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Diffstat (limited to 'buch')
132 files changed, 51650 insertions, 1083 deletions
diff --git a/buch/.gitignore b/buch/.gitignore deleted file mode 100644 index 5d8a46e..0000000 --- a/buch/.gitignore +++ /dev/null @@ -1,16 +0,0 @@ -*.aux -*.bbl -*.bib -*.blg -*.idx -*.ilg -*.ind -*.log -*.out -*.rpt -buch*.pdf -*.run.xml -*.toc -.build/ -*.synctex.gz -*.DS_Store diff --git a/buch/Makefile b/buch/Makefile index 722c177..1cd50dd 100755 --- a/buch/Makefile +++ b/buch/Makefile @@ -28,8 +28,19 @@ buch.ind: buch.idx separate: buch.aux buch.pdf bash splitpapers -numerik.pdf: - pdfjam --outfile numerik.pdf \ +matrizen.pdf: + pdfjam --outfile matrizen.pdf \ ../cover/front.pdf 1,{} \ buch.pdf 1-504 \ ../cover/back.pdf {},1 + +tests: test1.pdf test2.pdf test3.pdf + +test1.pdf: common/test-common.tex common/test1.tex aufgaben1.tex + pdflatex common/test1.tex + +test2.pdf: common/test-common.tex common/test1.tex aufgaben2.tex + pdflatex common/test2.tex + +test3.pdf: common/test-common.tex common/test1.tex aufgaben3.tex + pdflatex common/test3.tex diff --git a/buch/aufgaben1.tex b/buch/aufgaben1.tex new file mode 100644 index 0000000..9348019 --- /dev/null +++ b/buch/aufgaben1.tex @@ -0,0 +1,13 @@ +% +% aufgaben1.tex -- Aufgaben für Test 1 +% +% (c) 2012 Prof. Dr. Andreas Mueller, HSR +% + +\item +\input chapters/30-endlichekoerper/uebungsaufgaben/3003.tex +\item +\input chapters/30-endlichekoerper/uebungsaufgaben/3004.tex +\item +\input chapters/30-endlichekoerper/uebungsaufgaben/3005.tex + diff --git a/buch/aufgaben2.tex b/buch/aufgaben2.tex new file mode 100644 index 0000000..dc4fc59 --- /dev/null +++ b/buch/aufgaben2.tex @@ -0,0 +1,11 @@ +% +% aufgaben2.tex -- Aufgaben für Test 2 +% +% (c) 2021 Prof. Dr. Andreas Mueller, OST +% + +\item +\input chapters/40-eigenwerte/uebungsaufgaben/4004.tex +\item +\input chapters/40-eigenwerte/uebungsaufgaben/4005.tex + diff --git a/buch/aufgaben3.tex b/buch/aufgaben3.tex new file mode 100644 index 0000000..23c9153 --- /dev/null +++ b/buch/aufgaben3.tex @@ -0,0 +1,7 @@ +% +% aufgaben3.tex -- Aufgaben für Test 3 +% +% (c) 2021 Prof. Dr. Andreas Mueller, OST +% +\item +\input chapters/60-gruppen/uebungsaufgaben/6001.tex diff --git a/buch/chapters/70-graphen/wavelets.tex b/buch/chapters/70-graphen/wavelets.tex index ae065bc..ef1520e 100644 --- a/buch/chapters/70-graphen/wavelets.tex +++ b/buch/chapters/70-graphen/wavelets.tex @@ -174,14 +174,14 @@ die in der Umgebung eines Knotens wie die Konstante Funktion aussehen. Das Mutter-Wavelet einer Wavelet-Analyse zeichnet definiert, in welchem Mass sich Funktionen im Orts- und im Frequenzraum lokalisieren lassen. Die Standardbasis der Funktionen auf einem Graphen repräsentieren die -perfekte örtliche Lokalisierung, Eigenbasis der Laplace-Matrix repräsentiert +perfekte örtliche Lokalisierung, Eigenbasis der Laplace-Matrix $L$ repräsentiert die perfekte Lokalisierung im Frequenzraum. Sei $g(\lambda)\ge 0$ eine Funktion im Frequenzraum, die für $\lambda\to0$ und $\lambda\to\infty$ rasch abfällt mit einem Maximum irgendwo dazwischen (Abbildung~\ref{buch:graphs:fig:lokalisierung}). Sie kann als eine Lokalisierungsfunktion im Frequenzraum betrachtet werden. -Die Matrix $g(I)$ bildet entfernt aus einer Funktion die ganz hohen und +Die Matrix $g(L)$ bildet entfernt aus einer Funktion die ganz hohen und die ganz tiefen Frequenz, lokalisiert also die Funktionen im Frequenzraum. Die Standardbasisvektoren werden dabei zu Funktionen, die nicht mehr nur auf einem Knoten von $0$ verschieden sind, aber immer noch einigermassen @@ -190,15 +190,15 @@ Natürlich sind vor allem die Werte auf den Eigenwerten $\lambda_0 < \lambda_1\le \dots\le \lambda_n$ der Laplace-Matrix von Interesse. -Die Matrix $g(I)$ kann mit Hilfe der Spektraltheorie berechnet werden, +Die Matrix $g(L)$ kann mit Hilfe der Spektraltheorie berechnet werden, was im vorliegenden Fall naheliegend ist, weil ja die Eigenvektoren von der Laplace-Matrix bereits bekannt sind. Die Matrix $\chi^t$ bildet die Standardbasisvektoren in die Eigenbasis-Vektoren ab, also in eine Zerlegung im Frequenzraum ab, $\chi$ vermittelt die Umkehrabbildung. -Mit der Spektraltheorie findet man für die Abbildung $g(I)$ die Matrix +Mit der Spektraltheorie findet man für die Abbildung $g(L)$ die Matrix \begin{equation} -g(I) +g(L) = \chi \begin{pmatrix} @@ -214,7 +214,7 @@ g(\lambda_0)&0&\dots&0\\ \subsubsection{Dilatation} Die Dilatation um $a$ im Ortsraum wird zu einer Dilatation um $1/a$ im Frequenzraum. -Statt also nach einer echten Dilatation der Spaltenvektoren in $g(I)$ +Statt also nach einer echten Dilatation der Spaltenvektoren in $g(L)$ zu suchen, kann man sich darauf verlegen, Funktionen zu finden, deren Spektrum von einer Funktionen lokalisiert worden ist, die eine Dilatation von $g$ ist. @@ -225,9 +225,9 @@ Die zugehörigen Wavelet-Funktionen auf dem Graphen können wieder mit der Formel~\eqref{buch:graphen:eqn:mutterwavelet} berechnet werden, man erhält \begin{equation} -\tilde{D}_{1/a_i}g(I) +\tilde{D}_{1/a_i}g(L) = -g_i(I) +g_i(L) = \chi \begin{pmatrix} @@ -238,30 +238,30 @@ g(a_i\lambda_0)&0&\dots&0\\ \end{pmatrix} \chi^t . \end{equation} -Die Spalten von $g_i(I)$ bilden wieder eine Menge von Funktionen, die +Die Spalten von $g_i(L)$ bilden wieder eine Menge von Funktionen, die eine gemäss $g_i$ lokalisiertes Spektrum haben. \subsubsection{Vater-Wavelet} Wegen $g(0)=0$ wird die konstante Funktion, die Eigenvektor zum Eigenwert -$\lambda_0=0$ ist, von den Abbildungen $g_i(I)$ auf $0$ abgebildet. +$\lambda_0=0$ ist, von den Abbildungen $g_i(L)$ auf $0$ abgebildet. Andererseits ist diese Funktion nicht lokalisiert, man möchte Sie also für die Analyse nicht unbedingt verwenden. Man wählt daher eine Funktion $h(\lambda)$ mit $h(0)=1$ so, dass für $\lambda\to \infty$ der Wert $h(\lambda)$ genügend rasch gegen $0$ geht. -Die Matrix $h(I)$ bildet daher den konstanten Vektor nicht auf $0$ ab, +Die Matrix $h(L)$ bildet daher den konstanten Vektor nicht auf $0$ ab, sondern lokalisiert ihn im Ortsraum. -Wir erhalten daher in den Spalten von $h(I)$ Vektoren, die um die +Wir erhalten daher in den Spalten von $h(L)$ Vektoren, die um die einzelnen Knoten lokalisiert sind. \subsubsection{Rekonstruktion} -Die Operatoren $h(I)$ und $g_i(I)$ erzeugen analysieren eine Funktion +Die Operatoren $h(L)$ und $g_i(L)$ erzeugen analysieren eine Funktion nach den verschiedenen Frequenzen mit den Skalierungsfaktoren $a_i$, aber die Rekonstruktion ist noch nicht klar. Diese wäre einfacher, wenn die Operatoren zusammen die identische Abbildung ergäben, wenn also \[ -h(I) + \sum_{i}g_i(I)=I +h(L) + \sum_{i}g_i(L)=I \] gelten würde. Nach der Spektraltheorie gilt das nur, wenn für alle Eigenwerte @@ -301,14 +301,14 @@ B\|v\|^2 Die Zahlen $A$ und $B$ heissen die {\em Frame-Konstanten} des Frames. \end{definition} -Die oben gefundenen Vektoren, die Spalten Vektoren von $h(I)$ und $g_i(I)$ +Die oben gefundenen Vektoren, die Spalten Vektoren von $h(L)$ und $g_i(L)$ bilden daher ein Frame. Die Frame-Konstanten kann man unmittelbar ausrechnen. Der mittlere Term von \eqref{buch:graphen:eqn:frame} ist \[ -\|h(I) v\|^2 +\|h(L) v\|^2 + -\sum_{i} \|g_i(I)v\|^2, +\sum_{i} \|g_i(L)v\|^2, \] die durch die Funktion \[ diff --git a/buch/test3.tex b/buch/common/test-common.tex index 71b1529..289e59c 100644 --- a/buch/test3.tex +++ b/buch/common/test-common.tex @@ -1,9 +1,8 @@ % -% test3.tex -- Test 3 +% test.tex -- Gemeinsamer Rahmen für Kurztests % -% (c) 2021 Prof. Dr. Andreas Mueller, OST +% (c) 2021 Prof. Dr. Andreas Mueller, OST Ostschweizer Fachhochschule % -%\documentclass[a4paper,12pt]{book} \documentclass[a4paper,12pt]{article} \usepackage{geometry} \geometry{papersize={210mm,297mm},total={165mm,260mm}} @@ -72,20 +71,3 @@ \renewcommand{\qedsymbol}{} \begin{proof}[Hinweis]}{\end{proof}} -\begin{document} -{\parindent0pt\hbox to\hsize{% -Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} -\vspace{0.5cm} - -\section*{Kurztest 3} - -\begin{uebungsaufgaben} - -\item -\input chapters/60-gruppen/uebungsaufgaben/6001.tex -%\item -%\input chapters/60-gruppen/uebungsaufgaben/6002.tex - -\end{uebungsaufgaben} - -\end{document} diff --git a/buch/common/test1.tex b/buch/common/test1.tex new file mode 100644 index 0000000..1f5a155 --- /dev/null +++ b/buch/common/test1.tex @@ -0,0 +1,21 @@ +% +% test1.tex -- Test 1 +% +% (c) 2021 Prof. Dr. Andreas Mueller, OST +% +\input{common/test-common.tex} + +\begin{document} +{\parindent0pt\hbox to\hsize{% +Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} +\vspace{0.5cm} + +\section*{Kurztest 1} + +\begin{uebungsaufgaben} + +\input{aufgaben1.tex} + +\end{uebungsaufgaben} + +\end{document} diff --git a/buch/common/test2.tex b/buch/common/test2.tex new file mode 100644 index 0000000..0980e44 --- /dev/null +++ b/buch/common/test2.tex @@ -0,0 +1,21 @@ +% +% test2.tex -- Test 2 +% +% (c) 2012 Prof. Dr. Andreas Mueller, OST +% +\input{common/test-common.tex} + +\begin{document} +{\parindent0pt\hbox to\hsize{% +Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} +\vspace{0.5cm} + +\section*{Kurztest 2} + +\begin{uebungsaufgaben} + +\input{aufgaben2.tex} + +\end{uebungsaufgaben} + +\end{document} diff --git a/buch/common/test3.tex b/buch/common/test3.tex new file mode 100644 index 0000000..8b24262 --- /dev/null +++ b/buch/common/test3.tex @@ -0,0 +1,21 @@ +% +% test3.tex -- Test 3 +% +% (c) 2021 Prof. Dr. Andreas Mueller, OST +% +\input{common/test-common.tex} + +\begin{document} +{\parindent0pt\hbox to\hsize{% +Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} +\vspace{0.5cm} + +\section*{Kurztest 3} + +\begin{uebungsaufgaben} + +\input{aufgaben3.tex} + +\end{uebungsaufgaben} + +\end{document} diff --git a/buch/papers/clifford/0_ElevatorPitch.tex b/buch/papers/clifford/0_ElevatorPitch.tex new file mode 100644 index 0000000..0db5617 --- /dev/null +++ b/buch/papers/clifford/0_ElevatorPitch.tex @@ -0,0 +1,2 @@ +TODO... +GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains.
\ No newline at end of file diff --git a/buch/papers/clifford/10_Quaternionen.tex b/buch/papers/clifford/10_Quaternionen.tex new file mode 100644 index 0000000..8945ba8 --- /dev/null +++ b/buch/papers/clifford/10_Quaternionen.tex @@ -0,0 +1,61 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Quaternionen} +\rhead{Quaternionen} +Wie die komplexen Zahlen eine Erweiterung der reellen Zahlen sind, sind die Quaternionen eine Erweiterung der komplexen Zahlen für den 3 dimensionalen Raum. Sie haben, wie die komplexen Zahlen, eine dreh-streckende Eigenschaft. +Sie finden beispielsweise in der Computergraphik und in der Robotik Anwendung. +Die Quaternionen werden so definiert. +\begin{align} + q = w + xi + yj + zk; \quad w,x,y,z \in \mathbb{R};\enspace q \in \mathbb{H} +\end{align} +Eine Drehstreckung wird dabei mit dieser Formel erreicht. +\begin{align} \label{QuatRot} + \begin{split} + &v'' = qvq^{-1};\quad q,v,q^{-1} \in \mathbb{H}\\ + &Re(q) = Re(q^{-1});\enspace Im(q) = -Im(q^-1) + \end{split} +\end{align} +Die Quaternionen besitzen im Gegensatz zu dem komplexen Zahlen 3 imaginäre Einheiten $i,j,k$. Wieso 3? Weil es in der dritten Dimension 3 Drehachsen gibt, anstatt nur eine. Nun haben wir ein kleines Problem. Wie sollen wir die Quaternionen darstellen? Wir bräuchten 4 Achsen für die 3 Imaginären Einheiten und die eine reelle Einheit. Ein weiterer Nachteil in visueller Hinsicht entsteht beim Anwenden eines Quaternion auf einen Vektor. Sie befinden sich nicht im gleichen Raum und müssen zuerst ineinander umgewandelt werden, um damit zu rechnen, wie man bei $v$ in der Formel (\ref{QuatRot}) sieht. + +\subsection{geometrischen Algebra} +Die geometrische Algebra besitzt die Fähigkeit beide Probleme zu lösen. Die Quaternionen können, wie schon im 2 dimensionalen Fall durch die gerade Grade $\mathbb{G}_3^+ \cong \mathbb{H}$ dargestellt werden. Da wir uns jetzt aber in $\mathbb{G}_3$ befinden haben wir 3 Basisvektoren $e_1, e_2, e_3$ und können somit 3 Bivektoren bilden $e_{12}, e_{23}, e_{31}$. +\begin{align} + \mathbf{q} = w + x\mathbf{e_{12}} + y\mathbf{e_{23}} + z\mathbf{e_{31}}; \quad w,x,y,z \in \mathbb{R};\enspace q \in \mathbb{G}_3^+ +\end{align} +Die Probleme werden dadurch gelöst, da wir die Bivektoren im Raum nicht durch einzelne Achsen darstellen müssen, sondern sie als eine orientiere Fläche darstellen können. Anstatt die Vektoren in Quaternionen umzurechnen, können wir jetzt die Vektoren separat im gleichen Raum darstellen. +\\BILD VEKTOR, QUATERNION IN G3\\ +Wie schon im 2 dimensionalen Fall beschreibt ein Bivektor, um wie viel der um 90 grad gedrehte orginale Vektor gestreckt wird. Dabei dreht jeder Bivektor den Vektor um eine andere Achse. +\\BILD?\\ +In der Computergraphik und Robotik macht eine Drehstreckung aber nicht viel Sinn. Wieso sollte ein Objekt bei einer Drehung zusätzlich noch grösser werden? Darum verwendet man sogenannte Einheitsquaternion, welche den Betrag $|q|=1$ haben. Sie rotieren die Objekte bzw. Vektoren lediglich. +\begin{align} + \mathbf{q} = \cos(\alpha) + sin(\alpha)(x\mathbf{e_{12}} + y\mathbf{e_{23}} + z\mathbf{e_{31}}) +\end{align} +wobei definiert ist, dass $x^2+y^2+z^2=1$. Somit beträgt der Betrag immer 1. +\begin{align} + |q| = \sqrt{cos(\alpha)^2 + sin(\alpha)^2(x^2+y^2+z^2) } = \sqrt{cos(\alpha)^2 + sin(\alpha)^2} = 1 +\end{align} +Man verwendet um einen Vektor zu drehen wieder die gleiche Formel, wie auch schon im 2 dimensionalen Fall. +\begin{align} \label{QuatRot} + \begin{split} + &v'' = qvq^{-1}\\ + &Re(q) = Re(q^{-1});\enspace Im(q) = -Im(q^-1) + \end{split} +\end{align} +Es ist wichtig bei Quaternionen für eine reine Drehstreckung mit $q$ und $q^{-1}$ beidseitig zu multiplizieren, sonst werden die senkrechten Anteile zu den Bivektorebenen ebenfalls beeinflusst, wie man im Kapitel Rotation bei der Formel (\ref{RotAufPerpPar}) sehen kann + +\subsection{Gimbal-Lock und Interpolation} + +\subsection{Fazit} +andere Darstellungsweise. Besser für Verständnis => komplexe Zahlen erscheinen ähnlicher zu Quaternionen? Eine Sprache für alle Geometrische Probleme + + +\begin{tikzpicture} + \draw[thin,gray!40] (-3,-3) grid (3,3); + \draw[<->] (-3,0)--(3,0) node[right]{$x$}; + \draw[<->] (0,-3)--(0,3) node[above]{$y$}; + \draw[line width=2pt,blue,-stealth](0,0)--(1,1) node[anchor=south west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(-1,-1) node[anchor=north east]{$\boldsymbol{-u}$}; +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/clifford/1_Vektordarstellung.tex b/buch/papers/clifford/1_Vektordarstellung.tex new file mode 100644 index 0000000..88a5789 --- /dev/null +++ b/buch/papers/clifford/1_Vektordarstellung.tex @@ -0,0 +1,71 @@ +\section{Vektoroperationen\label{clifford:section:Vektoroperationen}} +\rhead{Vektoroperationen} +\subsection{Vektordarstellung\label{clifford:section:Vektordarstellung}} +Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus Basisvektoren dargestellt werden +\begin{equation} + \begin{split} + \textbf{a} + &= + \begin{pmatrix} + a_1 \\ a_2 \\ \vdots \\ a_n + \end{pmatrix} + = + a_1 \begin{pmatrix} + 1 \\ 0 \\ \vdots \\ 0 + \end{pmatrix} + + + a_2\begin{pmatrix} + 0 \\ 1 \\ \vdots \\ 0 + \end{pmatrix} + \dots + + + a_n\begin{pmatrix} + 0 \\ 0 \\ \vdots \\ 1 + \end{pmatrix} \\\ + &= + a_1\textbf{e}_1 + + + a_2\textbf{e}_2 + + + \dots + a_n\textbf{e}_n + = + \sum_{i=1}^{n} a_i \textbf{e}_i + \qquad + a_i \in \mathbb{R} + , \textbf{e}_i \in \mathbb{R}^n. + \end{split} +\end{equation} +Diese Basisvektoren sollen orthonormal sein und um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, ...$ ersetzt. +\begin{beispiel} +Linearkombination von Basisvektoren in $\mathbb{R}^4$ + \begin{equation} + \begin{pmatrix} + 42 \\ 2 \\ 1291 \\ 4 + \end{pmatrix} + = + 42 \begin{pmatrix} + 1 \\ 0 \\ 0 \\ 0 + \end{pmatrix} + + + 2 \begin{pmatrix} + 0 \\ 1 \\ 0 \\ 0 + \end{pmatrix} + + + 1291 + \begin{pmatrix} + 0 \\ 0 \\ 1 \\ 0 + \end{pmatrix} + + + 4 \begin{pmatrix} + 0 \\ 0 \\ 0 \\ 1 + \end{pmatrix} + = + 42\textbf{e}_1 + + + 2\textbf{e}_2 + + + 1291\textbf{e}_3 + + + 4\textbf{e}_4 + \end{equation} +\end{beispiel} +Wobei Beispiel für einen vier dimensionalen Vektor ist, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden.
