diff options
Diffstat (limited to '')
42 files changed, 1041 insertions, 132 deletions
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 Binary files differnew file mode 100644 index 0000000..aeb713e --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.jpg diff --git a/vorlesungen/slides/2/images/quotient1.pov b/vorlesungen/slides/2/images/quotient1.pov new file mode 100644 index 0000000..60bab7f --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.pov @@ -0,0 +1,8 @@ +// +// quotient1.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "quotient.inc" + +vektorraum(-1) diff --git a/vorlesungen/slides/2/images/quotient1.tex b/vorlesungen/slides/2/images/quotient1.tex new file mode 100644 index 0000000..30d82d2 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.tex @@ -0,0 +1,29 @@ +% +% quotient1.tex -- Vektorraumquotient +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.7,0,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=8cm]{quotient1.jpg}}; + +\node[color=blue] at (0.7,-1.3) {$v$}; +\node[color=darkgreen] at (-1.0,0.1) {$w$}; +\node[color=orange] at (2.5,0.1) {$v+w$}; +\node[color=darkred] at (-2.1,-0.9) {$0$}; +\node[color=darkred] at (-3.1,2.4) {$U$}; + +\end{tikzpicture} +\end{document} + diff --git a/vorlesungen/slides/2/images/quotient2.jpg b/vorlesungen/slides/2/images/quotient2.jpg Binary files differnew file mode 100644 index 0000000..345cf22 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.jpg diff --git a/vorlesungen/slides/2/images/quotient2.pov b/vorlesungen/slides/2/images/quotient2.pov new file mode 100644 index 0000000..771425d --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.pov @@ -0,0 +1,8 @@ +// +// quotient2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "quotient.inc" + +vektorraum(1) diff --git a/vorlesungen/slides/2/images/quotient2.tex b/vorlesungen/slides/2/images/quotient2.tex new file mode 100644 index 0000000..607fd03 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.tex @@ -0,0 +1,29 @@ +% +% quotient2.tex -- Vektorraumquotient +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.7,0,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=8cm]{quotient2.jpg}}; + +\node[color=blue] at (0.57,-0.94) {$v$}; +\node[color=darkgreen] at (-1.15,0.65) {$w$}; +\node[color=orange] at (2.15,1) {$v+w$}; +\node[color=darkred] at (-2.1,-0.9) {$0$}; +\node[color=darkred] at (-3.1,2.4) {$U$}; + +\end{tikzpicture} +\end{document} + diff --git a/vorlesungen/slides/2/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/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 Binary files differnew file mode 100644 index 0000000..1c513da --- /dev/null +++ b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg diff --git a/vorlesungen/slides/8/tokyo/tokyosubway.pdf b/vorlesungen/slides/8/tokyo/tokyosubway.pdf Binary files differnew file mode 100644 index 0000000..6b84a8d --- /dev/null +++ b/vorlesungen/slides/8/tokyo/tokyosubway.pdf diff --git a/vorlesungen/slides/8/tokyo/transportnetworkgraph.png b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png Binary files differnew file mode 100644 index 0000000..4a11183 --- /dev/null +++ b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc new file mode 100644 index 0000000..fa6c29b --- /dev/null +++ b/vorlesungen/slides/9/Makefile.inc @@ -0,0 +1,14 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter9 = \ + ../slides/9/google.tex \ + ../slides/9/markov.tex \ + ../slides/9/irreduzibel.tex \ + ../slides/9/stationaer.tex \ + ../slides/9/pf.tex \ + ../slides/9/chapter.tex + diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex new file mode 100644 index 0000000..9e26587 --- /dev/null +++ b/vorlesungen/slides/9/chapter.tex @@ -0,0 +1,14 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% + + +\folie{9/google.tex} +\folie{9/markov.tex} +\folie{9/stationaer.tex} +\folie{9/irreduzibel.tex} +\folie{9/pf.tex} + + diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/9/google.tex index d1ec31d..d1ec31d 100644 --- a/vorlesungen/slides/8/markov/google.tex +++ b/vorlesungen/slides/9/google.tex diff --git a/vorlesungen/slides/8/markov/irreduzibel.tex b/vorlesungen/slides/9/irreduzibel.tex index 87e90e4..87e90e4 100644 --- a/vorlesungen/slides/8/markov/irreduzibel.tex +++ b/vorlesungen/slides/9/irreduzibel.tex diff --git a/vorlesungen/slides/8/markov/markov.tex b/vorlesungen/slides/9/markov.tex index e92ff0f..e92ff0f 100644 --- a/vorlesungen/slides/8/markov/markov.tex +++ b/vorlesungen/slides/9/markov.tex diff --git a/vorlesungen/slides/8/markov/pf.tex b/vorlesungen/slides/9/pf.tex index da2ef2b..da2ef2b 100644 --- a/vorlesungen/slides/8/markov/pf.tex +++ b/vorlesungen/slides/9/pf.tex diff --git a/vorlesungen/slides/8/markov/stationaer.tex b/vorlesungen/slides/9/stationaer.tex index 92fab16..92fab16 100644 --- a/vorlesungen/slides/8/markov/stationaer.tex +++ b/vorlesungen/slides/9/stationaer.tex 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} |
