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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/erdbeben/teil1.tex b/buch/papers/erdbeben/teil1.tex index 0d21f84..bb3bdd4 100644 --- a/buch/papers/erdbeben/teil1.tex +++ b/buch/papers/erdbeben/teil1.tex @@ -10,13 +10,12 @@ % - \section{Kalman Filter} \subsection{Geschichte} Das Kalman Filter wurde 1960 von Rudolf Emil Kalman entdeckt und direkt von der NASA für die Appollo Mission benutzt. Der Filter kommt mit wenig Rechenleistung aus und war somit dafür geeignet die Rakete bei der Navigation zu unterstützen. Das Filter schätzt den Zustand eines Systems anhand von Messungen und kann den nächsten Zustand errechnen. Typische Anwendungen des Kalman-Filters sind die Glättung von verrauschten Daten und die Schätzung von Parametern und kommt heutzutage in jedem Satellit, Navigationssystem, Smartphones und Videospielen vor. \subsection{Wahrscheinlichkeit} -Das Kalman Filter versucht nichts anderes, als ein geeigneter Wert zwischen zwei Normalverteilungen zu schätzen. Die eine Kurve zeigt die errechnete Vorhersage des Zustands, bzw. deren Normal- Gauss-Verteilung. Die andere Kurve zeigt die verrauschte Messung des nächsten Zustand, bzw. deren Normal-Verteilung. Wie man in am Beispiel dieser zwei Gauss-Verteilungen sehen kann, ist sowohl der geschätzte Zustand als auch der gemessene Zustand nicht am selben Punkt. +Das Kalman-Filter schätzt den wahrscheinlichsten Wert zwischen zwei Normalverteilungen oder auch Gauss-Verteilung. Die eine Kurve zeigt die errechnete Vorhersage des Zustands, bzw. deren Normalverteilung. Die andere Kurve zeigt die verrauschte Messung des nächsten Zustand, bzw. deren Normalverteilung. Wie man am Beispiel dieser zwei Gauss-Verteilungen sehen kann, ist sowohl der geschätzte Zustand als auch der gemessene Zustand verteilt und haben unterschiedliche Standardabweichungen $\sigma$ und Erwartungswerte $\mu$. @@ -43,10 +42,10 @@ und für die Messung: Diesen werden nun Multipliziert und durch deren Fläche geteilt um sie wieder zu Normieren: \begin{equation} -{y_f}(x;{\mu_f},{\sigma_f})=\frac{ \frac{1}{\sqrt{2\pi\sigma_1^2}}e^{-\frac{(x-{\mu_1})^2}{2{\sigma_1}^2}} \cdot \frac{1}{\sqrt{2\pi\sigma_2^2}}e^{-\frac{(x-{\mu_2})^2}{2{\sigma_2}^2}}}{\int {y_1}*{y_2}\,} +{y_f}(x;{\mu_f},{\sigma_f})=\frac{ \frac{1}{\sqrt{2\pi\sigma_1^2}}e^{-\frac{(x-{\mu_1})^2}{2{\sigma_1}^2}} \cdot \frac{1}{\sqrt{2\pi\sigma_2^2}}e^{-\frac{(x-{\mu_2})^2}{2{\sigma_2}^2}}}{\int {y_1}*{y_2} dx\,} \end{equation} -Dadurch gleicht sich die neue Kurve den anderen an. Interessant daran ist, dass die fusionierte Kurve sich der genauere Normal-Verteilung anpasst. ist ${\sigma_2}$ klein und ${\sigma_1}$ gross, so wird sich die fusionierte Kurve näher an ${y_2}(x;{\mu_2},{\sigma_2})$ begeben. Sie ist also Gewichtet und die best mögliche Schätzung. +Dadurch gleicht sich die neue Kurve den anderen an. Interessant daran ist, dass die fusionierte Kurve sich der genauere Normal-Verteilung anpasst. ist ${\sigma_2}$ klein und ${\sigma_1}$ gross, so wird sich die fusionierte Kurve näher an ${y_2}(x;{\mu_2},{\sigma_2})$ begeben. Sie ist also gewichtet und die best mögliche Schätzung. \begin{figure} @@ -59,14 +58,13 @@ Dadurch gleicht sich die neue Kurve den anderen an. Interessant daran ist, dass Was in 2 Dimensionen erklärt wurde, funktioniert auch in mehreren Dimensionen. Dieses Prinzip mach sich der Kalman Filter zu nutze, und wird von uns für die Erdbeben Berechnung genutzt. -\subsection{Anwendungsgrenzen} -Nicht lineare Systeme %Noch nicht Fertig + \section{Aufbau} -Um ein Erdbeben kenntlich zumachen werden in der Regel Seismographen mit vielen Sensoren verwendet. -Ein Seismograph besteht im Grunde aus einer federgelagerten Masse. Wirkt eine Bodenerregung auf das Gerät ein, bleibt die gekoppelte Masse in der regel stehen und das Gehäuse schwingt mit.Relativbewegung des Bodens kann damit als Längenänderung im Zeitverlauf gemessen werden. In modernen Seismographen wird die Bodenbewegung in alle Richtungen gemessen, sowohl Horizontal als auch Vertikal. -Wir konstruieren uns eine einfachere Version eines Seismographen, welcher rein mechanisch funktioniert. Zudem kann er nur in eine Dimension Messwerte aufnehmen. Würde das System ausgebaut werden, um alle Horizontalbewegungen aufzunehmen, würde der Verwendung des Kalman-Filters zu kompliziert werden. Für zwei Dimensionen (x,y) würde der Pythagoras für das System benötigt werden. Da sich der Pythagoras bekanntlich nicht linear verhält, kann kein lineares Kalman-Filter implementiert werden. Da das Kalman-Filter besonders effektiv und einfach für lineare Abläufe geeignet ist, würde eine Zweidimensionale Betrachtung den Rahmen dieser Arbeit sprengen. Für ein nicht-lineares System werden Extended Kalman-Filter benötigt, bei denen die System-Matrix (A) durch die Jacobi-Matrix des System ersetzt wird. +Um ein Erdbeben kenntlich zu machen werden in der Regel Seismographen mit vielen Sensoren verwendet. +Ein Seismograph besteht im Grunde aus einer federgelagerten Masse. Wirkt eine Bodenerregung auf das Gerät ein, bleibt die gekoppelte Masse stehen und das Gehäuse schwingt mit.Relativbewegung des Bodens kann damit als Längenänderung im Zeitverlauf gemessen werden. In modernen Seismographen wird die Bodenbewegung in alle Richtungen gemessen, sowohl Horizontal als auch Vertikal. +Wir konstruieren uns eine einfachere Version eines Seismographen, welcher rein mechanisch funktioniert. Zudem kann er nur in eine Dimension Messwerte aufnehmen. Würde das System ausgebaut werden, um alle Horizontalbewegungen aufzunehmen, würde der Verwendung des Kalman-Filters zu kompliziert werden. Für zwei Dimensionen (x,y) würde der Pythagoras für das System benötigt werden. Da sich der Pythagoras bekanntlich nicht linear verhält, kann kein lineares Kalman-Filter implementiert werden. Da das Kalman-Filter besonders effektiv und einfach für lineare Abläufe geeignet ist, würde eine Zweidimensionale Betrachtung den Rahmen dieser Arbeit sprengen. Für ein nicht-lineares System werden Extended Kalman-Filter benötigt, bei denen die System-Matrix (A) durch die Jacobi-Matrix des System ersetzt wird. Einfachheitshalber beschränken wir uns aber auf den linearen Fall, da dadurch die wesentlichen punkte bereits aufgezeigt werden. \begin{figure} \begin{center} @@ -76,36 +74,20 @@ Wir konstruieren uns eine einfachere Version eines Seismographen, welcher rein m \end{figure} -\subsection{Optionen} -Wollte man einen 2D Seismographen aufbauen, ohne den Pythagroas zu verwenden, kann dies mit der Annahme, das die Feder sehr lang sind erfolgen. Da sich bei langen Federn die Auslenkungen verkleiner...!!Noch nicht fertig! \section{Systemgleichung} -Da das Kalman-Filter zum Schätzen des nächsten Zustand verwendet wird, wird eine Gleichung, welche das System beschreibt. Das Kalman-Filter benötigt eine Beschreibung der Systemdynamik. Im Fall unseres Seismographen, kann die Differentialgleichung zweiter Ordnung einer gedämpften Schwingung am harmonischen Oszillator verwendet werden. Diese lautet: +Da das Kalman-Filter zum Schätzen des nächsten Zustand verwendet wird, benötigt das Kalman-Filter eine Beschreibung der Systemdynamik. Im Fall unseres Seismographen, kann die Differentialgleichung zweiter Ordnung einer gedämpften Schwingung am harmonischen Oszillator verwendet werden. Diese lautet: \begin{equation} m\ddot x + 2k \dot x + Dx = f \end{equation} mit den Konstanten $m$ = Masse, $k$ = Dämpfungskonstante und $D$ = Federkonstante. -Um diese nun in die Systemmatrix umzuwandeln, wird aus der Differentialgleichung zweiter Ordnung durch eine Substitution eine DGL erster Ordnung: - -\begin{equation} -{x_1}=x, \qquad +Um diese nun in die Systemmatrix umzuwandeln, wird aus der Differentialgleichung zweiter Ordnung durch die Substitution
\[ {x_1}=x, \qquad {x_2}=\dot x, \qquad -{x_3}=\ddot x\qquad \mid \quad \text {Substitution} -\end{equation} - - -\begin{equation} -m{x_3}+ 2k{x_2} + D{x_1} = f\qquad \mid \quad \text {DGL 1. Ordnung} -\end{equation} +{x_3}=\ddot x\qquad\]
erhalten wir die Differentialgleichung
\[ m{x_3}+ 2k{x_2} + D{x_1} = f.\]
Diese können wir nun in der Form
\[ {x_3}=-\frac{D}{m} {x_1} -\frac{2k}{m} {x_2} + \frac{f} {m} \]
auch als Matrix-Vektor-Gleichung darstellen.
-\begin{equation} -{x_3}=-\frac{D}{m} {x_1} -\frac{2k}{m} {x_2} + \frac{f} {m} \qquad \mid \quad \text {nach} \quad{x_3} -\end{equation} -auch als Matrix-Vektor-Gleichung schreiben. -Hierbei beschreibt die Matrix $A$ die gesamte Systemdynamik in der Form, wie sie ein Kalman-Filter benötigt. -Um die lineare Differentialgleichung in das Kalman-Filter zu implementieren, muss dieses als Vektor-Gleichung umgewandelt werden. Dafür wird die Gleichung in die Zustände aufgeteilt. Die für uns relevanten Zustände sind die Position der Masse, die Geschwindigkeit der Masse und äussere Beschleunigung des ganzen System. Dabei muss unterschieden werden. um welche Beschleunigung es sich handelt. Das System beinhaltet sowohl eine Beschleunigung der Masse bzw. Feder (innere Beschleunigung), als auch eine Beschleunigung der ganzen Apparatur (äussere Beschleunigung). In unserem Fall wird die äusseren Beschleunigung gesucht, da diese der Erdbeben Anregung gleich kommt. +Dafür wird die Gleichung in die Zustände aufgeteilt. Die für uns relevanten Zustände sind die Position der Masse, die Geschwindigkeit der Masse und die äussere Beschleunigung des ganzen System. Dabei muss unterschieden werden, um welche Beschleunigung es sich handelt. Das System beinhaltet sowohl eine Beschleunigung der Masse bzw. Feder (innere Beschleunigung), als auch eine Beschleunigung der ganzen Apparatur (äussere Beschleunigung). In unserem Fall wird die äusseren Beschleunigung gesucht, da diese der Erdbeben Anregung gleich kommt. \begin{equation} @@ -134,7 +116,7 @@ Um den Kalman Filter zu starten, müssen gewisse Bedingungen definiert werden. I \subsection{Anfangsbedingungen} \subsubsection*{Anfangszustand $x$} -Das Filter benötigt eine Anfangsbedingung. In unserem Fall ist es die Ruhelage, die Masse bewegt sich nicht. Zudem erföhrt die Apparatur keine äussere Kraft. +Das Filter benötigt eine Anfangsbedingung. In unserem Fall ist es die Ruhelage, die Masse bewegt sich nicht. Zudem erfährt die Apparatur keine äussere Kraft. \begin{equation} {x_0 }= \left( \begin{array}{c} 0\\ 0\\ 0\end{array}\right) @@ -142,6 +124,8 @@ Das Filter benötigt eine Anfangsbedingung. In unserem Fall ist es die Ruhelage, \subsubsection*{Anfangsfehler / Kovarianzmatrix $P$} Da auch der Anfangszustand fehlerhaft sein kann, wird für den Filter einen Anfangsfehler eingeführt. Auf der Diagonalen werden die Varianzen eingesetzt, in den restlichen Felder stehen die Kovarianzen. +Zur Erinnerung: Die Varianz ist ein Mass für die Streuung eines Wertes, die Kovarianz hingegen beschreibt die Abhängigkeit der Streuungen zweier Werte.