\ No newline at end of file diff --git a/buch/papers/clifford/2_QuadratVektoren.tex b/buch/papers/clifford/2_QuadratVektoren.tex new file mode 100644 index 0000000..cfb05d6 --- /dev/null +++ b/buch/papers/clifford/2_QuadratVektoren.tex @@ -0,0 +1,110 @@ +\subsection{Quadrat von Vektoren} +Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. +Was soll es schon heissen zwei Vektoren miteinander zu multiplizieren? +\newline +Im Folgenden werden wir versuchen diese Operation ähnlich intuitiv darzustellen. +\newline +Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation ist. +Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen, auch so definiert ist. +\newline +Zusätzlich wollen wir auch das Assoziativgesetz und das Kommutativgesetz für Skalare beibehalten. Wobei das Kommutativgesetz leider, oder wie man sehen wird zum Glück, in der geometrischen Algebra im generellen nicht mehr gilt. Das heisst wir dürfen ausklammern \ref{eq:assoziativ} und die Position von Skalaren im Produkt ändern \ref{eq:kommSkalar}, allerdings nicht die Position der Vektoren \ref{eq:kommVector}. +\begin{equation} + \label{eq:assoziativ} + \textbf{e}_i(\textbf{e}_j + \textbf{e}_k) + = + \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k +\end{equation} +\begin{equation} + \label{eq:kommSkalar} + a\textbf{e}_ib\textbf{e}_j + = + ab\textbf{e}_i\textbf{e}_j +\end{equation} +\begin{equation} + \label{eq:kommVector} + \textbf{e}_i\textbf{e}_j + \neq + \textbf{e}_j\textbf{e}_i +\end{equation} +Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors. +\begin{align} + \textbf{a}^2 &= + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \label{eq:quad_a_1} + \\ + &= + \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} + + + \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j } + \label{eq:quad_a_2} + \\ + &= \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j}. + \label{eq:quad_a_3} +\end{align} + +\begin{beispiel} +Quadrat eines Vektors in $\mathbb{R}^2$ +\begin{equation} + \begin{split} + \textbf{a}^2 + &= (a_1\textbf{e}_1+a_2\textbf{e}_2)(a_1\textbf{e}_1+a_2\textbf{e}_2) \\\ + &= \textcolor{red}{a_1^2\textbf{e}_1^2 + a_2^2\textbf{e}_2^2} + + \textcolor{blue}{a_1\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} \\\ + & = \textcolor{cyan}{a_1^2 + a_2^2} + \textcolor{orange}{a_1b\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} + \end{split} +\end{equation} + +\end{beispiel} +Der Vektor wird in \ref{eq:quad_a_1} als Linearkombination geschrieben. +Das Quadrat kann, wie in \ref{eq:quad_a_2} gezeigt, in zwei Summen aufteilen werden , wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. +\newline +Da $\textbf{e}_i^2 = 1$ gilt, da zuvor vorausgesetzt wurde, dass man mit orthonormalen Einheitsvektoren arbeitet, wird dies nun eingesetzt ergibt sich \ref{eq:quad_a_3} +\newline +Die hellblaue Teil ist nun bereits Länge im Quadrat eines Vektors, also das Ziel der Multiplikation. +Daher muss der restliche Teil dieser Gleichung null ergeben. +Aus dieser Erkenntnis leiten wir in \ref{eq:Mischterme_Null} weitere Eigenschaften für die Multiplikation her. +\begin{equation} + \label{eq:Mischterme_Null} + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots = 0 +\end{equation} +Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_2$ gleich null gesetzt. Somit fallen alle Terme bis auf den blauen weg. Wird dies weiter vereinfacht ergibt sich +\begin{equation} +\begin{split} + a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1) &= 0 \\ + a_1a_2\textbf{e}_1\textbf{e}_2 &= -a_1a_2\textbf{e}_2\textbf{e}_1 \\ + \textbf{e}_1\textbf{e}_2 &= -\textbf{e}_2\textbf{e}_1. +\end{split} +\end{equation} +\begin{satz} + Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind. + \begin{equation} + \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \qquad \textbf{e}_i \perp \textbf{e}_j + \end{equation} +\end{satz} +Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in \ref{tab:multip_vec} gemacht wurde. +\begin{table} +\caption{Multiplikationstabelle für Vektoren} +\label{tab:multip_vec} +\begin{center} +\begin{tabular}{ |c|c|c|c|c|c| } + \hline + & $\textbf{e}_1$ & $\textbf{e}_2$ & $\dots$ & $\textbf{e}_{n-1}$ & $\textbf{e}_{n}$ \\ + \hline + $\textbf{e}_1$ & 1 & $\textbf{e}_1\textbf{e}_2$ & $\dots$ & $\textbf{e}_1\textbf{e}_{n-1}$ & $\textbf{e}_1\textbf{e}_{n}$ \\ + \hline + $\textbf{e}_2$ & $-\textbf{e}_1\textbf{e}_2$ & 1 & $\dots$ & $\textbf{e}_2\textbf{e}_{n-1}$ & $\textbf{e}_2\textbf{e}_{n}$ \\ + \hline + $\vdots$ & $\vdots$ & $\vdots$ & $\ddots$ & $\vdots$ & $\vdots$ \\ + \hline + $\textbf{e}_{n-1}$ & $-\textbf{e}_1\textbf{e}_{n-1}$ & $-\textbf{e}_2\textbf{e}_{n-1}$ & $\dots$ & $1$ & $\textbf{e}_{n-1}\textbf{e}_{n}$ \\ + \hline + $\textbf{e}_{n}$ & $-\textbf{e}_1\textbf{e}_{n}$ & $-\textbf{e}_2\textbf{e}_{n}$ & $\dots$ & $-\textbf{e}_{n-1}\textbf{e}_{n}$ & 1 \\ + \hline +\end{tabular} +\end{center} +\end{table}
\ No newline at end of file diff --git a/buch/papers/clifford/3_MultiplikationVektoren.tex b/buch/papers/clifford/3_MultiplikationVektoren.tex new file mode 100644 index 0000000..841dde4 --- /dev/null +++ b/buch/papers/clifford/3_MultiplikationVektoren.tex @@ -0,0 +1,175 @@ +\subsection{Multiplikation von Vektoren} +Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipliziert werden? +\begin{equation} + \textbf{u} = + \sum_{i=1}^{n} u_i \textbf{e}_i + \qquad + \textbf{v} = \sum_{i=1}^{n} v_i \textbf{e}_i +\end{equation} +\begin{equation} + \begin{split} + \textbf{u}\textbf{v} + = + \left ( + \sum_{i=1}^{n} u_i \textbf{e}_i + \right ) + \left ( + \sum_{i=1}^{n} v_i \textbf{e}_i + \right) + = + \sum_{i=1}^n u_iv_i\underbrace{\textbf{e}_i^2}_{1} + + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + \end{split} +\end{equation} +\begin{beispiel} + Multiplikation von Vektoren in $\mathbb{R}^2$ +\end{beispiel} +\begin{equation} + \begin{split} + \textbf{u}\textbf{v} + &= + (u_1\textbf{e}_1 + u_2\textbf{e}_2)(v_1\textbf{e}_1 + v_2\textbf{e}_2) + = + u_1v_1\textbf{e}_1^2 + + + u_2v_2\textbf{e}_2^2 + + + u_1v_2\textbf{e}_1\textbf{e}_2 + + + u_2v_1\underbrace{\textbf{e}_2\textbf{e}_1}_{-\textbf{e}_1\textbf{e}_2} + \\\ + &= + \underbrace{(u_1v_1 + u_2v_2)}_{\text{Skalarprodukt}} + + + \underbrace{(u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2}_{\text{Äusseres Produkt}} + \end{split} +\end{equation} +Der linke Teil dieser Multiplikation ergibt das Skalarprodukt der zwei Vektoren, der rechte Term ergibt etwas neues das sich das äussere Produkt der zwei Vektoren nennt. +\subsubsection{Äusseres Produkt} +Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt +\begin{equation} + \textbf{u}\wedge \textbf{v} + = + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j +\end{equation} +\begin{beispiel} +Äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$ +\end{beispiel} +\begin{equation} + \begin{split} + u \wedge v + &= + u_1v_2\textbf{e}_1\textbf{e}_2 + + + u_1v_3\textbf{e}_1\textbf{e}_3 + + + u_2v_2\textbf{e}_2\textbf{e}_3 + + + u_2v_1\textbf{e}_2\textbf{e}_1 + + + u_3v_1\textbf{e}_3\textbf{e}_1 + + + u_3v_2\textbf{e}_3\textbf{e}_2 \\\ + &= + (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 + + + (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 + + + (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3 + \end{split} +\end{equation} +Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \ref{eq:u_wedge_v}-\ref{eq:u_wedge_v_5} hergeleitet. +\begin{align} + \textbf{u}\wedge \textbf{v} + &= + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n + u_iv_j\textbf{e}_i\textbf{e}_j + \label{eq:u_wedge_v} + \\ + \label{eq:u_wedge_v_1} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + + + \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + \\ + \label{eq:u_wedge_v_2} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + + + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i + \\ + \label{eq:u_wedge_v_3} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + - + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j + \\ + \label{eq:u_wedge_v_4} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j + \\ + \label{eq:u_wedge_v_5} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}\textbf{e}_i\textbf{e}_j +\end{align} +Die Summe aus \ref{eq:u_wedge_v_1} wird in \ref{eq:u_wedge_v} in zwei verschiedene Summen aufgeteilt. +Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat. +\newline +Bei \ref{eq:u_wedge_v_2} werden die Indexe der zweiten Summe vertauscht, damit man nun bei beiden Teilen die gleiche Summe hat. +Danach werden in \ref{eq:u_wedge_v_3}, mit Hilfe der Antikommutativität, die Einheitsvektoren der zweiten Summe vertauscht. +\newline +Nun können die Summen, wie in \ref{eq:u_wedge_v_4} wieder in eine Summe zusammengefasst werden. +\newline +Der Term in der Klammer in \ref{eq:u_wedge_v_4} kann auch als Determinante einer 2x2 Matrix dargestellt werden, was in \ref{eq:u_wedge_v_5} gemacht wird. +\newline +Die Determinante einer Matrix beschreibt welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt. +\begin{figure} +\centering +\begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) ; + \draw[<->] (0,0)--(0,4) ; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} = u_iv_j - v_iu_j$}; +\end{tikzpicture} +\caption{Geometrische Interpretation der Determinante einer 2x2 Matrix\label{figure:det}} +\end{figure} +\newline +Das äussere Produkt besteht nun also aus der Summe + $\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n$ + von Flächen + $\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}$, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. +Dieses Produkt $\textbf{e}_i\textbf{e}_j$ der Basisvektoren interpretiert man als Umlaufrichtung. +Wobei die gebildete Fläche in Richtung des ersten Vektors umschritten wird. +Dies ist in \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. +Diese Fläche mit einer Richtung nennt man in der geometrischen Algebra einen Bivektor, da er eine Art zwei dimensionaler Vektor ist. +\begin{figure} +\centering +\begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) node[right]{$x$}; + \draw[<->] (0,0)--(0,4) node[above]{$y$}; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[->] (2.15,1.5) arc (0:310:0.3); + \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$u\wedge v = \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$}; +\end{tikzpicture} +\caption{Geometrische Interpretation des äusseren Produkt in $\mathbb{R}^2$\label{figure:wedge}} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/clifford/4_GeometrischesProdukt.tex b/buch/papers/clifford/4_GeometrischesProdukt.tex new file mode 100644 index 0000000..a19e983 --- /dev/null +++ b/buch/papers/clifford/4_GeometrischesProdukt.tex @@ -0,0 +1,59 @@ +\subsection{Geometrisches Produkt} +Die Multiplikation von zwei Vektoren nennt man in der Clifford Algebra das geometrische Produkt, dieses können wir nun als Summe aus dem Skalar- und dem äusseren Produkt darstellen +\begin{equation} + \textbf{u}\textbf{v} = \textbf{u}\cdot \textbf{v} + \textbf{u} \wedge \textbf{v}. +\end{equation} +Dieses Additionszeichen zwischen diesen zwei Produkten mag vielleicht ein wenig eigenartig wirken, da uns das Skalarprodukt ein Skalar und das äussere Produkt einen Bivektor zurück gibt. Was bedeutet es nun also diese beiden Elemente zu addieren? +Man kann sich die Addition wie bei den komplexen Zahlen vorstellen, wobei die imaginäre Einheit auch nicht explizit zu dem reelen Teil addiert werden kann, sondern die zwei Teile zusammen ein Objekt, eine komplexe Zahl bilden. +Dieses Objekt, also die Summe von verschiedenen Elemente der Clifford Algebra, wird Multivektor genannt. +\begin{definition} +Ein Multivektor besteht aus den verschiedenen Bauteilen, wie zum Beispiel Vektoren, Bivektoren oder Trivektoren (Volumen mit einer Richtung), der Clifford Algebra. +\begin{equation} + M = \sum \left ( \prod a_i\textbf{e}_j \right) +\end{equation} +\end{definition} +Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $\mathbb{G}^n$ beschrieben. +\begin{beispiel} +Allgemeiner Multivektor in $\mathbb{G}^3$ +\begin{equation} + M = a + + + \underbrace{b\textbf{e}_1 + c\textbf{e}_2 + d\textbf{e}_3}_{\text{Vektorteil}} + + + \underbrace{f\textbf{e}_1\textbf{e}_2 + g\textbf{e}_1\textbf{e}_3 + h\textbf{e}_2\textbf{e}_3 }_{\text{Bivektorteil}} + + + \underbrace{k\textbf{e}_1\textbf{e}_2\textbf{e}_3}_{\text{Trivektorteil}} +\end{equation} +\end{beispiel} +\begin{definition} +Um das Produkt von Basisvektoren in Zukunft darzustellen wird folgende Notation definiert + \begin{equation} + e_ie_j = e_{ij} + \end{equation} +\end{definition} +Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $\mathbb{G}^n$ erstellt werden. In \ref{tab:multip} ist dies für $\mathbb{G}^3$ gemacht. +\begin{table} + \caption{Multiplikationstabelle für $\mathbb{G^3}$} + \label{tab:multip} + \begin{center} + \begin{tabular}{ |c|c|c|c|c|c|c|c| } + \hline + 1 & $\textbf{e}_1$ & $\textbf{e}_2$ &$\textbf{e}_3$ & $\textbf{e}_{12}$ & $\textbf{e}_{13}$ & $\textbf{e}_{23}$ & $\textbf{e}_{123}$\\ + \hline + $\textbf{e}_1$ & 1 & $\textbf{e}_{12}$ & $\textbf{e}_{12}$ & $\textbf{e}_2$ & $\textbf{e}_3$ & $\textbf{e}_{123}$ & $\textbf{e}_{23}$\\ + \hline + $\textbf{e}_2$ & $-\textbf{e}_{12}$ & 1 & $\textbf{e}_{23}$ & $-\textbf{e}_1$ & $-\textbf{e}_{123}$ & $\textbf{e}_3$ & $-\textbf{e}_{13}$\\ + \hline + $\textbf{e}_3$ & $-\textbf{e}_{13}$ & $-\textbf{e}_{23}$ & 1 & $\textbf{e}_{123}$ & $-\textbf{e}_1$ & $-\textbf{e}_2$ & $\textbf{e}_{12}$\\ + \hline + $\textbf{e}_{12}$ & -$\textbf{e}_2$ & $\textbf{e}_1$& $\textbf{e}_{123}$ & -1 & $-\textbf{e}_{23}$ & $\textbf{e}_{13}$ & $-\textbf{e}_{3}$\\ + \hline + $\textbf{e}_{13}$ & $-\textbf{e}_{3}$ & $-\textbf{e}_{123}$ & $\textbf{e}_{1}$ & $\textbf{e}_{23}$ & -1 & $-\textbf{e}_{12}$ & $\textbf{e}_{2}$\\ + \hline + $\textbf{e}_{23}$ & $\textbf{e}_{123}$ & $-\textbf{e}_{3}$ & $\textbf{e}_{2}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & -1 & $-\textbf{e}_{1}$ \\ + \hline + $\textbf{e}_{123}$ & $\textbf{e}_{23}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & $-\textbf{e}_{3}$& $\textbf{e}_{2}$ & $-\textbf{e}_{1}$ & -1 \\ + \hline + \end{tabular} + \end{center} +\end{table} diff --git a/buch/papers/clifford/5_PolareDarstellung.tex b/buch/papers/clifford/5_PolareDarstellung.tex new file mode 100644 index 0000000..80fb49f --- /dev/null +++ b/buch/papers/clifford/5_PolareDarstellung.tex @@ -0,0 +1,29 @@ +\subsection{Polare Darstellung des geometrischen Produktes} +Beide Teile des geometrischen Produktes lassen sich durch trigonometrische Terme beschreiben. Das Skalarprodukt kann als +\begin{equation} + \textbf{u}\cdot \textbf{v} = |\textbf{u}||\textbf{v}|\cos{\alpha} +\end{equation} +beschrieben werden. Wobei $\alpha$ den Winkel zwischen den beiden Vektoren beschreibt. +\newline +Beim äusseren Produkt wurde bereits erwähnt, dass es aus dem Produkt der Fläche des von den zwei Vektoren aufgespannten Parallelogram und einer Umlaufrichtung beschrieben wird. Die Fläche eines Parallelograms lässt sich auch mit einen Sinus Term beschreiben +\begin{equation} + \textbf{u} \wedge \textbf{v} + = + \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}\textbf{e}_i\textbf{e}_j + = + \underbrace{|u||v|\sin{\alpha}}_{\text{Fläche}}\textbf{e}_i\textbf{e}_j +\end{equation} +Wobei die Fläche des Parallelogram auf der von $\textbf{e}_i$ und $\textbf{e}_j$ aufgespannten Ebene liegen.\newline +Nun kann man diese Terme wieder zum geometrischen Produkt vereinen +\begin{equation} + \textbf{u}\textbf{v} + = + |\textbf{u}||\textbf{v}|\cos{(\alpha)} + + + |\textbf{u}||\textbf{v}|\sin{(\alpha)} \textbf{e}_i\textbf{e}_j + = + |\textbf{u}||\textbf{v}|(\cos{(\alpha)} + \sin{(\alpha)}\textbf{e}_i\textbf{e}_j) +\end{equation}
\ No newline at end of file diff --git a/buch/papers/clifford/6_Dirac-Matrizen.tex b/buch/papers/clifford/6_Dirac-Matrizen.tex new file mode 100644 index 0000000..6417bb3 --- /dev/null +++ b/buch/papers/clifford/6_Dirac-Matrizen.tex @@ -0,0 +1,7 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Dirac-Matrizen} +\rhead{Dirac-Matrizen} diff --git a/buch/papers/clifford/7_Reflektion.tex b/buch/papers/clifford/7_Reflektion.tex new file mode 100644 index 0000000..d4942e0 --- /dev/null +++ b/buch/papers/clifford/7_Reflektion.tex @@ -0,0 +1,33 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Reflektion/ Spiegelung} +\rhead{Reflektion/ Spiegelung} +Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere, wie beispielsweise die später beschriebene Rotation, ableiten kann. Da die Geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Reflektion auch eine einfache, praktische Formulierung besitzen. \\HIER BILD +\subsection{linearen Algebra} +Aus der linearen Algebra ist bekannt, dass man eine Reflektion wie folgt beschreiben kann. +\begin{align} \label{RefLinAlg} + \mathbf{v^{'}} = \mathbf{v} - 2 \cdot \mathbf{v_{\perp u}} +\end{align} +Dabei stellt $\mathbf{u}$ die Spiegelachse dar. +Es scheint für diese Formel aber umständlich zu sein, weitere Reflektionen, mit weiteren Spiegelachsen, anzufügen. Man kann die Abbildung des Vektors auf den Reflektierten Vektor auch als Matrix schreiben, welche aus den Komponenten des zu der Spiegelachse orthonormalen Vektors $\mathbf{\hat{n}}$ besteht. +\\MATRIZEN O(2) und O(3) zeigen\\ +Diese Matrizen gehören der Matrizengruppe $O(n)$ an.... +\subsection{geometrischen Algebra} +Die Geometrische Algebra leitet aus der obigen Formel (\ref{RefLinAlg}) eine einfache und intuitive Form her, welche auch für weitere Operationen einfach erweitert werden kann. +\begin{align} + \mathbf{v'} = \mathbf{uvu^{-1}} +\end{align} +wobei die Inverse eines Vektors so definiert ist, dass multipliziert mit sich selbst das neutrale Element 1 ergibt. +\begin{align} + u^{-1} = \dfrac{u}{|u|^2} \Rightarrow uu^{-1} = 1 +\end{align} +verwendet man für $\mathbf{u}$ nur einen Einheitsvektor $\mathbf{\hat{u}}$, welcher die Länge 1 besitzt, wird somit die Formel reduziert zu einer beidseitigen Multiplikation von $\mathbf{\hat{u}}$. +\begin{align} + \mathbf{v'} = \mathbf{\hat{u}v\hat{u}} +\end{align} +Im Gegensatz zu den Abbildungen in der linearen Algebra, welche in jeder anderen Dimension durch andere Matrizen beschrieben werden müssen, ist es in der geometrischen Algebra immer der gleiche Vorgehensweise. +Zudem ist diese kompakte Schreibweise in der linearen Algebra nicht möglich, da keine Multiplikation von Vektoren definiert ist. +\\BEISPIEL?
\ No newline at end of file diff --git a/buch/papers/clifford/8_Rotation.tex b/buch/papers/clifford/8_Rotation.tex new file mode 100644 index 0000000..c2928bf --- /dev/null +++ b/buch/papers/clifford/8_Rotation.tex @@ -0,0 +1,100 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Rotation} +\rhead{Rotation} +Eine Rotation kann man aus zwei, aufeinanderfolgende Reflektionen bilden. Das war für mich zuerst eine verwirrende Aussage, da man aus den vorherig gezeigten Formeln annehmen könnte, dass die Reflektion schon für eine Drehung ausreicht. Obwohl sich die Längen, Winkel und Volumen sich bei einer Reflektion, wie bei einer Rotation, nicht ändert, sind sie doch verschieden, da die Orientierung bei der Reflektion invertiert wird. Stellt man sich beispielsweise ein Objekt in 3D vor und spiegelt dieses an einer Fläche, dann ist es unmöglich nur durch eine Rotation (egal an welchem Punkt) das ursprüngliche Objekt deckungsgleich auf das Gespiegelte zu drehen. Hingegen ist es wiederum möglich ein zweifach gespiegeltes Objekt durch eine Drehung zu erreichen. Das liegt daran, da die Orientierung zwei mal invertiert wurde. +\\BILD + +\subsection{linearen Algebra} +In der linearen Algebra haben wir Drehungen durch die Matrizen der Gruppe $SO(n)$ beschrieben. Die SO(2) werden beispielsweise auf diese Weise gebildet. +\begin{align} + D = + \begin{pmatrix} + cos(\alpha) & sin(\alpha) \\ + -sin(\alpha) & cos(\alpha) + \end{pmatrix} +\end{align} + +\subsection{geometrischen Algebra} +Da wir jetzt aus der Geometrie wissen, dass eine Rotation durch zwei Reflektionen gebildet werden kann, können wir die Rotation einfach herleiten. +\begin{align} \label{rotGA} + v'' = wv'w^{-1} = w(uvu^{-1})w^{-1} +\end{align} +Die Vektoren $\mathbf{w}$ und $\mathbf{u}$ bilden hier wiederum die Spiegelachsen. Diese versuchen wir jetzt noch zu verbessern. Dazu leiten wir zuerst die bekannte Polarform her. (Anmerkung: Hier wird eine Rotation auf der $\mathbf{e_{12}}$ Ebene hergeleitet. Weitere Drehungen können in höheren Dimensionen durch Linearkombinationen von Drehungen in den $\mathbf{e_{ij}}, i\not=j$ Ebenen erreicht werden) +\begin{align} + \mathbf{w} = |w| \left[\cos(\theta_w) e_1 + \sin(\theta_w) e_2\right] +\end{align} +Dabei können wir ausnützen, dass $e_1^2 = 1$ ist. Was nichts ändert wenn wir es einfügen. Zudem klammern wir dann $e_1$ aus. +\begin{align} + \mathbf{w} = |w| \left[\cos(\theta_w) e_1 + \sin(\theta_w) e_1e_1e_2\right] +\end{align} +\begin{align} \label{e1ausklammern} + \mathbf{w} = |w|e_1\left[\cos(\theta_w)+ \sin(\theta_w) e_{12}\right] +\end{align} +Durch die Reihenentwicklung ist es uns jetzt möglich den Term in eckigen Klammern mit der e-Funktion zu schreiben. +\begin{align} + \mathbf{w} = |w|\mathbf{e_1} e^{\theta_w \mathbf{e_{12}}} +\end{align} +Man kann es so interpretieren, dass der Einheitsvektor $e_1$ um die Länge w gestreckt und um $theta_w$ gedreht wird. +Nun werden wir den Effekt von zwei aneinandergereihten Vektoren $(wu)$ betrachten. +\begin{align} + \mathbf{wu} = |w|\mathbf{e_1} e^{\theta_w \mathbf{e_{12}}}||u||\mathbf{e_1} e^{\theta_u \mathbf{e_{12}}} +\end{align} +Um die beiden $\mathbf{e_1}$ zu kürzen, können wir die Reihenfolge des exponential Terms mit $\mathbf{e_1}$ wechseln, indem man bei der Gleichung (\ref{e1ausklammern}), anstatt mit $\mathbf{e_1e_1e_2}$ mit $\mathbf{e_2e_1e_1}$ erweitert. +\begin{align} + \mathbf{w} = |w|\left[\cos(\theta_w)+ \sin(\theta_w) \mathbf{e_2e_1}\right]\mathbf{e_1} +\end{align} +Da $\mathbf{e_2e_1 = -e_{12}}$ können wir einfach den Winkel negieren. +Jetzt können wir wieder $e_1e_1 = 1$ kürzen. Die Längen können als Skalare beliebig verschoben werden und die exponential Terme zusammengefasst werden. +\begin{align} + \mathbf{wu} = |w||u|e^{-\theta_w \mathbf{e_{12}}}\mathbf{e_1}\mathbf{e_1} e^{\theta_u \mathbf{e_{12}}} +\end{align} +\begin{align} + \mathbf{wu} = |w||u|e^{(\theta_u-\theta_w) \mathbf{e_{12}}} +\end{align} +der Term $\mathbf{u^{-1}w^{-1}}$ kann durch die selbe Methode zusammengefasst werden. +\begin{align} + \mathbf{u^{-1}w^{-1}} = \dfrac{1}{|w||u|}e^{(\theta_w-\theta_u) \mathbf{e_{12}}} +\end{align} +Dabei definieren wir den Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ als $\theta = \theta_w - \theta_u$. Setzten wir nun unsere neuen Erkenntnisse in die Gleichung (\ref{rotGA}) ein. +\begin{align} + \mathbf{v''} = |w||u|e^{-\theta \mathbf{e_{12}}} v \dfrac{1}{|w||u|}e^{\theta \mathbf{e_{12}}} +\end{align} +HIER DEFINITION/IST WICHTIGE FORMEL +\begin{align} + \mathbf{v''} = e^{-\theta \mathbf{e_{12}}} v e^{\theta \mathbf{e_{12}}} +\end{align} +Wir wissen nun, dass das diese beidseitige Multiplikation die Länge von $\mathbf{v}$ nicht verändert, da sich die Längen von $\mathbf{w}$ und $\mathbf{u}$ kürzen. Betrachten wir nun den Effekt der Exponentialterme auf $\mathbf{v}$. Dabei Teilen wir den Vektor $\mathbf{v}$ auf in einen Anteil $\mathbf{v_\parallel}$, welcher auf der Ebene $\mathbf{e_{12}}$ liegt, und einen Anteil $\mathbf{v_\perp}$, welcher senkrecht zu der Ebene steht. +\begin{align} \label{RotAufPerpPar} + \mathbf{v''} = e^{-\theta \mathbf{e_{12}}} (\mathbf{v_\perp + v_\parallel}) e^{\theta \mathbf{e_{12}}} +\end{align} +\begin{align} + \mathbf{v''} = e^{-\theta \mathbf{e_{12}}} \mathbf{v_\perp} e^{\theta \mathbf{e_{12}}} + e^{-\theta \mathbf{e_{12}}} \mathbf{v_\parallel} e^{\theta \mathbf{e_{12}}} +\end{align} +Auf eine allgemeine Herleitung wird hier zwar verzichtet, aber man kann zeigen, dass die Reihenfolge so vertauscht werden kann. Der Winkel wird dabei beim parallelen Term negiert. +\begin{align} + \mathbf{v''} = \mathbf{v_\perp} e^{-\theta \mathbf{e_{12}}} e^{\theta \mathbf{e_{12}}} + \mathbf{v_\parallel} e^{-(-\theta) \mathbf{e_{12}}} e^{\theta \mathbf{e_{12}}} +\end{align} +\begin{align} + \mathbf{v''} = \mathbf{v_\perp} + \mathbf{v_\parallel} e^{2\theta \mathbf{e_{12}}} +\end{align} +Man kann an dieser Gleichung sehen, dass nur der parallele Anteil des Vektors $\mathbf{v}$ auf der Ebene $\mathbf{e_{12}}$ um $2\theta$ gedreht wird. Der senkrechte Anteil bleibt gleich. Wichtig dabei zu sehen ist, dass nur der Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ von Bedeutung ist. Die Länge und Richtung der einzelnen Vektoren spielt keine Rolle. +\\BEISPIEL +\begin{align} + \begin{split} + &\mathbf{v} = 1\mathbf{e_1} + 2\mathbf{e_2} + 3\mathbf{e_3}\quad\Rightarrow\quad \mathbf{v_\parallel} = 1\mathbf{e_1} + 2\mathbf{e_2}; \quad \mathbf{v_\perp} = 3\mathbf{e_3}\\ &\mathbf{wu} = 1e^{(-\pi/2) \mathbf{e_{12}}} = 1[\cos(-\pi/2)\mathbf{e_1}+\sin(-\pi/2)\mathbf{e_2}] = -\mathbf{e_2}; \\ &\mathbf{u^{-1}w^{-1}} = 1e^{(\pi/2) \mathbf{e_{12}}} = \mathbf{e_2} + \end{split} +\end{align} +\begin{align} + \begin{split} + \mathbf{v''} = &\mathbf{(wu)v(u^{-1}w^{-1})} \\ + &-\mathbf{e_2} (1\mathbf{e_1} + 2\mathbf{e_2} + 3\mathbf{e_3}) \mathbf{e_2} \\ + & -1\mathbf{e_2e_1e_2} - 2\mathbf{e_2e_2e_2} - 3\mathbf{e_2e_3e_2} \\ + & 1\mathbf{e_2e_2e_1} - 2\mathbf{e_2} + 3\mathbf{e_2e_2e_3} \\ + & 1\mathbf{e_1} - 2\mathbf{e_2} + 3\mathbf{e_3} + \end{split} +\end{align} +Man sieht, dass sich der Vektor $\mathbf{v_\parallel}$ sich um $2\cdot90^\circ$ gedreht hat und der Vektor $\mathbf{v_\perp}$ unverändert blieb.
