From 77b274d37031641a160f11cb37f6bf03aead442f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 16 Feb 2021 18:41:02 +0100 Subject: start making slides --- vorlesungen/slides/0/Makefile.inc | 13 ++++ vorlesungen/slides/0/chapter.tex | 10 +++ vorlesungen/slides/0/intro.tex | 98 ++++++++++++++++++++++++++++++ vorlesungen/slides/0/latextipps.tex | 16 +++++ vorlesungen/slides/0/nextsteps.tex | 17 ++++++ vorlesungen/slides/0/resourcen.tex | 16 +++++ vorlesungen/slides/0/themen.tex | 66 ++++++++++++++++++++ vorlesungen/slides/Makefile | 25 ++++++++ vorlesungen/slides/Makefile.inc | 9 +++ vorlesungen/slides/common.tex | 25 ++++++++ vorlesungen/slides/slides-handout.tex | 12 ++++ vorlesungen/slides/slides-presentation.tex | 12 ++++ vorlesungen/slides/slides.tex | 13 ++++ vorlesungen/slides/test-handout.tex | 12 ++++ vorlesungen/slides/test-presentation.tex | 12 ++++ vorlesungen/slides/test.tex | 7 +++ 16 files changed, 363 insertions(+) create mode 100644 vorlesungen/slides/0/Makefile.inc create mode 100644 vorlesungen/slides/0/chapter.tex create mode 100644 vorlesungen/slides/0/intro.tex create mode 100644 vorlesungen/slides/0/latextipps.tex create mode 100644 vorlesungen/slides/0/nextsteps.tex create mode 100644 vorlesungen/slides/0/resourcen.tex create mode 100644 vorlesungen/slides/0/themen.tex create mode 100644 vorlesungen/slides/Makefile create mode 100644 vorlesungen/slides/Makefile.inc create mode 100644 vorlesungen/slides/common.tex create mode 100644 vorlesungen/slides/slides-handout.tex create mode 100644 vorlesungen/slides/slides-presentation.tex create mode 100644 vorlesungen/slides/slides.tex create mode 100644 vorlesungen/slides/test-handout.tex create mode 100644 vorlesungen/slides/test-presentation.tex create mode 100644 vorlesungen/slides/test.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc new file mode 100644 index 0000000..db4cdf0 --- /dev/null +++ b/vorlesungen/slides/0/Makefile.inc @@ -0,0 +1,13 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 20920 Prof Dr Andreas Müller, Hochschule Rapperswil +# +chapter0 = \ + ../slides/0/intro.tex \ + ../slides/0/resourcen.tex \ + ../slides/0/latextipps.tex \ + ../slides/0/themen.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..4f17162 --- /dev/null +++ b/vorlesungen/slides/0/chapter.tex @@ -0,0 +1,10 @@ +% +% chapter.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{0/intro.tex} +\folie{0/resourcen.tex} +\folie{0/latextipps.tex} +\folie{0/themen.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..eb2d9e5 --- /dev/null +++ b/vorlesungen/slides/0/nextsteps.tex @@ -0,0 +1,17 @@ +% +% nextsteps.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\begin{frame} +\frametitle{Nächste Schritte} + +\begin{enumerate} +\item +\item +\item +\item +\end{enumerate} + +\end{frame} diff --git a/vorlesungen/slides/0/resourcen.tex b/vorlesungen/slides/0/resourcen.tex new file mode 100644 index 0000000..9f55be3 --- /dev/null +++ b/vorlesungen/slides/0/resourcen.tex @@ -0,0 +1,16 @@ +% +% resourcen.tex +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame} +\frametitle{Resourcen} +\begin{block}{Moodle Modul MathSem} +\begin{enumerate} +\item Skript +\item Planung: Kurztests, Vorträge +\item Anleitung für die Seminar-Arbeit +\item Aufgabenstellungen +\end{enumerate} +\end{block} +\end{frame} diff --git a/vorlesungen/slides/0/themen.tex b/vorlesungen/slides/0/themen.tex new file mode 100644 index 0000000..74b6fb7 --- /dev/null +++ b/vorlesungen/slides/0/themen.tex @@ -0,0 +1,66 @@ +% +% themen.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 +\end{enumerate} +\end{frame} + +\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} + +\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/Makefile b/vorlesungen/slides/Makefile new file mode 100644 index 0000000..dd8b06e --- /dev/null +++ b/vorlesungen/slides/Makefile @@ -0,0 +1,25 @@ +# +# 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) + +slides-handout.pdf: slides-handout.tex slides.tex $(files) + pdflatex slides-handout.tex + +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..0909784 --- /dev/null +++ b/vorlesungen/slides/Makefile.inc @@ -0,0 +1,9 @@ +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +# +include ../slides/0/Makefile.inc + +slides = \ + $(chapter0) 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{% +\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..e4510c9 --- /dev/null +++ b/vorlesungen/slides/slides.tex @@ -0,0 +1,13 @@ +% +% slides.tex collection of all slides +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\title[Intro]{Intro} +\section{Intro} +\begin{frame} +\titlepage +\end{frame} +\subtitle{} +\input{0/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..b035710 --- /dev/null +++ b/vorlesungen/slides/test.tex @@ -0,0 +1,7 @@ +% +% test.tex collection of all slides +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\folie{0/intro.tex} -- cgit v1.2.1 From 69c5a77e6f008330327f2697f9c77cea6f2e1c3a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 16 Feb 2021 22:05:38 +0100 Subject: chapter 1 slides --- vorlesungen/slides/0/Makefile.inc | 1 + vorlesungen/slides/0/nextsteps.tex | 12 +++-- vorlesungen/slides/1/Makefile.inc | 18 ++++++++ vorlesungen/slides/1/algebrastruktur.tex | 73 ++++++++++++++++++++++++++++++ vorlesungen/slides/1/chapter.tex | 14 ++++++ vorlesungen/slides/1/dreieck.tex | 62 +++++++++++++++++++++++++ vorlesungen/slides/1/hadamard.tex | 51 +++++++++++++++++++++ vorlesungen/slides/1/j.tex | 63 ++++++++++++++++++++++++++ vorlesungen/slides/1/matrixalgebra.tex | 77 ++++++++++++++++++++++++++++++++ vorlesungen/slides/1/speziell.tex | 43 ++++++++++++++++++ vorlesungen/slides/1/strukturen.tex | 35 +++++++++++++++ vorlesungen/slides/1/vektorraum.tex | 54 ++++++++++++++++++++++ vorlesungen/slides/1/zahlensysteme.tex | 46 +++++++++++++++++++ vorlesungen/slides/Makefile.inc | 3 +- vorlesungen/slides/slides.tex | 8 ++++ vorlesungen/slides/test.tex | 10 ++++- 16 files changed, 564 insertions(+), 6 deletions(-) create mode 100644 vorlesungen/slides/1/Makefile.inc create mode 100644 vorlesungen/slides/1/algebrastruktur.tex create mode 100644 vorlesungen/slides/1/chapter.tex create mode 100644 vorlesungen/slides/1/dreieck.tex create mode 100644 vorlesungen/slides/1/hadamard.tex create mode 100644 vorlesungen/slides/1/j.tex create mode 100644 vorlesungen/slides/1/matrixalgebra.tex create mode 100644 vorlesungen/slides/1/speziell.tex create mode 100644 vorlesungen/slides/1/strukturen.tex create mode 100644 vorlesungen/slides/1/vektorraum.tex create mode 100644 vorlesungen/slides/1/zahlensysteme.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc index db4cdf0..99a2bfd 100644 --- a/vorlesungen/slides/0/Makefile.inc +++ b/vorlesungen/slides/0/Makefile.inc @@ -8,6 +8,7 @@ chapter0 = \ ../slides/0/intro.tex \ ../slides/0/resourcen.tex \ ../slides/0/latextipps.tex \ + ../slides/0/nextsteps.tex \ ../slides/0/themen.tex \ ../slides/0/chapter.tex diff --git a/vorlesungen/slides/0/nextsteps.tex b/vorlesungen/slides/0/nextsteps.tex index eb2d9e5..cb9a07e 100644 --- a/vorlesungen/slides/0/nextsteps.tex +++ b/vorlesungen/slides/0/nextsteps.tex @@ -8,10 +8,14 @@ \frametitle{Nächste Schritte} \begin{enumerate} -\item -\item -\item -\item +\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/1/Makefile.inc b/vorlesungen/slides/1/Makefile.inc new file mode 100644 index 0000000..3c1b5d4 --- /dev/null +++ b/vorlesungen/slides/1/Makefile.inc @@ -0,0 +1,18 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 20920 Prof Dr Andreas Müller, Hochschule Rapperswil +# +chapter1 = \ + ../slides/1/zahlensysteme.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..9647c04 --- /dev/null +++ b/vorlesungen/slides/1/algebrastruktur.tex @@ -0,0 +1,73 @@ +% +% 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} +\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; +\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; +\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 Halbgruppe}; + +\node at (0,-2.8) {Addition}; +\node at (0,-3.2) {\tiny Gruppe}; + +\node[color=blue] at (4.8,-0.5) [rotate=\a] {Ring\strut}; +\node[color=red] at (-4.8,-0.5) [rotate=-\a] {Vektorraum\strut}; + +\node[color=darkgreen] at (0,2.6) {$(\lambda a)b=\lambda(ab)$}; + +\node[color=red] at (-2.5,-0.5) {$\displaystyle +\begin{aligned} +\lambda(a+b)&=\lambda a + \lambda b\\ +(\lambda+\mu)a&=\lambda a +\mu a +\end{aligned}$}; + +\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/chapter.tex b/vorlesungen/slides/1/chapter.tex new file mode 100644 index 0000000..fec3330 --- /dev/null +++ b/vorlesungen/slides/1/chapter.tex @@ -0,0 +1,14 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{1/zahlensysteme.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..f4909e2 --- /dev/null +++ b/vorlesungen/slides/1/dreieck.tex @@ -0,0 +1,62 @@ +% +% dreieck.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Dreiecksmatrizen} +\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} +\begin{block}{Nilpotente Matrix} +\[ +N= +\begin{pmatrix} +*&*&*&\dots&*\\ +0&*&*&\dots&*\\ +0&0&*&\dots&*\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\dots&* +\end{pmatrix} +\] +$\Rightarrow N^n=0$ +\end{block} +\end{column} +\begin{column}{0.31\textwidth} +\begin{block}{Jordan-Matrix} +\[ +J_\lambda=\begin{pmatrix} +\lambda&1&0&\dots&0\\ +0&\lambda&1&\dots&0\\ +0&0&\lambda&\dots&0\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +0&0&0&\dots&\lambda +\end{pmatrix} +\] +$\Rightarrow J_\lambda -\lambda I$ ist nilpotent +\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/speziell.tex b/vorlesungen/slides/1/speziell.tex new file mode 100644 index 0000000..87b767a --- /dev/null +++ b/vorlesungen/slides/1/speziell.tex @@ -0,0 +1,43 @@ +% +% 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} +\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} +\begin{block}{Hadamard-Algebra} +Die Algebra der Diagonalmatrizen ist die Hadamard-Algebra +\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..4e66137 --- /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{}\uncover4->{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/Makefile.inc b/vorlesungen/slides/Makefile.inc index 0909784..3fdf7a4 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -4,6 +4,7 @@ # (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil # include ../slides/0/Makefile.inc +include ../slides/1/Makefile.inc slides = \ - $(chapter0) + $(chapter0) $(chapter1) diff --git a/vorlesungen/slides/slides.tex b/vorlesungen/slides/slides.tex index e4510c9..0dc6c6e 100644 --- a/vorlesungen/slides/slides.tex +++ b/vorlesungen/slides/slides.tex @@ -11,3 +11,11 @@ \subtitle{} \input{0/chapter.tex} +\title[Grundlagen]{Grundlagen} +\section{Grundlagen} +\begin{frame} +\titlepage +\end{frame} +\subtitle{} +\input{1/chapter.tex} + diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index b035710..5abeb50 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,4 +4,12 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{0/intro.tex} +%\folie{1/zahlensysteme.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} -- cgit v1.2.1 From ea812658422730db2440c458330f0487d9dc823e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 19 Feb 2021 16:45:39 +0100 Subject: new slides --- vorlesungen/00_template/Makefile | 26 +++++++++++++++++ vorlesungen/00_template/common.tex | 16 ++++++++++ vorlesungen/00_template/handout.tex | 11 +++++++ vorlesungen/00_template/presentation.tex | 14 +++++++++ vorlesungen/00_template/slides.tex | 6 ++++ vorlesungen/slides/0/Makefile.inc | 1 + vorlesungen/slides/0/chapter.tex | 1 + vorlesungen/slides/0/was.tex | 50 ++++++++++++++++++++++++++++++++ vorlesungen/slides/1/zahlensysteme.tex | 2 +- 9 files changed, 126 insertions(+), 1 deletion(-) create mode 100644 vorlesungen/00_template/Makefile create mode 100644 vorlesungen/00_template/common.tex create mode 100644 vorlesungen/00_template/handout.tex create mode 100644 vorlesungen/00_template/presentation.tex create mode 100644 vorlesungen/00_template/slides.tex create mode 100644 vorlesungen/slides/0/was.tex (limited to 'vorlesungen') diff --git a/vorlesungen/00_template/Makefile b/vorlesungen/00_template/Makefile new file mode 100644 index 0000000..fe6bbb7 --- /dev/null +++ b/vorlesungen/00_template/Makefile @@ -0,0 +1,26 @@ +# +# Makefile -- xxx +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: xxx-handout.pdf MathSem-yyy-xxx.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-yyy-xxx.pdf: MathSem-yyy-xxx.tex $(SOURCES) + pdflatex MathSem-yyy-xxx.tex + +xxx-handout.pdf: xxx-handout.tex $(SOURCES) + pdflatex xxx-handout.tex + +thumbnail: thumbnail.jpg + +thumbnail.pdf: MathSem-yyy-xxx.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-yyy-xxx.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + diff --git a/vorlesungen/00_template/common.tex b/vorlesungen/00_template/common.tex new file mode 100644 index 0000000..0700acf --- /dev/null +++ b/vorlesungen/00_template/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Titel]{Titel} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/00_template/handout.tex b/vorlesungen/00_template/handout.tex new file mode 100644 index 0000000..6a98c94 --- /dev/null +++ b/vorlesungen/00_template/handout.tex @@ -0,0 +1,11 @@ +% +% xxx-handout.tex -- Handout XXX +% +% (c) 2017 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/00_template/presentation.tex b/vorlesungen/00_template/presentation.tex new file mode 100644 index 0000000..599bed2 --- /dev/null +++ b/vorlesungen/00_template/presentation.tex @@ -0,0 +1,14 @@ +% +% MathSem-yyy-xxx.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/00_template/slides.tex b/vorlesungen/00_template/slides.tex new file mode 100644 index 0000000..5678823 --- /dev/null +++ b/vorlesungen/00_template/slides.tex @@ -0,0 +1,6 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + diff --git a/vorlesungen/slides/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc index 99a2bfd..ab91126 100644 --- a/vorlesungen/slides/0/Makefile.inc +++ b/vorlesungen/slides/0/Makefile.inc @@ -5,6 +5,7 @@ # (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 \ diff --git a/vorlesungen/slides/0/chapter.tex b/vorlesungen/slides/0/chapter.tex index 4f17162..a860830 100644 --- a/vorlesungen/slides/0/chapter.tex +++ b/vorlesungen/slides/0/chapter.tex @@ -4,6 +4,7 @@ % (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/themen.tex} 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/zahlensysteme.tex b/vorlesungen/slides/1/zahlensysteme.tex index 4e66137..9131cc6 100644 --- a/vorlesungen/slides/1/zahlensysteme.tex +++ b/vorlesungen/slides/1/zahlensysteme.tex @@ -22,7 +22,7 @@ inverses Element der Addition} \\ \mathbb{Z} &\phantom{}\raggedright\uncover<4->{zusätzlich: Multiplikation, neutrales Element $1$} -&\phantom{}\uncover4->{Ring} +&\phantom{}\uncover<4->{Ring} \\ \mathbb{Q} &\phantom{}\raggedright\uncover<5->{Addition und Multiplikation mit Inversen} -- cgit v1.2.1 From 918db721c225c4d7b891145ac73e3886d13f8166 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 19 Feb 2021 23:11:27 +0100 Subject: add new presentation --- vorlesungen/01_intro/Makefile | 26 ++++++++++++++++++++++++++ vorlesungen/01_intro/MathSem-01-intro.tex | 14 ++++++++++++++ vorlesungen/01_intro/common.tex | 16 ++++++++++++++++ vorlesungen/01_intro/intro-handout.tex | 11 +++++++++++ vorlesungen/01_intro/slides.tex | 22 ++++++++++++++++++++++ vorlesungen/slides/0/resourcen.tex | 30 +++++++++++++++++++++++++++--- vorlesungen/slides/0/themen.tex | 2 ++ vorlesungen/slides/test.tex | 4 +++- 8 files changed, 121 insertions(+), 4 deletions(-) create mode 100644 vorlesungen/01_intro/Makefile create mode 100644 vorlesungen/01_intro/MathSem-01-intro.tex create mode 100644 vorlesungen/01_intro/common.tex create mode 100644 vorlesungen/01_intro/intro-handout.tex create mode 100644 vorlesungen/01_intro/slides.tex (limited to 'vorlesungen') diff --git a/vorlesungen/01_intro/Makefile b/vorlesungen/01_intro/Makefile new file mode 100644 index 0000000..b75f884 --- /dev/null +++ b/vorlesungen/01_intro/Makefile @@ -0,0 +1,26 @@ +# +# Makefile -- intro +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: intro-handout.pdf MathSem-01-intro.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-01-intro.pdf: MathSem-01-intro.tex $(SOURCES) + pdflatex MathSem-01-intro.tex + +intro-handout.pdf: intro-handout.tex $(SOURCES) + pdflatex intro-handout.tex + +thumbnail: thumbnail.jpg + +thumbnail.pdf: MathSem-01-intro.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-01-intro.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + diff --git a/vorlesungen/01_intro/MathSem-01-intro.tex b/vorlesungen/01_intro/MathSem-01-intro.tex new file mode 100644 index 0000000..f8588e5 --- /dev/null +++ b/vorlesungen/01_intro/MathSem-01-intro.tex @@ -0,0 +1,14 @@ +% +% MathSem-01-intro.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/01_intro/common.tex b/vorlesungen/01_intro/common.tex new file mode 100644 index 0000000..6323d52 --- /dev/null +++ b/vorlesungen/01_intro/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Intro]{Einführung ins Mathematische Seminar} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/01_intro/intro-handout.tex b/vorlesungen/01_intro/intro-handout.tex new file mode 100644 index 0000000..b867c8d --- /dev/null +++ b/vorlesungen/01_intro/intro-handout.tex @@ -0,0 +1,11 @@ +% +% intro-handout.tex -- Handout XXX +% +% (c) 2017 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/01_intro/slides.tex b/vorlesungen/01_intro/slides.tex new file mode 100644 index 0000000..b840fdd --- /dev/null +++ b/vorlesungen/01_intro/slides.tex @@ -0,0 +1,22 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\folie{0/intro.tex} +\section{Admin} +\folie{0/was.tex} +\folie{0/resourcen.tex} +\folie{0/themen.tex} +\folie{0/nextsteps.tex} +\folie{0/latextipps.tex} +\section{Mathematik} +\folie{1/zahlensysteme.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/0/resourcen.tex b/vorlesungen/slides/0/resourcen.tex index 9f55be3..ddaa0c4 100644 --- a/vorlesungen/slides/0/resourcen.tex +++ b/vorlesungen/slides/0/resourcen.tex @@ -3,14 +3,38 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\begin{frame} +\begin{frame}[t] \frametitle{Resourcen} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} \begin{block}{Moodle Modul MathSem} \begin{enumerate} \item Skript -\item Planung: Kurztests, Vorträge -\item Anleitung für die Seminar-Arbeit +\begin{itemize} +\item Aktuellste Version in Github +\item regelmässige Updates in Moodle: \texttt{buch.pdf} +\end{itemize} +\item Informationen zur Planung: Kurztests, Vorträge +\item Anleitung für die Seminararbeit \item Aufgabenstellungen +\begin{itemize} +\item +\item +\end{itemize} \end{enumerate} \end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Weitere Quellen} +\begin{enumerate} +\item Zusätzliche Literaturhinweise in der Aufgabenbeschreibung im Moodle +\item Bibliothek +\item Google +\item Google Scholar +\item 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/themen.tex b/vorlesungen/slides/0/themen.tex index 74b6fb7..c50f32d 100644 --- a/vorlesungen/slides/0/themen.tex +++ b/vorlesungen/slides/0/themen.tex @@ -20,6 +20,8 @@ Vandermonde-Matrix Probabilistische Matrix-Produkt-Kontrolle \item Der Satz von Furrer-Hungerbühler-Jantschgi +\item +Clifford-Algebren \end{enumerate} \end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 5abeb50..7b0e02b 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,6 +4,8 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % +\folie{0/resourcen.tex} + %\folie{1/zahlensysteme.tex} %\folie{1/strukturen.tex} %\folie{1/j.tex} @@ -12,4 +14,4 @@ %\folie{1/algebrastruktur.tex} %\folie{1/speziell.tex} %\folie{1/dreieck.tex} -\folie{1/hadamard.tex} +%\folie{1/hadamard.tex} -- cgit v1.2.1 From 395d1e49d77c9991df3ba3fbcb3e0fe2a40cb63e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 19 Feb 2021 23:11:55 +0100 Subject: add missing files --- vorlesungen/common/common.tex | 9 +++++++++ vorlesungen/common/packages.tex | 28 ++++++++++++++++++++++++++++ 2 files changed, 37 insertions(+) create mode 100644 vorlesungen/common/common.tex create mode 100644 vorlesungen/common/packages.tex (limited to 'vorlesungen') diff --git a/vorlesungen/common/common.tex b/vorlesungen/common/common.tex new file mode 100644 index 0000000..901b4fd --- /dev/null +++ b/vorlesungen/common/common.tex @@ -0,0 +1,9 @@ +% +% common.tex +% +% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\newcommand{\folie}[1]{ +\input{../slides/#1} +} + diff --git a/vorlesungen/common/packages.tex b/vorlesungen/common/packages.tex new file mode 100644 index 0000000..df302a2 --- /dev/null +++ b/vorlesungen/common/packages.tex @@ -0,0 +1,28 @@ +% +% packages.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{ifthen} +\usepackage{lmodern} +\usepackage{amsmath} +\usepackage{adjustbox} +\usepackage{multimedia} +\usepackage{verbatim} +\usepackage{tikz} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{decorations.pathreplacing} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\input{../../common/linsys.tex} + -- cgit v1.2.1 From 4a7dd262ec4b3eb0e856e18af4f7db2360e3df70 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 21 Feb 2021 21:43:55 +0100 Subject: stream stuff --- vorlesungen/01_intro/MathSem-01-intro.tex | 4 +++ vorlesungen/stream/countdown.html | 40 ++++++++++++++++++++++++++++++ vorlesungen/stream/thumbnail.jpg | Bin 0 -> 309067 bytes 3 files changed, 44 insertions(+) create mode 100644 vorlesungen/stream/countdown.html create mode 100644 vorlesungen/stream/thumbnail.jpg (limited to 'vorlesungen') diff --git a/vorlesungen/01_intro/MathSem-01-intro.tex b/vorlesungen/01_intro/MathSem-01-intro.tex index f8588e5..b510238 100644 --- a/vorlesungen/01_intro/MathSem-01-intro.tex +++ b/vorlesungen/01_intro/MathSem-01-intro.tex @@ -9,6 +9,10 @@ \begin{document} \begin{frame} \titlepage +\vspace{-40pt} +\begin{center} +\includegraphics[width=6cm]{../../cover/buchcover.png} +\end{center} \end{frame} \input{slides.tex} \end{document} diff --git a/vorlesungen/stream/countdown.html b/vorlesungen/stream/countdown.html new file mode 100644 index 0000000..77f9ca9 --- /dev/null +++ b/vorlesungen/stream/countdown.html @@ -0,0 +1,40 @@ + + + + + + +
+ + + + + diff --git a/vorlesungen/stream/thumbnail.jpg b/vorlesungen/stream/thumbnail.jpg new file mode 100644 index 0000000..e9a5064 Binary files /dev/null and b/vorlesungen/stream/thumbnail.jpg differ -- cgit v1.2.1 From 80b724cc3edb6b815b03a4014d7c12cf7480a904 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 21 Feb 2021 22:17:21 +0100 Subject: add ende --- vorlesungen/stream/ende.html | 29 +++++++++++++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 vorlesungen/stream/ende.html (limited to 'vorlesungen') diff --git a/vorlesungen/stream/ende.html b/vorlesungen/stream/ende.html new file mode 100644 index 0000000..9dd27e1 --- /dev/null +++ b/vorlesungen/stream/ende.html @@ -0,0 +1,29 @@ + + + + + + +
+

+Vielen Dank für Ihren Besuch. +

+

+Fortsetzung der Seminar-Sitzung in +der BBB-Konferenz in Moodle. +

+
+ + + -- cgit v1.2.1 From 57839140bd44eba59e578e055f8a7720ea1a0865 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 22 Feb 2021 11:05:22 +0100 Subject: slides --- vorlesungen/01_intro/slides.tex | 2 ++ vorlesungen/slides/0/resourcen.tex | 29 +++++++++++++---------------- vorlesungen/slides/1/dreieck.tex | 14 ++++++++++---- vorlesungen/slides/1/speziell.tex | 1 + 4 files changed, 26 insertions(+), 20 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/01_intro/slides.tex b/vorlesungen/01_intro/slides.tex index b840fdd..e38e240 100644 --- a/vorlesungen/01_intro/slides.tex +++ b/vorlesungen/01_intro/slides.tex @@ -3,6 +3,7 @@ % % (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil % +\ifthenelse{\boolean{presentation}}{ \folie{0/intro.tex} \section{Admin} \folie{0/was.tex} @@ -10,6 +11,7 @@ \folie{0/themen.tex} \folie{0/nextsteps.tex} \folie{0/latextipps.tex} +}{} \section{Mathematik} \folie{1/zahlensysteme.tex} \folie{1/strukturen.tex} diff --git a/vorlesungen/slides/0/resourcen.tex b/vorlesungen/slides/0/resourcen.tex index ddaa0c4..02a3fb8 100644 --- a/vorlesungen/slides/0/resourcen.tex +++ b/vorlesungen/slides/0/resourcen.tex @@ -9,32 +9,29 @@ \begin{column}{0.48\textwidth} \begin{block}{Moodle Modul MathSem} \begin{enumerate} -\item Skript +\item<2-> Skript \begin{itemize} -\item Aktuellste Version in Github -\item regelmässige Updates in Moodle: \texttt{buch.pdf} -\end{itemize} -\item Informationen zur Planung: Kurztests, Vorträge -\item Anleitung für die Seminararbeit -\item Aufgabenstellungen -\begin{itemize} -\item -\item +\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 Zusätzliche Literaturhinweise in der Aufgabenbeschreibung im Moodle -\item Bibliothek -\item Google -\item Google Scholar -\item Paper ist nicht öffentlich zugänglich? $\rightarrow$ kann via +\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{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/1/dreieck.tex b/vorlesungen/slides/1/dreieck.tex index f4909e2..f38f78d 100644 --- a/vorlesungen/slides/1/dreieck.tex +++ b/vorlesungen/slides/1/dreieck.tex @@ -30,21 +30,25 @@ U&= \end{block} \end{column} \begin{column}{0.31\textwidth} +\uncover<2->{% \begin{block}{Nilpotente Matrix} \[ N= \begin{pmatrix} -*&*&*&\dots&*\\ 0&*&*&\dots&*\\ 0&0&*&\dots&*\\ +0&0&0&\dots&*\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ -0&0&0&\dots&* +0&0&0&\dots&0 \end{pmatrix} \] +\uncover<3->{% $\Rightarrow N^n=0$ -\end{block} +} +\end{block}} \end{column} \begin{column}{0.31\textwidth} +\uncover<4->{% \begin{block}{Jordan-Matrix} \[ J_\lambda=\begin{pmatrix} @@ -55,8 +59,10 @@ J_\lambda=\begin{pmatrix} 0&0&0&\dots&\lambda \end{pmatrix} \] +\uncover<5->{% $\Rightarrow J_\lambda -\lambda I$ ist nilpotent -\end{block} +} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/1/speziell.tex b/vorlesungen/slides/1/speziell.tex index 87b767a..83b3735 100644 --- a/vorlesungen/slides/1/speziell.tex +++ b/vorlesungen/slides/1/speziell.tex @@ -37,6 +37,7 @@ AI=IA=A \end{block} \begin{block}{Hadamard-Algebra} Die Algebra der Diagonalmatrizen ist die Hadamard-Algebra +(siehe später) \end{block} \end{column} \end{columns} -- cgit v1.2.1 From 95c9a63dbb58815be30d393a0a1891b0bddc8ae8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 22 Feb 2021 11:19:24 +0100 Subject: phases --- vorlesungen/slides/1/algebrastruktur.tex | 102 ++++++++++++++++++------------- vorlesungen/slides/1/dreieck.tex | 1 + vorlesungen/slides/1/speziell.tex | 6 +- 3 files changed, 66 insertions(+), 43 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/1/algebrastruktur.tex b/vorlesungen/slides/1/algebrastruktur.tex index 9647c04..d7794a3 100644 --- a/vorlesungen/slides/1/algebrastruktur.tex +++ b/vorlesungen/slides/1/algebrastruktur.tex @@ -13,34 +13,43 @@ \begin{tikzpicture}[>=latex,thick] \pgfmathparse{atan(7/4)} \xdef\a{\pgfmathresult} -\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; -\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; -\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; +\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}; @@ -49,22 +58,33 @@ \node at (0,-2.8) {Addition}; \node at (0,-3.2) {\tiny Gruppe}; -\node[color=blue] at (4.8,-0.5) [rotate=\a] {Ring\strut}; -\node[color=red] at (-4.8,-0.5) [rotate=-\a] {Vektorraum\strut}; +\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}; +} -\node[color=darkgreen] at (0,2.6) {$(\lambda a)b=\lambda(ab)$}; +\uncover<6->{ + \node[color=darkgreen] at (0,2.6) {$(\lambda a)b=\lambda(ab)$}; +} -\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<3->{ + \node[color=red] at (-2.5,-0.5) {$\displaystyle + \begin{aligned} + \lambda(a+b)&=\lambda a + \lambda b\\ + (\lambda+\mu)a&=\lambda a +\mu a + \end{aligned}$}; +} -\node[color=blue] at (2.5,-0.5) {$\displaystyle -\begin{aligned} -a(b+c)&=ab+ac\\ -(a+b)c&=ac+bc -\end{aligned}$}; +\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} diff --git a/vorlesungen/slides/1/dreieck.tex b/vorlesungen/slides/1/dreieck.tex index f38f78d..3797e4b 100644 --- a/vorlesungen/slides/1/dreieck.tex +++ b/vorlesungen/slides/1/dreieck.tex @@ -5,6 +5,7 @@ % \begin{frame}[t] \frametitle{Dreiecksmatrizen} +\vspace{-10pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.31\textwidth} \begin{block}{Dreiecksmatrix} diff --git a/vorlesungen/slides/1/speziell.tex b/vorlesungen/slides/1/speziell.tex index 83b3735..5b93da6 100644 --- a/vorlesungen/slides/1/speziell.tex +++ b/vorlesungen/slides/1/speziell.tex @@ -23,6 +23,7 @@ AI=IA=A \end{block} \end{column} \begin{column}{0.58\textwidth} +\uncover<2->{% \begin{block}{Diagonalmatrix} \[ \operatorname{diag}(\lambda_1,\lambda_2,\dots,\lambda_n) @@ -34,11 +35,12 @@ AI=IA=A 0&0&\dots&\lambda_n \end{pmatrix} \] -\end{block} +\end{block}} +\uncover<3->{% \begin{block}{Hadamard-Algebra} Die Algebra der Diagonalmatrizen ist die Hadamard-Algebra (siehe später) -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} -- cgit v1.2.1 From 278fed27ca07bff386d830e2142d207d904703fc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 23 Feb 2021 19:58:40 +0100 Subject: new slides chapter 3 --- vorlesungen/common/packages.tex | 1 + vorlesungen/slides/3/Makefile.inc | 10 ++++ vorlesungen/slides/3/chapter.tex | 7 +++ vorlesungen/slides/3/motivation.tex | 106 ++++++++++++++++++++++++++++++++++++ vorlesungen/slides/3/polynome.tex | 9 +++ vorlesungen/slides/Makefile.inc | 3 +- vorlesungen/slides/test.tex | 12 +--- 7 files changed, 137 insertions(+), 11 deletions(-) create mode 100644 vorlesungen/slides/3/Makefile.inc create mode 100644 vorlesungen/slides/3/chapter.tex create mode 100644 vorlesungen/slides/3/motivation.tex create mode 100644 vorlesungen/slides/3/polynome.tex (limited to 'vorlesungen') diff --git a/vorlesungen/common/packages.tex b/vorlesungen/common/packages.tex index df302a2..94a7c6f 100644 --- a/vorlesungen/common/packages.tex +++ b/vorlesungen/common/packages.tex @@ -11,6 +11,7 @@ \usepackage{ifthen} \usepackage{lmodern} \usepackage{amsmath} +\usepackage{mathtools} \usepackage{adjustbox} \usepackage{multimedia} \usepackage{verbatim} diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc new file mode 100644 index 0000000..a58ee05 --- /dev/null +++ b/vorlesungen/slides/3/Makefile.inc @@ -0,0 +1,10 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter3 = \ + ../slides/3/motivation.tex \ + ../slides/3/chapter.tex + diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex new file mode 100644 index 0000000..f8d422e --- /dev/null +++ b/vorlesungen/slides/3/chapter.tex @@ -0,0 +1,7 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{3/polynome.tex} +\folie{3/motivation.tex} diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex new file mode 100644 index 0000000..cf28d46 --- /dev/null +++ b/vorlesungen/slides/3/motivation.tex @@ -0,0 +1,106 @@ +% +% 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)&= \only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}} +\only<4->{X^2-2X\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +p(D) &= \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} +$p(x) = -x^3-5x^2+5x+1$ +\[ +\only<5-8>{ +A= +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*}} +\only<6-8>{ +\quad\Rightarrow\quad} +\uncover<6->{ +- +\only<-9>{A^3}\only<10->{ +\begin{pmatrix*}[r] +-169&-35&35\\ +-185&-39&40\\ + -45&-10&11 +\end{pmatrix*}} +-5 +\only<-8>{A^2}\only<9->{ +\begin{pmatrix*}[r] +29&6&-6\\ +32&6&-5\\ + 8&1& 0 +\end{pmatrix*}} ++5 +\only<-7>{A}\only<8->{ +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*}} ++ +\only<-6>{I}\only<7->{ +\begin{pmatrix*}[r] +1&0&0\\ +0&1&0\\ +0&0&1 +\end{pmatrix*}} +} +\uncover<11->{=0} +\] +\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/polynome.tex b/vorlesungen/slides/3/polynome.tex new file mode 100644 index 0000000..6ca0009 --- /dev/null +++ b/vorlesungen/slides/3/polynome.tex @@ -0,0 +1,9 @@ +% +% polynome.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame} +\frametitle{Polynome} + +\end{frame} diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 3fdf7a4..910f87e 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -5,6 +5,7 @@ # include ../slides/0/Makefile.inc include ../slides/1/Makefile.inc +include ../slides/3/Makefile.inc slides = \ - $(chapter0) $(chapter1) + $(chapter0) $(chapter1) $(chapter3) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 7b0e02b..56164b2 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,14 +4,6 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{0/resourcen.tex} -%\folie{1/zahlensysteme.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} +\folie{3/polynome.tex} +\folie{3/motivation.tex} -- cgit v1.2.1 From 7c20e14c7fe17b8a5489888d682e7b021c52a72f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 25 Feb 2021 11:32:52 +0100 Subject: new slides --- vorlesungen/slides/1/Makefile.inc | 3 ++ vorlesungen/slides/1/bruch.tex | 72 ++++++++++++++++++++++++++ vorlesungen/slides/1/chapter.tex | 3 ++ vorlesungen/slides/1/ganz.tex | 104 ++++++++++++++++++++++++++++++++++++++ vorlesungen/slides/1/peano.tex | 72 ++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 8 +-- 6 files changed, 259 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/1/bruch.tex create mode 100644 vorlesungen/slides/1/ganz.tex create mode 100644 vorlesungen/slides/1/peano.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/1/Makefile.inc b/vorlesungen/slides/1/Makefile.inc index 3c1b5d4..46bf6b3 100644 --- a/vorlesungen/slides/1/Makefile.inc +++ b/vorlesungen/slides/1/Makefile.inc @@ -6,6 +6,9 @@ # chapter1 = \ ../slides/1/zahlensysteme.tex \ + ../slides/1/peano.tex \ + ../slides/1/ganz.tex \ + ../slides/1/bruch.tex \ ../slides/1/strukturen.tex \ ../slides/1/j.tex \ ../slides/1/vektorraum.tex \ diff --git a/vorlesungen/slides/1/bruch.tex b/vorlesungen/slides/1/bruch.tex new file mode 100644 index 0000000..f6e551b --- /dev/null +++ b/vorlesungen/slides/1/bruch.tex @@ -0,0 +1,72 @@ +% +% 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 +\[ +(b,a)\sim (d,c) +\only<5>{ +\Leftrightarrow +\text{`` +$\displaystyle +\frac{b}{a}=\frac{d}{c} +$ +''} +} +\only<6->{ +\Leftrightarrow +bc=ad +} +\] +\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 index fec3330..1b971f7 100644 --- a/vorlesungen/slides/1/chapter.tex +++ b/vorlesungen/slides/1/chapter.tex @@ -4,6 +4,9 @@ % (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/strukturen.tex} \folie{1/j.tex} \folie{1/vektorraum.tex} diff --git a/vorlesungen/slides/1/ganz.tex b/vorlesungen/slides/1/ganz.tex new file mode 100644 index 0000000..196a495 --- /dev/null +++ b/vorlesungen/slides/1/ganz.tex @@ -0,0 +1,104 @@ +% +% 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 +\[ +(b,a)\sim (d,c) +\only<6>{\Leftrightarrow +\text{``\strut} +b-a=c-d +\text{\strut''}} +\only<7->{ +\Leftrightarrow +b+d=c+a} +\] +\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} +Halbgruppe $\only<11>{\mathbb{Z}}\only<12->{G}$ mit inversem Element +\[ +a\in \only<11>{\mathbb{Z}}\only<12->{G} +\Rightarrow +\only<11>{-a\in\mathbb{Z}}\only<12->{a^{-1}\in G} +\text{ mit } +\only<11>{ +a+(-a)=0 +} +\only<12->{ +\left\{ +\begin{aligned} +aa^{-1}&=e +\\ +a^{-1}a&=e +\end{aligned} +\right. +} +\] +\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/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/test.tex b/vorlesungen/slides/test.tex index 56164b2..e10b0ab 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,6 +4,8 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % - -\folie{3/polynome.tex} -\folie{3/motivation.tex} +%\folie{1/peano.tex} +\folie{1/ganz.tex} +\folie{1/bruch.tex} +%\folie{3/polynome.tex} +%\folie{3/motivation.tex} -- cgit v1.2.1 From 155981423fb5d7f6fd612b54ff90e3b084e1a678 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 25 Feb 2021 12:47:04 +0100 Subject: Ring --- vorlesungen/slides/1/Makefile.inc | 2 + vorlesungen/slides/1/chapter.tex | 2 + vorlesungen/slides/1/ring.tex | 58 ++++++++++++++++++++ vorlesungen/slides/1/schwierigkeiten.tex | 90 ++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 6 ++- 5 files changed, 156 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/slides/1/ring.tex create mode 100644 vorlesungen/slides/1/schwierigkeiten.