diff options
Diffstat (limited to 'vorlesungen')
168 files changed, 12006 insertions, 3536 deletions
diff --git a/vorlesungen/06_msegalois/Makefile b/vorlesungen/06_msegalois/Makefile index 4cdc3d1..a6e829d 100644 --- a/vorlesungen/06_msegalois/Makefile +++ b/vorlesungen/06_msegalois/Makefile @@ -1,33 +1,33 @@ -#
-# Makefile -- galois
-#
-# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
-#
-all: galois-handout.pdf MathSemMSE-06-galois.pdf
-
-include ../slides/Makefile.inc
-
-SOURCES = common.tex slides.tex $(slides)
-
-MathSemMSE-06-galois.pdf: MathSemMSE-06-galois.tex $(SOURCES)
- pdflatex MathSemMSE-06-galois.tex
-
-galois-handout.pdf: galois-handout.tex $(SOURCES)
- pdflatex galois-handout.tex
-
-thumbnail: thumbnail.jpg # fix1.jpg
-
-thumbnail.pdf: MathSemMSE-06-galois.pdf
- pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
- MathSemMSE-06-galois.pdf 1
-thumbnail.jpg: thumbnail.pdf
- convert -density 300 thumbnail.pdf \
- -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
-
-fix1.pdf: MathSemMSE-06-galois.pdf
- pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
- MathSemMSE-06-galois.pdf 1
-fix1.jpg: fix1.pdf
- convert -density 300 fix1.pdf \
- -resize 1920x1080 -units PixelsPerInch fix1.jpg
-
+# +# Makefile -- galois +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: galois-handout.pdf MathSemMSE-06-galois.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-06-galois.pdf: MathSemMSE-06-galois.tex $(SOURCES) + pdflatex MathSemMSE-06-galois.tex + +galois-handout.pdf: galois-handout.tex $(SOURCES) + pdflatex galois-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-06-galois.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-06-galois.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-06-galois.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-06-galois.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex b/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex index 1f6b354..2b8da6d 100644 --- a/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex +++ b/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex @@ -1,14 +1,14 @@ -%
-% MathSem-06-msegalois.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}
+% +% MathSem-06-msegalois.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_msegalois/common.tex b/vorlesungen/06_msegalois/common.tex index f88a87b..50adc4f 100644 --- a/vorlesungen/06_msegalois/common.tex +++ b/vorlesungen/06_msegalois/common.tex @@ -1,16 +1,16 @@ -%
-% common.tex -- gemeinsame definition
-%
-% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\input{../common/packages.tex}
-\input{../common/common.tex}
-\mode<beamer>{%
-\usetheme[hideothersubsections,hidetitle]{Hannover}
-}
-\beamertemplatenavigationsymbolsempty
-\title[Galois]{Galois-Theorie}
-\author[A.~Müller]{Prof. Dr. Andreas Müller}
-\date[]{}
-\newboolean{presentation}
-
+% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Galois]{Galois-Theorie} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/06_msegalois/galois-handout.tex b/vorlesungen/06_msegalois/galois-handout.tex index 54238f6..e3e80f8 100644 --- a/vorlesungen/06_msegalois/galois-handout.tex +++ b/vorlesungen/06_msegalois/galois-handout.tex @@ -1,11 +1,11 @@ -%
-% msegalois-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}
+% +% msegalois-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_msegalois/slides.tex b/vorlesungen/06_msegalois/slides.tex index 386d19f..95695c4 100644 --- a/vorlesungen/06_msegalois/slides.tex +++ b/vorlesungen/06_msegalois/slides.tex @@ -1,23 +1,23 @@ -%
-% slides.tex -- Slides für die kleine Einführung in die Galois-Theorie
-%
-% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-
-\section{Körpererweiterungen}
-\folie{4/galois/erweiterung.tex}
-
-\section{Geometrische Anwendungen}
-\folie{4/galois/konstruktion.tex}
-\folie{4/galois/wuerfel.tex}
-\folie{4/galois/winkeldreiteilung.tex}
-\folie{4/galois/quadratur.tex}
-
-\section{Galois-Gruppe}
-\folie{4/galois/automorphismus.tex}
-
-\section{Lösbarkeit durch Radikale}
-\folie{4/galois/radikale.tex}
-\folie{4/galois/aufloesbarkeit.tex}
-\folie{4/galois/sn.tex}
-
+% +% slides.tex -- Slides für die kleine Einführung in die Galois-Theorie +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Körpererweiterungen} +\folie{4/galois/erweiterung.tex} + +\section{Geometrische Anwendungen} +\folie{4/galois/konstruktion.tex} +\folie{4/galois/wuerfel.tex} +\folie{4/galois/winkeldreiteilung.tex} +\folie{4/galois/quadratur.tex} + +\section{Galois-Gruppe} +\folie{4/galois/automorphismus.tex} + +\section{Lösbarkeit durch Radikale} +\folie{4/galois/radikale.tex} +\folie{4/galois/aufloesbarkeit.tex} +\folie{4/galois/sn.tex} + diff --git a/vorlesungen/07_lie/Makefile b/vorlesungen/07_lie/Makefile index 7e925d8..1788301 100644 --- a/vorlesungen/07_lie/Makefile +++ b/vorlesungen/07_lie/Makefile @@ -1,33 +1,33 @@ -#
-# Makefile -- lie
-#
-# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
-#
-all: lie-handout.pdf MathSem-07-lie.pdf
-
-include ../slides/Makefile.inc
-
-SOURCES = common.tex slides.tex $(slides)
-
-MathSem-07-lie.pdf: MathSem-07-lie.tex $(SOURCES)
- pdflatex MathSem-07-lie.tex
-
-lie-handout.pdf: lie-handout.tex $(SOURCES)
- pdflatex lie-handout.tex
-
-thumbnail: thumbnail.jpg fix1.jpg
-
-thumbnail.pdf: MathSem-07-lie.pdf
- pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \
- MathSem-07-lie.pdf 1
-thumbnail.jpg: thumbnail.pdf
- convert -density 300 thumbnail.pdf \
- -resize 1920x1080 -units PixelsPerInch thumbnail.jpg
-
-fix1.pdf: MathSem-07-lie.pdf
- pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \
- MathSem-07-lie.pdf 205
-fix1.jpg: fix1.pdf
- convert -density 300 fix1.pdf \
- -resize 1920x1080 -units PixelsPerInch fix1.jpg
-
+# +# Makefile -- lie +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: lie-handout.pdf MathSem-07-lie.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-07-lie.pdf: MathSem-07-lie.tex $(SOURCES) + pdflatex MathSem-07-lie.tex + +lie-handout.pdf: lie-handout.tex $(SOURCES) + pdflatex lie-handout.tex + +thumbnail: thumbnail.jpg fix1.jpg + +thumbnail.pdf: MathSem-07-lie.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-07-lie.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-07-lie.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-07-lie.pdf 205 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/07_lie/MathSem-07-lie.tex b/vorlesungen/07_lie/MathSem-07-lie.tex index 6cf5bd3..8a5557d 100644 --- a/vorlesungen/07_lie/MathSem-07-lie.tex +++ b/vorlesungen/07_lie/MathSem-07-lie.tex @@ -1,18 +1,18 @@ -%
-% MathSem-07-lie.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{-1.5cm}
-\begin{center}
-\includegraphics[width=10cm]{../slides/7/images/rodriguez.jpg}
-\end{center}
-\end{frame}
-\input{slides.tex}
-\end{document}
+% +% MathSem-07-lie.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{-1.5cm} +\begin{center} +\includegraphics[width=10cm]{../slides/7/images/rodriguez.jpg} +\end{center} +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/07_lie/common.tex b/vorlesungen/07_lie/common.tex index 12f0700..8472b93 100644 --- a/vorlesungen/07_lie/common.tex +++ b/vorlesungen/07_lie/common.tex @@ -1,16 +1,16 @@ -%
-% common.tex -- gemeinsame definition
-%
-% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\input{../common/packages.tex}
-\input{../common/common.tex}
-\mode<beamer>{%
-\usetheme[hideothersubsections,hidetitle]{Hannover}
-}
-\beamertemplatenavigationsymbolsempty
-\title[Lie]{Lie-Gruppen und Lie-Algebren}
-\author[A.~Müller]{Prof. Dr. Andreas Müller}
-\date[]{}
-\newboolean{presentation}
-
+% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Lie]{Lie-Gruppen und Lie-Algebren} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/07_lie/lie-handout.tex b/vorlesungen/07_lie/lie-handout.tex index 43053b8..dbdb386 100644 --- a/vorlesungen/07_lie/lie-handout.tex +++ b/vorlesungen/07_lie/lie-handout.tex @@ -1,11 +1,11 @@ -%
-% lie-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}
+% +% lie-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/07_lie/slides.tex b/vorlesungen/07_lie/slides.tex index 7efc554..19131d8 100644 --- a/vorlesungen/07_lie/slides.tex +++ b/vorlesungen/07_lie/slides.tex @@ -1,26 +1,26 @@ -%
-% slides.tex -- Vorlesung über Lie-Theorie
-%
-% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Matrizen-Gruppen}
-\folie{7/symmetrien.tex}
-\folie{7/algebraisch.tex}
-\folie{7/parameter.tex}
-\folie{7/mannigfaltigkeit.tex}
-\folie{7/sl2.tex}
-\folie{7/drehung.tex}
-\ifthenelse{\boolean{presentation}}{
-\folie{7/drehanim.tex}
-}{}
-\folie{7/semi.tex}
-
-\section{Ableitungen}
-\folie{7/kurven.tex}
-\folie{7/einparameter.tex}
-\folie{7/ableitung.tex}
-\folie{7/liealgebra.tex}
-\folie{7/kommutator.tex}
-
-\section{Exponentialabbildung}
-\folie{7/dg.tex}
+% +% slides.tex -- Vorlesung über Lie-Theorie +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Matrizen-Gruppen} +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\ifthenelse{\boolean{presentation}}{ +\folie{7/drehanim.tex} +}{} +\folie{7/semi.tex} + +\section{Ableitungen} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/kommutator.tex} + +\section{Exponentialabbildung} +\folie{7/dg.tex} diff --git a/vorlesungen/09_mseliegruppen/slides.tex b/vorlesungen/09_mseliegruppen/slides.tex index 1ae259f..f63a6a4 100644 --- a/vorlesungen/09_mseliegruppen/slides.tex +++ b/vorlesungen/09_mseliegruppen/slides.tex @@ -16,15 +16,14 @@ \folie{7/semi.tex} % Zusammenhangskomponenten +\section{Zusammenhang} +\folie{7/zusammenhang.tex} -% XXX Hopf-Faserung für SO(2) -> SU(2) -> SO(3) - -% curled up dimensions in String theory - +\section{Quaternionen} +\folie{7/quaternionen.tex} +\folie{7/qdreh.tex} +\folie{7/ueberlagerung.tex} +\folie{7/hopf.tex} \section{Haar-Mass} -% Definition Haar-Mass -% Mittelung -% -% Méndez-Transformation - +\folie{7/haar.tex} diff --git a/vorlesungen/10_mseliealgebra/slides.tex b/vorlesungen/10_mseliealgebra/slides.tex index 0fceaff..936139a 100644 --- a/vorlesungen/10_mseliealgebra/slides.tex +++ b/vorlesungen/10_mseliealgebra/slides.tex @@ -9,13 +9,27 @@ \folie{7/einparameter.tex} \folie{7/ableitung.tex} \folie{7/liealgebra.tex} +% XXX Beispiele von Lie-Algebren +\folie{7/liealgbeispiel.tex} +% XXX Vektorprodukt als Lie-Algebra +\folie{7/vektorlie.tex} \folie{7/kommutator.tex} +% XXX kommutator rechnerisch +\folie{7/bch.tex} \section{Exponentialabbildung} \folie{7/dg.tex} +\folie{7/logarithmus.tex} +% Interpolation +\folie{7/interpolation.tex} + +\section{Integration} +\folie{7/haar.tex} +\ifthenelse{\boolean{presentation}}{ +\folie{7/mannigfaltigkeit.tex} +}{} +\folie{7/integration.tex} % LOG Reihe -% Interpolation % Mittelung auf einer Lie-Gruppe -% Vektorprodukt als Lie-Gruppe diff --git a/vorlesungen/11_msegraphen/Makefile b/vorlesungen/11_msegraphen/Makefile new file mode 100644 index 0000000..36f1877 --- /dev/null +++ b/vorlesungen/11_msegraphen/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- graphen +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: graphen-handout.pdf MathSemMSE-11-graphen.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-11-graphen.pdf: MathSemMSE-11-graphen.tex $(SOURCES) + pdflatex MathSemMSE-11-graphen.tex + +graphen-handout.pdf: graphen-handout.tex $(SOURCES) + pdflatex graphen-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-11-graphen.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-11-graphen.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-11-graphen.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-11-graphen.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex b/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex new file mode 100644 index 0000000..6d4ff2d --- /dev/null +++ b/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex @@ -0,0 +1,14 @@ +% +% MathSem-11-msegraphen.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/11_msegraphen/common.tex b/vorlesungen/11_msegraphen/common.tex new file mode 100644 index 0000000..67cf5e5 --- /dev/null +++ b/vorlesungen/11_msegraphen/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<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Graphen]{Spektrale Graphentheorie} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/11_msegraphen/graphen-handout.tex b/vorlesungen/11_msegraphen/graphen-handout.tex new file mode 100644 index 0000000..58d0aff --- /dev/null +++ b/vorlesungen/11_msegraphen/graphen-handout.tex @@ -0,0 +1,11 @@ +% +% msegraphen-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/11_msegraphen/slides.tex b/vorlesungen/11_msegraphen/slides.tex new file mode 100644 index 0000000..b3d1519 --- /dev/null +++ b/vorlesungen/11_msegraphen/slides.tex @@ -0,0 +1,35 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\folie{8/graph.tex} +\folie{8/dgraph.tex} +\folie{8/grad.tex} +\folie{8/inzidenz.tex} +\folie{8/inzidenzd.tex} +\folie{8/diffusion.tex} +\folie{8/laplace.tex} +\folie{8/produkt.tex} +\folie{8/fourier.tex} +\folie{8/spanningtree.tex} + +\folie{8/pfade/adjazenz.tex} +\folie{8/pfade/langepfade.tex} +\folie{8/pfade/beispiel.tex} +\folie{8/pfade/gf.tex} + +%\folie{8/floyd-warshall/problem.tex} +%\folie{8/floyd-warshall/rekursion.tex} +%\folie{8/floyd-warshall/iteration.tex} +%\folie{8/floyd-warshall/wegiteration.tex} +%\folie{8/floyd-warshall/wege.tex} + +\folie{8/chrind.tex} +\folie{8/chrindprop.tex} +\folie{8/chroma1.tex} +\folie{8/amax.tex} +\folie{8/subgraph.tex} +\folie{8/chrwilf.tex} +\folie{8/weitere.tex} diff --git a/vorlesungen/12_msewkeitsmatrizen/Makefile b/vorlesungen/12_msewkeitsmatrizen/Makefile new file mode 100644 index 0000000..ec420cd --- /dev/null +++ b/vorlesungen/12_msewkeitsmatrizen/Makefile @@ -0,0 +1,42 @@ +# +# Makefile -- wkeitsmatrizen +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: wkeitsmatrizen-handout.pdf MathSemMSE-12-wkeitsmatrizen.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-12-wkeitsmatrizen.pdf: MathSemMSE-12-wkeitsmatrizen.tex $(SOURCES) + pdflatex MathSemMSE-12-wkeitsmatrizen.tex + +wkeitsmatrizen-handout.pdf: wkeitsmatrizen-handout.tex $(SOURCES) + pdflatex wkeitsmatrizen-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-12-wkeitsmatrizen.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-12-wkeitsmatrizen.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-12-wkeitsmatrizen.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-12-wkeitsmatrizen.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + +parts: part1.pdf part2.pdf + +part1.pdf: MathSemMSE-12-wkeitsmatrizen.pdf + pdfjam --outfile part1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-12-wkeitsmatrizen.pdf 1-160 + +part2.pdf: MathSemMSE-12-wkeitsmatrizen.pdf + pdfjam --outfile part2.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-12-wkeitsmatrizen.pdf 161-211 diff --git a/vorlesungen/12_msewkeitsmatrizen/MathSemMSE-12-wkeitsmatrizen.tex b/vorlesungen/12_msewkeitsmatrizen/MathSemMSE-12-wkeitsmatrizen.tex new file mode 100644 index 0000000..e526731 --- /dev/null +++ b/vorlesungen/12_msewkeitsmatrizen/MathSemMSE-12-wkeitsmatrizen.tex @@ -0,0 +1,14 @@ +% +% MathSem-12-msewkeitsmatrizen.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/12_msewkeitsmatrizen/common.tex b/vorlesungen/12_msewkeitsmatrizen/common.tex new file mode 100644 index 0000000..2de59e8 --- /dev/null +++ b/vorlesungen/12_msewkeitsmatrizen/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<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[W'keitsmatrizen]{Wahrscheinlichkeitsmatrizen} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/12_msewkeitsmatrizen/slides.tex b/vorlesungen/12_msewkeitsmatrizen/slides.tex new file mode 100644 index 0000000..e0f8e9c --- /dev/null +++ b/vorlesungen/12_msewkeitsmatrizen/slides.tex @@ -0,0 +1,35 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Google} +\folie{9/google.tex} + + + +\section{Markov-Ketten} +\folie{9/markov.tex} +\folie{9/stationaer.tex} +\folie{9/irreduzibel.tex} + +\section{Perron-Frobenius} +\folie{9/pf/positiv.tex} +\folie{9/pf/primitiv.tex} +\folie{9/pf/trennung.tex} +\folie{9/pf/vergleich.tex} +\folie{9/pf/vergleich3d.tex} +\folie{9/pf/dreieck.tex} +\folie{9/pf/folgerungen.tex} +\folie{9/pf.tex} +\folie{9/potenz.tex} + +\section{Parrondo} +\folie{9/parrondo/uebersicht.tex} +\folie{9/parrondo/erwartung.tex} +\folie{9/parrondo/spiela.tex} +\folie{9/parrondo/spielb.tex} +\folie{9/parrondo/spielbmod.tex} +\folie{9/parrondo/kombiniert.tex} +\folie{9/parrondo/deformation.tex} + diff --git a/vorlesungen/12_msewkeitsmatrizen/wkeitsmatrizen-handout.tex b/vorlesungen/12_msewkeitsmatrizen/wkeitsmatrizen-handout.tex new file mode 100644 index 0000000..ae577c3 --- /dev/null +++ b/vorlesungen/12_msewkeitsmatrizen/wkeitsmatrizen-handout.tex @@ -0,0 +1,11 @@ +% +% msewkeitsmatrizen-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/13_msegraphwavelets/Makefile b/vorlesungen/13_msegraphwavelets/Makefile new file mode 100644 index 0000000..6dba66c --- /dev/null +++ b/vorlesungen/13_msegraphwavelets/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- graphwavelets +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: graphwavelets-handout.pdf MathSemMSE-13-graphwavelets.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-13-graphwavelets.pdf: MathSemMSE-13-graphwavelets.tex $(SOURCES) + pdflatex MathSemMSE-13-graphwavelets.tex + +graphwavelets-handout.pdf: graphwavelets-handout.tex $(SOURCES) + pdflatex graphwavelets-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-13-graphwavelets.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-13-graphwavelets.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-13-graphwavelets.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-13-graphwavelets.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/13_msegraphwavelets/MathSemMSE-13-graphwavelets.tex b/vorlesungen/13_msegraphwavelets/MathSemMSE-13-graphwavelets.tex new file mode 100644 index 0000000..112d952 --- /dev/null +++ b/vorlesungen/13_msegraphwavelets/MathSemMSE-13-graphwavelets.tex @@ -0,0 +1,14 @@ +% +% MathSem-13-msegraphwavelets.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/13_msegraphwavelets/common.tex b/vorlesungen/13_msegraphwavelets/common.tex new file mode 100644 index 0000000..b9799f0 --- /dev/null +++ b/vorlesungen/13_msegraphwavelets/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<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[SGWT]{Wavelets auf Graphen} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/13_msegraphwavelets/graphwavelets-handout.tex b/vorlesungen/13_msegraphwavelets/graphwavelets-handout.tex new file mode 100644 index 0000000..98789e5 --- /dev/null +++ b/vorlesungen/13_msegraphwavelets/graphwavelets-handout.tex @@ -0,0 +1,11 @@ +% +% msegraphwavelets-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/13_msegraphwavelets/slides.tex b/vorlesungen/13_msegraphwavelets/slides.tex new file mode 100644 index 0000000..3fd38f2 --- /dev/null +++ b/vorlesungen/13_msegraphwavelets/slides.tex @@ -0,0 +1,44 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +% Funktionen auf einem Graphen +\folie{8/wavelets/funktionen.tex} + +% Laplace-Basis auf dem Graphen +\folie{8/wavelets/laplacebasis.tex} + +% Fourier-Transformation auf einem Graphen +\folie{8/wavelets/fourier.tex} + +% Lokalisierung in Standardbasis und Fourier-Basis +\folie{8/wavelets/lokalisierungsvergleich.tex} + +% Lokalisierung im Frequenzraum +\folie{8/wavelets/frequenzlokalisierung.tex} + +% Dilatation im Frequenzraum +\folie{8/wavelets/dilatation.tex} + +% Dilatation in Matrixform +\folie{8/wavelets/matrixdilatation.tex} + +% Funktionen g und h +\folie{8/wavelets/gundh.tex} +\ifthenelse{\boolean{presentation}}{ +\folie{8/wavelets/dilbei.tex} +}{} + +% Wavelet Frame +\folie{8/wavelets/frame.tex} + +% Framekonstante +\folie{8/wavelets/framekonstanten.tex} + +% Kugel-Beispiel +\ifthenelse{\boolean{presentation}}{ +\folie{8/wavelets/beispiel.tex} +}{} + diff --git a/vorlesungen/14_msehilbertraum/Makefile b/vorlesungen/14_msehilbertraum/Makefile new file mode 100644 index 0000000..e5de69c --- /dev/null +++ b/vorlesungen/14_msehilbertraum/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- hilbertraum +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: hilbertraum-handout.pdf MathSemMSE-14-hilbertraum.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-14-hilbertraum.pdf: MathSemMSE-14-hilbertraum.tex $(SOURCES) + pdflatex MathSemMSE-14-hilbertraum.tex + +hilbertraum-handout.pdf: hilbertraum-handout.tex $(SOURCES) + pdflatex hilbertraum-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-14-hilbertraum.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-14-hilbertraum.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-14-hilbertraum.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-14-hilbertraum.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/14_msehilbertraum/MathSemMSE-14-hilbertraum.tex b/vorlesungen/14_msehilbertraum/MathSemMSE-14-hilbertraum.tex new file mode 100644 index 0000000..b06500c --- /dev/null +++ b/vorlesungen/14_msehilbertraum/MathSemMSE-14-hilbertraum.tex @@ -0,0 +1,14 @@ +% +% MathSem-14-msehilbertraum.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/14_msehilbertraum/common.tex b/vorlesungen/14_msehilbertraum/common.tex new file mode 100644 index 0000000..a9089bf --- /dev/null +++ b/vorlesungen/14_msehilbertraum/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<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Hilbertraum]{Hilbertraum} +\author[A.~Müller]{Prof.~Dr.~Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/14_msehilbertraum/hilbertraum-handout.tex b/vorlesungen/14_msehilbertraum/hilbertraum-handout.tex new file mode 100644 index 0000000..3dc7abf --- /dev/null +++ b/vorlesungen/14_msehilbertraum/hilbertraum-handout.tex @@ -0,0 +1,11 @@ +% +% msehilbertraum-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/14_msehilbertraum/slides.tex b/vorlesungen/14_msehilbertraum/slides.tex new file mode 100644 index 0000000..19925db --- /dev/null +++ b/vorlesungen/14_msehilbertraum/slides.tex @@ -0,0 +1,31 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Hilbertraum} +\folie{2/hilbertraum/definition.tex} +\folie{2/hilbertraum/l2beispiel.tex} +\folie{2/hilbertraum/basis.tex} +\folie{2/hilbertraum/plancherel.tex} + +\section{Beispiele} +\folie{2/hilbertraum/l2.tex} + +\section{Riesz-Darstellungssatz} +\folie{2/hilbertraum/riesz.tex} +\folie{2/hilbertraum/rieszbeispiel.tex} + +\section{$A^*$} +\folie{2/hilbertraum/adjungiert.tex} +\folie{2/hilbertraum/spektral.tex} + +\section{PDE und Hilbertraum} +\folie{2/hilbertraum/sturm.tex} +\folie{2/hilbertraum/laplace.tex} +\folie{2/hilbertraum/qm.tex} +\folie{2/hilbertraum/energie.tex} +\folie{2/hilbertraum/sobolev.tex} + + diff --git a/vorlesungen/99_vortraege/Makefile b/vorlesungen/99_vortraege/Makefile index 8a5751c..69ec665 100644 --- a/vorlesungen/99_vortraege/Makefile +++ b/vorlesungen/99_vortraege/Makefile @@ -15,7 +15,8 @@ MathSem-99-vortraege.pdf: MathSem-99-vortraege.tex $(SOURCES) vortraege-handout.pdf: vortraege-handout.tex $(SOURCES) pdflatex vortraege-handout.tex -thumbnail: thumbnail1.jpg thumbnail2.jpg thumbnail3.jpg thumbnail4.jpg +thumbnail: thumbnail1.jpg thumbnail2.jpg thumbnail3.jpg thumbnail4.jpg \ + thumbnail5.jpg thumbnail1.pdf: MathSem-99-vortraege.pdf pdfjam --outfile thumbnail1.pdf --papersize '{16cm,9cm}' \ @@ -45,6 +46,13 @@ thumbnail4.jpg: thumbnail4.pdf convert -density 300 thumbnail4.pdf \ -resize 1920x1080 -units PixelsPerInch thumbnail4.jpg +thumbnail5.pdf: MathSem-99-vortraege.pdf + pdfjam --outfile thumbnail5.pdf --papersize '{16cm,9cm}' \ + MathSem-99-vortraege.pdf 5 +thumbnail5.jpg: thumbnail5.pdf + convert -density 300 thumbnail5.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail5.jpg + fix1.pdf: MathSem-99-vortraege.pdf pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ MathSem-99-vortraege.pdf 1 diff --git a/vorlesungen/99_vortraege/MathSem-99-vortraege.tex b/vorlesungen/99_vortraege/MathSem-99-vortraege.tex index c962e90..ddcfcf8 100644 --- a/vorlesungen/99_vortraege/MathSem-99-vortraege.tex +++ b/vorlesungen/99_vortraege/MathSem-99-vortraege.tex @@ -61,5 +61,18 @@ Fabio Viecelli, Lukas Zogg: Erdbebenmessung \end{center} \end{frame} +\title[Vorträge]{31. Mai: Vorträge} +\begin{frame} +\titlepage +\vspace{-2cm} +\begin{center} +Marc Kühne: Munkres-Algorithmus +\phantom{blubb} \\ +\phantom{blubb} \\ +\phantom{blubb} \\ +Michael Schmid: Schnelle Matrixmultiplikation +\end{center} +\end{frame} + \input{slides.tex} \end{document} diff --git a/vorlesungen/common/README b/vorlesungen/common/README index 3edcf14..1ed40aa 100644 --- a/vorlesungen/common/README +++ b/vorlesungen/common/README @@ -1,28 +1,28 @@ -Die beiden Files
-
- presentation-template.tex
- slide-template.tex
-
-können als Basis für die eigene Präsentation verwendet werden.
-Dazu geht man wie folgt vor:
-
-1. In einem Arbeitsverzeichnis eine Kopie von presentation-template.tex
-anlegen und im file Author und Titel anpassen. Im Folgenden wird diese
-Kopie als beispiel-praesentation.tex bezeichnet.
-
-2. Für jede Folie der Präsentation im Arbeitsverzeichnis eine Kopie von
-slide-template.tex anlegen und den Inhalt anpassen.
-
-3. Die Slides mit Hilfe von Input-Befehlen, die in presentation-template.tex
-eingetragen werden, in die Präsentation importieren.
-
-4. Die Präsentation mit dem Befehl
-
- pdflatex beispiel-praesentation.tex
-
-erzeugen, es entsteht das File beispile-praesentation.pdf
-
-Diese Vorgehen erlaubt, die Reihenfolge der Folien während der Vorbereitung
-zu ändern oder zwecks Beschleunigung des pdflatex-Laufs während der
-Entwicklung auszukommentieren.
-
+Die beiden Files + + presentation-template.tex + slide-template.tex + +können als Basis für die eigene Präsentation verwendet werden. +Dazu geht man wie folgt vor: + +1. In einem Arbeitsverzeichnis eine Kopie von presentation-template.tex +anlegen und im file Author und Titel anpassen. Im Folgenden wird diese +Kopie als beispiel-praesentation.tex bezeichnet. + +2. Für jede Folie der Präsentation im Arbeitsverzeichnis eine Kopie von +slide-template.tex anlegen und den Inhalt anpassen. + +3. Die Slides mit Hilfe von Input-Befehlen, die in presentation-template.tex +eingetragen werden, in die Präsentation importieren. + +4. Die Präsentation mit dem Befehl + + pdflatex beispiel-praesentation.tex + +erzeugen, es entsteht das File beispile-praesentation.pdf + +Diese Vorgehen erlaubt, die Reihenfolge der Folien während der Vorbereitung +zu ändern oder zwecks Beschleunigung des pdflatex-Laufs während der +Entwicklung auszukommentieren. + diff --git a/vorlesungen/common/presentation-template.tex b/vorlesungen/common/presentation-template.tex index c872c58..9f92489 100644 --- a/vorlesungen/common/presentation-template.tex +++ b/vorlesungen/common/presentation-template.tex @@ -1,49 +1,49 @@ -%
-% presentation-template.tex -- Präsentation
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\documentclass[aspectratio=169]{beamer}
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{epic}
-\usepackage{color}
-\usepackage{array}
-\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}
-\usetikzlibrary{calc}
-\usetikzlibrary{arrows}
-\usetikzlibrary{3d}
-\usetikzlibrary{arrows,shapes,math,decorations.text,automata}
-\usepackage{pifont}
-\usepackage[all]{xy}
-\usepackage[many]{tcolorbox}
-\mode<beamer>{%
-\usetheme[hideothersubsections,hidetitle]{Hannover}
-}
-\beamertemplatenavigationsymbolsempty
-\title[Titel]{Titel}
-\author[A. Uthor]{A. Uthor}
-\date[]{}
-\newboolean{presentation}
-\setboolean{presentation}{true}
-\begin{document}
-
-\begin{frame}
-\titlepage
-\end{frame}
-
-%\input{slide.tex}
-
-\end{document}
+% +% presentation-template.tex -- Präsentation +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\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} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Titel]{Titel} +\author[A. Uthor]{A. Uthor} +\date[]{} +\newboolean{presentation} +\setboolean{presentation}{true} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\input{slide.tex} + +\end{document} diff --git a/vorlesungen/common/slide-template.tex b/vorlesungen/common/slide-template.tex index 2dd4db1..a1343f8 100644 --- a/vorlesungen/common/slide-template.tex +++ b/vorlesungen/common/slide-template.tex @@ -1,19 +1,19 @@ -%
-% template.tex -- slide template
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Template}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\end{column}
-\begin{column}{0.48\textwidth}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Template} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\end{column} +\begin{column}{0.48\textwidth} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/punktgruppen/.gitignore b/vorlesungen/punktgruppen/.gitignore new file mode 100644 index 0000000..3633a3d --- /dev/null +++ b/vorlesungen/punktgruppen/.gitignore @@ -0,0 +1,20 @@ +# directories +__pycache__ +media/Tex + +media/images/crystal +media/images/freezeframes + +media/videos +media/audio + +media/Punktgruppen +media/Punktgruppen.mp4 + +build + +# files +script.log +slides.log +slides.vrb +missfont.log diff --git a/vorlesungen/punktgruppen/Makefile b/vorlesungen/punktgruppen/Makefile new file mode 100644 index 0000000..302e976 --- /dev/null +++ b/vorlesungen/punktgruppen/Makefile @@ -0,0 +1,18 @@ +TEX=xelatex +TEXARGS=--output-directory=build --halt-on-error --shell-escape + +all: slides.pdf script.pdf media + +.PHONY: clean +clean: + @rm -rfv build + +%.pdf: %.tex + mkdir -p build + $(TEX) $(TEXARGS) $< + $(TEX) $(TEXARGS) $< + cp build/$@ . + +media: + poetry install + poetry run manim -ql crystals.py diff --git a/vorlesungen/punktgruppen/crystals.py b/vorlesungen/punktgruppen/crystals.py new file mode 100644 index 0000000..4a9836a --- /dev/null +++ b/vorlesungen/punktgruppen/crystals.py @@ -0,0 +1,611 @@ +from manim import * + +import math as m +import numpy as np +import itertools as it + +# configure style +config.background_color = '#202020' +config.tex_template.add_to_preamble( + r"\usepackage[p,osf]{scholax}" + r"\usepackage{amsmath}" + r"\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath}" +) + +# scenes +class Geometric2DSymmetries(Scene): + def construct(self): + self.wait(5) + + self.intro() + self.cyclic() + self.dihedral() + self.circle() + + def intro(self): + # create square + square = Square() + square.set_fill(PINK, opacity=.5) + self.play(SpinInFromNothing(square)) + self.wait() + + # the action of doing nothing + action = MathTex(r"\mathbb{1}") + self.play(Write(action)) + self.play(ApplyMethod(square.scale, 1.2)) + self.play(ApplyMethod(square.scale, 1/1.2)) + self.play(FadeOut(action)) + self.wait() + + # show some reflections + axis = DashedLine(2 * LEFT, 2 * RIGHT) + sigma = MathTex(r"\sigma") + sigma.next_to(axis, RIGHT) + + self.play(Create(axis)) + self.play(Write(sigma)) + self.play(ApplyMethod(square.flip, RIGHT)) + + for d in [UP + RIGHT, UP]: + self.play( + Rotate(axis, PI/4), + Rotate(sigma, PI/4, about_point=ORIGIN)) + + self.play(Rotate(sigma, -PI/4), run_time=.5) + self.play(ApplyMethod(square.flip, d)) + + self.play(FadeOutAndShift(sigma), Uncreate(axis)) + + # show some rotations + dot = Dot(UP + RIGHT) + figure = VGroup(square, dot) + + rot = MathTex(r"r") + self.play(Write(rot), Create(dot)) + + last = rot + for newrot in map(MathTex, [r"r", r"r^2", r"r^3"]): + self.play( + ReplacementTransform(last, newrot), + Rotate(figure, PI/2, about_point=ORIGIN)) + self.wait(.5) + last = newrot + + self.play(Uncreate(dot), FadeOut(square), FadeOut(last)) + + + def cyclic(self): + # create symmetric figure + figure = VGroup() + prev = [1.5, 0, 0] + for i in range(1,6): + pos = [ + 1.5*m.cos(2 * PI/5 * i), + 1.5*m.sin(2 * PI/5 * i), + 0 + ] + + if prev: + line = Line(prev, pos) + figure.add(line) + + dot = Dot(pos, radius=.1) + if i == 5: + dot.set_fill(RED) + + prev = pos + figure.add(dot) + + group = MathTex(r"G = \langle r \rangle") + self.play(Write(group), run_time = 2) + self.wait(3) + + self.play(ApplyMethod(group.to_edge, UP)) + + actions = map(MathTex, [ + r"\mathbb{1}", r"r", r"r^2", + r"r^3", r"r^4", r"\mathbb{1}", r"r"]) + + action = next(actions, MathTex(r"r")) + + self.play(Create(figure)) + self.play(Write(action)) + self.wait() + + for i in range(5): + newaction = next(actions, MathTex(r"r")) + self.play( + ReplacementTransform(action, newaction), + Rotate(figure, 2*PI/5, about_point=ORIGIN)) + action = newaction + + self.wait() + newaction = next(actions, MathTex(r"r")) + self.play( + ReplacementTransform(action, newaction), + Rotate(figure, 2*PI/5, about_point=ORIGIN)) + action = newaction + self.wait(2) + + self.play(Uncreate(figure), FadeOut(action)) + + whole_group = MathTex( + r"G = \langle r \rangle" + r"= \left\{\mathbb{1}, r, r^2, r^3, r^4 \right\}") + + self.play(ApplyMethod(group.move_to, ORIGIN)) + self.play(ReplacementTransform(group, whole_group)) + self.wait(5) + + cyclic = MathTex( + r"C_n = \langle r \rangle" + r"= \left\{\mathbb{1}, r, r^2, \dots, r^{n-1} \right\}") + + cyclic_title = Tex(r"Zyklische Gruppe") + cyclic_title.next_to(cyclic, UP * 2) + + cyclic.scale(1.2) + cyclic_title.scale(1.2) + + self.play(ReplacementTransform(whole_group, cyclic)) + self.play(FadeInFrom(cyclic_title, UP)) + + self.wait(5) + self.play(FadeOut(cyclic), FadeOut(cyclic_title)) + + def dihedral(self): + # create square + square = Square() + square.set_fill(PINK, opacity=.5) + + # generator equation + group = MathTex( + r"G = \langle \sigma, r \,|\,", + r"\sigma^2 = \mathbb{1},", + r"r^4 = \mathbb{1},", + r"(\sigma r)^2 = \mathbb{1} \rangle") + + self.play(Write(group), run_time = 2) + self.wait(5) + + self.play(ApplyMethod(group.to_edge, UP)) + self.play(FadeIn(square)) + self.wait() + + # flips + axis = DashedLine(2 * LEFT, 2 * RIGHT) + sigma = MathTex(r"\sigma^2 = \mathbb{1}") + sigma.next_to(axis, RIGHT) + self.play(Create(axis), Write(sigma)) + self.play(ApplyMethod(square.flip, RIGHT)) + self.play(ApplyMethod(square.flip, RIGHT)) + self.play(Uncreate(axis), FadeOut(sigma)) + + # rotations + dot = Dot(UP + RIGHT) + rot = MathTex(r"r^4 = \mathbb{1}") + rot.next_to(square, DOWN * 3) + + figure = VGroup(dot, square) + + self.play(Write(rot), Create(dot)) + for i in range(4): + self.play(Rotate(figure, PI/2)) + self.play(FadeOut(rot), Uncreate(dot)) + + # rotation and flip + action = MathTex(r"(\sigma r)^2 = \mathbb{1}") + action.next_to(square, DOWN * 5) + + dot = Dot(UP + RIGHT) + axis = DashedLine(2 * LEFT, 2 * RIGHT) + self.play(Create(dot), Create(axis), Write(action)) + + figure = VGroup(dot, square) + + for i in range(2): + self.play(Rotate(figure, PI/2)) + self.play(ApplyMethod(figure.flip, RIGHT)) + self.wait() + + self.play(Uncreate(dot), Uncreate(axis), FadeOut(action)) + self.play(FadeOut(square)) + + # equation for the whole + whole_group = MathTex( + r"G &= \langle \sigma, r \,|\," + r"\sigma^2 = r^4 = (\sigma r)^2 = \mathbb{1} \rangle \\" + r"&= \left\{" + r"\mathbb{1}, r, r^2, r^3, \sigma, \sigma r, \sigma r^2, \sigma r^3" + r"\right\}") + + self.play(ApplyMethod(group.move_to, ORIGIN)) + self.play(ReplacementTransform(group, whole_group)) + self.wait(2) + + dihedral = MathTex( + r"D_n &= \langle \sigma, r \,|\," + r"\sigma^2 = r^n = (\sigma r)^2 = \mathbb{1} \rangle \\" + r"&= \left\{" + r"\mathbb{1}, r, r^2, \dots, \sigma, \sigma r, \sigma r^2, \dots" + r"\right\}") + + dihedral_title = Tex(r"Diedergruppe: Symmetrien eines \(n\)-gons") + dihedral_title.next_to(dihedral, UP * 2) + + dihedral.scale(1.2) + dihedral_title.scale(1.2) + + self.play(ReplacementTransform(whole_group, dihedral)) + self.play(FadeInFrom(dihedral_title, UP)) + + self.wait(5) + self.play(FadeOut(dihedral), FadeOut(dihedral_title)) + + def circle(self): + circle = Circle(radius=2) + dot = Dot() + dot.move_to(2 * RIGHT) + + figure = VGroup(circle, dot) + group_name = MathTex(r"C_\infty") + + # create circle + self.play(Create(circle)) + self.play(Create(dot)) + + # move it around + self.play(Rotate(figure, PI/3)) + self.play(Rotate(figure, PI/6)) + self.play(Rotate(figure, -PI/3)) + + # show name + self.play(Rotate(figure, PI/4), Write(group_name)) + self.wait() + self.play(Uncreate(figure)) + + nsphere = MathTex(r"C_\infty \cong S^1 = \left\{z \in \mathbb{C} : |z| = 1\right\}") + nsphere_title = Tex(r"Kreisgruppe") + nsphere_title.next_to(nsphere, 2 * UP) + + nsphere.scale(1.2) + nsphere_title.scale(1.2) + + self.play(ReplacementTransform(group_name, nsphere)) + self.play(FadeInFrom(nsphere_title, UP)) + + self.wait(5) + self.play(FadeOut(nsphere_title), FadeOut(nsphere)) + self.wait(2) + + +class Geometric3DSymmetries(ThreeDScene): + def construct(self): + self.improper_rotation() + self.tetrahedron() + + def improper_rotation(self): + # changes the source of the light and camera + self.renderer.camera.light_source.move_to(3*IN) + self.set_camera_orientation(phi=0, theta=0) + + # initial square + square = Square() + square.set_fill(PINK, opacity=.5) + + self.play(SpinInFromNothing(square)) + self.wait(2) + + for i in range(4): + self.play(Rotate(square, PI/2)) + self.wait(.5) + + self.move_camera(phi= 75 * DEGREES, theta = -80 * DEGREES) + + # create rotation axis + axis = Line3D(start=[0,0,-2.5], end=[0,0,2.5]) + + axis_name = MathTex(r"r \in C_4") + # move to yz plane + axis_name.rotate(PI/2, axis = RIGHT) + axis_name.next_to(axis, OUT) + + self.play(Create(axis)) + self.play(Write(axis_name)) + self.wait() + + # create sphere from slices + cyclic_slices = [] + for i in range(4): + colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E] + cyclic_slices.append(ParametricSurface( + lambda u, v: np.array([ + np.sqrt(2) * np.cos(u) * np.cos(v), + np.sqrt(2) * np.cos(u) * np.sin(v), + np.sqrt(2) * np.sin(u) + ]), + v_min=PI/4 + PI/2 * i, + v_max=PI/4 + PI/2 * (i + 1), + u_min=-PI/2, u_max=PI/2, + checkerboard_colors=colors, resolution=(10,5))) + + self.play(FadeOut(square), *map(Create, cyclic_slices)) + + cyclic_sphere = VGroup(*cyclic_slices) + for i in range(4): + self.play(Rotate(cyclic_sphere, PI/2)) + self.wait() + + new_axis_name = MathTex(r"r \in D_4") + # move to yz plane + new_axis_name.rotate(PI/2, axis = RIGHT) + new_axis_name.next_to(axis, OUT) + self.play(ReplacementTransform(axis_name, new_axis_name)) + + # reflection plane + self.play(FadeOut(cyclic_sphere), FadeIn(square)) + plane = ParametricSurface( + lambda u, v: np.array([u, 0, v]), + u_min = -2, u_max = 2, + v_min = -2, v_max = 2, + fill_opacity=.3, resolution=(1,1)) + + plane_name = MathTex(r"\sigma \in D_4") + # move to yz plane + plane_name.rotate(PI/2, axis = RIGHT) + plane_name.next_to(plane, OUT + RIGHT) + + self.play(Create(plane)) + self.play(Write(plane_name)) + self.wait() + + self.move_camera(phi = 25 * DEGREES, theta = -75 * DEGREES) + self.wait() + + condition = MathTex(r"(\sigma r)^2 = \mathbb{1}") + condition.next_to(square, DOWN); + + self.play(Write(condition)) + self.play(Rotate(square, PI/2)) + self.play(Rotate(square, PI, RIGHT)) + + self.play(Rotate(square, PI/2)) + self.play(Rotate(square, PI, RIGHT)) + self.play(FadeOut(condition)) + + self.move_camera(phi = 75 * DEGREES, theta = -80 * DEGREES) + + # create sphere from slices + dihedral_slices = [] + for i in range(4): + for j in range(2): + colors = [PINK, RED] if i % 2 == 0 else [BLUE_D, BLUE_E] + dihedral_slices.append(ParametricSurface( + lambda u, v: np.array([ + np.sqrt(2) * np.cos(u) * np.cos(v), + np.sqrt(2) * np.cos(u) * np.sin(v), + np.sqrt(2) * np.sin(u) + ]), + v_min=PI/2 * j + PI/4 + PI/2 * i, + v_max=PI/2 * j + PI/4 + PI/2 * (i + 1), + u_min=-PI/2 if j == 0 else 0, + u_max=0 if j == 0 else PI/2, + checkerboard_colors=colors, resolution=(10,5))) + + dihedral_sphere = VGroup(*dihedral_slices) + + self.play(FadeOut(square), Create(dihedral_sphere)) + + for i in range(2): + self.play(Rotate(dihedral_sphere, PI/2)) + self.play(Rotate(dihedral_sphere, PI, RIGHT)) + self.wait() + + self.wait(2) + self.play(*map(FadeOut, [dihedral_sphere, plane, plane_name, new_axis_name]), FadeIn(square)) + self.wait(3) + self.play(*map(FadeOut, [square, axis])) + self.wait(3) + + def tetrahedron(self): + tet = Tetrahedron(edge_length=2) + self.play(FadeIn(tet)) + + self.move_camera(phi = 75 * DEGREES, theta = -100 * DEGREES) + self.begin_ambient_camera_rotation(rate=.1) + + axes = [] + for coord in tet.vertex_coords: + axes.append((-2 * coord, 2 * coord)) + + lines = [ + Line3D(start=s, end=e) for s, e in axes + ] + + self.play(*map(Create, lines)) + self.wait() + + for axis in axes: + self.play(Rotate(tet, 2*PI/3, axis=axis[1])) + self.play(Rotate(tet, 2*PI/3, axis=axis[1])) + + self.wait(5) + self.stop_ambient_camera_rotation() + self.wait() + self.play(*map(Uncreate, lines)) + self.play(FadeOut(tet)) + self.wait(5) + + +class AlgebraicSymmetries(Scene): + def construct(self): + self.wait(5) + self.cyclic() + # self.matrices() + + def cyclic(self): + # show the i product + product = MathTex( + r"1", r"\cdot i &= i \\", + r"i \cdot i &= -1 \\", + r"-1 \cdot i &= -i \\", + r"-i \cdot i &= 1") + product.scale(1.5) + + for part in product: + self.play(Write(part)) + self.wait() + + self.play(ApplyMethod(product.scale, 1/1.5)) + + # gather in group + group = MathTex(r"G = \left\{ 1, i, -1, -i \right\}") + self.play(ReplacementTransform(product, group)) + self.wait(2) + + # show C4 + grouppow = MathTex( + r"G &= \left\{ 1, i, i^2, i^3 \right\} \\", + r"C_4 &= \left\{ \mathbb{1}, r, r^2, r^3 \right\}") + self.play(ReplacementTransform(group, grouppow[0])) + self.wait(2) + + self.play(Write(grouppow[1])) + self.wait(4) + + self.play(ApplyMethod(grouppow.to_edge, UP)) + + # define morphisms + morphism = MathTex(r"\phi: C_4 \to G \\") + morphism.shift(UP) + self.play(Write(morphism)) + self.wait() + + # show an example + mappings = MathTex( + r"\phi(\mathbb{1}) &= 1 \\", + r"\phi(r) &= i \\", + r"\phi(r^2) &= i^2 \\", + r"\phi(r^3) &= i^3 \\") + mappings.next_to(morphism, 2 * DOWN) + + self.play(Write(mappings)) + self.wait(3) + self.play(FadeOutAndShift(mappings, DOWN)) + + # more general definition + homomorphism = MathTex( + r"\phi(r\circ \mathbb{1}) &= \phi(r)\cdot\phi(\mathbb{1}) \\", + r"&= i\cdot 1") + homomorphism.next_to(morphism, DOWN).align_to(morphism, LEFT) + for part in homomorphism: + self.play(Write(part)) + self.wait() + + hom_bracegrp = VGroup(morphism, homomorphism) + + self.play( + ApplyMethod(grouppow.shift, 3 * LEFT), + ApplyMethod(hom_bracegrp.shift, 3 * LEFT)) + + hom_brace = Brace(hom_bracegrp, direction=RIGHT) + hom_text = Tex("Homomorphismus").next_to(hom_brace.get_tip(), RIGHT) + hom_text_short = MathTex(r"\mathrm{Hom}(C_4, G)").next_to(hom_brace.get_tip(), RIGHT) + + self.play(Create(hom_brace)) + self.play(Write(hom_text)) + self.wait() + self.play(ReplacementTransform(hom_text, hom_text_short)) + self.wait() + + # add the isomorphism part + isomorphism = Tex(r"\(\phi\) ist bijektiv") + isomorphism.next_to(homomorphism, DOWN).align_to(homomorphism, LEFT) + self.play(Write(isomorphism)) + + iso_bracegrp = VGroup(hom_bracegrp, isomorphism) + + iso_brace = Brace(iso_bracegrp, RIGHT) + iso_text = Tex("Isomorphismus").next_to(iso_brace.get_tip(), RIGHT) + iso_text_short = MathTex("C_4 \cong G").next_to(iso_brace.get_tip(), RIGHT) + + self.play( + ReplacementTransform(hom_brace, iso_brace), + ReplacementTransform(hom_text_short, iso_text)) + self.wait() + + self.play(ReplacementTransform(iso_text, iso_text_short)) + self.wait() + + # create a group for the whole + morphgrp = VGroup(iso_bracegrp, iso_brace, iso_text_short) + + self.play( + ApplyMethod(grouppow.to_edge, LEFT), + ApplyMethod(morphgrp.to_edge, LEFT)) + + # draw a complex plane + plane = ComplexPlane(x_range = [-2.5, 2.5]) + coordinates = plane.get_coordinate_labels(1, -1, 1j, -1j) + + roots = list(map(lambda p: Dot(p, fill_color=PINK), ( + [1, 0, 0], [0, 1, 0], [-1, 0, 0], [0, -1, 0] + ))) + + arrow = CurvedArrow( + 1.5 * np.array([m.cos(10 * DEGREES), m.sin(10 * DEGREES), 0]), + 1.5 * np.array([m.cos(80 * DEGREES), m.sin(80 * DEGREES), 0])) + arrowtext = MathTex("\cdot i") + arrowtext.move_to(2 / m.sqrt(2) * (UP + RIGHT)) + + square = Square().rotate(PI/4).scale(1/m.sqrt(2)) + square.set_fill(PINK).set_opacity(.4) + + figuregrp = VGroup(plane, square, arrow, arrowtext, *coordinates, *roots) + figuregrp.to_edge(RIGHT) + + self.play(Create(plane)) + self.play( + *map(Create, roots), + *map(Write, coordinates)) + self.wait() + self.play(FadeIn(square), Create(arrow), Write(arrowtext)) + + for _ in range(4): + self.play(Rotate(square, PI/2)) + self.wait(.5) + + self.play( + *map(FadeOut, (square, arrow, arrowtext)), + *map(FadeOut, coordinates), + *map(FadeOut, roots)) + self.play(Uncreate(plane)) + self.play( + FadeOutAndShift(grouppow, RIGHT), + FadeOutAndShift(morphgrp, RIGHT)) + + modulo = MathTex( + r"\phi: C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\" + r"\phi(\mathbb{1} \circ r^2) &= 0 + 2 \pmod 4").scale(1.5) + self.play(Write(modulo)) + self.wait(2) + + self.play(FadeOut(modulo)) + self.wait(3) + + def matrices(self): + question = MathTex( + r"D_n &\cong \,? \\" + r"S_n &\cong \,? \\" + r"A_n &\cong \,?").scale(1.5) + + answer = MathTex( + r"D_n &\cong \,?\\" + r"S_4 &\cong \mathrm{Aut}(Q_8) \\" + r"A_5 &\cong \mathrm{PSL}_2 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= "" +authors = ["Nao Pross <np@0hm.ch>"] + +[tool.poetry.dependencies] +python = "^3.7" +manim = "^0.6.0" + +[tool.poetry.dev-dependencies] + +[build-system] +requires = ["poetry-core>=1.0.0"] +build-backend = "poetry.core.masonry.api" diff --git a/vorlesungen/punktgruppen/script.pdf b/vorlesungen/punktgruppen/script.pdf Binary files differnew file mode 100644 index 0000000..91993fb --- /dev/null +++ b/vorlesungen/punktgruppen/script.pdf diff --git a/vorlesungen/punktgruppen/script.tex b/vorlesungen/punktgruppen/script.tex new file mode 100644 index 0000000..bc50e21 --- /dev/null +++ b/vorlesungen/punktgruppen/script.tex @@ -0,0 +1,214 @@ +\documentclass[a4paper]{article} + +\usepackage{amsmath} +\usepackage{amssymb} + +\usepackage[cm]{manuscript} +\usepackage{xcolor} + +\newcommand{\scene}[1]{\par\noindent[ #1 ]\par} +\newenvironment{totranslate}{\color{blue!70!black}}{} + +\begin{document} +\section{Das sind wir} +\scene{Camera} + +\section{Ablauf} +Zuerst werden wir Symmetrien in 2 Dimensionen anschauen, dann \"uberlegen wir +kurz was es heisst f\"ur eine Symmetrie ``algebraisch'' zu sein. Von da aus +kommt die dritte Dimension hinzu, die man besser mit Matrizen verstehen kann. +Mit der aufgebauten Theorie werden wir versuchen Kristalle zu klassifizieren. +Und zum Schluss kommen wir zu Anwendungen, welche f\"ur Ingenieure von +Interesse sind. + +\section{intro} +\scene{Spontan} + +\section{2D Geometrie} +\scene{Intro} +Wir fangen mit den 2 dimensionalen Symmetrien an, da man sie sich am +einfachsten vorstellen kann. Eine Symmetrie eines Objektes beschreibt eine +Aktion, welche nachdem sie auf das Objekt wirkt, das Objekt wieder gleich +aussehen l\"asst. + +\scene{Viereck} +Die einfachste Aktion, ist das Viereck zu nehmen, und wieder hinzulegen. +Eine andere Aktion k\"onnte sein, das Objekt um eine Achse zu spiegeln, +oder eine Rotation um 90 Grad. + +\scene{Zyklische Gruppe} +Fokussieren wir uns auf die einfachste Klassen von Symmetrien: diejenigen die +von einer reinen Drehung generiert werden. Wir sammeln diese in einer Gruppe +\(G\), und notieren das sie von eine Rotation \(r\) generiert worden sind, mit +diesen spitzen Klammern. + +Nehmen wir als Beispiel dieses Pentagon. Wenn wir \(r\) 5-mal anwenden, ist es +dasselbe als wenn wir nichts gemacht h\"atten. Wenn wir es noch ein 6. mal +drehen, entspricht dies dasselbe wie \(r\) nur 1 mal zu nutzen. + +\scene{Notation} +So, die Gruppe setzt sich zusammen aus dem neutralen Element, und den Potenzen +1 bis 4 von \(r\). Oder im allgemein Gruppen mit dieser Struktur, in welcher die +Aktion \(n-1\) mal angewendet werden kann, heissen ``Zyklische Gruppe''. + +\scene{Diedergruppe} +Nehmen wir nun auch noch die Spiegeloperation \(\sigma\) dazu. Weil wir jetzt 2 +Operationen haben, m\"ussen wir auch im Generator schreiben wie sie +zusammenh\"angen. Schauen wir dann uns genauer diesen Ausdr\"uck an. Zweimal +Spielegeln ist \"aquivalent zum neutralen Element, sowie 4 mal um 90 Grad +drehen und 2 Drehspiegelungen, welche man auch Inversion nennt. + +\scene{Notation} +Daraus k\"onnen wir wieder die ganze Gruppe erzeugen, die im allgemeinen den +Symmetrien eines \(n\)-gons entsprechen. + +\scene{Kreisgruppe} +Bis jetzt hatten wir nur diskrete Symmetrien, was nicht zwingend der Fall sein +muss. Ein Ring kann man kontinuierlich drehen, und sieht dabei immer gleich +aus. + +Diese Symmetrie ist auch als Kreisgruppe bekannt, die man sch\"on mit dem +komplexen Einheitskreis definieren kann. + +\section{Algebra} +\scene{Produkt mit \(i\)} +\"Uberlegen wir uns eine spezielle algebraische Operation: Multiplikation mit +der imagin\"aren Einheit. \(1\) mal \(i\) ist gleich \(i\). Wieder mal \(i\) +ist \(-1\), dann \(-i\) und schliesslich kommen wir z\"uruck auf \(1\). Diese +fassen wir in eine Gruppe \(G\) zusammen. Oder sch\"oner geschrieben:. Sieht das +bekannt aus? + +\scene{Morphismen} +Das Gefühl, dass es sich um dasselbe handelt, kann wie folgt formalisiert +werden. Sei \(\phi\) eine Funktion von \(C_4\) zu \(G\) und ordnen wir zu +jeder Symmetrieoperation ein Element aus \(G\). Wenn man die Zuordnung richtig +definiert, dann sieht man die folgende Eigenschaft: Eine Operation nach eine +andere zu nutzen, und dann die Funktion des Resultats zu nehmen, ist gleich wie +die Funktion der einzelnen Operazionen zu nehmen und die Resultate zu +multiplizieren. Dieses Ergebnis ist so bemerkenswert, dass es in der Mathematik +einen Namen bekommen hat: Homorphismus, von griechisch "homos" dasselbe und +"morphe" Form. Manchmal auch so geschrieben. Ausserdem, wenn \(\phi\) eins zu +eins ist, heisst es \emph{Iso}morphismus: "iso" gleiche Form. Was man +typischerweise mit diesem Symbol schreibt. + +\scene{Animation} +Sie haben wahrscheinlich schon gesehen, worauf das hinausläuft. Dass die +zyklische Gruppe \(C_4\) und \(G\) isomorph sind ist nicht nur Fachjargon der +mathematik, sondern sie haben wirklich die selbe Struktur. + +\scene{Modulo} +Das Beispiel mit der komplexen Einheit, war wahrscheinlich nicht so +\"uberraschend. Aber was merkw\"urdig ist, ist das Beziehungen zwischen +Symmetrien und Algebra auch in Bereichen gefunden werden, welche auf den ersten +Blick, nicht geomerisch erscheinen. Ein R\"atsel für die Neugierigen: die Summe +in der Modulo-Arithmetik. Als Hinweis: Um die Geometrie zu finden denken Sie +an einer Uhr. + +\section{3D Geometrie} +2 Dimensionen sind einfacher zu zeichnen, aber leider leben wir im 3 +dimensionalen Raum. + +\scene{Zyklische Gruppe} +Wenn wir unser bekanntes Viereck mit seiner zyklischer Symmetrie in 3 +Dimensionen betrachten, k\"onnen wir seine Drehachse sehen. + +\scene{Diedergruppe} +Um auch noch die andere Symmetrie des Rechteckes zu sehen, ben\"otigen wir eine +Spiegelachse \(\sigma\), die hier eine Spiegelebene ist. + +\scene{Transition} +Um die Punktsymmetrien zu klassifizieren orientiert man sich an einer Achse, um +welche sich die meisten Symmetrien drehen. Das geht aber nicht immer, wie beim +Tetraeder. + +\scene{Tetraedergruppe} +Diese Geometrie hat 4 gleichwertige Symmetrieachsen, die eben eine +Symmetriegruppe aufbauen, welche kreativer weise Tetraedergruppe genannt wird. +Vielleicht fallen Ihnnen weitere Polygone ein mit dieser Eigenschaft, bevor wir +zum n\"achsten Thema weitergehen. + +\section{Matrizen} +\scene{Titelseite} +Nun gehen wir kurz auf den Thema unseres Seminars ein: Matrizen. Das man mit +Matrizen Dinge darstellen kann, ist keine Neuigkeit mehr, nach einem +Semester MatheSeminar. Also überrascht es wohl auch keinen, das man alle +punktsymmetrischen Operationen auch mit Matrizen Formulieren kann. + +\scene{Matrizen} + +Sei dann \(G\) unsere Symmetrie Gruppe, die unsere abstrakte Drehungen und +Spiegelungen enth\"ahlt. Die Matrix Darstellung dieser Gruppe, ist eine +Funktion gross \(\Phi\), von \(G\) zur orthogonalen Gruppe \(O(3)\), die zu +jeder Symmetrie Operation klein \(g\) eine Matrix gross \(\Phi_g\) zuordnet. + +Zur Erinnerung, die Orthogonale Gruppe ist definiert als die Matrizen, deren +transponierte auch die inverse ist. Da diese Volumen und Distanzen erhalten, +natuerlich nur bis zu einer Vorzeichenumkehrung, macht es Sinn, dass diese +Punksymmetrien genau beschreiben. + +Nehmen wir die folgende Operationen als Beispiele. Die Matrix der trivialen +Operation, dass heisst nichts zu machen, ist die Einheitsmatrix. Eine +Spiegelung ist dasselbe aber mit einem Minus, und Drehungen sind uns schon +dank Herrn M\"uller bekannt. + +\section{Kristalle} +\scene{Spontan} + +\section{Piezo} +\scene{Spontan} + +\section{Licht} +Als Finale, haben wir ein schwieriges Problem aus der Physik. Das Ziel dieser +Folie ist nicht jedes Zeichen zu versehen, sondern zu zeigen wie man von hier +weiter gehen kann. Wir mochten sehen wie sich Licht in einem Kristall verhaltet. +Genauer, wir m\"ochten die Amplitude einer +elektromagnetischer Welle in einem Kristall beschreiben. + +Das Beispiel richtet sich mehr an Elektrotechnik Studenten, aber die Theorie +ist die gleiche bei mechanischen Wellen in Materialien mit einer +Spannungstensor wie dem, den wir letzte Woche gesehen haben. +% Ganz grob gesagt, ersetzt man E durch Xi und epsilon durch das Sigma. + +Um eine Welle zu beschreiben, verwenden wir die Helmholtz-Gleichung, die einige +von uns bereits in anderen Kursen gel\"ost haben. Schwierig wird aber dieses +Problem, wenn der Term vor der Zeitableitung ein Tensor ist (f\"ur uns eine Matrix). + +Zur Vereinfachung werden wir eine ebene Welle verwenden. Setzt man dieses E in +die Helmholtz-Gleichung ein, erhält man folgendes zurück: ein Eigenwertproblem. + +Physikalisch bedeutet dies, dass die Welle in diesem Material ihre Amplitude in +Abhängigkeit von der Ausbreitungsrichtung ändert. Und die Eigenwerte sagen +aus, wie stark die Amplitude der Welle in jeder Richtung skaliert wird. + +Ich sagte, in jede Richtung skaliert, aber welche Richtungen genau? +Physikalisch hängt das von der kristallinen Struktur des Materials ab, aber +mathematisch können wir sagen: in Richtung der Eigenvektoren! Aber diesen +Eigenraum zu finden, in dem die Eigenvektoren wohnen, ist beliebig schwierig. + +Hier kommt unsere Gruppentheorie zu Hilfe. Wir können die Symmetrien unseres +Kristalls zur Hilfe nehmen. Zu jeder dieser Symmetrien lässt sich bekanntlich eine +einfache Matrix finden, deren Eigenraum ebenfalls relativ leicht zu finden ist. +Zum Beispiel ist der Eigenraum der Rotation \(r\), die Rotationsachse, für die +Reflexion \(\sigma\) eine Ebene, und so weiter. + +Nun ist die Frage, ob man diese Eingenraume der Symmetrienoperationen +kombinieren kann um den Eigenraum des physikalisches Problems zu finden. + +Aber leider ist meine Zeit abgelaufen in der Recherche, also müssen Sie mir 2 +Dingen einfach glauben, erstens dass es einen Weg gibt, und zweitens dass eher +nicht so schlimm ist, wenn man die Notation einmal gelernt hat. + +Nachdem wir an, wir haben den Eigenraum U gefunden, dann können wir einen +(Eigen)Vektor E daraus nehmen und in ihm direkt lambda ablesen. Das sagt uns, +wie die Amplitude der Welle, in diese Richtung gedämpft wurde. + +Diese Methode ist nicht spezifisch für dieses Problem, im Gegenteil, ich habe +gesehen, dass sie in vielen Bereichen eingesetzt wird, wie z.B.: +Kristallographie, Festkörperphysik, Molekülschwingungen in der Quantenchemie +und numerische Simulationen von Membranen. + +\section{Outro} +\scene{Camera} + +\end{document} +% vim:et ts=2 sw=2: diff --git a/vorlesungen/punktgruppen/slides.pdf b/vorlesungen/punktgruppen/slides.pdf Binary files differnew file mode 100644 index 0000000..e1769f8 --- /dev/null +++ b/vorlesungen/punktgruppen/slides.pdf diff --git a/vorlesungen/punktgruppen/slides.tex b/vorlesungen/punktgruppen/slides.tex new file mode 100644 index 0000000..cd3d7d7 --- /dev/null +++ b/vorlesungen/punktgruppen/slides.tex @@ -0,0 +1,895 @@ +\documentclass[12pt, xcolor, aspectratio=169, handout]{beamer} + +% language +\usepackage{polyglossia} +\setmainlanguage{german} + +% pretty drawings +\usepackage{tikz} +\usepackage{tikz-3dplot} + +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\usetikzlibrary{external} +\tikzexternalize[ + mode = graphics if exists, + figure list = true, + prefix=build/ +] + +% Theme +\beamertemplatenavigationsymbolsempty + +% set look +\usetheme{default} +\usecolortheme{fly} +\usefonttheme{serif} + +%% Set font +\usepackage[p,osf]{scholax} +\usepackage{amsmath} +\usepackage[scaled=1.075,ncf,vvarbb]{newtxmath} + +% set colors +\definecolor{background}{HTML}{202020} + +\setbeamercolor{normal text}{fg=white, bg=background} +\setbeamercolor{structure}{fg=white} + +\setbeamercolor{item projected}{use=item,fg=background,bg=item.fg!35} + +\setbeamercolor*{palette primary}{use=structure,fg=white,bg=structure.fg} +\setbeamercolor*{palette secondary}{use=structure,fg=white,bg=structure.fg!75} +\setbeamercolor*{palette tertiary}{use=structure,fg=white,bg=structure.fg!50} +\setbeamercolor*{palette quaternary}{fg=white,bg=background} + +\setbeamercolor*{block title}{parent=structure} +\setbeamercolor*{block body}{fg=background, bg=} + +\setbeamercolor*{framesubtitle}{fg=white} + +\setbeamertemplate{section page} +{ + \begin{center} + \Huge + \insertsection + \end{center} +} +\AtBeginSection{\frame{\sectionpage}} + +% Macros +\newcommand{\ten}[1]{#1} + +% Metadata +\title{\LARGE \scshape Punktgruppen und Kristalle} +\author[N. Pross, T. T\"onz]{Naoki Pross, Tim T\"onz} +\institute{Hochschule f\"ur Technik OST, Rapperswil} +\date{10. Mai 2021} + +% Slides +\begin{document} +\frame{ + \titlepage + \vfill + \begin{center} + \small \color{gray} + Slides: \texttt{s.0hm.ch/ctBsD} + \end{center} +} +\frame{\tableofcontents} + +\frame{ + \begin{itemize} + \item Was heisst \emph{Symmetrie} in der Mathematik? \pause + \item Wie kann ein Kristall modelliert werden? \pause + \item Aus der Physik: Licht, Piezoelektrizit\"at \pause + \end{itemize} + \begin{center} + \begin{tikzpicture} + \begin{scope}[ + node distance = 0cm + ] + \node[ + rectangle, fill = gray!40!background, + minimum width = 3cm, minimum height = 2cm, + ] (body) {\(\vec{E}_p = \vec{0}\)}; + + \node[ + draw, rectangle, thick, white, fill = red!50, + minimum width = 3cm, minimum height = 1mm, + above = of body + ] (pos) {}; + + \node[ + draw, rectangle, thick, white, fill = blue!50, + minimum width = 3cm, minimum height = 1mm, + below = of body + ] (neg) {}; + + \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {}; + \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {}; + + \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n); + \end{scope} + \begin{scope}[ + node distance = 0cm, + xshift = 7cm + ] + \node[ + rectangle, fill = gray!40!background, + minimum width = 3cm, minimum height = 1.5cm, + ] (body) {\(\vec{E}_p = \vec{0}\)}; + + \node[ + draw, rectangle, thick, white, fill = red!50, + minimum width = 3cm, minimum height = 1mm, + above = of body + ] (pos) {}; + + \node[ + draw, rectangle, thick, white, fill = blue!50, + minimum width = 3cm, minimum height = 1mm, + below = of body + ] (neg) {}; + + \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1); + \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1); + + \draw[white, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {}; + \draw[white, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {}; + + \draw[white, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n); + \end{scope} + \end{tikzpicture} + \end{center} +} + +\section{2D Symmetrien} +%% Made in video +{ + \usebackgroundtemplate{ + \includegraphics[height=\paperheight]{media/images/nosignal}} + \frame{} +} + +\section{Algebraische Symmetrien} +%% Made in video +\frame{ + \begin{columns}[T] + \begin{column}{.5\textwidth} + Produkt mit \(i\) + \begin{align*} + 1 \cdot i &= i \\ + i \cdot i &= -1 \\ + -1 \cdot i &= -i \\ + -i \cdot i &= 1 + \end{align*} + \pause + % + Gruppe + \begin{align*} + G &= \left\{ + 1, i, -1, -i + \right\} \\ + &= \left\{ + 1, i, i^2, i^3 + \right\} \\ + C_4 &= \left\{ + \mathbb{1}, r, r^2, r^3 + \right\} + \end{align*} + \pause + \end{column} + \begin{column}{.5\textwidth} + Darstellung \(\phi : C_4 \to G\) + \begin{align*} + \phi(\mathbb{1}) &= 1 & \phi(r^2) &= i^2 \\ + \phi(r) &= i & \phi(r^3) &= i^3 + \end{align*} + \pause + % + Homomorphismus + \begin{align*} + \phi(r \circ \mathbb{1}) &= \phi(r) \cdot \phi(\mathbb{1}) \\ + &= i \cdot 1 + \end{align*} + \pause + % + \(\phi\) ist bijektiv \(\implies C_4 \cong G\) + \pause + % + \begin{align*} + \psi : C_4 &\to (\mathbb{Z}/4\mathbb{Z}, +) \\ + \psi(\mathbb{1}\circ r^2) &= 0 + 2 \pmod{4} + \end{align*} + \end{column} + \end{columns} +} + +\section{3D Symmetrien} +%% Made in video +{ + \usebackgroundtemplate{ + \includegraphics[height=\paperheight]{media/images/nosignal}} + \frame{} +} + +\section{Matrizen} +\frame{ + \begin{columns}[T] + \begin{column}{.5\textwidth} + Symmetriegruppe + \[ + G = \left\{\mathbb{1}, r, \sigma, \dots \right\} + \] + \pause + Matrixdarstellung + \begin{align*} + \Phi : G &\to O(3) \\ + g &\mapsto \Phi_g + \end{align*} + \pause + Orthogonale Gruppe + \[ + O(n) = \left\{ Q : QQ^t = Q^tQ = I \right\} + \] + \end{column} + \pause + \begin{column}{.5\textwidth} + \begin{align*} + \Phi_\mathbb{1} &= \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} = I \\[1em] + \Phi_\sigma &= \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & -1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} \\[1em] + \Phi_r &= \begin{pmatrix} + \cos \alpha & -\sin \alpha & 0 \\ + \sin \alpha & \cos \alpha & 0 \\ + 0 & 0 & 1 + \end{pmatrix} + \end{align*} + \end{column} + \end{columns} +} + +\section{Kristalle} +\begin{frame}[fragile]{} + \begin{columns} + \onslide<1->{ + \begin{column}{.5\textwidth} + \begin{center} + \begin{tikzpicture}[ + dot/.style = { + draw, circle, thick, white, fill = gray!40!background, + minimum size = 2mm, + inner sep = 0pt, + outer sep = 1mm, + }, + ] + + \begin{scope} + \clip (-2,-2) rectangle (3,4); + \foreach \y in {-7,-6,...,7} { + \foreach \x in {-7,-6,...,7} { + \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {}; + } + } + \end{scope} + \draw[white, thick] (-2, -2) rectangle (3,4); + + \draw[red!80!background, thick, ->] + (N00) to node[midway, below] {\(\vec{a}_1\)} (N10); + \draw[cyan!80!background, thick, ->] + (N00) to node[midway, left] {\(\vec{a}_2\)} (N01); + \end{tikzpicture} + \end{center} + \end{column} + } + \begin{column}{.5\textwidth} + \onslide<2->{ + Kristallgitter: + \(n_i \in \mathbb{Z}\), + } + \onslide<3->{ + \(\vec{a}_i \in \mathbb{R}^3\) + } + \onslide<2->{ + \[ + \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 \onslide<3->{+ n_3 \vec{a}_3} + \] + } + \vspace{1cm} + + \onslide<4->{ + Invariant unter Translation + \[ + Q_i(\vec{r}) = \vec{r} + \vec{a}_i + \] + } + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{} + \begin{columns}[T] + \begin{column}{.5\textwidth} + \onslide<1->{ + Wie kombiniert sich \(Q_i\) mit der anderen Symmetrien? + } + \begin{center} + \begin{tikzpicture}[ + dot/.style = { + draw, circle, thick, white, fill = gray!40!background, + minimum size = 2mm, + inner sep = 0pt, + outer sep = 1mm, + }, + ] + + \onslide<2->{ + \node[dot] (A1) at (0,0) {}; + \node[below left] at (A1) {\(A\)}; + } + + \onslide<3->{ + \node[dot] (A2) at (2.5,0) {}; + \node[below right] at (A2) {\(A'\)}; + + \draw[red!80!background, thick, ->] + (A1) to node[midway, below] {\(\vec{Q}\)} (A2); + } + + \onslide<4->{ + \node[dot] (B1) at (120:2.5) {}; + \node[above left] at (B1) {\(B\)}; + + \draw[green!70!background, thick, ->] + (A1) ++(.5,0) arc (0:120:.5) + node[midway, above, xshift=1mm] {\(C_n\)}; + \draw[red!80!background, dashed, thick, ->] (A1) to (B1); + } + + \onslide<5->{ + \node[dot] (B2) at ($(A2)+(60:2.5)$) {}; + \node[above right] at (B2) {\(B'\)}; + + \draw[green!70!background, thick, dashed, ->] + (A2) ++(-.5,0) arc (180:60:.5); + \draw[red!80!background, dashed, thick, ->] (A2) to (B2); + } + + \onslide<6->{ + \draw[yellow!80!background, thick, ->] + (B1) to node[above, midway] {\(\vec{Q}'\)} (B2); + } + + \onslide<10->{ + \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {}; + \draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {}; + \node[above left, xshift=-2mm] at (Xl) {\(x\)}; + \node[above right, xshift= 2mm] at (Xr) {\(x\)}; + } + \end{tikzpicture} + \end{center} + \end{column} + \begin{column}{.5\textwidth} + \onslide<7->{ + Sei \(q = |\vec{Q}|\), \(\alpha = 2\pi/n\) und \(n \in \mathbb{N}\) + } + \begin{align*} + \onslide<9->{q' = n q \onslide<10->{&= q + 2x \\}} + \onslide<11->{nq &= q + 2q\sin(\alpha - \pi/2) \\} + \onslide<12->{n &= 1 - 2\cos\alpha} + \end{align*} + \onslide<13->{ + Somit muss + \begin{align*} + \alpha &= \cos^{-1}\left(\frac{1-n}{2}\right) \\[1em] + \alpha &\in \left\{ 0, 60^\circ, 90^\circ, 120^\circ, 180^\circ \right\} \\ + n &\in \left\{ 1, 2, 3, 4, 6 \right\} + \end{align*} + } + \end{column} + \end{columns} +\end{frame} + +\begin{frame}[fragile]{M\"ogliche Kristallstrukturen} + \begin{center} + \begin{tikzpicture}[] + \node[circle, dashed, draw = gray, + thick, fill = background, + minimum size = 4cm] {}; + \node[gray] at (.9,-1.2) {674}; + + \node[circle, draw = white, thick, + fill = orange!40!background, + xshift = -3mm, yshift = 2mm, + minimum size = 2.75cm, + outer sep = 1mm] (A) {}; + \node[white, yshift = 2mm] at (A) {230}; + \node[white, font=\large, above right = of A] (Al) {Raumgruppe}; + \draw[white, thick, ->] (Al.west) to[out=180, in=60] (A); + + \node[circle, draw = white, thick, + fill = red!20!background, + xshift = -5mm, yshift = -5mm, + minimum size = 1cm, + outer sep = 1mm] (B) {32}; + \node[white, font=\large, below left = of B, xshift=-4mm] (Bl) {Kristallklassen}; + \draw[white, thick, ->] (Bl.east) to[out = 0, in = 180] (B); + \end{tikzpicture} + \end{center} +\end{frame} + +{ + \usebackgroundtemplate[fragile]{ + \begin{tikzpicture}[ + overlay, + xshift = .45\paperwidth, + yshift = .47\paperheight, + classcirc/.style = { + draw = gray, thick, circle, + minimum size = 12mm, + inner sep = 0pt, outer sep = 0pt, + }, + classlabel/.style = { + below right = 5mm + }, + round/.style = { + draw = yellow, thick, circle, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt, + }, + cross/.style = { + cross out, draw = magenta, thick, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt + }, + ] + \matrix [row sep = 3mm, column sep = 0mm] { + \node[classcirc] (C1) {} node[classlabel] {\(C_{1}\)}; & + \node[classcirc] (C2) {} node[classlabel] {\(C_{2}\)}; & + \node[classcirc] (C3) {} node[classlabel] {\(C_{3}\)}; & + \node[classcirc] (Ci) {} node[classlabel] {\(C_{i}\)}; & + + \node[classcirc] (Cs) {} node[classlabel] {\(C_{s}\)}; & + \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; & + \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; & + \node[classcirc] (D2) {} node[classlabel] {\(D_{2}\)}; \\ + + \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; & + \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; & + \node[classcirc] (D3) {} node[classlabel] {\(D_{3}\)}; & + + \node[classcirc] (C4) {} node[classlabel] {\(C_{4}\)}; & + \node[classcirc] (C6) {} node[classlabel] {\(C_{6}\)}; & + \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (S4) {} node[classlabel] {\(S_{4}\)}; \\ + + \node[classcirc] (S3) {} node[classlabel] {\(S_{3}\)}; & + \node[classcirc, dashed] (T) {} node[classlabel] {\(T_{}\)}; & + \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; & + \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; & + + \node[classcirc, dashed] (Th) {} node[classlabel] {\(T_{h}\)}; & + \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; & + \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; & + \node[classcirc, dashed] (Td) {} node[classlabel] {\(T_{d}\)}; \\ + + \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; & + \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; & + \node[classcirc, dashed] (O) {} node[classlabel] {\(O_{}\)}; & + \node[classcirc] (D4) {} node[classlabel] {\(D_{4}\)}; & + + \node[classcirc] (D6) {} node[classlabel] {\(D_{6}\)}; & + \node[classcirc, dashed] (Oh) {} node[classlabel] {\(O_{h}\)}; & + \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; & + \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\ + }; + + + \node[cross] at ($(C1)+(4mm,0)$) {}; + + + \node[cross] at ($(C2)+(4mm,0)$) {}; + \node[cross] at ($(C2)-(4mm,0)$) {}; + + + \node[cross] at ($(C3)+( 0:4mm)$) {}; + \node[cross] at ($(C3)+(120:4mm)$) {}; + \node[cross] at ($(C3)+(240:4mm)$) {}; + + + \node[cross] at ($(Ci)+(4mm,0)$) {}; + \node[round] at ($(Ci)-(4mm,0)$) {}; + + + \node[cross] at ($(Cs)+(4mm,0)$) {}; + \node[round] at ($(Cs)+(4mm,0)$) {}; + + + \node[cross] at ($(C3i)+( 0:4mm)$) {}; + \node[cross] at ($(C3i)+(120:4mm)$) {}; + \node[cross] at ($(C3i)+(240:4mm)$) {}; + \node[round] at ($(C3i)+( 60:4mm)$) {}; + \node[round] at ($(C3i)+(180:4mm)$) {}; + \node[round] at ($(C3i)+(300:4mm)$) {}; + + + \node[cross] at ($(C2h)+(4mm,0)$) {}; + \node[cross] at ($(C2h)-(4mm,0)$) {}; + \node[round] at ($(C2h)+(4mm,0)$) {}; + \node[round] at ($(C2h)-(4mm,0)$) {}; + + + \node[cross] at ($(D2)+( 20:4mm)$) {}; + \node[cross] at ($(D2)+(200:4mm)$) {}; + \node[round] at ($(D2)+(160:4mm)$) {}; + \node[round] at ($(D2)+(340:4mm)$) {}; + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3d)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3d)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 180} { + \node[cross] at ($(C2v)+({\x+15}:4mm)$) {}; + \node[cross] at ($(C2v)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 180} { + \node[cross] at ($(D2h)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D2h)+({\x-15}:4mm)$) {}; + \node[round] at ($(D2h)+({\x+15}:4mm)$) {}; + \node[round] at ($(D2h)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3)+({\x+15}:4mm)$) {}; + \node[round] at ($(D3)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {}; + \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {}; + \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {}; + } + + + \node[cross] at ($(S4)+(4mm,0)$) {}; + \node[cross] at ($(S4)-(4mm,0)$) {}; + \node[round] at ($(S4)+(0,4mm)$) {}; + \node[round] at ($(S4)-(0,4mm)$) {}; + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(S3)+(\x:4mm)$) {}; + \node[round] at ($(S3)+(\x:4mm)$) {}; + } + + + %% TODO: T + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4h)+(\x:4mm)$) {}; + \node[round] at ($(C4h)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6h)+(\x:4mm)$) {}; + \node[round] at ($(C6h)+(\x:4mm)$) {}; + } + + + %% TODO: Th + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4v)+(\x+15:4mm)$) {}; + \node[cross] at ($(C4v)+(\x-15:4mm)$) {}; + } + + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6v)+(\x+10:4mm)$) {}; + \node[cross] at ($(C6v)+(\x-10:4mm)$) {}; + } + + + %% TODO: Td + + + \foreach \x in {0, 180} { + \node[cross] at ($(D2d)+({\x+15}:4mm)$) {}; + \node[round] at ($(D2d)+({\x-15}:4mm)$) {}; + + \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {}; + \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3h)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3h)+({\x-15}:4mm)$) {}; + \node[round] at ($(D3h)+({\x+15}:4mm)$) {}; + \node[round] at ($(D3h)+({\x-15}:4mm)$) {}; + } + + + %% TODO: O + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(D4)+({\x+15}:4mm)$) {}; + \node[round] at ($(D4)+({\x-15}:4mm)$) {}; + } + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(D6)+({\x+10}:4mm)$) {}; + \node[round] at ($(D6)+({\x-10}:4mm)$) {}; + } + + + % TODO Oh + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(D4h)+(\x+15:4mm)$) {}; + \node[cross] at ($(D4h)+(\x-15:4mm)$) {}; + \node[round] at ($(D4h)+(\x+15:4mm)$) {}; + \node[round] at ($(D4h)+(\x-15:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(D6h)+({\x+10}:4mm)$) {}; + \node[cross] at ($(D6h)+({\x-10}:4mm)$) {}; + \node[round] at ($(D6h)+({\x+10}:4mm)$) {}; + \node[round] at ($(D6h)+({\x-10}:4mm)$) {}; + } + \end{tikzpicture} + } + \begin{frame}[fragile]{} + \end{frame} +} + +\section{Anwendungen} +\begin{frame}[fragile]{} + \centering + \begin{tikzpicture}[ + box/.style = { + rectangle, thick, draw = white, fill = darkgray!50!background, + minimum height = 1cm, outer sep = 2mm, + }, + ] + + \matrix [nodes = {box, align = center}, column sep = 1cm, row sep = 1.5cm] { + & \node (A) {32 Kristallklassen}; \\ + \node (B) {11 Mit\\ Inversionszentrum}; & \node (C) {21 Ohne\\ Inversionszentrum}; \\ + & \node[fill=red!20!background] (D) {20 Piezoelektrisch}; & \node (E) {1 Nicht\\ piezoelektrisch}; \\ + }; + + \draw[thick, ->] (A.west) to[out=180, in=90] (B.north); + \draw[thick, ->] (A.south) to (C); + \draw[thick, ->] (C.south) to (D.north); + \draw[thick, ->] (C.east) to[out=0, in=90] (E.north); + \end{tikzpicture} +\end{frame} + +\begin{frame}[fragile]{} + \begin{tikzpicture}[ + overlay, xshift = 1.5cm, yshift = 1.5cm, + node distance = 2mm, + charge/.style = { + circle, draw = white, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum}; + \pause + + \begin{scope} + \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] { + \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\ + }; + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{scope} + \pause + + \begin{scope}[xshift=11cm] + \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} { + \node[charge, \t] (C\i) at (\x:1.5cm) {}; + } + + \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {}; + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + \end{scope} + \pause + + %% + \node[below = of title] {Polarisation Feld \(\vec{E}_p\)}; + + %% hex with vertical pressure + \begin{scope}[xshift=11cm, yshift=-4.5cm] + \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {}; + + \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \foreach \d in {C1, C2} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {C4, C5} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \node[white] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[red!50, right = of E] {\(+\)}; + \node[blue!50, left = of E] {\(-\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{scope} + \pause + + %% square with vertical pressure + \begin{scope}[yshift=-4.5cm] + \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { + \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ + }; + + \foreach \d in {NW, N, NE} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {SW, S, SE} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{scope} + \pause + + %% hex with horizontal pressure + \begin{scope}[xshift=5.5cm, yshift=-4.5cm] + \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {}; + + \draw[white] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \draw[orange, very thick, <-] (C6) to ++(.7,0); + \draw[orange, very thick, <-] (C3) to ++(-.7,0); + + \node[white] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{scope} + \pause + + + \end{tikzpicture} +\end{frame} + +\frame{ + \frametitle{Licht in Kristallen} + \begin{columns}[T] + \begin{column}{.45\textwidth} + \onslide<2->{ + Helmholtz Wellengleichung + \[ + \nabla^2 \vec{E} = \ten{\varepsilon}\mu + \frac{\partial^2}{\partial t^2} \vec{E} + \] + } + \onslide<3->{ + Ebene Welle + \[ + \vec{E} = \vec{E}_0 \exp\left[i + \left(\vec{k}\cdot\vec{r} - \omega t \right)\right] + \] + } + \onslide<4->{ + Anisotropisch Dielektrikum + \[ + (\ten{K}\ten{\varepsilon})\vec{E} + = \frac{k^2}{\mu \omega^2} \vec{E} + \implies + \Phi \vec{E} = \lambda \vec{E} + \] + } + \end{column} + \begin{column}{.55\textwidth} + \onslide<5->{ + Eingenraum + \begin{align*} + U_\lambda &= \left\{ v : \Phi v = \lambda v \right\} + = \mathrm{null}\left(\Phi - \lambda I\right) + \end{align*} + }\onslide<6->{ + Symmetriegruppe und Darstellung + \begin{align*} + G &= \left\{\mathbb{1}, r, \sigma, \dots \right\} \\ + &\Phi : G \to O(n) + \end{align*} + }\onslide<7->{ + Kann man \(U_\lambda\) von \(G\) herauslesen? + \only<7>{ + \[ + U_\lambda \stackrel{?}{=} f\left(\bigoplus_{g \in G} \Phi_g\right) + \] + }\only<8>{ + \begin{align*} + \mathrm{Tr}\left[\Phi_r(g)\right] + &= \sum_i n_i \mathrm{Tr}\left[\Psi_i(g)\right] \\ + |G| &= \sum_i\mathrm{Tr}\left[\Psi_i(\mathbb{1})\right] + \end{align*} + } + } + \end{column} + \end{columns} +} + +% \begin{frame}[fragile] +% \centering +% \tdplotsetmaincoords{70}{110} +% \begin{tikzpicture}[scale=2, tdplot_main_coords] +% \node[draw=white, thick, minimum size = 3cm, circle] {}; +% % \foreach \x in {0, 120, 240} { +% % } +% \end{tikzpicture} +% \end{frame} + + +\end{document} diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index c857fec..cbd4dfe 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -17,5 +17,19 @@ chapter2 = \ ../slides/2/frobeniusanwendung.tex \ ../slides/2/quotient.tex \ ../slides/2/quotientv.tex \ + ../slides/2/hilbertraum/definition.tex \ + ../slides/2/hilbertraum/l2beispiel.tex \ + ../slides/2/hilbertraum/basis.tex \ + ../slides/2/hilbertraum/plancherel.tex \ + ../slides/2/hilbertraum/l2.tex \ + ../slides/2/hilbertraum/riesz.tex \ + ../slides/2/hilbertraum/rieszbeispiel.tex \ + ../slides/2/hilbertraum/adjungiert.tex \ + ../slides/2/hilbertraum/spektral.tex \ + ../slides/2/hilbertraum/sturm.tex \ + ../slides/2/hilbertraum/laplace.tex \ + ../slides/2/hilbertraum/qm.tex \ + ../slides/2/hilbertraum/energie.tex \ + ../slides/2/hilbertraum/sobolev.