\ No newline at end of file diff --git a/buch/papers/clifford/9_KomplexeZahlen.tex b/buch/papers/clifford/9_KomplexeZahlen.tex new file mode 100644 index 0000000..4dbab2c --- /dev/null +++ b/buch/papers/clifford/9_KomplexeZahlen.tex @@ -0,0 +1,28 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{komplexe Zahlen} +\rhead{komplexe Zahlen} +Die komplexen Zahlen finden eine Vielzahl von Anwendungsgebiete in den Ingenieurwissenschaften. Das liegt daran, weil die komplexen Zahlen Rotationen und Schwingungen gut beschreiben können. Nachdem vorherigen Kapitel überrascht es wahrscheinlich nicht viele, dass es möglich ist Komplexe Zahlen in der geometrischen Algebra darzustellen. Sie können durch die geraden Grade der 2 Dimensionalen geometrischen Algebra vollständig beschrieben werden: $\mathbb{G}_2^+ \cong \mathbb{C}$. Das bedeutet eine komplexe Zahl kann durch ein Skalar (Grade 0) und einem Bivektor (Grade 2) dargestellt werden. Als Abkürzung nehme ich die Bezeichnung $g_n \in \mathbb{G}_2^+$. +\begin{align} + a_0 + a_1 j \cong a_0 + a_1 e_{12} = g_n;\quad a_0, a_1 \in \mathbb{R} +\end{align} +oder in Polarform. +\begin{align} + |r|e^{\theta j} \cong |r|e^{\theta e_{12}} = g_n; \quad r, \theta \in \mathbb{R} +\end{align} +Man beachte, dass wenn wir, wie bei den komplexen Zahlen, Elemente von $\mathbb{G}_2^+$ miteinander Multiplizieren, ist es nicht, wie im Kapitel Rotation bei der Formel (\ref{rotGA})beschrieben, eine Multiplikation von zwei $g_n$ mit einem Vektor. Im 2 dimensionalen bewirken beide Multiplikationen grundsätzlich das Gleiche (eine Drehstreckung), aber die Multiplikation von mehreren $g_n$ ist kommutativ, wie wir es von den komplexen zahlen kennen. +\begin{align} + \begin{split} + &(a + b \mathbf{e_{12}})(c + d \mathbf{e_{12}}) = (c + d \mathbf{e_{12}})(a + b \mathbf{e_{12}})\\ + &(a + b \mathbf{e_{12}})(x\mathbf{e_1}+y\mathbf{e_2})(c + d \mathbf{e_{12}}) \not= (a + b \mathbf{e_{12}})(c + d \mathbf{e_{12}})(x\mathbf{e_1}+y\mathbf{e_2}) + \end{split} +\end{align} +Um später die Auswirkung der Quaternionen besser zu verstehen, möchte ich kurz darauf eingehen, was ein $g_n$ für eine Auswirkung auf einen Vektor hat. +Wir kennen diesen Effekt schon von den komplexen Zahlen. Wenn eine komplexe Zahl $c_1=a+bj$ mit einer zweiten $c_2=c+dj$ multipliziert wird, dann kann man diese so aufteilen. +\begin{align} + c = (a + bj)(c + dj) = c\cdot(a+bj) + dj\cdot(a+bj) +\end{align} +Wobei $c\cdot(a+bj)$ die jetzige komplexe Zahl $c_1$ um den Faktor $c$ steckt und $dj\cdot(a+bj)$ die um 90° im gegenuhrzeigersinn gedrehte Zahl $c_1$ um den Faktor $d$ streckt. Diese Anteile addiert ergeben, dann den um $c_2$ drehgestreckten Vektor $c_1$. Die wirklichen Vorteile der geometrischen Algebra werden sich aber erst bei den Quaternionen zeigen. diff --git a/buch/papers/clifford/Makefile.inc b/buch/papers/clifford/Makefile.inc index 7b941b3..8cdd02e 100644 --- a/buch/papers/clifford/Makefile.inc +++ b/buch/papers/clifford/Makefile.inc @@ -3,12 +3,18 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-clifford = \ +dependencies-clifford = \ papers/clifford/packages.tex \ papers/clifford/main.tex \ - papers/clifford/references.bib \ - papers/clifford/teil0.tex \ - papers/clifford/teil1.tex \ - papers/clifford/teil2.tex \ - papers/clifford/teil3.tex - + papers/clifford/references.bib \ + papers/clifford/0_ElevatorPitch.tex \ + papers/clifford/1_Vektordarstellung.tex \ + papers/clifford/2_QuadratVektoren.tex \ + papers/clifford/3_MultiplikationVektoren.tex \ + papers/clifford/4_GeometrischesProdukt.tex \ + papers/clifford/5_PolareDarstellung.tex \ + papers/clifford/6_Dirac-Matrizen.tex \ + papers/clifford/7_Reflektion.tex \ + papers/clifford/8_Rotation.tex \ + papers/clifford/9_KomplexeZahlen.tex \ + papers/clifford/10_Quaternionen.tex diff --git a/buch/papers/clifford/main.tex b/buch/papers/clifford/main.tex index 5533c55..46d04bd 100644 --- a/buch/papers/clifford/main.tex +++ b/buch/papers/clifford/main.tex @@ -3,34 +3,23 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:clifford}} -\lhead{Thema} +\chapter{Clifford Algebra\label{chapter:clifford}} +\lhead{Clifford Algebra} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Thierry Schwaller, Marius Baumann} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} -\input{papers/clifford/teil0.tex} -\input{papers/clifford/teil1.tex} -\input{papers/clifford/teil2.tex} -\input{papers/clifford/teil3.tex} +\input{papers/clifford/0_ElevatorPitch.tex} +\input{papers/clifford/1_Vektordarstellung.tex} +\input{papers/clifford/2_QuadratVektoren.tex} +\input{papers/clifford/3_MultiplikationVektoren.tex} +\input{papers/clifford/4_GeometrischesProdukt.tex} +\input{papers/clifford/5_PolareDarstellung.tex} +\input{papers/clifford/6_Dirac-Matrizen.tex} +\input{papers/clifford/7_Reflektion.tex} +\input{papers/clifford/8_Rotation.tex} +\input{papers/clifford/9_KomplexeZahlen.tex} +\input{papers/clifford/10_Quaternionen.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/clifford/packages.tex b/buch/papers/clifford/packages.tex index 8abcef1..8fb4bd9 100644 --- a/buch/papers/clifford/packages.tex +++ b/buch/papers/clifford/packages.tex @@ -7,4 +7,3 @@ % if your paper needs special packages, add package commands as in the % following example %\usepackage{packagename} - diff --git a/buch/papers/clifford/papers/clifford/teil0.tex b/buch/papers/clifford/papers/clifford/teil0.tex new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/clifford/papers/clifford/teil0.tex diff --git a/buch/papers/clifford/teil0.tex b/buch/papers/clifford/teil0.tex deleted file mode 100644 index ac943f4..0000000 --- a/buch/papers/clifford/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{clifford:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{clifford:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/clifford/teil1.tex b/buch/papers/clifford/teil1.tex deleted file mode 100644 index 0674afb..0000000 --- a/buch/papers/clifford/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{clifford:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{clifford:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{clifford:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{clifford:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{clifford:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/clifford/teil2.tex b/buch/papers/clifford/teil2.tex deleted file mode 100644 index bbcefb0..0000000 --- a/buch/papers/clifford/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{clifford:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{clifford:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/clifford/teil3.tex b/buch/papers/clifford/teil3.tex deleted file mode 100644 index f50d42d..0000000 --- a/buch/papers/clifford/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{clifford:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{clifford:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/ifs/images/farn.eps b/buch/papers/ifs/images/farn.eps new file mode 100644 index 0000000..597745b --- /dev/null +++ b/buch/papers/ifs/images/farn.eps @@ -0,0 +1,2372 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. Version 9.7.0.1434023 \(R2019b\) Update 6. 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Undefined control sequence. +l.6 \section + {Teil 0\label{ifs:section:teil0}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.7 \rhead + {Teil 0} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! 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Undefined control sequence. +<write> ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.10 erat, sed diam voluptua \cite{ifs:bibtex} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `ifs:bibtex' on page undefined on input line 10. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! 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+Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--14 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 6--14 +[] + [] + +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 15--21 +[] + [] + +) (./teil1.tex +! Undefined control sequence. +l.6 \section + {Teil 1 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 1 in font nullfont! +! Undefined control sequence. +l.8 \rhead + {Problemstellung} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! 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Undefined control sequence. +l.34 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. +l.40 animi, id est laborum et dolorum fuga \eqref + {000tempmlate:equation1}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no : in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 1 in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 34--41 +[] + [] + +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +! Undefined control sequence. +<write> ...`ifs:section:loesung' on page \thepage + \space undefined\on@line . +l.43 \ref{ifs:section:loesung} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `ifs:section:loesung' on page undefined on input line + 43. + +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! 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Undefined control sequence. +<write> ...fs:section:folgerung' on page \thepage + \space undefined\on@line . +l.47 \ref{ifs:section:folgerung} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `ifs:section:folgerung' on page undefined on input li +ne 47. + +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! 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+Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 42--54 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 42--54 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 42--54 +[] + [] + +) (./teil2.tex +! Undefined control sequence. +l.6 \section + {Teil 2 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 2 in font nullfont! +! Undefined control sequence. +l.8 \rhead + {Teil 2} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--23 +[] + [] + +! Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 24--39 +[] + [] + +) (./teil3.tex +! Undefined control sequence. +l.6 \section + {Teil 3 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 3 in font nullfont! +! Undefined control sequence. +l.8 \rhead + {Teil 3} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--23 +[] + [] + +! Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! 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Emergency stop. +<*> main.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 37 strings out of 492609 + 427 string characters out of 6132818 + 62613 words of memory out of 5000000 + 4043 multiletter control sequences out of 15000+600000 + 4116 words of font info for 16 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 20i,3n,22p,104b,180s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/ifs/main.tex b/buch/papers/ifs/main.tex index 8d70951..cceaf87 100644 --- a/buch/papers/ifs/main.tex +++ b/buch/papers/ifs/main.tex @@ -3,34 +3,16 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:ifs}} -\lhead{Thema} +\chapter{Iterierte Funktionsschemata\label{chapter:ifs}} +\lhead{Iterierte Funktionschemata und ihre Anwendungen} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +\chapterauthor{Alain Keller} \input{papers/ifs/teil0.tex} \input{papers/ifs/teil1.tex} \input{papers/ifs/teil2.tex} \input{papers/ifs/teil3.tex} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/ifs/references.bib b/buch/papers/ifs/references.bib index 716857f..fbf75f4 100644 --- a/buch/papers/ifs/references.bib +++ b/buch/papers/ifs/references.bib @@ -13,14 +13,29 @@ day = {6} } -@book{ifs:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} +@online{ifs:chaos, + title = {Chaosspiel}, + url = {https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem#Chaosspiel}, + date = {20201-06-13}, + year = {2021}, + month = {6}, + day = {13} +} + +@online{ifs:barnsleyfern, + title = {Barnsley fern}, + url = {https://en.wikipedia.org/wiki/Barnsley_fern}, + date = {20201-06-13}, + year = {2021}, + month = {6}, + day = {13} +} +@book{ifs:fractal-geometry, + title = {Fractal Geometry}, + author = {Kenneth Falconer}, + publisher = {John Wiley \& Sons}, + year = {1900}, + isbn = {0-471-92287-0}, } @article{ifs:mendezmueller, @@ -33,3 +48,20 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@Inbook{ifs:Rousseau2012, + author= {Rousseau, Christiane + and Saint-Aubin, Yvan + and Stern, Manfred}, + title={Bildkompression: Iterierte Funktionensysteme}, + bookTitle={Mathematik und Technologie}, + year={2012}, + publisher={Springer Berlin Heidelberg}, + address={Berlin, Heidelberg}, + pages={341--386}, + abstract={Dieses Kapitel kann in ein bis zwei Wochen Vorlesungen behandelt werden. Steht nur eine Woche zur Verfügung, dann können Sie kurz die Einführung behandeln (Abschnitt 11.1) und anschlie{\ss}end ausf{\"u}hrlich den Begriff des Attraktors eines iterierten Funktionensystems betrachten (Abschnitt 11.3), wobei Sie sich auf das Sierpi{\'{n}}ski- Dreieck (Beispiel 11.5) konzentrieren. Beweisen Sie den Satz {\"u}ber die Konstruktion von affinen Transformationen, die drei Punkte der Ebene auf drei Punkte der Ebene abbilden und diskutieren Sie die speziellen affinen Transformationen, die h{\"a}ufig bei iterierten Funktionensystemen verwendet werden (Abschnitt 11.2).}, + isbn={978-3-642-30092-9}, + doi={10.1007/978-3-642-30092-9_11}, + url={https://doi.org/10.1007/978-3-642-30092-9_11} +} + + diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex index b605bfe..7cb218f 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -4,19 +4,9 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Teil 0\label{ifs:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{ifs:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\rhead{Was ist ein Iteriertes Funktionsschema} +Mit der Hilfe von Iterierten Funktionsschemata mit nur wenigen Funktionen, komplexe Bilder beschreiben. +In der Regel sind diese Bilder Fraktale. +Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir in diesem Kapitel untersuchen. diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index c824cb4..54089ec 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -3,53 +3,92 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 +\section{Fraktale \label{ifs:section:teil1}} \rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{ifs:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an. -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? -\subsection{De finibus bonorum et malorum -\label{ifs:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. +Über die genaue Definition von Fraktalen sind sich die Mathematiker nicht einig. +In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry \cite{ifs:fractal-geometry} beschreibt. +Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: +\begin{enumerate} + \item $F$ hat eine unendlich feine Struktur + \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. + \item Oftmals haf $F$ eine Form von Selbstähnlichkeit. + \item Die 'fraktale Dimension' ist grösser als die Topologische Dimension + \item Viele Fraktale lassen sich einfach beschrieben +\end{enumerate} +\subsection{Koch Kurve + \label{ifs:subsection:lilkoch}} +Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen. +In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht sie aus lauter kleineren Kopien von sich selber. +Den Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt. +Gestartet wird mit einer einzelnen Strecke der Länge $a$. +Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. +In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. +Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt. +Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. -Et harum quidem rerum facilis est et expedita distinctio -\ref{ifs:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{ifs:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\begin{figure} + \centering + \includegraphics{papers/ifs/images/koch8} + \caption{Koch Kurve} + \label{ifs:kochkurve8} +\end{figure} + +\begin{figure} + \centering + \subfigure[]{ + \label{ifs:kochconsta} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch0}} + \subfigure[]{ + \label{ifs:kochconstb} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch1}} + \subfigure[]{ + \label{kochconstc} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch2}} + \caption{(a) Start (b) 1. Iteration (c) 2. Iteration} + \label{ifs:kochconst} +\end{figure} + +Die resultierende Kurve hat ein paar interessante Eigenschaften. +Die Länge der Kurve lasst sich einfach berechnen. +\begin{align*} + l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad ... , \quad + l_n = a * \left( \frac{4}{3}\right)^n \quad + \Rightarrow \quad + \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty +\end{align*} +In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Somit divergiert die Länge gegen Unendlich. +Die Fläche unter der Kurve lässt sich folgendermassen berechnen +\begin{align*} + A_0 = 0 , \quad A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ + A_2 = A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\ + A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1 +\end{align*} +Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist. +Daraus resultiert eine konvergierende Geometrische Reihe. +\begin{align*} + A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\ + \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2 +\end{align*} +Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendlichem Umfang. +Zu guter Letzt bestimmen wir die Dimension der Kurve. +Es gibt viele verschiedene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. +Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten. +In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. +\begin{align*} + D = - \frac{log(N)}{log(\epsilon)} +\end{align*} +Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen. +Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge. +Somit hat das Dreieck die Dimension $D = 2$. +Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon = 1/3$. +\begin{align*} + D = - \frac{log(N)}{log(\epsilon)} = - \frac{log(4)}{log(1/3)} \approx 1.2619 +\end{align*} +Wie wir nun sehen besitzt die Kochkurve alle oben beschriebenen Eigenschaften von Fraktalen. +Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten. diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index bfd1684..143317a 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -3,38 +3,208 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 +\section{Fraktale mit IFS \label{ifs:section:teil2}} \rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale machen kann. +Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. +\begin{figure} + \centering + \includegraphics[width=0.5\textwidth]{papers/ifs/images/sierpinski} + \caption{Sierpinski-Dreieck} + \label{ifs:sierpinski10} +\end{figure} +Wenn man das Dreieck genau anschaut, erkennt man schnell, dass es aus drei kleineren Kopien seiner selbst besteht. +Es ist also ein Selbstähnliches Konstrukt. +Diese Eigenschaft wollen wir uns zunutze machen. -\subsection{De finibus bonorum et malorum + +Wir definieren das Dreieck mit Kantenlänge 1 als Menge $X$. +Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet +\begin{align*} + f_1(x,y) + = + \begin{pmatrix} + \frac{1}{2} & 0 \\ + 0 & \frac{1}{2} \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + ,\quad + f_2(x,y) + = + \begin{pmatrix} + \frac{1}{2} & 0 \\ + 0 & \frac{1}{2} \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + \frac{1}{2} \\ + 0 + \end{pmatrix} + , \quad + f_3(x,y) + = + \begin{pmatrix} + \frac{1}{2} & 0 \\ + 0 & \frac{1}{2} \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + \frac{1}{4} \\ + \frac{1}{2} + \end{pmatrix}\\ +\end{align*} +$f_1$ bildet das Dreieck auf das Teilstück unten links ab, $f_2$ auf das Teilstück unten rechts und $f_3$ auf das obere Teilstück. +Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an, entsteht also wieder ein Sierpinski-Dreieck. +\begin{align*} + X = \bigcup\limits_{i = 1}^{3} f_i(X) +\end{align*} +Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergiert die Menge gegen das Sierpinski-Dreieck. +\begin{figure} + \centering + \subfigure[]{ + \label{ifs:sierpconsta} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski1}} + \subfigure[]{ + \label{ifs:sierpconstb} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski2}} + \subfigure[]{ + \label{ifs:sierpconstc} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski3}} + \subfigure[]{ + \label{ifs:sierpconstd} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski6}} + \caption{Konstruktion eines Sierpinski-Dreiecks mit einem Schwarzen Quadrat als Start\\ + (a) 1. Iteration (b) 2. Iteration (c) 3. Iteration (d) 5. Iteration} + \label{ifs:sierpconst} +\end{figure} +Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird. +Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Iterationen gegen null. + +\subsection{Iterierte Funktionensysteme \label{ifs:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +In diesem Unterkapitel wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. + + +$S_1,...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt +\begin{align} + |S_i(x) - S_i(y)| \leq c_i|x - y| +\end{align} +für jedes i mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge $F$ für die gilt +\begin{equation} + F = \bigcup\limits_{i = 1}^{m} S_i(F) +\end{equation} +Weiter definieren wir die Transformation S auf kompakte Mengen ohne die leere Menge. +\begin{equation} + S(E) = \bigcup\limits_{i = 1}^m S_i(E) +\end{equation} +Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt +\begin{equation} + F = \bigcap\limits_{k = 1}^{\infty} S^k(E). +\end{equation} +In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. +Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet. +Auf den Beweis wird verzichtet. +\subsection{Beispiel: Barnsley-Farn} +Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein weiteres Fraktal, welches mit einem IFS generiert werden kann. +Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben. +\begin{align*} + {S_1(x,y)} + = + \begin{pmatrix} + 0 & 0 \\ + 0 & 0.16 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix}, \quad + {S_2(x,y)} + = + \begin{pmatrix} + 0.85 & 0.04 \\ + -0.04 & 0.85 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + 0 \\ + 1.6 + \end{pmatrix}\\ + {S_3(x,y)} + = + \begin{pmatrix} + 0.2 & -0.26 \\ + 0.23 & 0.22 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + 0 \\ + 1.6 + \end{pmatrix}, \quad + {S_4(x,y)} + = + \begin{pmatrix} + -0.15 & 0.28 \\ + 0.26 & 0.24 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + 0 \\ + 0.44 + \end{pmatrix}\\ +\end{align*} +In der Abbildung \ref{ifs:farncolor} sehen wir die vier Transformationen farblich dargestellt. +$S_1$ erstellt den Stiel des Farnblattes (rot). +Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab. +$S_2$ (grün) erstellt den Hauptteil des Farnes. +Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$. +$S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. +$S_4$ Spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. +Wir führen im Zusammenhang mit dem Barnsley-Farn \cite{ifs:barnsleyfern} noch eine weitere Methode ein, um IFS auszuführen. +Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden. +Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist diese Methode ziemlich rechenintensiv. +Eine Alternative ist das Chaosspiel \cite{ifs:chaos}. +Bei dieser Methode werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt. +Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein. +Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet. +Da, wie wir beim Barnsley-Farn gut sehen, dass nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet. +Die Gewichtung erfolgt über den Anteil der Gesamtmasse. +Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. +\begin{figure} + \centering + \makebox[\textwidth][c]{ + \includegraphics[width=1.4\textwidth]{papers/ifs/images/farn}} + \caption{Barnsley-Farn} + \label{ifs:farn} +\end{figure} +\begin{figure} + \centering + \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor} + \caption{Vier Transformationen des Barnsley-Farn} + \label{ifs:farncolor} +\end{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 23fabbc..24f0751 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -3,38 +3,141 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 +\section{Fraktale Bildkomprimierung \label{ifs:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum +\rhead{Fraktale Bildkomprimierung} +Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren. +Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat. +Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. +In diesem Unterkapitel wollen wir eine Methode dafür anschauen.\cite{ifs:Rousseau2012} + + +Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. +Dies muss jedoch nicht so sein. +Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. +Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. +Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen. +Es ist wohl nicht falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. +Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben? + +\subsection{das Kompressionsverfahren \label{ifs:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +In der Beschreibung des Verfahrens wird sich auf Graustufenbilder bezogen. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. + +In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$ +Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. +Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist. + +\subsubsection{Finden des ähnlichsten $D_j$} +Zuerst brauchen wir die Transformation um ein Element aus $D$ auf ein Element von $R$ Abzubilden. +\begin{align*} + T(x,y,z) = + \begin{pmatrix} + a & b & 0 \\ + c & d & 0 \\ + 0 & 0 & s + \end{pmatrix} + \begin{pmatrix} + x \\ + y \\ + z + \end{pmatrix} + + + \begin{pmatrix} + \alpha \\ + \beta \\ + g + \end{pmatrix} +\end{align*} +Diese Transformation bildet den Pixel $P$ auf Koordinate $(x,y)$ und Graustufe $z$ auf den Pixel $P'$ ab. + +Da wir mit Pixeln arbeiten, sind die Transformationen in der Ebene Beschränkt. +Diese wird durch die Parameter $a,b,c$ und $d$ bestimmt. +Mögliche Transformationen sind auf folgende Liste Beschränkt: +\begin{itemize} + \item Identische Transformation, keine Änderung + \item Drehung um 90, 180 oder 270 Grad. + \item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen. +\end{itemize} +$\alpha$ und $\beta$ verschieben den Pixel an die richtige Stelle. +Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $G_j$ um $1/2$ skalieren. +Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. +Skaliert und transformiert erhalten wir $\tilde{D_j}$ + +Die Parameter $s$ und $g$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Töne auf die richtige Helligkeit. +$s$ und $g$ werden mit der linearen Regression ermittelt. +\begin{align*} + z' = sz + g \\ + f(\tilde{D_j}) \text{, Funktion um das Bild eins Blockes zu erhalten} \\ + s = \frac{cov(f(R_i), f(\tilde{D_j}))}{var(\tilde{D_j})} \\ + g = E(f(R_i)) - s E(f(\tilde{D_j})) +\end{align*} +Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt. +Um zu beurteilen ob der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block $R_i$ genügend ähnlich ist, berechnet man den quadratischen Abstand $e$. +\begin{align*} + e = d(f(R_i), f(T(D_j))) +\end{align*} +Dieser Abstand sollte so klein wie möglich sein. +Die beste Kombination von $D_j$ und $T_i$ ist also diese, welche den kleinsten Abstand zum Block $R_i$ hat, und somit am ähnlichsten ist. + +Am Ende des Verfahrens haben wir also für jeden $R_i$ einen passenden $D_i$ mit der zugehörigen Abbildung $T_i$ gefunden. + +\subsubsection{Rekonstruktion des Bildes} +Mit den Gefundenen Abbildungen lässt sich das Bild generieren. +Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge. +In unserem Fall ist dieses ein Bild $f_0$ derselben Grösse. +Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$. +Dies wird verkürzt als Operator $W$ geschrieben. +So erhalten wir ein neues Bild $f_1 = W(f_0)$. +Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied feststellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht. + +\subsubsection{Farbbilder} +Dieses Verfahren mit Graustufenbilder lässt sich ganz einfach auf Farbbilder erweitern. +Jeder Pixel eines Farbbildes besteht aus einem Rot, Grün und Blauwert (RGB). +Teilt man ein Bild in die drei Farbkanäle auf, das heisst, es wird nur noch ein Farbwert benutzt, erhält man drei Bilder, welche wie ein Graustufenbild sind. +Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt nach der Rekonstruktion die Kanäle wieder zusammen. + +\subsubsection{Performance des Verfahren} +Dieser Grundalgorithmus der Fraktalen Bildkompression ist offensichtlich recht langsam und skaliert auch schlecht mit grösseren Bilder. +Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das jpeg verfahren. + +\subsection{Beispiel} +Kommen wir nun zu einem Beispiel. +Wir Verwenden dafür den oben beschriebenen Algorithmus. +Die Range-Blöcke wurden $4\times4$ gewählt und die Dommain dementsprechend $8\times8$. +Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Blöcke gebraucht. +Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}. +Der Algorithmus liefert uns für jeden Range-Block die benötigten Parameter. +Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren. +Als Startbild wird ein mittelgraues 360x360px Bild gewählt, Abbildung \ref{ifs:bild0}. +Nun lassen wir das IFS laufen. +Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar. +Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. +\begin{figure} + \centering + \includegraphics[width=0.4\textwidth]{papers/ifs/images/original} + \caption{Original Bild von Rapperswil} + \label{ifs:original} +\end{figure} +\begin{figure} + \centering + \includegraphics[width=0.4\textwidth]{papers/ifs/images/rapperswil} + \caption{Startbild} + \label{ifs:bild0} +\end{figure} +\begin{figure} + \centering + \subfigure[]{ + \label{ifs:rappirecoa} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil01}} + \subfigure[]{ + \label{ifs:rappirecob} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil001}} + \subfigure[]{ + \label{ifs:rappirecoc} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil04}} + \caption{(a) 1. Iteration (b) 2. Iteration (c) 5. Iteration} + \label{ifs:rappireco} +\end{figure} diff --git a/buch/papers/reedsolomon/.gitignor b/buch/papers/reedsolomon/.gitignor new file mode 100644 index 0000000..52a02ac --- /dev/null +++ b/buch/papers/reedsolomon/.gitignor @@ -0,0 +1,24 @@ +RS.aux +RS.bbl +RS.bib +RS.blg +RS.idx +RS.ilg +RS.ind +RS.log +RS.out +RS.pdf +RS.run.xml +RS.toc +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb diff --git a/buch/papers/reedsolomon/RS presentation/README.txt b/buch/papers/reedsolomon/RS presentation/README.txt new file mode 100644 index 0000000..4d0620f --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/README.txt @@ -0,0 +1 @@ +Dies ist die Presentation des Reed-Solomon-Code