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/1/Makefile.inc b/vorlesungen/slides/1/Makefile.inc index 46bf6b3..38b47b3 100644 --- a/vorlesungen/slides/1/Makefile.inc +++ b/vorlesungen/slides/1/Makefile.inc @@ -9,6 +9,8 @@ chapter1 = \ ../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 \ diff --git a/vorlesungen/slides/1/chapter.tex b/vorlesungen/slides/1/chapter.tex index 1b971f7..7bdda34 100644 --- a/vorlesungen/slides/1/chapter.tex +++ b/vorlesungen/slides/1/chapter.tex @@ -7,6 +7,8 @@ \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} 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/test.tex b/vorlesungen/slides/test.tex index e10b0ab..58b7a8c 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -5,7 +5,9 @@ % %\folie{1/peano.tex} -\folie{1/ganz.tex} -\folie{1/bruch.tex} +%\folie{1/ganz.tex} +%\folie{1/bruch.tex} +%\folie{1/ring.tex} +\folie{1/schwierigkeiten.tex} %\folie{3/polynome.tex} %\folie{3/motivation.tex} -- cgit v1.2.1 From 0df3581aa0b48f62059b8822febd11b8c112c6d9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 25 Feb 2021 12:53:08 +0100 Subject: add missing files --- vorlesungen/setup | 35 +++++++++++++++++++++++++++++++++++ 1 file changed, 35 insertions(+) create mode 100644 vorlesungen/setup (limited to 'vorlesungen') diff --git a/vorlesungen/setup b/vorlesungen/setup new file mode 100644 index 0000000..a6f5539 --- /dev/null +++ b/vorlesungen/setup @@ -0,0 +1,35 @@ +#! /bin/bash +# +# setup -- prepare a new topic +# +if [ $# -lt 2 ] +then + echo "not enough arguments" + echo "usage: setup number topic" + echo "Files will be in _, -handout, MathSem--.tex" + exit 1 +fi + +topic=${2} +number=`printf %02d ${1}` +directory=${number}_${topic} + +if [ -d ${directory} ] +then + echo ${directory} already exists + exit 1 +fi + +mkdir ${directory} + +cp 00_template/common.tex ${directory} +cp 00_template/slides.tex ${directory} + +sed -e "s/xxx/${topic}/g" -e "s/yyy/${number}/g" 00_template/Makefile \ + > ${directory}/Makefile +sed -e "s/xxx/${topic}/g" -e "s/yyy/${number}/g" 00_template/handout.tex \ + > ${directory}/${topic}-handout.tex +sed -e "s/xxx/${topic}/g" -e "s/yyy/${number}/g" 00_template/presentation.tex \ + > ${directory}/MathSem-${number}-${topic}.tex + + -- cgit v1.2.1 From c375af98f136f158bc00b8895e8843be8feaa758 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 25 Feb 2021 13:11:56 +0100 Subject: MathSemMSE slides session1 --- vorlesungen/01_mseintro/Makefile | 26 +++++++++++++++++++++ vorlesungen/01_mseintro/MathSemMSE-01-intro.tex | 14 +++++++++++ vorlesungen/01_mseintro/common.tex | 16 +++++++++++++ vorlesungen/01_mseintro/intro-handout.tex | 11 +++++++++ vorlesungen/01_mseintro/slides.tex | 31 +++++++++++++++++++++++++ vorlesungen/slides/1/bruch.tex | 3 ++- vorlesungen/slides/1/ganz.tex | 12 ++++++---- vorlesungen/slides/test.tex | 1 + 8 files changed, 108 insertions(+), 6 deletions(-) create mode 100644 vorlesungen/01_mseintro/Makefile create mode 100644 vorlesungen/01_mseintro/MathSemMSE-01-intro.tex create mode 100644 vorlesungen/01_mseintro/common.tex create mode 100644 vorlesungen/01_mseintro/intro-handout.tex create mode 100644 vorlesungen/01_mseintro/slides.tex (limited to 'vorlesungen') diff --git a/vorlesungen/01_mseintro/Makefile b/vorlesungen/01_mseintro/Makefile new file mode 100644 index 0000000..43fc0d6 --- /dev/null +++ b/vorlesungen/01_mseintro/Makefile @@ -0,0 +1,26 @@ +# +# Makefile -- intro +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: intro-handout.pdf MathSemMSE-01-intro.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-01-intro.pdf: MathSemMSE-01-intro.tex $(SOURCES) + pdflatex MathSemMSE-01-intro.tex + +intro-handout.pdf: intro-handout.tex $(SOURCES) + pdflatex intro-handout.tex + +thumbnail: thumbnail.jpg + +thumbnail.pdf: MathSemMSE-01-intro.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-01-intro.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + diff --git a/vorlesungen/01_mseintro/MathSemMSE-01-intro.tex b/vorlesungen/01_mseintro/MathSemMSE-01-intro.tex new file mode 100644 index 0000000..0f59cfe --- /dev/null +++ b/vorlesungen/01_mseintro/MathSemMSE-01-intro.tex @@ -0,0 +1,14 @@ +% +% MathSem-01-mse.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/01_mseintro/common.tex b/vorlesungen/01_mseintro/common.tex new file mode 100644 index 0000000..89f2520 --- /dev/null +++ b/vorlesungen/01_mseintro/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Einführung]{Zahlensysteme und Strukturen} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/01_mseintro/intro-handout.tex b/vorlesungen/01_mseintro/intro-handout.tex new file mode 100644 index 0000000..52ce936 --- /dev/null +++ b/vorlesungen/01_mseintro/intro-handout.tex @@ -0,0 +1,11 @@ +% +% mse-handout.tex -- Handout XXX +% +% (c) 2017 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/01_mseintro/slides.tex b/vorlesungen/01_mseintro/slides.tex new file mode 100644 index 0000000..5526276 --- /dev/null +++ b/vorlesungen/01_mseintro/slides.tex @@ -0,0 +1,31 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\folie{0/intro.tex} + +\section{Zahlensysteme} +\folie{1/peano.tex} +\folie{1/ganz.tex} +\folie{1/bruch.tex} +\folie{1/ring.tex} +\folie{1/schwierigkeiten.tex} +\folie{1/vektorraum.tex} +\folie{1/algebrastruktur.tex} + +\section{Übersicht} +\folie{1/zahlensysteme.tex} +\folie{1/strukturen.tex} + +\section{Spezielles} +\folie{1/j.tex} +\folie{1/dreieck.tex} +\folie{1/matrixalgebra.tex} +\folie{1/speziell.tex} + +\section{Hadamard-Algebra} + +\section{Hadamard-Algebra} +\folie{1/hadamard.tex} + diff --git a/vorlesungen/slides/1/bruch.tex b/vorlesungen/slides/1/bruch.tex index f6e551b..65521a2 100644 --- a/vorlesungen/slides/1/bruch.tex +++ b/vorlesungen/slides/1/bruch.tex @@ -28,6 +28,7 @@ Idee: $\displaystyle\frac{b}{a} = (b,a)$ \item<5-> Äquivalenzrelation \[ (b,a)\sim (d,c) +\ifthenelse{\boolean{presentation}}{ \only<5>{ \Leftrightarrow \text{`` @@ -35,7 +36,7 @@ $\displaystyle \frac{b}{a}=\frac{d}{c} $ ''} -} +}}{} \only<6->{ \Leftrightarrow bc=ad diff --git a/vorlesungen/slides/1/ganz.tex b/vorlesungen/slides/1/ganz.tex index 196a495..f1022f4 100644 --- a/vorlesungen/slides/1/ganz.tex +++ b/vorlesungen/slides/1/ganz.tex @@ -31,10 +31,11 @@ Idee: $b-a = (b,a)$ \item<5-> Äquivalenzrelation \[ (b,a)\sim (d,c) +\ifthenelse{\boolean{presentation}}{ \only<6>{\Leftrightarrow \text{``\strut} b-a=c-d -\text{\strut''}} +\text{\strut''}}}{} \only<7->{ \Leftrightarrow b+d=c+a} @@ -61,15 +62,16 @@ $\Rightarrow$ alle Differenzen in $\mathbb{Z}$} \setlength{\belowdisplayskip}{5pt} \uncover<11->{% \begin{block}{Gruppe} -Halbgruppe $\only<11>{\mathbb{Z}}\only<12->{G}$ mit inversem Element +Halbgruppe $\ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}{}\only<12->{G}$ mit inversem Element \[ -a\in \only<11>{\mathbb{Z}}\only<12->{G} +a\in \ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}\only<12->{G} \Rightarrow -\only<11>{-a\in\mathbb{Z}}\only<12->{a^{-1}\in G} +\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} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 58b7a8c..0cc5452 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -9,5 +9,6 @@ %\folie{1/bruch.tex} %\folie{1/ring.tex} \folie{1/schwierigkeiten.tex} +\folie{1/vektorraum.tex} %\folie{3/polynome.tex} %\folie{3/motivation.tex} -- cgit v1.2.1 From 3eb564bb021d51d6a8201ec3069e0acd08554f98 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 26 Feb 2021 13:07:53 +0100 Subject: typos --- vorlesungen/slides/1/algebrastruktur.tex | 2 +- vorlesungen/slides/1/ganz.tex | 6 +++--- 2 files changed, 4 insertions(+), 4 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/1/algebrastruktur.tex b/vorlesungen/slides/1/algebrastruktur.tex index d7794a3..fd474eb 100644 --- a/vorlesungen/slides/1/algebrastruktur.tex +++ b/vorlesungen/slides/1/algebrastruktur.tex @@ -53,7 +53,7 @@ \node at ({-3-0.5},2) {Skalarmultiplikation}; \node at (3.5,2.2) {Multiplikation}; -\node at (3.5,1.8) {\tiny Halbgruppe}; +\node at (3.5,1.8) {\tiny Monoid}; \node at (0,-2.8) {Addition}; \node at (0,-3.2) {\tiny Gruppe}; diff --git a/vorlesungen/slides/1/ganz.tex b/vorlesungen/slides/1/ganz.tex index f1022f4..7930826 100644 --- a/vorlesungen/slides/1/ganz.tex +++ b/vorlesungen/slides/1/ganz.tex @@ -62,9 +62,9 @@ $\Rightarrow$ alle Differenzen in $\mathbb{Z}$} \setlength{\belowdisplayskip}{5pt} \uncover<11->{% \begin{block}{Gruppe} -Halbgruppe $\ifthenelse{\boolean{presentation}}{\only<11>{\mathbb{Z}}}{}\only<12->{G}$ mit inversem Element +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} +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 } @@ -95,7 +95,7 @@ a+b=b+a \uncover<14->{% \begin{block}{Beispiele} \begin{itemize} -\item<15-> Brüche reelle Zahlen +\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 $1$: $\operatorname{SL}_n(\mathbb R)$ -- cgit v1.2.1 From cddeab74130c3814acd49c8ac7d03041b2a7b85d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 28 Feb 2021 22:37:26 +0100 Subject: add new slides --- vorlesungen/slides/3/Makefile.inc | 15 +++++++ vorlesungen/slides/3/adjunktion.tex | 30 +++++++++++++ vorlesungen/slides/3/chapter.tex | 15 ++++++- vorlesungen/slides/3/division.tex | 29 ++++++++++++ vorlesungen/slides/3/division2.tex | 32 +++++++++++++ vorlesungen/slides/3/drehfaktorisierung.tex | 70 +++++++++++++++++++++++++++++ vorlesungen/slides/3/drehmatrix.tex | 62 +++++++++++++++++++++++++ vorlesungen/slides/3/einsetzen.tex | 50 +++++++++++++++++++++ vorlesungen/slides/3/faktorisierung.tex | 42 +++++++++++++++++ vorlesungen/slides/3/faktorzerlegung.tex | 58 ++++++++++++++++++++++++ vorlesungen/slides/3/maximalergrad.tex | 67 +++++++++++++++++++++++++++ vorlesungen/slides/3/minimalbeispiel.tex | 34 ++++++++++++++ vorlesungen/slides/3/minimalpolynom.tex | 27 +++++++++++ vorlesungen/slides/3/operatoren.tex | 48 ++++++++++++++++++++ vorlesungen/slides/3/polynome.tex | 22 ++++++++- vorlesungen/slides/3/ringstruktur.tex | 50 +++++++++++++++++++++ vorlesungen/slides/3/teilbarkeit.tex | 44 ++++++++++++++++++ vorlesungen/slides/test.tex | 28 +++++++++--- 18 files changed, 714 insertions(+), 9 deletions(-) create mode 100644 vorlesungen/slides/3/adjunktion.tex create mode 100644 vorlesungen/slides/3/division.tex create mode 100644 vorlesungen/slides/3/division2.tex create mode 100644 vorlesungen/slides/3/drehfaktorisierung.tex create mode 100644 vorlesungen/slides/3/drehmatrix.tex create mode 100644 vorlesungen/slides/3/einsetzen.tex create mode 100644 vorlesungen/slides/3/faktorisierung.tex create mode 100644 vorlesungen/slides/3/faktorzerlegung.tex create mode 100644 vorlesungen/slides/3/maximalergrad.tex create mode 100644 vorlesungen/slides/3/minimalbeispiel.tex create mode 100644 vorlesungen/slides/3/minimalpolynom.tex create mode 100644 vorlesungen/slides/3/operatoren.tex create mode 100644 vorlesungen/slides/3/ringstruktur.tex create mode 100644 vorlesungen/slides/3/teilbarkeit.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index a58ee05..ac2c3a5 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -6,5 +6,20 @@ # chapter3 = \ ../slides/3/motivation.tex \ + ../slides/3/polynome.tex \ + ../slides/3/division.tex \ + ../slides/3/division2.tex \ + ../slides/3/ringstruktur.tex \ + ../slides/3/teilbarkeit.tex \ + ../slides/3/faktorisierung.tex \ + ../slides/3/faktorzerlegung.tex \ + ../slides/3/einsetzen.tex \ + ../slides/3/maximalergrad.tex \ + ../slides/3/minimalbeispiel.tex \ + ../slides/3/minimalpolynom.tex \ + ../slides/3/drehmatrix.tex \ + ../slides/3/drehfaktorisierung.tex \ + ../slides/3/operatoren.tex \ + ../slides/3/adjunktion.tex \ ../slides/3/chapter.tex diff --git a/vorlesungen/slides/3/adjunktion.tex b/vorlesungen/slides/3/adjunktion.tex new file mode 100644 index 0000000..3b55ab0 --- /dev/null +++ b/vorlesungen/slides/3/adjunktion.tex @@ -0,0 +1,30 @@ +% +% 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. +\[ +X^n = -m_{n-1}X^{n-1} - \dots - m_1X - m_0 +\] +Nullstelle $W$ als Operator betrachten: +\[ +W = \begin{pmatrix} + 0& 0& 0&\dots & 0& -m_0\\ + 1& 0& 0&\dots & 0& -m_1\\ + 0& 1& 0&\dots & 0& -m_2\\ + 0& 0& 1&\dots & 0& -m_3\\ +\vdots&\vdots&\vdots&\ddots&\vdots& \vdots\\ + 0& 0& 0&\dots & 1&-m_{n-1} +\end{pmatrix} +\] +Man kann nachrechnen, dass immer $m(W)=0$. +\medskip + +$\Rightarrow \Bbbk(W) = \{p(W)\;|\;p\in\Bbbk[X], \deg p<\deg m\}$ +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 index f8d422e..672b05a 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -3,5 +3,18 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % -\folie{3/polynome.tex} \folie{3/motivation.tex} +\folie{3/polynome.tex} +\folie{3/division.tex} +\folie{3/division2.tex} +\folie{3/ringstruktur.tex} +\folie{3/teilbarkeit.tex} +\folie{3/faktorisierung.tex} +\folie{3/faktorzerlegung.tex} +\folie{3/einsetzen.tex} +\folie{3/maximalergrad.tex} +\folie{3/minimalbeispiel.tex} +\folie{3/minimalpolynom.tex} +\folie{3/drehfaktorisierung.tex} +\folie{3/operatoren.tex} +\folie{3/adjunktion.tex} diff --git a/vorlesungen/slides/3/division.tex b/vorlesungen/slides/3/division.tex new file mode 100644 index 0000000..0932f48 --- /dev/null +++ b/vorlesungen/slides/3/division.tex @@ -0,0 +1,29 @@ +% +% 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} +\begin{block}{Lösung} +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcrcrcrcr} +X^4&-& X^3&-&7X^2&+& X&+&6&:&X^2&+&X&+&1&=&X^2&-&2X&-&6=q\\ +\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}& & & & & & & & & & & & & & & & \\ \cline{1-5} + &-&2X^3&-&8X^2&+& X& & & & & & & & & & & & & & \\ + &\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}& & & & & & & & & & & & & & \\ \cline{2-7} + & & &-&6X^2&+&3X&+&6& & & & & & & & & & & & \\ + & & &\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}& & & & & & & & & & & & \\ \cline{4-9} + & & & & & &9X&+&12\rlap{$\mathstrut=r$}& & & & & & & & & & & & \\ \cline{7-9} +\end{array} +\] +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..80d6a75 --- /dev/null +++ b/vorlesungen/slides/3/division2.tex @@ -0,0 +1,32 @@ +% +% 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} +\begin{block}{Lösung} +\[ +\arraycolsep=1.4pt +\renewcommand{\arraystretch}{1.2} +\begin{array}{rcrcrcrcrcrcrcrcrcrcr} +X^4&-& X^3&-& 7X^2&+& X&+& 6&:&2X^2&+&X&+&1&=&\frac12X^2&-&\frac34X&-\frac{27}{8} = q\\ +\llap{$-($}X^4&+&\frac12X^3&+& \frac12X^2\rlap{$)$}& & & & & & & & & & & & & & & \\ \cline{1-5} + &-&\frac32X^3&-&\frac{15}2X^2&+& X& & & & & & & & & & & & & \\ + &\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}& & & & & & & & & & & & & \\\cline{2-7} + & & &-&\frac{27}4X^2&+&\frac{ 7}4X&+& 6& & & & & & & & & & & \\ + & & &\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}& & & & & & & & & & & \\\cline{4-9} + & & & & & &\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}& & & & & & & & & & & \\ +\end{array} +\] +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..b44ca35 --- /dev/null +++ b/vorlesungen/slides/3/drehfaktorisierung.tex @@ -0,0 +1,70 @@ +% +% 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 +\[ +\biggl(X+\frac12+\frac{i\sqrt{3}}2\biggr) +\biggl(X+\frac12-\frac{i\sqrt{3}}2\biggr) += +X^2+X+\frac14 ++ +\frac34 += +X^2+X+1 +\] +\vspace{-10pt} +\begin{block}{Was ist $i\sqrt{3}$?} +Matrix mit Minimalpolynom $X^2+3$: +\[ +W=\begin{pmatrix}0&-3\\1&0\end{pmatrix} +\qquad\Rightarrow\qquad +W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I +\qquad\Rightarrow\qquad +W^2+3I=0 +\] +\end{block} +\vspace{-10pt} +\begin{block}{Faktorisierung von $X^2+X+1$} +\vspace{-10pt} +\begin{align*} +B_\pm +&= +-\frac12I\pm\frac12W +& +&\Rightarrow +& +(X+B_+)(X+B-) +&= +(X+\frac12I+\frac12W) +(X+\frac12I-\frac12W) +\\ +&= +\smash{ +{\textstyle\begin{pmatrix}-\frac12&-\frac32\\\frac12&-\frac12\end{pmatrix}} +} +& +& +& +&= +X^2+X + \frac14I - \frac14W^2 +\\ +& +& +&%\Rightarrow +& +&= +X^2+X + \frac14I + \frac34I += +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..bed0628 --- /dev/null +++ b/vorlesungen/slides/3/drehmatrix.tex @@ -0,0 +1,62 @@ +% +% drehmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Analyse einer Drehung um $120^\circ$} +$D$ eine Drehung um $120^\circ$ +\begin{enumerate} +\item +Drehwinkel = $120^\circ\quad\Rightarrow\quad D^3 = I$ +\item +$m_D(X)=X^3-1$ +\item +$m_D$ ist nicht irreduzibel, weil $m_D(1)=0$: +$ +m_D(X) = (X-1)(X^2+X+1) +$ +\item +Welche Matrix hat $X^2+X+1$ als Minimalpolynom +\[ +\arraycolsep=1.4pt +W += +\biggl(\begin{array}{cc} +-\frac12 & -\frac{\sqrt{3}}2 \\ + \frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) +\quad\Rightarrow\quad +W^2+W+I += +\biggl(\begin{array}{cc} +-\frac12 & -\frac{\sqrt{3}}2 \\ + \frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) ++ +\biggl(\begin{array}{cc} +-\frac12 & \frac{\sqrt{3}}2 \\ + -\frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) ++ +\biggl(\begin{array}{cc} +1&0\\0&1 +\end{array}\biggr) +=0 +\] +\item In einer geeigneten Basis hat $D$ die Form +\[ +D=\begin{pmatrix} +1&0&0\\ +0&-\frac12 & -\frac{\sqrt{3}}2 \\ +0&\frac{\sqrt{3}}2 & -\frac12 +\end{pmatrix} += +\begin{pmatrix} +1&0&0\\ +0&\cos 120^\circ & -\sin 120^\circ\\ +0&\sin 120^\circ & \cos 120^\circ +\end{pmatrix} +\] +\end{enumerate} +\end{frame} diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex new file mode 100644 index 0000000..936100d --- /dev/null +++ b/vorlesungen/slides/3/einsetzen.tex @@ -0,0 +1,50 @@ +% +% einsetzen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Matrix in ein Polynom einsetzen} +\vspace{-10pt} +\[ +\begin{array}{rcrcrcrcrcrcr} +p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\ +\bigg\downarrow\hspace*{4pt} & & +\bigg\downarrow\hspace*{4pt} & & +\bigg\downarrow\hspace*{10pt} & & & & +\bigg\downarrow\hspace*{4pt} & & +\bigg\downarrow\hspace*{2pt} & & +\bigg\downarrow\hspace*{0pt} \\ +p(A)&=&a_nA^n&+&a_{n-1}A^{n-1}&+&\dots&+&a_2A^2&+&a_1A&+&a_0 I +\end{array} +\] +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Nilpotente Matrizen} +$p(X) = (X-a)^n$ +\[ +p(A) = 0 +\quad\Rightarrow\quad +\text{$A$ ist nilpotent} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Eigenwerte} +$p(X) = (X-\lambda_1)(X-\lambda_2)$,\\ +$A$ eine $2\times 2$-Matrix +\[ +p(A)=0\quad\Rightarrow\quad +\left\{ +\begin{aligned} +&\text{$A-\lambda_1I$ ist singulär}\\ +&\text{$A-\lambda_2I$ ist singulär} +\end{aligned} +\right. +\] +\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..cbf7004 --- /dev/null +++ b/vorlesungen/slides/3/faktorisierung.tex @@ -0,0 +1,42 @@ +% +% 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*} +p&=7 +\\ +2021 &= 43 \cdot 47 +\\ +4095667&=2021\cdot 2027 +\\ +p&=43, 47, 1291, 2017, 2027 +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\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*} +p&=X-9 +\\ +X^2-1&= (X+1)(X-1) +\\ +X^2-2&\text{\; irreduzibel} +\\ +X^2-2&=(X-\sqrt{2})(X+\sqrt{2}) +\end{align*} +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..0e5a95b --- /dev/null +++ b/vorlesungen/slides/3/faktorzerlegung.tex @@ -0,0 +1,58 @@ +% +% 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} +\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} +\begin{block}{Polynomfaktorisierung hängt von den Koeffizienten ab} +Ist $X^2-2$ irreduzibel? +\vspace{-5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{in $\mathbb{Q}[X]$} +\[ +X^2-2\quad\text{ist irreduzibel} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\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/maximalergrad.tex b/vorlesungen/slides/3/maximalergrad.tex new file mode 100644 index 0000000..3d7e1a4 --- /dev/null +++ b/vorlesungen/slides/3/maximalergrad.tex @@ -0,0 +1,67 @@ +% +% 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*} +&\begin{pmatrix} +1&0&\dots&0\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix} +& +&\begin{pmatrix} +0&1&\dots&0\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix} +& +&\dots +& +&\begin{pmatrix} +0&0&\dots&1\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix} +\\ +&\begin{pmatrix} +0&0&\dots&0\\ +1&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix} +& +&\begin{pmatrix} +0&0&\dots&0\\ +0&1&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix} +& +&\dots +& +&\begin{pmatrix} +0&0&\dots&0\\ +0&0&\dots&1\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix} +\end{align*}} +\end{block} +\vspace{-10pt} +\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: +\[ +a_0I+a_1A+a_2A^2+\dots+a_{n^2-1}A^{n^2-1}+a_{n^2}A^{n^2} = 0 +\] +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} +$\Rightarrow$ $A$ über die Eigenschaften (Faktorisierung) von $p$ studieren +\end{frame} diff --git a/vorlesungen/slides/3/minimalbeispiel.tex b/vorlesungen/slides/3/minimalbeispiel.tex new file mode 100644 index 0000000..03909de --- /dev/null +++ b/vorlesungen/slides/3/minimalbeispiel.tex @@ -0,0 +1,34 @@ +% +% 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} +\[ +I ={\tiny\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}},\quad +A ={\tiny\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix}},\quad +A^2={\tiny\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix}},\quad +A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}},\quad +A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 & 2 \end{pmatrix}} +\] +\end{block} +\vspace{-5pt} +\begin{block}{linear abhängig} +Bereits die ersten $3$ sind linear abhängig: +\[ +-4I - A + A^2 += +-4\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} +-\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix} ++\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix} += +\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} +\] +$p(X) = X^2 - X - 4 \in \mathbb{Q}[X]$ 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..60d15f0 --- /dev/null +++ b/vorlesungen/slides/3/minimalpolynom.tex @@ -0,0 +1,27 @@ +% +% 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} +\begin{block}{Strategie} +Das Minimalpolynom ist eine ``Invariante'' der Matrix $A$ +\end{block} +\begin{block}{Satz von Cayley-Hamilton} +Für jede $n\times n$-Matrix $A\in M_n(\Bbbk)$ gilt $\chi_A(A)=0$ +\[ +\Downarrow +\] +Das Minimalpolynom $m_A\in \Bbbk[X]$ ist ein Teiler +des charakteristischen Polynoms $\chi_A\in \Bbbk[X]$ +\\ +$\Rightarrow $ +Faktorzerlegung on $\chi_A(X)$ ermitteln! +\end{block} +\end{frame} diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex new file mode 100644 index 0000000..20b7f3a --- /dev/null +++ b/vorlesungen/slides/3/operatoren.tex @@ -0,0 +1,48 @@ +% +% 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} +$a(X)=a_0+a_1X+\dots+a_nX^n$ +\[ +a(X) += +\begin{pmatrix} +a_0\\a_1\\a_2\\a_3\\\vdots\\a_nX^n +\end{pmatrix} +\] +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\begin{block}{Multiplikation mit $X$} +\strut +\[ +\begin{pmatrix} +1\\0\\0\\0\\\vdots\\0 +\end{pmatrix} +\mapsto +\begin{pmatrix} +0\\1\\0\\0\\\vdots\\0 +\end{pmatrix} +\mapsto +\begin{pmatrix} +0\\0\\1\\0\\\vdots\\0 +\end{pmatrix} +\mapsto +\begin{pmatrix} +0\\0\\0\\1\\\vdots\\0 +\end{pmatrix} +\mapsto\dots\mapsto +\begin{pmatrix} +0\\0\\0\\0\\\vdots\\1 +\end{pmatrix} +\] +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/polynome.tex b/vorlesungen/slides/3/polynome.tex index 6ca0009..161a4ed 100644 --- a/vorlesungen/slides/3/polynome.tex +++ b/vorlesungen/slides/3/polynome.tex @@ -3,7 +3,27 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % -\begin{frame} +\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 {\em Grad} des Polynoms: $\deg a(X) = \deg a = n$ +\item $\deg 0 = -\infty$ +\item {\em normiertes Polynom}: $a_n=1$ +\end{itemize} + \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..25e4fa6 --- /dev/null +++ b/vorlesungen/slides/3/teilbarkeit.tex @@ -0,0 +1,44 @@ +% +% 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 $a,b\in \mathbb{Z}$, $a>b$ gibt es +immer genau ein Paar $q,r\in\mathbb{Z}$ derart, dass +\begin{align*} +a&=bq+r +\\ +r&< b +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Teilen in $\mathbb{Q}[X]$} +Zu zwei Polynomen $a,b\in\mathbb{Q}[X]$, $\deg a > \deg b$ +gibt es +immer genau ein Paar $q,r\in\mathbb{Q}[X]$ derart, dass +\begin{align*} +a&=bq+r +\\ +\deg r&< \deg b +\end{align*} +\end{block} +\end{column} +\end{columns} +\begin{block}{Allgmein: 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/test.tex b/vorlesungen/slides/test.tex index 0cc5452..24d9ffa 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,11 +4,25 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{1/peano.tex} -%\folie{1/ganz.tex} -%\folie{1/bruch.tex} -%\folie{1/ring.tex} -\folie{1/schwierigkeiten.tex} -\folie{1/vektorraum.tex} -%\folie{3/polynome.tex} %\folie{3/motivation.tex} +%\folie{3/polynome.tex} +%\folie{3/division.tex} +%\folie{3/division2.tex} +%\folie{3/ringstruktur.tex} +%\folie{3/teilbarkeit.tex} +%\folie{3/faktorisierung.tex} +%\folie{3/faktorzerlegung.tex} +%\folie{3/einsetzen.tex} +%\folie{3/maximalergrad.tex} +%\folie{3/minimalbeispiel.tex} +%\folie{3/minimalpolynom.tex} +\folie{3/drehmatrix.tex} +\folie{3/drehfaktorisierung.tex} +\folie{3/operatoren.tex} +\folie{3/adjunktion.tex} +% XXX \folie{3/adjunktioni.tex} +% XXX \folie{3/adjunktionsqrt2.tex} +% XXX \folie{3/adjunktionphi.tex} +% Adjunktion von \cos(\pi/1291) und \cos(\pi/1291) +% XXX \folie{3/adj1291.tex} + -- cgit v1.2.1 From 67addea9bf154558ca3bbc14703c2b0fef8fc60c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 08:18:59 +0100 Subject: phases, typos --- vorlesungen/slides/3/Makefile.inc | 1 + vorlesungen/slides/3/chapter.tex | 2 ++ vorlesungen/slides/3/division.tex | 19 +++++++++++-------- vorlesungen/slides/3/division2.tex | 18 ++++++++++-------- vorlesungen/slides/3/operatoren.tex | 2 +- vorlesungen/slides/3/polynome.tex | 6 +++--- vorlesungen/slides/test.tex | 25 +++++++++++++------------ 7 files changed, 41 insertions(+), 32 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index ac2c3a5..3a55274 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -6,6 +6,7 @@ # chapter3 = \ ../slides/3/motivation.tex \ + ../slides/3/inverse.tex \ ../slides/3/polynome.tex \ ../slides/3/division.tex \ ../slides/3/division2.tex \ diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex index 672b05a..e63002c 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -4,6 +4,7 @@ % (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} @@ -15,6 +16,7 @@ \folie{3/maximalergrad.tex} \folie{3/minimalbeispiel.tex} \folie{3/minimalpolynom.tex} +\folie{3/drehmatrix.tex} \folie{3/drehfaktorisierung.tex} \folie{3/operatoren.tex} \folie{3/adjunktion.tex} diff --git a/vorlesungen/slides/3/division.tex b/vorlesungen/slides/3/division.tex index 0932f48..94df27b 100644 --- a/vorlesungen/slides/3/division.tex +++ b/vorlesungen/slides/3/division.tex @@ -11,19 +11,22 @@ $a= X^4- X^3-7X^2+ X+6\in\mathbb{Z}[X]$ und $b= X^2+X+1\in\mathbb{Z}[X]$ \end{block} +\uncover<2->{% \begin{block}{Lösung} \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcrcrcrcrcrcrcr} -X^4&-& X^3&-&7X^2&+& X&+&6&:&X^2&+&X&+&1&=&X^2&-&2X&-&6=q\\ -\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}& & & & & & & & & & & & & & & & \\ \cline{1-5} - &-&2X^3&-&8X^2&+& X& & & & & & & & & & & & & & \\ - &\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}& & & & & & & & & & & & & & \\ \cline{2-7} - & & &-&6X^2&+&3X&+&6& & & & & & & & & & & & \\ - & & &\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}& & & & & & & & & & & & \\ \cline{4-9} - & & & & & &9X&+&12\rlap{$\mathstrut=r$}& & & & & & & & & & & & \\ \cline{7-9} +\llap{$($}X^4&-& X^3&-&7X^2&+& X&+&6\rlap{$)$}&\;\mathstrut:\mathstrut&(X^2&+&X&+&1)&=&\uncover<3->{X^2}&\uncover<7->{-&2X}&\uncover<11->{-&6}=q\\ +\uncover<4->{\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}}& & & & & & & & & & & & & & & & \\ + &\uncover<5->{-&2X^3&-&8X^2}&\uncover<6->{+& X}& & & & & & & & & & & & & & \\ + &\uncover<8->{\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}}& & & & & & & & & & & & & & \\ + & & &\uncover<9->{-&6X^2&+&3X}&\uncover<10->{+&6}& & & & & & & & & & & & \\ + & & &\uncover<12->{\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}}& & & & & & & & & & & & \\ + & & & & & &\uncover<13->{9X&+&12\rlap{$\mathstrut=r$}}& & & & & & & & & & & & \end{array} \] +\uncover<14->{ Funktioniert weil $b$ normiert ist! -\end{block} +} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/3/division2.tex b/vorlesungen/slides/3/division2.tex index 80d6a75..0602598 100644 --- a/vorlesungen/slides/3/division2.tex +++ b/vorlesungen/slides/3/division2.tex @@ -12,21 +12,23 @@ $a(X) = X^4-X^3-7X^2+X+6$ und $b(X) = 2X^2+X+1$ \end{block} +\uncover<2->{% \begin{block}{Lösung} +\vspace{-15pt} \[ \arraycolsep=1.4pt \renewcommand{\arraystretch}{1.2} \begin{array}{rcrcrcrcrcrcrcrcrcrcr} -X^4&-& X^3&-& 7X^2&+& X&+& 6&:&2X^2&+&X&+&1&=&\frac12X^2&-&\frac34X&-\frac{27}{8} = q\\ -\llap{$-($}X^4&+&\frac12X^3&+& \frac12X^2\rlap{$)$}& & & & & & & & & & & & & & & \\ \cline{1-5} - &-&\frac32X^3&-&\frac{15}2X^2&+& X& & & & & & & & & & & & & \\ - &\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}& & & & & & & & & & & & & \\\cline{2-7} - & & &-&\frac{27}4X^2&+&\frac{ 7}4X&+& 6& & & & & & & & & & & \\ - & & &\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}& & & & & & & & & & & \\\cline{4-9} - & & & & & &\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}& & & & & & & & & & & \\ +\llap{$($}X^4&-& X^3&-& 7X^2&+& X&+& 6\rlap{$)$}&\mathstrut\;:\mathstrut&(2X^2&+&X&+&1)&=&\uncover<3->{\frac12X^2}&\uncover<7->{-&\frac34X}&\uncover<11->{-\frac{27}{8}} = q\\ +\uncover<4->{\llap{$-($}X^4&+&\frac12X^3&+& \frac12X^2\rlap{$)$}}& & & & & & & & & & & & & & & \\ + &\uncover<5->{-&\frac32X^3&-&\frac{15}2X^2}&\uncover<6->{+& X}& & & & & & & & & & & & & \\ + &\uncover<8->{\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}}& & & & & & & & & & & & & \\ + & & &\uncover<9->{-&\frac{27}4X^2&+&\frac{ 7}4X}&\uncover<10->{+& 6}& & & & & & & & & & & \\ + & & &\uncover<12->{\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}}& & & & & & & & & & & \\ + & & & & & &\uncover<13->{\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}}& & & & & & & & & & & \\ \end{array} \] Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann -\end{block} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex index 20b7f3a..161c330 100644 --- a/vorlesungen/slides/3/operatoren.tex +++ b/vorlesungen/slides/3/operatoren.tex @@ -13,7 +13,7 @@ $a(X)=a_0+a_1X+\dots+a_nX^n$ a(X) = \begin{pmatrix} -a_0\\a_1\\a_2\\a_3\\\vdots\\a_nX^n +a_0\\a_1\\a_2\\a_3\\\vdots\\a_n \end{pmatrix} \] \end{block} diff --git a/vorlesungen/slides/3/polynome.tex b/vorlesungen/slides/3/polynome.tex index 161a4ed..d7179a0 100644 --- a/vorlesungen/slides/3/polynome.tex +++ b/vorlesungen/slides/3/polynome.tex @@ -20,9 +20,9 @@ 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 {\em Grad} des Polynoms: $\deg a(X) = \deg a = n$ -\item $\deg 0 = -\infty$ -\item {\em normiertes Polynom}: $a_n=1$ +\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} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 24d9ffa..3c0348f 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,18 +4,19 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{3/motivation.tex} -%\folie{3/polynome.tex} -%\folie{3/division.tex} -%\folie{3/division2.tex} -%\folie{3/ringstruktur.tex} -%\folie{3/teilbarkeit.tex} -%\folie{3/faktorisierung.tex} -%\folie{3/faktorzerlegung.tex} -%\folie{3/einsetzen.tex} -%\folie{3/maximalergrad.tex} -%\folie{3/minimalbeispiel.tex} -%\folie{3/minimalpolynom.tex} +\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/faktorisierung.tex} +\folie{3/faktorzerlegung.tex} +\folie{3/einsetzen.tex} +\folie{3/maximalergrad.tex} +\folie{3/minimalbeispiel.tex} +\folie{3/minimalpolynom.tex} \folie{3/drehmatrix.tex} \folie{3/drehfaktorisierung.tex} \folie{3/operatoren.tex} -- cgit v1.2.1 From f42288021bcb0259edfed0f79cc58cb7058528f7 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 08:19:12 +0100 Subject: add new slides --- vorlesungen/slides/3/inverse.tex | 89 ++++++++++++++++++++++++++++++++++++++++ 1 file changed, 89 insertions(+) create mode 100644 vorlesungen/slides/3/inverse.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex new file mode 100644 index 0000000..28716bc --- /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 &= \only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}} +\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +0 &= \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} -- cgit v1.2.1 From 20d5294ab3401613076723ccb942dfbc484b74b9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 09:12:46 +0100 Subject: phases, new slides --- vorlesungen/slides/3/Makefile.inc | 1 + vorlesungen/slides/3/adjalgebra.tex | 43 +++++++++++++++++++++++++ vorlesungen/slides/3/adjunktion.tex | 11 +++++-- vorlesungen/slides/3/chapter.tex | 1 + vorlesungen/slides/3/drehfaktorisierung.tex | 49 ++++++++++++++++------------- vorlesungen/slides/3/drehmatrix.tex | 24 ++++++++------ vorlesungen/slides/3/einsetzen.tex | 30 ++++++++++-------- vorlesungen/slides/3/faktorisierung.tex | 23 ++++++++------ vorlesungen/slides/3/faktorzerlegung.tex | 14 ++++++--- vorlesungen/slides/3/maximalergrad.tex | 37 ++++++++++++---------- vorlesungen/slides/3/minimalbeispiel.tex | 12 ++++--- vorlesungen/slides/3/minimalpolynom.tex | 13 +++++--- vorlesungen/slides/3/operatoren.tex | 23 ++++++++------ vorlesungen/slides/3/teilbarkeit.tex | 21 +++++++------ vorlesungen/slides/test.tex | 1 + 15 files changed, 196 insertions(+), 107 deletions(-) create mode 100644 vorlesungen/slides/3/adjalgebra.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index 3a55274..ca6da41 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -22,5 +22,6 @@ chapter3 = \ ../slides/3/drehfaktorisierung.tex \ ../slides/3/operatoren.tex \ ../slides/3/adjunktion.tex \ + ../slides/3/adjalgebra.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 index 3b55ab0..a974a76 100644 --- a/vorlesungen/slides/3/adjunktion.tex +++ b/vorlesungen/slides/3/adjunktion.tex @@ -8,9 +8,12 @@ \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: \[ W = \begin{pmatrix} @@ -21,10 +24,12 @@ W = \begin{pmatrix} \vdots&\vdots&\vdots&\ddots&\vdots& \vdots\\ 0& 0& 0&\dots & 1&-m_{n-1} \end{pmatrix} -\] +\]} +\uncover<4->{% Man kann nachrechnen, dass immer $m(W)=0$. +} \medskip -$\Rightarrow \Bbbk(W) = \{p(W)\;|\;p\in\Bbbk[X], \deg p<\deg m\}$ -ist ein Körper, in dem $m(X)$ faktorisiert werden kann $m(X) = (X-W)q(X)$. +\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 index e63002c..2663bec 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -20,3 +20,4 @@ \folie{3/drehfaktorisierung.tex} \folie{3/operatoren.tex} \folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} diff --git a/vorlesungen/slides/3/drehfaktorisierung.tex b/vorlesungen/slides/3/drehfaktorisierung.tex index b44ca35..569f6e5 100644 --- a/vorlesungen/slides/3/drehfaktorisierung.tex +++ b/vorlesungen/slides/3/drehfaktorisierung.tex @@ -9,7 +9,8 @@ \frametitle{Faktorisierung von $X^2+X+1$} \vspace{-3pt} $X^2+X+1$ kann faktorisiert werden, wenn man $i\sqrt{3}$ -hinzufügt +hinzufügt: +\uncover<2->{% \[ \biggl(X+\frac12+\frac{i\sqrt{3}}2\biggr) \biggl(X+\frac12-\frac{i\sqrt{3}}2\biggr) @@ -17,54 +18,58 @@ hinzufügt X^2+X+\frac14 + \frac34 -= -X^2+X+1 -\] +\uncover<3->{= +X^2+X+1} +\]} \vspace{-10pt} +\uncover<4->{% \begin{block}{Was ist $i\sqrt{3}$?} Matrix mit Minimalpolynom $X^2+3$: \[ W=\begin{pmatrix}0&-3\\1&0\end{pmatrix} +\uncover<5->{% \qquad\Rightarrow\qquad -W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I +W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I} +\uncover<6->{% \qquad\Rightarrow\qquad -W^2+3I=0 +W^2+3I=0} \] -\end{block} +\end{block}} \vspace{-10pt} +\uncover<7->{% \begin{block}{Faktorisierung von $X^2+X+1$} \vspace{-10pt} \begin{align*} -B_\pm +\uncover<8->{B_\pm &= --\frac12I\pm\frac12W +-\frac12I\pm\frac12W} & -&\Rightarrow +&\uncover<10->{\Rightarrow & -(X+B_+)(X+B-) -&= +(X+B_+)(X+B-)} +&\uncover<11->{= (X+\frac12I+\frac12W) -(X+\frac12I-\frac12W) +(X+\frac12I-\frac12W)} \\ -&= +&\uncover<9->{= \smash{ {\textstyle\begin{pmatrix}-\frac12&-\frac32\\\frac12&-\frac12\end{pmatrix}} -} +}} & & & -&= -X^2+X + \frac14I - \frac14W^2 +&\uncover<12->{= +X^2+X + \frac14I - \frac14W^2} \\ & & &%\Rightarrow & -&= -X^2+X + \frac14I + \frac34I -= -X^2+X+I +&\uncover<13->{= +X^2+X + \frac14I + \frac34I} +\uncover<14->{= +X^2+X+I} \end{align*} -\end{block} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/3/drehmatrix.tex b/vorlesungen/slides/3/drehmatrix.tex index bed0628..9e5eb65 100644 --- a/vorlesungen/slides/3/drehmatrix.tex +++ b/vorlesungen/slides/3/drehmatrix.tex @@ -5,19 +5,23 @@ % \begin{frame}[t] \frametitle{Analyse einer Drehung um $120^\circ$} -$D$ eine Drehung um $120^\circ$ +$D$ eine Drehung des $\mathbb{R}^3$ um $120^\circ$ \begin{enumerate} -\item +\item<2-> Drehwinkel = $120^\circ\quad\Rightarrow\quad D^3 = I$ -\item +\uncover<3->{ +$\quad\Rightarrow\quad \chi_D(X)=X^3-1$ +} +\item<4-> $m_D(X)=X^3-1$ -\item +\item<5-> $m_D$ ist nicht irreduzibel, weil $m_D(1)=0$: $ m_D(X) = (X-1)(X^2+X+1) $ -\item -Welche Matrix hat $X^2+X+1$ als Minimalpolynom +\item<6-> +Welche Matrix hat $X^2+X+1$ als Minimalpolynom? +\uncover<7->{% \[ \arraycolsep=1.4pt W @@ -43,20 +47,20 @@ W^2+W+I 1&0\\0&1 \end{array}\biggr) =0 -\] -\item In einer geeigneten Basis hat $D$ die Form +\]} +\item<8-> In einer geeigneten Basis hat $D$ die Form \[ D=\begin{pmatrix} 1&0&0\\ 0&-\frac12 & -\frac{\sqrt{3}}2 \\ 0&\frac{\sqrt{3}}2 & -\frac12 \end{pmatrix} -= +\uncover<9->{= \begin{pmatrix} 1&0&0\\ 0&\cos 120^\circ & -\sin 120^\circ\\ 0&\sin 120^\circ & \cos 120^\circ -\end{pmatrix} +\end{pmatrix}} \] \end{enumerate} \end{frame} diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex index 936100d..7f54abb 100644 --- a/vorlesungen/slides/3/einsetzen.tex +++ b/vorlesungen/slides/3/einsetzen.tex @@ -9,41 +9,45 @@ \[ \begin{array}{rcrcrcrcrcrcr} p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\ -\bigg\downarrow\hspace*{4pt} & & -\bigg\downarrow\hspace*{4pt} & & -\bigg\downarrow\hspace*{10pt} & & & & -\bigg\downarrow\hspace*{4pt} & & -\bigg\downarrow\hspace*{2pt} & & -\bigg\downarrow\hspace*{0pt} \\ -p(A)&=&a_nA^n&+&a_{n-1}A^{n-1}&+&\dots&+&a_2A^2&+&a_1A&+&a_0 I +\uncover<2->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<3->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<4->{\bigg\downarrow\hspace*{10pt}} & & & & +\uncover<5->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<6->{\bigg\downarrow\hspace*{2pt}} & & +\uncover<7->{\bigg\downarrow\hspace*{0pt}} \\ +\uncover<2->{p(A)}&\uncover<3->{=&a_nA^n}&\uncover<4->{+&a_{n-1}A^{n-1}}&\uncover<5->{+&\dots&+&a_2A^2}&\uncover<6->{+&a_1A}&\uncover<7->{+&a_0 I} \end{array} \] \vspace{-10pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<8->{% \begin{block}{Nilpotente Matrizen} $p(X) = (X-a)^n$ \[ -p(A) = 0 +\uncover<9->{p(A) = 0} +\uncover<10->{ \quad\Rightarrow\quad -\text{$A$ ist nilpotent} +\text{$A-aI$ ist nilpotent}} \] -\end{block} +\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 \[ -p(A)=0\quad\Rightarrow\quad +\uncover<12->{p(A)=0} +\uncover<13->{\quad\Rightarrow\quad \left\{ \begin{aligned} &\text{$A-\lambda_1I$ ist singulär}\\ &\text{$A-\lambda_2I$ ist singulär} \end{aligned} -\right. +\right.