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 49e656a..d3714c3 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -15,3 +15,17 @@ \folie{2/frobeniusanwendung.tex} \folie{2/quotient.tex} \folie{2/quotientv.tex} +\folie{2/hilbertraum/definition.tex} +\folie{2/hilbertraum/l2beispiel.tex} +\folie{2/hilbertraum/basis.tex} +\folie{2/hilbertraum/plancherel.tex} +\folie{2/hilbertraum/l2.tex} +\folie{2/hilbertraum/riesz.tex} +\folie{2/hilbertraum/rieszbeispiel.tex} +\folie{2/hilbertraum/adjungiert.tex} +\folie{2/hilbertraum/spektral.tex} +\folie{2/hilbertraum/sturm.tex} +\folie{2/hilbertraum/laplace.tex} +\folie{2/hilbertraum/qm.tex} +\folie{2/hilbertraum/energie.tex} +\folie{2/hilbertraum/sobolev.tex} diff --git a/vorlesungen/slides/2/hilbertraum/adjungiert.tex b/vorlesungen/slides/2/hilbertraum/adjungiert.tex new file mode 100644 index 0000000..da41576 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/adjungiert.tex @@ -0,0 +1,83 @@ +% +% adjungiert.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Adjungierter Operator} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> +$A\colon H\to L$ lineare Abbildung zwischen Hilberträumen, $y\in L$ +\item<3-> +\[ +H\to\mathbb{C} +: +x\mapsto \langle y, Ax\rangle_L +\] +ist eine lineare Abbildung $H\to\mathbb{C}$ +\item<4-> +Nach dem Darstellungssatz gibt es $v\in H$ mit +\[ +\langle y,Ax\rangle_L = \langle v,x\rangle_H +\quad +\forall x\in H +\] +\end{itemize} +\uncover<5->{% +Die Abbildung +\[ +L\to H +: +y\mapsto v =: A^*y +\] +heisst {\em adjungierte Abbildung}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Endlichdimensional (Matrizen)} +\[ +A^* = \overline{A}^t +\] +\end{block}} +\vspace{-8pt} +\uncover<7->{% +\begin{block}{Selbstabbildungen} +Für Operatoren $A\colon H\to H$ ist $A^*\colon H\to H$ +\[ +\langle x,Ay\rangle += +\langle A^*x, y\rangle +\quad +\forall x,y\in H +\] +\end{block}} +\vspace{-8pt} +\uncover<9->{% +\begin{block}{Selbstadjungierte Operatoren} +\[ +A=A^* +\uncover<10->{\;\Leftrightarrow\; +\langle x,Ay \rangle += +\langle A^*x,y \rangle} +\uncover<11->{= +\langle Ax,y \rangle} +\] +\uncover<12->{Matrizen: +\begin{itemize} +\item<13-> hermitesch +\item<14-> für reelle Hilberträume: symmetrisch +\end{itemize}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/basis.tex b/vorlesungen/slides/2/hilbertraum/basis.tex new file mode 100644 index 0000000..022fa07 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/basis.tex @@ -0,0 +1,65 @@ +% +% basis.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hilbert-Basis} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Menge $\mathcal{B}=\{b_k|k>0\}$ ist eine Hilbertbasis, wenn +\begin{itemize} +\item<2-> $\mathcal{B}$ ist orthonormiert: $\langle b_k,b_l\rangle=\delta_{kl}$ +\item<3-> Der Unterraum $\langle b_k|k>0\rangle\subset H$ ist +dicht: +Jeder Vektor von $H$ kann beliebig genau durch Linearkombinationen von $b_k$ +approximiert werden. +\end{itemize} +\uncover<4->{% +Ein Hilbertraum mit einer Hilbertbasis heisst {\em separabel}} +\end{block} +\uncover<5->{% +\begin{block}{Endlichdimensional} +Der Algorithmus bricht nach endlich vielen Schritten ab. +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Konstruktion} +Iterativ: $\mathcal{B}_0=\emptyset$ +\begin{enumerate} +\item<7-> $V_k = \langle \mathcal{B}_k \rangle$ +\item<8-> Wenn $V_k\ne H$, wähle einen Vektor +\begin{align*} +x\in V_k^{\perp} +&= +\{ +x\in H\;|\; x\perp V_k +\} +\\ +&= +\{x\in H\;|\; +x\perp y\;\forall y\in V_k +\} +\end{align*} +\item<9-> $b_{k+1} = x/\|x\|$ +\[ +\mathcal{B}_{k+1} = \mathcal{B}_k\cup \{b_{k+1}\} +\] +\end{enumerate} +\uncover<10->{% +Wenn $H$ separabel ist, dann ist +\[ +\mathcal{B} = \bigcup_{k} \mathcal{B}_k +\] +eine Hilbertbasis für $H$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/definition.tex b/vorlesungen/slides/2/hilbertraum/definition.tex new file mode 100644 index 0000000..d101637 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/definition.tex @@ -0,0 +1,63 @@ +% +% definition.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Hilbertraum --- Definition} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{$\mathbb{C}$-Hilbertraum $H$} +\begin{enumerate} +\item<2-> $\mathbb{C}$-Vektorraum, muss nicht endlichdimensional sein +\item<3-> Sesquilineares Skalarprodukt +\[ +\langle \cdot,\cdot\rangle +\colon H \to \mathbb{C}: (x,y) \mapsto \langle x,y\rangle +\] +Dazugehörige Norm: +\[ +\|x\| = \sqrt{\langle x,x\rangle} +\] +\item<4-> Vollständigkeit: jede Cauchy-Folge konvergiert +\end{enumerate} +\uncover<5->{% +Ohne Vollständigkeit: {\em Prähilbertraum}} +\end{block} +\uncover<6->{% +\begin{block}{$\mathbb{R}$-Hilbertraum} +Vollständiger $\mathbb{R}$-Vektorraum mit bilinearem Skalarprodukt +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Vollständigkeit} +\begin{itemize} +\item<8-> $(x_n)_{n\in\mathbb{N}}$ ist eine Cauchy-Folge: +Für alle $\varepsilon>0$ gibt es $N>0$ derart, dass +\[ +\| x_n-x_m\| < \varepsilon\quad\forall n,m>N +\] +\item<9-> Grenzwert existiert: $\exists x\in H$ derart, dass es für alle +$\varepsilon >0$ ein $N>0$ gibt derart, dass +\[ +\|x_n-x\|<\varepsilon\quad\forall n>N +\] +\end{itemize} +\end{block}} +\uncover<10->{% +\begin{block}{Cauchy-Schwarz-Ungleichung} +\[ +|\langle x,y\rangle| +\le \|x\| \cdot \|y\| +\] +Gleichheit für linear abhängige $x$ und $y$ +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/energie.tex b/vorlesungen/slides/2/hilbertraum/energie.tex new file mode 100644 index 0000000..202a7c5 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/energie.tex @@ -0,0 +1,67 @@ +% +% energie.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Energie --- Zeitentwicklung --- Schrödinger} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.30\textwidth} +\uncover<2->{% +\begin{block}{Totale Energie} +Hamilton-Funktion +\begin{align*} +H +&= +\frac12mv^2 + V(x) +\\ +&= +\frac{p^2}{2m} + V(x) +\end{align*} +\end{block}} +\uncover<3->{% +\begin{block}{Quantisierungsregel} +\begin{align*} +\text{Variable}&\to \text{Operator} +\\ +x_k & \to x_k +\\ +p_k & \to \frac{\hbar}{i} \frac{\partial}{\partial x_k} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.66\textwidth} +\uncover<4->{% +\begin{block}{Energie-Operator} +\[ +H += +-\frac{\hbar^2}{2m}\Delta + V(x) +\] +\end{block}} +\uncover<5->{% +\begin{block}{Eigenwertgleichung} +\[ +-\frac{\hbar^2}{2m}\Delta\psi(x,t) + V(x)\psi(x,t) = E\psi(x,t) +\] +Zeitunabhängige Schrödingergleichung +\end{block}} +\uncover<6->{% +\begin{block}{Zeitabhängigkeit = Schrödingergleichung} +\[ +-\frac{\hbar}{i} +\frac{\partial}{\partial t} +\psi(x,t) += +-\frac{\hbar^2}{2m}\Delta\psi(x,t) + V(x)\psi(x,t) +\] +\uncover<7->{Eigenwertgleichung durch Separation von $t$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/l2.tex b/vorlesungen/slides/2/hilbertraum/l2.tex new file mode 100644 index 0000000..bd744ab --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/l2.tex @@ -0,0 +1,61 @@ +% +% l2.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$L^2$-Hilbertraum} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> +Vektorraum: Funktionen +\[ +f\colon [a,b] \to \mathbb{C} +\] +\item<3-> +Sesquilineares Skalarprodukt +\[ +\langle f,g\rangle += +\int_a^b \overline{f(x)}\, g(x) \,dx +\] +\item<4-> +Norm: +\[ +\|f\|^2 = \int_a^b |f(x)|^2\,dx +\] +\item<5-> +Vollständigkeit? +\uncover<6->{$\rightarrow$ +Lebesgue Konvergenz-Satz} +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Vollständigkeit} +\begin{itemize} +\item +Funktioniert nicht für Riemann-Integral +\item<8-> +Erweiterung des Integrals auf das sogenannte Lebesgue-Integral (nach +Henri Lebesgue) +\item<9-> +Abzählbare Mengen spielen keine Rolle $\rightarrow$ Nullmengen +\item<10-> +Funktionen $\rightarrow$ Klassen von Funktionen, die sich auf einer Nullmenge +unterscheiden +\item<11-> +Konvergenz-Satz von Lebesgue $\rightarrow$ es funktioniert +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/l2beispiel.tex b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex new file mode 100644 index 0000000..3ae44af --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/l2beispiel.tex @@ -0,0 +1,82 @@ +% +% l2beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Beispiele: $\mathbb{R},\mathbb{R}^2,\dots,\mathbb{R}^n,\dots,l^2$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> Quadratsummierbare Folgen von komplexen Zahlen +\[ +l^2 += +\biggl\{ +(x_k)_{k\in\mathbb{N}}\,\bigg|\, \sum_{k=0}^\infty |x_k|^2 < \infty +\biggr\} +\] +\item<3-> Skalarprodukt: +\begin{align*} +\langle x,y\rangle +&= +\sum_{k=0}^\infty \overline{x}_ky_k, +& +\uncover<4->{\|x\|^2 = \sum_{k=0}^\infty |x_k|^2} +\end{align*} +\item<5-> Vollständigkeit, +Konvergenz: Cauchy-Schwarz-Ungleichung +\[ +\biggl| +\sum_{k=0}^\infty \overline{x}_ky_k +\biggr| +\le +\sum_{k=0}^\infty |x_k|^2 +\sum_{l=0}^\infty |y_l|^2 +\] +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Standardbasisvektoren} +\begin{align*} +e_i +&= +(0,\dots,0,\underset{\underset{\textstyle i}{\textstyle\uparrow}}{1},0,\dots) +\\ +\uncover<7->{(e_i)_k &= \delta_{ik}} +\end{align*} +\uncover<8->{sind orthonormiert: +\begin{align*} +\langle e_i,e_j\rangle +&= +\sum_k \overline{\delta}_{ik}\delta_{jk} +\uncover<9->{= +\delta_{ij}} +\end{align*}} +\end{block}} +\vspace{-16pt} +\uncover<10->{% +\begin{block}{Analyse} +$x_k$ kann mit Skalarprodukten gefunden werden: +\begin{align*} +\hat{x}_i += +\langle e_i,x\rangle +&\uncover<11->{= +\sum_{k=0}^\infty \overline{\delta}_{ik} x_k} +\uncover<12->{= +x_i} +\end{align*} +\uncover<13->{(Fourier-Koeffizienten)} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/laplace.tex b/vorlesungen/slides/2/hilbertraum/laplace.tex new file mode 100644 index 0000000..8f6b196 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/laplace.tex @@ -0,0 +1,66 @@ +% +% laplace.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Höhere Dimension} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Problem} +Gegeben: $\Omega\subset\mathbb{R}^n$ ein Gebiet +\\ +Gesucht: Lösungen von $\Delta u=0$ mit $u_{|\partial\Omega}=0$ +\end{block} +\uncover<2->{% +\begin{block}{Funktionen} +Hilbertraum $H$ der Funktionen $f:\overline{\Omega}\to\mathbb{C}$ +mit $f_{|\partial\Omega}=0$ +\end{block}} +\uncover<3->{% +\begin{block}{Skalarprodukt} +\[ +\langle f,g\rangle += +\int_{\Omega} \overline{f}(x) g(x)\,d\mu(x) +\] +\end{block}} +\uncover<4->{% +\begin{block}{Laplace-Operator} +\[ +\Delta \psi = \operatorname{div}\operatorname{grad}\psi +\] +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<5->{% +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,\Delta g\rangle +&\uncover<6->{= +\int_{\Omega} \overline{f}(x)\operatorname{div}\operatorname{grad}g(x)\,d\mu(x)} +\\ +&\uncover<7->{= +\int_{\partial\Omega} +\underbrace{\overline{f}(x)}_{\displaystyle=0}\operatorname{grad}g(x)\,d\nu(x)} +\\ +&\uncover<7->{\qquad +- +\int_{\Omega} +\operatorname{grad}\overline{f}(x)\cdot \operatorname{grad}g(x) +\,d\mu(x)} +\\ +&\uncover<8->{=\int_{\Omega}\operatorname{div}\operatorname{grad}\overline{f}(x)g(x)\,d\mu(x)} +\\ +&\uncover<9->{= +\langle \Delta f,g\rangle} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/plancherel.tex b/vorlesungen/slides/2/hilbertraum/plancherel.tex new file mode 100644 index 0000000..73dd46b --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/plancherel.tex @@ -0,0 +1,102 @@ +% +% plancherel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Plancherel-Gleichung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Hilbertraum mit Hilbert-Basis} +$H$ Hilbertraum mit Hilbert-Basis +$\mathcal{B}=\{b_k\;|\; k>0\}$, $x\in H$ +\end{block} +\uncover<2->{% +\begin{block}{Analyse: Fourier-Koeffizienten} +\begin{align*} +a_k = \hat{x}_k &=\langle b_k, x\rangle +\\ +\uncover<3->{\hat{x}&=\mathcal{F}x} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Synthese: Fourier-Reihe} +\begin{align*} +\tilde{x} +&= +\sum_k a_k b_k +\uncover<5->{= +\sum_k \langle x,b_k\rangle b_k} +\end{align*} +\end{block}} +\vspace{-6pt} +\uncover<6->{% +\begin{block}{Analyse von $\tilde{x}$} +\begin{align*} +\langle b_l,\tilde{x}\rangle +&= +\biggl\langle +b_l,\sum_{k}\langle b_k,x\rangle b_k +\biggr\rangle +\uncover<7->{= +\sum_k \langle b_k,x\rangle\langle b_l,b_k\rangle} +\uncover<8->{= +\sum_k \langle b_k,x\rangle\delta_{kl}} +\uncover<9->{= +\langle b_l,x\rangle} +\uncover<10->{= +\hat{x}_l} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Plancherel-Gleichung} +\begin{align*} +\|\tilde{x}\|^2 +&= +\langle \tilde{x},\tilde{x}\rangle += +\biggl\langle +\sum_k \hat{x}_kb_k, +\sum_l \hat{x}_lb_l +\biggr\rangle +\\ +&\uncover<12->{= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\langle b_k,b_l\rangle} +\uncover<13->{= +\sum_{k,l} \overline{\hat{x}}_k\hat{x}_l\delta_{kl}} +\\ +\uncover<14->{ +\|\tilde{x}\|^2 +&= +\sum_k |\hat{x}_k|^2} +\uncover<15->{= +\|\hat{x}\|_{l^2}^2} +\uncover<16->{= +\|\mathcal{F}x\|_{l^2}^2} +\end{align*} +\end{block}} +\vspace{-12pt} +\uncover<17->{% +\begin{block}{Isometrie} +\begin{align*} +\mathcal{F} +\colon +H \to l^2 +\colon +x\mapsto \hat{x} +\end{align*} +\uncover<18->{Alle separablen Hilberträume sind isometrisch zu $l^2$ via +%Fourier-Transformation +$\mathcal{F}$} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/qm.tex b/vorlesungen/slides/2/hilbertraum/qm.tex new file mode 100644 index 0000000..a108121 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/qm.tex @@ -0,0 +1,90 @@ +% +% qm.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Anwendung: Quantenmechanik} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zustände (Wellenfunktion)} +$L^2$-Funktionen auf $\mathbb{R}^3$ +\[ +\psi\colon\mathbb{R}^3\to\mathbb{C} +\] +\end{block} +\vspace{-6pt} +\uncover<2->{% +\begin{block}{Wahrscheinlichkeitsinterpretation} +\[ +|\psi(x)|^2 = \left\{ +\begin{minipage}{4.6cm}\raggedright +Wahrscheinlichkeitsdichte für Position $x$ des Teilchens +\end{minipage}\right. +\] +\end{block}} +\vspace{-6pt} +\uncover<3->{% +\begin{block}{Skalarprodukt} +\[ +\langle\psi,\psi\rangle += +\int_{\mathbb{R}^3} |\psi(x)|^2\,dx = 1 +\] +\end{block}} +\vspace{-6pt} +\uncover<4->{% +\begin{block}{Messgrösse $A$} +Selbstadjungierter Operator $A$ +\\ +\uncover<5->{$\rightarrow$ +Hilbertbasis $|i\rangle$ von EV von $A$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Überlagerung} +\begin{align*} +|\psi\rangle +&= +\sum_i +w_i|i\rangle +\\ +\uncover<7->{\langle \psi|\psi\rangle +&= +\sum_i |w_i|^2 \qquad\text{(Plancherel)}} +\end{align*} +\uncover<8->{% +$|w_i|^2=|\langle \psi|i\rangle|^2$ Wahrscheinlichkeit für Zustand $|i\rangle$ +} +\end{block}} +\uncover<9->{% +\begin{block}{Erwartungswert} +\begin{align*} +E(A) +&\uncover<10->{= +\sum_i |w_i|^2 \alpha_i} +\uncover<11->{= +\sum_i \overline{w}_i\alpha_i w_i } +\hspace{5cm} +\\ +&\only<12>{= +\sum_{i,j} \overline{w}_j\alpha_i w_i \langle j|i\rangle} +\uncover<13->{= +\sum_{i} \overline{w}_j\langle j| \sum_i \alpha_i w_i |i\rangle} +\\ +&\uncover<14->{= +\sum_{i,j} \overline{w}_j w_i \langle j| +A|i\rangle} +\uncover<15->{= +\langle \psi| A |\psi\rangle} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/riesz.tex b/vorlesungen/slides/2/hilbertraum/riesz.tex new file mode 100644 index 0000000..437fb3c --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/riesz.tex @@ -0,0 +1,76 @@ +% +% riesz.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Darstellungssatz von Riesz} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Dualraum} +$V$ ein Vektorraum, $V^*$ der Raum aller Linearformen +\[ +f\colon V\to \mathbb{C} +\] +\end{block} +\uncover<3->{% +\begin{block}{Beispiel: $l^\infty$} +$l^\infty=\text{beschränkte Folgen in $\mathbb{C}$}$, +Linearformen: +\begin{align*} +\uncover<4->{ +f(x) +&= +\sum_{i=0}^\infty f_ix_i} +\\ +\uncover<5->{ +\|f\| +&= +\sup_{\|x\|_{\infty}\le 1} +|f(x)|} +\uncover<6->{= +\sum_{k\in\mathbb{N}} |f_k|} +\\ +\uncover<7->{ +\Rightarrow +l^{\infty*} +&= +l^1} +\uncover<9->{\qquad(\ne l^2)} +\\ +\uncover<8->{ +&=\{\text{summierbare Folgen in $\mathbb{C}$}\} +} +\end{align*} + +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiel: $\mathbb{C}^n$} +${\mathbb{C}^n}^* = \mathbb{C}^n$ +\end{block}} +\uncover<10->{% +\begin{theorem}[Riesz] +Zu einer stetigen Linearform $f\colon H\to\mathbb{C}$ gibt es $v\in H$ mit +\[ +f(x) = \langle v,x\rangle +\quad\forall x\in H +\] +und $\|f\| = \|v\|$ +\end{theorem}} +\uncover<11->{% +\begin{block}{Dualraum von $H$} +$H^*=H$ +\end{block}}% +\uncover<12->{% +Der Hilbertraum ist die ``intuitiv richtige, unendlichdimensionale'' +Verallgemeinerung von $\mathbb{C}^n$} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex new file mode 100644 index 0000000..de9383f --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/rieszbeispiel.tex @@ -0,0 +1,107 @@ +% +% rieszbeispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Linearform auf $L^2$-Funktionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Linearform auf $\mathbb{C}^n$} +\begin{align*} +{\color{blue}x}&=\begin{pmatrix}x_1\\x_2\\\vdots\\x_n\end{pmatrix}, +& +f({\color{blue}x}) +&= +\begin{pmatrix}f_1&f_2&\dots&f_n\end{pmatrix} {\color{blue}x} +\\ +\uncover<2->{ +{\color{red}v}&= +\rlap{$ +\begin{pmatrix} +\overline{f}_1&\overline{f}_2&\dots&\overline{f}_n +\end{pmatrix}^t +\uncover<3->{\;\Rightarrow\; +f({\color{blue}x})=\langle {\color{red}v},{\color{blue}x}\rangle} +$}} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Linearform auf $L^2([a,b])$} +\begin{align*} +{\color{red}x}&\in L^2([a,b]) +\\ +\uncover<5->{ +f&\colon L^2([a,b]) \to \mathbb{C} +: {\color{red}x} \mapsto f({\color{red}x})} +\intertext{\uncover<6->{Riesz-Darstellungssatz: $\exists {\color{blue}v}\in L^2([a,b])$}} +\uncover<7->{f({\color{red}x}) +&= +\int_a^b {\color{blue}\overline{v}(t)}{\color{red}x(t)}\,dt} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope}[xshift=-3.5cm] +\def\s{0.058} +\foreach \n in {0,...,5}{ +\uncover<3->{ + \draw[color=red,line width=3pt] + ({\n+\s},{1/(\n+0.5)}) -- ({\n+\s},0); + \node[color=red] at ({\n},{-0.2+1/(\n+0.5)}) + [above right] {$v_\n\mathstrut$}; +} + \draw[color=blue,line width=3pt] + ({\n-\s},{0.4+0.55*sin(200*\n)+0.25*\n}) -- ({\n-\s},0); + \node[color=blue] at ({\n},{-0.2+0.4+0.55*sin(200*\n)+0.25*\n}) + [above left] {$x_\n\mathstrut$}; +} +\draw[->] (-0.6,0) -- (6,0) coordinate[label={$n$}]; +\draw[->] (-0.5,-0.1) -- (-0.5,2.5) coordinate[label={right:$x$}]; +\foreach \n in {0,...,5}{ + \fill (\n,0) circle[radius=0.08]; + \node at (\n,0) [below] {$\n$\strut}; +} +\node at (5.6,0) [below] {$\cdots$\strut}; +\end{scope} +\uncover<4->{ +\begin{scope}[xshift=3.5cm] +\uncover<7->{ +\fill[color=red!40,opacity=0.5] + plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)}) + -- + (5,0) -- (0,0) -- cycle; +} +\fill[color=blue!40,opacity=0.5] + plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x}) + -- (5,0) -- (0,0) -- cycle; +\uncover<7->{ +\draw[color=red,line width=1.4pt] + plot[domain=0:5,samples=100] (\x,{1/(\x+0.5)}); +\node[color=red] at (0,2) [right] {$x(t)$}; +} + +\draw[color=blue,line width=1.4pt] + plot[domain=0:5,samples=100] (\x,{0.4+0.55*sin(200*\x)+0.25*\x}); +\node[color=blue] at (4.5,2) [right]{$v(t)$}; + +\draw[->] (-0.6,0) -- (6.0,0) coordinate[label={$t$}]; +\draw[->] (-0.5,-0.1) -- (-0.5,2.5) coordinate[label={right:$x$}]; +\draw (0.0,-0.1) -- (0.0,0.1); +\node at (0.0,0) [below] {$a$\strut}; +\draw (5.0,-0.1) -- (5.0,0.1); +\node at (5.0,0) [below] {$b$\strut}; +\end{scope} +} +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/sobolev.tex b/vorlesungen/slides/2/hilbertraum/sobolev.tex new file mode 100644 index 0000000..828d34d --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/sobolev.tex @@ -0,0 +1,51 @@ +% +% sobolev.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Sobolev-Raum} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Vektorrraum $W$} +Funktionen $f\colon \Omega\to\mathbb{C}$ +\begin{itemize} +\item<2-> +$f\in L^2(\Omega)$ +\item<3-> +$\nabla f\in L^2(\Omega)$ +\item<4-> +homogene Randbedingungen: +$f_{|\partial \Omega}=0$ +\end{itemize} +\end{block} +\uncover<5->{% +\begin{block}{Skalarprodukt} +\begin{align*} +\langle f,g\rangle_W +&\uncover<6->{= +\int_\Omega \overline{\nabla f}(x)\cdot\nabla g(x)\,d\mu(x)} +\\ +&\uncover<7->{\qquad + \int_{\Omega} \overline{f}(x)\,g(x)\,d\mu(x)} +\\ +&\uncover<8->{=\langle f,-\Delta g + g\rangle_{L^2(\Omega)}} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<9->{% +\begin{block}{Vollständigkeit} +\dots +\end{block}} +\uncover<10->{% +\begin{block}{Anwendung} +``Ein Hilbertraum für jedes partielle Differentialgleichungsproblem'' +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/spektral.tex b/vorlesungen/slides/2/hilbertraum/spektral.tex new file mode 100644 index 0000000..b561b69 --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/spektral.tex @@ -0,0 +1,91 @@ +% +% spektral.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spektraltheorie für selbstadjungierte Operatoren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Voraussetzungen} +\begin{itemize} +\item +Hilbertraum $H$ +\item +$A\colon H\to H$ linear +\end{itemize} +\end{block} +\uncover<2->{% +\begin{block}{Eigenwerte} +$x\in H$ ein EV von $A$ zum EW $\lambda\ne 0$ +\begin{align*} +\uncover<3->{\langle x,x\rangle +&= +\frac1{\lambda} +\langle x,\lambda x\rangle} +\uncover<3->{= +\frac1{\lambda} +\langle x,Ax\rangle} +\\ +&\uncover<4->{= +\frac1{\lambda} +\langle Ax,x\rangle} +\uncover<5->{= +\frac{\overline{\lambda}}{\lambda} +\langle x,x\rangle} +\\ +\uncover<6->{\frac{\overline{\lambda}}{\lambda}&=1 +\quad\Rightarrow\quad +\overline{\lambda} = \lambda} +\uncover<7->{\quad\Rightarrow\quad +\lambda\in\mathbb{R}} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Orthogonalität} +$u,v$ EV zu EW $\mu,\lambda\in \mathbb{R}\setminus\{0\}$, $\overline{\mu}=\mu\ne\lambda$ +\begin{align*} +\uncover<9->{ +\langle u,v\rangle +&= +\frac{1}{\mu} +\langle \mu u,v\rangle} +\uncover<10->{= +\frac{1}{\mu} +\langle Au,v\rangle} +\\ +&\uncover<11->{= +\frac{1}{\mu} +\langle u,Av\rangle} +\uncover<12->{= +\frac{1}{\mu} +\langle u,\lambda v\rangle} +\uncover<13->{= +\frac{\lambda}{\mu} +\langle u,v\rangle} +\\ +\uncover<14->{\Rightarrow +\; +0 +&= +\underbrace{\biggl(\frac{\lambda}{\mu}-1\biggr)}_{\displaystyle \ne 0} +\langle u,v\rangle} +\uncover<15->{\;\Rightarrow\; +\langle u,v\rangle = 0} +\end{align*} +\uncover<16->{EV zu verschiedenen EW sind orthogonal} +\end{block}} +\end{column} +\end{columns} +\uncover<17->{% +\begin{block}{Spektralsatz} +Es gibt eine Hilbertbasis von $H$ aus Eigenvektoren von $A$ +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/hilbertraum/sturm.tex b/vorlesungen/slides/2/hilbertraum/sturm.tex new file mode 100644 index 0000000..a6865ab --- /dev/null +++ b/vorlesungen/slides/2/hilbertraum/sturm.tex @@ -0,0 +1,58 @@ +% +% sturm.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Sturm-Liouville-Problem} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Wellengleichung} +Saite mit variabler Massedichte führt auf die DGL +\[ +-y''(t) + q(t) y(t) = \lambda y(t), +\quad +q(t) > 0 +\] +mit Randbedingungen $y(0)=y(1)=0$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Sturm-Liouville-Operator} +\[ +A=-\frac{d^2}{dt^2} + q(t) = -D^2 + p +\] +auf differenzierbaren Funktionen $\Omega=[0,1]\to\mathbb{C}$ mit Randwerten +\[ +f(0)=f(1)=0 +\] +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Selbstadjungiert} +\begin{align*} +\langle f,Ag \rangle +&\uncover<4->{= +\langle f,-D^2 g\rangle + \langle f,qg\rangle += +- +\int_0^1 \overline{f}(t) \frac{d^2}{dt^2}g(t)\,dt ++\langle f,qg\rangle} +\\ +&\uncover<5->{=-\underbrace{[\overline{f}(t)g'(t)]_0^1}_{\displaystyle=0} ++\int_0^1 \overline{f}'(t)g'(t)\,dt ++\langle f,qg\rangle} +\uncover<6->{=-\int_0^1 \overline{f}''(t)g(t)\,dt ++\langle qf,g\rangle} +\\ +&\uncover<7->{=\langle Af,g\rangle} +\end{align*} +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 1ab27fa..5aac429 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -1,36 +1,36 @@ -
-#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-chapter4 = \
- ../slides/4/ggt.tex \
- ../slides/4/euklidmatrix.tex \
- ../slides/4/euklidbeispiel.tex \
- ../slides/4/euklidtabelle.tex \
- ../slides/4/fp.tex \
- ../slides/4/division.tex \
- ../slides/4/gauss.tex \
- ../slides/4/dh.tex \
- ../slides/4/divisionpoly.tex \
- ../slides/4/euklidpoly.tex \
- ../slides/4/polynomefp.tex \
- ../slides/4/schieberegister.tex \
- ../slides/4/charakteristik.tex \
- ../slides/4/char2.tex \
- ../slides/4/frobenius.tex \
- ../slides/4/qundr.tex \
- ../slides/4/alpha.tex \
- ../slides/4/galois/erweiterung.tex \
- ../slides/4/galois/automorphismus.tex \
- ../slides/4/galois/konstruktion.tex \
- ../slides/4/galois/wuerfel.tex \
- ../slides/4/galois/winkeldreiteilung.tex \
- ../slides/4/galois/quadratur.tex \
- ../slides/4/galois/radikale.tex \
- ../slides/4/galois/aufloesbarkeit.tex \
- ../slides/4/galois/sn.tex \
- ../slides/4/chapter.tex
-
-
+ +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter4 = \ + ../slides/4/ggt.tex \ + ../slides/4/euklidmatrix.tex \ + ../slides/4/euklidbeispiel.tex \ + ../slides/4/euklidtabelle.tex \ + ../slides/4/fp.tex \ + ../slides/4/division.tex \ + ../slides/4/gauss.tex \ + ../slides/4/dh.tex \ + ../slides/4/divisionpoly.tex \ + ../slides/4/euklidpoly.tex \ + ../slides/4/polynomefp.tex \ + ../slides/4/schieberegister.tex \ + ../slides/4/charakteristik.tex \ + ../slides/4/char2.tex \ + ../slides/4/frobenius.tex \ + ../slides/4/qundr.tex \ + ../slides/4/alpha.tex \ + ../slides/4/galois/erweiterung.tex \ + ../slides/4/galois/automorphismus.tex \ + ../slides/4/galois/konstruktion.tex \ + ../slides/4/galois/wuerfel.tex \ + ../slides/4/galois/winkeldreiteilung.tex \ + ../slides/4/galois/quadratur.tex \ + ../slides/4/galois/radikale.tex \ + ../slides/4/galois/aufloesbarkeit.tex \ + ../slides/4/galois/sn.tex \ + ../slides/4/chapter.tex + + diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 3015e7c..0691e39 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -1,31 +1,31 @@ -%
-% chapter.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
-%
-\folie{4/ggt.tex}
-\folie{4/euklidmatrix.tex}
-\folie{4/euklidbeispiel.tex}
-\folie{4/euklidtabelle.tex}
-\folie{4/fp.tex}
-\folie{4/division.tex}
-\folie{4/gauss.tex}
-\folie{4/dh.tex}
-\folie{4/divisionpoly.tex}
-\folie{4/euklidpoly.tex}
-\folie{4/polynomefp.tex}
-\folie{4/alpha.tex}
-\folie{4/schieberegister.tex}
-\folie{4/charakteristik.tex}
-\folie{4/char2.tex}
-\folie{4/frobenius.tex}
-\folie{4/qundr.tex}
-\folie{4/galois/erweiterung.tex}
-\folie{4/galois/automorphismus.tex}
-\folie{4/galois/konstruktion.tex}
-\folie{4/galois/wuerfel.tex}
-\folie{4/galois/winkeldreiteilung.tex}
-\folie{4/galois/quadratur.tex}
-\folie{4/galois/radikale.tex}
-\folie{4/galois/aufloesbarkeit.tex}
-\folie{4/galois/sn.tex}
+% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{4/ggt.tex} +\folie{4/euklidmatrix.tex} +\folie{4/euklidbeispiel.tex} +\folie{4/euklidtabelle.tex} +\folie{4/fp.tex} +\folie{4/division.tex} +\folie{4/gauss.tex} +\folie{4/dh.tex} +\folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} +\folie{4/polynomefp.tex} +\folie{4/alpha.tex} +\folie{4/schieberegister.tex} +\folie{4/charakteristik.tex} +\folie{4/char2.tex} +\folie{4/frobenius.tex} +\folie{4/qundr.tex} +\folie{4/galois/erweiterung.tex} +\folie{4/galois/automorphismus.tex} +\folie{4/galois/konstruktion.tex} +\folie{4/galois/wuerfel.tex} +\folie{4/galois/winkeldreiteilung.tex} +\folie{4/galois/quadratur.tex} +\folie{4/galois/radikale.tex} +\folie{4/galois/aufloesbarkeit.tex} +\folie{4/galois/sn.tex} diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex index 3d52b00..ef5902b 100644 --- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex +++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex @@ -1,120 +1,120 @@ -%
-% aufloesbarkeit.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Auflösbarkeit}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Radikalerweiterung}
-Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$
-einer Radikalerweiterung
-\[
-\Bbbk \subset \Bbbk(\alpha)
-\]
-sind festgelegt durch Wahl von $f(\alpha)$.
-
-\begin{itemize}
-\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt
-\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein
-\end{itemize}
-
-\end{block}}
-\uncover<8->{%
-\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$}
-$\mathbb{Q}\subset \Bbbk$,
-$n$ verschiedene Nullstellen $\mathbb{C}$:
-\[
-\uncover<9->{
-\operatorname{Gal}(\Bbbk/\mathbb{Q})
-\cong
-S_n}
-\uncover<10->{
-\quad
-\text{auflösbar?}}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{\uncover<5->{Galois-Gruppen}}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{1.2}
-
-\uncover<2->{
-\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3});
-\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen};
-}
-
-\node at (0,0) {$\mathbb{Q}$};
-\node at (0,{1*\s}) {$E_1$};
-\node at (0,{2*\s}) {$E_2$};
-\node at (0,{3*\s}) {$E_3$};
-\node at (0,{4*\s}) {$\vdots\mathstrut$};
-\node at (0,{5*\s}) {$\Bbbk$};
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s});
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s});
-
-\begin{scope}[xshift=0.5cm]
-\uncover<7->{
-\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3});
-\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe};
-}
-\uncover<5->{
-\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$};
-\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$};
-\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$};
-\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$};
-\node at (1,{4*\s}) {$\vdots\mathstrut$};
-\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$};
-\node at (1,{0.5*\s}) {$\cap\mathstrut$};
-\node at (1,{1.5*\s}) {$\cap\mathstrut$};
-\node at (1,{2.5*\s}) {$\cap\mathstrut$};
-\node at (1,{3.5*\s}) {$\cap\mathstrut$};
-\node at (1,{4.5*\s}) {$\cap\mathstrut$};
-}
-
-\uncover<6->{
-\begin{scope}[xshift=2.5cm]
-\node at (0,{0*\s}) {$G_n$};
-\node at (0,{1*\s}) {$G_{n-1}$};
-\node at (0,{2*\s}) {$G_{n-2}$};
-\node at (0,{3*\s}) {$G_{n-3}$};
-\node at (0,{5*\s}) {$G_0=\{e\}$};
-\node at (0,{0.5*\s}) {$\cap\mathstrut$};
-\node at (0,{1.5*\s}) {$\cap\mathstrut$};
-\node at (0,{2.5*\s}) {$\cap\mathstrut$};
-\node at (0,{3.5*\s}) {$\cap\mathstrut$};
-\node at (0,{4.5*\s}) {$\cap\mathstrut$};
-}
-
-\uncover<7->{
-\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$};
-\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch};
-
-\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$};
-\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch};
-
-\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$};
-\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch};
-}
-
-\end{scope}
-\end{scope}
-
-
-
-\end{tikzpicture}
-\end{center}
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
+% +% aufloesbarkeit.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Auflösbarkeit} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Radikalerweiterung} +Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$ +einer Radikalerweiterung +\[ +\Bbbk \subset \Bbbk(\alpha) +\] +sind festgelegt durch Wahl von $f(\alpha)$. + +\begin{itemize} +\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt +\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein +\end{itemize} + +\end{block}} +\uncover<8->{% +\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$} +$\mathbb{Q}\subset \Bbbk$, +$n$ verschiedene Nullstellen $\mathbb{C}$: +\[ +\uncover<9->{ +\operatorname{Gal}(\Bbbk/\mathbb{Q}) +\cong +S_n} +\uncover<10->{ +\quad +\text{auflösbar?}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{\uncover<5->{Galois-Gruppen}} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{1.2} + +\uncover<2->{ +\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3}); +\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen}; +} + +\node at (0,0) {$\mathbb{Q}$}; +\node at (0,{1*\s}) {$E_1$}; +\node at (0,{2*\s}) {$E_2$}; +\node at (0,{3*\s}) {$E_3$}; +\node at (0,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) {$\Bbbk$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s}); + +\begin{scope}[xshift=0.5cm] +\uncover<7->{ +\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3}); +\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe}; +} +\uncover<5->{ +\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$}; +\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$}; +\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$}; +\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$}; +\node at (1,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$}; +\node at (1,{0.5*\s}) {$\cap\mathstrut$}; +\node at (1,{1.5*\s}) {$\cap\mathstrut$}; +\node at (1,{2.5*\s}) {$\cap\mathstrut$}; +\node at (1,{3.5*\s}) {$\cap\mathstrut$}; +\node at (1,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<6->{ +\begin{scope}[xshift=2.5cm] +\node at (0,{0*\s}) {$G_n$}; +\node at (0,{1*\s}) {$G_{n-1}$}; +\node at (0,{2*\s}) {$G_{n-2}$}; +\node at (0,{3*\s}) {$G_{n-3}$}; +\node at (0,{5*\s}) {$G_0=\{e\}$}; +\node at (0,{0.5*\s}) {$\cap\mathstrut$}; +\node at (0,{1.5*\s}) {$\cap\mathstrut$}; +\node at (0,{2.5*\s}) {$\cap\mathstrut$}; +\node at (0,{3.5*\s}) {$\cap\mathstrut$}; +\node at (0,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<7->{ +\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$}; +\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$}; +\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$}; +\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch}; +} + +\end{scope} +\end{scope} + + + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex index e59f9b9..6051813 100644 --- a/vorlesungen/slides/4/galois/automorphismus.tex +++ b/vorlesungen/slides/4/galois/automorphismus.tex @@ -1,118 +1,118 @@ -%
-% automorphismus.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{4pt}
-\setlength{\belowdisplayskip}{4pt}
-\frametitle{Galois-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.40\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{3.0}
-\begin{scope}[xshift=-1.5cm]
-\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut};
-\node at (0,{\s}) {$G$};
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0);
-\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0);
-\node at (0,{-0.5*\s}) [left] {$[F:E]$};
-\node at (0,{0.5*\s}) [left] {$[G:F]$};
-\node at (0,0) {$F$};
-\node at (0,{-\s}) {$E$};
-\end{scope}
-\uncover<3->{
-\begin{scope}[xshift=1.8cm]
-\node at (0,{\s+0.1}) [above] {Gruppe\strut};
-\fill (0,{-\s}) circle[radius=0.06];
-\fill (0,0) circle[radius=0.06];
-\fill (0,{\s}) circle[radius=0.06];
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
- (0,{-\s}) to[out=100,in=-100] (0,{\s});
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
- (0,{-\s}) to[out=80,in=-80] (0,0);
-\draw[shorten >= 0.1cm,shorten <= 0.1cm]
- (0,0) to[out=80,in=-80] (0,{\s});
-\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$};
-\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$};
-\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$};
-\end{scope}
-\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s});
-\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$};
-}
-\uncover<4->{
-\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s});
-\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<2->{%
-\begin{block}{Automorphismus}
-\vspace{-10pt}
-\[
-\operatorname{Aut}(F)
-=
-\left\{
-f\colon F\to F
-\left|
-\begin{aligned}
-f(x+y)&=f(x)+f(y)\\
-f(xy)&=f(x)f(y)
-\end{aligned}
-\right.
-\right\}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<3->{%
-\begin{block}{Galois-Gruppe}
-Automorphismen, die $E$ festlassen
-\[
-{\color{red}
-\operatorname{Gal}(F/E)
-}
-=
-\left\{
-\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E
-\right\}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<4->{%
-\begin{block}{Fixkörper}
-$H\subset \operatorname{Aut}(F)$:
-\begin{align*}
-{\color{blue}F^H}
-&=
-\{x\in F\;|\; hx = x\forall h\in H\}
-=\operatorname{Fix}(H)
-\end{align*}
-\end{block}}
-\vspace{-13pt}
-\uncover<5->{%
-\begin{block}{Beispiel}
-\begin{itemize}
-\item<6->
-\(
-\operatorname{Gal}(\mathbb{C}/\mathbb{R})
-=
-\{
-\operatorname{id}_{\mathbb{C}},
-\operatorname{conj}\colon z\mapsto\overline{z}
-\}
-\)
-\item<7->
-\(
-\mathbb{C}^{\operatorname{conj}}
-=
-\mathbb{R}
-\)
-\end{itemize}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% automorphismus.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{4pt} +\setlength{\belowdisplayskip}{4pt} +\frametitle{Galois-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{3.0} +\begin{scope}[xshift=-1.5cm] +\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut}; +\node at (0,{\s}) {$G$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0); +\node at (0,{-0.5*\s}) [left] {$[F:E]$}; +\node at (0,{0.5*\s}) [left] {$[G:F]$}; +\node at (0,0) {$F$}; +\node at (0,{-\s}) {$E$}; +\end{scope} +\uncover<3->{ +\begin{scope}[xshift=1.8cm] +\node at (0,{\s+0.1}) [above] {Gruppe\strut}; +\fill (0,{-\s}) circle[radius=0.06]; +\fill (0,0) circle[radius=0.06]; +\fill (0,{\s}) circle[radius=0.06]; +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=100,in=-100] (0,{\s}); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=80,in=-80] (0,0); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,0) to[out=80,in=-80] (0,{\s}); +\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$}; +\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$}; +\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$}; +\end{scope} +\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s}); +\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$}; +} +\uncover<4->{ +\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s}); +\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<2->{% +\begin{block}{Automorphismus} +\vspace{-10pt} +\[ +\operatorname{Aut}(F) += +\left\{ +f\colon F\to F +\left| +\begin{aligned} +f(x+y)&=f(x)+f(y)\\ +f(xy)&=f(x)f(y) +\end{aligned} +\right. +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Galois-Gruppe} +Automorphismen, die $E$ festlassen +\[ +{\color{red} +\operatorname{Gal}(F/E) +} += +\left\{ +\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Fixkörper} +$H\subset \operatorname{Aut}(F)$: +\begin{align*} +{\color{blue}F^H} +&= +\{x\in F\;|\; hx = x\forall h\in H\} +=\operatorname{Fix}(H) +\end{align*} +\end{block}} +\vspace{-13pt} +\uncover<5->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<6-> +\( +\operatorname{Gal}(\mathbb{C}/\mathbb{R}) += +\{ +\operatorname{id}_{\mathbb{C}}, +\operatorname{conj}\colon z\mapsto\overline{z} +\} +\) +\item<7-> +\( +\mathbb{C}^{\operatorname{conj}} += +\mathbb{R} +\) +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex index 20b278e..6909849 100644 --- a/vorlesungen/slides/4/galois/erweiterung.tex +++ b/vorlesungen/slides/4/galois/erweiterung.tex @@ -1,65 +1,65 @@ -%
-% erweiterung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Körpererweiterungen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Körpererweiterung}
-$E,F$ Körper: $E\subset F$
-\end{block}
-\uncover<6->{%
-\begin{block}{Vektorraum}
-$F$ ist ein Vektorraum über $E$
-\end{block}}
-\uncover<7->{%
-\begin{block}{Endliche Körpererweiterung}
-$\dim_E F < \infty$
-\end{block}}
-\uncover<8->{%
-\begin{block}{Adjunktion eines $\alpha$}
-$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und
-$\alpha$ enthält.
