots&*\\ +0&0&*&\dots&*\\ +0&0&0&\dots&*\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\dots&0 +\end{pmatrix} +\] +\uncover<3->{% +$\Rightarrow N^n=0$ +} +\end{block}} +\end{column} +\begin{column}{0.31\textwidth} +\uncover<4->{% +\begin{block}{Jordan-Matrix} +\[ +J_\lambda=\begin{pmatrix} +\lambda&1&0&\dots&0\\ +0&\lambda&1&\dots&0\\ +0&0&\lambda&\dots&0\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\dots&\lambda +\end{pmatrix} +\] +\uncover<5->{% +$\Rightarrow J_\lambda -\lambda I$ ist nilpotent +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/ganz.tex b/vorlesungen/slides/1/ganz.tex new file mode 100644 index 0000000..7930826 --- /dev/null +++ b/vorlesungen/slides/1/ganz.tex @@ -0,0 +1,106 @@ +% +% ganz.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Ganze Zahlen: Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{block}{Subtrahieren} +Nicht für alle $a,b\in \mathbb{N}$ hat die +Gleichung +\[ +a+x=b +\uncover<2->{ +\quad +\Rightarrow +\quad +x=b-a} +\] +eine Lösung in $\mathbb{N}$\uncover<2->{, nämlich wenn $a>b$}% +\end{block} +\uncover<3->{% +\begin{block}{Ganze Zahlen = Paare} +Idee: $b-a = (b,a)$ +\begin{enumerate} +\item<4-> $(b,a)=\mathbb{N}\times\mathbb{N}$ +\item<5-> Äquivalenzrelation +\[ +(b,a)\sim (d,c) +\ifthenelse{\boolean{presentation}}{ +\only<6>{\Leftrightarrow +\text{``\strut} +b-a=c-d +\text{\strut''}}}{} +\only<7->{ +\Leftrightarrow +b+d=c+a} +\] +\end{enumerate} +\vspace{-10pt} +\uncover<8->{% +Ganze Zahlen: +\( +\mathbb{Z} += +\mathbb{N}\times\mathbb{N}/\sim +\)} +\\ +\uncover<9->{% +$z\in\mathbb{Z}$, $z=\mathstrut$ Paare $(u,v)$ mit +``gleicher Differenz''} +\uncover<10->{% +$\Rightarrow$ alle Differenzen in $\mathbb{Z}$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\uncover<11->{% +\begin{block}{Gruppe} +Monoid $\ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}{}\only<12->{G}$ mit inversem Element +\[ +a\in \ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}{}\only<12->{G} +\Rightarrow +\ifthenelse{\boolean{presentation}}{\only<11>{-a\in\mathbb{Z}}}{}\only<12->{a^{-1}\in G} +\text{ mit } +\ifthenelse{\boolean{presentation}}{ +\only<11>{ +a+(-a)=0 +}}{} +\only<12->{ +\left\{ +\begin{aligned} +aa^{-1}&=e +\\ +a^{-1}a&=e +\end{aligned} +\right. +} +\] +\end{block}} +\vspace{-15pt} +\uncover<13->{% +\begin{block}{Abelsche Gruppe} +Verknüpfung ist kommutativ: +\[ +a+b=b+a +\] +\end{block}} +\vspace{-12pt} +\uncover<14->{% +\begin{block}{Beispiele} +\begin{itemize} +\item<15-> Brüche, reelle Zahlen +\item<16-> invertierbare Matrizen: $\operatorname{GL}_n(\mathbb{R})$ +\item<17-> Drehmatrizen: $\operatorname{SO}(n)$ +\item<18-> Matrizen mit Determinante $1$: $\operatorname{SL}_n(\mathbb R)$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/hadamard.tex b/vorlesungen/slides/1/hadamard.tex new file mode 100644 index 0000000..5cb692a --- /dev/null +++ b/vorlesungen/slides/1/hadamard.tex @@ -0,0 +1,51 @@ +% +% hadamard.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Hadamard-Algebra} +\begin{block}{Alternatives Produkt: Hadamard-Produkt} +\[ +\begin{pmatrix} +a_{11}&\dots&a_{1n}\\ +\vdots&\ddots&\vdots\\ +a_{m1}&\dots&a_{mn}\\ +\end{pmatrix} +\odot +\begin{pmatrix} +b_{11}&\dots&b_{1n}\\ +\vdots&\ddots&\vdots\\ +b_{m1}&\dots&b_{mn}\\ +\end{pmatrix} += +\begin{pmatrix} +a_{11}b_{11}&\dots&a_{1n}b_{1n}\\ +\vdots&\ddots&\vdots\\ +a_{m1}b_{m1}&\dots&a_{mn}b_{mn}\\ +\end{pmatrix} +\] +\end{block} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.58\textwidth} +\uncover<2->{% +\begin{block}{Algebra} +\begin{itemize} +\item<3-> $M_{mn}(\Bbbk)$ ist eine Algebra mit +$\odot$ als Produkt +\item<4-> Neutrales Element $U$: Matrix aus lauter Einsen +\item<5-> Anwendung: Wahrscheinlichkeitsmatrizen +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.38\textwidth} +\uncover<6->{% +\begin{block}{Nicht so interessant} +Die Hadamard-Algebra ist kommutativ +\uncover<7->{$\Rightarrow$ +kann ``keine'' interessanten algebraischen Relationen darstellen} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/j.tex b/vorlesungen/slides/1/j.tex new file mode 100644 index 0000000..132f1d0 --- /dev/null +++ b/vorlesungen/slides/1/j.tex @@ -0,0 +1,63 @@ +% +% j.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Beispiele} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Imaginäre Einheit $i$} +Gibt es eine Zahl $i$ mit $i^2=-1$? +\end{block} +\uncover<2->{% +\begin{block}{Matrixlösung} +Die Matrix +\[ +J += +\begin{pmatrix}0&-1\\1&0\end{pmatrix} +\] +erfüllt +\[ +J^2 += +%\begin{pmatrix}0&-1\\1&0\end{pmatrix} +%\begin{pmatrix}0&-1\\1&0\end{pmatrix} +%= +\begin{pmatrix}-1&0\\0&-1\end{pmatrix} += +-I +\] +$\Rightarrow$ $J$ ist eine Matrixdarstellung von $i$ + +Drehmatrix mit Winkel $90^\circ$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Quadratwurzel $\sqrt{2}$} +Gibt es eine Zahl $\sqrt{2}$ derart, dass $(\sqrt{2})^2=2$? +\end{block}} +\uncover<4->{% +\begin{block}{Matrixlösung} +%\setlength{\abovedisplayskip}{5pt} +%\setlength{\belowdisplayskip}{5pt} +Die Matrix +\[ +W += +\begin{pmatrix}0&2\\1&0\end{pmatrix} +\] +erfüllt +\[ +W^2 += +\begin{pmatrix}2&0\\0&2\end{pmatrix} = 2I +\] +$\Rightarrow$ $W$ ist eine Matrixdarstellung von $\sqrt{2}$ +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/matrixalgebra.tex b/vorlesungen/slides/1/matrixalgebra.tex new file mode 100644 index 0000000..a3c3a76 --- /dev/null +++ b/vorlesungen/slides/1/matrixalgebra.tex @@ -0,0 +1,77 @@ +% +% matrixalgebra.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\newtcbox{\myboxA}{blank,boxsep=0mm, +clip upper,minipage, +width=31.0mm,height=17.0mm,nobeforeafter, +borderline={0.0pt}{0.0pt}{white}, +} +\definecolor{magenta}{rgb}{0.8,0.2,0.8} + +\begin{frame}[t] +\frametitle{Matrix-Algebra} +\vspace{-10pt} +\[ +\begin{pmatrix} +a_{11}&\dots &a_{1n}\\ +\vdots&\ddots&\vdots\\ +a_{m1}&\dots &a_{mn} +\end{pmatrix} ++ +\begin{pmatrix} +b_{11}&\dots &b_{1n}\\ +\vdots&\ddots&\vdots\\ +b_{m1}&\dots &b_{mn} +\end{pmatrix} += +\begin{pmatrix} +a_{11}+b_{11}&\dots &a_{1n}+b_{1n}\\ +\vdots&\ddots&\vdots\\ +a_{m1}+b_{m1}&\dots &a_{mn}+b_{mn} +\end{pmatrix} +\] +\[ +\lambda +\begin{pmatrix} +a_{11}&\dots &a_{1n}\\ +\vdots&\ddots&\vdots\\ +a_{m1}&\dots &a_{mn} +\end{pmatrix} += +\begin{pmatrix} +\lambda a_{11}&\dots &\lambda a_{1n}\\ +\vdots&\ddots&\vdots\\ +\lambda a_{m1}&\dots &\lambda a_{mn} +\end{pmatrix} +\] +\uncover<2->{% +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope}[xshift=-4.5cm] +\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; +\draw[color=red,line width=3pt] (0,2) -- (3,2); +\draw (0,0) rectangle (3,3); +\end{scope} +\node at (-0.75,1.5) {$\mathstrut\cdot\mathstrut$}; +\begin{scope}[xshift=0cm] +\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; +\draw[color=blue,line width=3pt] (2.7,0) -- (2.7,3); +\draw (0,0) rectangle (3,3); +\end{scope} +\node at (3.75,1.5) {$\mathstrut=\mathstrut$}; +\begin{scope}[xshift=4.5cm] +\node at (1.5,1.53) {$\left(\myboxA{}\right)$}; +\draw[color=gray,line width=1pt] (2.7,0) -- (2.7,3); +\draw[color=gray,line width=1pt] (0,2) -- (3,2); +\fill[color=magenta] (2.7,2) circle[radius=0.12]; +\draw (0,0) rectangle (3,3); +\end{scope} +\end{tikzpicture} +\end{center}} +\end{frame} + +\egroup diff --git a/vorlesungen/slides/1/peano.tex b/vorlesungen/slides/1/peano.tex new file mode 100644 index 0000000..219c853 --- /dev/null +++ b/vorlesungen/slides/1/peano.tex @@ -0,0 +1,72 @@ +% +% peano.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Natürliche Zahlen\uncover<2->{: Peano}} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zählen} +Mit den natürlichen Zahlen zählt man: +\[ +\mathbb{N} += +\left\{ +\begin{minipage}{5cm} +\raggedright +Äquivalenzklassen von gleich mächtigen +endlichen Mengen +\end{minipage} +\right\} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Peano-Axiome} +\begin{enumerate} +\item<3-> $0\in\mathbb{N}$ +\item<4-> $n\in\mathbb{N}\Rightarrow \text{Nachfolger }n'\in\mathbb{N}$ +\item<5-> $0$ ist nicht Nachfolger +\item<6-> $n,m\in\mathbb{N}\wedge n'=m'\Rightarrow n=m$ +\item<7-> $X\subset \mathbb{N}\wedge 0\in X\wedge \forall n\in X(n'\in X) +\Rightarrow +\mathbb{N}=X +$ +\end{enumerate} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Monoid} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +Menge $\only<8-10>{\mathbb{N}}\only<11->{M}$ mit einer +zweistelligen Verknüpfung $a\only<8-10>{+}\only<11->{*}b$ +\begin{enumerate} +\item<9-> Assoziativ: $a,b,c\in M$ +\[ +(a\only<8-10>{+}\only<11->{*}b)\only<8-10>{+}\only<11->{*}c=a\only<8-10>{+}\only<11->{*}(b\only<8-10>{+}\only<11->{*}c) +\] +\item<10-> Neutrales Element: $\only<8-10>{0}\only<11->{e}\in M$ +\[ +\only<8-10>{0+}\only<11->{e*} a += +a \only<8-10>{+0}\only<11->{*e} +\] +\end{enumerate} +\end{block}}% +\vspace{-15pt} +\uncover<12->{% +\begin{block}{Axiom 5 = Vollständige Induktion} +$X=\{n\in\mathbb{N}\;|\; \text{$P(n)$ ist wahr}\}$ +\begin{enumerate} +\item<13-> Verankerung: $0\in X$ +\item<14-> Induktionsannahme: $n\in X$ +\item<15-> Induktionsschritt: $n'\in X$ +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/ring.tex b/vorlesungen/slides/1/ring.tex new file mode 100644 index 0000000..9641975 --- /dev/null +++ b/vorlesungen/slides/1/ring.tex @@ -0,0 +1,58 @@ +% +% ring.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Ring\only<15->{/Körper}} +\vspace{-10pt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Addition und Multiplikation} +$\mathbb{Z}$ und $\mathbb{Q}$ +haben zwei Verknüpfungen: +\begin{enumerate} +\item<2-> Addition +\[ +a,b\in R\Rightarrow a+b\in R +\] +\item<3-> Multiplikation +\[ +a,b\in R\Rightarrow a\cdot b=ab\in R +\] +\end{enumerate} +\vspace{-5pt} +\uncover<4->{% +Gilt auch für +\begin{itemize} +\item<5-> Polynome +\item<6-> $M_{n}(\mathbb{R})$ +\item<7-> $\mathbb{R}^3$ mit Vektorprodukt +\end{itemize}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Definition} +Ein Ring\only<15->{/{\color{red}Körper}} ist eine Menge $R$ mit zwei +Verknüpfungen $+$ und $\cdot$: +\begin{enumerate} +\item<9-> +$R$ mit $+$ ist eine abelsche Gruppe +\item<10-> +$R$ mit $\cdot$ ist ein Monoid\only<15->{/{\color{red}eine Gruppe}} +\item<11-> +Verträglichkeit: Distributivgesetz +\begin{align*} +\uncover<12->{a(b+c)&=ab+bc} +\\ +\uncover<13->{(a+b)c&=ac+bc} +\end{align*} +\uncover<14->{(Ausmultiplizieren)} +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/schwierigkeiten.tex b/vorlesungen/slides/1/schwierigkeiten.tex new file mode 100644 index 0000000..fb22e58 --- /dev/null +++ b/vorlesungen/slides/1/schwierigkeiten.tex @@ -0,0 +1,90 @@ +% +% schwierigkeiten.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Schwierigkeiten} +\vspace{-15pt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Nullteiler} +Elemente $a,b$ mit $ab=0$ +$\Rightarrow$ nicht invertierbar +\begin{itemize} +\item<3-> Projektionen +\[ +\begin{pmatrix} +1&0\\0&0 +\end{pmatrix} +\begin{pmatrix} +0&0\\0&1 +\end{pmatrix} += +0 +\] +\item<4-> Nilpotente Matrizen +\[ +\begin{pmatrix} +0&1&0\\ +0&0&1\\ +0&0&0 +\end{pmatrix}^3 +=0 +\] +\item<5-> +In $\mathbb{Z}/15\mathbb{Z}$ (modulo 15): +\[ +3\cdot 5 = 15 \equiv 0\mod 15 +\] +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Invertierbarkeit} +\begin{itemize} +\item<7-> +$7\in\mathbb{Z}$, aber $7^{-1}\not\in\mathbb{Z}$, $7^{-1}\in\mathbb{Q}$ +\item<8-> +$A$ regulär heisst nicht $A^{-1}\in M_n(\mathbb{Z})$ +\[ +A=\begin{pmatrix} +1&-1\\ +1&1 +\end{pmatrix} +\;\Rightarrow\; +A^{-1} += +\begin{pmatrix} +\frac12&\frac12\\ +-\frac12&\frac12 +\end{pmatrix} +\] +\item<9-> +$A\in\operatorname{SL}_n(\mathbb{Z})$ invertierbar in +$M_n(\mathbb{Z})$: +\[ +A= +\begin{pmatrix} +5&4\\4&3 +\end{pmatrix} +\; +\Rightarrow +\; +A^{-1}= +\begin{pmatrix} +-3&4\\4&-5 +\end{pmatrix} +\] +\end{itemize} +\uncover<10->{% +Invertierbarkeit erreichen durch ``vergrössern'' des Ringes +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/speziell.tex b/vorlesungen/slides/1/speziell.tex new file mode 100644 index 0000000..5b93da6 --- /dev/null +++ b/vorlesungen/slides/1/speziell.tex @@ -0,0 +1,46 @@ +% +% speziell.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\frametitle{Diagonalmatrizen} +\begin{block}{Einheitsmatrix} +\[ +I=\begin{pmatrix} +1&0&\dots&0\\ +0&1&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +0&0&\dots&1 +\end{pmatrix} +\] +Neutrales Element der Matrixmultiplikation: +\[ +AI=IA=A +\] +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<2->{% +\begin{block}{Diagonalmatrix} +\[ +\operatorname{diag}(\lambda_1,\lambda_2,\dots,\lambda_n) += +\begin{pmatrix} +\lambda_1&0&\dots&0\\ +0&\lambda_2&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +0&0&\dots&\lambda_n +\end{pmatrix} +\] +\end{block}} +\uncover<3->{% +\begin{block}{Hadamard-Algebra} +Die Algebra der Diagonalmatrizen ist die Hadamard-Algebra +(siehe später) +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/strukturen.tex b/vorlesungen/slides/1/strukturen.tex new file mode 100644 index 0000000..a5fc09a --- /dev/null +++ b/vorlesungen/slides/1/strukturen.tex @@ -0,0 +1,35 @@ +% +% strukturen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Strukturen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.42\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/10-vektorenmatrizen/images/strukturen.pdf} +\end{center} +\end{column} +\begin{column}{0.54\textwidth} +\begin{itemize}[<+->] +\item Gruppen: Drehungen, Symmetrien +\item Vektorraum: Geometrie +\item Ring (mit Eins) +\item Algebra: Vektorraum und Ring +\item Algebra mit Eins: Vektorraum und Ring mit Eins +\item Körper +\end{itemize} +\uncover<7->{% +\begin{block}{Matrizen} +Jede beliebige Struktur lässt sich mit Matrizen darstellen: +\begin{itemize} +\item<8-> Permutationsmatrizen +\item<9-> Wahrscheinlichkeitsmatrizen +\item<10-> Wurzeln +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/vektorraum.tex b/vorlesungen/slides/1/vektorraum.tex new file mode 100644 index 0000000..2566085 --- /dev/null +++ b/vorlesungen/slides/1/vektorraum.tex @@ -0,0 +1,54 @@ +% +% vektorraum.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Vektorraum} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Operationen} +Addition: +\[ +\begin{pmatrix}a_1\\\vdots\\a_n \end{pmatrix} ++ +\begin{pmatrix}b_1\\\vdots\\b_n \end{pmatrix} += +\begin{pmatrix}a_1+b_1\\\vdots\\a_n+b_n \end{pmatrix} +\] +Skalarmultiplikation: +\[ +\lambda\begin{pmatrix}a_1\\\vdots\\a_n \end{pmatrix} += +\begin{pmatrix}\lambda a_1\\\vdots\\\lambda a_n \end{pmatrix} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Additive Gruppe} +$\mathbb{R}^n$ ist eine Gruppe bezüglich der Addition +mit +\[ +0=\begin{pmatrix}0\\\vdots\\0\end{pmatrix}, +\qquad +-a += +-\begin{pmatrix}a_1\\\vdots\\a_n\end{pmatrix} += +\begin{pmatrix}-a_1\\\vdots\\-a_n\end{pmatrix} +\] +\end{block}} +\vspace{-5pt} +\uncover<3->{% +\begin{block}{Skalarmultiplikation} +Distributivgesetz +\begin{align*} +(\lambda+\mu)a&=\lambda a + \mu a\\ +\lambda (a+b)&=\lambda a + \lambda b +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/1/zahlensysteme.tex b/vorlesungen/slides/1/zahlensysteme.tex new file mode 100644 index 0000000..9131cc6 --- /dev/null +++ b/vorlesungen/slides/1/zahlensysteme.tex @@ -0,0 +1,46 @@ +% +% zahlensysteme.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Zahlensysteme} +\begin{center} +\begin{tabular}{|>{$}c<{$}|p{7cm}|p{3cm}|} +\hline +\text{Zahlenmenge}&\text{Eigenschaften}&\text{Struktur} +\\ +\hline +\mathbb{N} +&\phantom{}\raggedright\uncover<2->{Addition, neutrales Element $0$} +&\phantom{}\uncover<2->{Monoid} +\\ +\mathbb{Z} +&\phantom{}\raggedright\uncover<3->{Addition, neutrales Element $0$, +inverses Element der Addition} +&\phantom{}\uncover<3->{Gruppe} +\\ +\mathbb{Z} +&\phantom{}\raggedright\uncover<4->{zusätzlich: Multiplikation, neutrales Element $1$} +&\phantom{}\uncover<4->{Ring} +\\ +\mathbb{Q} +&\phantom{}\raggedright\uncover<5->{Addition und Multiplikation mit Inversen} +&\phantom{}\uncover<5->{Körper} +\\ +\mathbb{R} +&\phantom{}\raggedright\uncover<6->{zusätzlich: Ordnungsrelation, Vollständigkeit} +&\phantom{}\uncover<6->{Körper mit Ordnung} +\\ +\mathbb{C} +&\phantom{}\raggedright\uncover<7->{zusätzlich: Alle Wurzeln} +&\phantom{}\uncover<7->{algebraisch abgeschlossener Körper} +\\ +\uncover<8->{\mathbb{H}} +&\phantom{}\raggedright\uncover<8->{höhere Dimension, nichtkommutativ} +&\phantom{}\uncover<8->{Schiefkörper} +\\ +\hline +\end{tabular} +\end{center} +\end{frame} diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc new file mode 100644 index 0000000..c857fec --- /dev/null +++ b/vorlesungen/slides/2/Makefile.inc @@ -0,0 +1,21 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter2 = \ + ../slides/2/norm.tex \ + ../slides/2/skalarprodukt.tex \ + ../slides/2/cauchyschwarz.tex \ + ../slides/2/polarformel.tex \ + ../slides/2/funktionenraum.tex \ + ../slides/2/operatornorm.tex \ + ../slides/2/linearformnormen.tex \ + ../slides/2/funktionenalgebra.tex \ + ../slides/2/frobeniusnorm.tex \ + ../slides/2/frobeniusanwendung.tex \ + ../slides/2/quotient.tex \ + ../slides/2/quotientv.tex \ + ../slides/2/chapter.tex + diff --git a/vorlesungen/slides/2/cauchyschwarz.tex b/vorlesungen/slides/2/cauchyschwarz.tex new file mode 100644 index 0000000..a24ada8 --- /dev/null +++ b/vorlesungen/slides/2/cauchyschwarz.tex @@ -0,0 +1,94 @@ +% +% cauchyschwarz.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.5,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Cauchy-Schwarz-Ungleichung} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz (Cauchy-Schwarz)} +$\langle\;,\;\rangle$ eine positiv definite, hermitesche Sesquilinearform +\[ +{\color{darkgreen} +|\operatorname{Re}\langle u,v\rangle| +\le +|\langle u,v\rangle| +\le +\|u\|_2\cdot \|v\|_2 +} +\] +Gleichheit genau dann, wenn $u$ und $v$ linear abhängig sind +\end{block} +\begin{block}{Dreiecksungleichung} +\vspace{-12pt} +\begin{align*} +\|u+v\|_2^2 +&= +\|u\|_2^2 + 2\operatorname{Re}\langle u,v\rangle + \|v\|_2^2 +\\ +&\le +\|u\|_2^2 + 2{\color{darkgreen}|\langle u,v\rangle|} + \|v\|_2^2 +\\ +&\le +\|u\|_2^2 + 2{\color{darkgreen}\|u\|_2\cdot \|v\|_2} + \|v\|_2^2 +\\ +&=(\|u\|_2 + \|v\|_2)^2 +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{proof}[Beweis] +Die quadratische Funktion +\begin{align*} +Q(t) +&= +\langle u+tv,u+tv\rangle \ge 0 +\\ +\uncover<3->{ +Q(t) +&= +\|u\|_2^2 + 2t\operatorname{Re}\langle u,v\rangle + t^2\|v\|_2^2} +\end{align*} +\uncover<4->{hat ihr Minimum bei}% +\begin{align*} +\uncover<5->{ +t&= +-\operatorname{Re}\langle u,v\rangle/\|v\|_2^2} +\intertext{\uncover<6->{mit Wert}} +\uncover<7->{ +Q(t) +&= +\|u\|_2^2 +-2\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2} +\\ +\uncover<7->{ +&\qquad + \operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2} +\\ +\uncover<8->{ +0 +&\le +\|u\|_2^2-\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2} +\\ +\uncover<9->{ +\operatorname{Re}\langle u,v\rangle^2 +&\le +\|u\|_2^2\cdot\|v\|_2^2} +\\ +\uncover<10->{ +\operatorname{Re}\langle u,v\rangle +&\le +\|u\|_2\cdot\|v\|_2} +\qedhere +\end{align*} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex new file mode 100644 index 0000000..49e656a --- /dev/null +++ b/vorlesungen/slides/2/chapter.tex @@ -0,0 +1,17 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{2/norm.tex} +\folie{2/skalarprodukt.tex} +\folie{2/cauchyschwarz.tex} +\folie{2/polarformel.tex} +\folie{2/funktionenraum.tex} +\folie{2/operatornorm.tex} +\folie{2/linearformnormen.tex} +\folie{2/funktionenalgebra.tex} +\folie{2/frobeniusnorm.tex} +\folie{2/frobeniusanwendung.tex} +\folie{2/quotient.tex} +\folie{2/quotientv.tex} diff --git a/vorlesungen/slides/2/frobeniusanwendung.tex b/vorlesungen/slides/2/frobeniusanwendung.tex new file mode 100644 index 0000000..277d600 --- /dev/null +++ b/vorlesungen/slides/2/frobeniusanwendung.tex @@ -0,0 +1,80 @@ +% +% frobeniusanwendung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Anwendung der Frobenius-Norm} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung nach $X\in M_{m\times n}(\mathbb{R})$} +Die Ableitung $Df=\partial f/\partial X$ der Funktion +$f\colon M_{m\times n}(\mathbb{R})\to \mathbb{R}$ ist die Matrix +mit Einträgen +\begin{align*} +\biggl( +\frac{\partial f}{\partial X} +\biggr)_{ij} +&= +\frac{\partial f}{\partial x_{ij}} += +D_{ij}f +\end{align*} +\end{block} +\uncover<2->{% +\begin{block}{Richtungsableitung} +\uncover<5->{Die Matrix $Df$ ist ein Gradient:} +\begin{align*} +\frac{\partial}{\partial t}f(X+tY)\bigg|_{t=0} +&=\uncover<3->{ +\sum_{i,j} +D_{ij} f(X) \cdot y_{ij}} +\\ +&\uncover<4->{= +\langle D_{ij}f(X), Y\rangle_F} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Quadratische Minimalprobleme} +$A=A^t,B,X\in M_n(\mathbb{R})$, Minimum von +\begin{align*} +f(X)&=\langle X,AX\rangle_F + \langle B,X\rangle_F +\intertext{\uncover<7->{Folgerungen:}} +\uncover<8->{ +\langle X,AY\rangle_F&=\langle AX,Y\rangle_F +} +\\ +\uncover<9->{ +D\langle B,\mathstrut\cdot\mathstrut\rangle_F +&= +B +} +\\ +\uncover<10->{ +D_X\langle X, AY\rangle_F +&=AY +} +\\ +\uncover<11->{ +D_Y\langle X, AY\rangle_F +&=AX +} +\\ +\uncover<12->{ +Df &= 2AX + B +} +\intertext{\uncover<13->{Minimum:}} +\uncover<14->{ +X&=-\frac12 A^{-1}B +} +\end{align*} +\uncover<15->{(Kalman-Filter)} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/frobeniusnorm.tex b/vorlesungen/slides/2/frobeniusnorm.tex new file mode 100644 index 0000000..461005a --- /dev/null +++ b/vorlesungen/slides/2/frobeniusnorm.tex @@ -0,0 +1,96 @@ +% +% frobeniusnorm.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Frobenius-Norm} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Skalarprodukt} +$A,B\in M_{m\times n}(\mathbb{C})$ +\begin{align*} +\langle A,B\rangle_F +&\uncover<2->{= +\sum_{i,j} \overline{a}_{ik}b_{ik}} +\uncover<3->{= +\operatorname{Spur} A^*B} +\\ +\uncover<4->{ +\|A\|_F^2 +&= +\langle A,A\rangle} +\uncover<5->{= +\sum_{i,k} |a_{ik}|^2} +\end{align*} +\uncover<6->{% +$\Rightarrow M_{m\times n}(\mathbb{C})$ ist ein normierter Raum} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Singulärwertzerlegung} +\vspace{-12pt} +\begin{align*} +\uncover<13->{ +A +&= +U\Sigma V^*} +\\ +\uncover<14->{ +A^*A +&= +V\Sigma^*U^*U\Sigma V^*} +\uncover<15->{= +V\Sigma^*\Sigma V^*} +\\ +\uncover<16->{% +\operatorname{Spur}{A^*A} +&= +\operatorname{Spur}V\Sigma^*\Sigma V^*} +\\ +\uncover<17->{% +&= +\operatorname{Spur}V^*V\Sigma^*\Sigma} +\\ +\uncover<18->{% +&= +\operatorname{Spur}\Sigma^*\Sigma} +\uncover<19->{= +\sum_{i} |\sigma_i|^2} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<7->{% +\begin{block}{Produkt} +\vspace{-10pt} +\begin{align*} +\|AB\|_F +\uncover<8->{= +\sum_{i,j} +\biggl| +\sum_{k} +a_{ik}b_{kj} +\biggr|^2} +&\uncover<9->{\le +\sum_{i,j} +\biggl( +\sum_k |a_{ik}|^2 +\biggr) +\biggl( +\sum_l |b_{lj}|^2 +\biggr)} +\\ +\uncover<10->{ +&= +\sum_{i,k} |a_{ik}|^2 +\sum_{l,j} |b_{lj}|^2} +\uncover<11->{= +\|A\|_F\cdot \|B\|_F} +\end{align*} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex new file mode 100644 index 0000000..9116be4 --- /dev/null +++ b/vorlesungen/slides/2/funktionenalgebra.tex @@ -0,0 +1,88 @@ +% +% funktionenalgebra.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Funktionenalgebra} +\vspace{-17pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Algebra $C([0,1])$} +Funktionenraum +\[ +C([0,1]) += +\{f\colon[0,1]\to\mathbb{C}\;|\;\text{$f$ stetig}\} +\] +mit Supremum-Norm\uncover<2->{ und punktweisem Produkt +\[ +(f\cdot g)(x) += +f(x)\cdot g(x) +\]} +\end{block} +\vspace{-8pt} +\uncover<3->{% +\begin{block}{Algebranorm} +\vspace{-12pt} +\begin{align*} +\|f\cdot g\|_\infty +&= +\sup_{x\in[0,1]} |f(x)g(x)| +\\ +\uncover<4->{ +&\le +\sup_{x\in[0,1]}|f(x)| +\sup_{y\in[0,1]}|g(y)| +} +\\ +\uncover<5->{ +&= +\|f\|_\infty \cdot \|g\|_\infty +} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Faltungs-Algebra $L^2([0,1])$} +Funktionenraum +\[ +L^2=\{f\colon \mathbb{R}\to\mathbb{C}\;|\;\text{$f$ $1$-periodisch}\} +\] +mit $L^2$-Skalarprodukt\uncover<7->{ und Faltungsprodukt +\[ +f*g(x) += +\int_0^1 +\underbrace{f(x-t)}_{(=\gamma_x\check{f})(t)} g(t)\,dx +\]} +\end{block}} +\vspace{-21pt} +\uncover<8->{% +\begin{block}{Norm} +\vspace{-12pt} +\begin{align*} +\|f*g\|_2^2 +&\uncover<9->{=\int_0^1 | +\langle \gamma_x\check{f},g\rangle +|^2\,dx} +\\ +\uncover<10->{ +&\le +\int_0^1 +\|\gamma_t\check{f}\|_2^2 +\|g\|_2^2 +\,dx} +\\ +\uncover<11->{ +&=\|f\|_2^2\cdot \|g\|_2^2 +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/funktionenraum.tex b/vorlesungen/slides/2/funktionenraum.tex new file mode 100644 index 0000000..f7733cc --- /dev/null +++ b/vorlesungen/slides/2/funktionenraum.tex @@ -0,0 +1,70 @@ +% +% funktionenraum.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Funktionenraum} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Supremum-Norm} +Vektorraum +\[ +C([a,b]) += +\{f\colon[a,b]\to\mathbb{R}\;|\; \text{$f$ stetig}\} +\] +\only<2->{wird Banachraum }% +mit der Norm +\(\displaystyle +\|f\| += +\|f\|_{\infty} += +\sup_{x\in[a,b]} |f(x)| +\) +\end{block} +\uncover<3->{% +\begin{block}{$L^1$-Norm} +Vektorraum +\[ +L^1([a,b]) += +\{f\colon[a,b]\;|\;\text{$f$ integrierbar}\} +\] +\only<4->{wird Banachraum }% +mit der Norm +\[ +\|f\|_1 += +\int_a^b |f(x)|\,dx +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{$L^2$-Norm} +Vektorraum +\[ +L^2([a,b]) += +\{f\colon[a,b]\to\mathbb{R}\;|\; \|f\|_2^2<\infty\} +\] +mit Skalarprodukt +\begin{align*} +\langle f,g\rangle +&= +\int_a^b \overline{f}(x)g(x)\,dx +\\ +\|f\|_2^2 +&= +\int_a^b |f(x)|^2\,dx +\end{align*} +\uncover<6->{ist ein Banachraum} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/images/Makefile b/vorlesungen/slides/2/images/Makefile new file mode 100644 index 0000000..8bce5c9 --- /dev/null +++ b/vorlesungen/slides/2/images/Makefile @@ -0,0 +1,32 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: quotient1.jpg quotient2.jpg quotient1.pdf quotient2.pdf + +quotient1.png: quotient1.pov quotient.inc + povray +A0.1 +W1920 +H1080 -Oquotient1.png quotient1.pov + +quotient1.jpg: quotient1.png Makefile + convert -extract 1360x1040+330+20 quotient1.png \ + -density 300 -units PixelsPerInch quotient1.jpg + +quotient2.png: quotient2.pov quotient.inc + povray +A0.1 +W1920 +H1080 -Oquotient2.png quotient2.pov + +quotient2.jpg: quotient2.png Makefile + convert -extract 1360x1040+330+20 quotient2.png \ + -density 300 -units PixelsPerInch quotient2.jpg + +quotient: quotient.ini quotient.inc quotient.pov + rm -rf quotient + mkdir quotient + povray +A0.1 -Oquotient/0.png -W1920 -H1080 quotient.ini + +quotient1.pdf: quotient1.tex quotient1.jpg + pdflatex quotient1.tex + +quotient2.pdf: quotient2.tex quotient2.jpg + pdflatex quotient2.tex + diff --git a/vorlesungen/slides/2/images/quotient.inc b/vorlesungen/slides/2/images/quotient.inc new file mode 100644 index 0000000..3fa49d1 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient.inc @@ -0,0 +1,186 @@ +// +// quotient.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.035; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <8, 15, -50> + look_at <0.4, 0.2, 0.4> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-4, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#macro kasten() + box { <-0.5,-0.5,-0.5>, <1.5,1,1.5> } +#end + + +arrow(<-0.6,0,0>, <1.6,0,0>, at, White) +arrow(<0,0,-0.6>, <0,0,1.6>, at, White) +arrow(<0,-0.6,0>, <0,1.2,0>, at, White) + +#declare U = <-1,3,-0.5>; +#declare V1 = <1,0.2,0>; +#declare V2 = <0,0.2,1>; + +#macro gerade(richtung, farbe) + intersection { + kasten() + cylinder { -U + richtung, U + richtung, at } + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } + } +#end + +#declare A = <0.8, -0.2, 0>; +#declare B = <0.2, 0.8, 0>; + +#macro ebene(vektor1, vektor2) +#declare n = vcross(vektor1,vektor2); + + +intersection { + kasten() + plane { n, 0.005 } + plane { -n, 0.005 } + pigment { + color rgbf<0.8,0.8,1,0.7> + } + finish { + specular 0.9 + metallic + } +} + +intersection { + kasten() + union { + #declare Xstep = 0.45; + #declare X = -5 * Xstep; + #while (X < 5.5 * Xstep) + cylinder { X*vektor1 - 5*vektor2, X*vektor1 + 5*vektor2, at/2 } + #declare X = X + Xstep; + #end + #declare Ystep = 0.45; + #declare Y = -5 * Ystep; + #while (Y < 5.5 * Ystep) + cylinder { -5*vektor1 + Y*vektor2, 5*vektor1 + Y*vektor2, at/2 } + #declare Y = Y + Ystep; + #end + } + pigment { + color rgb<0.9,0.9,1> + } + finish { + specular 0.9 + metallic + } +} +#end + + +gerade(O, Red) + +#declare gruen = rgb<0.2,0.4,0.2>; +#declare blau = rgb<0,0.4,0.8>; +#declare rot = rgb<1,0.4,0.0>; + +#macro repraesentanten(vektor1, vektor2) + +#declare d1 = A.x*vektor1 + A.y*vektor2; +#declare d2 = B.x*vektor1 + B.y*vektor2; + +arrow(0, d1 + d2, at, rot) +gerade(d1 + d2, rot) + +gerade(d1, blau) +arrow(O, d1, at, blau) +cylinder { d1, d1 + d2, 0.6 * at + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } +} + +gerade(d2, gruen) +arrow(O, d2, at, gruen) +cylinder { d2, d1 + d2, 0.6 * at + pigment { + color blau + } + finish { + specular 0.9 + metallic + } +} + +#end + +#macro vektorraum(s) +#declare b1 = V1 + s * 0.03 * U; +#declare b2 = V2 + s * 0.03 * U; + +ebene(b1, b2) +repraesentanten(b1, b2) +#end + diff --git a/vorlesungen/slides/2/images/quotient.ini b/vorlesungen/slides/2/images/quotient.ini new file mode 100644 index 0000000..f62b21a --- /dev/null +++ b/vorlesungen/slides/2/images/quotient.ini @@ -0,0 +1,7 @@ +Input_File_Name="quotient.pov" +Initial_Frame=0 +Final_Frame=100 +Initial_Clock=-1 +Final_Clock=1 +Cyclic_Animation=off +Pause_when_Done=off diff --git a/vorlesungen/slides/2/images/quotient1.jpg b/vorlesungen/slides/2/images/quotient1.jpg Binary files differnew file mode 100644 index 0000000..aeb713e --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.jpg diff --git a/vorlesungen/slides/2/images/quotient1.pov b/vorlesungen/slides/2/images/quotient1.pov new file mode 100644 index 0000000..60bab7f --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.pov @@ -0,0 +1,8 @@ +// +// quotient1.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "quotient.inc" + +vektorraum(-1) diff --git a/vorlesungen/slides/2/images/quotient1.tex b/vorlesungen/slides/2/images/quotient1.tex new file mode 100644 index 0000000..30d82d2 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.tex @@ -0,0 +1,29 @@ +% +% quotient1.tex -- Vektorraumquotient +% +% (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} +\definecolor{darkred}{rgb}{0.7,0,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=8cm]{quotient1.jpg}}; + +\node[color=blue] at (0.7,-1.3) {$v$}; +\node[color=darkgreen] at (-1.0,0.1) {$w$}; +\node[color=orange] at (2.5,0.1) {$v+w$}; +\node[color=darkred] at (-2.1,-0.9) {$0$}; +\node[color=darkred] at (-3.1,2.4) {$U$}; + +\end{tikzpicture} +\end{document} + diff --git a/vorlesungen/slides/2/images/quotient2.jpg b/vorlesungen/slides/2/images/quotient2.jpg Binary files differnew file mode 100644 index 0000000..345cf22 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.jpg diff --git a/vorlesungen/slides/2/images/quotient2.pov b/vorlesungen/slides/2/images/quotient2.pov new file mode 100644 index 0000000..771425d --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.pov @@ -0,0 +1,8 @@ +// +// quotient2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "quotient.inc" + +vektorraum(1) diff --git a/vorlesungen/slides/2/images/quotient2.tex b/vorlesungen/slides/2/images/quotient2.tex new file mode 100644 index 0000000..607fd03 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.tex @@ -0,0 +1,29 @@ +% +% quotient2.tex -- Vektorraumquotient +% +% (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} +\definecolor{darkred}{rgb}{0.7,0,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=8cm]{quotient2.jpg}}; + +\node[color=blue] at (0.57,-0.94) {$v$}; +\node[color=darkgreen] at (-1.15,0.65) {$w$}; +\node[color=orange] at (2.15,1) {$v+w$}; +\node[color=darkred] at (-2.1,-0.9) {$0$}; +\node[color=darkred] at (-3.1,2.4) {$U$}; + +\end{tikzpicture} +\end{document} + diff --git a/vorlesungen/slides/2/linearformnormen.tex b/vorlesungen/slides/2/linearformnormen.tex new file mode 100644 index 0000000..8993f66 --- /dev/null +++ b/vorlesungen/slides/2/linearformnormen.tex @@ -0,0 +1,76 @@ +% +% linearformnormen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Linearformen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Linearformen $\varphi\colon L^1\to\mathbb{R}$} +Beispiel: $g\in C([a,b])$ +\[ +\varphi(f) += +\int_a^b g(x)f(x)\,dx +\] +\uncover<2->{% +erfüllt +\begin{align*} +|\varphi(f)| +&= +\biggl|\int_a^b g(x)f(x)\,dx\biggr| +\\ +\uncover<3->{ +&\le \|g\|_\infty\cdot \|f\|_1 +} +\end{align*}} +\uncover<4->{% +und hat daher die Operatornorm +\[ +\|\varphi\|_{C([a,b])^*} += +\|g\|_\infty +\]} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Linearformen $\varphi\colon L^2\to\mathbb{R}$} +\uncover<5->{% +Darstellungssatz von Riesz: $\exists g\in L^2$ +\[ +\varphi(f) = \langle g,f\rangle +\]} +\uncover<6->{% +erfüllt Cauchy-Schwarz} +\begin{align*} +\uncover<7->{ +|\varphi(f)| +&= +|\langle g,f\rangle|} +\\ +\uncover<8->{ +&\le +\|g\|_2 \cdot \|f\|_2 +} +\end{align*} +\uncover<9->{% +und hat daher die Operatornorm +\[ +\|\varphi\|_{L^2([a,b])^*} += \|g\|_2 +\]} +\end{block} +\end{column} +\end{columns} + +\vspace{8pt} +{\usebeamercolor[fg]{title} +\uncover<10->{% +$\Rightarrow$ +Operatornorm hängt von den Vektorraumnormen ab} +} +\end{frame} diff --git a/vorlesungen/slides/2/norm.tex b/vorlesungen/slides/2/norm.tex new file mode 100644 index 0000000..35d2513 --- /dev/null +++ b/vorlesungen/slides/2/norm.tex @@ -0,0 +1,58 @@ +% +% norm.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Norm} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Wozu} +Ziel: Konvergenz von Folgen, Grenzwert in einem Vektorraum +\end{block} +\uncover<7->{% +\begin{block}{Cauchy-Folge} +Eine Folge $(x_n)_{n\in\mathbb{N}}$ von Vektoren in $V$ heisst +{\em Cauchy-Folge}, +wenn es für alle $\varepsilon >0$ ein $N$ gibt mit +\[ +\|x_n-x_m\| < \varepsilon\; \forall n,m>N +\] +\end{block}} +\vspace{-8pt} +\uncover<8->{% +\begin{block}{Grenzwert} +$x\in V$ heisst Grenzwert der Folge $x_n$, wenn es für alle $\varepsilon>0$ +ein $N$ gibt mit +\[ +\| x-x_n\| < \varepsilon \;\forall n>N +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Definition} +$V$ ein $\mathbb{R}$-Vektorraum. +Eine Funktion +\[ +\|\cdot\| \colon V \to \mathbb{R}_{\ge 0} : v \mapsto \|v\| +\] +heisst eine {\em Norm}, wenn +\begin{itemize} +\item<3-> $\| v \|>0$ für $v\ne 0$ +\item<4-> $\|\lambda v\| = |\lambda|\cdot\|v\|$ +\item<5-> $\| u + v \| \le \|u\| + \|v\|$ (Dreiecksungleichung) +\end{itemize} +\uncover<6->{% +Ein Vektorraum mit einer Norm heisst {\em normierter Raum}} +\end{block}} +\uncover<9->{% +\begin{block}{Banach-Raum} +Normierter Raum, in dem jede Cauchy-Folge einen Grenwzert hat +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/operatornorm.tex b/vorlesungen/slides/2/operatornorm.tex new file mode 100644 index 0000000..d20461a --- /dev/null +++ b/vorlesungen/slides/2/operatornorm.tex @@ -0,0 +1,59 @@ +% +% operatorname.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Operatornorm} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Lineare Operatoren} +$A\colon U\to V$ lineare Abbildung mit $U$, $V$ normiert +\end{block}} +\uncover<3->{% +\begin{block}{Operatornorm} +eines linearen Operators $A$: +\[ +\|A\| += +\sup_{\|x\|_U\le 1} \|Ax\|_V +\] +\uncover<4->{$\Rightarrow \|Ax\| \le \| A \|\cdot \|x\|$} +\end{block}} +\uncover<5->{% +\begin{block}{Stetigkeit} +Wenn $\|A\|<\infty$, dann ist $A$ stetig, d.~h. +\[ +\lim_{n\to\infty} Ax_n += +A\lim_{n\to\infty} x_n +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Algebranorm} +$A$ ein normierter Raum, der auch ein Algebra ist. +Dann heisst $A$ eine normierte Algebra, wenn +\[ +\| ab\| \le \| a\|\cdot \|b\| +\quad\forall a,b\in A +\] +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Operatoralgebra} +$U$ ein normierter Raum, dann ist die Algebra der linearen Operatoren +$A\colon U\to U$ mit der Operatornorm eine normierte Algebra +\end{block}} +\uncover<8->{% +\begin{block}{Banach-Algebra} +Ein Banach-Raum, der auch eine normierte Algebra ist +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/polarformel.tex b/vorlesungen/slides/2/polarformel.tex new file mode 100644 index 0000000..ebdbf81 --- /dev/null +++ b/vorlesungen/slides/2/polarformel.tex @@ -0,0 +1,113 @@ +% +% polarformel.tex +% +% (c) 2021 Prod Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkcolor}{rgb}{0,0.6,0} +\def\yone{-2.1} +\def\ytwo{-3.55} +\def\ythree{-5.0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Polarformel} +\vspace{-5pt} +\begin{block}{Aufgabe} +$\langle x,y\rangle$ aus Werten von $\|\cdot\|_2$ rekonstruieren: + +\end{block} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\node at (0,0) {$ +\begin{aligned} +\uncover<2->{ +\|x+ty\|_2^2 +&= +\|x\|_2^2 ++t\langle x,y\rangle ++\overline{t}\langle y,x\rangle ++ \|y\|_2^2} +\\ +\uncover<3->{ +&= +\|x\|_2^2 ++t\langle x,y\rangle ++\overline{t\langle x,y\rangle} ++ \|y\|_2^2} +\\ +\uncover<4->{ +&= +\|x\|_2^2 ++2\operatorname{Re}(t\langle x,y\rangle) ++ \|y\|_2^2} +\end{aligned}$}; + +\uncover<5->{ + \draw[->] (-1,-0.9) -- (-3.3,{\yone+0.25}); + \node at (-3.5,\yone) {$ + \|x\pm y\|_2^2 + = + \|x\|_2^2 + \pm2\operatorname{Re}\langle x,y\rangle + + + \|y\|_2^2 + $}; +} + +\uncover<8->{ + \draw[->] (1,-0.9) -- (3.3,{\yone+0.25}); + \node at (3.5,\yone) {$ + \|x\pm iy\|_2^2 + = + \|x\|_2^2 + \pm2i\operatorname{Im}\langle x,y\rangle + + + \|y\|_2^2 + $}; +} + +\uncover<6->{ + \draw[->] (-3.5,{\yone-0.2}) -- (-3.5,{\ytwo+0.2}); + \node at (-3.5,\ytwo) {$\operatorname{Re}\langle x,y\rangle + = + \frac12\bigl( + \|x+y\|_2^2-\|x-y\|_2^2 + \bigr) + $}; +} + +\uncover<9->{ + \draw[->] (3.5,{\yone-0.2}) -- (3.5,{\ytwo+0.2}); + \node at (3.5,\ytwo) {$ + \operatorname{Im}\langle x,y\rangle + = + \frac1{2i}\bigl( + \|x+iy\|_2^2-\|x-iy\|_2^2 + \bigr) + $}; +} + +\uncover<7->{ + \draw[->] (-3.3,{\ytwo-0.25}) -- (-1.5,{\ythree+0.25}); + \node at (0,\ythree) {$ + \langle x,y\rangle + = + \frac12\bigl( + \|x+y\|_2^2-\|x-y\|_2^2 + \uncover<10->{ + + + \|x+iy\|_2^2-\|x-iy\|_2^2 + } + \bigr)$}; +} + +\uncover<10->{ + \draw[->] (3.3,{\ytwo-0.25}) -- (1.5,{\ythree+0.25}); +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/quotient.tex b/vorlesungen/slides/2/quotient.tex new file mode 100644 index 0000000..24b0523 --- /dev/null +++ b/vorlesungen/slides/2/quotient.tex @@ -0,0 +1,110 @@ +% +% quotient.