} \] -\end{block} +\end{block}} \end{column} \end{columns} diff --git a/vorlesungen/slides/3/faktorisierung.tex b/vorlesungen/slides/3/faktorisierung.tex index cbf7004..b4ea1d5 100644 --- a/vorlesungen/slides/3/faktorisierung.tex +++ b/vorlesungen/slides/3/faktorisierung.tex @@ -11,32 +11,37 @@ 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*} -p&=7 +\uncover<2->{p&=7} \\ -2021 &= 43 \cdot 47 +\uncover<3->{2021 &= 43 \cdot 47} \\ -4095667&=2021\cdot 2027 +\uncover<4->{2048 &= 2^{11}} \\ -p&=43, 47, 1291, 2017, 2027 +\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*} -p&=X-9 +\uncover<8->{p&=X-9} \\ -X^2-1&= (X+1)(X-1) +\uncover<9->{X^2-1&= (X+1)(X-1)} \\ -X^2-2&\text{\; irreduzibel} +\uncover<10->{X^2-2&\text{\; irreduzibel}} \\ -X^2-2&=(X-\sqrt{2})(X+\sqrt{2}) +\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{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/3/faktorzerlegung.tex b/vorlesungen/slides/3/faktorzerlegung.tex index 0e5a95b..eb44cf3 100644 --- a/vorlesungen/slides/3/faktorzerlegung.tex +++ b/vorlesungen/slides/3/faktorzerlegung.tex @@ -17,6 +17,7 @@ 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 @@ -32,27 +33,30 @@ p_2^{n_2} \cdot p_k^{n_k} \] -\end{block} +\end{block}} \end{column} \end{columns} -\begin{block}{Polynomfaktorisierung hängt von den Koeffizienten ab} +\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{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{block}} \end{column} \end{columns} -\end{block} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/3/maximalergrad.tex b/vorlesungen/slides/3/maximalergrad.tex index 3d7e1a4..d33ddc0 100644 --- a/vorlesungen/slides/3/maximalergrad.tex +++ b/vorlesungen/slides/3/maximalergrad.tex @@ -13,55 +13,60 @@ Der Ring $M_{n}(\Bbbk)$ ist ein $n^2$-dimensionaler Vektorraum mit Basis {\tiny \begin{align*} -&\begin{pmatrix} +&\uncover<2->{\begin{pmatrix} 1&0&\dots&0\\ 0&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ -\end{pmatrix} +\end{pmatrix}} & -&\begin{pmatrix} +&\uncover<3->{\begin{pmatrix} 0&1&\dots&0\\ 0&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ -\end{pmatrix} +\end{pmatrix}} & -&\dots +&\uncover<4->{\dots} & -&\begin{pmatrix} +&\uncover<5->{\begin{pmatrix} 0&0&\dots&1\\ 0&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ -\end{pmatrix} +\end{pmatrix}} \\ -&\begin{pmatrix} +&\uncover<6->{\begin{pmatrix} 0&0&\dots&0\\ 1&0&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ -\end{pmatrix} +\end{pmatrix}} & -&\begin{pmatrix} +&\uncover<7->{\begin{pmatrix} 0&0&\dots&0\\ 0&1&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ -\end{pmatrix} +\end{pmatrix}} & -&\dots +&\uncover<8->{\dots} & -&\begin{pmatrix} +&\uncover<9->{\begin{pmatrix} 0&0&\dots&0\\ 0&0&\dots&1\\ \vdots&\vdots&\ddots&\vdots\\ -\end{pmatrix} +\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: \[ +\uncover<11->{ a_0I+a_1A+a_2A^2+\dots+a_{n^2-1}A^{n^2-1}+a_{n^2}A^{n^2} = 0 +} \] -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<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/minimalbeispiel.tex b/vorlesungen/slides/3/minimalbeispiel.tex index 03909de..f94cf8d 100644 --- a/vorlesungen/slides/3/minimalbeispiel.tex +++ b/vorlesungen/slides/3/minimalbeispiel.tex @@ -12,12 +12,13 @@ \[ I ={\tiny\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}},\quad A ={\tiny\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix}},\quad -A^2={\tiny\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix}},\quad -A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}},\quad -A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 & 2 \end{pmatrix}} +\uncover<2->{A^2={\tiny\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix}}} +\uncover<3->{,\quad A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}}} +\uncover<4->{,\quad A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 & 2 \end{pmatrix}}} \] \end{block} \vspace{-5pt} +\uncover<5->{% \begin{block}{linear abhängig} Bereits die ersten $3$ sind linear abhängig: \[ @@ -29,6 +30,7 @@ Bereits die ersten $3$ sind linear abhängig: = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \] -$p(X) = X^2 - X - 4 \in \mathbb{Q}[X]$ hat die Eigenschaft $p(A)=0$ -\end{block} +\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 index 60d15f0..2b36a65 100644 --- a/vorlesungen/slides/3/minimalpolynom.tex +++ b/vorlesungen/slides/3/minimalpolynom.tex @@ -10,18 +10,21 @@ 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} +\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 $\chi_A\in \Bbbk[X]$ +des charakteristischen Polynoms $\chi_A\in \Bbbk[X]$} \\ -$\Rightarrow $ -Faktorzerlegung on $\chi_A(X)$ ermitteln! -\end{block} +\uncover<5->{$\Rightarrow $ +Faktorzerlegung on $\chi_A(X)$ ermitteln!} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex index 161c330..d646353 100644 --- a/vorlesungen/slides/3/operatoren.tex +++ b/vorlesungen/slides/3/operatoren.tex @@ -9,40 +9,43 @@ \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 \[ \begin{pmatrix} 1\\0\\0\\0\\\vdots\\0 \end{pmatrix} -\mapsto +\uncover<4->{\overset{\cdot X}{\mapsto} \begin{pmatrix} 0\\1\\0\\0\\\vdots\\0 -\end{pmatrix} -\mapsto +\end{pmatrix}} +\uncover<5->{\overset{\cdot X}{\mapsto} \begin{pmatrix} 0\\0\\1\\0\\\vdots\\0 -\end{pmatrix} -\mapsto +\end{pmatrix}} +\uncover<6->{\overset{\cdot X}{\mapsto} \begin{pmatrix} 0\\0\\0\\1\\\vdots\\0 -\end{pmatrix} -\mapsto\dots\mapsto +\end{pmatrix}} +\uncover<7->{\overset{\cdot X}{\mapsto}\dots} +\uncover<8->{\overset{\cdot X}{\mapsto} \begin{pmatrix} 0\\0\\0\\0\\\vdots\\1 -\end{pmatrix} +\end{pmatrix}} \] -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/3/teilbarkeit.tex b/vorlesungen/slides/3/teilbarkeit.tex index 25e4fa6..a5ea9b9 100644 --- a/vorlesungen/slides/3/teilbarkeit.tex +++ b/vorlesungen/slides/3/teilbarkeit.tex @@ -9,29 +9,32 @@ \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Teilen in $\mathbb{Z}$} -Zu zwei Zahlen $a,b\in \mathbb{Z}$, $a>b$ gibt es -immer genau ein Paar $q,r\in\mathbb{Z}$ derart, dass +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 \\ -r&< b +\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 $a,b\in\mathbb{Q}[X]$, $\deg a > \deg b$ +Zu zwei Polynomen $a,b\in\mathbb{Q}[X]$, \only<4->{{\color<4>{red}$\deg a > \deg b$},} gibt es -immer genau ein Paar $q,r\in\mathbb{Q}[X]$ derart, dass +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 \\ -\deg r&< \deg b +\uncover<4->{{\color<4>{red}\deg r}&{\color<4>{red}< \deg b}} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} -\begin{block}{Allgmein: euklidischer Ring} +\uncover<5->{% +\begin{block}{Allgemein: euklidischer Ring} Nullteilerfreier Ring $R$ mit einer Funktion $d\colon R\setminus{0}\to\mathbb{N}$ mit \begin{itemize} @@ -40,5 +43,5 @@ $d\colon R\setminus{0}\to\mathbb{N}$ mit $x=qy +r$ mit $d(y)>d(r)$ \end{itemize} Euklidische Ringe haben ähnliche Eigenschaften wie Polynomringe -\end{block} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 3c0348f..01c4ed9 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -21,6 +21,7 @@ \folie{3/drehfaktorisierung.tex} \folie{3/operatoren.tex} \folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} % XXX \folie{3/adjunktioni.tex} % XXX \folie{3/adjunktionsqrt2.tex} % XXX \folie{3/adjunktionphi.tex} -- cgit v1.2.1 From 37f6831ca6c7be6ecec6ce75b1d688a8fcfbb05c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 09:56:51 +0100 Subject: add new files --- vorlesungen/slides/3/Makefile.inc | 1 + vorlesungen/slides/3/chapter.tex | 1 + vorlesungen/slides/3/fibonacci.tex | 71 ++++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 37 ++++++++++---------- 4 files changed, 92 insertions(+), 18 deletions(-) create mode 100644 vorlesungen/slides/3/fibonacci.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index ca6da41..7f52cb1 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -17,6 +17,7 @@ chapter3 = \ ../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 \ diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex index 2663bec..0f049e7 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -15,6 +15,7 @@ \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} diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex new file mode 100644 index 0000000..e26175e --- /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/test.tex b/vorlesungen/slides/test.tex index 01c4ed9..9a657d1 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,24 +4,25 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\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/faktorisierung.tex} -\folie{3/faktorzerlegung.tex} -\folie{3/einsetzen.tex} -\folie{3/maximalergrad.tex} -\folie{3/minimalbeispiel.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/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/faktorisierung.tex} +%\folie{3/faktorzerlegung.tex} +%\folie{3/einsetzen.tex} +%\folie{3/maximalergrad.tex} +%\folie{3/minimalbeispiel.tex} +%\folie{3/minimalpolynom.tex} +%\folie{3/drehmatrix.tex} +%\folie{3/drehfaktorisierung.tex} +\folie{3/fibonacci.tex} +%\folie{3/operatoren.tex} +%\folie{3/adjunktion.tex} +%\folie{3/adjalgebra.tex} % XXX \folie{3/adjunktioni.tex} % XXX \folie{3/adjunktionsqrt2.tex} % XXX \folie{3/adjunktionphi.tex} -- cgit v1.2.1 From 5e981c99268ba53faddb74ed89091e6a2eec408a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 10:09:58 +0100 Subject: handout capability --- vorlesungen/slides/3/inverse.tex | 4 ++-- vorlesungen/slides/3/motivation.tex | 7 ++++--- 2 files changed, 6 insertions(+), 5 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex index 28716bc..4ad22d2 100644 --- a/vorlesungen/slides/3/inverse.tex +++ b/vorlesungen/slides/3/inverse.tex @@ -48,10 +48,10 @@ D&=\begin{pmatrix} \sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} \end{pmatrix} \\ -0 &= \only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}} +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 &= \only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}} +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} diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex index cf28d46..91f27dd 100644 --- a/vorlesungen/slides/3/motivation.tex +++ b/vorlesungen/slides/3/motivation.tex @@ -42,10 +42,10 @@ D&=\begin{pmatrix} \sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} \end{pmatrix} \\ -p(X)&= \only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}} +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) &= \only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}} +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}} @@ -56,6 +56,7 @@ p(D) &= \only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}} \begin{block}{3D-Beispiel} $p(x) = -x^3-5x^2+5x+1$ \[ +\ifthenelse{\boolean{presentation}}{ \only<5-8>{ A= \begin{pmatrix*}[r] @@ -64,7 +65,7 @@ A= -1&-1&2 \end{pmatrix*}} \only<6-8>{ -\quad\Rightarrow\quad} +\quad\Rightarrow\quad}}{} \uncover<6->{ - \only<-9>{A^3}\only<10->{ -- cgit v1.2.1 From bb1a5867c5a5be41de7fa3953d6e599c32a3afdf Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 14:01:04 +0100 Subject: typos --- vorlesungen/slides/3/fibonacci.tex | 2 +- vorlesungen/slides/3/motivation.tex | 5 +++-- 2 files changed, 4 insertions(+), 3 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex index e26175e..3d01020 100644 --- a/vorlesungen/slides/3/fibonacci.tex +++ b/vorlesungen/slides/3/fibonacci.tex @@ -63,7 +63,7 @@ v_{n}&=v_{n-1}+v_{n-2} \uncover<12->{= X^2-X +\Phi(I-\Phi)} \uncover<13->{= -X^2-X -\underbrace{\Phi^2-\Phi}_{\displaystyle=I} +X^2-X -(\underbrace{\Phi^2-\Phi}_{\displaystyle=I}) } \] \end{block}} diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex index 91f27dd..048e6a2 100644 --- a/vorlesungen/slides/3/motivation.tex +++ b/vorlesungen/slides/3/motivation.tex @@ -42,10 +42,11 @@ D&=\begin{pmatrix} \sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} \end{pmatrix} \\ -p(X)&= \ifthenelse{\boolean{presentation}\only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +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}}}}{} +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}} -- cgit v1.2.1 From 37f57e7328eca4340589968b9e93a4229707611f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 15:37:14 +0100 Subject: add week 3 presentation --- vorlesungen/02_polynome/Makefile | 26 ++++++++++++++ vorlesungen/02_polynome/MathSem-02-polynome.tex | 18 ++++++++++ vorlesungen/02_polynome/common.tex | 16 +++++++++ vorlesungen/02_polynome/meme.jpg | Bin 0 -> 46843 bytes vorlesungen/02_polynome/polynome-handout.tex | 11 ++++++ vorlesungen/02_polynome/slides.tex | 39 +++++++++++++++++++++ vorlesungen/03_endlichekoerper/Makefile | 26 ++++++++++++++ .../MathSem-03-endlichekoerper.tex | 14 ++++++++ vorlesungen/03_endlichekoerper/common.tex | 16 +++++++++ .../03_endlichekoerper/endlichekoerper-handout.tex | 11 ++++++ vorlesungen/03_endlichekoerper/slides.tex | 29 +++++++++++++++ vorlesungen/slides/test.tex | 23 +++++++++++- 12 files changed, 228 insertions(+), 1 deletion(-) create mode 100644 vorlesungen/02_polynome/Makefile create mode 100644 vorlesungen/02_polynome/MathSem-02-polynome.tex create mode 100644 vorlesungen/02_polynome/common.tex create mode 100644 vorlesungen/02_polynome/meme.jpg create mode 100644 vorlesungen/02_polynome/polynome-handout.tex create mode 100644 vorlesungen/02_polynome/slides.tex create mode 100644 vorlesungen/03_endlichekoerper/Makefile create mode 100644 vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex create mode 100644 vorlesungen/03_endlichekoerper/common.tex create mode 100644 vorlesungen/03_endlichekoerper/endlichekoerper-handout.tex create mode 100644 vorlesungen/03_endlichekoerper/slides.tex (limited to 'vorlesungen') diff --git a/vorlesungen/02_polynome/Makefile b/vorlesungen/02_polynome/Makefile new file mode 100644 index 0000000..75409d2 --- /dev/null +++ b/vorlesungen/02_polynome/Makefile @@ -0,0 +1,26 @@ +# +# Makefile -- polynome +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: polynome-handout.pdf MathSem-02-polynome.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-02-polynome.pdf: MathSem-02-polynome.tex $(SOURCES) + pdflatex MathSem-02-polynome.tex + +polynome-handout.pdf: polynome-handout.tex $(SOURCES) + pdflatex polynome-handout.tex + +thumbnail: thumbnail.jpg + +thumbnail.pdf: MathSem-02-polynome.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-02-polynome.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + diff --git a/vorlesungen/02_polynome/MathSem-02-polynome.tex b/vorlesungen/02_polynome/MathSem-02-polynome.tex new file mode 100644 index 0000000..6da53dd --- /dev/null +++ b/vorlesungen/02_polynome/MathSem-02-polynome.tex @@ -0,0 +1,18 @@ +% +% MathSem-02-polynome.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\vspace{-40pt} +\begin{center} +\includegraphics[width=6cm]{meme.jpg} +\end{center} +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/02_polynome/common.tex b/vorlesungen/02_polynome/common.tex new file mode 100644 index 0000000..022edab --- /dev/null +++ b/vorlesungen/02_polynome/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Polynome]{Polynome und Matrizen} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/02_polynome/meme.jpg b/vorlesungen/02_polynome/meme.jpg new file mode 100644 index 0000000..8670e1a Binary files /dev/null and b/vorlesungen/02_polynome/meme.jpg differ diff --git a/vorlesungen/02_polynome/polynome-handout.tex b/vorlesungen/02_polynome/polynome-handout.tex new file mode 100644 index 0000000..62ff0ad --- /dev/null +++ b/vorlesungen/02_polynome/polynome-handout.tex @@ -0,0 +1,11 @@ +% +% polynome-handout.tex -- Handout XXX +% +% (c) 2017 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/02_polynome/slides.tex b/vorlesungen/02_polynome/slides.tex new file mode 100644 index 0000000..56efe3c --- /dev/null +++ b/vorlesungen/02_polynome/slides.tex @@ -0,0 +1,39 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Polynome und Matrizen} +\folie{3/motivation.tex} +\folie{3/inverse.tex} +\folie{3/polynome.tex} +\folie{3/division.tex} +\folie{3/division2.tex} + +\section{Polynomering} +\folie{3/ringstruktur.tex} +\folie{3/teilbarkeit.tex} +\folie{3/faktorisierung.tex} +\folie{3/faktorzerlegung.tex} +\folie{3/einsetzen.tex} + +\section{Minimalpolynom} +\folie{3/maximalergrad.tex} +\folie{3/minimalbeispiel.tex} +\folie{3/minimalpolynom.tex} + +\section{Wurzeln hinzufügen} +\folie{3/drehmatrix.tex} +\folie{3/drehfaktorisierung.tex} +\folie{3/fibonacci.tex} +\folie{3/operatoren.tex} +\folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} +% XXX \folie{3/adjunktioni.tex} +% XXX \folie{3/adjunktionsqrt2.tex} +% XXX \folie{3/adjunktionphi.tex} +% Adjunktion von \cos(\pi/1291) und \cos(\pi/1291) +% XXX \folie{3/adj1291.tex} + +% XXX \folie{3/ diff --git a/vorlesungen/03_endlichekoerper/Makefile b/vorlesungen/03_endlichekoerper/Makefile new file mode 100644 index 0000000..316e749 --- /dev/null +++ b/vorlesungen/03_endlichekoerper/Makefile @@ -0,0 +1,26 @@ +# +# Makefile -- endlichekoerper +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: endlichekoerper-handout.pdf MathSem-03-endlichekoerper.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-03-endlichekoerper.pdf: MathSem-03-endlichekoerper.tex $(SOURCES) + pdflatex MathSem-03-endlichekoerper.tex + +endlichekoerper-handout.pdf: endlichekoerper-handout.tex $(SOURCES) + pdflatex endlichekoerper-handout.tex + +thumbnail: thumbnail.jpg + +thumbnail.pdf: MathSem-03-endlichekoerper.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-03-endlichekoerper.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + diff --git a/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex b/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex new file mode 100644 index 0000000..31229d7 --- /dev/null +++ b/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex @@ -0,0 +1,14 @@ +% +% MathSem-03-endlichekoerper.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/03_endlichekoerper/common.tex b/vorlesungen/03_endlichekoerper/common.tex new file mode 100644 index 0000000..3a279f2 --- /dev/null +++ b/vorlesungen/03_endlichekoerper/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Endliche Körper]{Rechnen in endlichen Körpern} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/03_endlichekoerper/endlichekoerper-handout.tex b/vorlesungen/03_endlichekoerper/endlichekoerper-handout.tex new file mode 100644 index 0000000..2d6c5b2 --- /dev/null +++ b/vorlesungen/03_endlichekoerper/endlichekoerper-handout.tex @@ -0,0 +1,11 @@ +% +% endlichekoerper-handout.tex -- Handout XXX +% +% (c) 2017 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/03_endlichekoerper/slides.tex b/vorlesungen/03_endlichekoerper/slides.tex new file mode 100644 index 0000000..67c97d6 --- /dev/null +++ b/vorlesungen/03_endlichekoerper/slides.tex @@ -0,0 +1,29 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Euklidischer Algorithmus} +% XXX \folie{4/ggt.tex} +% XXX \folie{4/euklidmatrix.tex} + +\section{Die Galois-Körper $\mathbb{F}_p$} +% XXX \folie{4/fp.tex} +% XXX \folie{4/division.tex} +% XXX \folie{4/gauss.tex} +% XXX \folie{4/dh.tex} +% XXX ? \folie{4/polynomefp.tex} +% XXX \folie{4/frobenius.tex} + +\section{Rechnen in $\mathbb{F}_p(\alpha)$} +% XXX \folie{4/ggtpoly.tex} +% XXX \folie{4/divisionpoly.tex} +% XXX \folie{4/euklidpoly.tex} + +\section{$\mathbb{F}_2(\alpha)$} +% XXX \folie{4/f2.tex} +% XXX \folie{4/schieberegister.tex} + +\section{Anwendung: elliptische Kurven} +% XXX Idee der elliptischen Kurve +% XXX \folie{4/ecidee.tex} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 9a657d1..7e97995 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -19,7 +19,7 @@ %\folie{3/minimalpolynom.tex} %\folie{3/drehmatrix.tex} %\folie{3/drehfaktorisierung.tex} -\folie{3/fibonacci.tex} +%\folie{3/fibonacci.tex} %\folie{3/operatoren.tex} %\folie{3/adjunktion.tex} %\folie{3/adjalgebra.tex} @@ -29,3 +29,24 @@ % Adjunktion von \cos(\pi/1291) und \cos(\pi/1291) % XXX \folie{3/adj1291.tex} +% XXX \folie{4/ggt.tex} +% XXX \folie{4/euklidmatrix.tex} + +% XXX \folie{4/fp.tex} +% XXX \folie{4/division.tex} +% XXX \folie{4/gauss.tex} +% XXX \folie{4/dh.tex} +% XXX ? \folie{4/polynomefp.tex} +% XXX \folie{4/frobenius.tex} + +% XXX \folie{4/ggtpoly.tex} +% XXX \folie{4/divisionpoly.tex} +% XXX \folie{4/euklidpoly.tex} + +% XXX \folie{4/f2.tex} +% XXX \folie{4/schieberegister.tex} + +% XXX Idee der elliptischen Kurve +% XXX \folie{4/ecidee.tex} +~ + -- cgit v1.2.1 From 6a56afa0e79fb4ac6f04c1d3c3a8e6314331989b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 16:04:13 +0100 Subject: new slides --- vorlesungen/slides/4/Makefile.inc | 10 ++++++ vorlesungen/slides/4/chapter.tex | 6 ++++ vorlesungen/slides/4/euklidmatrix.tex | 9 +++++ vorlesungen/slides/4/ggt.tex | 62 +++++++++++++++++++++++++++++++++++ vorlesungen/slides/Makefile.inc | 3 +- vorlesungen/slides/test.tex | 2 +- 6 files changed, 90 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/slides/4/Makefile.inc create mode 100644 vorlesungen/slides/4/chapter.tex create mode 100644 vorlesungen/slides/4/euklidmatrix.tex create mode 100644 vorlesungen/slides/4/ggt.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc new file mode 100644 index 0000000..dabdb7c --- /dev/null +++ b/vorlesungen/slides/4/Makefile.inc @@ -0,0 +1,10 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter4 = \ + ../slides/4/ggt.tex \ + ../slides/4/chapter.tex + diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex new file mode 100644 index 0000000..1e04e9f --- /dev/null +++ b/vorlesungen/slides/4/chapter.tex @@ -0,0 +1,6 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{4/ggt.tex} diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex new file mode 100644 index 0000000..2090c0a --- /dev/null +++ b/vorlesungen/slides/4/euklidmatrix.tex @@ -0,0 +1,9 @@ +% +% euklidmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Matrixform} + +\end{frame} diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex new file mode 100644 index 0000000..77b2a1d --- /dev/null +++ b/vorlesungen/slides/4/ggt.tex @@ -0,0 +1,62 @@ +% +% 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} +\begin{block}{Euklidischer Algorithmus} +$a_0 = a$, $b_0=b$ +\begin{align*} +a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0 \\ +a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1 \\ +a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2 \\ + &\;\vdots & & & & \\ +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}); +} +\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 $\x$}; +} +\foreach \x in {0,...,4}{ + \foreach \y in {0,...,2}{ + \fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8]; + } +} +\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/Makefile.inc b/vorlesungen/slides/Makefile.inc index 910f87e..61dc3c6 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -6,6 +6,7 @@ include ../slides/0/Makefile.inc include ../slides/1/Makefile.inc include ../slides/3/Makefile.inc +include ../slides/4/Makefile.inc slides = \ - $(chapter0) $(chapter1) $(chapter3) + $(chapter0) $(chapter1) $(chapter3) $(chapter4) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 7e97995..2faf3b3 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -29,7 +29,7 @@ % Adjunktion von \cos(\pi/1291) und \cos(\pi/1291) % XXX \folie{3/adj1291.tex} -% XXX \folie{4/ggt.tex} +\folie{4/ggt.tex} % XXX \folie{4/euklidmatrix.tex} % XXX \folie{4/fp.tex} -- cgit v1.2.1 From ca12778acdf6f2da84aec311c5ab63cde5d847cd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 1 Mar 2021 21:20:36 +0100 Subject: add slides --- vorlesungen/common/packages.tex | 3 + vorlesungen/slides/4/Makefile.inc | 3 + vorlesungen/slides/4/chapter.tex | 3 + vorlesungen/slides/4/euklidbeispiel.tex | 44 ++++++++++ vorlesungen/slides/4/euklidmatrix.tex | 86 +++++++++++++++++- vorlesungen/slides/4/fp.tex | 150 ++++++++++++++++++++++++++++++++ vorlesungen/slides/4/ggt.tex | 2 +- vorlesungen/slides/test.tex | 6 +- 8 files changed, 292 insertions(+), 5 deletions(-) create mode 100644 vorlesungen/slides/4/euklidbeispiel.tex create mode 100644 vorlesungen/slides/4/fp.tex (limited to 'vorlesungen') diff --git a/vorlesungen/common/packages.tex b/vorlesungen/common/packages.tex index 94a7c6f..d71438b 100644 --- a/vorlesungen/common/packages.tex +++ b/vorlesungen/common/packages.tex @@ -11,10 +11,13 @@ \usepackage{ifthen} \usepackage{lmodern} \usepackage{amsmath} +\usepackage{amssymb} \usepackage{mathtools} \usepackage{adjustbox} \usepackage{multimedia} \usepackage{verbatim} +\usepackage{wasysym} +\usepackage{stmaryrd} \usepackage{tikz} \usetikzlibrary{shapes.geometric} \usetikzlibrary{decorations.pathreplacing} diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index dabdb7c..24e4a80 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -6,5 +6,8 @@ # chapter4 = \ ../slides/4/ggt.tex \ + ../slides/4/euklidmatrix.tex \ + ../slides/4/euklidbeispiel.tex \ + ../slides/4/fp.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 1e04e9f..4fec776 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -4,3 +4,6 @@ % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % \folie{4/ggt.tex} +\folie{4/euklidmatrix.tex} +\folie{4/euklidbeispiel.tex} +\folie{4/fp.tex} diff --git a/vorlesungen/slides/4/euklidbeispiel.tex b/vorlesungen/slides/4/euklidbeispiel.tex new file mode 100644 index 0000000..cbc3137 --- /dev/null +++ b/vorlesungen/slides/4/euklidbeispiel.tex @@ -0,0 +1,44 @@ +% +% euklidmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{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 &25&=15 \cdot {\color{red} 1} + 10 &q_0 &= {\color{red}1} & r_0 &= 10\\ +a_1&=15 & b_1 &= 10 &15&=10 \cdot {\color{darkgreen}1} + \phantom{0}5 &q_1 &= {\color{darkgreen}1} & r_1 &= \phantom{0}5 \\ +a_2&=10 & b_2 &= \phantom{0}5 &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} +\begin{block}{Matrix-Operationen} +\begin{align*} +Q +&= +Q({\color{blue}2}) Q({\color{darkgreen}1}) Q({\color{red}1}) += +\begin{pmatrix}0&1\\1&-{\color{blue}2}\end{pmatrix} +\begin{pmatrix}0&1\\1&-{\color{darkgreen}1}\end{pmatrix} +\begin{pmatrix}0&1\\1&-{\color{red}1}\end{pmatrix} +=\begin{pmatrix} +-1&2\\3&-5 +\end{pmatrix} +\end{align*} +\end{block} +\vspace{-5pt} +\begin{block}{Relationen ablesen} +\begin{align*} +\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 = -1\cdot {\usebeamercolor[fg]{title}25} + 2\cdot {\usebeamercolor[fg]{title}15} \\ + 0 &= \phantom{5=-}3\cdot {\usebeamercolor[fg]{title}25} -5\cdot {\usebeamercolor[fg]{title}15} +\end{align*} +\end{block} + +\end{frame} diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex index 2090c0a..6ffa4c2 100644 --- a/vorlesungen/slides/4/euklidmatrix.tex +++ b/vorlesungen/slides/4/euklidmatrix.tex @@ -4,6 +4,90 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule % \begin{frame}[t] -\frametitle{Matrixform} +\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} +\[ +a_k = b_kq_k + r_k +\;\Rightarrow\; +\left\{ +\begin{aligned} +a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\ +b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k +\end{aligned} +\right. +\] +\end{block} +\end{column} +\begin{column}{0.44\textwidth} +\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} += +\underbrace{\begin{pmatrix}0&1\\1&-q_k\end{pmatrix}}_{\displaystyle =Q(q_k)} +\begin{pmatrix} +a_k\\b_k +\end{pmatrix} +\end{align*} +\end{block} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{block}{Ende des Algorithmus} +\vspace{-10pt} +\begin{align*} +\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} += +\underbrace{Q(q_n) +\dots +Q(q_1) +Q(q_0)}_{\displaystyle =Q} +\begin{pmatrix} a_0\\ b_0\end{pmatrix} += +Q\begin{pmatrix}a\\b\end{pmatrix} +\end{align*} +\end{block} +\begin{block}{Konsequenzen} +\[ +Q=\begin{pmatrix} +q_{11}&q_{12}\\ +a_{21}&q_{22} +\end{pmatrix} +\quad\Rightarrow\quad +\left\{ +\quad +\begin{aligned} +\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b \\ + 0 &= q_{21}a + q_{22}b +\end{aligned} +\right. +\] +\end{block} \end{frame} diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex new file mode 100644 index 0000000..a893238 --- /dev/null +++ b/vorlesungen/slides/4/fp.tex @@ -0,0 +1,150 @@ +% +% 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}); +} +\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} +\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} +\begin{block}{Galois-Körper $\mathbb{F}_p\mathstrut$} +$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\mathstrut$ +\end{block} +\begin{block}{$n$ prim} +Für $n=p$ prim ist $\mathbb{Z}/n\mathbb{Z}$ nullteilerfrei +\medskip + +$\Rightarrow \quad \mathbb{F}_p$ ist ein Körper +\end{block} +\end{column} +\end{columns} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.45] +\begin{scope}[xshift=-7cm] +\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} +\begin{scope}[xshift=7cm] +\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} +\end{scope} +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex index 77b2a1d..e3c55e6 100644 --- a/vorlesungen/slides/4/ggt.tex +++ b/vorlesungen/slides/4/ggt.tex @@ -22,7 +22,7 @@ a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0 \\ a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1 \\ a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2 \\ &\;\vdots & & & & \\ -a_n&=b_nq_n + r_n & r_n &= 0 & r_{n-1}&)=\operatorname{ggT}(a,b) +a_n&=b_nq_n + r_n & r_n &= 0 & r_{n-1}&=\operatorname{ggT}(a,b) \end{align*} \end{block} \end{column} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 2faf3b3..e7108ec 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -30,9 +30,9 @@ % XXX \folie{3/adj1291.tex} \folie{4/ggt.tex} -% XXX \folie{4/euklidmatrix.tex} - -% XXX \folie{4/fp.tex} +\folie{4/euklidmatrix.tex} +\folie{4/euklidbeispiel.tex} +\folie{4/fp.tex} % XXX \folie{4/division.tex} % XXX \folie{4/gauss.tex} % XXX \folie{4/dh.tex} -- cgit v1.2.1 From a04efb8ee3eef6ec32b1381801ee244744db1201 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 2 Mar 2021 10:23:02 +0100 Subject: division in F_p --- vorlesungen/slides/4/Makefile.inc | 2 ++ vorlesungen/slides/4/chapter.tex | 2 ++ vorlesungen/slides/4/division.tex | 62 ++++++++++++++++++++++++++++++++ vorlesungen/slides/4/euklidtabelle.tex | 64 ++++++++++++++++++++++++++++++++++ vorlesungen/slides/4/fp.tex | 25 +++++++++++-- vorlesungen/slides/test.tex | 3 +- 6 files changed, 155 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/4/division.tex create mode 100644 vorlesungen/slides/4/euklidtabelle.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 24e4a80..0adf913 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -8,6 +8,8 @@ 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/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 4fec776..1c23783 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -6,4 +6,6 @@ \folie{4/ggt.tex} \folie{4/euklidmatrix.tex} \folie{4/euklidbeispiel.tex} +\folie{4/euklidtabelle.tex} \folie{4/fp.tex} +\folie{4/division.tex} diff --git a/vorlesungen/slides/4/division.tex b/vorlesungen/slides/4/division.tex new file mode 100644 index 0000000..aeab1e3 --- /dev/null +++ b/vorlesungen/slides/4/division.tex @@ -0,0 +1,62 @@ +% +% 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*} +&& a{\color{red}b} &\equiv 1 \mod p +\\ +&\Leftrightarrow& a{\color{red}b}&=1 + {\color{red}n}p +\\ +&&a{\color{red}b}-{\color{red}n}p&=1 +\end{align*} +Wegen +$\operatorname{ggT}(a,p)=1$ gibt es +$s$ und $t$ mit +\[ +sa+tb=1 +\Rightarrow +{\color{red}b}=s,\; +{\color{red}n}=-t +\] +$\Rightarrow$ Die Inverse kann mit dem euklidischen Algorithmus +berechnet werden +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\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& 0& 0& 1\\ +1&1291& 47& 27& 1& 0\\ +2& 47& 22& 2& -27& 1\\ +3& 22& 3& 7& 55& -2\\ +4& 3& 1& 3&{\color{red}-412}&{\color{red}15}\\ +5& 1& 0& & 1291& -47\\ +\hline +\end{tabular} +\end{center} +\[ +{\color{red}-412}\cdot 47 +{\color{red}15}\cdot 1291 = 1 +\;\Rightarrow\; +47^{-1}={\color{red}879} +\] +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/euklidtabelle.tex b/vorlesungen/slides/4/euklidtabelle.tex new file mode 100644 index 0000000..2d67823 --- /dev/null +++ b/vorlesungen/slides/4/euklidtabelle.tex @@ -0,0 +1,64 @@ +% +% 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 $k=0,1,\dots,n$ +\begin{block}{Multiplikation mit $Q(q_k)$} +\vspace{-12pt} +\begin{align*} +Q(q_k) +%\begin{pmatrix} +%0&1\\1&-q_k +%\end{pmatrix} +\begin{pmatrix} +u&v\\c&d +\end{pmatrix} +&= +\begin{pmatrix} +c&d\\ +u-q_kc&v-q_kd +\end{pmatrix} +&&\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} +\begin{equation*} +\begin{tabular}{|>{\tiny$}r<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|} +\hline +k &q_k & c_k & d_k \\ +\hline +-1 & & 1 & 0 \\ + 0 &q_0 & 0 & 1 \\ + 1 &q_1 &c_{-1} -q_0 \cdot c_0 &d_{-1} -q_0 \cdot d_0 \\ + 2 &q_2 &c_0 -q_1 \cdot c_1 &d_0 -q_1 \cdot d_1 \\ +\vdots&\vdots&\vdots &\vdots \\ + n &q_n &c_{n-2}-q_{n-1}\cdot c_{n-1}&d_{n-2}-q_{n-1}\cdot d_{n-1}\\ +n+1& &c_{n-1}-q_{n} \cdot c_{n} &d_{n-1}-q_{n} \cdot d_{n} \\ +\hline +\end{tabular} +\Rightarrow +\left\{ +\begin{aligned} +\rlap{$c_{n}$}\phantom{c_{n+1}} a + \rlap{$d_n$}\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 index a893238..caa6ceb 100644 --- a/vorlesungen/slides/4/fp.tex +++ b/vorlesungen/slides/4/fp.tex @@ -13,6 +13,13 @@ \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} @@ -53,7 +60,9 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper \vspace{-20pt} \begin{center} \begin{tikzpicture}[>=latex,thick,scale=0.45] -\begin{scope}[xshift=-7cm] +\fill[color=white] (-12,0) circle[radius=0.1]; +\fill[color=white] (12,0) circle[radius=0.1]; +\begin{scope}[xshift=-8cm] \rot{2}{3} \rot{4}{3} \rot{3}{2} @@ -98,7 +107,13 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper \feld{4}{5}{2} \feld{5}{4}{2} \feld{5}{5}{1} \end{scope} -\begin{scope}[xshift=7cm] +\begin{scope}[xshift=6cm] +\gruen{1}{1} +\gruen{2}{4} +\gruen{3}{5} +\gruen{4}{2} +\gruen{5}{3} +\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}{ @@ -143,6 +158,12 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper \geld{5}{5}{4} \geld{6}{5}{2} \geld{5}{6}{2} \geld{6}{6}{1} +\inverse{1}{1} +\inverse{2}{4} +\inverse{3}{5} +\inverse{4}{2} +\inverse{5}{3} +\inverse{6}{6} \end{scope} \end{tikzpicture} \end{center} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index e7108ec..8d3d490 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -32,8 +32,9 @@ \folie{4/ggt.tex} \folie{4/euklidmatrix.tex} \folie{4/euklidbeispiel.tex} +\folie{4/euklidtabelle.tex} \folie{4/fp.tex} -% XXX \folie{4/division.tex} +\folie{4/division.tex} % XXX \folie{4/gauss.tex} % XXX \folie{4/dh.tex} % XXX ? \folie{4/polynomefp.tex} -- cgit v1.2.1 From e65f01b8c2a8c70662c7fd712d0678ded1be0540 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 2 Mar 2021 11:30:24 +0100 Subject: =?UTF-8?q?vereinfachte=20Durchf=C3=BChrung?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- vorlesungen/slides/3/drehfaktorisierung.