-\end{block}}
-\uncover<9->{%
-\begin{block}{Algebraische Erweiterung}
-$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von
-$m(X)\in\Bbbk[X]$
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Beispiele}
-\begin{enumerate}
-\item<3->
-$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$
-\item<4->
-$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$
-\item<5->
-$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$
-\end{enumerate}
-\end{block}}
-\uncover<7->{%
-\begin{block}{Grad}
-$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls
-\[
-\dim_E F = n =: [F:E]
-\]
-\uncover<8->{%
-Gleichbedeutend: $\deg m(X) = n$}
-\uncover<10->{%
-\[
-E\subset F\subset G
-\Rightarrow
-[G:E] = [G:F]\cdot [F:E]
-\]
-(in unseren Fällen)}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% erweiterung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Körpererweiterungen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Körpererweiterung} +$E,F$ Körper: $E\subset F$ +\end{block} +\uncover<6->{% +\begin{block}{Vektorraum} +$F$ ist ein Vektorraum über $E$ +\end{block}} +\uncover<7->{% +\begin{block}{Endliche Körpererweiterung} +$\dim_E F < \infty$ +\end{block}} +\uncover<8->{% +\begin{block}{Adjunktion eines $\alpha$} +$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und +$\alpha$ enthält. +\end{block}} +\uncover<9->{% +\begin{block}{Algebraische Erweiterung} +$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von +$m(X)\in\Bbbk[X]$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiele} +\begin{enumerate} +\item<3-> +$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$ +\item<4-> +$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$ +\item<5-> +$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$ +\end{enumerate} +\end{block}} +\uncover<7->{% +\begin{block}{Grad} +$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls +\[ +\dim_E F = n =: [F:E] +\] +\uncover<8->{% +Gleichbedeutend: $\deg m(X) = n$} +\uncover<10->{% +\[ +E\subset F\subset G +\Rightarrow +[G:E] = [G:F]\cdot [F:E] +\] +(in unseren Fällen)} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile index fd197ce..444944e 100644 --- a/vorlesungen/slides/4/galois/images/Makefile +++ b/vorlesungen/slides/4/galois/images/Makefile @@ -1,12 +1,12 @@ -#
-# Makefile
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-all: wuerfel2.png wuerfel.png
-
-wuerfel.png: wuerfel.pov common.inc
- povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov
-
-wuerfel2.png: wuerfel2.pov common.inc
- povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov
+# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: wuerfel2.png wuerfel.png + +wuerfel.png: wuerfel.pov common.inc + povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov + +wuerfel2.png: wuerfel2.pov common.inc + povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc index 44ee4c8..6cfcabe 100644 --- a/vorlesungen/slides/4/galois/images/common.inc +++ b/vorlesungen/slides/4/galois/images/common.inc @@ -1,89 +1,89 @@ -//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-#include "textures.inc"
-#include "stones.inc"
-
-global_settings {
- assumed_gamma 1
-}
-
-#declare imagescale = 0.133;
-#declare O = <0, 0, 0>;
-#declare E = <1, 1, 1>;
-#declare a = pow(2, 1/3);
-#declare at = 0.02;
-
-camera {
- location <3, 2, 12>
- look_at E * (a / 2) * 0.93
- right x * imagescale
- up y * imagescale
-}
-
-light_source {
- <11, 20, 16> color White
- area_light <1,0,0> <0,0,1>, 10, 10
- adaptive 1
- jitter
-}
-
-sky_sphere {
- pigment {
- color rgb<1,1,1>
- }
-}
-
-#macro wuerfelgitter(A, AT)
- cylinder { O, <A, 0, 0>, AT }
- cylinder { O, <0, A, 0>, AT }
- cylinder { O, <0, 0, A>, AT }
- cylinder { <A, 0, 0>, <A, A, 0>, AT }
- cylinder { <A, 0, 0>, <A, 0, A>, AT }
- cylinder { <0, A, 0>, <A, A, 0>, AT }
- cylinder { <0, A, 0>, <0, A, A>, AT }
- cylinder { <0, 0, A>, <A, 0, A>, AT }
- cylinder { <0, 0, A>, <0, A, A>, AT }
- cylinder { <A, A, 0>, <A, A, A>, AT }
- cylinder { <A, 0, A>, <A, A, A>, AT }
- cylinder { <0, A, A>, <A, A, A>, AT }
- sphere { <0, 0, 0>, AT }
- sphere { <A, 0, 0>, AT }
- sphere { <0, A, 0>, AT }
- sphere { <0, 0, A>, AT }
- sphere { <A, A, 0>, AT }
- sphere { <A, 0, A>, AT }
- sphere { <0, A, A>, AT }
- sphere { <A, A, A>, AT }
-#end
-
-#macro wuerfel()
- union {
- box { O, E }
- wuerfelgitter(1, 0.5*at)
- texture {
- T_Grnt24
- }
- finish {
- specular 0.9
- metallic
- }
- }
-#end
-
-#macro wuerfel2()
- union {
- wuerfelgitter(a, at)
- pigment {
- color rgb<0.8,0.4,0.4>
- }
- finish {
- specular 0.9
- metallic
- }
- }
-#end
+// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" +#include "textures.inc" +#include "stones.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.133; +#declare O = <0, 0, 0>; +#declare E = <1, 1, 1>; +#declare a = pow(2, 1/3); +#declare at = 0.02; + +camera { + location <3, 2, 12> + look_at E * (a / 2) * 0.93 + right x * imagescale + up y * imagescale +} + +light_source { + <11, 20, 16> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro wuerfelgitter(A, AT) + cylinder { O, <A, 0, 0>, AT } + cylinder { O, <0, A, 0>, AT } + cylinder { O, <0, 0, A>, AT } + cylinder { <A, 0, 0>, <A, A, 0>, AT } + cylinder { <A, 0, 0>, <A, 0, A>, AT } + cylinder { <0, A, 0>, <A, A, 0>, AT } + cylinder { <0, A, 0>, <0, A, A>, AT } + cylinder { <0, 0, A>, <A, 0, A>, AT } + cylinder { <0, 0, A>, <0, A, A>, AT } + cylinder { <A, A, 0>, <A, A, A>, AT } + cylinder { <A, 0, A>, <A, A, A>, AT } + cylinder { <0, A, A>, <A, A, A>, AT } + sphere { <0, 0, 0>, AT } + sphere { <A, 0, 0>, AT } + sphere { <0, A, 0>, AT } + sphere { <0, 0, A>, AT } + sphere { <A, A, 0>, AT } + sphere { <A, 0, A>, AT } + sphere { <0, A, A>, AT } + sphere { <A, A, A>, AT } +#end + +#macro wuerfel() + union { + box { O, E } + wuerfelgitter(1, 0.5*at) + texture { + T_Grnt24 + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro wuerfel2() + union { + wuerfelgitter(a, at) + pigment { + color rgb<0.8,0.4,0.4> + } + finish { + specular 0.9 + metallic + } + } +#end diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov index a0466f3..a5db465 100644 --- a/vorlesungen/slides/4/galois/images/wuerfel.pov +++ b/vorlesungen/slides/4/galois/images/wuerfel.pov @@ -1,9 +1,9 @@ -//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-
+// +// wuerfel.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +wuerfel() + diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov index a11bab0..ac32b2f 100644 --- a/vorlesungen/slides/4/galois/images/wuerfel2.pov +++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov @@ -1,9 +1,9 @@ -//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-wuerfel2()
+// +// wuerfel.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +wuerfel() +wuerfel2() diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex index b461d44..094b570 100644 --- a/vorlesungen/slides/4/galois/konstruktion.tex +++ b/vorlesungen/slides/4/galois/konstruktion.tex @@ -1,147 +1,147 @@ -%
-% konstruktion.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Konstruktion mit Zirkel und Lineal}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Strahlensatz}
-\uncover<6->{%
-Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$
-kann mit Zirkel und Lineal konstruiert werden.}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Kreis--Gerade}
-Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$
-\uncover<13->{%
-$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}
-\end{block}}
-\end{column}
-\end{columns}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{0.5}
-\def\t{0.45}
-
-\coordinate (A) at (0,0);
-\coordinate (B) at ({10*\t},0);
-
-\uncover<2->{
- \draw (0,0) -- (30:{10.5*\s});
-}
-
-\uncover<3->{
- \foreach \x in {0,...,10}{
- \fill (30:{\x*\s}) circle[radius=0.03];
- }
- \foreach \x in {0,1,2,3,4,7,8,9}{
- \node at (30:{\x*\s}) [above] {\tiny $\x$};
- }
- \node at (30:{10*\s}) [above right] {$q=10$};
-}
-
-\uncover<4->{
- \foreach \x in {1,...,10}{
- \fill (0:{\x*\t}) circle[radius=0.03];
- \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});
- }
-}
-
-\draw (A) -- (0:{10.5*\t});
-\node at (A) [below left] {$A$};
-\node at (B) [below right] {$B$};
-\fill (A) circle[radius=0.05];
-\fill (B) circle[radius=0.05];
-
-\uncover<5->{
- \node at (30:{6*\s}) [above left] {$p=6$};
- \draw[line width=0.2pt] (0,0) -- (0,-0.4);
- \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);
- \draw[<->] (0,-0.3) -- ({6*\t},-0.3);
- \node at ({3*\t},-0.4) [below]
- {$\displaystyle\frac{p}{q}\cdot\overline{AB}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<8->{%
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-%\foreach \x in {8,...,14}{
-% \only<\x>{\node at (4,4) {$\x$};}
-%}
-
-\def\r{4}
-\def\a{50}
-
-\coordinate (A) at ({\r*cos(\a)},0);
-
-\uncover<10->{
- \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;
- \fill[color=gray]
- (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;
-}
-
-\draw[->] (0,0) -- (95:\r);
-\node at (95:{0.5*\r}) [left] {$c$};
-
-\begin{scope}
- \clip (-1,-0.3) rectangle (4.5,4.1);
- \uncover<10->{
- \draw (-1,0) -- (5,0);
- \draw[->] (0,0) -- (\r,0);
- \draw (0,0) circle[radius=\r];
- \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);
- }
-\end{scope}
-
-\uncover<11->{
- \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;
-}
-
-\uncover<9->{
- \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;
- \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];
- \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);
-}
-
-\uncover<11->{
- \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);
- \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};
-}
-
-\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);
-\fill[color=blue] (0,0) circle[radius=0.04];
-\fill[color=blue] (A) circle[radius=0.04];
-\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};
-
-\uncover<12->{
- \fill[color=white,opacity=0.8]
- ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})
- rectangle
- ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});
-
- \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]
- {$b=\sqrt{c^2-a^2}$};
- \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);
- \fill[color=red] (\a:\r) circle[radius=0.05];
- \fill[color=red] (A) circle[radius=0.05];
-}
-
-\end{tikzpicture}
-\end{center}}
-\end{column}
-\end{columns}
-\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:}
-Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$}
-\end{frame}
+% +% konstruktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Konstruktion mit Zirkel und Lineal} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Strahlensatz} +\uncover<6->{% +Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$ +kann mit Zirkel und Lineal konstruiert werden.} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Kreis--Gerade} +Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$ +\uncover<13->{% +$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden} +\end{block}} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{0.5} +\def\t{0.45} + +\coordinate (A) at (0,0); +\coordinate (B) at ({10*\t},0); + +\uncover<2->{ + \draw (0,0) -- (30:{10.5*\s}); +} + +\uncover<3->{ + \foreach \x in {0,...,10}{ + \fill (30:{\x*\s}) circle[radius=0.03]; + } + \foreach \x in {0,1,2,3,4,7,8,9}{ + \node at (30:{\x*\s}) [above] {\tiny $\x$}; + } + \node at (30:{10*\s}) [above right] {$q=10$}; +} + +\uncover<4->{ + \foreach \x in {1,...,10}{ + \fill (0:{\x*\t}) circle[radius=0.03]; + \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t}); + } +} + +\draw (A) -- (0:{10.5*\t}); +\node at (A) [below left] {$A$}; +\node at (B) [below right] {$B$}; +\fill (A) circle[radius=0.05]; +\fill (B) circle[radius=0.05]; + +\uncover<5->{ + \node at (30:{6*\s}) [above left] {$p=6$}; + \draw[line width=0.2pt] (0,0) -- (0,-0.4); + \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4); + \draw[<->] (0,-0.3) -- ({6*\t},-0.3); + \node at ({3*\t},-0.4) [below] + {$\displaystyle\frac{p}{q}\cdot\overline{AB}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +%\foreach \x in {8,...,14}{ +% \only<\x>{\node at (4,4) {$\x$};} +%} + +\def\r{4} +\def\a{50} + +\coordinate (A) at ({\r*cos(\a)},0); + +\uncover<10->{ + \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle; + \fill[color=gray] + (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle; +} + +\draw[->] (0,0) -- (95:\r); +\node at (95:{0.5*\r}) [left] {$c$}; + +\begin{scope} + \clip (-1,-0.3) rectangle (4.5,4.1); + \uncover<10->{ + \draw (-1,0) -- (5,0); + \draw[->] (0,0) -- (\r,0); + \draw (0,0) circle[radius=\r]; + \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5); + } +\end{scope} + +\uncover<11->{ + \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle; +} + +\uncover<9->{ + \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle; + \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07]; + \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1); +} + +\uncover<11->{ + \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r); + \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$}; +} + +\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0); +\fill[color=blue] (0,0) circle[radius=0.04]; +\fill[color=blue] (A) circle[radius=0.04]; +\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$}; + +\uncover<12->{ + \fill[color=white,opacity=0.8] + ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25}) + rectangle + ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25}); + + \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right] + {$b=\sqrt{c^2-a^2}$}; + \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r); + \fill[color=red] (\a:\r) circle[radius=0.05]; + \fill[color=red] (A) circle[radius=0.05]; +} + +\end{tikzpicture} +\end{center}} +\end{column} +\end{columns} +\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:} +Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$} +\end{frame} diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex index f9510ba..f5763b9 100644 --- a/vorlesungen/slides/4/galois/quadratur.tex +++ b/vorlesungen/slides/4/galois/quadratur.tex @@ -1,66 +1,66 @@ -%
-% quadratur.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Quadratur des Kreises}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{center}
-\uncover<2->{%
-\begin{tikzpicture}[>=latex,thick]
-
-\def\r{2.8}
-\pgfmathparse{sqrt(3.14159)*\r/2}
-\xdef\s{\pgfmathresult}
-
-\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);
-\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];
-
-\uncover<3->{
- \draw[->,color=red] (0,0) -- (50:\r);
- \fill[color=red] (0,0) circle[radius=0.04];
- \node[color=red] at (50:{0.5*\r}) [below right] {$r$};
-}
-
-\uncover<4->{
- \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});
- \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});
- \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});
- \node[color=blue] at (0,{-\s-0.6}) [below] {$l$};
-}
-
-\uncover<5->{
- \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}
- \;\Rightarrow\;
- {\color{blue}l}={\color{red}\sqrt{\pi}r}$};
-}
-
-\end{tikzpicture}}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere ein zu einem Kreis flächengleiches Quadrat
-\end{block}
-\uncover<6->{%
-\begin{block}{Modifizierte Aufgabe}
-Konstruiere eine Strecke, deren Länge Lösung der Gleichung
-$x^2-\pi=0$ ist.
-\end{block}}
-\uncover<7->{%
-\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]
-\begin{itemize}
-\item<8-> Lösung in einem Erweiterungskörper
-\item<9-> Lösung ist Nullstelle eines Polynoms
-\item<10-> Lösung ist algebraisch
-\item<11-> $\pi$ ist {\bf nicht} algebraisch
-\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% quadratur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Quadratur des Kreises} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{center} +\uncover<2->{% +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.8} +\pgfmathparse{sqrt(3.14159)*\r/2} +\xdef\s{\pgfmathresult} + +\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s); +\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r]; + +\uncover<3->{ + \draw[->,color=red] (0,0) -- (50:\r); + \fill[color=red] (0,0) circle[radius=0.04]; + \node[color=red] at (50:{0.5*\r}) [below right] {$r$}; +} + +\uncover<4->{ + \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7}); + \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7}); + \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6}); + \node[color=blue] at (0,{-\s-0.6}) [below] {$l$}; +} + +\uncover<5->{ + \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2} + \;\Rightarrow\; + {\color{blue}l}={\color{red}\sqrt{\pi}r}$}; +} + +\end{tikzpicture}} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Aufgabe} +Konstruiere ein zu einem Kreis flächengleiches Quadrat +\end{block} +\uncover<6->{% +\begin{block}{Modifizierte Aufgabe} +Konstruiere eine Strecke, deren Länge Lösung der Gleichung +$x^2-\pi=0$ ist. +\end{block}} +\uncover<7->{% +\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch] +\begin{itemize} +\item<8-> Lösung in einem Erweiterungskörper +\item<9-> Lösung ist Nullstelle eines Polynoms +\item<10-> Lösung ist algebraisch +\item<11-> $\pi$ ist {\bf nicht} algebraisch +\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})} +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex index cb08dca..e9e4ce8 100644 --- a/vorlesungen/slides/4/galois/radikale.tex +++ b/vorlesungen/slides/4/galois/radikale.tex @@ -1,69 +1,69 @@ -%
-% radikale.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Lösung durch Radikale}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Problemstellung}
-Finde Nullstellen eines Polynomes
-\[
-p(X)
-=
-a_nX^n + a_{n-1}X^{n-1}
-+\dots+
-a_1X+a_0
-\]
-$p\in\mathbb{Q}[X]$
-\end{block}
-\uncover<2->{%
-\begin{block}{Radikale}
-Geschachtelte Wurzelausdrücke
-\[
-\sqrt[3]{
--\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
-}
-+
-\sqrt[3]{
--\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
-}
-\]
-\uncover<3->{(Lösung von $x^3+px+q=0$)}
-\end{block}}
-\uncover<4->{%
-\begin{block}{Lösbar durch Radikale}
-Nullstelle von $p(X)$ ist ein Radikal
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<5->{%
-\begin{block}{Algebraische Formulierung}
-Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$,
-finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart,
-dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{:
-$\Bbbk = \mathbb{Q}[X]/(p)$}
-\end{block}}
-\uncover<7->{%
-\begin{block}{Radikalerweiterung}
-Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften
-\begin{itemize}
-\item<8-> $\alpha$ ist eine Einheitswurzel
-\item<9-> $\alpha^k\in\Bbbk$
-\end{itemize}
-\end{block}}
-\vspace{-5pt}
-\uncover<10->{%
-\begin{block}{Lösbar durch Radikale}
-Radikalerweiterungen
-\[
-\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% radikale.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lösung durch Radikale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Problemstellung} +Finde Nullstellen eines Polynomes +\[ +p(X) += +a_nX^n + a_{n-1}X^{n-1} ++\dots+ +a_1X+a_0 +\] +$p\in\mathbb{Q}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Radikale} +Geschachtelte Wurzelausdrücke +\[ +\sqrt[3]{ +-\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} ++ +\sqrt[3]{ +-\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} +\] +\uncover<3->{(Lösung von $x^3+px+q=0$)} +\end{block}} +\uncover<4->{% +\begin{block}{Lösbar durch Radikale} +Nullstelle von $p(X)$ ist ein Radikal +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Algebraische Formulierung} +Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$, +finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart, +dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{: +$\Bbbk = \mathbb{Q}[X]/(p)$} +\end{block}} +\uncover<7->{% +\begin{block}{Radikalerweiterung} +Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften +\begin{itemize} +\item<8-> $\alpha$ ist eine Einheitswurzel +\item<9-> $\alpha^k\in\Bbbk$ +\end{itemize} +\end{block}} +\vspace{-5pt} +\uncover<10->{% +\begin{block}{Lösbar durch Radikale} +Radikalerweiterungen +\[ +\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex index f340825..1cae3fa 100644 --- a/vorlesungen/slides/4/galois/sn.tex +++ b/vorlesungen/slides/4/galois/sn.tex @@ -1,87 +1,87 @@ -%
-% sn.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Nichtauflösbarkeit von $S_n$}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Die symmetrische Gruppe $S_n$}
-Permutationen auf $n$ Elementen
-\[
-\sigma
-=
-\begin{pmatrix}
-1&2&3&\dots&n\\
-\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n)
-\end{pmatrix}
-\]
-\end{block}
-\vspace{-10pt}
-\uncover<2->{%
-\begin{block}{Signum}
-$t(\sigma)=\mathstrut$ Anzahl Transpositionen
-\[
-\operatorname{sgn}(\sigma)
-=
-(-1)^{t(\sigma)}
-=
-\begin{cases}
-\phantom{-}1&\text{$t(\sigma)$ gerade}
-\\
--1&\text{$t(\sigma)$ ungerade}
-\end{cases}
-\]
-Homomorphismus!
-\end{block}}
-\uncover<3->{%
-\begin{block}{Die alternierende Gruppe $A_n$}
-\vspace{-12pt}
-\[
-A_n = \ker \operatorname{sgn}
-=
-\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Normale Untergruppe}
-\begin{itemize}
-\item
-$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$
-\item
-$G/N$ ist wohldefiniert
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<5->{%
-\begin{block}{Einfache Gruppe}
-$G$ einfach $\Leftrightarrow$
-\[
-H\triangleleft G
-\;
-\Rightarrow
-\;
-\text{$H=\{e\}$ oder $H=G$}
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<6->{%
-\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$}
-\begin{enumerate}
-\item<7-> Zeigen, dass $A_5$ einfach ist
-\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach
-\end{enumerate}
-\uncover<9->{%
-$\Rightarrow$ i.~A.~keine Lösung der
-einer Polynomgleichung vom Grad $\ge 5$ durch Radikale
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% sn.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nichtauflösbarkeit von $S_n$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Die symmetrische Gruppe $S_n$} +Permutationen auf $n$ Elementen +\[ +\sigma += +\begin{pmatrix} +1&2&3&\dots&n\\ +\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n) +\end{pmatrix} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Signum} +$t(\sigma)=\mathstrut$ Anzahl Transpositionen +\[ +\operatorname{sgn}(\sigma) += +(-1)^{t(\sigma)} += +\begin{cases} +\phantom{-}1&\text{$t(\sigma)$ gerade} +\\ +-1&\text{$t(\sigma)$ ungerade} +\end{cases} +\] +Homomorphismus! +\end{block}} +\uncover<3->{% +\begin{block}{Die alternierende Gruppe $A_n$} +\vspace{-12pt} +\[ +A_n = \ker \operatorname{sgn} += +\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Normale Untergruppe} +\begin{itemize} +\item +$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$ +\item +$G/N$ ist wohldefiniert +\end{itemize} +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Einfache Gruppe} +$G$ einfach $\Leftrightarrow$ +\[ +H\triangleleft G +\; +\Rightarrow +\; +\text{$H=\{e\}$ oder $H=G$} +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$} +\begin{enumerate} +\item<7-> Zeigen, dass $A_5$ einfach ist +\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach +\end{enumerate} +\uncover<9->{% +$\Rightarrow$ i.~A.~keine Lösung der +einer Polynomgleichung vom Grad $\ge 5$ durch Radikale +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex index 28c07fe..54b941b 100644 --- a/vorlesungen/slides/4/galois/winkeldreiteilung.tex +++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex @@ -1,94 +1,94 @@ -%
-% winkeldreiteilung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Winkeldreiteilung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.43\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\r{5}
-\def\a{25}
-
-\uncover<3->{
- \draw[line width=0.7pt] (\r,0) arc (0:90:\r);
-}
-
-\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
-\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
-
-\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
-
-\uncover<2->{
- \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
- \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
- \node[color=red] at ({0.5*\a}:{0.7*\r})
- {$\displaystyle\frac{\alpha}{3}$};
-}
-
-\uncover<3->{
- \fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
- \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
-
- \fill[color=red] ({\a}:\r) circle[radius=0.05];
- \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
-
- \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
- \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
-}
-
-
-\uncover<4->{
-\node at ({0.5*\r},-0.5) [below] {$\displaystyle
-\cos{\color{blue}\alpha}
-=
-4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
-$};
-}
-
-\uncover<5->{
- \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
- \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.53\textwidth}
-\begin{block}{Aufgabe}
-Teile einen Winkel in drei gleiche Teile
-\end{block}
-\vspace{-2pt}
-\uncover<6->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere $x$ aus $a$ derart, dass
-\[
-p(x)
-=
-x^3-\frac34 x -a = 0
-\]
-\uncover<7->{%
-$a=0$:}
-\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
-\end{block}}
-\vspace{-2pt}
-\uncover<10->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<11->
-$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel
-\item<12->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<13->
-Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% winkeldreiteilung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Winkeldreiteilung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.43\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{5} +\def\a{25} + +\uncover<3->{ + \draw[line width=0.7pt] (\r,0) arc (0:90:\r); +} + +\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle; +\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$}; + +\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r); + +\uncover<2->{ + \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle; + \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r); + \node[color=red] at ({0.5*\a}:{0.7*\r}) + {$\displaystyle\frac{\alpha}{3}$}; +} + +\uncover<3->{ + \fill[color=blue] ({3*\a}:\r) circle[radius=0.05]; + \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1); + + \fill[color=red] ({\a}:\r) circle[radius=0.05]; + \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1); + + \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}]; + \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}]; +} + + +\uncover<4->{ +\node at ({0.5*\r},-0.5) [below] {$\displaystyle +\cos{\color{blue}\alpha} += +4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3} +$}; +} + +\uncover<5->{ + \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$}; + \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.53\textwidth} +\begin{block}{Aufgabe} +Teile einen Winkel in drei gleiche Teile +\end{block} +\vspace{-2pt} +\uncover<6->{% +\begin{block}{Algebraisierte Aufgabe} +Konstruiere $x$ aus $a$ derart, dass +\[ +p(x) += +x^3-\frac34 x -a = 0 +\] +\uncover<7->{% +$a=0$:} +\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$} +\end{block}} +\vspace{-2pt} +\uncover<10->{% +\begin{proof}[Unmöglichkeitsbeweis] +\begin{itemize} +\item<11-> +$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel +\item<12-> +$p(x)$ definiert eine Körpererweiterung vom Grad $3$ +\item<13-> +Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$ +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex index 907d60a..ada6079 100644 --- a/vorlesungen/slides/4/galois/wuerfel.tex +++ b/vorlesungen/slides/4/galois/wuerfel.tex @@ -1,64 +1,64 @@ -%
-% wuerfel.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Würfelverdoppelung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};
-\uncover<2->{
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};
-}
-
-\uncover<3->{
- \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);
- \node[color=blue] at (0.75,-2.3) [above] {$a$};
-}
-
-\uncover<4->{
- \begin{scope}[yshift=0.03cm]
- \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);
- \draw[color=red] (2.85,-2.7) -- (2.85,-3.0);
- \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);
- \end{scope}
- \node[color=red] at (0.36,-2.9) [below] {$b$};
-}
-
-\uncover<5->{
-\node at (0,-4) {$
- 2{\color{blue}a}^3={\color{red}b}^3
- \uncover<6->{\;\Rightarrow\;
- \frac{b}{a} = \sqrt[3]{2}}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere einen Würfel mit doppeltem Volumen
-\end{block}
-\uncover<7->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere eine Nullstelle von $p(x)=x^3-2$
-\end{block}}
-\uncover<8->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<9->
-$p(x)$ irreduzibel
-\item<10->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<11->
-Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+% +% wuerfel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Würfelverdoppelung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}}; +\uncover<2->{ +\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}}; +} + +\uncover<3->{ + \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25); + \node[color=blue] at (0.75,-2.3) [above] {$a$}; +} + +\uncover<4->{ + \begin{scope}[yshift=0.03cm] + \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19); + \draw[color=red] (2.85,-2.7) -- (2.85,-3.0); + \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9); + \end{scope} + \node[color=red] at (0.36,-2.9) [below] {$b$}; +} + +\uncover<5->{ +\node at (0,-4) {$ + 2{\color{blue}a}^3={\color{red}b}^3 + \uncover<6->{\;\Rightarrow\; + \frac{b}{a} = \sqrt[3]{2}}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Aufgabe} +Konstruiere einen Würfel mit doppeltem Volumen +\end{block} +\uncover<7->{% +\begin{block}{Algebraisierte Aufgabe} +Konstruiere eine Nullstelle von $p(x)=x^3-2$ +\end{block}} +\uncover<8->{% +\begin{proof}[Unmöglichkeitsbeweis] +\begin{itemize} +\item<9-> +$p(x)$ irreduzibel +\item<10-> +$p(x)$ definiert eine Körpererweiterung vom Grad $3$ +\item<11-> +Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 2391099..ffd5091 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -1,22 +1,35 @@ -#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-chapter5 = \
- ../slides/7/symmetrien.tex \
- ../slides/7/algebraisch.tex \
- ../slides/7/parameter.tex \
- ../slides/7/mannigfaltigkeit.tex \
- ../slides/7/sl2.tex \
- ../slides/7/drehung.tex \
- ../slides/7/drehanim.tex \
- ../slides/7/semi.tex \
- ../slides/7/kurven.tex \
- ../slides/7/einparameter.tex \
- ../slides/7/ableitung.tex \
- ../slides/7/liealgebra.tex \
- ../slides/7/kommutator.tex \
- ../slides/7/dg.tex \
- ../slides/7/chapter.tex
-
+# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter5 = \ + ../slides/7/symmetrien.tex \ + ../slides/7/algebraisch.tex \ + ../slides/7/parameter.tex \ + ../slides/7/mannigfaltigkeit.tex \ + ../slides/7/sl2.tex \ + ../slides/7/drehung.tex \ + ../slides/7/drehanim.tex \ + ../slides/7/semi.tex \ + ../slides/7/kurven.tex \ + ../slides/7/einparameter.tex \ + ../slides/7/ableitung.tex \ + ../slides/7/liealgebra.tex \ + ../slides/7/liealgbeispiel.tex \ + ../slides/7/vektorlie.tex \ + ../slides/7/kommutator.tex \ + ../slides/7/bch.tex \ + ../slides/7/dg.tex \ + ../slides/7/interpolation.tex \ + ../slides/7/exponentialreihe.tex \ + ../slides/7/logarithmus.tex \ + ../slides/7/zusammenhang.tex \ + ../slides/7/quaternionen.tex \ + ../slides/7/qdreh.tex \ + ../slides/7/ueberlagerung.tex \ + ../slides/7/hopf.tex \ + ../slides/7/haar.tex \ + ../slides/7/integration.tex \ + ../slides/7/chapter.tex + diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex index 5a4b94e..12f9084 100644 --- a/vorlesungen/slides/7/ableitung.tex +++ b/vorlesungen/slides/7/ableitung.tex @@ -1,68 +1,68 @@ -%
-% ableitung.tex -- Ableitung in der Lie-Gruppe
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Ableitung in der Matrix-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Ableitung in $\operatorname{O}(n)$}
-\uncover<2->{%
-$s \mapsto A(s)\in\operatorname{O}(n)$
-}
-\begin{align*}
-\uncover<3->{I
-&=
-A(s)^tA(s)}
-\\
-\uncover<4->{0
-=
-\frac{d}{ds} I
-&=
-\frac{d}{ds} (A(s)^t A(s))}
-\\
-&\uncover<5->{=
-\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)}
-\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}}
-\uncover<7->{0
-&=
-\dot{A}(0)^t
-+
-\dot{A}(0)}
-\\
-\uncover<8->{\Leftrightarrow
-\qquad
-\dot{A}(0)^t &= -\dot{A}(0)}
-\end{align*}
-\uncover<9->{%
-``Tangentialvektoren'' sind antisymmetrische Matrizen
-}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$}
-\uncover<2->{%
-$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$
-}
-\begin{align*}
-\uncover<3->{1 &= \det A(t)}
-\\
-\uncover<10->{0
-=
-\frac{d}{dt}1
-&=
-\frac{d}{dt} \det A(t)}
-\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}}
-\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)}
-\end{align*}
-\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen}
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% ableitung.tex -- Ableitung in der Lie-Gruppe +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ableitung in der Matrix-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{O}(n)$} +\uncover<2->{% +$s \mapsto A(s)\in\operatorname{O}(n)$ +} +\begin{align*} +\uncover<3->{I +&= +A(s)^tA(s)} +\\ +\uncover<4->{0 += +\frac{d}{ds} I +&= +\frac{d}{ds} (A(s)^t A(s))} +\\ +&\uncover<5->{= +\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)} +\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}} +\uncover<7->{0 +&= +\dot{A}(0)^t ++ +\dot{A}(0)} +\\ +\uncover<8->{\Leftrightarrow +\qquad +\dot{A}(0)^t &= -\dot{A}(0)} +\end{align*} +\uncover<9->{% +``Tangentialvektoren'' sind antisymmetrische Matrizen +} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} +\uncover<2->{% +$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ +} +\begin{align*} +\uncover<3->{1 &= \det A(t)} +\\ +\uncover<10->{0 += +\frac{d}{dt}1 +&= +\frac{d}{dt} \det A(t)} +\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}} +\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)} +\end{align*} +\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex index fba42cf..31d209a 100644 --- a/vorlesungen/slides/7/algebraisch.tex +++ b/vorlesungen/slides/7/algebraisch.tex @@ -1,115 +1,115 @@ -%
-% algebraisch.tex -- algebraische Definition der Symmetrien
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Erhaltungsgrössen und Algebra}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Längen und Winkel}
-Längenmessung mit Skalarprodukt
-\begin{align*}
-\|\vec{v}\|^2
-&=
-\langle \vec{v},\vec{v}\rangle
-=
-\vec{v}\cdot \vec{v}
-\uncover<2->{=
-\vec{v}^t\vec{v}}
-\end{align*}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<3->{%
-\begin{block}{Flächeninhalt/Volumen}
-$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$
-\\
-Volumen des Parallelepipeds: $\det V$
-\end{block}}
-\end{column}
-\end{columns}
-%
-\vspace{-7pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Längenerhaltende Transformationen}
-$A\in\operatorname{GL}_n(\mathbb{R})$
-\begin{align*}
-\vec{x}^t\vec{y}
-&=
-(A\vec{x})
-\cdot
-(A\vec{y})
-\uncover<5->{=
-(A\vec{x})^t
-(A\vec{y})}
-\\
-\uncover<6->{
-\vec{x}^tI\vec{y}
-&=
-\vec{x}^tA^tA\vec{y}}
-\uncover<7->{
-\Rightarrow I=A^tA}
-\end{align*}
-\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<9->{%
-\begin{block}{Volumenerhaltende Transformationen}
-$A\in\operatorname{GL}_n(\mathbb{R})$
-\begin{align*}
-\det(V)
-&=
-\det(AV)
-\uncover<10->{=
-\det(A)\det(V)}
-\\
-\uncover<11->{
-1&=\det(A)}
-\end{align*}
-\uncover<10->{
-(Produktsatz für Determinante)
-}
-\end{block}}
-\end{column}
-\end{columns}
-%
-\vspace{-3pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<12->{%
-\begin{block}{Orthogonale Matrizen}
-Längentreue Abbildungen = orthogonale Matrizen:
-\[
-O(n)
-=
-\{
-A \in \operatorname{GL}_n(\mathbb{R})
-\;|\;
-A^tA=I
-\}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<13->{%
-\begin{block}{``Spezielle'' Matrizen}
-Volumen-/Orientierungserhaltende Transformationen:
-\[
-\operatorname{SL}_n(\mathbb R)
-=
-\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\}
-\]
-\end{block}}
-\end{column}
-\end{columns}
-
-\end{frame}
-\egroup
+% +% algebraisch.tex -- algebraische Definition der Symmetrien +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Erhaltungsgrössen und Algebra} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Längen und Winkel} +Längenmessung mit Skalarprodukt +\begin{align*} +\|\vec{v}\|^2 +&= +\langle \vec{v},\vec{v}\rangle += +\vec{v}\cdot \vec{v} +\uncover<2->{= +\vec{v}^t\vec{v}} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Flächeninhalt/Volumen} +$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ +\\ +Volumen des Parallelepipeds: $\det V$ +\end{block}} +\end{column} +\end{columns} +% +\vspace{-7pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Längenerhaltende Transformationen} +$A\in\operatorname{GL}_n(\mathbb{R})$ +\begin{align*} +\vec{x}^t\vec{y} +&= +(A\vec{x}) +\cdot +(A\vec{y}) +\uncover<5->{= +(A\vec{x})^t +(A\vec{y})} +\\ +\uncover<6->{ +\vec{x}^tI\vec{y} +&= +\vec{x}^tA^tA\vec{y}} +\uncover<7->{ +\Rightarrow I=A^tA} +\end{align*} +\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<9->{% +\begin{block}{Volumenerhaltende Transformationen} +$A\in\operatorname{GL}_n(\mathbb{R})$ +\begin{align*} +\det(V) +&= +\det(AV) +\uncover<10->{= +\det(A)\det(V)} +\\ +\uncover<11->{ +1&=\det(A)} +\end{align*} +\uncover<10->{ +(Produktsatz für Determinante) +} +\end{block}} +\end{column} +\end{columns} +% +\vspace{-3pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Orthogonale Matrizen} +Längentreue Abbildungen = orthogonale Matrizen: +\[ +O(n) += +\{ +A \in \operatorname{GL}_n(\mathbb{R}) +\;|\; +A^tA=I +\} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<13->{% +\begin{block}{``Spezielle'' Matrizen} +Volumen-/Orientierungserhaltende Transformationen: +\[ +\operatorname{SL}_n(\mathbb R) += +\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} +\] +\end{block}} +\end{column} +\end{columns} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/bch.tex b/vorlesungen/slides/7/bch.tex new file mode 100644 index 0000000..0148dc4 --- /dev/null +++ b/vorlesungen/slides/7/bch.tex @@ -0,0 +1,76 @@ +% +% bch.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Baker-Campbell-Hausdorff-Formel} +$g(t),h(t)\in G +\uncover<2->{\Rightarrow \exists A,B\in LG\text{ mit } +g(t)=\exp At, h(t)=\exp Bt}$ +\uncover<3->{% +\begin{align*} +g(t) +&= +I + At + \frac{A^2t^2}{2!} + \frac{A^3t^3}{3!} + \dots, +& +h(t) +&= +I + Bt + \frac{B^2t^2}{2!} + \frac{B^3t^3}{3!} + \dots +\end{align*}} +\uncover<5->{% +\begin{block}{Kommutator in G: $c(t) = g(t)h(t)g(t)^{-1}h(t)^{-1}$} +\begin{align*} +\uncover<6->{c(t) +&= +\biggl( + {\color<7,9-11,13-15,19-21>{red}I} + + {\color<8,16-19>{red}A}t + + \frac{{\color<12>{red}A^2}t^2}{2!} + + \dots +\biggr) +\biggl( + {\color<7,8,10-12,14-15,17-18,21>{red}I} + + {\color<9,16,19-20>{red}B}t + + \frac{{\color<13>{red}B^2}t^2}{2!} + + \dots +\biggr) +\exp(-{\color<10,14,17,19,21>{red}A}t) +\exp(-{\color<11,15,18,20-21>{red}B}t) +} +\\ +&\uncover<7->{={\color<7>{red}I}} +\uncover<8->{+t( + \uncover<8->{ {\color<8>{red}A}} + \uncover<9->{+ {\color<9>{red}B}} + \uncover<10->{- {\color<10>{red}A}} + \uncover<11->{- {\color<11>{red}B}} +)} +\uncover<12->{+\frac{t^2}{2!}( + \uncover<12->{ {\color<12>{red}A^2}} + \uncover<13->{+ {\color<13>{red}B^2}} + \uncover<14->{+ {\color<14>{red}A^2}} + \uncover<15->{+ {\color<15>{red}B^2}} +)} +\\ +&\phantom{\mathstrut=I} +\uncover<12->{+t^2( + \uncover<16->{ {\color<16>{red}AB}} + \uncover<17->{- {\color<17>{red}A^2}} + \uncover<18->{- {\color<18>{red}AB}} + \uncover<19->{- {\color<19>{red}BA}} + \uncover<20->{- {\color<20>{red}B^2}} + \uncover<21->{+ {\color<21>{red}AB}} +)} +\uncover<22->{+t^3(\dots)+\dots} +\\ +&\uncover<23->{= +I + \frac{t^2}{2}[A,B] + o(t^3) +} +\end{align*}} +\end{block} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 0f14a9a..3736e0f 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -1,19 +1,32 @@ -%
-% chapter.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
-%
-\folie{7/symmetrien.tex}
-\folie{7/algebraisch.tex}
-\folie{7/parameter.tex}
-\folie{7/mannigfaltigkeit.tex}
-\folie{7/sl2.tex}
-\folie{7/drehung.tex}
-\folie{7/drehanim.tex}
-\folie{7/semi.tex}
-\folie{7/kurven.tex}
-\folie{7/einparameter.tex}
-\folie{7/ableitung.tex}
-\folie{7/liealgebra.tex}
-\folie{7/kommutator.tex}
-\folie{7/dg.tex}
+% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\folie{7/drehanim.tex} +\folie{7/semi.tex} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/liealgbeispiel.tex} +\folie{7/vektorlie.tex} +\folie{7/kommutator.tex} +\folie{7/bch.tex} +\folie{7/dg.tex} +\folie{7/interpolation.tex} +\folie{7/exponentialreihe.tex} +\folie{7/logarithmus.tex} +\folie{7/zusammenhang.tex} +\folie{7/quaternionen.tex} +\folie{7/qdreh.tex} +\folie{7/ueberlagerung.tex} +\folie{7/hopf.tex} +\folie{7/haar.tex} +\folie{7/integration.tex} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 446b2ab..f9528a4 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -1,92 +1,92 @@ -%