Kovarianz: Cov(x, y) undVarianz: Var(x) = Cov(x, x) + In unserem Fall ist der Anfangszustand gut bekannt. Wir gehen davon aus, dass das System in Ruhe und in Abwesenheit eines Erdbeben startet, somit kann die Matrix mit Nullen bestückt werden. Somit ergibt sich für die Kovarianzmatrix \begin{equation} @@ -183,16 +167,16 @@ Q = \left( \end{array}\right) \end{equation} -Die Standabweichungen müssten Statistisch ermittelt werden, da der Fehler nicht vom Sensor kommt und somit nicht vom Hersteller gegeben ist. Das Bedeutet wiederum dass $Q$ die Unsicherheit des Prozesses beschreibt, und die Messung. +Die Standabweichungen müssten Statistisch ermittelt werden, da der Fehler nicht vom Sensor kommt und somit nicht vom Hersteller gegeben ist. Das Bedeutet wiederum dass $Q$ die Unsicherheit des Prozesses beschreibt, und die der Messung. \subsubsection*{Messmatrix $H$} -Die Messmatrix gibt an, welcher Parameter gemessen werden soll. In unserem Fall ist es nur die Position der Massen. +Die Messmatrix gibt an, welche Parameter gemessen werden soll. In unserem Falle ist es nur die Position der Massen. \[ H = (1, 0, 0) \] \subsubsection*{Messrauschkovarianz $R$} -Die Messrauschkovarianzmatrix beinhaltet, wie der Name es schon sagt, das Rauschen der Positionssensoren. In unserem Fall wird nur die Position der Masse gemessen. Da wir keine anderen Sensoren haben ist dies lediglich: +Die Messrauschkovarianzmatrix beinhaltet, wie der Name es schon sagt, das Rauschen der Positionssensoren. In unserem Fall wird nur die Position der Masse gemessen. Da wir keine anderen Sensoren haben ist $R$ lediglich: \begin{equation} R= ({\sigma_x}^2). \end{equation} @@ -203,17 +187,14 @@ Nachdem alle Parameter aufgestellt sind, wird der Filter initialisiert und wird \subsubsection*{Vorhersage} -Im Filterschritt Vorhersage wird der nächste Zustand anhand des Anfangszustand und der Systemmatrix berechnet. Dies funktioniert ganz Trivial mit dem Rechenschritt: +Im Filterschritt Vorhersage wird der nächste Zustand anhand des Anfangszustand und der Systemmatrix berechnet. Dies funktioniert mit dem Rechenschritt: \begin{equation} {x_{t+1}}=A\cdot{x_t}. \end{equation} -Die Kovarianz $P_{pred}$ wird ebenfalls neu berechnet, da die Unsicherheit im Vorhersage grösser wird als im Aktuellen. Da wir ein mehrdimensionales System haben, kommt noch die Messunsicherheit $Q$ dazu, so dass die Unsicherheit des Anfangsfehlers $P$ immer grösser wird. Dies funktioniert durch multiplizieren der Systemmatrix, deren Ableitung und mit dem aktualisierten Anfangsfehler. Dazu wird noch die Messunsicherheit addiert, somit entsteht die Gleichung - -\begin{equation} -{P_{pred}}=APA^T+Q. -\end{equation} +Die Kovarianz $P_{pred}$ wird ebenfalls neu berechnet. Da wir ein mehrdimensionales System haben, kommt noch die Prozessunsicherheit $Q$ dazu, so dass die Unsicherheit des Anfangsfehlers $P$ laufend verändert. Dies funktioniert durch multiplizieren der Systemmatrix mit dem aktualisierten Anfangsfehler. Dazu wird noch die Prozessunsicherheit addiert, somit entsteht die Gleichung +\[ P_\mathrm{pred} = A P A^T + Q . \] wird dieser Vorgang wiederholt, schaut der Filter wie genau die letzte Anpassung von $P$ zur Messung stimmt. Ist der Unterschied klein, wird die Kovarianz $P$ kleiner, ist der Unterschied gross, wird auch die Kovarianz grösser. Das Filter passt sich selber an und korrigiert sich bei grosser Abweichung. @@ -223,41 +204,164 @@ Hier bei wird lediglich die Messung mit dem Fehler behaftet, und die Messmatrix \begin{equation} w=Z-(H\cdot x) \end{equation} -Die Innovation ist der Teil der Messung, die nicht durch die Systemdynamik erklärt werden kann. -Innovation = Messung - Vorhersage. Dies ist Intuitiv logisch, eine Innovation von 0 bedeutet, dass die Messung nichts Neues hervorbrachte. +Die Innovation ist der Teil der Messung, die nicht durch die Systemdynamik erklärt werden kann. Die Hilfsgröße Innovation beschreibt, wie genau der vorhergesagte Mittelwert den aktuellen Messwert mittels der Beobachtungsgleichung beschreiben kann. Für eine schlechte Vorhersage wird die dazugehörige Innovation gross, für eine genaue Vorhersage dagegen klein sein. Entsprechende Korrekturen müssen dann gross bzw. nur gering ausfallen. Innovation = Messung - Vorhersage. Dies ist intuitiv logisch, eine Innovation von 0 bedeutet, dass die Messung nichts Neues hervorbrachte. Im nächsten Schritt wir analysiert, mit welcher Kovarianz weiter gerechnet wird. +Hierbei wird die Unsicherheit $P$, die Messmatrix $H$ und die Messunsicherheit $R$ miteinander verrechnet. +\begin{equation} +S=Z-(H\cdot P\cdot H`+R) +\end{equation} + + +\subsubsection*{Aktualisieren} +Im nächsten Schritt kommt nun die Wahrscheinlichkeit nach Gauss dazu. + +\begin{equation} +K= \frac{P \cdot H`}S +\end{equation} +Dieser Vorgang wird Kalman-Gain genannt. Er sagt aus, welcher Kurve mehr Vertraut werden soll, dem Messwert oder der Systemdynamik. +Das Kalman-Gain wird geringer wen der Messwert dem vorhergesagten Systemzustand entspricht. Sind die Messwerte komplett anders als die Vorhersage, wo werden die Elemente in der Matrix $K$ grösser. -\subsubsection*{Korrigieren} -Udpdate -\section{Anfügen der Schwingung} +Anhand der Informationen aus dem Kalman-Gain $K$ wird das System geupdated. + +\begin{equation} +x=x+(K \cdot w) +\end{equation} -Ein Erdbeben breitet sich im Boden wellenartig aus und bringt Objekte, wie zum Beispiel ein Gebäude, in Schwingung. -Diese Schwingungen pflanzen sich im Gebäude mit gleicher Amplitude, Geschwindigkeit und Beschleunigung in horizontaler und vertikaler Bewegung fort. -Wir möchten herauszufinden, wie gross die Massenbeschleunigung infolge eines Erdbeben ist. -Mit Hilfe von fiktiven Sensoren, die eine Ortsveränderung des Gebäude messen, können wir mit Anwendung von Matrizen und dem Kalman-Filter die Beschleunigung berechnen. +Dazu kommt eine neue Kovarianz für den nächste Vorhersageschritt: \begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{erdbeben:equation1} +P=(I-(K \cdot H)) \cdot P \end{equation} -\section{Erreger-Schwingung} -Wir möchten mit einer gedämpften harmonischen Schwingung ein einfaches Erdbeben simulieren, die im Kalman Filter eingespeist wird. -Die Gleichung lautet +Der ganze Ablauf wird nun zum Algorithmus und beginnt wieder mit der Vorhersage + +\begin{equation} +{x_{t+1}}=e^{A\Delta t}{ x_t}. +\end{equation} + + +\subsection{Zusammenfassung } +Zusammenfassend kann das Kalman-Filter in offizieller Typus dargestellt werden. Dabei beginnt das Filter mit dem Anfangszustand für $k=0$ + +1. Nächster Zustand vorhersagen +\begin{equation} +{x_{k|k-1}}={A_{k-1}}{x_{k-1}}+{B_{k-1}}{u_{k-1}} +\end{equation} + +2. Nächste Fehlerkovarianz vorhersagen +\begin{equation} +{P_{k|k-1}}={A_{k-1}}{P_{k-1}}{A_{k-1}^T}+{Q_{k-1}} +\end{equation} + +3. Das Kalman Filter anwenden \begin{equation} -x(t)=Ae^{t/2}sin(t). +{K_k}={P_{k-1}}{H_{k}^T({H_k}{P_{k|k-1}}{H_k}^T}+{R_k})^{-1} \end{equation} -Mit dieser Schwingung können wir ein einachsiger Seismograph simulieren, der eine Ortsverschiebung auf der x-Achse durchführt. -Die Dämpfung der Schwingung ist relevant, da das System beim Schwingungsvorgang durch die Federkonstante und der Reibung, Energie verliert. +4. Schätzung aktualisieren +\begin{equation} +{x_k}={x_{k|k-1}}+{K_k}({z_k}-{H_k}{x_{k|k-1}}) +\end{equation} + +5. Fehlerkovarianz aktualisieren +\begin{equation} +{P_k}=(I-{K_k}{H_k}){P_{k|k-1}} +\end{equation} + +6. Die Outputs von $k$ werden die Inputs für ${k-1}$ und werden wieder im Schritt 1 verwendet + + +\section{Matlab-Code} +Um das simulierte Erdbeben auf die theoretische Apparatur zu bringen und mit dem Kalman-Filter Resultate zu generieren, wurde in Matlab der Algorithmus programmiert. +\begin{lstlisting} +%% Initialisierte Werte +t0 = 0.00; % Anfangszeit +deltat = 0.01; % Zeitschritt +tend = 50.00; % Endzeit + +% Standard-Abweichungen Prozess +sigmax = 0.05e-3; % Position +sigmav = 0.01e-3; % Geschwindigkeit +sigmaf = 1; % (Äussere) Kraft + +% Standard-Abweichung Messung +sigmam = 0.01e-3; + +% Systemparameter +m = 1.00; % Masse +D = 0.30; % Federkonstante +k = 0.10; % Dämpfung + + +%% Kalmanfilter +% Initialisierung + + +% Anfangszustand (Position, Geschwindigkeit, Kraft) +x0 = [0; 0; 0]; + +% Unsicherheit des Anfangszustand +P0 = [0, 0, 0; ... + 0, 0, 0; ... + 0, 0, 0]; + +% Systemmatrizen +A = [0, 1, 0;... % Dynamikmatrix + -D/m, -2*k/m, 1;... + 0, 0, 0]; % Ableitungen von f(t) unbekant. Annahme: 0 +A = expm(A * deltat); + +Q = [sigmax^2, 0, 0;... + 0, sigmav^2, 0;... + 0, 0, sigmaf^2]; % Prozessrauschen (Covarianz) + +% Messprozess +H = [1, 0, 0]; % Messmatrix +R = sigmam^2; % Messrauschen (Könnte durch Versuche bestimmt werden) + +I = eye(3); % Identity matrix (Einheitsmatrix) + +% Filterprozess + +% Initialisieren der Variablen +N = length(t); % Anzahl Punkte im Einheitsvektor (= Anzahl Messwerte) +xhat = zeros(3, N); % Matrix mit geschätzten Zuständen + +% Index ':' bedeutet: 'alles' +% Index '(1, :)' bedeutet: 'alles aus der 1. Zeile' + +% Anfangszustand setzen +xhat(:, 1) = x0; +P = P0; + +% Kalman-Matrizen konvergiert. Vorab-Berechnung in 'genügenden' Iterationen +for idx = 1:100 + Ppred = A * P * A' + Q; % Prädizieren der Kovarianz + S = (H * Ppred * H' + R); % Innovationskovarianz + K = Ppred * H' / S; % Filter-Matrix (Kalman-Gain) + P = (I - K * H) * Ppred; % Aktualisieren der Kovarianz +end + +% Anfangszustand gegeben +% Erster zu berechnender Wert ist der zweite +for idx = 2:N + % Vorhersage + xpred = A * xhat(:, idx-1); % Prädizierter Zustand aus Bisherigem und System + % Ppred = A * P * A' + Q; % Prädizieren der Kovarianz + + % Korrektur + y = xt(idx) - H * xpred; % Messungen/ Kraft aus System - Vohersage + % S = (H * Ppred * H' + R); % Innovationskovarianz + % K = Ppred * H' / S; + + xhat(:, idx) = xpred + K * y; % Aktualisieren des Systemzustands + % P = (I - K * H) * Ppred; % Aktualisieren der Kovarianz +end +\end{lstlisting} + -Die Ergebnisse dieser Schwingung setzen wir in die Messmatrix ein und können den Kalman-Filter starten. diff --git a/buch/papers/ifs/images/FIC.pdf b/buch/papers/ifs/images/FIC.pdf new file mode 100644 index 0000000..1c76dfe --- /dev/null +++ b/buch/papers/ifs/images/FIC.pdf @@ -0,0 +1,2003 @@ +%PDF-1.6
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License terms: http://www.apache.org/licenses/LICENSE-2.0) +/BeginEPSF { %def +/b4_Inc_state save def % Save state for cleanup +/dict_count countdictstack def % Count objects on dict stack +/op_count count 1 sub def % Count objects on operand stack +userdict begin % Push userdict on dict stack +/showpage { } def % Redefine showpage, { } = null proc +0 setgray 0 setlinecap % Prepare graphics state +1 setlinewidth 0 setlinejoin +10 setmiterlimit [ ] 0 setdash newpath +/languagelevel where % If level not equal to 1 then +{pop languagelevel % set strokeadjust and +1 ne % overprint to their defaults. +{false setstrokeadjust false setoverprint +} if +} if +} bd +/EndEPSF { %def +count op_count sub {pop} repeat % Clean up stacks +countdictstack dict_count sub {end} repeat +b4_Inc_state restore +} bd +%%EndResource +%%EndProlog +%%Page: 1 1 +%%PageBoundingBox: 0 0 420 315 +%%BeginPageSetup +[1 0 0 -1 0 315] CT +%%EndPageSetup +GS +[0.75 0 0 0.75 0 0] CT +1 GC +N +0 0 560 420 re +f +GR +GS 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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 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! +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! 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+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! 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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 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! +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! 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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..817c5a4 100644 --- a/buch/papers/ifs/references.bib +++ b/buch/papers/ifs/references.bib @@ -4,32 +4,59 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{ifs:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@online{ifs:chaos, + title = {Chaosspiel}, + url = {https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem#Chaosspiel}, + date = {20201-06-13}, + year = {2021}, + month = {6}, + day = {13} } -@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: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}, +} + +@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} } -@article{ifs: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} +@article{ifs:pifs, + title = {Applications of Partitioned Iterated Function Systems in Image and Video Compression}, + journal = {Journal of Visual Communication and Image Representation}, + volume = 7, + number = {2}, + pages = {144-154}, + year = 1996, + issn = {1047-3203}, + doi = {https://doi.org/10.1006/jvci.1996.0014}, + url = {https://www.sciencedirect.com/science/article/pii/S1047320396900140}, + author = {Guojun Lu and Toon Lin Yew}, + abstract = {Iterated function systems (IFS) have been used to compress image data. Because of difficulty in finding IFS in natural images, a technique based on partitioned IFS (PIFS) has been proposed for image compression. In this technique, an image to be compressed is divided into nonoverlapping blocks. For each block an affine transformation is found in the image. This set of affine transformations (called PIFS) corresponds to a unique image. In the simplest case, images are partitioned into fixed size blocks. In this paper, we investigate image and video compression techniques using variable block sizes based on the quadtree partition. One property of images generated using PIFS is scalability: they have fine detail in any scale. We exploit this property to reduce required compression time and improve compression performance. There are large amounts of temporal redundancy between fames of a video sequence. We describe a method to remove temporal redundancies effectively using a quadtree partitioning technique. We have implemented the above schemes to compress image and video sequences and will report our experimental results.} } + diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex index b605bfe..833748c 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -3,20 +3,10 @@ % % (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. +\section{Einleitung \label{ifs:section:teil0}} +\rhead{Was ist ein Iteriertes Funktionsschema} +Mit der Hilfe von Iterierten Funktionsschemata (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden. +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..a75b529 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -3,53 +3,132 @@ % % (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. - -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}. - -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. +Bevor wir die IFS ansehen, schauen wir uns Fraktale genauer an. +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 hat $F$ eine Form von Selbstähnlichkeit. + \item Die 'fraktale Dimension' ist grösser als die topologische Dimension + \item Viele Fraktale lassen sich auf eine simple Art definieren. Es genügen zum Beispiel nur wenige Funktionen, welche rekursiv ausgeführt werden, um ein Fraktal zu definieren. +\end{enumerate} +\subsection{Koch Kurve + \label{ifs:subsection:lilkoch}} +Diese Eigenschaften möchten wir nun am Beispiel der Koch Kurve näher anschauen. +In Abbildung \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Sie besteht 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 durch 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 der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. +Man spricht von einer selbstähnlichen Menge, wenn sich diese Menge überdecken lässt mit echten Teilmengen, die zur ganzen Menge ähnlich sind. + + +\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 der jeweiligen Iteration lässt sich mit +\begin{align*} + l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad \cdots , \quad + l_n = a \cdot \left( \frac{4}{3}\right)^n \quad + \Rightarrow \quad + \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty +\end{align*} +berechnen. +In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Daraus resultiert, dass die Länge gegen $\infty$ divergiert. + + +Die Fläche unter der Kurve lässt sich folgendermassen berechnen +\begin{align*} + A_0 &= 0 \\ + 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. +Die Gesamtfläche ist daher gegeben durch die 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 \\ +\end{align*} +mit dem Grenzwert +\begin{align*} + \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 Koch-Kurve ein Objekt mit endlicher Fläche, aber unendlichem Umfang. + + +Zu guter Letzt bestimmen wir die Dimension der Kurve. +Es gibt viele verschiedene Methoden die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. +Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur unterschiedliche Arten. +In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. +Die Ähnlichkeits-Dimension $D$ ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$ + +\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 $l$, Abbildung \ref{ifs:trinagle}. +Somit hat das Dreieck die Dimension $D = 2$. +Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon =l \cdot 1/3$. +Ihre Ähnlichkeits-Dimension ist somit +\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 Koch-Kurve 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. +\begin{figure} + \centering + \begin{tikzpicture} + + % draw the background + \draw [line width=1.5pt, fill=gray!2] (0,0) -- (60:4) -- (4,0) -- cycle; + + \coordinate[label=left:$A$] (A) at (0,0); + \coordinate[label=right:$B$] (B) at (4,0); + \coordinate[label=above:$C$] (C) at (2,3.464); + + \coordinate[label=below:$l$](c) at ($ (A)!.5!(B) $); + \coordinate[label=left:$l$] (b) at ($ (A)!.5!(C) $); + \coordinate[label=right:$l$](a) at ($ (B)!.5!(C) $); + + \coordinate[label=below:$l/2$](d) at ($ (b)!.5!(a)$); + + % the triangle + \draw [line width=1.5pt] (A) -- (B) -- (C) -- cycle; + \draw [line width=0.5pt] (a) -- (b); + \draw [line width=0.5pt] (a) -- (c); + \draw [line width=0.5pt] (c) -- (b); + + \end{tikzpicture} + \caption{Selbstähnlichkeit eines gleichseitigen Dreiecks} + \label{ifs:trinagle} +\end{figure} + diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index bfd1684..fd10634 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -3,38 +3,254 @@ % % (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? - -\subsection{De finibus bonorum et malorum -\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. +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} +Es besteht aus drei kleineren Kopien von sich selbst. +Es ist also ein Selbstähnliches Gebilde. +Diese Eigenschaft wollen wir uns zunutze machen. +Wir definieren das Dreieck mit Kantenlänge 1 als Menge $X$. +Ausserdem bestimmen wir drei Funktionen +\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*} +welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet. +$f_1$ 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 +\begin{align*} + X = \bigcup\limits_{i = 1}^{3} f_i(X), +\end{align*} +entsteht also wieder ein Sierpinski-Dreieck. +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 gegen null. + +\subsection{Iterierte Funktionensysteme +\label{ifs:subsection:IteratedFunktionensysteme}} +In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. + + +$S_1,\dots,S_n$ 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 $c_i < 1$. +Der Banachsche Fixpunktsatz besagt, dass für solche Kontraktionen ein Eindeutiges $A$ existiert, für das $S(A) = A$ gilt. +Den Beweis kann man in \cite{ifs:Rousseau2012} nachlesen. +Hat man nicht nur eine sondern mehrere Kontraktionen, 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 $E$ ohne die leere Menge +\begin{equation} + S(E) = \bigcup\limits_{i = 1}^m S_i(E). + \label{ifs:transformation} +\end{equation} +Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, gilt +\begin{equation} + F = \bigcap\limits_{k = 1}^{\infty} S^k(E). + \label{ifs:ifsForm} +\end{equation} +In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. +Diese Menge ist auch als Attraktor eines IFS bekannt. +Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. + +\subsection{Beispiel: Barnsley-Farn} +Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines 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. +Die vier affinen Transformationen +\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}\\ + \label{ifs:farnFormel} +\end{align} +, welche für die konstruktion des Farns benötigt werden sind in der Abbildung \ref{ifs:farncolor} farblich dargestellt. +Das gesamte Farnblatt ist in der schwarzen Box. +Auf diese werden die Transformationen angewendet +$S_1$ 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. +\subsection{Erzeugung eines Bildes zu einem IFS} +Es gibt zwei verschiedene Methoden um das Bild zu einem IFS zu erzeugen. +Die erste Methode ist wahrscheinlich die intuitivste. +Wir beginnen mit einm Startbild, zum Beispiel ein Schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab. +Das neue Bild, dass entsteht, ist die nächste Iterierte. +Dieses wird wieder mit den Transformationen abgebildet. +Wir wiederholen den letzten schritt, bis wir zufrieden mit der neusten Iterierten sind. + +Diesen Vorgang haben wir beim Sierpinski-Dreieck in Abbildung \ref{ifs:sierpconst} gebraucht. +In Abbildung \ref{ifs:sierpinski10} ist die zehnte Iterierte zu sehen. +Weitere Iterationen hätten in dieser Darstellungsgrösse kaum mehr einen Unterschied gemacht. + + +Die zweite Methode ist das Chaosspiel \cite{ifs:chaos}. +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 die erste Methode ziemlich rechenintensiv. +Beim Chaosspiel 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, nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet. +Je mehr eine Transformation kontrahiert, desto weniger Punkte braucht es um die resultierende Teilabbildung darzustellen. +Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. +Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grüne Hauptteil des Blattes, $S_2$. + +In Abbildung \ref{ifs:farnNoWeight} wurden die vier gleich stark gewichtet. +Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn nicht so gut abgebildet wird. + +Am besten sieht man den Effekt einer schlechten Gewichtung in Abbildung \ref{ifs:farnrightWeight}. +Hier wurde $S_4$, welches für das rechte untere Teilblatt zuständig ist, mit nur $1\%$ statt $7\%$ gewichtet. +Man sieht, wie sich der Mangel an Punkten auf die anderen Abbildungen das Farnblattes auswirkt. +In jeder Kopie des ganzen Farns fehlen die Punkte für dieses rechte untere Teilblatt. + + + +\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=\textwidth]{papers/ifs/images/farncolor2} + \caption{Vier Transformationen des Barnsley-Farn in unterschiedlichen Farben} + \label{ifs:farncolor} +\end{figure} + +\begin{figure} + \centering + \subfigure[]{ + \label{ifs:farnNoWeight} + \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnnotweight}} + \subfigure[]{ + \label{ifs:farnrightWeight} + \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnrightwight}} + \caption{(a) Chaosspiel ohne Gewichtung (b) $S_4$ zu wenig gewichtet} + \label{ifs:farnweight} +\end{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 23fabbc..78fb935 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -3,38 +3,183 @@ % % (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, wie sie in \cite{ifs:Rousseau2012} beschrieben ist. + +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. +Ein IFS, wie wir es in \ref{ifs:subsection:IteratedFunktionensysteme} definiert haben, wird uns also nicht weiter helfen. +Die Lösung dazu sind Partitionierte IFS (PIFS) \cite{ifs:pifs}. +In \ref{ifs:transformation} wurde definiert, dass die Kontraktionen $S_i$ bei IFS auf die gesamte Menge $E$ angewendet werden. +Bei einem PIFS wird der Attraktor in disjunkte Teilmengen aufgeteilt. +Für jede dieser Teilmengen $R_i$ braucht es dann eine grössere Teilmenge, welche mit einer affinen Transformation eine zu $R_i$ ähnliche Menge bildet. +Wir müssen nicht mehr Ähnlichkeiten zum ganzen Bild finden, sondern zwischen Teilen des Bildes. +Doch wie finden wir das PIFS, welches das Bild als Attraktor hat? + +\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. +Wir beschränken das Verfahren für Graustufenbilder. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. +Ein Graustufenbild kann man als Pixelraster mit einer x und y Achse verstehen. +Jedem dieser Pixel wird ein Grauwert zugeordnet. +Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert $z$ zuweist +\begin{align*} + z = f(x,y). +\end{align*} + +Wir suchen ein PIFS welches das zu komprimierende Bild als Attraktor hat. +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. +Zwei Beispiele wie solche Domain-, und Range-Block Paare aussehen können, sehen wir in Abbildung \ref{ifs:FIC} + +\subsubsection{Finden des ähnlichsten $D_j$} +Zuerst brauchen wir die Transformation +\begin{align*} + T_i(x,y,z) = + \begin{pmatrix} + a_i & b_i & 0 \\ + c_i & d_i & 0 \\ + 0 & 0 & s_i + \end{pmatrix} + \begin{pmatrix} + x \\ + y \\ + z + \end{pmatrix} + + + \begin{pmatrix} + \alpha_i \\ + \beta_i \\ + g_i + \end{pmatrix} +\end{align*} +um ein Element aus $D$ auf ein Element von $R$ Abzubilden. +Wenn wir die Grauwerte ausser acht lassen, haben wir die affine Abbildung +\begin{align} + t_i(x,y) = + \begin{pmatrix} + a_i & b_i \\ + c_i & d_i + \end{pmatrix} + \begin{pmatrix} + x \\ + y + \end{pmatrix} + + + \begin{pmatrix} + \alpha_i \\ + \beta_i + \end{pmatrix}. +\label{ifs:affTrans} +\end{align} +Da wir mit Pixeln arbeiten, ist die Auswahl der möglichen Abbildungen begrenzt. +Wir sind auf folgende acht Abbildungen 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} +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. + + +Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Grautöne auf die richtige Helligkeit, sie bilden die lineare Funktion +\begin{align*} + z' = s_i z + g_i. +\end{align*} +Für die Bestimmung dieser Parameter führen wir zuerst die Bildfunktionen $f_{R_i}$ und $\tilde{f_{R_i}}$ ein. +$f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst Skalierten und dann mit \ref{ifs:affTrans} transformierten Domain-Blocks $D_j$. + +Wir suchen $s_i$ und $g_i$ so das +\begin{align*} + f_{R_i} = s_i \tilde{f_{R_i}} + g_i = \bar{f_{R_i}}. +\end{align*} +Die Parameter lassen sich mit +\begin{align*} + s = \frac{\operatorname{cov}(f_{R_i}), f(\tilde{f_{R_i}}))}{\operatorname{var}(\tilde{f_{R_i}})} \\ + g = E(f_{R_i}) - s E(f(\tilde{f_{R_i}})) +\end{align*} +berechnen. +Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt. +Um zu beurteilen wie ähnlich der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block ist, berechnet man den quadratischen Abstand +\begin{align*} + e = d(f_{R_i}, \bar{f_{R_i}}). +\end{align*} +Dieser Abstand sollte so klein wie möglich sein. + +Wir bestimmen die Parameter $s$ und $g$ für jede der acht möglichen affinen Abbildungen und das mit jedem Domain-Block. +Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist die beste. + +Diese Schritte führen wir für jeden Range-Block $R_i$ aus. +Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und Transformation gefunden. + +\begin{figure} + \centering + \includegraphics[width=\textwidth]{papers/ifs/images/FIC} + \caption{Domain-, und Range-Block Paare in Grün und Rot} + \label{ifs:FIC} +\end{figure} + +\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 Attraktor, 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 recht langsam und skaliert auch schlecht für grössere Bilder. +Dies resultiert aus eigenen Experimenten. +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 nicht so schnell wie zum Beispiel das JPEG-Verfahren. +Es wurden bessere Algorithmen der fraktalen Bildkompression entwickelt, doch auch diese können, vor allem in der Laufzeit, noch nicht mit herkömmlichen Komprimierungsverfahren mithalten. +\subsection{Beispiel} +Wir Verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert. +Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren. +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}. +Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}. +Es kann jedoch ein beliebiges Startbild +Nun lassen wir das PIFS 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/punktgruppen/.gitignore b/buch/papers/punktgruppen/.gitignore new file mode 100644 index 0000000..6827d9f --- /dev/null +++ b/buch/papers/punktgruppen/.gitignore @@ -0,0 +1 @@ +standalone diff --git a/buch/papers/punktgruppen/Makefile b/buch/papers/punktgruppen/Makefile index 0274594..f92dc95 100644 --- a/buch/papers/punktgruppen/Makefile +++ b/buch/papers/punktgruppen/Makefile @@ -3,7 +3,39 @@ # # (c) 2020 Prof Dr Andreas Mueller # +SOURCES := \ + crystals.tex \ + intro.tex \ + main.tex \ + piezo.tex \ + symmetry.tex -images: - @echo "no images to be created in punktgruppen" +TIKZFIGURES := \ + tikz/combine-symmetries.tex \ + tikz/lattice.tex \ + tikz/piezo-atoms.tex \ + tikz/piezo.tex \ + tikz/projections.tex \ + tikz/symmetric-shapes.tex +FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES)) + +.PHONY: images +images: $(FIGURES) + +figures/%.pdf: tikz/%.tex + mkdir -p figures + pdflatex --output-directory=figures $< + +.PHONY: standalone +standalone: standalone.tex $(SOURCES) + mkdir -p standalone + cd ../..; \ + pdflatex \ + --halt-on-error \ + --shell-escape \ + --output-directory=papers/punktgruppen/standalone \ + papers/punktgruppen/standalone.tex; + cd standalone; \ + bibtex standalone; \ + makeindex standalone; diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index b6a76c1..8cde9d7 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -10,5 +10,9 @@ dependencies-punktgruppen = \ papers/punktgruppen/symmetry.tex \ papers/punktgruppen/crystals.tex \ papers/punktgruppen/piezo.tex \ - papers/punktgruppen/references.bib - + papers/punktgruppen/references.bib \ + papers/punktgruppen/tikz/combine-symmetries.tex \ + papers/punktgruppen/tikz/lattice.tex \ + papers/punktgruppen/tikz/piezo-atoms.tex \ + papers/punktgruppen/tikz/piezo.tex \ + papers/punktgruppen/tikz/projections.tex diff --git a/buch/papers/punktgruppen/crystals.tex b/buch/papers/punktgruppen/crystals.tex index 6de2bca..1aec16f 100644 --- a/buch/papers/punktgruppen/crystals.tex +++ b/buch/papers/punktgruppen/crystals.tex @@ -1,4 +1,5 @@ \section{Kristalle} +%einleitung sollte noch an das ende von der Symmetrie angepasst werden Unter dem Begriff Kristall sollte sich jeder ein Bild machen können. Wir werden uns aber nicht auf sein Äusseres fokussieren, sondern was ihn im Inneren ausmacht. Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. @@ -6,11 +7,122 @@ Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. Ein Kristall besteht aus Atomen, welche sich in einem Muster arrangieren, welches sich in drei Dimensionen periodisch wiederholt. \end{definition} +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/lattice} + \caption{ + Zweidimensionales Kristallgitter. + \texttt{TODO: make wider and shorter} + \label{fig:punktgruppen:lattice} + } +\end{figure} +\subsection{Kristallgitter} +Ein zweidimensionales Beispiel eines solchen Muster ist Abbildung \ref{fig:punktgruppen:lattice}. +Für die Überschaubarkeit haben wir ein simples Motiv eines einzelnen grauen Punktes gewählt und betrachten dies nur in Zwei Dimensionen. +Die eingezeichneten Vektoren $\vec{a}$ und $\vec{b}$ sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. +Wird ein beliebiger grauer Gitterpunkt in \ref{fig:punktgruppen:lattice} gewählt +und um eine ganzzahlige Linearkombination von $\vec{a}$ und $\vec{b}$ verschoben, +endet er zwangsweise auf einem Gitterpunkt, wenn nicht wieder am selben Ort. +Im Dreidimensionalen-Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor $\vec{c}$ also +\[ + \vec{r} = n_1 \vec{a} + n_2 \vec{b} + n_3 \vec{c} +\] +erreicht werden sofern $\{n_1,n_2,n_3\} \in \mathbb{Z}$ sind. +Sind die Vektoren $\vec{a}$ , $\vec{b}$ , $\vec{c}$ gegeben , +ist ein Kristallgitter eindeutig beschrieben, weswegen sie auch als Grundvektoren bekannt sind. + +\subsection{Translationssymmetrie} +Da sich das ganze Kristallgitter wiederholt, wiederholen sich auch dessen Eigenschaften periodisch mit den Grundvektoren. +Sollte man sich auf einem Gitterpunkt in einem Kristall aufhalten, ist es unmöglich zu wissen, auf welchem Gitterpunkt man sich befindet, +da die Umgebungen aller Punkte Identisch sind. +Mit anderen worten: Jedes Kristallgitter $ G $ ist \emph{Translationssymmetrisch} in der Translation +\[ + Q_i(G) = G + \vec{a_i} +\] wobei der Vektor $a_i$ ein Grundvektor sein muss. +Da die Translationssymmetrie beliebig oft mit allen Grundvektoren angewendet werden kann, +können wir auch sagen, dass alle Verschiebungen um eine Linearkombination +der Vektoren $\vec{a}$ , $\vec{b}$ und $\vec{c}$ erlaubt sind oder kurz, um $\vec{r}$. +Verschiebungen um $\vec{r}$ bewirken demnach keine Veränderungen, +solange wir ein unendlich grosses Kristallgitter verschieben. + +\subsection{Limitierte Kristallsymmetrien} + Die Translationssymmetrie ist wohl keine grosse Überraschung, wenn man die Abbildung \ref{fig:punktgruppen:lattice} betrachtet. + Was nicht direkt ersichtlich ist, ist das auch wenn die Grundvektoren frei gewählt werden können, + können nur Rotationssymmetrische Kristalle bestimmter Rotationswinkel erzeugt werden. + +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/combine-symmetries} + \caption{ + Translations und Rotationssymmetrisches Kristallgitter + \texttt{TODO: make wider and change color (yellow)} + } + \label{fig:punktgruppen:rot-geometry} +\end{figure} + + \subsubsection{Translationssymmetrie $Q$ in Kombination mit Rotationssymmetrie $C_\alpha$} % Müssen uns auf eine schreibweise für Symmetrie Operationen einigen oder sicher am Ende überprüfen + In Abbildung \ref{fig:punktgruppen:rot-geometry} Sehen wir Gitterpunkte und deren Zusammenhänge. + + \begin{itemize} + \item $A$ ist unser erster Gitterpunkt. + + \item $A'$ ist gegeben, weil wir $A$ mit der Translation $Q$ um einen Grundvektor verschieben und wir wissen, + dass nach einer Translation wieder ein Gitterpunkt an der Verschobenen Stelle sein muss. + \item $B$ entsteht, weil wir die Rotationssymmetrie $C_\alpha$ auf den Punkt $A$ anwenden. + Dadurch dreht sich das ganze Gitter um den Winkel $\alpha$. + Für uns bedeutet dies lediglich, dass unser zweiter Punkt $A'$ abgedreht wird. + An der neuen Position von $A'$ muss also auch ein Punkt sein, um die Rotationssymmetrie zu erfüllen. + \item $B$ ist unser Name für diesen neuen Punkt. + Da auch die Eigenschaften des Kristallgittes periodisch mit dem Gitter sein müssen, dürfen wir $C_\alpha$ auch auf $A'$ anwenden. + Also wenden wir $C_\alpha$ invertiert + \footnote{Eine Rotationssymmetrie muss auch in die inverse Richtung funktionieren. + Genauere Überlegungen hierzu werden dem Leser überlassen, da sich die Autoren nicht explizit mit dieser Frage Auseinander gesetzt haben.} + auch auf $A'$ an. + Dies dreht $A$ auf einen neuen Punkt. + \item $B'$ ist kein zufälliger Name für diesen neuen Punkt, denn wir wissen, dass zwischen allen Punkten eine Translationssymmetrie bestehen muss. + Die Translationssymmetrie zwischen $B$ und $B'$ ist hier als $Q'$ bezeichnet. + \end{itemize} + Mit den gegebenen Punkten lassen sich geometrische Folgerungen ziehen. + Wir beginnen, indem wir die Länge der Translation $Q$ mit jener von $Q'$ vergleichen. + Aus Abbildung \ref{fig:punktgruppen:rot-geometry} ist ersichtlich, dass $|Q| = |Q'|+ 2x$. + Ist $Q$ ein Grundvektor so muss $|Q'|$ ein ganzes vielfaches von $|Q|$ sein. Also + \[ + |Q'| = n|Q| = |Q| + 2x + \] + Die Strecke $x$ lässt sich auch mit hilfe der Trigonometrie und dem angenommenen Rotationswinkel $\alpha$ ausdrücken: + \[ + n|Q| = |Q| + 2|Q|\sin(\alpha - \pi/2) + \] + Wir können mit $|Q|$ dividieren um unabhängig von der Läge des Grundvektors zu werden, + was auch Sinn macht, da eine Skalierung eines Kristalles seine Symmetrieeigenschaften nicht tangieren soll. + Zusätzlich können wir den Sinusterm vereinfachen. + \[ + n = 1 - 2\cos\alpha + \alpha = \cos^{-1}\left(\frac{1-n}{2}\right) + \] + Dies schränkt die möglichen Rotationssymmetrien auf + \[ + \alpha \in \left\{ 0^\circ, 60^\circ, 90^\circ, 120^\circ, 180^\circ\right\} + \] +ein. + +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/projections} + \caption{Kristallklassen mit zugehöriger Schönfliesnotation} + \label{fig:punktgruppen:Kristallkassen} +\end{figure} + +\subsection{Kristallklassen} +Vorgehend wurde gezeigt, dass in einem zweidimensionalen Kristallgitter nicht alle Symmetrien möglich sind. +Mit weiteren ähnlichen überlegungen gezeigt werden kann, dass Kristalle im dreidimensionalen Raum +\footnote{Alle $17$ möglichen zweidimensionalen Symmetrien sind als Wandmustergruppen bekannt} +nur auf genau $32$ Arten punktsymmetrisch sein können. +Diese $32$ möglichen Punktsymmetrien scheinen durchaus relevant zu sein, denn sie werden unter anderem als Kristallklassen bezeichnet. +Eine mögliche Art, die Klassen zu benennen ist nacht dem Mathematiker Arthur Moritz Schönflies, +welcher sich mit der Klasifizierung dieser Symmetrien auseinandergesetzt hat. +Auf der Abbildung \ref{fig:punktgruppen:Kristallkassen} sind die möglichen Punktsymmetrien mit deren Schönfliesnotation aufgelistet. +Als Darstellungsmethode wurde die stereographische Projektion gewählt, wobei $5$ Klassen aus Gründen der Überschaubarkeit nicht gezeichnet wurden. + + -Ein Zweidimensionales Beispiel eines solchen Muster ist Abbildung \ref{fig:punktgruppen:lattce-grid}. -Für die Überschaubarkeit haben wir ein simples Muster eines einzelnen XgrauenX Punktes gewählt in nur Zwei Dimensionen. -Die eingezeichneten Vektoren a und b sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. -Dadurch können von einem einzelnen XGrauenX Gitterpunkt in \ref{fig:punktgruppen:lattce-grid} können mit einer ganzzahligen Linearkombination von a und b alle anderen Gitterpunkte des Kristalles erreicht werden. -Ein Kristallgitter kann eindeutig mit a und b und deren winkeln beschrieben werden weswegen a und b auch Gitterparameter genannt werden. -Im Dreidimensionalen-Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor also FRMEL FÜR TRANSLATIONSVEKTOR erreicht werden. -Da sich das Ganze Kristallgitter wiederholt, wiederholen sich auch die Eigenschaften eines Gitterpunktes Periodisch mit eiem diff --git a/buch/papers/punktgruppen/figures/combine-symmetries.pdf 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sehen, wie eine Symmetrie eines Objektes weit -2.ter versuch: -Die Kristallographie ist ein grosses Thema, Symmetrien auch. -Für beide bestehen schon bewährte Mathematische Modelle und Definitionen. -Die +Es gibt viele Möglichkeiten sich in Kristallen zu verlieren. +Auch wen man nur die mathematischen Betrachtunngsweisen berüksichtigt, hat man noch viel zu viele Optionen sich mit Kristallen zu beschäftigen. +In diesem Kapitel ist daher der Fokus ``nur'' auf die Symmetrie gelegt. +Zu beginn werden wir zeigen was eine Symmetrie ausmacht und dass sie noch weit mehr in sich verbirgt als nur schön auszusehen. +Die vorgestellten Symmetrien sind äusserst gut geeignet um die Grundeigenschaften eines Kristalles zu Beschreiben. +Mit etwas kiffligen geometrischen Überlegungen kann man zeigen wass in der Welt der Kristallographie alles möglich ist oder nicht. +Die Einschränkungen sind durchaus wilkommen, dank ihnen halten sich die möglichen Kristallgitter in Grenzen und Lassen sich Kategorisieren. +Kategorien sind nicht nur für einen besseren Überblich nützlich, sondern kann man aus ihnen auch auf Physikalische Eigenschaften schliessen, als spannendes Beispiel: Die Piezoelektrizität. +Die Piezoelektrizität ist vielleicht noch nicht jedem bekannt, sie versteckt sich aber in diversen Altagsgegenständen zum Beispiel sorgen sie in den meisten Feuerzeugen für die Zündung. +Ein Funken Interesse ist hoffentlich geweckt um sich mit dem scheinbar trivialen thema der Symmetrie auseinander zu setzten. + + diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index d88e221..a6e246c 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -3,10 +3,12 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Crystal M\rotatebox[origin=c]{180}{a}th\label{chapter:punktgruppen}} -\lhead{Crystal M\rotatebox[origin=c]{180}{a}th} +\newcommand{\flippedA}{\raisebox{\fontcharht\font`a}{\scalebox{-1}[-1]{a}}} + +\chapter[Crystal Math]{Crystal M\flippedA{}th\label{chapter:punktgruppen}} +\lhead{Crystal M\flippedA{}th} \begin{refsection} -\chapterauthor{Tim T\"onz, Naoki Pross} +\chapterauthor{Naoki Pross, Tim T\"onz} \input{papers/punktgruppen/intro} \input{papers/punktgruppen/symmetry} diff --git a/buch/papers/punktgruppen/piezo.tex b/buch/papers/punktgruppen/piezo.tex index 7ee4174..e6b595a 100644 --- a/buch/papers/punktgruppen/piezo.tex +++ b/buch/papers/punktgruppen/piezo.tex @@ -1 +1,74 @@ -\section{Piezoelektrizit\"at} +\section{Piezoelektrizität} +Die Piezoelektrizität ist per Definition spannend. +Sie beschreibt die Eigenschaft, dass gewisse Kristalle eine elektrische Spannung erzeugen, wenn machanischer Druck auf sie ausgeübt wird. + +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/piezo} %das Efeld mit Naoki disskutieren, müssen sicher gehen, dass es mit jenen in Abbildung Piezo aufbau übereinstimmt + \caption{Piezoelektrisches Material in Ruhe und unter Druck} + \label{fig:punktgruppen:basicPiezo} +\end{figure} + +\subsection{Polarisierung} +Piezoelektrizität basiert darauf, dass zwischen den Oberflächen des Kristalles ein Ladungsungleichgewicht entsteht siehe Abbildung\ref{fig:punktgruppen:basicPiezo}. +Dieses Ungleichgewicht resultiert, +weil durch den mechanischen Druck auf der einen Oberfläche des Kristalles positiv Ione näher an die Oberfläche gelangen, +wärend auf der gegenüberliegenden Oberfläche sich mehr negative Ionen Sammeln. +Das sich die atomare Struktur eines Kristalles unter Druck genau so verformt ist nicht bei jedem Kristall gegeben. +Der Aufbau und somit auch die Symmetrie des Kristalles sind daher relevant für die Entstehung dieses Effektes. + +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/piezo-atoms} + \caption{ + Kristallstrukturen mit und ohne piezoelektrischer Eigenschaft. + \texttt{TODO: adapt figure for paper with subfigure markers.} + } + \label{fig:punktgruppen:atomPiezo} +\end{figure} + +\subsection{Atomarer Aufbau} +Die Polarisation resultiert über eine gesamte Oberfläche eines Kristalles, entscheidend ist aber der atomare Aufbau. +Wir wollen dazu die verschiedenen Kristallstrukturen auf Abbildung \ref{fig:punktgruppen:atomPiezo} diskutieren. +In Abbildung \ref{fig:punktgruppen:atomPiezo} gilt für alle Strukturen, dass rote Kreise Positive Ionen und blaue negative Ionen repräsentieren. +%liste oder anderes format?.. +Struktur$(a)$ zeigt ein piezoelektrisches Material in Ruhe. Struktur $(b)$ ist dasselbe Kristallgitter, jedoch wird es senkrecht belastet. +Eingezeichnet ist auch das elektrische Feld, welches entsteht, weil mitlleren Ladungsträger weiter auseinander gerdrückt werden. +Als hilfe zur Vorstellung kann man $(b)$ zwischen zwei leitende Platten setzen, +so wird ersichtlich, dass mit wachsendem Druck eine negative Ladung an die rechte Platte gedrückt wird, +während sich die positiven Ionen weiter entfernen. +$(d)$ ist nicht piezoelektrisch. +Dies wird ersichtlich, wenn man $(d)$ unterdruck setzt und sich die Struktur zu $(e)$ verformt. +Setzt man $(e)$ gedanklich auch zwischen zwei leitende Platten scheint es als würden rechts mehr Positive Ionen in die Platte gedrückt werden +und links umgekehrt. +Dies ist aber nicht mehr der Fall, wenn der Kristall nach oben und periodisch wiederholt. +Struktur $(c)$ zeigt $(a)$ in unter horizontaler Belastung. +Was in zwischen $(b)$ und $(c)$ zu beobachten ist, ist dass das entstandene Ladungsdifferenz orthogonal zu der angelegten Kraft entsteht, +im Gegensatz zu $(b)$. +Daraus kann man schlissen, dass $(a)$ keine Rotationssymmetrie von $90^\circ$ besitzen kann, weil die Eigenschaften ändern bei einer $90^\circ$ Drehung. +Das Fehlen dieser Rotationssymmetrie kann mit betrachten von $(a)$ bestätigt werden. + +\subsection{Punktsymmetrie}\footnote{In der Literatur wird ein Punktsymmetrisches Kristallgitter oft als Kristallgitter mit Inversionszentrum bezeichnet.