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkred}{rgb}{0.7,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\s{0.3} +\def\punkt#1#2{({#1-3*#2},{8*#2})} +\def\gerade#1{ +\draw[darkgreen,line width=1.4pt] + \punkt{#1}{1} + -- + \punkt{#1}{-1}; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quotientenraum} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Einen Unterraum ``ignorieren''} +{\usebeamercolor[fg]{title}Gegeben:} $U\subset V$ ein Unterraum +\\ +{\usebeamercolor[fg]{title}Gesucht:} Eine Projektion auf einen Vektorraum, +in dem die Richtungen in $U$ zu $0$ gemacht werden +\end{block} +\uncover<2->{% +\begin{block}{Projektion} +In $V$ Klassen bilden: +\[ +\pi +\colon +v\mapsto +\llbracket v\rrbracket += +v+U +\] +\end{block}} +\vspace{-12pt} +\uncover<3->{% +\begin{block}{Quotientenraum} +\vspace{-12pt} +\begin{align*} +V/U +&= +\{ v+U\;|\; v\in V \} +\\ +\uncover<4->{\pi(\lambda v)&=\lambda v+U= \lambda \pi(v)} +\\ +\uncover<5->{\pi(v+w) +&= +v+w+U} +\ifthenelse{\boolean{presentation}}{ +\only<6>{= +v+U+w+U}}{} +\uncover<7->{= +\pi(v) + \pi(w)} +\phantom{blubb} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (U) at (-3,8); +\def\t{0.03} +\begin{scope} +\clip (-2,-2) rectangle (4,4.8); +\draw[color=darkred,line width=2pt] (-3,8) -- (1.5,-4); +\node[color=darkred] at (-1.45,4.6) {$U$}; +\node[color=darkred] at (-0.05,-0.05) [above left] {$0$}; + +\gerade{2.5} + +\ifthenelse{\boolean{presentation}}{ + \foreach \n in {8,...,25}{ + \pgfmathparse{(\n-12)*0.04} + \xdef\s{\pgfmathresult} + \only<\n>{ + \draw[color=blue,line width=1.2pt] + \punkt{-5}{-2*\s} -- \punkt{5}{2*\s}; + \draw[->,color=blue,line width=2pt] + (0,0) -- \punkt{2.5}{\s}; + \node[color=blue] at \punkt{2.5}{\s} + [above right] {$v'$}; + } + } +}{ + \xdef\s{0.35} + \draw[color=blue,line width=1.2pt] + \punkt{-5}{-2*\s} -- \punkt{5}{2*\s}; + \draw[->,color=blue,line width=2pt] (0,0) -- \punkt{2.5}{\s}; + \node[color=blue] at \punkt{2.5}{\s} [above right] {$v'$}; +} + +\draw[->,color=darkgreen,line width=1.4pt] (0,0) -- \punkt{2.5}{0.1}; + +\node[color=darkgreen] at \punkt{2.5}{0.1} [above right] {$v$}; + +\end{scope} +\draw[->] (-2,0) -- (4,0) coordinate[label={$x$}]; +\draw[->] (0,-2) -- (0,5) coordinate[label={right:$x$}]; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/quotientv.tex b/vorlesungen/slides/2/quotientv.tex new file mode 100644 index 0000000..dc01f21 --- /dev/null +++ b/vorlesungen/slides/2/quotientv.tex @@ -0,0 +1,62 @@ +% +% quotientv.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkred}{rgb}{0.7,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Quotient} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.33\textwidth} +\begin{block}{Repräsentanten} +Jeder Unterraum $W\subset V$ mit +$W\cap U = \{0\}$ +kann als Menge von Repräsentanten +für +\begin{align*} +V/U +&= +\{v+U\;|\;v \in V\} +\\ +&\simeq W +\end{align*} +dienen. +\end{block} +\uncover<3->{% +\begin{block}{Orthogonalraum} +Mit Skalarprodukt ist +$W=U^\perp$ eine bevorzugte Wahl +\end{block}} +\end{column} +\begin{column}{0.66\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\only<1>{ + \node at (0,0) + {\includegraphics[width=8.5cm]{../slides/2/images/quotient1.jpg}}; + \node[color=darkgreen] at (-0.5,0.3) {$v$}; + \node[color=blue] at (0.7,-1.4) {$w$}; + \node[color=orange] at (2.7,0.1) {$v+w$}; + \fill[color=white,opacity=0.5] (3.7,1.0) circle[radius=0.25]; + \node at (3.7,1.0) {$W$}; +} +\only<2->{ + \node at (0,0) + {\includegraphics[width=8.5cm]{../slides/2/images/quotient2.jpg}}; + \node[color=darkgreen] at (-0.75,0.95) {$v$}; + \node[color=blue] at (0.6,-1.05) {$w$}; + \node[color=orange] at (2.36,1.05) {$v+w$}; + \fill[color=white,opacity=0.5] (3.7,2.9) circle[radius=0.25]; + \node at (3.7,2.9) {$W$}; +} +\node[color=darkred] at (-3.3,2.6) {$U$}; +\node[color=darkred] at (-2.25,-1.0) {$0$}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex new file mode 100644 index 0000000..99d8a73 --- /dev/null +++ b/vorlesungen/slides/2/skalarprodukt.tex @@ -0,0 +1,96 @@ +% +% skalarprodukt.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Skalarprodukt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Positiv definite, symmetrische Bilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$ +\begin{itemize} +\item<2-> +Bilinear: +\begin{align*} +\langle \alpha u+\beta v,w\rangle +&= +\alpha\langle u,w\rangle ++ +\beta\langle v,w\rangle +\\ +\langle u,\alpha v+\beta w\rangle +&= +\alpha\langle u,v\rangle ++ +\beta\langle u,w\rangle +\end{align*} +\item<3-> +Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$ +\item<4-> +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Positive definite, hermitesche Sesquilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$ +\begin{itemize} +\item<6-> +Sesquilinear: +\begin{align*} +\langle \alpha u+\beta v,w\rangle +&= +\overline{\alpha}\langle u,w\rangle ++ +\overline{\beta}\langle v,w\rangle +\\ +\langle u,\alpha v+\beta w\rangle +&= +\alpha\langle u,v\rangle ++ +\beta\langle u,w\rangle +\end{align*} +\item<7-> +Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$ +\item<8-> +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.28\textwidth} +\uncover<9->{% +\begin{block}{$2$-Norm} +$\|v\|_2^2 = \langle v,v\rangle$ +\\ +$\|v\|_2 = \sqrt{\langle v,v\rangle}$ +\end{block}} +\end{column} +\begin{column}{0.78\textwidth} +\uncover<10->{% +\begin{itemize} +\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ +\item<12-> $\| \lambda v \|_2 += +\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut} += +\sqrt{\overline{\lambda}\lambda\langle v,v\rangle} += +|\lambda|\cdot \|v\|_2$ +\item<13-> +\raisebox{-8pt}{ +$\begin{aligned} +\|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2 +\\ +(\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2 +\end{aligned}$} +\end{itemize}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc new file mode 100644 index 0000000..442bd15 --- /dev/null +++ b/vorlesungen/slides/3/Makefile.inc @@ -0,0 +1,37 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter3 = \ + ../slides/3/motivation.tex \ + ../slides/3/inverse.tex \ + ../slides/3/polynome.tex \ + ../slides/3/division.tex \ + ../slides/3/division2.tex \ + ../slides/3/ringstruktur.tex \ + ../slides/3/teilbarkeit.tex \ + ../slides/3/ideal.tex \ + ../slides/3/nichthauptideal.tex \ + ../slides/3/nichthauptideal2.tex \ + ../slides/3/idealverband.tex \ + ../slides/3/maximalideal.tex \ + ../slides/3/quotientenring.tex \ + ../slides/3/faktorisierung.tex \ + ../slides/3/faktorzerlegung.tex \ + ../slides/3/einsetzen.tex \ + ../slides/3/maximalergrad.tex \ + ../slides/3/minimalbeispiel.tex \ + ../slides/3/fibonacci.tex \ + ../slides/3/minimalpolynom.tex \ + ../slides/3/drehmatrix.tex \ + ../slides/3/drehfaktorisierung.tex \ + ../slides/3/operatoren.tex \ + ../slides/3/adjunktion.tex \ + ../slides/3/adjalgebra.tex \ + ../slides/3/wurzel2.tex \ + ../slides/3/phi.tex \ + ../slides/3/multiplikation.tex \ + ../slides/3/chapter.tex + diff --git a/vorlesungen/slides/3/adjalgebra.tex b/vorlesungen/slides/3/adjalgebra.tex new file mode 100644 index 0000000..e65b621 --- /dev/null +++ b/vorlesungen/slides/3/adjalgebra.tex @@ -0,0 +1,43 @@ +% +% adjalgebra.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Adjunktion einer Nullstelle, abstrakt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom. + +\uncover<2->{% +\begin{block}{Existenz} +Es gibt ein ``Objekt'' $\alpha$ mit +\( +m(\alpha) = 0 +\) +\end{block}} + +\uncover<3->{% +\begin{block}{Körpererweiterung} +Der kleinste Körper, der $\Bbbk$ und $\alpha$ enthält ist +\[ +\Bbbk(\alpha) += +\left +\{ p(\alpha) +\;\left|\; +\begin{minipage}{8cm}\raggedright +$p\in\Bbbk[X]$ ein Polynom vom Grad +$\deg p<\deg m$ +\end{minipage} +\right. +\right\} +\] +\uncover<4->{Das Polynom $m$ definiert, wie mit $\alpha$ gerechnet werden +muss: +\[ +\alpha^n = -m_0-m_1\alpha-m_2\alpha^2 - \dots - m_{n-1}\alpha^{n-1} +\]} +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/adjunktion.tex b/vorlesungen/slides/3/adjunktion.tex new file mode 100644 index 0000000..a974a76 --- /dev/null +++ b/vorlesungen/slides/3/adjunktion.tex @@ -0,0 +1,35 @@ +% +% adjunktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Adjunktion einer Nullstelle von $m(X)$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom. +\uncover<2->{% +\[ +X^n = -m_{n-1}X^{n-1} - \dots - m_1X - m_0 +\] +}% +\uncover<3->{% +Nullstelle $W$ als Operator betrachten: +\[ +W = \begin{pmatrix} + 0& 0& 0&\dots & 0& -m_0\\ + 1& 0& 0&\dots & 0& -m_1\\ + 0& 1& 0&\dots & 0& -m_2\\ + 0& 0& 1&\dots & 0& -m_3\\ +\vdots&\vdots&\vdots&\ddots&\vdots& \vdots\\ + 0& 0& 0&\dots & 1&-m_{n-1} +\end{pmatrix} +\]} +\uncover<4->{% +Man kann nachrechnen, dass immer $m(W)=0$. +} +\medskip + +\uncover<5->{$\Rightarrow \Bbbk(W) = \{p(W)\;|\;p\in\Bbbk[X], \deg p<\deg m\}$ +ist ein Körper, in dem $m(X)$ faktorisiert werden kann $m(X) = (X-W)q(X)$.} +\end{frame} diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex new file mode 100644 index 0000000..3fbc3fd --- /dev/null +++ b/vorlesungen/slides/3/chapter.tex @@ -0,0 +1,33 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{3/motivation.tex} +\folie{3/inverse.tex} +\folie{3/polynome.tex} +\folie{3/division.tex} +\folie{3/division2.tex} +\folie{3/ringstruktur.tex} +\folie{3/teilbarkeit.tex} +\folie{3/ideal.tex} +\folie{3/nichthauptideal.tex} +\folie{3/nichthauptideal2.tex} +\folie{3/maximalideal.tex} +\folie{3/idealverband.tex} +\folie{3/quotientenring.tex} +\folie{3/faktorisierung.tex} +\folie{3/faktorzerlegung.tex} +\folie{3/einsetzen.tex} +\folie{3/maximalergrad.tex} +\folie{3/minimalbeispiel.tex} +\folie{3/fibonacci.tex} +\folie{3/minimalpolynom.tex} +\folie{3/drehmatrix.tex} +\folie{3/drehfaktorisierung.tex} +\folie{3/operatoren.tex} +\folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} +\folie{3/wurzel2.tex} +\folie{3/phi.tex} +\folie{3/multiplikation.tex} diff --git a/vorlesungen/slides/3/division.tex b/vorlesungen/slides/3/division.tex new file mode 100644 index 0000000..94df27b --- /dev/null +++ b/vorlesungen/slides/3/division.tex @@ -0,0 +1,32 @@ +% +% division.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Polynomdivision} +\begin{block}{Aufgabe} +Finde Quotient und Rest für +$a= X^4- X^3-7X^2+ X+6\in\mathbb{Z}[X]$ +und +$b= X^2+X+1\in\mathbb{Z}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Lösung} +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcrcrcrcr} +\llap{$($}X^4&-& X^3&-&7X^2&+& X&+&6\rlap{$)$}&\;\mathstrut:\mathstrut&(X^2&+&X&+&1)&=&\uncover<3->{X^2}&\uncover<7->{-&2X}&\uncover<11->{-&6}=q\\ +\uncover<4->{\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}}& & & & & & & & & & & & & & & & \\ + &\uncover<5->{-&2X^3&-&8X^2}&\uncover<6->{+& X}& & & & & & & & & & & & & & \\ + &\uncover<8->{\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}}& & & & & & & & & & & & & & \\ + & & &\uncover<9->{-&6X^2&+&3X}&\uncover<10->{+&6}& & & & & & & & & & & & \\ + & & &\uncover<12->{\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}}& & & & & & & & & & & & \\ + & & & & & &\uncover<13->{9X&+&12\rlap{$\mathstrut=r$}}& & & & & & & & & & & & +\end{array} +\] +\uncover<14->{ +Funktioniert weil $b$ normiert ist! +} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/division2.tex b/vorlesungen/slides/3/division2.tex new file mode 100644 index 0000000..0602598 --- /dev/null +++ b/vorlesungen/slides/3/division2.tex @@ -0,0 +1,34 @@ +% +% division2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Division in $\Bbbk[X]$} +\vspace{-5pt} +\begin{block}{Aufgabe} +Finde Quotienten und Rest der Polynome +$a(X) = X^4-X^3-7X^2+X+6$ +und +$b(X) = 2X^2+X+1$ +\end{block} +\uncover<2->{% +\begin{block}{Lösung} +\vspace{-15pt} +\[ +\arraycolsep=1.4pt +\renewcommand{\arraystretch}{1.2} +\begin{array}{rcrcrcrcrcrcrcrcrcrcr} +\llap{$($}X^4&-& X^3&-& 7X^2&+& X&+& 6\rlap{$)$}&\mathstrut\;:\mathstrut&(2X^2&+&X&+&1)&=&\uncover<3->{\frac12X^2}&\uncover<7->{-&\frac34X}&\uncover<11->{-\frac{27}{8}} = q\\ +\uncover<4->{\llap{$-($}X^4&+&\frac12X^3&+& \frac12X^2\rlap{$)$}}& & & & & & & & & & & & & & & \\ + &\uncover<5->{-&\frac32X^3&-&\frac{15}2X^2}&\uncover<6->{+& X}& & & & & & & & & & & & & \\ + &\uncover<8->{\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}}& & & & & & & & & & & & & \\ + & & &\uncover<9->{-&\frac{27}4X^2&+&\frac{ 7}4X}&\uncover<10->{+& 6}& & & & & & & & & & & \\ + & & &\uncover<12->{\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}}& & & & & & & & & & & \\ + & & & & & &\uncover<13->{\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}}& & & & & & & & & & & \\ +\end{array} +\] +Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/drehfaktorisierung.tex b/vorlesungen/slides/3/drehfaktorisierung.tex new file mode 100644 index 0000000..64418d5 --- /dev/null +++ b/vorlesungen/slides/3/drehfaktorisierung.tex @@ -0,0 +1,75 @@ +% +% drehfaktorisierung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{4pt} +\setlength{\belowdisplayskip}{4pt} +\frametitle{Faktorisierung von $X^2+X+1$} +\vspace{-3pt} +$X^2+X+1$ kann faktorisiert werden, wenn man $i\sqrt{3}$ +hinzufügt: +\uncover<2->{% +\[ +\biggl(X+\frac12+\frac{i\sqrt{3}}2\biggr) +\biggl(X+\frac12-\frac{i\sqrt{3}}2\biggr) += +X^2+X+\frac14 ++ +\frac34 +\uncover<3->{= +X^2+X+1} +\]} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Was ist $i\sqrt{3}$?} +Matrix mit Minimalpolynom $X^2+3$: +\[ +W=\begin{pmatrix}0&-3\\1&0\end{pmatrix} +\uncover<5->{% +\qquad\Rightarrow\qquad +W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I} +\uncover<6->{% +\qquad\Rightarrow\qquad +W^2+3I=0} +\] +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Faktorisierung von $X^2+X+1$} +\vspace{-10pt} +\begin{align*} +\uncover<8->{B_\pm +&= +-\frac12I\pm\frac12W} +& +&\uncover<10->{\Rightarrow +& +(X+B_+)(X+B_-)} +&\uncover<11->{= +(X+\frac12I+\frac12W) +(X+\frac12I-\frac12W)} +\\ +&\uncover<9->{= +\smash{ +{\textstyle\begin{pmatrix}-\frac12&-\frac32\\\frac12&-\frac12\end{pmatrix}} +}} +& +& +& +&\uncover<12->{= +X^2+X + \frac14I - \frac14W^2} +\\ +& +& +&%\Rightarrow +& +&\uncover<13->{= +X^2+X + \frac14I + \frac34I} +\uncover<14->{= +X^2+X+I} +\end{align*} +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/drehmatrix.tex b/vorlesungen/slides/3/drehmatrix.tex new file mode 100644 index 0000000..9e5eb65 --- /dev/null +++ b/vorlesungen/slides/3/drehmatrix.tex @@ -0,0 +1,66 @@ +% +% drehmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Analyse einer Drehung um $120^\circ$} +$D$ eine Drehung des $\mathbb{R}^3$ um $120^\circ$ +\begin{enumerate} +\item<2-> +Drehwinkel = $120^\circ\quad\Rightarrow\quad D^3 = I$ +\uncover<3->{ +$\quad\Rightarrow\quad \chi_D(X)=X^3-1$ +} +\item<4-> +$m_D(X)=X^3-1$ +\item<5-> +$m_D$ ist nicht irreduzibel, weil $m_D(1)=0$: +$ +m_D(X) = (X-1)(X^2+X+1) +$ +\item<6-> +Welche Matrix hat $X^2+X+1$ als Minimalpolynom? +\uncover<7->{% +\[ +\arraycolsep=1.4pt +W += +\biggl(\begin{array}{cc} +-\frac12 & -\frac{\sqrt{3}}2 \\ + \frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) +\quad\Rightarrow\quad +W^2+W+I += +\biggl(\begin{array}{cc} +-\frac12 & -\frac{\sqrt{3}}2 \\ + \frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) ++ +\biggl(\begin{array}{cc} +-\frac12 & \frac{\sqrt{3}}2 \\ + -\frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) ++ +\biggl(\begin{array}{cc} +1&0\\0&1 +\end{array}\biggr) +=0 +\]} +\item<8-> In einer geeigneten Basis hat $D$ die Form +\[ +D=\begin{pmatrix} +1&0&0\\ +0&-\frac12 & -\frac{\sqrt{3}}2 \\ +0&\frac{\sqrt{3}}2 & -\frac12 +\end{pmatrix} +\uncover<9->{= +\begin{pmatrix} +1&0&0\\ +0&\cos 120^\circ & -\sin 120^\circ\\ +0&\sin 120^\circ & \cos 120^\circ +\end{pmatrix}} +\] +\end{enumerate} +\end{frame} diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex new file mode 100644 index 0000000..7f54abb --- /dev/null +++ b/vorlesungen/slides/3/einsetzen.tex @@ -0,0 +1,54 @@ +% +% einsetzen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Matrix in ein Polynom einsetzen} +\vspace{-10pt} +\[ +\begin{array}{rcrcrcrcrcrcr} +p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\ +\uncover<2->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<3->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<4->{\bigg\downarrow\hspace*{10pt}} & & & & +\uncover<5->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<6->{\bigg\downarrow\hspace*{2pt}} & & +\uncover<7->{\bigg\downarrow\hspace*{0pt}} \\ +\uncover<2->{p(A)}&\uncover<3->{=&a_nA^n}&\uncover<4->{+&a_{n-1}A^{n-1}}&\uncover<5->{+&\dots&+&a_2A^2}&\uncover<6->{+&a_1A}&\uncover<7->{+&a_0 I} +\end{array} +\] +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Nilpotente Matrizen} +$p(X) = (X-a)^n$ +\[ +\uncover<9->{p(A) = 0} +\uncover<10->{ +\quad\Rightarrow\quad +\text{$A-aI$ ist nilpotent}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Eigenwerte} +$p(X) = (X-\lambda_1)(X-\lambda_2)$,\\ +$A$ eine $2\times 2$-Matrix +\[ +\uncover<12->{p(A)=0} +\uncover<13->{\quad\Rightarrow\quad +\left\{ +\begin{aligned} +&\text{$A-\lambda_1I$ ist singulär}\\ +&\text{$A-\lambda_2I$ ist singulär} +\end{aligned} +\right.} +\] +\end{block}} +\end{column} +\end{columns} + +\end{frame} diff --git a/vorlesungen/slides/3/faktorisierung.tex b/vorlesungen/slides/3/faktorisierung.tex new file mode 100644 index 0000000..b4ea1d5 --- /dev/null +++ b/vorlesungen/slides/3/faktorisierung.tex @@ -0,0 +1,47 @@ +% +% faktorisierung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Faktorisierung} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Primzahlen\strut} +Eine Zahl $p\in\mathbb{Z}$, $p>1$ heisst Primzahl, wenn sie nicht als Produkt +$p=ab$ mit $a,b\in\mathbb{Z},a>1, b>1$ geschrieben werden kann. +\begin{align*} +\uncover<2->{p&=7} +\\ +\uncover<3->{2021 &= 43 \cdot 47} +\\ +\uncover<4->{2048 &= 2^{11}} +\\ +\uncover<5->{4095667&=2021\cdot 2027} +\\ +\uncover<6->{p&=43, 47, 1291, 2017, 2027} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Irreduzible Polynome in $\mathbb{Q}[X]$} +Ein Polynome $p\in\mathbb{Q}[X]$, $\deg p>0$ wenn es nicht als Produkt +$p=ab$ mit $a,b\in\mathbb{Q}[X]$, $\deg a>0$, $\deg b>0$ geschrieben +werden kann. +\begin{align*} +\uncover<8->{p&=X-9} +\\ +\uncover<9->{X^2-1&= (X+1)(X-1)} +\\ +\uncover<10->{X^2-2&\text{\; irreduzibel}} +\\ +\uncover<11->{X^2-2&=(X-\sqrt{2})(X+\sqrt{2})} +\end{align*} +\uncover<12->{% +aber: $X\pm\sqrt{2}\not\in\mathbb{Q}[X]$ +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/faktorzerlegung.tex b/vorlesungen/slides/3/faktorzerlegung.tex new file mode 100644 index 0000000..eb44cf3 --- /dev/null +++ b/vorlesungen/slides/3/faktorzerlegung.tex @@ -0,0 +1,62 @@ +% +% faktorzerlegung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Faktorzerlegung} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{in $\mathbb{Z}$} +Jede Zahl kann eindeutig in Primfaktoren zerlegt werden: +\[ +z = p_1^{n_1}\cdot p_2^{n_2} \cdot\dots\cdot p_k^{n_k} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{in $\mathbb{Q}[X]$} +Jedes Polynom $p\in\mathbb{Q}[X]$ +kann eindeutig faktorisiert werden in irreduzible, normierte Polynome +\[ +p += +a_n +p_1^{n_1} +\cdot +p_2^{n_2} +\cdot +\dots +\cdot +p_k^{n_k} +\] +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Polynomfaktorisierung hängt vom Koeffizientenring ab} +Ist $X^2-2$ irreduzibel? +\vspace{-5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{in $\mathbb{Q}[X]$} +\[ +X^2-2\quad\text{ist irreduzibel} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{in $\mathbb{R}[X]$} +\[ +X^2-2 = (X-\sqrt{2})(X+\sqrt{2}) +\] +\end{block}} +\end{column} +\end{columns} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex new file mode 100644 index 0000000..3d01020 --- /dev/null +++ b/vorlesungen/slides/3/fibonacci.tex @@ -0,0 +1,71 @@ +% +% fibonacci.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\begin{frame}[t] +\frametitle{Fibonacci} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{block}{Fibonacci-Rekursion} +$x_i$ Fibonacci-Zahlen\uncover<2->{, d.~h.~$x_{n+1\mathstrut}=x_{n\mathstrut}+x_{n-1\mathstrut}$} +\[ +\uncover<3->{ +v_n += +\begin{pmatrix} +x_{n+1}\\ +x_n +\end{pmatrix}} +\uncover<4->{ +\quad\Rightarrow\quad +v_n = +\underbrace{ +\begin{pmatrix} +1&1\\ +1&0 +\end{pmatrix} +}_{\displaystyle=\Phi} +v_{n-1}} +\uncover<5->{ +\quad\Rightarrow\quad +v_n += +\Phi^n +v_0}\uncover<6->{, +\; +v_0 = \begin{pmatrix} 1\\0\end{pmatrix}} +\] +\end{block} +\vspace{-5pt} +\uncover<7->{% +\begin{block}{Rekursionsformel für $\Phi$} +\vspace{-12pt} +\begin{align*} +v_{n}&=v_{n-1}+v_{n-2} +&&\uncover<8->{\Rightarrow& +\Phi^n v_0 &= \Phi^{n-1} v_0 + \Phi^{n-2}v_0} +&&\uncover<9->{\Rightarrow& +\Phi^{n-2}(\Phi^2-\Phi-I)v_0&=0} +\\ +\end{align*} +\vspace{-22pt}% + +\uncover<10->{$\Phi$ ist $\chi_\Phi(X)=m_\Phi(X) = X^2-X-1$, irreduzibel} +\end{block}} + +\uncover<11->{% +\begin{block}{Faktorisierung} +\vspace{-12pt} +\[ +(X-\Phi)(X-(I-\Phi)) +\uncover<12->{= +X^2-X +\Phi(I-\Phi)} +\uncover<13->{= +X^2-X -(\underbrace{\Phi^2-\Phi}_{\displaystyle=I}) +} +\] +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/ideal.tex b/vorlesungen/slides/3/ideal.tex new file mode 100644 index 0000000..f7f432e --- /dev/null +++ b/vorlesungen/slides/3/ideal.tex @@ -0,0 +1,63 @@ +% +% ideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ideal} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Voraussetzungen} +$R$ ein Ring, $r\in R$ +\end{block} +\uncover<2->{% +\begin{block}{Vielfache\uncover<4->{ = Hauptideal}} +Die Menge aller Elemente, die durch $r$ teilbar sind\uncover<3->{: +\[ +(r)=rR +\]} +\uncover<4->{heisst {\em Hauptideal}} +\end{block}} +\uncover<5->{% +\begin{block}{Ideal} +$I\subset R$ mit +\(RI\subset I\), \(I+I\subset I\) +\end{block}} +\uncover<6->{% +\begin{block}{Hauptidealring} +Jedes Ideal von $R$ ist ein Hauptideal +\\ +\uncover<7->{{\usebeamercolor[fg]{title}Beispiele:} +$\mathbb{Z}$, +$\Bbbk[X]$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Grösster gemeinsamer Teiler} +$a,b\in R$ +\begin{align*} +\uncover<9->{(a) + (b) +&= aR + bR} +\intertext{\uncover<10->{ist eine Ideal }\uncover<11->{$\Rightarrow$ ein Hauptideal}} +&\uncover<12->{= cR}\uncover<13->{ = \operatorname{ggT}(a,b)R} +\end{align*} +\uncover<14->{Existenz des $\operatorname{ggT}(a,b)$ ist eine +gemeinsame Eigenschaft} +\end{block}} +\uncover<15->{% +\begin{block}{Allgemein} +\begin{itemize} +\item<16-> +Alle euklidischen Ringe sind Hauptidealringe +\item<17-> +Alle solchen Ringe verwenden den gleichen Algorithmus +für $\operatorname{ggT}(a,b)$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/idealverband.tex b/vorlesungen/slides/3/idealverband.tex new file mode 100644 index 0000000..3434868 --- /dev/null +++ b/vorlesungen/slides/3/idealverband.tex @@ -0,0 +1,78 @@ +% +% idealverband.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Idealverband} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {$\mathbb{Z}$}; + +\uncover<2->{ +\node at (-6,-2) {$2\mathbb{Z}$}; +\node at (-2,-2) {$3\mathbb{Z}$}; +\node at (2,-2) {$5\mathbb{Z}$}; +\node at (6,-2) {$7\mathbb{Z}$}; +\node at (7,-2) {$\dots$}; +} + +\uncover<3->{ +\node at (-4,-4) {$6\mathbb{Z}$}; +\node at (-2,-4) {$10\mathbb{Z}$}; +\node at (0,-4) {$15\mathbb{Z}$}; +\node at (2,-4) {$21\mathbb{Z}$}; +\node at (4,-4) {$35\mathbb{Z}$}; +\node at (6,-4) {$\dots$}; +} + +\uncover<4->{ +\node at (-2,-6) {$30\mathbb{Z}$}; +\node at (0,-6) {$70\mathbb{Z}$}; +\node at (2,-6) {$105\mathbb{Z}$}; +} + +\uncover<5->{ + \node at (-5,-6) {$\dots$}; + \node at (5,-6) {$\dots$}; +} + +\uncover<2->{ +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-6,-2); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-2,-2); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (2,-2); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (6,-2); +} + +\uncover<3->{ +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-4,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-2,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (-4,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (0,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (2,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (-2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (0,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (4,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (4,-4); +} + +\uncover<4->{ +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-4,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (0,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (-2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (4,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (0,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (4,-4); +} + +\end{tikzpicture} +\end{center} +\end{frame} diff --git a/vorlesungen/slides/3/images/Makefile b/vorlesungen/slides/3/images/Makefile new file mode 100644 index 0000000..e338fcf --- /dev/null +++ b/vorlesungen/slides/3/images/Makefile @@ -0,0 +1,55 @@ +# +# Makefile -- build images +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: hauptideal.jpg nichthauptideal.jpg hauptideal2.jpg hauptidealX.jpg \ + hauptidealR.jpg hauptidealXR.jpg ring.jpg + +ring.png: ring.pov common.inc + povray +A0.1 +W1920 +H1080 -Oring.png ring.pov +ring.jpg: ring.png + convert ring.png -density 300 -units PixelsPerInch ring.jpg + +hauptideal.png: hauptideal.pov common.inc + povray +A0.1 +W1920 +H1080 -Ohauptideal.png hauptideal.pov +hauptideal.jpg: hauptideal.png + convert hauptideal.png -density 300 -units PixelsPerInch \ + hauptideal.jpg + +hauptidealR.png: hauptidealR.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealR.png hauptidealR.pov +hauptidealR.jpg: hauptidealR.png + convert hauptidealR.png -density 300 -units PixelsPerInch \ + hauptidealR.jpg + +hauptideal2.png: hauptideal2.pov common.inc + povray +A0.1 +W1920 +H1080 -Ohauptideal2.png hauptideal2.pov +hauptideal2.jpg: hauptideal2.png + convert hauptideal2.png -density 300 -units PixelsPerInch \ + hauptideal2.jpg + +hauptidealX.png: hauptidealX.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealX.png hauptidealX.pov +hauptidealX.jpg: hauptidealX.png + convert hauptidealX.png -density 300 -units PixelsPerInch \ + hauptidealX.jpg + +hauptidealXR.png: hauptidealXR.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealXR.png hauptidealXR.pov +hauptidealXR.jpg: hauptidealXR.png + convert hauptidealXR.png -density 300 -units PixelsPerInch \ + hauptidealXR.jpg + +nichthauptideal.png: nichthauptideal.pov common.inc + povray +A0.1 +W1920 +H1080 -Onichthauptideal.png nichthauptideal.pov +nichthauptideal.jpg: nichthauptideal.png + convert nichthauptideal.png -density 300 -units PixelsPerInch \ + nichthauptideal.jpg + +ideal: ideal.pov ideal.ini common.inc + rm -rf ideal + mkdir ideal + povray +A0.1 +W1920 +H1080 -Oideal/i.png ideal.ini + + diff --git a/vorlesungen/slides/3/images/common.inc b/vorlesungen/slides/3/images/common.inc new file mode 100644 index 0000000..36c4e6b --- /dev/null +++ b/vorlesungen/slides/3/images/common.inc @@ -0,0 +1,277 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.40; +#declare O = <0, 0, 0>; +#declare at = 0.10; + +#declare Xunten = -10; +#declare Xoben = 10; +#declare Yunten = -8; +#declare Yoben = 8; +#declare Zunten = 0; +#declare Zoben = 20; + +#declare phi0 = 2 * pi * 290 / 360; + +camera { + location <60 * cos(2*pi*T+phi0), 15, 60 * sin(2*pi*T+phi0) + 10> + look_at <0, -2, 10> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-14, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +light_source { + <41, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +arrow(< -12.0, 0.0, 0 >, < 12.0, 0.0, 0.0 >, at, Gray) +arrow(< 0.0, 0.0, -2.0>, < 0.0, 0.0, 22.0 >, at, Gray) +arrow(< 0.0, -10.0, 0 >, < 0.0, 10.0, 0.0 >, at, Gray) + +#macro kasten() + box { <-10.5,-8.5,-0.5>, <10.5,8.5,20.5> } +#end + +#declare gruen = rgb<0.2,0.4,0.2>; +#declare blau = rgb<0.0,0.4,0.8>; +#declare rot = rgb<1.0,0.4,0.0>; + +#declare r = 0.4; + +#macro Zring() + union { + #declare X = Xunten; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X, Y, Z>, r } + + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 1; + #end + pigment { + color rot + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Hauptideal() + union { + #declare A = Xunten; + #while (A <= Xoben + 0.5) + #declare B = Zunten; + #while (B <= Zoben + 0.5) + #declare Y = A + B; + #if ((Y >= Yunten - 0.5) & (Y <= Yoben + 0.5)) + sphere { <A, Y, B>, r } + #end + #declare B = B + 1; + #end + #declare A = A + 1; + #end + pigment { + color blau + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro HauptidealR() + intersection { + kasten() + #declare n = vnormalize(< 1, -2, 1 >); + plane { n, 0.1 } + plane { -n, 0.1 } + pigment { + color blau + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Ideal2() + union { + #declare X = Xunten; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X, Y, Z>, r } + #declare Z = Z + 2; + #end + #declare Y = Y + 2; + #end + #declare X = X + 2; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro IdealX() + union { + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <0, Y, Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro IdealXR() + intersection { + kasten() + plane { <1,0,0>, 0.1 } + plane { <-1,0,0>, 0.1 } + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Nichthauptideal() + union { + #declare X = Xunten/2; + #while (X <= Xoben/2 + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = 0; + #while (Z <= Zoben + 0.5) + sphere { <2*X,Y,Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 1; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro NichthauptidealKomplement() + union { + #declare X = Xunten + 1; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X,Y,Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 2; + #end + pigment { + color rot + } + finish { + specular 0.9 + metallic + } + } +#end + + + + + + + diff --git a/vorlesungen/slides/3/images/hauptideal.jpg b/vorlesungen/slides/3/images/hauptideal.jpg Binary files differnew file mode 100644 index 0000000..769f53c --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal.jpg diff --git a/vorlesungen/slides/3/images/hauptideal.pov b/vorlesungen/slides/3/images/hauptideal.pov new file mode 100644 index 0000000..a934e57 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal.pov @@ -0,0 +1,10 @@ +// +// hauptideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Hauptideal() + diff --git a/vorlesungen/slides/3/images/hauptideal2.jpg b/vorlesungen/slides/3/images/hauptideal2.jpg Binary files differnew file mode 100644 index 0000000..51823f3 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal2.jpg diff --git a/vorlesungen/slides/3/images/hauptideal2.pov b/vorlesungen/slides/3/images/hauptideal2.pov new file mode 100644 index 0000000..9da5a1a --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal2.pov @@ -0,0 +1,10 @@ +// +// hauptideal2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Ideal2() + diff --git a/vorlesungen/slides/3/images/hauptidealR.jpg b/vorlesungen/slides/3/images/hauptidealR.jpg Binary files differnew file mode 100644 index 0000000..fae5840 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealR.jpg diff --git a/vorlesungen/slides/3/images/hauptidealR.pov b/vorlesungen/slides/3/images/hauptidealR.pov new file mode 100644 index 0000000..330e523 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealR.pov @@ -0,0 +1,10 @@ +// +// hauptidealR.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +HauptidealR() + diff --git a/vorlesungen/slides/3/images/hauptidealX.jpg b/vorlesungen/slides/3/images/hauptidealX.jpg Binary files differnew file mode 100644 index 0000000..f9b4540 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealX.jpg diff --git a/vorlesungen/slides/3/images/hauptidealX.pov b/vorlesungen/slides/3/images/hauptidealX.pov new file mode 100644 index 0000000..d0045f9 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealX.pov @@ -0,0 +1,10 @@ +// +// hauptidealX.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +IdealX() + diff --git a/vorlesungen/slides/3/images/hauptidealXR.jpg b/vorlesungen/slides/3/images/hauptidealXR.jpg Binary files differnew file mode 100644 index 0000000..d8906c8 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealXR.jpg diff --git a/vorlesungen/slides/3/images/hauptidealXR.pov b/vorlesungen/slides/3/images/hauptidealXR.pov new file mode 100644 index 0000000..5daa3e6 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealXR.pov @@ -0,0 +1,10 @@ +// +// hauptidealXR.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +IdealXR() + diff --git a/vorlesungen/slides/3/images/ideal.ini b/vorlesungen/slides/3/images/ideal.ini new file mode 100644 index 0000000..66aa191 --- /dev/null +++ b/vorlesungen/slides/3/images/ideal.ini @@ -0,0 +1,7 @@ +Input_File_Name=ideal.pov +Initial_Frame=0 +Final_Frame=2500 +Initial_Clock=0 +Final_Clock=5 +Cyclic_Animation=off +Pause_when_Done=off diff --git a/vorlesungen/slides/3/images/ideal.pov b/vorlesungen/slides/3/images/ideal.pov new file mode 100644 index 0000000..88afaf7 --- /dev/null +++ b/vorlesungen/slides/3/images/ideal.pov @@ -0,0 +1,26 @@ +// +// ideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = clock; +#include "common.inc" + +#if (T < 1) +Zring() +#else + #if (T < 2) + Hauptideal() + #else + #if (T < 3) + Ideal2() + #else + #if (T < 4) + IdealX() + #else + Nichthauptideal() + NichthauptidealKomplement() + #end + #end + #end +#end diff --git a/vorlesungen/slides/3/images/nichthauptideal.jpg b/vorlesungen/slides/3/images/nichthauptideal.jpg Binary files differnew file mode 100644 index 0000000..55858d0 --- /dev/null +++ b/vorlesungen/slides/3/images/nichthauptideal.jpg diff --git a/vorlesungen/slides/3/images/nichthauptideal.pov b/vorlesungen/slides/3/images/nichthauptideal.pov new file mode 100644 index 0000000..72a6330 --- /dev/null +++ b/vorlesungen/slides/3/images/nichthauptideal.pov @@ -0,0 +1,10 @@ +// +// hauptideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Nichthauptideal() +NichthauptidealKomplement() diff --git a/vorlesungen/slides/3/images/ring.jpg b/vorlesungen/slides/3/images/ring.jpg Binary files differnew file mode 100644 index 0000000..27721b1 --- /dev/null +++ b/vorlesungen/slides/3/images/ring.jpg diff --git a/vorlesungen/slides/3/images/ring.pov b/vorlesungen/slides/3/images/ring.pov new file mode 100644 index 0000000..f854335 --- /dev/null +++ b/vorlesungen/slides/3/images/ring.pov @@ -0,0 +1,10 @@ +// +// ring.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Zring() + diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex new file mode 100644 index 0000000..4ad22d2 --- /dev/null +++ b/vorlesungen/slides/3/inverse.tex @@ -0,0 +1,89 @@ +% +% inverse.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Inverse Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.24\textwidth} +\begin{block}{Imaginäre Einheit} +\vspace{-15pt} +\begin{align*} +J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix} +\\ +0&= +J^2 + I +\\ +0&= +J+J^{-1} +\\ +J^{-1}&=-J +\end{align*} +\end{block} +\end{column} +\begin{column}{0.25\textwidth} +\uncover<2->{% +\begin{block}{Wurzel $\sqrt{2}$} +\vspace{-15pt} +\begin{align*} +W&=\begin{pmatrix}0&2\\1&0\end{pmatrix} +\\ +0 &= X^2-2 +\\ +0 &= W-2W^{-1} +\\ +W^{-1}&=\frac12 W +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.41\textwidth} +\uncover<3->{% +\begin{block}{Drehmatrix} +\vspace{-15pt} +\begin{align*} +D&=\begin{pmatrix} +\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\ +\sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} +\end{pmatrix} +\\ +0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{D-2\cos\frac{\pi\mathstrut}{1291\mathstrut}+D^{-1}} +\\ +D^{-1} +&= \only<-3>{-D^{1290}\phantom{+\frac{\mathstrut}{\mathstrut}}}% +\only<4->{-D+2I\cos\frac{\pi\mathstrut}{1291\mathstrut}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-25pt} +\uncover<5->{ +\begin{block}{3D-Beispiel} +$p(x) = -x^3-5x^2+5x+1$ +\[ +A= +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*} +\quad\Rightarrow\quad +A^{-1} += +A^2+5A-5I += +\begin{pmatrix*}[r] +-1& 1&-1\\ + 7&-9&10\\ + 3&-4& 5 +\end{pmatrix*} +\] +\end{block}} +\vspace{-10pt} + +\end{frame} diff --git a/vorlesungen/slides/3/maximalergrad.tex b/vorlesungen/slides/3/maximalergrad.tex new file mode 100644 index 0000000..d33ddc0 --- /dev/null +++ b/vorlesungen/slides/3/maximalergrad.tex @@ -0,0 +1,72 @@ +% +% maximalergrad.