-% dg.tex -- Differentialgleichung für die Exponentialabbildung
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Zurück zur Lie-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Tangentialvektor im Punkt $\gamma(t)$}
-Ableitung von $\gamma(t)$ an der Stelle $t$:
-\begin{align*}
-\dot{\gamma}(t)
-&\uncover<2->{=
-\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t}
-}
-\\
-&\uncover<3->{=
-\frac{d}{ds}
-\gamma(t+s)
-\bigg|_{s=0}
-}
-\\
-&\uncover<4->{=
-\frac{d}{ds}
-\gamma(t)\gamma(s)
-\bigg|_{s=0}
-}
-\\
-&\uncover<5->{=
-\gamma(t)
-\frac{d}{ds}
-\gamma(s)
-\bigg|_{s=0}
-}
-\uncover<6->{=
-\gamma(t) \dot{\gamma}(0)
-}
-\end{align*}
-\end{block}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Differentialgleichung}
-\vspace{-10pt}
-\[
-\dot{\gamma}(t) = \gamma(t) A
-\quad
-\text{mit}
-\quad
-A=\dot{\gamma}(0)\in LG
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.50\textwidth}
-\uncover<8->{%
-\begin{block}{Lösung}
-Exponentialfunktion
-\[
-\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k
-\]
-\end{block}}
-\vspace{-5pt}
-\uncover<9->{%
-\begin{block}{Kontrolle: Tangentialvektor berechnen}
-\vspace{-10pt}
-\begin{align*}
-\frac{d}{dt}e^{At}
-&\uncover<10->{=
-\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!}
-}
-\\
-&\uncover<11->{=
-\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A
-}
-\\
-&\uncover<12->{=
-\sum_{k=0} A^k\frac{t^k}{k!}
-A
-}
-\uncover<13->{=
-e^{At} A
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% dg.tex -- Differentialgleichung für die Exponentialabbildung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zurück zur Lie-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Tangentialvektor im Punkt $\gamma(t)$} +Ableitung von $\gamma(t)$ an der Stelle $t$: +\begin{align*} +\dot{\gamma}(t) +&\uncover<2->{= +\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} +} +\\ +&\uncover<3->{= +\frac{d}{ds} +\gamma(t+s) +\bigg|_{s=0} +} +\\ +&\uncover<4->{= +\frac{d}{ds} +\gamma(t)\gamma(s) +\bigg|_{s=0} +} +\\ +&\uncover<5->{= +\gamma(t) +\frac{d}{ds} +\gamma(s) +\bigg|_{s=0} +} +\uncover<6->{= +\gamma(t) \dot{\gamma}(0) +} +\end{align*} +\end{block} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Differentialgleichung} +%\vspace{-10pt} +\[ +\dot{\gamma}(t) = \gamma(t) A +\quad +\text{mit} +\quad +A=\dot{\gamma}(0)\in LG +\] +\end{block}} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<8->{% +\begin{block}{Lösung} +Exponentialfunktion +\[ +\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k +\] +\end{block}} +\vspace{-5pt} +\uncover<9->{% +\begin{block}{Kontrolle: Tangentialvektor berechnen} +%\vspace{-10pt} +\begin{align*} +\frac{d}{dt}e^{At} +&\uncover<10->{= +\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!} +} +\\ +&\uncover<11->{= +\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A +} +\\ +&\uncover<12->{= +\sum_{k=0} A^k\frac{t^k}{k!} +A +} +\uncover<13->{= +e^{At} A +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex index 776617f..ac136f1 100644 --- a/vorlesungen/slides/7/drehanim.tex +++ b/vorlesungen/slides/7/drehanim.tex @@ -1,155 +1,155 @@ -%
-% template.tex -- slide template
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
-
-\makeatletter
-\hoffset=-2cm
-\advance\textwidth2cm
-\hsize\textwidth
-\columnwidth\textwidth
-\makeatother
-
-\begin{frame}[t,plain]
-\vspace{-5pt}
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-\fill[color=white] (-4,-4) rectangle (9,4.5);
-
-\def\a{60}
-
-\pgfmathparse{tan(\a)}
-\xdef\T{\pgfmathresult}
-
-\pgfmathparse{-sin(\a)*cos(\a)}
-\xdef\S{\pgfmathresult}
-
-\pgfmathparse{1/cos(\a)}
-\xdef\E{\pgfmathresult}
-
-\def\N{20}
-\pgfmathparse{2*\N}
-\xdef\Nzwei{\pgfmathresult}
-\pgfmathparse{3*\N}
-\xdef\Ndrei{\pgfmathresult}
-
-\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
-\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$}
-\begin{itemize}
-\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized
-as shear mappings and axis rescalings
-\[
-A=
-\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix}
-\begin{pmatrix}1&s\\0&1\end{pmatrix}
-\begin{pmatrix}1&0\\t&1\end{pmatrix}
-\]
-\item Most rotations can be decomposed into a product of
-shear mappings and axis rescalings
-\end{itemize}
-\end{block}
-\end{minipage}};
-
-\foreach \d in {1,2,...,\Ndrei}{
- % Scherung in Y-Richtung
- \ifnum \d>\N
- \pgfmathparse{\T}
- \else
- \pgfmathparse{\T*(\d-1)/(\N-1)}
- \fi
- \xdef\t{\pgfmathresult}
-
- % Scherung in X-Richtung
- \ifnum \d>\Nzwei
- \xdef\s{\S}
- \else
- \ifnum \d<\N
- \xdef\s{0}
- \else
- \ifnum \d=\N
- \xdef\s{0}
- \else
- \pgfmathparse{\S*(\d-\N-1)/(\N-1)}
- \xdef\s{\pgfmathresult}
- \fi
- \fi
- \fi
-
- % Reskalierung der Achsen
- \ifnum \d>\Nzwei
- \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))}
- \else
- \pgfmathparse{1}
- \fi
- \xdef\e{\pgfmathresult}
-
- % Matrixelemente
- \pgfmathparse{(\e)*((\s)*(\t)+1)}
- \xdef\A{\pgfmathresult}
-
- \pgfmathparse{(\e)*(\s)}
- \xdef\B{\pgfmathresult}
-
- \pgfmathparse{(\t)/(\e)}
- \xdef\C{\pgfmathresult}
-
- \pgfmathparse{1/(\e)}
- \xdef\D{\pgfmathresult}
-
- \only<\d>{
- \node at (5.0,-0.9) [below right] {$
- \begin{aligned}
- t &= \t \\
- s &= \s \\
- d &= \e \\
- D &= \begin{pmatrix}
- \A&\B\\
- \C&\D
- \end{pmatrix}
- \qquad
- \only<60>{\checkmark}
- \end{aligned}
- $};
- }
-
- \begin{scope}
- \clip (-4.05,-4.05) rectangle (4.05,4.05);
- \only<\d>{
- \foreach \x in {-6,...,6}{
- \draw[color=blue,line width=0.5pt]
- \punkt{\x}{-12} -- \punkt{\x}{12};
- }
- \foreach \y in {-12,...,12}{
- \draw[color=darkgreen,line width=0.5pt]
- \punkt{-6}{\y} -- \punkt{6}{\y};
- }
-
- \foreach \r in {1,2,3,4}{
- \draw[color=red] plot[domain=0:359,samples=360]
- ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))})
- --
- cycle;
- }
- }
- \end{scope}
-
-% \uncover<\d>{
-% \node at (5,4) {\d};
-% }
-}
-
-\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}];
-\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}];
-
-\end{tikzpicture}
-\end{center}
-\end{frame}
-\egroup
+% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) } + +\makeatletter +\hoffset=-2cm +\advance\textwidth2cm +\hsize\textwidth +\columnwidth\textwidth +\makeatother + +\begin{frame}[t,plain] +\vspace{-5pt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=white] (-4,-4) rectangle (9,4.5); + +\def\a{60} + +\pgfmathparse{tan(\a)} +\xdef\T{\pgfmathresult} + +\pgfmathparse{-sin(\a)*cos(\a)} +\xdef\S{\pgfmathresult} + +\pgfmathparse{1/cos(\a)} +\xdef\E{\pgfmathresult} + +\def\N{20} +\pgfmathparse{2*\N} +\xdef\Nzwei{\pgfmathresult} +\pgfmathparse{3*\N} +\xdef\Ndrei{\pgfmathresult} + +\node at (4.2,4.2) [below right] {\begin{minipage}{7cm} +\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$} +\begin{itemize} +\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized +as shear mappings and axis rescalings +\[ +A= +\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix} +\begin{pmatrix}1&s\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\t&1\end{pmatrix} +\] +\item Most rotations can be decomposed into a product of +shear mappings and axis rescalings +\end{itemize} +\end{block} +\end{minipage}}; + +\foreach \d in {1,2,...,\Ndrei}{ + % Scherung in Y-Richtung + \ifnum \d>\N + \pgfmathparse{\T} + \else + \pgfmathparse{\T*(\d-1)/(\N-1)} + \fi + \xdef\t{\pgfmathresult} + + % Scherung in X-Richtung + \ifnum \d>\Nzwei + \xdef\s{\S} + \else + \ifnum \d<\N + \xdef\s{0} + \else + \ifnum \d=\N + \xdef\s{0} + \else + \pgfmathparse{\S*(\d-\N-1)/(\N-1)} + \xdef\s{\pgfmathresult} + \fi + \fi + \fi + + % Reskalierung der Achsen + \ifnum \d>\Nzwei + \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))} + \else + \pgfmathparse{1} + \fi + \xdef\e{\pgfmathresult} + + % Matrixelemente + \pgfmathparse{(\e)*((\s)*(\t)+1)} + \xdef\A{\pgfmathresult} + + \pgfmathparse{(\e)*(\s)} + \xdef\B{\pgfmathresult} + + \pgfmathparse{(\t)/(\e)} + \xdef\C{\pgfmathresult} + + \pgfmathparse{1/(\e)} + \xdef\D{\pgfmathresult} + + \only<\d>{ + \node at (5.0,-0.9) [below right] {$ + \begin{aligned} + t &= \t \\ + s &= \s \\ + d &= \e \\ + D &= \begin{pmatrix} + \A&\B\\ + \C&\D + \end{pmatrix} + \qquad + \only<60>{\checkmark} + \end{aligned} + $}; + } + + \begin{scope} + \clip (-4.05,-4.05) rectangle (4.05,4.05); + \only<\d>{ + \foreach \x in {-6,...,6}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{-12} -- \punkt{\x}{12}; + } + \foreach \y in {-12,...,12}{ + \draw[color=darkgreen,line width=0.5pt] + \punkt{-6}{\y} -- \punkt{6}{\y}; + } + + \foreach \r in {1,2,3,4}{ + \draw[color=red] plot[domain=0:359,samples=360] + ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))}) + -- + cycle; + } + } + \end{scope} + +% \uncover<\d>{ +% \node at (5,4) {\d}; +% } +} + +\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}]; +\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}]; + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index e7b4a92..02201d4 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -1,132 +1,132 @@ -%
-% drehung.tex -- Drehung aus streckungen
-%
-% (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{Drehung aus Streckungen und Scherungen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.38\textwidth}
-\begin{block}{Drehung}
-{\color{blue}Längen}, {\color<2->{blue}Winkel},
-{\color<2->{darkgreen}Orientierung}
-erhalten
-\uncover<2->{
-\[
-\operatorname{SO}(2)
-=
-{\color{blue}\operatorname{O}(2)}
-\cap
-{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
-\]}
-\vspace{-20pt}
-\end{block}
-\uncover<3->{%
-\begin{block}{Zusammensetzung}
-Eine Drehung muss als Zusammensetzung geschrieben werden können:
-\[
-D_{\alpha}
-=
-\begin{pmatrix}
-\cos\alpha & -\sin\alpha\\
-\sin\alpha &\phantom{-}\cos\alpha
-\end{pmatrix}
-=
-DST
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<12->{%
-\begin{block}{Beispiel}
-\vspace{-12pt}
-\[
-D_{60^\circ}
-=
-{\tiny
-\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
-\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
-\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
-}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.58\textwidth}
-\uncover<4->{%
-\begin{block}{Ansatz}
-\vspace{-12pt}
-\begin{align*}
-DST
-&=
-\begin{pmatrix}
-c^{-1}&0\\
- 0 &c
-\end{pmatrix}
-\begin{pmatrix}
-1&-s\\
-0&1
-\end{pmatrix}
-\begin{pmatrix}
-1&0\\
-t&1
-\end{pmatrix}
-\\
-&\uncover<5->{=
-\begin{pmatrix}
-c^{-1}&0\\
- 0 &c
-\end{pmatrix}
-\begin{pmatrix}
-1-st&-s\\
- t& 1
-\end{pmatrix}
-}
-\\
-&\uncover<6->{=
-\begin{pmatrix}
-{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
-{\color<9->{blue}ct}&{\color<8->{red}c}
-\end{pmatrix}}
-\uncover<7->{=
-\begin{pmatrix}
-{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\
-{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha}
-\end{pmatrix}}
-\end{align*}
-\end{block}}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Koeffizientenvergleich}
-\vspace{-15pt}
-\begin{align*}
-\uncover<8->{
-{\color{red} c}
-&=
-{\color{red}\cos\alpha }}
-&&
-&
-\uncover<9->{
-{\color{blue}
-t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\
-\uncover<10->{
-{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
-&
-&\Rightarrow&
-{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
-}
-\\
-\uncover<11->{
-{\color{orange} (1-st)c^{-t}}
-&=
-\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% drehung.tex -- Drehung aus streckungen +% +% (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{Drehung aus Streckungen und Scherungen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Drehung} +{\color{blue}Längen}, {\color<2->{blue}Winkel}, +{\color<2->{darkgreen}Orientierung} +erhalten +\uncover<2->{ +\[ +\operatorname{SO}(2) += +{\color{blue}\operatorname{O}(2)} +\cap +{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})} +\]} +\vspace{-20pt} +\end{block} +\uncover<3->{% +\begin{block}{Zusammensetzung} +Eine Drehung muss als Zusammensetzung geschrieben werden können: +\[ +D_{\alpha} += +\begin{pmatrix} +\cos\alpha & -\sin\alpha\\ +\sin\alpha &\phantom{-}\cos\alpha +\end{pmatrix} += +DST +\] +\end{block}} +\vspace{-10pt} +\uncover<12->{% +\begin{block}{Beispiel} +\vspace{-12pt} +\[ +D_{60^\circ} += +{\tiny +\begin{pmatrix}2&0\\0&\frac12\end{pmatrix} +\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix} +} +\] +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<4->{% +\begin{block}{Ansatz} +%\vspace{-12pt} +\begin{align*} +DST +&= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +1&-s\\ +0&1 +\end{pmatrix} +\begin{pmatrix} +1&0\\ +t&1 +\end{pmatrix} +\\ +&\uncover<5->{= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +1-st&-s\\ + t& 1 +\end{pmatrix} +} +\\ +&\uncover<6->{= +\begin{pmatrix} +{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<9->{blue}ct}&{\color<8->{red}c} +\end{pmatrix}} +\uncover<7->{= +\begin{pmatrix} +{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\ +{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha} +\end{pmatrix}} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Koeffizientenvergleich} +%\vspace{-15pt} +\begin{align*} +\uncover<8->{ +{\color{red} c} +&= +{\color{red}\cos\alpha }} +&& +& +\uncover<9->{ +{\color{blue} +t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ +\uncover<10->{ +{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} +& +&\Rightarrow& +{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha +} +\\ +\uncover<11->{ +{\color{orange} (1-st)c^{-t}} +&= +\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex index e9699a6..a32affd 100644 --- a/vorlesungen/slides/7/einparameter.tex +++ b/vorlesungen/slides/7/einparameter.tex @@ -1,93 +1,93 @@ -%
-% einparameter.tex -- Einparameter Untergruppen
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Einparameter-Untergruppen}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Definition}
-Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$,
-die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit}
-\[
-\uncover<3->{
-\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R}
-}
-\]
-\end{block}
-\uncover<4->{%
-\begin{block}{Drehungen}
-Drehmatrizen bilden Einparameter- Untergruppen
-\begin{align*}
-t \mapsto D_{x,t}
-&=
-\begin{pmatrix}
-1&0&0\\
-0&\cos t&-\sin t\\
-0&\sin t& \cos t
-\end{pmatrix}
-\\
-D_{x,t}D_{x,s}
-&=
-D_{x,t+s}
-\end{align*}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<5->{%
-\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
-\[
-\begin{pmatrix}
-1&s\\
-0&1
-\end{pmatrix}
-\begin{pmatrix}
-1&t\\
-0&1
-\end{pmatrix}
-=
-\begin{pmatrix}
-1&s+t\\
-0&1
-\end{pmatrix}
-\]
-\end{block}}
-\vspace{-12pt}
-\uncover<6->{%
-\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
-\[
-\begin{pmatrix}
-e^s&0\\0&e^{-s}
-\end{pmatrix}
-\begin{pmatrix}
-e^t&0\\0&e^{-t}
-\end{pmatrix}
-=
-\begin{pmatrix}
-e^{t+s}&0\\0&e^{-(t+s)}
-\end{pmatrix}
-\]
-\end{block}}
-\vspace{-12pt}
-\uncover<7->{%
-\begin{block}{Gemischt}
-\vspace{-12pt}
-\begin{gather*}
-A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
-\\
-\text{dank}\quad
-\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2
-=I
-\end{gather*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% einparameter.tex -- Einparameter Untergruppen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Einparameter-Untergruppen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$, +die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit} +\[ +\uncover<3->{ +\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R} +} +\] +\end{block} +\uncover<4->{% +\begin{block}{Drehungen} +Drehmatrizen bilden Einparameter- Untergruppen +\begin{align*} +t \mapsto D_{x,t} +&= +\begin{pmatrix} +1&0&0\\ +0&\cos t&-\sin t\\ +0&\sin t& \cos t +\end{pmatrix} +\\ +D_{x,t}D_{x,s} +&= +D_{x,t+s} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} +%\vspace{-12pt} +\[ +\begin{pmatrix} +1&s\\ +0&1 +\end{pmatrix} +\begin{pmatrix} +1&t\\ +0&1 +\end{pmatrix} += +\begin{pmatrix} +1&s+t\\ +0&1 +\end{pmatrix} +\] +\end{block}} +\vspace{-12pt} +\uncover<6->{% +\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} +%\vspace{-12pt} +\[ +\begin{pmatrix} +e^s&0\\0&e^{-s} +\end{pmatrix} +\begin{pmatrix} +e^t&0\\0&e^{-t} +\end{pmatrix} += +\begin{pmatrix} +e^{t+s}&0\\0&e^{-(t+s)} +\end{pmatrix} +\] +\end{block}} +\vspace{-12pt} +\uncover<7->{% +\begin{block}{Gemischt} +%\vspace{-12pt} +\begin{gather*} +A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t +\\ +\text{dank}\quad +\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 +=I +\end{gather*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/exponentialreihe.tex b/vorlesungen/slides/7/exponentialreihe.tex new file mode 100644 index 0000000..b1aeda6 --- /dev/null +++ b/vorlesungen/slides/7/exponentialreihe.tex @@ -0,0 +1,24 @@ +% +% exponentialreihe.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Exponentialreihe} +\begin{align*} +h(s) &= \exp(tA_0 + sB) = \sum_{k=0}^\infty \frac{1}{k!} (tA_0 + sB)^k +\\ +&= +I + (tA_0 + sB) + \frac{1}{2!}(t^2A_0^2 + ts(A_0B + BA_0) + s^2B^2) ++ \frac{1}{3!}(t^3A_0^3 + t^2s(A_0^2B + A_0BA_0 + BA_0^2) + \dots) ++ \dots +\\ +\frac{dg(s)}{ds} +&= +B + \frac1{2!}t(A_0B+BA_0) + \frac{1}{3!}t^2(A_0^2B+A_0BA_0+BA_0^2) + \dots +\end{align*} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/haar.tex b/vorlesungen/slides/7/haar.tex new file mode 100644 index 0000000..454dd69 --- /dev/null +++ b/vorlesungen/slides/7/haar.tex @@ -0,0 +1,84 @@ +% +% haar.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Haar-Mass} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Invariantes Mass} +Auf jeder lokalkompakten Gruppe $G$ gibt es ein \only<2->{invariantes }% +Integral +\begin{align*} +\uncover<2->{\text{rechts:}}&& +\int_G f(g)\,d\mu(g) +&\uncover<2->{= +\int_G f(gh)\,d\mu(g)} +\\ +\uncover<3->{ +\text{links:}&& +\int_G f(g)\,d\mu(g) +&= +\int_G f(hg)\,d\mu(g)} +\end{align*} + +\end{block} +\uncover<7->{% +\begin{block}{Modulus-Funktion} +$\mu$ linksinvariant, dann ist die Rechtsverschiebung ebenfalls +linksinvariant +\[ +\int_G f(gh) \, d\mu(g) +\uncover<8->{ += +\int_G f(g) \Delta(h)\, d\mu(g) +} +\] +\uncover<9->{$\Delta(h)$ heisst Modulus-Funktion} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Beispiel: $G=\mathbb{R}$} +\[ +\int_Gf(g)\,d\mu(g) += +\int_{-\infty}^{\infty} f(x)\,dx +\] +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Beispiel: $\operatorname{SO}(2)$} +\[ +\int_{\operatorname{SO}(2)} +f(g)\,d\mu(g) += +\frac{1}{2\pi} +\int_{0}^{2\pi} f(D_{\alpha})\,d\alpha +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{Beispiel: $G$ endlich} +\[ +\int_G f(g)\,d\mu(g) = \frac{1}{|G|}\sum_{g\in G}f(g) +\] +\end{block}} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Unimodular} +$\Delta(h)=1$ heisst rechtsinvariant = linksinvariant +\\ +\uncover<11->{% +$G$ kompakt $\Rightarrow$ unimodular +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/hopf.tex b/vorlesungen/slides/7/hopf.tex new file mode 100644 index 0000000..a90737f --- /dev/null +++ b/vorlesungen/slides/7/hopf.tex @@ -0,0 +1,69 @@ +% +% hopf.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Orbit-Räume} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aktion von $\operatorname{SO}(3)$ auf $S^2$} +\begin{align*} +S^2 &= \{x\in\mathbb{R}^3\;|\; |x|=1\} +\\ +\operatorname{SO}(3) \times S^2 &\to S^2: (g, x) \mapsto gx +\end{align*} +\uncover<2->{% +Allgemein: Aktion von $G$ auf $X$ +\begin{align*} +\text{links:}&& +G\times X \to X &: (g,x) \mapsto gx +\\ +\text{rechts:}&& +X\times G \to X &: (x,g) \mapsto xg +\end{align*}} +\end{block} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Stabilisator} +Zu $x\in X$ gibt es eine Untergruppe +\begin{align*} +G_x = \{g\in G\;|\; gx=x\}, +\end{align*} +der {\em Stabilisator} von $x$. + +\uncover<4->{% +Der Stabilisator von $v\in S^2$ ist die Gruppe der Drehungen um +die Achse $v$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Quotient} +$G$ operiert von rechts auf $X$ +\[ +X/G = \{ xG \;|\; x\in X\} +\] +heisst Quotient +\end{block}} +\uncover<6->{ +\begin{block}{$\operatorname{SO}(3)/\operatorname{SO}(2)$} +Wähle $\operatorname{SO}(2)$ als Drehungen um die $z$-Achse: +\[ +\operatorname{SO}(3) \to S^2 +: +g \mapsto \text{letzte Spalte von $g$} +\] +\uncover<7->{Daher +\[ +S^2 \cong \operatorname{SO}(3) / \operatorname{SO}(2) +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile index 9de1c34..6f99bc3 100644 --- a/vorlesungen/slides/7/images/Makefile +++ b/vorlesungen/slides/7/images/Makefile @@ -1,19 +1,29 @@ -#
-# Makefile -- Illustrationen zu
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-all: rodriguez.jpg
-
-rodriguez.png: rodriguez.pov
- povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
-
-rodriguez.jpg: rodriguez.png
- convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
-
-commutator: commutator.ini commutator.pov common.inc
- povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
-jpg:
- for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
-
-
+# +# Makefile -- Illustrationen zu +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: rodriguez.jpg test.png + +rodriguez.png: rodriguez.pov + povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov + +rodriguez.jpg: rodriguez.png + convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg + +commutator: commutator.ini commutator.pov common.inc + povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini +jpg: + for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done + +dreibein/timestamp: interpolation.m + octave interpolation.m + touch dreibein/timestamp + +test.png: test.pov drehung.inc dreibein/d025.inc dreibein/timestamp + povray +A0.1 -W1080 -H1080 -Otest.png test.pov + +dreibein/d025.inc: dreibein/timestamp + +animation: + povray +A0.1 -W1080 -H1080 -Ointerpolation/i.png interpolation.ini diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc index b028956..0e27c9a 100644 --- a/vorlesungen/slides/7/images/common.inc +++ b/vorlesungen/slides/7/images/common.inc @@ -1,70 +1,70 @@ -//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
- assumed_gamma 1
-}
-
-#declare imagescale = 0.025;
-#declare O = <0, 0, 0>;
-#declare at = 0.015;
-
-camera {
- location <18, 15, -50>
- look_at <0.0, 0.5, 0>
- right 16/9 * x * imagescale
- up y * imagescale
-}
-
-light_source {
- <-40, 30, -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
-
-#declare l = 1.2;
-
-arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
-arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
-arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
-
+// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.025; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <18, 15, -50> + look_at <0.0, 0.5, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-40, 30, -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 + +#declare l = 1.2; + +arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) +arrow(< 0, 0, -l >, < 0, 0, l >, at, White) +arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) + diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini index 44a5ac5..8c2211e 100644 --- a/vorlesungen/slides/7/images/commutator.ini +++ b/vorlesungen/slides/7/images/commutator.ini @@ -1,8 +1,8 @@ -Input_File_Name=commutator.pov
-Initial_Frame=1
-Final_Frame=60
-Initial_Clock=1
-Final_Clock=60
-Cyclic_Animation=off
-Pause_when_Done=off
-
+Input_File_Name=commutator.pov +Initial_Frame=1 +Final_Frame=60 +Initial_Clock=1 +Final_Clock=60 +Cyclic_Animation=off +Pause_when_Done=off + diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m index 3f5ea17..5a448db 100644 --- a/vorlesungen/slides/7/images/commutator.m +++ b/vorlesungen/slides/7/images/commutator.m @@ -1,111 +1,111 @@ -#
-# commutator.m
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-
-X = [
- 0, 0, 0;
- 0, 0, -1;
- 0, 1, 0
-];
-
-Y = [
- 0, 0, 1;
- 0, 0, 0;
- -1, 0, 0
-];
-
-Z = [
- 0, -1, 0;
- 1, 0, 0;
- 0, 0, 0
-];
-
-function retval = Dx(alpha)
- retval = [
- 1, 0, 0 ;
- 0, cos(alpha), -sin(alpha);
- 0, sin(alpha), cos(alpha)
- ];
-end
-
-function retval = Dy(beta)
- retval = [
- cos(beta), 0, sin(beta);
- 0, 1, 0 ;
- -sin(beta), 0, cos(beta)
- ];
-end
-
-t = 0.9;
-P = Dx(t) * Dy(t)
-Q = Dy(t) * Dx(t)
-P - Q
-(P - Q) * [0;0;1]
-
-function retval = kurven(filename, t)
- retval = -1;
- N = 20;
- fn = fopen(filename, "w");
- fprintf(fn, "//\n");
- fprintf(fn, "// %s\n", filename);
- fprintf(fn, "//\n");
- fprintf(fn, "#macro XYkurve()\n");
- for i = (0:N-1)
- v1 = Dx(t * i / N) * [0;0;1];
- v2 = Dx(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- for i = (0:N-1)
- v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
- v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v2(1,1), v2(3,1), v2(2,1));
- fprintf(fn, "#end\n");
- fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
- v2(1,1), v2(3,1), v2(2,1));
- fprintf(fn, "#macro YXkurve()\n");
- for i = (0:N-1)
- v1 = Dy(t * i / N) * [0;0;1];
- v2 = Dy(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- for i = (0:N-1)
- v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
- v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1));
- fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
- v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
- end
- fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
- v2(1,1), v2(3,1), v2(2,1));
- fprintf(fn, "#end\n");
- fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
- v2(1,1), v2(3,1), v2(2,1));
-
- fclose(fn);
- retval = 0;
-end
-
-function retval = kurve(i)
- n = pi / 180;
- filename = sprintf("f/%04d.inc", i);
- kurven(filename, n * i);
-end
-
-for i = (1:60)
- kurve(i);
-end
+# +# commutator.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +X = [ + 0, 0, 0; + 0, 0, -1; + 0, 1, 0 +]; + +Y = [ + 0, 0, 1; + 0, 0, 0; + -1, 0, 0 +]; + +Z = [ + 0, -1, 0; + 1, 0, 0; + 0, 0, 0 +]; + +function retval = Dx(alpha) + retval = [ + 1, 0, 0 ; + 0, cos(alpha), -sin(alpha); + 0, sin(alpha), cos(alpha) + ]; +end + +function retval = Dy(beta) + retval = [ + cos(beta), 0, sin(beta); + 0, 1, 0 ; + -sin(beta), 0, cos(beta) + ]; +end + +t = 0.9; +P = Dx(t) * Dy(t) +Q = Dy(t) * Dx(t) +P - Q +(P - Q) * [0;0;1] + +function retval = kurven(filename, t) + retval = -1; + N = 20; + fn = fopen(filename, "w"); + fprintf(fn, "//\n"); + fprintf(fn, "// %s\n", filename); + fprintf(fn, "//\n"); + fprintf(fn, "#macro XYkurve()\n"); + for i = (0:N-1) + v1 = Dx(t * i / N) * [0;0;1]; + v2 = Dx(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + for i = (0:N-1) + v1 = Dx(t) * Dy(t * i / N) * [0;0;1]; + v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#end\n"); + fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#macro YXkurve()\n"); + for i = (0:N-1) + v1 = Dy(t * i / N) * [0;0;1]; + v2 = Dy(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + for i = (0:N-1) + v1 = Dy(t) * Dx(t * i / N) * [0;0;1]; + v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#end\n"); + fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n", + v2(1,1), v2(3,1), v2(2,1)); + + fclose(fn); + retval = 0; +end + +function retval = kurve(i) + n = pi / 180; + filename = sprintf("f/%04d.inc", i); + kurven(filename, n * i); +end + +for i = (1:60) + kurve(i); +end diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov index 8229a06..9ae11b9 100644 --- a/vorlesungen/slides/7/images/commutator.pov +++ b/vorlesungen/slides/7/images/commutator.pov @@ -1,59 +1,59 @@ -//
-// commutator.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-sphere { O, 0.99
- pigment {
- color rgbt<1,1,1,0.5>
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-#declare filename = concat("f/", str(clock, -4, 0), ".inc");
-
-#include filename
-
-#declare n1 = vcross(<0,1,0>, finalXY);
-#declare n2 = vcross(<0,1,0>, finalYX);
-
-intersection {
- sphere { O, 1 }
- plane { -n1, 0 }
- plane { n2, 0 }
- pigment {
- color rgb<0,0.4,0.1>
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-union {
- XYkurve()
- pigment {
- color Red
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-union {
- YXkurve()
- pigment {
- color Blue
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
+// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +sphere { O, 0.99 + pigment { + color rgbt<1,1,1,0.5> + } + finish { + specular 0.9 + metallic + } +} + +#declare filename = concat("f/", str(clock, -4, 0), ".inc"); + +#include filename + +#declare n1 = vcross(<0,1,0>, finalXY); +#declare n2 = vcross(<0,1,0>, finalYX); + +intersection { + sphere { O, 1 } + plane { -n1, 0 } + plane { n2, 0 } + pigment { + color rgb<0,0.4,0.1> + } + finish { + specular 0.9 + metallic + } +} + +union { + XYkurve() + pigment { + color Red + } + finish { + specular 0.9 + metallic + } +} + +union { + YXkurve() + pigment { + color Blue + } + finish { + specular 0.9 + metallic + } +} + diff --git a/vorlesungen/slides/7/images/drehung.inc b/vorlesungen/slides/7/images/drehung.inc new file mode 100644 index 0000000..c9b4bb7 --- /dev/null +++ b/vorlesungen/slides/7/images/drehung.inc @@ -0,0 +1,142 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.23; +#declare O = <0, 0, 0>; +#declare at = 0.02; + +camera { + location <8.5, 2, 6.5> + look_at <0, 0, 0> + right 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, 10> 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 = 1.0; + +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 farbeX = rgb<1.0,0.2,0.6>; +#declare farbeY = rgb<0.0,0.8,0.4>; +#declare farbeZ = rgb<0.4,0.6,1.0>; + +#declare farbex = rgb<1.0,0.0,0.0>; +#declare farbey = rgb<0.0,0.6,0.0>; +#declare farbez = rgb<0.0,0.0,1.0>; + +#macro quadrant(X, Y, Z) + intersection { + sphere { O, 0.5 } + plane { -X, 0 } + plane { -Y, 0 } + plane { -Z, 0 } + pigment { + color rgb<1.0,0.6,0.2> + } + finish { + specular 0.95 + metallic + } + } + arrow(O, X, 1.1*at, farbex) + arrow(O, Y, 1.1*at, farbey) + arrow(O, Z, 1.1*at, farbez) +#end + +#macro drehung(X, Y, Z) +// intersection { +// sphere { O, 0.5 } +// plane { -X, 0 } +// plane { -Y, 0 } +// plane { -Z, 0 } +// pigment { +// color Gray +// } +// finish { +// specular 0.95 +// metallic +// } +// } + arrow(O, 1.1*X, 0.9*at, farbeX) + arrow(O, 1.1*Y, 0.9*at, farbeY) + arrow(O, 1.1*Z, 0.9*at, farbeZ) +#end + +#macro achse(H) + cylinder { H, -H, at + pigment { + color rgb<0.6,0.4,0.2> + } + finish { + specular 0.95 + metallic + } + } + cylinder { 0.003 * H, -0.003 * H, 1 + pigment { + color rgbt<0.6,0.4,0.2,0.5> + } + finish { + specular 0.95 + metallic + } + } +#end diff --git a/vorlesungen/slides/7/images/interpolation.ini b/vorlesungen/slides/7/images/interpolation.ini new file mode 100644 index 0000000..f07c079 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.ini @@ -0,0 +1,8 @@ +Input_File_Name=interpolation.pov +Initial_Frame=0 +Final_Frame=50 +Initial_Clock=0 +Final_Clock=50 +Cyclic_Animation=off +Pause_when_Done=off + diff --git a/vorlesungen/slides/7/images/interpolation.m b/vorlesungen/slides/7/images/interpolation.m new file mode 100644 index 0000000..31554e8 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.m @@ -0,0 +1,54 @@ +# +# interpolation.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 50; +global A; +global B; + +A = (pi / 2) * [ + 0, 0, 0; + 0, 0, -1; + 0, 1, 0 +]; +g0 = expm(A) + +B = (pi / 2) * [ + 0, 0, 1; + 0, 0, 0; + -1, 0, 0 +]; +g1 = expm(B) + +function retval = g(t) + global A; + global B; + retval = expm((1-t)*A+t*B); +endfunction + +function dreibein(fn, M, funktion) + fprintf(fn, "%s(<%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>)\n", + funktion, + M(1,1), M(3,1), M(2,1), + M(1,2), M(3,2), M(2,2), + M(1,3), M(3,3), M(2,3)); +endfunction + +G = g1 * inverse(g0); +[V, lambda] = eig(G); +H = real(V(:,3)); + +D = logm(g1*inverse(g0)); + +for i = (0:N) + filename = sprintf("dreibein/d%03d.inc", i); + fn = fopen(filename, "w"); + t = i/N; + dreibein(fn, g(t), "quadrant"); + dreibein(fn, expm(t*D)*g0, "drehung"); + fprintf(fn, "achse(<%.4f,%.4f,%.4f>)\n", H(1,1), H(3,1), H(2,1)); + fclose(fn); +endfor + diff --git a/vorlesungen/slides/7/images/interpolation.pov b/vorlesungen/slides/7/images/interpolation.pov new file mode 100644 index 0000000..71e0257 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.pov @@ -0,0 +1,10 @@ +// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "drehung.inc" + +#declare filename = concat("dreibein/d", str(clock, -3, 0), ".inc"); +#include filename + diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov index 62306f8..07aec19 100644 --- a/vorlesungen/slides/7/images/rodriguez.pov +++ b/vorlesungen/slides/7/images/rodriguez.pov @@ -1,118 +1,118 @@ -//
-// rodriguez.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-
-global_settings {
- assumed_gamma 1
-}
-
-#declare imagescale = 0.020;
-#declare O = <0, 0, 0>;
-#declare at = 0.015;
-
-camera {
- location <8, 15, -50>
- look_at <0.1, 0.475, 0>
- 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
-
-#declare K = vnormalize(<0.2,1,0.1>);
-#declare X = vnormalize(<1.1,1,-1.2>);
-#declare O = <0,0,0>;
-
-#declare r = vlength(vcross(K, X)) / vlength(K);
-
-#declare l = 1.0;
-
-arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
-arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
-arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
-
-arrow(O, X, at, Red)
-arrow(O, K, at, Blue)
-
-#macro punkt(H,phi)
- ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
-#end
-
-cone { vdot(K, X) * K, r, O, 0
- pigment {
- color rgbt<0.6,0.6,0.6,0.5>
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-
-union {
- #declare phistep = pi / 100;
- #declare phi = 0;
- #while (phi < 2 * pi - phistep/2)
- sphere { punkt(K, phi), at/2 }
- cylinder {
- punkt(X, phi),
- punkt(X, phi + phistep),
- at/2
- }
- #declare phi = phi + phistep;
- #end
- pigment {
- color Orange
- }
- finish {
- specular 0.9
- metallic
- }
-}
-
-arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
-#declare Darkgreen = rgb<0,0.5,0>;
-arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
+// +// rodriguez.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.020; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <8, 15, -50> + look_at <0.1, 0.475, 0> + 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 + +#declare K = vnormalize(<0.2,1,0.1>); +#declare X = vnormalize(<1.1,1,-1.2>); +#declare O = <0,0,0>; + +#declare r = vlength(vcross(K, X)) / vlength(K); + +#declare l = 1.0; + +arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) +arrow(< 0, 0, -l >, < 0, 0, l >, at, White) +arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) + +arrow(O, X, at, Red) +arrow(O, K, at, Blue) + +#macro punkt(H,phi) + ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K) +#end + +cone { vdot(K, X) * K, r, O, 0 + pigment { + color rgbt<0.6,0.6,0.6,0.5> + } + finish { + specular 0.9 + metallic + } +} + + +union { + #declare phistep = pi / 100; + #declare phi = 0; + #while (phi < 2 * pi - phistep/2) + sphere { punkt(K, phi), at/2 } + cylinder { + punkt(X, phi), + punkt(X, phi + phistep), + at/2 + } + #declare phi = phi + phistep; + #end + pigment { + color Orange + } + finish { + specular 0.9 + metallic + } +} + +arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow) +#declare Darkgreen = rgb<0,0.5,0>; +arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen) diff --git a/vorlesungen/slides/7/images/test.pov b/vorlesungen/slides/7/images/test.pov new file mode 100644 index 0000000..5707be1 --- /dev/null +++ b/vorlesungen/slides/7/images/test.pov @@ -0,0 +1,7 @@ +// +// test.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "drehung.inc" +#include "dreibein/d025.inc" diff --git a/vorlesungen/slides/7/integration.tex b/vorlesungen/slides/7/integration.tex new file mode 100644 index 0000000..525e6de --- /dev/null +++ b/vorlesungen/slides/7/integration.tex @@ -0,0 +1,66 @@ +% +% integration.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Invariante Integration} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Koordinatenwechsel} +Die Koordinatentransformation +$f\colon\mathbb{R}^n\to\mathbb{R}^n:x\to y$ +hat die Ableitungsmatrix +\[ +t_{ij} += +\frac{\partial y_i}{\partial x_j} +\] +\uncover<2->{% +$n$-faches Integral +\begin{gather*} +\int\dots\int +h(f(x)) +\det +\biggl( +\frac{\partial y_i}{\partial x_j} +\biggr) +\,dx_1\,\dots dx_n +\\ += +\int\dots\int +h(y) +\,dy_1\,\dots dy_n +\end{gather*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{auf einer Lie-Gruppe} +Koordinatenwechsel sind Multiplikationen mit einer +Matrix $g\in G$ +\end{block}} +\uncover<4->{% +\begin{block}{Volumenelement in $I$} +Man muss nur das Volumenelement in $I$ in einem beliebigen +Koordinatensystem definieren: +\[ +dV = dy_1\,\dots\,dy_n +\] +\end{block}} +\uncover<5->{% +\begin{block}{Volumenelement in $g$} +\[ +\text{``\strut}g\cdot dV\text{\strut''} += +\det(g) \, dy_1\,\dots\,dy_n +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/interpolation.tex b/vorlesungen/slides/7/interpolation.tex new file mode 100644 index 0000000..249ee26 --- /dev/null +++ b/vorlesungen/slides/7/interpolation.tex @@ -0,0 +1,112 @@ +% +% interpolation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{\only<#1|handout:0>{\includegraphics[width=\textwidth]{../slides/7/images/interpolation/#2.png}}} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Interpolation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aufgabe} +Finde einen Weg $g(t)\in \operatorname{SO}(3)$ zwischen +$g_0\in\operatorname{SO}(3)$ +und +$g_1\in\operatorname{SO}(3)$: +\[ +g_0=g(0) +\quad\wedge\quad +g_1=g(1) +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Lösung} +$g_i=\exp(A_i) \uncover<3->{\Rightarrow A_i^t=-A_i}$ +\begin{align*} +\uncover<4->{A(t) &= (1-t)A_0 + tA_1}\uncover<8->{ \in \operatorname{so}(3)} +\\ +\uncover<5->{A(t)^t +&=(1-t)A_0^t + tA_1^t} +\\ +&\uncover<6->{= +-(1-t)A_0 - t A_1} +\uncover<7->{= +-A(t)} +\\ +\uncover<9->{\Rightarrow +g(t) &= \exp A(t) \in \operatorname{SO}(3)} +\\ +&\uncover<10->{\ne +\exp (\log(g_1g_0^{-1})t) g_0} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Animation} +\centering +\ifthenelse{\boolean{presentation}}{ +\bild{12}{i00} +\bild{13}{i01} +\bild{14}{i02} +\bild{15}{i03} +\bild{16}{i04} +\bild{17}{i05} +\bild{18}{i06} +\bild{19}{i07} +\bild{20}{i08} +\bild{21}{i09} +\bild{22}{i10} +\bild{23}{i11} +\bild{24}{i12} +\bild{25}{i13} +\bild{26}{i14} +\bild{27}{i15} +\bild{28}{i16} +\bild{29}{i17} +\bild{30}{i18} +\bild{31}{i19} +\bild{32}{i20} +\bild{33}{i21} +\bild{34}{i22} +\bild{35}{i23} +\bild{36}{i24} +\bild{37}{i25} +\bild{38}{i26} +\bild{39}{i27} +\bild{40}{i28} +\bild{41}{i29} +\bild{42}{i30} +\bild{43}{i31} +\bild{44}{i32} +\bild{45}{i33} +\bild{46}{i34} +\bild{47}{i35} +\bild{48}{i36} +\bild{49}{i37} +\bild{50}{i38} +\bild{51}{i39} +\bild{52}{i40} +\bild{53}{i41} +\bild{54}{i42} +\bild{55}{i43} +\bild{56}{i44} +\bild{57}{i45} +\bild{58}{i46} +\bild{59}{i47} +\bild{60}{i48} +\bild{61}{i49} +\bild{62}{i50} +}{ +\includegraphics[width=\textwidth]{../slides/7/images/interpolation/i25.png} +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index 9000160..84bf034 100644 --- a/vorlesungen/slides/7/kommutator.tex +++ b/vorlesungen/slides/7/kommutator.tex @@ -1,166 +1,166 @@ -%
-% template.tex -- slide template
-%
-% (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{Kommutator in $\operatorname{SO}(3)$}
-\vspace{-20pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\t{14.0cm}
-\ifthenelse{\boolean{presentation}}{
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-\coordinate (C) at (-2.1,-1.2);
-\draw[->,color=red,line width=1.4pt]
- (A)
- to[out=-143,in=60]
- (B)
- to[out=-120,in=80]
- (C);
-%\fill[color=red] (B) circle[radius=0.08];
-\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$};
-\coordinate (D) at (0.3,3.2);
-\coordinate (E) at (1.8,2.8);
-\coordinate (F) at (5.2,-0.3);
-\draw[->,color=blue,line width=1.4pt]
- (D)
- to[out=-10,in=157]
- (E)
- to[out=-23,in=120]
- (F);
-\fill[color=blue] (E) circle[radius=0.08];
-\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$};
-\draw[->,color=darkgreen,line width=1.4pt]
- (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6);
-\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$};
-\end{tikzpicture}
-\end{center}
-\end{frame}
-\egroup