} +Piezoelektrische Kristalle können nicht Punktsymmetrisch sein. +Kristallgitter, bei welchen eine Punktspiegelung eine symmetrische Operation ist, können keine piezoelektrische Kristalle bilden. +Auf Abbildung \ref{fig:punktgruppen:atomPiezo} ist bewusst $(a)$ ein nicht Punktsymmetrischer Kristall mit einem Punktsymmetrischen $(d)$ verglichen worden. +Als vereinfachte Erklärung kann mann sich wieder das Bild vor augen führen, eines Kristalles, +welcher unter Druck auf der einen Seite negative und der anderen Seite positive Ionen an seine Oberfläche verdrängt. +Spiegelt man nun den Kristall um den Gitterpunkt in der mitte des Kristalles, so würden die negativen Ionen auf den Positiven auf der anderen seite landen, +was der Definition einer Symmetrie deutlich widerspricht. + +\subsection{Vom Kristall zum Feuer} +Piezoelektrizität hat durchaus nutzen im Alltag. +Feuerzeuge welche nicht auf dem Prinzip beruhen einen Zündstein abzuschleifen, +sonder ohne Verschleiss auf Knopfdruck einen Zündfunken erzeugen, basieren auf dem Prinzip der Piezoelektrizität. +Drückt der Nutzende auf den Zündknopf spannt sich eine Feder bis zu einer Konfigurierten Spannung. +Wird vom Nutzenden weiter gedrückt entspannt sich die Feder schlagartig und beschleunigt mit der gespeicherten Energie ein Hammer, +welcher auf das Piezoelement aufschlägt. +Der augenblicklich hohe Druck sorgt an den Piezokontakten für eine eben so Kurze aber hohe elekrische Spannung. +Die Spannung reicht aus, um eine Funkenstrecke zu überwinden und so eine entflammbares Gas zu entzünden. +Sollten Sie also eines Tages in die Situation geraten, in welcher Sie zwei verschiedene Kristalle vor sich haben +und ein piezoelektrisches Feuerzeug bauen müssen, +wobei Sie aber wissen, dass einer eine Punktsymmetrie aufweist, +versuche sie es mit dem anderen. +Ich muss aber anmerken, dass aus den $21$ möglichen Kristallsymmetrien ohne Punktsymmetrie einer nicht piezoelektrisch ist. +ein wenig glück brauchen Sie also immer noch. diff --git a/buch/papers/punktgruppen/standalone.tex b/buch/papers/punktgruppen/standalone.tex new file mode 100644 index 0000000..3317318 --- /dev/null +++ b/buch/papers/punktgruppen/standalone.tex @@ -0,0 +1,30 @@ +\documentclass{book} + +\input{common/packages.tex} + +% additional packages used by the individual papers, add a line for +% each paper +\input{papers/common/addpackages.tex} + +% workaround for biblatex bug +\makeatletter +\def\blx@maxline{77} +\makeatother +\addbibresource{chapters/references.bib} + +% Bibresources for each article +\input{papers/common/addbibresources.tex} + +% make sure the last index starts on an odd page +\AtEndDocument{\clearpage\ifodd\value{page}\else\null\clearpage\fi} +\makeindex + +%\pgfplotsset{compat=1.12} +\setlength{\headheight}{15pt} % fix headheight warning +\DeclareGraphicsRule{*}{mps}{*}{} + +\begin{document} + \input{common/macros.tex} + \def\chapterauthor#1{{\large #1}\bigskip\bigskip} + \input{papers/punktgruppen/main.tex} +\end{document} diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index db05ff5..1dc6f98 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -1,8 +1,8 @@ \section{Symmetrie} Das Wort Symmetrie ist sehr alt und hat sich seltsamerweise von seinem ursprünglichen griechischen Wort -\(\mathrm{\sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\) -\footnote{\emph{Simmetr\'ia}: ein gemeinsames Mass habend, gleichmässig, +\(\mathrm{\Sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\) +\footnote{\emph{Symmetr\'ia}: ein gemeinsames Mass habend, gleichmässig, verhältnismässig} fast nicht verändert. In der Alltagssprache mag es ein locker definierter Begriff sein, aber in der Mathematik hat Symmetrie eine sehr präzise Bedeutung. @@ -10,99 +10,85 @@ präzise Bedeutung. Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer bestimmten Operation invariant ist. \end{definition} +Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit +einigen geometrischen Beispielen beginnen. Wie wir jedoch später sehen werden, +ist das Konzept der Symmetrie eigentlich viel allgemeiner. -Wenn der Leser noch nicht mit der Gruppentheorie in Berührung gekommen ist, ist -vielleicht nicht ganz klar, was eine Operation ist, aber die Definition sollte -trotzdem Sinn machen. Die Formalisierung dieser Idee wird bald kommen, aber -zunächst wollen wir eine Intuition aufbauen. - -\begin{figure}[h] +\begin{figure} \centering - \begin{tikzpicture}[ - node distance = 2cm, - shapetheme/.style = { - very thick, draw = black, fill = magenta!20!white, - minimum size = 2cm, - }, - line/.style = {thick, draw = darkgray}, - axis/.style = {line, dashed}, - dot/.style = { - circle, draw = darkgray, fill = darkgray, - minimum size = 1mm, inner sep = 0, outer sep = 0, - }, - ] - - \node[ - shapetheme, - rectangle - ] (R) {}; - \node[dot] at (R) {}; - \draw[axis] (R) ++(-1.5, 0) to ++(3, 0) node[right] {\(\sigma\)}; - - \node[ - shapetheme, - regular polygon, - regular polygon sides = 5, - right = of R, - ] (Ps) {}; - \node[dot] (P) at (Ps) {}; - \draw[line, dotted] (P) to ++(18:1.5); - \draw[line, dotted] (P) to ++(90:1.5); - \draw[line, ->] (P) ++(18:1.2) - arc (18:90:1.2) node[midway, above right] {\(r, 72^\circ\)}; - - \node[ - shapetheme, - circle, right = of P - ] (Cs) {}; - \node[dot] (C) at (Cs) {}; - \draw[line, dotted] (C) to ++(1.5,0); - \draw[line, dotted] (C) to ++(60:1.5); - \draw[line, ->] (C) ++(1.2,0) - arc (0:60:1.2) node[midway, above right] {\(r, \alpha\)}; - - \end{tikzpicture} + \includegraphics{papers/punktgruppen/figures/symmetric-shapes} \caption{ Beispiele für geometrisch symmetrische Formen. \label{fig:punktgruppen:geometry-example} } \end{figure} -Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit -einigen geometrischen Beispielen beginnen. Wie wir jedoch später sehen werden, -ist das Konzept der Symmetrie eigentlich viel allgemeiner. In Abbildung -\ref{fig:punktgruppen:geometry-example} haben wir einige Formen, die -offensichtlich symmetrisch sind. Zum Beispiel hat ein Quadrat viele Achsen, um -die es gedreht werden kann, ohne sein Aussehen zu verändern. Regelmässige -Polygone mit \(n\) Seiten sind gute Beispiele, um eine diskrete +\subsection{Geometrische Symmetrien} + +In Abbildung \ref{fig:punktgruppen:geometry-example} haben wir einige Formen, +die offensichtlich symmetrisch sind. Zum Beispiel hat das Quadrat eine Gerade, an +deren es gespiegelt werden kann, ohne sein Aussehen zu verändern. Regelmässige +Polygone mit \(n\) Seiten sind auch gute Beispiele, um eine diskrete Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um -einen Punkt um einen bestimmten Winkel \(360^\circ/n\) sie unverändert lässt. -Das letzte Beispiel auf der rechten Seite ist eine unendliche +einen Punkt um einen bestimmten Winkel \(360^\circ/n\) die Figur unverändert +lässt. Das letzte Beispiel auf der rechten Seite ist eine unendliche Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für \(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen. Dies ist hoffentlich ausreichend, um die Bedeutung hinter der Notation zu verstehen, die nun eingeführt wird. +% Vieleicht eine kurze Einführung in für die Definition, ich habe das gefühl, dass in der Definition die Symmetrie-Operation und die Gruppe auf einmal erklährt wird +\subsubsection{Symetriegruppe} +\texttt{TODO: review this paragraph, explain what is \(\mathds{1}\).} +Ein Objekt kann mehr als nur eine Symmetrie aufweisen. +Als Beispiel, kann das Quadrat in Abbildung \ref{fig:punktgruppen:geometry-example} +nicht nur um $\sigma$ sondern auch Diagonal gespiegelt werden oder um $90^\circ$ gedreht werden. +Fässt man die möglichen Symmetrien zusammen, entsteht eine Symmetriegruppe. + \begin{definition}[Symmetriegruppe] Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt. Bei einer anderen Operation \(h\) definieren wir die Komposition \(h\circ g\) als die Anwendung der Operationen nacheinander. Alle Operationen bilden unter Komposition eine Gruppe, die Symmetriegruppe genannt wird. +\end{definition} % ich lese diese Definition ein wenig holprig, vieleicht können wir sie zusammen anschauen + +% Nach meinem Geschmack könne es hier auch eine einleitung wie mein Beispiel geben dammit man den Text flüssiger lesen kann +\begin{definition}[Zyklische Untergruppe, Erzeuger] + Sei \(g\) ein Element einer Symmetriegruppe \(G\). Alle möglichen + Kompositionen von \(g\) und \(g^{-1}\) bilden eine sogenannte zyklische + Untergruppe von \(G\), und \(g\) wird ihr Erzeuger genannt. Die erzeugte + Untergruppe \(\langle g \rangle\) wird mit spitzen Klammern um den Erzeuger + bezeichnet. \end{definition} -Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. Wenn wir -\(r\) eine Drehung von \(2\pi/n\) sein lassen, gibt es eine wohlbekannte Symmetriegruppe +Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. +Bezeichnen wir mit \(r\) eine Drehung im Gegenuhrzeigersinn von \(360^\circ/n\) +um einen Punkt. Diese Definition reicht aus, um die gesamte Symmetriegruppe \[ C_n = \langle r \rangle = \left\{\mathds{1}, r, r^2, \ldots, r^{n-1}\right\} - = \mathbb{Z}/n\mathbb{Z}, \] -die Zyklische Gruppe heisst. Hier die Potenzen von \(r\) sind als wiederholte -Komposition gemeint, d.h. \(r^n = r\circ r \circ \cdots r\circ r\). Die -Schreibweise mit den spitzen Klammern wird als Erzeugendensystem bezeichnet. -Das liegt daran, dass alle Elemente der Symmetriegruppe aus Kombinationen einer -Teilmenge erzeugt werden, die als erzeugende Elemente bezeichnet werden. Die -Reflexionssymmetriegruppe ist nicht so interessant, da sie nur +der Drehungen eines \(n\)-Gons zu definieren. Das liegt daran, +dass wir durch die mehrfache Verwendung von \(r\) jeden Winkel erzeugen, der +die Rotationssymmetrie bewahrt. Hier die Potenzen von \(r\) sind als +wiederholte Komposition gemeint, dass heisst \(r^n = r\circ r \circ \cdots +r\circ r\). Wenn wir diese Idee nun erweitern, können wir mit einem +Erzeugendensystemen komplexere Strukturen aufbauen. + +\begin{definition}[Erzeugendensysteme] + % please fix this unreadable mess + Jede Gruppe kann durch eines oder mehrere ihrer Elemente generiert werden. + Wir lassen \(g_1, g_2, \ldots, g_n\) erzeugenden Elemente einer + Symmetriegruppe sein. Da es mehrere Erzeuger gibt, müssen auch die + sogenannte Definitionsgleichungen gegeben werden, die die + Multiplikationstabelle vollständig definieren. Die Gleichungen sind ebenfalls + in den Klammern angegeben. Die erzeugende Elementen zusammen mit der + Definitionsgleichungen bauen ein Erzeugendensysteme. +\end{definition} + +\texttt{TODO: should put examples for generators?} \\ + +Die Reflexionssymmetriegruppe ist nicht so interessant, da sie nur \(\left\{\mathds{1}, \sigma\right\}\) enthält. Kombiniert man sie jedoch mit der Rotation, erhält man die so genannte Diedergruppe \[ @@ -111,22 +97,48 @@ der Rotation, erhält man die so genannte Diedergruppe \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} \right\}. \] -Diesmal muss die Generator-Notation die Beziehungen zwischen den beiden -Operationen beinhalten. Die ersten beiden sind leicht zu erkennen, für die -letzte empfehlen wir, sie an einem 2D-Quadrat auszuprobieren. +Die Symmetrieoperationen, die wir bis jetzt besprochen haben, haben immer +mindestens einen Punkt gehabt, der wieder auf sich selbst abgebildet wird. Im +Fall der Rotation war es der Drehpunkt, bei der Spiegelung die Punkte der +Spiegelachse. Dies ist jedoch keine Voraussetzung für eine Symmetrie, da es +Symmetrien gibt, die jeden Punkt zu einem anderen Punkt verschieben können. +Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, nennt man +Punktsymmetrie. +\begin{definition}[Punktgruppe] + Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens + einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine + Punktgruppe ist. +\end{definition} + +\subsection{Algebraische Symmetrien} Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich -möglich ist, eine nicht kommutative Algebra zu erstellen. Die naheliegende -Frage ist dann, könnte es sein, dass wir bereits etwas haben, das dasselbe tut? -Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. -\begin{definition}[Darstellung einer Gruppe, Gruppenhomomorphismus] +möglich ist, Gleichungen zu schreiben. Die naheliegende Frage ist dann, könnte +es sein, dass wir bereits etwas haben, das dasselbe tut? Natürlich, ja. +Um es formaler zu beschreiben, werden wir einige Begriffe einführen. +\begin{definition}[Gruppenhomomorphismus] Seien \(G\) und \(H\) Gruppe mit unterschiedlicher Operation \(\diamond\) bzw. \(\star\). Ein Homomorphismus\footnote{ Für eine ausführlichere Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt \(f(a\diamond b) = f(a) \star f(b)\). Man sagt, dass der Homomorphismus - \(f\) \(G\) in \(H\) transformiert, oder dass \(H\) eine Darstellung von - \(G\) ist. + \(f\) \(G\) in \(H\) transformiert. +\end{definition} +\begin{beispiel} + Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen + Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem + komplexen Einheitskreis. Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) + ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. +\end{beispiel} + +\begin{definition}[Darstellung einer Gruppe] + Die Darstellung einer Gruppe ist ein Homomorphismus, der eine Symmetriegruppe + auf eine Menge von Matrizen abbildet. + \[ + \Phi: G \to \operatorname{GL}_n(\mathbb{R}). + \] + Äquivalent kann man sagen, dass ein Element aus der Symmetriegruppe auf einen + Vektorraum \(V\) wirkt, indem man definiert \(\Phi : G \times V \to V\). \end{definition} \begin{beispiel} Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine @@ -142,40 +154,21 @@ Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. die zweite die Matrixmultiplikation. Man kann überprüfen, dass \(\Phi(r^2 \circ r) = \Phi(r^2)\Phi(r)\). \end{beispiel} -\begin{beispiel} - Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen - Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem - komplexen Einheitskreis. Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) - ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. -\end{beispiel} -Die Symmetrien, die wir bis jetzt besprochen haben, haben immer mindestens -einen Punkt unbesetzt gelassen. Im Fall der Rotation war es der Drehpunkt, bei -der Spiegelung die Achse. Dies ist jedoch keine Voraussetzung für eine -Symmetrie, da es Symmetrien gibt, die jeden Punkt zu einem anderen Punkt -verschieben können. Ein aufmerksamer Leser wird bemerken, dass die -unveränderten Punkte zum Eigenraum\footnote{Zur Erinnerung \(E_\lambda = -\mathrm{null}(\Phi - \lambda I)\), \(\vec{v}\in E_\lambda \implies \Phi \vec{v} -= \lambda\vec{v}\)} der Matrixdarstellung der Symmetrieoperation gehören. -Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, nennt man -Punktsymmetrie. -\begin{definition}[Punktgruppe] - Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens - einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine - Punktgruppe ist. -\end{definition} -Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren: -eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr -nützlich ist, nämlich die Translationssymmetrie. Von einem mathematischen -Objekt \(U\) wird gesagt, dass es eine Translationssymmetrie \(Q(x) = x + a\) -hat, wenn es die Gleichung -\[ - U(x) = U(Q(x)) = U(x + a), -\] -für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche -Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine -zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\) -dann \(\vec{v}(t) = \vec{v}(t + \tau)\). +\texttt{TODO: rewrite section on translational symmetry.