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Jede Matrix hat eine Polynomrelation} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-5pt} +\begin{block}{Dimension des Matrizenrings} +Der Ring $M_{n}(\Bbbk)$ ist ein $n^2$-dimensionaler Vektorraum mit +Basis +{\tiny +\begin{align*} +&\uncover<2->{\begin{pmatrix} +1&0&\dots&0\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<3->{\begin{pmatrix} +0&1&\dots&0\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<4->{\dots} +& +&\uncover<5->{\begin{pmatrix} +0&0&\dots&1\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +\\ +&\uncover<6->{\begin{pmatrix} +0&0&\dots&0\\ +1&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<7->{\begin{pmatrix} +0&0&\dots&0\\ +0&1&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<8->{\dots} +& +&\uncover<9->{\begin{pmatrix} +0&0&\dots&0\\ +0&0&\dots&1\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +\end{align*}} +\end{block} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Potenzen von $A$} +Die $n^2+1$ Matrizen $I,A,A^2,\dots,A^{n^2-1},A^{n^2}$ müssen linear abhängig +sein: +\[ +\uncover<11->{ +a_0I+a_1A+a_2A^2+\dots+a_{n^2-1}A^{n^2-1}+a_{n^2}A^{n^2} = 0 +} +\] +\uncover<12->{d.~h.~$p(X) = a_0+a_1X+a_2X^2+\dots +a_{n^2-1}X^{n^2-1}+a_{n^2}A^{n^2}\in\Bbbk[X]$ ist ein Polynom mit $p(A)=0$.} +\end{block}} +\uncover<13->{% +$\Rightarrow$ $A$ über die Eigenschaften (Faktorisierung) von $p$ studieren +} +\end{frame} diff --git a/vorlesungen/slides/3/maximalideal.tex b/vorlesungen/slides/3/maximalideal.tex new file mode 100644 index 0000000..21a945a --- /dev/null +++ b/vorlesungen/slides/3/maximalideal.tex @@ -0,0 +1,64 @@ +% +% maximalideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Maximale Ideale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Teilbarkeit} +$a|b$ +\uncover<2->{$\Rightarrow$ +$b\in aR$} +\uncover<3->{$\Rightarrow$ +$bR\subset aR$} +\end{block} +\uncover<4->{% +\begin{block}{Nicht mehr teilbar} +$a\in R$ nicht faktorisierbar +\\ +\uncover<5->{$\Rightarrow$ +\\ +es gibt kein Ideal zwischen $aR$ und $R$} +\\ +\uncover<6->{$\Leftrightarrow$ +\\ +$J$ ein Ideal +$aR \subset J \subset R$, dann ist +$J=aR$ oder $J=R$} +\end{block}} +\uncover<7->{ +\begin{block}{maximales Ideal} +$I\subset R$ heisst maximal, wenn für jedes Ideal $J$ +mit $I\subset J\subset R$ gilt +$I=J$ oder $J=R$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{ +\begin{block}{Beispiele} +\begin{itemize} +\item Primzahlen $p$ erzeugen maximale Ideale in $\mathbb{Z}$ +\item<9-> Irreduzible Polynome erzeugen maximale Ideale in $\Bbbk[X]$ +\end{itemize} +\end{block}} +\uncover<10->{% +\begin{block}{Körper} +$M\subset R$ ein maximales Ideal, dann ist +$R/M$ ein Körper +\end{block}} +\uncover<11->{% +\begin{block}{Beispiel} +\begin{itemize} +\item +$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$ +\item<12-> +$m$ ein irreduzibles Polynom: +$\Bbbk[X]/ (m)$ ist ein Körper +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/minimalbeispiel.tex b/vorlesungen/slides/3/minimalbeispiel.tex new file mode 100644 index 0000000..f94cf8d --- /dev/null +++ b/vorlesungen/slides/3/minimalbeispiel.tex @@ -0,0 +1,36 @@ +% +% minimalbeispiel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Beispiel für $p(A)=0$} +\begin{block}{Potenzen einer $2\times 2$-Matrix $A$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-10pt} +\[ +I ={\tiny\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}},\quad +A ={\tiny\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix}},\quad +\uncover<2->{A^2={\tiny\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix}}} +\uncover<3->{,\quad A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}}} +\uncover<4->{,\quad A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 & 2 \end{pmatrix}}} +\] +\end{block} +\vspace{-5pt} +\uncover<5->{% +\begin{block}{linear abhängig} +Bereits die ersten $3$ sind linear abhängig: +\[ +-4I - A + A^2 += +-4\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} +-\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix} ++\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix} += +\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} +\] +\uncover<6->{$p(X) = X^2 - X - 4 \in \mathbb{Q}[X]$ hat die Eigenschaft +$p(A)=0$} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/minimalpolynom.tex b/vorlesungen/slides/3/minimalpolynom.tex new file mode 100644 index 0000000..2b36a65 --- /dev/null +++ b/vorlesungen/slides/3/minimalpolynom.tex @@ -0,0 +1,30 @@ +% +% minimalpolynom.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Minimalpolynom} +\begin{block}{Definition} +Zu jeder $n\times n$-Matrix $A$ +gibt es ein Polynom $m_A(X)\in\Bbbk[X]$ minimalen Grades $\deg m_A\le n^2$ +derart, dass $m_A(A)=0$. +\end{block} +\uncover<2->{% +\begin{block}{Strategie} +Das Minimalpolynom ist eine ``Invariante'' der Matrix $A$ +\end{block}} +\uncover<3->{% +\begin{block}{Satz von Cayley-Hamilton} +Für jede $n\times n$-Matrix $A\in M_n(\Bbbk)$ gilt $\chi_A(A)=0$ +\uncover<4->{% +\[ +\Downarrow +\] +Das Minimalpolynom $m_A\in \Bbbk[X]$ ist ein Teiler +des charakteristischen Polynoms $\chi_A\in \Bbbk[X]$} +\\ +\uncover<5->{$\Rightarrow $ +Faktorzerlegung on $\chi_A(X)$ ermitteln!} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex new file mode 100644 index 0000000..048e6a2 --- /dev/null +++ b/vorlesungen/slides/3/motivation.tex @@ -0,0 +1,108 @@ +% +% motivation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Motivation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.24\textwidth} +\begin{block}{Imaginäre Einheit} +\vspace{-15pt} +\begin{align*} +J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix} +\\ +p(X) &= X^2 + 1 +\\ +p(J) &= J^2 + I = 0 +\end{align*} +\end{block} +\end{column} +\begin{column}{0.25\textwidth} +\uncover<2->{% +\begin{block}{Wurzel $\sqrt{2}$} +\vspace{-15pt} +\begin{align*} +W&=\begin{pmatrix}0&2\\1&0\end{pmatrix} +\\ +p(X) &= X^2-2 +\\ +p(W) &= W^2-2I=0 +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.41\textwidth} +\uncover<3->{% +\begin{block}{Drehmatrix} +\vspace{-15pt} +\begin{align*} +D&=\begin{pmatrix} +\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\ +\sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} +\end{pmatrix} +\\ +p(X)&= +\ifthenelse{\boolean{presentation}}{\only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{X^2-2X\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +p(D) &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-20pt} +\uncover<5->{ +\begin{block}{3D-Beispiel} +$p(x) = -x^3-5x^2+5x+1$ +\[ +\ifthenelse{\boolean{presentation}}{ +\only<5-8>{ +A= +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*}} +\only<6-8>{ +\quad\Rightarrow\quad}}{} +\uncover<6->{ +- +\only<-9>{A^3}\only<10->{ +\begin{pmatrix*}[r] +-169&-35&35\\ +-185&-39&40\\ + -45&-10&11 +\end{pmatrix*}} +-5 +\only<-8>{A^2}\only<9->{ +\begin{pmatrix*}[r] +29&6&-6\\ +32&6&-5\\ + 8&1& 0 +\end{pmatrix*}} ++5 +\only<-7>{A}\only<8->{ +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*}} ++ +\only<-6>{I}\only<7->{ +\begin{pmatrix*}[r] +1&0&0\\ +0&1&0\\ +0&0&1 +\end{pmatrix*}} +} +\uncover<11->{=0} +\] +\end{block}} +\vspace{-10pt} +\uncover<12->{% +{\usebeamercolor[fg]{title}$\Rightarrow$ +Rechenregeln von Matrizen können durch Polynome ausgedrückt werden} +} +\end{frame} diff --git a/vorlesungen/slides/3/multiplikation.tex b/vorlesungen/slides/3/multiplikation.tex new file mode 100644 index 0000000..13f4e03 --- /dev/null +++ b/vorlesungen/slides/3/multiplikation.tex @@ -0,0 +1,180 @@ +% +% multiplikation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\N{21} +\begin{frame}[t,fragile] +\frametitle{Multiplikation mit $\alpha$ in $\mathbb{Z}(\alpha)$} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] + +\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\sqrt{2})$}; +\node[color=blue] at (3.5,3.2) [above left] {$\sqrt{2}\mathbb{Z}(\sqrt{2})$}; + +\pgfmathparse{sqrt(2)} +\xdef\a{\pgfmathresult} +\pgfmathparse{-int(3.2/\a)} +\xdef\ymin{\pgfmathresult} +\pgfmathparse{int(3.2/\a)} +\xdef\ymax{\pgfmathresult} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\sqrt{2}$}]; + +\def\punkt#1#2#3{ + ({(1-(#3))*(#1)+2*(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1))*\a}) +} + +\foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.5pt] + \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08]; + } +} +\foreach \y in {\ymin,...,\ymax}{ + \draw[color=red,line width=0.5pt] + \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0}; +} + + +\def\bildnetz#1{ + \pgfmathparse{(#1-1)/(\N-1)} + \xdef\t{\pgfmathresult} + \only<#1>{ + \uncover<2->{ + \draw[->,color=blue,line width=1.4pt] + (0,\a) -- \punkt{0}{1}{\t}; + \draw[->,color=blue,line width=1.4pt] + (1,0) -- \punkt{1}{0}{\t}; + } + \foreach \x in {-3,...,3}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=blue] + \punkt{\x}{\y}{\t} + circle[radius=0.06]; + } + } + \foreach \y in {\ymin,...,\ymax}{ + \draw[color=blue,line width=0.5pt] + \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t}; + } + } +} + +\begin{scope} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\ifthenelse{\boolean{presentation}}{ + \foreach \T in {1,...,\N}{ + \bildnetz{\T} + } +}{ + \bildnetz{\N} +} +\end{scope} + +\uncover<\N->{ +\begin{scope}[yshift=-2.5cm] +\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8); +\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8); +\node at (0,0) {$\displaystyle W=\begin{pmatrix}0&2\\1&0\end{pmatrix}$}; +\end{scope} +} + +\node at (0,-3.7) {$\alpha^2 = 2$}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] + +\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\varphi)$}; +\node[color=blue] at (3.5,3.2) [above left] {$\varphi\mathbb{Z}(\varphi)$}; + +\pgfmathparse{(sqrt(5)+1)/2} +\xdef\a{\pgfmathresult} +\pgfmathparse{-int(3.3/\a)} +\xdef\ymin{\pgfmathresult} +\pgfmathparse{int(3.3/\a)} +\xdef\ymax{\pgfmathresult} +\def\punkt#1#2#3{ + ({(1-(#3))*(#1)+(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1+#2))*\a}) +} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\varphi$}]; + +\foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.5pt] + \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08]; + } +} +\foreach \y in {\ymin,...,\ymax}{ + \draw[color=red,line width=0.5pt] + \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0}; +} + +\def\bildnetz#1{ + \pgfmathparse{(#1-1)/(\N-1)} + \xdef\t{\pgfmathresult} + \only<#1>{ + \uncover<2->{ + \draw[->,color=blue,line width=1.4pt] + (0,\a) -- \punkt{0}{1}{\t}; + \draw[->,color=blue,line width=1.4pt] + (1,0) -- \punkt{1}{0}{\t}; + } + \foreach \x in {-3,...,3}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=blue] \punkt{\x}{\y}{\t} + circle[radius=0.06]; + } + } + \foreach \y in {\ymin,...,\ymax}{ + \draw[color=blue,line width=0.5pt] + \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t}; + } + } +} + +\begin{scope} + +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\ifthenelse{\boolean{presentation}}{ + \foreach \T in {1,...,\N}{ + \bildnetz{\T} + } +}{ + \bildnetz{\N} +} +\end{scope} + +\uncover<\N->{ +\begin{scope}[yshift=-2.5cm] +\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8); +\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8); +\node at (0,0) {$\displaystyle \Phi=\begin{pmatrix}0&1\\1&1\end{pmatrix}$}; +\end{scope} +} + +\node at (0,-3.7) {$\alpha^2 = \alpha + 1$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/nichthauptideal.tex b/vorlesungen/slides/3/nichthauptideal.tex new file mode 100644 index 0000000..46074b9 --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal.tex @@ -0,0 +1,78 @@ +% +% nichthauptideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nicht-Hauptideal in $\mathbb{Z}[X]$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Hauptideal\uncover<2->{ = ``Gerade''}} +\vspace{-10pt} +\begin{align*} +\langle X+1\rangle&=(X+1) = {\color{red}(X+1)\cdot\mathbb{Z}[X]} +\end{align*} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] +\draw[->] (-6.3,0) -- (6.8,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-6.2) -- (0,6.6) coordinate[label={right:$\mathbb{Z}X$}]; +\foreach \x in {-6,...,6}{ + \fill[color=red] (\x,\x) circle[radius=0.12]; +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Ideal mit zwei Erzeugenden} +\vspace{-10pt} +\begin{align*} +\uncover<6->{ +{\color{darkgreen} +\langle 2,X\rangle +} +&=} +\uncover<5->{ +{\color{red}2\cdot\mathbb{Z}[X]} +} +\uncover<6->{+} +\uncover<4->{ +{\color{blue}X\cdot\mathbb{Z}[X]} +} +\end{align*} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] +\draw[->] (-6.3,0) -- (6.9,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-6.2) -- (0,7.0) coordinate[label={right:$\mathbb{Z}X$}]; +\uncover<6->{ + \foreach \x in {-6,-4,...,6}{ + \foreach \y in {-6,...,6}{ + \fill[color=darkgreen] (\x,\y) circle[radius=0.20]; + } + } +} +\uncover<5->{ + \foreach \x in {-6,-4,...,6}{ + \foreach \y in {-6,-4,...,6}{ + \fill[color=red] (\x,\y) circle[radius=0.16]; + } + } +} +\uncover<4->{ + \foreach \y in {-6,...,6}{ + \fill[color=blue] (0,\y) circle[radius=0.12]; + } +} +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex new file mode 100644 index 0000000..e1424ff --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal2.tex @@ -0,0 +1,95 @@ +% +% nichthauptideal2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Das Ideal $\langle 2,X\rangle \subset \mathbb{Z}[X]$} +\vspace{-12pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\c{\clip (-2.8,-2.0) rectangle (2.8,2.0);} + +\def\labels{ + \fill[color=white,opacity=0.5] (1.5,-0.1) circle[radius=0.2]; + \node at (1.5,-0.1) {$1$}; + \fill[color=white,opacity=0.5] (-0.9,1.7) circle[radius=0.2]; + \node at (-0.9,1.7) {$X$}; + \fill[color=white,opacity=0.5] (0.8,0.7) circle[radius=0.2]; + \node at (0.8,0.7) {$X^2$}; +} + +\only<-3>{ +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/ring.jpg}}; + \end{scope} + \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; +\end{scope} +} + +\uncover<2->{ +\begin{scope}[xshift=-3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptideal.jpg}}; + \end{scope} + \node[color=blue] at (-0.2,-1.2) {$(X+1)\cdot\mathbb{Z}[X]$}; + \labels +\end{scope} +} + +\uncover<3->{ +\begin{scope}[xshift=-3.0cm,yshift=-1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptideal2.jpg}}; + \end{scope} + \node[color=darkgreen] at (-3.0,-0.8) {$2\cdot\mathbb{Z}[X]$}; +\end{scope} + +\begin{scope}[xshift=3.0cm,yshift=-1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptidealX.jpg}}; + \end{scope} + \node[color=darkgreen] at (2.5,-0.8) {$X\cdot\mathbb{Z}[X]$}; + \labels +\end{scope} +} + +\uncover<4->{ +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/nichthauptideal.jpg}}; + \end{scope} + \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; + \node[color=darkgreen] at (1.9,-0.4) [right] {$\langle 2,X\rangle$}; +\end{scope} +} + +\draw[color=gray!50] (-6.6,0) -- (6.4,0); +\draw[color=gray!50] (0,-3.8) -- (0,3.8); + +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \fill[color=white,opacity=0.5] (1.5,-0.6) circle[radius=0.2]; + \node at (1.5,-0.6) {$1$}; + \fill[color=white,opacity=0.5] (-0.4,1.7) circle[radius=0.2]; + \node at (-0.4,1.7) {$X$}; +\end{scope} + +\end{tikzpicture} +\end{center} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex new file mode 100644 index 0000000..d646353 --- /dev/null +++ b/vorlesungen/slides/3/operatoren.tex @@ -0,0 +1,51 @@ +% +% operatoren.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{$X$ als Operator} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Polynome} +$a(X)=a_0+a_1X+\dots+a_nX^n$ +\uncover<2->{% +\[ +a(X) += +\begin{pmatrix} +a_0\\a_1\\a_2\\a_3\\\vdots\\a_n +\end{pmatrix} +\]} +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<3->{% +\begin{block}{Multiplikation mit $X$} +\strut +\[ +\begin{pmatrix} +1\\0\\0\\0\\\vdots\\0 +\end{pmatrix} +\uncover<4->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\1\\0\\0\\\vdots\\0 +\end{pmatrix}} +\uncover<5->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\0\\1\\0\\\vdots\\0 +\end{pmatrix}} +\uncover<6->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\0\\0\\1\\\vdots\\0 +\end{pmatrix}} +\uncover<7->{\overset{\cdot X}{\mapsto}\dots} +\uncover<8->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\0\\0\\0\\\vdots\\1 +\end{pmatrix}} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/phi.tex b/vorlesungen/slides/3/phi.tex new file mode 100644 index 0000000..ee0814c --- /dev/null +++ b/vorlesungen/slides/3/phi.tex @@ -0,0 +1,85 @@ +% +% phi.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\mathbb{Q}(\varphi)=\mathbb{Q}[X]/(X^2-X-1)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Der Ring $\mathbb{Z}(\varphi)$} +$\mathbb{Z}(\varphi)$ als Teilrung: +{\color{blue} +\[ +R=\{a+b\varphi\;|\; a,b\in\mathbb{Z}\} +\]}% +\uncover<2->{$\varphi\not\in\mathbb{Q}$}\uncover<3->{ +$\Rightarrow$ +$1$ und $\varphi$ sind inkommensurabel}\uncover<4->{ +$\Rightarrow$ +$R$ dicht in $\mathbb{R}$} +\end{block} +\uncover<5->{% +\begin{block}{Algebraische Konstruktion} +\uncover<8->{% +Das Polynom $X^2-X-1$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$} +\[ +\uncover<8->{\mathbb{Q}[X]/(X^2-X-1) +=} +{\color{red}\{a+b\varphi\;|\;a,b\in\mathbb{Z}\}} +\]\uncover<7->{% +mit der Rechenregel: $X^2=X+1$} +\end{block}} +\uncover<9->{% +\begin{block}{Körper} +$\mathbb{Q}(\varphi) = \mathbb{Q}[X]/(X^2+X+1)$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] +\begin{scope} +\pgfmathparse{(sqrt(5)-1))/2} +\xdef\gphi{\pgfmathresult} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\foreach \x in {-10,...,10}{ + \pgfmathparse{int(\x/\gphi)-10} + \xdef\s{\pgfmathresult} + \pgfmathparse{int(\x/\gphi)+10} + \xdef\t{\pgfmathresult} + \foreach \y in {\s,...,\t}{ + \uncover<4->{ + \fill[color=blue] ({\x-\y*\gphi},0) + circle[radius=0.05]; + } + \uncover<6->{ + \draw[color=blue,line width=0.1pt] + ({\x-\y*\gphi-3.2},3.2) + -- + ({\x-\y*\gphi+3.2},-3.2); + } + } +} +\end{scope} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; + +\uncover<5->{ + \draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}]; + + \foreach \x in {-3,...,3}{ + \foreach \y in {-5,...,5}{ + \fill[color=red] + ({\x},{\y*\gphi}) circle[radius=0.08]; + } + } +} + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/polynome.tex b/vorlesungen/slides/3/polynome.tex new file mode 100644 index 0000000..d7179a0 --- /dev/null +++ b/vorlesungen/slides/3/polynome.tex @@ -0,0 +1,29 @@ +% +% polynome.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Polynome} +$R$ ein Ring, z.~B.~$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ + +\begin{definition} +Polynome in $X$ mit Koeffizienten in $R$: +\[ +R[X] += +\{ +a(X)\;|\; +a(X) = a_nX^n+a_{n-1}X^{n-1} + \dots a_2X^2+a_1X + a_0, a_k\in R +\} +\] +\end{definition} + +\begin{itemize} +\item<2-> {\em Grad} des Polynoms: $\deg a(X) = \deg a = n$ +\item<3-> $\deg 0 = -\infty$ +\item<4-> {\em normiertes Polynom}: $a_n=1$ +\end{itemize} + + +\end{frame} diff --git a/vorlesungen/slides/3/quotientenring.tex b/vorlesungen/slides/3/quotientenring.tex new file mode 100644 index 0000000..4aa9e43 --- /dev/null +++ b/vorlesungen/slides/3/quotientenring.tex @@ -0,0 +1,59 @@ +% +% Quotientenring.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quotientenring} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quotientenring} +$I\subset R$ ein Ideal +\\ +\uncover<2->{ +$R/I$ hat eine Ringstruktur: +\begin{align*} +\uncover<3->{\pi(s)&=s+I} +\\ +\uncover<4->{\pi(s)\pi(r)&= (s+I)(r+I)}\\ + &\uncover<5->{= sr +\underbrace{sI + rI}_{\subset RI\subset I} + II = sr+I} +\\ +\uncover<6->{\pi(s)+\pi(r)&= (s+I)+(r+I)}\\ + &\uncover<7->{=s+r+I=\pi(s+r)} +\end{align*}} +\end{block} +\vspace{-15pt} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Beispiele} +\begin{itemize} +\item +$\mathbb{Z}/(n)=\mathbb{Z}/n\mathbb{Z}$, +$\mathbb{F}_p=\mathbb{Z}/(p)=\mathbb{Z}/p\mathbb{Z}$ +\item<8-> +$p\in\Bbbk[X]$ +$\Rightarrow$ +$\Bbbk[X]/(p)$ ist ein Ring +\end{itemize} +\end{block}} +\uncover<9->{% +\begin{block}{Algebraideal} +$I\subset A$ +\begin{itemize} +\item<10-> +$I$ ein Unterraum von $A$ als Vektorraum +\item<11-> +$I$ ein Ideal von $A$ als Ring +\end{itemize} +\end{block}} +\uncover<12->{% +\begin{block}{Quotientenalgebra} +$A/I$ ist eine Algebra +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/ringstruktur.tex b/vorlesungen/slides/3/ringstruktur.tex new file mode 100644 index 0000000..d653300 --- /dev/null +++ b/vorlesungen/slides/3/ringstruktur.tex @@ -0,0 +1,50 @@ +% +% ringstruktur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Ringstruktur} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.46\textwidth} +\begin{block}{Ring} +Menge $R$ mit zwei zweistelligen Verknüfpungen $+$ und $\cdot$ +mit +\begin{enumerate} +\item<3-> +$R$ ist abelsche Gruppe bezüglich $+$ +\item<5-> +$R\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$ +\item<7-> +Für alle $a,b,c\in R$ +\begin{align*} +a(b+c) &= ab+ac +\\ +(a+b)c &= ac+bc +\end{align*} +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<2->{% +\begin{block}{Polynomring} +$R$ ein Ring, $R[X]$ ``erbt'' Addition und Multiplikation mit +\begin{enumerate} +\item<4-> +$R[X]$ ist abelsche Gruppe bezüglich $+$ +\item<6-> +$R[X]\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$ +\item<8-> +Für alle $a,b,c\in R[X]$ +\begin{align*} +a(b+c) &= ab+ac +\\ +(a+b)c &= ac+bc +\end{align*} +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/teilbarkeit.tex b/vorlesungen/slides/3/teilbarkeit.tex new file mode 100644 index 0000000..a5ea9b9 --- /dev/null +++ b/vorlesungen/slides/3/teilbarkeit.tex @@ -0,0 +1,47 @@ +% +% teilbarkeit.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Teilen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Teilen in $\mathbb{Z}$} +Zu zwei Zahlen $a,b\in \mathbb{Z}$, \only<3->{{\color<3-4>{red}$a>b$}, }gibt es +immer \only<3->{{\color<3-4>{red}genau}} ein Paar $q,r\in\mathbb{Z}$ derart, dass +\begin{align*} +a&=bq+r +\\ +\uncover<3->{{\color<3-4>{red}r}&{\color<3-4>{red}< b}} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Teilen in $\mathbb{Q}[X]$} +Zu zwei Polynomen $a,b\in\mathbb{Q}[X]$, \only<4->{{\color<4>{red}$\deg a > \deg b$},} +gibt es +immer \only<4->{{\color<4>{red}bis auf eine Einheit genau }}% +ein Paar $q,r\in\mathbb{Q}[X]$ derart, dass +\begin{align*} +a&=bq+r +\\ +\uncover<4->{{\color<4>{red}\deg r}&{\color<4>{red}< \deg b}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<5->{% +\begin{block}{Allgemein: euklidischer Ring} +Nullteilerfreier Ring $R$ mit einer Funktion +$d\colon R\setminus{0}\to\mathbb{N}$ mit +\begin{itemize} +\item Für $x,y\in R$ gilt $d(xy) \ge d(x)$. +\item Für $x,y\in R$ gibt es $q,r\in R$ derart +$x=qy +r$ mit $d(y)>d(r)$ +\end{itemize} +Euklidische Ringe haben ähnliche Eigenschaften wie Polynomringe +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex new file mode 100644 index 0000000..d20bfc4 --- /dev/null +++ b/vorlesungen/slides/3/wurzel2.tex @@ -0,0 +1,83 @@ +% +% wurzel2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\mathbb{Z}(\sqrt{2})\only<7->{ = \mathbb{Z}[X]/(X^2-2)}$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Der Ring $\mathbb{Z}(\sqrt{2})$} +$\mathbb{Z}(\sqrt{2})$ als Teilring: +{\color{blue} +\[ +R=\{ a+b\sqrt{2}\;|\; a,b\in\mathbb{Z} \} \subset \mathbb{R} +\]}% +\uncover<2->{$\sqrt{2}\not\in\mathbb{Q}$}\uncover<3->{ +$\Rightarrow$ +$1$ und $\sqrt{2}$ sind inkommensurabel}\uncover<4->{ +$\Rightarrow$ +$R$ dicht in $\mathbb{R}$} +\end{block} +\uncover<5->{% +\begin{block}{Algebraische Konstruktion} +\uncover<8->{% +Das Polynom $X^2-2$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$} +\[ +\uncover<8->{\mathbb{Z}[X]/(X^2-2) +=} +{\color{red}\{a+bX\;|\;a,b\in\mathbb{Z}\}} +\]\uncover<7->{% +mit Rechenregel: $X^2=2$} +\end{block}} +\uncover<9->{% +\begin{block}{Körper} +$\mathbb{Q}(\sqrt{2}) = \mathbb{Q}[X]/(X^2-2)$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] +\begin{scope} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\foreach \x in {-10,...,10}{ + \pgfmathparse{int(\x/sqrt(2))-5} + \xdef\s{\pgfmathresult} + \pgfmathparse{int(\x/sqrt(2))+5} + \xdef\t{\pgfmathresult} + \foreach \y in {\s,...,\t}{ + \uncover<4->{ + \fill[color=blue] ({\x-\y*sqrt(2)},0) + circle[radius=0.05]; + } + \uncover<6->{ + \draw[color=blue,line width=0.1pt] + ({\x-\y*sqrt(2)-3.2},3.2) + -- + ({\x-\y*sqrt(2)+3.2},-3.2); + } + } +} +\end{scope} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; + +\uncover<5->{ + \draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}]; + + \foreach \x in {-3,...,3}{ + \foreach \y in {-2,...,2}{ + \fill[color=red] + ({\x},{\y*sqrt(2)}) circle[radius=0.08]; + } + } +} + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc new file mode 100644 index 0000000..ad1081e --- /dev/null +++ b/vorlesungen/slides/4/Makefile.inc @@ -0,0 +1,22 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter4 = \ + ../slides/4/ggt.tex \ + ../slides/4/euklidmatrix.tex \ + ../slides/4/euklidbeispiel.tex \ + ../slides/4/euklidtabelle.tex \ + ../slides/4/fp.tex \ + ../slides/4/division.tex \ + ../slides/4/gauss.tex \ + ../slides/4/dh.tex \ + ../slides/4/divisionpoly.tex \ + ../slides/4/euklidpoly.tex \ + ../slides/4/polynomefp.tex \ + ../slides/4/schieberegister.tex \ + ../slides/4/alpha.tex \ + ../slides/4/chapter.tex + diff --git a/vorlesungen/slides/4/alpha.tex b/vorlesungen/slides/4/alpha.tex new file mode 100644 index 0000000..3cd54c0 --- /dev/null +++ b/vorlesungen/slides/4/alpha.tex @@ -0,0 +1,54 @@ +% +% alpha.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\frametitle{Was ist $\alpha$?} +$m(X)$ ein irreduzibles Polynome in $\mathbb{F}_2[X]$ + +Beispiel: $m(X) = X^8{\color{red}\mathstrut+X^4+X^3+X^2+1}\in\mathbb{F}_2[X]$ +\begin{columns}[t] +\begin{column}{0.40\textwidth} +\uncover<2->{% +\begin{block}{Abstrakt} +$\alpha$ ist ein ``imaginäres'' +Objekt mit der Rechenregel $m(\alpha)=0$ +\begin{align*} +\alpha^8 &= {\color{red}\alpha^4+\alpha^3+\alpha^2+1}\\ +\uncover<3->{ +\alpha^9 &= \alpha^5+\alpha^4+\alpha^3+\alpha}\\ +\uncover<4->{ +\alpha^{10}&= \alpha^6+\alpha^5+\alpha^4+\alpha^2}\\ +\uncover<5->{ +\alpha^{11}&= \alpha^7+\alpha^6+\alpha^5+\alpha^3}\\ +\uncover<6->{ +\alpha &= \alpha^7+\alpha^3+\alpha^2+\alpha} +\\ +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.54\textwidth} +\uncover<7->{% +\begin{block}{Matrix} +Eine konkretes Element in $M_n(\mathbb{F}_2)$ +\[ +\alpha += +\begin{pmatrix} +0& 0& 0& 0& 0& 0& 0& {\color{red}1}\\ +1& 0& 0& 0& 0& 0& 0& {\color{red}0}\\ +0& 1& 0& 0& 0& 0& 0& {\color{red}1}\\ +0& 0& 1& 0& 0& 0& 0& {\color{red}1}\\ +0& 0& 0& 1& 0& 0& 0& {\color{red}1}\\ +0& 0& 0& 0& 1& 0& 0& {\color{red}0}\\ +0& 0& 0& 0& 0& 1& 0& {\color{red}0}\\ +0& 0& 0& 0& 0& 0& 1& {\color{red}0} +\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} + +\end{frame} diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex new file mode 100644 index 0000000..a10712a --- /dev/null +++ b/vorlesungen/slides/4/chapter.tex @@ -0,0 +1,18 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{4/ggt.tex} +\folie{4/euklidmatrix.tex} +\folie{4/euklidbeispiel.tex} +\folie{4/euklidtabelle.tex} +\folie{4/fp.tex} +\folie{4/division.tex} +\folie{4/gauss.tex} +\folie{4/dh.tex} +\folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} +\folie{4/polynomefp.tex} +\folie{4/alpha.tex} +\folie{4/schieberegister.tex} diff --git a/vorlesungen/slides/4/dh.tex b/vorlesungen/slides/4/dh.tex new file mode 100644 index 0000000..b0a88e5 --- /dev/null +++ b/vorlesungen/slides/4/dh.tex @@ -0,0 +1,62 @@ +% +% dh.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Diffie-Hellmann Schlüsselaushandlung} + +\begin{center} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\skala{0.95} +\begin{tikzpicture}[>=latex,thick,scale=\skala] +\def\l{2.5} +\fill[color=blue!20] (-7,-6.5) rectangle (7,0.5); +\fill[color=red!20] (-\l,-6.5) rectangle (\l,0.501); +\node[color=red] at (0,-1.5) {öffentliches Netzwerk}; +\node[color=blue] at (-7,0.2) [right] {privat}; +\node[color=blue] at (7,0.2) [left] {privat}; +\coordinate (A) at (-\l,-2.5); +\coordinate (C) at (\l,-5.0); +\coordinate (B) at (\l,-2.5); +\coordinate (D) at (-\l,-5.0); +\node at (0,0) {$p\in\mathbb{N},g\in\mathbb{F}_p$ aushandeln}; +\fill[color=white] (-\l,-0.7) circle[radius=0.3]; +\draw (-\l,-0.7) circle[radius=0.3]; +\fill[color=white] (\l,-0.7) circle[radius=0.3]; +\draw (\l,-0.7) circle[radius=0.3]; +\node at (-\l,-0.7) {$A$}; +\node at (\l,-0.7) {$B$}; +\uncover<2->{ + \node at (-\l,-1.5) [left] {$a$ auswählen\strut}; + \node at (-\l,-2.0) [left] {$x=g^a\in\mathbb{F}_p$\strut}; + \node at (\l,-1.5) [right] {$b$ auswählen\strut}; + \node at (\l,-2.0) [right] {$y=g^b\in\mathbb{F}_p$\strut}; +} +\draw[->] (-\l,-1) -- (-\l,-6); +\draw[->] (\l,-1) -- (\l,-6); +\uncover<3->{ + \draw[->] (A) -- (C); + \draw[->] (B) -- (D); + \fill (A) circle[radius=0.08]; + \fill (B) circle[radius=0.08]; + \node at ($0.8*(A)+0.2*(C)+(-0.4,0)$) [above right] {$x=g^a$}; + \node at ($0.8*(B)+0.2*(D)+(0.4,0)$) [above left] {$y=g^b$}; +} +\uncover<4->{ + \node at (-\l,-5.0) [left] {$s=g^{ab}=y^a\in\mathbb{F}_p$}; + \node at (-\l,-5.5) [left] {ausrechnen}; + \node at (\l,-5.0) [right] {$s=g^{ab}=x^b\in\mathbb{F}_p$}; + \node at (\l,-5.5) [right] {ausrechnen}; +} +\uncover<5->{ + \fill[rounded corners=0.3cm,color=darkgreen!20] + ({-\l-1.7},-7) rectangle ({\l+1.7},-6); + \draw[rounded corners=0.3cm] ({-\l-1.7},-7) rectangle ({\l+1.7},-6); + \node at (0,-6.5) {$A$ und $B$ haben den gemeinsamen Schlüssel $s$}; +} +\end{tikzpicture} + +\end{center} + +\end{frame} diff --git a/vorlesungen/slides/4/division.tex b/vorlesungen/slides/4/division.tex new file mode 100644 index 0000000..846738f --- /dev/null +++ b/vorlesungen/slides/4/division.tex @@ -0,0 +1,65 @@ +% +% division.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Division in $\mathbb{F}_p$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Inverse {\bf berechnen}} +Gegeben $a\in\mathbb{F}_p$, finde $b=a^{-1}\in\mathbb{F}_p$ +\begin{align*} +\uncover<2->{&& a{\color{blue}b} &\equiv 1 \mod p} +\\ +\uncover<3->{&\Leftrightarrow& a{\color{blue}b}&=1 + {\color{blue}n}p} +\\ +\uncover<4->{&&a{\color{blue}b}-{\color{blue}n}p&=1} +\end{align*} +\uncover<5->{Wegen +$\operatorname{ggT}(a,p)=1$ gibt es +$s$ und $t$ mit +\[ +{\color{red}s}a+{\color{red}t}p=1 +\Rightarrow +{\color{blue}b}={\color{red}s},\; +{\color{blue}n}=-{\color{red}t} +\]} +\uncover<6->{% +$\Rightarrow$ Die Inverse kann mit dem euklidischen Algorithmus +berechnet werden} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Beispiel in $\mathbb{F}_{1291}$} +Finde $47^{-1}\in\mathbb{F}_{1291}$ +%\vspace{-10pt} +\begin{center} +\begin{tabular}{|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|} +\hline +k& a_k& b_k&q_k& c_k& d_k\\ +\hline + & & & & 1& 0\\ +0& 47&1291&\uncover<8->{ 0}& 0& 1\\ +1&\uncover<9->{ 1291& 47}&\uncover<11->{ 27}&\uncover<10->{ 1& 0}\\ +2&\uncover<12->{ 47& 22}&\uncover<14->{ 2}&\uncover<13->{ -27& 1}\\ +3&\uncover<15->{ 22& 3}&\uncover<17->{ 7}&\uncover<16->{ 55& -2}\\ +4&\uncover<18->{ 3& 1}&\uncover<20->{ 3}&\uncover<19->{{\color{red}-412}&{\color{red}15}}\\ +5&\uncover<21->{ 1& 0}& &\uncover<22->{ 1291& -47}\\ +\hline +\end{tabular} +\end{center} +\uncover<23->{% +\[ +{\color{red}-412}\cdot 47 +{\color{red}15}\cdot 1291 = 1 +\uncover<24->{\;\Rightarrow\; +47^{-1}={\color{red}879}} +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/divisionpoly.tex b/vorlesungen/slides/4/divisionpoly.tex new file mode 100644 index 0000000..5e71c95 --- /dev/null +++ b/vorlesungen/slides/4/divisionpoly.tex @@ -0,0 +1,37 @@ +% +% divisionpoly.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Polynomdivision in $\mathbb{F}_3[X]$} +Rechenregeln in $\mathbb{F}_3$: $1+2=0$, $2\cdot 2 = 1$ +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcrcrcrc} +\llap{$ ($}X^4&+&X^3&+& X^2&+& X&+&1\rlap{$)$}&\;\;:&(X^2&+&X&+&2)&=&\uncover<2->{X^2}&\uncover<5->{+&2=q}\\ +\uncover<3->{\llap{$-($}X^4&+&X^3&+&2X^2\rlap{$)$}}& & & & & & & & & & & & & & & \\ +\uncover<4->{ & & & &2X^2&+& X&+& 1} & & & & & & & & & & \\ +\uncover<6->{ & & & &\llap{$-($}2X^2&+&2X&+& 2\rlap{$)$}}& & & & & & & & & & \\ +\uncover<7->{ & & & & & &2X&+&2\rlap{$\mathstrut=r$}& & & & & & & & & &} +\end{array} +\] +\uncover<8->{% +Kontrolle: +\[ +\arraycolsep=1.4pt +\begin{array}{rclcrcr} +(\underbrace{X^2+2}_{\displaystyle=q}) +(X^2+X+2) + &=&\rlap{$\uncover<9->{X^4+X^3+2X^2}\uncover<10->{ + 2X^2+2X+2}$} +\\ +\uncover<11->{&=& X^4+X^3+X^2&+&2X&+&2} +\\ +\uncover<12->{& & &&\llap{$r=\mathstrut$}2X&+&2} +\\ +\uncover<13->{&=& X^4+X^3+X^2&+&1X&+&1} +\end{array} +\] +} + +\end{frame} diff --git a/vorlesungen/slides/4/euklidbeispiel.tex b/vorlesungen/slides/4/euklidbeispiel.tex new file mode 100644 index 0000000..366a7a6 --- /dev/null +++ b/vorlesungen/slides/4/euklidbeispiel.tex @@ -0,0 +1,78 @@ +% +% euklidmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Euklidischer Algorithmus: Beispiel} +\setlength{\abovedisplayskip}{0pt} +\setlength{\belowdisplayskip}{0pt} +\vspace{-0pt} +\begin{block}{Finde $\operatorname{ggT}(25,15)$} +\vspace{-12pt} +\begin{align*} +a_0&=25 & b_0 &= 15 &\uncover<2->{25&=15 \cdot {\color{orange} 1} + 10 &q_0 &= {\color{orange}1} & r_0 &= 10}\\ +\uncover<3->{a_1&=15 & b_1 &= 10}&\uncover<4->{15&=10 \cdot {\color{darkgreen}1} + \phantom{0}5 &q_1 &= {\color{darkgreen}1} & r_1 &= \phantom{0}5}\\ +\uncover<5->{a_2&=10 & b_2 &= \phantom{0}5}&\uncover<6->{10&=\phantom{0}5 \cdot {\color{blue} 2} + \phantom{0}0 &q_2 &= {\color{blue}2} & r_2 &= \phantom{0}0 } +\end{align*} +\end{block} +\vspace{-5pt} +\uncover<7->{% +\begin{block}{Matrix-Operationen} +\begin{align*} +Q +&= +\uncover<9->{Q({\color{blue}2})} +\uncover<8->{Q({\color{darkgreen}1})} +Q({\color{orange}1}) += +\uncover<9->{ +\begin{pmatrix*}[r]0&1\\1&-{\color{blue}2}\end{pmatrix*} +} +\uncover<8->{ +\begin{pmatrix*}[r]0&1\\1&-{\color{darkgreen}1}\end{pmatrix*} +} +\begin{pmatrix*}[r]0&1\\1&-{\color{orange}1}\end{pmatrix*} += +\ifthenelse{\boolean{presentation}}{ +\only<7>{ +\begin{pmatrix*}[r]\phantom{-}0&1\\1&-1\end{pmatrix*} +} +\only<8>{ +\begin{pmatrix*}[r] +1&-1\\-1&2 +\end{pmatrix*} +} +}{} +\only<9->{ +\begin{pmatrix*}[r] +{\color{red}-1}&{\color{red}2}\\3&-5 +\end{pmatrix*}} +\end{align*} +\end{block}} +\vspace{-5pt} +\uncover<10->{% +\begin{block}{Relationen ablesen} +\[ +\begin{pmatrix} +\operatorname{ggT}(a,b)\\0 +\end{pmatrix} += +Q +\begin{pmatrix}a\\b\end{pmatrix} +\uncover<11->{% +\quad +\Rightarrow\quad +\left\{ +\begin{aligned} +\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 = +{\color{red}-1}\cdot {\usebeamercolor[fg]{title}25} + {\color{red}2}\cdot {\usebeamercolor[fg]{title}15} \\ + 0 &= \phantom{5=-}3\cdot {\usebeamercolor[fg]{title}25} -5\cdot {\usebeamercolor[fg]{title}15} +\end{aligned} +\right.} +\] +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex new file mode 100644 index 0000000..be5b3ca --- /dev/null +++ b/vorlesungen/slides/4/euklidmatrix.tex @@ -0,0 +1,108 @@ +% +% euklidmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Matrixform des euklidischen Algorithmus} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.52\textwidth} +\begin{block}{Einzelschritt} +\vspace{-10pt} +\[ +a_k = b_kq_k + r_k +\uncover<2->{ +\;\Rightarrow\; +\left\{ +\begin{aligned} +a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\ +b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k +\end{aligned} +\right.} +\] +\end{block} +\end{column} +\begin{column}{0.44\textwidth} +\uncover<3->{% +\begin{block}{Matrixschreibweise} +\vspace{-10pt} +\begin{align*} +\begin{pmatrix} +a_{k+1}\\ +b_{k+1} +\end{pmatrix} +&= +\begin{pmatrix} +b_k\\r_k +\end{pmatrix} += +\uncover<4->{ +\underbrace{\begin{pmatrix} +\uncover<5->{0&1}\\ +\uncover<6->{1&-q_k} +\end{pmatrix}}_{\uncover<7->{\displaystyle =Q(q_k)}} +} +\begin{pmatrix} +a_k\\b_k +\end{pmatrix} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-10pt} +\uncover<8->{% +\begin{block}{Ende des Algorithmus} +\vspace{-10pt} +\begin{align*} +\uncover<9->{ +\begin{pmatrix} +a_{n+1}\\ +b_{n+1}\\ +\end{pmatrix} +&=} +\begin{pmatrix} +r_{n-1}\\ +r_{n} +\end{pmatrix} += +\begin{pmatrix} +\operatorname{ggT}(a,b) \\ +0 +\end{pmatrix} +\uncover<11->{ += +\underbrace{\uncover<15->{Q(q_n)} +\uncover<14->{\dots} +\uncover<13->{Q(q_1)} +\uncover<12->{Q(q_0)}}_{\displaystyle =Q}} +\uncover<10->{ +\begin{pmatrix} a_0\\ b_0\end{pmatrix} +\uncover<6->{ += +Q\begin{pmatrix}a\\b\end{pmatrix} +} +} +\end{align*} +\end{block}} +\uncover<16->{% +\begin{block}{Konsequenzen} +\[ +Q=\begin{pmatrix} +q_{11}&q_{12}\\ +q_{21}&q_{22} +\end{pmatrix} +\quad\Rightarrow\quad +\left\{ +\quad +\begin{aligned} +\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b = {\color{red}s}a+{\color{red}t}b\\ + 0 &= q_{21}a + q_{22}b +\end{aligned} +\right. +\] +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/4/euklidpoly.tex b/vorlesungen/slides/4/euklidpoly.tex new file mode 100644 index 0000000..432b6b4 --- /dev/null +++ b/vorlesungen/slides/4/euklidpoly.tex @@ -0,0 +1,47 @@ +% +% euklidpoly.