+% +% template.tex -- slide template +% +% (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{Kommutator in $\operatorname{SO}(3)$} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\t{14.0cm} +\ifthenelse{\boolean{presentation}}{ +\only<1>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};} +\only<2>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};} +\only<3>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};} +\only<4>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};} +\only<5>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};} +\only<6>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};} +\only<7>{\node at (0,0) { 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+\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};} +\only<52>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};} +\only<53>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};} +\only<54>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};} +\only<55>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};} +\only<56>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};} +\only<57>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};} +\only<58>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};} +\only<59>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};} +}{} +\only<60>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};} +\coordinate (A) at (-0.3,3); +\coordinate (B) at (-1.1,2); +\coordinate (C) at (-2.1,-1.2); +\draw[->,color=red,line width=1.4pt] + (A) + to[out=-143,in=60] + (B) + to[out=-120,in=80] + (C); +%\fill[color=red] (B) circle[radius=0.08]; +\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$}; +\coordinate (D) at (0.3,3.2); +\coordinate (E) at (1.8,2.8); +\coordinate (F) at (5.2,-0.3); +\draw[->,color=blue,line width=1.4pt] + (D) + to[out=-10,in=157] + (E) + to[out=-23,in=120] + (F); +\fill[color=blue] (E) circle[radius=0.08]; +\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$}; +\draw[->,color=darkgreen,line width=1.4pt] + (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6); +\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$}; +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex index bca8417..e0690eb 100644 --- a/vorlesungen/slides/7/kurven.tex +++ b/vorlesungen/slides/7/kurven.tex @@ -1,104 +1,104 @@ -%
-% kurven.tex -- slide template
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Kurven und Tangenten}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Kurven}
-Kurve in $\mathbb{R}^n$:
-\vspace{-12pt}
-\[
-\gamma
-\colon
-I=[a,b] \to \mathbb{R}^n
-:
-t\mapsto \gamma(t)
-\uncover<2->{
-=
-\begin{pmatrix}
-x_1(t)\\
-x_2(t)\\
-\vdots\\
-x_n(t)
-\end{pmatrix}
-}
-\]
-\vspace{-15pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\coordinate (A) at (1,0.5);
-\coordinate (B) at (4,0.5);
-\coordinate (C) at (2,2.2);
-\coordinate (D) at (5,2);
-\coordinate (E) at ($(C)+(80:2)$);
-
-\draw[color=red,line width=1.4pt]
- (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D);
-\fill[color=red] (C) circle[radius=0.06];
-\node[color=red] at (C) [left] {$\gamma(t)$};
-
-\uncover<4->{
- \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
- \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
-}
-
-\uncover<2->{
- \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}];
- \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
-}
-\end{tikzpicture}
-\end{center}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<4->{%
-\begin{block}{Tangenten}
-Ableitung
-\[
-\frac{d}{dt}\gamma(t)
-=
-\dot{\gamma}(t)
-=
-\begin{pmatrix}
-\dot{x}_1(t)\\
-\dot{x}_2(t)\\
-\vdots\\
-\dot{x}_n(t)
-\end{pmatrix}
-\]
-\uncover<5->{%
-Lineare Approximation:
-\[
-\gamma(t+h)
-=
-\gamma(t)
-+
-\dot{\gamma}(t) \cdot h
-+
-o(h)
-\]}%
-\vspace{-10pt}
-\begin{itemize}
-\item<6->
-Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
-in $\mathbb{R}^n$ liegen
-\item<7->
-Gilt auch für
-\[
-\operatorname{GL}_n(\mathbb{R})
-\uncover<8->{\subset M_n(\mathbb{R})}
-\uncover<9->{ = \mathbb{R}^{n\times n}}
-\]
-\end{itemize}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% kurven.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kurven und Tangenten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Kurven} +Kurve in $\mathbb{R}^n$: +\vspace{-12pt} +\[ +\gamma +\colon +I=[a,b] \to \mathbb{R}^n +: +t\mapsto \gamma(t) +\uncover<2->{ += +\begin{pmatrix} +x_1(t)\\ +x_2(t)\\ +\vdots\\ +x_n(t) +\end{pmatrix} +} +\] +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (1,0.5); +\coordinate (B) at (4,0.5); +\coordinate (C) at (2,2.2); +\coordinate (D) at (5,2); +\coordinate (E) at ($(C)+(80:2)$); + +\draw[color=red,line width=1.4pt] + (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D); +\fill[color=red] (C) circle[radius=0.06]; +\node[color=red] at (C) [left] {$\gamma(t)$}; + +\uncover<4->{ + \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); + \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; +} + +\uncover<2->{ + \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; + \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Tangenten} +Ableitung +\[ +\frac{d}{dt}\gamma(t) += +\dot{\gamma}(t) += +\begin{pmatrix} +\dot{x}_1(t)\\ +\dot{x}_2(t)\\ +\vdots\\ +\dot{x}_n(t) +\end{pmatrix} +\] +\uncover<5->{% +Lineare Approximation: +\[ +\gamma(t+h) += +\gamma(t) ++ +\dot{\gamma}(t) \cdot h ++ +o(h) +\]}% +\vspace{-10pt} +\begin{itemize} +\item<6-> +Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ +in $\mathbb{R}^n$ liegen +\item<7-> +Gilt auch für +\[ +\operatorname{GL}_n(\mathbb{R}) +\uncover<8->{\subset M_n(\mathbb{R})} +\uncover<9->{ = \mathbb{R}^{n\times n}} +\] +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/liealgbeispiel.tex b/vorlesungen/slides/7/liealgbeispiel.tex new file mode 100644 index 0000000..a17de40 --- /dev/null +++ b/vorlesungen/slides/7/liealgbeispiel.tex @@ -0,0 +1,78 @@ +% +% liealgbeispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra Beispiele} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{sl}_2(\mathbb{R})$} +Spurlose Matrizen: +\[ +\operatorname{sl}_2(\mathbb{R}) += +\{A\in M_n(\mathbb{R})\;|\; \operatorname{Spur}A=0\} +\] +\end{block} +\begin{block}{Lie-Algebra?} +Nachrechnen: $[A,B]\in \operatorname{sl}_2(\mathbb{R})$: +\begin{align*} +\operatorname{Spur}([A,B]) +&= +\operatorname{Spur}(AB-BA) +\\ +&= +\operatorname{Spur}(AB)-\operatorname{Spur}(BA) +\\ +&= +\operatorname{Spur}(AB)-\operatorname{Spur}(AB) +\\ +&=0 +\end{align*} +$\Rightarrow$ $\operatorname{sl}_2(\mathbb{R})$ ist eine Lie-Algebra +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{so}(n)$} +Antisymmetrische Matrizen: +\[ +\operatorname{so}(n) += +\{A\in M_n(\mathbb{R}) +\;|\; +A=-A^t +\} +\] +\end{block} +\begin{block}{Lie-Algebra?} +Nachrechnen: $A,B\in \operatorname{so}(n)$ +\begin{align*} +[A,B]^t +&= +(AB-BA)^t +\\ +&= +B^tA^t - A^tB^t +\\ +&= +(-B)(-A)-(-A)(-B) +\\ +&= +BA-AB += +-(AB-BA) +\\ +&= +-[A,B] +\end{align*} +$\Rightarrow$ $\operatorname{so}(n)$ ist eine Lie-Algebra +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 59c9121..574467b 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -1,85 +1,85 @@ -%
-% liealgebra.tex -- Lie-Algebra
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Lie-Algebra}
-\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}}
-\begin{block}{Vektorraum}
-Tangentialvektoren im Punkt $I$:
-\begin{center}
-\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}}
-\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\
-\hline
-\uncover<2->{
-\operatorname{GL}_n(\mathbb{R})
-& beliebige Matrizen
-& M_n(\mathbb{R})
-}
-\\
-\uncover<3->{
-\operatorname{O(n)}
-& antisymmetrische Matrizen
-& \operatorname{o}(n)
-}
-\\
-\uncover<4->{
-\operatorname{SL}_n(\mathbb{R})
-& spurlose Matrizen
-& \operatorname{sl}_2(\mathbb{R})
-}
-\\
-\uncover<5->{
-\operatorname{U(n)}
-& antihermitesche Matrizen
-& \operatorname{u}(n)
-}
-\\
-\uncover<6->{
-\operatorname{SU(n)}
-& spurlose, antihermitesche Matrizen
-& \operatorname{su}(n)
-}
-\end{tabular}
-\end{center}
-\end{block}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.40\textwidth}
-\uncover<7->{%
-\begin{block}{Lie-Klammer}
-Kommutator: $[A,B] = AB-BA$
-\end{block}}
-\uncover<8->{%
-\begin{block}{Nachprüfen}
-$[A,B]\in LG$
-für $A,B\in LG$
-\end{block}}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<9->{%
-\begin{block}{Algebraische Eigenschaften}
-\begin{itemize}
-\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
-\item<11-> Jacobi-Identität
-\[
-[A,[B,C]]+
-[B,[C,A]]+
-[C,[A,B]]
-= 0
-\]
-\end{itemize}
-\vspace{-13pt}
-\uncover<12->{%
-{\usebeamercolor[fg]{title}
-Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% liealgebra.tex -- Lie-Algebra +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra} +\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}} +\begin{block}{Vektorraum} +Tangentialvektoren im Punkt $I$: +\begin{center} +\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} +\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ +\hline +\uncover<2->{ +\operatorname{GL}_n(\mathbb{R}) +& beliebige Matrizen +& M_n(\mathbb{R}) +} +\\ +\uncover<3->{ +\operatorname{O(n)} +& antisymmetrische Matrizen +& \operatorname{o}(n) +} +\\ +\uncover<4->{ +\operatorname{SL}_n(\mathbb{R}) +& spurlose Matrizen +& \operatorname{sl}_2(\mathbb{R}) +} +\\ +\uncover<5->{ +\operatorname{U(n)} +& antihermitesche Matrizen +& \operatorname{u}(n) +} +\\ +\uncover<6->{ +\operatorname{SU(n)} +& spurlose, antihermitesche Matrizen +& \operatorname{su}(n) +} +\end{tabular} +\end{center} +\end{block} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\uncover<7->{% +\begin{block}{Lie-Klammer} +Kommutator: $[A,B] = AB-BA$ +\end{block}} +\uncover<8->{% +\begin{block}{Nachprüfen} +$[A,B]\in LG$ +für $A,B\in LG$ +\end{block}} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<9->{% +\begin{block}{Algebraische Eigenschaften} +\begin{itemize} +\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ +\item<11-> Jacobi-Identität +\[ +[A,[B,C]]+ +[B,[C,A]]+ +[C,[A,B]] += 0 +\] +\end{itemize} +\vspace{-13pt} +\uncover<12->{% +{\usebeamercolor[fg]{title} +Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex new file mode 100644 index 0000000..58065d7 --- /dev/null +++ b/vorlesungen/slides/7/logarithmus.tex @@ -0,0 +1,82 @@ +% +% logarithmus.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmus} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Taylor-Reihe} +\begin{align*} +\frac{d}{dx}\log(1+x) +&= \frac{1}{1+x} +\\ +\uncover<2->{ +\Rightarrow\quad +\log (1+x) +&= +\int_0^x \frac{1}{1+t}\,dt} +\end{align*} +\begin{align*} +\uncover<3->{\frac{1}{1+t} +&= +1-t+t^2-t^3+\dots} +\\ +\uncover<4->{\log(1+x) +&=\int_0^x +1-t+t^2-t^3+\dots +\,dt} +\\ +&\only<5>{= +x-\frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}4 + \dots} +\uncover<6->{= +\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}} +\\ +\uncover<7->{\log (I+A) +&= +\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Konvergenzradius} +Polstelle bei $x=-1$ +\( +\varrho =1 +\) +\end{block}} +\vspace{-5pt} +\begin{block}{\uncover<9->{Alternative: Spektraltheorie}} +\uncover<9->{ +Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$ +definiert:} +\vspace{-15pt} +\uncover<8->{ +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1); +} +\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}]; +\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}]; +\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1]; +\draw[color=blue] (1,0) circle[radius=1]; +\uncover<9->{ + \draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0); +} +\fill (1,0) circle[radius=0.08]; +\node at (2.3,1.9) {$\mathbb{C}$}; +\node at (1,0) [below] {$1$}; +\end{tikzpicture} +\end{center}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex index f88042a..077dc9d 100644 --- a/vorlesungen/slides/7/mannigfaltigkeit.tex +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -1,46 +1,46 @@ -%
-% mannigfaltigkeit.tex -- Mannigfaltigkeit
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Mannigfaltigkeit}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf}
-\end{center}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{Definition}
-\begin{itemize}
-\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
-\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
-\[
-\varphi_\beta\circ\varphi_\alpha^{-1}
-\colon
-\varphi_\alpha(U_\alpha\cap U_\beta)
-\to
-\varphi_\beta(U_\alpha\cap U_\beta)
-\]
-\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken
-\end{itemize}
-\end{block}
-\vspace{-7pt}
-\uncover<5->{%
-\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$}
-Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus
-\end{block}}
-\vspace{-3pt}
-\uncover<6->{%
-\begin{block}{Lie-Gruppe}
-Gruppe und Mannigfaltigkeit
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% mannigfaltigkeit.tex -- Mannigfaltigkeit +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Mannigfaltigkeit} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ +\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet +\[ +\varphi_\beta\circ\varphi_\alpha^{-1} +\colon +\varphi_\alpha(U_\alpha\cap U_\beta) +\to +\varphi_\beta(U_\alpha\cap U_\beta) +\] +\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken +\end{itemize} +\end{block} +\vspace{-7pt} +\uncover<5->{% +\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$} +Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus +\end{block}} +\vspace{-3pt} +\uncover<6->{% +\begin{block}{Lie-Gruppe} +Gruppe und Mannigfaltigkeit +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index afc67c5..f3579a3 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -1,107 +1,107 @@ -%
-% parameter.tex -- Parametrisierung der Matrizen
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-\definecolor{darkyellow}{rgb}{1,0.8,0}
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Drehungen Parametrisieren}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.4\textwidth}
-\begin{block}{Drehung um Achsen}
-\vspace{-12pt}
-\begin{align*}
-\uncover<2->{
-D_{x,\alpha}
-&=
-\begin{pmatrix}
-1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
-\end{pmatrix}
-}
-\\
-\uncover<3->{
-D_{y,\beta}
-&=
-\begin{pmatrix}
-\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta
-\end{pmatrix}
-}
-\\
-\uncover<4->{
-D_{z,\gamma}
-&=
-\begin{pmatrix}
-\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
-\end{pmatrix}
-}
-\intertext{\uncover<5->{beliebige Drehung:}}
-\uncover<5->{
-D
-&=
-D_{x,\alpha}
-D_{y,\beta}
-D_{z,\gamma}
-}
-\end{align*}
-\end{block}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<6->{%
-\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional}
-\uncover<7->{%
-$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
-}
-\\
-\uncover<8->{%
-$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
-}
-\[
-\uncover<9->{
-{\color{red}\vec{x}}
-\mapsto
-}
-\uncover<10->{
-({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})
-\cos\omega
-+
-}
-\uncover<11->{
-({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega
-+
-}
-\uncover<9->{
-{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x})
-}
-\]
-\vspace{-40pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\uncover<9->{
- \node at (0,0)
- {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}};
- \node[color=red] at (1.6,-0.9) {$\vec{x}$};
- \node[color=blue] at (0.5,2) {$\vec{k}$};
-}
-\uncover<11->{
- \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$};
-}
-\uncover<10->{
- \node[color=yellow] at (2.2,-0.2)
- {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$};
-}
-\end{tikzpicture}
-\end{center}
-\end{block}}
-\end{column}
-\end{columns}
-\vspace{-15pt}
-\uncover<5->{%
-{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
-dreidimensionale Gruppe}
-\end{frame}
-\egroup
+% +% parameter.tex -- Parametrisierung der Matrizen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkyellow}{rgb}{1,0.8,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen Parametrisieren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.4\textwidth} +\begin{block}{Drehung um Achsen} +%\vspace{-12pt} +\begin{align*} +\uncover<2->{ +D_{x,\alpha} +&= +\begin{pmatrix} +1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha +\end{pmatrix} +} +\\ +\uncover<3->{ +D_{y,\beta} +&= +\begin{pmatrix} +\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta +\end{pmatrix} +} +\\ +\uncover<4->{ +D_{z,\gamma} +&= +\begin{pmatrix} +\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 +\end{pmatrix} +} +\intertext{\uncover<5->{beliebige Drehung:}} +\uncover<5->{ +D +&= +D_{x,\alpha} +D_{y,\beta} +D_{z,\gamma} +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<6->{% +\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional} +\uncover<7->{% +$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel +} +\\ +\uncover<8->{% +$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse +} +\[ +\uncover<9->{ +{\color{red}\vec{x}} +\mapsto +} +\uncover<10->{ +({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) +\cos\omega ++ +} +\uncover<11->{ +({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega ++ +} +\uncover<9->{ +{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) +} +\] +\vspace{-40pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \node at (0,0) + {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; + \node[color=red] at (1.6,-0.9) {$\vec{x}$}; + \node[color=blue] at (0.5,2) {$\vec{k}$}; +} +\uncover<11->{ + \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; +} +\uncover<10->{ + \node[color=yellow] at (2.2,-0.2) + {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +} +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\vspace{-15pt} +\uncover<5->{% +{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine +dreidimensionale Gruppe} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/qdreh.tex b/vorlesungen/slides/7/qdreh.tex new file mode 100644 index 0000000..8ed512a --- /dev/null +++ b/vorlesungen/slides/7/qdreh.tex @@ -0,0 +1,110 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen mit Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Drehung?} +Abbildung von $\vec{x}$ mit $\operatorname{Re}\vec{x}=0$: +\[ +\varrho_{q} +\colon +\vec{x}\mapsto q\vec{x}q^{-1} = q\vec{x}\overline{q} +\] +\end{block} +\uncover<2->{% +\begin{block}{Achse} +\begin{align*} +\varrho_q(q) +&= +qq\overline{q} +\uncover<3->{= +q(qq^{-1})} +\uncover<4->{= +q} +\end{align*} +\end{block}} +\uncover<4->{% +\begin{block}{Norm} +\begin{align*} +|\varrho_q(\vec{x})|^2 +&= +q\vec{x}\overline{q}\overline{(q\vec{x}\overline{q})} +\uncover<5->{= +q\vec{x}\overline{q}\overline{\overline{q}}\overline{\vec{x}}\overline{q} +} +\\ +&\uncover<6->{= +q\vec{x}(\overline{q}q)\overline{\vec{x}}\overline{q}} +\uncover<7->{= +q(\vec{x}\overline{\vec{x}})\overline{q}} +\uncover<8->{= +q\overline{q}|\vec{x}|^2} +\\ +&\uncover<9->{= +|\vec{x}|^2} +\end{align*} +\uncover<10->{% +$\Rightarrow$ $\varrho_q\in\operatorname{O}(3)$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Drehung!} +$\vec{a},\vec{b},\vec{n}$ bilden ein on.~Rechtssystem +\begin{align*} +\uncover<12->{ +qa +&= +c\vec{a}+s\vec{n}\times \vec{a}} +\uncover<13->{= +c\vec{a} + s\vec{b}} +\\ +\uncover<14->{ +q\vec{a}\overline{q} +&= +(c\vec{a}+s\vec{b}) c +-(c\vec{a}+s\vec{b})\times s\vec{n}} +\\ +&\uncover<15->{= +c^2 \vec{a}+ sc\vec{b} ++sc\vec{b} - s^2 \vec{a}} +\\ +&\uncover<16->{= +\vec{a} \cos\alpha +\vec{b} \sin\alpha } +\end{align*} +\vspace{-5pt} +\uncover<17->{wegen +%\vspace{-5pt} +\[ +\begin{aligned} +\cos\alpha &= \cos^2\frac{\alpha}2 - \sin^2\frac{\alpha}2 &&=c^2-s^2 +\\ +\sin\alpha &= 2\cos\frac{\alpha}2\sin\frac{\alpha}2&&=2cs +\end{aligned}\]} +\end{block}} +\vspace{-18pt} +\uncover<18->{% +\begin{block}{Matrix} +\[ +D += +\tiny +\begin{pmatrix} +1-2(q_2^2+q_3^2)&-2q_0q_3+2q_1q_2&-2q_0q_2+2q_1q_3\\ + 2q_0q_3+2q_1q_2&1-2(q_1^2+q_3^2)&-2q_0q_1+2q_2q_3\\ +-2q_0q_2+2q_1q_3& 2q_0q_1+2q_2q_3&1-2(q_1^2+q_2^2) +\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/quaternionen.tex b/vorlesungen/slides/7/quaternionen.tex new file mode 100644 index 0000000..f526366 --- /dev/null +++ b/vorlesungen/slides/7/quaternionen.tex @@ -0,0 +1,74 @@ +% +% quaternionen.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quaternionen} +$4$-dimensionaler $\mathbb{R}$-Vektorraum +\[ +\mathbb{H} += +\langle 1,i,j,k\rangle_{\mathbb{R}} +\] +mit Rechenregeln +\[ +i^2=j^2=k^2=ijk=-1 +\] +$x=x_0+x_1i+x_2j+x_3k\in\mathbb{H}$ +\begin{itemize} +\item<2-> Realteil: $\operatorname{Re}x=x_0$ +\item<3-> Vektorteil: $\operatorname{Im}x=x_1i+x_2j+x_3k$ +\item<4-> Konjugation: $\overline{x}=\operatorname{Re}x-\operatorname{Im}x$ +\item<5-> Norm: $|x|^2 = x\overline{x} = x_0^2+x_1^2+x_2^2+x_3^2$ +\item<6-> Inverse: $x^{1}= \overline{x}/x\overline{x}$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<7->{% +\begin{block}{Skalarprodukt und Vektorprodukt} +\begin{align*} +pq +&= +\operatorname{Re}p \operatorname{Re}q +- +\operatorname{Im}p\cdot \operatorname{Im}q +\\ +&\phantom{=} ++ +\operatorname{Re}p\operatorname{Im}q ++ +\operatorname{Im}p\operatorname{Re}q ++ +\operatorname{Im}p\times\operatorname{Im}q +\end{align*} +\end{block}} +\uncover<8->{% +\begin{block}{Einheitsquaternionen} +$q\in \mathbb{H}$, $|q|=1, q^{-1}=\overline{q}$ +\end{block}} +\uncover<9->{% +\begin{block}{Polardarstellung} +\[ +q = \cos\frac{\alpha}2 + \vec{n} \sin\frac{\alpha}2 +\] +\vspace{-8pt} +\begin{itemize} +\item<10-> +Drehmatrix: 9 Parameter, 6 Bedingungen +\item<11-> +Quaternionen: 4 Parameter, 1 Bedingung +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index d74b7d0..cd974c9 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -1,117 +1,117 @@ -%
-% semi.tex -- Beispiele: semidirekte Produkte
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Drehung/Skalierung und Verschiebung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Skalierung und Verschiebung}
-Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$
-\\
-Wirkung auf $\mathbb{R}$:
-\[
-x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t
-\]
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{Drehung und Verschiebung}
-Gruppe
-$G=
-\{ (\alpha,\vec{t})
-\;|\;
-\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2
-\}$
-Wirkung auf $\mathbb{R}^2$:
-\[
-\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t}
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\vspace{-15pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<3->{%
-\begin{block}{Verknüpfung}
-\vspace{-15pt}
-\begin{align*}
-(e^{s_1},t_1)(e^{s_2},t_2)x
-&\uncover<4->{=
-(e^{s_1},t_1)(e^{s_2}x+t_2)}
-\\
-&\uncover<5->{=
-e^{s_1+s_2}x + e^{s_1}t_2+t_1}
-\\
-\uncover<6->{
-(e^{s_1},t_1)(e^{s_2},t_2)
-&=
-(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)}
-\end{align*}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Verknüpfung}
-\vspace{-15pt}
-\begin{align*}
-(\alpha_1,\vec{t}_1)
-(\alpha_2,\vec{t}_2)
-\vec{x}
-&\uncover<8->{=
-(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
-\\
-&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1}
-\\
-\uncover<10->{
-(\alpha_1,\vec{t}_1)
-(\alpha_2,\vec{t}_2)
-&=
-(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1)
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\vspace{-10pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<11->{%
-\begin{block}{Matrixschreibweise}
-\vspace{-12pt}
-\[
-g=(e^s,t) =
-\begin{pmatrix}
-e^s&t\\
-0&1
-\end{pmatrix}
-\quad\text{auf}\quad
-\begin{pmatrix}x\\1\end{pmatrix}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<12->{%
-\begin{block}{Matrixschreibweise}
-\vspace{-12pt}
-\[
-g=(\alpha,\vec{t}) =
-\begin{pmatrix}
-D_{\alpha}&\vec{t}\\
-0&1
-\end{pmatrix}
-\quad\text{auf}\quad
-\begin{pmatrix}\vec{x}\\1\end{pmatrix}
-\]
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% semi.tex -- Beispiele: semidirekte Produkte +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehung/Skalierung und Verschiebung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Skalierung und Verschiebung} +Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$ +\\ +Wirkung auf $\mathbb{R}$: +\[ +x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Drehung und Verschiebung} +Gruppe +$G= +\{ (\alpha,\vec{t}) +\;|\; +\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2 +\}$ +Wirkung auf $\mathbb{R}^2$: +\[ +\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} +\] +\end{block}} +\end{column} +\end{columns} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Verknüpfung} +%\vspace{-15pt} +\begin{align*} +(e^{s_1},t_1)(e^{s_2},t_2)x +&\uncover<4->{= +(e^{s_1},t_1)(e^{s_2}x+t_2)} +\\ +&\uncover<5->{= +e^{s_1+s_2}x + e^{s_1}t_2+t_1} +\\ +\uncover<6->{ +(e^{s_1},t_1)(e^{s_2},t_2) +&= +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Verknüpfung} +%\vspace{-15pt} +\begin{align*} +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +\vec{x} +&\uncover<8->{= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} +\\ +&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1} +\\ +\uncover<10->{ +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +&= +(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Matrixschreibweise} +%\vspace{-12pt} +\[ +g=(e^s,t) = +\begin{pmatrix} +e^s&t\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}x\\1\end{pmatrix} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Matrixschreibweise} +%\vspace{-12pt} +\[ +g=(\alpha,\vec{t}) = +\begin{pmatrix} +D_{\alpha}&\vec{t}\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}\vec{x}\\1\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex index 58e87a1..a65b4f6 100644 --- a/vorlesungen/slides/7/sl2.tex +++ b/vorlesungen/slides/7/sl2.tex @@ -1,242 +1,242 @@ -%
-% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\begin{frame}[t,fragile]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{block}{Determinante}
-\[
-A=\begin{pmatrix}
-a&b\\
-c&d
-\end{pmatrix}
-\;\Rightarrow\;
-\det A = ad-bc
-\]
-\end{block}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Dimension}
-\[
-4\; \text{Variablen}
--
-1\; \text{Bedingung}
-=
-3\; \text{Dimensionen}
-\]
-\end{block}
-\end{column}
-\end{columns}
-\vspace{-10pt}
-\uncover<3->{%
-\begin{columns}[t,onlytextwidth]
-\def\s{0.94}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \fill[color=blue!20] (-1,-1) rectangle (1,1);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {4,...,10}{
- \only<\d>{
- \pgfmathparse{1+(\d-4)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- ({-\t},{-1/\t}) rectangle (\t,{1/\t});
- \foreach \x in {-2,...,2}{
- \draw[color=red,line width=0.3pt]
- ({\x*\t},-3) -- ({\x*\t},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{\y/\t}) -- (3,{\y/\t});
- }
- }
- }
- }{}
- \uncover<11->{
- \xdef\t{1.6}
- \fill[color=red!40,opacity=0.5]
- ({-\t},{-1/\t}) rectangle (\t,{1/\t});
- \foreach \x in {-2,...,2}{
- \draw[color=red,line width=0.3pt]
- ({\x*\t},-3) -- ({\x*\t},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{\y/\t}) -- (3,{\y/\t});
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<3->{%
- \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- D
- =
- \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {11,...,17}{
- \only<\d>{
- \pgfmathparse{(\d-11)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- ({-1+\t*(-1)},{-1})
- --
- ({1+\t*(-1)},{-1})
- --
- ({1+\t},{1})
- --
- ({-1+\t},{1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
- }
- }
- }
- }{}
- \uncover<18->{
- \xdef\t{0.6}
- \fill[color=red!40,opacity=0.5]
- ({-1+\t*(-1)},{-1})
- --
- ({1+\t*(-1)},{-1})
- --
- ({1+\t},{1})
- --
- ({-1+\t},{1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- ({-3+\t*\y},\y) -- ({3+\t*\y},\y);
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<11->{
- \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- S
- =
- \begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
- \clip (-2.1,-2.1) rectangle (2.3,2.3);
- \foreach \x in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
- }
- \foreach \y in {-2,...,2}{
- \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
- }
- \ifthenelse{\boolean{presentation}}{
- \foreach \d in {18,...,24}{
- \only<\d>{
- \pgfmathparse{(\d-18)/10}
- \xdef\t{\pgfmathresult}
- \fill[color=red!40,opacity=0.5]
- (-1,{\t*(-1)-1})
- --
- (1,{\t*1-1})
- --
- (1,{\t*1+1})
- --
- (-1,{\t*(-1)+1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
- }
- }
- }
- }{}
- \uncover<25->{
- \xdef\t{0.6}
- \fill[color=red!40,opacity=0.5]
- (-1,{\t*(-1)-1})
- --
- (1,{\t*1-1})
- --
- (1,{\t*1+1})
- --
- (-1,{\t*(-1)+1})
- -- cycle;
- \foreach \x in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (\x,{\x*\t-3}) -- (\x,{\x*\t+3});
- }
- \foreach \y in {-3,...,3}{
- \draw[color=red,line width=0.3pt]
- (-3,{-3*\t+\y}) -- (3,{3*\t+\y});
- }
- }
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<18->{%
-\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
- \node at (0,-2.1) {$
- T
- =
- \begin{pmatrix} 1&0\\t&1\end{pmatrix}
- $};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\end{columns}}
-\end{frame}
-\egroup
+% +% sl2.tex -- Beispiel: Parametrisierung von SL_2(R) +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t,fragile] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Determinante} +\[ +A=\begin{pmatrix} +a&b\\ +c&d +\end{pmatrix} +\;\Rightarrow\; +\det A = ad-bc +\] +\end{block} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Dimension} +\[ +4\; \text{Variablen} +- +1\; \text{Bedingung} += +3\; \text{Dimensionen} +\] +\end{block} +\end{column} +\end{columns} +\vspace{-10pt} +\uncover<3->{% +\begin{columns}[t,onlytextwidth] +\def\s{0.94} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \fill[color=blue!20] (-1,-1) rectangle (1,1); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \ifthenelse{\boolean{presentation}}{ + \foreach \d in {4,...,10}{ + \only<\d>{ + \pgfmathparse{1+(\d-4)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + ({-\t},{-1/\t}) rectangle (\t,{1/\t}); + \foreach \x in {-2,...,2}{ + \draw[color=red,line width=0.3pt] + ({\x*\t},-3) -- ({\x*\t},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{\y/\t}) -- (3,{\y/\t}); + } + } + } + }{} + \uncover<11->{ + \xdef\t{1.6} + \fill[color=red!40,opacity=0.5] + ({-\t},{-1/\t}) rectangle (\t,{1/\t}); + \foreach \x in {-2,...,2}{ + \draw[color=red,line width=0.3pt] + ({\x*\t},-3) -- ({\x*\t},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{\y/\t}) -- (3,{\y/\t}); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<3->{% + \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + D + = + \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\fill[color=blue!20] (-1,-1) rectangle (1,1); +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \ifthenelse{\boolean{presentation}}{ + \foreach \d in {11,...,17}{ + \only<\d>{ + \pgfmathparse{(\d-11)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + ({-1+\t*(-1)},{-1}) + -- + ({1+\t*(-1)},{-1}) + -- + ({1+\t},{1}) + -- + ({-1+\t},{1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({-3+\t*\y},\y) -- ({3+\t*\y},\y); + } + } + } + }{} + \uncover<18->{ + \xdef\t{0.6} + \fill[color=red!40,opacity=0.5] + ({-1+\t*(-1)},{-1}) + -- + ({1+\t*(-1)},{-1}) + -- + ({1+\t},{1}) + -- + ({-1+\t},{1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({-3+\t*\y},\y) -- ({3+\t*\y},\y); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<11->{ + \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + S + = + \begin{pmatrix} 1&s\\ 0&1\end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\fill[color=blue!20] (-1,-1) rectangle (1,1); +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \ifthenelse{\boolean{presentation}}{ + \foreach \d in {18,...,24}{ + \only<\d>{ + \pgfmathparse{(\d-18)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + (-1,{\t*(-1)-1}) + -- + (1,{\t*1-1}) + -- + (1,{\t*1+1}) + -- + (-1,{\t*(-1)+1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); + } + } + } + }{} + \uncover<25->{ + \xdef\t{0.6} + \fill[color=red!40,opacity=0.5] + (-1,{\t*(-1)-1}) + -- + (1,{\t*1-1}) + -- + (1,{\t*1+1}) + -- + (-1,{\t*(-1)+1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<18->{% +\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + T + = + \begin{pmatrix} 1&0\\t&1\end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\end{columns}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex index 8931a24..35d62d8 100644 --- a/vorlesungen/slides/7/symmetrien.tex +++ b/vorlesungen/slides/7/symmetrien.tex @@ -1,145 +1,145 @@ -%
-% symmetrien.tex -- Symmetrien
-%
-% (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{Symmetrien}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Diskrete Symmetrien}
-\begin{itemize}
-\item<2->
-Ebenen-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
-{\tiny
-\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\uncover<4->{\!,\;
-\vec{x}
-\mapsto
-\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
-}
-\]
-\vspace{-10pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\a{10}
-\def\b{50}
-\def\r{2}
-\coordinate (O) at (0,0);
-\coordinate (A) at (\b:\r);
-\coordinate (B) at ({180+2*\a-\b}:\r);
-\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
-\coordinate (N) at (\a:2);
-\coordinate (D) at (\a:{\r*cos(\b-\a)});
-\uncover<3->{
-\clip (-2.5,-0.45) rectangle (2.5,1.95);
-
- \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)
- -- cycle;
- \draw[->,color=darkgreen] (O) -- (N);
- \node[color=darkgreen] at (N) [above] {$\vec{n}$};
-
-
- \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)
- -- cycle;
- \fill[color=red] (O) circle[radius=0.06];
- \draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
- \fill[color=blue] (C) circle[radius=0.06];
- \draw[color=blue,line width=0.1pt] (A) -- (D);
- \node[color=darkgreen] at (D) [below,rotate=\a]
- {$(\vec{n}\cdot\vec{x})\vec{n}$};
- \draw[color=blue,line width=0.5pt] (A)--(B);
-
- \node[color=blue] at (A) [above right] {$\vec{x}$};
- \node[color=blue] at (B) [above left] {$\vec{x}'$};
-
- \node[color=red] at (O) [below left] {$O$};
-
- \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);
- \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);
-}
-
-\end{tikzpicture}
-\end{center}
-\vspace{-5pt}
-$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
-\item<5->
-Punkt-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
--
-{\tiny
-\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\]
-\end{itemize}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<6->{%
-\begin{block}{Kontinuierliche Symmetrien}
-\begin{itemize}
-\item<7-> Translation:
-\(
-\vec{x} \mapsto \vec{x} + \vec{t}
-\)
-\item<8-> Drehung:
-\vspace{-3pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\a{25}
-\def\r{1.3}
-\coordinate (O) at (0,0);
-\begin{scope}
-\clip (-1.1,-0.1) rectangle (2.3,2.3);
-\draw[color=red] (O) circle[radius=2];
-\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
-\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
-\node at ({0.5*\a}:1) {$\alpha$};
-\node at ({90+0.5*\a}:1) {$\alpha$};
-\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);
-\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);
-\end{scope}
-\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
-\end{tikzpicture}
-\end{center}
-\[
-\uncover<9->{%
-\begin{pmatrix}x\\y\end{pmatrix}
-\mapsto
-\begin{pmatrix}
-{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
-{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
-\end{pmatrix}
-\begin{pmatrix}x\\y\end{pmatrix}
-}
-\]
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<10->{%
-\begin{block}{Definition}
-Längen/Winkel bleiben erhalten
-\\
-\uncover<11->{%
-$\Rightarrow$ $\exists$ Erhaltungsgrösse}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+% +% symmetrien.tex -- Symmetrien +% +% (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{Symmetrien} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Diskrete Symmetrien} +\begin{itemize} +\item<2-> +Ebenen-Spiegelung: +\[ +{\tiny +\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} +} +\mapsto +{\tiny +\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*} +} +\uncover<4->{\!,\; +\vec{x} +\mapsto +\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} +} +\] +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{10} +\def\b{50} +\def\r{2} +\coordinate (O) at (0,0); +\coordinate (A) at (\b:\r); +\coordinate (B) at ({180+2*\a-\b}:\r); +\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); +\coordinate (N) at (\a:2); +\coordinate (D) at (\a:{\r*cos(\b-\a)}); +\uncover<3->{ +\clip (-2.5,-0.45) rectangle (2.5,1.95); + + \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) + -- cycle; + \draw[->,color=darkgreen] (O) -- (N); + \node[color=darkgreen] at (N) [above] {$\vec{n}$}; + + + \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) + -- cycle; + \fill[color=red] (O) circle[radius=0.06]; + \draw[color=red] ({\a-90}:2) -- ({\a+90}:2); + \fill[color=blue] (C) circle[radius=0.06]; + \draw[color=blue,line width=0.1pt] (A) -- (D); + \node[color=darkgreen] at (D) [below,rotate=\a] + {$(\vec{n}\cdot\vec{x})\vec{n}$}; + \draw[color=blue,line width=0.5pt] (A)--(B); + + \node[color=blue] at (A) [above right] {$\vec{x}$}; + \node[color=blue] at (B) [above left] {$\vec{x}'$}; + + \node[color=red] at (O) [below left] {$O$}; + + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A); + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B); +} + +\end{tikzpicture} +\end{center} +\vspace{-5pt} +$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ +\item<5-> +Punkt-Spiegelung: +\[ +{\tiny +\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} +} +\mapsto +- +{\tiny +\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*} +} +\] +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Kontinuierliche Symmetrien} +\begin{itemize} +\item<7-> Translation: +\( +\vec{x} \mapsto \vec{x} + \vec{t} +\) +\item<8-> Drehung: +\vspace{-3pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{25} +\def\r{1.3} +\coordinate (O) at (0,0); +\begin{scope} +\clip (-1.1,-0.1) rectangle (2.3,2.3); +\draw[color=red] (O) circle[radius=2]; +\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle; +\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; +\node at ({0.5*\a}:1) {$\alpha$}; +\node at ({90+0.5*\a}:1) {$\alpha$}; +\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2); +\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2); +\end{scope} +\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; +\end{tikzpicture} +\end{center} +\[ +\uncover<9->{% +\begin{pmatrix}x\\y\end{pmatrix} +\mapsto +\begin{pmatrix} +{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\ +{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} +\end{pmatrix} +\begin{pmatrix}x\\y\end{pmatrix} +} +\] +\end{itemize} +\end{block}} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Definition} +Längen/Winkel bleiben erhalten +\\ +\uncover<11->{% +$\Rightarrow$ $\exists$ Erhaltungsgrösse} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/ueberlagerung.tex b/vorlesungen/slides/7/ueberlagerung.tex new file mode 100644 index 0000000..426641a --- /dev/null +++ b/vorlesungen/slides/7/ueberlagerung.tex @@ -0,0 +1,98 @@ +% +% ueberlagerung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$S^3$, $\operatorname{SU}(2)$ und $\operatorname{SO}(3)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\uncover<6->{% +\begin{block}{Überlagerung} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (0,0); +\coordinate (B) at (2,0); +\coordinate (C) at (2,-2); +\coordinate (D) at (0,-2); + +\uncover<7->{ +\node at (A) {$\{\pm 1\}\mathstrut$}; +} +\uncover<6->{ +\node at (B) {$S^3\mathstrut$}; +\node at ($(B)+(0.1,0)$) [right] {$=\operatorname{SU}(2)\mathstrut$}; +} +\uncover<7->{ +\node at (C) {$\operatorname{SO}(3)\mathstrut$}; +\node at (D) {$\{I\}\mathstrut$}; +} + +\uncover<7->{ +\draw[->,shorten >= 0.3cm,shorten <= 0.5cm] (A) -- (B); +\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (D); +\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C); +\draw[->,shorten >= 0.6cm,shorten <= 0.3cm] (D) -- (C); +} + +\end{tikzpicture} +\end{center} +\begin{itemize} +\item<7-> +$\pm q\in S^3$ $\Rightarrow$ $\varrho_{q}=\varrho_{-q}$ +\item<8-> +In der Nähe von $I$ sehen die Gruppen +$\operatorname{SO}(3)$ +und +$\operatorname{SU}(2)$ +``gleich'' aus +\item<9-> +$\operatorname{SU}(2)$ ist geometrisch ``einfacher'' +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\begin{block}{Pauli-Matrizen} +Quaternionen als $2\times 2$-Matrizen schreiben +\begin{align*} +1&=\begin{pmatrix}1&0\\0&1\end{pmatrix}=\sigma_0, +& +i&=\begin{pmatrix}0&i\\i&0\end{pmatrix}=-i\sigma_1 +\\ +j&=\begin{pmatrix}0&-1\\1&0\end{pmatrix}=-i\sigma_2, +& +k&=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=-i\sigma_3 +\end{align*} +\uncover<2->{% +erfüllen $i^2=j^2=k^2=ijk=-1$.