} +%% TODO: title / fix continuity +% Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren: +% eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr +% nützlich ist, nämlich die Translationssymmetrie. Von einem mathematischen +% Objekt \(U\) wird gesagt, dass es eine Translationssymmetrie \(Q(x) = x + a\) +% hat, wenn es die Gleichung +% \[ +% U(x) = U(Q(x)) = U(x + a), +% \] +% für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche +% Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine +% zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\) +% dann \(\vec{v}(t) = \vec{v}(t + \tau)\). % \subsection{Sch\"onflies notation} diff --git a/buch/papers/punktgruppen/tikz/combine-symmetries.tex b/buch/papers/punktgruppen/tikz/combine-symmetries.tex new file mode 100644 index 0000000..84e0a76 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/combine-symmetries.tex @@ -0,0 +1,56 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} +\begin{tikzpicture}[ + dot/.style = { + draw, circle, thick, black, fill = gray!40!white, + minimum size = 2mm, + inner sep = 0pt, + outer sep = 1mm, + }, + ] + + \node[dot] (A1) at (0,0) {}; + \node[below left] at (A1) {\(A\)}; + + \node[dot] (A2) at (2.5,0) {}; + \node[below right] at (A2) {\(A'\)}; + + \draw[red!80!black, thick, ->] + (A1) to node[midway, below] {\(\vec{Q}\)} (A2); + + \node[dot] (B1) at (120:2.5) {}; + \node[above left] at (B1) {\(B\)}; + + \draw[green!70!black, thick, ->] + (A1) ++(.5,0) arc (0:120:.5) + node[midway, above, xshift=1mm] {\(C_n\)}; + \draw[red!80!black, dashed, thick, ->] (A1) to (B1); + + \node[dot] (B2) at ($(A2)+(60:2.5)$) {}; + \node[above right] at (B2) {\(B'\)}; + + \draw[green!70!black, thick, dashed, ->] + (A2) ++(-.5,0) arc (180:60:.5); + \draw[red!80!black, dashed, thick, ->] (A2) to (B2); + + \draw[yellow!50!orange, thick, ->] + (B1) to node[above, midway] {\(\vec{Q}'\)} (B2); + + \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {}; + \draw[gray, dashed, thick] (A2) to (A2 |- B2) node (Xr) {}; + \node[above left, xshift=-2mm] at (Xl) {\(x\)}; + \node[above right, xshift= 2mm] at (Xr) {\(x\)}; +\end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/lattice.tex b/buch/papers/punktgruppen/tikz/lattice.tex new file mode 100644 index 0000000..9c05af3 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/lattice.tex @@ -0,0 +1,39 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} +\begin{tikzpicture}[ + dot/.style = { + draw, circle, thick, black, fill = gray!40!white, + minimum size = 2mm, + inner sep = 0pt, + outer sep = 1mm, + }, + ] + + \begin{scope} + \clip (-2,-2) rectangle (3,4); + \foreach \y in {-7,-6,...,7} { + \foreach \x in {-7,-6,...,7} { + \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {}; + } + } + \end{scope} + \draw[black, thick] (-2, -2) rectangle (3,4); + + \draw[red!80!black, thick, ->] + (N00) to node[midway, below] {\(\vec{a}_1\)} (N10); + \draw[cyan!80!black, thick, ->] + (N00) to node[midway, left] {\(\vec{a}_2\)} (N01); +\end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/piezo-atoms.tex b/buch/papers/punktgruppen/tikz/piezo-atoms.tex new file mode 100644 index 0000000..82a2710 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/piezo-atoms.tex @@ -0,0 +1,121 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[font = {\large\bfseries}, align = center] (title) at (5.5,0) {Mit und Ohne\\ Symmetriezentrum}; + + \begin{scope} + \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] { + \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\ + }; + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{scope} + + \begin{scope}[xshift=11cm] + \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} { + \node[charge, \t] (C\i) at (\x:1.5cm) {}; + } + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {}; + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + \end{scope} + + %% + \node[below = of title] {Polarisation Feld \(\vec{E}_p\)}; + + %% hex with vertical pressure + \begin{scope}[xshift=11cm, yshift=-4.5cm] + \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {}; + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \foreach \d in {C1, C2} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {C4, C5} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \node[black] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[red!50, right = of E] {\(+\)}; + \node[blue!50, left = of E] {\(-\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{scope} + + %% square with vertical pressure + \begin{scope}[yshift=-4.5cm] + \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { + \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ + }; + + \foreach \d in {NW, N, NE} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {SW, S, SE} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{scope} + + %% hex with horizontal pressure + \begin{scope}[xshift=5.5cm, yshift=-4.5cm] + \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {}; + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \draw[orange, very thick, <-] (C6) to ++(.7,0); + \draw[orange, very thick, <-] (C3) to ++(-.7,0); + + \node[black] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{scope} + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/piezo.tex b/buch/papers/punktgruppen/tikz/piezo.tex new file mode 100644 index 0000000..1d16ab7 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/piezo.tex @@ -0,0 +1,71 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} +\begin{tikzpicture} + \begin{scope}[ + node distance = 0cm + ] + \node[ + rectangle, fill = gray!60!white, + minimum width = 3cm, minimum height = 2cm, + ] (body) {\(\vec{E}_p = \vec{0}\)}; + + \node[ + draw, rectangle, thick, black, fill = red!50, + minimum width = 3cm, minimum height = 1mm, + above = of body + ] (pos) {}; + + \node[ + draw, rectangle, thick, black, fill = blue!50, + minimum width = 3cm, minimum height = 1mm, + below = of body + ] (neg) {}; + + \draw[black, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {}; + \draw[black, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {}; + + \draw[black, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U = 0\)} (n); + \end{scope} + \begin{scope}[ + node distance = 0cm, + xshift = 7cm + ] + \node[ + rectangle, fill = gray!40!white, + minimum width = 3cm, minimum height = 1.5cm, + ] (body) {\(\vec{E}_p = \vec{0}\)}; + + \node[ + draw, rectangle, thick, black, fill = red!50, + minimum width = 3cm, minimum height = 1mm, + above = of body + ] (pos) {}; + + \node[ + draw, rectangle, thick, black, fill = blue!50, + minimum width = 3cm, minimum height = 1mm, + below = of body + ] (neg) {}; + + \draw[orange, very thick, <-] (pos.north) to node[near end, right] {\(\vec{F}\)} ++(0,1); + \draw[orange, very thick, <-] (neg.south) to node[near end, right] {\(\vec{F}\)} ++(0,-1); + + \draw[black, very thick, -Circle] (pos.east) to ++ (1,0) node (p) {}; + \draw[black, very thick, -Circle] (neg.east) to ++ (1,0) node (n) {}; + + \draw[black, thick, ->] (p) to[out = -70, in = 70] node[midway, right] {\(U > 0\)} (n); + \end{scope} +\end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/projections.tex b/buch/papers/punktgruppen/tikz/projections.tex new file mode 100644 index 0000000..a763e77 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/projections.tex @@ -0,0 +1,257 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} +\begin{tikzpicture}[ + classcirc/.style = { + draw = gray, thick, circle, + minimum size = 12mm, + inner sep = 0pt, outer sep = 0pt, + }, + classlabel/.style = { + below right = 5mm + }, + round/.style = { + draw = orange, thick, circle, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt, + }, + cross/.style = { + cross out, draw = magenta, thick, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt + }, + ] + \matrix [row sep = 3mm, column sep = 0mm] { + \node[classcirc] (C1) {} node[classlabel] {\(C_{1}\)}; & + \node[classcirc] (C2) {} node[classlabel] {\(C_{2}\)}; & + \node[classcirc] (C3) {} node[classlabel] {\(C_{3}\)}; & + \node[classcirc] (Ci) {} node[classlabel] {\(C_{i}\)}; & + + \node[classcirc] (Cs) {} node[classlabel] {\(C_{s}\)}; & + \node[classcirc] (C3i) {} node[classlabel] {\(C_{3i}\)}; & + \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; & + \node[classcirc] (D2) {} node[classlabel] {\(D_{2}\)}; \\ + + \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; & + \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; & + \node[classcirc] (D3) {} node[classlabel] {\(D_{3}\)}; & + + \node[classcirc] (C4) {} node[classlabel] {\(C_{4}\)}; & + \node[classcirc] (C6) {} node[classlabel] {\(C_{6}\)}; & + \node[classcirc] (D3dP) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (S4) {} node[classlabel] {\(S_{4}\)}; \\ + + \node[classcirc] (S3) {} node[classlabel] {\(S_{3}\)}; & + \node[classcirc, dashed] (T) {} node[classlabel] {\(T_{}\)}; & + \node[classcirc] (C4h) {} node[classlabel] {\(C_{4h}\)}; & + \node[classcirc] (C6h) {} node[classlabel] {\(C_{6h}\)}; & + + \node[classcirc, dashed] (Th) {} node[classlabel] {\(T_{h}\)}; & + \node[classcirc] (C4v) {} node[classlabel] {\(C_{4v}\)}; & + \node[classcirc] (C6v) {} node[classlabel] {\(C_{6v}\)}; & + \node[classcirc, dashed] (Td) {} node[classlabel] {\(T_{d}\)}; \\ + + \node[classcirc] (D2d) {} node[classlabel] {\(D_{2d}\)}; & + \node[classcirc] (D3h) {} node[classlabel] {\(D_{3h}\)}; & + \node[classcirc, dashed] (O) {} node[classlabel] {\(O_{}\)}; & + \node[classcirc] (D4) {} node[classlabel] {\(D_{4}\)}; & + + \node[classcirc] (D6) {} node[classlabel] {\(D_{6}\)}; & + \node[classcirc, dashed] (Oh) {} node[classlabel] {\(O_{h}\)}; & + \node[classcirc] (D4h) {} node[classlabel] {\(D_{4h}\)}; & + \node[classcirc] (D6h) {} node[classlabel] {\(D_{6h}\)}; \\ + }; + + + \node[cross] at ($(C1)+(4mm,0)$) {}; + + + \node[cross] at ($(C2)+(4mm,0)$) {}; + \node[cross] at ($(C2)-(4mm,0)$) {}; + + + \node[cross] at ($(C3)+( 0:4mm)$) {}; + \node[cross] at ($(C3)+(120:4mm)$) {}; + \node[cross] at ($(C3)+(240:4mm)$) {}; + + + \node[cross] at ($(Ci)+(4mm,0)$) {}; + \node[round] at ($(Ci)-(4mm,0)$) {}; + + + \node[cross] at ($(Cs)+(4mm,0)$) {}; + \node[round] at ($(Cs)+(4mm,0)$) {}; + + + \node[cross] at ($(C3i)+( 0:4mm)$) {}; + \node[cross] at ($(C3i)+(120:4mm)$) {}; + \node[cross] at ($(C3i)+(240:4mm)$) {}; + \node[round] at ($(C3i)+( 60:4mm)$) {}; + \node[round] at ($(C3i)+(180:4mm)$) {}; + \node[round] at ($(C3i)+(300:4mm)$) {}; + + + \node[cross] at ($(C2h)+(4mm,0)$) {}; + \node[cross] at ($(C2h)-(4mm,0)$) {}; + \node[round] at ($(C2h)+(4mm,0)$) {}; + \node[round] at ($(C2h)-(4mm,0)$) {}; + + + \node[cross] at ($(D2)+( 20:4mm)$) {}; + \node[cross] at ($(D2)+(200:4mm)$) {}; + \node[round] at ($(D2)+(160:4mm)$) {}; + \node[round] at ($(D2)+(340:4mm)$) {}; + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3d)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3d)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 180} { + \node[cross] at ($(C2v)+({\x+15}:4mm)$) {}; + \node[cross] at ($(C2v)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 180} { + \node[cross] at ($(D2h)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D2h)+({\x-15}:4mm)$) {}; + \node[round] at ($(D2h)+({\x+15}:4mm)$) {}; + \node[round] at ($(D2h)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3)+({\x+15}:4mm)$) {}; + \node[round] at ($(D3)+({\x-15}:4mm)$) {}; + } + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3dP)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3dP)+({\x-15}:4mm)$) {}; + \node[round] at ($(D3dP)+({\x+15+60}:4mm)$) {}; + \node[round] at ($(D3dP)+({\x-15+60}:4mm)$) {}; + } + + + \node[cross] at ($(S4)+(4mm,0)$) {}; + \node[cross] at ($(S4)-(4mm,0)$) {}; + \node[round] at ($(S4)+(0,4mm)$) {}; + \node[round] at ($(S4)-(0,4mm)$) {}; + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(S3)+(\x:4mm)$) {}; + \node[round] at ($(S3)+(\x:4mm)$) {}; + } + + + %% TODO: T + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4h)+(\x:4mm)$) {}; + \node[round] at ($(C4h)+(\x:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6h)+(\x:4mm)$) {}; + \node[round] at ($(C6h)+(\x:4mm)$) {}; + } + + + %% TODO: Th + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(C4v)+(\x+15:4mm)$) {}; + \node[cross] at ($(C4v)+(\x-15:4mm)$) {}; + } + + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(C6v)+(\x+10:4mm)$) {}; + \node[cross] at ($(C6v)+(\x-10:4mm)$) {}; + } + + + %% TODO: Td + + + \foreach \x in {0, 180} { + \node[cross] at ($(D2d)+({\x+15}:4mm)$) {}; + \node[round] at ($(D2d)+({\x-15}:4mm)$) {}; + + \node[round] at ($(D2d)+({\x+15+90}:4mm)$) {}; + \node[cross] at ($(D2d)+({\x-15+90}:4mm)$) {}; + } + + + \foreach \x in {0, 120, 240} { + \node[cross] at ($(D3h)+({\x+15}:4mm)$) {}; + \node[cross] at ($(D3h)+({\x-15}:4mm)$) {}; + \node[round] at ($(D3h)+({\x+15}:4mm)$) {}; + \node[round] at ($(D3h)+({\x-15}:4mm)$) {}; + } + + + %% TODO: O + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(D4)+({\x+15}:4mm)$) {}; + \node[round] at ($(D4)+({\x-15}:4mm)$) {}; + } + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(D6)+({\x+10}:4mm)$) {}; + \node[round] at ($(D6)+({\x-10}:4mm)$) {}; + } + + + % TODO Oh + + + \foreach \x in {0, 90, 180, 270} { + \node[cross] at ($(D4h)+(\x+15:4mm)$) {}; + \node[cross] at ($(D4h)+(\x-15:4mm)$) {}; + \node[round] at ($(D4h)+(\x+15:4mm)$) {}; + \node[round] at ($(D4h)+(\x-15:4mm)$) {}; + } + + + \foreach \x in {0, 60, 120, 180, 240, 300} { + \node[cross] at ($(D6h)+({\x+10}:4mm)$) {}; + \node[cross] at ($(D6h)+({\x-10}:4mm)$) {}; + \node[round] at ($(D6h)+({\x+10}:4mm)$) {}; + \node[round] at ($(D6h)+({\x-10}:4mm)$) {}; + } +\end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/symmetric-shapes.tex b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex new file mode 100644 index 0000000..b2c051f --- /dev/null +++ b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex @@ -0,0 +1,59 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + node distance = 2cm, + shapetheme/.style = { + very thick, draw = black, fill = magenta!20!white, + minimum size = 2cm, + }, + line/.style = {thick, draw = darkgray}, + axis/.style = {line, dashed}, + dot/.style = { + circle, draw = darkgray, fill = darkgray, + minimum size = 1mm, inner sep = 0, outer sep = 0, + }, + ] + + \node[ + shapetheme, rectangle + ] (R) {}; + \node[dot] at (R) {}; + \draw[axis] (R) ++(-1.5, 0) to ++(3, 0) node[right] {\(\sigma\)}; + + \node[ + shapetheme, + regular polygon, + regular polygon sides = 5, + right = of R, + ] (Ps) {}; + \node[dot] (P) at (Ps) {}; + \draw[line, dotted] (P) to ++(18:1.5); + \draw[line, dotted] (P) to ++(90:1.5); + \draw[line, ->] (P) ++(18:1.2) + arc (18:90:1.2) node[midway, above right] {\(r, 72^\circ\)}; + + \node[ + shapetheme, + circle, right = of P + ] (Cs) {}; + \node[dot] (C) at (Cs) {}; + \draw[line, dotted] (C) to ++(1.5,0); + \draw[line, dotted] (C) to ++(60:1.5); + \draw[line, ->] (C) ++(1.2,0) + arc (0:60:1.2) node[midway, above right] {\(r, \alpha\)}; + + \end{tikzpicture} +\end{document} 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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b/buch/papers/reedsolomon/RS presentation/RS.snm @@ -0,0 +1 @@ +\beamer@slide {ft_discrete}{21} diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz Binary files differnew file mode 100644 index 0000000..04bd239 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex new file mode 100644 index 0000000..6ee6cc7 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -0,0 +1,934 @@ +\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} + \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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Dabei müssen wir die \textcolor{red}{Definition 4.6 (verweis auf eine Definition im Buch ohne label)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. +Wir legen für unser Beispiel den endlichen Körper $\mathbb{F}_{q}$ 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 die Resultate der arithmetischen Operationen im Körper $\mathbb{F}_{11}$, die 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} + +Die Menge uns zur Verfügung stehender Zahlen legt auch fest, 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 für das Beispiel 