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Euklidischer Algorithmus in $\mathbb{F}_2[X]$} +Gegeben: $m(X)=X^4+X+1$, $b(X) = {\color{blue}X^2+1}$ +\\ +\uncover<2->{Berechne $s,t\in\mathbb{F}_2[X]$ derart, dass $sm+tb=1$} +\uncover<3->{% +\begin{center} +\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|} +\hline +k& a_k& b_k& q_k&r_k& c_k& d_k\\ +\hline + & & & & & 1& 0\\ +0&X^4+X+1&{\color{blue}X^2+1}&\uncover<4->{X^2+1}&\uncover<4->{X}& 0& 1\\ +1&\uncover<5->{X^2+1 }&\uncover<5->{X}&\uncover<5->{X}&\uncover<5->{1}&\uncover<5->{1}&\uncover<5->{X^2+1}\\ +2&\uncover<6->{X }&\uncover<6->{1}&\uncover<6->{X}&\uncover<6->{0}&\uncover<6->{{\color{red}X}}&\uncover<6->{{\color{red}X^3+X+1}}\\ +3&\uncover<7->{1 }&\uncover<7->{0}&&&\uncover<7->{X^2+1}&\uncover<7->{X^4+X+1} \\ +\hline +\end{tabular} +\end{center}} +\ifthenelse{\boolean{presentation}}{ +\only<8->{% +\begin{block}{Kontrolle} +\vspace{-10pt} +\begin{align*} +{\color{red}X}\cdot (X^4+X+1) + ({\color{red}X^3+X+1})({\color{blue}X^2+1}) +&\uncover<9->{= +(X^5+X^2+X)}\\ +&\qquad \uncover<10->{+ (X^5+X^3+X^2+X^3+X+1)} +\\ +&\uncover<11->{=(X^5+X^2+X) + (X^5+X^2+X+1)} +\\ +&\uncover<12->{=1} +\end{align*} +\end{block}}}{} +\begin{block}{Rechenregeln in $\mathbb{F}_2$} +$1+1=0$, +$2=0$, $+1=-1$. +\end{block} + +\end{frame} diff --git a/vorlesungen/slides/4/euklidtabelle.tex b/vorlesungen/slides/4/euklidtabelle.tex new file mode 100644 index 0000000..3f1b8d7 --- /dev/null +++ b/vorlesungen/slides/4/euklidtabelle.tex @@ -0,0 +1,73 @@ +% +% euklidtabelle.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Durchführung des euklidischen Algorithmus} +Problem: Berechnung der Produkte $Q(q_k)\cdots Q(q_1)Q(q_0)$ für $k=0,1,\dots,n$ +\uncover<2->{% +\begin{block}{Multiplikation mit $Q(q_k)$} +\vspace{-12pt} +\begin{align*} +Q(q_k) +\ifthenelse{\boolean{presentation}}{ +\only<-3>{ +\begin{pmatrix} +u&v\\c&d +\end{pmatrix} +=\begin{pmatrix} +0&1\\1&-q_k +\end{pmatrix} +}}{} +\begin{pmatrix} +u&v\\c&d +\end{pmatrix} +&\uncover<3->{= +\begin{pmatrix} +c&d\\ +u-q_kc&v-q_kd +\end{pmatrix}} +&&\uncover<5->{\Rightarrow& +\begin{pmatrix} +c_k&d_k\\c_{k+1}&d_{k+1} +\end{pmatrix} +&= +Q(q_k) +%\begin{pmatrix} +%0&1\\1&-q_k +%\end{pmatrix} +\begin{pmatrix} +c_{k-1}&d_{k-1}\\c_{k}&d_{k} +\end{pmatrix}} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{equation*} +\begin{tabular}{|>{\tiny$}r<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|} +\hline +k &q_k & c_k & d_k \\ +\hline +-1 & & 1 & 0 \\ + 0 &\uncover<7->{q_0 }& 0 & 1 \\ + 1 &\uncover<9->{q_1 }&\uncover<8->{c_{-1} -q_0 \cdot c_0 &d_{-1} -q_0 \cdot d_0 }\\ + 2 &\uncover<11->{q_2 }&\uncover<10->{c_0 -q_1 \cdot c_1 &d_0 -q_1 \cdot d_1 }\\ +\vdots&\uncover<12->{\vdots}&\uncover<12->{\vdots &\vdots }\\ + n &\uncover<14->{q_n }&\uncover<13->{{\color{red}c_{n-2}-q_{n-1}\cdot c_{n-1}}&{\color{red}d_{n-2}-q_{n-1}\cdot d_{n-1}}}\\ +n+1& &\uncover<15->{c_{n-1}-q_{n} \cdot c_{n} &d_{n-1}-q_{n} \cdot d_{n} }\\ +\hline +\end{tabular} +\uncover<16->{ +\Rightarrow +\left\{ +\begin{aligned} +\rlap{${\color{red}c_{n}}$}\phantom{c_{n+1}} a + \rlap{${\color{red}d_n}$}\phantom{d_{n+1}}b &= \operatorname{ggT}(a,b) +\\ +c_{n+1} a + d_{n+1} b &= 0 +\end{aligned} +\right.} +\end{equation*}} +\end{frame} diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex new file mode 100644 index 0000000..968b777 --- /dev/null +++ b/vorlesungen/slides/4/fp.tex @@ -0,0 +1,178 @@ +% +% fp.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\feld#1#2#3{ + \node at ({#1},{5-#2}) {$#3$}; +} +\def\geld#1#2#3{ + \node at ({#1},{6-#2}) {$#3$}; +} +\def\rot#1#2{ + \fill[color=red!20] ({#1-0.5},{5-#2-0.5}) rectangle ({#1+0.5},{5-#2+0.5}); +} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\gruen#1#2{ + \fill[color=darkgreen!20] ({#1-0.5},{6-#2-0.5}) rectangle ({#1+0.5},{6-#2+0.5}); +} +\def\inverse#1#2{ + \node at (9,{6-#1}) {$#1^{-1}=#2\mathstrut$}; +} +\begin{frame}[t] +\frametitle{Galois-Körper} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Restklassenring$\mathstrut$} +$\mathbb{Z}/n\mathbb{Z} +=\{ \llbracket r\rrbracket\;|\; 0\le r < n \} \mathstrut$ +ist ein Ring +\end{block} +\uncover<2->{% +\begin{block}{Nullteiler} +Falls $n=n_1n_2$, dann sind $\llbracket n_1\rrbracket$ und +$\llbracket n_2\rrbracket$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$: +\[ +\llbracket n_1\rrbracket +\llbracket n_2\rrbracket += +\llbracket n_1n_2 \rrbracket += +\llbracket n\rrbracket += +\llbracket 0 \rrbracket +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Galois-Körper $\mathbb{F}_p\mathstrut$} +$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\mathstrut$ +\end{block}} +\uncover<4->{% +\begin{block}{$n$ prim} +Für $n=p$ prim ist $\mathbb{Z}/n\mathbb{Z}$ nullteilerfrei +\medskip + +\uncover<5->{ +$\Rightarrow \quad \mathbb{F}_p$ ist ein Körper +} +\end{block}} +\end{column} +\end{columns} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.45] +\fill[color=white] (-12,0) circle[radius=0.1]; +\fill[color=white] (12,0) circle[radius=0.1]; +\uncover<3->{ +\begin{scope}[xshift=-8cm] +\rot{2}{3} +\rot{4}{3} +\rot{3}{2} +\rot{3}{4} +\fill[color=gray!40] (-0.5,5.5) rectangle (5.5,6.5); +\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,5.5); +\foreach \x in {-0.5,5.5}{ + \draw (\x,-0.5) -- (\x,6.5); +} +\foreach \x in {0.5,...,4.5}{ + \draw[line width=0.3pt] (\x,-0.5) -- (\x,6.5); +} +\foreach \y in {0.5,...,5.5}{ + \draw[line width=0.3pt] (-1.5,\y) -- (5.5,\y); +} +\foreach \y in {-0.5,5.5}{ + \draw (-1.5,\y) -- (5.5,\y); +} +\draw (-1.5,-0.5) -- (-1.5,5.5); +\draw (-0.5,6.5) -- (5.5,6.5); +\foreach \x in {0,...,5}{ + \node at (\x,6) {$\x$}; + \node at (-1,{5-\x}) {$\x$}; +} +\foreach \x in {0,...,5}{ + \feld{\x}{0}{0} + \feld{0}{\x}{0} +} +\foreach \x in {2,...,5}{ + \feld{\x}{1}{\x} + \feld{1}{\x}{\x} +} +\feld{1}{1}{1} +\feld{2}{2}{4} +\feld{2}{3}{0} \feld{3}{2}{0} +\feld{2}{4}{2} \feld{4}{2}{2} +\feld{2}{5}{4} \feld{5}{2}{4} +\feld{3}{3}{3} +\feld{4}{3}{0} \feld{3}{4}{0} +\feld{5}{3}{3} \feld{3}{5}{3} +\feld{4}{4}{4} +\feld{4}{5}{2} \feld{5}{4}{2} +\feld{5}{5}{1} +\end{scope}} +\uncover<6->{ +\begin{scope}[xshift=6cm] +\uncover<7->{ \gruen{1}{1} } +\uncover<8->{ \gruen{4}{2} } +\uncover<9->{ \gruen{5}{3} } +\uncover<10->{ \gruen{2}{4} } +\uncover<11->{ \gruen{3}{5} } +\uncover<12->{ \gruen{6}{6} } +\fill[color=gray!40] (-0.5,6.5) rectangle (6.5,7.5); +\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,6.5); +\foreach \x in {-0.5,6.5}{ + \draw (\x,-0.5) -- (\x,7.5); +} +\foreach \x in {0.5,...,5.5}{ + \draw[line width=0.3pt] (\x,-0.5) -- (\x,7.5); +} +\foreach \y in {0.5,...,6.5}{ + \draw[line width=0.3pt] (-1.5,\y) -- (6.5,\y); +} +\foreach \y in {-0.5,6.5}{ + \draw (-1.5,\y) -- (6.5,\y); +} +\draw (-1.5,-0.5) -- (-1.5,6.5); +\draw (-0.5,7.5) -- (6.5,7.5); +\foreach \x in {0,...,6}{ + \node at (\x,7) {$\x$}; + \node at (-1,{6-\x}) {$\x$}; +} +\foreach \x in {0,...,6}{ + \geld{\x}{0}{0} + \geld{0}{\x}{0} +} +\foreach \x in {2,...,6}{ + \geld{\x}{1}{\x} + \geld{1}{\x}{\x} +} +\geld{1}{1}{1} +\geld{2}{2}{4} +\geld{2}{3}{6} \geld{3}{2}{6} +\geld{2}{4}{1} \geld{4}{2}{1} +\geld{2}{5}{3} \geld{5}{2}{3} +\geld{2}{6}{5} \geld{6}{2}{5} +\geld{3}{3}{2} +\geld{4}{3}{5} \geld{3}{4}{5} +\geld{5}{3}{1} \geld{3}{5}{1} +\geld{6}{3}{4} \geld{3}{6}{4} +\geld{4}{4}{2} +\geld{5}{4}{6} \geld{4}{5}{6} +\geld{6}{4}{3} \geld{4}{6}{3} +\geld{5}{5}{4} +\geld{6}{5}{2} \geld{5}{6}{2} +\geld{6}{6}{1} +\uncover<7->{ \inverse{1}{1} } +\uncover<8->{ \inverse{2}{4} } +\uncover<9->{ \inverse{3}{5} } +\uncover<10->{ \inverse{4}{2} } +\uncover<11->{ \inverse{5}{3} } +\uncover<12->{ \inverse{6}{6} } +\end{scope}} +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex new file mode 100644 index 0000000..23cdfee --- /dev/null +++ b/vorlesungen/slides/4/gauss.tex @@ -0,0 +1,143 @@ +% +% gauss.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\def\ds{0.5} +\def\punkt#1#2{({(#1)*\ds},{-(#2)*\ds})} +\def\tabelle{ + \foreach \x in {-0.5,0.5,3.5}{ + \draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5}; + \draw \punkt{-0.5}{\x} -- \punkt{3.5}{\x}; + } + \node at \punkt{0}{1} {$0$}; + \node at \punkt{0}{2} {$1$}; + \node at \punkt{0}{3} {$2$}; + \node at \punkt{1}{0} {$0$}; + \node at \punkt{2}{0} {$1$}; + \node at \punkt{3}{0} {$2$}; +} +\begin{frame}[t] +\frametitle{Gauss-Algorithmus in $\mathbb{F}_3$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Additions-/Multiplikationstabelle} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope}[xshift=-1.6cm] +\tabelle +\node at \punkt{0}{0} {$+$}; +\node at \punkt{1}{1} {$0$}; +\node at \punkt{1}{2} {$1$}; +\node at \punkt{1}{3} {$2$}; +\node at \punkt{2}{1} {$1$}; +\node at \punkt{2}{2} {$2$}; +\node at \punkt{2}{3} {$0$}; +\node at \punkt{3}{1} {$2$}; +\node at \punkt{3}{2} {$0$}; +\node at \punkt{3}{3} {$1$}; +\end{scope} +\begin{scope}[xshift=1.6cm] +\tabelle +\node at \punkt{0}{0} {$\cdot$}; +\node at \punkt{1}{1} {$0$}; +\node at \punkt{1}{2} {$0$}; +\node at \punkt{1}{3} {$0$}; +\node at \punkt{2}{1} {$0$}; +\node at \punkt{2}{2} {$1$}; +\node at \punkt{2}{3} {$2$}; +\node at \punkt{3}{1} {$0$}; +\node at \punkt{3}{2} {$2$}; +\node at \punkt{3}{3} {$1$}; +\end{scope} +\end{tikzpicture} +\end{center} + +\end{block} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<2->{% +\begin{block}{Gleichungssystem\uncover<9->{/Lösung}} +\[ +\left. +\begin{array}{rcrcrcrcr} + x&+&y&+2z&=&1\\ +2x& & &+ z&=&2\\ + x&+&y& &=&2 +\end{array} +\uncover<9->{ +\right\} +\Rightarrow +\left\{ +\begin{aligned} +x&=2\\ +y&=0\\ +z&=1 +\end{aligned} +\right.} +\] +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Gauss-Algorithmus} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {\begin{minipage}{13cm}% +\[ +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 1 & 2 & 1 \\ + 2 & 0 & 1 & 2 \\ + 1 & 1 & 0 & 2 \\ +\hline +\end{tabular} +\uncover<5->{% +\to +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 1 & 2 & 1 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 1 \\ +\hline +\end{tabular}} +\uncover<7->{% +\to +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 1 & 0 & 2 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 1 \\ +\hline +\end{tabular}} +\uncover<9->{% +\to +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 0 & 0 & 2 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 1 \\ +\hline +\end{tabular}} +\] +\end{minipage}}; +\begin{scope}[yshift=0.2cm] +\uncover<4->{ +\draw[color=red] (-5.6,0.3) circle[radius=0.2]; +\draw[color=blue] (-5.4,-0.8) -- (-5.4,-0.2) arc (0:180:0.2) -- (-5.8,-0.8); +} +\uncover<6->{ +\draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5); +} +\uncover<8->{ +\draw[color=blue] (1.05,0.5) -- (1.05,0.2) arc (180:360:0.2) -- (1.45,0.5); +} +\end{scope} +\end{tikzpicture} +\end{center} +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex new file mode 100644 index 0000000..ef97182 --- /dev/null +++ b/vorlesungen/slides/4/ggt.tex @@ -0,0 +1,75 @@ +% +% ggt.tex -- GGT, Definition und Algorithmus +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschuöe +% +\begin{frame}[t] +\frametitle{Grösster gemeinsamer Teiler} +\vspace{-15pt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gegeben: $a,b\in\mathbb Z$ +\\ +Gesucht: grösster gemeinsamer Teiler $\operatorname{ggT}(a,b)$ +\end{block} +\uncover<4->{% +\begin{block}{Euklidischer Algorithmus} +$a_0 = a$, $b_0=b$ +\begin{align*} +\uncover<5->{ +a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0}\\ +\uncover<6->{ +a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1}\\ +\uncover<7->{ +a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2}\\ +\uncover<8->{ + &\;\vdots & & & & }\\ +\uncover<9->{ +a_n&=b_nq_n + r_n & r_n &= 0 & r_{n-1}&=\operatorname{ggT}(a,b)} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{ggT}(15,25) = 5$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.09] +\draw[->] (-1,0) -- (65,0) coordinate[label={$a$}]; +\draw[->] (0,-1) -- (0,65) coordinate[label={right:$b$}]; +\begin{scope} +\clip (-1,-1) rectangle (65,65); +\foreach \x in {0,...,4}{ + \draw[line width=0.2pt] ({\x*15},-2) -- ({\x*15},65); +} +\foreach \y in {0,...,2}{ + \draw[line width=0.2pt] (-2,{\y*25}) -- (65,{\y*25}); +} +\uncover<3->{ + \foreach \x in {0,5,...,120}{ + \draw[color=blue] ({\x+2},-2) -- ({\x+2-70},{-2+70}); + \node[color=blue] at ({0.5*\x-0.5},{0.5*\x-0.5}) + [rotate=-45,above] {\tiny $\x$}; + } +} +\uncover<2->{ + \foreach \x in {0,...,4}{ + \foreach \y in {0,...,2}{ + \fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8]; + } + } +} +\uncover<3->{ + \foreach \x in {0,5,...,60}{ + \fill[color=blue] (\x,0) circle[radius=0.5]; + \node at (\x,0) [below] {\tiny $\x$}; + } +} +\end{scope} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/polynomefp.tex b/vorlesungen/slides/4/polynomefp.tex new file mode 100644 index 0000000..1db50e1 --- /dev/null +++ b/vorlesungen/slides/4/polynomefp.tex @@ -0,0 +1,62 @@ +% +% polynomefp.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Polynome über $\mathbb{F}_p[X]$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Polynomring} +$\mathbb{F}_p[X]$ sind Polynome +\[ +p(X) += +a_0+a_1X+\dots+a_nX^n +\] +mit $a_i\in\mathbb{F}_p$. +\uncover<2->{ObdA: $a_n=1$}% + +\end{block} +\uncover<3->{% +\begin{block}{Irreduzible Polynome} +$m(X)$ ist irreduzibel, wenn es keine Faktorisierung +$m(X)=p(X)q(X)$ mit $p,q\in\mathbb{F}_p[X]$ gibt +\end{block}} +\uncover<4->{% +\begin{block}{Rest modulo $m(X)$} +$X^{n+k}$ kann immer reduziert werden: +\[ +X^{n+k} = -(a_0+a_1X+\dots+a_{n-1}X^{n-1})X^k +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Körper $\mathbb{F}_p/(m(X))$} +Wenn $m(X)$ irreduzibel ist, dann ist +$\mathbb{F}_p[X]$ nullteilerfrei. +\medskip + +\uncover<6->{$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist} +\begin{enumerate} +\item<7-> +$\operatorname{ggT}(a,m) = 1$ +\item<8-> +Es gibt $s,t\in\mathbb{F}_p[X]$ mit +\[ +s(X)m(X)+t(X)a(X) = 1 +\] +(aus dem euklidischen Algorithmus) +\item<9-> +$a^{-1} = t(X)$ +\end{enumerate} +\uncover<9->{$\Rightarrow$ $\mathbb{F}_p[X]/(m(X))$ ist ein Körper +mit genau $p^n$ Elementen} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/schieberegister.tex b/vorlesungen/slides/4/schieberegister.tex new file mode 100644 index 0000000..f349337 --- /dev/null +++ b/vorlesungen/slides/4/schieberegister.tex @@ -0,0 +1,120 @@ +% +% schieberegister.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\ds{0.7} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\punkt#1#2{({(#1)*\ds},{(#2)*\ds})} +\def\rahmen{ + \draw ({-0.5*\ds},{-0.5*\ds}) rectangle ({7.5*\ds},{0.5*\ds}); + \foreach \x in {0.5,1.5,...,6.5}{ + \draw ({\x*\ds},{-0.5*\ds}) rectangle ({\x*\ds},{0.5*\ds}); + } +} +\def\polynom#1#2#3#4#5#6#7#8{ + \node at \punkt{0}{0} {$#1$}; + \node at \punkt{1}{0} {$#2$}; + \node at \punkt{2}{0} {$#3$}; + \node at \punkt{3}{0} {$#4$}; + \node at \punkt{4}{0} {$#5$}; + \node at \punkt{5}{0} {$#6$}; + \node at \punkt{6}{0} {$#7$}; + \node at \punkt{7}{0} {$#8$}; +} +\begin{frame}[t] +\frametitle{Implementation der Multiplikation in $\mathbb{F}_2(\alpha)$\uncover<10->{: Schieberegister}} +Rechnen in $\mathbb{F}_2[X]$\only<5->{ und $\mathbb{F}_2(\alpha)$} +ist speziell einfach +\\ +Minimalpolynom von $\alpha$: ${\color{darkgreen}m(X) = X^8 + X^4+X^3+X+1}$ +(aus dem AES Standard) + +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<4->{ + \fill[color=blue!20] + \punkt{-0.5}{-0.5} rectangle \punkt{7.5}{0.5}; +} + +\uncover<2->{ +\begin{scope} + \rahmen + \node at \punkt{-0.5}{1} [left] {$p(X)=\mathstrut$}; + \node at \punkt{0}{1} {$X^7$\strut}; + \node at \punkt{2.5}{1}{$+$\strut}; + \node at \punkt{3}{1} {$X^4$\strut}; + \node at \punkt{4.5}{1}{$+$\strut}; + \node at \punkt{5}{1} {$X^2$\strut}; + \node at \punkt{6.5}{1}{$+$\strut}; + \node at \punkt{7}{1} {$1$\strut}; + \polynom10010101 +\end{scope}} + +\uncover<3->{ + \draw[->] ({7.7*\ds},-0.2) to[out=-45,in=45] ({7.7*\ds},-1.8); + \node at ({8*\ds},-1) [right] {$\mathstrut\cdot X = \text{Shift}$}; +} +\uncover<4->{ + \foreach \x in {0,...,7}{ + \draw[->,color=blue!40] + ({\x*\ds},{-0.6*\ds}) -- ({(\x-1)*\ds},{-2+0.6*\ds}); + } +} + +\fill[color=white] (-4.65,0) circle[radius=0.01]; + +\uncover<3->{ + \begin{scope}[yshift=-2cm] + \uncover<4->{ + \fill[color=blue!20] + \punkt{-1.5}{-0.5} rectangle \punkt{6.5}{0.5}; + \rahmen + \polynom00101010 + } + \node at \punkt{2}{1} {$X^5$\strut}; + \node at \punkt{3.5}{1}{$+$\strut}; + \node at \punkt{4}{1} {$X^3$\strut}; + \node at \punkt{5.5}{1}{$+$\strut}; + \node at \punkt{6}{1} {$X$\strut}; + \begin{scope}[xshift=0.4cm] + \node at \punkt{-1}{1} [left] + {$\uncover<5->{{\color{darkgreen}\alpha^4+\alpha^3+\alpha+1=\alpha^8}}\only<-4>{X^8}$\strut}; + \end{scope} + \node at \punkt{-1}{0} {$1$\strut}; + \end{scope} +} + +\uncover<6->{ + {\color<8->{red} + \draw[->] (-2.5,-1.5) to[out=-90,in=180] (-0.5,-2.7); + } + \begin{scope}[yshift=-2.7cm] + \rahmen + {\color{darkgreen} + \polynom00011011 + } + \end{scope} +} + +\uncover<7->{ + \node at ({3.5*\ds},-3.45) {$\|$}; + + \begin{scope}[yshift=-4.2cm] + \rahmen + \polynom00110001 + \node at \punkt{7.6}{0} [right] {$\mathstrut=\alpha\cdot p(\alpha)$}; + \end{scope} +} + +\uncover<8->{ + \node[color=red] at (-3.0,-2.5) {Feedback}; +} + +\end{tikzpicture} +\end{center} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/Aiteration.tex b/vorlesungen/slides/5/Aiteration.tex new file mode 100644 index 0000000..3078c55 --- /dev/null +++ b/vorlesungen/slides/5/Aiteration.tex @@ -0,0 +1,59 @@ +% +% Aiteration.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Iteration von $A$} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.34\textwidth} +\begin{block}{$\varrho(A) > 1\uncover<4->{\Rightarrow \|A^k\|\to\infty}$} +\uncover<2->{% +Eigenvektor $v$, $\|v\|=1$, zum Eigenwert $\lambda$ mit $|\lambda| > 1$} +\uncover<3->{% +\[ +\|A^kv\| = |\lambda|^k\to \infty +\]} +\uncover<4->{$\Rightarrow \|A\|^k\to\infty$} + +\end{block} +\end{column} +\begin{column}{0.63\textwidth} +\begin{block}{$\varrho(A) < 1\uncover<12->{\Rightarrow \|A\|^k\to 0}$} +\uncover<5->{% +$A$ setzt sich zusammen aus Jordanblöcken: +\[ +J(\lambda)^k += +\renewcommand{\arraystretch}{1.2} +\begin{pmatrix} +\lambda^k&\binom{k}{1}\lambda^{k-1}&\binom{k}{2}\lambda^{k-2} + &\dots&\binom{k}{n-1}\lambda^{k-n+1}\\ + 0 &\lambda^k&\binom{k}{1}\lambda^{k-1} + &\dots&\binom{k}{n-2}\lambda^{k-n+2}\\ + 0 & 0 &\lambda^k&\dots &\binom{k}{n-3}\lambda^{k-n+3}\\ + \vdots & \vdots & \vdots &\ddots &\vdots\\ + 0 & 0 & 0 &\dots &\lambda^k +\end{pmatrix} +\]} +\uncover<6->{Alle Matrixelemente konvergieren gegen $0$:} +\[ +\uncover<7->{\binom{k}{s} \le k^s} +\uncover<8->{\Rightarrow +\underbrace{\binom{k}{s}}_{\text{\uncover<9->{polynomiell $\to \infty$}}} +\underbrace{\lambda^{k-s}}_{\text{\uncover<10->{exponentiell $\to 0$}}} +} +\uncover<11->{\to 0} +\] +\end{block} +\end{column} +\end{columns} +\uncover<13->{% +{\usebeamercolor[fg]{title}Folgerung:} +Es gibt $m,M$ derart, dass +$m\varrho(A)^k \le \|A^k\| \le M \varrho(A)^k$ +} +\end{frame} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc new file mode 100644 index 0000000..4ca3de4 --- /dev/null +++ b/vorlesungen/slides/5/Makefile.inc @@ -0,0 +1,44 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter5 = \ + ../slides/5/verzerrung.tex \ + ../slides/5/motivation.tex \ + ../slides/5/charpoly.tex \ + ../slides/5/kernbildintro.tex \ + ../slides/5/kernbilder.tex \ + ../slides/5/kernbild.tex \ + ../slides/5/ketten.tex \ + ../slides/5/dimension.tex \ + ../slides/5/folgerungen.tex \ + ../slides/5/injektiv.tex \ + ../slides/5/nilpotent.tex \ + ../slides/5/eigenraeume.tex \ + ../slides/5/zerlegung.tex \ + ../slides/5/normalnilp.tex \ + ../slides/5/bloecke.tex \ + ../slides/5/jordanblock.tex \ + ../slides/5/jordan.tex \ + ../slides/5/reellenormalform.tex \ + ../slides/5/cayleyhamilton.tex \ + \ + ../slides/5/spektrum.tex \ + ../slides/5/normal.tex \ + ../slides/5/unitaer.tex \ + \ + ../slides/5/konvergenzradius.tex \ + ../slides/5/krbeispiele.tex \ + ../slides/5/spektralgelfand.tex \ + ../slides/5/Aiteration.tex \ + ../slides/5/satzvongelfand.tex \ + \ + ../slides/5/stoneweierstrass.tex \ + ../slides/5/potenzreihenmethode.tex \ + ../slides/5/logarithmusreihe.tex \ + ../slides/5/exponentialfunktion.tex \ + ../slides/5/hyperbolisch.tex \ + ../slides/5/chapter.tex + diff --git a/vorlesungen/slides/5/beispiele/Makefile b/vorlesungen/slides/5/beispiele/Makefile new file mode 100644 index 0000000..05bd5b5 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/Makefile @@ -0,0 +1,32 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: kern bild kb kombiniert.jpg leer.jpg drei.jpg + +kern: kern1.jpg kern2.jpg +bild: bild1.jpg bild2.jpg +kb: kernbild1.jpg kernbild2.jpg + +JK1.inc: kernbild.m + octave kernbild.m + +kernbild1.png: JK1.inc common.inc kernbild1.pov +kernbild2.png: JK1.inc common.inc kernbild2.pov +bild1.png: JK1.inc common.inc bild1.pov +bild2.png: JK1.inc common.inc bild2.pov +kern1.png: JK1.inc common.inc kern1.pov +kern2.png: JK1.inc common.inc kern2.pov +kombiniert.png: JK1.inc common.inc kombiniert.pov +leer.png: JK1.inc common.inc leer.pov +drei.png: JK1.inc common.inc drei.pov + +%.png: %.pov + povray +A0.1 -W1920 -H1080 -O$@ $< + +%.jpg: %.png + convert -extract 1080x1080+420+0 $< $@ + +clean: + rm -f *.png *.jpg diff --git a/vorlesungen/slides/5/beispiele/bild1.jpg b/vorlesungen/slides/5/beispiele/bild1.jpg Binary files differnew file mode 100644 index 0000000..879fae8 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/bild1.jpg diff --git a/vorlesungen/slides/5/beispiele/bild1.pov b/vorlesungen/slides/5/beispiele/bild1.pov new file mode 100644 index 0000000..fd814f1 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/bild1.pov @@ -0,0 +1,13 @@ +// +// bild1.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j11, at, orange1) +arrow(O, j12, at, orange1) +ebene(j11, j12, orange1) + diff --git a/vorlesungen/slides/5/beispiele/bild2.jpg b/vorlesungen/slides/5/beispiele/bild2.jpg Binary files differnew file mode 100644 index 0000000..2597c95 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/bild2.jpg diff --git a/vorlesungen/slides/5/beispiele/bild2.pov b/vorlesungen/slides/5/beispiele/bild2.pov new file mode 100644 index 0000000..6e3c6dd --- /dev/null +++ b/vorlesungen/slides/5/beispiele/bild2.pov @@ -0,0 +1,17 @@ +// +// bild2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j11, 0.7 * at, orange1) +arrow(O, j12, 0.7 * at, orange1) +ebene(j11, j12, orange1) + +arrow(O, j21, at, orange2) +gerade(j21, orange2) + + diff --git a/vorlesungen/slides/5/beispiele/common.inc b/vorlesungen/slides/5/beispiele/common.inc new file mode 100644 index 0000000..ffcff60 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/common.inc @@ -0,0 +1,134 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.25; +#declare O = <0, 0, 0>; +#declare at = 0.02; + +camera { + location <3, 2, -10> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +//light_source { +// <-14, 20, -50> color White +// area_light <1,0,0> <0,0,1>, 10, 10 +// adaptive 1 +// jitter +//} + +light_source { + <41, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.0 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end +#declare r = 1.1; + +arrow(< -r-0.2, 0.0, 0 >, < r+0.2, 0.0, 0.0 >, at, Gray) +arrow(< 0.0, 0.0, -r-0.2>, < 0.0, 0.0, r+0.2 >, at, Gray) +arrow(< 0.0, -r-0.2, 0 >, < 0.0, r+0.2, 0.0 >, at, Gray) + +#declare gruen1 = rgb<0.0,0.4,0.0>; +#declare gruen2 = rgb<0.0,0.4,0.8>; +#declare orange1 = rgb<1.0,0.6,0.0>; +#declare orange2 = rgb<0.8,0.0,0.4>; + +#macro ebene(v1, v2, farbe) + intersection { + box { <-r,-r,-r>, <r,r,r> } + plane { vnormalize(vcross(v1, v2)), 0.004 } + plane { vnormalize(-vcross(v1, v2)), 0.004 } + pigment { + color rgbt<farbe.x, farbe.y, farbe.z, 0.5> + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro gerade(v1, farbe) + intersection { + box { <-r,-r,-r>, <r,r,r> } + cylinder { -2 * r * vnormalize(v1), + 2 * r * vnormalize(v1), 0.80*at } + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro kasten() + difference { + box { <-r-0.01,-r-0.01,-r-0.01>, <r+0.01,r+0.01,r+0.01> } + union { + box { < -r, -r, -r >, + < r, r, r > } + box { <-2*r, -r+0.03, -r+0.03>, + < 2*r, r-0.03, r-0.03> } + box { < -r+0.03, -2*r, -r+0.03>, + < r-0.03, 2*r, r-0.03> } + box { < -r+0.03, -r+0.03, -2*r >, + < r-0.03, r-0.03, 2*r > } + } + pigment { + color rgb<0.8,0.8,0.8> + } + finish { + specular 0.9 + metallic + } + } +#end + diff --git a/vorlesungen/slides/5/beispiele/drei.jpg b/vorlesungen/slides/5/beispiele/drei.jpg Binary files differnew file mode 100644 index 0000000..35f9034 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/drei.jpg diff --git a/vorlesungen/slides/5/beispiele/drei.pov b/vorlesungen/slides/5/beispiele/drei.pov new file mode 100644 index 0000000..bdc9630 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/drei.pov @@ -0,0 +1,22 @@ +// +// drei.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j21, at, orange2) +//arrow(O, k21, at, gruen2) +//arrow(O, k22, at, gruen2) +gerade(j21, orange2) +//ebene(k21, k22, gruen2) + +#declare at = 0.7 * at; + +arrow(O, j11, at, orange1) +arrow(O, j12, at, orange1) +arrow(O, k11, at, gruen1) +ebene(j11, j12, orange1) + diff --git a/vorlesungen/slides/5/beispiele/kern1.jpg b/vorlesungen/slides/5/beispiele/kern1.jpg Binary files differnew file mode 100644 index 0000000..5c99664 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kern1.jpg diff --git a/vorlesungen/slides/5/beispiele/kern1.pov b/vorlesungen/slides/5/beispiele/kern1.pov new file mode 100644 index 0000000..8e61d8d --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kern1.pov @@ -0,0 +1,12 @@ +// +// kern1.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, k11, at, gruen1) +gerade(k11, gruen1) + diff --git a/vorlesungen/slides/5/beispiele/kern2.jpg b/vorlesungen/slides/5/beispiele/kern2.jpg Binary files differnew file mode 100644 index 0000000..87d18ac --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kern2.jpg diff --git a/vorlesungen/slides/5/beispiele/kern2.pov b/vorlesungen/slides/5/beispiele/kern2.pov new file mode 100644 index 0000000..70127a2 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kern2.pov @@ -0,0 +1,17 @@ +// +// kern2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, k21, at, gruen2) +arrow(O, k22, at, gruen2) +ebene(k21, k22, gruen2) + +#declare at = 0.7 * at; +arrow(O, k11, at, gruen1) +gerade(k11, gruen1) + diff --git a/vorlesungen/slides/5/beispiele/kernbild.m b/vorlesungen/slides/5/beispiele/kernbild.m new file mode 100644 index 0000000..28cd552 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kernbild.m @@ -0,0 +1,79 @@ +# +# kernbild.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +rand("seed", 1291) +rand("seed", 4711) + +lambda1 = 1; +lambda2 = 1.8; + +A = [ + lambda1, 0, 0; + 0, lambda2, 1; + 0, 0, lambda2 +]; + +B = eye(3) + rand(3,3); +det(B) + + +C = B*A*inverse(B) +rank(C) + +# Eigenwert lambda1 +E2 = C - lambda1 * eye(3) +rref(E2) + +# Eigenwert lambda2, k = 1 +E1 = C - lambda2 * eye(3) +D = rref(E1); +K1 = [ + -D(1,3); + -D(2,3); + 1 +]; +K1(:,1) = K1(:,1) / norm(K1(:,1)); +K1 + +f = fopen("JK.inc", "w"); +fprintf(f, "//\n// JK.inc\n//\n// (c) 2021 Prof Dr Andreas Müller\n//\n\n"); +fprintf(f, "// Kern und Bild von C - %.3f I\n", lambda2); +fprintf(f, "#declare k11 = < %.5f, %.5f, %.5f>;\n", K1(1,1), K1(2,1), K1(3,1)); +fprintf(f, "#declare j11 = < %.5f, %.5f, %.5f>;\n", E1(1,1), E1(2,1), E1(3,1)); +fprintf(f, "#declare j12 = < %.5f, %.5f, %.5f>;\n", E1(1,2), E1(2,2), E1(3,2)); +fprintf(f, "\n"); + +# k = 2 +E12 = E1 * E1 +D = rref(E12); +K2 = [ + -D(1,2), -D(1,3); + 1, 0; + 0, 1 +] +K2(:,1) = K2(:,1) / norm(K2(:,1)); +K2(:,2) = K2(:,2) / norm(K2(:,2)); +K2 + +fprintf(f, "// Kern und Bild von (C - %.3f I)^2\n", lambda2); +fprintf(f, "#declare k21 = < %.5f, %.5f, %.5f>;\n", K2(1,1), K2(2,1), K2(3,1)); +fprintf(f, "#declare k22 = < %.5f, %.5f, %.5f>;\n", K2(1,2), K2(2,2), K2(3,2)); +fprintf(f, "#declare j21 = < %.5f, %.5f, %.5f>;\n", E12(1,1), E12(2,1), E12(3,1)); +fprintf(f, "\n"); + +fclose(f); + +# Verifikation +x = K2 \ K1 +K2 * x + +eig(C) + +[U, S, V] = svd(C) + + +s = rand("seed") + diff --git a/vorlesungen/slides/5/beispiele/kernbild1.jpg b/vorlesungen/slides/5/beispiele/kernbild1.jpg Binary files differnew file mode 100644 index 0000000..87e874e --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kernbild1.jpg diff --git a/vorlesungen/slides/5/beispiele/kernbild1.pov b/vorlesungen/slides/5/beispiele/kernbild1.pov new file mode 100644 index 0000000..425f299 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kernbild1.pov @@ -0,0 +1,15 @@ +// +// kernbild1.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j11, at, orange1) +arrow(O, j12, at, orange1) +arrow(O, k11, at, gruen1) +ebene(j11, j12, orange1) + +//kasten() diff --git a/vorlesungen/slides/5/beispiele/kernbild2.jpg b/vorlesungen/slides/5/beispiele/kernbild2.jpg Binary files differnew file mode 100644 index 0000000..1160b31 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kernbild2.jpg diff --git a/vorlesungen/slides/5/beispiele/kernbild2.pov b/vorlesungen/slides/5/beispiele/kernbild2.pov new file mode 100644 index 0000000..ae67ea1 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kernbild2.pov @@ -0,0 +1,21 @@ +// +// kernbild2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j21, at, orange2) +arrow(O, k21, at, gruen2) +arrow(O, k22, at, gruen2) +gerade(j21, orange2) +ebene(k21, k22, gruen2) + +//arrow(O, j11, at, orange1) +//arrow(O, j12, at, orange1) +//arrow(O, k11, at, gruen1) +//gerade(k11, gruen1) +//ebene(j11, j12, orange1) + diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg Binary files differnew file mode 100644 index 0000000..9cb789c --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kombiniert.jpg diff --git a/vorlesungen/slides/5/beispiele/kombiniert.pov b/vorlesungen/slides/5/beispiele/kombiniert.pov new file mode 100644 index 0000000..c187d08 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kombiniert.pov @@ -0,0 +1,22 @@ +// +// kombiniert.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j21, at, orange2) +arrow(O, k21, at, gruen2) +arrow(O, k22, at, gruen2) +gerade(j21, orange2) +ebene(k21, k22, gruen2) + +#declare at = 0.7 * at; + +arrow(O, j11, at, orange1) +arrow(O, j12, at, orange1) +arrow(O, k11, at, gruen1) +ebene(j11, j12, orange1) + diff --git a/vorlesungen/slides/5/beispiele/leer.jpg b/vorlesungen/slides/5/beispiele/leer.jpg Binary files differnew file mode 100644 index 0000000..9789887 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/leer.jpg diff --git a/vorlesungen/slides/5/beispiele/leer.pov b/vorlesungen/slides/5/beispiele/leer.pov new file mode 100644 index 0000000..f4653d9 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/leer.pov @@ -0,0 +1,9 @@ +// +// leer.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + diff --git a/vorlesungen/slides/5/bloecke.tex b/vorlesungen/slides/5/bloecke.tex new file mode 100644 index 0000000..974f238 --- /dev/null +++ b/vorlesungen/slides/5/bloecke.tex @@ -0,0 +1,141 @@ +% +% bloecke.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\sx{1} +\def\sy{0.1} +\def\block#1#2{ + \fill[color=red] ({#1},{-#1}) rectangle ({#1+#2},{-#1-#2}); +} +\def\kreuz#1{ + \draw[color=white,line width=0.1pt] (0,{-#1})--(60,{-#1}); + \draw[color=white,line width=0.1pt] (#1,0)--(#1,-60); +} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Blockgrössen aus $\dim\mathcal{K}^k(A)$ ablesen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.56\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (A) at ({1*\sx},{20*\sy}); +\coordinate (B) at ({2*\sx},{(20+15)*\sy}); +\coordinate (C) at ({3*\sx},{(20+15+10)*\sy}); +\coordinate (D) at ({4*\sx},{(20+15+10+8)*\sy}); +\coordinate (E) at ({5*\sx},{(20+15+10+8+5)*\sy}); +\coordinate (F) at ({6*\sx},{(20+15+10+8+5+2)*\sy}); +\fill[color=darkgreen!20] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F) + -- ({6*\sx},0) -- cycle; + +\fill[color=darkgreen!40] (0,0) -- ({1*\sx},0) -- (A) -- cycle; +\fill[color=darkgreen!40] (A) -- ({2*\sx},{20*\sy}) -- (B) -- cycle; +\fill[color=darkgreen!40] (B) -- ({3*\sx},{(20+15)*\sy}) -- (C) -- cycle; +\fill[color=darkgreen!40] (C) -- ({4*\sx},{(20+15+10)*\sy}) -- (D) -- cycle; +\fill[color=darkgreen!40] (D) -- ({5*\sx},{(20+15+10+8)*\sy}) -- (E) -- cycle; +\fill[color=darkgreen!40] (E) -- ({6*\sx},{(20+15+10+8+5)*\sy}) -- (F) -- cycle; + +\draw[color=darkgreen,line width=1.4pt] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F); + +\draw[color=gray] (A) -- (0,{20*\sy}); +\draw[color=gray] (B) -- (0,{(20+15)*\sy}); +\draw[color=gray] (C) -- (0,{(20+15+10)*\sy}); +\draw[color=gray] (D) -- (0,{(20+15+10+8)*\sy}); +\draw[color=gray] (E) -- (0,{(20+15+10+8+5)*\sy}); +\draw[color=gray] (F) -- (0,{(20+15+10+8+5+2)*\sy}); + +\node at ({0.5*\sx},{0.5*20*\sy}) + [right] {$d_1 = \dim\mathcal{K}^1(A)-\dim\mathcal{K}^0(A)$}; +\node at ({1.5*\sx},{0.5*(20+20+15)*\sy}) + [right] {$d_2 = \dim\mathcal{K}^2(A)-\dim\mathcal{K}^1(A)$}; +\node at ({2.5*\sx},{0.5*(2*20+2*15+1*10)*\sy}) [right] {$d_3$}; +\node at ({3.5*\sx},{0.5*(2*20+2*15+2*10+8)*\sy}) [right] {$d_4$}; +\node at ({4.5*\sx-0.1},{0.5*(2*20+2*15+2*10+2*8+5)*\sy+0.2}) [below right] {$d_5$}; +\node at ({5.5*\sx},{0.5*(2*20+2*15+2*10+2*8+2*5+2)*\sy+0.1}) [below] {$d_6$}; + +\fill (A) circle[radius=0.08]; +\fill (B) circle[radius=0.08]; +\fill (C) circle[radius=0.08]; +\fill (D) circle[radius=0.08]; +\fill (E) circle[radius=0.08]; +\fill (F) circle[radius=0.08]; + +\draw[->] (-0.1,0) -- ({6*\sx+1},0) coordinate[label={$k$}]; +\draw[->] (0,-0.1) -- (0,6.5) coordinate[label={right:$\dim\mathcal{K}^k(A)$}]; + +\foreach \x in {0,1,...,6}{ + \draw ({\sx*\x},{-0.05}) -- ({\sx*\x},0.05); + \node at ({\sx*\x},{-0.1}) [below] {$\x$}; +} + +\node at (0,{60*\sy}) [left] {\llap{$n$}}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.43\textwidth} +\vspace{-10pt} +\begin{center} +\begin{tabular}{>{$}c<{$}|>{$}r<{$}|>{$}c<{$}|>{$}c<{$}} +k&d_k&\# M_k(\Bbbk)\text{-Blöcke}&\text{Beispiel}\\ +\hline +0& 0& &\\ +1& 20& d_1-d_2&5\\ +2& 15& d_2-d_3&5\\ +3& 10& d_3-d_4&2\\ +4& 8& d_4-d_5&3\\ +5& 5& d_5-d_6&3\\ +6& 2& d_6 &2\\ +\end{tabular} +\end{center} +\vspace{-13pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.05] +\fill[color=gray!40] (0,0) rectangle (60,-60); +\node[color=white] at (30,-30) [scale=6] {$A$}; +\kreuz{5} +\kreuz{15} +\kreuz{21} +\kreuz{33} +\kreuz{48} +\node at (0,-2.5) [left] {$k=1$}; +\node at (60,-2.5) [right] {$5$ Blöcke}; +\node at (0,-10) [left] {$k=2$}; +\node at (60,-10) [right] {$5$ Blöcke}; +\node at (0,-18) [left] {$k=3$}; +\node at (60,-18) [right] {$2$ Blöcke}; +\node at (0,-27) [left] {$k=4$}; +\node at (60,-27) [right] {$3$ Blöcke}; +\node at (0,-40.5) [left] {$k=5$}; +\node at (60,-40.5) [right] {$3$ Blöcke}; +\node at (0,-54) [left] {$k=6$}; +\node at (60,-54) [right] {$2$ Blöcke}; +\block{0}{1} +\block{1}{1} +\block{2}{1} +\block{3}{1} +\block{4}{1} +\block{5}{2} +\block{7}{2} +\block{9}{2} +\block{11}{2} +\block{13}{2} +\block{15}{3} +\block{18}{3} +\block{21}{4} +\block{25}{4} +\block{29}{4} +\block{33}{5} +\block{38}{5} +\block{43}{5} +\block{48}{6} +\block{54}{6} +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/cayleyhamilton.tex b/vorlesungen/slides/5/cayleyhamilton.tex new file mode 100644 index 0000000..c0813be --- /dev/null +++ b/vorlesungen/slides/5/cayleyhamilton.tex @@ -0,0 +1,91 @@ +% +% cayleyhamilton.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Satz von Cayley-Hamilton} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ein Eigenwert $\lambda$\strut} +$A$ besteht aus +$b$ Blöcken $J_\lambda$ mit maximaler Dimension $l$: +\phantom{blubb\strut} +\begin{align*} +\uncover<2->{ +\chi_{A}(X) +&= +\det (A-XI) = (\lambda-X)^n +} +\\ +\uncover<3->{ +m_{A}(X) +&= +(\lambda-X)^l +} +\\ +\uncover<4->{ +b&= \ker A +} +\end{align*} +\uncover<5->{% +Wegen $l \le n$ folgt +\[ +m_A(X) | \chi_A(X) +\uncover<6->{\quad\Rightarrow\quad +\chi_A(A) = 0} +\]} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{$A=A_1\oplus\dots\oplus A_k$} +\uncover<8->{% +$A_i\in M_{n_i}(\Bbbk)$ mit EW $\lambda_i$, +$A_i$ besteht aus +$b_i$ Blöcken $J_{\lambda_i}$ mit max.~Dimension $l_i$\strut:} +\begin{align*} +\uncover<9->{ +\chi_A(X) +&= +(\lambda_1-X)^{n_1} +\dots +(\lambda_k-X)^{n_k} +} +\\ +\uncover<10->{ +m_A(X) +&= +(\lambda_1-X)^{l_1} +\dots +(\lambda_k-X)^{l_k} +} +\\ +\uncover<11->{ +b_i &= \ker (A-\lambda_iI) +} +\end{align*} +\uncover<12->{% +$A=A_1\oplus\dots\oplus A_k$} +\begin{align*} +\uncover<13->{ +\chi_{A_i}(A_i)&=0\;\forall i +} +\\ +\uncover<14->{% +\chi_A(A) &= +\chi_{A_1}(A)\dots\chi_{A_k}(A) + = 0} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<15->{% +\begin{block}{Satz} +Für jede Matrix $A\in M_n(\Bbbk)$ gilt +$m_A(X) | \chi_A(X)$ oder $\chi_A(A)=0$ +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex new file mode 100644 index 0000000..96eea29 --- /dev/null +++ b/vorlesungen/slides/5/chapter.tex @@ -0,0 +1,36 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{5/verzerrung.tex} +\folie{5/motivation.tex} +\folie{5/charpoly.tex} +\folie{5/kernbildintro.tex} +\folie{5/kernbilder.tex} +\folie{5/kernbild.tex} +\folie{5/ketten.tex} +\folie{5/dimension.tex} +\folie{5/folgerungen.tex} +\folie{5/injektiv.tex} +\folie{5/nilpotent.tex} +\folie{5/eigenraeume.tex} +\folie{5/zerlegung.tex} +\folie{5/normalnilp.tex} +\folie{5/bloecke.tex} +\folie{5/jordanblock.tex} +\folie{5/jordan.tex} +\folie{5/reellenormalform.tex} +\folie{5/cayleyhamilton.tex} +\folie{5/konvergenzradius.tex} +\folie{5/krbeispiele.tex} +\folie{5/spektralgelfand.tex} +\folie{5/Aiteration.tex} +\folie{5/satzvongelfand.tex} +\folie{5/stoneweierstrass.tex} +\folie{5/potenzreihenmethode.tex} +\folie{5/logarithmusreihe.tex} +\folie{5/exponentialfunktion.tex} +\folie{5/hyperbolisch.tex} +\folie{5/spektrum.tex} +\folie{5/normal.tex} diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex new file mode 100644 index 0000000..63bfee5 --- /dev/null +++ b/vorlesungen/slides/5/charpoly.tex @@ -0,0 +1,78 @@ +% +% charpoly.