} +\end{block} +\uncover<3->{% +\begin{block}{$S^3 = \operatorname{SU}(2)$} +\[ +a+bi+cj+dk += +\begin{pmatrix} +a+id&-c+bi\\ +c+ib&a-id +\end{pmatrix} += +A +\] +\begin{align*} +\uncover<4->{ +\det A &= a^2 + b^2 + c^2 + d^2 = 1 +} +\\ +\uncover<5->{ +A^* &= a - ib - jc - kd +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/vektorlie.tex b/vorlesungen/slides/7/vektorlie.tex new file mode 100644 index 0000000..621a832 --- /dev/null +++ b/vorlesungen/slides/7/vektorlie.tex @@ -0,0 +1,206 @@ +% +% viktorlie.tex -- slide template +% +% (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{Vektorprodukt als Lie-Algebra} +%\vspace{-10pt} +\centering +\begin{tikzpicture}[>=latex,thick] +\arraycolsep=2.4pt +\def\Ax{0} +\def\Ux{4.1} +\def\Kx{7.2} +\def\Rx{13.1} + +\def\Lx{2.2} +\def\Ly{0} +\def\Lz{-2.2} + +\fill[color=red!20] (\Ax,{\Lx-1.55}) rectangle ({\Ux-0.1},{\Lx+0.55}); +\fill[color=red!20] (\Ux,{\Lx-1.55}) rectangle ({\Kx-0.1},{\Lx+0.55}); +\fill[color=red!20] (\Kx,{\Lx-1.55}) rectangle ({\Rx},{\Lx+0.55}); + +\fill[color=darkgreen!20] (\Ax,{\Ly-1.55}) rectangle ({\Ux-0.1},{\Ly+0.55}); +\fill[color=darkgreen!20] (\Ux,{\Ly-1.55}) rectangle ({\Kx-0.1},{\Ly+0.55}); +\fill[color=darkgreen!20] (\Kx,{\Ly-1.55}) rectangle ({\Rx},{\Ly+0.55}); + +\fill[color=blue!20] (\Ax,{\Lz-1.55}) rectangle ({\Ux-0.1},{\Lz+0.55}); +\fill[color=blue!20] (\Ux,{\Lz-1.55}) rectangle ({\Kx-0.1},{\Lz+0.55}); +\fill[color=blue!20] (\Kx,{\Lz-1.55}) rectangle ({\Rx},{\Lz+0.55}); + +\coordinate (A) at (\Ax,3.2); +\coordinate (Ax) at (\Ax,\Lx); +\coordinate (Ay) at (\Ax,\Ly); +\coordinate (Az) at (\Ax,\Lz); + +\node at (A) [right] + {\usebeamercolor[fg]{title}Drehmatrix, $\operatorname{SO}(n)$\strut}; + +\node at (Ax) [right] {$\displaystyle\tiny +D_{x,\alpha}=\begin{pmatrix} +1&0&0\\ +0&\cos\alpha&-\sin\alpha\\ +0&\sin\alpha&\cos\alpha +\end{pmatrix}$}; + +\node at (Ay) [right] {$\displaystyle\tiny +D_{y,\alpha}=\begin{pmatrix} +\cos\alpha&0&\sin\alpha\\ +0&1&0\\ +-\sin\alpha&0&\cos\alpha +\end{pmatrix}$}; + +\node at (Az) [right] {$\displaystyle\tiny +D_{z,\alpha}=\begin{pmatrix} +\cos\alpha&-\sin\alpha&0\\ +\sin\alpha&\cos\alpha&0\\ +0&0&1 +\end{pmatrix}$}; + +\coordinate (U) at (\Ux,3.2); +\coordinate (Ux) at (\Ux,\Lx); +\coordinate (Uy) at (\Ux,\Ly); +\coordinate (Uz) at (\Ux,\Lz); +\coordinate (Ex) at (\Ux,{\Lx-1}); +\coordinate (Ey) at (\Ux,{\Ly-1}); +\coordinate (Ez) at (\Ux,{\Lz-1}); + +\uncover<2->{ +\node at (U) [right] + {\usebeamercolor[fg]{title}Ableitung, $\operatorname{so}(n)$\strut}; + +\node at (Ux) [right] {$\displaystyle\tiny +U_x=\begin{pmatrix*}[r] +0&0&0\\ +0&0&-1\\ +0&1&0 +\end{pmatrix*} +$}; + +\node at (Uy) [right] {$\displaystyle\tiny +U_y=\begin{pmatrix*}[r] +0&0&1\\ +0&0&0\\ +-1&0&0 +\end{pmatrix*} +$}; + +\node at (Uz) [right] {$\displaystyle\tiny +U_z=\begin{pmatrix*}[r] +0&-1&0\\ +1&0&0\\ +0&0&0 +\end{pmatrix*} +$}; +} + +\uncover<9->{ +\node at (Ex) [right] {$\displaystyle +\, e_x = \tiny\begin{pmatrix}1\\0\\0\end{pmatrix} +$}; + +\node at (Ey) [right] {$\displaystyle +\, e_y = \tiny\begin{pmatrix}0\\1\\0\end{pmatrix} +$}; + +\node at (Ez) [right] {$\displaystyle +\, e_z = \tiny\begin{pmatrix}0\\0\\1\end{pmatrix} +$}; +} + +\coordinate (K) at (\Kx,3.2); +\coordinate (Kx) at (\Kx,\Lx); +\coordinate (Ky) at (\Kx,\Ly); +\coordinate (Kz) at (\Kx,\Lz); +\coordinate (Vx) at (\Kx,{\Lx-1}); +\coordinate (Vy) at (\Kx,{\Ly-1}); +\coordinate (Vz) at (\Kx,{\Lz-1}); + +\uncover<3->{ +\node at (K) [right] + {\usebeamercolor[fg]{title}Kommutator\strut}; + +\node at (Kx) [right] {$\displaystyle +\begin{aligned} +[U_y,U_z] &\uncover<4->{= +{\tiny +\begin{pmatrix} +0&0&0\\ +0&0&0\\ +0&1&0 +\end{pmatrix}} +\uncover<5->{\mathstrut- +\tiny +\begin{pmatrix} +0&0&0\\ +0&0&1\\ +0&0&0 +\end{pmatrix}}} +\uncover<6->{=U_x} +\end{aligned} +$}; +} + +\uncover<7->{ +\node at (Ky) [right] {$\displaystyle +\begin{aligned} +[U_z,U_x] &= +{\tiny +\begin{pmatrix} +0&0&1\\ +0&0&0\\ +0&0&0 +\end{pmatrix} +- +\begin{pmatrix} +0&0&0\\ +0&0&0\\ +1&0&0 +\end{pmatrix}} +=U_y +\end{aligned} +$}; +} + +\uncover<8->{ +\node at (Kz) [right] {$\displaystyle +\begin{aligned} +[U_x,U_y] &= +{\tiny +\begin{pmatrix} +0&0&0\\ +1&0&0\\ +0&0&0 +\end{pmatrix} +- +\begin{pmatrix} +0&1&0\\ +0&0&0\\ +0&0&0 +\end{pmatrix}} +=U_z +\end{aligned} +$}; +} + +\uncover<10->{ +\node at (Vx) [right] {$\displaystyle \phantom{]}e_y\times e_z = e_x$}; +} + +\uncover<11->{ +\node at (Vy) [right] {$\displaystyle \phantom{]}e_z\times e_x = e_y$}; +} + +\uncover<12->{ +\node at (Vz) [right] {$\displaystyle \phantom{]}e_x\times e_y = e_z$}; +} + +\end{tikzpicture} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/zusammenhang.tex b/vorlesungen/slides/7/zusammenhang.tex new file mode 100644 index 0000000..6a43cd8 --- /dev/null +++ b/vorlesungen/slides/7/zusammenhang.tex @@ -0,0 +1,99 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zusammenhang} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zusammenhängend --- oder nicht} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\ds{2.4} +\coordinate (A) at (0,0); +\coordinate (B) at (\ds,0); +\coordinate (C) at ({2*\ds},0); + +\node at (A) {$\operatorname{SO}(n)$}; +\node at (B) {$\operatorname{O}(n)$}; +\node at (C) {$\{\pm 1\}$}; + +\draw[->,shorten <= 0.6cm,shorten >= 0.5cm] (A) -- (B); +\draw[->,shorten <= 0.5cm,shorten >= 0.5cm] (B) -- (C); +\node at ($0.5*(B)+0.5*(C)$) [above] {$\det$}; + +\coordinate (A2) at (0,-1.0); +\coordinate (B2) at (\ds,-1.0); +\coordinate (C2) at ({2*\ds},-1.0); + +\draw[color=blue] (A2) ellipse (1cm and 0.3cm); +\draw[color=blue] (B2) ellipse (1cm and 0.3cm); +\node[color=blue] at (C2) {$+1$}; + +\coordinate (A3) at (0,-1.7); +\coordinate (B3) at (\ds,-1.7); +\coordinate (C3) at ({2*\ds},-1.7); + +\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B2) -- (C2); +\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B3) -- (C3); + +\draw[color=red] (B3) ellipse (1cm and 0.3cm); +\node[color=red] at (C3) {$-1$}; + +\end{tikzpicture} +\end{center} +\end{block} +\begin{block}{Zusammenhangskomponente von $e$} +$G_e\subset G$ grösste zusammenhängende Menge, die $e$ enthält: +\begin{align*} +\operatorname{SO}(n)&\subset \operatorname{O}(n) +\\ +\{A\in\operatorname{GL}_n(\mathbb{R})\,|\, \det A > 0\} + &\subset \operatorname{GL}_n(\mathbb{R}) +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Eigenschaften} +\begin{itemize} +\item +{\bf Untergruppe}: $\gamma_i(t)$ Weg von $e$ nach $g_i$, +dann ist +\begin{itemize} +\item +$\gamma_1(t)\gamma_2(t)$ ein Weg von $e$ nach $g_1g_2$ +\item +$\gamma_1(t)^{-1}$ Weg von $e$ nach $g_1^{-1}$ +\end{itemize} +\item +{\bf Normalteiler}: $\gamma(t)$ ein Weg von $e$ nach $g$, dann +ist $h\gamma(t)h^{-1}$ ein Weg von $h$ nach $hgh^{-1}$ +$\Rightarrow hG_eh^{-1}\subset G_e$ +\end{itemize} +\end{block} +\begin{block}{Quotient} +$G/G_e$ ist eine diskrete Gruppe +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (0,0); +\coordinate (B) at (2,0); +\coordinate (C) at (4,0); +\node at (A) {$G_e$}; +\node at (B) {$G$}; +\node at (C) {$G/G_e$}; +\draw [->,shorten <= 0.3cm,shorten >= 0.3cm] (A) -- (B); +\draw [->,shorten <= 0.3cm,shorten >= 0.5cm] (B) -- (C); +\end{tikzpicture} +\end{center} +\vspace{-7pt} +$\Rightarrow$ $G_e$ und $G/G_e$ separat studieren +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index d46dc7f..6ac5665 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -28,5 +28,25 @@ chapter8 = \ ../slides/8/tokyo/bahn0.tex \ ../slides/8/tokyo/bahn1.tex \ ../slides/8/tokyo/bahn2.tex \ + ../slides/8/chrind.tex \ + ../slides/8/chrindprop.tex \ + ../slides/8/chroma1.tex \ + ../slides/8/amax.tex \ + ../slides/8/subgraph.tex \ + ../slides/8/chrwilf.tex \ + ../slides/8/weitere.tex \ + ../slides/8/wavelets/funktionen.tex \ + ../slides/8/wavelets/laplacebasis.tex \ + ../slides/8/wavelets/vektoren.tex \ + ../slides/8/wavelets/fourier.tex \ + ../slides/8/wavelets/lokalisierungsvergleich.tex \ + ../slides/8/wavelets/frequenzlokalisierung.tex \ + ../slides/8/wavelets/dilatation.tex \ + ../slides/8/wavelets/matrixdilatation.tex \ + ../slides/8/wavelets/gundh.tex \ + ../slides/8/wavelets/dilbei.tex \ + ../slides/8/wavelets/frame.tex \ + ../slides/8/wavelets/framekonstanten.tex \ + ../slides/8/wavelets/beispiel.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/amax.tex b/vorlesungen/slides/8/amax.tex new file mode 100644 index 0000000..951400a --- /dev/null +++ b/vorlesungen/slides/8/amax.tex @@ -0,0 +1,86 @@ +% +% amax.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\alpha_{\text{max}}$ und $d$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Definition} +$\alpha_{\text{max}}$ ist der grösste Eigenwert der Adjazenzmatrix +\end{block} +\uncover<2->{ +\begin{block}{Fakten} +\begin{itemize} +\item<3-> +Der Eigenwert $\alpha_{\text{max}}$ ist einfach +\item<4-> +Es gibt einen positiven Eigenvektor $f$ zum Eigenwert $\alpha_{\text{max}}$ +\item<5-> +$f$ maximiert +\[ +\frac{\langle Af,f\rangle}{\langle f,f\rangle} += +\alpha_{\text{max}} +\] +\end{itemize} +Herkunft: Perron-Frobenius-Theorie positiver Matrizen (nächste Woche) +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<6->{% +\begin{block}{Mittlerer Grad} +\[ +\overline{d} += +\frac1{n} \sum_{v} \operatorname{deg}(v) +\le +\alpha_{\text{max}} +\le +d +\] +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{proof}[Beweis] +\begin{itemize} +\item Konstante Funktion $1$ anstelle von $f$: +\[ +\frac{\langle A1,1\rangle}{\langle 1,1\rangle} +\uncover<8->{= +\frac{\sum_v \operatorname{deg}(v)}{n}} +\uncover<9->{= +\overline{d}} +\uncover<10->{\le +\alpha_{\text{max}}} +\] +\item<11-> Komponenten von $Af$ summieren: +\begin{align*} +\uncover<12->{ +\alpha_{\text{max}} +f(v) &= (Af)(v)}\uncover<13->{ = \sum_{u\sim v} f(u)} +\\ +\uncover<14->{\alpha_{\text{max}} +\sum_{v}f(v) +&= +\sum_v +\operatorname{deg}(v) f(v)} +\\ +&\uncover<15->{\le +d\sum_v f(v)} +\; +\uncover<16->{\Rightarrow +\; +\alpha_{\text{max}} \le d} +\end{align*} +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 6a0b13f..69b7231 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -30,3 +30,24 @@ \folie{8/tokyo/bahn1.tex} \folie{8/tokyo/bahn2.tex} +\folie{8/chrind.tex} +\folie{8/chrindprop.tex} +\folie{8/chroma1.tex} +\folie{8/amax.tex} +\folie{8/subgraph.tex} +\folie{8/chrwilf.tex} +\folie{8/weitere.tex} + +\folie{8/wavelets/funktionen.tex} +\folie{8/wavelets/laplacebasis.tex} +\folie{8/wavelets/fourier.tex} +\folie{8/wavelets/lokalisierungsvergleich.tex} +\folie{8/wavelets/frequenzlokalisierung.tex} +\folie{8/wavelets/dilatation.tex} +\folie{8/wavelets/matrixdilatation.tex} +\folie{8/wavelets/gundh.tex} +\folie{8/wavelets/frame.tex} +\folie{8/wavelets/dilbei.tex} +\folie{8/wavelets/framekonstanten.tex} +\folie{8/wavelets/beispiel.tex} + diff --git a/vorlesungen/slides/8/chrind.tex b/vorlesungen/slides/8/chrind.tex new file mode 100644 index 0000000..bd406ab --- /dev/null +++ b/vorlesungen/slides/8/chrind.tex @@ -0,0 +1,231 @@ +% +% chrind.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Chromatische Zahl und Unabhängigkeitszahl} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Chromatische Zahl} +$\operatorname{chr}(G)=\mathstrut$ +minimale Anzahl Farben, die zum Einfärben eines Graphen $G$ nötig sind derart, +dass benachbarte Knoten verschiedene Farben haben. +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\Ra{2} +\def\Ri{1} +\def\e{1.0} +\def\r{0.2} + +\definecolor{rot}{rgb}{0.8,0,0.8} +\definecolor{gruen}{rgb}{0.2,0.6,0.2} +\definecolor{blau}{rgb}{1,0.6,0.2} + +\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)}); +\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)}); +\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)}); +\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)}); +\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)}); + +\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)}); +\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)}); +\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)}); +\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)}); +\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)}); + +\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle; +\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle; +\draw (PA)--(QA); +\draw (PB)--(QB); +\draw (PC)--(QC); +\draw (PD)--(QD); +\draw (PE)--(QE); + +\only<1>{ + \fill[color=white] (PA) circle[radius=\r]; + \fill[color=white] (PB) circle[radius=\r]; + \fill[color=white] (PC) circle[radius=\r]; + \fill[color=white] (PD) circle[radius=\r]; + \fill[color=white] (PE) circle[radius=\r]; + \fill[color=white] (QA) circle[radius=\r]; + \fill[color=white] (QB) circle[radius=\r]; + \fill[color=white] (QC) circle[radius=\r]; + \fill[color=white] (QD) circle[radius=\r]; + \fill[color=white] (QE) circle[radius=\r]; +} + +\only<2->{ + \fill[color=blau] (PA) circle[radius=\r]; + \fill[color=rot] (PB) circle[radius=\r]; + \fill[color=rot] (PC) circle[radius=\r]; + \fill[color=gruen] (PD) circle[radius=\r]; + \fill[color=gruen] (PE) circle[radius=\r]; + + \fill[color=rot] (QA) circle[radius=\r]; + \fill[color=blau] (QB) circle[radius=\r]; + \fill[color=gruen] (QC) circle[radius=\r]; + \fill[color=rot] (QD) circle[radius=\r]; + \fill[color=blau] (QE) circle[radius=\r]; +} + +\draw (PA) circle[radius=\r]; +\draw (PB) circle[radius=\r]; +\draw (PC) circle[radius=\r]; +\draw (PD) circle[radius=\r]; +\draw (PE) circle[radius=\r]; + +\draw (QA) circle[radius=\r]; +\draw (QB) circle[radius=\r]; +\draw (QC) circle[radius=\r]; +\draw (QD) circle[radius=\r]; +\draw (QE) circle[radius=\r]; + +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{chr} G = 3$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Unabhängigkeitszahl} +$\operatorname{ind}(G)=\mathstrut$ +maximale Anzahl nicht benachbarter Knoten +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\Ra{2} +\def\Ri{1} +\def\e{1.0} +\def\r{0.2} + +\definecolor{rot}{rgb}{0.8,0,0.8} +\definecolor{gruen}{rgb}{0.2,0.6,0.2} +\definecolor{blau}{rgb}{1,0.6,0.2} +\definecolor{gelb}{rgb}{0,0,1} + +\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)}); +\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)}); +\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)}); +\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)}); +\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)}); + +\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)}); +\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)}); +\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)}); +\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)}); +\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)}); + +\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle; +\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle; +\draw (PA)--(QA); +\draw (PB)--(QB); +\draw (PC)--(QC); +\draw (PD)--(QD); +\draw (PE)--(QE); + +\foreach \n in {1,...,7}{ + \only<\n>{\node[color=white] at (1,2.9) {$\n$};} +} + +\fill[color=white] (PA) circle[radius=\r]; +\fill[color=white] (PB) circle[radius=\r]; +\fill[color=white] (PC) circle[radius=\r]; +\fill[color=white] (PD) circle[radius=\r]; +\fill[color=white] (PE) circle[radius=\r]; +\fill[color=white] (QA) circle[radius=\r]; +\fill[color=white] (QB) circle[radius=\r]; +\fill[color=white] (QC) circle[radius=\r]; +\fill[color=white] (QD) circle[radius=\r]; +\fill[color=white] (QE) circle[radius=\r]; + +\only<4->{ + \fill[color=rot] (QA) circle[radius={1.5*\r}]; + \fill[color=rot!40] (QB) circle[radius=\r]; + \fill[color=rot!40] (QE) circle[radius=\r]; + \fill[color=rot!40] (PA) circle[radius=\r]; +} + +\only<5->{ + \fill[color=blau] (PB) circle[radius={1.5*\r}]; + \fill[color=blau!40] (PD) circle[radius=\r]; + \fill[color=blau!40] (PE) circle[radius=\r]; + \fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r]; +} + +\only<6->{ + \fill[color=gruen] (PC) circle[radius={1.5*\r}]; + \fill[color=gruen!40] (QC) circle[radius=\r]; + \fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r]; + \fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r]; +} + +\only<7->{ + \fill[color=gelb] (QD) circle[radius={1.5*\r}]; + \fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r]; + \fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r]; + \fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r]; +} + +\only<-3|handout:0>{ + \draw (QA) circle[radius=\r]; +} +\only<4->{ + \draw (QA) circle[radius={1.5*\r}]; +} + +\only<-4|handout:0>{ + \draw (PB) circle[radius=\r]; +} +\only<5->{ + \draw (PB) circle[radius={1.5*\r}]; +} + +\only<-5|handout:0>{ + \draw (PC) circle[radius=\r]; +} +\only<6->{ + \draw (PC) circle[radius={1.5*\r}]; +} + +\only<-6|handout:0>{ + \draw (QD) circle[radius=\r]; +} +\only<7->{ + \draw (QD) circle[radius={1.5*\r}]; +} + +\draw (PA) circle[radius=\r]; +\draw (PD) circle[radius=\r]; +\draw (PE) circle[radius=\r]; + +\draw (QB) circle[radius=\r]; +\draw (QC) circle[radius=\r]; +\draw (QE) circle[radius=\r]; + +\only<4|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 1$}; +} +\only<5|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 2$}; +} +\only<6|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 3$}; +} +\only<7->{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 4$}; +} + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chrindprop.tex b/vorlesungen/slides/8/chrindprop.tex new file mode 100644 index 0000000..094588c --- /dev/null +++ b/vorlesungen/slides/8/chrindprop.tex @@ -0,0 +1,62 @@ +% +% chrindprop.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zusammenhang zwischen $\operatorname{chr}G$ und $\operatorname{ind}G$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Proposition} +Ist $G$ ein Graph mit $n$ Knoten, dann gilt +\[ +\operatorname{chr}G +\cdot +\operatorname{ind}G +\ge n +\] +\end{block} +\uncover<2->{% +\begin{block}{Beispiel} +Peterson-Graph $K$ hat $n=10$ Knoten: +\[ +\operatorname{chr}(K) +\cdot +\operatorname{ind}(K) += +3\cdot 4 +\ge +10 += +n +\] +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<3->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<4-> eine minimale Färbung hat $\operatorname{chr}(G)$ Farben +\item<5-> Sie teilt die Knoten in $\operatorname{chr}(G)$ +gleichfarbige Mengen auf +\item<6-> Jede einfarbige Menge von Knoten ist unabhängig, d.~h.~sie +besteht aus Knoten, die nicht miteinander verbunden sind. +\item<7-> Jede einfarbige Menge enthält höchstens $\operatorname{ind}(G)$ +\item<8-> Die Gesamtzahl der Knoten ist +\[ +n\uncover<9->{=\sum_{\text{Farbe}}\underbrace{|V_{\text{Farbe}}|}_{\le \operatorname{ind}(G)}} +\uncover<10->{\le +\operatorname{chr}(G) +\cdot +\operatorname{ind}(G)} +\] +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chroma1.tex b/vorlesungen/slides/8/chroma1.tex new file mode 100644 index 0000000..6a55704 --- /dev/null +++ b/vorlesungen/slides/8/chroma1.tex @@ -0,0 +1,56 @@ +% +% chroma1.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Schranke für $\operatorname{chr}(G)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Proposition} +Ist $G$ ein Graph mit maximalem Grad $d$, dann gilt +\[ +\operatorname{chr}(G) \le d + 1 +\] +\end{block} +\uncover<2->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<3-> +Peterson-Graph $G$: maximaler Grad ist $d=3$, aber +\[ +\operatorname{chr}(G) += +3 +< d+1=4 +\] +\item<4-> +Voller Graph $V$: maximaler Grad ist $d=n-1$, +\[ +\operatorname{chr}(V) = n = d+1 +\] +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<4->{% +\begin{proof}[Beweis] +Mit vollständiger Induktion, d.~h.~Annahme: Graphen mit $<n$ Knoten und +maximalem Grad $d$ lassen sich mit höchstens $d+1$ Farben färben. +\begin{itemize} +\item<5-> $X$ ein Graph mit $n$ Knoten +\item<6-> entferne den Knoten $v\in X$, $X'=X\setminus\{v\}$ +\item<7-> $X'$ lässt sich mit höchstens $d+1$ Farben einfärben +\item<8-> $v$ hat höchstens $d$ Nachbarn, die höchsten $d$ verschiedene +Farben haben +\item<9-> Es bleibt eine Farbe für $v$ +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chrwilf.tex b/vorlesungen/slides/8/chrwilf.tex new file mode 100644 index 0000000..7edb10e --- /dev/null +++ b/vorlesungen/slides/8/chrwilf.tex @@ -0,0 +1,115 @@ +% +% chrwilf.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\def\kante#1#2{ + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (#1) -- (#2); +} +\def\knoten#1#2{ + \uncover<8->{ + \fill[color=#2!30] (#1) circle[radius=0.2]; + \draw[color=#2] (#1) circle[radius=0.2]; + } + \only<-7>{ + \draw (#1) circle[radius=0.2]; + } +} +\def\R{1.5} +\definecolor{rot}{rgb}{1,0,0} +\definecolor{gruen}{rgb}{0,0.6,0} +\definecolor{blau}{rgb}{0,0,1} +\begin{frame}[t] +\frametitle{Schranke für die chromatische Zahl} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz (Wilf)} +$\uncover<2->{\operatorname{chr}(X) \le 1+}\alpha_{\text{max}} \le\uncover<2->{ 1 + }d$ +\end{block} +\uncover<3->{% +\begin{block}{Beispiel} +\begin{align*} +\uncover<4->{d&= 4} +&&\uncover<5->{\Rightarrow& \operatorname{chr}(G) &\le 5}\\ +\uncover<6->{\alpha_{\text{max}} &= +2.9565} +&&\uncover<7->{\Rightarrow& \operatorname{chr}(G) &\le 3}\\ +\uncover<4->{\overline{d} &= \frac{24}{9}=\rlap{$2.6666$}} +\end{align*} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (A) at (0:\R); +\coordinate (B) at (40:\R); +\coordinate (C) at (80:\R); +\coordinate (D) at (120:\R); +\coordinate (E) at (160:\R); +\coordinate (F) at (200:\R); +\coordinate (G) at (240:\R); +\coordinate (H) at (280:\R); +\coordinate (I) at (320:\R); + +\knoten{A}{rot} +\knoten{B}{blau} +\knoten{C}{gruen} +\knoten{D}{blau} +\knoten{E}{rot} +\knoten{F}{blau} +\knoten{G}{rot} +\knoten{H}{gruen} +\knoten{I}{blau} + +\kante{A}{B} +\kante{B}{C} +\kante{C}{D} +\kante{D}{E} +\kante{E}{F} +\kante{F}{G} +\kante{G}{H} +\kante{H}{I} +\kante{I}{A} + +\kante{A}{C} +\kante{A}{D} +\kante{D}{G} + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\begin{column}{0.52\textwidth} +\uncover<9->{% +\begin{proof}[Beweis] +Induktion nach der Grösse $n$ des Graphen. +\begin{itemize} +\item<10-> +Entferne $v\in X$ mit minimalem Grad: $X'=X\setminus \{v\}$ +\item<11-> +Induktionsannahme: +\[ +\operatorname{chr}(X') +\le +1+ +\alpha_{\text{max}}' +\] +\item<12-> +$X'$ kann mit höhcstens $1+\alpha_{\text{max}}'\le 1+\alpha_{\text{max}}$ +Farben eingefärbt werden. +\item<13-> +Wegen +\( +\deg(v) \le \overline{d} \le \alpha_{\text{max}} +\) +hat $v$ höchstens $\alpha_{\text{max}}$ Nachbarn, um $v$ zu färben, +braucht man also höchstens $1+\alpha_{\text{max}}$ Farben. +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex index 952c85b..10f88cd 100644 --- a/vorlesungen/slides/8/inzidenz.tex +++ b/vorlesungen/slides/8/inzidenz.tex @@ -5,6 +5,8 @@ % \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \begin{frame}[t] \frametitle{Inzidenz- und Adjazenzmatrix} \vspace{-20pt} @@ -67,7 +69,7 @@ \vspace{-10pt} \uncover<5->{% \begin{block}{Definition} -\vspace{-15pt} +%\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 $j$} diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex index 5f2f51a..43e5330 100644 --- a/vorlesungen/slides/8/inzidenzd.tex +++ b/vorlesungen/slides/8/inzidenzd.tex @@ -5,6 +5,8 @@ % \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \begin{frame}[t] \frametitle{Inzidenz- und Adjazenz-Matrix} \vspace{-20pt} @@ -67,7 +69,7 @@ \vspace{-15pt} \uncover<5->{% \begin{block}{Definition} -\vspace{-20pt} +%\vspace{-20pt} \begin{align*} B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\ B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\ diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex index 1d8b725..93333bc 100644 --- a/vorlesungen/slides/8/produkt.tex +++ b/vorlesungen/slides/8/produkt.tex @@ -56,7 +56,7 @@ \end{center} \vspace{-15pt} \begin{block}{Berechne} -\vspace{-20pt} +%\vspace{-20pt} \begin{align*} \uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t \end{align*} diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex index 425fe1c..62180d9 100644 --- a/vorlesungen/slides/8/spanningtree.tex +++ b/vorlesungen/slides/8/spanningtree.tex @@ -3,6 +3,7 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % +\bgroup \begin{frame} \frametitle{Spannbäume} @@ -121,7 +122,7 @@ Wieviele Spannbäume gibt es? \begin{column}{0.56\hsize} \uncover<5->{% \begin{block}{Laplace-Matrix} -\vspace{-15pt} +%\vspace{-15pt} \[ L= \tiny @@ -162,3 +163,4 @@ L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{arr \end{columns} \end{frame} +\egroup diff --git a/vorlesungen/slides/8/subgraph.tex b/vorlesungen/slides/8/subgraph.tex new file mode 100644 index 0000000..f3005f9 --- /dev/null +++ b/vorlesungen/slides/8/subgraph.tex @@ -0,0 +1,60 @@ +% +% subgraph.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\alpha_{\text{max}}$ eines Untergraphen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$X'$ ein echter Untergraph von $X$ mit Adjazenzmatrix $A'$ und grösstem +Eigenwert $\alpha_{\text{max}}'$ +\[ +\alpha_{\text{max}}' \le \alpha_{\text{max}} +\] +\end{block} +\uncover<2->{$V'$ die Knoten von $X'$} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<4-> +$f'$ der positive Eigenvektor von $A'$ +\item<5-> +Definiere +\[ +g(v) += +\begin{cases} +f'(v) &\qquad v\in V'\\ +0 &\qquad \text{sonst} +\end{cases} +\] +\item<6-> Skalarprodukte: +\begin{align*} +\uncover<7->{\langle f',f'\rangle &= \langle g,g\rangle} +\\ +\uncover<8->{\langle A'f',f'\rangle &\le \langle Ag,g\rangle} +\end{align*} +\item<9-> Vergleich +\[ +\alpha_{\text{max}}' += +\frac{\langle A'f',f'\rangle}{\langle f',f'\rangle} +\uncover<10->{\le +\frac{\langle Ag,g\rangle}{\langle g,g\rangle}} +\uncover<11->{\le +\alpha_{\text{max}}} +\] +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/Makefile b/vorlesungen/slides/8/wavelets/Makefile new file mode 100644 index 0000000..3b4a5ce --- /dev/null +++ b/vorlesungen/slides/8/wavelets/Makefile @@ -0,0 +1,8 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +vektoren.tex: ev.m + octave ev.m diff --git a/vorlesungen/slides/8/wavelets/beispiel.tex b/vorlesungen/slides/8/wavelets/beispiel.tex new file mode 100644 index 0000000..dcc33d4 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/beispiel.tex @@ -0,0 +1,44 @@ +% +% beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{ +\node at (0,0) [rotate=-90] +{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}}; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wavelets auf einer Kugel} +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\only<1>{ \bild{0.6}{wavelets-phi-sphere-334.pdf} } + +\only<2>{ \bild{0.6}{wavelets-psi-5-sphere-334.pdf} } +\only<3>{ \bild{0.6}{wavelets-psi-4-sphere-334.pdf} } +\only<4>{ \bild{0.6}{wavelets-psi-3-sphere-334.pdf} } +\only<5>{ \bild{0.6}{wavelets-psi-2-sphere-334.pdf} } +\only<6>{ \bild{0.6}{wavelets-psi-1-sphere-334.pdf} } + +\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; } +\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; } +\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; } +\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; } +\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; } +\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; } + +\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; } +\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; } +\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; } +\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; } +\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; } +\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; } + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/dilatation.tex b/vorlesungen/slides/8/wavelets/dilatation.tex new file mode 100644 index 0000000..881f760 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/dilatation.tex @@ -0,0 +1,62 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Dilatation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Dilatation in $\mathbb{R}$} +$f\colon \mathbb{R}\to\mathbb{R}$ +Definition im Ortsraum: +\[ +(D_af)(x) += +\frac{1}{\sqrt{|a|}} +f\biggl(\frac{x}{a}\biggr) +\] +\uncover<2->{% +Dilatation im Frequenzraum: +\[ +\widehat{D_af}(\omega) += +D_{1/a}\hat{f}(\omega) +\]} +\uncover<3->{% +Spektrum wird mit $1/a$ skaliert!} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{``Dilatation'' auf einem Graphen} +\begin{itemize} +\item<5-> Dilatation auf dem Graphen gibt es nicht +\item<6-> Dilatation im Spektrum $\{\lambda_1,\dots,\lambda_n\}$ gibt es nicht +\item<7-> ``Spektrale Dilatation'' verwenden +\begin{enumerate} +\item<8-> Start: $e_k$ +\item<9-> Fourier-Transformation: $\chi^te_k$ +\item<10-> Spektrum skalieren: mit +$D_{1/a}g$ filtern +\item<11-> Rücktransformation +\[ +D_{g,a}e_k += +\chi +\uncover<12->{\operatorname{diag}(\tilde{D}_{1/a}g(\lambda_*)) +\chi^t e_k} +\] +\end{enumerate} +\end{itemize} + + +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/dilbei.tex b/vorlesungen/slides/8/wavelets/dilbei.tex new file mode 100644 index 0000000..fc66a0a --- /dev/null +++ b/vorlesungen/slides/8/wavelets/dilbei.tex @@ -0,0 +1,46 @@ +% +% beispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{ +\node at (0,0) [rotate=-90] +{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}}; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wavelets einer Strecke} +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\only<1>{ \bild{0.6}{wavelets-psi-line-5-10.pdf} } +\only<2>{ \bild{0.6}{wavelets-psi-line-4-10.pdf} } +\only<3>{ \bild{0.6}{wavelets-psi-line-3-10.pdf} } +\only<4>{ \bild{0.6}{wavelets-psi-line-2-10.pdf} } +\only<5>{ \bild{0.6}{wavelets-psi-line-1-10.pdf} } + +\only<6>{ \bild{0.6}{wavelets-phi-line-10.pdf} } + +\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; } +\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; } +\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; } +\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; } +\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; } +\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; } + + +\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; } +\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; } +\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; } +\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; } +\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; } + +\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; } + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/ev.m b/vorlesungen/slides/8/wavelets/ev.m new file mode 100644 index 0000000..7f4dd55 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/ev.m @@ -0,0 +1,97 @@ +# +# ev.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +L = [ + 2, -1, 0, -1, 0; + -1, 4, -1, -1, -1; + 0, -1, 2, 0, -1; + -1, -1, 0, 3, -1; + 0, -1, -1, -1, 3 +]; + +[v, lambda] = eig(L); + +function knoten(fn, wert, punkt) + if (wert > 0) + farbe = sprintf("red!%02d", round(100 * wert)); + else + farbe = sprintf("blue!%02d", round(-100 * wert)); + end + fprintf(fn, "\t\\fill[color=%s] %s circle[radius=0.25];\n", + farbe, punkt); + fprintf(fn, "\t\\draw %s circle[radius=0.25];\n", punkt); +endfunction + +function vektor(fn, v, name, lambda) + fprintf(fn, "\\def\\%s{\n", name); + fprintf(fn, "\t\\coordinate (A) at ({0*\\a},0);\n"); + fprintf(fn, "\t\\coordinate (B) at ({1*\\a},0);\n"); + fprintf(fn, "\t\\coordinate (C) at ({2*\\a},0);\n"); + fprintf(fn, "\t\\coordinate (D) at ({0.5*\\a},{-\\b});\n"); + fprintf(fn, "\t\\coordinate (E) at ({1.5*\\a},{-\\b});\n"); + fprintf(fn, "\t\\draw (A) -- (B);\n"); + fprintf(fn, "\t\\draw (A) -- (D);\n"); + fprintf(fn, "\t\\draw (B) -- (C);\n"); + fprintf(fn, "\t\\draw (B) -- (D);\n"); + fprintf(fn, "\t\\draw (B) -- (E);\n"); + fprintf(fn, "\t\\draw (C) -- (E);\n"); + fprintf(fn, "\t\\draw (D) -- (E);\n"); + fprintf(fn, "\t\\node at (-2.8,{-0.5*\\b}) [right] {$\\lambda=%.4f$};\n", + round(1000 * abs(lambda)) / 10000); + w = v / max(abs(v)); + knoten(fn, w(1,1), "(A)"); + knoten(fn, w(2,1), "(B)"); + knoten(fn, w(3,1), "(C)"); + knoten(fn, w(4,1), "(D)"); + knoten(fn, w(5,1), "(E)"); + fprintf(fn, "}\n"); +endfunction + +function punkt(fn, x, wert) + fprintf(fn, "({%.4f*\\c},{%.4f*\\d})", x, wert); +endfunction + +function funktion(fn, v, name, lambda) + fprintf(fn, "\\def\\%s{\n", name); + fprintf(fn, "\t\\draw[color=red,line width=1.4pt]\n\t\t"); + punkt(fn, -2, v(1,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, -1, v(4,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, 0, v(2,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, 1, v(5,1)); + fprintf(fn, " --\n\t\t"); + punkt(fn, 2, v(3,1)); + fprintf(fn, ";\n"); + fprintf(fn, "\t\\draw[->] ({-2.1*\\c},0) -- ({2.1*\\c},0);\n"); + fprintf(fn, "\t\\draw[->] (0,{-1.1*\\d}) -- (0,{1.1*\\d});\n"); + for x = (-2:2) + fprintf(fn, "\t\\fill ({%d*\\c},0) circle[radius=0.05];\n", x); + endfor + fprintf(fn, "}\n"); +endfunction + +fn = fopen("vektoren.tex", "w"); + +vektor(fn, v(:,1), "vnull", lambda(1,1)); +funktion(fn, v(:,1), "fnull", lambda(1,1)); + +vektor(fn, v(:,2), "vone", lambda(2,2)); +funktion(fn, v(:,2), "fone", lambda(2,2)); + +vektor(fn, v(:,3), "vtwo", lambda(3,3)); +funktion(fn, v(:,3), "ftwo", lambda(3,3)); + +vektor(fn, v(:,4), "vthree", lambda(4,4)); +funktion(fn, v(:,4), "fthree", lambda(4,4)); + +vektor(fn, v(:,5), "vfour", lambda(5,5)); +funktion(fn, v(:,5), "ffour", lambda(5,5)); + +fclose(fn); + + diff --git a/vorlesungen/slides/8/wavelets/fourier.tex b/vorlesungen/slides/8/wavelets/fourier.tex new file mode 100644 index 0000000..3195ec8 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/fourier.tex @@ -0,0 +1,86 @@ +% +% fourier.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\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}{Aufgabe} +Gegeben: Funktion $f$ auf dem Graphen +\\ +\uncover<2->{% +Gesucht: Koeffizienten $\hat{f}$ der Darstellung in der Laplace-Basis} +\end{block} +\uncover<3->{% +\begin{block}{Definition $\chi$-Matrix} +Eigenwerte $0=\lambda_1<\lambda_2\le \dots \le \lambda_n$ von $L$ +\vspace{-10pt} +\begin{center} +\begin{tikzpicture} +\node at (-1.9,0) [left] {$\chi=\mathstrut$}; +\node at (0,0) {$\left(\raisebox{0pt}[1.7cm][1.7cm]{\hspace{3.5cm}}\right)$}; + +\fill[color=blue!20] (-1.7,-1.7) rectangle (-1.1,1.7); +\draw[color=blue] (-1.7,-1.7) rectangle (-1.1,1.7); +\node at (-1.4,0) [rotate=90] {$v_1=\mathstrut$EV zum EW $\lambda_1$\strut}; + +\fill[color=blue!20] (-1.0,-1.7) rectangle (-0.4,1.7); +\draw[color=blue] (-1.0,-1.7) rectangle (-0.4,1.7); +\node at (-0.7,0) [rotate=90] {$v_2=\mathstrut$EV zum EW $\lambda_2$\strut}; + +\fill[color=blue!20] (1.1,-1.7) rectangle (1.7,1.7); +\draw[color=blue] (1.1,-1.7) rectangle (1.7,1.7); +\node at (1.4,0) [rotate=90] {$v_n=\mathstrut$EV zum EW $\lambda_n$\strut}; + +\node at (0.4,0) {$\dots$}; + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Transformation} +$L$ symmetrisch +\\ +\uncover<5->{$\Rightarrow$ +Die Eigenvektoren von $L$ können orthonormiert gewählt werden} +\\ +\uncover<6->{$\Rightarrow$ +Koeffizienten können durch Skalarprodukte ermittelt werden:} +\uncover<7->{% +\[ +\hat{f}(k) += +\hat{f}(\lambda_k) +\uncover<8->{= +\langle v_k, f\rangle +\quad\Rightarrow\quad +\hat{f}} +\uncover<9->{= +\chi^tf} +\]} +\uncover<10->{% +$\chi$ ist die {\em Fourier-Transformation}} +\end{block}} +\uncover<11->{% +\begin{block}{Rücktransformation} +Eigenvektoren orthonormiert +\\ +\uncover<12->{$\Rightarrow$ +$\chi$ orthogonal} +\uncover<13->{ +\[ +\chi\chi^t = I +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/frame.tex b/vorlesungen/slides/8/wavelets/frame.tex new file mode 100644 index 0000000..4d0c7d1 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/frame.tex @@ -0,0 +1,66 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Graph Wavelet Frame} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Frame-Vektoren} +Zu Dilatationsfaktoren $A=\{a_i\,|\,i=1,\dots,N\}$ +konstruiere das Frame +\begin{align*} +F= +\{&D_he_1,\dots,D_he_n,\\ + &Dg_1e_1,\dots,Dg_1e_n,\\ + &Dg_2e_1,\dots,Dg_2e_n,\\ + &\dots\\ + &Dg_Ne_1,\dots,Dg_Ne_n\} +\end{align*} +\uncover<2->{Notation: +\begin{align*} +v_{0,k} +&= +D_he_k +\\ +v_{i,k} +&= +Dg_ie_k +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Frameoperator} +\begin{align*} +\mathcal{T}\colon \mathbb{R}^n\to\mathbb{R}^{nN} +: +v +&\mapsto +\begin{pmatrix} +\uncover<4->{\langle D_he_1,v\rangle}\\ +\uncover<4->{\vdots}\\ +\uncover<4->{\langle D_he_n,v\rangle}\\ +\hline +\uncover<5->{\langle D_{g_1}e_1,v\rangle}\\ +\uncover<5->{\vdots}\\ +\uncover<5->{\langle D_{g_1}e_n,v\rangle}\\ +\hline +\uncover<6->{\vdots}\\ +\uncover<6->{\vdots}\\ +\hline +\uncover<7->{\langle D_{g_N}e_1,v\rangle}\\ +\uncover<7->{\vdots}\\ +\uncover<7->{\langle D_{g_N}e_n,v\rangle} +\end{pmatrix} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/framekonstanten.tex b/vorlesungen/slides/8/wavelets/framekonstanten.tex new file mode 100644 index 0000000..a436536 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/framekonstanten.tex @@ -0,0 +1,71 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +%\setlength{\abovedisplayskip}{5pt} +%\setlength{\belowdisplayskip}{5pt} +\frametitle{Framekonstanten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Menge $\mathcal{F}$ von Vektoren heisst ein Frame, +falls es Konstanten $A$ und $B$ gibt derart, dass +\[ +A\|v\|^2 +\le +\|\mathcal{T}v\|^2 +\sum_{b\in\mathcal{F}} |\langle b,v\rangle|^2 +\le +B\|v\|^2 +\] +\uncover<2->{$A>0$ garantiert Invertierbarkeit} +\end{block} +\uncover<3->{% +\begin{block}{$\|\mathcal{T}v\|$ für Graph-Wavelets} +\begin{align*} +\|\mathcal{T}v\|^2 +&= +\sum_k |\langle D_he_k,v\rangle|^2 ++ +\sum_{i,k} |\langle D_{g_i}e_k, v\rangle|^2 +\\ +&\uncover<4->{= +\sum_k |h(\lambda_k) \hat{v}(k)|^2 ++ +\sum_{k,i} |g_i(\lambda_k) \hat{v}(k)|^2} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{$A$ und $B$} +Frame-Norm-Funktion +\begin{align*} +f(\lambda) +&= +h(\lambda) ++ +\sum_i g_i(\lambda) +\\ +&\uncover<6->{= +h(\lambda) ++ +\sum_i g(a_i\lambda)} +\end{align*} +\uncover<7->{Abschätzung für Frame-Konstanten +\begin{align*} +A&\uncover<8->{= +\min_{i} f(\lambda_i)} +\\ +B&\uncover<9->{= +\max_{i} f(\lambda_i)} +\end{align*}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex new file mode 100644 index 0000000..c78e6dd --- /dev/null +++ b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex @@ -0,0 +1,78 @@ +% +% frequenzlokalisierung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\def\kurve#1#2{ + \draw[color=#2,line width=1.4pt] + plot[domain=0:6.3,samples=400] + ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1}); +} +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lokalisierung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Bandpass} +Gegeben durch $g(\lambda)\ge 0$: +\begin{align*} +g(0) &= 0\\ +\lim_{\lambda\to\infty}g(\lambda)&= 0 +\end{align*} +\vspace{-10pt} +\begin{enumerate} +\item<3-> Fourier-transformieren +\item<4-> Amplituden mit $g(\lambda)$ multiplizieren +\item<5-> Rücktransformieren +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Tiefpass} +Gegeben durch $h(\lambda)\ge0$: +\begin{align*} +h(0) &= 1\\ +\lim_{\lambda\to\infty}h(\lambda)&= 0 +\end{align*} +\vspace{-10pt} +\begin{enumerate} +\item<8-> Fourier-Transformation +\item<9-> Amplituden mit $h(\lambda)$ multiplizieren +\item<10-> Rücktransformation +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.8] + +\uncover<2->{ +\begin{scope}[xshift=-4.5cm] +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; +\kurve{3}{red} +\draw[->] (0,-0.1) -- (0,3.3); +\end{scope} +} + +\uncover<7->{ +\begin{scope}[xshift=4.5cm] +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; +\draw[color=darkgreen,line width=1.4pt] + plot[domain=0:6.3,samples=100] + ({\x},{3*exp(-(\x/0.5)*(\x/0.5)}); + +\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}]; +\end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/funktionen.tex b/vorlesungen/slides/8/wavelets/funktionen.tex new file mode 100644 index 0000000..2e3ae9b --- /dev/null +++ b/vorlesungen/slides/8/wavelets/funktionen.tex @@ -0,0 +1,78 @@ +% +% funktionen.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\knoten#1#2{ + \draw #1 circle[radius=0.25]; + \node at #1 {$#2$}; +} +\def\kante#1#2{ + \draw[shorten >= 0.25cm,shorten <= 0.25cm] #1 -- #2; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Funktionen auf einem Graphen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Ein Graph $G=(V,E)$, eine Funktion auf dem Graphen ist +\[ +f\colon V \to \mathbb{R} : v\mapsto f(v) +\] +Knoten: $V=\{1,\dots,n\}$ +\\ +\uncover<2->{% +Vektorschreibweise +\[ +f = \begin{pmatrix} +f(1)\\f(2)\\\vdots\\f(n) +\end{pmatrix} +\]} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Matrizen} +Adjazenz-, Grad- und Laplace-Matrix operieren auf Funktionen auf Graphen: +\[ +L += +\begin{pmatrix*}[r] + 2&-1& 0&-1& 0\\ +-1& 4&-1&-1&-1\\ + 0&-1& 2& 0&-1\\ +-1&-1& 0& 3&-1\\ + 0&-1&-1&-1& 3\\ +\end{pmatrix*} +\] +\end{block} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{2} +\coordinate (A) at (0,0); +\coordinate (B) at (\a,0); +\coordinate (C) at ({2*\a},0); +\coordinate (D) at ({0.5*\a},{-0.5*sqrt(3)*\a}); +\coordinate (E) at ({1.5*\a},{-0.5*sqrt(3)*\a}); +\knoten{(A)}{1} +\knoten{(B)}{2} +\knoten{(C)}{3} +\knoten{(D)}{4} +\knoten{(E)}{5} +\kante{(A)}{(B)} +\kante{(B)}{(C)} +\kante{(A)}{(D)} +\kante{(B)}{(D)} +\kante{(B)}{(E)} +\kante{(C)}{(E)} +\kante{(D)}{(E)} +\end{tikzpicture} +\end{center}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/gundh.tex b/vorlesungen/slides/8/wavelets/gundh.tex new file mode 100644 index 0000000..2d6c677 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/gundh.tex @@ -0,0 +1,85 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\kurve#1#2{ + \draw[color=#2,line width=1.4pt] + plot[domain=0:6.3,samples=400] + ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1}); +} + +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Wavelets} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Mutterwavelets + Dilatation} +Eine Menge von Dilatationsfaktoren +\[ +A= \{a_1,a_2,\dots,a_N\} +\] +wählen\uncover<2->{, und mit Funktionen +\[ +{\color{blue}g_i} = \tilde{D}_{1/a_i}{\color{red}g} +\] +die Standardbasisvektoren filtern} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{ +\begin{block}{Vaterwavelets} +Tiefpass mit Funktion ${\color{darkgreen}h(\lambda)}$, +Standardbasisvektoren mit ${\color{darkgreen}h}$ filtern: +\[ +D_{\color{darkgreen}h}e_k +\] +\end{block}} +\end{column} +\end{columns} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\begin{scope} + +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; + +\kurve{1}{red} +\uncover<4->{ +\foreach \k in {0,...,4}{ + \pgfmathparse{0.30*exp(ln(2)*\k)} + \xdef\l{\pgfmathresult} + \kurve{\l}{blue} +} +} + +\node[color=red] at ({0.7*1},3) [above] {$g(\lambda)$}; +\uncover<4->{ +\node[color=blue] at ({0.7*0.3*16},3) [above] {$g_i(\lambda)$}; +} + +\draw[->] (0,-0.1) -- (0,3.3); +\end{scope} + +\begin{scope}[xshift=7cm] + +\uncover<6->{ +\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}]; + +\draw[color=darkgreen,line width=1.4pt] + plot[domain=0:6.3,samples=100] + ({\x},{3*exp(-(\x/0.5)*(\x/0.5)}); + +\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}]; +} + +\end{scope} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/laplacebasis.tex b/vorlesungen/slides/8/wavelets/laplacebasis.tex new file mode 100644 index 0000000..ced4c09 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/laplacebasis.tex @@ -0,0 +1,62 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\a{2} +\def\b{0.8} +\def\c{1} +\def\d{0.6} +\input{../slides/8/wavelets/vektoren.tex} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Laplace-Basis} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\begin{scope}[yshift=-0.4cm,xshift=-5.5cm] +\fnull +\end{scope} + +\begin{scope}[yshift=-1.8cm,xshift=-5.5cm] +\fone +\end{scope} + +\begin{scope}[yshift=-3.2cm,xshift=-5.5cm] +\ftwo +\end{scope} + +\begin{scope}[yshift=-4.6cm,xshift=-5.5cm] +\fthree +\end{scope} + +\begin{scope}[yshift=-6.0cm,xshift=-5.5cm] +\ffour +\end{scope} + +\begin{scope}[yshift=0cm] +\vnull +\end{scope} + +\begin{scope}[yshift=-1.4cm] +\vone +\end{scope} + +\begin{scope}[yshift=-2.8cm] +\vtwo +\end{scope} + +\begin{scope}[yshift=-4.2cm] +\vthree +\end{scope} + +\begin{scope}[yshift=-5.6cm] +\vfour +\end{scope} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex new file mode 100644 index 0000000..d6575d0 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex @@ -0,0 +1,46 @@ +% +% lokalisierungsvergleich.