diff options
| -rw-r--r-- | vorlesungen/slides/2/funktionenalgebra.tex | 6 | ||||
| -rw-r--r-- | vorlesungen/slides/2/skalarprodukt.tex | 27 | ||||
| -rw-r--r-- | vorlesungen/slides/5/unitaer.tex | 6 | ||||
| -rw-r--r-- | vorlesungen/slides/8/Makefile.inc | 11 | ||||
| -rw-r--r-- | vorlesungen/slides/8/chapter.tex | 7 | ||||
| -rw-r--r-- | vorlesungen/slides/8/dgraph.tex | 100 | ||||
| -rw-r--r-- | vorlesungen/slides/8/graph.tex | 117 | ||||
| -rw-r--r-- | vorlesungen/slides/Makefile.inc | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/test.tex | 115 |
9 files changed, 260 insertions, 132 deletions
diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex index e3339c3..9116be4 100644 --- a/vorlesungen/slides/2/funktionenalgebra.tex +++ b/vorlesungen/slides/2/funktionenalgebra.tex @@ -31,12 +31,12 @@ f(x)\cdot g(x) \begin{align*} \|f\cdot g\|_\infty &= -\sup_{x\in[0,1]} f(x)g(x) +\sup_{x\in[0,1]} |f(x)g(x)| \\ \uncover<4->{ &\le -\sup_{x\in[0,1]}f(x) -\sup_{y\in[0,1]}g(y) +\sup_{x\in[0,1]}|f(x)| +\sup_{y\in[0,1]}|g(y)| } \\ \uncover<5->{ diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex index 2a9784f..99d8a73 100644 --- a/vorlesungen/slides/2/skalarprodukt.tex +++ b/vorlesungen/slides/2/skalarprodukt.tex @@ -13,7 +13,7 @@ \begin{block}{Positiv definite, symmetrische Bilinearform} $\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$ \begin{itemize} -\item +\item<2-> Bilinear: \begin{align*} \langle \alpha u+\beta v,w\rangle @@ -28,18 +28,19 @@ Bilinear: + \beta\langle u,w\rangle \end{align*} -\item +\item<3-> Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$ -\item +\item<4-> $\langle x,x\rangle >0 \quad\forall x\ne 0$ \end{itemize} \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<5->{% \begin{block}{Positive definite, hermitesche Sesquilinearform} $\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$ \begin{itemize} -\item +\item<6-> Sesquilinear: \begin{align*} \langle \alpha u+\beta v,w\rangle @@ -54,40 +55,42 @@ Sesquilinear: + \beta\langle u,w\rangle \end{align*} -\item +\item<7-> Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$ -\item +\item<8-> $\langle x,x\rangle >0 \quad\forall x\ne 0$ \end{itemize} -\end{block} +\end{block}} \end{column} \end{columns} \begin{columns}[t,onlytextwidth] \begin{column}{0.28\textwidth} +\uncover<9->{% \begin{block}{$2$-Norm} $\|v\|_2^2 = \langle v,v\rangle$ \\ $\|v\|_2 = \sqrt{\langle v,v\rangle}$ -\end{block} +\end{block}} \end{column} \begin{column}{0.78\textwidth} +\uncover<10->{% \begin{itemize} -\item $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ -\item $\| \lambda v \|_2 +\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ +\item<12-> $\| \lambda v \|_2 = \sqrt{\langle \lambda v,\lambda v\rangle\mathstrut} = \sqrt{\overline{\lambda}\lambda\langle v,v\rangle} = |\lambda|\cdot \|v\|_2$ -\item +\item<13-> \raisebox{-8pt}{ $\begin{aligned} \|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2 \\ (\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2 \end{aligned}$} -\end{itemize} +\end{itemize}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex index 36e3be2..f0c4401 100644 --- a/vorlesungen/slides/5/unitaer.tex +++ b/vorlesungen/slides/5/unitaer.tex @@ -20,11 +20,11 @@ $U$ unitär lässt das Skalarprodukt invariant \uncover<2->{% $\lambda$ ein Eigenwert mit Eigenvektor $v$: \begin{align*} -\langle u,v\rangle +\langle v,v\rangle &= \langle Uu,Uv\rangle -\uncover<3->{= \langle \lambda u,\lambda v\rangle} -\uncover<4->{= |\lambda|^2 \langle u,v\rangle} +\uncover<3->{= \langle \lambda v,\lambda v\rangle} +\uncover<4->{= |\lambda|^2 \langle v,v\rangle} \\ \uncover<5->{\Rightarrow\;|\lambda|&=1} \end{align*}} diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc new file mode 100644 index 0000000..e8c7502 --- /dev/null +++ b/vorlesungen/slides/8/Makefile.inc @@ -0,0 +1,11 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter8 = \ + ../slides/8/dgraph.tex \ + ../slides/8/graph.tex \ + ../slides/8/chapter.tex + diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex new file mode 100644 index 0000000..761ea63 --- /dev/null +++ b/vorlesungen/slides/8/chapter.tex @@ -0,0 +1,7 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{8/graph.tex} +\folie{8/dgraph.tex} diff --git a/vorlesungen/slides/8/dgraph.tex b/vorlesungen/slides/8/dgraph.tex new file mode 100644 index 0000000..6b5864a --- /dev/null +++ b/vorlesungen/slides/8/dgraph.tex @@ -0,0 +1,100 @@ +% +% dgraph.