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+ %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Reed-Solomon-Code:} + \begin{itemize} + \visible<1->{\item Für Übertragung von Daten} + \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} + \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- +\section{Polynom Ansatz} + \begin{frame} + \begin{itemize} + \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} + \end{frame} + \begin{frame} + \frametitle{Beispiel} + Übertragen von + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + + + Versende $ (p(1),p(2),\dots,p(7))$ + \visible<2->{ = (\textcolor{green}{8},} + \only<2>{\textcolor{green}{15},} + \only<3>{\textcolor{red}{50},} + \only<2>{\textcolor{green}{26},} + \only<3>{\textcolor{red}{37},} + \visible<2->{\textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})} + \only<2>{\includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<3>{\includegraphics[scale = 1.2]{images/polynom2.pdf}} + \visible<3>{ + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ +\visible<1->{3}& +\visible<1->{3}& +\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ + &&\\ +\visible<1->{$k$} & +\visible<1->{$t$} & +\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ + \hline + &&\\ + &&\\ + \multicolumn{3}{l} { + \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + } + \end{tabular} + \end{center} + \end{frame} + +%------------------------------------------------------------------------------- + +\section{Diskrete Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + \visible<1->{ + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]} + \visible<2->{ + \item Ersetzte + \[ + w = e^{-\frac{2\pi j}{N} k} + \]} + \visible<3->{ + \item Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \]} + \end{itemize} + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = \frac{1}{N} + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] + \end{frame} +%------------------------------------------------------------------------------- + + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \visible<2>{ + \newline + Indem in einem endlichen Körper gerechnet wird. + } + \end{frame} + +%------------------------------------------------------------------------------- + + +\section{Reed-Solomon in Endlichen Körpern} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \onslide<1->{\item Warum endliche Körper?} + + \onslide<2->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} + + \onslide<3->{\qquad digitale Fehlerkorrektur} + + %\onslide<4->{\qquad bessere Laufzeit} + + \vspace{10pt} + + \onslide<4->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} + + \vspace{10pt} + + \onslide<5->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} + + \onslide<6->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \onslide<1->{\item endlicher Körper $q = 11$} + + \onslide<2->{ist eine Primzahl} + + \onslide<3->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} + + \vspace{10pt} + + \onslide<4->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen} + + \onslide<5->{$n = q - 1 = 10$ Zahlen} + + \vspace{10pt} + + \onslide<6->{\item Max.~Fehler $t = 2$} + + \onslide<7->{maximale Anzahl von Fehler, die wir noch korrigieren können} + + \vspace{10pt} + + \onslide<8->{\item Nutzlast $k = n -2t = 6$ Zahlen} + + \onslide<9->{Fehlerkorrkturstellen $2t = 4$ Zahlen} + + \onslide<10->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} + + \onslide<11->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \onslide<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} + + \vspace{10pt} + + \onslide<2->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} + + \vspace{10pt} + + \onslide<3->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken} + + \onslide<4->{$\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} + + \vspace{10pt} + + \onslide<5->{\item Wir wählen $a = 8$} + + \onslide<6->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} + + \onslide<7->{$8$ ist eine primitive Einheitswurzel} + + \vspace{10pt} + + \onslide<8->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} + + \onslide<9->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \onslide<1->{\item Übertragungsvektor $v$} + + \onslide<2->{\item $v = A \cdot m$} + + \end{itemize} + + \[ + \onslide<3->{ + v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + } + \] + + \begin{itemize} + \onslide<4->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \onslide<1->{\item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$} + + \vspace{10pt} + + \onslide<2->{\item Wir suchen die Inverse der Matrix $A$} + + \vspace{10pt} + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.55\textwidth} + \onslide<3->{ Inverse der Fouriertransformation} + \vspace{10pt} + \onslide<4->{ + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + } + \vspace{10pt} + \onslide<5->{ + \[ + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + } + \end{column} + \begin{column}{0.45\textwidth} + \onslide<6->{Inverse von $a$} + + \vspace{10pt} + + \onslide<7->{ + \[ + 8^{1} \Rightarrow 8^{-1} + \] + } + + \onslide<8->{Inverse finden wir über den Eulkidischen Algorithmus} + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + \onslide<1->{ + \begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor<2->{blue}{$-4$}& \textcolor<2->{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + } + + \vspace{10pt} + + \begin{tabular}{rcl} + \onslide<3->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ + \onslide<4->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ + \onslide<5->{$8^{-1}$ &$=$& $7$} + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Inverser Matrix} + + \begin{itemize} + \onslide<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} + + \onslide<2->{\item $m = 1/10 \cdot A^{-1} \cdot v$} + + \onslide<3->{\item $m = 10 \cdot A^{-1} \cdot v$} + + \end{itemize} + \onslide<4->{ + \[ + m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + } + + \begin{itemize} + \onslide<5->{\item $m = [0,0,0,0,4,7,2,5,8,1]$} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \onslide<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} + + \onslide<2->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + + \onslide<3->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} + + \end{itemize} + + \onslide<4->{Wie finden wir die Fehler?} + + \begin{itemize} + \onslide<5->{\item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + + \onslide<6->{\item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$} + + %\only<7->{\item $e(X) = r(X) - m(X)$} + + \onslide<7->{\item $e(X) = r(X) - m(X)$} + + \end{itemize} + + \begin{center} + \onslide<8->{ + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& \onslide<9->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ + $m(a^{i})$& \onslide<10->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ + $e(a^{i})$& \onslide<11->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ + \hline + \end{tabular} + } + \end{center} + + + \begin{itemize} + \onslide<12->{\item Alle Stellen, die nicht Null sind, sind Fehler} + \end{itemize} + + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} + + \vspace{10pt} + + \onslide<2->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} + + \vspace{10pt} + + \onslide<3->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} + + \vspace{10pt} + + \onslide<4->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} + + \vspace{10pt} + + \onslide<5->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} + + \vspace{10pt} + + \onslide<6->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} + + \vspace{10pt} + + \onslide<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$} + + \vspace{10pt} + + \onslide<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} + + \vspace{10pt} + + \onslide<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} + + \vspace{10pt} + + \onslide<1->{\item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können} + + \vspace{10pt} + + \onslide<2->{$\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$} + + \onslide<3->{$= d(X) \cdot e(X)$} + + \vspace{10pt} + + \onslide<4->{\item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Kennen wir $e(X)$?} + + \begin{itemize} + + \onslide<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} + + \vspace{10pt} + + \onslide<2->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} + + \vspace{10pt} + + \onslide<3->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat} + + \vspace{10pt} + + \onslide<4->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} + + \vspace{10pt} + + \onslide<5->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} + + \vspace{10pt} + + \onslide<6->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + \onslide<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} + \onslide<2->{ + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + } + \onslide<3->{ + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + } + \vspace{10pt} + + \onslide<4->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} + + \vspace{10pt} + + \onslide<5->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor<2->{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + \onslide<3->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} + \onslide<4->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} + \onslide<5->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} + \end{tabular} + + \vspace{10pt} + + \onslide<6->{ + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + } + + \onslide<7->{$d(X) = (X-a^3)(X-a^8)$} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \onslide<1->{\item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + + \onslide<2->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} + + \end{itemize} + \onslide<3->{ + \[ + \textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor<4->{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor<4->{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor<4->{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor<4->{red}{1} \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^3}& \textcolor<4->{red}{8^6}& \textcolor<4->{red}{8^9}& \textcolor<4->{red}{8^{12}}& \textcolor<4->{red}{8^{15}}& \textcolor<4->{red}{8^{18}}& \textcolor<4->{red}{8^{21}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^8}& \textcolor<4->{red}{8^{16}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{32}}& \textcolor<4->{red}{8^{40}}& \textcolor<4->{red}{8^{48}}& \textcolor<4->{red}{8^{56}}& \textcolor<4->{red}{8^{64}}& \textcolor<4->{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + } + + \begin{itemize} + \onslide<5->{\item Fehlerstellen entfernen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<4->{green}{8^6}& \textcolor<4->{green}{8^7}& \textcolor<4->{green}{8^8}& \textcolor<4->{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<4->{green}{8^{12}}& \textcolor<4->{green}{8^{14}}& \textcolor<4->{green}{8^{16}}& \textcolor<4->{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<4->{green}{8^{24}}& \textcolor<4->{green}{8^{28}}& \textcolor<4->{green}{8^{32}}& \textcolor<4->{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<4->{green}{8^{30}}& \textcolor<4->{green}{8^{35}}& \textcolor<4->{green}{8^{40}}& \textcolor<4->{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<4->{green}{8^{36}}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{48}}& \textcolor<4->{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{49}}& \textcolor<4->{green}{8^{56}}& \textcolor<4->{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<4->{green}{8^{54}}& \textcolor<4->{green}{8^{63}}& \textcolor<4->{green}{8^{72}}& \textcolor<4->{green}{8^{81}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\ + \end{pmatrix} + \] + + \begin{itemize} + \onslide<3->{\item Nullstellen entfernen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<3->{red}{7} \\ \textcolor<3->{red}{4} \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^7}& \textcolor<3->{red}{8^{14}}& \textcolor<3->{red}{8^{21}}& \textcolor<3->{red}{8^{28}}& \textcolor<3->{red}{8^{35}}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^9}& \textcolor<3->{red}{8^{18}}& \textcolor<3->{red}{8^{27}}& \textcolor<3->{red}{8^{36}}& \textcolor<3->{red}{8^{45}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \onslide<2->{\item Matrix in eine Quadratische Form bringen} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \onslide<2->{\item Matrix Invertieren} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + \onslide<2->{$\Downarrow$} + \end{center} + \[ + \onslide<3->{ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + } + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \onslide<2->{\item $m = [4,7,2,5,8,1]$} + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + +\end{document} diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc new file mode 100644 index 0000000..095b5e6 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -0,0 +1,9 @@ +\babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{5}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{13}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{27}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{29}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{31}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{36}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{43}{0}{8} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.aux 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presentation/RS_handout.tex @@ -0,0 +1,907 @@ +\documentclass[11pt,aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[ngerman]{babel} +\usepackage{tikz} +\usetheme{Hannover} + +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Reed-Solomon-Code:} + \begin{itemize} + \item Für Übertragung von Daten + \item Ermöglicht Korrektur von Übertragungsfehler + \item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc. + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- +\section{Polynom Ansatz} + \begin{frame} + \begin{itemize} + \item $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} + \frametitle{Beispiel} + Übertragen von + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + \newline + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern. + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3& 3& 9 Werte eines Polynoms vom Grad 2 \\ + &&\\ + $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ + \hline + &&\\ + &&\\ + \multicolumn{3}{l} { + Ausserdem können bis zu $2t$ Fehler erkannt werden! + } + \end{tabular} + \end{center} + \end{frame} + +%------------------------------------------------------------------------------- + +\section{Diskrete Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \] + + \item Ersetzte + \[ + w = e^{-\frac{2\pi j}{N} k} + \] + + \item Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] + \end{itemize} + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = \frac{1}{N} + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] + \end{frame} +%------------------------------------------------------------------------------- + + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \newline + Indem in einem endlichen Körper gerechnet wird. + + \end{frame} + +%------------------------------------------------------------------------------- + + +\section{Reed-Solomon in Endlichen Körpern} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \item Warum endliche Körper? + + \qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler + + \qquad digitale Fehlerkorrektur + + %\onslide<4->{\qquad bessere Laufzeit} + + \vspace{10pt} + + \item Nachricht = Nutzdaten + Fehlerkorrekturteil + + \vspace{10pt} + + \item aus Fehlerkorrekturteil die Fehlerstellen finden + + \qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \item endlicher Körper $q = 11$ + + ist eine Primzahl + + beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen + + $n = q - 1 = 10$ Zahlen + + \vspace{10pt} + + \item Max.~Fehler $t = 2$ + + maximale Anzahl von Fehler, die wir noch korrigieren können + + \vspace{10pt} + + \item Nutzlast $k = n -2t = 6$ Zahlen + + Fehlerkorrkturstellen $2t = 4$ Zahlen + + Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$ + + als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation + + \vspace{10pt} + + \item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$ + + \vspace{10pt} + + \item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken + + $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$ + + \vspace{10pt} + + \item Wir wählen $a = 8$ + + $\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$ + + $8$ ist eine primitive Einheitswurzel + + \vspace{10pt} + + \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + + $\Rightarrow$ \qquad können wir auch als Matrix schreiben + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Übertragungsvektor $v$ + + \item $v = A \cdot m$ + + \end{itemize} + + \[ + v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$ + + \vspace{10pt} + + \item Wir suchen die Inverse der Matrix $A$ + + \vspace{10pt} + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.55\textwidth} + Inverse der Fouriertransformation + \vspace{10pt} + + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + + \vspace{10pt} + + \[ + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + + \end{column} + \begin{column}{0.45\textwidth} + Inverse von $a$ + + \vspace{10pt} + + \[ + 8^{1} \Rightarrow 8^{-1} + \] + + Inverse finden wir über den Eulkidischen Algorithmus + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + + \begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + + + \vspace{10pt} + + \begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Inverser Matrix} + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item $m = 1/10 \cdot A^{-1} \cdot v$ + + \item $m = 10 \cdot A^{-1} \cdot v$ + + \end{itemize} + + \[ + m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$ + + \end{itemize} + + Wie finden wir die Fehler? + + \begin{itemize} + \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + + \item $e(X) = r(X) - m(X)$ + + \end{itemize} + + \begin{center} + + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline + \end{tabular} + + \end{center} + + \begin{itemize} + \item Alle Stellen, die nicht Null sind, sind Fehler + \end{itemize} + + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + + \vspace{10pt} + + $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + + $= d(X) \cdot e(X)$ + + \vspace{10pt} + + \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Kennen wir $e(X)$?} + + \begin{itemize} + + \item $e(X)$ ist unbekannt auf der Empfängerseite + + \vspace{10pt} + + \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + + \vspace{10pt} + + \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + + \vspace{10pt} + + \item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + + \vspace{10pt} + + \item $f(X) = X^{10} - 1 = X^{10} + 10$ + + \vspace{10pt} + + \item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + $\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$ + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + + \vspace{10pt} + + $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ + Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ + Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \end{tabular} + + \vspace{10pt} + + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + + + $d(X) = (X-a^3)(X-a^8)$ + + \end{frame} +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item $d(X) = (X-\textcolor{red}{a^3})(X-\textcolor{red}{a^8})$ + + \end{itemize} + + \[ + \textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Fehlerstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{green}{8^6}& \textcolor{green}{8^7}& \textcolor{green}{8^8}& \textcolor{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Nullstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix in eine Quadratische Form bringen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix Invertieren + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + $\Downarrow$ + \end{center} + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + +\end{document} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.toc b/buch/papers/reedsolomon/RS presentation/RS_handout.toc new file mode 100644 index 0000000..ce1bdc2 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.toc @@ -0,0 +1,9 @@ +\babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{5}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{16}{0}{4} +\beamer@sectionintoc {5}{Codierung eines 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index 0000000..578f2a3 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1.tex b/buch/papers/reedsolomon/RS presentation/images/polynom1.tex new file mode 100644 index 0000000..db83daa --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1.tex @@ -0,0 +1,59 @@ +% polynome1 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\teiler}{40} +\begin{document} + %Übertragen von den Zahlen + %\textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} + %als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline + %Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + % \textcolor{green}{15}, \textcolor{green}{26}, + % \textcolor{green}{ 41}, 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Dabei müssen wir die \textcolor{red}{Definition 4.6 (wie verweist man auf eine definition?)