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/3/drehfaktorisierung.tex b/vorlesungen/slides/3/drehfaktorisierung.tex index 569f6e5..64418d5 100644 --- a/vorlesungen/slides/3/drehfaktorisierung.tex +++ b/vorlesungen/slides/3/drehfaktorisierung.tex @@ -46,7 +46,7 @@ W^2+3I=0} & &\uncover<10->{\Rightarrow & -(X+B_+)(X+B-)} +(X+B_+)(X+B_-)} &\uncover<11->{= (X+\frac12I+\frac12W) (X+\frac12I-\frac12W)} -- cgit v1.2.1 From 6c0c557cc85dd8617505687176b0826d4f81fa2e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 2 Mar 2021 14:37:55 +0100 Subject: new presentation for spectral theory --- vorlesungen/02_msespektral/Makefile | 26 ++++++++++++++++++++++ .../02_msespektral/MathSemMSE-02-spektral.tex | 14 ++++++++++++ vorlesungen/02_msespektral/common.tex | 16 +++++++++++++ vorlesungen/02_msespektral/slides.tex | 6 +++++ vorlesungen/02_msespektral/spektral-handout.tex | 11 +++++++++ vorlesungen/slides/5/Makefile.inc | 9 ++++++++ vorlesungen/slides/5/chapter.tex | 5 +++++ vorlesungen/slides/Makefile.inc | 1 + 8 files changed, 88 insertions(+) create mode 100644 vorlesungen/02_msespektral/Makefile create mode 100644 vorlesungen/02_msespektral/MathSemMSE-02-spektral.tex create mode 100644 vorlesungen/02_msespektral/common.tex create mode 100644 vorlesungen/02_msespektral/slides.tex create mode 100644 vorlesungen/02_msespektral/spektral-handout.tex create mode 100644 vorlesungen/slides/5/Makefile.inc create mode 100644 vorlesungen/slides/5/chapter.tex (limited to 'vorlesungen') diff --git a/vorlesungen/02_msespektral/Makefile b/vorlesungen/02_msespektral/Makefile new file mode 100644 index 0000000..6451d8a --- /dev/null +++ b/vorlesungen/02_msespektral/Makefile @@ -0,0 +1,26 @@ +# +# Makefile -- spektral +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: spektral-handout.pdf MathSemMSE-02-spektral.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-02-spektral.pdf: MathSemMSE-02-spektral.tex $(SOURCES) + pdflatex MathSemMSE-02-spektral.tex + +spektral-handout.pdf: spektral-handout.tex $(SOURCES) + pdflatex spektral-handout.tex + +thumbnail: thumbnail.jpg + +thumbnail.pdf: MathSem-02-spektral.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-02-spektral.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + diff --git a/vorlesungen/02_msespektral/MathSemMSE-02-spektral.tex b/vorlesungen/02_msespektral/MathSemMSE-02-spektral.tex new file mode 100644 index 0000000..386b231 --- /dev/null +++ b/vorlesungen/02_msespektral/MathSemMSE-02-spektral.tex @@ -0,0 +1,14 @@ +% +% MathSem-02-msespektral.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/02_msespektral/common.tex b/vorlesungen/02_msespektral/common.tex new file mode 100644 index 0000000..ddc3e5d --- /dev/null +++ b/vorlesungen/02_msespektral/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Spektraltheorie]{Spektraltheorie} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex new file mode 100644 index 0000000..5678823 --- /dev/null +++ b/vorlesungen/02_msespektral/slides.tex @@ -0,0 +1,6 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + diff --git a/vorlesungen/02_msespektral/spektral-handout.tex b/vorlesungen/02_msespektral/spektral-handout.tex new file mode 100644 index 0000000..4b8f061 --- /dev/null +++ b/vorlesungen/02_msespektral/spektral-handout.tex @@ -0,0 +1,11 @@ +% +% msespektral-handout.tex -- Handout XXX +% +% (c) 2017 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/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc new file mode 100644 index 0000000..d690514 --- /dev/null +++ b/vorlesungen/slides/5/Makefile.inc @@ -0,0 +1,9 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter5 = \ + ../slides/5/chapter.tex + diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex new file mode 100644 index 0000000..884732f --- /dev/null +++ b/vorlesungen/slides/5/chapter.tex @@ -0,0 +1,5 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 61dc3c6..441589b 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -7,6 +7,7 @@ include ../slides/0/Makefile.inc include ../slides/1/Makefile.inc include ../slides/3/Makefile.inc include ../slides/4/Makefile.inc +include ../slides/5/Makefile.inc slides = \ $(chapter0) $(chapter1) $(chapter3) $(chapter4) -- cgit v1.2.1 From 4d16eab7a381a8379bc139ed2744912683661bab Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 2 Mar 2021 21:25:06 +0100 Subject: add slides --- vorlesungen/02_msespektral/slides.tex | 28 ++++++ vorlesungen/slides/5/Makefile.inc | 7 ++ vorlesungen/slides/5/chapter.tex | 7 ++ vorlesungen/slides/5/charpoly.tex | 68 +++++++++++++ vorlesungen/slides/5/dimension.tex | 68 +++++++++++++ vorlesungen/slides/5/folgerungen.tex | 31 ++++++ vorlesungen/slides/5/kernbild.tex | 68 +++++++++++++ vorlesungen/slides/5/ketten.tex | 78 +++++++++++++++ vorlesungen/slides/5/motivation.tex | 63 ++++++++++++ vorlesungen/slides/5/nilpotent.tex | 176 ++++++++++++++++++++++++++++++++++ vorlesungen/slides/Makefile.inc | 2 +- vorlesungen/slides/test.tex | 47 ++++++--- 12 files changed, 629 insertions(+), 14 deletions(-) create mode 100644 vorlesungen/slides/5/charpoly.tex create mode 100644 vorlesungen/slides/5/dimension.tex create mode 100644 vorlesungen/slides/5/folgerungen.tex create mode 100644 vorlesungen/slides/5/kernbild.tex create mode 100644 vorlesungen/slides/5/ketten.tex create mode 100644 vorlesungen/slides/5/motivation.tex create mode 100644 vorlesungen/slides/5/nilpotent.tex (limited to 'vorlesungen') diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex index 5678823..33d5f90 100644 --- a/vorlesungen/02_msespektral/slides.tex +++ b/vorlesungen/02_msespektral/slides.tex @@ -3,4 +3,32 @@ % % (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil % +\section{Matrixrelationen} +\folie{3/maximalergrad.tex} +\folie{3/minimalbeispiel.tex} +\folie{3/minimalpolynom.tex} + +\section{Eigenwertproblem} +% XXX Motivation: beliebige Funktionen f(A) berechnen +\folie{5/motivation.tex} +\folie{5/charpoly.tex} + +\section{Invariante Unterräume} +\folie{5/kernbild.tex} +\folie{5/ketten.tex} +\folie{5/dimension.tex} +\folie{5/nilpotent.tex} +% XXX \folie{5/eigenraeume.tex} + +% Jordan Normalform +\section{Jordan-Normalform} +% XXX Diagonalform +% XXX \folie{5/diagonalform.tex} +% XXX \folie{5/jordannormalform.tex} +% XXX \folie{5/minimalpolynom.tex} +% XXX \folie{5/reellenormalform.tex} +% XXX \folie{5/hessenberg.tex} + +\section{Satz von Cayley-Hamilton} +% XXX \folie{5/cayleyhamilton.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index d690514..d081b29 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -5,5 +5,12 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapter5 = \ + ../slides/5/motivation.tex \ + ../slides/5/charpoly.tex \ + ../slides/5/kernbild.tex \ + ../slides/5/ketten.tex \ + ../slides/5/dimension.tex \ + ../slides/5/folgerungen.tex \ + ../slides/5/nilpotent.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 884732f..4bcee8e 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,3 +3,10 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % +folie{5/motivation.tex} +folie{5/charpoly.tex} +folie{5/kernbild.tex} +folie{5/ketten.tex} +folie{5/dimension.tex} +folie{5/folgerungen.tex} +folie{5/nilpotent.tex} diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex new file mode 100644 index 0000000..1211b43 --- /dev/null +++ b/vorlesungen/slides/5/charpoly.tex @@ -0,0 +1,68 @@ +% +% 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} +\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} +\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} +\begin{block}{Minimalpolynom} +Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen +\end{block} +\begin{proof}[Beweis] +\begin{enumerate} +\item +$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$ +\item +$A-\lambda I$ ist regulär +\end{enumerate} +\begin{align*} +&\Rightarrow& +m_A(A)&=0 +\\ +&& +(A-\lambda)^{-1}m_A(A) &=0 +\\ +&& +\prod_{i\in I}(A-\mu_i)&=0, +\end{align*} +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/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex new file mode 100644 index 0000000..96efd7f --- /dev/null +++ b/vorlesungen/slides/5/folgerungen.tex @@ -0,0 +1,31 @@ +% +% folgerungen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Folgerungen} +\begin{columns}[t] +\begin{column}{0.48\textwidth} +\begin{block}{Zunahme} +Für alle $k=latex,thick,scale=0.42] + +\only<2->{ + \feld{0} + \feld{1} + \feld{2} + \feld{3} +} +\only<2->{ \feld{4} } +\only<3->{ \feld{5} } +\only<2->{ \feld{6} } +\only<3->{ \feld{7} } +\only<4->{ \feld{8} } +\only<5->{ \feld{9} } +\only<6->{ \feld{10} } +\only<7->{ \feld{11} } + +\only<1>{ \node at (6,0) [below] {$k=0$}; } +\only<2>{ \node at (6,0) [below] {$k=1$}; } +\only<3>{ \node at (6,0) [below] {$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); +\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}) {$1$}; + } +} +\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) {$1$}; +} +\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]; +\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] {$6$}; +} +\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/Makefile.inc b/vorlesungen/slides/Makefile.inc index 441589b..90bb388 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -10,4 +10,4 @@ include ../slides/4/Makefile.inc include ../slides/5/Makefile.inc slides = \ - $(chapter0) $(chapter1) $(chapter3) $(chapter4) + $(chapter0) $(chapter1) $(chapter3) $(chapter4) $(chapter5) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 8d3d490..5c1b5af 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -23,18 +23,13 @@ %\folie{3/operatoren.tex} %\folie{3/adjunktion.tex} %\folie{3/adjalgebra.tex} -% XXX \folie{3/adjunktioni.tex} -% XXX \folie{3/adjunktionsqrt2.tex} -% XXX \folie{3/adjunktionphi.tex} -% Adjunktion von \cos(\pi/1291) und \cos(\pi/1291) -% XXX \folie{3/adj1291.tex} - -\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/ggt.tex} +%\folie{4/euklidmatrix.tex} +%\folie{4/euklidbeispiel.tex} +%\folie{4/euklidtabelle.tex} +%\folie{4/fp.tex} +%\folie{4/division.tex} % XXX \folie{4/gauss.tex} % XXX \folie{4/dh.tex} % XXX ? \folie{4/polynomefp.tex} @@ -49,5 +44,31 @@ % XXX Idee der elliptischen Kurve % XXX \folie{4/ecidee.tex} -~ + + +\section{Eigenwertproblem} +% XXX Motivation: beliebige Funktionen f(A) berechnen +\folie{5/motivation.tex} +\folie{5/charpoly.tex} + +\section{Invariante Unterräume} +\folie{5/kernbild.tex} +\folie{5/ketten.tex} +\folie{5/dimension.tex} +\folie{5/folgerungen.tex} +\folie{5/nilpotent.tex} +% XXX \folie{5/eigenraeume.tex} + +% Jordan Normalform +\section{Jordan-Normalform} +% XXX Diagonalform +% XXX \folie{5/diagonalform.tex} +% XXX \folie{5/jordannormalform.tex} +% XXX \folie{5/minimalpolynom.tex} +% XXX \folie{5/reellenormalform.tex} +% XXX \folie{5/hessenberg.tex} + +\section{Satz von Cayley-Hamilton} +% XXX \folie{5/cayleyhamilton.tex} + -- cgit v1.2.1 From 98a00850977e96d68c2612ca218543bec95b8cd6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 2 Mar 2021 21:39:49 +0100 Subject: typos --- vorlesungen/02_msespektral/slides.tex | 1 + vorlesungen/slides/5/nilpotent.tex | 16 +++++++++++++++- 2 files changed, 16 insertions(+), 1 deletion(-) (limited to 'vorlesungen') diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex index 33d5f90..dc34236 100644 --- a/vorlesungen/02_msespektral/slides.tex +++ b/vorlesungen/02_msespektral/slides.tex @@ -17,6 +17,7 @@ \folie{5/kernbild.tex} \folie{5/ketten.tex} \folie{5/dimension.tex} +\folie{5/folgerungen.tex} \folie{5/nilpotent.tex} % XXX \folie{5/eigenraeume.tex} diff --git a/vorlesungen/slides/5/nilpotent.tex b/vorlesungen/slides/5/nilpotent.tex index 9b7ded1..ca38c40 100644 --- a/vorlesungen/slides/5/nilpotent.tex +++ b/vorlesungen/slides/5/nilpotent.tex @@ -24,8 +24,9 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \feld{3} } \only<2->{ \feld{4} } -\only<3->{ \feld{5} } \only<2->{ \feld{6} } +\ifthenelse{\boolean{presentation}}{ +\only<3->{ \feld{5} } \only<3->{ \feld{7} } \only<4->{ \feld{8} } \only<5->{ \feld{9} } @@ -33,19 +34,24 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \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}) {$0$}; @@ -56,6 +62,7 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \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}) {$0$}; @@ -101,6 +108,7 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \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] @@ -108,9 +116,11 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \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} @@ -136,14 +146,17 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \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); @@ -167,6 +180,7 @@ ${\color{red}\dim\mathcal{K}^k(A)}$ \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} -- cgit v1.2.1 From f297ac1a4e95db255ff6c8e892904c1743ed57e5 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 3 Mar 2021 22:51:16 +0100 Subject: add new slides --- vorlesungen/slides/5/Makefile.inc | 4 ++ vorlesungen/slides/5/chapter.tex | 4 ++ vorlesungen/slides/5/eigenraeume.tex | 42 +++++++++++++++++++++ vorlesungen/slides/5/folgerungen.tex | 5 ++- vorlesungen/slides/5/injektiv.tex | 9 +++++ vorlesungen/slides/5/normalnilp.tex | 8 ++++ vorlesungen/slides/5/zerlegung.tex | 73 ++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 5 ++- 8 files changed, 147 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/5/eigenraeume.tex create mode 100644 vorlesungen/slides/5/injektiv.tex create mode 100644 vorlesungen/slides/5/normalnilp.tex create mode 100644 vorlesungen/slides/5/zerlegung.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index d081b29..0975571 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -11,6 +11,10 @@ chapter5 = \ ../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/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 4bcee8e..245b31d 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -9,4 +9,8 @@ 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} diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex new file mode 100644 index 0000000..5192cbc --- /dev/null +++ b/vorlesungen/slides/5/eigenraeume.tex @@ -0,0 +1,42 @@ +% +% 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) +\\ +&= +\{v\in V\;|\; f(v) = \lambda v\} +\end{align*} +{\em Eigenraum} von $f$ zum Eigenwert $\lambda$. +\end{block} +$E_\lambda(f)\subset V$ ist ein Unterraum + +\begin{block}{Eigenwert} +Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$. +\end{block} + +\end{column} +\begin{column}{0.48\textwidth} +\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/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex index 96efd7f..a1fa6bf 100644 --- a/vorlesungen/slides/5/folgerungen.tex +++ b/vorlesungen/slides/5/folgerungen.tex @@ -24,8 +24,9 @@ Ausserdem ist $l\le n$ \end{block} \end{column} \begin{column}{0.48\textwidth} -\begin{block}{} -\end{block} +\begin{proof}[Beweis] +XXX +\end{proof} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex new file mode 100644 index 0000000..e08cafd --- /dev/null +++ b/vorlesungen/slides/5/injektiv.tex @@ -0,0 +1,9 @@ +% +% injektiv.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{$f$ injektiv auf $\mathcal{J}(f)$} +XXX +\end{frame} diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex new file mode 100644 index 0000000..a7af682 --- /dev/null +++ b/vorlesungen/slides/5/normalnilp.tex @@ -0,0 +1,8 @@ +% +% normalnilp.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Normalform einer nilpotenten Matrix} +\end{frame} diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex new file mode 100644 index 0000000..9c20c60 --- /dev/null +++ b/vorlesungen/slides/5/zerlegung.tex @@ -0,0 +1,73 @@ +% +% 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] +\fill[color=blue!20] (0,11) rectangle (4,15); +\fill[color=red!20] (4,0) rectangle (15,11); +\fill[color=red!40] (9,0) rectangle (15,6); +\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11); +\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6); +\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3); +\draw[line width=0.1pt] (0,11) -- (15,11); +\draw[line width=0.1pt] (0,6) -- (15,6); +\draw[line width=0.1pt] (0,3) -- (15,3); +\draw[line width=0.1pt] (4,0) -- (4,15); +\draw[line width=0.1pt] (9,0) -- (9,15); +\draw[line width=0.1pt] (12,0) -- (12,15); +\draw (0,0) rectangle (15,15); +\node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$}; +\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$}; +\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$}; +\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$}; +\node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$}; +\node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$}; +\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$}; +\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*} +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] +f_1 &= f_{|V_1} +\\[10pt] +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] +f_1 &= f_{|V_1} +\\ +&\phantom{0}\vdots +\end{align*} +$\Rightarrow$ $f$ 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/test.tex b/vorlesungen/slides/test.tex index 5c1b5af..03601bd 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -56,8 +56,11 @@ \folie{5/ketten.tex} \folie{5/dimension.tex} \folie{5/folgerungen.tex} +\folie{5/injektiv.tex} \folie{5/nilpotent.tex} -% XXX \folie{5/eigenraeume.tex} +\folie{5/eigenraeume.tex} +\folie{5/zerlegung.tex} +\folie{5/normalnilp.tex} % Jordan Normalform \section{Jordan-Normalform} -- cgit v1.2.1 From 5dcf196bce586d3d3d0d51759fcf86a293a4deca Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 3 Mar 2021 22:54:31 +0100 Subject: add new slides --- vorlesungen/slides/5/Makefile.inc | 1 + vorlesungen/slides/5/chapter.tex | 1 + vorlesungen/slides/5/jordan.tex | 10 ++++++++++ vorlesungen/slides/test.tex | 1 + 4 files changed, 13 insertions(+) create mode 100644 vorlesungen/slides/5/jordan.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 0975571..4bed49b 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -16,5 +16,6 @@ chapter5 = \ ../slides/5/eigenraeume.tex \ ../slides/5/zerlegung.tex \ ../slides/5/normalnilp.tex \ + ../slides/5/jordan.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 245b31d..5083abf 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -14,3 +14,4 @@ folie{5/nilpotent.tex} folie{5/eigenraeume.tex} folie{5/zerlegung.tex} folie{5/normalnilp.tex} +folie{5/jordan.tex} diff --git a/vorlesungen/slides/5/jordan.tex b/vorlesungen/slides/5/jordan.tex new file mode 100644 index 0000000..ad0e31d --- /dev/null +++ b/vorlesungen/slides/5/jordan.tex @@ -0,0 +1,10 @@ +% +% jordan.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Jordan Normalform} +XXX + +\end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 03601bd..c2d361f 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -64,6 +64,7 @@ % Jordan Normalform \section{Jordan-Normalform} +\folie{5/jordan.tex} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} % XXX \folie{5/jordannormalform.tex} -- cgit v1.2.1 From 38f326c35243942285c43310751025d2c16801dd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 4 Mar 2021 20:30:33 +0100 Subject: Spektraltheorie 1. Teil --- vorlesungen/02_msespektral/slides.tex | 11 +- vorlesungen/slides/5/Makefile.inc | 3 + vorlesungen/slides/5/bloecke.tex | 141 +++++++++++++++++++++ vorlesungen/slides/5/chapter.tex | 3 + vorlesungen/slides/5/charpoly.tex | 26 ++-- vorlesungen/slides/5/eigenraeume.tex | 14 ++- vorlesungen/slides/5/folgerungen.tex | 62 ++++++++- vorlesungen/slides/5/injektiv.tex | 74 ++++++++++- vorlesungen/slides/5/jordan.tex | 130 ++++++++++++++++++- vorlesungen/slides/5/kernbild.tex | 28 ++++- vorlesungen/slides/5/ketten.tex | 3 +- vorlesungen/slides/5/motivation.tex | 12 +- vorlesungen/slides/5/normalnilp.tex | 229 ++++++++++++++++++++++++++++++++++ vorlesungen/slides/5/zerlegung.tex | 44 ++++++- vorlesungen/slides/test.tex | 28 ++--- 15 files changed, 756 insertions(+), 52 deletions(-) create mode 100644 vorlesungen/slides/5/bloecke.tex (limited to 'vorlesungen') diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex index dc34236..9be6ce1 100644 --- a/vorlesungen/02_msespektral/slides.tex +++ b/vorlesungen/02_msespektral/slides.tex @@ -18,18 +18,23 @@ \folie{5/ketten.tex} \folie{5/dimension.tex} \folie{5/folgerungen.tex} +\folie{5/injektiv.tex} \folie{5/nilpotent.tex} -% XXX \folie{5/eigenraeume.tex} +\folie{5/eigenraeume.tex} +\folie{5/zerlegung.tex} +\folie{5/normalnilp.tex} +\folie{5/bloecke.tex} % Jordan Normalform \section{Jordan-Normalform} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} -% XXX \folie{5/jordannormalform.tex} +\folie{5/jordanblock.tex} +\folie{5/jordan.tex} % XXX \folie{5/minimalpolynom.tex} % XXX \folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} -% XXX \folie{5/cayleyhamilton.tex} +\folie{5/cayleyhamilton.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 4bed49b..872798e 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -16,6 +16,9 @@ chapter5 = \ ../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/cayleyhamilton.tex \ ../slides/5/chapter.tex 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/chapter.tex b/vorlesungen/slides/5/chapter.tex index 5083abf..7698f01 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -14,4 +14,7 @@ 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/cayleyhamilton.tex} diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex index 1211b43..63bfee5 100644 --- a/vorlesungen/slides/5/charpoly.tex +++ b/vorlesungen/slides/5/charpoly.tex @@ -20,6 +20,7 @@ $A-\mu I$ singulär ist: \] $\Rightarrow$ $\mu$ 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 $\chi_A(X)$: \[ @@ -27,33 +28,42 @@ $\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$: = (X-\mu_1)\dots (X-\mu_n) \] -\end{block} +\end{block}} +\uncover<3->{% \begin{block}{Fundamentalsatz der Algebra} Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in Linearfaktoren -\end{block} +\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} +\end{block}} +\uncover<5->{% \begin{proof}[Beweis] \begin{enumerate} -\item +\item<6-> $m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$ -\item +\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*} +} +\end{align*}} +\uncover<11->{% d.~h.~\( \displaystyle \overline{m}_A(X) @@ -61,8 +71,8 @@ d.~h.~\( \prod_i{i\in I}(X-\mu_i) \in \mathbb{C}[X] -\) -\end{proof} +\)} +\end{proof}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex index 5192cbc..fd4803c 100644 --- a/vorlesungen/slides/5/eigenraeume.tex +++ b/vorlesungen/slides/5/eigenraeume.tex @@ -15,19 +15,25 @@ E_\lambda(f) &= \ker (f-\lambda) \\ +\uncover<2->{ &= \{v\in V\;|\; f(v) = \lambda v\} +} \end{align*} -{\em Eigenraum} von $f$ zum Eigenwert $\lambda$. +\uncover<3->{% +{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.} \end{block} -$E_\lambda(f)\subset V$ ist ein Unterraum +\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{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<6->{% \begin{block}{verallgemeinerter Eigenraum} Für $\lambda\in \Bbbk$ heisst \[ @@ -36,7 +42,7 @@ Für $\lambda\in \Bbbk$ heisst \mathcal{K}(f-\lambda) \] verallgemeinerter Eigenraum -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex index a1fa6bf..4a8dbe6 100644 --- a/vorlesungen/slides/5/folgerungen.tex +++ b/vorlesungen/slides/5/folgerungen.tex @@ -3,10 +3,14 @@ % % (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}{0.48\textwidth} +\begin{column}{0.30\textwidth} \begin{block}{Zunahme} Für alle $k=latex,thick] +\def\pfad{ + ({0*\sx},6) -- + ({1*\sx},4.5) -- + ({2*\sx},3.5) -- + ({3*\sx},2.9) -- + ({4*\sx},2.6) -- + ({5*\sx},2.4) -- + ({6*\sx},2.4) +} + +\fill[color=orange!20] \pfad -- ({6*\sx},0) -- (0,0) -- cycle; +\fill[color=darkgreen!20] \pfad -- ({6*\sx},6) -- cycle; +\fill[color=orange!40] ({5*\sx},0) rectangle ({6*\sx},2.4); +\fill[color=darkgreen!40] ({5*\sx},6) rectangle ({6*\sx},2.4); + +\draw[color=darkgreen,line width=2pt] ({3*\sx},6) -- ({3*\sx},2.9); +\node[color=darkgreen] at ({3*\sx},4.45) [rotate=90,above] {$\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$}; + +\end{tikzpicture} +\end{center} \end{column} \end{columns} \end{frame} +\egroup diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex index e08cafd..90cbcd6 100644 --- a/vorlesungen/slides/5/injektiv.tex +++ b/vorlesungen/slides/5/injektiv.tex @@ -3,7 +3,79 @@ % % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % +\bgroup +\def\sx{1.05} \begin{frame}[t] \frametitle{$f$ injektiv auf $\mathcal{J}(f)$} -XXX +\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 index ad0e31d..e6ece47 100644 --- a/vorlesungen/slides/5/jordan.tex +++ b/vorlesungen/slides/5/jordan.tex @@ -3,8 +3,136 @@ % % (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} -XXX +\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/kernbild.tex b/vorlesungen/slides/5/kernbild.tex index f0bd6fa..3890717 100644 --- a/vorlesungen/slides/5/kernbild.tex +++ b/vorlesungen/slides/5/kernbild.tex @@ -10,29 +10,38 @@ \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} +\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{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% \begin{block}{Bild} Lineare Abbildung $f\colon V\to V$ \[ @@ -42,27 +51,36 @@ Lineare Abbildung $f\colon V\to V$ = \{f(v)\;|\; v\in V\} \] -\end{block} +\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\} +} \\ -&\mathcal{J}^{k-1}(f) +\uncover<11->{ +&=\mathcal{J}^{k-1}(f) +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex index 759d964..1116a83 100644 --- a/vorlesungen/slides/5/ketten.tex +++ b/vorlesungen/slides/5/ketten.tex @@ -33,6 +33,7 @@ Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt: \] \end{block} \vspace{-20pt} +\uncover<2->{% \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Abildung der Kerne} @@ -74,5 +75,5 @@ f\mathcal{J}(f)&= \mathcal{J}(f) \end{align*} \end{block} \end{column} -\end{columns} +\end{columns}} \end{frame} diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex index 4e8142d..b0a1d82 100644 --- a/vorlesungen/slides/5/motivation.tex +++ b/vorlesungen/slides/5/motivation.tex @@ -13,6 +13,7 @@ Matrix $A$ mit Minimalpolynom $m_A(X)$ vom Grad $s$ \end{block} +\uncover<2->{% \begin{block}{Faktorisieren} Minimalpolynom faktorisieren \[ @@ -20,7 +21,8 @@ m_A(X) = (X-\mu_1)(X-\mu_2)\dots(X-\mu_s) \] -\end{block} +\end{block}} +\uncover<3->{% \begin{block}{Vertauschen} $\sigma\in S_s$ eine Permutation von $1,\dots,s$ ist @@ -39,16 +41,18 @@ m_A(X) \dots (A-\mu_{\sigma(s)}) \end{align*} -\end{block} +\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} +\end{block}} +\uncover<5->{% \begin{block}{Eigenwerte} Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$ gibt mit @@ -57,7 +61,7 @@ gibt mit \Rightarrow Av = {\color{red}\mu} v \] Eigenwertbedingung -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex index a7af682..9457136 100644 --- a/vorlesungen/slides/5/normalnilp.tex +++ b/vorlesungen/slides/5/normalnilp.tex @@ -3,6 +3,235 @@ % % (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/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex index 9c20c60..a734d69 100644 --- a/vorlesungen/slides/5/zerlegung.tex +++ b/vorlesungen/slides/5/zerlegung.tex @@ -14,27 +14,48 @@ \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] (0,6) -- (15,6); -\draw[line width=0.1pt] (0,3) -- (15,3); \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 at (7,15) [above] {$\mathcal{E}_{\lambda_2}$}; -\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$}; -\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$}; \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} @@ -42,30 +63,41 @@ \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{align*}} \end{block} \end{column} \end{columns} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index c2d361f..2538f29 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -48,31 +48,31 @@ \section{Eigenwertproblem} % XXX Motivation: beliebige Funktionen f(A) berechnen -\folie{5/motivation.tex} -\folie{5/charpoly.tex} +%\folie{5/motivation.tex} +%\folie{5/charpoly.tex} \section{Invariante Unterräume} -\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/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} % Jordan Normalform \section{Jordan-Normalform} +\folie{5/jordanblock.tex} \folie{5/jordan.tex} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} -% XXX \folie{5/jordannormalform.tex} -% XXX \folie{5/minimalpolynom.tex} % XXX \folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} -% XXX \folie{5/cayleyhamilton.tex} +\folie{5/cayleyhamilton.tex} -- cgit v1.2.1 From 0dbc80dcaaf314f5f270cf7ffe11a410caec92d9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 5 Mar 2021 14:01:28 +0100 Subject: more slides --- vorlesungen/02_msespektral/slides.tex | 2 +- vorlesungen/slides/5/Makefile.inc | 1 + vorlesungen/slides/5/cayleyhamilton.tex | 91 +++++++++++++++++++++++ vorlesungen/slides/5/chapter.tex | 1 + vorlesungen/slides/5/jordanblock.tex | 68 ++++++++++++++++++ vorlesungen/slides/5/reellenormalform.tex | 115 ++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 8 +-- 7 files changed, 281 insertions(+), 5 deletions(-) create mode 100644 vorlesungen/slides/5/cayleyhamilton.tex create mode 100644 vorlesungen/slides/5/jordanblock.tex create mode 100644 vorlesungen/slides/5/reellenormalform.tex (limited to 'vorlesungen') diff --git a/vorlesungen/02_msespektral/slides.tex b/vorlesungen/02_msespektral/slides.tex index 9be6ce1..5e45a20 100644 --- a/vorlesungen/02_msespektral/slides.tex +++ b/vorlesungen/02_msespektral/slides.tex @@ -31,8 +31,8 @@ % XXX \folie{5/diagonalform.tex} \folie{5/jordanblock.tex} \folie{5/jordan.tex} +\folie{5/reellenormalform.tex} % XXX \folie{5/minimalpolynom.tex} -% XXX \folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 872798e..617e09f 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -19,6 +19,7 @@ chapter5 = \ ../slides/5/bloecke.tex \ ../slides/5/jordanblock.tex \ ../slides/5/jordan.tex \ + ../slides/5/reellenormalform.tex \ ../slides/5/cayleyhamilton.tex \ ../slides/5/chapter.tex 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 index 7698f01..1ce9d26 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -17,4 +17,5 @@ folie{5/normalnilp.tex} folie{5/bloecke.tex} folie{5/jordanblock.tex} folie{5/jordan.tex} +folie{5/reellenormalform.tex} folie{5/cayleyhamilton.tex} 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/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/test.tex b/vorlesungen/slides/test.tex index 2538f29..05946f0 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -65,14 +65,14 @@ % Jordan Normalform \section{Jordan-Normalform} -\folie{5/jordanblock.tex} -\folie{5/jordan.tex} +%\folie{5/jordanblock.tex} +%\folie{5/jordan.tex} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} -% XXX \folie{5/reellenormalform.tex} +\folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} -\folie{5/cayleyhamilton.tex} +%\folie{5/cayleyhamilton.tex} -- cgit v1.2.1 From 9243858393d79e761f03b4454547310056d7fcea Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 7 Mar 2021 21:00:59 +0100 Subject: Gauss --- vorlesungen/03_endlichekoerper/slides.tex | 8 +- vorlesungen/slides/4/Makefile.inc | 1 + vorlesungen/slides/4/chapter.tex | 1 + vorlesungen/slides/4/gauss.tex | 143 ++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 4 +- 5 files changed, 151 insertions(+), 6 deletions(-) create mode 100644 vorlesungen/slides/4/gauss.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_endlichekoerper/slides.tex b/vorlesungen/03_endlichekoerper/slides.tex index 67c97d6..19a9ab0 100644 --- a/vorlesungen/03_endlichekoerper/slides.tex +++ b/vorlesungen/03_endlichekoerper/slides.tex @@ -4,12 +4,12 @@ % (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Euklidischer Algorithmus} -% XXX \folie{4/ggt.tex} -% XXX \folie{4/euklidmatrix.tex} +\folie{4/ggt.tex} +\folie{4/euklidmatrix.tex} \section{Die Galois-Körper $\mathbb{F}_p$} -% XXX \folie{4/fp.tex} -% XXX \folie{4/division.tex} +\folie{4/fp.tex} +\folie{4/division.tex} % XXX \folie{4/gauss.tex} % XXX \folie{4/dh.tex} % XXX ? \folie{4/polynomefp.tex} diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 0adf913..91767a9 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -11,5 +11,6 @@ chapter4 = \ ../slides/4/euklidtabelle.tex \ ../slides/4/fp.tex \ ../slides/4/division.tex \ + ../slides/4/gauss.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 1c23783..27884a1 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -9,3 +9,4 @@ \folie{4/euklidtabelle.tex} \folie{4/fp.tex} \folie{4/division.tex} +\folie{4/gauss.tex} diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex new file mode 100644 index 0000000..960e8e1 --- /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}% +\[ +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 1 & 2 & 1 \\ + 2 & 0 & 1 & 2 \\ + 1 & 1 & 0 & 2 \\ +\hline +\end{tabular} +\uncover<4->{% +\to +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 1 & 2 & 1 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 1 \\ +\hline +\end{tabular}} +\uncover<6->{% +\to +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 1 & 0 & 2 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 1 \\ +\hline +\end{tabular}} +\uncover<8->{% +\to +\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} +\hline + 1 & 0 & 0 & 2 \\ + 0 & 1 & 0 & 0 \\ + 0 & 0 & 1 & 1 \\ +\hline +\end{tabular}} +\] +\end{minipage}}; +\begin{scope}[yshift=0.2cm] +\uncover<3->{ +\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<5->{ +\draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5); +} +\uncover<7->{ +\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/test.tex b/vorlesungen/slides/test.tex index 05946f0..07fb6a8 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -30,7 +30,7 @@ %\folie{4/euklidtabelle.tex} %\folie{4/fp.tex} %\folie{4/division.tex} -% XXX \folie{4/gauss.tex} +%\folie{4/gauss.tex} % XXX \folie{4/dh.tex} % XXX ? \folie{4/polynomefp.tex} % XXX \folie{4/frobenius.tex} @@ -69,7 +69,7 @@ %\folie{5/jordan.tex} % XXX Diagonalform % XXX \folie{5/diagonalform.tex} -\folie{5/reellenormalform.tex} +%\folie{5/reellenormalform.tex} % XXX \folie{5/hessenberg.tex} \section{Satz von Cayley-Hamilton} -- cgit v1.2.1 From a5eaa4838f81fe396b428708702bc392189e7484 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 7 Mar 2021 21:15:31 +0100 Subject: Diffie-Hellmann --- vorlesungen/slides/4/Makefile.inc | 1 + vorlesungen/slides/4/chapter.tex | 1 + vorlesungen/slides/4/dh.tex | 62 +++++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 2 +- 4 files changed, 65 insertions(+), 1 deletion(-) create mode 100644 vorlesungen/slides/4/dh.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 91767a9..4eae5b0 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -12,5 +12,6 @@ chapter4 = \ ../slides/4/fp.tex \ ../slides/4/division.tex \ ../slides/4/gauss.tex \ + ../slides/4/dh.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 27884a1..38e90eb 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -10,3 +10,4 @@ \folie{4/fp.tex} \folie{4/division.tex} \folie{4/gauss.tex} +\folie{4/dh.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/test.tex b/vorlesungen/slides/test.tex index 07fb6a8..9d1f6ef 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -31,7 +31,7 @@ %\folie{4/fp.tex} %\folie{4/division.tex} %\folie{4/gauss.tex} -% XXX \folie{4/dh.tex} +\folie{4/dh.tex} % XXX ? \folie{4/polynomefp.tex} % XXX \folie{4/frobenius.tex} -- cgit v1.2.1 From ccac1b9539544b1f782ea8351dea314512d19c4b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 7 Mar 2021 22:01:23 +0100 Subject: add new slides --- vorlesungen/slides/4/Makefile.inc | 2 ++ vorlesungen/slides/4/chapter.tex | 2 ++ vorlesungen/slides/4/divisionpoly.tex | 37 +++++++++++++++++++++++ vorlesungen/slides/4/polynomefp.tex | 57 +++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 6 ++-- 5 files changed, 101 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/4/divisionpoly.tex create mode 100644 vorlesungen/slides/4/polynomefp.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 4eae5b0..13de58c 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -13,5 +13,7 @@ chapter4 = \ ../slides/4/division.tex \ ../slides/4/gauss.tex \ ../slides/4/dh.tex \ + ../slides/4/divisionpoly.tex \ + ../slides/4/polynomefp.