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame} +\frametitle{Gerichteter Graph} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{2.4} + +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\uncover<3->{ + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); + \draw[color=white,line width=5pt] (B) -- (D); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); + + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); +} + +\uncover<2->{ + \draw (A) circle[radius=0.2]; + \draw (B) circle[radius=0.2]; + \draw (C) circle[radius=0.2]; + \draw (D) circle[radius=0.2]; + \draw (E) circle[radius=0.2]; + + \node at (A) {$1$}; + \node at (B) {$2$}; + \node at (C) {$3$}; + \node at (D) {$4$}; + \node at (E) {$5$}; +} +\node at (0,0) {$G$}; + +\uncover<3->{ + \node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$}; + \node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$}; + \node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$}; + \node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$}; + \node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$}; + \node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$}; + \node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$}; +} + +\uncover<7->{ + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=red] + (E) to[out=-18,in=-126,distance=2cm] (E); +} + +\uncover<9->{ + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=darkgreen] + (D) to[out=120,in=-120] (C); +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Definition} +Ein gerichteter Graph $G=(V,E)$ ist +\begin{enumerate} +\item<2-> Eine Menge $V$ von Knoten (Vertizes) +$V=\{v_1,v_2,\dots\}$ +\item<3-> +Eine Menge $E$ von gerichteten Kanten +(Edges) +\[ +E\subset \{ (v_1,v_2)\;|\; v_i\in V\} +\] +\end{enumerate} +\end{block} +\vspace{-30pt} +\uncover<6->{% +\begin{block}{Achtung} +\begin{itemize} +\item<6-> Kanten sind {\em geordnete} Paare +\uncover<7->{$\Rightarrow$ {\color{red}Schleifen} sind möglich} +\item<8-> Kanten sind immer ``Einbahnstrassen'' +\item<9-> {\color{darkgreen}Gegenrichtung explizit angeben} +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/graph.tex b/vorlesungen/slides/8/graph.tex new file mode 100644 index 0000000..32150af --- /dev/null +++ b/vorlesungen/slides/8/graph.tex @@ -0,0 +1,117 @@ +% +% graph.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Graph} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{2.4} + +\begin{scope} +\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)}); +\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)}); +\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)}); +\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)}); +\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)}); + +\uncover<3->{ + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); + \draw[color=white,line width=5pt] (B) -- (D); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D); + + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); + \draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); +} + +\uncover<2->{ + \draw (A) circle[radius=0.2]; + \draw (B) circle[radius=0.2]; + \draw (C) circle[radius=0.2]; + \draw (D) circle[radius=0.2]; + \draw (E) circle[radius=0.2]; + + \node at (A) {$1$}; + \node at (B) {$2$}; + \node at (C) {$3$}; + \node at (D) {$4$}; + \node at (E) {$5$}; +} +\node at (0,0) {$G$}; + +\uncover<3->{ + \node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) + [above right] {$\scriptstyle 1$}; + \node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) + [above left] {$\scriptstyle 2$}; + \node at ($0.5*(C)+0.5*(D)+(0.05,0)$) + [left] {$\scriptstyle 3$}; + \node at ($0.5*(D)+0.5*(E)$) + [below] {$\scriptstyle 4$}; + \node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) + [below right] {$\scriptstyle 5$}; + \node at ($0.6*(A)+0.4*(C)$) + [above] {$\scriptstyle 6$}; + \node at ($0.4*(B)+0.6*(D)$) + [left] {$\scriptstyle 7$}; +} + +\uncover<8->{ + \draw[shorten >= 0.2cm,shorten <= 0.2cm] + (E) to[out=-18,in=-126,distance=2cm] (E); + + \draw[color=red,line width=4pt] ($(E)+(-0.5,-0.5)+(0,-0.5)$) + -- ($(E)+(0.5,0.5)+(0,-0.5)$); + \draw[color=red,line