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Charakteristisches Polynom über $\mathbb{C}$} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Eigenwerte} +Nur diejenigen $\mu$ kommen in Frage, für die +$A-\mu I$ singulär ist: +\[ +\chi_{A}(\mu) += +\det (A-\mu I) = 0 +\] +$\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Zerlegung in Linearfaktoren} +$\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$: +\[ +\chi_A(X) += +(X-\mu_1)\dots (X-\mu_n) +\] +\end{block}} +\uncover<3->{% +\begin{block}{Fundamentalsatz der Algebra} +Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in +Linearfaktoren +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Minimalpolynom} +Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen +\end{block}} +\uncover<5->{% +\begin{proof}[Beweis] +\begin{enumerate} +\item<6-> +$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$ +\item<7-> +$A-\lambda I$ ist regulär +\end{enumerate} +\uncover<8->{% +\begin{align*} +&\Rightarrow& +m_A(A)&=0 +\\ +&& +\uncover<9->{ +(A-\lambda)^{-1}m_A(A) &=0 +} +\\ +&& +\uncover<10->{ +\prod_{i\in I}(A-\mu_i)&=0, +} +\end{align*}} +\uncover<11->{% +d.~h.~\( +\displaystyle +\overline{m}_A(X) += +\prod_i{i\in I}(X-\mu_i) +\in +\mathbb{C}[X] +\)} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/dimension.tex b/vorlesungen/slides/5/dimension.tex new file mode 100644 index 0000000..ff687b3 --- /dev/null +++ b/vorlesungen/slides/5/dimension.tex @@ -0,0 +1,68 @@ +% +% dimension.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Dimension von $\mathcal{K}^k(f)$ und $\mathcal{J}^k(f)$} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\pfad{ + (0,0) -- (1,0.3) -- (2,0.9) + -- + (4,2.4) -- (5,2.7) -- (6,3.3) + -- + (8,3.7) -- (9,4) -- (10,4) -- (11,4) -- (12,4) +} + +\fill[color=darkgreen!20] \pfad -- (12,0) -- cycle; +\fill[color=orange!20] \pfad -- (12,6) -- (0,6) -- cycle; + +\fill[color=darkgreen!40] (9,0) -- (12,0) -- (12,4) -- (9,4) -- cycle; +\fill[color=orange!40] (9,4) -- (12,4) -- (12,6) -- (9,6) -- cycle; + +\node[color=orange] at (10.5,5) {$\mathcal{J}(f)$}; +\node[color=darkgreen] at (10.5,2) {$\mathcal{K}(f)$}; + +\node[color=orange] at (5.5,4.5) {$\mathcal{J}^k(f)\supset\mathcal{J}^{k+1}(f)$}; +\node[color=darkgreen] at (5.5,1.5) {$\mathcal{K}^k(f)\subset\mathcal{K}^{k+1}(f)$}; + +\draw[line width=1.4pt] \pfad; + +\draw[->] (-0.1,6) -- (12.5,6) coordinate[label={$k$}]; +\draw[->] (-0.1,0) -- (12.5,0) coordinate[label={$k$}]; +\node at (-0.1,6) [left] {$n$}; +\node at (-0.1,0) [left] {$0$}; +\foreach \x in {0,1,2,4,5,6,8,9,10,11,12}{ + \fill (\x,0) circle[radius=0.05]; + \fill (\x,6) circle[radius=0.05]; +} +\node at (0,0) [below] {$0$}; +\node at (1,0) [below] {$1$}; +\node at (2,0) [below] {$2$}; + +\node at (4,0) [below] {$k-1$}; +\node at (5,0) [below] {$k$}; +\node at (6,0) [below] {$k+1$}; + +\node at (8,0) [below] {$l-1$}; +\node at (9,0) [below] {$l$}; +\node at (10,0) [below] {$l+1$}; +\node at (11,0) [below] {$l+2$}; +\node at (12,0) [below] {$l+3$}; + +\fill (9,4) circle[radius=0.05]; + +\node[color=orange] at (-0.2,3) [rotate=90] {$\dim\mathcal{J}^k(f)$}; +\node[color=darkgreen] at (12.2,2) [rotate=-90] {$\dim\mathcal{K}^k(f)$}; + +\node[color=orange] at (9,5) [rotate=-90] {$\dim\mathcal{J}(f)$}; +\node[color=darkgreen] at (9,2) [rotate=-90] {$\dim\mathcal{K}(f)$}; + +\end{tikzpicture} +\end{center} + +\end{frame} diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex new file mode 100644 index 0000000..fd4803c --- /dev/null +++ b/vorlesungen/slides/5/eigenraeume.tex @@ -0,0 +1,48 @@ +% +% eigenraeume.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Eigenräume} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Eigenraum} +Für $\lambda\in\Bbbk$ heisst +\begin{align*} +E_\lambda(f) +&= +\ker (f-\lambda) +\\ +\uncover<2->{ +&= +\{v\in V\;|\; f(v) = \lambda v\} +} +\end{align*} +\uncover<3->{% +{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.} +\end{block} +\uncover<4->{% +$E_\lambda(f)\subset V$ ist ein Unterraum} + +\uncover<5->{% +\begin{block}{Eigenwert} +Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$. +\end{block}} + +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{verallgemeinerter Eigenraum} +Für $\lambda\in \Bbbk$ heisst +\[ +\mathcal{E}_\lambda(f) += +\mathcal{K}(f-\lambda) +\] +verallgemeinerter Eigenraum +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/exponentialfunktion.tex b/vorlesungen/slides/5/exponentialfunktion.tex new file mode 100644 index 0000000..caae16b --- /dev/null +++ b/vorlesungen/slides/5/exponentialfunktion.tex @@ -0,0 +1,131 @@ +% +% exponentialfunktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Exponentialfunktion} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\only<1-6>{% +\ifthenelse{\boolean{presentation}}{ +\begin{column}{0.48\textwidth} +\begin{block}{$x(t) \in\mathbb{R}$} +\vspace{-10pt} +\begin{align*} +\frac{d}{dt}x(t) &= ax(t) &a&\in\mathbb{R} +\\ +x(0) &= c&&\in\mathbb{R} +\intertext{\uncover<2->{Lösung:}} +\uncover<2->{x(t) &= ce^{at}} +\end{align*} +\end{block} +\end{column}}{}} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{$X(t) \in M_n(\mathbb{R})$} +\vspace{-10pt} +\begin{align*} +\frac{d}{dt}X(t) +&= +A +X(t)&A&\in M_n(\mathbb{R}) +\\ +X(0)&=C&&\in M_n(\mathbb{R}) +\intertext{\uncover<4->{gekoppelte Differentialgleichung für +vier Funktionen $x_{ij}(t)$}} +\uncover<5->{\dot{x}_{11} &= \rlap{$a_{11} x_{11}(t) + a_{12} x_{21}(t)$}}\\ +\uncover<5->{\dot{x}_{12} &= \rlap{$a_{11} x_{12}(t) + a_{12} x_{22}(t)$}}\\ +\uncover<5->{\dot{x}_{21} &= \rlap{$a_{21} x_{11}(t) + a_{22} x_{21}(t)$}}\\ +\uncover<5->{\dot{x}_{22} &= \rlap{$a_{21} x_{12}(t) + a_{22} x_{22}(t)$}}\\ +\intertext{\uncover<6->{Lösung:}} +\uncover<6->{X(t) &= \exp(At) C} +\end{align*} +\end{block}} +\end{column} +\only<7-9>{% +\ifthenelse{\boolean{presentation}}{ +\begin{column}{0.48\textwidth} +\begin{block}{Beispiel: Diagonalmatrix} +%$D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$ +\begin{align*} +\frac{d}{dt}X&=DX &&\uncover<8->{\Rightarrow &\dot{x}_{ij}(t) &= \lambda_i x_{ij}(t)} +\\ +X(0)&=C +&&\uncover<8->{\Rightarrow&x_{ij}(t)&=c_{ij}} +\end{align*} +\uncover<9->{% +Lösung: +\[ +x_{ij}(t) =c_{ij}e^{\lambda_i t} +\]} +\end{block} +\end{column}}{}} +\uncover<10->{% +\begin{column}{0.48\textwidth} +\begin{block}{Beispiel: Jordan-Block} +\vspace{-10pt} +\begin{align*} +A&=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix} +\rlap{$\displaystyle,\; +X(t) += +\ifthenelse{\boolean{presentation}}{ +\only<22>{ + e^{\lambda t} + \begin{pmatrix} 1&t/\lambda\\ 0&1 \end{pmatrix} +}}{} +\only<23>{ + \frac{e^{\lambda t}}{\lambda} + \begin{pmatrix} \lambda&t\\ 0&\lambda \end{pmatrix} +} +C +$} +\\ +\uncover<11->{ +\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + \phantom{\lambda}x_{2i}(t),&&x_{1i}(0)&=c_{1i} +} +\\ +\uncover<12->{ +\dot{x}_{2i}(t)&=\phantom{\lambda x_{1i}(t)+\mathstrut}\lambda x_{2i}(t),&&x_{2i}(0)&=c_{2i} +} +\end{align*} +\uncover<13->{% +Lösung:} +\begin{align*} +\uncover<14->{ +x_{2i}(t)&=c_{2i}e^{\lambda t} +} +\\ +\uncover<15->{ +\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + c_{2i}e^{\lambda t} +} +\\ +\ifthenelse{\boolean{presentation}}{ +\only<16-17>{x_{1i\only<16>{,h}}(t)}}{} +\only<18->{\dot{x}_{1i}(t)} +& +\ifthenelse{\boolean{presentation}}{ +\only<16-17>{=c\only<17>{(t)}\lambda e^{\lambda t}} +\only<18>{=\dot{c}(t)\lambda e^{\lambda t} ++ +c(t)\lambda^2 e^{\lambda t}} +}{} +\only<19->{=\lambda x_{1i}(t) + \dot{c}(t)\lambda e^{\lambda t}} +\\ +\uncover<20->{\Rightarrow +\dot{c}(t)&= c_{2i}/\lambda +\Rightarrow +c(t) = c_{2i}(0) +tc_{2i}/\lambda +} +\\ +\uncover<21->{ +x_{1i}(t) & =c_{1i}e^{\lambda t} + t(c_{2i}/\lambda)e^{\lambda t} +} +\end{align*} +\end{block} +\end{column}} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex new file mode 100644 index 0000000..4a8dbe6 --- /dev/null +++ b/vorlesungen/slides/5/folgerungen.tex @@ -0,0 +1,84 @@ +% +% folgerungen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\sx{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Folgerungen} +\vspace{-10pt} +\begin{columns}[t] +\begin{column}{0.30\textwidth} +\begin{block}{Zunahme} +Für alle $k<l$ gilt +\begin{align*} +\mathcal{J}^k(f) &\supsetneq \mathcal{J}^{k+1}(f) +\\ +\mathcal{K}^k(f) &\subsetneq \mathcal{K}^{k+1}(f) +\end{align*} +Für $k\ge l$ gilt +\begin{align*} +\mathcal{J}^k(f) &= \mathcal{J}^{k+1}(f) +\\ +\mathcal{K}^k(f) &= \mathcal{K}^{k+1}(f) +\end{align*} +Ausserdem ist $l\le n$ +\end{block} +\end{column} +\begin{column}{0.66\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\pfad{ + ({0*\sx},6) -- + ({1*\sx},4.5) -- + ({2*\sx},3.5) -- + ({3*\sx},2.9) -- + ({4*\sx},2.6) -- + ({5*\sx},2.4) -- + ({6*\sx},2.4) +} + +\fill[color=orange!20] \pfad -- ({6*\sx},0) -- (0,0) -- cycle; +\fill[color=darkgreen!20] \pfad -- ({6*\sx},6) -- cycle; +\fill[color=orange!40] ({5*\sx},0) rectangle ({6*\sx},2.4); +\fill[color=darkgreen!40] ({5*\sx},6) rectangle ({6*\sx},2.4); + +\draw[color=darkgreen,line width=2pt] ({3*\sx},6) -- ({3*\sx},2.9); +\node[color=darkgreen] at ({3*\sx},4.45) [rotate=90,above] {$\dim\mathcal{K}^k(A)$}; +\draw[color=orange,line width=2pt] ({3*\sx},0) -- ({3*\sx},2.9); +\node[color=orange] at ({3*\sx},1.45) [rotate=90,above] {$\dim\mathcal{J}^k(A)$}; + +\node[color=orange] at ({5.5*\sx},1.2) [rotate=90] {bijektiv}; +\node[color=darkgreen] at ({5.5*\sx},4.2) [rotate=90] {konstant}; + +\fill ({0*\sx},6) circle[radius=0.08]; +\fill ({1*\sx},4.5) circle[radius=0.08]; +\fill ({2*\sx},3.5) circle[radius=0.08]; +\fill ({3*\sx},2.9) circle[radius=0.08]; +\fill ({4*\sx},2.6) circle[radius=0.08]; +\fill ({5*\sx},2.4) circle[radius=0.08]; +\fill ({6*\sx},2.4) circle[radius=0.08]; + +\draw \pfad; + +\draw[->] (-0.1,0) -- ({6*\sx+0.5},0) coordinate[label={$k$}]; +\draw[->] (-0.1,6) -- ({6*\sx+0.5},6); + +\foreach \x in {0,...,6}{ + \draw (\x,-0.05) -- (\x,0.05); +} +\foreach \x in {0,...,3}{ + \node at ({\x*\sx},-0.05) [below] {$\x$}; +} +\node at ({4*\sx},-0.05) [below] {$\dots\mathstrut$}; +\node at ({5*\sx},-0.05) [below] {$l$}; +\node at ({6*\sx},-0.05) [below] {$l+1$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/hyperbolisch.tex b/vorlesungen/slides/5/hyperbolisch.tex new file mode 100644 index 0000000..905082a --- /dev/null +++ b/vorlesungen/slides/5/hyperbolisch.tex @@ -0,0 +1,105 @@ +% +% hyperbolisch.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hyperbolische Funktionen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Differentialgleichung} +\vspace{-10pt} +\begin{align*} +\ddot{y} &= y +\;\Rightarrow\; +\frac{d}{dt} +\begin{pmatrix}y\\y_1\end{pmatrix} += +\begin{pmatrix}0&1\\1&0\end{pmatrix} +\begin{pmatrix}y\\y_1\end{pmatrix} +\\ +y(0)&=a,\qquad y'(0)=b +\end{align*} +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Lösung} +\vspace{-13pt} +\begin{align*} +\lambda^2-1&=0 +\uncover<3->{ +\qquad\Rightarrow\qquad \lambda=\pm 1 +} +\\ +\uncover<4->{ +y(t)&=Ae^t+Be^{-t}} +\uncover<5->{ +\Rightarrow +\left\{ +\arraycolsep=1.4pt +\begin{array}{rcrcr} +A&+&B&=&a\\ +A&-&B&=&b +\end{array} +\right.} +\\ +&\uncover<6->{ +=\frac{a+b}2e^t + \frac{a-b}2e^{-t}} +\\ +&\uncover<7->{= +a{\color{darkgreen}\frac{e^t+e^{-t}}2} + b{\color{red}\frac{e^t-e^{-t}}2}} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.49\textwidth} +\uncover<8->{% +\begin{block}{Potenzreihe} +\vspace{-12pt} +\begin{align*} +K&=\begin{pmatrix}0&1\\1&0\end{pmatrix} +\uncover<10->{\quad\Rightarrow\quad K^2=I} +\\ +\uncover<9->{ +e^{Kt} +&= +I+K+\frac1{2!}K^2 + \frac{1}{3!}K^3 + \frac{1}{4!}K^4+\dots +} +\\ +\uncover<11->{ +&= +\biggl( 1+\frac{t^2}{2!} + \frac{t^4}{4!}+\dots \biggr)I +} +\\ +\uncover<11->{ +&\qquad ++\biggl(t+\frac{t^3}{3!}+\frac{t^5}{5!}+\dots\biggr)K +} +\\ +\uncover<12->{ +&= +I{\,\color{darkgreen}\cosh t} + K{\,\color{red}\sinh t} +} +\\ +\uncover<13->{ +\begin{pmatrix}y(t)\\y_1(t)\end{pmatrix} +&= +e^{Kt}\begin{pmatrix}a\\b\end{pmatrix} +} +\uncover<14->{ += +\begin{pmatrix} +a{\,\color{darkgreen}\cosh t} + b{\,\color{red}\sinh t}\\ +a{\,\color{red}\sinh t} + b{\,\color{darkgreen}\cosh t} +\end{pmatrix} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex new file mode 100644 index 0000000..90cbcd6 --- /dev/null +++ b/vorlesungen/slides/5/injektiv.tex @@ -0,0 +1,81 @@ +% +% injektiv.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\sx{1.05} +\begin{frame}[t] +\frametitle{$f$ injektiv auf $\mathcal{J}(f)$} +\setlength{\abovedisplayskip}{8pt} +\setlength{\belowdisplayskip}{8pt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.58\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=orange!20] + ({0*\sx},-3.0) -- ({1*\sx},-2.0) -- ({2*\sx},-1.5) -- + ({3*\sx},-1.1) -- ({4*\sx},-0.9) -- ({5*\sx},-0.8) -- + ({6*\sx},-0.8) -- + ({6*\sx},0.8) -- ({5*\sx},0.8) -- ({4*\sx},0.9) -- + ({3*\sx},1.1) -- ({2*\sx},1.5) -- ({1*\sx},2.0) -- + ({0*\sx},3.0) -- cycle; +\fill[color=orange!40] (0,-0.8) rectangle ({6*\sx},0.8); + +\foreach \x in {0,...,6}{ + \draw[color=gray,line width=3pt] ({\x*\sx},-3)--({\sx*\x},3); +} +\foreach \x in {0,1,2,3}{ + \node at ({\sx*\x},-3) [below] {$\x$}; +} +\node at ({\sx*5},-3) [below] {$l$}; +\node at ({\sx*6},-3) [below] {$l+1$}; +\draw[->] (-0.1,-3.5) -- ({6*\sx+0.4},-3.5) coordinate[label={below:$k$}]; + +\draw[line width=3pt,color=orange] ({0*\sx},-3.0) -- ({0*\sx},3.0); +\draw[line width=3pt,color=orange] ({1*\sx},-2.0) -- ({1*\sx},2.0); +\draw[line width=3pt,color=orange] ({2*\sx},-1.5) -- ({2*\sx},1.5); +\draw[line width=3pt,color=orange] ({3*\sx},-1.1) -- ({3*\sx},1.1); +\draw[line width=3pt,color=orange] ({4*\sx},-0.9) -- ({4*\sx},0.9); +\draw[line width=3pt,color=orange] ({5*\sx},-0.8) -- ({5*\sx},0.8); +\draw[line width=3pt,color=orange] ({6*\sx},-0.8) -- ({6*\sx},0.8); + +\foreach \x in {0,1,2,3}{ + \node at ({\x*\sx},0) [rotate=90] {$\mathcal{J}^{\x}(A)$}; +} +\node at ({4*\sx},0) {$\cdots$}; +\node at ({5*\sx},0) [rotate=90] {$\mathcal{J}^{l}(A)$}; +\node at ({6*\sx},0) [rotate=90] {$\mathcal{J}^{l+1}(A)$}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.38\textwidth} +\begin{block}{stationär} +$l$ der $k$-Wert, ab dem gilt +\begin{align*} +\mathcal{J}^l(A) &= \mathcal{J}^{l+1}(A) = A\mathcal{J}^l(A) +\end{align*} +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Dimension} +\vspace{-10pt} +\[ +\dim \mathcal{J}^l(A) = \dim\mathcal{J}^{l+1}(A) +\] +\uncover<3->{% +d.~h.~$A$ ist bijektiv als Selbstabbildung von +$\mathcal{J}(A)$} +\uncover<4->{% +\[ +\Downarrow +\] +$A|\mathcal{J}(A)$ ist injektiv} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/jordan.tex b/vorlesungen/slides/5/jordan.tex new file mode 100644 index 0000000..e6ece47 --- /dev/null +++ b/vorlesungen/slides/5/jordan.tex @@ -0,0 +1,138 @@ +% +% jordan.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\L#1{ + \node at ({#1-0.5},{0.5-#1}) {$\lambda$}; +} +\def\E#1{ + \node at ({#1-0.5},{1.5-#1}) {$1$}; +} + +\begin{frame}[t] +\frametitle{Jordan Normalform} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Wahl der Basis} +\begin{enumerate} +\item<2-> Zerlegung in verallgemeinerte Eigenräume +\begin{align*} +V +&= +\mathcal{E}_{{\color{blue}\lambda}}(A) +\oplus +\mathcal{E}_{{\color{darkgreen}\lambda}}(A) +\oplus +\mathcal{E}_{{\color{red}\lambda}}(A) +%\oplus +%\dots +\\ +\llap{$A\mathcal{E}_{{\color{blue}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{blue}\lambda}}(A) +\\ +\llap{$A\mathcal{E}_{{\color{darkgreen}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{darkgreen}\lambda}}(A) +\\ +\llap{$A\mathcal{E}_{{\color{red}\lambda}}$}(A) +&\subset +\mathcal{E}_{{\color{red}\lambda}}(A), +\dots +\end{align*} +\item<3-> In jedem Eigenraum: Zerlegung in Jordan-Blöcke +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.33] +\fill[color=gray!20] (0,-20) rectangle (20,0); +\node[color=white] at (10,-10) [scale=12] {$A$}; + +\uncover<2->{ + \fill[color=blue!20,opacity=0.5] (0,0) rectangle (8,-8); + \fill[color=darkgreen!20,opacity=0.5] (8,-8) rectangle (15,-15); + \fill[color=red!20,opacity=0.5] (15,-15) rectangle (20,-20); + \fill[color=blue!20] (0,0) rectangle (8,2); + \fill[color=blue!20] (-2,-8) rectangle (0,0); + \fill[color=darkgreen!20] (8,0) rectangle (15,2); + \fill[color=darkgreen!20] (-2,-15) rectangle (0,-8); + \fill[color=red!20] (15,0) rectangle (20,2); + \fill[color=red!20] (-2,-20) rectangle (0,-15); +} + +\uncover<3->{ + \draw[color=gray] (0,0) rectangle (5,-5); + \draw[color=gray] (5,-5) rectangle (8,-8); + \draw[color=gray] (8,-8) rectangle (15,-15); + \draw[color=gray] (15,-15) rectangle (16,-16); + \draw[color=gray] (16,-16) rectangle (17,-17); + \draw[color=gray] (17,-17) rectangle (20,-20); +} + +\uncover<2->{ + \draw[color=gray] (8,0) -- (8,-20); + \draw[color=gray] (15,0) -- (15,-20); + \draw[color=gray] (0,-8) -- (20,-8); + \draw[color=gray] (0,-15) -- (20,-15); + + \node at (0,-4) [above,rotate=90] + {$\mathcal{E}_{{\color{blue}\lambda}}(A)$}; + \node at (4,0) [above] + {$\mathcal{E}_{{\color{blue}\lambda}}(A)$}; + \node at (0,-11.5) [above,rotate=90] + {$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$}; + \node at (11.5,0) [above] + {$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$}; + \node at (0,-18.5) [above,rotate=90] + {$\mathcal{E}_{{\color{red}\lambda}}(A)$}; + \node at (18.5,0) [above] + {$\mathcal{E}_{{\color{red}\lambda}}(A)$}; +} + +\uncover<2->{ + {\color{blue} + \foreach \x in {1,...,8}{ \L{\x} } + } + {\color{darkgreen} + \foreach \x in {9,...,15}{ \L{\x} } + } + {\color{red} + \foreach \x in {16,...,20}{ \L{\x} } + } +} + +\uncover<3->{ +\E{2} +\E{3} +\E{4} +\E{5} + +\E{7} +\E{8} + +\E{10} +\E{11} +\E{12} +\E{13} +\E{14} +\E{15} + +\E{19} +\E{20} +} + +\draw (0,-20) rectangle (20,0); +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} + +\egroup diff --git a/vorlesungen/slides/5/jordanblock.tex b/vorlesungen/slides/5/jordanblock.tex new file mode 100644 index 0000000..1c3bce9 --- /dev/null +++ b/vorlesungen/slides/5/jordanblock.tex @@ -0,0 +1,68 @@ +% +% jordanblock.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\def\NL{ +\ifthenelse{\boolean{presentation}}{ +\only<-8>{\phantom{\lambda}\llap{$0$}}\only<9->{\lambda} +}{ +\lambda +} +} + +\begin{frame}[t] +\frametitle{Jordan-Block} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Gegeben} +Matrix $A\in M_n(\Bbbk)$ derart, dass +\begin{itemize} +\item<2-> +$A-\lambda I$ nilpotent +\item<5-> +$A^{n-1}\ne 0$ +\end{itemize} +\end{block} +\vspace{-5pt} +\uncover<3->{ +\begin{block}{Folgerungen} +Es gibt eine Basis derart, dass +\begin{enumerate} +\item<4-> +$A-\lambda I$ hat Normalform einer nilpotenten Matrix +\item<6-> +Es gibt nur einen Block, da $\dim\ker(A-\lambda I)=1$ +\end{enumerate} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{\ifthenelse{\boolean{presentation}}{\only<-8>{Normalform einer nilpotenten Matrix\strut}}{}\only<9->{Normalform: genau ein Eigenwert\strut}} +\[ +A\uncover<-8>{-\lambda I}=\begin{pmatrix} +\NL &1& & & & & & & \\ + &\NL &1& & & & & & \\ + & &\NL &\uncover<7->{{\color<7>{red}1}}& & & & & \\ + & & &\NL &1& & & & \\ + & & & &\NL &1& & & \\ + & & & & &\NL &1& & \\ + & & & & & &\NL &\uncover<7->{{\color<7>{red}1}}& \\ + & & & & & & &\NL &\uncover<7->{{\color<7>{red}1}}\\ + & & & & & & & &\NL +\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} +\vspace{-5pt} +\uncover<8->{% +\begin{block}{Jordan-Normalform} +In dieser Basis hat $A$ Jordan-Normalform +\end{block}} +\end{frame} + +\egroup diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex new file mode 100644 index 0000000..3890717 --- /dev/null +++ b/vorlesungen/slides/5/kernbild.tex @@ -0,0 +1,86 @@ +% +% kernbild.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kern und Bild} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<1->{% +\begin{block}{Kern} +Lineare Abbildung $f\colon V\to V$ +\[ +\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\} +\] +\end{block}} +\uncover<3->{% +\begin{block}{Kern von $A^k$} +\[ +\mathcal{K}^k(f) = \operatorname{ker} f^k +\] +\begin{align*} +\uncover<5->{ +\mathcal{K}^k(f) +&= +\{v\in V\;|\; f^{k}(v)=0\} +} +\\ +\uncover<6->{ +&\subset +\{v\in V\;|\; f^{k+1}(v)=0\} +} +\\ +\uncover<7->{ +&=\mathcal{K}^{k+1}(f) +} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Bild} +Lineare Abbildung $f\colon V\to V$ +\[ +\operatorname{im}f += +\mathcal{J}(f) += +\{f(v)\;|\; v\in V\} +\] +\end{block}} +\uncover<4->{% +\begin{block}{Bild von $A^k$} +\[ +\mathcal{J}^k(f) = \operatorname{im}f^k +\] +\begin{align*} +\uncover<8->{ +\mathcal{J}^k(f) +&= +\operatorname{im}f^k += +\operatorname{im}(f^{k}\circ f) +} +\\ +\uncover<9->{ +&= +\{f^{k-1} w\;|\; w = f(v)\} +} +\\ +\uncover<10->{ +&\subset +\{f^{k-1} w\;|\; w \in V\} +} +\\ +\uncover<11->{ +&=\mathcal{J}^{k-1}(f) +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/kernbilder.tex b/vorlesungen/slides/5/kernbilder.tex new file mode 100644 index 0000000..08581ff --- /dev/null +++ b/vorlesungen/slides/5/kernbilder.tex @@ -0,0 +1,68 @@ +% +% kernbilder.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{grueneins}{rgb}{0.0,0.4,0.0} +\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8} +\definecolor{orangeeins}{rgb}{1.0,0.6,0.0} +\definecolor{orangezwei}{rgb}{0.8,0.0,0.4} + +\begin{frame}[t] +\frametitle{Kerne und Bilder} +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<2->{ +\begin{scope}[xshift=-4cm,yshift=1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}}; +\node[color=orangeeins] at (1.6,1.3) [right] {$\mathcal{J}^1(A)$}; +\end{scope} +} + +\uncover<3->{ +\begin{scope}[xshift=-4cm,yshift=-1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}}; +\node[color=orangezwei] at (0.9,0.5) {$\mathcal{J}^2(A)$}; +\end{scope} +} + +\begin{scope}[xshift=0cm,yshift=0cm] +\uncover<1>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/leer.jpg}}; +} +\uncover<2>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}}; +} +\uncover<3>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}}; +} +\uncover<4>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/drei.jpg}}; +} +\uncover<5->{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kombiniert.jpg}}; +} +\end{scope} + +\uncover<4->{ +\begin{scope}[xshift=4cm,yshift=1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern1.jpg}}; +\node[color=grueneins] at (1.0,1.3) [right] {$\mathcal{K}^1(A)$}; +\end{scope} +} + +\uncover<5->{ +\begin{scope}[xshift=4cm,yshift=-1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern2.jpg}}; +\node[color=gruenzwei] at (0.7,-0.6) {$\mathcal{K}^2(A)$}; +\end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/kernbildintro.tex b/vorlesungen/slides/5/kernbildintro.tex new file mode 100644 index 0000000..9fd7849 --- /dev/null +++ b/vorlesungen/slides/5/kernbildintro.tex @@ -0,0 +1,89 @@ +% +% kernbildintro.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{grueneins}{rgb}{0.0,0.4,0.0} +\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8} +\definecolor{orangeeins}{rgb}{1.0,0.6,0.0} +\definecolor{orangezwei}{rgb}{0.8,0.0,0.4} + +\begin{frame}[t] +\frametitle{Bilder und Kerne} +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\begin{scope}[xshift=-3.4cm] + +\only<1>{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/leer.jpg}}; +} +\only<2-3>{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild1.jpg}}; +} +\uncover<4->{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild2.jpg}}; +} +\uncover<2->{ + \fill[color=white,opacity=0.7] (0.1,2.18) rectangle (4,2.64); + \node[color=orangeeins] at (0,2.4) [right] + {$\operatorname{im} A = \{Av\;|v\in\mathbb{R}^n\}$}; +} +\uncover<4->{ + \node[color=orangezwei] at (4,0.7) [left] + {$\operatorname{im} A^2 = \{A^2v\;|v\in\mathbb{R}^n\}$}; +} +\end{scope} + +\begin{scope}[xshift=3.4cm] + +\uncover<2->{ +\fill[color=orangeeins!40] (-1,0.5) rectangle (1.8,2); +} +\uncover<4->{ +\fill[color=orangezwei!40] (-1.1,-1.7) rectangle (-0.,-0.3); +} + +\node at (0,0) {\begin{minipage}{6cm} +\begin{align*} +A&={\scriptstyle\begin{pmatrix*}[r] + -0.979& -0.142& 0.917\\ + -0.260& -0.643& 1.069\\ + -0.285& -0.449& 0.823 +\end{pmatrix*}} +\\ +\operatorname{Rang}A&=2 +\\ +\uncover<3->{ +A^2&={\scriptstyle\begin{pmatrix*}[r] + 0.734& -0.181& -0.295\\ + 0.118& -0.029& -0.047\\ + 0.161& -0.039& -0.065 +\end{pmatrix*}}}\\ +\uncover<3->{ +\operatorname{Rang}A^2&=1} +\end{align*} +\end{minipage}}; + +\only<5>{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern1.jpg}}; +} + +\uncover<6->{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern2.jpg}}; +\node[color=gruenzwei] at (-1.35,-3.0) [right] {$\ker A^2 = \{v\;|\; A^2v=0\}$}; +} + +\uncover<5->{ +\node[color=grueneins] at (-0.9,3.1) [right] {$\ker A = \{v\;|\; Av=0\}$}; +} + +\end{scope} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex new file mode 100644 index 0000000..1116a83 --- /dev/null +++ b/vorlesungen/slides/5/ketten.tex @@ -0,0 +1,79 @@ +% +% ketten.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ketten von Unterräumen} +\begin{block}{Schachtelung} +Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt: +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcccc} +0 &=&\mathcal{K}^0(f) + &\subset&\mathcal{K}^1(f) + &\subset&\dots + &\subset&\mathcal{K}^k(f) + &\subset&\mathcal{K}^{k+1}(f) + &\subset&\dots + &\subset&\displaystyle\bigcup_{k=0}^\infty \mathcal{K}^k(f) + &=:&\mathcal{K}(f) +\\[14pt] +\Bbbk^n &=&\mathcal{J}^0(f) + &\supset&\mathcal{J}^1(f) + &\supset&\dots + &\supset&\mathcal{J}^{k}(f) + &\supset&\mathcal{J}^{k+1}(f) + &\supset&\dots + &\supset&\displaystyle\bigcap_{k=0}^\infty \mathcal{J}^k(f) + &=:&\mathcal{J}(f) +\end{array} +\] +\end{block} +\vspace{-20pt} +\uncover<2->{% +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Abildung der Kerne} +\vspace{-10pt} +\begin{align*} +f \mathcal{K}^k(f) +&= +\{f(v)\;|\; f^k(v) = 0\} +\\ +&\subset +\{ v\;|\; f^{k+1}(v)=0\} +\\ +&= +\mathcal{K}^{k+1}(f) +\\ +\Rightarrow +f\mathcal{K}(f)&= f\mathcal{K}(f) +\quad\text{invariant} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Abbildung der Bild} +\vspace{-10pt} +\begin{align*} +f\mathcal{J}^k(f) +&= +\{f(f^{k}(v))\;|\; v\in V\} +\\ +&= +\{f^{k+1}(v)\;|\; v\in V\} +\\ +&= +\mathcal{J}^{k+1}(f) +\\ +\Rightarrow +f\mathcal{J}(f)&= \mathcal{J}(f) +\quad\text{invariant} +\end{align*} +\end{block} +\end{column} +\end{columns}} +\end{frame} diff --git a/vorlesungen/slides/5/konvergenzradius.tex b/vorlesungen/slides/5/konvergenzradius.tex new file mode 100644 index 0000000..a0b4b3a --- /dev/null +++ b/vorlesungen/slides/5/konvergenzradius.tex @@ -0,0 +1,109 @@ +% +% konvergenzradius.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\setbeamercolor{column}{bg=blue!20} +\def\punkt#1{ + \fill[color=blue!30] #1 circle[radius=0.05]; + \draw[color=blue] #1 circle[radius=0.05]; +} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Konvergenzradius} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Potenzreihen} +$f\colon\mathbb{C}\to\mathbb{C}$ (komplex differenzierbar) +\begin{equation} +f(z) = \sum_{k=0}^\infty a_kz^k +\label{reihe} +\end{equation} +\end{block} +\vspace{-8pt} +\uncover<2->{% +\begin{block}{Konvergenz} +\eqref{reihe} konvergiert für $|z| < {\color{darkgreen}R}$, +\[ +\frac{1}{{\color{darkgreen}R}} += +\limsup_{k\to\infty} |a_k|^{\frac1k} +\] +\end{block}} +\uncover<3->{% +\begin{block}{Polstellen} +{\color{darkgreen}$R$} ist der Radius des grössten Kreises um $O$, +auf dessen Rand eine +{\color{blue}Polstelle} der Funktion $f(z)$ liegt +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{2.5} +\uncover<2->{ + \fill[color=red!20] (0,0) circle[radius=\r]; + \draw[color=red] (0,0) circle[radius=\r]; +} +\draw[->] (-2.6,0) -- (2.9,0) coordinate[label={$\operatorname{Re}z$}]; +\draw[->] (0,-2.6) -- (0,2.9) coordinate[label={$\operatorname{Im}z$}]; + +\uncover<2->{ + \draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (100:\r); + \draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (220:\r); + \node[color=darkgreen] at ($0.5*(100:\r)$) [left] {$R$}; + \node[color=darkgreen] at ($0.5*(220:\r)+(-0.1,0.1)$) + [below right] {$R$}; + + \fill[color=white] (0,0) circle[radius=0.05]; + \draw (0,0) circle[radius=0.05]; +} + +\node at (2.8,2.8) {$\mathbb{C}$}; + +\uncover<3->{ + \punkt{(100:\r)} + \punkt{(220:\r)} + + \begin{scope} + \clip (-2.6,-2.6) rectangle (2.9,2.9); + + \punkt{(144.2527:2.7232)} + %\punkt{(226.1822:2.5164)} + \punkt{(173.7501:3.4140)} + \punkt{(267.4103,2.7668)} + \punkt{(137.7328:3.1683)} + %\punkt{(30.1155:3.3629)} + %\punkt{(139.1036:2.5366)} + \punkt{(167.4964:3.0503)} + \punkt{(289.2650:3.4324)} + \punkt{(120.1911:3.2966)} + %\punkt{(292.3422:2.7550)} + \punkt{(141.4877:2.6494)} + \punkt{(70.8326:2.9005)} + \punkt{(56.0758:3.2098)} + \punkt{(99.0585:3.2340)} + \punkt{(299.7242:2.5990)} + \punkt{(158.8802:2.6539)} + \punkt{(235.2721:2.9476)} + \punkt{(108.0584:2.8344)} + \punkt{(220.0117:2.7679)} + + \end{scope} + + \begin{scope}[yshift=-3.2cm,xshift=-1.0cm] + \punkt{(0,-0.05)} + \node at (0,0) [right] {$=$ Polstelle}; + \end{scope} +} + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/krbeispiele.tex b/vorlesungen/slides/5/krbeispiele.tex new file mode 100644 index 0000000..b51df78 --- /dev/null +++ b/vorlesungen/slides/5/krbeispiele.tex @@ -0,0 +1,99 @@ +% +% krbeispiele.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Konvergenzradius --- Beispiele} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Exponentialreihe} +\vspace{-20pt} +\begin{align*} +e^z &= \sum_{k=0}^\infty \frac{z^k}{k!} +\\ +\uncover<2->{ +\frac1k\log k! +} +&\uncover<3->{=\frac1k\sum_{x=1}^k {\color{blue}\log x}} +\uncover<6->{>\frac1k\int_1^k{\color{red}\log x}\,dx} +\\ +& +\ifthenelse{\boolean{presentation}}{ +\only<7>{=\frac1k[x\log x -x]_1^k} +}{} +\only<8->{= +\log k -1 +\frac1k} +\uncover<9->{\to \infty\phantom{\frac1k}} +\\ +\uncover<10->{(k!)^{\frac1k} +&\to\infty}\uncover<11->{ \quad\Rightarrow\quad R = \infty} +\end{align*} +\vspace{-40pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.7] +\uncover<4->{ +\foreach \x in {2,...,9}{ + \fill[color=blue!20] ({\x-1},0) rectangle ({\x},{ln(\x)}); + \draw[color=blue] ({\x-1},0) rectangle ({\x},{ln(\x)}); + \node at ({\x-0.5},{ln(\x)}) [above] {\tiny $\log\x$}; + \draw (\x,-0.1) -- (\x,0.1); + \node at (\x,0) [below] {\tiny$\x$}; +} +\draw (1,-0.1) -- (1,0.1); +\uncover<5->{ +\begin{scope} + \clip (0,-1) rectangle (9.5,2.5); + \fill[color=red!40,opacity=0.5] (0,0) -- (0,-1) + -- plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)}) + -- (9.1,0) -- cycle; + \draw[color=red] plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)}); +\end{scope} +} +\draw[->] (-0.2,0) -- (9.4,0) coordinate[label={$x$}]; +\draw[->] (0,-1) -- (0,2.5) coordinate[label={right:$y$}]; +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Geometrische Reihe} +\vspace{-15pt} +\begin{align*} +\uncover<13->{ +\frac{1}{{\color{blue}1}-z} +&= +\sum_{k=0}^\infty +z^k} +\\ +\uncover<14->{ +a_k&=1} +\uncover<15->{\quad\Rightarrow\quad +|a_k|^{\frac1k}=1} +\\ +\uncover<16->{ +\limsup_{k\to\infty} &= |a_k|^{\frac1k}=1}\uncover<17->{ = \frac1R} +\uncover<18->{\quad\Rightarrow\quad R=1} +\end{align*} +%\uncover<19->{Polstelle bei $z=1$ limitiert Konvergenzradius} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope} +\clip (-2.2,-1.5) rectangle (2.2,1.5); +\fill[color=red!20] (0,0) circle[radius=2]; +\draw[color=red] (0,0) circle[radius=2]; +\end{scope} +\draw[->] (-2.2,0) -- (2.5,0) coordinate[label={$\operatorname{Re}z$}]; +\draw[->] (0,-1.6) -- (0,1.8) coordinate[label={right:$\operatorname{Im}z$}]; +\fill[color=blue!20] (2,0) circle[radius=0.08]; +\draw[color=blue] (2,0) circle[radius=0.08]; +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/logarithmusreihe.tex b/vorlesungen/slides/5/logarithmusreihe.tex new file mode 100644 index 0000000..85ba0ef --- /dev/null +++ b/vorlesungen/slides/5/logarithmusreihe.tex @@ -0,0 +1,53 @@ +% +% logarithmus.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmusreihe} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Integralgleichung} +\vspace{-5pt} +\begin{align*} +\log(1+x)&=\int_0^x \frac{1}{1+t}\,dt +\\ +&\uncover<5->{= +\int_0^x +1-t+t^2-t^3+\dots\,dt +} +\\ +\uncover<6->{ +&= +x-\frac{x^2}2+\frac{x^3}{3}-\frac{x^4}{4}+\dots +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Geometrische Reihe} +\vspace{-5pt} +\begin{align*} +\frac{1}{1-q}&=1+q+q^2+q^3+\dots +\\ +\uncover<3->{ +\frac{1}{1+q}&=1-q+q^2-q^3+\dots +} +\end{align*} +\uncover<4->{Konvergenzradius $1$} +\end{block}} +\end{column} +\end{columns} +\uncover<7->{% +\begin{block}{Matrix-Logarithmus} +Für $\operatorname{Sp}(A)\subset \{z\in\mathbb{C}\;|\;|z-1|<1\}$ konvergiert +\[ +\log A += +(A-I) - \frac12(A-I)^2 + \frac13(A-I)^3 - \frac14(A-I)^4 + \dots +\] +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex new file mode 100644 index 0000000..b0a1d82 --- /dev/null +++ b/vorlesungen/slides/5/motivation.tex @@ -0,0 +1,67 @@ +% +% movitation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Motivation} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Matrix $A$ analysieren} +Matrix $A$ mit Minimalpolynom $m_A(X)$ vom +Grad $s$ +\end{block} +\uncover<2->{% +\begin{block}{Faktorisieren} +Minimalpolynom faktorisieren +\[ +m_A(X) += +(X-\mu_1)(X-\mu_2)\dots(X-\mu_s) +\] +\end{block}} +\uncover<3->{% +\begin{block}{Vertauschen} +$\sigma\in S_s$ eine Permutation von $1,\dots,s$ +ist +\begin{align*} +m_A(X) +&= +(X-\mu_{\sigma(1)}) +%(X-\mu_{\sigma(2)}) +\dots +(X-\mu_{\sigma(s)}) +\\ +0 +&= +(A-\mu_{\sigma(1)}) +%(A-\mu_{\sigma(2)}) +\dots +(A-\mu_{\sigma(s)}) +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Bedingung für $\mu_k$} +Permutation wählen so dass $\mu_k$ an erster Stelle steht: +\[ +0=(A-\mu_k) \prod_{i\ne k}(A-\mu_i) v +\] +für alle $v\in\Bbbk^n$. +\end{block}} +\uncover<5->{% +\begin{block}{Eigenwerte} +Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$ +gibt mit +\[ +(A-\mu)v = 0 +\Rightarrow Av = {\color{red}\mu} v +\] +Eigenwertbedingung +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/nilpotent.tex b/vorlesungen/slides/5/nilpotent.tex new file mode 100644 index 0000000..ca38c40 --- /dev/null +++ b/vorlesungen/slides/5/nilpotent.tex @@ -0,0 +1,190 @@ +% +% nilpotent.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\feld#1{ + \fill[color=red!20] (#1,0) rectangle ({#1+1},12); +} +\begin{frame}[t] +\frametitle{$\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ für nilpotente Matrizen} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.42\textwidth} +Matrix mit dem dargestellten Verlauf von +${\color{red}\dim\mathcal{K}^k(A)}$ +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.42] + +\only<2->{ + \feld{0} + \feld{1} + \feld{2} + \feld{3} +} +\only<2->{ \feld{4} } +\only<2->{ \feld{6} } +\ifthenelse{\boolean{presentation}}{ +\only<3->{ \feld{5} } +\only<3->{ \feld{7} } +\only<4->{ \feld{8} } +\only<5->{ \feld{9} } +\only<6->{ \feld{10} } +\only<7->{ \feld{11} } + +\only<1>{ \node at (6,0) [below] {$k=0$}; } +}{} +\only<2>{ \node at (6,0) [below] {$k=1$}; } +\ifthenelse{\boolean{presentation}}{ +\only<3>{ \node at (6,0) [below] {$k=2$}; } +\only<4>{ \node at (6,0) [below] {$k=3$}; } +\only<5>{ \node at (6,0) [below] {$k=4$}; } +\only<6>{ \node at (6,0) [below] {$k=5$}; } +\only<7>{ \node at (6,0) [below] {$k=6$}; } +}{} + +\draw (0,0) rectangle (12,12); +\ifthenelse{\boolean{presentation}}{ +\only<1>{ + \foreach \x in {1,...,12}{ + \node at ({\x-0.5},{12-\x+0.5}) {$1$}; + } +} +}{} +\only<2->{ + \foreach \x in {1,...,12}{ + \node at ({\x-0.5},{12-\x+0.5}) {$0$}; + } +} +\only<2>{ + \foreach \x in {7,...,11}{ + \node at ({\x+0.5},{12-\x+0.5}) {$1$}; + } +} +\ifthenelse{\boolean{presentation}}{ +\only<3->{ + \foreach \x in {7,...,11}{ + \node at ({\x+0.5},{12-\x+0.5}) {$0$}; + } +} +\only<3>{ + \foreach \x in {8,...,11}{ + \node at ({\x+0.5},{13-\x+0.5}) {$1$}; + } +} +\only<4->{ + \foreach \x in {8,...,11}{ + \node at ({\x+0.5},{13-\x+0.5}) {$0$}; + } +} +\only<4>{ + \foreach \x in {9,...,11}{ + \node at ({\x+0.5},{14-\x+0.5}) {$1$}; + } +} +\only<5->{ + \foreach \x in {9,...,11}{ + \node at ({\x+0.5},{14-\x+0.5}) {$0$}; + } +} +\only<5>{ + \foreach \x in {10,...,11}{ + \node at ({\x+0.5},{15-\x+0.5}) {$1$}; + } +} +\only<6->{ + \foreach \x in {10,...,11}{ + \node at ({\x+0.5},{15-\x+0.5}) {$0$}; + } +} +\only<6>{ + \foreach \x in {11,...,11}{ + \node at ({\x+0.5},{16-\x+0.5}) {$1$}; + } +} +\only<7->{ + \foreach \x in {11,...,11}{ + \node at ({\x+0.5},{16-\x+0.5}) {$0$}; + } +} +}{} +\draw[line width=0.1pt] + (0,11) -- (2,11) -- (2,9) -- (4,9) -- (4,6) -- (12,6); +\draw[line width=0.1pt] + (1,12) -- (1,10) -- (3,10) -- (3,8) -- (6,8) -- (6,0); +\only<2>{ + \node at (5.5,7.5) {$1$}; +} +\ifthenelse{\boolean{presentation}}{ +\only<3->{ + \node at (5.5,7.5) {$0$}; +} +}{} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.56\textwidth} +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\pfad{ + (0,0) -- (1,3) -- (2,4) -- (3,4.5) -- (4,5) -- (5,5.5) -- (6,6) +} +\fill[color=orange!20] \pfad -- (0,6) -- cycle; +\fill[color=darkgreen!20] \pfad -- (6,0) -- cycle; +\foreach \y in {0.5,1,...,5.75}{ + \draw[line width=0.1pt] (0,\y) -- (6,\y); +} +\draw[line width=1.4pt] \pfad; +\draw[->] (-0.1,6) -- (6.5,6); \node at (-0.1,6) [left] {$n$}; +\draw[->] (-0.1,0) -- (6.5,0); \node at (-0.1,0) [left] {$0$}; +\fill (0,0) circle[radius=0.05]; +\fill (1,3) circle[radius=0.05]; +\fill (2,4) circle[radius=0.05]; +\fill (3,4.5) circle[radius=0.05]; +\fill (4,5) circle[radius=0.05]; +\fill (5,5.5) circle[radius=0.05]; +\fill (6,6) circle[radius=0.05]; +\ifthenelse{\boolean{presentation}}{ +\only<1>{ + \fill[color=red] (0,0) circle[radius=0.08]; +} +}{} +\only<2>{ + \fill[color=red] (1,3) circle[radius=0.08]; + \draw[color=red] (0,3) -- (1,3); + \node[color=red] at (0,3) [left] {$6$}; +} +\ifthenelse{\boolean{presentation}}{ +\only<3>{ + \fill[color=red] (2,4) circle[radius=0.08]; + \draw[color=red] (0,4) -- (2,4); + \node[color=red] at (0,4) [left] {$8$}; +} +\only<4>{ + \fill[color=red] (3,4.5) circle[radius=0.08]; + \draw[color=red] (0,4.5) -- (3,4.5); + \node[color=red] at (0,4.5) [left] {$9$}; +} +\only<5>{ + \fill[color=red] (4,5.0) circle[radius=0.08]; + \draw[color=red] (0,5.0) -- (4,5.0); + \node[color=red] at (0,5.0) [left] {$10$}; +} +\only<6>{ + \fill[color=red] (5,5.5) circle[radius=0.08]; + \draw[color=red] (0,5.5) -- (5,5.5); + \node[color=red] at (0,5.5) [left] {$11$}; +} +\only<7>{ + \fill[color=red] (6,6.0) circle[radius=0.08]; +} +}{} +\draw[color=white] (-0.7,0) -- (-0.7,6); +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/normal.tex b/vorlesungen/slides/5/normal.tex new file mode 100644 index 0000000..7245608 --- /dev/null +++ b/vorlesungen/slides/5/normal.tex @@ -0,0 +1,69 @@ +% +% normal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Normale Operatoren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frage} +$f,g\colon \mathbb{C}\to\mathbb{C}$. +\\ +In welchen Punkten müssen $f$ und $g$ übereinstimmen, damit +$f(A)=g(A)$? +\end{block} +\uncover<2->{% +\begin{block}{Definition $f(A)$} +$f$ durch eine Folge von Polynomen +appoximieren: $p_n\to f$ +\[ +f(A) = \lim_{n\to\infty}p_n(A) +\] +\end{block}} +\vspace{-15pt} +\uncover<3->{% +\begin{block}{Vermutung} +Falls $f(z)=g(z)$ für $z\in\operatorname{Sp}(A)$, +dann ist $f(A)=g(A)$ + +\smallskip +\uncover<4->{% +{\usebeamercolor[fg]{title}Stimmt für: } $A$ diagonalisierbar +} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Beispiel} +\[ +A=\begin{pmatrix}2&1\\0&2\end{pmatrix}, \quad +\operatorname{Sp}(A)=\{2\} +\] +\uncover<6->{% +\begin{align*} +f(z)&=(z-2)^2 &g(z)&=z-2 +\\ +\uncover<7->{ +f(A)&=0&g(A)&=\begin{pmatrix}0&1\\0&0\end{pmatrix} +} +\end{align*}} +\end{block}} +\vspace{-18pt} +\uncover<8->{% +\begin{block}{Normal} +$A$ heisst {\em normal}, wenn $AA^*=A^*A$ +\begin{itemize} +\item<9-> +symmetrische Matrizen: $A=A^*$ +\item<10-> +unitäre Matrizen: $A^*=A^{-1}\Rightarrow +AA^*=AA^{-1}=A^{-1}A=A^*A$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex new file mode 100644 index 0000000..9457136 --- /dev/null +++ b/vorlesungen/slides/5/normalnilp.tex @@ -0,0 +1,237 @@ +% +% normalnilp.