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 38e90eb..84b1f8f 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -11,3 +11,5 @@ \folie{4/division.tex} \folie{4/gauss.tex} \folie{4/dh.tex} +\folie{4/divisionpoly.tex} +\folie{4/polynomefp.tex} 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/polynomefp.tex b/vorlesungen/slides/4/polynomefp.tex new file mode 100644 index 0000000..fe514dd --- /dev/null +++ b/vorlesungen/slides/4/polynomefp.tex @@ -0,0 +1,57 @@ +% +% 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 $a_i\in\mathbb{F}_p$. +ObdA: $a_n=1$ + +\end{block} +\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} +\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} +\begin{block}{Körper $\mathbb{F}_p/(m(X))$} +Wenn $m(X)$ irreduzibel ist, dann ist +$\mathbb{F}_p[X]$ nullteilerfrei. +\medskip + +$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist +\begin{enumerate} +\item +$\operatorname{ggT}(a,m) = 1$ +\item +Es gibt $s,t\in\mathbb{F}_p[X]$ mit +\[ +s(X)m(X)+t(X)a(X) = 1 +\] +(aus dem euklidischen Algorithmus) +\item +$a^{-1} = t(X)$ +\end{enumerate} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 9d1f6ef..2df5421 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -31,13 +31,13 @@ %\folie{4/fp.tex} %\folie{4/division.tex} %\folie{4/gauss.tex} -\folie{4/dh.tex} -% XXX ? \folie{4/polynomefp.tex} +% \folie{4/dh.tex} % XXX \folie{4/frobenius.tex} +\folie{4/divisionpoly.tex} % XXX \folie{4/ggtpoly.tex} -% XXX \folie{4/divisionpoly.tex} % XXX \folie{4/euklidpoly.tex} +%\folie{4/polynomefp.tex} % XXX \folie{4/f2.tex} % XXX \folie{4/schieberegister.tex} -- cgit v1.2.1 From f7987f342c3f9153a0f05c2f940f370aa61960ee Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 7 Mar 2021 22:34:49 +0100 Subject: euklidslide --- vorlesungen/slides/4/Makefile.inc | 1 + vorlesungen/slides/4/chapter.tex | 1 + vorlesungen/slides/4/euklidpoly.tex | 47 +++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 9 ++++--- 4 files changed, 53 insertions(+), 5 deletions(-) create mode 100644 vorlesungen/slides/4/euklidpoly.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 13de58c..2329e4a 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -14,6 +14,7 @@ chapter4 = \ ../slides/4/gauss.tex \ ../slides/4/dh.tex \ ../slides/4/divisionpoly.tex \ + ../slides/4/euklidpoly.tex \ ../slides/4/polynomefp.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 84b1f8f..830537e 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -12,4 +12,5 @@ \folie{4/gauss.tex} \folie{4/dh.tex} \folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} \folie{4/polynomefp.tex} 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/test.tex b/vorlesungen/slides/test.tex index 2df5421..b7b696a 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -26,17 +26,16 @@ %\folie{4/ggt.tex} %\folie{4/euklidmatrix.tex} -%\folie{4/euklidbeispiel.tex} -%\folie{4/euklidtabelle.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} % XXX \folie{4/frobenius.tex} -\folie{4/divisionpoly.tex} -% XXX \folie{4/ggtpoly.tex} -% XXX \folie{4/euklidpoly.tex} +%\folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} %\folie{4/polynomefp.tex} % XXX \folie{4/f2.tex} -- cgit v1.2.1 From f2454006fa4e2a0b4093507300fab8a29e3b5901 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 09:40:32 +0100 Subject: final preparation --- vorlesungen/00_template/Makefile | 9 +- vorlesungen/03_endlichekoerper/Makefile | 11 ++- .../MathSem-03-endlichekoerper.tex | 4 + vorlesungen/03_endlichekoerper/dh.jpg | Bin 0 -> 154552 bytes vorlesungen/03_endlichekoerper/slides.tex | 16 ++-- vorlesungen/slides/4/Makefile.inc | 2 + vorlesungen/slides/4/alpha.tex | 54 ++++++++++++ vorlesungen/slides/4/chapter.tex | 2 + vorlesungen/slides/4/division.tex | 41 +++++---- vorlesungen/slides/4/euklidbeispiel.tex | 66 ++++++++++---- vorlesungen/slides/4/euklidmatrix.tex | 39 +++++--- vorlesungen/slides/4/euklidtabelle.tex | 43 +++++---- vorlesungen/slides/4/fp.tex | 41 +++++---- vorlesungen/slides/4/gauss.tex | 12 +-- vorlesungen/slides/4/ggt.tex | 43 +++++---- vorlesungen/slides/4/polynomefp.tex | 21 +++-- vorlesungen/slides/4/schieberegister.tex | 98 +++++++++++++++++++++ vorlesungen/slides/test.tex | 9 +- 18 files changed, 387 insertions(+), 124 deletions(-) create mode 100644 vorlesungen/03_endlichekoerper/dh.jpg create mode 100644 vorlesungen/slides/4/alpha.tex create mode 100644 vorlesungen/slides/4/schieberegister.tex (limited to 'vorlesungen') diff --git a/vorlesungen/00_template/Makefile b/vorlesungen/00_template/Makefile index fe6bbb7..14c254d 100644 --- a/vorlesungen/00_template/Makefile +++ b/vorlesungen/00_template/Makefile @@ -15,7 +15,7 @@ MathSem-yyy-xxx.pdf: MathSem-yyy-xxx.tex $(SOURCES) xxx-handout.pdf: xxx-handout.tex $(SOURCES) pdflatex xxx-handout.tex -thumbnail: thumbnail.jpg +thumbnail: thumbnail.jpg # fix1.jpg thumbnail.pdf: MathSem-yyy-xxx.pdf pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ @@ -24,3 +24,10 @@ thumbnail.jpg: thumbnail.pdf convert -density 300 thumbnail.pdf \ -resize 1920x1080 -units PixelsPerInch thumbnail.jpg +fix1.pdf: MathSem-yyy-xxx.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-yyy-xxx.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/03_endlichekoerper/Makefile b/vorlesungen/03_endlichekoerper/Makefile index 316e749..cc2ded5 100644 --- a/vorlesungen/03_endlichekoerper/Makefile +++ b/vorlesungen/03_endlichekoerper/Makefile @@ -15,7 +15,7 @@ MathSem-03-endlichekoerper.pdf: MathSem-03-endlichekoerper.tex $(SOURCES) endlichekoerper-handout.pdf: endlichekoerper-handout.tex $(SOURCES) pdflatex endlichekoerper-handout.tex -thumbnail: thumbnail.jpg +thumbnail: thumbnail.jpg fix1.jpg thumbnail.pdf: MathSem-03-endlichekoerper.pdf pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ @@ -24,3 +24,12 @@ thumbnail.jpg: thumbnail.pdf convert -density 300 thumbnail.pdf \ -resize 1920x1080 -units PixelsPerInch thumbnail.jpg +fix1.pdf: MathSem-03-endlichekoerper.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-03-endlichekoerper.pdf 76 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg +dh.jpg: fix1.jpg + convert -extract 1598x860+256+186 fix1.jpg dh.jpg + diff --git a/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex b/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex index 31229d7..9ff79e6 100644 --- a/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex +++ b/vorlesungen/03_endlichekoerper/MathSem-03-endlichekoerper.tex @@ -9,6 +9,10 @@ \begin{document} \begin{frame} \titlepage +\vspace{-1.6cm} +\begin{center} +\includegraphics[width=11cm]{dh.jpg} +\end{center} \end{frame} \input{slides.tex} \end{document} diff --git a/vorlesungen/03_endlichekoerper/dh.jpg b/vorlesungen/03_endlichekoerper/dh.jpg new file mode 100644 index 0000000..8bd07c3 Binary files /dev/null and b/vorlesungen/03_endlichekoerper/dh.jpg differ diff --git a/vorlesungen/03_endlichekoerper/slides.tex b/vorlesungen/03_endlichekoerper/slides.tex index 19a9ab0..2111463 100644 --- a/vorlesungen/03_endlichekoerper/slides.tex +++ b/vorlesungen/03_endlichekoerper/slides.tex @@ -10,19 +10,19 @@ \section{Die Galois-Körper $\mathbb{F}_p$} \folie{4/fp.tex} \folie{4/division.tex} -% XXX \folie{4/gauss.tex} -% XXX \folie{4/dh.tex} -% XXX ? \folie{4/polynomefp.tex} +\folie{4/gauss.tex} +\folie{4/dh.tex} % XXX \folie{4/frobenius.tex} \section{Rechnen in $\mathbb{F}_p(\alpha)$} -% XXX \folie{4/ggtpoly.tex} -% XXX \folie{4/divisionpoly.tex} -% XXX \folie{4/euklidpoly.tex} +\folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} +\folie{4/polynomefp.tex} +\folie{4/alpha.tex} -\section{$\mathbb{F}_2(\alpha)$} +\section{Spezialfall $\mathbb{F}_2(\alpha)$} % XXX \folie{4/f2.tex} -% XXX \folie{4/schieberegister.tex} +\folie{4/schieberegister.tex} \section{Anwendung: elliptische Kurven} % XXX Idee der elliptischen Kurve diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 2329e4a..ad1081e 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -16,5 +16,7 @@ chapter4 = \ ../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 index 830537e..a10712a 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -14,3 +14,5 @@ \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/division.tex b/vorlesungen/slides/4/division.tex index aeab1e3..846738f 100644 --- a/vorlesungen/slides/4/division.tex +++ b/vorlesungen/slides/4/division.tex @@ -13,26 +13,28 @@ \begin{block}{Inverse {\bf berechnen}} Gegeben $a\in\mathbb{F}_p$, finde $b=a^{-1}\in\mathbb{F}_p$ \begin{align*} -&& a{\color{red}b} &\equiv 1 \mod p +\uncover<2->{&& a{\color{blue}b} &\equiv 1 \mod p} \\ -&\Leftrightarrow& a{\color{red}b}&=1 + {\color{red}n}p +\uncover<3->{&\Leftrightarrow& a{\color{blue}b}&=1 + {\color{blue}n}p} \\ -&&a{\color{red}b}-{\color{red}n}p&=1 +\uncover<4->{&&a{\color{blue}b}-{\color{blue}n}p&=1} \end{align*} -Wegen +\uncover<5->{Wegen $\operatorname{ggT}(a,p)=1$ gibt es $s$ und $t$ mit \[ -sa+tb=1 +{\color{red}s}a+{\color{red}t}p=1 \Rightarrow -{\color{red}b}=s,\; -{\color{red}n}=-t -\] +{\color{blue}b}={\color{red}s},\; +{\color{blue}n}=-{\color{red}t} +\]} +\uncover<6->{% $\Rightarrow$ Die Inverse kann mit dem euklidischen Algorithmus -berechnet werden +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} @@ -42,21 +44,22 @@ Finde $47^{-1}\in\mathbb{F}_{1291}$ k& a_k& b_k&q_k& c_k& d_k\\ \hline & & & & 1& 0\\ -0& 47&1291& 0& 0& 1\\ -1&1291& 47& 27& 1& 0\\ -2& 47& 22& 2& -27& 1\\ -3& 22& 3& 7& 55& -2\\ -4& 3& 1& 3&{\color{red}-412}&{\color{red}15}\\ -5& 1& 0& & 1291& -47\\ +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->{% \[ {\color{red}-412}\cdot 47 +{\color{red}15}\cdot 1291 = 1 -\;\Rightarrow\; -47^{-1}={\color{red}879} -\] -\end{block} +\uncover<24->{\;\Rightarrow\; +47^{-1}={\color{red}879}} +\]} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/4/euklidbeispiel.tex b/vorlesungen/slides/4/euklidbeispiel.tex index cbc3137..366a7a6 100644 --- a/vorlesungen/slides/4/euklidbeispiel.tex +++ b/vorlesungen/slides/4/euklidbeispiel.tex @@ -6,39 +6,73 @@ \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} \begin{frame}[t] -\frametitle{Beispiel} +\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 &25&=15 \cdot {\color{red} 1} + 10 &q_0 &= {\color{red}1} & r_0 &= 10\\ -a_1&=15 & b_1 &= 10 &15&=10 \cdot {\color{darkgreen}1} + \phantom{0}5 &q_1 &= {\color{darkgreen}1} & r_1 &= \phantom{0}5 \\ -a_2&=10 & b_2 &= \phantom{0}5 &10&=\phantom{0}5 \cdot {\color{blue} 2} + \phantom{0}0 &q_2 &= {\color{blue}2} & r_2 &= \phantom{0}0 +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 &= -Q({\color{blue}2}) Q({\color{darkgreen}1}) Q({\color{red}1}) +\uncover<9->{Q({\color{blue}2})} +\uncover<8->{Q({\color{darkgreen}1})} +Q({\color{orange}1}) = -\begin{pmatrix}0&1\\1&-{\color{blue}2}\end{pmatrix} -\begin{pmatrix}0&1\\1&-{\color{darkgreen}1}\end{pmatrix} -\begin{pmatrix}0&1\\1&-{\color{red}1}\end{pmatrix} -=\begin{pmatrix} --1&2\\3&-5 -\end{pmatrix} +\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} +\end{block}} \vspace{-5pt} +\uncover<10->{% \begin{block}{Relationen ablesen} -\begin{align*} -\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 = -1\cdot {\usebeamercolor[fg]{title}25} + 2\cdot {\usebeamercolor[fg]{title}15} \\ +\[ +\begin{pmatrix} +\operatorname{ggT}(a,b)\\0 +\end{pmatrix} += +Q +\begin{pmatrix}a\\b\end{pmatrix} +\uncover<11->{% +\quad +\Rightarrow\quad +\left\{ +\begin{aligned} +\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 = +{\color{red}-1}\cdot {\usebeamercolor[fg]{title}25} + {\color{red}2}\cdot {\usebeamercolor[fg]{title}15} \\ 0 &= \phantom{5=-}3\cdot {\usebeamercolor[fg]{title}25} -5\cdot {\usebeamercolor[fg]{title}15} -\end{align*} -\end{block} +\end{aligned} +\right.} +\] +\end{block}} \end{frame} diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex index 6ffa4c2..be5b3ca 100644 --- a/vorlesungen/slides/4/euklidmatrix.tex +++ b/vorlesungen/slides/4/euklidmatrix.tex @@ -14,17 +14,19 @@ \vspace{-10pt} \[ a_k = b_kq_k + r_k +\uncover<2->{ \;\Rightarrow\; \left\{ \begin{aligned} a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\ b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k \end{aligned} -\right. +\right.} \] \end{block} \end{column} \begin{column}{0.44\textwidth} +\uncover<3->{% \begin{block}{Matrixschreibweise} \vspace{-10pt} \begin{align*} @@ -37,23 +39,30 @@ b_{k+1} b_k\\r_k \end{pmatrix} = -\underbrace{\begin{pmatrix}0&1\\1&-q_k\end{pmatrix}}_{\displaystyle =Q(q_k)} +\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{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} @@ -63,31 +72,37 @@ r_{n} \operatorname{ggT}(a,b) \\ 0 \end{pmatrix} +\uncover<11->{ = -\underbrace{Q(q_n) -\dots -Q(q_1) -Q(q_0)}_{\displaystyle =Q} +\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} +\end{block}} +\uncover<16->{% \begin{block}{Konsequenzen} \[ Q=\begin{pmatrix} q_{11}&q_{12}\\ -a_{21}&q_{22} +q_{21}&q_{22} \end{pmatrix} \quad\Rightarrow\quad \left\{ \quad \begin{aligned} -\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b \\ +\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b = {\color{red}s}a+{\color{red}t}b\\ 0 &= q_{21}a + q_{22}b \end{aligned} \right. \] -\end{block} +\end{block}} \end{frame} diff --git a/vorlesungen/slides/4/euklidtabelle.tex b/vorlesungen/slides/4/euklidtabelle.tex index 2d67823..3f1b8d7 100644 --- a/vorlesungen/slides/4/euklidtabelle.tex +++ b/vorlesungen/slides/4/euklidtabelle.tex @@ -8,22 +8,29 @@ \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) -%\begin{pmatrix} -%0&1\\1&-q_k -%\end{pmatrix} +\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} -&&\Rightarrow& +\end{pmatrix}} +&&\uncover<5->{\Rightarrow& \begin{pmatrix} c_k&d_k\\c_{k+1}&d_{k+1} \end{pmatrix} @@ -34,31 +41,33 @@ Q(q_k) %\end{pmatrix} \begin{pmatrix} c_{k-1}&d_{k-1}\\c_{k}&d_{k} -\end{pmatrix} +\end{pmatrix}} \end{align*} -\end{block} +\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 &q_0 & 0 & 1 \\ - 1 &q_1 &c_{-1} -q_0 \cdot c_0 &d_{-1} -q_0 \cdot d_0 \\ - 2 &q_2 &c_0 -q_1 \cdot c_1 &d_0 -q_1 \cdot d_1 \\ -\vdots&\vdots&\vdots &\vdots \\ - n &q_n &c_{n-2}-q_{n-1}\cdot c_{n-1}&d_{n-2}-q_{n-1}\cdot d_{n-1}\\ -n+1& &c_{n-1}-q_{n} \cdot c_{n} &d_{n-1}-q_{n} \cdot d_{n} \\ + 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{$c_{n}$}\phantom{c_{n+1}} a + \rlap{$d_n$}\phantom{d_{n+1}}b &= \operatorname{ggT}(a,b) +\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*} +\right.} +\end{equation*}} \end{frame} diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex index caa6ceb..968b777 100644 --- a/vorlesungen/slides/4/fp.tex +++ b/vorlesungen/slides/4/fp.tex @@ -30,6 +30,7 @@ $\mathbb{Z}/n\mathbb{Z} =\{ \llbracket r\rrbracket\;|\; 0\le r < n \} \mathstrut$ 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 $\mathbb{Z}/n\mathbb{Z}$: @@ -43,18 +44,22 @@ $\llbracket n_2\rrbracket$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$: = \llbracket 0 \rrbracket \] -\end{block} +\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} +\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{block}} \end{column} \end{columns} \vspace{-20pt} @@ -62,6 +67,7 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper \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} @@ -106,14 +112,15 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper \feld{4}{4}{4} \feld{4}{5}{2} \feld{5}{4}{2} \feld{5}{5}{1} -\end{scope} +\end{scope}} +\uncover<6->{ \begin{scope}[xshift=6cm] -\gruen{1}{1} -\gruen{2}{4} -\gruen{3}{5} -\gruen{4}{2} -\gruen{5}{3} -\gruen{6}{6} +\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}{ @@ -158,13 +165,13 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper \geld{5}{5}{4} \geld{6}{5}{2} \geld{5}{6}{2} \geld{6}{6}{1} -\inverse{1}{1} -\inverse{2}{4} -\inverse{3}{5} -\inverse{4}{2} -\inverse{5}{3} -\inverse{6}{6} -\end{scope} +\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} diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex index 960e8e1..23cdfee 100644 --- a/vorlesungen/slides/4/gauss.tex +++ b/vorlesungen/slides/4/gauss.tex @@ -95,7 +95,7 @@ z&=1 1 & 1 & 0 & 2 \\ \hline \end{tabular} -\uncover<4->{% +\uncover<5->{% \to \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} \hline @@ -104,7 +104,7 @@ z&=1 0 & 0 & 1 & 1 \\ \hline \end{tabular}} -\uncover<6->{% +\uncover<7->{% \to \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} \hline @@ -113,7 +113,7 @@ z&=1 0 & 0 & 1 & 1 \\ \hline \end{tabular}} -\uncover<8->{% +\uncover<9->{% \to \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|} \hline @@ -125,14 +125,14 @@ z&=1 \] \end{minipage}}; \begin{scope}[yshift=0.2cm] -\uncover<3->{ +\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<5->{ +\uncover<6->{ \draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5); } -\uncover<7->{ +\uncover<8->{ \draw[color=blue] (1.05,0.5) -- (1.05,0.2) arc (180:360:0.2) -- (1.45,0.5); } \end{scope} diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex index e3c55e6..ef97182 100644 --- a/vorlesungen/slides/4/ggt.tex +++ b/vorlesungen/slides/4/ggt.tex @@ -15,16 +15,22 @@ 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*} -a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0 \\ -a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1 \\ -a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2 \\ - &\;\vdots & & & & \\ -a_n&=b_nq_n + r_n & r_n &= 0 & r_{n-1}&=\operatorname{ggT}(a,b) +\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{block}} \end{column} \begin{column}{0.48\textwidth} \begin{block}{$\operatorname{ggT}(15,25) = 5$} @@ -40,18 +46,25 @@ a_n&=b_nq_n + r_n & r_n &= 0 & r_{n-1}&=\operatorname{ggT}(a,b) \foreach \y in {0,...,2}{ \draw[line width=0.2pt] (-2,{\y*25}) -- (65,{\y*25}); } -\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<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 $\x$}; + } } -\foreach \x in {0,...,4}{ - \foreach \y in {0,...,2}{ - \fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8]; +\uncover<2->{ + \foreach \x in {0,...,4}{ + \foreach \y in {0,...,2}{ + \fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8]; + } } } -\foreach \x in {0,5,...,60}{ - \fill[color=blue] (\x,0) circle[radius=0.5]; - \node at (\x,0) [below] {\tiny $\x$}; +\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} diff --git a/vorlesungen/slides/4/polynomefp.tex b/vorlesungen/slides/4/polynomefp.tex index fe514dd..1db50e1 100644 --- a/vorlesungen/slides/4/polynomefp.tex +++ b/vorlesungen/slides/4/polynomefp.tex @@ -18,40 +18,45 @@ p(X) a_0+a_1X+\dots+a_nX^n \] mit $a_i\in\mathbb{F}_p$. -ObdA: $a_n=1$ +\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} +\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{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 -$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist +\uncover<6->{$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist} \begin{enumerate} -\item +\item<7-> $\operatorname{ggT}(a,m) = 1$ -\item +\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 +\item<9-> $a^{-1} = t(X)$ \end{enumerate} -\end{block} +\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..6914c79 --- /dev/null +++ b/vorlesungen/slides/4/schieberegister.tex @@ -0,0 +1,98 @@ +% +% schieberegister.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\ds{0.7} +\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{Schieberegister} +Rechnen mit Polynomen in $\mathbb{F}_2(\alpha)$ ist speziell einfach +\\ +Minimalpolynom von $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard) + +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<2->{ +\begin{scope} + \rahmen + \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<3->{ + \begin{scope}[yshift=-2cm] + \uncover<4->{ + \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] + {$\uncover<5->{X^4+X^3+X+1=}X^8$\strut}; + \end{scope} + \node at \punkt{-1}{0} {$1$\strut}; + \end{scope} +} + +\uncover<6->{ + {\color<8->{red} + \draw[->] (-3,-1.5) to[out=-90,in=180] (-0.5,-2.7); + } + \begin{scope}[yshift=-2.7cm] + \rahmen + \polynom00011011 + \end{scope} +} + +\uncover<7->{ + \node at ({3.5*\ds},-3.45) {$\|$}; + + \begin{scope}[yshift=-4.2cm] + \rahmen + \polynom00110111 + \end{scope} +} + +\uncover<8->{ + \node[color=red] at (-3.5,-2.7) {Feedback}; +} + +\end{tikzpicture} +\end{center} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index b7b696a..659be92 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -26,8 +26,8 @@ %\folie{4/ggt.tex} %\folie{4/euklidmatrix.tex} -\folie{4/euklidbeispiel.tex} -\folie{4/euklidtabelle.tex} +%\folie{4/euklidbeispiel.tex} +%\folie{4/euklidtabelle.tex} %\folie{4/fp.tex} %\folie{4/division.tex} %\folie{4/gauss.tex} @@ -35,11 +35,12 @@ % XXX \folie{4/frobenius.tex} %\folie{4/divisionpoly.tex} -\folie{4/euklidpoly.tex} +%\folie{4/euklidpoly.tex} %\folie{4/polynomefp.tex} +%\folie{4/alpha.tex} % XXX \folie{4/f2.tex} -% XXX \folie{4/schieberegister.tex} +\folie{4/schieberegister.tex} % XXX Idee der elliptischen Kurve % XXX \folie{4/ecidee.tex} -- cgit v1.2.1 From 18c273cb55ea5f54b125d8d6e3032b25cf56f8f3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 10:27:04 +0100 Subject: besser strukturierung --- vorlesungen/slides/4/schieberegister.tex | 38 +++++++++++++++++++++++++------- 1 file changed, 30 insertions(+), 8 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/slides/4/schieberegister.tex b/vorlesungen/slides/4/schieberegister.tex index 6914c79..f349337 100644 --- a/vorlesungen/slides/4/schieberegister.tex +++ b/vorlesungen/slides/4/schieberegister.tex @@ -5,6 +5,7 @@ % \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}); @@ -23,17 +24,25 @@ \node at \punkt{7}{0} {$#8$}; } \begin{frame}[t] -\frametitle{Schieberegister} -Rechnen mit Polynomen in $\mathbb{F}_2(\alpha)$ ist speziell einfach +\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 $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard) +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}; @@ -48,10 +57,20 @@ Minimalpolynom von $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard) \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 } @@ -62,7 +81,7 @@ Minimalpolynom von $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard) \node at \punkt{6}{1} {$X$\strut}; \begin{scope}[xshift=0.4cm] \node at \punkt{-1}{1} [left] - {$\uncover<5->{X^4+X^3+X+1=}X^8$\strut}; + {$\uncover<5->{{\color{darkgreen}\alpha^4+\alpha^3+\alpha+1=\alpha^8}}\only<-4>{X^8}$\strut}; \end{scope} \node at \punkt{-1}{0} {$1$\strut}; \end{scope} @@ -70,11 +89,13 @@ Minimalpolynom von $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard) \uncover<6->{ {\color<8->{red} - \draw[->] (-3,-1.5) to[out=-90,in=180] (-0.5,-2.7); + \draw[->] (-2.5,-1.5) to[out=-90,in=180] (-0.5,-2.7); } \begin{scope}[yshift=-2.7cm] \rahmen - \polynom00011011 + {\color{darkgreen} + \polynom00011011 + } \end{scope} } @@ -83,12 +104,13 @@ Minimalpolynom von $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard) \begin{scope}[yshift=-4.2cm] \rahmen - \polynom00110111 + \polynom00110001 + \node at \punkt{7.6}{0} [right] {$\mathstrut=\alpha\cdot p(\alpha)$}; \end{scope} } \uncover<8->{ - \node[color=red] at (-3.5,-2.7) {Feedback}; + \node[color=red] at (-3.0,-2.5) {Feedback}; } \end{tikzpicture} -- cgit v1.2.1 From 8080f48d82554f32afb2c935a155d723bce1c118 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 15:22:21 +0100 Subject: start working on MSE week 3 --- vorlesungen/03_mseanalysis/Makefile | 33 +++++++++++++++ .../03_mseanalysis/MathSemMSE-03-analysis.tex | 14 +++++++ vorlesungen/03_mseanalysis/analysis-handout.tex | 11 +++++ vorlesungen/03_mseanalysis/common.tex | 16 ++++++++ vorlesungen/03_mseanalysis/slides.tex | 48 ++++++++++++++++++++++ vorlesungen/slides/2/Makefile.inc | 9 ++++ vorlesungen/slides/2/chapter.tex | 6 +++ vorlesungen/slides/Makefile.inc | 4 +- vorlesungen/slides/test.tex | 1 - 9 files changed, 140 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/03_mseanalysis/Makefile create mode 100644 vorlesungen/03_mseanalysis/MathSemMSE-03-analysis.tex create mode 100644 vorlesungen/03_mseanalysis/analysis-handout.tex create mode 100644 vorlesungen/03_mseanalysis/common.tex create mode 100644 vorlesungen/03_mseanalysis/slides.tex create mode 100644 vorlesungen/slides/2/Makefile.inc create mode 100644 vorlesungen/slides/2/chapter.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/Makefile b/vorlesungen/03_mseanalysis/Makefile new file mode 100644 index 0000000..4f8e482 --- /dev/null +++ b/vorlesungen/03_mseanalysis/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- analysis +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: analysis-handout.pdf MathSemMSE-03-analysis.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-03-analysis.pdf: MathSemMSE-03-analysis.tex $(SOURCES) + pdflatex MathSemMSE-03-analysis.tex + +analysis-handout.pdf: analysis-handout.tex $(SOURCES) + pdflatex analysis-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-03-analysis.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-03-analysis.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-03-analysis.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-03-analysis.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/03_mseanalysis/MathSemMSE-03-analysis.tex b/vorlesungen/03_mseanalysis/MathSemMSE-03-analysis.tex new file mode 100644 index 0000000..174562e --- /dev/null +++ b/vorlesungen/03_mseanalysis/MathSemMSE-03-analysis.tex @@ -0,0 +1,14 @@ +% +% MathSem-03-mseanalysis.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/03_mseanalysis/analysis-handout.tex b/vorlesungen/03_mseanalysis/analysis-handout.tex new file mode 100644 index 0000000..4986d7d --- /dev/null +++ b/vorlesungen/03_mseanalysis/analysis-handout.tex @@ -0,0 +1,11 @@ +% +% mseanalysis-handout.tex -- Handout XXX +% +% (c) 2017 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/03_mseanalysis/common.tex b/vorlesungen/03_mseanalysis/common.tex new file mode 100644 index 0000000..6c73e87 --- /dev/null +++ b/vorlesungen/03_mseanalysis/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Analysis]{Analysis mit Matrizen} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex new file mode 100644 index 0000000..185b67b --- /dev/null +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -0,0 +1,48 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Matrixnormen} +% XXX Vektornormen +% XXX \folie{2/norm.tex} +% XXX Skalarprodukt und L^2-Norm +% XXX \folie{2/skalarprodukt.tex} +% XXX Normen, die sich aus der Vektornorm ableiten lassen +% XXX \folie{2/operatornorm.tex} +% XXX Frobenius-Norm und Hadamard-Algebra +% XXX \folie{2/frobeniusnorm.tex} + +\section{Approximation mit Polynomen} +% XXX Stone-Weierstrass +% XXX \folie{5/stoneweierstrass.tex} +% XXX Spektrum einer Matrix +% XXX \folie{5/spektrum.tex} +% XXX Approximation einer Funktion auf dem Spektrum +% XXX \folie{5/spektrumapproximation.tex} +% XXX Approximation einer Matrix in der erzeugten Algebra +% XXX \folie{5/matrixapproximation.tex} +% XXX Gelfand-Transformation +% XXX \folie{5/gelfandtransformation.tex} + +\section{Potenzreihen} +% XXX Konvergenzradius +% XXX \folie{5/konvergenzradius.tex} +% XXX Gelfand-Radius +% XXX \folie{5/gelfandradius.tex} +% XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) +% XXX \folie{5/satzvongelfand.tex} + +\section{Differentialgleichungen} +% XXX Potenzreihenmethode zur Lösung von Differentialgleichungen +% XXX \folie{5/potenzreihenmethode.tex} +% XXX Exponentialfunktion +% XXX \folie{5/exponentialfunktion.tex} +% XXX Exponentialreihe +% XXX \folie{5/exponentialreihe.tex} +% XXX Logarithmus +% XXX \folie{5/logarithmusreihe.tex} +% XXX Sinus und Cosinus, Eulerscher Satz +% XXX \folie{5/sinuscosinus.tex} + diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc new file mode 100644 index 0000000..644e6ae --- /dev/null +++ b/vorlesungen/slides/2/Makefile.inc @@ -0,0 +1,9 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter2 = \ + ../slides/2/chapter.tex + diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex new file mode 100644 index 0000000..003d95d --- /dev/null +++ b/vorlesungen/slides/2/chapter.tex @@ -0,0 +1,6 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +%\folie{2/norm.tex} diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 90bb388..20929e4 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -5,9 +5,11 @@ # 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 slides = \ - $(chapter0) $(chapter1) $(chapter3) $(chapter4) $(chapter5) + $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ + $(chapter5) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 659be92..38d481b 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -75,4 +75,3 @@ \section{Satz von Cayley-Hamilton} %\folie{5/cayleyhamilton.tex} - -- cgit v1.2.1 From 72e7b8b04c01440fe8511204f0542fa0e4b81abe Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 19:22:42 +0100 Subject: new slides --- vorlesungen/03_endlichekoerper/slides.tex | 2 ++ vorlesungen/slides/2/Makefile.inc | 2 ++ vorlesungen/slides/2/chapter.tex | 3 ++- vorlesungen/slides/test.tex | 44 ++++++++++++++++++++++++++++++- 4 files changed, 49 insertions(+), 2 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/03_endlichekoerper/slides.tex b/vorlesungen/03_endlichekoerper/slides.tex index 2111463..c4b06b4 100644 --- a/vorlesungen/03_endlichekoerper/slides.tex +++ b/vorlesungen/03_endlichekoerper/slides.tex @@ -6,6 +6,8 @@ \section{Euklidischer Algorithmus} \folie{4/ggt.tex} \folie{4/euklidmatrix.tex} +\folie{4/euklidbeispiel.tex} +\folie{4/euklidtabelle.tex} \section{Die Galois-Körper $\mathbb{F}_p$} \folie{4/fp.tex} diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 644e6ae..93bcbd5 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -5,5 +5,7 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapter2 = \ + ../slides/2/norm.tex \ + ../slides/2/skalarprodukt.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 003d95d..cfca9ab 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -3,4 +3,5 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % -%\folie{2/norm.tex} +\folie{2/norm.tex} +\folie{2/skalarprodukt.tex} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 38d481b..eea9bec 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -40,7 +40,7 @@ %\folie{4/alpha.tex} % XXX \folie{4/f2.tex} -\folie{4/schieberegister.tex} +%\folie{4/schieberegister.tex} % XXX Idee der elliptischen Kurve % XXX \folie{4/ecidee.tex} @@ -75,3 +75,45 @@ \section{Satz von Cayley-Hamilton} %\folie{5/cayleyhamilton.tex} +\section{Matrixnormen} +% XXX Vektornormen +\folie{2/norm.tex} +% XXX Skalarprodukt und L^2-Norm +\folie{2/skalarprodukt.tex} +% XXX Normen, die sich aus der Vektornorm ableiten lassen +% XXX \folie{2/operatornorm.tex} +% XXX Frobenius-Norm und Hadamard-Algebra +% XXX \folie{2/frobeniusnorm.tex} + +\section{Approximation mit Polynomen} +% XXX Stone-Weierstrass +% XXX \folie{5/stoneweierstrass.tex} +% XXX Spektrum einer Matrix +% XXX \folie{5/spektrum.tex} +% XXX Approximation einer Funktion auf dem Spektrum +% XXX \folie{5/spektrumapproximation.tex} +% XXX Approximation einer Matrix in der erzeugten Algebra +% XXX \folie{5/matrixapproximation.tex} +% XXX Gelfand-Transformation +% XXX \folie{5/gelfandtransformation.tex} + +\section{Potenzreihen} +% XXX Konvergenzradius +% XXX \folie{5/konvergenzradius.tex} +% XXX Gelfand-Radius +% XXX \folie{5/gelfandradius.tex} +% XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) +% XXX \folie{5/satzvongelfand.tex} + +\section{Differentialgleichungen} +% XXX Potenzreihenmethode zur Lösung von Differentialgleichungen +% XXX \folie{5/potenzreihenmethode.tex} +% XXX Exponentialfunktion +% XXX \folie{5/exponentialfunktion.tex} +% XXX Exponentialreihe +% XXX \folie{5/exponentialreihe.tex} +% XXX Logarithmus +% XXX \folie{5/logarithmusreihe.tex} +% XXX Sinus und Cosinus, Eulerscher Satz +% XXX \folie{5/sinuscosinus.tex} + -- cgit v1.2.1 From 8ca5b82187860699d4ab3f3b9771c1b12aed7370 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 20:22:40 +0100 Subject: new slides --- vorlesungen/slides/2/Makefile.inc | 1 + vorlesungen/slides/2/cauchyschwarz.tex | 94 ++++++++++++++++++++++++++++++++++ vorlesungen/slides/2/chapter.tex | 1 + vorlesungen/slides/2/norm.tex | 58 +++++++++++++++++++++ vorlesungen/slides/2/skalarprodukt.tex | 93 +++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 4 ++ 6 files changed, 251 insertions(+) create mode 100644 vorlesungen/slides/2/cauchyschwarz.tex create mode 100644 vorlesungen/slides/2/norm.tex create mode 100644 vorlesungen/slides/2/skalarprodukt.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 93bcbd5..2eb3ce9 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -7,5 +7,6 @@ chapter2 = \ ../slides/2/norm.tex \ ../slides/2/skalarprodukt.tex \ + ../slides/2/cauchyschwarz.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 index cfca9ab..4c86f39 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -5,3 +5,4 @@ % \folie{2/norm.tex} \folie{2/skalarprodukt.tex} +\folie{2/cauchyschwarz.tex} 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/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex new file mode 100644 index 0000000..2a9784f --- /dev/null +++ b/vorlesungen/slides/2/skalarprodukt.tex @@ -0,0 +1,93 @@ +% +% 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 +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 +Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$ +\item +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Positive definite, hermitesche Sesquilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$ +\begin{itemize} +\item +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 +Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$ +\item +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.28\textwidth} +\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} +\begin{itemize} +\item $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ +\item $\| \lambda v \|_2 += +\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut} += +\sqrt{\overline{\lambda}\lambda\langle v,v\rangle} += +|\lambda|\cdot \|v\|_2$ +\item +\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/test.tex b/vorlesungen/slides/test.tex index eea9bec..b71c64b 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -80,6 +80,10 @@ \folie{2/norm.tex} % XXX Skalarprodukt und L^2-Norm \folie{2/skalarprodukt.tex} +% XXX Cauchy-Schwarz-Ungleichung +\folie{2/cauchyschwarz.tex} +% XXX Polarformel +% XXX \folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen % XXX \folie{2/operatornorm.tex} % XXX Frobenius-Norm und Hadamard-Algebra -- cgit v1.2.1 From d705a0c3bec5e691cbee064cf4a3ba663e927754 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 21:21:07 +0100 Subject: new slides --- vorlesungen/slides/2/Makefile.inc | 1 + vorlesungen/slides/2/chapter.tex | 1 + vorlesungen/slides/2/polarformel.tex | 113 +++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 10 ++-- 4 files changed, 120 insertions(+), 5 deletions(-) create mode 100644 vorlesungen/slides/2/polarformel.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 2eb3ce9..7c4dfb8 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -8,5 +8,6 @@ chapter2 = \ ../slides/2/norm.tex \ ../slides/2/skalarprodukt.tex \ ../slides/2/cauchyschwarz.tex \ + ../slides/2/polarformel.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 4c86f39..7b968d1 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -6,3 +6,4 @@ \folie{2/norm.tex} \folie{2/skalarprodukt.tex} \folie{2/cauchyschwarz.tex} +\folie{2/polarformel.tex} 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/test.tex b/vorlesungen/slides/test.tex index b71c64b..ab2ec45 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -77,13 +77,13 @@ \section{Matrixnormen} % XXX Vektornormen -\folie{2/norm.tex} -% XXX Skalarprodukt und L^2-Norm -\folie{2/skalarprodukt.tex} +%\folie{2/norm.tex} +%% XXX Skalarprodukt und L^2-Norm +%\folie{2/skalarprodukt.tex} % XXX Cauchy-Schwarz-Ungleichung -\folie{2/cauchyschwarz.tex} +%\folie{2/cauchyschwarz.tex} % XXX Polarformel -% XXX \folie{2/polarformel.tex} +\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen % XXX \folie{2/operatornorm.tex} % XXX Frobenius-Norm und Hadamard-Algebra -- cgit v1.2.1 From 28d9457049cf9bee9935aef3868a2cc8e7cf0d33 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 21:47:39 +0100 Subject: add slide --- vorlesungen/slides/2/Makefile.inc | 1 + vorlesungen/slides/2/chapter.tex | 1 + vorlesungen/slides/2/operatornorm.tex | 59 +++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 6 ++-- 4 files changed, 64 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/2/operatornorm.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 7c4dfb8..95fde20 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -9,5 +9,6 @@ chapter2 = \ ../slides/2/skalarprodukt.tex \ ../slides/2/cauchyschwarz.tex \ ../slides/2/polarformel.tex \ + ../slides/2/operatornorm.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 7b968d1..cb141a4 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -7,3 +7,4 @@ \folie{2/skalarprodukt.tex} \folie{2/cauchyschwarz.tex} \folie{2/polarformel.tex} +\folie{2/operatornorm.tex} 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/test.tex b/vorlesungen/slides/test.tex index ab2ec45..8881606 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -77,15 +77,15 @@ \section{Matrixnormen} % XXX Vektornormen -%\folie{2/norm.tex} +\folie{2/norm.tex} %% XXX Skalarprodukt und L^2-Norm %\folie{2/skalarprodukt.tex} % XXX Cauchy-Schwarz-Ungleichung %\folie{2/cauchyschwarz.tex} % XXX Polarformel -\folie{2/polarformel.tex} +%\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen -% XXX \folie{2/operatornorm.tex} +\folie{2/operatornorm.tex} % XXX Frobenius-Norm und Hadamard-Algebra % XXX \folie{2/frobeniusnorm.tex} -- cgit v1.2.1 From 846f9ed03605abab2219f077478e833a11c4f459 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 8 Mar 2021 22:28:57 +0100 Subject: add frobeniusnorm --- vorlesungen/slides/2/Makefile.inc | 1 + vorlesungen/slides/2/chapter.tex | 1 + vorlesungen/slides/2/frobeniusnorm.tex | 96 ++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 6 +-- 4 files changed, 101 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/2/frobeniusnorm.