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 Nachricht 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 Abschnitt \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 wählen deshalb eine Zahl $a$, die die gleichen Aufgaben haben soll wie $e^{\frac{j}{2 \pi}}$ in der diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ ist. Dazu soll die Potenz von $a$ 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 c c c c c c} +$a = 1$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 2$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 3$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 4$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 5$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 6$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 7$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 8$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 9$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 10$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +\end{tabular} +\end{center} +%\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 Einheitswurzel auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a = 8$. + +\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} +als unser Übertragungsvektor. + +\subsection{Allgemeine Codierung + \label{reedsolomon:subsection:algCod}} +Um das Ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen, die unsere Transformationsmatrix $A$ bildet. + +Für die allgemeine 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..a46d7da --- /dev/null +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -0,0 +1,319 @@ +% +% 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{Decodierung mit Fehler} +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. +Als Empfänger wissen wir 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 berechnen wir die Differenz $d$ des empfangenen und dem gesendeten Übertragungsvektor mit +%Alle Stellen in $d$, die nicht null sind sind demnach fehler. +% +%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} +und nennen $d(X)$ als unseres Fehlerstellenpolynom. Dieses Polynom soll uns sagen, welche Stellen korrekt und welche fehlerhaft sind. + +Durch das verwenden von $m(X)$ stossen wir auf weitere Probleme, da wir den Nachrichtenvektor auf der Empfängerseite nicht kennen (unser Ziel ist es ja genau diesen zu finden). Dieses Problem betrachten wir im Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} genauer. Um die Überlegungen in den folgenden Abschnitten besser zu verstehen sei $m(X)$ bekannt auf der Empfängerseite. + +%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 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 den Satz von Fermat, dessen Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass +\[ +f(X) = X^{q-1} -1 = 0 +\] +gilt für jedes $X$. Setzen wir 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 Faktorisierung 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 $p(X)$ 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, auf der Empfängerseite aber nicht kennen. + +\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 +\[ +m = [0,0,0,0,?,?,?,?,?,?]. +\] +Nach der Definition des Reed-Solomon-Codes soll an genau diesen vier Stellen auch die Information befinden, 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. + +\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 +\[ +l(X) = 2X^2 + 5 \qquad \rightarrow \qquad l(X) = 2(X-5)(X-6). +\] +\subsubsection{Schritt 3: Fehlerstellen bestimmen} +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..0470db0 --- /dev/null +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -0,0 +1,207 @@ +% +% 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{Decodierung ohne Fehler} + +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:sfaktor} 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 Abschnitt \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: +\[ +\begin{pmatrix} + 8^0 & 8^0 & 8^0 & 8^0 & \dots & 8^0 \\ + 8^0 & 8^{-1} & 8^{-2} & 8^{-3} & \dots & 8^{-9} \\ + 8^0 & 8^{-2} & 8^{-4} & 8^{-6} & \dots & 8^{-18} \\ + 8^0 & 8^{-3} & 8^{-6} & 8^{-9} & \dots & 8^{-27} \\ + \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ + 8^0 & 8^{-9} & 8^{-18} & 8^{-27} & \dots & 8^{-81} \\ +\end{pmatrix} +\qquad +\Rightarrow +\qquad +\begin{pmatrix} + 7^0 & 7^0 & 7^0 & 7^0 & \dots & 7^0 \\ + 7^0 & 7^{1} & 7^{2} & 7^{3} & \dots & 7^{9} \\ + 7^0 & 7^{2} & 7^{4} & 7^{6} & \dots & 7^{18} \\ + 7^0 & 7^{3} & 7^{6} & 7^{9} & \dots & 7^{27} \\ + \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ + 7^0 & 7^{9} & 7^{18} & 7^{27} & \dots & 7^{81} \\ +\end{pmatrix} +\] + +\subsection{Der Faktor $s$ + \label{reedsolomon:subsection:sfaktor}} +Die diskrete Fouriertransformation benötigt für die Inverse einen Vorfaktor von $\frac{1}{2\pi}$. +Primitiv nehmen wir an, dass wir für die Inverse Transformationsmatrix ebenfalls einen benötigen. +Nur stellt sich jetzt die Frage, wie wir diesen Vorfaktor in unserem Fall ermitteln können. +Dafür betrachten wir eine Regel aus der Linearen Algebra, nämlich dass + +\[ +A \cdot A^{-1} = E +\] +entsprechen muss. +Ist dies nicht der Fall, so benötigt $A^{-1}$ eben genau diesen Korrekturfaktor und ändert die Gleichung so zu +\begin{equation} + A \cdot s \cdot A^{-1} = E. + \label{reedsolomon:equation:sfaktor} +\end{equation} +%\[ +%A \cdot s \cdot A^{-1} = E. +%\] +Somit sollte es für uns ein leichtes Spiel sein, $s$ für unser Beispiel zu ermitteln: +\[ +\begin{pmatrix} + 8^0 & 8^0 & 8^0 & \dots & 8^0 \\ + 8^0 & 8^1 & 8^2 & \dots & 8^9 \\ + 8^0 & 8^2 & 8^4 & \dots & 8^{18} \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 8^0 & 8^9 & 8^{18} & \dots & 8^{81} \\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 7^0 & 7^0 & 7^0 & \dots & 7^0 \\ + 7^0 & 7^{1} & 7^{2} & \dots & 7^{9} \\ + 7^0 & 7^{2} & 7^{4} & \dots & 7^{18} \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 7^0 & 7^{9} & 7^{18} & \dots & 7^{81} \\ +\end{pmatrix} += +\begin{pmatrix} + 10 & 0 & 0 & \dots & 0 \\ + 0 & 10 & 0 & \dots & 0 \\ + 0 & 0 & 10 & \dots & 0 \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & 0 & \dots & 10 \\ +\end{pmatrix} +\] +Aus der letzten Matrix folgt, dass wir +\[ +s = \dfrac{1}{10} +\] +als unseren Vorfaktor setzen müssen um die Gleichung \ref{reedsolomon:equation:sfaktor} zu erfüllen. Da wir in $\mathbb{F}_{11}$ nur mit ganzen Zahlen arbeiten schreiben wir $\frac{1}{10}$ in $10^{-1}$ um und bestimmen diese Inverse erneut mit dem euklidischen Algorithmus und erhalten für $10^{-1} = 10$ als unseren Vorfaktor in $\mathbb{F}_{11}$. +% +%erfüllt wird. Wir schreiben den Bruch um in $\frac{1}{10} = 10^{-1}$ und wenden darauf erneut den euklidischen Algorithmus an und erhalten somit den Vorfaktor $10^{-1} = 10 = s$ in $\mathbb{F}_{11}$. +% +%Um $s$ eindeutig zu bestimmen müssen wir $\frac{1}{10}$ nur noch in den Bereich von $\mathbb{F}_{11}$ verschieben. Wie sich herausstellt können wir das recht einfach bewerkstelligen, da $\frac{1}{10} = 10^{-1}$ entspricht. Daraus können wir $s$ mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. +% +%Da $s$ jetzt ein Bruch ist brauchen wir ihn nur noch in $\mathbb{F}_{11}$ zu schieben. Praktischerweise können wir $\frac{1}{10} = 10^{-1}$ darstellen +% +%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. +% +%Daher nehmen wir an, dass wir für die Inverse Transformationsmatrix ebenfalls ein solcher Vorfaktor benötigen. Dieser Faktor hat seinen Ursprung in der Gleichung +%\[ +%A \cdot A^{-1} = E. +%\] +%Sollte diese Gleichung nicht aufgehen, so muss die Inverse mit +\subsection{Allgemeine Decodierung + \label{reedsolomon:subsection:algdec}} + +Wir haben jetzt alles für eine erfolgreiche Rücktransformation vom empfangenen Nachrichtenvektor beisammen. Die allgemeine Gleichung für die Rücktransformation lautet +\[ +m = s \cdot A^{-1} \cdot v. +\] +Setzen wir nun die Werte ein in +% +%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..19e5dd4 --- /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{Reed-Solomon in endlichen Körpern} +\[ +\textcolor{red}{\text{TODO: (warten auf den 1. Teil)}} +\] +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..b006f21 --- /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{Hilfstabellen für $\mathbb{F}_{11}$ + \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..4e2fd60 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -1,10 +1,10 @@ % % main.tex -- Paper zum Thema <reedsolomon> % -% (c) 2020 Hochschule Rapperswil +% (c) 2021 Joshua Bär und Michael Steiner, Hochschule Rapperswil % -\chapter{Thema\label{chapter:reedsolomon}} -\lhead{Thema} +\chapter{Reed-Solomon-Code\label{chapter:reedsolomon}} +\lhead{Reed-Solomon-Code} \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/zusammenfassung} +%\input{papers/reedsolomon/anwendungen} -> geplant +\input{papers/reedsolomon/hilfstabellen} + +\nocite{reedsolomon:weitz} +\nocite{reedsolomon:informationkommunikation} +%\nocite{reedsolomon:mendezmueller} + \printbibliography[heading=subbibliography] \end{refsection} 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..04e748c --- /dev/null +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -0,0 +1,188 @@ +% +% rekonstruktion.tex +% Autor: Michael Steiner +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Nachricht Rekonstruieren +\label{reedsolomon:section:rekonstruktion}} +\rhead{Rekonstruktion der Nachricht} +Im letzten Abschnitt 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 keinen 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. + +Wir möchten noch anmerken, dass es mehrere Wege für die Rekonstruktion des Nutzdatenteils gibt, diese aber alle auf dem Lokatorpolynom basieren. + 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/reedsolomon/zusammenfassung.tex b/buch/papers/reedsolomon/zusammenfassung.tex new file mode 100644 index 0000000..3624f16 --- /dev/null +++ b/buch/papers/reedsolomon/zusammenfassung.tex @@ -0,0 +1,15 @@ +\section{Zusammenfassung +\label{reedsolomon:section:zf}} +\rhead{Zusammenfassung} +Dieser Abschnitt beinhaltet eine Übersicht über die Funktionsweise eines Reed-Solomon-Codes für beliebige endliche Körper. + +TODO: + +\subsubsection{Schritt 1: primitives Element} + +\subsubsection{Schritt 2: Codierung} + +\subsubsection{Schritt 3: Decodierung ohne Fehler} + +\subsubsection{Schritt 4: Decodierung mit Fehler} + diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index cf6e916..b1588ff 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -1,15 +1,18 @@ \section{Einleitung\label{spannung:section:Einleitung}} +\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. -Dieses beschreibt die Beziehung von Spannung und Dehnung von linear elastischen Materialien im Eindimensionalen. Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen. -Jeder erdenkliche Punkt im Dreidimensionalen beschreibt daher einen entsprechenden individuellen Spannungszustand. +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. -Diese allgemeine Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. +Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden. +Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. 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}. \section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} \rhead{Spannungsausbreitung} @@ -21,30 +24,34 @@ Belastet man den Boden mit einer Spannung \sigma = \frac{F}{A} +, \] -, so wird diese in den Boden geleitet und von diesem kompensiert. -Im Boden entstehen unterschiedlich hohe Zusatzspannung. -Die Zusatzspannung scheint sich räumlich und berechenbar im Boden auszubreiten. -Im Falle einer konstanten Flächenlast $\sigma$ (siehe Abbildung 1.1) breitet sich die Zusatzspannung zwiebelartig aus. -Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung 1.2). -Wie diese Geometrie der Ausbreitung ist wird durch viele Modelle und Ansätze näherungsweise beschrieben. -Diese Zusatzspannung $\sigma$ ist aber sicher abhängig von $(x,y,t)$. +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~\ref{spannung:Bild4} breitet sich die Zusatzspannung zwiebelartig aus. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} - \caption{Ausbreitung der Zusatzspannung im Boden} + \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} +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{spannung:Bild5}). +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=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} - \caption{Funktionen Spannung und Dehnung} + \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 einher, welche eine Setzung bedeutet. +Bei jeder dieser Zusatzspannung geht eine entsprechende Zusatzdehnung des Bodens einher, welche eine Setzung bedeutet. Im einfachsten Fall kann modellhaft mit \[ \varepsilon @@ -58,43 +65,25 @@ s \int_{0}^{\infty}\varepsilon\enspace dt \] berechnet werden mit: -\[ -\varepsilon -= -\text{Dehnung [$-$]} -\] -\[ -\sigma -= -\text{Spannung [\si{\kilo\pascal}]} -\] -\[ -E -= -\text{Elastizitätsmodul; Young-Modul [\si{\kilo\pascal}]} -\] -\[ -t -= -\text{Tiefe [\si{\meter}]} -\] -\[ -s -= -\text{Setzung, Absenkung [m]} -\] - +\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). +Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{spannung:Bild3}). 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 Dehnung mit Hilfe einer Spannungsgleichung berechnet werden. -Diese beruht auf Annahmen nach Hooke auf einem linear elastischen Boden. +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. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} - \caption{Beispiel Lastauftrag auf Boden} + \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}