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\sx{1.9} +\def\sy{0.6} +\def\punkt#1#2#3{ + \foreach \y in {0,...,#2}{ + } +} +\def\block#1#2{ + \fill[rounded corners=2pt,color=white] + ({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05}); + \draw[rounded corners=2pt] + ({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05}); +} +\def\teilmenge#1#2#3{ + \fill[rounded corners=2pt,color=white] + ({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); + \draw[rounded corners=2pt,color=gray] + ({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); +} +\def\rot#1#2#3{ + \fill[rounded corners=2pt,color=red!20] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); + \draw[rounded corners=2pt,color=red] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); +} +\def\hellblau#1#2#3{ + \fill[rounded corners=2pt,color=blue!20] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); + \draw[rounded corners=2pt,color=blue!40] + ({-#1*\sx-0.35},{#2*\sy+0.00}) + rectangle ({-#1*\sx+0.35},{#3*\sy+0.00}); +} +\def\punkt#1#2{ + \fill[color=blue] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08]; +} +\def\bildpunkt#1#2{ + \fill[color=blue!40] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08]; +} +\def\pfeil#1#2#3{ + \draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.1cm] + ({-#1*\sx},{(#2-0.5)*\sy}) + -- + ({-(#1-1)*\sx},{(#3-0.5)*\sy}) ; +} +\begin{frame}[t] +\frametitle{Normalform einer nilpotenten Matrix} +{\usebeamercolor[fg]{title}$A^l=0$ $\Rightarrow$ finde eine ``gute'' Basis} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\vspace{-25pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=darkgreen!20,rounded corners=2pt] + ({-3*\sx+0.35},0) -- (-0.35,0) + -- + ({-1*\sx+0.35},{4*\sy}) -- ({-1*\sx-0.35},{4*\sy}) + -- + ({-2*\sx+0.35},{7*\sy}) -- ({-2*\sx-0.35},{7*\sy}) + -- + ({-3*\sx+0.35},{8*\sy}) -- cycle; + +\block{0}{0} + +\block{1}{4} +\uncover<10->{ + \rot{1}{0}{1} + \node[color=red] at ({-1*\sx-0.28},{0.5*\sy}) [left] {$\mathcal{C}_{l-2}$}; +} +\uncover<8->{ + \hellblau{1}{1}{3} +} +\uncover<4->{ + \hellblau{1}{3}{4} +} + +\block{2}{7} +\uncover<4->{ + \hellblau{2}{6}{7} +} +\uncover<6->{ + \rot{2}{4}{6} + \node[color=red] at ({-2*\sx-0.28},{5*\sy}) [left] {$\mathcal{C}_{l-1}$}; +} +\teilmenge{2}{0}{4} + +\block{3}{8} +\uncover<2->{ + \rot{3}{7}{8} + \node[color=red] at ({-3*\sx-0.28},{7.5*\sy}) [left] {$\mathcal{C}_l$}; +} +\teilmenge{3}{0}{7} + +\uncover<3->{ + \punkt{3}{8} +} +\uncover<4->{ + \pfeil{3}{8}{7} + \bildpunkt{2}{7} + \pfeil{2}{7}{4} + \bildpunkt{1}{4} +} + +\uncover<7->{ + \punkt{2}{5} + \punkt{2}{6} +} +\uncover<8->{ + \pfeil{2}{5}{2} + \bildpunkt{1}{3} + \pfeil{2}{6}{3} + \bildpunkt{1}{2} +} + +\uncover<11->{ +\punkt{1}{1} +} + +\node at ({-3*\sx},0) [below] {$\mathcal{K}^l(A)\mathstrut$}; +\node at ({-2*\sx},0) [below] {$\mathcal{K}^{l-1}(A)\mathstrut$}; +\node at ({-1.45*\sx},0) [below] {$\dots\mathstrut$}; +\node at ({-1*\sx},0) [below] {$\mathcal{K}^1(A)\mathstrut$}; +\node at ({-0*\sx},0) [below] {$0=\mathcal{K}^0(A)\mathstrut$}; +\node[color=gray] at ({-2*\sx},{2*\sy}) [rotate=90] {$\mathcal{K}^1(A)$}; +\node[color=gray] at ({-3*\sx},{3.5*\sy}) [rotate=90] {$\mathcal{K}^{l-1}(A)$}; +\foreach \x in {0,1,2}{ + \draw[->,shorten >= 0.1cm, shorten <= 0.1cm] + ({-(\x+1)*\sx},{8.7*\sy}) -- ({-(\x)*\sx},{8.7*\sy}); + \node at ({-(\x+0.5)*\sx},{8.7*\sy}) [above] {$A$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\vspace{-30pt} +\begin{enumerate} +\item<2-> \( + \mathcal{K}^l(A)=\mathcal{K}^{l-1}\oplus {\color{red}\mathcal{C}_l} + \) +\item<3-> \( + {\color{blue}b_1}\in{\color{red}\mathcal{C}_l} + \) +\item<4-> \( + \mathcal{B}_l + = + \{{\color{blue}b_1},{\color{blue!40}Ab_1},{\color{blue!40}A^2b_1},\dots, + {\color{blue!40}A^{l-1}b_1}\} + \) +\item<5-> \( + \mathcal{K}^{l-1}(A) + = + \mathcal{K}^{l-2}(A) + \oplus + {\color{red}\mathcal{C}_{l-1}} + \oplus + {\color{blue}A\mathcal{C}_l} + \) +\item<6-> \( + {\color{blue}b_2},{\color{blue}b_3}\in{\color{red}\mathcal{C}_{l-1}} + \) +\item<7-> \( + \mathcal{B}_{l-1} + = + \{ + {\color{blue}b_2},{\color{blue}b_3}, + {\color{blue!40}Ab_2}, {\color{blue!40}Ab_3},\dots + \} + \) +\item<8-> \dots +\end{enumerate} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] + +\uncover<2-4>{ + \fill[color=red!20] (2,0) rectangle (3,8); +} +\uncover<4->{ + \fill[color=blue!20] (0,6) rectangle (2,8); +} +\uncover<5->{ + \fill[color=red!20] (2,5) rectangle (3,8); + \node[color=blue] at (2.5,6.5) {$1$}; + \node[color=blue] at (1.5,7.5) {$1$}; + \node[color=gray] at (0.5,7.5) {$0$}; + \node[color=gray] at (1.5,6.5) {$0$}; + \node[color=gray] at (2.5,5.5) {$0$}; + \draw[color=gray] (0.05,5.05) rectangle (2.95,7.95); +} + +\uncover<6-8>{ + \fill[color=red!20] (4,0) rectangle (5,8); + \fill[color=red!20] (6,0) rectangle (7,8); +} +\uncover<8->{ + \fill[color=blue!20] (3,4) rectangle (4,5); + \fill[color=blue!20] (5,2) rectangle (6,3); +} +\uncover<9->{ + \fill[color=red!20] (4,3) rectangle (5,5); + \node[color=blue] at (4.5,4.5) {$1$}; + \node[color=gray] at (3.5,4.5) {$0$}; + \node[color=gray] at (4.5,3.5) {$0$}; + \draw[color=gray] (3.05,3.05) rectangle (4.95,4.95); + \fill[color=red!20] (6,1) rectangle (7,3); + \node[color=blue] at (6.5,2.5) {$1$}; + \node[color=gray] at (5.5,2.5) {$0$}; + \node[color=gray] at (6.5,1.5) {$0$}; + \draw[color=gray] (5.05,1.05) rectangle (6.95,2.95); +} + +\uncover<10>{ + \fill[color=red!20] (7,0) rectangle (8,8); +} +\uncover<11->{ + \fill[color=red!20] (7,0) rectangle (8,1); + \node[color=gray] at (7.5,0.5) {$0$}; + \draw[color=gray] (7.05,0.05) rectangle (7.95,0.95); +} + +\draw (0,0) rectangle (8,8); +\node at (-0.1,4) [left] {$A=$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/potenzreihenmethode.tex b/vorlesungen/slides/5/potenzreihenmethode.tex new file mode 100644 index 0000000..0c3503d --- /dev/null +++ b/vorlesungen/slides/5/potenzreihenmethode.tex @@ -0,0 +1,93 @@ +% +% potenzreihenmethode.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Potenzreihenmethode} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Lineare Differentialgleichung} +\vspace{-12pt} +\begin{align*} +y'&=ay&&\Rightarrow&y'-ay&=0 +\\ +y(0)&=C +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Potenzreihenansatz} +\vspace{-12pt} +\begin{align*} +y(x) +&= +a_0+ a_1x + a_2x^2 + \dots +\\ +y(0)&=a_0=C +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Lösung} +\vspace{-12pt} +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcr} +\uncover<3->{ y'(x)} + \uncover<5->{ + &=&\phantom{(} a_1\phantom{\mathstrut-aa_0)} + &+& 2a_2\phantom{\mathstrut-aa_1)}x + &+& 3a_3\phantom{\mathstrut-aa_2)}x^2 + &+& 4a_4\phantom{\mathstrut-aa_3)}x^3 + &+& \dots}\\ +\uncover<3->{-ay(x)} + \uncover<6->{ + &=&\mathstrut-aa_0 \phantom{)} + &-& aa_1\phantom{)}x + &-& aa_2\phantom{)}x^2 + &-& aa_3\phantom{)}x^3 + &-& \dots}\\[2pt] +\hline +\\[-10pt] +\uncover<3->{0} + \uncover<7->{ + &=&(a_1-aa_0) + &+& (2a_2-aa_1)x + &+& (3a_3-aa_2)x^2 + &+& (4a_4-aa_3)x^3 + &+& \dots}\\ +\end{array} +\] +\begin{align*} +\uncover<4->{ +a_0&=C}\uncover<8->{, +\quad +a_1=aa_0=aC}\uncover<9->{, +\quad +a_2=\frac12a^2C}\uncover<10->{, +\quad +a_3=\frac16a^3C}\uncover<11->{, +\dots +a_k=\frac1{k!}a^kC} +\hspace{3cm} +\\ +\uncover<4->{ +\Rightarrow y(x) &= C}\uncover<8->{+Cax}\uncover<9->{ + C\frac12(ax)^2} +\uncover<10->{ + C \frac16(ac)^3} +\uncover<11->{ + \dots+C\frac{1}{k!}(ax)^k+\dots} +\ifthenelse{\boolean{presentation}}{ +\only<12>{ += +C\sum_{k=0}^\infty \frac{(ax)^k}{k!}} +}{} +\uncover<13->{= +Ce^{ax}} +\end{align*} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/5/reellenormalform.tex b/vorlesungen/slides/5/reellenormalform.tex new file mode 100644 index 0000000..4ceabe9 --- /dev/null +++ b/vorlesungen/slides/5/reellenormalform.tex @@ -0,0 +1,115 @@ +% +% reellenormalform.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Reelle Normalform} +$A\in M_n(\mathbb{R})\subset M_n(\mathbb{C})$ hat reelle und Paare von +konjugiert komplexen Eigenwerten +\medskip + +$\Rightarrow$ Konjugiert komplexe Eigenvektoren $v$ und $\overline{v}$, +$x=\operatorname{Re}v$ und $y=\operatorname{Im}v$ +\begin{align*} +\only<-2>{ +\begin{pmatrix} +Av\\ +A\overline v +\end{pmatrix} += +\begin{pmatrix} +Ax+Ay J \\ +Ax-Ay J +\end{pmatrix} +&= +\begin{pmatrix} +\lambda v\\ +\overline{\lambda}\overline{v} +\end{pmatrix} += +\begin{pmatrix} +a+bJ & 0 \\ + 0 & a-bJ +\end{pmatrix} +\begin{pmatrix} +x+ yJ\\ +x- yJ +\end{pmatrix} +\\ +} +\only<2-3>{ +\begin{pmatrix} +Ax&-Ay\\ +Ay& Ax\\ +Ax& Ay\\ +-Ay&Ax +\end{pmatrix} +&= +\begin{pmatrix} +a&-b& 0& 0\\ +b& a& 0& 0\\ +0& 0& a& b\\ +0& 0&-b& a +\end{pmatrix} +\begin{pmatrix} +x&-y\\ +y& x\\ +x& y\\ +-y&x +\end{pmatrix} +\\ +} +\only<3-4>{ +\ifthenelse{\boolean{presentation}}{ +\begin{pmatrix} +Ax&-Ay\\ +Ax& Ay\\ +Ay& Ax\\ +-Ay&Ax +\end{pmatrix} +& += +\begin{pmatrix} +a& 0&-b& 0\\ +0& a& 0& b\\ +b& 0& a& 0\\ +0&-b& 0& a +\end{pmatrix} +\begin{pmatrix} +x&-y\\ +x& y\\ +y& x\\ +-y&x +\end{pmatrix} +\Rightarrow +\\ +}{} +} +\only<4->{ +Ax &= ax -by \\ +Ay &= bx +ay +} +\end{align*} +\uncover<5->{% +D.h. in Basis $x=\operatorname{Re}v,y=\operatorname{Im}v$ hat $A$ die Matrix +$\begin{pmatrix}a&-b\\b&a\end{pmatrix}$} +\uncover<6->{% +\[ +\text{ +Reeller +Jordan-Block: +} +\qquad +J_{\lambda,\overline{\lambda}} += +\begin{pmatrix} +a&-b&1& 0&0& 0\\ +b& a&0& 1&0& 0\\ + & &a&-b&1& 0\\ + & &b& a&0& 1\\ + & & & &a&-b\\ + & & & &b& a +\end{pmatrix} +\]} +\end{frame} diff --git a/vorlesungen/slides/5/satzvongelfand.tex b/vorlesungen/slides/5/satzvongelfand.tex new file mode 100644 index 0000000..3cf8710 --- /dev/null +++ b/vorlesungen/slides/5/satzvongelfand.tex @@ -0,0 +1,89 @@ +% +% satzvongelfand.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{0pt} +\setlength{\belowdisplayskip}{0pt} +\setbeamercolor{block body}{bg=blue!20} +\setbeamercolor{block title}{bg=blue!20} +\frametitle{Satz von Gelfand} +{\usebeamercolor[fg]{title}Behauptung:} $\varrho(A)=\pi(A)$\uncover<2->{, +$A(\varepsilon) = \displaystyle\frac{A}{\varrho(A)+\varepsilon}$}\uncover<3->{, +$\varrho(A(\varepsilon))=\displaystyle\frac{\varrho(A)}{\varrho(A)+\varepsilon} +\uncover<4->{=\frac{1}{1+\varepsilon/\varrho(A)}}$} + +\uncover<5->{% +%{\usebeamercolor[fg]{title}Beweisidee:} +%$\displaystyle\pi\biggl(\frac{A}{\varrho(A)+\epsilon}\biggr) +%= +%\frac{\pi(A)}{\varrho(A)+\epsilon}$ berechnen +\vspace{-5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{$\varepsilon < 0$} +\vspace{-10pt} +\begin{align*} +\uncover<6->{ +\varrho(A(\varepsilon))&>1}\uncover<7->{\quad\Rightarrow\quad \|A(\varepsilon)^k\|\to \infty} +\\ +\uncover<8->{\|A(\varepsilon)^k\| &\ge m\varrho(A(\varepsilon))^k} +\\ +\uncover<9->{\|A(\varepsilon)^k\|^{\frac1k} &\ge m^{\frac1k} \varrho(A(\varepsilon))} +\\ +\uncover<10->{\pi(A) &\ge \lim_{k\to\infty}m^{\frac1k}\varrho(A(\varepsilon))} +\\ +&\uncover<11->{= \varrho(A(\varepsilon))}\uncover<12->{ > 1} +\\ +\uncover<13->{\frac{ \pi(A(\varepsilon))}{\varrho(A)+\varepsilon} &> 1} +\\ +\uncover<14->{ +\pi(A) &> \varrho(A)+\varepsilon +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{$\varepsilon > 0$} +\vspace{-10pt} +\begin{align*} +\uncover<16->{ +\varrho(A(\varepsilon)) &<1} +\uncover<17->{\quad\Rightarrow\quad \|A(\varepsilon)^k\| \to 0} +\\ +\uncover<18->{\|A(\varepsilon)^k\| +&\le M\varrho(A(\varepsilon))^k} +\\ +\uncover<19->{ +\|A(\varepsilon)^k\|^{\frac1k} +&\le M^{\frac1k}\varrho(A(\varepsilon)) +} +\\ +\uncover<20->{ +\pi(A(\varepsilon)) +&\le +\varrho(A(\varepsilon)) \lim_{k\to\infty} M^{\frac1k} +} +\\ +&\uncover<21->{= \varrho(A(\varepsilon))} +\uncover<22->{ < 1} +\\ +\uncover<23->{\frac{\pi(A)}{\varrho(A)+\varepsilon}&< 1} +\\ +\uncover<24->{\pi(A)&< \varrho(A) + \varepsilon} +\end{align*} +\end{block} +\end{column} +\end{columns}} +\uncover<15->{% +\vspace{2pt} +{\usebeamercolor[fg]{title}Folgerung:} +$\varrho(A)-\varepsilon < \pi(A) \uncover<25->{< \varrho(A)+\varepsilon}\quad\forall\varepsilon>0 +\uncover<26->{ +\qquad\Rightarrow\qquad +\varrho(A)=\pi(A)}$ +} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/spektralgelfand.tex b/vorlesungen/slides/5/spektralgelfand.tex new file mode 100644 index 0000000..9342cd6 --- /dev/null +++ b/vorlesungen/slides/5/spektralgelfand.tex @@ -0,0 +1,190 @@ +% +% spektralgelfand.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\eigenwert#1#2{ + \fill[color=blue!30] (#1:#2) circle[radius=0.05]; + \draw[color=blue] (#1:#2) circle[radius=0.05]; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spektral- und Gelfand-Radius} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\begin{block}{Spektralradius} +\vspace{-10pt} +\[ +\varrho(A) += +\sup\{|\lambda|\;|\; \text{{\color{blue}$\lambda$} ist EW von $A$}\} +\] +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<5->{ + \fill[color=red!30] (0,0) circle[radius=2.2]; + \draw[color=red] (0,0) circle[radius=2.2]; +} + +\uncover<3->{ + \eigenwert{190.46}{1.3365} + %\eigenwert{52.663}{2.1819} + \eigenwert{281.94}{1.7305} + \eigenwert{21.29}{1.0406} + \eigenwert{69.511}{1.56} + \eigenwert{63.365}{1.3535} + \eigenwert{281.43}{0.31994} + \eigenwert{313.1}{1.5419} + \eigenwert{118.14}{1.1966} + \eigenwert{195.75}{0.41156} + \eigenwert{233.42}{1.5613} + \eigenwert{25.203}{1.1936} + \eigenwert{53.375}{1.4886} + \eigenwert{346.13}{2.1073} + \eigenwert{246.47}{1.124} + \eigenwert{35.451}{1.99} + \eigenwert{212.43}{1.9708} + \eigenwert{58.479}{0.61602} + \eigenwert{344.37}{1.6107} + \eigenwert{305.42}{1.7075} + \eigenwert{29.693}{0.28791} + \eigenwert{195.82}{0.63079} + \eigenwert{209.71}{0.25669} + \eigenwert{51.355}{0.7247} + \eigenwert{356.43}{1.0867} + \eigenwert{33.119}{0.7328} + \eigenwert{73.131}{1.5021} + \eigenwert{345.67}{0.37564} + \eigenwert{76.52}{0.71763} + %\eigenwert{197.04}{2.1431} + \eigenwert{217.87}{1.7704} + \eigenwert{172.93}{1.1204} + \eigenwert{339.19}{1.5305} + \eigenwert{272.86}{2.04} + \eigenwert{168.8}{1.6289} + \eigenwert{248.68}{0.70879} + \eigenwert{237.98}{0.71097} + \eigenwert{81.411}{1.8461} + \eigenwert{224.65}{1.0827} + \eigenwert{357.54}{0.291} + \eigenwert{325.26}{1.2778} + \eigenwert{150.97}{0.32358} + \eigenwert{260.68}{1.4077} + \eigenwert{116.29}{1.0715} + \eigenwert{358.25}{0.99667} + \eigenwert{276.2}{0.077375} + \eigenwert{316.16}{0.77763} + \eigenwert{69.398}{1.2818} + \eigenwert{353.5}{0.74099} + \eigenwert{4.7935}{1.391} + \eigenwert{136.98}{1.7572} + \eigenwert{45.62}{1.9649} + \eigenwert{299.96}{0.19199} + \eigenwert{187.32}{0.63805} + \eigenwert{272.88}{1.1467} + \eigenwert{231.85}{1.5763} + \eigenwert{124.24}{0.77024} + \eigenwert{196.24}{2.0375} + \eigenwert{186.33}{1.0656} + %\eigenwert{22.812}{2.1616} + \eigenwert{37.982}{0.038956} + \eigenwert{142.36}{1.7944} + \eigenwert{56.863}{1.8952} + \eigenwert{4.6281}{1.1857} + \eigenwert{71.674}{0.07642} + \eigenwert{94.049}{1.8985} + \eigenwert{97.294}{0.23412} + \eigenwert{84.739}{0.31209} + \eigenwert{147.42}{1.8434} + \eigenwert{160.67}{0.76956} + \eigenwert{292.5}{0.85697} + \eigenwert{308.1}{1.7061} + \eigenwert{68.669}{2.111} + \eigenwert{86.866}{1.1271} + \eigenwert{124.72}{1.3019} + \eigenwert{267.36}{0.7462} + \eigenwert{295.78}{1.0425} + \eigenwert{44.972}{0.65363} + \eigenwert{34.534}{1.2817} + \eigenwert{357.78}{2.0592} + \eigenwert{147.52}{0.020535} + %\eigenwert{28.502}{2.1964} + \eigenwert{343.48}{2.0968} + \eigenwert{129.96}{0.80371} + \eigenwert{254.75}{1.5775} + \eigenwert{89.91}{0.88605} + \eigenwert{20.35}{0.66065} + \eigenwert{60.382}{1.7585} + \eigenwert{158.87}{0.68399} + \eigenwert{328.44}{1.504} + \eigenwert{189.41}{0.33879} + \eigenwert{273.47}{0.11109} + \eigenwert{285.99}{0.66704} + \eigenwert{311.42}{2.0266} + \eigenwert{32.636}{0.5713} + \eigenwert{221.35}{2.1329} + \eigenwert{50.983}{1.1957} + \eigenwert{53.298}{1.2982} + \eigenwert{101.4}{1.9051} + \eigenwert{71.999}{0.25671} +} + +\uncover<2->{ + \draw[->] (-2.4,0) -- (2.7,0) + coordinate[label={$\operatorname{Re}z$}]; + \draw[->] (0,-2.4) -- (0,2.5) + coordinate[label={right:$\operatorname{Im}z$}]; +} +\uncover<4->{ + \fill[color=darkgreen] (0,0) circle[radius=0.05]; + \draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (150:2.2); + \node[color=darkgreen] at ($(150:1.85)+(0.4,0)$) + [below left] {$\varrho(A)$}; +} +\uncover<3->{ + \eigenwert{150}{2.2} +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Gelfand-Radius} +\[ +\pi(A) += +\lim_{k\to\infty} \|A^k\|^{\frac{1}{k}} +\] +\end{block}} +\vspace{-8pt} +\uncover<7->{% +\begin{block}{Konvergenz der Neumann-Reihe} +$ +\uncover<8->{t<1/\pi(A)\;} +\uncover<10->{\Rightarrow\; \exists q} +\uncover<11->{,N}$ +\begin{align*} +\uncover<9->{ t\pi(A) & \only<10->{< q} < 1 } +\\ +\uncover<11->{ \|(tA)^k\|^{\frac1k} &\le q } +\\ +\uncover<12->{ +\|(tA)^k\| +&\le +(t\pi(A))^k<q^k +} +\end{align*} +\uncover<11->{für $k>N$.} +\uncover<13->{ +$\Rightarrow$ +$(1-tA)^{-1}=\displaystyle\sum_{k=0}^\infty (tA)^k$ konvergiert für $t<1/\pi(A)$ +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/spektrum.tex b/vorlesungen/slides/5/spektrum.tex new file mode 100644 index 0000000..6cbdd7f --- /dev/null +++ b/vorlesungen/slides/5/spektrum.tex @@ -0,0 +1,76 @@ +% +% spektrum.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spektrum} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +$A\colon V\to V$ beschränkter Operator zwischen Banach-Räumen +\[ +\operatorname{Sp}A += +\left\{ +\lambda\in\mathbb{C} +\;\left|\; +\begin{minipage}{2cm}\raggedright +$A-\lambda I$ nicht invertierbar +\end{minipage} +\right. +\right\} +\] +\end{block} +\uncover<2->{% +\begin{block}{Endlichdimensionale Räume} +\vspace{-15pt} +\begin{align*} +&\lambda\in\operatorname{Sp}A +\\ +\uncover<3->{ +\Leftrightarrow\quad&\text{$(A-\lambda I)$ nicht invertierbar} +} +\\ +\uncover<4->{ +\Leftrightarrow\quad&\text{$(A-\lambda I)$ singulär} +} +\\ +\uncover<5->{ +\Leftrightarrow\quad&\ker(A-\lambda I)\ne 0 +} +\\ +\uncover<6->{ +\Leftrightarrow\quad&\exists v\in V, v\ne 0, Av=\lambda v +} +\end{align*} +\uncover<7->{% +$\Rightarrow$ $\operatorname{Sp}A$ ist die Menge der Eigenwerte +} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Unendlichdimensional} +Es gibt eine Folge $x_n\in V$ von Einheitsvektoren +$\|x_n\|=1$ +mit +\begin{align*} +\lim_{n\to\infty} (A - \lambda)x_n &= 0 +\end{align*} +\end{block}} +\uncover<9->{% +\begin{block}{Spektrum und Norm} +\[ +\operatorname{Sp}(A) +\subset +\{\lambda\in\mathbb{C}\;|\; +|\lambda|\le \|A\|\} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/stoneweierstrass.tex b/vorlesungen/slides/5/stoneweierstrass.tex new file mode 100644 index 0000000..3f9cab5 --- /dev/null +++ b/vorlesungen/slides/5/stoneweierstrass.tex @@ -0,0 +1,11 @@ +% +% stoneweierstrass.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Stone-Weierstrass} + +TODO XXX + +\end{frame} diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex new file mode 100644 index 0000000..f0c4401 --- /dev/null +++ b/vorlesungen/slides/5/unitaer.tex @@ -0,0 +1,75 @@ +% +% unitaer.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Unitäre Matrizen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Eigenwerte} +$U$ unitär lässt das Skalarprodukt invariant +\[ +\langle Ux,Uy\rangle += +\langle x,y\rangle +\] +\uncover<2->{% +$\lambda$ ein Eigenwert mit Eigenvektor $v$: +\begin{align*} +\langle v,v\rangle +&= +\langle Uu,Uv\rangle +\uncover<3->{= \langle \lambda v,\lambda v\rangle} +\uncover<4->{= |\lambda|^2 \langle v,v\rangle} +\\ +\uncover<5->{\Rightarrow\;|\lambda|&=1} +\end{align*}} +\end{block} +\uncover<6->{% +\begin{block}{Diagonalisierbar} +Unitäre Matrizen sind über $\mathbb{C}$ diagonalisierbar +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Grosse Jordan-Blöcke?} +Falls es Vektoren $v,w$ gibt mit +\begin{align*} +\uncover<7->{ Uv&=\lambda v} +\\ +\uncover<8->{ Uw&=\lambda w + v} +\intertext{\uncover<9->{Skalarprodukt:}} +\uncover<10->{ +\langle v,w\rangle +&= +\langle Uv,Uw\rangle} +\\ +\uncover<11->{ +&= +\langle \lambda v,\lambda w\rangle ++ +\langle\lambda v,v\rangle} +\\ +\uncover<12->{ +&= +|\lambda|^2 \langle v,w\rangle ++ +\langle\lambda v,v\rangle} +\\ +\uncover<13->{ +&= +\langle v,w\rangle ++ +\lambda \| v\|^2} +\\ +\uncover<14->{ +\Rightarrow\quad +0&=\|v\|^2\quad\Rightarrow\quad \|v\|=0} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/verzerrung.tex b/vorlesungen/slides/5/verzerrung.tex new file mode 100644 index 0000000..8d6514c --- /dev/null +++ b/vorlesungen/slides/5/verzerrung.tex @@ -0,0 +1,121 @@ +% +% verzerrung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\r{1.10} +\def\s{1.12} +\def\q{1.23} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Verzerrung} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.49\textwidth} +\begin{block}{Abbildung $A\colon v\mapsto Av$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=2.5] +\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1]; + +\coordinate (a1) at (0.974,0.171); +\coordinate (a2) at (0.037,1.018); + +\coordinate (v1) at (-0.5216,0.8532); +\coordinate (v2) at (-0.3343,-0.9425); + +\foreach \a in {0,5,...,355}{ + \draw[color=red,line width=1.2pt] + ($cos(\a)*(a1)+sin(\a)*(a2)$) -- + ($cos(\a+5)*(a1)+sin(\a+5)*(a2)$); +} +\foreach \a in {1,...,144}{ + \only<\a>{ + \fill[color=red,line width=1.4pt] + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03]; + \draw[->,color=red,line width=1.4pt] (0,0) -- + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1); + \fill[color=blue] ({5*\a}:1) circle[radius=0.03]; + \node[color=blue] at ({5*\a}:\r) {$v$}; + \node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$) + {$Av$}; + } +} + +\begin{scope} +\clip (-1.2,-1.1) rectangle (1.2,1.1); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2); +\end{scope} + +\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}]; +\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}]; + +\node[color=darkgreen] at (v1) [above left] {$v_1$}; +\node[color=darkgreen] at (v2) [below left] {$v_2$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.49\textwidth} +\uncover<73->{% +\begin{block}{Abbildung $A\colon v\mapsto (A-\lambda)v$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=2.5] +\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1]; + +\coordinate (a1) at (0.121,0.343); +\coordinate (a2) at (0.074,0.209); + +\coordinate (v1) at (-0.5216,0.8532); +\coordinate (v2) at (-0.3343,-0.9425); + +\begin{scope} +\clip (-1.2,-1.2) rectangle (1.2,1.2); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$); +\end{scope} + +\foreach \a in {0,5,...,355}{ + \draw[color=red!60,line width=4pt] + ($cos(\a)*(a1)+sin(\a)*(a2)$) -- + ($cos(\a+5)*(a1)+sin(\a+5)*(a2)$); +} +\foreach \a in {73,...,144}{ + \only<\a>{ + \fill[color=red,line width=1.4pt] + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03]; + \draw[->,color=red,line width=1.4pt] (0,0) -- + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1); + \fill[color=blue] ({5*\a}:1) circle[radius=0.03]; + \node[color=blue] at ({5*\a}:\r) {$v$}; + \node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$) + {$(A-\lambda)v$}; + } +} + +\begin{scope} +\clip (-1.2,-1.1) rectangle (1.2,1.1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2); +\end{scope} + +\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}]; +\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}]; + +\node[color=darkgreen] at (v1) [above left] {$v_1$}; +\node[color=darkgreen] at (v2) [below left] {$v_2$}; + +\end{tikzpicture} +\end{center} + +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/verzerrung/verzerrung.m b/vorlesungen/slides/5/verzerrung/verzerrung.m new file mode 100644 index 0000000..028e7f9 --- /dev/null +++ b/vorlesungen/slides/5/verzerrung/verzerrung.m @@ -0,0 +1,13 @@ +# +# verzerrung.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +rand("seed", 4712); + +A = eye(2) + 1.0 * (rand(2,2) - 0.5 * ones(2,2)) + +[V, lambda] = eig(A) + +B = A - lambda(1,1) * eye(2) diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex new file mode 100644 index 0000000..a734d69 --- /dev/null +++ b/vorlesungen/slides/5/zerlegung.tex @@ -0,0 +1,105 @@ +% +% zerlegung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Zerlegung in Eigenräume} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.38] +\uncover<2->{ +\fill[color=blue!20] (0,11) rectangle (4,15); +\fill[color=red!20] (4,0) rectangle (15,11); +} +\uncover<3->{ +\fill[color=red!40] (9,0) rectangle (15,6); +\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11); +} +\uncover<4->{ +\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6); +\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3); +} + +\uncover<2->{ +\draw[line width=0.1pt] (0,11) -- (15,11); +\draw[line width=0.1pt] (4,0) -- (4,15); +} + +\uncover<3->{ +\draw[line width=0.1pt] (0,6) -- (15,6); +\draw[line width=0.1pt] (9,0) -- (9,15); +} + +\uncover<4->{ +\draw[line width=0.1pt] (0,3) -- (15,3); +\draw[line width=0.1pt] (12,0) -- (12,15); +} +\draw (0,0) rectangle (15,15); +\uncover<2->{ +\node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$}; +\node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$}; +\node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$}; +} +\uncover<3->{ +\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$}; +\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$}; +} +\uncover<4->{ +\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$}; +\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$}; +\node at (10.5,4.5) {$(f_2)_{|\mathcal{E}_{\lambda_3}}$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Iteration} +$\Lambda=\{\lambda_1,\dots,\lambda_s\}$ Eigenwerte +\begin{align*} +\uncover<2->{ +V +&= +\mathcal{K}(f-\lambda_1) +\oplus +\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!20] (0,0) rectangle (1.83,1.1);}}}} +\underbrace{\mathcal{J}(f-\lambda_1)}_{\displaystyle=V_1} +} +\\[-15pt] +\uncover<2->{ +f_1 &= f_{|V_1} +} +\\[10pt] +\uncover<3->{ +V_1 +&= +\mathcal{K}(f_1-\lambda_2) +\oplus +\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!40] (0,0) rectangle (1.9,1.1);}}}} +\underbrace{\mathcal{J}(f_1-\lambda_2)}_{\displaystyle=V_2} +} +\\[-15pt] +\uncover<3->{ +f_1 &= f_{|V_1} +} +\\ +\uncover<4->{ +&\phantom{0}\vdots +} +\end{align*} +\uncover<5->{% +$\Rightarrow$ $f$ hat {\color{blue}Blockdiagonalform} für die Zerlegung +\begin{align*} +V&=\bigoplus_{\lambda\in\Lambda} \mathcal{E}_{\lambda} +\end{align*}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc new file mode 100644 index 0000000..d46dc7f --- /dev/null +++ b/vorlesungen/slides/8/Makefile.inc @@ -0,0 +1,32 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter8 = \ + ../slides/8/dgraph.tex \ + ../slides/8/graph.tex \ + ../slides/8/grad.tex \ + ../slides/8/inzidenz.tex \ + ../slides/8/inzidenzd.tex \ + ../slides/8/diffusion.tex \ + ../slides/8/laplace.tex \ + ../slides/8/produkt.tex \ + ../slides/8/fourier.tex \ + ../slides/8/spanningtree.tex \ + ../slides/8/pfade/adjazenz.tex \ + ../slides/8/pfade/langepfade.tex \ + ../slides/8/pfade/beispiel.tex \ + ../slides/8/pfade/gf.tex \ + ../slides/8/floyd-warshall/problem.tex \ + ../slides/8/floyd-warshall/rekursion.tex \ + ../slides/8/floyd-warshall/iteration.tex \ + ../slides/8/floyd-warshall/wegiteration.tex \ + ../slides/8/floyd-warshall/wege.tex \ + ../slides/8/tokyo/google.tex \ + ../slides/8/tokyo/bahn0.tex \ + ../slides/8/tokyo/bahn1.tex \ + ../slides/8/tokyo/bahn2.tex \ + ../slides/8/chapter.tex + diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex new file mode 100644 index 0000000..6a0b13f --- /dev/null +++ b/vorlesungen/slides/8/chapter.tex @@ -0,0 +1,32 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{8/graph.tex} +\folie{8/dgraph.tex} +\folie{8/grad.tex} +\folie{8/inzidenz.tex} +\folie{8/inzidenzd.tex} +\folie{8/diffusion.tex} +\folie{8/laplace.tex} +\folie{8/produkt.tex} +\folie{8/fourier.tex} +\folie{8/spanningtree.tex} + +\folie{8/pfade/adjazenz.tex} +\folie{8/pfade/langepfade.tex} +\folie{8/pfade/beispiel.tex} +\folie{8/pfade/gf.tex} + +\folie{8/floyd-warshall/problem.tex} +\folie{8/floyd-warshall/rekursion.tex} +\folie{8/floyd-warshall/iteration.tex} +\folie{8/floyd-warshall/wegiteration.tex} +\folie{8/floyd-warshall/wege.tex} + +\folie{8/tokyo/google.tex} +\folie{8/tokyo/bahn0.tex} +\folie{8/tokyo/bahn1.tex} +\folie{8/tokyo/bahn2.tex} + diff --git a/vorlesungen/slides/8/dgraph.tex b/vorlesungen/slides/8/dgraph.tex new file mode 100644 index 0000000..6b5864a --- /dev/null +++ b/vorlesungen/slides/8/dgraph.tex @@ -0,0 +1,100 @@ +% +% dgraph.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame} +\frametitle{Gerichteter Graph} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{2.4} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\uncover<3->{ + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); + \draw[color=white,line width=5pt] (B) -- (D); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); + + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); +} + +\uncover<2->{ + \draw (A) circle[radius=0.2]; + \draw (B) circle[radius=0.2]; + \draw (C) circle[radius=0.2]; + \draw (D) circle[radius=0.2]; + \draw (E) circle[radius=0.2]; + + \node at (A) {$1$}; + \node at (B) {$2$}; + \node at (C) {$3$}; + \node at (D) {$4$}; + \node at (E) {$5$}; +} +\node at (0,0) {$G$}; + +\uncover<3->{ + \node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$}; + \node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$}; + \node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$}; + \node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$}; + \node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$}; + \node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$}; + \node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; +} + +\uncover<7->{ + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=red] + (E) to[out=-18,in=-126,distance=2cm] (E); +} + +\uncover<9->{ + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=darkgreen] + (D) to[out=120,in=-120] (C); +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Definition} +Ein gerichteter Graph $G=(V,E)$ ist +\begin{enumerate} +\item<2-> Eine Menge $V$ von Knoten (Vertizes) +$V=\{v_1,v_2,\dots\}$ +\item<3-> +Eine Menge $E$ von gerichteten Kanten +(Edges) +\[ +E\subset \{ (v_1,v_2)\;|\; v_i\in V\} +\] +\end{enumerate} +\end{block} +\vspace{-30pt} +\uncover<6->{% +\begin{block}{Achtung} +\begin{itemize} +\item<6-> Kanten sind {\em geordnete} Paare +\uncover<7->{$\Rightarrow$ {\color{red}Schleifen} sind möglich} +\item<8-> Kanten sind immer ``Einbahnstrassen'' +\item<9-> {\color{darkgreen}Gegenrichtung explizit angeben} +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/diffusion.tex b/vorlesungen/slides/8/diffusion.tex new file mode 100644 index 0000000..0d07a27 --- /dev/null +++ b/vorlesungen/slides/8/diffusion.tex @@ -0,0 +1,89 @@ +% +% diffusion.