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lokalisierung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ortsraum} +Ortsraum$\mathstrut=V$ +\begin{itemize} +\item<3-> Standardbasis +\item<5-> lokalisiert in den Knoten +\item<7-> die meisten $\hat{f}(k)$ gross +\item<9-> vollständig delokalisiert im Frequenzraum +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Frequenzraum} +\uncover<2->{Frequenzraum $\mathstrut=\{\lambda_1,\lambda_2,\dots,\lambda_n\}$} +\begin{itemize} +\item<4-> Laplace-Basis +\item<6-> lokalisiert in den Eigenwerten +\item<8-> die meisten Komponenten gross +\item<10-> vollständig delokalisiert im Ortsraum +\end{itemize} +\end{block} +\end{column} +\end{columns} +\uncover<11->{% +\begin{block}{Plan} +Gesucht sind Funktionen auf dem Graphen derart, die +\begin{enumerate} +\item<12-> in der Nähe einzelner Knoten konzentriert/lokalisiert sind und +\item<13-> deren Fourier-Transformation in der Nähe einzelner Eigenwerte +konzentriert/lokalisiert ist +\end{enumerate} +\end{block}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/matrixdilatation.tex b/vorlesungen/slides/8/wavelets/matrixdilatation.tex new file mode 100644 index 0000000..3536736 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/matrixdilatation.tex @@ -0,0 +1,39 @@ +% +% matrixdilatation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Dilatation in Matrixform} +Dilatationsfaktor $a$, skaliertes Wavelet beim Knoten $k$ mit Spektrum +$\tilde{D}_{1/a}g$ +\begin{align*} +D_{g,a}e_k +&= +\chi +\begin{pmatrix} +g(a\lambda_1)& 0 & \dots & 0 \\ + 0 &g(a\lambda_2)& \dots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \dots &g(a\lambda_n) +\end{pmatrix} +\chi^t +e_k +\intertext{\uncover<2->{``verschmierter'' Standardbasisvektor am Knoten $k$}} +\uncover<2->{D_he_k +&= +\chi +\begin{pmatrix} +h(\lambda_1)& 0 & \dots & 0 \\ + 0 &h(\lambda_2)& \dots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \dots &h(\lambda_n) +\end{pmatrix} +\chi^t +e_k} +\end{align*} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wavelets/vektoren.tex b/vorlesungen/slides/8/wavelets/vektoren.tex new file mode 100644 index 0000000..2315d53 --- /dev/null +++ b/vorlesungen/slides/8/wavelets/vektoren.tex @@ -0,0 +1,200 @@ +\def\vnull{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.0000$}; + \fill[color=red!100] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=red!100] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!100] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=red!100] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!100] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\fnull{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.4472*\d}) -- + ({-1.0000*\c},{0.4472*\d}) -- + ({0.0000*\c},{0.4472*\d}) -- + ({1.0000*\c},{0.4472*\d}) -- + ({2.0000*\c},{0.4472*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vone{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.1586$}; + \fill[color=blue!100] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=blue!00] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!100] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=blue!41] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!41] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\fone{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{-0.6533*\d}) -- + ({-1.0000*\c},{-0.2706*\d}) -- + ({0.0000*\c},{-0.0000*\d}) -- + ({1.0000*\c},{0.2706*\d}) -- + ({2.0000*\c},{0.6533*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vtwo{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.3000$}; + \fill[color=red!100] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=blue!00] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!100] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=blue!100] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=blue!100] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\ftwo{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.5000*\d}) -- + ({-1.0000*\c},{-0.5000*\d}) -- + ({0.0000*\c},{-0.0000*\d}) -- + ({1.0000*\c},{-0.5000*\d}) -- + ({2.0000*\c},{0.5000*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vthree{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.4414$}; + \fill[color=red!41] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=red!00] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=blue!41] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=blue!100] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!100] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\fthree{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.2706*\d}) -- + ({-1.0000*\c},{-0.6533*\d}) -- + ({0.0000*\c},{0.0000*\d}) -- + ({1.0000*\c},{0.6533*\d}) -- + ({2.0000*\c},{-0.2706*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} +\def\vfour{ + \coordinate (A) at ({0*\a},0); + \coordinate (B) at ({1*\a},0); + \coordinate (C) at ({2*\a},0); + \coordinate (D) at ({0.5*\a},{-\b}); + \coordinate (E) at ({1.5*\a},{-\b}); + \draw (A) -- (B); + \draw (A) -- (D); + \draw (B) -- (C); + \draw (B) -- (D); + \draw (B) -- (E); + \draw (C) -- (E); + \draw (D) -- (E); + \node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.5000$}; + \fill[color=red!25] (A) circle[radius=0.25]; + \draw (A) circle[radius=0.25]; + \fill[color=blue!100] (B) circle[radius=0.25]; + \draw (B) circle[radius=0.25]; + \fill[color=red!25] (C) circle[radius=0.25]; + \draw (C) circle[radius=0.25]; + \fill[color=red!25] (D) circle[radius=0.25]; + \draw (D) circle[radius=0.25]; + \fill[color=red!25] (E) circle[radius=0.25]; + \draw (E) circle[radius=0.25]; +} +\def\ffour{ + \draw[color=red,line width=1.4pt] + ({-2.0000*\c},{0.2236*\d}) -- + ({-1.0000*\c},{0.2236*\d}) -- + ({0.0000*\c},{-0.8944*\d}) -- + ({1.0000*\c},{0.2236*\d}) -- + ({2.0000*\c},{0.2236*\d}); + \draw[->] ({-2.1*\c},0) -- ({2.1*\c},0); + \draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d}); + \fill ({-2*\c},0) circle[radius=0.05]; + \fill ({-1*\c},0) circle[radius=0.05]; + \fill ({0*\c},0) circle[radius=0.05]; + \fill ({1*\c},0) circle[radius=0.05]; + \fill ({2*\c},0) circle[radius=0.05]; +} diff --git a/vorlesungen/slides/8/weitere.tex b/vorlesungen/slides/8/weitere.tex new file mode 100644 index 0000000..46a3da0 --- /dev/null +++ b/vorlesungen/slides/8/weitere.tex @@ -0,0 +1,43 @@ +% +% weitere.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Weitere Resultate der spektralen Graphentheorie} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz (Hoffmann)} +\[ +\operatorname{chr} X \ge 1 + \frac{\alpha_{\text{max}}}{-\alpha_{\text{min}}} +\] +\end{block} +\uncover<2->{% +\begin{block}{Satz (Hoffmann)} +\[ +\operatorname{ind} X \le n \biggl(1-\frac{d_{\text{min}}}{\lambda_{\text{max}}}\biggr) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Korollar} +Für einen regulären Graphen mit $n$ Knoten gilt +\begin{align*} +\operatorname{ind} X +&\le +\frac{n}{\displaystyle 1-\frac{d}{\alpha_{\text{min}}}} +\\ +\operatorname{chr} X +&\ge +1-\frac{d}{\alpha_{\text{min}}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/wilf.m b/vorlesungen/slides/8/wilf.m new file mode 100644 index 0000000..49dc161 --- /dev/null +++ b/vorlesungen/slides/8/wilf.m @@ -0,0 +1,22 @@ +# +# wilf.m -- chromatische Zahl für einen Graphen +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +N = 9; +A = zeros(N,N); + +for i = (1:N) + j = 1 + rem(i, N) + A(i,j) = 1; +endfor +for i = (1:3:N-3) + j = 1 + rem(i + 2, N) + A(i,j) = 1; +endfor + +A(1,3) = 1; + +A = A + A' + +eig(A) diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc index fa6c29b..2257810 100644 --- a/vorlesungen/slides/9/Makefile.inc +++ b/vorlesungen/slides/9/Makefile.inc @@ -10,5 +10,20 @@ chapter9 = \ ../slides/9/irreduzibel.tex \ ../slides/9/stationaer.tex \ ../slides/9/pf.tex \ + ../slides/9/potenz.tex \ + ../slides/9/pf/positiv.tex \ + ../slides/9/pf/primitiv.tex \ + ../slides/9/pf/trennung.tex \ + ../slides/9/pf/vergleich.tex \ + ../slides/9/pf/vergleich3d.tex \ + ../slides/9/pf/dreieck.tex \ + ../slides/9/pf/folgerungen.tex \ + ../slides/9/parrondo/uebersicht.tex \ + ../slides/9/parrondo/erwartung.tex \ + ../slides/9/parrondo/spiela.tex \ + ../slides/9/parrondo/spielb.tex \ + ../slides/9/parrondo/spielbmod.tex \ + ../slides/9/parrondo/kombiniert.tex \ + ../slides/9/parrondo/deformation.tex \ ../slides/9/chapter.tex diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex index 9e26587..cbab0f0 100644 --- a/vorlesungen/slides/9/chapter.tex +++ b/vorlesungen/slides/9/chapter.tex @@ -10,5 +10,21 @@ \folie{9/stationaer.tex} \folie{9/irreduzibel.tex} \folie{9/pf.tex} +\folie{9/potenz.tex} +\folie{9/pf/positiv.tex} +\folie{9/pf/primitiv.tex} +\folie{9/pf/trennung.tex} +\folie{9/pf/vergleich.tex} +\folie{9/pf/vergleich3d.tex} +\folie{9/pf/dreieck.tex} +\folie{9/pf/folgerungen.tex} + +\folie{9/parrondo/uebersicht.tex} +\folie{9/parrondo/erwartung.tex} +\folie{9/parrondo/spiela.tex} +\folie{9/parrondo/spielb.tex} +\folie{9/parrondo/spielbmod.tex} +\folie{9/parrondo/kombiniert.tex} +\folie{9/parrondo/deformation.tex} diff --git a/vorlesungen/slides/9/parrondo/deformation.tex b/vorlesungen/slides/9/parrondo/deformation.tex new file mode 100644 index 0000000..40d2eb9 --- /dev/null +++ b/vorlesungen/slides/9/parrondo/deformation.tex @@ -0,0 +1,45 @@ +% +% deformation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Deformation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Verlustspiele} +Durch Deformation (Parameter $e$ und $\varepsilon$) kann man +aus $A_e$ und $B_\varepsilon$ Spiele mit negativer Gewinnerwartung machen +\uncover<2->{% +\begin{align*} +E(X)&=0&&\rightarrow&E(X_e)&<0\\ +E(Y)&=0&&\rightarrow&E(Y_\varepsilon)&<0\\ +\end{align*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Kombiniertes Spiel} +\uncover<3->{% +Die Deformation für das Spiel $C$ startet mit Erwartungswert $\frac{18}{709}$}% +\begin{align*} +\uncover<4->{E(Z)&=\frac{18}{709}>0} +&&\uncover<5->{\rightarrow& +E(Z_*)&>0} +\end{align*} +\uncover<6->{Wegen Stetigkeit!} +\\ +\uncover<5->{Die Deformation ist immer noch ein Gewinnspiel (für Parameter klein genug)} +\end{block} +\uncover<7->{% +\begin{block}{Parrondo-Paradoxon} +Zufällig zwischen zwei Verlustspielen auswählen kann trotzdem ein +Gewinnspiel ergeben +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/erwartung.tex b/vorlesungen/slides/9/parrondo/erwartung.tex new file mode 100644 index 0000000..b58c37f --- /dev/null +++ b/vorlesungen/slides/9/parrondo/erwartung.tex @@ -0,0 +1,81 @@ +% +% erwartung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Erwartung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zufallsvariable} +\begin{center} +\[ +\begin{array}{c|c} +\text{Werte $X$}&\text{Wahrscheinlichkeit $p$}\\ +\hline +x_1&p_1=P(X=x_1)\\ +x_2&p_2=P(X=x_2)\\ +\vdots&\vdots\\ +x_n&p_n=P(X=x_n) +\end{array} +\] +\end{center} +\end{block} +\uncover<4->{% +\begin{block}{Einervektoren/-matrizen} +\[ +U=\begin{pmatrix} +1&1&\dots&1\\ +1&1&\dots&1\\ +\vdots&\vdots&\ddots&\vdots\\ +1&1&\dots&1 +\end{pmatrix} +\in +M_{n\times m}(\Bbbk) +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Erwartungswerte} +\begin{align*} +E(X) +&= +\sum_i x_ip_i += +x^tp +\uncover<5->{= +U^t x\odot p} +\hspace*{3cm} +\\ +\uncover<2->{E(X^2) +&= +\sum_i x_i^2p_i} +\ifthenelse{\boolean{presentation}}{ +\only<6>{= +(x\odot x)^tp}}{} +\uncover<7->{= +U^t (x\odot x) \odot p} +\\ +\uncover<3->{E(X^k) +&= +\sum_i x_i^kp_i} +\uncover<8->{= +U^t x^{\odot k}\odot p} +\end{align*} +\uncover<9->{% +Substitution: +\begin{align*} +\uncover<10->{\sum_i &\to U^t}\\ +\uncover<11->{x_i^k &\to x^{\odot k}} +\end{align*}}% +\uncover<12->{Kann für Übergangsmatrizen von Markov-Ketten verallgemeinert werden} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/kombiniert.tex b/vorlesungen/slides/9/parrondo/kombiniert.tex new file mode 100644 index 0000000..5012d06 --- /dev/null +++ b/vorlesungen/slides/9/parrondo/kombiniert.tex @@ -0,0 +1,73 @@ +% +% kombiniert.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kombiniertes Spiel $C$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Ein fairer Münzwurf entscheidet, ob +Spiel $A$ oder Spiel $B$ gespielt wird +\end{block} +\uncover<2->{% +\begin{block}{Übergangsmatrix} +Münzwurf $X$ +\begin{align*} +C +&= +P(X=\text{Kopf})\cdot A ++ +P(X=\text{Zahl})\cdot B +\\ +&\uncover<3->{= +\begin{pmatrix} + 0&\frac{3}{8}&\frac{5}{8}\\ +\frac{3}{10}& 0&\frac{3}{8}\\ +\frac{7}{10}&\frac{5}{8}& 0 +\end{pmatrix}} +\end{align*} +\end{block}} +\vspace{-8pt} +\uncover<4->{% +\begin{block}{Gewinnerwartung im Einzelspiel} +\[ +p=\frac13U +\Rightarrow +U^t(G\odot C)p +\uncover<5->{= +-\frac{1}{30}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Iteriertes Spiel} +\[ +\overline{p}=C\overline{p} +\quad +\uncover<7->{\Rightarrow +\quad +\overline{p}=\frac{1}{709}\begin{pmatrix}245\\180\\284\end{pmatrix}} +\] +\end{block}} +\uncover<8->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +E(Z) +&= +U^t (G\odot C) \overline{p} +\uncover<9->{= +\frac{18}{709}} +\end{align*} +\uncover<10->{$C$ ist ein Gewinnspiel!} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/spiela.tex b/vorlesungen/slides/9/parrondo/spiela.tex new file mode 100644 index 0000000..629586f --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spiela.tex @@ -0,0 +1,52 @@ +% +% spiela.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spiel $A$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gewinn = Zufallsvariable $X$ mit Werten $\pm 1$ +\begin{align*} +P(X=\phantom{+}1) +&= +\frac12\uncover<2->{+e} +\\ +P(X= - 1) +&= +\frac12\uncover<2->{-e} +\end{align*} +Bernoulli-Experiment mit $p=\frac12\uncover<2->{+e}$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{ +\begin{block}{Gewinnerwartung} +\begin{align*} +E(X) +&=\uncover<4->{ +P(X=1)\cdot (1)} +\\ +&\qquad +\uncover<4->{+ +P(X=-1)\cdot (-1)} +\\ +&\uncover<5->{= +\biggl(\frac12+e\biggr)\cdot 1 ++ +\biggl(\frac12-e\biggr)\cdot (-1)} +\\ +&\uncover<6->{=2e} +\end{align*} +\uncover<7->{$\Rightarrow$ {\usebeamercolor[fg]{title}Verlustspiel für $e<0$}} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/spielb.tex b/vorlesungen/slides/9/parrondo/spielb.tex new file mode 100644 index 0000000..f65564f --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spielb.tex @@ -0,0 +1,100 @@ +% +% spielb.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Spiel $B$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des +aktuellen Kapitals $K$: +\begin{center} +\uncover<2->{% +\begin{tikzpicture}[>=latex,thick] +\coordinate (A0) at (90:2); +\coordinate (A1) at (210:2); +\coordinate (A2) at (330:2); + +\node at (A0) {$0$}; +\node at (A1) {$1$}; +\node at (A2) {$2$}; + +\draw (A0) circle[radius=0.4]; +\draw (A1) circle[radius=0.4]; +\draw (A2) circle[radius=0.4]; + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2); + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1); + +\def\R{1.9} +\def\r{0.7} + +\node at (30:\r) {$\frac{9}{10}$}; +\node at (150:\r) {$\frac1{10}$}; +\node at (270:\r) {$\frac34$}; + +\node at (30:\R) {$\frac{3}{4}$}; +\node at (150:\R) {$\frac1{4}$}; +\node at (270:\R) {$\frac14$}; + +\end{tikzpicture}} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Markov-Kette $Y$} +Übergangsmatrix +\[ +B=\begin{pmatrix} +0&\frac14&\frac34\\ +\frac{1}{10}&0&\frac14\\ +\frac{9}{10}&\frac34&0 +\end{pmatrix} +\] +\vspace{-10pt} + +\uncover<4->{% +Gewinnmatrix: +\vspace{-2pt} +\[ +G=\begin{pmatrix*}[r] +0&-1&1\\ +1&0&-1\\ +-1&1&0 +\end{pmatrix*} +\]} +\end{block}} +\vspace{-12pt} +\uncover<5->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +&&&& +E(Y) +&= +U^t(G\odot B)p +\\ +p&={\textstyle\frac13}U +&&\Rightarrow& +E(Y)&={\textstyle\frac1{15}} +\\ +\overline{p}&={\tiny\frac{1}{13}\begin{pmatrix}5\\2\\6\end{pmatrix}} +&&\Rightarrow& +E(Y)&=0 +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex new file mode 100644 index 0000000..66d39bc --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spielbmod.tex @@ -0,0 +1,103 @@ +% +% spielb.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Modifiziertes Spiel $\tilde{B}$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des +aktuellen Kapitals $K$: +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A0) at (90:2); +\coordinate (A1) at (210:2); +\coordinate (A2) at (330:2); + +\node at (A0) {$0$}; +\node at (A1) {$1$}; +\node at (A2) {$2$}; + +\draw (A0) circle[radius=0.4]; +\draw (A1) circle[radius=0.4]; +\draw (A2) circle[radius=0.4]; + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2); + +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0); +\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1); + +\def\R{1.9} +\def\r{0.7} + +\node at (30:{0.9*\r}) {\tiny $\frac{9}{10}\uncover<2->{+\varepsilon}$}; +\node at (150:{0.9*\r}) {\tiny $\frac1{10}\uncover<2->{-\varepsilon}$}; +\node at (270:\r) {$\frac34\uncover<2->{-\varepsilon}$}; + +\node at (30:{1.1*\R}) {$\frac{3}{4}\uncover<2->{-\varepsilon}$}; +\node at (150:{1.1*\R}) {$\frac1{4}\uncover<2->{+\varepsilon}$}; +\node at (270:\R) {$\frac14\uncover<2->{+\varepsilon}$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Markov-Kette $\tilde{Y}$} +Übergangsmatrix +\[ +\tilde{B}= +B\uncover<2->{+\varepsilon F} +\uncover<3->{= +B+\varepsilon\begin{pmatrix*}[r] +0&1&-1\\ +-1&0&1\\ +1&-1&0 +\end{pmatrix*}} +\] +\vspace{-12pt} + +\uncover<4->{% +Gewinnmatrix: +\[ +G=\begin{pmatrix*}[r] +0&-1&1\\ +1&0&-1\\ +-1&1&0 +\end{pmatrix*} +\]} +\end{block} +\vspace{-12pt} +\uncover<5->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +\uncover<6->{E(\tilde{Y}) +&= +U^t(G\odot \tilde{B})p} +\\ +&\uncover<7->{= +E(Y) + \varepsilon U^t(G\odot F)p} +\uncover<8->{= +{\textstyle\frac1{15}}+2\varepsilon} +\\ +\uncover<9->{ +\text{rep.} +&= +-{\textstyle\frac{294}{169}}\varepsilon+O(\varepsilon^2) +\quad\text{Verlustspiel} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/parrondo/uebersicht.tex b/vorlesungen/slides/9/parrondo/uebersicht.tex new file mode 100644 index 0000000..2f3597a --- /dev/null +++ b/vorlesungen/slides/9/parrondo/uebersicht.tex @@ -0,0 +1,17 @@ +% +% uebersicht.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Parrondo-Paradoxon} +\begin{center} +\Large +Zufällige +Wahl zwischen zwei Verlustspielen = Gewinnspiel? +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/dreieck.tex b/vorlesungen/slides/9/pf/dreieck.tex new file mode 100644 index 0000000..0a572f3 --- /dev/null +++ b/vorlesungen/slides/9/pf/dreieck.tex @@ -0,0 +1,44 @@ +% +% dreieck.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Verallgemeinerte Dreiecksungleichung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.32\textwidth} +\begin{block}{Satz} +\[ +|u+v|\le |u|+|v| +\] +Gleichheit wenn lin.~abh. +\end{block} +\begin{block}{Satz} +\[ +\biggl|\sum_i u_i\biggr| +\le +\sum_i |u_i| +\] +Gleichheit wenn $u_i = \lambda_i u$ +\end{block} +\begin{block}{Satz} +\[ +\biggl|\sum_i z_i\biggr| +\le +\sum_i |z_i| +\] +Gleichheit, wenn $z_i=|z_i|c$, $c\in\mathbb{C}$ +\end{block} +\end{column} +\begin{column}{0.68\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/dreieck.pdf} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/folgerungen.tex b/vorlesungen/slides/9/pf/folgerungen.tex new file mode 100644 index 0000000..5042c78 --- /dev/null +++ b/vorlesungen/slides/9/pf/folgerungen.tex @@ -0,0 +1,203 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Folgerungen für $A>0$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$u\ge 0$ ein EV zum EW $ \lambda\ne 0$, +dann ist $u>0$ und $\lambda >0$ +\end{block} +\uncover<6->{% +\begin{block}{Satz} +$v$ ein EV zum EW $\lambda$ mit $|\lambda| = \varrho(A)$, +dann ist $u=|v|$ mit $u_i=|v_i|$ ein EV mit EW $\varrho(A)$ +\end{block}} +\uncover<29->{% +\begin{block}{Satz} +$v$ ein EV zum EW $\lambda$ mit $|\lambda|=\varrho(A)$, +dann ist $\lambda=\varrho(A)$ +\end{block}} +\uncover<46->{% +\begin{block}{Satz} +Der \only<57->{verallgemeinerte }Eigenraum zu EW $\varrho(A)$ +ist eindimensional +\end{block} +} +\end{column} +\ifthenelse{\boolean{presentation}}{ +\only<-6>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{itemize} +\item<3-> +Vergleich: $Au>0$ +\item<4-> +$Au=\lambda u > 0$ +\item<5-> +$\lambda >0$ und $u>0$ +\end{itemize} +\end{proof} +\end{column}} +\only<7-20>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{align*} +(Au)_i +&\only<-8>{= +\sum_j a_{ij}u_j} +\only<8-9>{= +\sum_j |a_{ij}v_j|} +\only<9->{\ge} +\only<9-10>{ +\biggl|\sum_j a_{ij}v_j\biggr|} +\only<10>{=} +\only<10-11>{ +|(Av)_i|} +\only<11>{=} +\only<11-12>{ +|\lambda v_i|} +\only<12>{=} +\only<12-13>{ +\varrho(A) |v_i|} +\only<13>{=} +\uncover<13->{ +\varrho(A) u_i} +\hspace*{5cm} +\\ +\uncover<14->{Au&\ge \varrho(A)u} +\intertext{\uncover<15->{Vergleich}} +\uncover<16->{A^2u&> \varrho(A)Au} +\intertext{\uncover<17->{Trennung: $\exists \vartheta >1$ mit}} +\uncover<18->{A^2u&\ge \vartheta \varrho(A) Au }\\ +\uncover<19->{A^3u&\ge (\vartheta \varrho(A))^2 Au }\\ +\uncover<20->{A^ku&\ge (\vartheta \varrho(A))^{k-1} Au }\\ +\end{align*} +\end{proof} +\end{column}} +\only<21-29>{% +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis, Fortsetzung] +Abschätzung der Operatornorm: +\begin{align*} +\|A^k\|\, |Au| +\ge +\|A^{k+1}u\| +\uncover<22->{ +\ge +(\vartheta\varrho(A))^k |Au|} +\end{align*} +\uncover<23->{Abschätzung des Spektralradius} +\begin{align*} +\uncover<24->{\|A^k\| &\ge (\vartheta\varrho(A))^k} +\\ +\uncover<25->{\|A^k\|^{\frac1k} &\ge \vartheta \varrho(A)} +\\ +\uncover<26->{\lim_{k\to\infty}\|A^k\|^{\frac1k} &\ge \vartheta \varrho(A)} +\\ +\uncover<27->{\varrho(A) &\ge \underbrace{\vartheta}_{>1} \varrho(A)} +\end{align*} +\uncover<28->{Widerspruch: $u=v$} +\end{proof} +\end{column}} +\only<30-46>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +$u$ ist EV mit EW $\varrho(A)$: +\[ +Au=\varrho(A)u +\uncover<31->{\Rightarrow +\sum_j a_{ij}|v_j| = {\color<38->{red}\varrho(A) |v_i|}} +\] +\uncover<33->{Andererseits: $Av=\lambda v$} +\[ +\uncover<34->{\sum_{j}a_{ij}v_j=\lambda v_i} +\] +\uncover<35->{Betrag} +\begin{align*} +\uncover<36->{\biggl|\sum_j a_{ij}v_j\biggr| +&= +|\lambda v_i|} +\uncover<37->{= +{\color<38->{red}\varrho(A) |v_i|}} +\uncover<39->{= +\sum_j a_{ij}|v_j|} +\end{align*} +\uncover<40->{Dreiecksungleichung: $v_j=|v_j|c, c\in\mathbb{C}$} +\[ +\uncover<41->{\lambda v = Av} +\uncover<42->{= Acu} +\uncover<43->{= c\varrho(A) u} +\uncover<44->{= \varrho(A)v} +\] +\uncover<45->{$\Rightarrow +\lambda=\varrho(A) +$} +\end{proof} +\end{column}} +\only<47-57>{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{itemize} +\item<48-> $u>0$ ein EV zum EW $\varrho(A)$ +\item<49-> $v$ ein weiterer EV, man darf $v\in\mathbb{R}^n$ annehmen +\item<50-> Da $u>0$ gibt es $c>0$ mit $u\ge cv$ aber $u\not > cv$ +\item<51-> $u-cv\ge 0$ aber $u-cv\not > 0$ +\item<52-> $A$ anwenden: +\[ +\begin{array}{ccc} +\uncover<53->{A(u-cv)}&\uncover<54->{>&0} +\\ +\uncover<53->{\|}&& +\\ +\uncover<53->{\varrho(A)(u-cv)}&\uncover<55->{\not>&0} +\end{array} +\] +\uncover<56->{Widerspruch: $v$ existiert nicht} +\end{itemize} +\end{proof} +\end{column}} +\only<58->{ +\begin{column}{0.48\textwidth} +\begin{proof}[Beweis] +\begin{itemize} +\item<59-> $Au=\varrho(A)u$ und $A^tp^t=\varrho(A)p^t$ +\item<60-> $u>0$ und $p>0$ $\Rightarrow$ $up>0$ +\item<61-> $px=0$, dann ist +\[ +\uncover<62->{pAx} +\only<62-63>{= +(A^tp^t)^t x} +\only<63-64>{= +\varrho(A) (p^t)^t x} +\uncover<64->{= +\varrho(A) px} +\uncover<65->{= 0} +\] +\uncover<66->{also ist $\{x\in\mathbb{R}^n\;|\; px=0\}$ +invariant} +\item<67-> Annahme: $v\in \mathcal{E}_{\varrho(A)}$ +\item<68-> Dann muss es einen EV zum EW $\varrho(A)$ in +$\mathcal{E}_{\varrho(A)}$ geben +\item<69-> Widerspruch: der Eigenraum ist eindimensional +\end{itemize} +\end{proof} +\end{column}} +}{ +\begin{column}{0.48\textwidth} +\begin{block}{} +\usebeamercolor[fg]{title} +Beweise: Buch Abschnitt 9.3 +\end{block} +\end{column} +} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/positiv.tex b/vorlesungen/slides/9/pf/positiv.tex new file mode 100644 index 0000000..d7e833d --- /dev/null +++ b/vorlesungen/slides/9/pf/positiv.tex @@ -0,0 +1,64 @@ +% +% positiv.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Positive und nichtnegative Matrizen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Positive Matrix\strut} +Eine Matrix $A$ heisst positiv, wenn +\[ +a_{ij} > 0\quad\forall i,j +\] +Man schreibt $A>0\mathstrut$ +\end{block} +\uncover<2->{% +\begin{block}{Relation $>\mathstrut$} +Man schreibt $A>B$ wenn $A-B > 0\mathstrut$ +\end{block}} +\uncover<5->{% +\begin{block}{Wahrscheinlichkeitsmatrix} +\[ +W=\begin{pmatrix} +0.7&0.2&0.1\\ +0.2&0.6&0.1\\ +0.1&0.2&0.8 +\end{pmatrix} +\] +Spaltensumme$\mathstrut=1$, Zeilensumme$\mathstrut=?$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Nichtnegative Matrix\strut} +Eine Matrix $A$ heisst nichtnegativ, wenn +\[ +a_{ij} \ge 0\quad\forall i,j +\] +Man schreibt $A\ge 0\mathstrut$ +\end{block}} +\uncover<4->{% +\begin{block}{Relation $\ge\mathstrut$} +Man schreibt $A\ge B$ wenn $A-B \ge 0\mathstrut$ +\end{block}} +\uncover<6->{% +\begin{block}{Permutationsmatrix} +\[ +P=\begin{pmatrix} +0&0&1\\ +1&0&0\\ +0&1&0 +\end{pmatrix} +\] +Genau eine $1$ in jeder Zeile/Spalte +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/primitiv.tex b/vorlesungen/slides/9/pf/primitiv.tex new file mode 100644 index 0000000..961b1d5 --- /dev/null +++ b/vorlesungen/slides/9/pf/primitiv.tex @@ -0,0 +1,84 @@ +% +% primitiv.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Primitive Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +$A\ge 0$ heisst primitiv, wenn es ein $n>0$ gibt mit $A^n>0$ +\end{block} +\uncover<9->{% +\begin{block}{Intuition} +\begin{itemize} +\item<10-> +Markov-Ketten: $a_{ij} > 0$ bedeutet, $i$ von $j$ aus erreichbar. +\item<11-> +Band: {\em alle} Verbindung mit allen Nachbarn +\item<12-> +$n$-te Potenz: Pfade der Länge $n$ +\item<13-> +Durchmesser: wenn $n>\text{Durchmesser des Zustandsdiagramms}$, +dann ist $A^n>0$ +\end{itemize} +\end{block} +} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiel: Reduzible W'keitsmatrix} +\vspace{-5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\fill[color=gray!40] (-1,0) rectangle (0,1); +\fill[color=gray!40] (0,-1) rectangle (1,0); +\draw[line width=0.3pt] (0,-1) -- (0,1); +\draw[line width=0.3pt] (-1,0) -- (1,0); +%\draw (-1,-1) rectangle (1,1); +\node at (0,0) {$\left( \raisebox{0pt}[1cm][1cm]{\hspace*{2cm}} \right)$}; +\node at (-1.3,0) [left] {$\mathstrut W=$}; +\node at (0.5,0.5) {$0$}; +\node at (-0.5,-0.5) {$0$}; +\end{tikzpicture} +\end{center} +\vspace{-10pt} + +$\Rightarrow$ $W$ ist nicht primitiv +\end{block}} +\uncover<3->{% +\begin{block}{Beispiel: Bandmatrix} +\centering +\begin{tikzpicture}[>=latex,thick] +\begin{scope} +\clip (-1,-1) rectangle (1,1); +\foreach \n in {3,...,8}{ + \pgfmathparse{0.3*(\n-2)} + \xdef\x{\pgfmathresult} + \only<\n>{ + \fill[color=gray!40] + ({-1.2-\x},1) -- (1,{-1.2-\x}) -- (1,{-0.8+\x}) + -- ({-0.8+\x},1) -- cycle; + } +} +\fill[color=gray] (-1.2,1) -- (1,-1.2) -- (1,-0.8) -- (-0.8,1) -- cycle; +\end{scope} +\foreach \n in {2,...,8}{ + \uncover<\n>{ + \pgfmathparse{int(\n-2)} + \xdef\k{\pgfmathresult} + \node at (-1.3,0) [left] {$\mathstrut B^{\k}=$}; + } +} +\node at (0,0) {$\left( \raisebox{0pt}[1cm][1cm]{\hspace*{2cm}} \right)$}; +\end{tikzpicture} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/trennung.tex b/vorlesungen/slides/9/pf/trennung.tex new file mode 100644 index 0000000..9c85849 --- /dev/null +++ b/vorlesungen/slides/9/pf/trennung.tex @@ -0,0 +1,99 @@ +% +% trennung.tex -- slide template +% +% (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{Trennung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\coordinate (u) at (3.5,4.5); +\coordinate (v) at (2.5,2); +\coordinate (va) at ({(3.5/2.5)*2.5},{(3.5/2.5)*2}); + +\uncover<3->{ +\fill[color=darkgreen!20] (0,0) rectangle (5.3,5.3); +\node[color=darkgreen] at (1.5,4.9) {$u\not\ge w$}; +\node[color=darkgreen] at (4.4,0.6) {$u\not\ge w$}; +} + +\uncover<5->{ +\begin{scope} +\clip (0,0) rectangle (5.3,5.3); +\draw[color=darkgreen] (0,0) -- ($3*(v)$); +\end{scope} + +\node[color=darkgreen] at ($1.2*(va)$) + [below,rotate={atan(2/2.5)}] {$(1+\mu)v$}; +} + +\uncover<2->{ + \fill[color=red!20] (0,0) rectangle (u); +} + +\fill[color=red] (u) circle[radius=0.08]; +\node[color=red] at (u) [above right] {$u$}; + +\uncover<4->{ + \fill[color=blue!40,opacity=0.5] (0,0) rectangle (v); +} + +\uncover<2->{ + \fill[color=blue] (v) circle[radius=0.08]; + \node[color=blue] at (v) [above] {$v$}; +} + +\uncover<4->{ + \draw[color=blue] (0,0) -- (va); + + \fill[color=blue] (va) circle[radius=0.08]; + \node[color=blue] at (va) [above left] {$(1+\varepsilon)v$}; +} + +\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}]; +\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}]; + +\uncover<2->{ + \draw[->,color=red] (3.0,-0.2) -- (3.0,1.5); + \node[color=red] at (3.0,-0.2) [below] + {$\{w\in\mathbb{R}^n\;|\; w<u\}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$u>v\ge 0$\uncover<4->{, dann gibt es $\varepsilon>0$ mit +\[ +u\ge (1+\varepsilon)v +\]}% +\uncover<5->{und für $\mu>\varepsilon$ ist +\[ +u \not\ge (1+\mu)v +\]} +\uncover<6->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<7-> +$u>v$ $\Rightarrow$ $u_i/v_i>1$ falls $v_i>0$ +\item<8-> +\[ +\vartheta = \min_{v_i\ne 0} \frac{u_i}{v_i} > 1 +\] +\uncover<9->{$\varepsilon = \vartheta - 1$} +\end{itemize} +\end{proof}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/vergleich.tex b/vorlesungen/slides/9/pf/vergleich.tex new file mode 100644 index 0000000..c1a1f7a --- /dev/null +++ b/vorlesungen/slides/9/pf/vergleich.tex @@ -0,0 +1,113 @@ +% +% vergleich.tex -- slide template +% +% (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{Vergleich} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\a{1.2} \def\b{0.35} +\def\c{0.5} \def\d{1.25} +\def\r{4} + +\coordinate (u) at (3.5,0); +\coordinate (v) at (2.5,0); + +\coordinate (Au) at ({3.5*\a},{3.5*\c}); +\coordinate (Av) at ({2.5*\a},{2.5*\c}); + +\uncover<2->{ + \begin{scope} + \clip (0,0) rectangle (5,5); + \fill[color=red!20] (0,0) circle[radius=4]; + \end{scope} + \node[color=red] at (0,4) [below right] {$\mathbb{R}^n$}; + + \fill[color=blue!40,opacity=0.5] (0,0) -- ({\a*\r},{\c*\r}) + -- plot[domain=0:90,samples=100] + ({\r*(\a*cos(\x)+\b*sin(\x))},{\r*(\c*cos(\x)+\d*sin(\x))}) + -- ({\b*\r},{\d*\r}) -- cycle; + \node[color=blue] at ({\r*\b},{\r*\d}) [below right] {$A\mathbb{R}^n$}; +} + +\draw[->] (-0.1,0) -- (5.5,0) coordinate[label={$x_1$}]; +\draw[->] (0,-0.1) -- (0,5.5) coordinate[label={right:$x_2$}]; + +\uncover<3->{ + \fill[color=darkgreen!30,opacity=0.5] + (0,0) rectangle ({3.5*\a},{3.5*\c}); + \draw[color=white,line width=0.7pt] + ({3.5*\a},0) -- ({3.5*\a},{3.5*\c}) -- (0,{3.5*\c}); +} + +\uncover<2->{ + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\a},{\r*\c}); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({\r*\b},{\r*\d}); + + \draw[->,color=red,line width=1.4pt] (0,0) -- (4,0); + \draw[->,color=red,line width=1.4pt] (0,0) -- (0,4); +} + +\draw[color=darkgreen,line width=2pt] (u) -- (v); +\fill[color=darkgreen] (u) circle[radius=0.08]; +\fill[color=darkgreen] (v) circle[radius=0.08]; + +\node[color=darkgreen] at (u) [below right] {$u$}; +\node[color=darkgreen] at (v) [below left] {$v$}; +\node[color=darkgreen] at ($0.5*(u)+0.5*(v)$) [above] {$v\le u$}; + +\uncover<3->{ + \draw[color=darkgreen,line width=2pt] (Au) -- (Av); + \fill[color=darkgreen] (Au) circle[radius=0.08]; + \fill[color=darkgreen] (Av) circle[radius=0.08]; + + \node[color=darkgreen] at (Au) [above left] {$Au$}; + \node[color=darkgreen] at (Av) [above left] {$Av$}; + + \node[color=darkgreen] at ($0.5*(Au)+0.5*(Av)$) + [below,rotate={atan(\c/\a)}] {$Av<Au$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Satz} +$u\ge v\ge 0$ \uncover<2->{und $A > 0$}\uncover<3->{ $\Rightarrow$ $Au>Av$} +\end{block} +\uncover<4->{% +\begin{block}{intuitiv} +$A>0$ befördert $\ge$ zu $>$ +\end{block}} +\uncover<5->{% +\begin{proof}[Beweis] +$d=u-v\ge 0$ +\begin{align*} +(Ad)_i +\uncover<6->{= +\sum_{j} +\underbrace{a_{ij}}_{>0}d_j} +\uncover<7->{> +0} +\uncover<8->{\quad\Rightarrow\quad +Au > Av} +\end{align*} +\uncover<7->{da mindestens ein $d_j>0$ ist} +\end{proof}} +\uncover<9->{% +\begin{block}{Korollar} +$A>0$ und $d\ge 0$ $\Rightarrow$ $Ad > 0$ +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/pf/vergleich3d.tex b/vorlesungen/slides/9/pf/vergleich3d.tex new file mode 100644 index 0000000..1c019a6 --- /dev/null +++ b/vorlesungen/slides/9/pf/vergleich3d.tex @@ -0,0 +1,26 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Vergleich} + +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.57\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/vergleich.pdf} +\end{center} +\end{column} +\begin{column}{0.38\textwidth} +\begin{block}{Satz} +$u\ge v\ge 0$ $\Rightarrow$ $Au>Av$ +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/9/potenz.tex b/vorlesungen/slides/9/potenz.tex new file mode 100644 index 0000000..2c3afa3 --- /dev/null +++ b/vorlesungen/slides/9/potenz.tex @@ -0,0 +1,15 @@ +% +% potenz.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Potenzmethode} +\begin{center} +\includegraphics[width=0.9\textwidth]{../../buch/chapters/80-wahrscheinlichkeit/images/positiv.pdf} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 130fa28..a9d72be 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -1,21 +1,21 @@ -#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-#
-include ../slides/0/Makefile.inc
-include ../slides/1/Makefile.inc
-include ../slides/2/Makefile.inc
-include ../slides/3/Makefile.inc
-include ../slides/4/Makefile.inc
-include ../slides/5/Makefile.inc
-include ../slides/6/Makefile.inc
-include ../slides/7/Makefile.inc
-include ../slides/8/Makefile.inc
-include ../slides/9/Makefile.inc
-include ../slides/a/Makefile.inc
-
-slides = \
- $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \
- $(chapter5) $(chapter6) $(chapter7) $(chapter8) $(chapter9) \
- $(chaptera)
+# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +# +include ../slides/0/Makefile.inc +include ../slides/1/Makefile.inc +include ../slides/2/Makefile.inc +include ../slides/3/Makefile.inc +include ../slides/4/Makefile.inc +include ../slides/5/Makefile.inc +include ../slides/6/Makefile.inc +include ../slides/7/Makefile.inc +include ../slides/8/Makefile.inc +include ../slides/9/Makefile.inc +include ../slides/a/Makefile.inc + +slides = \ + $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ + $(chapter5) $(chapter6) $(chapter7) $(chapter8) $(chapter9) \ + $(chaptera) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index ce63ae7..4289c44 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -1,23 +1,28 @@ -%
-% test.tex collection of all slides
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-
-%\folie{a/dc/prinzip.tex}
-%\folie{a/dc/effizient.tex}
-%\folie{a/dc/beispiel.tex}
-
-%\folie{a/ecc/gruppendh.tex}
-%\folie{a/ecc/kurve.tex}
-%\folie{a/ecc/inverse.tex}
-%\folie{a/ecc/operation.tex}
-%\folie{a/ecc/quadrieren.tex}
-%\folie{a/ecc/oakley.tex}
-
-%\folie{a/aes/bytes.tex}
-%\folie{a/aes/sinverse.tex}
-%\folie{a/aes/blocks.tex}
-\folie{a/aes/keys.tex}
-%\folie{a/aes/runden.tex}
-
+% +% test.tex collection of all slides +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +%\folie{9/google.tex} +%\folie{9/markov.tex} +%\folie{9/stationaer.tex} +%\folie{9/irreduzibel.tex} +%\folie{9/pf.tex} + +%\folie{9/pf/positiv.tex} +%\folie{9/pf/primitiv.tex} +%\folie{9/pf/trennung.tex} +%\folie{9/pf/vergleich.tex} +%\folie{9/pf/vergleich3d.tex} +%\folie{9/pf/dreieck.tex} +%\folie{9/pf/folgerungen.tex} +%\folie{9/potenz.tex} + +\folie{9/parrondo/erwartung.tex} +%\folie{9/parrondo/uebersicht.tex} +\folie{9/parrondo/spiela.tex} +\folie{9/parrondo/spielb.tex} +\folie{9/parrondo/spielbmod.tex} +\folie{9/parrondo/kombiniert.tex} +\folie{9/parrondo/deformation.tex} + diff --git a/vorlesungen/stream/countdown.html b/vorlesungen/stream/countdown.html index 739b39d..12f99ac 100644 --- a/vorlesungen/stream/countdown.html +++ b/vorlesungen/stream/countdown.html @@ -1,40 +1,40 @@ -<!DOCTYPE HTML>
-<html>
-<head>
-<style>
-p {
-text-align: center;
-font-size: 60px;
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-font-family: "Courier";
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-var x = setInterval(function() {
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-var t = deadline - now;
-if (t > 0) {
- var days = Math.floor(t / (1000 * 60 * 60 * 24));
- var hours = Math.floor((t%(1000 * 60 * 60 * 24))/(1000 * 60 * 60));
- var minutes = Math.floor((t % (1000 * 60 * 60)) / (1000 * 60));
- var seconds = Math.floor((t % (1000 * 60)) / 1000);
- document.getElementById("demo").innerHTML
- = ("00" + minutes).slice(-2) + ":" + ("00" + seconds).slice(-2);
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-</html>
-
+<!DOCTYPE HTML> +<html> +<head> +<style> +p { +text-align: center; +font-size: 60px; +} +div { +font-family: "Courier"; +text-align: center; +font-size: 60px; +color: #990000; +} +</style> +</head> +<body> +<div id="demo"></div> +<script> +var deadline = new Date("May 17, 2021 17:00:00").getTime(); +var x = setInterval(function() { +var now = new Date().getTime(); +var t = deadline - now; +if (t > 0) { + var days = Math.floor(t / (1000 * 60 * 60 * 24)); + var hours = Math.floor((t%(1000 * 60 * 60 * 24))/(1000 * 60 * 60)); + var minutes = Math.floor((t % (1000 * 60 * 60)) / (1000 * 60)); + var seconds = Math.floor((t % (1000 * 60)) / 1000); + document.getElementById("demo").innerHTML + = ("00" + minutes).slice(-2) + ":" + ("00" + seconds).slice(-2); +} else { + document.getElementById("demo").innerHTML = "00:00"; +} +}, 1000); + +</script> + +</body> +</html> + diff --git a/vorlesungen/stream/ende.html b/vorlesungen/stream/ende.html index cfd9e99..ee25dcf 100644 --- a/vorlesungen/stream/ende.html +++ b/vorlesungen/stream/ende.html @@ -1,30 +1,30 @@ -<!DOCTYPE HTML>
-<html>
-<head>
-<style>
-p {
-text-align: center;
-font-size: 60px;
-}
-div {
-font-family: "Courier";
-text-align: center;
-font-size: 60px;
-color: #990000;
-}
-</style>
-</head>
-<body>
-<div id="demo">
-<p>
-Vielen Dank für Ihren Besuch.
-</p>
-<div style="margin: 200px;"/> </div>
-<p>
-Fortsetzung der Seminar-Sitzung in
-der BBB-Konferenz in Moodle.
-</p>
-</div>
-</body>
-</html>
-
+<!DOCTYPE HTML> +<html> +<head> +<style> +p { +text-align: center; +font-size: 60px; +} +div { +font-family: "Courier"; +text-align: center; +font-size: 60px; +color: #990000; +} +</style> +</head> +<body> +<div id="demo"> +<p> +Vielen Dank für Ihren Besuch. +</p> +<div style="margin: 200px;"/> </div> +<p> +Fortsetzung der Seminar-Sitzung in +der BBB-Konferenz in Moodle. +</p> +</div> +</body> +</html> + |