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. +Wir legen für unser Beispiel den endlichen Körper mit $q = 11$ fest. +Zur Hilfestellung können dazu die beiden Tabellen \ref{reedsolomon:subsection:adtab} und +\ref{reedsolomon:subsection:mptab} hinzugezogen werden. Diese Tabellen enthalten sämtliche Resultate aller gültigen Operationen \textcolor{red}{(Notiz: nach meinem Wissen gibt es ja nur addition und multiplikation als gültige operationen)}, die in diesem Körper durchgeführt werden können. +Aus der Definition der Endlichen Körper (ersichtlich auch in den Tabellen) folgt, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur verfügung stehen und somit $11 = 0$ gelten muss. + +% OLD TEXT +%Alle folgenden Berechnungen wurden mit den beiden Restetabellen \ref{reedsolomon:subsection:adtab} und \ref{reedsolomon:subsection:mptab} durchgeführt. +%Aus den Tabellen folgt auch, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen. + +% die beiden Restetabellen von F_11 +%\input{papers/reedsolomon/restetabelle1} +%\input{papers/reedsolomon/restetabelle2} + +Anhand der Menge uns zur Verfügung stehenden Zahlen wird auch festgelegt, wie viele Zahlen ein Nachrichtenblock $n$, bestehend aus Nutzdatenteil und Fehlerkorrekturteil, umfassen kann. +Der Nachrichtenblock im Beispiel besteht aus +\[ +n = q - 1 = 10 \text{ Zahlen}, +\] +wobei die null weggelassen wird. Wenn wir versuchen würden, mit der null zu codieren, so stellen wir fest, dass wir wieder null an der gleichen Stelle erhalten und somit wäre die Codierung nicht eindeutig. + +% Notes +%Da bei allen Codes, die codiert werden wird an der gleichen Stelle eine Nullstelle auftreten. + +% Old Text +%Die grösse des endlichen Körpers legt auch fest, wie gross unsere Nachricht $n$ bestehend aus Nutzdatenteil und Fehlerkorrekturteil sein kann und beträgt in unserem Beispiel +%\[ +%n = q - 1 = 10 \text{ Zahlen}. +%\] + +Im nächsten Schritt bestimmen wir, wie viele Fehler $t$ maximal während der Übertragung auftreten dürfen, damit wir sie noch korrigieren können. +Unser Beispielcode sollte in der Lage sein +\[ +t = 2 +\] +Fehlerstellen korrigieren zu können. + +Die Grösse des Nutzdatenteils hängt von der Grösse des Nachrichtenblocks sowie der Anzahl der Fehlerkorrekturstellen ab. Je robuster der Code sein muss, desto weniger Platz für Nutzdaten $k$ bleibt in der Nachricht übrig. +Bei maximal 2 Fehler können wir noch +\[ +k = n - 2t = 6\text{ Zahlen} +\] +übertragen. + +Zusammenfassend haben wir einen Nachrichtenblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten zu rekonstruieren. Zudem werden wir im weiteren feststellen, dass dieser Code maximal vier Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. + +Wir legen nun die Nachricht +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Stellen für die Fehlerkorrektur zuständig sind. +Solange diese Stellen vor dem Codieren und nach dem Decodieren den Wert null haben, so ist die Nachricht Fehlerfrei übertragen worden. + +Da wir in den folgenden Abschnitten mit Polynomen arbeiten, stellen wir die Nachicht auch noch als Polynom +\[ +m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 +\] +dar. + +% Old Text +%Die Nachricht können wir auch als Polynom +%\[ +%m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 +%\] +%darstellen. + +\subsection{Der Ansatz der diskreten Fouriertransformation + \label{reedsolomon:subsection:diskFT}} + +In einem vorherigen Kapitel \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. +Wir legen deshalb die Zahl $a$ fest. Diese Zahl soll die gleichen aufgaben haben, wie $e^{\frac{j}{2 \pi}}$ in der Diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ existiert. Dazu soll $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken, um +\[ +\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} +\] +in +\[ +\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. +\] +umzuschreiben. +% Old Text +%Wir suchen also eine Zahl $a$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann. +%Dazu schreiben wir +%\[ +%\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} +%\] +%um in +%\[ +%\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. +%\] +% +%Wenn wir alle möglichen Werte für $a$ einsetzen, also +%\begin{align} +%a = 0 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\} \\ +%a = 1 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\} \\ +%a = 2 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\} \\ +%a = 3 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\} \\ +%a = 4 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\} \\ +%a = 5 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\} \\ +%a = 6 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\} \\ +%a = 7 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\} \\ +%a = 8 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\} \\ +%a = 9 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\} \\ +%a = 10 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\} +%\end{align} + +\subsubsection{Die primitiven Einheitswurzeln + \label{reedsolomon:subsection:primsqrt}} + +Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen +\begin{center} +\begin{tabular}{c r c l} +%$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\ +$a = 1 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ \\ +$a = 2 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ \\ +$a = 3 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ \\ +$a = 4 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ \\ +$a = 5 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ \\ +$a = 6 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ \\ +$a = 7 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ \\ +$a = 8 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ \\ +$a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ \\ +$a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ +\end{tabular} +\end{center} +so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. +Wenden wir diese Vorgehensweise auch für andere Endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a^i = 8$. + +\subsubsection{Bildung einer Transformationsmatrix + \label{reedsolomon:subsection:transMat}} + +Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu Codieren. Daraus sollen wir dann einen Übertragungsvektor $v$ erhalten, den wir an den Empfänger schicken können. Für die Codierung müssen wir alle $a^i$ in das Polynom $m(X)$ einsetzen. Da wir $a^i = 8^i$ gewählt haben ergibt sich daraus +% +%Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. +% +\begin{center} + \begin{tabular}{c} + $m(8^0) = 4 \cdot 1^5 + 7 \cdot 1^4 + 2 \cdot 1^3 + 5 \cdot 1^2 + 8 \cdot 1^1 + 1 = 5$ \\ + $m(8^1) = 4 \cdot 8^5 + 7 \cdot 8^4 + 2 \cdot 8^3 + 5 \cdot 8^2 + 8 \cdot 8^1 + 1 = 3$ \\ + \vdots \\ + $m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$ + \end{tabular} +\end{center} +unser Übertragungsvektor. Um das ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen und bildet so unsere Transformationsmatrix $A$. + +\subsection{Allgemeine Codierung + \label{reedsolomon:subsection:algCod}} + +Für die Codierung benötigen wir die Nachricht $m$, die Codiert werden soll sowie die Transformationsmatrix $A$. Daraus erhalten wir den Übertragungsvektor $v$. Setzen wir die Zahlen aus dem Beispiel ein erhalten wir folgende Darstellung. +\[ +v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ +\end{pmatrix} +\] +Für unseren Übertragungsvektor resultiert +\[ +v = [5,3,6,5,2,10,2,7,10,4], +\] +den wir jetzt über einen beliebigen Nachrichtenkanal versenden können. diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex new file mode 100644 index 0000000..feaa027 --- /dev/null +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -0,0 +1,312 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Decodierung: Ansatz mit Fehlerkorrektur +\label{reedsolomon:section:decmitfehler}} +\rhead{fehlerhafte rekonstruktion} +Bisher haben wir die Decodierung unter der Bedingung durchgeführt, dass der Übertragungsvektor fehlerlos versendet und empfangen wurde. +In der realen Welt müssen wir uns jedoch damit abfinden, dass kein Übertragungskanal garantiert fehlerfrei ist und das wir früher oder später mit Fehlern rechnen müssen. +Genau für dieses Problem wurden Fehler korrigierende Codes, wie der Reed-Solomon-Code, entwickelt. +In diesem Abschnitt betrachten wir somit die Idee der Fehlerkorrektur und wie wir diese auf unser Beispiel anwenden können. +Der Übertragungskanal im Beispiel weisst jetzt den Fehlervektor +\[ +u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] +\] +auf. + +Senden wir jetzt unser Übertragungsvektor $v$ durch diesen Kanal addiert sich der Fehlervektor $u$ auf unsere Übertragung und wir erhalten +\begin{center} + + \begin{tabular}{c | c r } + $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ + $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ + \hline + $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ + \end{tabular} + + % alternative design + %\begin{tabular}{c | c cccccccccccc } + % $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ + % $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ + % \hline + % $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ + %\end{tabular} + +\end{center} +als neuen, fehlerbehafteten Übertragungsvektor $w$ auf der Empfängerseite. +% Old Text +%In diesem Abschnitt gehen wir genauer darauf ein, wie der Reed-Solomon-Code eine solche Feherkorrektur vornimt. +% +%In diesem Abschnitt betrachten wir das Problem, dass während der Übertragung des Übertragungsvektors von unserem Beispiel +% +% +%Zu diesem Zweck wurden Fehler korrigierende Codes entwickelt. +% +%Dieser Optimalfall kann jedoch mit keinem Übertragungskanal garantiert werden +% +% +%Im zweiten Teil zur Decodierung betrachten wir den Fall, dass unser Übertragungskanal nicht fehlerfrei ist. +%Wir legen daher den Fehlervektor +%\[ +%u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] +%\] +%fest, den wir zu unserem Übertragungsvektor als Fehler dazu addieren und somit +% +%\begin{center} +% +%\begin{tabular}{c | c r } +% $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ +% $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ +% \hline +% $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ +%\end{tabular} +% +%% alternative design +%%\begin{tabular}{c | c cccccccccccc } +%% $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ +%% $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ +%% \hline +%% $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ +%%\end{tabular} +% +%\end{center} +%als Übertragungsvektor auf der Empfängerseite erhalten. +Wir jetzt als Empfänger wissen jedoch nicht, dass der erhaltene Übertragungsvektor jetzt fehlerbehaftet ist und werden dementsprechend den Ansatz aus Abschnitt \ref{reedsolomon:section:decohnefehler} anwenden. +Wir stellen jedoch recht schnell fest, dass am decodierten Nachrichtenblock +\[ +r = [\underbrace{5,7,4,10,}_{\text{Syndrom}}5,4,5,7,6,7]. +\] +etwas nicht in Ordnung ist, denn die vorderen vier Fehlerkorrekturstellen haben nicht mehr den Wert null. +Der Nachrichtenblock weisst jetzt ein \em Syndrom \em auf, welches anzeigt, dass der Übertragungsvektor fehlerhaft empfangen wurde. +% Old Text +%Wenn wir den Übertragungsvektor jetzt Rücktransformieren wie im vorherigen Kapitel erhalten wir +%\[ +%r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]. +%\] +Jetzt stellt sich natürlich die Frage, wie wir daraus den ursprünglich gesendeten Nachrichtenvektor zurückerhalten sollen. Laut der Definition über die Funktionsweise eines Reed-Solomon-Codes können wir aus den Fehlerkorrekturstellen ein ``Lokatorpolynom'' berechnen, welches die Information enthält, welche stellen innerhalb des empfangenen Übertragungsvektors fehlerhaft sind. + +\subsection{Das Fehlerstellenpolynom $d(X)$ + \label{reedsolomon:subsection:fehlerpolynom}} +Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Möglichkeit finden, die Fehlerhaften von den Korrekten Stellen im Übertragungsvektor unterscheiden zu können. In einem ersten Versuch könnten wir $d$ berechnen mit +\begin{center} +\begin{tabular}{r c l} + $m(X)$ & $=$ & $4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \\ + $r(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ \\ + $d(X)$ & $=$ & $r(X) - m(X)$. +\end{tabular} +\end{center} +Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, wir werden werden deshalb erst in Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} darauf zurückkommen. + +Setzen wir jetzt noch unsere Einheitswurzel aus dem Beispiel ein so erhalten wir +% Old Text +%\begin{align} +% m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\ +% r(X) & = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7 \\ +% e(X) & = r(X) - m(X). +%\end{align} +%Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir +\begin{center} +\begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $d(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline +\end{tabular} +\end{center} +und damit die Information, dass allen Stellen, die nicht Null sind, Fehler enthalten. +Aus der Tabelle lesen wir, das in unserem Beispiel die Fehler an der Stelle drei und acht zu finden sind. + +Für das einfache Bestimmen von Hand mag dies ja noch ausreichen, jedoch können wir mit diesen Stellen nicht das Lokatorpolynom bestimmen, denn dafür bräuchten wir alle Nullstellen, an denen es Fehler gegeben hat (also sozusagen genau das umgekehrte). Um dies zu erreichen wenden wir eine andere Herangehensweise und nehmen uns den Satz von Fermat sowie den kleinsten gemeinsamen Teiler zur Hilfe. + +\subsection{Mit dem grössten gemeinsamen Teiler auf Nullstellenjagd +\label{reedsolomon:subsection:ggT}} + +Zuerst betrachten wir mal den Satz von Fermat deren Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass für +\[ +f(X) = X^{q-1} -1 = 0 +\] +wobei dies für jedes $q$ gilt. Setzen wir also das $q$ von unserem Beispiel ein +\[ +f(X) = X^{10}-1 = 0 \qquad \text{für } X = \{1,2,3,4,5,6,7,8,9,10\} +\] +und stellen dies als Nullstellenform (\textcolor{red}{richtiger name für die Schreibweise?}) dar. So ergibt sich die Darstellung +\[ +f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9). +\] +Zur Überprüfung können wir unsere Einheitswurzel in $a$ einsetzen und werden sehen, dass wir für $f(X) = 0$ erhalten werden. + +Wir können jetzt auch $d(X)$ nach der gleichen Überlegung darstellen als +\[ +d(X) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) \cdot p(x), +\] +wobei diese Darstellung nicht mehr alle Nullstellen umfasst wie es noch in $f(X)$ der Fall war. +Dies liegt daran, dass wir ja zwei Fehlerstellen (grau markiert) haben, die nicht Null sind. Diese fassen wir zum Restpolynom $p(X)$ (\textcolor{red}{eventuell farblich kennzeichnen?}) zusammen. +Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $d(X)$ berechnen, so erhalten wir mit +\[ +\operatorname{ggT}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) +\] +eine Liste von Nullstellen, an denen es keine Fehler gegeben hat. +Dies scheint zuerst nicht sehr hilfreich zu sein, da wir für das Lokatorpolynom ja eine Liste der Nullstellen suchen, an denen es Fehler gegeben hat. Aus diesem Grund berechnen wir im nächsten Schritt das kleinste gemeinsame Vielfache von $f(X)$ und $d(X)$. + +%Wir werden auch feststellen, das unsere Bemühungen bisher nicht umsonst waren. + +\subsection{Mit dem kgV fehlerhafte Nullstellen finden + \label{reedsolomon:subsection:kgV}} + +Das kgV hat nämlich die Eigenschaft sämtliche Nullstellen zu finden, also nicht nur die fehlerhaften sondern auch die korrekten, was in +\[ +\operatorname{kgV}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9) \cdot q(X). +\] +ersichtlich ist. +Aus dem vorherigen Abschnitt wissen wir auch, dass $d(X)$ alle korrekten Nullstellen beinhaltet. Teilen wir das kgV jetzt auf in +\[ +\operatorname{kgV}(f(X),d(X)) = d(X) \cdot l(X) +\] +sollten wir für $l(X)$ eine Liste mit allen fehlerhaften Nullstellen erhalten. +Somit ist +\[ +l(X) = (X-a^3)(X-a^8) +\] +unser gesuchtes Lokatorpolynom. +Es scheint so als müssten wir nur noch an den besagten Stellen den Übertragungsvektor korrigieren und wir währen fertig mit der Fehlerkorrektur. +Jedoch haben wir noch ein grundlegendes Problem, dass zu beginn aufgetaucht ist, wir aber beiseite geschoben haben. Die Rede ist natürlich vom Nachrichtenvektor $m(X)$, mit dem wir in erster Linie das wichtige Fehlerstellenpolynom $d(X)$ berechnet haben. + +\subsection{Der problematische Nachrichtenvektor $m(X)$ + \label{reedsolomon:subsection:nachrichtenvektor}} + +In Abschnitt \ref{reedsolomon:section:decmitfehler} haben wir +\[ +d(X) = r(X) - m(X) +\] +in Abhängigkeit von $m(X)$ berechnet. +Jedoch haben wir ausser acht gelassen, dass $m(X)$ auf der Empfängerseite nicht existiert und somit gänzlich unbekannt ist. +Es scheint so als würde dieser Lösungsansatz, den wir bisher verfolgt haben, nicht funktioniert. +Wir könnten uns höchstens noch fragen, ob wir tatsächlich nichts über den Nachrichtenvektor im Beispiel wissen. Wenn wir noch einmal den Vektor betrachten als +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +fällt uns aber auf, dass wir doch etwas über diesen Vektor wissen, nämlich den Wert der ersten 2t (im Beispiel vier) stellen. +Im Normalfall sollen diese nämlich den Wert null betragen und somit sind nur die letzten k stellen (im Beispiel sechs) für uns unbekannt, dargestellt als +\[ +m = [0,0,0,0,?,?,?,?,?,?]. +\] +Wie der Zufall es so will liegt an diesen vier Stellen auch die Information, wo die Fehlerstellen liegen. Daher reicht es auch aus +% darum werden die stellen auch als fehlerkorrekturstellen bezeichnet +\[ +d(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +\] +so zu berechnen, dass wir die wichtigen vier Stellen kennen, der Rest des Polynoms jedoch im unbekannten Restpolynom $p(X)$ enthalten ist. + +\textcolor{red}{ist das wechseln zwischen 2t,k aus dem allgemeinfall und vier,sechs aus dem beispiel zu verwirrend?} + +\subsection{Die Berechnung der Fehlerstellen + \label{reedsolomon:subsection:nachrichtenvektor}} + +Um die Fehlerstellen zu berechnen wenden wir die gleiche Vorgehensweise wie zuvor an, also zuerst den ggT, danach berechnen wir das kgV um am Ende das Lokatorpolynom zu erhalten. + +\subsubsection{Schritt 1: ggT} + +Wir berechnen den ggT von $f(X)$ und $d(X)$ mit +\begin{center} +\begin{tabular}{r c l} + $f(X)$ & $=$ & $X^{10} - 1 = X^{10} + 10$ \\ + $d(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ +\end{tabular} +\end{center} +% +% +% +%Das einzige Problem was jetzt noch bleibt ist, dass wir $e(X)$ berechnet haben aus +%\[ +%e(X) = r(X) - m(X), +%\] +%wobei $m(X)$ auf der Empfängerseite unbekannt ist. +%Es sieht danach aus, das wir diesen Lösungsansatz nicht verwenden können, da uns ein entscheidender Teil fehlt. +%Bei einer näheren Betrachtung von $m(X)$ fällt uns aber auf, dass wir doch etwas über $m(X)$ wissen. +%Wir kennen nämlich die ersten vier Stellen, da diese für die Fehlerkorrektur zuständig sind und daher Null sein müssen. +%\[ +%m = [0,0,0,0,?,?,?,?,?,?] +%\] +%An genau diesen Stellen liegt auch die Information, wo unsere Fehlerstellen liegen, was uns ermöglicht, den Teil von $e(X)$ zu berechnen, der uns auch interessiert. +% +%Wir können $e(X)$ also bestimmen als +%\[ +%e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +%\] +%wobei $p(X)$ wiederum ein unbekanntes Restpolynom ist und +%\[ +%f(X) = X^{10} - 1 = X^{10} + 10 +%\] +%ist können wir so in einer ersten Instanz den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen. +%Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berechnen so +% +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ +\end{array} +\] + +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ +\end{array} +\] +und erhalten +\[ +\operatorname{ggT}(f(X),e(X)) = 6X^8. +\] + +\subsubsection{Schritt 2: kgV} + +Mit dem Resultat das wir vom ggT erhalten haben können wir jetzt das kgV berechnen. Dazu können wir jetzt den erweiterten Euklidischen Algorithmus verwenden, den wir in Abschnitt \ref{buch:subsection:daskgv} kennengelernt haben. +% +%Mit den Resultaten, die wir vom Rechenweg des grössten gemeinsamen Teiler erhalten haben können wir jetzt auch das kleinste Gemeinsame Vielfache berechnen. Eine detailliertere Vorgehensweise findet man in Kapitel ???. +% +%Aus diesem erweiterten Euklidischen Algorithmus erhalten wir +\begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + +\end{center} +Daraus erhalten wir die Faktoren +\[ +l(X) = 2X^2 + 5 \qquad \rightarrow \qquad l(X) = 2(X-5)(X-6). +\] +Unser gesuchtes Lokatorpolynom hat also die Form +\[ +l(X) = (X-a^i)(X-a^j). +\] +Also brauchen wir nur noch $i$ und $j$ zu berechnen und wir haben unsere gesuchten Fehlerstellen. +Diese bekommen wir recht einfach mit +\begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^j = 6 \qquad \Rightarrow \qquad j = 8$. +\end{center} +Schlussendlich erhalten wir +\[ +d(X) = (X-a^3)(X-a^8) +\] +als unser Lokatorpolynom mit den fehlerhaften Stellen. diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex new file mode 100644 index 0000000..3b709f3 --- /dev/null +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -0,0 +1,116 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Decodierung: Ansatz ohne Fehler +\label{reedsolomon:section:decohnefehler}} +\rhead{fehlerlose rekonstruktion} + +In diesem Abschnitt betrachten wie die Überlegung, wie wir auf der Empfängerseite die Nachricht aus dem empfangenen Übertragungsvektor erhalten. Nach einer einfachen Überlegung müssen wir den Übertragungsvektor decodieren, was auf den ersten Blick nicht allzu kompliziert sein sollte, solange wir davon ausgehen können, dass es während der Übertragung keine Fehler gegeben hat. Wir betrachten deshalb den Übertragungskanal als fehlerfrei. + +Der Übertragungsvektor empfangen wir also als +\[ +v = [5,3,6,5,2,10,2,7,10,4]. +\] +% Old Text +%Im ersten Teil zur Decodierung des Übertragungsvektor betrachten wir den Übertragungskanal als fehlerfrei. +%Wir erhalten also unseren Übertragungsvektor +%\[ +%v = [5,3,6,5,2,10,2,7,10,4]. +%\] +Nach einem banalen Ansatz ist die Decodierung die Inverse der Codierung. Dank der Matrixschreibweise lässt sich dies relativ einfach umsetzen. +% Old Text +%Gesucht ist nun einen Weg, mit dem wir auf unseren Nachrichtenvektor zurückrechnen können. +%Ein banaler Ansatz ist das Invertieren der Glechung +\[ +v = A \cdot m \qquad \Rightarrow \qquad m = A^{-1} \cdot v +\] +Nur stellt sich jetzt die Frage, wie wir die Inverse von $A$ berechnen. +Dazu können wir wiederum den Ansatz der Fouriertransformation uns zur Hilfe nehmen, +jedoch betrachten wir jetzt deren Inverse. +Definiert ist sie als +\[ +F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \qquad \Rightarrow \qquad \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega. +\] +Damit beschäftigen wir uns im Abschnitt \ref{reedsolomon:subsection:algdec} weiter, konkret suchen wir momentan aber eine Inverse für unsere primitive Einheitswurzel $a$. +\[ +8^1 \qquad \rightarrow \qquad 8^{-1} +\] +Mit einem solchen Problem haben wir uns bereits in Abschnitt \ref{buch:section:euklid} befasst und so den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. + +% Old Text +%Im Abschnitt \textcolor{red}{4.1} haben wir den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. + +\subsection{Inverse der primitiven Einheitswurzel +\label{reedsolomon:subsection:invEinh}} + +Die Funktionsweise des euklidischen Algorithmus ist im Kapitel \ref{buch:section:euklid} ausführlich beschrieben. +Für unsere Anwendung wählen wir die Parameter $a = 8$ und $b = 11$ ($\mathbb{F}_{11}$). +Daraus erhalten wir + +\begin{center} + +\begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline +\end{tabular} + +\end{center} +\begin{center} + +\begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + +\end{tabular} + +\end{center} +als Inverse der primitiven Einheitswurzel. Die inverse Transformationsmatrix $A^{-1}$ bilden wir indem wir jetzt die inverse primitive Einheitswurzel anstelle der primitiven Einheitswurzel in die Matrix einsetzen. + +\subsection{Allgemeine Decodierung + \label{reedsolomon:subsection:algdec}} + +Wir haben jetzt fast alles für eine erfolgreiche Rücktransformation beisammen. Wir haben aber noch nicht alle Aspekte der inversen diskreten Fouriertransformation befolgt, so fehlt uns noch einen Vorfaktor +\[ +m = \textcolor{red}{s} \cdot A^{-1} \cdot v +\] +den wir noch bestimmen müssen. +Glücklicherweise lässt der sich analog wie bei der inversen diskreten Fouriertransformation bestimmen und beträgt +\[ +s = \frac{1}{10}. +\] +Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. Somit lässt sich der Nachrichtenvektor einfach bestimmen mit +\[ +m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ +\end{pmatrix} +\] +und wir erhalten +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +als unsere Nachricht zurück.
\ No newline at end of file diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex new file mode 100644 index 0000000..146067a --- /dev/null +++ b/buch/papers/reedsolomon/endlichekoerper.tex @@ -0,0 +1,23 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Reed-Solomon in Endlichen Körpern +\label{reedsolomon:section:endlichekoerper}} +\rhead{Problemstellung} + +\textcolor{red}{TODO: (warten auf den 1. Teil)} + +Das Rechnen in endlichen Körpern bietet einige Vorteile: + +\begin{itemize} + \item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten. + + \item Digitale Fehlerkorrektur: lässt sich nur in endlichen Körpern umsetzen. + +\end{itemize} + +Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen. + +Nun stellt sich die Frage, wie wir eine fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes ``Lokatorpolynom'' generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. diff --git a/buch/papers/reedsolomon/hilfstabellen.tex b/buch/papers/reedsolomon/hilfstabellen.tex new file mode 100644 index 0000000..4e39de5 --- /dev/null +++ b/buch/papers/reedsolomon/hilfstabellen.tex @@ -0,0 +1,19 @@ +% +% hilfstabellen.tex +% Autor: Michael Steiner +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{$\mathbb{F}_{11}$ Hilfstabellen + \label{reedsolomon:section:hilfstabellen}} +\rhead{Hilfstabellen} + +Um das rechnen zu erleichtern findet man in diesem Abschnitt die Resultate, die bei der Addition und der Multiplikation in $\mathbb{F}_{11}$ resultieren. + +\subsection{Additionstabelle + \label{reedsolomon:subsection:adtab}} +\input{papers/reedsolomon/restetabelle1.tex} + +\subsection{Multiplikationstabelle + \label{reedsolomon:subsection:mptab}} +\input{papers/reedsolomon/restetabelle2.tex}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index 8219b63..fa20936 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -3,7 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:reedsolomon}} +\chapter{Reed-Solomon-Code\label{chapter:reedsolomon}} \lhead{Thema} \begin{refsection} \chapterauthor{Joshua Bär und Michael Steiner} @@ -27,10 +27,25 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. \end{itemize} +% Joshua \input{papers/reedsolomon/teil0.tex} \input{papers/reedsolomon/teil1.tex} \input{papers/reedsolomon/teil2.tex} \input{papers/reedsolomon/teil3.tex} +% Michael +\input{papers/reedsolomon/endlichekoerper} +\input{papers/reedsolomon/codebsp} +\input{papers/reedsolomon/decohnefehler} +\input{papers/reedsolomon/decmitfehler} +\input{papers/reedsolomon/rekonstruktion} +\input{papers/reedsolomon/nachschlagewerk} +\input{papers/reedsolomon/hilfstabellen} +%\input{papers/reedsolomon/anwendungen} -> geplant + +\nocite{reedsolomon:weitz} +\nocite{reedsolomon:informationkommunikation} +%\nocite{reedsolomon:mendezmueller} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/reedsolomon/nachschlagewerk.tex b/buch/papers/reedsolomon/nachschlagewerk.tex new file mode 100644 index 0000000..60b857e --- /dev/null +++ b/buch/papers/reedsolomon/nachschlagewerk.tex @@ -0,0 +1,4 @@ +\section{Nachschlagewerk + \label{reedsolomon:section:nachschlagen}} +\rhead{nachschlagewerk} +todo: auflistung von z.b nachrichtenvektor, übertragungsvektor usw. inklusiver erklärung was es ist falls man beim lesen den faden verliert