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Diffusion} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C); +\draw[color=white,line width=5pt] (B) -- (D); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D); + +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A); + +\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm] + (A) -- (E); +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (A) -- (B); +\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm] + (A) -- (C); +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (B) -- (C); +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (C) -- (D); +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (D) -- (E); +\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm] + (B) -- (D); + +\fill[color=red] (A) circle[radius=0.3]; +\fill[color=red!50] (B) circle[radius=0.3]; +\fill[color=white] (C) circle[radius=0.3]; +\fill[color=blue!50] (D) circle[radius=0.3]; +\fill[color=blue] (E) circle[radius=0.3]; + +\draw (A) circle[radius=0.3]; +\draw (B) circle[radius=0.3]; +\draw (C) circle[radius=0.3]; +\draw (D) circle[radius=0.3]; +\draw (E) circle[radius=0.3]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; +\node at (0,0) {$G$}; + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +\begin{block}{Knotenfunktion} +$f\colon V\to \mathbb{R}$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Fluss} +Je grösser die Differenz zu den Nachbarn, desto grösser der Fluss in +den Knoten: +\begin{align*} +\frac{df(v)}{dt} +&= +\kappa \sum_{\text{$v'$ Nachbar von $v$}} (f(v')-f(v)) +\end{align*} +``Wärmeleitungsgleichung'' +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/floyd-warshall/burgerking.png b/vorlesungen/slides/8/floyd-warshall/burgerking.png Binary files differnew file mode 100644 index 0000000..cf4211d --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/burgerking.png diff --git a/vorlesungen/slides/8/floyd-warshall/fw.tex b/vorlesungen/slides/8/floyd-warshall/fw.tex new file mode 100644 index 0000000..99929fb --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/fw.tex @@ -0,0 +1,680 @@ +% +% fw.tex -- Durchführung des Floyd-Warshall Algorithmus +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup + +\definecolor{wegelb}{rgb}{1,0.6,0} +\definecolor{weghell}{rgb}{1,0.9,0.6} +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\begin{columns}[t] +\begin{column}{0.44\hsize} +\begin{center} +\begin{tikzpicture}[>=latex] + + +\def\r{2.2} +\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)}); +\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)}); +\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)}); +\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)}); +\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)}); + +\def\knoten#1#2#3{ + \fill[color=#3] #1 circle[radius=0.3]; + \draw[line width=1pt] #1 circle[radius=0.3]; + \node at #1 {$#2$}; +} + +\def\kante#1#2#3{ + \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] #1 -- #2; + \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22]; + \node at ($0.5*#1+0.5*#2$) {$#3$}; +} + +% Wege über 1 +% 3--1--5 +\only<4>{ + \draw[->,line width=7pt,color=red!50] + (C)--(A)--(E); +} + +% Wege über 2 +% 5--2--3 +\only<6>{ + \draw[->,line width=7pt,color=red!50] + (E)--(B)--(C); +} +% 5--2--4 +\only<7>{ + \draw[->,line width=7pt,color=red!50] + (E)--(B)--(D); + \draw[->,line width=7pt,color=blue!50,dashed] + (E)--(D); +} + +% Wege über 3 +% 2--3--1 +\only<9>{ + \draw[->,line width=7pt,color=red!50] + (B)--(C)--(A); +} +% 2--3--5 +\only<10>{ + \draw[->,line width=7pt,color=red!50] + (B)--(C)--(A)--(E); +} +% 4--3--1 +\only<11>{ + \draw[->,line width=7pt,color=red!50] + (D)--(C)--(A); +} +% 4--3--5 +\only<12>{ + \draw[->,line width=7pt,color=red!50] + (D)--(C)--(A)--(E); +} +% 5--3--1 +\only<13>{ + \draw[->,line width=7pt,color=red!50] + (E)--(B)--(C)--(A); +} +% 5--3--5 +\only<14>{ + \draw[->,line width=7pt,color=red!50] + (E)--(B)--(C)--(A)--(E); +} + +% Wege über 4 +% 2--4--1 +\only<16>{ + \draw[->,line width=7pt,color=red!50] + (B)--(D)--(C)--(A); + \draw[->,line width=7pt,color=blue!50,dashed] + (B)--(C)--(A); +} +% 2--4--3 +\only<17>{ + \draw[->,line width=7pt,color=red!50] + (B)--(D)--(C); + \draw[->,line width=7pt,color=blue!50,dashed] + (B)--(C); +} +% 2--4--5 +\only<18>{ + \draw[->,line width=7pt,color=red!50] + (B)--(D)--(C)--(A)--(E); + \draw[->,line width=7pt,color=blue!50,dashed] + (B)--(C)--(A)--(E); +} +% 5--4--1 +\only<19>{ + \draw[->,line width=7pt,color=red!50] + (E)--(D)--(C)--(A); + \draw[->,line width=7pt,color=blue!50,dashed] + (E)--(B)--(C)--(A); +} +% 5--4--3 +\only<20>{ + \draw[->,line width=7pt,color=red!50] + (E)--(D)--(C); + \draw[->,line width=7pt,color=blue!50,dashed] + (E)--(B)--(C); +} +% 5--4--5 +\only<21>{ + \draw[->,line width=7pt,color=red!50] + (E)--(D)--(C)--(A)--(E); + \draw[->,line width=7pt,color=blue!50,dashed] + (E)--(B)--(C)--(A)--(E); +} +% 1--5--1 +\only<23>{ + \draw[->,line width=7pt,color=red!50] + (A)--(E)--(B)--(C)--(A); +} +% 1--5--2 +\only<24>{ + \draw[->,line width=7pt,color=red!50] + (A)--(E)--(B); +} +% 1--5--3 +\only<25>{ + \draw[->,line width=7pt,color=red!50] + (A)--(E)--(B)--(C); +} +% 1--5--4 +\only<26>{ + \draw[->,line width=7pt,color=red!50] + (A)--(E)--(D); +} +% 2--5--1 +\only<27>{ + \draw[->,line width=7pt,color=red!50] + (B)--(C)--(A)--(E)--(D)--(C)--(A); + \draw[->,line width=7pt,color=blue!50,dashed] + (B)--(C)--(A); +} +% 2--5--2 +\only<28>{ + \draw[->,line width=7pt,color=red!50] + (B)--(C)--(A)--(E)--(B); +} +% 2--5--3 +\only<29>{ + \draw[->,line width=7pt,color=red!50] + (B)--(C)--(A)--(E)--(D)--(C); + \draw[->,line width=7pt,color=blue!50,dashed] + (B)--(C); +} +% 2--5--4 +\only<30>{ + \draw[->,line width=7pt,color=red!50] + (B)--(C)--(A)--(E)--(D); + \draw[->,line width=7pt,color=blue!50,dashed] + (B)--(D); +} +% 3--5--1 +\only<31>{ + \draw[->,line width=7pt,color=red!50] + (C)--(A)--(E)--(D)--(C)--(A); + \draw[->,line width=7pt,color=blue!50,dashed] + (C)--(A); +} +% 3--5--2 +\only<32>{ + \draw[->,line width=7pt,color=red!50] + (C)--(A)--(E)--(B); +} +% 3--5--3 +\only<33>{ + \draw[->,line width=7pt,color=red!50] + (C)--(A)--(E)--(B)--(C); +} +% 3--5--4 +\only<34>{ + \draw[->,line width=7pt,color=red!50] + (C)--(A)--(E)--(D); +} +% 4--5--1 +\only<35>{ + \draw[->,line width=7pt,color=red!50] + (D)--(C)--(A)--(E)--(D)--(C)--(A); + \draw[->,line width=7pt,color=blue!50,dashed] + (D)--(C)--(A); +} +% 4--5--2 +\only<36>{ + \draw[->,line width=7pt,color=red!50] + (D)--(C)--(A)--(E)--(B); +} +% 4--5--3 +\only<37>{ + \draw[->,line width=7pt,color=red!50] + (D)--(C)--(A)--(E)--(D)--(C); + \draw[->,line width=7pt,color=blue!50,dashed] + (D)--(C); +} +% 4--5--4 +\only<38>{ + \draw[->,line width=7pt,color=red!50] + (D)--(C)--(A)--(E)--(D); +} + + +\uncover<40>{ + \draw[->,color=red!50,line width=7pt] + (B)--(C)--(A)--(E)--(D); +} + +\kante{(A)}{(E)}{1} +\kante{(B)}{(C)}{2} +\kante{(B)}{(D)}{13} +\kante{(C)}{(A)}{3} +\kante{(D)}{(C)}{6} +\kante{(E)}{(B)}{5} +\kante{(E)}{(D)}{6} + +\only<1>{ + \knoten{(A)}{}{white} + \knoten{(B)}{}{white} + \knoten{(C)}{}{white} + \knoten{(D)}{}{white} + \knoten{(E)}{}{white} +} + +\only<2->{ + \knoten{(A)}{1}{white} + \knoten{(B)}{2}{white} + \knoten{(C)}{3}{white} + \knoten{(D)}{4}{white} + \knoten{(E)}{5}{white} +} + +\only<4>{ + \knoten{(A)}{1}{darkgreen!50} +} +\only<6-7>{ + \knoten{(B)}{2}{darkgreen!50} +} +\only<9-14>{ + \knoten{(C)}{3}{darkgreen!50} +} +\only<16-21>{ + \knoten{(D)}{4}{darkgreen!50} +} +\only<23-38>{ + \knoten{(E)}{5}{darkgreen!50} +} + +\end{tikzpicture} +\end{center} +\begin{block}{Aufgabe} +Finde den kürzesten Weg von 2 nach 4 +\end{block} +\end{column} +\begin{column}{0.5\hsize} +\begin{center} +\begin{tikzpicture}[>=latex,scale=0.8] + +\def\punkt#1#2{ + ({#2-0.5},{0.5-(#1)}) +} +\def\punktoff#1#2{ + ({#2-0.7},{0.7-(#1)}) +} +\def\feld#1#2#3{ + \ifthenelse{\boolean{wegweiser}}{ + \fill[color=white] + ({#2-1},{1-#1}) rectangle ({#2-0.45},{0.45-#1}); + \node at \punktoff{#1}{#2} {$#3$}; + }{ + \fill[color=white] + ({#2-1},{1-#1}) rectangle ({#2},{-#1}); + \node at \punkt{#1}{#2} {$#3$}; + } +} +\def\verbindung#1#2#3{ + \draw[->,line width=5pt,color=red!20,shorten >= 0.2cm,shorten <= 0.2cm] + \punkt{#1}{#2}--\punkt{#2}{#3}; + \node at (5,-5.5) [left] {$#1\rightsquigarrow #2\rightsquigarrow #3$\strut}; +} +\def\Infty{{}} +\def\wegweiser#1#2#3{ + \ifthenelse{\boolean{wegweiser}}{ + \ifnum #2 = #3 + \fill[color=weghell] + ({#2-0.45},{0.45-#1}) rectangle ({#2-0.05},{0.05-#1}); + \else + \fill[color=wegelb] + ({#2-0.45},{0.45-#1}) rectangle ({#2-0.05},{0.05-#1}); + \fi + \node at ({#2-0.25},{0.25-#1}) {\tiny #3}; + }{} +} + +% direkte Wege +\uncover<3->{ + \feld{1}{1}{\Infty} + \feld{1}{2}{\Infty} + \feld{1}{3}{\Infty} + \feld{1}{4}{\Infty} + \feld{1}{5}{1} + \wegweiser{1}{5}{5} + + \feld{2}{1}{\Infty} + \feld{2}{2}{\Infty} + \feld{2}{3}{2} + \wegweiser{2}{3}{3} + \feld{2}{4}{13} + \wegweiser{2}{4}{4} + \feld{2}{5}{\Infty} + + \feld{3}{1}{3} + \wegweiser{3}{1}{1} + \feld{3}{2}{\Infty} + \feld{3}{3}{\Infty} + \feld{3}{4}{\Infty} + \feld{3}{5}{\Infty} + + \feld{4}{1}{\Infty} + \feld{4}{2}{\Infty} + \feld{4}{3}{6} + \wegweiser{4}{3}{3} + \feld{4}{4}{\Infty} + \feld{4}{5}{\Infty} + + \feld{5}{1}{\Infty} + \feld{5}{2}{5} + \wegweiser{5}{2}{2} + \feld{5}{3}{\Infty} + \feld{5}{4}{6} + \wegweiser{5}{4}{4} + \feld{5}{5}{\Infty} +} + +\uncover<3-3>{ + \node at (-0.8,-5.5) [right] {direkte Verbindungen}; +} + +\uncover<4-4>{ + \node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $1$:\strut}; +} + +% Wege über 1 +% 3-1-5 +\uncover<4>{ + \verbindung{3}{1}{5} + \feld{3}{5}{\color{red}4} + \wegweiser{3}{5}{1} +} +\uncover<5->{ + \feld{3}{5}{4} + \wegweiser{3}{5}{1} +} + +\uncover<6-7>{ + \node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $2$:\strut}; +} + +% Wege über 2 +% 5-2-3 +\uncover<6>{ + \verbindung{5}{2}{3} + \feld{5}{3}{\color{red}7} + \wegweiser{5}{3}{2} +} +\uncover<7->{ + \feld{5}{3}{7} + \wegweiser{5}{3}{2} +} +% 5-2-4 +\uncover<7>{ + \verbindung{5}{2}{4} + \feld{5}{4}{\color{blue}6} +} + +\uncover<9-14>{ + \node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $3$:\strut}; +} + +% Wege über 3 +% 2-3-1 +\uncover<9>{ + \verbindung{2}{3}{1} + \feld{2}{1}{\color{red}5} + \wegweiser{2}{1}{3} +} +\uncover<10->{ + \feld{2}{1}{5} + \wegweiser{2}{1}{3} +} +% 2-3-5 +\uncover<10>{ + \verbindung{2}{3}{5} + \feld{2}{5}{\color{red}6} + \wegweiser{2}{5}{3} +} +\uncover<11->{ + \feld{2}{5}{6} + \wegweiser{2}{5}{3} +} +% 4-3-1 +\uncover<11>{ + \verbindung{4}{3}{1} + \feld{4}{1}{\color{red}9} + \wegweiser{4}{1}{3} +} +\uncover<12->{ + \feld{4}{1}{9} + \wegweiser{4}{1}{3} +} +% 4-3-5 +\uncover<12>{ + \verbindung{4}{3}{5} + \feld{4}{5}{\color{red}10} + \wegweiser{4}{5}{3} +} +\uncover<13->{ + \feld{4}{5}{10} + \wegweiser{4}{5}{3} +} +% 5-3-1 +\uncover<13>{ + \verbindung{5}{3}{1} + \feld{5}{1}{\color{red}10} + \wegweiser{5}{1}{2} +} +\uncover<14->{ + \feld{5}{1}{10} + \wegweiser{5}{1}{2} +} +% 5-3-5 +\uncover<14>{ + \verbindung{5}{3}{5} + \feld{5}{5}{\color{red}11} + \wegweiser{5}{5}{2} +} +\uncover<15->{ + \feld{5}{5}{11} + \wegweiser{5}{5}{2} +} + +\uncover<16-21>{ + \node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $4$:\strut}; +} + +% Wege über 4 +% 2-4-1 +\uncover<16>{ + \verbindung{2}{4}{1} + \feld{2}{1}{\color{blue}5} +} +% 2-4-3 +\uncover<17>{ + \verbindung{2}{4}{3} + \feld{2}{3}{\color{blue}2} +} +% 2-4-5 +\uncover<18>{ + \verbindung{2}{4}{5} + \feld{2}{5}{\color{blue}6} +} +% 5-4-1 +\uncover<19>{ + \verbindung{5}{4}{1} + \feld{5}{1}{\color{blue}10} +} +% 5-4-3 +\uncover<20>{ + \verbindung{5}{4}{3} + \feld{5}{3}{\color{blue}7} +} +% 5-4-5 +\uncover<21>{ + \verbindung{5}{4}{5} + \feld{5}{5}{\color{blue}11} +} + +% Wege über 5 +\uncover<23-38>{ + \node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $5$:\strut}; +} + +% Wege über 5 +% 1-5-1 +\uncover<23>{ + \verbindung{1}{5}{1} + \feld{1}{1}{\color{red}11} + \wegweiser{1}{1}{5} +} +\uncover<24->{ + \feld{1}{1}{11} + \wegweiser{1}{1}{5} +} +% 1-5-2 +\uncover<24>{ + \verbindung{1}{5}{2} + \feld{1}{2}{\color{red}6} + \wegweiser{1}{2}{5} +} +\uncover<25->{ + \feld{1}{2}{6} + \wegweiser{1}{2}{5} +} +% 1-5-3 +\uncover<25>{ + \verbindung{1}{5}{3} + \feld{1}{3}{\color{red}8} + \wegweiser{1}{3}{5} +} +\uncover<26->{ + \feld{1}{3}{8} + \wegweiser{1}{3}{5} +} +% 1-5-4 +\uncover<26>{ + \verbindung{1}{5}{4} + \feld{1}{4}{\color{red}7} + \wegweiser{1}{4}{5} +} +\uncover<27->{ + \feld{1}{4}{7} + \wegweiser{1}{4}{5} +} +% 2-5-1 +\uncover<27>{ + \verbindung{2}{5}{1} + \feld{2}{1}{\color{blue}5} +} +% 2-5-2 +\uncover<28>{ + \verbindung{2}{5}{2} + \feld{2}{2}{\color{red}11} + \wegweiser{2}{2}{3} +} +\uncover<29->{ + \feld{2}{2}{11} + \wegweiser{2}{2}{3} +} +% 2-5-3 +\uncover<29>{ + \verbindung{2}{5}{3} + \feld{2}{3}{\color{blue}2} +} +% 2-5-4 +\uncover<30>{ + \verbindung{2}{5}{4} + \feld{2}{4}{\color{red}12} + \wegweiser{2}{4}{3} +} +\uncover<31->{ + \feld{2}{4}{12} + \wegweiser{2}{4}{3} +} +% 3-5-1 +\uncover<31>{ + \verbindung{3}{5}{1} + \feld{3}{1}{\color{blue}3} +} +% 3-5-2 +\uncover<32>{ + \verbindung{3}{5}{2} + \feld{3}{2}{\color{red}9} + \wegweiser{3}{2}{1} +} +\uncover<33->{ + \feld{3}{2}{9} + \wegweiser{3}{2}{1} +} +% 3-5-3 +\uncover<33>{ + \verbindung{3}{5}{3} + \feld{3}{3}{\color{red}11} + \wegweiser{3}{3}{1} +} +\uncover<34->{ + \feld{3}{3}{11} + \wegweiser{3}{3}{1} +} +% 3-5-4 +\uncover<34>{ + \verbindung{3}{5}{4} + \feld{3}{4}{\color{red}10} + \wegweiser{3}{4}{1} +} +\uncover<35->{ + \feld{3}{4}{10} + \wegweiser{3}{4}{1} +} +% 4-5-1 +\uncover<35>{ + \verbindung{4}{5}{1} + \feld{4}{1}{\color{blue}9} +} +% 4-5-2 +\uncover<36>{ + \verbindung{4}{5}{2} + \feld{4}{2}{\color{red}15} + \wegweiser{4}{2}{3} +} +\uncover<37->{ + \feld{4}{2}{15} + \wegweiser{4}{2}{3} +} +% 4-5-3 +\uncover<37>{ + \verbindung{4}{5}{3} + \feld{4}{3}{\color{blue}6} +} +% 4-5-4 +\uncover<38>{ + \verbindung{4}{5}{4} + \feld{4}{4}{\color{red}16} + \wegweiser{4}{4}{3} +} +\uncover<39->{ + \feld{4}{4}{16} + \wegweiser{4}{4}{3} +} + + +\uncover<3->{ + + \foreach \x in {0,...,5}{ + \draw[line width=0.7pt] (\x,0.8)--(\x,-5); + } + \foreach \y in {0,...,-5}{ + \draw[line width=0.7pt] (-0.8,\y)--(5,\y); + } + \draw[line width=1.4pt] (0,0)--(5,0)--(5,-5)--(0,-5)--cycle; + + \node at (0.5,0.5) {$1$}; + \node at (1.5,0.5) {$2$}; + \node at (2.5,0.5) {$3$}; + \node at (3.5,0.5) {$4$}; + \node at (4.5,0.5) {$5$}; + + \node at (-0.5,-0.5) {$1$}; + \node at (-0.5,-1.5) {$2$}; + \node at (-0.5,-2.5) {$3$}; + \node at (-0.5,-3.5) {$4$}; + \node at (-0.5,-4.5) {$5$}; +} + +\end{tikzpicture} +\end{center} + +\uncover<40>{ + \vspace{-22pt} + \begin{block}{Lösung} + Der kürzeste Weg von 2 nach 4 ist 2---3---1---5---4 + \end{block} +} + +\end{column} +\end{columns} + +\egroup diff --git a/vorlesungen/slides/8/floyd-warshall/iteration.tex b/vorlesungen/slides/8/floyd-warshall/iteration.tex new file mode 100644 index 0000000..d7e782d --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/iteration.tex @@ -0,0 +1,14 @@ +% +% iteration.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\newboolean{wegweiser} +\begin{frame}[fragile] +\frametitle{Floyd-Warshall: Iteration} +\setboolean{wegweiser}{false} +\input{../slides/8/floyd-warshall/fw.tex} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/floyd-warshall/macdonalds.png b/vorlesungen/slides/8/floyd-warshall/macdonalds.png Binary files differnew file mode 100644 index 0000000..c442dfb --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/macdonalds.png diff --git a/vorlesungen/slides/8/floyd-warshall/problem.tex b/vorlesungen/slides/8/floyd-warshall/problem.tex new file mode 100644 index 0000000..93f8229 --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/problem.tex @@ -0,0 +1,146 @@ +% +% graph.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[fragile] +\frametitle{Problem: Kürzeste Wege} +\begin{center} +\begin{tikzpicture}[>=latex] + +\def\blob#1#2#3{ + \fill[color=#3] #1 circle[radius=0.2]; + \draw[line width=0.7pt] #1 circle[radius=0.2]; + \node at #1 {{\tiny #2}}; +} + +\def\kante#1#2{ + \draw[line width=0.7pt,shorten >= 0.2,shorten >= 0.2] #1 -- #2 ; +} + +\def\a{72} +\def\r{1.0} +\def\R{2.0} +\def\RR{3.5} + +\coordinate (A1) at ({\r*cos(0*\a)},{\r*sin(0*\a)}); +\coordinate (B1) at ({\r*cos(1*\a)},{\r*sin(1*\a)}); +\coordinate (C1) at ({\r*cos(2*\a)},{\r*sin(2*\a)}); +\coordinate (D1) at ({\r*cos(3*\a)},{\r*sin(3*\a)}); +\coordinate (E1) at ({\r*cos(4*\a)},{\r*sin(4*\a)}); + +\coordinate (F1) at ({\R*cos(0*\a)},{\R*sin(0*\a)}); +\coordinate (G1) at ({\R*cos(1*\a)},{\R*sin(1*\a)}); +\coordinate (H1) at ({\R*cos(2*\a)},{\R*sin(2*\a)}); +\coordinate (I1) at ({\R*cos(3*\a)},{\R*sin(3*\a)}); +\coordinate (J1) at ({\R*cos(4*\a)},{\R*sin(4*\a)}); + +\coordinate (K1) at ({\RR*cos(0.5*\a)},{\RR*sin(0.5*\a)}); +\coordinate (L1) at ({\RR*cos(1.5*\a)},{\RR*sin(1.5*\a)}); +\coordinate (M1) at ({\RR*cos(2.5*\a)},{\RR*sin(2.5*\a)}); +\coordinate (N1) at ({\RR*cos(3.5*\a)},{\RR*sin(3.5*\a)}); +\coordinate (O1) at ({\RR*cos(4.5*\a)},{\RR*sin(4.5*\a)}); + +\kante{(A1)}{(C1)} +\kante{(C1)}{(E1)} +\kante{(E1)}{(B1)} +\kante{(B1)}{(D1)} +\kante{(D1)}{(A1)} + +\kante{(F1)}{(G1)} +\kante{(G1)}{(H1)} +\kante{(H1)}{(I1)} +\kante{(I1)}{(J1)} +\kante{(J1)}{(F1)} + +\kante{(A1)}{(F1)} +\kante{(B1)}{(G1)} +\kante{(C1)}{(H1)} +\kante{(D1)}{(I1)} +\kante{(E1)}{(J1)} + +\kante{(K1)}{(L1)} +\kante{(L1)}{(M1)} +\kante{(M1)}{(N1)} +\kante{(N1)}{(O1)} +\kante{(O1)}{(K1)} + +\kante{(F1)}{(K1)} +\kante{(G1)}{(L1)} +\kante{(H1)}{(M1)} +\kante{(I1)}{(N1)} +\kante{(J1)}{(O1)} + +\kante{(F1)}{(O1)} +\kante{(G1)}{(K1)} +\kante{(H1)}{(L1)} +\kante{(I1)}{(M1)} +\kante{(J1)}{(N1)} + +\uncover<2>{ + \draw[line width=2pt,color=red] (M1)--(H1)--(G1)--(B1); +} + +\uncover<3>{ + \draw[line width=2pt,color=red] (M1)--(L1)--(G1)--(B1); +} + +\uncover<4>{ + \draw[line width=2pt,color=red] (M1)--(I1)--(D1)--(B1); +} + +\uncover<5>{ + \draw[line width=2pt,color=red] (M1)--(I1)--(D1)--(A1)--(F1); +} + +\uncover<6->{ + \draw[line width=2pt,color=red] (M1)--(I1)--(J1)--(F1); +} + +\uncover<2-4>{ + \blob{(B1)}{1}{red!20} + \blob{(M1)}{12}{red!20} +} +\uncover<5-8>{ + \blob{(M1)}{1}{red!20} + \blob{(F1)}{12}{red!20} +} + +\blob{(A1)}{0}{white} +\uncover<1>{ + \blob{(B1)}{1}{white} +} +\uncover<5-8>{ + \blob{(B1)}{12}{white} +} +\blob{(C1)}{2}{white} +\blob{(D1)}{3}{white} +\blob{(E1)}{4}{white} +\uncover<1-4>{ + \blob{(F1)}{5}{white} +} +\blob{(G1)}{6}{white} +\blob{(H1)}{7}{white} +\blob{(I1)}{8}{white} +\blob{(J1)}{9}{white} +\blob{(K1)}{10}{white} +\blob{(L1)}{11}{white} +\uncover<1>{ + \blob{(M1)}{12}{white} +} +\blob{(N1)}{13}{white} +\blob{(O1)}{14}{white} + +\node at (6,0) {\begin{minipage}{5cm} +\begin{itemize} +\item<3-> Nicht eindeutig +\item<5-> geradeste Wege sind nicht unbedingt die kürzesten +\item<7-> Gewichtung der Kanten +($\text{Schnellstrassen}\ne\text{Feldwege}$) +\item<8-> Orientierung der Kanten? +\end{itemize} +\end{minipage}}; + +\end{tikzpicture} +\end{center} +\end{frame} diff --git a/vorlesungen/slides/8/floyd-warshall/rekursion.tex b/vorlesungen/slides/8/floyd-warshall/rekursion.tex new file mode 100644 index 0000000..c664e41 --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/rekursion.tex @@ -0,0 +1,108 @@ +% +% rekursion.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[fragile] +\frametitle{Rekursion} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex] + +\def\blob#1#2#3{ + \fill[color=#3] #1 circle[radius=0.3]; + \draw[line width=0.7pt] #1 circle[radius=0.3]; + \node at #1 {{#2}}; +} + +\def\kante#1#2{ + \draw[line width=0.7pt,shorten >= 0.3,shorten >= 0.3] #1 -- #2 ; +} + +\coordinate (A) at (0,0); +\coordinate (B1) at (2,2); +\coordinate (B2) at (2,1); +\coordinate (B3) at (2,0); +\coordinate (B4) at (2,-1); +\coordinate (B5) at (2,-2); + +\draw[line width=1.9pt,color=gray] (A)--(B1); +\draw[line width=1.9pt,color=gray] (A)--(B2); +\draw[line width=1.9pt,color=gray] (A)--(B3); +\draw[line width=1.9pt,color=gray] (A)--(B4); +\draw[line width=1.9pt,color=gray] (A)--(B5); + +\coordinate (Z) at (10,0); + +\begin{scope} +\clip (2,-2.3) rectangle (10,2.3); +\foreach \y in{-10,...,10}{ + \draw[line width=1.9pt,color=gray] + (2,\y)--(10,{\y-8}); + \draw[line width=1.9pt,color=gray] + (2,\y)--(10,{\y+8}); +} +\end{scope} + +\uncover<2>{ +\draw[line width=4pt,color=red] (A)--(B1)--(5,-1)--(8,2)--(Z); +} + +\uncover<3>{ +\draw[line width=4pt,color=red] (A)--(B2)--(3,0)--(4,1)--(5,0)--(6,1)--(8.5,-1.5)--(Z); +} + +\uncover<4>{ +\draw[line width=4pt,color=red] (A)--(B3)--(2.5,0.5)--(3.5,-0.5)--(5,1.0)--(7,-1)--(9,1)--(Z); +} + +\uncover<5>{ +\draw[line width=4pt,color=red] (A)--(B4)--(3,0)--(4,1)--(5,0)--(6,1)--(7,0) + --(6.0,-1.0)--(7,-2)--(7.5,-1.5)--(7,-1)--(7.5,-0.5) + --(8.5,-1.5)--(Z); +} + +\uncover<6->{ + \draw[line width=4pt,color=red] (A)--(B5)--(6,2); +} +\uncover<7->{ + \draw[line width=4pt,color=red] (6,2)--(7,1)--(5,-1); +} +\uncover<8->{ + \draw[line width=4pt,color=red] (5,-1)--(6,-2)--(8,0)--(9,-1); +} +\uncover<9->{ + \draw[line width=4pt,color=red] (9,-1)--(Z); +} + +\blob{(A)}{$A$}{red!20} +\blob{(B1)}{$B_1$}{white} +\blob{(B2)}{$B_2$}{white} +\blob{(B3)}{$B_3$}{white} +\blob{(B4)}{$B_4$}{white} +\blob{(B5)}{$B_5$}{white} + +\blob{(Z)}{$Z$}{red!20} + +\uncover<6->{ + \node at (6,2) {\includegraphics[width=1.5cm]{../slides/8/floyd-warshall/macdonalds.png}}; +} + +\uncover<7->{ + \node at (5,-1) {\includegraphics[width=1.5cm]{../slides/8/floyd-warshall/starbucks.png}}; +} + +\uncover<8->{ + \node at (9,-1) {\includegraphics[width=2cm]{../slides/8/floyd-warshall/burgerking.png}}; +} + +\end{tikzpicture} +\end{center} + +\begin{block}{Abstieg} +Für den kürzesten Weg von $A$ nach $Z$ suche denjenigen Nachbarn $B_i$ +von $A$, der den kürzesten Weg von $B_i$ nach $Z$ hat. +\uncover<7->{$\Rightarrow$ wir brauchen {\color{red}alle} kürzesten Wege!} +\end{block} + +\end{frame} diff --git a/vorlesungen/slides/8/floyd-warshall/starbucks.png b/vorlesungen/slides/8/floyd-warshall/starbucks.png Binary files differnew file mode 100644 index 0000000..a28dbf7 --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/starbucks.png diff --git a/vorlesungen/slides/8/floyd-warshall/wege.tex b/vorlesungen/slides/8/floyd-warshall/wege.tex new file mode 100644 index 0000000..7ff62a1 --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/wege.tex @@ -0,0 +1,26 @@ +% +% wege.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\frametitle{Wege statt Weglänge?} +\begin{columns}[t] +\begin{column}{0.48\hsize} +\begin{block}{Wege speichern?} +\uncover<3->{Es reicht, einen Wegweiser zum nächsten Knoten zu speichern} +\end{block} +\begin{center} +\begin{tikzpicture}[>=latex] + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\hsize} +\uncover<2->{% +\begin{center} +\includegraphics[width=\hsize]{../slides/8/floyd-warshall/wegweiser.jpg} +\end{center}}% +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/8/floyd-warshall/wegiteration.tex b/vorlesungen/slides/8/floyd-warshall/wegiteration.tex new file mode 100644 index 0000000..84ec679 --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/wegiteration.tex @@ -0,0 +1,13 @@ +% +% wegiteration.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\newboolean{wegweiser} +\begin{frame}[fragile] +\frametitle{Floyd-Warshall: Wegweiser} +\setboolean{wegweiser}{true} +\input{../slides/8/floyd-warshall/fw.tex} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg Binary files differnew file mode 100644 index 0000000..33aebe3 --- /dev/null +++ b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg diff --git a/vorlesungen/slides/8/fourier.tex b/vorlesungen/slides/8/fourier.tex new file mode 100644 index 0000000..86d8086 --- /dev/null +++ b/vorlesungen/slides/8/fourier.tex @@ -0,0 +1,83 @@ +% +% fourier.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Fourier-Transformation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Algebra} +Die Laplace-Matrix eines Graphen ist symmetrisch +\uncover<2->{% + +$\Rightarrow$ +Es gibt eine Basis aus Eigenvektoren $g_i\in\mathbb{R}^n$ von $L(G)$: +\begin{align*} +L(G)g_i&=\lambda_i g_i +\end{align*}} +\end{block} +\uncover<12->{% +\vspace{-20pt} +\begin{block}{Fourier-Transformation} +Jedes $f\in\mathbb{R}^n$ kann durch die $g_i$ ausgedrückt werden +\begin{align*} +\uncover<13->{ +f&= a_1 g_1 + \dots + a_n g_n +} +\\ +\uncover<14->{ +&= \hat{f}_1 g_1 + \dots + \hat{f}_ng_n = \sum_{k=1}^n \hat{f}_kg_k +} +\end{align*} +\uncover<15->{% +Zerlegung nach Zeitkonstante $\lambda_i$ +} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Anwendung} +Wärmeleitungsgleichung +\begin{align*} +\uncover<4->{ +\frac{d}{dt}f &= L(G) f +} +\intertext{\uncover<5->{{\usebeamercolor[fg]{title}Ansatz:}}} +\uncover<6->{ +f&=a_1g_1T_1(t)+\dots + a_ng_nT_n(t) +} +\\ +\uncover<7->{ +\frac{d}{dt}f +&= +a_1g_1\dot{T}_1(t) + \dots + a_1g_1 \dot{T}_n(t) +} +\\ +\uncover<8->{ +&= +a_1Lg_1 + \dots + a_nLg_n +} +\\ +\uncover<9->{ +&= +a_1\lambda_1 g_1 + \dots + a_n\lambda_n g_n +} +\\ +\uncover<10->{ +\dot{T}_i(t) &= \lambda_i T_i(t) +} +\uncover<11->{ +\quad +\Rightarrow +\quad +T_i(t) = e^{\lambda_it} \uncover<-9>{T_i(0)} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/8/grad.tex b/vorlesungen/slides/8/grad.tex new file mode 100644 index 0000000..a232828 --- /dev/null +++ b/vorlesungen/slides/8/grad.tex @@ -0,0 +1,84 @@ +% +% grad.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Grad} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); +\draw[color=white,line width=5pt] (B) -- (D); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); + +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); +%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); + +\draw (A) circle[radius=0.2]; +\draw (B) circle[radius=0.2]; +\draw (C) circle[radius=0.2]; +\draw (D) circle[radius=0.2]; +\draw (E) circle[radius=0.2]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; +\node at (0,0) {$G$}; + +%\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$}; +%\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$}; +%\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$}; +%\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$}; +%\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$}; +%\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$}; +%\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; + +\end{tikzpicture} +\end{center} +\begin{block}{Definition} +Der Grad +$\deg v$ +eines Knotens $v\in V$ ist die Anzahl der Kanten mit Ende in $v$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Gradmatrix} +Diagonalmatrix mit $d_{ii}=\deg v_i$ +\[ +D(G) += +\begin{pmatrix} +3&0&0&0&0\\ +0&3&0&0&0\\ +0&0&3&0&0\\ +0&0&0&2&0\\ +0&0&0&0&1 +\end{pmatrix} +\] +\end{block} +\begin{block}{Satz} +Die Summe der Grade ist gerade: +\[ +\sum_{i=1}^n\deg v_i = \operatorname{Spur} D(G) \equiv 0 \mod 2 +\] +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/8/graph.tex b/vorlesungen/slides/8/graph.tex new file mode 100644 index 0000000..32150af --- /dev/null +++ b/vorlesungen/slides/8/graph.tex @@ -0,0 +1,117 @@ +% +% graph.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Graph} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{2.4} + +\begin{scope} +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\uncover<3->{ + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); + \draw[color=white,line width=5pt] (B) -- (D); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); + + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); +} + +\uncover<2->{ + \draw (A) circle[radius=0.2]; + \draw (B) circle[radius=0.2]; + \draw (C) circle[radius=0.2]; + \draw (D) circle[radius=0.2]; + \draw (E) circle[radius=0.2]; + + \node at (A) {$1$}; + \node at (B) {$2$}; + \node at (C) {$3$}; + \node at (D) {$4$}; + \node at (E) {$5$}; +} +\node at (0,0) {$G$}; + +\uncover<3->{ + \node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) + [above right] {$\scriptstyle 1$}; + \node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) + [above left] {$\scriptstyle 2$}; + \node at ($0.5*(C)+0.5*(D)+(0.05,0)$) + [left] {$\scriptstyle 3$}; + \node at ($0.5*(D)+0.5*(E)$) + [below] {$\scriptstyle 4$}; + \node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) + [below right] {$\scriptstyle 5$}; + \node at ($0.6*(A)+0.4*(C)$) + [above] {$\scriptstyle 6$}; + \node at ($0.4*(B)+0.6*(D)$) + [left] {$\scriptstyle 7$}; +} + +\uncover<8->{ + \draw[shorten >= 0.2cm,shorten <= 0.2cm] + (E) to[out=-18,in=-126,distance=2cm] (E); + + \draw[color=red,line width=4pt] ($(E)+(-0.5,-0.5)+(0,-0.5)$) + -- ($(E)+(0.5,0.5)+(0,-0.5)$); + \draw[color=red,line width=4pt] ($(E)+(-0.5,0.5)+(0,-0.5)$) + -- ($(E)+(0.5,-0.5)+(0,-0.5)$); +} + +\end{scope} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Ein Graph $G=(V,E)$ ist +\begin{enumerate} +\item<2-> +Eine Menge $V$ von Knoten (Vertizes): +$V=\{v_1,v_2,\dots\}$ +\item<3-> +Eine Menge $E$ von Kanten (Edges): +\[ +E\subset +\left\{ e = \{v_1,v_2\}\;\left|\; \begin{minipage}{1.3cm}\raggedright +$v_i\in V$\\ +$v_1\ne v_2$ +\end{minipage} +\right. +\right\} +\] +\end{enumerate} +\end{block} +\vspace{-20pt} +\uncover<5->{% +\begin{block}{Achtung:} +\begin{itemize} +\item<6-> Kanten sind Mengen +\uncover<7->{$\Rightarrow$ zwei verschiedene Knoten} +\uncover<8->{$\Rightarrow$ Keine Schleifen} +\item<9-> Kanten sind ungerichtet, keine ``Einbahnstrassen'' +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex new file mode 100644 index 0000000..952c85b --- /dev/null +++ b/vorlesungen/slides/8/inzidenz.tex @@ -0,0 +1,150 @@ +% +% inzidenz.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Inzidenz- und Adjazenzmatrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); +\draw[color=white,line width=5pt] (B) -- (D); +{\color<2->{darkgreen} +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); +} + +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); +%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); +\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); + +\only<-2>{ +\fill[color=white] (B) circle[radius=0.2]; +} +\only<3->{ +\fill[color=red!20] (B) circle[radius=0.2]; +} + +\draw (A) circle[radius=0.2]; +\draw (B) circle[radius=0.2]; +\draw (C) circle[radius=0.2]; +\draw (D) circle[radius=0.2]; +\draw (E) circle[radius=0.2]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; +\node at (0,0) {$G$}; + +\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$}; +\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$}; +\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$}; +\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 4$}; +\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 5$}; +{\color<2->{darkgreen} +\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 6$}; +} + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Definition} +\vspace{-15pt} +\begin{align*} +B(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante $j$ endet in Knoten $i$}\\ +A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\dy{0.48} +\def\dx{0.54} + + +\begin{scope} +\uncover<3->{ +\fill[color=red!20] (1.8,1.8) rectangle (4.75,2.15); +} +\uncover<2->{ +\fill[color=darkgreen!40,opacity=0.5] (4.46,0.36) rectangle (4.79,2.65); +} +\foreach \y in {1,...,5}{ + \node[color=gray] at (5.3,{2.45-(\y-1)*\dy}) {\tiny $\y$}; +} +\foreach \y in {1,...,6}{ + \node[color=gray] at ({1.92+(\y-1)*\dx},2.90) {\tiny $\y$}; +} +\draw[color=gray] (1.8,2.75) -- (4.7,2.75); +\draw[color=gray] (5.2,2.55) -- (5.2,0.45); +\node[color=gray] at ({1.92+2.5*\dx},3.1) {\tiny Kanten}; +\node[color=gray] at (5.3,{2.45-2*\dy}) [above,rotate=-90] {\tiny Knoten}; +\end{scope} + +\uncover<4->{ +\begin{scope} +\uncover<3->{ +\fill[color=red!20] (1.8,-1.16) rectangle (4.25,-0.77); +\fill[color=red!20] (2.3,-2.6) rectangle (2.63,-0.29); +} +\foreach \y in {1,...,5}{ + \node[color=gray] at (4.7,{-0.5-(\y-1)*\dy}) {\tiny $\y$}; + \node[color=gray] at ({1.92+(\y-1)*\dx},-0.1) {\tiny $\y$}; +} +\draw[color=gray] (1.8,-0.22) -- (4.2,-0.22); +\draw[color=gray] (4.6,-0.4) -- (4.6,-2.55); +\node[color=gray] at ({1.92+2*\dx},0.1) {\tiny Knoten}; +\node[color=gray] at (4.7,{-0.5-2*\dy}) [above,rotate=-90] {\tiny Knoten}; +\end{scope} +} + +\node (0,0) [right] {$\displaystyle +\begin{aligned} +B(G) +&= +\begin{pmatrix} +1&0&0&1&1&0\\ +1&1&0&0&0&1\\ +0&1&1&0&1&0\\ +0&0&1&0&0&1\\ +0&0&0&1&0&0 +\end{pmatrix} +\\[12pt] +\uncover<4->{ +A(G) +&= +\begin{pmatrix} +0&1&1&0&1\\ +1&0&1&1&0\\ +1&1&0&1&0\\ +0&1&1&0&0\\ +1&0&0&0&0 +\end{pmatrix} +\end{aligned}}$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex new file mode 100644 index 0000000..5f2f51a --- /dev/null +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -0,0 +1,164 @@ +% +% inzidenzd.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Inzidenz- und Adjazenz-Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); +\draw[color=white,line width=5pt] (B) -- (D); +{\color<2->{darkgreen} +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); +} + +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); + +\draw (A) circle[radius=0.2]; +\only<-2>{ +\fill[color=white] (B) circle[radius=0.2]; +} +\only<3->{ +\fill[color=red!20] (B) circle[radius=0.2]; +} +\draw (B) circle[radius=0.2]; +\draw (C) circle[radius=0.2]; +\draw (D) circle[radius=0.2]; +\draw (E) circle[radius=0.2]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; +\node at (0,0) {$G$}; + +\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$}; +\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$}; +\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$}; +\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$}; +\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$}; +\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$}; +{\color<2->{darkgreen} +\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; +} + +\end{tikzpicture} +\end{center} +\vspace{-15pt} +\uncover<5->{% +\begin{block}{Definition} +\vspace{-20pt} +\begin{align*} +B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\ +B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\ +A(G)_{ij}&=\phantom{-}1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\dx{0.84} +\def\dy{0.48} + +\begin{scope}[xshift=4cm,yshift=3cm] +\uncover<3->{ +\fill[color=red!20] +({-0.67-(7-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2}) +rectangle +({-0.67-(7-7)*\dx+0.2},{-0.38-(2-1)*\dy+0.16}); +} +\uncover<2->{ +\fill[color=darkgreen!40,opacity=0.5] +({-0.67-(7-7)*\dx-0.4},{-0.38-(5-1)*\dy-0.2}) +rectangle +({-0.67-(7-7)*\dx+0.2},{-0.38-(1-1)*\dy+0.16}); +} +%\draw (0,0) circle[radius=0.05]; +\foreach \x in {1,...,7}{ + \node[color=gray] at ({-0.67-(7-\x)*\dx},0.0) {\tiny $\x$}; +} +\draw[color=gray] ({-0.72-6*\dx},-0.1) -- (-0.6,-0.1); +\foreach \y in {1,...,5}{ + \node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$}; +} +\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4); +\node[color=gray] at ({-0.67-(7-4)*\dx},0.04) [above] {\tiny Kanten}; +\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy}) + [above,rotate=-90] {\tiny Knoten}; +\end{scope} + +\uncover<4->{ +\begin{scope}[xshift=2.32cm,yshift=-0.24cm] +%\draw (0,0) circle[radius=0.05]; +\fill[color=red!20] +({-0.67-(5-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2}) +rectangle +({-0.67-(5-5)*\dx+0.2},{-0.38-(2-1)*\dy+0.16}); +\fill[color=red!20] +({-0.67-(5-2)*\dx-0.4},{-0.38-(5-1)*\dy-0.2}) +rectangle +({-0.67-(5-2)*\dx+0.2},{-0.38-(1-1)*\dy+0.16}); +\foreach \x in {1,...,5}{ + \node[color=gray] at ({-0.67-(5-\x)*\dx},0.0) {\tiny $\x$}; +} +\draw[color=gray] ({-0.72-4*\dx},-0.1) -- (-0.6,-0.1); +\foreach \y in {1,...,5}{ + \node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$}; +} +\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4); +\node[color=gray] at ({-0.67-(5-3)*\dx},0.04) [above] {\tiny Knoten}; +\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy}) + [above,rotate=-90] {\tiny Knoten}; +\end{scope} +} + +\node at (0,0) {$\displaystyle +\begin{aligned} +B(G) +&= +\begin{pmatrix*}[r] +-1& 0& 0& 0&+1&-1& 0\\ ++1&-1& 0& 0& 0& 0&-1\\ + 0&+1&-1& 0& 0&+1& 0\\ + 0& 0&+1&-1& 0& 0&+1\\ + 0& 0& 0&+1&-1& 0& 0 +\end{pmatrix*} +\\[20pt] +\uncover<4->{ +A(G) +&= +\begin{pmatrix*}[r] + 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\ +\phantom{-}1& 0&\phantom{-}1&\phantom{-}1& 0\\ +\phantom{-}1&\phantom{-}1& 0&\phantom{-}1& 0\\ + 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\ +\phantom{-}1& 0& 0&\phantom{-}1& 0 +\end{pmatrix*}} +\end{aligned}$}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/8/laplace.tex b/vorlesungen/slides/8/laplace.tex new file mode 100644 index 0000000..a1c364d --- /dev/null +++ b/vorlesungen/slides/8/laplace.tex @@ -0,0 +1,213 @@ +% +% laplace.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Laplace-Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C); +\draw[color=white,line width=5pt] (B) -- (D); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D); + +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A); + +\uncover<2-4>{ +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (A) -- (B); +} + +\uncover<3-7>{ +\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm] + (A) -- (C); +} + +\uncover<4-13>{ +\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm] + (A) -- (E); +} + +\uncover<5->{ +\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (A) -- (B); +} + +\uncover<6-8>{ +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (B) -- (C); +} + +\uncover<7-10>{ +\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm] + (B) -- (D); +} + +\uncover<8->{ +\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm] + (A) -- (C); +} + +\uncover<9->{ +\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (B) -- (C); +} + +\uncover<10-11>{ +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (C) -- (D); +} + +\uncover<11->{ +\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm] + (B) -- (D); +} + +\uncover<12->{ +\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (C) -- (D); +} + +\uncover<13-14>{ +\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (D) -- (E); +} + +\uncover<14->{ +\draw[<->,color=darkgreen,line width=8pt,shorten <= 0cm,shorten >= 0cm] + (A) -- (E); +} + +\uncover<15->{ +\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm] + (D) -- (E); +} + +\fill[color=red] (A) circle[radius=0.3]; +\fill[color=red!50] (B) circle[radius=0.3]; +\fill[color=white] (C) circle[radius=0.3]; +\fill[color=blue!50] (D) circle[radius=0.3]; +\fill[color=blue] (E) circle[radius=0.3]; + +\draw (A) circle[radius=0.3]; +\draw (B) circle[radius=0.3]; +\draw (C) circle[radius=0.3]; +\draw (D) circle[radius=0.3]; +\draw (E) circle[radius=0.3]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; + +\end{tikzpicture} +\end{center} +\uncover<16->{% +\begin{block}{Definition} +Laplace-Matrix +\[ +L(G) = D(G) - A(G) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{align*} +f +&= +\begin{pmatrix} +f(1)\\ +f(2)\\ +f(3)\\ +f(4)\\ +f(5) +\end{pmatrix} +\\ +\frac{df}{dt} +&= +-\kappa +\begin{pmatrix*}[r] +\only<1>{\phantom{-0}} + \only<2>{\phantom{-}1} + \only<3>{\phantom{-}2} + \only<4->{\phantom{-}3} + &\only<1>{\phantom{-0}}\only<2->{{-1}}% + &\only<-2>{\phantom{-0}}\only<3->{{-1}}% + &\uncover<16->{ 0} + &\only<-3>{\phantom{-0}}\only<4->{{-1}}\\ +\only<-4>{\phantom{-0}}\only<5->{{-1}} + &\only<-4>{\phantom{-0}} + \only<5>{\phantom{-}1} + \only<6>{\phantom{-}2} + \only<7->{\phantom{-}3} + &\only<-5>{\phantom{-0}}\only<6->{{-1}} + &\only<-6>{\phantom{-0}}\only<7->{{-1}} + &\uncover<16->{ 0}\\ +\only<-7>{\phantom{-0}}\only<8->{{-1}} + &\only<-8>{\phantom{-0}}\only<9->{{-1}} + &\only<-7>{\phantom{-0}} + \only<8>{\phantom{-}1} + \only<9>{\phantom{-}2} + \only<10->{\phantom{-}3} + &\only<-9>{\phantom{-0}}\only<10->{{-1}} + &\uncover<16->{ 0}\\ +\uncover<16->{ 0} + &\only<-10>{\phantom{-0}}\only<11->{{-1}} + &\only<-11>{\phantom{-0}}\only<12->{{-1}} + &\only<-10>{\phantom{-0}} + \only<11>{\phantom{-}1} + \only<12>{\phantom{-}2} + \only<13->{\phantom{-}3} + &\only<-12>{\phantom{-0}}\only<13->{{-1}}\\ +\only<-13>{\phantom{-0}}\only<14->{{-1}} + &\uncover<16->{ 0} + &\uncover<16->{ 0} + &\only<-14>{\phantom{-0}}\only<15->{{-1}} + &\only<-13>{\phantom{-0}} + \only<14>{\phantom{-}1} + \only<15->{\phantom{-}2} +\end{pmatrix*} +\begin{pmatrix} +f(1)\\ +f(2)\\ +f(3)\\ +f(4)\\ +f(5) +\end{pmatrix} +\\ +\uncover<17->{ +&= +-\kappa L f} +\end{align*} +\vspace{-20pt} +\uncover<18->{% +\begin{block}{Rekonstruktion} +Der Graph lässt sich aus $L$ rekonstruieren +\end{block}} +\end{column} +\end{columns} +\end{frame} + +\egroup + + diff --git a/vorlesungen/slides/8/pfade/adjazenz.tex b/vorlesungen/slides/8/pfade/adjazenz.tex new file mode 100644 index 0000000..f923262 --- /dev/null +++ b/vorlesungen/slides/8/pfade/adjazenz.tex @@ -0,0 +1,97 @@ +% +% adjazenz.