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 95fde20..37800e5 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -10,5 +10,6 @@ chapter2 = \ ../slides/2/cauchyschwarz.tex \ ../slides/2/polarformel.tex \ ../slides/2/operatornorm.tex \ + ../slides/2/frobeniusnorm.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index cb141a4..f13cf74 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -8,3 +8,4 @@ \folie{2/cauchyschwarz.tex} \folie{2/polarformel.tex} \folie{2/operatornorm.tex} +\folie{2/frobeniusnorm.tex} 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/test.tex b/vorlesungen/slides/test.tex index 8881606..ebf336e 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -78,16 +78,16 @@ \section{Matrixnormen} % XXX Vektornormen \folie{2/norm.tex} -%% XXX Skalarprodukt und L^2-Norm +% XXX Skalarprodukt und L^2-Norm %\folie{2/skalarprodukt.tex} % XXX Cauchy-Schwarz-Ungleichung %\folie{2/cauchyschwarz.tex} % XXX Polarformel %\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen -\folie{2/operatornorm.tex} +%\folie{2/operatornorm.tex} % XXX Frobenius-Norm und Hadamard-Algebra -% XXX \folie{2/frobeniusnorm.tex} +\folie{2/frobeniusnorm.tex} \section{Approximation mit Polynomen} % XXX Stone-Weierstrass -- cgit v1.2.1 From f015066601bbc0f5a7f81b2ed9f12aeea227b715 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Mar 2021 08:25:24 +0100 Subject: new slide --- vorlesungen/03_mseanalysis/slides.tex | 14 +++-- vorlesungen/slides/2/Makefile.inc | 1 + vorlesungen/slides/2/chapter.tex | 1 + vorlesungen/slides/2/frobeniusanwendung.tex | 79 +++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 3 +- 5 files changed, 89 insertions(+), 9 deletions(-) create mode 100644 vorlesungen/slides/2/frobeniusanwendung.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex index 185b67b..53019ba 100644 --- a/vorlesungen/03_mseanalysis/slides.tex +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -5,14 +5,12 @@ % \section{Matrixnormen} -% XXX Vektornormen -% XXX \folie{2/norm.tex} -% XXX Skalarprodukt und L^2-Norm -% XXX \folie{2/skalarprodukt.tex} -% XXX Normen, die sich aus der Vektornorm ableiten lassen -% XXX \folie{2/operatornorm.tex} -% XXX Frobenius-Norm und Hadamard-Algebra -% XXX \folie{2/frobeniusnorm.tex} +%\folie{2/norm.tex} +%\folie{2/skalarprodukt.tex} +%\folie{2/cauchyschwarz.tex} +%\folie{2/polarformel.tex} +%\folie{2/operatornorm.tex} +%\folie{2/frobeniusnorm.tex} \section{Approximation mit Polynomen} % XXX Stone-Weierstrass diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 37800e5..774a440 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -11,5 +11,6 @@ chapter2 = \ ../slides/2/polarformel.tex \ ../slides/2/operatornorm.tex \ ../slides/2/frobeniusnorm.tex \ + ../slides/2/frobeniusanwendung.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index f13cf74..78b9519 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -9,3 +9,4 @@ \folie{2/polarformel.tex} \folie{2/operatornorm.tex} \folie{2/frobeniusnorm.tex} +\folie{2/frobeniusanwendung.tex} diff --git a/vorlesungen/slides/2/frobeniusanwendung.tex b/vorlesungen/slides/2/frobeniusanwendung.tex new file mode 100644 index 0000000..a38a7a3 --- /dev/null +++ b/vorlesungen/slides/2/frobeniusanwendung.tex @@ -0,0 +1,79 @@ +% +% 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{-15pt} +\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*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index ebf336e..19b9ef6 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -77,7 +77,7 @@ \section{Matrixnormen} % XXX Vektornormen -\folie{2/norm.tex} +%\folie{2/norm.tex} % XXX Skalarprodukt und L^2-Norm %\folie{2/skalarprodukt.tex} % XXX Cauchy-Schwarz-Ungleichung @@ -88,6 +88,7 @@ %\folie{2/operatornorm.tex} % XXX Frobenius-Norm und Hadamard-Algebra \folie{2/frobeniusnorm.tex} +\folie{2/frobeniusanwendung.tex} \section{Approximation mit Polynomen} % XXX Stone-Weierstrass -- cgit v1.2.1 From c865d9d887cca53b2bbf5ab4282974aa5695871f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Mar 2021 08:29:49 +0100 Subject: comment --- vorlesungen/03_mseanalysis/slides.tex | 1 + vorlesungen/slides/2/frobeniusanwendung.tex | 3 ++- vorlesungen/slides/test.tex | 2 +- 3 files changed, 4 insertions(+), 2 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex index 53019ba..62915f2 100644 --- a/vorlesungen/03_mseanalysis/slides.tex +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -11,6 +11,7 @@ %\folie{2/polarformel.tex} %\folie{2/operatornorm.tex} %\folie{2/frobeniusnorm.tex} +%\folie{2/frobeniusanwendung.tex} \section{Approximation mit Polynomen} % XXX Stone-Weierstrass diff --git a/vorlesungen/slides/2/frobeniusanwendung.tex b/vorlesungen/slides/2/frobeniusanwendung.tex index a38a7a3..277d600 100644 --- a/vorlesungen/slides/2/frobeniusanwendung.tex +++ b/vorlesungen/slides/2/frobeniusanwendung.tex @@ -7,7 +7,7 @@ \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} \frametitle{Anwendung der Frobenius-Norm} -\vspace{-15pt} +\vspace{-18pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Ableitung nach $X\in M_{m\times n}(\mathbb{R})$} @@ -73,6 +73,7 @@ Df &= 2AX + B X&=-\frac12 A^{-1}B } \end{align*} +\uncover<15->{(Kalman-Filter)} \end{block}} \end{column} \end{columns} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 19b9ef6..a556570 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -86,7 +86,7 @@ %\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen %\folie{2/operatornorm.tex} -% XXX Frobenius-Norm und Hadamard-Algebra +% XXX Frobenius-Norm \folie{2/frobeniusnorm.tex} \folie{2/frobeniusanwendung.tex} -- cgit v1.2.1 From 990f93a2ea8e6fce226bd5a56ed54c01f6306f10 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Mar 2021 10:04:24 +0100 Subject: algebranormen --- vorlesungen/03_mseanalysis/slides.tex | 3 + vorlesungen/slides/2/Makefile.inc | 3 + vorlesungen/slides/2/chapter.tex | 3 + vorlesungen/slides/2/funktionenalgebra.tex | 88 ++++++++++++++++++++++++++++++ vorlesungen/slides/2/funktionenraum.tex | 70 ++++++++++++++++++++++++ vorlesungen/slides/2/linearformnormen.tex | 76 ++++++++++++++++++++++++++ vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 34 ++++++------ vorlesungen/slides/5/stoneweierstrass.tex | 11 ++++ vorlesungen/slides/test.tex | 7 ++- 10 files changed, 279 insertions(+), 18 deletions(-) create mode 100644 vorlesungen/slides/2/funktionenalgebra.tex create mode 100644 vorlesungen/slides/2/funktionenraum.tex create mode 100644 vorlesungen/slides/2/linearformnormen.tex create mode 100644 vorlesungen/slides/5/stoneweierstrass.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex index 62915f2..7a49363 100644 --- a/vorlesungen/03_mseanalysis/slides.tex +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -9,7 +9,10 @@ %\folie{2/skalarprodukt.tex} %\folie{2/cauchyschwarz.tex} %\folie{2/polarformel.tex} +%\folie{2/funktionenraum.tex} %\folie{2/operatornorm.tex} +%\folie{2/funktionenalgebra.tex} +%\folie{2/linearformnormen.tex} %\folie{2/frobeniusnorm.tex} %\folie{2/frobeniusanwendung.tex} diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 774a440..34dadf6 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -9,7 +9,10 @@ chapter2 = \ ../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/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 78b9519..38eb21f 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -7,6 +7,9 @@ \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} diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex new file mode 100644 index 0000000..e3339c3 --- /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/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/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 617e09f..1b13fb0 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -21,5 +21,7 @@ chapter5 = \ ../slides/5/jordan.tex \ ../slides/5/reellenormalform.tex \ ../slides/5/cayleyhamilton.tex \ + \ + ../slides/5/stoneweierstrass.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 1ce9d26..1ee570c 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,19 +3,21 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % -folie{5/motivation.tex} -folie{5/charpoly.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/motivation.tex} +\folie{5/charpoly.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/stoneweierstrass.tex} 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/test.tex b/vorlesungen/slides/test.tex index a556570..fb0333e 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -82,13 +82,16 @@ %\folie{2/skalarprodukt.tex} % XXX Cauchy-Schwarz-Ungleichung %\folie{2/cauchyschwarz.tex} +\folie{2/funktionenraum.tex} % XXX Polarformel %\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen %\folie{2/operatornorm.tex} +\folie{2/funktionenalgebra.tex} +\folie{2/linearformnormen.tex} % XXX Frobenius-Norm -\folie{2/frobeniusnorm.tex} -\folie{2/frobeniusanwendung.tex} +%\folie{2/frobeniusnorm.tex} +%\folie{2/frobeniusanwendung.tex} \section{Approximation mit Polynomen} % XXX Stone-Weierstrass -- cgit v1.2.1 From ecc99689218d54f550a3e393c6536592d3678e93 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Mar 2021 11:10:02 +0100 Subject: potenzreihenmethode --- vorlesungen/slides/5/Makefile.inc | 1 + vorlesungen/slides/5/chapter.tex | 1 + vorlesungen/slides/5/potenzreihenmethode.tex | 93 ++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 6 +- 4 files changed, 98 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/5/potenzreihenmethode.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 1b13fb0..230df72 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -23,5 +23,6 @@ chapter5 = \ ../slides/5/cayleyhamilton.tex \ \ ../slides/5/stoneweierstrass.tex \ + ../slides/5/potenzreihenmethode.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 1ee570c..6544294 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -21,3 +21,4 @@ \folie{5/cayleyhamilton.tex} \folie{5/stoneweierstrass.tex} +\folie{5/potenzreihenmethode.tex} 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/test.tex b/vorlesungen/slides/test.tex index fb0333e..5b3c8f6 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -87,8 +87,8 @@ %\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen %\folie{2/operatornorm.tex} -\folie{2/funktionenalgebra.tex} -\folie{2/linearformnormen.tex} +%\folie{2/funktionenalgebra.tex} +%\folie{2/linearformnormen.tex} % XXX Frobenius-Norm %\folie{2/frobeniusnorm.tex} %\folie{2/frobeniusanwendung.tex} @@ -115,7 +115,7 @@ \section{Differentialgleichungen} % XXX Potenzreihenmethode zur Lösung von Differentialgleichungen -% XXX \folie{5/potenzreihenmethode.tex} +\folie{5/potenzreihenmethode.tex} % XXX Exponentialfunktion % XXX \folie{5/exponentialfunktion.tex} % XXX Exponentialreihe -- cgit v1.2.1 From 5f146c5b38ad91022dd0854bc52d1b5086bf9e13 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Mar 2021 21:34:40 +0100 Subject: add two slides --- vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 2 + vorlesungen/slides/5/exponentialfunktion.tex | 125 +++++++++++++++++++++++++++ vorlesungen/slides/5/logarithmusreihe.tex | 53 ++++++++++++ vorlesungen/slides/test.tex | 12 +-- 5 files changed, 189 insertions(+), 5 deletions(-) create mode 100644 vorlesungen/slides/5/exponentialfunktion.tex create mode 100644 vorlesungen/slides/5/logarithmusreihe.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 230df72..03ce407 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -24,5 +24,7 @@ chapter5 = \ \ ../slides/5/stoneweierstrass.tex \ ../slides/5/potenzreihenmethode.tex \ + ../slides/5/logarithmusreihe.tex \ + ../slides/5/exponentialfunktion.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 6544294..31c6d25 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -22,3 +22,5 @@ \folie{5/stoneweierstrass.tex} \folie{5/potenzreihenmethode.tex} +\folie{5/logarithmusreihe.tex} +\folie{5/exponentialfunktion.tex} diff --git a/vorlesungen/slides/5/exponentialfunktion.tex b/vorlesungen/slides/5/exponentialfunktion.tex new file mode 100644 index 0000000..698d8a5 --- /dev/null +++ b/vorlesungen/slides/5/exponentialfunktion.tex @@ -0,0 +1,125 @@ +% +% 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>{% +\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>{% +\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{$\displaystyle,\; +X(t) += +\only<22>{ + e^{\lambda t} + \begin{pmatrix} 1&t/\lambda\\ 0&1 \end{pmatrix} +} +\only<23>{ + \frac{e^{\lambda t}}{\lambda} + \begin{pmatrix} \lambda&t\\ 0&\lambda \end{pmatrix} +} +C +$} +\\ +\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} +} +\\ +\only<16-17>{x_{1i\only<16>{,h}}(t)} +\only<18->{\dot{x}_{1i}(t)} +& +\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/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/test.tex b/vorlesungen/slides/test.tex index 5b3c8f6..e398f84 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -82,7 +82,7 @@ %\folie{2/skalarprodukt.tex} % XXX Cauchy-Schwarz-Ungleichung %\folie{2/cauchyschwarz.tex} -\folie{2/funktionenraum.tex} +%\folie{2/funktionenraum.tex} % XXX Polarformel %\folie{2/polarformel.tex} % XXX Normen, die sich aus der Vektornorm ableiten lassen @@ -110,18 +110,20 @@ % XXX \folie{5/konvergenzradius.tex} % XXX Gelfand-Radius % XXX \folie{5/gelfandradius.tex} +% XXX Spektralradius +% XXX \folie{5/spektralradius.tex} % XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) % XXX \folie{5/satzvongelfand.tex} +% XXX Logarithmus +\folie{5/logarithmusreihe.tex} \section{Differentialgleichungen} % XXX Potenzreihenmethode zur Lösung von Differentialgleichungen -\folie{5/potenzreihenmethode.tex} +%\folie{5/potenzreihenmethode.tex} % XXX Exponentialfunktion -% XXX \folie{5/exponentialfunktion.tex} +\folie{5/exponentialfunktion.tex} % XXX Exponentialreihe % XXX \folie{5/exponentialreihe.tex} -% XXX Logarithmus -% XXX \folie{5/logarithmusreihe.tex} % XXX Sinus und Cosinus, Eulerscher Satz % XXX \folie{5/sinuscosinus.tex} -- cgit v1.2.1 From 2189e00f55b3585575354adbc5e2c2ddc57b350c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 11 Mar 2021 09:15:18 +0100 Subject: new slide --- vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 2 +- vorlesungen/slides/5/spektralgelfand.tex | 190 +++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 1 + 4 files changed, 194 insertions(+), 1 deletion(-) create mode 100644 vorlesungen/slides/5/spektralgelfand.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 03ce407..00c8337 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -22,6 +22,8 @@ chapter5 = \ ../slides/5/reellenormalform.tex \ ../slides/5/cayleyhamilton.tex \ \ + ../slides/5/spektralgelfand.tex \ + \ ../slides/5/stoneweierstrass.tex \ ../slides/5/potenzreihenmethode.tex \ ../slides/5/logarithmusreihe.tex \ diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 31c6d25..6f3228d 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -19,7 +19,7 @@ \folie{5/jordan.tex} \folie{5/reellenormalform.tex} \folie{5/cayleyhamilton.tex} - +\folie{5/spektralgelfand.tex} \folie{5/stoneweierstrass.tex} \folie{5/potenzreihenmethode.tex} \folie{5/logarithmusreihe.tex} 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{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/test.tex b/vorlesungen/slides/test.tex index e398f84..a46cd04 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -112,6 +112,7 @@ % XXX \folie{5/gelfandradius.tex} % XXX Spektralradius % XXX \folie{5/spektralradius.tex} +\folie{5/spektralgelfand.tex} % XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) % XXX \folie{5/satzvongelfand.tex} % XXX Logarithmus -- cgit v1.2.1 From a893dbae9b4a6c90fe837712e4e72be42154d54e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 11 Mar 2021 11:46:57 +0100 Subject: add new slides --- vorlesungen/slides/5/Aiteration.tex | 59 ++++++++++++++++++++++ vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 2 + vorlesungen/slides/5/satzvongelfand.tex | 89 +++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 19 +++---- 5 files changed, 160 insertions(+), 11 deletions(-) create mode 100644 vorlesungen/slides/5/Aiteration.tex create mode 100644 vorlesungen/slides/5/satzvongelfand.tex (limited to 'vorlesungen') 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 index 00c8337..e635c42 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -23,6 +23,8 @@ chapter5 = \ ../slides/5/cayleyhamilton.tex \ \ ../slides/5/spektralgelfand.tex \ + ../slides/5/Aiteration.tex \ + ../slides/5/satzvongelfand.tex \ \ ../slides/5/stoneweierstrass.tex \ ../slides/5/potenzreihenmethode.tex \ diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 6f3228d..fab6a28 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -20,6 +20,8 @@ \folie{5/reellenormalform.tex} \folie{5/cayleyhamilton.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} 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/test.tex b/vorlesungen/slides/test.tex index a46cd04..07b5379 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -108,21 +108,18 @@ \section{Potenzreihen} % XXX Konvergenzradius % XXX \folie{5/konvergenzradius.tex} -% XXX Gelfand-Radius -% XXX \folie{5/gelfandradius.tex} -% XXX Spektralradius -% XXX \folie{5/spektralradius.tex} -\folie{5/spektralgelfand.tex} +%\folie{5/spektralgelfand.tex} % XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) -% XXX \folie{5/satzvongelfand.tex} -% XXX Logarithmus -\folie{5/logarithmusreihe.tex} +\folie{5/Aiteration.tex} +\folie{5/satzvongelfand.tex} +% Logarithmus +%\folie{5/logarithmusreihe.tex} \section{Differentialgleichungen} -% XXX Potenzreihenmethode zur Lösung von Differentialgleichungen +% Potenzreihenmethode zur Lösung von Differentialgleichungen %\folie{5/potenzreihenmethode.tex} -% XXX Exponentialfunktion -\folie{5/exponentialfunktion.tex} +% Exponentialfunktion +%\folie{5/exponentialfunktion.tex} % XXX Exponentialreihe % XXX \folie{5/exponentialreihe.tex} % XXX Sinus und Cosinus, Eulerscher Satz -- cgit v1.2.1 From 31cddc1e5890d5acf33e12078c7cdc311e92031c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 11 Mar 2021 17:11:17 +0100 Subject: add new slides --- vorlesungen/03_mseanalysis/slides.tex | 41 +++++----- vorlesungen/slides/5/Makefile.inc | 3 + vorlesungen/slides/5/chapter.tex | 3 + vorlesungen/slides/5/exponentialfunktion.tex | 14 +++- vorlesungen/slides/5/hyperbolisch.tex | 105 ++++++++++++++++++++++++++ vorlesungen/slides/5/konvergenzradius.tex | 109 +++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 22 ++---- 7 files changed, 253 insertions(+), 44 deletions(-) create mode 100644 vorlesungen/slides/5/hyperbolisch.tex create mode 100644 vorlesungen/slides/5/konvergenzradius.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex index 7a49363..5dbfb2a 100644 --- a/vorlesungen/03_mseanalysis/slides.tex +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -5,16 +5,16 @@ % \section{Matrixnormen} -%\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/funktionenalgebra.tex} -%\folie{2/linearformnormen.tex} -%\folie{2/frobeniusnorm.tex} -%\folie{2/frobeniusanwendung.tex} +\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/funktionenalgebra.tex} +\folie{2/linearformnormen.tex} +\folie{2/frobeniusnorm.tex} +\folie{2/frobeniusanwendung.tex} \section{Approximation mit Polynomen} % XXX Stone-Weierstrass @@ -29,22 +29,15 @@ % XXX \folie{5/gelfandtransformation.tex} \section{Potenzreihen} -% XXX Konvergenzradius -% XXX \folie{5/konvergenzradius.tex} -% XXX Gelfand-Radius -% XXX \folie{5/gelfandradius.tex} -% XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) -% XXX \folie{5/satzvongelfand.tex} +\folie{5/konvergenzradius.tex} +\folie{5/krbeispiele.tex} +\folie{5/spektralgelfand.tex} +\folie{5/satzvongelfand.tex} \section{Differentialgleichungen} -% XXX Potenzreihenmethode zur Lösung von Differentialgleichungen -% XXX \folie{5/potenzreihenmethode.tex} -% XXX Exponentialfunktion -% XXX \folie{5/exponentialfunktion.tex} -% XXX Exponentialreihe -% XXX \folie{5/exponentialreihe.tex} -% XXX Logarithmus -% XXX \folie{5/logarithmusreihe.tex} +\folie{5/potenzreihenmethode.tex} +\folie{5/exponentialfunktion.tex} +\folie{5/logarithmusreihe.tex} % XXX Sinus und Cosinus, Eulerscher Satz % XXX \folie{5/sinuscosinus.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index e635c42..1707e67 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -22,6 +22,8 @@ chapter5 = \ ../slides/5/reellenormalform.tex \ ../slides/5/cayleyhamilton.tex \ \ + ../slides/5/konvergenzradius.tex \ + ../slides/5/krbeispiele.tex \ ../slides/5/spektralgelfand.tex \ ../slides/5/Aiteration.tex \ ../slides/5/satzvongelfand.tex \ @@ -30,5 +32,6 @@ chapter5 = \ ../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/chapter.tex b/vorlesungen/slides/5/chapter.tex index fab6a28..46597e5 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -19,6 +19,8 @@ \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} @@ -26,3 +28,4 @@ \folie{5/potenzreihenmethode.tex} \folie{5/logarithmusreihe.tex} \folie{5/exponentialfunktion.tex} +\folie{5/hyperbolisch.tex} diff --git a/vorlesungen/slides/5/exponentialfunktion.tex b/vorlesungen/slides/5/exponentialfunktion.tex index 698d8a5..caae16b 100644 --- a/vorlesungen/slides/5/exponentialfunktion.tex +++ b/vorlesungen/slides/5/exponentialfunktion.tex @@ -10,6 +10,7 @@ \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} @@ -21,7 +22,7 @@ x(0) &= c&&\in\mathbb{R} \uncover<2->{x(t) &= ce^{at}} \end{align*} \end{block} -\end{column}} +\end{column}}{}} \begin{column}{0.48\textwidth} \uncover<3->{% \begin{block}{$X(t) \in M_n(\mathbb{R})$} @@ -45,6 +46,7 @@ vier Funktionen $x_{ij}(t)$}} \end{block}} \end{column} \only<7-9>{% +\ifthenelse{\boolean{presentation}}{ \begin{column}{0.48\textwidth} \begin{block}{Beispiel: Diagonalmatrix} %$D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$ @@ -60,7 +62,7 @@ Lösung: x_{ij}(t) =c_{ij}e^{\lambda_i t} \]} \end{block} -\end{column}} +\end{column}}{}} \uncover<10->{% \begin{column}{0.48\textwidth} \begin{block}{Beispiel: Jordan-Block} @@ -70,10 +72,11 @@ 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} @@ -100,13 +103,16 @@ x_{2i}(t)&=c_{2i}e^{\lambda t} \dot{x}_{1i}(t)&=\lambda x_{1i}(t) + c_{2i}e^{\lambda t} } \\ -\only<16-17>{x_{1i\only<16>{,h}}(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 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/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/test.tex b/vorlesungen/slides/test.tex index 07b5379..66f5ade 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -76,20 +76,14 @@ %\folie{5/cayleyhamilton.tex} \section{Matrixnormen} -% XXX Vektornormen %\folie{2/norm.tex} -% XXX Skalarprodukt und L^2-Norm %\folie{2/skalarprodukt.tex} -% XXX Cauchy-Schwarz-Ungleichung %\folie{2/cauchyschwarz.tex} %\folie{2/funktionenraum.tex} -% XXX Polarformel %\folie{2/polarformel.tex} -% XXX Normen, die sich aus der Vektornorm ableiten lassen %\folie{2/operatornorm.tex} %\folie{2/funktionenalgebra.tex} %\folie{2/linearformnormen.tex} -% XXX Frobenius-Norm %\folie{2/frobeniusnorm.tex} %\folie{2/frobeniusanwendung.tex} @@ -106,22 +100,18 @@ % XXX \folie{5/gelfandtransformation.tex} \section{Potenzreihen} -% XXX Konvergenzradius -% XXX \folie{5/konvergenzradius.tex} +% Konvergenzradius +%\folie{5/konvergenzradius.tex} +%\folie{5/krbeispiele.tex} %\folie{5/spektralgelfand.tex} -% XXX Gleichheit von Konvergenz-Radius und Gelfand-Radius (braucht JNF) -\folie{5/Aiteration.tex} -\folie{5/satzvongelfand.tex} -% Logarithmus +%\folie{5/Aiteration.tex} +%\folie{5/satzvongelfand.tex} %\folie{5/logarithmusreihe.tex} \section{Differentialgleichungen} -% Potenzreihenmethode zur Lösung von Differentialgleichungen %\folie{5/potenzreihenmethode.tex} -% Exponentialfunktion %\folie{5/exponentialfunktion.tex} -% XXX Exponentialreihe -% XXX \folie{5/exponentialreihe.tex} % XXX Sinus und Cosinus, Eulerscher Satz % XXX \folie{5/sinuscosinus.tex} +\folie{5/hyperbolisch.tex} -- cgit v1.2.1 From 1a026e2f7deb2e890c2d3751fc7bae0f37eac3de Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 11 Mar 2021 18:49:40 +0100 Subject: spectrum slides --- vorlesungen/03_mseanalysis/slides.tex | 1 + vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/krbeispiele.tex | 99 +++++++++++++++++++++++++++++++++++ vorlesungen/slides/5/spektrum.tex | 63 ++++++++++++++++++++++ vorlesungen/slides/test.tex | 4 +- 5 files changed, 167 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/slides/5/krbeispiele.tex create mode 100644 vorlesungen/slides/5/spektrum.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex index 5dbfb2a..01be58a 100644 --- a/vorlesungen/03_mseanalysis/slides.tex +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -40,4 +40,5 @@ \folie{5/logarithmusreihe.tex} % XXX Sinus und Cosinus, Eulerscher Satz % XXX \folie{5/sinuscosinus.tex} +\folie{5/hyperbolisch.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 1707e67..76b9032 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -22,6 +22,8 @@ chapter5 = \ ../slides/5/reellenormalform.tex \ ../slides/5/cayleyhamilton.tex \ \ + ../slides/5/spektrum.tex \ + \ ../slides/5/konvergenzradius.tex \ ../slides/5/krbeispiele.tex \ ../slides/5/spektralgelfand.tex \ 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/spektrum.tex b/vorlesungen/slides/5/spektrum.tex new file mode 100644 index 0000000..f427c9a --- /dev/null +++ b/vorlesungen/slides/5/spektrum.tex @@ -0,0 +1,63 @@ +% +% 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} +\begin{block}{Endlichdimensionale Räume} +\vspace{-15pt} +\begin{align*} +&\lambda\in\operatorname{Sp}A +\\ +\Leftrightarrow\quad&\text{$(A-\lambda I)$ nicht invertierbar} +\\ +\Leftrightarrow\quad&\text{$(A-\lambda I)$ singulär} +\\ +\Leftrightarrow\quad&\ker(A-\lambda I)\ne 0 +\\ +\Leftrightarrow\quad&\exists v\in V, v\ne 0, Av=\lambda v +\end{align*} +$\Rightarrow$ $\operatorname{Sp}A$ ist die Menge der Eigenwerte +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\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} +\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/test.tex b/vorlesungen/slides/test.tex index 66f5ade..7137c5a 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -91,7 +91,7 @@ % XXX Stone-Weierstrass % XXX \folie{5/stoneweierstrass.tex} % XXX Spektrum einer Matrix -% XXX \folie{5/spektrum.tex} +\folie{5/spektrum.tex} % XXX Approximation einer Funktion auf dem Spektrum % XXX \folie{5/spektrumapproximation.tex} % XXX Approximation einer Matrix in der erzeugten Algebra @@ -113,5 +113,5 @@ %\folie{5/exponentialfunktion.tex} % XXX Sinus und Cosinus, Eulerscher Satz % XXX \folie{5/sinuscosinus.tex} -\folie{5/hyperbolisch.tex} +%\folie{5/hyperbolisch.tex} -- cgit v1.2.1 From 6b0d0429cea0741f7b90db507a34b35574dd36d4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 11 Mar 2021 21:07:31 +0100 Subject: new slides --- vorlesungen/03_mseanalysis/slides.tex | 6 ++- vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 2 + vorlesungen/slides/5/normal.tex | 69 ++++++++++++++++++++++++++++++++ vorlesungen/slides/5/spektrum.tex | 19 +++++++-- vorlesungen/slides/5/unitaer.tex | 75 +++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 4 +- 7 files changed, 171 insertions(+), 6 deletions(-) create mode 100644 vorlesungen/slides/5/normal.tex create mode 100644 vorlesungen/slides/5/unitaer.tex (limited to 'vorlesungen') diff --git a/vorlesungen/03_mseanalysis/slides.tex b/vorlesungen/03_mseanalysis/slides.tex index 01be58a..7151909 100644 --- a/vorlesungen/03_mseanalysis/slides.tex +++ b/vorlesungen/03_mseanalysis/slides.tex @@ -17,6 +17,9 @@ \folie{2/frobeniusanwendung.tex} \section{Approximation mit Polynomen} +\folie{5/spektrum.tex} +\folie{5/normal.tex} +\folie{5/unitaer.tex} % XXX Stone-Weierstrass % XXX \folie{5/stoneweierstrass.tex} % XXX Spektrum einer Matrix @@ -32,13 +35,12 @@ \folie{5/konvergenzradius.tex} \folie{5/krbeispiele.tex} \folie{5/spektralgelfand.tex} +\folie{5/Aiteration.tex} \folie{5/satzvongelfand.tex} \section{Differentialgleichungen} \folie{5/potenzreihenmethode.tex} \folie{5/exponentialfunktion.tex} \folie{5/logarithmusreihe.tex} -% XXX Sinus und Cosinus, Eulerscher Satz -% XXX \folie{5/sinuscosinus.tex} \folie{5/hyperbolisch.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 76b9032..0858369 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -23,6 +23,8 @@ chapter5 = \ ../slides/5/cayleyhamilton.tex \ \ ../slides/5/spektrum.tex \ + ../slides/5/normal.tex \ + ../slides/5/unitaer.tex \ \ ../slides/5/konvergenzradius.tex \ ../slides/5/krbeispiele.tex \ diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 46597e5..c19222c 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -29,3 +29,5 @@ \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/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/spektrum.tex b/vorlesungen/slides/5/spektrum.tex index f427c9a..6cbdd7f 100644 --- a/vorlesungen/slides/5/spektrum.tex +++ b/vorlesungen/slides/5/spektrum.tex @@ -25,23 +25,35 @@ $A-\lambda I$ nicht invertierbar \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{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<8->{% \begin{block}{Unendlichdimensional} Es gibt eine Folge $x_n\in V$ von Einheitsvektoren $\|x_n\|=1$ @@ -49,7 +61,8 @@ mit \begin{align*} \lim_{n\to\infty} (A - \lambda)x_n &= 0 \end{align*} -\end{block} +\end{block}} +\uncover<9->{% \begin{block}{Spektrum und Norm} \[ \operatorname{Sp}(A) @@ -57,7 +70,7 @@ mit \{\lambda\in\mathbb{C}\;|\; |\lambda|\le \|A\|\} \] -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex new file mode 100644 index 0000000..36e3be2 --- /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 u,v\rangle +&= +\langle Uu,Uv\rangle +\uncover<3->{= \langle \lambda u,\lambda v\rangle} +\uncover<4->{= |\lambda|^2 \langle u,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/test.tex b/vorlesungen/slides/test.tex index 7137c5a..5f08a8f 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -91,7 +91,9 @@ % XXX Stone-Weierstrass % XXX \folie{5/stoneweierstrass.tex} % XXX Spektrum einer Matrix -\folie{5/spektrum.tex} +%\folie{5/spektrum.tex} +\folie{5/normal.tex} +\folie{5/unitaer.tex} % XXX Approximation einer Funktion auf dem Spektrum % XXX \folie{5/spektrumapproximation.tex} % XXX Approximation einer Matrix in der erzeugten Algebra -- cgit v1.2.1 From 972eba4c4cb38eb330bd053500e627085a3f4328 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 12 Mar 2021 17:27:08 +0100 Subject: add new slides --- vorlesungen/slides/2/funktionenalgebra.tex | 6 +- vorlesungen/slides/2/skalarprodukt.tex | 27 ++++--- vorlesungen/slides/5/unitaer.tex | 6 +- vorlesungen/slides/8/Makefile.inc | 11 +++ vorlesungen/slides/8/chapter.tex | 7 ++ vorlesungen/slides/8/dgraph.tex | 100 ++++++++++++++++++++++++ vorlesungen/slides/8/graph.tex | 117 +++++++++++++++++++++++++++++ vorlesungen/slides/Makefile.inc | 3 +- vorlesungen/slides/test.tex | 115 +--------------------------- 9 files changed, 260 insertions(+), 132 deletions(-) create mode 100644 vorlesungen/slides/8/Makefile.inc create mode 100644 vorlesungen/slides/8/chapter.tex create mode 100644 vorlesungen/slides/8/dgraph.tex create mode 100644 vorlesungen/slides/8/graph.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex index e3339c3..9116be4 100644 --- a/vorlesungen/slides/2/funktionenalgebra.tex +++ b/vorlesungen/slides/2/funktionenalgebra.tex @@ -31,12 +31,12 @@ f(x)\cdot g(x) \begin{align*} \|f\cdot g\|_\infty &= -\sup_{x\in[0,1]} f(x)g(x) +\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) +\sup_{x\in[0,1]}|f(x)| +\sup_{y\in[0,1]}|g(y)| } \\ \uncover<5->{ diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex index 2a9784f..99d8a73 100644 --- a/vorlesungen/slides/2/skalarprodukt.tex +++ b/vorlesungen/slides/2/skalarprodukt.tex @@ -13,7 +13,7 @@ \begin{block}{Positiv definite, symmetrische Bilinearform} $\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$ \begin{itemize} -\item +\item<2-> Bilinear: \begin{align*} \langle \alpha u+\beta v,w\rangle @@ -28,18 +28,19 @@ Bilinear: + \beta\langle u,w\rangle \end{align*} -\item +\item<3-> Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$ -\item +\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 +\item<6-> Sesquilinear: \begin{align*} \langle \alpha u+\beta v,w\rangle @@ -54,40 +55,42 @@ Sesquilinear: + \beta\langle u,w\rangle \end{align*} -\item +\item<7-> Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$ -\item +\item<8-> $\langle x,x\rangle >0 \quad\forall x\ne 0$ \end{itemize} -\end{block} +\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{block}} \end{column} \begin{column}{0.78\textwidth} +\uncover<10->{% \begin{itemize} -\item $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ -\item $\| \lambda v \|_2 +\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 +\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{itemize}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex index 36e3be2..f0c4401 100644 --- a/vorlesungen/slides/5/unitaer.tex +++ b/vorlesungen/slides/5/unitaer.tex @@ -20,11 +20,11 @@ $U$ unitär lässt das Skalarprodukt invariant \uncover<2->{% $\lambda$ ein Eigenwert mit Eigenvektor $v$: \begin{align*} -\langle u,v\rangle +\langle v,v\rangle &= \langle Uu,Uv\rangle -\uncover<3->{= \langle \lambda u,\lambda v\rangle} -\uncover<4->{= |\lambda|^2 \langle u,v\rangle} +\uncover<3->{= \langle \lambda v,\lambda v\rangle} +\uncover<4->{= |\lambda|^2 \langle v,v\rangle} \\ \uncover<5->{\Rightarrow\;|\lambda|&=1} \end{align*}} diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc new file mode 100644 index 0000000..e8c7502 --- /dev/null +++ b/vorlesungen/slides/8/Makefile.inc @@ -0,0 +1,11 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter8 = \ + ../slides/8/dgraph.tex \ + ../slides/8/graph.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..761ea63 --- /dev/null +++ b/vorlesungen/slides/8/chapter.tex @@ -0,0 +1,7 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{8/graph.tex} +\folie{8/dgraph.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/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/Makefile.inc b/vorlesungen/slides/Makefile.inc index 20929e4..fe22282 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -9,7 +9,8 @@ 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 slides = \ $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ - $(chapter5) + $(chapter5) $(chapter8) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 5f08a8f..30aef37 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,117 +3,6 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % - -%\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/faktorisierung.tex} -%\folie{3/faktorzerlegung.tex} -%\folie{3/einsetzen.tex} -%\folie{3/maximalergrad.tex} -%\folie{3/minimalbeispiel.tex} -%\folie{3/minimalpolynom.tex} -%\folie{3/drehmatrix.tex} -%\folie{3/drehfaktorisierung.tex} -%\folie{3/fibonacci.tex} -%\folie{3/operatoren.tex} -%\folie{3/adjunktion.tex} -%\folie{3/adjalgebra.tex} - -%\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} -% XXX \folie{4/frobenius.tex} - -%\folie{4/divisionpoly.tex} -%\folie{4/euklidpoly.tex} -%\folie{4/polynomefp.tex} -%\folie{4/alpha.tex} - -% XXX \folie{4/f2.tex} -%\folie{4/schieberegister.tex} - -% XXX Idee der elliptischen Kurve -% XXX \folie{4/ecidee.tex} - - -\section{Eigenwertproblem} -% XXX Motivation: beliebige Funktionen f(A) berechnen -%\folie{5/motivation.tex} -%\folie{5/charpoly.tex} - -\section{Invariante Unterräume} -%\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} - -% Jordan Normalform -\section{Jordan-Normalform} -%\folie{5/jordanblock.tex} -%\folie{5/jordan.tex} -% XXX Diagonalform -% XXX \folie{5/diagonalform.tex} -%\folie{5/reellenormalform.tex} -% XXX \folie{5/hessenberg.tex} - -\section{Satz von Cayley-Hamilton} -%\folie{5/cayleyhamilton.tex} - -\section{Matrixnormen} -%\folie{2/norm.tex} -%\folie{2/skalarprodukt.tex} -%\folie{2/cauchyschwarz.tex} -%\folie{2/funktionenraum.tex} -%\folie{2/polarformel.tex} -%\folie{2/operatornorm.tex} -%\folie{2/funktionenalgebra.tex} -%\folie{2/linearformnormen.tex} -%\folie{2/frobeniusnorm.tex} -%\folie{2/frobeniusanwendung.tex} - -\section{Approximation mit Polynomen} -% XXX Stone-Weierstrass -% XXX \folie{5/stoneweierstrass.tex} -% XXX Spektrum einer Matrix -%\folie{5/spektrum.tex} -\folie{5/normal.tex} -\folie{5/unitaer.tex} -% XXX Approximation einer Funktion auf dem Spektrum -% XXX \folie{5/spektrumapproximation.tex} -% XXX Approximation einer Matrix in der erzeugten Algebra -% XXX \folie{5/matrixapproximation.tex} -% XXX Gelfand-Transformation -% XXX \folie{5/gelfandtransformation.tex} - -\section{Potenzreihen} -% Konvergenzradius -%\folie{5/konvergenzradius.tex} -%\folie{5/krbeispiele.tex} -%\folie{5/spektralgelfand.tex} -%\folie{5/Aiteration.tex} -%\folie{5/satzvongelfand.tex} -%\folie{5/logarithmusreihe.tex} - -\section{Differentialgleichungen} -%\folie{5/potenzreihenmethode.tex} -%\folie{5/exponentialfunktion.tex} -% XXX Sinus und Cosinus, Eulerscher Satz -% XXX \folie{5/sinuscosinus.tex} -%\folie{5/hyperbolisch.tex} +\folie{8/dgraph.tex} +\folie{8/graph.tex} -- cgit v1.2.1 From 0a0f140762a5be23e2475eceecf51d6cb6f84399 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 12 Mar 2021 21:13:21 +0100 Subject: add new slides --- vorlesungen/slides/8/Makefile.inc | 2 + vorlesungen/slides/8/chapter.tex | 2 + vorlesungen/slides/8/grad.tex | 84 ++++++++++++++++++++++++ vorlesungen/slides/8/inzidenz.tex | 131 ++++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 6 +- 5 files changed, 223 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/slides/8/grad.tex create mode 100644 vorlesungen/slides/8/inzidenz.