width=4pt] ($(E)+(-0.5,0.5)+(0,-0.5)$) + -- ($(E)+(0.5,-0.5)+(0,-0.5)$); +} + +\end{scope} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Ein Graph $G=(V,E)$ ist +\begin{enumerate} +\item<2-> +Eine Menge $V$ von Knoten (Vertizes): +$V=\{v_1,v_2,\dots\}$ +\item<3-> +Eine Menge $E$ von Kanten (Edges): +\[ +E\subset +\left\{ e = \{v_1,v_2\}\;\left|\; \begin{minipage}{1.3cm}\raggedright +$v_i\in V$\\ +$v_1\ne v_2$ +\end{minipage} +\right. +\right\} +\] +\end{enumerate} +\end{block} +\vspace{-20pt} +\uncover<5->{% +\begin{block}{Achtung:} +\begin{itemize} +\item<6-> Kanten sind Mengen +\uncover<7->{$\Rightarrow$ zwei verschiedene Knoten} +\uncover<8->{$\Rightarrow$ Keine Schleifen} +\item<9-> Kanten sind ungerichtet, keine ``Einbahnstrassen'' +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 20929e4..fe22282 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -9,7 +9,8 @@ include ../slides/2/Makefile.inc include ../slides/3/Makefile.inc include ../slides/4/Makefile.inc include ../slides/5/Makefile.inc +include ../slides/8/Makefile.inc slides = \ $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ - $(chapter5) + $(chapter5) $(chapter8) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 5f08a8f..30aef37 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,117 +3,6 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % - -%\folie{3/motivation.tex} -%\folie{3/inverse.tex} -%\folie{3/polynome.tex} -%\folie{3/division.tex} -%\folie{3/division2.tex} -%\folie{3/ringstruktur.tex} -%\folie{3/teilbarkeit.tex} -%\folie{3/faktorisierung.tex} -%\folie{3/faktorzerlegung.tex} -%\folie{3/einsetzen.tex} -%\folie{3/maximalergrad.tex} -%\folie{3/minimalbeispiel.tex} -%\folie{3/minimalpolynom.tex} -%\folie{3/drehmatrix.tex} -%\folie{3/drehfaktorisierung.tex} -%\folie{3/fibonacci.tex} -%\folie{3/operatoren.tex} -%\folie{3/adjunktion.tex} -%\folie{3/adjalgebra.tex} - -%\folie{4/ggt.tex} -%\folie{4/euklidmatrix.tex} -%\folie{4/euklidbeispiel.tex} -%\folie{4/euklidtabelle.tex} -%\folie{4/fp.tex} -%\folie{4/division.tex} -%\folie{4/gauss.tex} -% \folie{4/dh.tex} -% XXX \folie{4/frobenius.tex} - -%\folie{4/divisionpoly.tex} -%\folie{4/euklidpoly.tex} -%\folie{4/polynomefp.tex} -%\folie{4/alpha.tex} - -% XXX \folie{4/f2.tex} -%\folie{4/schieberegister.tex} - -% XXX Idee der elliptischen Kurve -% XXX \folie{4/ecidee.tex} - - -\section{Eigenwertproblem} -% XXX Motivation: beliebige Funktionen f(A) berechnen -%\folie{5/motivation.tex} -%\folie{5/charpoly.tex} - -\section{Invariante Unterräume} -%\folie{5/kernbild.tex} -%\folie{5/ketten.tex} -%\folie{5/dimension.tex} -%\folie{5/folgerungen.tex} -%\folie{5/injektiv.tex} -%\folie{5/nilpotent.tex} -%\folie{5/eigenraeume.tex} -%\folie{5/zerlegung.tex} -%\folie{5/normalnilp.tex} -%\folie{5/bloecke.tex} - -% Jordan Normalform -\section{Jordan-Normalform} -%\folie{5/jordanblock.tex} -%\folie{5/jordan.tex} -% XXX Diagonalform -% XXX \folie{5/diagonalform.tex} -%\folie{5/reellenormalform.tex} -% XXX \folie{5/hessenberg.tex} - -\section{Satz von Cayley-Hamilton} -%\folie{5/cayleyhamilton.tex} - -\section{Matrixnormen} -%\folie{2/norm.tex} -%\folie{2/skalarprodukt.tex} -%\folie{2/cauchyschwarz.tex} -%\folie{2/funktionenraum.tex} -%\folie{2/polarformel.tex} -%\folie{2/operatornorm.tex} -%\folie{2/funktionenalgebra.tex} -%\folie{2/linearformnormen.tex} -%\folie{2/frobeniusnorm.tex} -%\folie{2/frobeniusanwendung.tex} - -\section{Approximation mit Polynomen} -% XXX Stone-Weierstrass -% XXX \folie{5/stoneweierstrass.tex} -% XXX Spektrum einer Matrix -%\folie{5/spektrum.tex} -\folie{5/normal.tex} -\folie{5/unitaer.tex} -% XXX Approximation einer Funktion auf dem Spektrum -% XXX \folie{5/spektrumapproximation.tex} -% XXX Approximation einer Matrix in der erzeugten Algebra -% XXX \folie{5/matrixapproximation.tex} -% XXX Gelfand-Transformation -% XXX \folie{5/gelfandtransformation.tex} - -\section{Potenzreihen} -% Konvergenzradius -%\folie{5/konvergenzradius.tex} -%\folie{5/krbeispiele.tex} -%\folie{5/spektralgelfand.tex} -%\folie{5/Aiteration.tex} -%\folie{5/satzvongelfand.tex} -%\folie{5/logarithmusreihe.tex} - -\section{Differentialgleichungen} -%\folie{5/potenzreihenmethode.tex} -%\folie{5/exponentialfunktion.tex} -% XXX Sinus und Cosinus, Eulerscher Satz -% XXX \folie{5/sinuscosinus.tex} -%\folie{5/hyperbolisch.tex} +\folie{8/dgraph.tex} +\folie{8/graph.tex} |