\ No newline at end of file +\end{figure} 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 be837ac..7647252 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,48 +1,47 @@ -\section{Einachsiger Spannungszustand\label{spannung:section:Einachsiger Spannungsustand}} -\rhead{Einachsiger Spannungszustand} -Ein Spannungszustand beschreibt alle Spannungen, welche in einem beliebigen Punkt im Körper wirken (siehe Abbildung 1.4). +\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~\ref{spannung:Bild2}). 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/Bild2.png} + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild2.png} \caption{Infinitesimales Bodenelement mit den 9 Spannungen} - \label{fig:infintesimaler-wurfel} + \label{fig:Bild2} \end{figure} -Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten die selben Spannungen aufweisen. -Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$ muss sich zwingend im Gleichgewicht befinden. -So sind insgesamt 9 verschiedene Spannungen möglich, wobei 3 Normal- und 6 Schubspannungen sind. -Normalspannung wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. -Alle Beträge dieser 9 Spannungen am Elementarwürfel bilden den Spannungszustand. +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. -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} - \caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe} - \label{fig:infintesimaler-wurfel} -\end{figure} -Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung). +\section{Spannungszustand\label{spannung:section:Spannungsustand}} +\rhead{Spannungszustand} + +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{spannung:Bild1}). Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. Nach Hooke gilt: \[ F \sim \Delta l -\] . -Teilt man beide Seiten mit den Konstanten $A$ und $l_0$ erhält man +\] +Teilt man beide Seiten durch die Konstanten $A$ und $l_0$, erhält man \[ \frac{F}{A} = \sigma \sim -\] -\[ \varepsilon = \frac{\Delta l}{l_0} @@ -52,22 +51,21 @@ und somit \sigma \sim \varepsilon +, \] -. -Mit: -\[ -l_0 -= -\text{Länge zu Beginn [\si{\meter}]} -\] -\[ -A -= -\text{Fläche [\si{\meter\squared}]} -\] - -Diese Beziehung gilt bei linear elastischen Materialien, welche reversibel sind und nicht dauerhaft verformt werden. +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=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} + \caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe} + \label{fig:Bild1} +\end{figure} Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich der eindimensionale Fall mit \[ \sigma @@ -75,7 +73,7 @@ Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich E\cdot\varepsilon \] beschreiben. -Im Falle, dass der E-Modul nicht konstant ist, kann dieser näherungsweise mit +Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch \[ E = diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 3b40ee9..74516c1 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,17 +1,24 @@ \section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}} \rhead{Skalare, Vektoren, Matrizen und 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 +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. + +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 8be0bdc..6326eab 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,16 +1,22 @@ \section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}} \rhead{Dreiachsiger Spannungszustand} Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen. -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. +\begin{figure} + \centering + \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 9 Spannungen. -Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als 3x3 Matrix mit +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 \[ \overline{\sigma} = @@ -23,13 +29,12 @@ Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als 3x3 Matrix mit \end{pmatrix} \] dargestellt werden und beschreibt somit den gesamten Spannungszustand. -Die Dehnungen wirken adäquat zu den Spannungen und sind durch die zwei Indizes +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 +Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\times3$ Matrix mit \[ \overline{\varepsilon} = @@ -43,14 +48,7 @@ Der Dehnungstensor ist ebenfalls ein Tensor 2.Stufe und kann somit auch als $3\t \] dargestellt werden und beschreibt den gesamten Dehnungszustand. -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} - \caption{Infinitesimales Bodenteilchen} - \label{fig:infintesimaler-wurfel} -\end{figure} - -Der Spannungs- und Dehnungstensor 2.Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. +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 @@ -108,22 +106,22 @@ und Dehnungsvektor \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} -\]. - -Um die Beziehung von Spannung und Dehnung, welche mit Tensoren 2.Stufen ausgedrückt werden, zu beschreiben, wird ein Elastizitätstensor 4.Stufe benötigt. -Dieser ist im 1D Spannungszustand ein Tensor 0.Stufe und somit ein Skalar. -Dieses Skalar ist das Elastizitätsmodul $E$. +. +\] +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 ein Vektor mit einer Matrix operiert. +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 mit +die zueinander verknüpft werden müssen, zu begründen ist. +Es ergeben sich $3^4$ Einträge, sprich eine $9\times9$ Matrix, welche allgemein \[ \overline{\overline{C}} = @@ -141,25 +139,37 @@ C_{3211} & C_{3212} & C_{3213} & C_{3221} & C_{3222} & C_{3223} & C_{3231} & C_{ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{3332} & C_{3333} \end{pmatrix} \] -ausgedrückt wird. +geschrieben werden kann. Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. Folglich gilt: \[ \overline{\overline{C}} = \overline{\overline{C}}~^{T} -\]. - +. +\] Die allgemeine Spannungsgleichung lautet nun: \[ \vec\sigma = \overline{\overline{C}}\cdot\vec{\varepsilon} -\]. +. +\] + +Als Indexnotation +\[ +\sigma_{ij} += +\sum_{k=1}^3 +\sum_{l=1}^3 +C_{ijkl}\cdot\varepsilon_{kl} +\] +kann dies ebenfalls geschrieben werden. 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 eine weitere Kennzahl eingeführt werden. -Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche mit +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 \[ \nu = @@ -168,17 +178,11 @@ Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche mit \frac{\Delta b}{b_0} \] und -\[ -\varepsilon -= -\text{Längsdehnung [$-$]} -\] -\[ -\varepsilon_q -= -\text{Querdehnung [$-$]} -\] -definiert ist. Trägt man die Konstanten in die Matrix ein ergibt sich +\begin{align*} + \varepsilon &= \text{Längsdehnung [$-$]} \\ + \varepsilon_q &= \text{Querdehnung [$-$]} +\end{align*} +definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich \[ \begin{pmatrix} \sigma_{11}\\ @@ -215,19 +219,9 @@ definiert ist. Trägt man die Konstanten in die Matrix ein ergibt sich \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} +. \] - -, welche ebenfalls als Indexnotation mit -\[ -\sigma_{ij} -= -\sum_{k=1}^3 -\sum_{l=1}^3 -C_{ijkl}\cdot\varepsilon_{kl} -\] -ausgedrückt werden können. -Die Normalspannung $\sigma_{11}$ lässt sich exemplarisch mit - +Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als \[ \sigma_{22} = @@ -247,10 +241,12 @@ Diese Symmetrie setzt daher voraus, dass = \sigma_{21} , +\qquad \sigma_{13} = \sigma_{31} , +\qquad \sigma_{23} = \sigma_{32} @@ -261,16 +257,18 @@ und folglich auch = \varepsilon_{21} , +\qquad \varepsilon_{13} = \varepsilon_{31} , +\qquad \varepsilon_{23} = \varepsilon_{32} \] gilt. -Diese Eigenschaft wird durch die Voigt'sche Notation ausgenutzt um die Gleichung vereinfachen zu können. +Diese Eigenschaft wird durch die Voigt'sche Notation \cite{spannung:Voigtsche-Notation} ausgenutzt, um die Gleichung vereinfachen zu können. Durch diese Symmetrie gilt \[ \overline{\sigma} @@ -284,7 +282,7 @@ Durch diese Symmetrie gilt \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ & \sigma_{22} & \sigma_{23} \\ - sym & & \sigma_{33} + \text{sym} & & \sigma_{33} \end{pmatrix} \qquad \Rightarrow @@ -312,7 +310,7 @@ und entsprechend = \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ - & \varepsilon_{22} & \varepsilon_{23} \\ + & \varepsilon_{22} & \varepsilon_{23} \\ \text{sym} & & \varepsilon_{33} \end{pmatrix} \qquad @@ -328,9 +326,10 @@ und entsprechend \varepsilon_{13} \\ \varepsilon_{12} \end{pmatrix} -\]. +. +\] -Aus den Vereinfachungen der Voigt'schen Notation lassen sich die Spannungs- und Dehnungstensoren als Spaltenvektoren mit je 6 Einträgen darstellen. +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 \[ @@ -350,12 +349,12 @@ beziehungsweise \end{pmatrix} = \begin{pmatrix} - C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ - C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ - C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} \\ - C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} \\ - C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} \\ - C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} + C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\ + C_{2211} & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\ + C_{3311} & C_{3322} & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ + C_{2311} & C_{2322} & C_{2333} & C_{2323} & C_{2313} & C_{2312} \\ + C_{1311} & C_{1322} & C_{1333} & C_{1323} & C_{1313} & C_{1312} \\ + C_{1211} & C_{1222} & C_{1233} & C_{1223} & C_{1213} & C_{1212} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ @@ -367,9 +366,9 @@ beziehungsweise \end{pmatrix} \] beschreiben. -Die Spannung $\sigma_{11}$ beispielsweise besteht so aus der Summe aller 6 Produkte der Konstanten $C$ und Dehnungen $\varepsilon$. +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. -Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit +Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit \[ \begin{pmatrix} @@ -382,12 +381,12 @@ Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit \end{pmatrix} = \begin{pmatrix} - C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ - & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ - & & C_{33} & C_{34} & C_{35} & C_{36} \\ - & & & C_{44} & C_{45} & C_{46} \\ - & & & & C_{55} & C_{56} \\ - \text{sym} & & & & & C_{66} + C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\ + & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\ + & & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ + & & & C_{2323} & C_{2313} & C_{2312} \\ + & & & & C_{1313} & C_{1312} \\ + \text{sym} & & & & & C_{1212} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ @@ -399,10 +398,9 @@ Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit \end{pmatrix} \] beschreiben. -Die Konstanten $C$ und $\nu$ werden wieder nach dem Hook'schen Gesetz definiert. -Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: - -\[ +Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert. +Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist: +\begin{equation} \begin{pmatrix} \sigma_{11}\\ \sigma_{22}\\ @@ -429,10 +427,12 @@ Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \varepsilon_{13}\\ \varepsilon_{12} \end{pmatrix} -\]. +. +\label{spannung:Spannungsgleichung} +\end{equation} Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, -dass diese jeweiligen Konstanten keinen Einfluss auf unsere Spannung haben. +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. @@ -477,27 +477,18 @@ Dadurch erhält man die Dehnungsgleichung: \sigma_{13}\\ \sigma_{12} \end{pmatrix} -\]. - +. +\] 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 mit +Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man die Dehnung \[ \varepsilon_{22} = \frac{1}{E}\sigma_{22} - \frac{\nu}{E}\sigma_{11} - \frac{\nu}{E}\sigma_{33} = \frac{1}{E}\cdot(\sigma_{22}-\nu\cdot\sigma_{11}-\nu\cdot\sigma_{33}) +. \] - -die Dehnung $\varepsilon_{22}$. 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 -\[ -\nu -= -0.5 -\]. - - - +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 e5574b8..3e456c3 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -1,81 +1,89 @@ -\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 +\begin{equation} p = \frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3} -\] - -oder durch Vereinfachung, da $\sigma_{22}=\sigma_{33}$ : - +\label{spannung:Invariante_p} +, +\end{equation} +welche als arithmetisches Mittel aller Normalspannungen im infinitesimalen Würfel definiert ist. +Die zweite Invariante ist die deviatorische Spannung +\begin{equation} +q += +\sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{11}-\sigma_{33})^{2}+(\sigma_{22}-\sigma_{33})^{2}}{2}} +\label{spannung:Invariante_q} +. +\end{equation} +Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt. +Die hydrostatische Spannung $p$ kann gemäss Gleichung \eqref{spannung:Invariante_p} als \[ p = \frac{\sigma_{11}+2\sigma_{33}}{3} \] - +vereinfacht werden. +Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q}als \[ q = \sigma_{11}-\sigma_{33} \] -. - -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 - -\[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} -= -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{\nu}} -\] +vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten. +Die Invarianten können mit der Spannungsformel \eqref{spannung:Spannungsgleichung} 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 \[ -\varepsilon_{\nu} +\varepsilon_{v} += +(\varepsilon_{11} - 2\varepsilon_{33}) +\qquad +\text{und} +\qquad +\varepsilon_{s} = -\text{Deviatorische Dehnung} [-] +\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33}) \] - -werden. - -Diese Komponenten kann man nun in die Vereinfachte Matrix -\[ +eingeführt werden, mit +\begin{align*} + \varepsilon_{v} &= \text{Hydrostatische Dehnung [-]} \\ + \varepsilon_{s} &= \text{Deviatorische Dehnung [-].} +\end{align*} +Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglichen werden. +Die deviatorische Dehnung $\varepsilon_{s}$ kann mit einer Verzerrung verglichen werden. + +Diese zwei Gleichungen kann man durch die Matrixschreibweise +\begin{equation} \begin{pmatrix} q\\ p @@ -87,12 +95,14 @@ Diese Komponenten kann man nun in die Vereinfachte Matrix \end{pmatrix} \begin{pmatrix} \varepsilon_{s}\\ - \varepsilon_{\nu} + \varepsilon_{v} \end{pmatrix} -\] -einsetzen. -Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. - -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. -Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben.
\ No newline at end of file +\label{spannung:Matrixschreibweise} +\end{equation} +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 \eqref{spannung:Matrixschreibweise} 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 60f2518..2f2e4ce 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -1,16 +1,16 @@ \section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}} \rhead{Oedometer-Versuch} -Mit dem Oedometer-Versuch kann der Oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. +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 steig verdichten. +Die Probe wird sich so stetig verdichten. Das Volumen nimmt ab und die Dehnung nimmt immer mehr zu. -Unter diesen Bedingungen wird das Oedometrische E-Modul mit steigender Dehnung zunehmen. +Unter diesen Bedingungen wird der oedometrische Elastizitätsmodul mit steigender Dehnung zunehmen. -Da im Boden das umgebende Material ähnliche eine seitliche Verformung verhindert, -gibt dieser Oedometrische E-Modul die Realität besser als der gewöhnliche E-Modul wieder. +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 \[ \varepsilon_{22} @@ -25,15 +25,16 @@ aber auch = \sigma_{33} \neq 0 +. \] -Die Spannung $\sigma_{11}$ wird durch durch die aufgebrachte Kraft mit +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 Randbedingen können in die vereinfachte Gleichung eingesetzt. +Diese Randbedingungen können in die vereinfachte Gleichung \eqref{spannung:Matrixschreibweise} eingesetzt werden. Diese lautet nun: \[ \begin{pmatrix} @@ -42,28 +43,37 @@ Diese lautet nun: \end{pmatrix} = \begin{pmatrix} - \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{(1-2\nu)} + \frac{E_{OED}}{(1+\nu)} & 0 \\ + 0 & \frac{E_{OED}}{3(1-2\nu)} \end{pmatrix} \begin{pmatrix} \varepsilon_{11}\\ \varepsilon_{11} \end{pmatrix} -\] . - -Daraus lässt sich bei jedem Setzungsgrad das Oedometrische E-Modul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen - -GLEICHUNGEN... - +\] +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. -Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung 1.7). +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~\ref{spannung:DiagrammOedometer-Versuch}). 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. \begin{figure} \centering \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} - \caption{Diagramm Oedometer-Versuch} - \label{fig:Diagramm Oedometer-Versuch} + \caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material} + \label{fig:DiagrammOedometer-Versuch} \end{figure}
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