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index e8c7502..903fce2 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -7,5 +7,7 @@ chapter8 = \ ../slides/8/dgraph.tex \ ../slides/8/graph.tex \ + ../slides/8/grad.tex \ + ../slides/8/inzidenz.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 761ea63..23d9aac 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -5,3 +5,5 @@ % \folie{8/graph.tex} \folie{8/dgraph.tex} +\folie{8/grad.tex} +\folie{8/inzidenz.tex} 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/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex new file mode 100644 index 0000000..5911593 --- /dev/null +++ b/vorlesungen/slides/8/inzidenz.tex @@ -0,0 +1,131 @@ +% +% 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); +\draw[color=darkgreen,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); + +\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] {$\scriptstyle 5$}; +\node[color=darkgreen] at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 6$}; + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +\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} +\fill[color=red!20] (1.8,1.8) rectangle (4.75,2.15); +\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} + +\begin{scope} +\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] {$\displaystyle +\begin{aligned} +B(G) +&= +\begin{pmatrix} +1&0&0&1&1&0\\ +1&1&0&0&0&1\\ +0&1&1&0&1&0\\ +0&0&1&0&0&0\\ +0&0&0&1&0&1 +\end{pmatrix} +\\[12pt] +A(G) +&= +\begin{pmatrix} +0&1&1&0&1\\ +1&0&1&1&0\\ +1&1&0&1&0\\ +0&1&1&0&0\\ +1&0&0&0&0 +\end{pmatrix} +\end{aligned}$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 30aef37..c5237d1 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,6 +3,8 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{8/dgraph.tex} -\folie{8/graph.tex} +%\folie{8/dgraph.tex} +%\folie{8/graph.tex} +%\folie{8/grad.tex} +\folie{8/inzidenz.tex} -- cgit v1.2.1 From 33baa215192f04f27d7c389c83e7f2fd9f481abd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 12 Mar 2021 22:15:39 +0100 Subject: new slides --- vorlesungen/slides/8/Makefile.inc | 1 + vorlesungen/slides/8/chapter.tex | 1 + vorlesungen/slides/8/inzidenzd.tex | 145 +++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 1 + 4 files changed, 148 insertions(+) create mode 100644 vorlesungen/slides/8/inzidenzd.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index 903fce2..68e2e27 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -9,5 +9,6 @@ chapter8 = \ ../slides/8/graph.tex \ ../slides/8/grad.tex \ ../slides/8/inzidenz.tex \ + ../slides/8/inzidenzd.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 23d9aac..6ac3e98 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -7,3 +7,4 @@ \folie{8/dgraph.tex} \folie{8/grad.tex} \folie{8/inzidenz.tex} +\folie{8/inzidenzd.tex} diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex new file mode 100644 index 0000000..19da619 --- /dev/null +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -0,0 +1,145 @@ +% +% 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); +\draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=darkgreen] (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]; +\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] {$\scriptstyle 6$}; +\node[color=darkgreen] at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; + +\end{tikzpicture} +\end{center} +\vspace{-15pt} +\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}&=\pm 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] +\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}); +\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}; +\end{scope} + +\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) {$\displaystyle +\begin{aligned} +B(G) +&= +\begin{pmatrix*}[r] +-1& 0& 0& 0& 1&-1& 0\\ + 1&-1& 0& 0& 0& 0&-1\\ + 0& 1&-1& 0& 0& 1& 0\\ + 0& 0& 1&-1& 0& 0& 1\\ + 0& 0& 0& 1&-1& 0& 0 +\end{pmatrix*} +\\[20pt] +A(G) +&= +\begin{pmatrix*}[r] + 0&-1&-1& 0&-1\\ +-1& 0&-1&-1& 0\\ +-1&-1& 0&-1& 0\\ + 0&-1&-1& 0&-1\\ +-1& 0& 0&-1& 0 +\end{pmatrix*} +\end{aligned}$}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index c5237d1..d4ebfd3 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -7,4 +7,5 @@ %\folie{8/graph.tex} %\folie{8/grad.tex} \folie{8/inzidenz.tex} +\folie{8/inzidenzd.tex} -- cgit v1.2.1 From ac7015feefad911eb609c5ab032d234cb654ef4d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 13 Mar 2021 10:48:55 +0100 Subject: add new slides --- vorlesungen/slides/8/Makefile.inc | 11 + vorlesungen/slides/8/chapter.tex | 18 + vorlesungen/slides/8/diffusion.tex | 89 +++ vorlesungen/slides/8/floyd-warshall/burgerking.png | Bin 0 -> 121512 bytes vorlesungen/slides/8/floyd-warshall/fw.tex | 680 +++++++++++++++++++++ vorlesungen/slides/8/floyd-warshall/iteration.tex | 14 + vorlesungen/slides/8/floyd-warshall/macdonalds.png | Bin 0 -> 129601 bytes vorlesungen/slides/8/floyd-warshall/problem.tex | 146 +++++ vorlesungen/slides/8/floyd-warshall/rekursion.tex | 108 ++++ vorlesungen/slides/8/floyd-warshall/starbucks.png | Bin 0 -> 160194 bytes vorlesungen/slides/8/floyd-warshall/wege.tex | 26 + .../slides/8/floyd-warshall/wegiteration.tex | 13 + vorlesungen/slides/8/floyd-warshall/wegweiser.jpg | Bin 0 -> 431961 bytes vorlesungen/slides/8/inzidenzd.tex | 24 +- vorlesungen/slides/8/laplace.tex | 213 +++++++ vorlesungen/slides/8/pfade/adjazenz.tex | 97 +++ vorlesungen/slides/8/pfade/beispiel.tex | 404 ++++++++++++ vorlesungen/slides/8/pfade/gf.tex | 54 ++ vorlesungen/slides/8/pfade/langepfade.tex | 59 ++ vorlesungen/slides/test.tex | 20 +- 20 files changed, 1964 insertions(+), 12 deletions(-) create mode 100644 vorlesungen/slides/8/diffusion.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/burgerking.png create mode 100644 vorlesungen/slides/8/floyd-warshall/fw.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/iteration.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/macdonalds.png create mode 100644 vorlesungen/slides/8/floyd-warshall/problem.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/rekursion.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/starbucks.png create mode 100644 vorlesungen/slides/8/floyd-warshall/wege.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/wegiteration.tex create mode 100644 vorlesungen/slides/8/floyd-warshall/wegweiser.jpg create mode 100644 vorlesungen/slides/8/laplace.tex create mode 100644 vorlesungen/slides/8/pfade/adjazenz.tex create mode 100644 vorlesungen/slides/8/pfade/beispiel.tex create mode 100644 vorlesungen/slides/8/pfade/gf.tex create mode 100644 vorlesungen/slides/8/pfade/langepfade.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index 68e2e27..0a20f88 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -10,5 +10,16 @@ chapter8 = \ ../slides/8/grad.tex \ ../slides/8/inzidenz.tex \ ../slides/8/inzidenzd.tex \ + ../slides/8/diffusion.tex \ + ../slides/8/laplace.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/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 6ac3e98..bd1c18f 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -8,3 +8,21 @@ \folie{8/grad.tex} \folie{8/inzidenz.tex} \folie{8/inzidenzd.tex} +\folie{8/diffusion.tex} +\folie{8/laplace.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} + + + + + 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 new file mode 100644 index 0000000..cf4211d Binary files /dev/null and b/vorlesungen/slides/8/floyd-warshall/burgerking.png differ 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 new file mode 100644 index 0000000..c442dfb Binary files /dev/null and b/vorlesungen/slides/8/floyd-warshall/macdonalds.png differ 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 new file mode 100644 index 0000000..a28dbf7 Binary files /dev/null and b/vorlesungen/slides/8/floyd-warshall/starbucks.png differ 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 new file mode 100644 index 0000000..33aebe3 Binary files /dev/null and b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg differ diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex index 19da619..18feeae 100644 --- a/vorlesungen/slides/8/inzidenzd.tex +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -61,7 +61,7 @@ \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}&=\pm 1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$} +A(G)_{ij}&=\phantom{-}1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$} \end{align*} \end{block} \end{column} @@ -91,6 +91,8 @@ rectangle } \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} \begin{scope}[xshift=2.32cm,yshift=-0.24cm] @@ -121,21 +123,21 @@ rectangle 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 +-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] A(G) &= \begin{pmatrix*}[r] - 0&-1&-1& 0&-1\\ --1& 0&-1&-1& 0\\ --1&-1& 0&-1& 0\\ - 0&-1&-1& 0&-1\\ --1& 0& 0&-1& 0 + 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} 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/test.tex b/vorlesungen/slides/test.tex index d4ebfd3..ca85ddd 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -7,5 +7,23 @@ %\folie{8/graph.tex} %\folie{8/grad.tex} \folie{8/inzidenz.tex} -\folie{8/inzidenzd.tex} +%\folie{8/inzidenzd.tex} +% XXX Diffusion +\folie{8/diffusion.tex} +\folie{8/laplace.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/wege.tex} +%\folie{8/floyd-warshall/wegiteration.tex} + +% Markov-Ketten +% - Google-Matrix +% - Stationär -- cgit v1.2.1 From a0a5f0f799be512684e31d13cdedc2ef1c404792 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 13 Mar 2021 21:09:35 +0100 Subject: add additional paper --- vorlesungen/04_graphen/title.jpg | Bin 0 -> 113703 bytes vorlesungen/slides/8/Makefile.inc | 11 +++ vorlesungen/slides/8/chapter.tex | 12 ++- vorlesungen/slides/8/inzidenz.tex | 27 +++++- vorlesungen/slides/8/inzidenzd.tex | 25 ++++- vorlesungen/slides/8/markov/google.tex | 123 +++++++++++++++++++++++++ vorlesungen/slides/8/markov/irreduzibel.tex | 136 ++++++++++++++++++++++++++++ vorlesungen/slides/8/markov/markov.tex | 111 +++++++++++++++++++++++ vorlesungen/slides/8/markov/pf.tex | 53 +++++++++++ vorlesungen/slides/8/markov/stationaer.tex | 57 ++++++++++++ vorlesungen/slides/8/tokyo/bahn0.tex | 11 +++ vorlesungen/slides/8/tokyo/bahn1.tex | 28 ++++++ vorlesungen/slides/8/tokyo/bahn2.tex | 12 +++ vorlesungen/slides/8/tokyo/google.tex | 11 +++ vorlesungen/slides/test.tex | 24 ++++- 15 files changed, 628 insertions(+), 13 deletions(-) create mode 100644 vorlesungen/04_graphen/title.jpg create mode 100644 vorlesungen/slides/8/markov/google.tex create mode 100644 vorlesungen/slides/8/markov/irreduzibel.tex create mode 100644 vorlesungen/slides/8/markov/markov.tex create mode 100644 vorlesungen/slides/8/markov/pf.tex create mode 100644 vorlesungen/slides/8/markov/stationaer.tex create mode 100644 vorlesungen/slides/8/tokyo/bahn0.tex create mode 100644 vorlesungen/slides/8/tokyo/bahn1.tex create mode 100644 vorlesungen/slides/8/tokyo/bahn2.tex create mode 100644 vorlesungen/slides/8/tokyo/google.tex (limited to 'vorlesungen') diff --git a/vorlesungen/04_graphen/title.jpg b/vorlesungen/04_graphen/title.jpg new file mode 100644 index 0000000..c5ab581 Binary files /dev/null and b/vorlesungen/04_graphen/title.jpg differ diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index 0a20f88..cc2c497 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -12,6 +12,8 @@ chapter8 = \ ../slides/8/inzidenzd.tex \ ../slides/8/diffusion.tex \ ../slides/8/laplace.tex \ + ../slides/8/fourier.tex \ + ../slides/8/spanningtree.tex \ ../slides/8/pfade/adjazenz.tex \ ../slides/8/pfade/langepfade.tex \ ../slides/8/pfade/beispiel.tex \ @@ -21,5 +23,14 @@ chapter8 = \ ../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/markov/google.tex \ + ../slides/8/markov/markov.tex \ + ../slides/8/markov/irreduzibel.tex \ + ../slides/8/markov/stationaer.tex \ + ../slides/8/markov/pf.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index bd1c18f..b29145f 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -10,6 +10,8 @@ \folie{8/inzidenzd.tex} \folie{8/diffusion.tex} \folie{8/laplace.tex} +\folie{8/fourier.tex} +\folie{8/spanningtree.tex} \folie{8/pfade/adjazenz.tex} \folie{8/pfade/langepfade.tex} @@ -22,7 +24,15 @@ \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} - +\folie{8/markov/google.tex} +\folie{8/markov/markov.tex} +\folie{8/markov/stationaer.tex} +\folie{8/markov/irreduzibel.tex} +\folie{8/markov/pf.tex} diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex index 5911593..87578df 100644 --- a/vorlesungen/slides/8/inzidenz.tex +++ b/vorlesungen/slides/8/inzidenz.tex @@ -23,7 +23,9 @@ \draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); \draw[color=white,line width=5pt] (B) -- (D); -\draw[color=darkgreen,shorten >= 0.2cm,shorten <= 0.2cm] (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); @@ -31,7 +33,12 @@ %\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]; @@ -51,18 +58,21 @@ \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] {$\scriptstyle 5$}; -\node[color=darkgreen] at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 6$}; +{\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{block}} \end{column} \begin{column}{0.48\textwidth} \begin{center} @@ -73,8 +83,12 @@ A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$} \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 $\y$}; } @@ -87,9 +101,12 @@ A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$} \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 $\y$}; @@ -99,6 +116,7 @@ A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$} \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] {$\displaystyle \begin{aligned} @@ -112,6 +130,7 @@ B(G) 0&0&0&1&0&1 \end{pmatrix} \\[12pt] +\uncover<4->{ A(G) &= \begin{pmatrix} @@ -121,7 +140,7 @@ A(G) 0&1&1&0&0\\ 1&0&0&0&0 \end{pmatrix} -\end{aligned}$}; +\end{aligned}}$}; \end{tikzpicture} \end{center} diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex index 18feeae..5f2f51a 100644 --- a/vorlesungen/slides/8/inzidenzd.tex +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -23,7 +23,9 @@ \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); \draw[color=white,line width=5pt] (B) -- (D); -\draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=darkgreen] (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); @@ -32,7 +34,12 @@ \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]; @@ -51,11 +58,14 @@ \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[color=darkgreen] at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; +{\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*} @@ -63,7 +73,7 @@ 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{block}} \end{column} \begin{column}{0.58\textwidth} \begin{center} @@ -73,14 +83,18 @@ A(G)_{ij}&=\phantom{-}1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$} \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 $\x$}; @@ -95,6 +109,7 @@ rectangle [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] @@ -117,6 +132,7 @@ rectangle \node[color=gray] at ({0.00},{-0.38-(3-1)*\dy}) [above,rotate=-90] {\tiny Knoten}; \end{scope} +} \node at (0,0) {$\displaystyle \begin{aligned} @@ -130,6 +146,7 @@ B(G) 0& 0& 0&+1&-1& 0& 0 \end{pmatrix*} \\[20pt] +\uncover<4->{ A(G) &= \begin{pmatrix*}[r] @@ -138,7 +155,7 @@ A(G) \phantom{-}1&\phantom{-}1& 0&\phantom{-}1& 0\\ 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\ \phantom{-}1& 0& 0&\phantom{-}1& 0 -\end{pmatrix*} +\end{pmatrix*}} \end{aligned}$}; \end{tikzpicture} \end{center} diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/8/markov/google.tex new file mode 100644 index 0000000..d1ec31d --- /dev/null +++ b/vorlesungen/slides/8/markov/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/8/markov/irreduzibel.tex b/vorlesungen/slides/8/markov/irreduzibel.tex new file mode 100644 index 0000000..87e90e4 --- /dev/null +++ b/vorlesungen/slides/8/markov/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/8/markov/markov.tex b/vorlesungen/slides/8/markov/markov.tex new file mode 100644 index 0000000..e92ff0f --- /dev/null +++ b/vorlesungen/slides/8/markov/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/8/markov/pf.tex b/vorlesungen/slides/8/markov/pf.tex new file mode 100644 index 0000000..da2ef2b --- /dev/null +++ b/vorlesungen/slides/8/markov/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/8/markov/stationaer.tex b/vorlesungen/slides/8/markov/stationaer.tex new file mode 100644 index 0000000..92fab16 --- /dev/null +++ b/vorlesungen/slides/8/markov/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/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/test.tex b/vorlesungen/slides/test.tex index ca85ddd..6c5d5ff 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -7,10 +7,15 @@ %\folie{8/graph.tex} %\folie{8/grad.tex} \folie{8/inzidenz.tex} -%\folie{8/inzidenzd.tex} -% XXX Diffusion -\folie{8/diffusion.tex} -\folie{8/laplace.tex} +\folie{8/inzidenzd.tex} +%\folie{8/diffusion.tex} +%\folie{8/laplace.tex} +%\folie{8/fourier.tex} +%\folie{8/spanningtree.tex} + +% XXX \folie{8/schwingungen.tex} +% XXX \folie{8/eigenvektorbasis.tex} +% XXX \folie{8/spektralegraphentheorie.tex} %\folie{8/pfade/adjazenz.tex} %\folie{8/pfade/langepfade.tex} @@ -27,3 +32,14 @@ % - Google-Matrix % - Stationär +%\folie{8/tokyo/google.tex} +%\folie{8/tokyo/bahn0.tex} +%\folie{8/tokyo/bahn1.tex} +%\folie{8/tokyo/bahn2.tex} + + +%\folie{8/markov/google.tex} +%\folie{8/markov/markov.tex} +%\folie{8/markov/irreduzibel.tex} +%\folie{8/markov/stationaer.tex} +%\folie{8/markov/pf.tex} -- cgit v1.2.1 From 1c1979d3d94a712af2b1c59a78bbf53ac020a86f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 15 Mar 2021 13:25:12 +0100 Subject: add new slide --- vorlesungen/slides/8/Makefile.inc | 1 + vorlesungen/slides/8/chapter.tex | 1 + vorlesungen/slides/8/produkt.tex | 100 ++++++++++++++++++++++++++++++++++++++ vorlesungen/slides/test.tex | 5 +- 4 files changed, 105 insertions(+), 2 deletions(-) create mode 100644 vorlesungen/slides/8/produkt.tex (limited to 'vorlesungen') diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index cc2c497..233835a 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -12,6 +12,7 @@ chapter8 = \ ../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 \ diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index b29145f..ac06775 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -10,6 +10,7 @@ \folie{8/inzidenzd.tex} \folie{8/diffusion.tex} \folie{8/laplace.tex} +\folie{8/produkt.tex} \folie{8/fourier.tex} \folie{8/spanningtree.tex} 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/test.tex b/vorlesungen/slides/test.tex index 6c5d5ff..411414e 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -6,10 +6,11 @@ %\folie{8/dgraph.tex} %\folie{8/graph.tex} %\folie{8/grad.tex} -\folie{8/inzidenz.tex} +%\folie{8/inzidenz.tex} \folie{8/inzidenzd.tex} %\folie{8/diffusion.tex} -%\folie{8/laplace.tex} +\folie{8/laplace.tex} +\folie{8/produkt.tex} %\folie{8/fourier.tex} %\folie{8/spanningtree.tex} -- cgit v1.2.1 From 4614294614e6f6b38e0ca86e77871e75b4c26071 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 16 Mar 2021 15:48:10 +0100 Subject: add new slides --- vorlesungen/01_intro/slides.tex | 4 +- vorlesungen/04_msepolynome/Makefile | 33 ++++ .../04_msepolynome/MathSem-04-msepolynome.tex | 14 ++ vorlesungen/04_msepolynome/common.tex | 16 ++ vorlesungen/04_msepolynome/msepolynome-handout.tex | 11 ++ vorlesungen/04_msepolynome/slides.tex | 34 ++++ vorlesungen/05_spektral1/Makefile | 33 ++++ vorlesungen/05_spektral1/MathSem-05-spektral1.tex | 14 ++ vorlesungen/05_spektral1/common.tex | 16 ++ vorlesungen/05_spektral1/slides.tex | 28 ++++ vorlesungen/05_spektral1/spektral1-handout.tex | 11 ++ vorlesungen/06_spektral2/Makefile | 33 ++++ vorlesungen/06_spektral2/MathSem-06-spektral2.tex | 14 ++ vorlesungen/06_spektral2/common.tex | 16 ++ vorlesungen/06_spektral2/slides.tex | 22 +++ vorlesungen/06_spektral2/spektral2-handout.tex | 11 ++ vorlesungen/slides/0/Makefile.inc | 4 +- vorlesungen/slides/0/chapter.tex | 4 +- vorlesungen/slides/0/themen.tex | 68 -------- vorlesungen/slides/0/themen1.tex | 27 +++ vorlesungen/slides/0/themen2.tex | 27 +++ vorlesungen/slides/0/themen3.tex | 26 +++ vorlesungen/slides/2/Makefile.inc | 2 + vorlesungen/slides/2/chapter.tex | 2 + vorlesungen/slides/2/images/Makefile | 32 ++++ vorlesungen/slides/2/images/quotient.inc | 186 +++++++++++++++++++++ vorlesungen/slides/2/images/quotient.ini | 7 + vorlesungen/slides/2/images/quotient1.jpg | Bin 0 -> 181755 bytes vorlesungen/slides/2/images/quotient1.pov | 8 + vorlesungen/slides/2/images/quotient1.tex | 29 ++++ vorlesungen/slides/2/images/quotient2.jpg | Bin 0 -> 206065 bytes vorlesungen/slides/2/images/quotient2.pov | 8 + vorlesungen/slides/2/images/quotient2.tex | 29 ++++ vorlesungen/slides/2/quotient.tex | 110 ++++++++++++ vorlesungen/slides/2/quotientv.tex | 62 +++++++ vorlesungen/slides/3/Makefile.inc | 2 + vorlesungen/slides/3/chapter.tex | 2 + vorlesungen/slides/3/ideal.tex | 63 +++++++ vorlesungen/slides/3/quotientenring.tex | 59 +++++++ vorlesungen/slides/8/Makefile.inc | 5 - vorlesungen/slides/8/chapter.tex | 7 - vorlesungen/slides/8/fourier.tex | 83 +++++++++ vorlesungen/slides/8/inzidenz.tex | 4 +- vorlesungen/slides/8/markov/google.tex | 123 -------------- vorlesungen/slides/8/markov/irreduzibel.tex | 136 --------------- vorlesungen/slides/8/markov/markov.tex | 111 ------------ vorlesungen/slides/8/markov/pf.tex | 53 ------ vorlesungen/slides/8/markov/stationaer.tex | 57 ------- vorlesungen/slides/8/spanningtree.tex | 164 ++++++++++++++++++ vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg | Bin 0 -> 231575 bytes vorlesungen/slides/8/tokyo/tokyosubway.pdf | Bin 0 -> 1016965 bytes .../slides/8/tokyo/transportnetworkgraph.png | Bin 0 -> 114239 bytes vorlesungen/slides/9/Makefile.inc | 14 ++ vorlesungen/slides/9/chapter.tex | 14 ++ vorlesungen/slides/9/google.tex | 123 ++++++++++++++ vorlesungen/slides/9/irreduzibel.tex | 136 +++++++++++++++ vorlesungen/slides/9/markov.tex | 111 ++++++++++++ vorlesungen/slides/9/pf.tex | 53 ++++++ vorlesungen/slides/9/stationaer.tex | 57 +++++++ vorlesungen/slides/Makefile | 9 + vorlesungen/slides/Makefile.inc | 3 +- vorlesungen/slides/slides.tex | 70 +++++++- vorlesungen/slides/test.tex | 43 +---- 63 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delete mode 100644 vorlesungen/slides/8/markov/google.tex delete mode 100644 vorlesungen/slides/8/markov/irreduzibel.tex delete mode 100644 vorlesungen/slides/8/markov/markov.tex delete mode 100644 vorlesungen/slides/8/markov/pf.tex delete mode 100644 vorlesungen/slides/8/markov/stationaer.tex create mode 100644 vorlesungen/slides/8/spanningtree.tex create mode 100644 vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg create mode 100644 vorlesungen/slides/8/tokyo/tokyosubway.pdf create mode 100644 vorlesungen/slides/8/tokyo/transportnetworkgraph.png create mode 100644 vorlesungen/slides/9/Makefile.inc create mode 100644 vorlesungen/slides/9/chapter.tex create mode 100644 vorlesungen/slides/9/google.tex create mode 100644 vorlesungen/slides/9/irreduzibel.tex create mode 100644 vorlesungen/slides/9/markov.tex create mode 100644 vorlesungen/slides/9/pf.tex create mode 100644 vorlesungen/slides/9/stationaer.tex (limited to 'vorlesungen') diff --git a/vorlesungen/01_intro/slides.tex b/vorlesungen/01_intro/slides.tex index e38e240..298dde3 100644 --- a/vorlesungen/01_intro/slides.tex +++ b/vorlesungen/01_intro/slides.tex @@ -8,7 +8,9 @@ \section{Admin} \folie{0/was.tex} \folie{0/resourcen.tex} -\folie{0/themen.tex} +\folie{0/themen1.tex} +\folie{0/themen2.tex} +\folie{0/themen3.tex} \folie{0/nextsteps.tex} \folie{0/latextipps.tex} }{} diff --git a/vorlesungen/04_msepolynome/Makefile b/vorlesungen/04_msepolynome/Makefile new file mode 100644 index 0000000..a20a980 --- /dev/null +++ b/vorlesungen/04_msepolynome/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- msepolynome +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: msepolynome-handout.pdf MathSem-04-msepolynome.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-04-msepolynome.pdf: MathSem-04-msepolynome.tex $(SOURCES) + pdflatex MathSem-04-msepolynome.tex + +msepolynome-handout.pdf: msepolynome-handout.tex $(SOURCES) + pdflatex msepolynome-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSem-04-msepolynome.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-04-msepolynome.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-04-msepolynome.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-04-msepolynome.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex b/vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex new file mode 100644 index 0000000..8486fc6 --- /dev/null +++ b/vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex @@ -0,0 +1,14 @@ +% +% MathSem-04-msepolynome.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/04_msepolynome/common.tex b/vorlesungen/04_msepolynome/common.tex new file mode 100644 index 0000000..3d78995 --- /dev/null +++ b/vorlesungen/04_msepolynome/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Polynome]{Polynome} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/04_msepolynome/msepolynome-handout.tex b/vorlesungen/04_msepolynome/msepolynome-handout.tex new file mode 100644 index 0000000..df10dde --- /dev/null +++ b/vorlesungen/04_msepolynome/msepolynome-handout.tex @@ -0,0 +1,11 @@ +% +% msepolynome-handout.tex -- Handout XXX +% +% (c) 2017 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/04_msepolynome/slides.tex b/vorlesungen/04_msepolynome/slides.tex new file mode 100644 index 0000000..401688f --- /dev/null +++ b/vorlesungen/04_msepolynome/slides.tex @@ -0,0 +1,34 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Quotient} +\folie{2/quotient.tex} +\folie{2/quotientv.tex} + +\section{Polynome} +\folie{3/motivation.tex} +\folie{3/polynome.tex} +\folie{3/operatoren.tex} +\folie{3/division.tex} +\folie{3/division2.tex} +\folie{3/teilbarkeit.tex} +% XXX Ideal als algebraische Struktur für Teilbarkeit +\folie{3/ideal.tex} +\folie{3/faktorisierung.tex} +\folie{3/faktorzerlegung.tex} +\folie{3/einsetzen.tex} +\folie{3/minimalpolynom.tex} + +\section{Adjunktion} +% XXX Quotientenring +% XXX \folie{3/quotientenring.tex} +% XXX Maximales Ideal und Körper +% XXX \folie{3/maximalideal.tex} +\folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} +% XXX Beispiel: Adjunktion einer Wurzel +% XXX Beispiel: Adjunktion von \varphi +\folie{3/fibonacci.tex} + diff --git a/vorlesungen/05_spektral1/Makefile b/vorlesungen/05_spektral1/Makefile new file mode 100644 index 0000000..3e195bd --- /dev/null +++ b/vorlesungen/05_spektral1/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- spektral1 +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: spektral1-handout.pdf MathSem-05-spektral1.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-05-spektral1.pdf: MathSem-05-spektral1.tex $(SOURCES) + pdflatex MathSem-05-spektral1.tex + +spektral1-handout.pdf: spektral1-handout.tex $(SOURCES) + pdflatex spektral1-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSem-05-spektral1.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-05-spektral1.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-05-spektral1.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-05-spektral1.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/05_spektral1/MathSem-05-spektral1.tex b/vorlesungen/05_spektral1/MathSem-05-spektral1.tex new file mode 100644 index 0000000..568707c --- /dev/null +++ b/vorlesungen/05_spektral1/MathSem-05-spektral1.tex @@ -0,0 +1,14 @@ +% +% MathSem-05-spektral1.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/05_spektral1/common.tex b/vorlesungen/05_spektral1/common.tex new file mode 100644 index 0000000..376ab55 --- /dev/null +++ b/vorlesungen/05_spektral1/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Spektraltheorie~1]{Spektraltheorie 1: Eigenwerte/Eigenvektoren} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/05_spektral1/slides.tex b/vorlesungen/05_spektral1/slides.tex new file mode 100644 index 0000000..05be1aa --- /dev/null +++ b/vorlesungen/05_spektral1/slides.tex @@ -0,0 +1,28 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\folie{5/motivation.tex} +% XXX there is a bug here! +\folie{5/charpoly.tex} +% XXX Visualisierung von Kern und Bild +% \folie{5/kernbildintro.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} +% XXX Visualisierung der Jordan-Normalform +% XXX \folie{5/eigen3d.tex} +\folie{5/eigenraeume.tex} +\folie{5/zerlegung.tex} +\folie{5/normalnilp.tex} +\folie{5/jordanblock.tex} +\folie{5/jordan.tex} +% XXX Visualisierung Cayley-Hamilton-Produkte +% XXX \folie{5/chvisual.tex} +\folie{5/cayleyhamilton.tex} + diff --git a/vorlesungen/05_spektral1/spektral1-handout.tex b/vorlesungen/05_spektral1/spektral1-handout.tex new file mode 100644 index 0000000..98f796e --- /dev/null +++ b/vorlesungen/05_spektral1/spektral1-handout.tex @@ -0,0 +1,11 @@ +% +% spektral1-handout.tex -- Handout XXX +% +% (c) 2017 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/06_spektral2/Makefile b/vorlesungen/06_spektral2/Makefile new file mode 100644 index 0000000..7f92ec7 --- /dev/null +++ b/vorlesungen/06_spektral2/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- spektral2 +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: spektral2-handout.pdf MathSem-06-spektral2.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-06-spektral2.pdf: MathSem-06-spektral2.tex $(SOURCES) + pdflatex MathSem-06-spektral2.tex + +spektral2-handout.pdf: spektral2-handout.tex $(SOURCES) + pdflatex spektral2-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSem-06-spektral2.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-06-spektral2.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-06-spektral2.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-06-spektral2.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/06_spektral2/MathSem-06-spektral2.tex b/vorlesungen/06_spektral2/MathSem-06-spektral2.tex new file mode 100644 index 0000000..196fcf0 --- /dev/null +++ b/vorlesungen/06_spektral2/MathSem-06-spektral2.tex @@ -0,0 +1,14 @@ +% +% MathSem-06-spektral2.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/06_spektral2/common.tex b/vorlesungen/06_spektral2/common.tex new file mode 100644 index 0000000..a3d5d01 --- /dev/null +++ b/vorlesungen/06_spektral2/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Spektraltheorie~2]{Spektraltheorie 2: Funktion $f(A)$ einer Matrix} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/06_spektral2/slides.tex b/vorlesungen/06_spektral2/slides.tex new file mode 100644 index 0000000..b049f2f --- /dev/null +++ b/vorlesungen/06_spektral2/slides.tex @@ -0,0 +1,22 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Normen} +\folie{2/norm.tex} +\folie{2/skalarprodukt.tex} +\folie{2/operatornorm.tex} + +\section{Matrix-Analysis} +\folie{5/konvergenzradius.tex} +\folie{5/krbeispiele.tex} +\folie{5/spektralgelfand.tex} +\folie{5/Aiteration.tex} +\folie{5/satzvongelfand.tex} +% XXX stone weierstrass incomplete +\folie{5/stoneweierstrass.tex} +\folie{5/spektrum.tex} +% XXX polynome auf dem spektrum +% XXX Motiviation für *-Operation +\folie{5/normal.tex} diff --git a/vorlesungen/06_spektral2/spektral2-handout.tex b/vorlesungen/06_spektral2/spektral2-handout.tex new file mode 100644 index 0000000..40ce63b --- /dev/null +++ b/vorlesungen/06_spektral2/spektral2-handout.tex @@ -0,0 +1,11 @@ +% +% spektral2-handout.tex -- Handout XXX +% +% (c) 2017 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/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc index ab91126..a6bb320 100644 --- a/vorlesungen/slides/0/Makefile.inc +++ b/vorlesungen/slides/0/Makefile.inc @@ -10,6 +10,8 @@ chapter0 = \ ../slides/0/resourcen.tex \ ../slides/0/latextipps.tex \ ../slides/0/nextsteps.tex \ - ../slides/0/themen.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 index a860830..6e09557 100644 --- a/vorlesungen/slides/0/chapter.tex +++ b/vorlesungen/slides/0/chapter.tex @@ -7,5 +7,7 @@ \folie{0/was.tex} \folie{0/resourcen.tex} \folie{0/latextipps.tex} -\folie{0/themen.tex} +\folie{0/themen1.tex} +\folie{0/themen2.tex} +\folie{0/themen3.tex} \folie{0/nextsteps.tex} diff --git a/vorlesungen/slides/0/themen.tex b/vorlesungen/slides/0/themen.tex deleted file mode 100644 index c50f32d..0000000 --- a/vorlesungen/slides/0/themen.tex +++ /dev/null @@ -1,68 +0,0 @@ -% -% themen.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} - -\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} - -\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/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/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 34dadf6..c857fec 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -15,5 +15,7 @@ chapter2 = \ ../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/chapter.tex b/vorlesungen/slides/2/chapter.tex index 38eb21f..49e656a 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -13,3 +13,5 @@ \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/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 new file mode 100644 index 0000000..aeb713e Binary files /dev/null and b/vorlesungen/slides/2/images/quotient1.jpg differ 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 new file mode 100644 index 0000000..345cf22 Binary files /dev/null and b/vorlesungen/slides/2/images/quotient2.jpg differ 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/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/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index 7f52cb1..f2edc80 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -12,6 +12,8 @@ chapter3 = \ ../slides/3/division2.tex \ ../slides/3/ringstruktur.tex \ ../slides/3/teilbarkeit.tex \ + ../slides/3/ideal.tex \ + ../slides/3/quotientenring.tex \ ../slides/3/faktorisierung.tex \ ../slides/3/faktorzerlegung.tex \ ../slides/3/einsetzen.tex \ diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex index 0f049e7..deec12e 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -10,6 +10,8 @@ \folie{3/division2.tex} \folie{3/ringstruktur.tex} \folie{3/teilbarkeit.tex} +\folie{3/ideal.tex} +\folie{3/quotientenring.tex} \folie{3/faktorisierung.tex} \folie{3/faktorzerlegung.tex} \folie{3/einsetzen.tex} 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/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/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index 233835a..d46dc7f 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -28,10 +28,5 @@ chapter8 = \ ../slides/8/tokyo/bahn0.tex \ ../slides/8/tokyo/bahn1.tex \ ../slides/8/tokyo/bahn2.tex \ - ../slides/8/markov/google.tex \ - ../slides/8/markov/markov.tex \ - ../slides/8/markov/irreduzibel.tex \ - ../slides/8/markov/stationaer.tex \ - ../slides/8/markov/pf.