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\definecolor{darkred}{rgb}{0.5,0,0} +\begin{frame}[fragile] +\newboolean{pfeilspitzen} +\setboolean{pfeilspitzen}{true} +\frametitle{Adjazenz-Matrix} + +\begin{columns}[t] +\begin{column}{0.48\hsize} +\begin{center} +\begin{tikzpicture}[>=latex] + +\def\r{2.2} +\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)}); +\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)}); +\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)}); +\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)}); +\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)}); + +\def\knoten#1#2{ + \fill[color=white] #1 circle[radius=0.3]; + \draw[line width=1pt] #1 circle[radius=0.3]; + \node at #1 {$#2$}; +} + +\def\kante#1#2#3{ + \ifthenelse{\boolean{pfeilspitzen}}{ + \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] + #1 -- #2; + }{ + \draw[line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] + #1 -- #2; + } +% \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22]; +% \node at ($0.5*#1+0.5*#2$) {$#3$}; +} + +\kante{(A)}{(E)}{1} +\kante{(B)}{(C)}{2} +\kante{(B)}{(D)}{13} +\kante{(C)}{(A)}{3} +\kante{(D)}{(C)}{6} +\kante{(E)}{(B)}{5} +\kante{(E)}{(D)}{6} + +\knoten{(A)}{1} +\knoten{(B)}{2} +\knoten{(C)}{3} +\knoten{(D)}{4} +\knoten{(E)}{5} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\hsize} +\[ +a_{{\color{darkred}i}{\color{blue}j}} += +\begin{cases} +1&\quad\text{\# Kanten von ${\color{blue}j}$ nach ${\color{darkred}i}$}\\ +0&\quad\text{sonst} +\end{cases} +\] +\begin{center} +\begin{tikzpicture}[>=latex] +\node at (0,0) {$\displaystyle +A= +\begin{pmatrix} +0&0&1&0&0\\ +0&0&0&0&1\\ +0&1&0&1&0\\ +0&1&0&0&1\\ +1&0&0&0&0 +\end{pmatrix} +$}; +\def\s{0.54} +\foreach \x in {1,...,5}{ + \node[color=blue] at ({-0.71+(\x-1)*\s},1.4) {\tiny $\x$}; +} +\node[color=blue] at ({-0.71+2*\s},1.7) {von}; +\def\r{0.48} +\foreach \y in {1,...,5}{ + \node[color=darkred] at ({-0.71+5*\s},{0.02+(3-\y)*\r}) {\tiny $\y$}; +} +\node[color=darkred] at ({-0.4+5*\s},{0.02}) [rotate=90] {nach}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/pfade/beispiel.tex b/vorlesungen/slides/8/pfade/beispiel.tex new file mode 100644 index 0000000..43685f3 --- /dev/null +++ b/vorlesungen/slides/8/pfade/beispiel.tex @@ -0,0 +1,404 @@ +% +% beispiel.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\newboolean{pfeilspitzen} + +\def\knoten#1#2{ + \fill[color=white] #1 circle[radius=0.3]; + \draw[line width=1pt] #1 circle[radius=0.3]; + \node at #1 {$#2$}; +} + +\def\kante#1#2#3{ + \ifthenelse{\boolean{pfeilspitzen}}{ + \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] + #1 -- #2; + }{ + \draw[line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] + #1 -- #2; + } +% \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22]; +% \node at ($0.5*#1+0.5*#2$) {$#3$}; +} + +\begin{frame} +\setboolean{pfeilspitzen}{true} +\frametitle{Beispiel} +\begin{columns}[t] +\begin{column}{0.37\hsize} +\begin{center} +\begin{tikzpicture}[>=latex] + +\def\r{2.2} +\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)}); +\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)}); +\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)}); +\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)}); +\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)}); + +\kante{(A)}{(E)}{1} +\kante{(B)}{(C)}{2} +\kante{(B)}{(D)}{13} +\kante{(C)}{(A)}{3} +\kante{(D)}{(C)}{6} +\kante{(E)}{(B)}{5} +\kante{(E)}{(D)}{6} + +\knoten{(A)}{1} +\knoten{(B)}{2} +\knoten{(C)}{3} +\knoten{(D)}{4} +\knoten{(E)}{5} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.59\hsize} + +\only<1>{ +\begin{block}{Pfade der Länge 1} +\[ +A= +\begin{pmatrix} +0&0&1&0&0\\ +0&0&0&0&1\\ +0&1&0&1&0\\ +0&1&0&0&1\\ +1&0&0&0&0 +\end{pmatrix} +\] +\end{block} +} + +\only<2>{ +\begin{block}{Pfade der Länge 2} +\[ +A^2=\begin{pmatrix} + 0 & 1 & 0 & 1 & 0 \\ + 1 & 0 & 0 & 0 & 0 \\ + 0 & 1 & 0 & 0 & 2 \\ + 1 & 0 & 0 & 0 & 1 \\ + 0 & 0 & 1 & 0 & 0 +\end{pmatrix} +\] +\end{block} +} + +\only<3>{ +\begin{block}{Pfade der Länge 3} +\[ +A^3=\begin{pmatrix} + 0 & 1 & 0 & 0 & 2 \\ + 0 & 0 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 0 \\ + 2 & 0 & 0 & 0 & 1 \\ + 1 & 0 & 1 & 0 & 0 \\ + 0 & 1 & 0 & 1 & 0 +\end{pmatrix} +\] +\end{block} +} + +\only<4>{ +\begin{block}{Pfade der Länge 4} +\[ +A^4=\begin{pmatrix} + 2% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 0 & 0 & 1 \\ + 0 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 1 & 0 \\ + 1 & 0 & 2% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 0 \\ + 0 & 1 & 1 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 \\ + 0 & 1 & 0 & 0 & 2% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +\end{pmatrix} +\] +\end{block} +} + +\only<5>{ +\begin{block}{Pfade der Länge 5} +\[ +A^5=\begin{pmatrix} + 1 & 0 & 2 & 0 & 0 \\ + 0 & 1 & 0 & 0 & 2 \\ + 0 & 2 & 1 & 2 & 0 \\ + 0 & 2 & 0 & 1 & 2 \\ + 2 & 0 & 0 & 0 & 1 +\end{pmatrix} +\] +\end{block} +} + +\only<6>{ +\begin{block}{Pfade der Länge 6} +\[ +A^6=\begin{pmatrix} + 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 2 & 1 & 2 & 0 \\ + 2 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 0 & 1 \\ + 0 & 3 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 1 & 4 \\ + 2 & 1 & 0 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 3 \\ + 1 & 0 & 2 & 0 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +\end{pmatrix} +\] +\end{block} +} + +\only<7>{ +\begin{block}{Pfade der Länge 7} +\[ +A^7=\begin{pmatrix} + 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 3 & 0 & 1 & 4 \\ + 1 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 2 & 0 & 0 \\ + 4 & 1 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 4 \\ + 3 & 0 & 2 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 1 \\ + 0 & 2 & 1 & 2 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +\end{pmatrix} +\] +\end{block} +} + +\only<8>{ +\begin{block}{Pfade der Länge 8} +\[ +A^8=\begin{pmatrix} + 4 & 1 & 0 & 0 & 4 \\ + 0 & 2 & 1 & 2 & 0 \\ + 4 & 0 & 4 & 0 & 1 \\ + 1 & 2 & 3 & 2 & 0 \\ + 0 & 3 & 0 & 1 & 4 +\end{pmatrix} +\] +\end{block} +} + +\only<9>{ +\begin{block}{Pfade der Länge 9} +\[ +A^9=\begin{pmatrix} + 4 & 0 & 4 & 0 & 1 \\ + 0 & 3 & 0 & 1 & 4 \\ + 1 & 4 & 4 & 4 & 0 \\ + 0 & 5 & 1 & 3 & 4 \\ + 4 & 1 & 0 & 0 & 4 +\end{pmatrix} +\] +\end{block} +} + +\only<10>{ +\begin{block}{Pfade der Länge 10} +\[ +A^{10}=\begin{pmatrix} + 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 4 & 4 & 4 & 0 \\ + 4 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 0 & 0 & 4 \\ + 0 & 8 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 4 & 8 \\ + 4 & 4 & 0 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 8 \\ + 4 & 0 & 4 & 0 & 1% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +\end{pmatrix} +\] +\end{block} +} + +\only<11>{ +\begin{block}{Pfade der Länge 15} +\[ +A^{15}=\begin{pmatrix} + 1 & 20 & 6 & 12 & 16 \\ + 12 & 1 & 8 & 0% +\begin{picture}(0,0) +\color{red}\put(-3,4){\circle{12}} +\end{picture}% +& 6 \\ + 16 & 18 & 1 & 6 & 32 \\ + 20 & 6 & 8 & 1 & 18 \\ + 6 & 8 & 12 & 8 & 1 +\end{pmatrix} +\] +\end{block} +} + +\only<12>{ +\begin{block}{Pfade der Länge 20} +\[ +A^{20}=\begin{pmatrix} + 33 & 56 & 8 & 24 & 80 \\ + 24 & 17 & 32 & 16 & 8 \\ + 80 & 32 & 33 & 8 & 80 \\ + 56 & 24 & 48 & 17 & 32 \\ + 8 & 48 & 24 & 32 & 33 +\end{pmatrix} +\] +\end{block} +} + +\only<13>{ +\begin{block}{Pfade der Länge 25} +\[ +A^{25}=\begin{pmatrix} + 193 & 120 & 74 & 40 & 240 \\ + 40 & 113 & 80 & 80 & 74 \\ + 240 & 114 & 193 & 74 & 160 \\ + 120 & 154 & 160 & 113 & 114 \\ + 74 & 160 & 40 & 80 & 193 +\end{pmatrix} +\] +\end{block} +} + +\only<14>{ +\begin{block}{Pfade der Länge 30} +\[ +A^{30}=\begin{pmatrix} + 673 & 348 & 460 & 188 & 560 \\ + 188 & 433 & 160 & 240 & 460 \\ + 560 & 648 & 673 & 460 & 536 \\ + 348 & 700 & 400 & 433 & 648 \\ + 460 & 400 & 188 & 160 & 673 +\end{pmatrix} +\] +\end{block} +} + +\only<15>{ +\begin{block}{Pfade der Länge 35} +\[ +A^{35}=\begin{pmatrix} + 1793% +\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3) +& 1644 & 1806 & 1108 & 1632 \\ + 1108 & 1233 & 536 & 560 & 1806 \\ + 1632 & 2914 & 1793% +\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3) +& 1806 & 2752 \\ + 1644 & 2366 & 1096 & 1233 & 2914 \\ + 1806 & 1096 & 1108 & 536 & 1793% +\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3) +\end{pmatrix} +\] +\end{block} +} + +\end{column} +\end{columns} +\vbox to2cm{ +\vfill +\only<3>{ + \begin{block}{Kürzester Verbindung von 3 nach 2} + Der Weg 3---1---6---2 ist die kürzeste Verbindung von 3 nach 2 + \end{block} +} +\only<4>{ + \begin{block}{Kürzeste Zyklen} + Jeder Knoten liegt auf einem Zyklus der Länge 4, + dies sind die kürzesten Zyklen. + 1, 3 und 5 liegen auf beiden Zyklen, 2 und 4 nur auf einem. + \end{block} +} +\only<6>{ + \begin{block}{Zyklen der Länge 6} + {\em Keine} Zyklen der Länge 6 + \end{block} +} +\only<7>{ + \begin{block}{Zyklen der Länge 7} + {\em Keine} Zyklen der Länge 7 + \end{block} +} +\only<10>{ + \begin{block}{Zyklen der Länge 10} + Genau ein Zyklus der Länge 10 + \end{block} +} +\only<11>{ + \begin{block}{Verbindung von 4 nach 2} + {\em Keine} Verbindung der Länge 15 von 4 nach 2 + \end{block} +} +\only<15>{ + \begin{block}{Zyklen der Länge 35} + Es gibt 1793 Zyklen, die 1 enthalten, und sie enthalten alle auch 3 und 5 + \end{block} +} +} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/pfade/gf.tex b/vorlesungen/slides/8/pfade/gf.tex new file mode 100644 index 0000000..e89a1fb --- /dev/null +++ b/vorlesungen/slides/8/pfade/gf.tex @@ -0,0 +1,54 @@ +% +% gf.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\definecolor{darkred}{rgb}{0.8,0,0} +\frametitle{Erzeugende Funktion} +Alle Weglängen zusammen: +\[ +\uncover<7->{f({\color{darkred}t})=} +\uncover<4->{E+} +A +\uncover<3->{{\color{darkred}t}} +\uncover<2->{+} +\uncover<5->{\frac{1}{2!}} +A^2 +\uncover<3->{{\color{darkred}t^2}} +\uncover<2->{+} +\uncover<5->{\frac{1}{3!}} +A^3 +\uncover<3->{{\color{darkred}t^3}} +\uncover<2->{+} +\uncover<5->{\frac{1}{4!}} +A^4 +\uncover<3->{{\color{darkred}t^4}} +\uncover<2->{+} +\uncover<5->{\frac{1}{5!}} +A^5 +\uncover<3->{{\color{darkred}t^5}} +\uncover<2->{+} +\uncover<5->{\frac{1}{6!}} +A^6 +\uncover<3->{{\color{darkred}t^6}} +\uncover<2->{+} +\uncover<5->{\frac{1}{7!}} +A^7 +\uncover<3->{{\color{darkred}t^7}} +\dots +\uncover<6->{= e^{A{\color{darkred}t}}} +\] +\uncover<4->{% +heisst {\em\usebeamercolor[fg]{title} \only<5->{exponentiell} erzeugende Funktion} +der Wege-Anzahlen} + +\begin{itemize} +\item<8-> +Begriff der Entropie auf einem Graphen +\item<9-> +Wahrscheinlichkeit, dass ein Zufallsspaziergänger auf einem Graphen an +einem bestimmten Knoten vorbeikommt +\end{itemize} + +\end{frame} diff --git a/vorlesungen/slides/8/pfade/langepfade.tex b/vorlesungen/slides/8/pfade/langepfade.tex new file mode 100644 index 0000000..8c0dd0d --- /dev/null +++ b/vorlesungen/slides/8/pfade/langepfade.tex @@ -0,0 +1,59 @@ +% +% langepfade.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\bgroup +\definecolor{darkred}{rgb}{0.5,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame} +\frametitle{Wieviele Pfade der Länge $k$?} +\begin{definition} +Anzahl Pfade der Länge $k$ zwischen zwei Knoten +\[ +a_{{\color{darkred}i}{\color{blue}j}}^{(k)} += +\#\{\text{Pfade der Länge $k$ von $\color{blue}j$ nach $\color{darkred}i$}\}, +\qquad +A^{(k)} += +\left( +a_{{\color{darkred}i}{\color{blue}j}}^{(k)} +\right) +\] +\end{definition} +\uncover<2->{ +{\usebeamercolor[fg]{title}Spezialfall:} $A^{(1)}=A$. +} + +\uncover<3->{ +\begin{block}{Rekursionsformel} +\vspace{-25pt} +\begin{align*} +a_{{\color{darkred}i}{\color{blue}{\color{blue}j}}}^{(k)} +&\uncover<4->{= +\sum_{{\color{darkgreen}l}=1}^n +\#\{\text{Pfade der Länge $1$ von $\color{darkgreen}l$ nach $\color{darkred}i$}\}} +\\[-11pt] +&\uncover<4->{\qquad\qquad\times +\#\{\text{Pfade der Länge $k-1$ von $\color{blue}j$ nach $\color{darkgreen}l$}\}} +\\ +&\uncover<5->{= +\sum_{{\color{darkgreen}l}=1}^n +a_{{\color{darkred}i}{\color{darkgreen}l}}^{(1)} +\cdot +a_{{\color{darkgreen}l}{\color{blue}j}}^{(k-1)}} +\\ +\uncover<6->{ +\Rightarrow\qquad +A^{(k)}} +&\uncover<6->{= +A\;A^{(k-1)}} +\uncover<7->{ +\qquad\Rightarrow\qquad +A^{(k)} = A^k} +\end{align*} +\end{block} } + +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex new file mode 100644 index 0000000..1d8b725 --- /dev/null +++ b/vorlesungen/slides/8/produkt.tex @@ -0,0 +1,100 @@ +% +% produkt.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Inzidenz- und Laplace-Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); +\draw[color=white,line width=5pt] (B) -- (D); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); + +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); + +\draw (A) circle[radius=0.2]; +\draw (B) circle[radius=0.2]; +\draw (C) circle[radius=0.2]; +\draw (D) circle[radius=0.2]; +\draw (E) circle[radius=0.2]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; +\node at (0,0) {$G$}; + +\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$}; +\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$}; +\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$}; +\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$}; +\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$}; +\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$}; +\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; + +\end{tikzpicture} +\end{center} +\vspace{-15pt} +\begin{block}{Berechne} +\vspace{-20pt} +\begin{align*} +\uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t +\end{align*} +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\dx{0.84} +\def\dy{0.48} + +\node at (0,0) {$\displaystyle +\begin{aligned} +B(G) +&= +\begin{pmatrix*}[r] +-1& 0& 0& 0&+1&-1& 0\\ ++1&-1& 0& 0& 0& 0&-1\\ + 0&+1&-1& 0& 0&+1& 0\\ + 0& 0&+1&-1& 0& 0&+1\\ + 0& 0& 0&+1&-1& 0& 0 +\end{pmatrix*} +\\[20pt] +\uncover<2->{ +L(G) +&= +\begin{pmatrix*}[r] + 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{-1}\\ +\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}\\ +\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{ 0}\\ +\uncover<3->{ 0}&\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}\\ +\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{ 0}&\uncover<3->{-1}& 2 +\end{pmatrix*}} +\end{aligned}$}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex new file mode 100644 index 0000000..425fe1c --- /dev/null +++ b/vorlesungen/slides/8/spanningtree.tex @@ -0,0 +1,164 @@ +% +% spanningtree.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\frametitle{Spannbäume} + +\vspace{-16pt} + +\begin{columns}[t] + +\begin{column}{0.40\hsize} +\begin{block}{Netzwerk} +Alle Knoten erreichen, Schleifen vermeiden $\Rightarrow$ Spannbaum +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,scale=0.18] + +\coordinate (A) at ( 1.2927,-15.0076); +\coordinate (B) at ( 5.0261,- 7.7143); +\coordinate (C) at ( 4.9260,-13.0335); +\coordinate (D) at (12.2094,-22.9960); +\coordinate (F) at (17.8334,-13.4687); +\coordinate (G) at ( 6.4208,-10.2438); +\coordinate (H) at (17.2367,- 3.1047); +\coordinate (K) at (24.3760,- 3.0293); +\coordinate (L) at (23.2834,- 1.3563); +\coordinate (M) at (28.7093,- 4.0627); + +\fill (A) circle[radius=0.5]; +\fill (B) circle[radius=0.5]; +\fill (C) circle[radius=0.5]; +\fill (D) circle[radius=0.5]; +\fill (F) circle[radius=0.5]; +\fill (G) circle[radius=0.5]; +\fill (H) circle[radius=0.5]; +\fill (K) circle[radius=0.5]; +\fill (L) circle[radius=0.5]; +\fill (M) circle[radius=0.5]; + +%\uncover<1-4>{ +%\node at (A) [above] {$A$}; +%\node at (B) [above] {$B$}; +%\node at (C) [below] {$C$}; +%\node at (D) [below] {$D$}; +%\node at (F) [below right] {$F$}; +%\node at (G) [above] {$G$}; +%\node at (H) [above] {$H$}; +%\node at (K) [above right] {$K$}; +%\node at (L) [above] {$L$}; +%\node at (M) [above] {$M$}; +%} + +\uncover<5->{ +\node at (A) [above] {$1$}; +\node at (B) [above] {$2$}; +\node at (C) [below] {$3$}; +\node at (D) [below] {$4$}; +\node at (F) [below right] {$5$}; +\node at (G) [above] {$6$}; +\node at (H) [above] {$7$}; +\node at (K) [above right] {$8$}; +\node at (L) [above] {$9$}; +\node at (M) [above] {$10$}; +} + +\draw (L)--(H); +\draw (L)--(K); +\draw (L)--(M); + +\draw (H)--(B); +\draw (H)--(G); +\draw (H)--(F); +\draw (H)--(K); + +\draw (K)--(F); +\draw (K)--(M); + +\draw (M)--(F); +\draw (M)--(D); + +\draw (B)--(A); +\draw (B)--(C); +\draw (B)--(G); + +\draw (G)--(C); +\draw (G)--(F); + +\draw (F)--(D); + +\draw (C)--(F); +\draw (C)--(A); +\draw (C)--(D); + +\draw (A)--(D); + +\uncover<2>{ +\draw[line width=2pt,join=round] + (A)--(D)--(C)--(F)--(G)--(B)--(H)--(K)--(L)--(M); +} + +\uncover<3>{ +\draw[line width=2pt,join=round] + (M)--(D)--(A)--(C)--(G)--(B)--(H)--(L)--(K)--(F); +} + +\uncover<4->{ +\draw[line width=2pt] (M)--(K)--(L)--(H)--(F)--(D); +\draw[line width=2pt] (F)--(G)--(C)--(A); +\draw[line width=2pt] (G)--(B); +} + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +Wieviele Spannbäume gibt es? +\end{block} +\end{column} + +\begin{column}{0.56\hsize} +\uncover<5->{% +\begin{block}{Laplace-Matrix} +\vspace{-15pt} +\[ +L= +\tiny +\begin{pmatrix} + 3&-1&-1&-1& 0& 0& 0& 0& 0& 0\\ +-1& 4&-1& 0& 0&-1&-1& 0& 0& 0\\ +-1&-1& 5&-1&-1&-1& 0& 0& 0& 0\\ +-1& 0&-1& 4&-1& 0& 0& 0& 0&-1\\ + 0& 0&-1&-1& 6&-1&-1&-1& 0&-1\\ + 0&-1&-1& 0&-1& 4&-1& 0& 0& 0\\ + 0&-1& 0& 0&-1&-1& 5&-1&-1& 0\\ + 0& 0& 0& 0&-1& 0&-1& 4&-1&-1\\ + 0& 0& 0& 0& 0& 0&-1&-1& 3&-1\\ + 0& 0& 0&-1&-1& 0& 0&-1&-1& 4\\ +\end{pmatrix} +\] +\end{block}} +\vspace{-15pt} +\uncover<6->{% +\begin{block}{Satz von Kirchhoff} +Die Anzahl der Spannbäume eines Netzwerkes ist ein Kofaktor +des Laplaceoperators +\vspace{-5pt} +\[ +\det L_{ij} = +\left| +L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{array}\right. +\right| +\] +\end{block}} +\vspace{-12pt} +\uncover<7->{% +{\usebeamercolor[fg]{title}Beispiel:} 41524 +} + +\end{column} + +\end{columns} + +\end{frame} diff --git a/vorlesungen/slides/8/tokyo/bahn0.tex b/vorlesungen/slides/8/tokyo/bahn0.tex new file mode 100644 index 0000000..9c39712 --- /dev/null +++ b/vorlesungen/slides/8/tokyo/bahn0.tex @@ -0,0 +1,11 @@ +% +% bahn.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\begin{center} +\includegraphics[width=\hsize]{../slides/8/tokyo/tokyosubway.pdf} +\end{center} +\end{frame} + diff --git a/vorlesungen/slides/8/tokyo/bahn1.tex b/vorlesungen/slides/8/tokyo/bahn1.tex new file mode 100644 index 0000000..6ac3344 --- /dev/null +++ b/vorlesungen/slides/8/tokyo/bahn1.tex @@ -0,0 +1,28 @@ +% +% bahn.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\begin{frame} +\frametitle{Tokyo Bahn-Netz} +\begin{center} +\begin{tabular}{rl} +882&Bahnstationen\\ +108&Bahnlinien\\ +29&Bahngesellschaften\\ +20\,000\,000&Passagiere täglich\\ +7\,000\,000&Passagiere alleine in Shinjuku\\ +\end{tabular} +\end{center} +\uncover<2->{ +\begin{block}{Dirichlet-Zerlegung und Bahnhöfe} +\begin{center} +\uncover<3->{Passagiere wählen den nächsten Bahnhöfe}\\ +\uncover<4->{$\Downarrow$}\\ +\uncover<5->{Bahnhöfe definieren eine Dirichletzerlegung der Stadt} +\end{center} +\end{block} +} +\end{frame} + diff --git a/vorlesungen/slides/8/tokyo/bahn2.tex b/vorlesungen/slides/8/tokyo/bahn2.tex new file mode 100644 index 0000000..4adc1bf --- /dev/null +++ b/vorlesungen/slides/8/tokyo/bahn2.tex @@ -0,0 +1,12 @@ +% +% bahn.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\begin{frame} +\begin{center} +\includegraphics[width=\hsize]{../slides/8/tokyo/shinjuku-subway-map.jpg} +\end{center} +\end{frame} + diff --git a/vorlesungen/slides/8/tokyo/google.tex b/vorlesungen/slides/8/tokyo/google.tex new file mode 100644 index 0000000..d37c98d --- /dev/null +++ b/vorlesungen/slides/8/tokyo/google.tex @@ -0,0 +1,11 @@ +% +% google.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\begin{center} +\includegraphics[width=\hsize]{../slides/8/tokyo/transportnetworkgraph.png} +\end{center} +\end{frame} + diff --git a/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg Binary files differnew file mode 100644 index 0000000..1c513da --- /dev/null +++ b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg diff --git a/vorlesungen/slides/8/tokyo/tokyosubway.pdf b/vorlesungen/slides/8/tokyo/tokyosubway.pdf Binary files differnew file mode 100644 index 0000000..6b84a8d --- /dev/null +++ b/vorlesungen/slides/8/tokyo/tokyosubway.pdf diff --git a/vorlesungen/slides/8/tokyo/transportnetworkgraph.png b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png Binary files differnew file mode 100644 index 0000000..4a11183 --- /dev/null +++ b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc new file mode 100644 index 0000000..fa6c29b --- /dev/null +++ b/vorlesungen/slides/9/Makefile.inc @@ -0,0 +1,14 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter9 = \ + ../slides/9/google.tex \ + ../slides/9/markov.tex \ + ../slides/9/irreduzibel.tex \ + ../slides/9/stationaer.tex \ + ../slides/9/pf.tex \ + ../slides/9/chapter.tex + diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex new file mode 100644 index 0000000..9e26587 --- /dev/null +++ b/vorlesungen/slides/9/chapter.tex @@ -0,0 +1,14 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% + + +\folie{9/google.tex} +\folie{9/markov.tex} +\folie{9/stationaer.tex} +\folie{9/irreduzibel.tex} +\folie{9/pf.tex} + + diff --git a/vorlesungen/slides/9/google.tex b/vorlesungen/slides/9/google.tex new file mode 100644 index 0000000..d1ec31d --- /dev/null +++ b/vorlesungen/slides/9/google.tex @@ -0,0 +1,123 @@ +% +% google.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Google-Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.4} +\coordinate (A) at (0,0); +\coordinate (B) at (0:\r); +\coordinate (C) at (60:\r); +\coordinate (D) at (120:\r); +\coordinate (E) at (180:\r); + +\foreach \a in {2,...,5}{ + \fill[color=white] ({60*(\a-2)}:\r) circle[radius=0.2]; + \draw ({60*(\a-2)}:\r) circle[radius=0.2]; + \node at ({60*(\a-2)}:\r) {$\a$}; +} +\fill[color=white] (A) circle[radius=0.2]; +\draw (A) circle[radius=0.2]; +\node at (A) {$1$}; + +{\color<6>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); +} + +{\color<7>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) to[out=-150,in=-30] (E); +} + +{\color<8>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-90,in=30] (A); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-30,in=90] (B); +} + +{\color<9>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (C); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (A); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); +} + +{\color<10>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) to[out=90,in=-150] (D); +} + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +\renewcommand{\arraystretch}{1.1} +\uncover<5->{ +\begin{align*} +H&=\begin{pmatrix} +\uncover<6->{0 } + &\uncover<7->{0 } + &\uncover<8->{{\color<8>{red}\frac{1}{2}}} + &\uncover<9->{{\color<9>{red}\frac{1}{3}}} + &\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\ +\uncover<6->{{\color<6>{red}\frac{1}{2}}} + &\uncover<7->{0 } + &\uncover<8->{{\color<8>{red}\frac{1}{2}}} + &\uncover<9->{0 } + &\uncover<10->{0 }\\ +\uncover<6->{{\color<6>{red}\frac{1}{2}}} + &\uncover<7->{{\color<7>{red}\frac{1}{2}}} + &\uncover<8->{0 } + &\uncover<9->{{\color<9>{red}\frac{1}{3}}} + &\uncover<10->{0 }\\ +\uncover<6->{0 } + &\uncover<7->{0 } + &\uncover<8->{0 } + &\uncover<9->{0 } + &\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\ +\uncover<6->{0 } + &\uncover<7->{{\color<7>{red}\frac{1}{2}}} + &\uncover<8->{0 } + &\uncover<9->{{\color<9>{red}\frac{1}{3}}} + &\uncover<10->{0 } +\end{pmatrix} +\\ +\uncover<11->{ +h_{ij} +&= +\frac{1}{\text{Anzahl Links ausgehend von $j$}} +} +\end{align*}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Aufgabe} +Bestimme die Wahrscheinlichkeit $p(i)$, mit der sich ein Surfer +auf der Website $i$ befindet +\end{block} +\uncover<2->{ +\begin{block}{Navigation} +$p(i) = P(i,\text{vor Navigation})$, +\uncover<3->{$p'(i)=P(i,\text{nach Navigation})$} +\uncover<4->{ +\[ +p'(i) = \sum_{j=1}^n h_{ij} p(j) +\]} +\end{block}} +\vspace{-15pt} +\begin{block}{Freier Wille} +\vspace{-12pt} +\[ +G = \alpha H + (1-\alpha)\frac{UU^t}{n} +\] +Google-Matrix +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/9/irreduzibel.tex b/vorlesungen/slides/9/irreduzibel.tex new file mode 100644 index 0000000..87e90e4 --- /dev/null +++ b/vorlesungen/slides/9/irreduzibel.tex @@ -0,0 +1,136 @@ +% +% irreduzibel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Irreduzible Markovkette} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{2} +\coordinate (A) at ({\r*cos(0*60)},{\r*sin(0*60)}); +\coordinate (B) at ({\r*cos(1*60)},{\r*sin(1*60)}); +\coordinate (C) at ({\r*cos(2*60)},{\r*sin(2*60)}); +\coordinate (D) at ({\r*cos(3*60)},{\r*sin(3*60)}); +\coordinate (E) at ({\r*cos(4*60)},{\r*sin(4*60)}); +\coordinate (F) at ({\r*cos(5*60)},{\r*sin(5*60)}); + +\uncover<-2>{ +\draw (A) -- (B); +\draw (A) -- (C); +\draw (A) -- (D); +\draw (A) -- (E); +\draw (A) -- (F); + +\draw (B) -- (A); +\draw (B) -- (C); +\draw (B) -- (D); +\draw (B) -- (E); +\draw (B) -- (F); + +\draw (C) -- (A); +\draw (C) -- (B); +\draw (C) -- (D); +\draw (C) -- (E); +\draw (C) -- (F); + +\draw (D) -- (A); +\draw (D) -- (B); +\draw (D) -- (C); +\draw (D) -- (E); +\draw (D) -- (F); + +\draw (E) -- (A); +\draw (E) -- (B); +\draw (E) -- (C); +\draw (E) -- (D); +\draw (E) -- (F); + +\draw (F) -- (A); +\draw (F) -- (B); +\draw (F) -- (C); +\draw (F) -- (D); +\draw (F) -- (E); +} + +\uncover<3->{ + +\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] (A) to[out=90,in=-30] (B); +\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] (A) -- (C); +\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) -- (A); +\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] (B) to[out=150,in=30] (C); +\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) to[out=-90,in=-150,distance=1cm] (B); +\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) to[out=-60,in=180] (A); +\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) -- (B); + +\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt] + (D) to[out=-90,in=150] (E); +\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] + (E) -- (D); +\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] + (E) to[out=-30,in=-150] (F); +\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt] + (F) -- (E); +\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] + (F) to[out=120,in=0] (D); +\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] + (D) -- (F); +} + +\fill[color=white] (A) circle[radius=0.2]; +\fill[color=white] (B) circle[radius=0.2]; +\fill[color=white] (C) circle[radius=0.2]; +\fill[color=white] (D) circle[radius=0.2]; +\fill[color=white] (E) circle[radius=0.2]; +\fill[color=white] (F) circle[radius=0.2]; + +\draw (A) circle[radius=0.2]; +\draw (B) circle[radius=0.2]; +\draw (C) circle[radius=0.2]; +\draw (D) circle[radius=0.2]; +\draw (E) circle[radius=0.2]; +\draw (F) circle[radius=0.2]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; +\node at (F) {$6$}; + +\end{tikzpicture} +\end{center} +\uncover<2->{% +\begin{block}{Irreduzibel} +Graph zusammenhängend $\Rightarrow$ +Keine Zerlegung in Teilgraphen möglich +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Reduzibel} +Die Zustandsmenge zerfällt in zwei disjunkte Teilmengen $V=V_1\cup V_2$ +und es gibt keine Übergängen zwischen den Mengen: +\uncover<4->{% +\begin{align*} +P +&= +\begin{pmatrix*}[l] +0 &0.2&0.5& & & \\ +0.3&0.2&0.5& & & \\ +0.7&0.6&0 & & & \\ + & & &0 &0.3&0.4\\ + & & &0.4&0 &0.6\\ + & & &0.6&0.7&0 +\end{pmatrix*} +\end{align*}}% +\uncover<5->{% +$P$ zerfällt in zwei Blöcke die unabhängig voneinander analysiert werden können +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/9/markov.tex b/vorlesungen/slides/9/markov.tex new file mode 100644 index 0000000..e92ff0f --- /dev/null +++ b/vorlesungen/slides/9/markov.tex @@ -0,0 +1,111 @@ +% +% markov.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{frame}[t] +\frametitle{Markovketten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.2} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!40] + (A) -- (C); +\draw[color=white,line width=8pt] (B) -- (D); +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!80] + (B) -- (D); + +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!60] + (A) -- (B); +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!20] + (B) -- (C); +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black] + (C) -- (D); +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black] + (D) -- (E); +\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black] + (E) -- (A); + +\fill[color=white] (A) circle[radius=0.2]; +\fill[color=white] (B) circle[radius=0.2]; +\fill[color=white] (C) circle[radius=0.2]; +\fill[color=white] (D) circle[radius=0.2]; +\fill[color=white] (E) circle[radius=0.2]; + +\draw (A) circle[radius=0.2]; +\draw (B) circle[radius=0.2]; +\draw (C) circle[radius=0.2]; +\draw (D) circle[radius=0.2]; +\draw (E) circle[radius=0.2]; + +\node at (A) {$1$}; +\node at (B) {$2$}; +\node at (C) {$3$}; +\node at (D) {$4$}; +\node at (E) {$5$}; + +\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 0.6$}; +\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 0.2$}; +\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 1$}; +\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 1$}; +\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 1$}; +\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 0.4$}; +\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 0.8$}; + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Verteilung} +\begin{itemize} +\item<8-> +Welche stationäre Verteilung auf den Knoten stellt sich ein? +\item<9-> +$P(i)=?$ +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{\strut\mbox{Übergang\only<3->{s-/Wahrscheinlichkeit}smatrix}} +$P_{ij} = P(i | j)$, Wahrscheinlichkeit, in den Zustand $i$ überzugehen, +\begin{align*} +P +&= +\begin{pmatrix} + & & & &1\phantom{.0}\\ +0.6& & & & \\ +0.4&0.2& & & \\ + &0.8&1\phantom{.0}& & \\ + & & &1\phantom{.0}& +\end{pmatrix} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Eigenschaften} +\begin{itemize} +\item<5-> $P_{ij}\ge 0\;\forall i,j$ +\item<6-> Spaltensumme: +\( +\displaystyle +\sum_{i=1}^n P_{ij} = 1\;\forall j +\) +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/9/pf.tex b/vorlesungen/slides/9/pf.tex new file mode 100644 index 0000000..da2ef2b --- /dev/null +++ b/vorlesungen/slides/9/pf.tex @@ -0,0 +1,53 @@ +% +% pf.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Perron-Frobenius-Theorie} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Positive Matrizen und Vektoren} +$P\in M_{m\times n}(\mathbb{R})$ +\begin{itemize} +\item<2-> +$P$ heisst positiv, $P>0$, wenn $p_{ij}>0\;\forall i,j$ +\item<3-> +$P\ge 0$, wenn $p_{ij}\ge 0\;\forall i,j$ +\end{itemize} +\end{block} +\uncover<4->{% +\begin{block}{Beispiele} +\begin{itemize} +\item<5-> +Adjazenzmatrix $A(G)$ +\item<6-> +Gradmatrix $D(G)$ +\item<7-> +Wahrscheinlichkeitsmatrizen +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Satz} +Es gibt einen positiven Eigenvektor $p$ von $P$ zum Eigenwert $1$ +\end{block}} +\uncover<9->{% +\begin{block}{Satz} +$P$ irreduzible Matrix, $P\ge 0$, hat einen Eigenvektor $p$, $p\ge 0$, +zum Eigenwert $1$ +\end{block}} +\uncover<10->{% +\begin{block}{Potenzmethode} +Falls $P\ge 0$ einen eindeutigen Eigenvektor $p$ hat\uncover<11->{, +dann konveriert die rekursiv definierte Folge +\[ +p_{n+1}=\frac{Pp_n}{\|Pp_n\|}, p_0 \ge 0, p_0\ne 0 +\]}% +\uncover<12->{$\displaystyle\lim_{n\to\infty} p_n = p$} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/9/stationaer.tex b/vorlesungen/slides/9/stationaer.tex new file mode 100644 index 0000000..92fab16 --- /dev/null +++ b/vorlesungen/slides/9/stationaer.tex @@ -0,0 +1,57 @@ +% +% stationaer.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Stationäre Verteilung} +%\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zeitentwicklung} +\begin{itemize} +\item<2-> +$P$ eine Wahrscheinlichkeitsmatrix +\item<3-> +$p_0\in\mathbb{R}^n$ Verteilung zur Zeit $t=0$ bekannt +\item<4-> +$p_k\in\mathbb{R}^n$ Verteilung zur Zeit $t=k$ +\end{itemize} +\uncover<5->{% +Entwicklungsgesetz +\begin{align*} +P(i,t=k) +&= +\sum_{j=1}^n P_{ij} P(j,t=k-1) +\\ +\uncover<6->{ +p_k &= Pp_{k-1} +} +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Stationär} +Bedingung: $p_{k\mathstrut} = p_{k-1}$ +\uncover<8->{ +\begin{align*} +\Rightarrow +Pp &= p +\end{align*}}\uncover<9->{% +Eigenvektor zum Eigenwert $1$} +\end{block}} +\uncover<10->{% +\begin{block}{Fragen} +\begin{enumerate} +\item<11-> +Gibt es eine stationäre Verteilung? +\item<12-> +Gibt es einen Eigenvektor mit Einträgen $\ge 0$? +\item<13-> +Gibt es mehr als eine Verteilung? +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/Makefile b/vorlesungen/slides/Makefile new file mode 100644 index 0000000..d3c7a17 --- /dev/null +++ b/vorlesungen/slides/Makefile @@ -0,0 +1,34 @@ +# +# Makefile -- build the slide collection +# +# (c) 2019 Prof Dr Andreas Müller, Hochschule Rapeprswil +# +test: test-handout.pdf test-presentation.pdf + +slides: slides-handout.pdf slides-presentation.pdf + +include Makefile.inc + +files = common.tex $(slides) + +catalog: slides-catalog.pdf +presentation: slides-presentation.pdf +handout: slides-handout.pdf + +slides-handout.pdf: slides-handout.tex slides.tex $(files) + pdflatex slides-handout.tex + +slides-catalog.pdf: slides-handout.pdf + pdfjam --outfile slides-catalog.pdf \ + --paper a4paper --nup 2x5 \ + slides-handout.pdf + +slides-presentation.pdf: slides-presentation.tex slides.tex $(files) + pdflatex slides-presentation.tex + +test-handout.pdf: test-handout.tex test.tex $(files) + pdflatex test-handout.tex + +test-presentation.pdf: test-presentation.tex test.tex $(files) + pdflatex test-presentation.tex + diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc new file mode 100644 index 0000000..4bf9348 --- /dev/null +++ b/vorlesungen/slides/Makefile.inc @@ -0,0 +1,17 @@ +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +# +include ../slides/0/Makefile.inc +include ../slides/1/Makefile.inc +include ../slides/2/Makefile.inc +include ../slides/3/Makefile.inc +include ../slides/4/Makefile.inc +include ../slides/5/Makefile.inc +include ../slides/8/Makefile.inc +include ../slides/9/Makefile.inc + +slides = \ + $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ + $(chapter5) $(chapter8) $(chapter9) diff --git a/vorlesungen/slides/common.tex b/vorlesungen/slides/common.tex new file mode 100644 index 0000000..866bab1 --- /dev/null +++ b/vorlesungen/slides/common.tex @@ -0,0 +1,25 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Seminar]{Seminar} +\subtitle{Foliensammlung} +\author[A.~Müller]{Andreas Müller} +\date[]{} +\newboolean{presentation} + +\def\folie#1{ +%\subsection{#1} +\begin{frame} +\begin{center} +\tt #1 +\end{center} +\end{frame} +\input{#1} +} diff --git a/vorlesungen/slides/slides-handout.tex b/vorlesungen/slides/slides-handout.tex new file mode 100644 index 0000000..d834053 --- /dev/null +++ b/vorlesungen/slides/slides-handout.tex @@ -0,0 +1,12 @@ +% +% slides-handout.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{slides.tex} +\end{document} + diff --git a/vorlesungen/slides/slides-presentation.tex b/vorlesungen/slides/slides-presentation.tex new file mode 100644 index 0000000..ff80a11 --- /dev/null +++ b/vorlesungen/slides/slides-presentation.tex @@ -0,0 +1,12 @@ +% +% slides-presentation.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\input{slides.tex} +\end{document} + diff --git a/vorlesungen/slides/slides.tex b/vorlesungen/slides/slides.tex new file mode 100644 index 0000000..b606375 --- /dev/null +++ b/vorlesungen/slides/slides.tex @@ -0,0 +1,79 @@ +% +% slides.tex collection of all slides +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\def\titel{ +% title slide for this chapter +\begin{frame} +\titlepage +\end{frame} +\ifthenelse{\boolean{presentation}}{}{ +% add an empty slide for alignment in the slide catalog +\begin{frame} +\end{frame} +} +} + +\title[MathSem]{Mathematisches Seminar: Matrizen} +\section{Intro} +\titel +\author[]{} +\subtitle{} +\input{0/chapter.tex} + +\title[Grundlagen]{Grundlagen} +\section{Grundlagen} +\titel +\input{1/chapter.tex} + +\title[Vektoren/Matrizen]{Vektoren und Matrizen} +\section{Vektoren und Matrizen} +\titel +\input{2/chapter.tex} + +\title[Polynome]{Polynome} +\section{Polynome} +\titel +\input{3/chapter.tex} + +\title[Endliche Körper]{Endliche Körper} +\section{Endliche Körper} +\titel +\input{4/chapter.tex} + +\title[EW/EV]{Eigenwerte und Eigenvektoren} +\section{Eigenwerte und Eigenvektoren} +\titel +\input{5/chapter.tex} + +%\title[Permutationen]{Permutationen} +%\section{Permutationen} +%\titel +%\input{6/chapter.tex} + +%\title[Matrizengruppen]{Matrizengruppen} +%\section{Matrizengruppen} +%\titel +%\input{7/chapter.tex} + +\title[Graphen]{Graphen} +\section{Graphen} +\titel +\input{8/chapter.tex} + +\title[Wahrscheinlichkeit]{Wahrscheinlichkeitsmatrizen} +\section{Wahrscheinlichkeitsmatrizen} +\titel +\input{9/chapter.tex} + +%\title[Krypto]{Anwendungen in Kryptographie und Codierungstheorie} +%\section{Krypto} +%\titel +%\input{a/chapter.tex} + +%\title[Homologie]{Homologie} +%\section{Homologie} +%\titel +%\input{b/chapter.tex} + diff --git a/vorlesungen/slides/test-handout.tex b/vorlesungen/slides/test-handout.tex new file mode 100644 index 0000000..63f41e4 --- /dev/null +++ b/vorlesungen/slides/test-handout.tex @@ -0,0 +1,12 @@ +% +% test-handout.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{test.tex} +\end{document} + diff --git a/vorlesungen/slides/test-presentation.tex b/vorlesungen/slides/test-presentation.tex new file mode 100644 index 0000000..2cf9816 --- /dev/null +++ b/vorlesungen/slides/test-presentation.tex @@ -0,0 +1,12 @@ +% +% test-presentation.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\input{test.tex} +\end{document} + diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex new file mode 100644 index 0000000..e4b9ad7 --- /dev/null +++ b/vorlesungen/slides/test.tex @@ -0,0 +1,17 @@ +% +% test.tex collection of all slides +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +%\folie{5/verzerrung.tex} + +% XXX Visualisierung Cayley-Hamilton-Produkte +% XXX \folie{5/chvisual.tex} + +% XXX stone weierstrass incomplete +%\folie{5/stoneweierstrass.tex} + +% XXX polynome auf dem spektrum +% XXX Motiviation für *-Operation +%\folie{5/normal.tex} + |