\ No newline at end of file diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib index 38613bd..731bd35 100644 --- a/buch/papers/reedsolomon/references.bib +++ b/buch/papers/reedsolomon/references.bib @@ -4,32 +4,22 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{reedsolomon:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@online{reedsolomon:weitz, + title = {Fehlerkorrektur mit Reed-Solomon-Codes}, + url = {https://youtu.be/uOLW43OIZJ0}, + date = {2021-06-10}, + year = {2021}, + month = {6}, + day = {10} } -@book{reedsolomon:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{reedsolomon:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@book{reedsolomon:informationkommunikation, + title = {Information und Kommunikation}, + author = {Markus Hufschmid}, + publisher = {Teubner}, + year = {2007}, + isbn = {978-3-8351-0122-7}, + inseries = {}, + volume = {1} } diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex new file mode 100644 index 0000000..89a700f --- /dev/null +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -0,0 +1,185 @@ +% +% rekonstruktion.tex +% Autor: Michael Steiner +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Nachricht Rekonstruieren +\label{reedsolomon:section:rekonstruktion}} +\rhead{Rekonstruktion} +Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können. +Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren. +Das Lokatorpolynom +\[ +l(X) = (X - a^3)(X-a^8) +\] +markiert dabei diese fehlerhaften Stellen im Übertragungsvektor +\[ +w = [5,3,6,8,2,10,2,7,1,4]. +\] +Als Ausgangslage verwenden wir die Matrix, mit der wir den Nachrichtenvektor ursprünglich codiert haben. +Unser Ziel ist es wie auch schon im Abschnitt \ref{reedsolomon:section:decohnefehler} eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. +Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Abschnitt \ref{reedsolomon:section:decohnefehler} angewendet haben. + +Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen. +\[ +\textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ +\end{pmatrix}} +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ +\end{pmatrix} +\] +Die rot markierten Stellen im Übertragungsvektor enthalten Fehler und bringt uns daher kein weiterer Nutzen. +Aus diesem Grund werden diese Stellen aus dem Vektor entfernt, was wir hier ohne Probleme machen können, da dieser Code ja über Fehlerkorrekturstellen verfügt, deren Aufgabe es ist, eine bestimmte Anzahl an Fehler kompensieren zu können. +Die dazugehörigen Zeilen in der Matrix werden ebenfalls entfernt, da die Matrix gleich viele Zeilen wie im Übertragungsvektor aufweisen muss, damit man ihn decodieren kann. + +Daraus resultiert +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ +\end{pmatrix} +. +\] +Die Matrix ist jedoch nicht mehr quadratisch, was eine Rekonstruktion durch Inversion ausschliesst. +Um die quadratische Form wieder herzustellen müssen wir zwei Spalten aus der Matrix entfernen. +Wir kennen aber das Resultat aus den letzten vier Spalten, da wir wissen, das die Nachricht aus Nutzdatenteil und Fehlerkorrekturteil besteht, wobei der letzteres bekanntlich aus lauter Nullstellen besteht. +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{darkgreen}{8^6}& \textcolor{darkgreen}{8^7}& \textcolor{darkgreen}{8^8}& \textcolor{darkgreen}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{darkgreen}{8^{12}}& \textcolor{darkgreen}{8^{14}}& \textcolor{darkgreen}{8^{16}}& \textcolor{darkgreen}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{darkgreen}{8^{24}}& \textcolor{darkgreen}{8^{28}}& \textcolor{darkgreen}{8^{32}}& \textcolor{darkgreen}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{darkgreen}{8^{30}}& \textcolor{darkgreen}{8^{35}}& \textcolor{darkgreen}{8^{40}}& \textcolor{darkgreen}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{darkgreen}{8^{36}}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{48}}& \textcolor{darkgreen}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{49}}& \textcolor{darkgreen}{8^{56}}& \textcolor{darkgreen}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{darkgreen}{8^{54}}& \textcolor{darkgreen}{8^{63}}& \textcolor{darkgreen}{8^{72}}& \textcolor{darkgreen}{8^{81}}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{darkgreen}{m_6} \\ \textcolor{darkgreen}{m_7} \\ \textcolor{darkgreen}{m_8} \\ \textcolor{darkgreen}{m_9} \\ +\end{pmatrix} +\] +Wir nehmen die entsprechenden Spalten aus der Matrix heraus und erhalten so das Überbestimmte Gleichungssystem +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +. +\] +Die roten Zeilen können wir aufgrund der Überbestimmtheit ebenfalls entfernen und erhalten so die gesuchte quadratische Matrix +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +. +\] +Nun können wir den Gauss-Algorithmus anwenden um die Matrix zu Invertieren. +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} += +\begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +\qquad +\Rightarrow +\qquad +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} += +\begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} +\] +Multiplizieren wir nun aus, erhalten wir unseren Nutzdatenteil +\[ +m = [4,7,2,5,8,1] +\] +zurück, den wir ursprünglich versendet haben. + diff --git a/buch/papers/reedsolomon/restetabelle1.tex b/buch/papers/reedsolomon/restetabelle1.tex new file mode 100644 index 0000000..3969ef2 --- /dev/null +++ b/buch/papers/reedsolomon/restetabelle1.tex @@ -0,0 +1,176 @@ +% created by Michael Steiner +% +% Restetabelle von F_11: Addition + +% alternatives design +%\begin{figure} +%\begin{center} +%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +%\hline +%+&0&1&2&3&4&5&6&7&8&9&10\\ +%\hline +%0&0&1&2&3&4&5&6&7&8&9&10\\ +%1&1&2&3&4&5&6&7&8&9&10&0\\ +%2&2&3&4&5&6&7&8&9&10&0&1\\ +%3&3&4&5&6&7&8&9&10&0&1&2\\ +%4&4&5&6&7&8&9&10&0&1&2&3\\ +%5&5&6&7&8&9&10&0&1&2&3&4\\ +%6&6&7&8&9&10&0&1&2&3&4&5\\ +%7&7&8&9&10&0&1&2&3&4&5&6\\ +%8&8&9&10&0&1&2&3&4&5&6&7\\ +%9&9&10&0&1&2&3&4&5&6&7&8\\ +%10&10&0&1&2&3&4&5&6&7&8&9\\ +%\hline +%\end{tabular} +%\end{center} +%\end{figure} + +\begin{center} + +\begin{tikzpicture}[>=latex,thick,scale=0.45] +\fill[color=gray!40] (0,0) rectangle (18,-1.5); +\fill[color=gray!40] (0,0) rectangle (1.5,-18); +\draw[step = 1.5, gray,very thin] (0,0) grid (18,-18); +\draw[very thick] (0,0) rectangle (18,-18); +\draw[very thick] (0,-1.5) -- (18,-1.5); +\draw[very thick] (1.5,0) -- (1.5,-18); +\node at (0.75,-0.75) {$+$}; +\foreach \x in {0,...,10} + \node at (2.25+\x*1.5,-0.75) {$\x$}; +\foreach \y in {0,...,10} + \node at (0.75,-2.25+\y*-1.5) {$\y$}; +% Row 0 +\node at ( 2.25,-2.25) {$0$}; +\node at ( 3.75,-2.25) {$1$}; +\node at ( 5.25,-2.25) {$2$}; +\node at ( 6.75,-2.25) {$3$}; +\node at ( 8.25,-2.25) {$4$}; +\node at ( 9.75,-2.25) {$5$}; +\node at (11.25,-2.25) {$6$}; +\node at (12.75,-2.25) {$7$}; +\node at (14.25,-2.25) {$8$}; +\node at (15.75,-2.25) {$9$}; +\node at (17.25,-2.25) {$10$}; +% Row 1 +\node at ( 2.25,-3.75) {$1$}; +\node at ( 3.75,-3.75) {$2$}; +\node at ( 5.25,-3.75) {$3$}; +\node at ( 6.75,-3.75) {$4$}; +\node at ( 8.25,-3.75) {$5$}; +\node at ( 9.75,-3.75) {$6$}; +\node at (11.25,-3.75) {$7$}; +\node at (12.75,-3.75) {$8$}; +\node at (14.25,-3.75) {$9$}; +\node at (15.75,-3.75) {$10$}; +\node at (17.25,-3.75) {$0$}; +% Row 2 +\node at ( 2.25,-5.25) {$2$}; +\node at ( 3.75,-5.25) {$3$}; +\node at ( 5.25,-5.25) {$4$}; +\node at ( 6.75,-5.25) {$5$}; +\node at ( 8.25,-5.25) {$6$}; +\node at ( 9.75,-5.25) {$7$}; +\node at (11.25,-5.25) {$8$}; +\node at (12.75,-5.25) {$9$}; +\node at (14.25,-5.25) {$10$}; +\node at (15.75,-5.25) {$0$}; +\node at (17.25,-5.25) {$1$}; +% Row 3 +\node at ( 2.25,-6.75) {$3$}; +\node at ( 3.75,-6.75) {$4$}; +\node at ( 5.25,-6.75) {$5$}; +\node at ( 6.75,-6.75) {$6$}; +\node at ( 8.25,-6.75) {$7$}; +\node at ( 9.75,-6.75) {$8$}; +\node at (11.25,-6.75) {$9$}; +\node at (12.75,-6.75) {$10$}; +\node at (14.25,-6.75) {$0$}; +\node at (15.75,-6.75) {$1$}; +\node at (17.25,-6.75) {$2$}; +% Row 4 +\node at ( 2.25,-8.25) {$4$}; +\node at ( 3.75,-8.25) {$5$}; +\node at ( 5.25,-8.25) {$6$}; +\node at ( 6.75,-8.25) {$7$}; +\node at ( 8.25,-8.25) {$8$}; +\node at ( 9.75,-8.25) {$9$}; +\node at (11.25,-8.25) {$10$}; +\node at (12.75,-8.25) {$0$}; +\node at (14.25,-8.25) {$1$}; +\node at (15.75,-8.25) {$2$}; +\node at (17.25,-8.25) {$3$}; +% Row 5 +\node at ( 2.25,-9.75) {$5$}; +\node at ( 3.75,-9.75) {$6$}; +\node at ( 5.25,-9.75) {$7$}; +\node at ( 6.75,-9.75) {$8$}; +\node at ( 8.25,-9.75) {$9$}; +\node at ( 9.75,-9.75) {$10$}; +\node at (11.25,-9.75) {$0$}; +\node at (12.75,-9.75) {$1$}; +\node at (14.25,-9.75) {$2$}; +\node at (15.75,-9.75) {$3$}; +\node at (17.25,-9.75) {$4$}; +% Row 6 +\node at ( 2.25,-11.25) {$6$}; +\node at ( 3.75,-11.25) {$7$}; +\node at ( 5.25,-11.25) {$8$}; +\node at ( 6.75,-11.25) {$9$}; +\node at ( 8.25,-11.25) {$10$}; +\node at ( 9.75,-11.25) {$0$}; +\node at (11.25,-11.25) {$1$}; +\node at (12.75,-11.25) {$2$}; +\node at (14.25,-11.25) {$3$}; +\node at (15.75,-11.25) {$4$}; +\node at (17.25,-11.25) {$5$}; +% Row 7 +\node at ( 2.25,-12.75) {$7$}; +\node at ( 3.75,-12.75) {$8$}; +\node at ( 5.25,-12.75) {$9$}; +\node at ( 6.75,-12.75) {$10$}; +\node at ( 8.25,-12.75) {$0$}; +\node at ( 9.75,-12.75) {$1$}; +\node at (11.25,-12.75) {$2$}; +\node at (12.75,-12.75) {$3$}; +\node at (14.25,-12.75) {$4$}; +\node at (15.75,-12.75) {$5$}; +\node at (17.25,-12.75) {$6$}; +% Row 8 +\node at ( 2.25,-14.25) {$8$}; +\node at ( 3.75,-14.25) {$9$}; +\node at ( 5.25,-14.25) {$10$}; +\node at ( 6.75,-14.25) {$0$}; +\node at ( 8.25,-14.25) {$1$}; +\node at ( 9.75,-14.25) {$2$}; +\node at (11.25,-14.25) {$3$}; +\node at (12.75,-14.25) {$4$}; +\node at (14.25,-14.25) {$5$}; +\node at (15.75,-14.25) {$6$}; +\node at (17.25,-14.25) {$7$}; +% Row 9 +\node at ( 2.25,-15.75) {$9$}; +\node at ( 3.75,-15.75) {$10$}; +\node at ( 5.25,-15.75) {$0$}; +\node at ( 6.75,-15.75) {$1$}; +\node at ( 8.25,-15.75) {$2$}; +\node at ( 9.75,-15.75) {$3$}; +\node at (11.25,-15.75) {$4$}; +\node at (12.75,-15.75) {$5$}; +\node at (14.25,-15.75) {$6$}; +\node at (15.75,-15.75) {$7$}; +\node at (17.25,-15.75) {$8$}; +% Row 10 +\node at ( 2.25,-17.25) {$10$}; +\node at ( 3.75,-17.25) {$0$}; +\node at ( 5.25,-17.25) {$1$}; +\node at ( 6.75,-17.25) {$2$}; +\node at ( 8.25,-17.25) {$3$}; +\node at ( 9.75,-17.25) {$4$}; +\node at (11.25,-17.25) {$5$}; +\node at (12.75,-17.25) {$6$}; +\node at (14.25,-17.25) {$7$}; +\node at (15.75,-17.25) {$8$}; +\node at (17.25,-17.25) {$9$}; +\end{tikzpicture} + +\end{center} diff --git a/buch/papers/reedsolomon/restetabelle2.tex b/buch/papers/reedsolomon/restetabelle2.tex new file mode 100644 index 0000000..1a9815c --- /dev/null +++ b/buch/papers/reedsolomon/restetabelle2.tex @@ -0,0 +1,176 @@ +% created by Michael Steiner +% +% Restetabelle von F_11: Multiplikation + +% alternatives design +%\begin{figure} +%\begin{center} +%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +%\hline +%\cdot&0&1&2&3&4&5&6&7&8&9&10\\ +%\hline +%0&0&0&0&0&0&0&0&0&0&0&0\\ +%1&0&1&2&3&4&5&6&7&8&9&10\\ +%2&0&2&4&6&8&10&1&3&5&7&9\\ +%3&0&3&6&9&1&4&7&10&2&5&8\\ +%4&0&4&8&1&5&9&2&6&10&3&7\\ +%5&0&5&10&4&9&3&8&2&7&1&6\\ +%6&0&6&1&7&2&8&3&9&4&10&5\\ +%7&0&7&3&10&6&2&9&5&1&8&4\\ +%8&0&8&5&2&10&7&4&1&9&6&3\\ +%9&0&9&7&5&3&1&10&8&6&4&2\\ +%10&0&10&9&8&7&6&5&4&3&2&1\\ +%\hline +%\end{tabular} +%\end{center} +%\end{figure} + +\begin{center} + + \begin{tikzpicture}[>=latex,thick,scale=0.45] + \fill[color=gray!40] (0,0) rectangle (18,-1.5); + \fill[color=gray!40] (0,0) rectangle (1.5,-18); + \draw[step = 1.5, gray,very thin] (0,0) grid (18,-18); + \draw[very thick] (0,0) rectangle (18,-18); + \draw[very thick] (0,-1.5) -- (18,-1.5); + \draw[very thick] (1.5,0) -- (1.5,-18); + \node at (0.75,-0.75) {$\cdot$}; + \foreach \x in {0,...,10} + \node at (2.25+\x*1.5,-0.75) {$\x$}; + \foreach \y in {0,...,10} + \node at (0.75,-2.25+\y*-1.5) {$\y$}; + % Row 0 + \node at ( 2.25,-2.25) {$0$}; + \node at ( 3.75,-2.25) {$0$}; + \node at ( 5.25,-2.25) {$0$}; + \node at ( 6.75,-2.25) {$0$}; + \node at ( 8.25,-2.25) {$0$}; + \node at ( 9.75,-2.25) {$0$}; + \node at (11.25,-2.25) {$0$}; + \node at (12.75,-2.25) {$0$}; + \node at (14.25,-2.25) {$0$}; + \node at (15.75,-2.25) {$0$}; + \node at (17.25,-2.25) {$0$}; + % Row 1 + \node at ( 2.25,-3.75) {$0$}; + \node at ( 3.75,-3.75) {$1$}; + \node at ( 5.25,-3.75) {$2$}; + \node at ( 6.75,-3.75) {$3$}; + \node at ( 8.25,-3.75) {$4$}; + \node at ( 9.75,-3.75) {$5$}; + \node at (11.25,-3.75) {$6$}; + \node at (12.75,-3.75) {$7$}; + \node at (14.25,-3.75) {$8$}; + \node at (15.75,-3.75) {$9$}; + \node at (17.25,-3.75) {$10$}; + % Row 2 + \node at ( 2.25,-5.25) {$0$}; + \node at ( 3.75,-5.25) {$2$}; + \node at ( 5.25,-5.25) {$4$}; + \node at ( 6.75,-5.25) {$6$}; + \node at ( 8.25,-5.25) {$8$}; + \node at ( 9.75,-5.25) {$10$}; + \node at (11.25,-5.25) {$1$}; + \node at (12.75,-5.25) {$3$}; + \node at (14.25,-5.25) {$5$}; + \node at (15.75,-5.25) {$7$}; + \node at (17.25,-5.25) {$9$}; + % Row 3 + \node at ( 2.25,-6.75) {$0$}; + \node at ( 3.75,-6.75) {$3$}; + \node at ( 5.25,-6.75) {$6$}; + \node at ( 6.75,-6.75) {$9$}; + \node at ( 8.25,-6.75) {$1$}; + \node at ( 9.75,-6.75) {$4$}; + \node at (11.25,-6.75) {$7$}; + \node at (12.75,-6.75) {$10$}; + \node at (14.25,-6.75) {$2$}; + \node at (15.75,-6.75) {$5$}; + \node at (17.25,-6.75) {$8$}; + % Row 4 + \node at ( 2.25,-8.25) {$0$}; + \node at ( 3.75,-8.25) {$4$}; + \node at ( 5.25,-8.25) {$8$}; + \node at ( 6.75,-8.25) {$1$}; + \node at ( 8.25,-8.25) {$5$}; + \node at ( 9.75,-8.25) {$9$}; + \node at (11.25,-8.25) {$2$}; + \node at (12.75,-8.25) {$6$}; + \node at (14.25,-8.25) {$10$}; + \node at (15.75,-8.25) {$3$}; + \node at (17.25,-8.25) {$7$}; + % Row 5 + \node at ( 2.25,-9.75) {$0$}; + \node at ( 3.75,-9.75) {$5$}; + \node at ( 5.25,-9.75) {$10$}; + \node at ( 6.75,-9.75) {$4$}; + \node at ( 8.25,-9.75) {$9$}; + \node at ( 9.75,-9.75) {$3$}; + \node at (11.25,-9.75) {$8$}; + \node at (12.75,-9.75) {$2$}; + \node at (14.25,-9.75) {$7$}; + \node at (15.75,-9.75) {$1$}; + \node at (17.25,-9.75) {$6$}; + % Row 6 + \node at ( 2.25,-11.25) {$0$}; + \node at ( 3.75,-11.25) {$6$}; + \node at ( 5.25,-11.25) {$1$}; + \node at ( 6.75,-11.25) {$7$}; + \node at ( 8.25,-11.25) {$2$}; + \node at ( 9.75,-11.25) {$8$}; + \node at (11.25,-11.25) {$3$}; + \node at (12.75,-11.25) {$9$}; + \node at (14.25,-11.25) {$4$}; + \node at (15.75,-11.25) {$10$}; + \node at (17.25,-11.25) {$5$}; + % Row 7 + \node at ( 2.25,-12.75) {$0$}; + \node at ( 3.75,-12.75) {$7$}; + \node at ( 5.25,-12.75) {$3$}; + \node at ( 6.75,-12.75) {$10$}; + \node at ( 8.25,-12.75) {$6$}; + \node at ( 9.75,-12.75) {$2$}; + \node at (11.25,-12.75) {$9$}; + \node at (12.75,-12.75) {$5$}; + \node at (14.25,-12.75) {$1$}; + \node at (15.75,-12.75) {$8$}; + \node at (17.25,-12.75) {$4$}; + % Row 8 + \node at ( 2.25,-14.25) {$0$}; + \node at ( 3.75,-14.25) {$8$}; + \node at ( 5.25,-14.25) {$5$}; + \node at ( 6.75,-14.25) {$2$}; + \node at ( 8.25,-14.25) {$10$}; + \node at ( 9.75,-14.25) {$7$}; + \node at (11.25,-14.25) {$4$}; + \node at (12.75,-14.25) {$1$}; + \node at (14.25,-14.25) {$9$}; + \node at (15.75,-14.25) {$6$}; + \node at (17.25,-14.25) {$3$}; + % Row 9 + \node at ( 2.25,-15.75) {$0$}; + \node at ( 3.75,-15.75) {$9$}; + \node at ( 5.25,-15.75) {$7$}; + \node at ( 6.75,-15.75) {$5$}; + \node at ( 8.25,-15.75) {$3$}; + \node at ( 9.75,-15.75) {$1$}; + \node at (11.25,-15.75) {$10$}; + \node at (12.75,-15.75) {$8$}; + \node at (14.25,-15.75) {$6$}; + \node at (15.75,-15.75) {$4$}; + \node at (17.25,-15.75) {$2$}; + % Row 10 + \node at ( 2.25,-17.25) {$0$}; + \node at ( 3.75,-17.25) {$10$}; + \node at ( 5.25,-17.25) {$9$}; + \node at ( 6.75,-17.25) {$8$}; + \node at ( 8.25,-17.25) {$7$}; + \node at ( 9.75,-17.25) {$6$}; + \node at (11.25,-17.25) {$5$}; + \node at (12.75,-17.25) {$4$}; + \node at (14.25,-17.25) {$3$}; + \node at (15.75,-17.25) {$2$}; + \node at (17.25,-17.25) {$1$}; + \end{tikzpicture} + +\end{center}
\ No newline at end of file diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index 37c2ec2..0cb1433 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -1,118 +1,89 @@ \section{Einleitung\label{spannung:section:Einleitung}} -In diesem Kapitel geht es darum die Matrix im dreidimensionalen Spannungszustand genauer zu untersuchen. -In der Geotechnik wendet man solche Matrizen an, um Spannungen im Boden zu berechnen. -Mit diesen Grundlagen dimensioniert man beispielsweise Böschungen, Fundationen, Dämme und Tunnels. -Ebenfalls benötigt man diese Matrix, um aus Versuchen Kennzahlen über den anstehenden Boden zu gewinnen. -Besonderes Augenmerk liegt dabei auf dem Oedometer - Versuch. +\rhead{Einleitung} +Das Hook'sche Gesetz beschreibt die Beziehung von Spannung und Dehnung von linear-elastischen Materialien im Eindimensionalen. +In diesem Kapitel geht es darum das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. +Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen. +In jedem erdenklichen Punkt im Dreidimensionalen herrscht daher ein entsprechender individueller Spannungszustand. +Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus. +Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen. +Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden. +Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. -Bei dieser Untersuchung der zugehörigen Berechnungen hat man es mit Vektoren, Matrizen und Tensoren zu tun. Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. -Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen, -damit sich den Berechnungen schlüssig folgen lässt. +Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. +In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein, +wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden, z.~B.~\cite{spannung:Grundlagen-der-Geotechnik}. -In diesem Kapitel hat man es insbesondere mit Spannungen und Dehnungen zu tun. -Mit einer Spannung ist hier jedoch keine elektrische Spannung gemeint, -sondern eine Kraft geteilt durch Fläche. +\section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} +\rhead{Spannungsausbreitung} +Die Geotechnik ist eine Ingenieurdisziplin, bei welcher man Erdbau und den Erdbau tangierende Bauwerke dimensioniert. +Sie beinhaltet aber auch die statische Beurteilung von Boden und Fels. -\section{Einführung wichtige Begriffe\label{spannung:section:Wichtige Begriffe}} -\[ -l_0 -= -\text{Ausgangslänge [\si{\meter}]} -\] -\[ -\Delta l -= -\text{Längenänderung nach Kraftauftrag [\si{\meter}]} -\] -\[ -\Delta b -= -\text{Längenänderung in Querrichtung nach Kraftauftrag [\si{\meter}]} -\] -\[ -\varepsilon -= -\text{Dehnung [$-$]} -\] +Belastet man den Boden mit einer Spannung \[ \sigma = -\text{Spannung [\si{\kilo\pascal}]} -\] -\[ -E -= -\text{Elastizitätsmodul [\si{\kilo\pascal}]} -\] -\[ -\nu -= -\text{Querdehnungszahl; Poissonzahl [$-$]} -\] -\[ -F -= -\text{Kraft [\si{\kilo\newton}]} -\] -\[ -A -= -\text{Fläche [\si{\meter\squared}]} -\] -\[ -t -= -\text{Tiefe [\si{\meter}]} -\] -\[ -s -= -\text{Setzung, Absenkung [m]} +\frac{F}{A} +, \] +so wird diese in den Boden geleitet und von diesem kompensiert. +Im Boden entstehen unterschiedlich hohe Zusatzspannungen. +Diese Zusatzspannung breitet sich räumlich im Boden aus. +Im Falle einer konstanten Flächenlast $\sigma$ (siehe Abbildung 1.1) breitet sich die Zusatzspannung zwiebelartig aus. + +\begin{figure} + \centering + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} + \caption{Ausbreitung der Zusatzspannung im Boden infolge einfacher Flächenlast} + \label{fig:Bild4} +\end{figure} -Beziehungen +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung 1.2). +Wie diese Geometrie der Ausbreitung ist, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. +Diese Zusatzspannung $\sigma$ ist im Wesentlichen abhängig von $(x,y,t)$. +Je nach Modell werden noch andere Parameter berücksichtigt. +Das können beispielsweise jenste Bodenkennwerte oder auch der Wassergehalt sein. + +\begin{figure} + \centering + \includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} + \caption{Funktionen der Spannung und Dehnung im Zusammenhang mit der Tiefe} + \label{fig:Bild5} +\end{figure} + +Bei jeder dieser Zusatzspannung geht eine entsprechende Zusatzdehnung des Bodens einher, welche eine Setzung bedeutet. +Im einfachsten Fall kann modellhaft mit \[ \varepsilon = -\frac{\Delta l}{l_0} -\] -\[ -\varepsilon_q -= -\frac{\Delta b}{l_0} -= -\varepsilon\cdot\nu -\] -\[ -\sigma -= -\frac{N}{A} -\] -\[ -F -= -\int_{A} \sigma dA +\frac{\sigma}{E} \] +die Setzung an einem Punkt an der Bodenoberfläche mit \[ -\varepsilon^{\prime} +s = -\frac{1}{l_0} -\] +\int_{0}^{\infty}\varepsilon\enspace dt +\] +berechnet werden mit: +\begin{align*} + \varepsilon &= \text{Dehnung [$-$]} \\ + \sigma &= \text{Spannung [\si{\kilo\pascal}]} \\ + E &= \text{Elastizitätsmodul; Young-Modul [\si{\kilo\pascal}]}\\ + t &= \text{Tiefe [\si{\meter}]} \\ + s &= \text{Setzung, Absenkung [m].} +\end{align*} +Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen-der-Geotechnik}] beschrieben. +In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. +Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung 1.3). +Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. +Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$. +Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnungen mit Hilfe einer Spannungsgleichung berechnet werden. +Diese beruht auf Annahmen nach Hooke auf einem linear-elastischen Boden. +Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. -\section{Einführung wichtige Begriffe\label{spannung:section:Tensoren}} -Tensoren wurden als erstes in der Elastizitätstheorie eingesetzt. (Quelle Herr Müller) -In der Elastizitätstheorie geht es darum viele verschiedene Komponenten zu beschreiben. -Mit einer Matrix oder einem Vektor kann man dies nicht mehr bewerkstelligen. -Wenn man den dreidimensionalen Spannungszustand abbilden möchte, müsste man mehrere Vektoren haben. -Deshalb wurden 1840 von Rowan Hamilton Tensoren in die Mathematik eingeführt. -Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. -Albert Einstein hat Tensoren zudem in der allgemeinen Relativitätstheorie benutzt. -Tensor sind eine Stufe höher als Matrizen. Matrizen sind 2. Stufe. -Da Tensoren eine Stufe höher sind, kann man auch Matrizen, Vektoren und Skalare als Tensoren bezeichnen. -Der Nachteil von den Tensoren ist, dass man die gewohnten Rechenregeln, die man bei Vektoren oder Matrizen kennt, -nicht darauf anwenden kann. Man ist deshalb bestrebt die Tensoren als Vektoren und Matrizen darzustellen, -damit man die gewohnten Rechenregeln darauf anwenden kann. (Quelle Wikipedia) -In der vorliegenden Arbeit sind bereits alle Tensoren als Matrizen 2. Stufe abgebildet. -Trotzdem kann man diese Matrizen wie vorher beschrieben als Tensor bezeichnen. -Da diese als Matrizen abgebildet sind, dürfen wir die bekannten Rechenregeln auf unsere Tensoren anwenden.
\ No newline at end of file +\begin{figure} + \centering + \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} + \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat} + \label{fig:Bild3} +\end{figure} diff --git a/buch/papers/spannung/Grafiken/Bild1.png b/buch/papers/spannung/Grafiken/Bild1.png Binary files differnew file mode 100644 index 0000000..32b627e --- /dev/null +++ b/buch/papers/spannung/Grafiken/Bild1.png diff --git a/buch/papers/spannung/Grafiken/Bild2.png b/buch/papers/spannung/Grafiken/Bild2.png Binary files differnew file mode 100644 index 0000000..d1321a4 --- /dev/null +++ b/buch/papers/spannung/Grafiken/Bild2.png diff --git a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png Binary files differindex 398529c..2c359e6 100644 --- a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png +++ b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png diff --git a/buch/papers/spannung/references.bib b/buch/papers/spannung/references.bib index ed5703c..02f8d09 100644 --- a/buch/papers/spannung/references.bib +++ b/buch/papers/spannung/references.bib @@ -4,27 +4,46 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{spannung:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, +@online{spannung:Tensor, + title = {Tensor}, + url = {https://de.wikipedia.org/wiki/Tensor}, + date = {2021-05-29}, + year = {2021}, + month = {5}, day = {6} } -@book{spannung:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} +@online{spannung:Voigtsche-Notation, + title = {Voigtsche Notation}, + url = {https://de.wikipedia.org/wiki/Voigtsche_Notation}, + date = {2021-05-29}, + year = {2021}, + month = {5}, + day = {6} +} + +@book{spannung:Grundlagen-der-Geotechnik, + title = {Grundlagen der Geotechnik}, + author = {Hans-Henning Schmidt and Roland F. Buchmaier and Carola Vogt-Breyer}, + publisher = {Springer Fachmedien Wiesbaden GmbH}, + year = {2017}, + isbn = {978-3-658-14930-7}, + inseries = {Geotechnik nach Eurocode}, + volume = {5} +} + +@book{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik, + title = {Stoffgesetze und numerische Modellierung in der Geotechnik}, + author = {Carlo Rabaiotti and Alessio Höttges}, + publisher = {Hochschule Rapperswil}, + year = {2021}, + isbn = {}, + inseries = {}, + volume = {} } @article{spannung:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, + author = { Tabea Méndez and Andreas Müller }, title = { Noncommutative harmonic analysis and image registration }, journal = { Appl. Comput. Harmon. Anal.}, year = 2019, diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index 2f4d23b..ffc9009 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,56 +1,82 @@ -\section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} -\rhead{Spannungsausbreitung} -Anhand untenstehendem Bild kann ein einfaches Beispiel betrachtet werden. -Es gibt eine Flächenlast (Kraft), diese wird auf den Boden abgetragen. -Diese Last muss dann vom Boden aufgenommen werden. -Im Boden entsteht nebst der Eigenspannung eine weitere Spannung durch diese Last (Zusatzspannung). -Diese Zusatzspannung $\sigma$ ist abhängig von $(x,y,t)$. -Je nach dem, wo man sich im Boden befindet variert die Spannung. -Mit der Tiefe wird die Zusatzspannung geringer. -Die Ausbreitung der Zusatzspannung im Boden hat die Form einer Zwiebel. -Durch Untersuchung der Spannung an verschiedenen Punkten im Boden, kann man eine Funktion abtragen. -Dasselbe macht man auch mit der Dehnung. Es zeigt sich, dass die Form der beiden Funktionen gleich ist. -Dies erklärt sich dadurch, dass die Spannung und die Dehnung proportional zueinander sind. -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} - \caption{Ausbreitung der Spannung im Boden} - \label{fig:Bild4} -\end{figure} +\section{Der Spannungszustand\label{spannung:section:Der Spannungsustand}} +\rhead{Der Spannungszustand} +Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung 1.4). +Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material. +Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt. +Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} - \caption{Funktionen Spannung und Dehnung} - \label{fig:Bild5} + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild2.png} + \caption{Infinitesimales Bodenelement mit den 9 Spannungen} + \label{fig:Bild2} \end{figure} -Anhand eines etwas schwierigeren Beispiels sieht man, -dass die Spannungsausbreitung nicht immer ganz einfach ist. -Man hat hier eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen wurde. -Was aber immer noch gilt ist, dass die Spannung $\sigma$ von drei Variablen abhängig ist $(x,y,t)$. -Ansätze um die Spannungsausbreitung zu berechnen gibt es je nach Bodentyp verschiedene. +Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten im Betrag die selben Spannungen aufweisen, +sodass der Elementarwürfel im Gleichgewicht ist. +Wäre dieses Gleichgewicht nicht vorhanden, käme es zu Verschiebungen und Drehungen. +Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$. +Veränderungen der Normalspannungen können durch Schubspannungen kompensiert werden und umgekehrt. +So sind insgesamt neun verschiedene Spannungen möglich, wobei drei Normal- und sechs Schubspannungen sind. +Normalspannungen wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. +Alle Beträge dieser neun Spannungen am Elementarwürfel bilden den Spannungszustand. +Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetz berechnet werden. +Daher gibt es auch den entsprechenden Dehnungszustand. + + +\section{Spannungszustand\label{spannung:section:Spannungsustand}} +\rhead{Spannungszustand} +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung 1.5). +Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. +Nach Hooke gilt: +\[ +F +\sim +\Delta l +. +\] +Teilt man beide Seiten durch die Konstanten $A$ und $l_0$, erhält man +\[ +\frac{F}{A} += +\sigma +\sim +\varepsilon += +\frac{\Delta l}{l_0} +\] +und somit +\[ +\sigma +\sim +\varepsilon +, +\] +mit +\begin{align*} + l_0 &= \text{Länge zu Beginn [\si{\meter}]} \\ + A &= \text{Fläche [\si{\meter\squared}].} +\end{align*} +Diese Beziehung gilt bei linear-elastischen Materialien, welche reversible Verformungen zulassen. +Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} - \caption{Beispiel Lastauftrag auf Boden} - \label{fig:Bild3} + \includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} + \caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe} + \label{fig:Bild1} \end{figure} - -Die Spannungsausbreitung ist uns jedoch gegeben, es geht nicht darum, dies genauer zu untersuchen. -Durch die Spannungsausbreitung und das Elastizitätsmodul kann man eine Dehnung berechnen. -Anhand dieser Dehnung kann man mit einem Integral wiederum die Setzung berechnen. +Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich der eindimensionale Fall mit \[ -\varepsilon +\sigma = -\frac{\sigma}{E} +E\cdot\varepsilon \] +beschreiben. +Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch \[ -s +E = -\int_{0}^{\infty}\varepsilon\enspace dt +\frac{\Delta\sigma}{\Delta\varepsilon} \] -Die Setzung zu bestimmen ist in der Geotechnik sehr wichtig. -Besonders ungleichmässige Setzungen können bei Bauwerken Probleme ergeben. -Es gilt also die Bauwerke so zu dimensionieren, dass es verträgliche Setzungen gibt.