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index ac06775..6a0b13f 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -30,10 +30,3 @@ \folie{8/tokyo/bahn1.tex} \folie{8/tokyo/bahn2.tex} -\folie{8/markov/google.tex} -\folie{8/markov/markov.tex} -\folie{8/markov/stationaer.tex} -\folie{8/markov/irreduzibel.tex} -\folie{8/markov/pf.tex} - - 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/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex index 87578df..952c85b 100644 --- a/vorlesungen/slides/8/inzidenz.tex +++ b/vorlesungen/slides/8/inzidenz.tex @@ -126,8 +126,8 @@ B(G) 1&0&0&1&1&0\\ 1&1&0&0&0&1\\ 0&1&1&0&1&0\\ -0&0&1&0&0&0\\ -0&0&0&1&0&1 +0&0&1&0&0&1\\ +0&0&0&1&0&0 \end{pmatrix} \\[12pt] \uncover<4->{ diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/8/markov/google.tex deleted file mode 100644 index d1ec31d..0000000 --- a/vorlesungen/slides/8/markov/google.tex +++ /dev/null @@ -1,123 +0,0 @@ -% -% 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/8/markov/irreduzibel.tex b/vorlesungen/slides/8/markov/irreduzibel.tex deleted file mode 100644 index 87e90e4..0000000 --- a/vorlesungen/slides/8/markov/irreduzibel.tex +++ /dev/null @@ -1,136 +0,0 @@ -% -% 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/8/markov/markov.tex b/vorlesungen/slides/8/markov/markov.tex deleted file mode 100644 index e92ff0f..0000000 --- a/vorlesungen/slides/8/markov/markov.tex +++ /dev/null @@ -1,111 +0,0 @@ -% -% 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/8/markov/pf.tex b/vorlesungen/slides/8/markov/pf.tex deleted file mode 100644 index da2ef2b..0000000 --- a/vorlesungen/slides/8/markov/pf.tex +++ /dev/null @@ -1,53 +0,0 @@ -% -% 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/8/markov/stationaer.tex b/vorlesungen/slides/8/markov/stationaer.tex deleted file mode 100644 index 92fab16..0000000 --- a/vorlesungen/slides/8/markov/stationaer.tex +++ /dev/null @@ -1,57 +0,0 @@ -% -% 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/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/shinjuku-subway-map.jpg b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg new file mode 100644 index 0000000..1c513da Binary files /dev/null and b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg differ diff --git a/vorlesungen/slides/8/tokyo/tokyosubway.pdf b/vorlesungen/slides/8/tokyo/tokyosubway.pdf new file mode 100644 index 0000000..6b84a8d Binary files /dev/null and b/vorlesungen/slides/8/tokyo/tokyosubway.pdf differ diff --git a/vorlesungen/slides/8/tokyo/transportnetworkgraph.png b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png new file mode 100644 index 0000000..4a11183 Binary files /dev/null and b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png differ 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 index dd8b06e..d3c7a17 100644 --- a/vorlesungen/slides/Makefile +++ b/vorlesungen/slides/Makefile @@ -11,9 +11,18 @@ 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 diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index fe22282..4bf9348 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -10,7 +10,8 @@ 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) + $(chapter5) $(chapter8) $(chapter9) diff --git a/vorlesungen/slides/slides.tex b/vorlesungen/slides/slides.tex index 0dc6c6e..b606375 100644 --- a/vorlesungen/slides/slides.tex +++ b/vorlesungen/slides/slides.tex @@ -3,19 +3,77 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\title[Intro]{Intro} -\section{Intro} +\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} -\begin{frame} -\titlepage -\end{frame} -\subtitle{} +\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.tex b/vorlesungen/slides/test.tex index 411414e..2423a3f 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,44 +3,5 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{8/dgraph.tex} -%\folie{8/graph.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} - -% XXX \folie{8/schwingungen.tex} -% XXX \folie{8/eigenvektorbasis.tex} -% XXX \folie{8/spektralegraphentheorie.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/wege.tex} -%\folie{8/floyd-warshall/wegiteration.tex} - -% Markov-Ketten -% - Google-Matrix -% - Stationär - -%\folie{8/tokyo/google.tex} -%\folie{8/tokyo/bahn0.tex} -%\folie{8/tokyo/bahn1.tex} -%\folie{8/tokyo/bahn2.tex} - - -%\folie{8/markov/google.tex} -%\folie{8/markov/markov.tex} -%\folie{8/markov/irreduzibel.tex} -%\folie{8/markov/stationaer.tex} -%\folie{8/markov/pf.tex} +\folie{3/ideal.tex} +\folie{3/quotientenring.tex} -- cgit v1.2.1 From 755f1805249a01bcd39390565f080032c109d2ca Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 16 Mar 2021 20:19:36 +0100 Subject: add new slides --- vorlesungen/04_msepolynome/Makefile | 20 +++--- .../04_msepolynome/MathSem-04-msepolynome.tex | 14 ---- .../04_msepolynome/MathSemMSE-04-polynome.tex | 14 ++++ vorlesungen/04_msepolynome/msepolynome-handout.tex | 11 --- vorlesungen/04_msepolynome/polynome-handout.tex | 11 +++ vorlesungen/04_msepolynome/slides.tex | 10 ++- vorlesungen/slides/3/Makefile.inc | 3 + vorlesungen/slides/3/chapter.tex | 3 + vorlesungen/slides/3/idealverband.tex | 78 +++++++++++++++++++++ vorlesungen/slides/3/maximalideal.tex | 64 ++++++++++++++++++ vorlesungen/slides/3/wurzel2.tex | 79 ++++++++++++++++++++++ vorlesungen/slides/test.tex | 3 +- 12 files changed, 267 insertions(+), 43 deletions(-) delete mode 100644 vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex create mode 100644 vorlesungen/04_msepolynome/MathSemMSE-04-polynome.tex delete mode 100644 vorlesungen/04_msepolynome/msepolynome-handout.tex create mode 100644 vorlesungen/04_msepolynome/polynome-handout.tex create mode 100644 vorlesungen/slides/3/idealverband.tex create mode 100644 vorlesungen/slides/3/maximalideal.tex create mode 100644 vorlesungen/slides/3/wurzel2.tex (limited to 'vorlesungen') diff --git a/vorlesungen/04_msepolynome/Makefile b/vorlesungen/04_msepolynome/Makefile index a20a980..50a9ce2 100644 --- a/vorlesungen/04_msepolynome/Makefile +++ b/vorlesungen/04_msepolynome/Makefile @@ -1,32 +1,32 @@ # -# Makefile -- msepolynome +# Makefile -- polynome # # (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil # -all: msepolynome-handout.pdf MathSem-04-msepolynome.pdf +all: polynome-handout.pdf MathSemMSE-04-polynome.pdf include ../slides/Makefile.inc SOURCES = common.tex slides.tex $(slides) -MathSem-04-msepolynome.pdf: MathSem-04-msepolynome.tex $(SOURCES) - pdflatex MathSem-04-msepolynome.tex +MathSemMSE-04-polynome.pdf: MathSemMSE-04-polynome.tex $(SOURCES) + pdflatex MathSemMSE-04-polynome.tex -msepolynome-handout.pdf: msepolynome-handout.tex $(SOURCES) - pdflatex msepolynome-handout.tex +polynome-handout.pdf: polynome-handout.tex $(SOURCES) + pdflatex polynome-handout.tex thumbnail: thumbnail.jpg # fix1.jpg -thumbnail.pdf: MathSem-04-msepolynome.pdf +thumbnail.pdf: MathSemMSE-04-polynome.pdf pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ - MathSem-04-msepolynome.pdf 1 + MathSemMSE-04-polynome.pdf 1 thumbnail.jpg: thumbnail.pdf convert -density 300 thumbnail.pdf \ -resize 1920x1080 -units PixelsPerInch thumbnail.jpg -fix1.pdf: MathSem-04-msepolynome.pdf +fix1.pdf: MathSemMSE-04-polynome.pdf pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ - MathSem-04-msepolynome.pdf 1 + MathSemMSE-04-polynome.pdf 1 fix1.jpg: fix1.pdf convert -density 300 fix1.pdf \ -resize 1920x1080 -units PixelsPerInch fix1.jpg diff --git a/vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex b/vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex deleted file mode 100644 index 8486fc6..0000000 --- a/vorlesungen/04_msepolynome/MathSem-04-msepolynome.tex +++ /dev/null @@ -1,14 +0,0 @@ -% -% MathSem-04-msepolynome.tex -- Präsentation -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\documentclass[aspectratio=169]{beamer} -\input{common.tex} -\setboolean{presentation}{true} -\begin{document} -\begin{frame} -\titlepage -\end{frame} -\input{slides.tex} -\end{document} diff --git a/vorlesungen/04_msepolynome/MathSemMSE-04-polynome.tex b/vorlesungen/04_msepolynome/MathSemMSE-04-polynome.tex new file mode 100644 index 0000000..8486fc6 --- /dev/null +++ b/vorlesungen/04_msepolynome/MathSemMSE-04-polynome.tex @@ -0,0 +1,14 @@ +% +% MathSem-04-msepolynome.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/04_msepolynome/msepolynome-handout.tex b/vorlesungen/04_msepolynome/msepolynome-handout.tex deleted file mode 100644 index df10dde..0000000 --- a/vorlesungen/04_msepolynome/msepolynome-handout.tex +++ /dev/null @@ -1,11 +0,0 @@ -% -% msepolynome-handout.tex -- Handout XXX -% -% (c) 2017 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/04_msepolynome/polynome-handout.tex b/vorlesungen/04_msepolynome/polynome-handout.tex new file mode 100644 index 0000000..df10dde --- /dev/null +++ b/vorlesungen/04_msepolynome/polynome-handout.tex @@ -0,0 +1,11 @@ +% +% msepolynome-handout.tex -- Handout XXX +% +% (c) 2017 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/04_msepolynome/slides.tex b/vorlesungen/04_msepolynome/slides.tex index 401688f..f8a7ba2 100644 --- a/vorlesungen/04_msepolynome/slides.tex +++ b/vorlesungen/04_msepolynome/slides.tex @@ -14,21 +14,19 @@ \folie{3/division.tex} \folie{3/division2.tex} \folie{3/teilbarkeit.tex} -% XXX Ideal als algebraische Struktur für Teilbarkeit \folie{3/ideal.tex} +\folie{3/idealverband.tex} \folie{3/faktorisierung.tex} \folie{3/faktorzerlegung.tex} \folie{3/einsetzen.tex} \folie{3/minimalpolynom.tex} \section{Adjunktion} -% XXX Quotientenring -% XXX \folie{3/quotientenring.tex} -% XXX Maximales Ideal und Körper -% XXX \folie{3/maximalideal.tex} +\folie{3/quotientenring.tex} +\folie{3/maximalideal.tex} \folie{3/adjunktion.tex} \folie{3/adjalgebra.tex} -% XXX Beispiel: Adjunktion einer Wurzel +\folie{3/wurzel2.tex} % XXX Beispiel: Adjunktion von \varphi \folie{3/fibonacci.tex} diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index f2edc80..a70f73b 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -13,6 +13,8 @@ chapter3 = \ ../slides/3/ringstruktur.tex \ ../slides/3/teilbarkeit.tex \ ../slides/3/ideal.tex \ + ../slides/3/idealverband.tex \ + ../slides/3/maximalideal.tex \ ../slides/3/quotientenring.tex \ ../slides/3/faktorisierung.tex \ ../slides/3/faktorzerlegung.tex \ @@ -26,5 +28,6 @@ chapter3 = \ ../slides/3/operatoren.tex \ ../slides/3/adjunktion.tex \ ../slides/3/adjalgebra.tex \ + ../slides/3/wurzel2.tex \ ../slides/3/chapter.tex diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex index deec12e..ea2718d 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -11,6 +11,8 @@ \folie{3/ringstruktur.tex} \folie{3/teilbarkeit.tex} \folie{3/ideal.tex} +\folie{3/maximalideal.tex} +\folie{3/idealverband.tex} \folie{3/quotientenring.tex} \folie{3/faktorisierung.tex} \folie{3/faktorzerlegung.tex} @@ -24,3 +26,4 @@ \folie{3/operatoren.tex} \folie{3/adjunktion.tex} \folie{3/adjalgebra.tex} +\folie{3/wurzel2.tex} 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/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/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex new file mode 100644 index 0000000..48cc210 --- /dev/null +++ b/vorlesungen/slides/3/wurzel2.tex @@ -0,0 +1,79 @@ +% +% 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}} +\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/test.tex b/vorlesungen/slides/test.tex index 2423a3f..9402625 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,5 +3,4 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{3/ideal.tex} -\folie{3/quotientenring.tex} +\folie{3/wurzel2.tex} -- cgit v1.2.1 From 385db80750dce2b87fb49784814ee660123ca7ff Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 18 Mar 2021 13:21:18 +0100 Subject: =?UTF-8?q?slides=20=C3=BCber=20Ideale?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- vorlesungen/04_msepolynome/slides.tex | 1 + vorlesungen/slides/3/Makefile.inc | 2 + vorlesungen/slides/3/chapter.tex | 2 + vorlesungen/slides/3/images/Makefile | 55 +++++ vorlesungen/slides/3/images/common.inc | 277 ++++++++++++++++++++++++ vorlesungen/slides/3/images/hauptideal.jpg | Bin 0 -> 218404 bytes vorlesungen/slides/3/images/hauptideal.pov | 10 + vorlesungen/slides/3/images/hauptideal2.jpg | Bin 0 -> 474455 bytes vorlesungen/slides/3/images/hauptideal2.pov | 10 + vorlesungen/slides/3/images/hauptidealR.jpg | Bin 0 -> 89610 bytes vorlesungen/slides/3/images/hauptidealR.pov | 10 + vorlesungen/slides/3/images/hauptidealX.jpg | Bin 0 -> 169673 bytes vorlesungen/slides/3/images/hauptidealX.pov | 10 + vorlesungen/slides/3/images/hauptidealXR.jpg | Bin 0 -> 52538 bytes vorlesungen/slides/3/images/hauptidealXR.pov | 10 + vorlesungen/slides/3/images/ideal.ini | 7 + vorlesungen/slides/3/images/ideal.pov | 26 +++ vorlesungen/slides/3/images/nichthauptideal.jpg | Bin 0 -> 641055 bytes vorlesungen/slides/3/images/nichthauptideal.pov | 10 + vorlesungen/slides/3/images/ring.jpg | Bin 0 -> 681411 bytes vorlesungen/slides/3/images/ring.pov | 10 + vorlesungen/slides/3/nichthauptideal.tex | 78 +++++++ vorlesungen/slides/3/nichthauptideal2.tex | 92 ++++++++ vorlesungen/slides/test.tex | 4 +- 24 files changed, 613 insertions(+), 1 deletion(-) create mode 100644 vorlesungen/slides/3/images/Makefile create mode 100644 vorlesungen/slides/3/images/common.inc create mode 100644 vorlesungen/slides/3/images/hauptideal.jpg create mode 100644 vorlesungen/slides/3/images/hauptideal.pov create mode 100644 vorlesungen/slides/3/images/hauptideal2.jpg create mode 100644 vorlesungen/slides/3/images/hauptideal2.pov create mode 100644 vorlesungen/slides/3/images/hauptidealR.jpg create mode 100644 vorlesungen/slides/3/images/hauptidealR.pov create mode 100644 vorlesungen/slides/3/images/hauptidealX.jpg create mode 100644 vorlesungen/slides/3/images/hauptidealX.pov create mode 100644 vorlesungen/slides/3/images/hauptidealXR.jpg create mode 100644 vorlesungen/slides/3/images/hauptidealXR.pov create mode 100644 vorlesungen/slides/3/images/ideal.ini create mode 100644 vorlesungen/slides/3/images/ideal.pov create mode 100644 vorlesungen/slides/3/images/nichthauptideal.jpg create mode 100644 vorlesungen/slides/3/images/nichthauptideal.pov create mode 100644 vorlesungen/slides/3/images/ring.jpg create mode 100644 vorlesungen/slides/3/images/ring.pov create mode 100644 vorlesungen/slides/3/nichthauptideal.tex create mode 100644 vorlesungen/slides/3/nichthauptideal2.tex (limited to 'vorlesungen') diff --git a/vorlesungen/04_msepolynome/slides.tex b/vorlesungen/04_msepolynome/slides.tex index f8a7ba2..46ba8bf 100644 --- a/vorlesungen/04_msepolynome/slides.tex +++ b/vorlesungen/04_msepolynome/slides.tex @@ -15,6 +15,7 @@ \folie{3/division2.tex} \folie{3/teilbarkeit.tex} \folie{3/ideal.tex} +\folie{3/nichthauptideal.tex} \folie{3/idealverband.tex} \folie{3/faktorisierung.tex} \folie{3/faktorzerlegung.tex} diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index a70f73b..d6ce6b7 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -13,6 +13,8 @@ chapter3 = \ ../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 \ diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex index ea2718d..5287ffe 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -11,6 +11,8 @@ \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} 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 { , 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 { , 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 { , 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 { , 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 new file mode 100644 index 0000000..769f53c Binary files /dev/null and b/vorlesungen/slides/3/images/hauptideal.jpg differ 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 new file mode 100644 index 0000000..51823f3 Binary files /dev/null and b/vorlesungen/slides/3/images/hauptideal2.jpg differ 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 new file mode 100644 index 0000000..fae5840 Binary files /dev/null and b/vorlesungen/slides/3/images/hauptidealR.jpg differ 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 new file mode 100644 index 0000000..f9b4540 Binary files /dev/null and b/vorlesungen/slides/3/images/hauptidealX.jpg differ 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 new file mode 100644 index 0000000..d8906c8 Binary files /dev/null and b/vorlesungen/slides/3/images/hauptidealXR.jpg differ 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 new file mode 100644 index 0000000..55858d0 Binary files /dev/null and b/vorlesungen/slides/3/images/nichthauptideal.jpg differ 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 new file mode 100644 index 0000000..27721b1 Binary files /dev/null and b/vorlesungen/slides/3/images/ring.jpg differ 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/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..f362a4a --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal2.tex @@ -0,0 +1,92 @@ +% +% 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] {$\mathbb{Z}[X]$}; + \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} +} + +\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); + +\end{tikzpicture} +\end{center} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 9402625..276b978 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,4 +3,6 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{3/wurzel2.tex} +%\folie{3/wurzel2.tex} +\folie{3/nichthauptideal.tex} +\folie{3/nichthauptideal2.tex} -- cgit v1.2.1 From ed94764e29c2a9269ff89d915de80422e027a689 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 18 Mar 2021 16:36:45 +0100 Subject: new slides --- vorlesungen/04_msepolynome/slides.tex | 10 +- vorlesungen/slides/3/Makefile.inc | 2 + vorlesungen/slides/3/chapter.tex | 2 + vorlesungen/slides/3/multiplikation.tex | 180 ++++++++++++++++++++++++++++++++ vorlesungen/slides/3/phi.tex | 85 +++++++++++++++ vorlesungen/slides/3/wurzel2.tex | 4 + vorlesungen/slides/test.tex | 6 +- 7 files changed, 285 insertions(+), 4 deletions(-) create mode 100644 vorlesungen/slides/3/multiplikation.tex create mode 100644 vorlesungen/slides/3/phi.tex (limited to 'vorlesungen') diff --git a/vorlesungen/04_msepolynome/slides.tex b/vorlesungen/04_msepolynome/slides.tex index 46ba8bf..618ea0e 100644 --- a/vorlesungen/04_msepolynome/slides.tex +++ b/vorlesungen/04_msepolynome/slides.tex @@ -13,10 +13,15 @@ \folie{3/operatoren.tex} \folie{3/division.tex} \folie{3/division2.tex} + +\section{Teilbarkeit} \folie{3/teilbarkeit.tex} \folie{3/ideal.tex} -\folie{3/nichthauptideal.tex} \folie{3/idealverband.tex} +\folie{3/nichthauptideal.tex} +\folie{3/nichthauptideal2.tex} + +\section{Faktorisierung} \folie{3/faktorisierung.tex} \folie{3/faktorzerlegung.tex} \folie{3/einsetzen.tex} @@ -28,6 +33,7 @@ \folie{3/adjunktion.tex} \folie{3/adjalgebra.tex} \folie{3/wurzel2.tex} -% XXX Beispiel: Adjunktion von \varphi +\folie{3/phi.tex} +\folie{3/multiplikation.tex} \folie{3/fibonacci.tex} diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index d6ce6b7..442bd15 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -31,5 +31,7 @@ chapter3 = \ ../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/chapter.tex b/vorlesungen/slides/3/chapter.tex index 5287ffe..3fbc3fd 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -29,3 +29,5 @@ \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/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/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/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex index 48cc210..d20bfc4 100644 --- a/vorlesungen/slides/3/wurzel2.tex +++ b/vorlesungen/slides/3/wurzel2.tex @@ -33,6 +33,10 @@ Das Polynom $X^2-2$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$} \]\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} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 276b978..c482bee 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -4,5 +4,7 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % %\folie{3/wurzel2.tex} -\folie{3/nichthauptideal.tex} -\folie{3/nichthauptideal2.tex} +%\folie{3/phi.tex} +%\folie{3/nichthauptideal.tex} +%\folie{3/nichthauptideal2.tex} +\folie{3/multiplikation.tex} -- cgit v1.2.1 From 752637d19f3415bb148563aeb7b9658e0a490190 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Thu, 18 Mar 2021 16:44:34 +0100 Subject: new slides --- vorlesungen/04_msepolynome/slides.tex | 4 ++-- vorlesungen/slides/3/nichthauptideal2.tex | 11 +++++++---- vorlesungen/slides/test.tex | 4 ++-- 3 files changed, 11 insertions(+), 8 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/04_msepolynome/slides.tex b/vorlesungen/04_msepolynome/slides.tex index 618ea0e..25bee83 100644 --- a/vorlesungen/04_msepolynome/slides.tex +++ b/vorlesungen/04_msepolynome/slides.tex @@ -30,10 +30,10 @@ \section{Adjunktion} \folie{3/quotientenring.tex} \folie{3/maximalideal.tex} -\folie{3/adjunktion.tex} -\folie{3/adjalgebra.tex} \folie{3/wurzel2.tex} \folie{3/phi.tex} \folie{3/multiplikation.tex} \folie{3/fibonacci.tex} +\folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex index f362a4a..e1424ff 100644 --- a/vorlesungen/slides/3/nichthauptideal2.tex +++ b/vorlesungen/slides/3/nichthauptideal2.tex @@ -30,10 +30,6 @@ {\includegraphics[width=7cm]{../slides/3/images/ring.jpg}}; \end{scope} \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; - \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} } @@ -85,6 +81,13 @@ \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} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index c482bee..9a4f0c3 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -6,5 +6,5 @@ %\folie{3/wurzel2.tex} %\folie{3/phi.tex} %\folie{3/nichthauptideal.tex} -%\folie{3/nichthauptideal2.tex} -\folie{3/multiplikation.tex} +\folie{3/nichthauptideal2.tex} +%\folie{3/multiplikation.tex} -- cgit v1.2.1 From 654d9ee4aad5389798a37f521a7a4238be3fdc34 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 21 Mar 2021 17:21:45 +0100 Subject: JK bilder --- vorlesungen/slides/5/beispiele/Makefile | 30 ++++++ vorlesungen/slides/5/beispiele/bild1.jpg | Bin 0 -> 76321 bytes vorlesungen/slides/5/beispiele/bild1.pov | 13 +++ vorlesungen/slides/5/beispiele/bild2.jpg | Bin 0 -> 101222 bytes vorlesungen/slides/5/beispiele/bild2.pov | 17 ++++ vorlesungen/slides/5/beispiele/common.inc | 134 ++++++++++++++++++++++++++ vorlesungen/slides/5/beispiele/kern1.jpg | Bin 0 -> 61750 bytes vorlesungen/slides/5/beispiele/kern1.pov | 12 +++ vorlesungen/slides/5/beispiele/kern2.jpg | Bin 0 -> 88123 bytes vorlesungen/slides/5/beispiele/kern2.pov | 17 ++++ vorlesungen/slides/5/beispiele/kernbild.m | 78 +++++++++++++++ vorlesungen/slides/5/beispiele/kernbild1.jpg | Bin 0 -> 84631 bytes vorlesungen/slides/5/beispiele/kernbild1.pov | 15 +++ vorlesungen/slides/5/beispiele/kernbild2.jpg | Bin 0 -> 90826 bytes vorlesungen/slides/5/beispiele/kernbild2.pov | 21 ++++ vorlesungen/slides/5/beispiele/kombiniert.jpg | Bin 0 -> 116884 bytes vorlesungen/slides/test.tex | 22 ++++- 17 files changed, 354 insertions(+), 5 deletions(-) create mode 100644 vorlesungen/slides/5/beispiele/Makefile create mode 100644 vorlesungen/slides/5/beispiele/bild1.jpg create mode 100644 vorlesungen/slides/5/beispiele/bild1.pov create mode 100644 vorlesungen/slides/5/beispiele/bild2.jpg create mode 100644 vorlesungen/slides/5/beispiele/bild2.pov create mode 100644 vorlesungen/slides/5/beispiele/common.inc create mode 100644 vorlesungen/slides/5/beispiele/kern1.jpg create mode 100644 vorlesungen/slides/5/beispiele/kern1.pov create mode 100644 vorlesungen/slides/5/beispiele/kern2.jpg create mode 100644 vorlesungen/slides/5/beispiele/kern2.pov create mode 100644 vorlesungen/slides/5/beispiele/kernbild.m create mode 100644 vorlesungen/slides/5/beispiele/kernbild1.jpg create mode 100644 vorlesungen/slides/5/beispiele/kernbild1.pov create mode 100644 vorlesungen/slides/5/beispiele/kernbild2.jpg create mode 100644 vorlesungen/slides/5/beispiele/kernbild2.pov create mode 100644 vorlesungen/slides/5/beispiele/kombiniert.jpg (limited to 'vorlesungen') diff --git a/vorlesungen/slides/5/beispiele/Makefile b/vorlesungen/slides/5/beispiele/Makefile new file mode 100644 index 0000000..c03ea62 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/Makefile @@ -0,0 +1,30 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: kern bild kb kombiniert.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 + +%.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 new file mode 100644 index 0000000..e8bdc1c Binary files /dev/null and b/vorlesungen/slides/5/beispiele/bild1.jpg differ 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 new file mode 100644 index 0000000..8e5dde5 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/bild2.jpg differ 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>, } + plane { vnormalize(vcross(v1, v2)), 0.004 } + plane { vnormalize(-vcross(v1, v2)), 0.004 } + pigment { + color rgbt + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro gerade(v1, farbe) + intersection { + box { <-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>, } + 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/kern1.jpg b/vorlesungen/slides/5/beispiele/kern1.jpg new file mode 100644 index 0000000..f5fcf7d Binary files /dev/null and b/vorlesungen/slides/5/beispiele/kern1.jpg differ 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 new file mode 100644 index 0000000..c3b4b79 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/kern2.jpg differ 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..ca6bd6a --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kernbild.m @@ -0,0 +1,78 @@ +# +# 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 +D = rref(E1 * E1); +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", E2(1,1), E2(2,1), E2(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 new file mode 100644 index 0000000..bda5ce4 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/kernbild1.jpg differ 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 new file mode 100644 index 0000000..4139918 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/kernbild2.jpg differ 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 new file mode 100644 index 0000000..c214596 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/kombiniert.jpg differ diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 9a4f0c3..2c0746b 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,8 +3,20 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{3/wurzel2.tex} -%\folie{3/phi.tex} -%\folie{3/nichthauptideal.tex} -\folie{3/nichthauptideal2.tex} -%\folie{3/multiplikation.tex} + +% XXX Visualisierung von Kern und Bild +\folie{5/kernbildintro.tex} + +% XXX Visualisierung der Jordan-Normalform +% XXX \folie{5/eigen3d.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} + -- cgit v1.2.1 From e2737868b67b183152c4436ececf12e68b5a145b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 21 Mar 2021 20:37:06 +0100 Subject: new slides --- vorlesungen/05_spektral1/Makefile | 6 +- vorlesungen/05_spektral1/MathSem-05-spektral1.tex | 4 + vorlesungen/05_spektral1/jordan.jpg | Bin 0 -> 81986 bytes vorlesungen/05_spektral1/slides.tex | 22 ++-- vorlesungen/slides/5/Makefile.inc | 3 + vorlesungen/slides/5/beispiele/Makefile | 4 +- vorlesungen/slides/5/beispiele/bild1.jpg | Bin 76321 -> 76315 bytes vorlesungen/slides/5/beispiele/bild2.jpg | Bin 101222 -> 87846 bytes vorlesungen/slides/5/beispiele/drei.jpg | Bin 0 -> 95383 bytes vorlesungen/slides/5/beispiele/drei.pov | 22 ++++ vorlesungen/slides/5/beispiele/kern1.jpg | Bin 61750 -> 61717 bytes vorlesungen/slides/5/beispiele/kern2.jpg | Bin 88123 -> 87289 bytes vorlesungen/slides/5/beispiele/kernbild.m | 5 +- vorlesungen/slides/5/beispiele/kernbild1.jpg | Bin 84631 -> 84647 bytes vorlesungen/slides/5/beispiele/kernbild2.jpg | Bin 90826 -> 76111 bytes vorlesungen/slides/5/beispiele/kombiniert.jpg | Bin 116884 -> 109739 bytes vorlesungen/slides/5/beispiele/kombiniert.pov | 22 ++++ vorlesungen/slides/5/beispiele/leer.jpg | Bin 0 -> 23984 bytes vorlesungen/slides/5/beispiele/leer.pov | 9 ++ vorlesungen/slides/5/chapter.tex | 3 + vorlesungen/slides/5/kernbilder.tex | 68 ++++++++++++ vorlesungen/slides/5/kernbildintro.tex | 89 ++++++++++++++++ vorlesungen/slides/5/verzerrung.tex | 121 ++++++++++++++++++++++ vorlesungen/slides/5/verzerrung/verzerrung.m | 13 +++ vorlesungen/slides/test.tex | 11 +- 25 files changed, 380 insertions(+), 22 deletions(-) create mode 100644 vorlesungen/05_spektral1/jordan.jpg create mode 100644 vorlesungen/slides/5/beispiele/drei.jpg create mode 100644 vorlesungen/slides/5/beispiele/drei.pov create mode 100644 vorlesungen/slides/5/beispiele/kombiniert.pov create mode 100644 vorlesungen/slides/5/beispiele/leer.jpg create mode 100644 vorlesungen/slides/5/beispiele/leer.pov create mode 100644 vorlesungen/slides/5/kernbilder.tex create mode 100644 vorlesungen/slides/5/kernbildintro.tex create mode 100644 vorlesungen/slides/5/verzerrung.tex create mode 100644 vorlesungen/slides/5/verzerrung/verzerrung.m (limited to 'vorlesungen') diff --git a/vorlesungen/05_spektral1/Makefile b/vorlesungen/05_spektral1/Makefile index 3e195bd..a72a5a8 100644 --- a/vorlesungen/05_spektral1/Makefile +++ b/vorlesungen/05_spektral1/Makefile @@ -15,7 +15,7 @@ MathSem-05-spektral1.pdf: MathSem-05-spektral1.tex $(SOURCES) spektral1-handout.pdf: spektral1-handout.tex $(SOURCES) pdflatex spektral1-handout.tex -thumbnail: thumbnail.jpg # fix1.jpg +thumbnail: thumbnail.jpg fix1.jpg thumbnail.pdf: MathSem-05-spektral1.pdf pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ @@ -26,8 +26,10 @@ thumbnail.jpg: thumbnail.pdf fix1.pdf: MathSem-05-spektral1.pdf pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ - MathSem-05-spektral1.pdf 1 + MathSem-05-spektral1.pdf 92 fix1.jpg: fix1.pdf convert -density 300 fix1.pdf \ -resize 1920x1080 -units PixelsPerInch fix1.jpg +jordan.jpg: fix1.jpg Makefile + convert -extract 873x873+967+199 fix1.jpg jordan.jpg diff --git a/vorlesungen/05_spektral1/MathSem-05-spektral1.tex b/vorlesungen/05_spektral1/MathSem-05-spektral1.tex index 568707c..7eeacd9 100644 --- a/vorlesungen/05_spektral1/MathSem-05-spektral1.tex +++ b/vorlesungen/05_spektral1/MathSem-05-spektral1.tex @@ -9,6 +9,10 @@ \begin{document} \begin{frame} \titlepage +\vspace{-1.5cm} +\begin{center} +\includegraphics[width=6cm]{jordan.jpg} +\end{center} \end{frame} \input{slides.tex} \end{document} diff --git a/vorlesungen/05_spektral1/jordan.jpg b/vorlesungen/05_spektral1/jordan.jpg new file mode 100644 index 0000000..7c26f19 Binary files /dev/null and b/vorlesungen/05_spektral1/jordan.jpg differ diff --git a/vorlesungen/05_spektral1/slides.tex b/vorlesungen/05_spektral1/slides.tex index 05be1aa..1360eea 100644 --- a/vorlesungen/05_spektral1/slides.tex +++ b/vorlesungen/05_spektral1/slides.tex @@ -3,26 +3,30 @@ % % (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil % - -\folie{5/motivation.tex} -% XXX there is a bug here! +\section{Eigenwerte} +\ifthenelse{\boolean{presentation}}{ +\folie{5/verzerrung.tex} +}{} +\folie{5/spektrum.tex} +\folie{5/kernbildintro.tex} +\folie{5/kernbilder.tex} \folie{5/charpoly.tex} -% XXX Visualisierung von Kern und Bild -% \folie{5/kernbildintro.tex} + +\section{Ketten} \folie{5/kernbild.tex} \folie{5/ketten.tex} \folie{5/dimension.tex} \folie{5/folgerungen.tex} \folie{5/injektiv.tex} + +\section{Nilpotent} \folie{5/nilpotent.tex} -% XXX Visualisierung der Jordan-Normalform -% XXX \folie{5/eigen3d.tex} \folie{5/eigenraeume.tex} \folie{5/zerlegung.tex} \folie{5/normalnilp.tex} + +\section{Jordan Normalform} \folie{5/jordanblock.tex} \folie{5/jordan.tex} -% XXX Visualisierung Cayley-Hamilton-Produkte -% XXX \folie{5/chvisual.tex} \folie{5/cayleyhamilton.tex} diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 0858369..4ca3de4 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -5,8 +5,11 @@ # (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 \ diff --git a/vorlesungen/slides/5/beispiele/Makefile b/vorlesungen/slides/5/beispiele/Makefile index c03ea62..05bd5b5 100644 --- a/vorlesungen/slides/5/beispiele/Makefile +++ b/vorlesungen/slides/5/beispiele/Makefile @@ -3,7 +3,7 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: kern bild kb kombiniert.jpg +all: kern bild kb kombiniert.jpg leer.jpg drei.jpg kern: kern1.jpg kern2.jpg bild: bild1.jpg bild2.jpg @@ -19,6 +19,8 @@ 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$@ $< diff --git a/vorlesungen/slides/5/beispiele/bild1.jpg b/vorlesungen/slides/5/beispiele/bild1.jpg index e8bdc1c..879fae8 100644 Binary files a/vorlesungen/slides/5/beispiele/bild1.jpg and b/vorlesungen/slides/5/beispiele/bild1.jpg differ diff --git a/vorlesungen/slides/5/beispiele/bild2.jpg b/vorlesungen/slides/5/beispiele/bild2.jpg index 8e5dde5..2597c95 100644 Binary files a/vorlesungen/slides/5/beispiele/bild2.jpg and b/vorlesungen/slides/5/beispiele/bild2.jpg differ diff --git a/vorlesungen/slides/5/beispiele/drei.jpg b/vorlesungen/slides/5/beispiele/drei.jpg new file mode 100644 index 0000000..35f9034 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/drei.jpg differ 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 index f5fcf7d..5c99664 100644 Binary files a/vorlesungen/slides/5/beispiele/kern1.jpg and b/vorlesungen/slides/5/beispiele/kern1.jpg differ diff --git a/vorlesungen/slides/5/beispiele/kern2.jpg b/vorlesungen/slides/5/beispiele/kern2.jpg index c3b4b79..87d18ac 100644 Binary files a/vorlesungen/slides/5/beispiele/kern2.jpg and b/vorlesungen/slides/5/beispiele/kern2.jpg differ diff --git a/vorlesungen/slides/5/beispiele/kernbild.m b/vorlesungen/slides/5/beispiele/kernbild.m index ca6bd6a..28cd552 100644 --- a/vorlesungen/slides/5/beispiele/kernbild.m +++ b/vorlesungen/slides/5/beispiele/kernbild.m @@ -47,7 +47,8 @@ fprintf(f, "#declare j12 = < %.5f, %.5f, %.5f>;\n", E1(1,2), E1(2,2), E1(3,2)); fprintf(f, "\n"); # k = 2 -D = rref(E1 * E1); +E12 = E1 * E1 +D = rref(E12); K2 = [ -D(1,2), -D(1,3); 1, 0; @@ -60,7 +61,7 @@ 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", E2(1,1), E2(2,1), E2(3,1)); +fprintf(f, "#declare j21 = < %.5f, %.5f, %.5f>;\n", E12(1,1), E12(2,1), E12(3,1)); fprintf(f, "\n"); fclose(f); diff --git a/vorlesungen/slides/5/beispiele/kernbild1.jpg b/vorlesungen/slides/5/beispiele/kernbild1.jpg index bda5ce4..87e874e 100644 Binary files a/vorlesungen/slides/5/beispiele/kernbild1.jpg and b/vorlesungen/slides/5/beispiele/kernbild1.jpg differ diff --git a/vorlesungen/slides/5/beispiele/kernbild2.jpg b/vorlesungen/slides/5/beispiele/kernbild2.jpg index 4139918..1160b31 100644 Binary files a/vorlesungen/slides/5/beispiele/kernbild2.jpg and b/vorlesungen/slides/5/beispiele/kernbild2.jpg differ diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg index c214596..9cb789c 100644 Binary files a/vorlesungen/slides/5/beispiele/kombiniert.jpg and b/vorlesungen/slides/5/beispiele/kombiniert.jpg differ 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 new file mode 100644 index 0000000..9789887 Binary files /dev/null and b/vorlesungen/slides/5/beispiele/leer.jpg differ 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/chapter.tex b/vorlesungen/slides/5/chapter.tex index c19222c..96eea29 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,8 +3,11 @@ % % (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} 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/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/test.tex b/vorlesungen/slides/test.tex index 2c0746b..e4b9ad7 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,20 +3,15 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % +%\folie{5/verzerrung.tex} -% XXX Visualisierung von Kern und Bild -\folie{5/kernbildintro.tex} - -% XXX Visualisierung der Jordan-Normalform -% XXX \folie{5/eigen3d.tex} - % XXX Visualisierung Cayley-Hamilton-Produkte % XXX \folie{5/chvisual.tex} % XXX stone weierstrass incomplete -\folie{5/stoneweierstrass.tex} +%\folie{5/stoneweierstrass.tex} % XXX polynome auf dem spektrum % XXX Motiviation für *-Operation -\folie{5/normal.tex} +%\folie{5/normal.tex} -- cgit v1.2.1 From 68bcdf838c435a2315c3dcb9a18d78e2a68fe96f Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 22 Mar 2021 13:41:14 +0100 Subject: add title image --- vorlesungen/06_spektral2/Makefile | 4 +++- vorlesungen/06_spektral2/MathSem-06-spektral2.tex | 4 ++++ vorlesungen/06_spektral2/spektralradius.jpg | Bin 0 -> 91054 bytes 3 files changed, 7 insertions(+), 1 deletion(-) create mode 100644 vorlesungen/06_spektral2/spektralradius.jpg (limited to 'vorlesungen') diff --git a/vorlesungen/06_spektral2/Makefile b/vorlesungen/06_spektral2/Makefile index 7f92ec7..58534f0 100644 --- a/vorlesungen/06_spektral2/Makefile +++ b/vorlesungen/06_spektral2/Makefile @@ -26,8 +26,10 @@ thumbnail.jpg: thumbnail.pdf fix1.pdf: MathSem-06-spektral2.pdf pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ - MathSem-06-spektral2.pdf 1 + MathSem-06-spektral2.pdf 65 fix1.jpg: fix1.pdf convert -density 300 fix1.pdf \ -resize 1920x1080 -units PixelsPerInch fix1.jpg +spektralradius.jpg: fix1.jpg Makefile + convert -extract 660x620+300+400 fix1.jpg spektralradius.jpg diff --git a/vorlesungen/06_spektral2/MathSem-06-spektral2.tex b/vorlesungen/06_spektral2/MathSem-06-spektral2.tex index 196fcf0..b03d2d6 100644 --- a/vorlesungen/06_spektral2/MathSem-06-spektral2.tex +++ b/vorlesungen/06_spektral2/MathSem-06-spektral2.tex @@ -9,6 +9,10 @@ \begin{document} \begin{frame} \titlepage +\vspace{-1.7cm} +\begin{center} +\includegraphics[width=6.6cm]{spektralradius.jpg} +\end{center} \end{frame} \input{slides.tex} \end{document} diff --git a/vorlesungen/06_spektral2/spektralradius.jpg b/vorlesungen/06_spektral2/spektralradius.jpg new file mode 100644 index 0000000..4e952cd Binary files /dev/null and b/vorlesungen/06_spektral2/spektralradius.jpg differ -- cgit v1.2.1