\ No newline at end of file +ausgedrückt werden.
\ No newline at end of file diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 9467d21..74516c1 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,41 +1,24 @@ -\section{Proportionalität Spannung-Dehnung\label{spannung:section:Proportionalität Spannung-Dehnung}} -\rhead{Proportionalität Spannung-Dehnung} -Das Hook'sche Gesetz beschreibt die elastische Längenänderung von Festkörpern im Zusammenhang mit einer Krafteinwirkung. -Die Längenänderung $\Delta l$ ist proportional zur Krafteinwirkung $F$. -\[ -F -\sim -\Delta l -\] -Man kann dies nur im Bereich vom linearen-elastischen Materialverhalten anwenden. -Das heisst, dass alle Verformungen reversibel sind, sobald man die Kraft wegnimmt. -Es findet somit keine dauernde Verformung statt. -Da es sehr praktisch ist die Längenänderung nicht absolut auszudrücken haben wir $\varepsilon$. -Die Dehnung $\varepsilon$ beschreibt die relative Längenänderung. -Die Dehnung $\varepsilon$ ist wiederum proportional zu der aufgebrachten Spannung. -Im Bauingenieurwesen hat man es oft mit grösseren Teilen oder grösseren Betrachtungsräumen zu tun. -Da ist es nun natürlich sehr sinnvoll, wenn wir nicht mit absoluten Zahlen rechnen, -sondern unabhängig von der Länge den Zustand mit Dehnung $\varepsilon$ beschreiben können. -Mithilfe vom E-Modul, (steht für Elastizitätsmodul) einer Proportionalitätskonstante, -kann man das in eine Gleichung bringen, wie man hier sieht. Das E-Modul beschreibt, -das Verhältnis von Kraftaufnahme eines Werkstoffes und dessen zusammenhängender Längenveränderung. -(Quelle Wikipedia) -\[ -\sigma -= -E\cdot\varepsilon -\] -\[ -E -= -\frac{\Delta\sigma}{\Delta\varepsilon} -= -const. -\] +\section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}} +\rhead{Skalare, Vektoren, Matrizen und Tensoren} +Der Begriff Tensor kann als Überbegriff, der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden. +Allerdings sind noch höhere Stufen dieser Objekte beinhaltet. +Ein Skalar, ein Vektor oder eine Matrix ist daher auch ein Tensor. +Ein Skalar ist ein Tensor 0. Stufe. +Mit einem Vektor können mehrere Skalare auf einmal beschrieben werden. +Ein Vektor hat daher die Stufe 1 und ist höherstufig als ein Skalar. +Mit einer Matrix können wiederum mehrere Vektoren auf einmal beschrieben werden. +Eine Matrix hat daher die Stufe 2 und ist noch höherstufig als ein Vektor. +Versteht man diese Stufen, so versteht man den Sinn des Begriffs Tensor. -Aus diesem Verhältnis kann man das E-Modul berechnen. -Je nach Material ist dies verschieden. -Das E-Modul lässt sich nur im linearen-elastischen Materialverhalten anwenden. -Für Bodenmaterial gibt es ein spezielles E-Modul. Dieses wird mit dem Oedometer-Versuch ermittelt. -Es wird mit $E_{OED}$ ausgedrückt. Dieser Versuch wird später noch beschrieben. -Der Oedometer-Versuch ist abhängig von den diesem Kapitel zu untersuchenden Matrizen.
\ No newline at end of file +Jede Stufe von Tensoren verlangt andere Rechenregeln. +So zeigt sich auch der Nachteil von Tensoren mit Stufen höher als 2. +Man ist also bestrebt höherstufige Tensoren mit Skalaren, Vektoren oder Matrizen zu beschreiben. + +Der Begriff Tensor wurde 1840 von Rowan Hamilton in die Mathematik eingeführt. +James Clerk Maxwell hat bereits mit Tensoren operiert, ohne den Begriff Tensor gekannt zu haben. +Erst Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. +Er hat in der Elastizitätstheorie als erstes Tensoren eingesetzt und beschrieben. +Auch Albert Einstein hat solche Tensoren eingesetzt, +um in der Relativitätstheorie die Änderung der 4D Raumzeit beschreiben zu können. +\cite{spannung:Tensor} +\cite{spannung:Voigtsche-Notation} diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 7dcf65f..921d2b8 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,34 +1,57 @@ \section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}} -\rhead{Proportionalität Spannung-Dehnung} -Wie im Kapitel Spannungsausbreitung beschrieben herrscht in jedem Punkt ein anderer Spannungszustand. -Um die Spannung im Boden genauer untersuchen zu können, führt man einen infinitesimales Bodenteilchen ein. -Das Bodenteilchen ist geometrisch gesehen ein Würfel. -An diesem Bodenteilchen trägt man die Spannungen ein in alle Richtungen. - +\rhead{Dreiachsiger Spannungszustand} +Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} - \caption{Infinitesimales Bodenteilchen} - \label{fig:infintesimaler-wurfel} + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} + \caption{Beispiel eines Spannungszustandes; Vergrösserung eines infinitesimalen Bodenteilchen} + \label{fig:infinitesimalerWuerfel} \end{figure} +Ein Tensor 0. Stufe, sprich ein Skalar, kann lediglich den 1D Spannungszustand beschreiben. +Um den 3D Spannungszustandes als ein mathematisches Objekt darstellen zu können, wird ein Tensor 2. Stufe, sprich eine Matrix, eingesetzt. +Die Spannungen sind durch die zwei Indizes +\[ +i, j\in\left\{1, 2, 3\right\} +\] +definiert. +Daher ergeben sich die neun Spannungen. +Die nachfolgenden Zusammenhänge sind in \cite{spannung:Voigtsche-Notation} beschrieben. +Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als $3\times3$ Matrix mit +\[ +\overline{\sigma} += +\sigma_{ij} += +\begin{pmatrix} + \sigma_{11} & \sigma_{12} & \sigma_{13} \\ + \sigma_{21} & \sigma_{22} & \sigma_{23} \\ + \sigma_{31} & \sigma_{32} & \sigma_{33} +\end{pmatrix} +\] +dargestellt werden und beschreibt somit den gesamten Spannungszustand. +Die Dehnungen wirken in die gleichen Richtungen wie die korrespondierenden Spannungen und sind durch die zwei Indizes +\[ +k, l\in\left\{1, 2, 3\right\} +\] +definiert. +Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\times3$ Matrix mit +\[ +\overline{\varepsilon} += +\varepsilon_{kl} += +\begin{pmatrix} + \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ + \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ + \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} +\end{pmatrix} +\] +dargestellt werden und beschreibt den gesamten Dehnungszustand. -An diesem infinitesimalen Bodenteilchen hat man ein räumliches Koordinatensystem, die Achsen $(1,2,3)$. -Die Achsen vom Koordinatensystem zeigen aus den 3 ersichtlichen Flächen heraus. -Pro ersichtliche Fläche haben wir eine Normalspannung und zwei Schubspannungen. -Im Gegensatz zum eindimensionalen Zustand entstehen bei einer Belastung des Bodenteilchens eine Vielzahl an Spannungen. -Es entstehen diverse Normal- und Schubspannungen. -Die Schubspannungen befinden sich an der Fläche, sie gehen rechtwinklig von den Achsen weg. -Die Schubspannungen auf einer Fläche stehen im 90 Grad Winkel zueinander. -Geschrieben werden diese mit $\sigma$, mit jeweils zwei Indizes. -Die Indizes geben uns an, in welche Richtung die Spannungen zeigen. -Der erste Index ist die Fläche auf welcher man sich befindet. -Der zweite Index gibt an, in welche Richtung die Spannung zeigt, dabei referenzieren die Indizes auch auf die Achsen $(1,2,3)$. -Bei den Spannungen sind immer positive als auch negative Spannungen möglich. -Es können also Druck- oder Zugspannungen sein. - -Zunächst wird untenstehend der allgemeine Spannungszustand betrachtet. +Der Spannungs- und Dehnungstensor 2. Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. +Gemäss der Hadamard-Algebra dürfen Zeile um Zeile in eine Spalte notiert werden, sodass es einen Spaltenvektor ergibt. +So ergibt sich der Spannungsvektor -Spannungstensor 2. Stufe i,j $\in$ {1,2,3} \[ \overline{\sigma} = @@ -39,7 +62,6 @@ Spannungstensor 2. Stufe i,j $\in$ {1,2,3} \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix} -= \qquad \Rightarrow \qquad @@ -57,9 +79,7 @@ Spannungstensor 2. Stufe i,j $\in$ {1,2,3} \sigma_{33} \end{pmatrix} \] - -Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} - +und Dehnungsvektor \[ \overline{\varepsilon} = @@ -70,7 +90,6 @@ Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \end{pmatrix} -= \qquad \Rightarrow \qquad @@ -87,13 +106,22 @@ Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} +. \] - -Bei diesen zwei obenstehenden Formeln kann man sehen wie Matrizen zu einem Vektor umgewandelt wurden. -Unter dem Kapitel Hadamard-Algebra kann man sehen, dass man dabei Zeile um Zeile in eine Spalte schreiben kann, -sodass es einen Vektor ergibt. - -Elastizitätstensor 4. Stufe i,j,k,l $\in$ {1,2,3} +Um die Beziehung von Spannung und Dehnung, welche mit Tensoren 2. Stufe ausgedrückt werden, zu beschreiben, wird ein Elastizitätstensor 4. Stufe benötigt. +Dieser ist im 1D Spannungszustand ein Tensor 0. Stufe und somit ein Skalar, der Elastizitätsmodul $E$. + +Dieser Elastizitätstensor 4. Stufe kann als Tensor 2. Stufe, sprich als Matrix, dargestellt werden. +So wird die Spannungsgleichung stark vereinfacht, da nun eine Matrix auf einen Vektor operiert. +Dieser Tensor muss für eine Spannung jeden Einfluss aus allen 9 Dehnungen mit Konstanten erfassen. +Dies bedeutet um eine von 9 Spannungen berechnen zu können müssen alle 9 Dehnung mit unterschiedlichen Faktoren summiert werden. +Es ergeben sich $9^2$ Einträge, welches mit den 4 Indizes +\[ +i, j, k, l\in\left\{1, 2, 3\right\} +, +\] +die zueinander verknüpft werden müssen, zu begründen ist. +Es ergeben sich $3^4$ Einträge, sprich eine $9\times9$ Matrix, welche allgemein \[ \overline{\overline{C}} = @@ -104,32 +132,46 @@ C_{1111} & C_{1112} & C_{1113} & C_{1121} & C_{1122} & C_{1123} & C_{1131} & C_{ C_{1211} & C_{1212} & C_{1213} & C_{1221} & C_{1222} & C_{1223} & C_{1231} & C_{1232} & C_{1233} \\ C_{1311} & C_{1312} & C_{1313} & C_{1321} & C_{1322} & C_{1323} & C_{1331} & C_{1332} & C_{1333} \\ C_{2111} & C_{2112} & C_{2113} & C_{2121} & C_{2122} & C_{2123} & C_{2131} & C_{2132} & C_{2133} \\ -C_{2211} & C_{2212} & C_{1113} & C_{2221} & C_{2222} & C_{2223} & C_{2231} & C_{2232} & C_{2233} \\ +C_{2211} & C_{2212} & C_{2213} & C_{2221} & C_{2222} & C_{2223} & C_{2231} & C_{2232} & C_{2233} \\ C_{2311} & C_{2312} & C_{2313} & C_{2321} & C_{2322} & C_{2323} & C_{2331} & C_{2332} & C_{2333} \\ C_{3111} & C_{3112} & C_{3113} & C_{3121} & C_{3122} & C_{3123} & C_{3131} & C_{3132} & C_{3133} \\ C_{3211} & C_{3212} & C_{3213} & C_{3221} & C_{3222} & C_{3223} & C_{3231} & C_{3232} & C_{3233} \\ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{3332} & C_{3333} \end{pmatrix} \] - -Dieser Elastizitätstensor muss eine quadratische Matrix mit $3^{4}$ Einträgen ergeben, -da die Basis mit den drei Richtungen $1, 2, 3$ und die Potenz mit den 4 Indizes mit je $1, 2, 3$ definiert sind. -Dies gibt daher eine 9 x 9 Matrix, welche zudem symmetrisch ist. - +geschrieben werden kann. +Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. Folglich gilt: \[ \overline{\overline{C}} = \overline{\overline{C}}~^{T} +. \] - -Allgemeine Spannungsgleichung (mit Vektoren und Tensor) +Die allgemeine Spannungsgleichung lautet nun: \[ \vec\sigma = \overline{\overline{C}}\cdot\vec{\varepsilon} +. \] - +Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert. +Da dieser Modul durch die eindimensionale Betrachtung definiert ist, +muss für die dreidimensionale Betrachtung eine weitere Kennzahl eingeführt werden. +Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche durch +\[ +\nu += +\frac{\varepsilon_q}{\varepsilon} += +\frac{\Delta b}{b_0} +\] +und +\begin{align*} + \varepsilon &= \text{Längsdehnung [$-$]} \\ + \varepsilon_q &= \text{Querdehnung [$-$]} +\end{align*} +definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich \[ \begin{pmatrix} \sigma_{11}\\ @@ -166,34 +208,66 @@ Allgemeine Spannungsgleichung (mit Vektoren und Tensor) \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} +, \] - -Man kann das zudem auch als Indexnotation aufschreiben. - +welche ebenfalls als Indexnotation mit \[ \sigma_{ij} = -= -\sum_k=1^3 -\sum_l=1^3 +\sum_{k=1}^3 +\sum_{l=1}^3 C_{ijkl}\cdot\varepsilon_{kl} \] - -Um die Berechnung an einem Beispiel zu veranschaulichen: - +ausgedrückt werden kann. +Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als \[ \sigma_{22} = \frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{11}+\frac{E}{(1+\nu)}\cdot\varepsilon_{22}+\frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{33} \] +berechnen. -Anhand dem Tensor der allgemeinen Spannungsgleichung kann man zwar eine Symmetrie erkennen. -Die verschiedenen Einträge wechseln sich aber mit einander ab und es gibt keine klaren Blöcke mit nur einem gleichen Eintrag. -Man greift deshalb auf die Voigt'sche Notation zurück. - - -Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: +Man betrachte nun die Eigenschaften des Elastizitätstensors. +Dieser ist quadratisch und symmetrisch, die verschiedenen Einträge wechseln sich aber miteinander ab. +Es ergeben sich keine Blöcke mit einheitlichen Einträgen. +Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch Elastizitätstensor symmetrisch sind. +Wäre dem nicht so, würde sich das Material je nach Richtung unterschiedlich elastisch verhalten. +Diese Symmetrie setzt daher voraus, dass +\[ +\sigma_{12} += +\sigma_{21} +, +\qquad +\sigma_{13} += +\sigma_{31} +, +\qquad +\sigma_{23} += +\sigma_{32} +\] +und folglich auch +\[ +\varepsilon_{12} += +\varepsilon_{21} +, +\qquad +\varepsilon_{13} += +\varepsilon_{31} +, +\qquad +\varepsilon_{23} += +\varepsilon_{32} +\] +gilt. +Diese Eigenschaft wird durch die Voigt'sche Notation \cite{spannung:Voigtsche-Notation} ausgenutzt, um die Gleichung vereinfachen zu können. +Durch diese Symmetrie gilt \[ \overline{\sigma} = @@ -206,9 +280,11 @@ Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ & \sigma_{22} & \sigma_{23} \\ - sym & & \sigma_{33} + \text{sym} & & \sigma_{33} \end{pmatrix} +\qquad \Rightarrow +\qquad \vec{\sigma} = \begin{pmatrix} @@ -220,22 +296,7 @@ Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: \sigma_{12} \end{pmatrix} \] - -In der Voigt'sche Notation hat man die Reihenfolge von der Ecke links oben, diagonal zur Ecke rechts unten. -Danach ist noch $\sigma_{23}$, $\sigma_{13}$ und $\sigma_{12}$ aufzuschreiben um den Vektor zu erhalten. - -Eine weitere Besonderheit ist die Symmetrie der Matrix. -So entspricht $\sigma_{23}$ dem Wert $\sigma_{32}$ und $\sigma_{13}$ dem Wert $\sigma_{31}$. -Dies ist dadurch bedingt, dass die Kräfte in seitlicher Richtung im Boden die gleichen Werte annehmen. -Man hat in dieser Berechnung ein isotropes Material. -Im infinitesimalen Körper muss ein Gleichgewicht vorherrschen. -Ist kein Gleichgewicht vorhanden, würde sich der Körper zu drehen beginnen. -Es macht somit keinen Unterschied, ob man auf der Achse 2 in Richtung 3 geht, -oder auf der Achse 3 in Richtung 2. - -Da die Spannung proportional zur Dehnung ist, kann man die ganze Voigt'sche Notation auch mit der Dehnung ausdrücken. -Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. - +und entsprechend \[ \overline{\varepsilon} = @@ -247,7 +308,7 @@ Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. = \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ - & \varepsilon_{22} & \varepsilon_{23} \\ + & \varepsilon_{22} & \varepsilon_{23} \\ \text{sym} & & \varepsilon_{33} \end{pmatrix} \qquad @@ -263,31 +324,18 @@ Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. \varepsilon_{13} \\ \varepsilon_{12} \end{pmatrix} +. \] - -Mit der hergeleiteten Beziehung für die Spannungsgleichung anhand vom E-Modul, -der allgemeinen linearen Spannungsgleichung kann man diese Beziehungen neu aufschreiben. -Man benötigt dazu den zuvor berechneten Dehnungsvektor. -Die Gleichung besagt: -\[ -\text{Spannungsvektor} -= -\text{Elastizitätstensor}\cdot\text{Dehnungsvektor} -\] +Aus den Vereinfachungen der Voigt'schen Notation lassen sich die Spannungs- und Dehnungstensoren als Spaltenvektoren mit je sechs Einträgen darstellen. +Der Elastizitätstensor kann entsprechend auf eine $6\times6$ Matrix reduziert werden. +Es lässt sich nun eine reduzierte allgemeine Spannungsgleichung mit \[ \vec{\sigma} = \overline{\overline{C}}\cdot\vec{\varepsilon} \] - -Die Vektoren haben je 6 Einträge. Um das ganze auszudrücken braucht es einen 6 x 6 Elastizitätstensor. -Der Tensor hat sich also im Vergleich zum 9 x 9 Tensor verkleinert. -Dies ist deshalb der Fall, da man in den Achsen 2 und 3 Symmetrien hat. -Dadurch kann man die Einträge $(\varepsilon_{21}=\varepsilon_{12}; \varepsilon_{31}=\varepsilon_{13}; \varepsilon_{32}=\varepsilon_{23})$ -zusammenfassen und drei Einträge verschwinden, da drei Dehnungen gleich sind. -Das ganze sieht dann wie folgt aus: - +beziehungsweise \[ \begin{pmatrix} \sigma_{11} \\ @@ -299,12 +347,12 @@ Das ganze sieht dann wie folgt aus: \end{pmatrix} = \begin{pmatrix} - C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ - C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ - C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} \\ - C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} \\ - C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} \\ - C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} + C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\ + C_{2211} & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\ + C_{3311} & C_{3322} & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ + C_{2311} & C_{2322} & C_{2333} & C_{2323} & C_{2313} & C_{2312} \\ + C_{1311} & C_{1322} & C_{1333} & C_{1323} & C_{1313} & C_{1312} \\ + C_{1211} & C_{1222} & C_{1233} & C_{1223} & C_{1213} & C_{1212} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ @@ -315,11 +363,10 @@ Das ganze sieht dann wie folgt aus: \varepsilon_{12} \end{pmatrix} \] - -Die Spannung $\sigma_{11}$ besteht somit aus Anteilen von all diesen sechs Konstanten und den verschiedenen Dehnungen. -Zuvor bei der Voigt'schen Notation hat man jedoch gesehen, dass die Tensoren symmetrisch sind. -Folglich muss auch dieser Elastizitätstensor symmetrisch sein. -Das sind folgendermassen aus: +beschreiben. +Die Spannung $\sigma_{11}$ beispielsweise erhält man, wenn man die sechs Produkte aus den Konstanten $C$ und Dehnungen $\varepsilon$ summiert. +Die Symmetrieeigenschaft des Elastizitätstensors bleibt auch hier erhalten. +Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit \[ \begin{pmatrix} @@ -332,12 +379,12 @@ Das sind folgendermassen aus: \end{pmatrix} = \begin{pmatrix} - C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ - & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ - & & C_{33} & C_{34} & C_{35} & C_{36} \\ - & & & C_{44} & C_{45} & C_{46} \\ - & & & & C_{55} & C_{56} \\ - \text{sym} & & & & & C_{66} + C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\ + & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\ + & & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ + & & & C_{2323} & C_{2313} & C_{2312} \\ + & & & & C_{1313} & C_{1312} \\ + \text{sym} & & & & & C_{1212} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ @@ -348,10 +395,9 @@ Das sind folgendermassen aus: \varepsilon_{12} \end{pmatrix} \] - -Die Konstanten $C$ kann man nun anders ausdrücken. -Und zwar bewerkstelligt man dies mithilfe vom Hook'schen Gesetz. - +beschreiben. +Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert. +Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \[ \begin{pmatrix} \sigma_{11}\\ @@ -379,25 +425,26 @@ Und zwar bewerkstelligt man dies mithilfe vom Hook'schen Gesetz. \varepsilon_{13}\\ \varepsilon_{12} \end{pmatrix} +. \] -Mithilfe der Poissonzahl, welche uns die Querdehnung angibt, -sprich wie viel sich der Körper in Querrichtung verformt und dem E-Modul kann man alle Konstanten ausdrücken. -Bei einigen fällt auf, dass diese 0 werden. Der Tensor besagt also, -dass diese jeweiligen Konstanten keinen Einfluss auf unsere Spannung haben. -Man sieht nun auch ganz gut, dass sich im Vergleich bei der allgemeinen Darstellung der Spannungsgleichung, -die Einträge verschoben haben. Man hat nun eine sehr vorteilhafte Anordnung der verschiedenen Blöcke im Tensor. -Als Beispiel kann man sich $\sigma_{33}$ anschauen. -Es ist ersichtlich, dass die Konstante $C_{31}$, $C_{32}$, $C_{33}$, $C_{35}$ und $C_{36}$ keinen Einfluss auf $\sigma_{33}$ haben. -Dies kann wie folgt erklärt werden. Auf Achse 3 geht $\sigma_{33}$ in Richtung 3. -Der Einfluss von $C_{31}$, Achse 3 in Richtung 1 hat keinen Einfluss auf $\sigma_{33}$. +Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, +dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben. +Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben. +Da nach Voigt zuerst die Normalspannungen und anschliessend die Schubspannungen notiert worden sind, ergeben sich die $3\times3$ Blöcke. -Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\overline{C}}~^{-1}$ stellt sich die ganze Gleichung um. +Man betrachte als Beispiel die Berechnung von $\sigma_{33}$. +Es ist ersichtlich, dass die Schubdehnungen keinen Einfluss auf $\sigma_{33}$ haben. +Der Einfluss der zu $\sigma_{33}$ äquivalenten Dehnung $\varepsilon_{33}$ hat den grössten Einfluss. +Die anderen Normalspannungen $\sigma_{11}$ und $\sigma_{22}$ haben einen unter anderem mit $\nu$ korrigierten Einfluss. + +Von $\overline{\overline{C}}$ bildet man noch die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$ um die Gleichung umstellen zu können. +Dadurch erhält man die Dehnungsgleichung: \[ \vec{\varepsilon} = -\overline{\overline{C}}~^{-1}\cdot \vec{\sigma} +\overline{\overline{C}}\mathstrut^{-1}\cdot \vec{\sigma} \] \[ @@ -427,25 +474,18 @@ Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\ove \sigma_{13}\\ \sigma_{12} \end{pmatrix} +. \] - -Die zwei Blöcke links unten und rechts oben sind immer noch vorhanden. -Im Vergleich wo wir die Inverse noch nicht gemacht haben hat sich das nicht geändert. -Um die Einflüsse der Parameter zu veranschaulichen schreibt man folgende Gleichung. - +Die zwei $3\times3$ Blöcke links unten und rechts oben sind folglich noch vorhanden. +Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man die Dehnung \[ \varepsilon_{22} = \frac{1}{E}\sigma_{22} - \frac{\nu}{E}\sigma_{11} - \frac{\nu}{E}\sigma_{33} += +\frac{1}{E}\cdot(\sigma_{22}-\nu\cdot\sigma_{11}-\nu\cdot\sigma_{33}) +. \] - -$\varepsilon_{22}$ beschreibt die Dehnung in Achse 2 und in Richtung 2. -In erster Linie hängt $\varepsilon_{22}$ von $\sigma_{22}$ ab. -Wenn die Poisson - Zahl grösser wird oder $\sigma_{11}$ oder $\sigma_{33}$, dann wird dadurch die Dehnung $\varepsilon_{22}$ kleiner. -Das heisst, auf Kosten von Verformung in anderer Richtung als Achse 2 Richtung 2 erfolgt die Verformung an anderer Stelle. -Wiederum hat die Schubspannung auf $\sigma_{11}$ keinen Einfluss. - -Nun kennt man die Beziehung der 6 Dehnungen mit den 6 Spannungen. -In der Geotechnik wäre das aufgrund der vielen Komponenten sehr umständlich um damit Berechnungen zu machen. -Es braucht daher eine Vereinfachung mit Invarianten, welche im nächsten Kapitel beschrieben sind. - +Diese hängt wieder am meisten von $\sigma_{22}$ ab. +Ist die Querdehnung $\nu$ grösser, so wird die Dehnung $\varepsilon_{22}$ reduziert. +Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich $\nu=0.5$. diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 500c404..8d99733 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -1,78 +1,86 @@ -\section{Spannungsausbreitung\label{spannung:section:Invarianten}} -\rhead{Invarianten} -Trotz der Vereinfachung lässt sich mit den Invarianten die Realität adäquat abbilden. -Als erste Bedingung stellt man folgendes Verhältnis auf: +\section{Die geotechnischen Invarianten\label{spannung:section:Die geotechnischen Invarianten}} +\rhead{Die geotechnischen Invarianten} +In vielen Fällen in der Geotechnik und auch in Versuchen hat man gleichmässige Belastungen über eine grössere Fläche. +Durch eine solche Belastung auf den Boden, entstehen gleichermassen Spannungen in Richtung $2$ und $3$, +wenn man von einem isotropen Bodenmaterial ausgeht. +Folglich gilt: \[ \sigma_{22} = \sigma_{33} +. \] - -Dies deshalb, da man von einem isotropen Bodenmaterial ausgeht. -In Achse 22, Richtung 22 hat man den gleichen Boden wie in Achse 33 und Richtung 33. -Das Verhalten bezüglich Kraftaufnahme, Dehnung Spannung ist somit dasselbe. - -Man führt die zwei Werte p als hydrostatische Spannung und q als deviatorische Spannung ein. -Die Berechnung von p und q sieht wie folgt aus: - +Dadurch wird der Spannungszustand vereinfacht. +Diesen vereinfachten Spannungszustand kann man mit den zwei geotechnischen Invarianten abbilden. +Die erste Invariante ist die volumetrische Spannung \[ p = \frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3} +, \] - -oder durch Vereinfachung, da $\sigma_{22}=\sigma_{33}$ : - +welche als arithmetisches Mittel aller Normalspannungen im infinitesimalen Würfel definiert ist. +Die zweite Invariante ist die deviatorische Spannung +\[ +q += +\sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{11}-\sigma_{33})^{2}+(\sigma_{22}-\sigma_{33})^{2}}{2}} +. +\] +Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt. +Die hydrostatische Spannung $p$ kann gemäss Gleichung (Nr) als \[ p = \frac{\sigma_{11}+2\sigma_{33}}{3} \] - +vereinfacht werden. +Die deviatorische Spannung $q$ wird gemäss Gleichung (Nr) als \[ q = \sigma_{11}-\sigma_{33} \] +vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten. -p ist das arithmetische Mittel von der Spannung im infinitesimalen Würfel. -q ist die Differenz zwischen der Spannung in vertikaler Richtung und der Spannung in Richtung 2 und 3. -Man kann p als Druckspannung und q als Schubspannung anschauen. - -Aus der Formel vom vorherigen Kapitel konnten wir die Spannungen berechnen. -Deshalb kann man nun p und q in die Gleichung einsetzen. -Die Dehnungen werden mit neuen Variablen eingeführt. -Die Deviatorische Dehnung kann mit einer Schubdehnung verglichen werden. -Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen werden. - -\[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} -= -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{\nu}} -\] - +Die Invarianten können mit der Spannungsformel (Nr..xxx) berechnet werden. +Durch geschickte Umformung dieser Gleichung, lassen sich die Module als Faktor separieren. +Dabei entstehen spezielle Faktoren mit den Dehnungskomponenten. +So ergibt sich \[ \overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} = -\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{s}} +\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{v}} \] - +und \[ -\varepsilon_{s} +\overbrace{\sigma_{11}-\sigma_{33}}^{q} = -\text{Hydrostatische Dehnung} [-] +\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{s}} +. \] - +Die Faktoren mit den Dehnungskomponenten können so mit \[ -\varepsilon_{\nu} +\varepsilon_{v} += +(\varepsilon_{11} - 2\varepsilon_{33}) +\qquad +\text{und} +\qquad +\varepsilon_{s} = -\text{Deviatorische Dehnung} [-] +\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33}) \] - -Diese Komponenten kann man nun in die Vereinfachte Matrix einsetzen. -Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. - +eingeführt werden, mit +\begin{align*} + \varepsilon_{v} &= \text{Hydrostatische Dehnung [-]} \\ + \varepsilon_{s} &= \text{Deviatorische Dehnung [-].} +\end{align*} +Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglichen werden. +Die deviatorische Dehnung $\varepsilon_{s}$ kann mit einer Verzerrung verglichen werden. + +Diese zwei Gleichungen kann man durch die Matrixschreibweise \[ \begin{pmatrix} q\\ @@ -85,10 +93,13 @@ Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor \end{pmatrix} \begin{pmatrix} \varepsilon_{s}\\ - \varepsilon_{\nu} + \varepsilon_{v} \end{pmatrix} \] +(sollte nummeriert sein) vereinfachen. +Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. +Änderungen des Spannungszustandes können mit dieser Gleichung vollumfänglich erfasst werden. -Mit dieser Formel lassen sich verschieden Parameter von Versuchen analysieren und berechnen. -Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird ist der Oedometer-Versuch. +Mit dieser Formel lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. +Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index 85e9b1b..d524f13 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -1,68 +1,79 @@ -\section{Spannungsausbreitung\label{spannung:section:Oedometer - Versuch}} -\rhead{Oedometer - Versuch} -Beim Oedometer - Versucht hat man einen Stahlring mit einer Filterplatte am Boden. -In diesen Stahlring wird eine Bodenprobe eingefüllt. -Anschliessend wir mit einer Platte das Bodenmaterial mit einer ansteigenden Kraft belastet. - -Die Probe wird sich so verdichten. Das Volumen nimmt ab. -Der Stahlring verhindert ein seitliches ausbrechen oder entweichen der Bodenprobe. -Die Dehnung auf der Seite beträgt somit 0. -Mit dem Wert der Kraft und der Fläche lässt sich die Spannung berechnen. -Anhand der Volumenabnahme errechnet man die Dehnung. -Aus diesen Werten lässt sich wiederum das E-Modul bestimmen. -Beim Oedometer Versuch ist das E-Modul als $E_{OED}$ bezeichnet. - -Das $E_{OED}$ hat man speziell in der Geotechnik. -Dies aufgrund der speziellen Situation wo man sich mit dem infinitesimalen Würfel befindet. -Mit dem Stahlring, der verhindert das Material seitlich entweichen kann hat man ganz ähnliche Verhältnisse wie tief im Untergrund. -Auch dort kann das Material bei einer Belastung nicht seitlich entweichen. - -Wichtig ist nochmals zu betonen, dass alle diese beschriebenen Berechnungen ausschliesslich im linear-elastischen Materialverhalten funktionieren. -So ist es auch beim Oedometer - Versuch. -Den Versuch kann man auf einem $\sigma$ und $\varepsilon$ Diagramm abtragen. - -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} - \caption{Diagramm Oedometer - Versuch} - \label{fig:Diagramm Oedometer - Versuch} -\end{figure} - -Bei einem Versuch mit anderem Baumaterial wie beispielsweise Holz nimmt die Dehnung im Laufe des Versuchs stärker zu, obwohl weniger Spannung abgetragen wird. -Bei den meisten Böden ist dies anders. Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. - -Man kann die Dehnung in unsere vereinfachte Matrix einsetzen. Das E-Modul ersetzt man mit dem $E_{OED}$. +\section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}} +\rhead{Oedometer-Versuch} +Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. +Dieser beschreibt ebenfalls das Verhältnis zwischen Spannung und Dehnung, allerdings unter anderen Bedingungen. +Diese Bedingung ist das Verhindern der seitlichen Verformung, sprich der Dehnung in Richtung $1$ und $2$. +Es wird ein Probeelement mit immer grösseren Gewichten belastet, welche gleichmässig auf das Material drücken. +Die seitliche Verschiebung des Materials wird durch einen Stahlring verhindert. +Die Probe wird sich so stetig verdichten. +Das Volumen nimmt ab und die Dehnung nimmt immer mehr zu. +Unter diesen Bedingungen wird der oedometrische Elastizitätsmodul mit steigender Dehnung zunehmen. +Da im Boden das umgebende Material ähnlich eine seitliche Verformung verhindert, +bildet dieser oedometrische Elastizitätsmodul die Realität besser ab, als der gewöhnliche Elastizitätsmodul. +Durch dieses Verhindern des seitlichen Ausbrechens ist \[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} +\varepsilon_{22} = -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - 0)}^{\varepsilon_{\nu}} +\varepsilon_{33} += +0 \] - +aber auch \[ -\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} +\sigma_{22} = -\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\cdot0)}^{\varepsilon_{s}} +\sigma_{33} +\neq 0 +. \] - +Die Spannung $\sigma_{11}$ wird durch die aufgebrachte Kraft mit +\[ +\sigma_{11} += +\frac{F}{A} +\] +und die Dehnung $\varepsilon_{11}$ jeweils mit den entsprechenden Setzungen berechnet. +Diese Randbedingungen können in die vereinfachte Gleichung (Nrxxx) eingesetzt werden. +Diese lautet nun: \[ \begin{pmatrix} \sigma_{11}-\sigma_{33} \\ \sigma_{11}+2\sigma_{33} \end{pmatrix} = -\begin{bmatrix} - \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{(1-2\nu)} -\end{bmatrix} +\begin{pmatrix} + \frac{E_{OED}}{(1+\nu)} & 0 \\ + 0 & \frac{E_{OED}}{3(1-2\nu)} +\end{pmatrix} \begin{pmatrix} \varepsilon_{11}\\ \varepsilon_{11} \end{pmatrix} +. \] +Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen +\[ +\sigma_{11}-\sigma_{33} += +\frac{E_{OED}}{(1+\nu)}\cdot\varepsilon_{11} +\] +und +\[ +\sigma_{11}+2\sigma_{33} += +\frac{E_{OED}}{3(1-2\nu)}\cdot\varepsilon_{11} +\] +berechnen. +Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen. +Die Poisson-Zahl muss als Kennwert gemäss der Bodenklasse gewählt werden. +Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung 1.7). +Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. +Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. -An einem geeigneten Punkt, wo man noch im linear-elastischen Materialverhalten ist, kann man nun das $E_{OED}$ abtragen. -Es wird nur ein Delta betrachtet um $E_{OED}$ zu berechnen. -Man darf die Dehnung nicht über den gesamten Verlauf betrachten um $E_{OED}$ zu berechnen. - -Mit diesem ermittelten E-Modul kann man nun weitere Berechnungen für die Geotechnik durchführen. +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} + \caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material} + \label{fig:DiagrammOedometer-Versuch} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex index 638d9dd..6a5dc28 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -1,55 +1,70 @@ -\section{Versuchsreihe} -\label{section:verkehr/versuchsreihe} +\section{Einführung} +\label{section:verkehr/einfuehrung} -Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. -Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. -Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. -Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. +\subsection{Verkehrsnetze} +Das Verkehrsnetz besteht aus allen Anlagen, auf oder unter der Erdoberfläche, auf denen eine räumliche Fortbewegung von Personen oder auch Gütern stattfindet. Verkehrsnetze sind ein Bestandteil der Verkehrsinfrastruktur, die auf topografischen Karten festgehalten werden. Sie umfassen den Schienenverkehr, alle Strassen und Wege, wie auch Flugplätze und alle dazugehörigen Bauwerke. +Aus verkehrsgeografischer Sicht besteht das Verkehrsnetz aus Kanten, Knotenpunkten und dem Hinterland. Die Knotenpunkte werden auch hier durch die Kanten verbunden, die den Verkehrsstrom aufnehmen, wobei das Hinterland durch einzelne Knoten versorgt wird. Die Aufteilung in Kanten und Knotenpunkte ermöglicht eine Vereinfachung komplexer Verkehrsnetze, damit sie mittels der Graphentheorie untersucht werden können. +Grundsätzlich können kurze Wege zwischen den Knotenpunkten das Ziel beim +Aufbau eines Verkehrsnetzes sein. Es kann aber auch versucht werden, die Bau- und Unterhaltskosten des Verkehrsnetzes in einem gewissen Rahmen zu halten. Aus diesen Vorgaben ergibt sich dann, je nach dem was gewünscht wird, eine grob- oder feinmaschige Struktur des Netzes. +Ziel ist aber ein möglichst wirtschaftliches und optimales Verkehrsnetz. -Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. +\subsection{Suchalgorithmen} -\subsection{Einfluss der Knotenzahl auf die Rechenzeit} -\label{verkehr:Knotenzahl} +\subsubsection{Dijkstra-Algorithmus} +Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Infomratikprofessor Edsger Dijkstra. Den Algorithmus hat er im Jahr 1959 erfunden. +Der Algorithmus von Dijkstra ist ein Greedy-Algorithmus (gieriger Algorithmus), der schrittweise einen Folgezustand auswählt, damit beim Zeitpunkt der Wahl der grösste Gewinn bzw. das beste Ergebnis erzielt werden kann. +Trotz der Schnelligkeit der Greedy-Algorithmen, können viele Probleme nicht optimal gelöst werden. +Vereinfacht wird beim Dijkstra-Algorithmus, ausgehend von einem Startknoten so lange dem kürzesten Pfad gefolgt, bis der Zielknoten erreicht wird. Dabei muss für jeden besuchten Knoten die Kostenfunktion als auch der Pfad dahin (vorheriger Knoten) gespeichert werden. +Dadurch wird hingegen garantiert, dass, wenn der Zielknoten erreicht wird, auch der kürzeste Pfad gefunden wurde. +Grundlegende Voraussetzung für den Dijkstra-Algorithmus ist die strikte Positivität der Kantengewichte. Andernfalls würde ein wiederholtes Ablaufen einer Kante mit negativem Gewicht zu einer stetigen Reduktion der Kostenfunktion führen, was zu einer unendlichen Schlaufe führen würde. -\begin{figure} -\centering -\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr1.png} +\subsubsection{A*-Algorithmus} +Suchalgorithmen werden nach einfachen (uninformierte) und heuristischen (informierten) Algorithmen unterschieden. Während einfache Algorithmen den Suchraum intuitiv durchsuchen, beziehen heuristische Algorithmen Wissen über den Suchraum mit ein. +Der A*-Algorithmus geht auf seine Erfinder Peter Hart, Nils Nilsson und Bertram Raphael zurück, die den Algorithmus erstmals im Jahr 1968 beschrieben. +Der A*-Algorithmus ist ein heuristischer Suchalgorithmus, der den kürzesten Pfad zwischen zwei Knoten in einem Graphen mit positiven Kantengewichten berechnet. +Im Gegensatz zu einfachen Suchalgorithmen, wird beim A*-Algorithmus eine Schätzfunktion, die sogenannte Heuristik, verwendet. Dies ermöglicht ein zielgerichtetes Suchen und gleichzeitig wird die Laufzeit verringert. +Ausserdem findet der A*-Algorithmus immer eine optimale Lösung, sofern eine vorhanden ist. +Der A*-Algorithmus wird als Verallgemeinerung gehandhabt und gilt als Erweiterung des Dijkstra-Algorithmus. +======= -\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr1} -\end{figure} +\subsubsection{Floyd-Warshall-Algorithmus} +Der Floyd-Warshall-Algorithmus wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt. +Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die kürzesten , beziehungsweise die optimalsten Wege zwischen allen Paaren von Knoten berechnet, sofern der Graph keinen negativen Kreis (Zyklus) aufweist. +Ein Kreis in einem Graphen ist ein Weg, bei dem Start- und Endpunkt den gleichen Knoten aufweisen. Dieser wird negativ, wenn die Summe der gewichteten Kanten kleiner als Null wird. -In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. -Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. -Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. +\subsubsection{Euklidische Heuristik} +Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar. +Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant. -\begin{figure} -\centering -\includegraphics[width=12cm]{papers/verkehr/figures/chart_pathDiff.png} +\subsection{PageRank-Algorithmus} +Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc.. +Beim PageRank-Algorithmus handelt es sich um den Algorithmus von Google, aus dem die Google-Matrix abgeleitet wird. +Die Google-Matrix ist eine immens grosse Matrix mit Millionen Zeilen und Spalten, die für die schnelle und vor allem exakte Bestimmung der PageRanks (Gewichtung) eine grosse Bedeutung hat. +Der PageRank-Algorithmus analysiert und gewichtet beispielsweise die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur. +Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\ +Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche gilt. -\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} -\label{verkehr:pathDifference} -\end{figure} +%THEORIE... +Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen. +\begin{equation} +A_{i,j}=\left\{ \begin{matrix} +1 & \text{Kante von $j$ nach $i$} \\ 0 & \text{keine Kante von $j$ nach $i$} +\end{matrix} + \right. +\label{verkehr:Adja} +\end{equation} -\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} -\begin{figure} -\centering -\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}\\ -\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr2} -\end{figure} - -Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ -Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. - -\begin{figure} -\centering -\includegraphics[width=6cm]{papers/verkehr/figures/network_dij.png}\qquad -\includegraphics[width=6cm]{papers/verkehr/figures/network_aStar.png} -\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} -\label{verkehr:Comparison} -\end{figure} - -In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. +Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1...n\right\}\end{equation} +Beim PageRank-Algorithmus wird eine abgewandelte Form der Adjazenz-Matrix verwendet. +Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt. +\begin{equation} P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \end{equation} +Anschliessend multipliziert man diese Matrix $P$ mit einem Spaltenvektor $\Vec{r_0}$ mit $n$ Einträgen, für welchen gilt: +\begin{equation} \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1...n\right\} \end{equation} +Dieser Vektor stellt ein neutrales Ranking dar. Alle Knoten werden gleich gewichtet. +Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das "erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt. +\begin{equation} \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\end{equation} +somit +\begin{equation} \Vec{r_i} = P^i\cdot\Vec{r_0}\end{equation} +Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ und stellt das abschliessende Ranking dar. diff --git a/buch/test1.tex b/buch/test1.tex deleted file mode 100644 index 8345230..0000000 --- a/buch/test1.tex +++ /dev/null @@ -1,93 +0,0 @@ -% -% test1.tex -- Test 1 -% -% (c) 2012 Prof. Dr. Andreas Mueller, HSR -% -%\documentclass[a4paper,12pt]{book} -\documentclass[a4paper,12pt]{article} -\usepackage{geometry} -\geometry{papersize={210mm,297mm},total={165mm,260mm}} -\usepackage{ngerman} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{times} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{amsfonts} -\usepackage{amsthm} -\usepackage{graphicx} -\usepackage{fancyhdr} -\usepackage{textcomp} -\usepackage[all]{xy} -\usepackage{txfonts} -\usepackage{alltt} -\usepackage{verbatim} -\usepackage{paralist} -\usepackage{makeidx} -\usepackage{array} -\usepackage{hyperref} -\usepackage{caption} -\usepackage{subcaption} -\usepackage{standalone} -\usepackage{environ} -\usepackage{tikz} -\input{../common/linsys.tex} -\newcounter{beispiel} -\newenvironment{beispiele}{ -\bgroup\smallskip\parindent0pt\bf Beispiele\egroup - -\begin{list}{\arabic{beispiel}.} - {\usecounter{beispiel} - \setlength{\labelsep}{5mm} - \setlength{\rightmargin}{0pt} -}}{\end{list}} -\newcounter{uebungsaufgabe} -% environment fuer uebungsaufgaben -\newenvironment{uebungsaufgaben}{ -\begin{list}{\arabic{uebungsaufgabe}.} - {\usecounter{uebungsaufgabe} - \setlength{\labelwidth}{2cm} - \setlength{\leftmargin}{0pt} - \setlength{\labelsep}{5mm} - \setlength{\rightmargin}{0pt} - \setlength{\itemindent}{0pt} -}}{\end{list}\vfill\pagebreak} -\newenvironment{teilaufgaben}{ -\begin{enumerate} -\renewcommand{\labelenumi}{\alph{enumi})} -}{\end{enumerate}} -% Loesung -\NewEnviron{loesung}{% -\begin{proof}[L"osung]% -\renewcommand{\qedsymbol}{$\bigcirc$} -\BODY -\end{proof}} -\NewEnviron{bewertung}{\relax} -\NewEnviron{diskussion}{ -\BODY -} -\RenewEnviron{loesung}{\relax} -\RenewEnviron{diskussion}{\relax} -\newenvironment{hinweis}{% -\renewcommand{\qedsymbol}{} -\begin{proof}[Hinweis]}{\end{proof}} - -\begin{document} -{\parindent0pt\hbox to\hsize{% -Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} -\vspace{0.5cm} - -\section*{Kurztest 1} - -\begin{uebungsaufgaben} - -\item -\input chapters/30-endlichekoerper/uebungsaufgaben/3003.tex -\item -\input chapters/30-endlichekoerper/uebungsaufgaben/3004.tex -\item -\input chapters/30-endlichekoerper/uebungsaufgaben/3005.tex - -\end{uebungsaufgaben} - -\end{document} diff --git a/buch/test2.tex b/buch/test2.tex deleted file mode 100644 index ea842ce..0000000 --- a/buch/test2.tex +++ /dev/null @@ -1,91 +0,0 @@ -% -% test2.tex -- Test 2 -% -% (c) 2012 Prof. Dr. Andreas Mueller, HSR -% -%\documentclass[a4paper,12pt]{book} -\documentclass[a4paper,12pt]{article} -\usepackage{geometry} -\geometry{papersize={210mm,297mm},total={165mm,260mm}} -\usepackage{ngerman} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{times} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{amsfonts} -\usepackage{amsthm} -\usepackage{graphicx} -\usepackage{fancyhdr} -\usepackage{textcomp} -\usepackage[all]{xy} -\usepackage{txfonts} -\usepackage{alltt} -\usepackage{verbatim} -\usepackage{paralist} -\usepackage{makeidx} -\usepackage{array} -\usepackage{hyperref} -\usepackage{caption} -\usepackage{subcaption} -\usepackage{standalone} -\usepackage{environ} -\usepackage{tikz} -\input{../common/linsys.tex} -\newcounter{beispiel} -\newenvironment{beispiele}{ -\bgroup\smallskip\parindent0pt\bf Beispiele\egroup - -\begin{list}{\arabic{beispiel}.} - {\usecounter{beispiel} - \setlength{\labelsep}{5mm} - \setlength{\rightmargin}{0pt} -}}{\end{list}} -\newcounter{uebungsaufgabe} -% environment fuer uebungsaufgaben -\newenvironment{uebungsaufgaben}{ -\begin{list}{\arabic{uebungsaufgabe}.} - {\usecounter{uebungsaufgabe} - \setlength{\labelwidth}{2cm} - \setlength{\leftmargin}{0pt} - \setlength{\labelsep}{5mm} - \setlength{\rightmargin}{0pt} - \setlength{\itemindent}{0pt} -}}{\end{list}\vfill\pagebreak} -\newenvironment{teilaufgaben}{ -\begin{enumerate} -\renewcommand{\labelenumi}{\alph{enumi})} -}{\end{enumerate}} -% Loesung -\NewEnviron{loesung}{% -\begin{proof}[L"osung]% -\renewcommand{\qedsymbol}{$\bigcirc$} -\BODY -\end{proof}} -\NewEnviron{bewertung}{\relax} -\NewEnviron{diskussion}{ -\BODY -} -\RenewEnviron{loesung}{\relax} -\RenewEnviron{diskussion}{\relax} -\newenvironment{hinweis}{% -\renewcommand{\qedsymbol}{} -\begin{proof}[Hinweis]}{\end{proof}} - -\begin{document} -{\parindent0pt\hbox to\hsize{% -Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} -\vspace{0.5cm} - -\section*{Kurztest 2} - -\begin{uebungsaufgaben} - -\item -\input chapters/40-eigenwerte/uebungsaufgaben/4004.tex -\item -\input chapters/40-eigenwerte/uebungsaufgaben/4005.tex - -\end{uebungsaufgaben} - -\end{document} |
