143 files changed, 50233 insertions, 1468 deletions
diff --git a/buch/papers/erdbeben/Gausskurve2.pdf b/buch/papers/erdbeben/Gausskurve2.pdf
index bee3bc0..5e4afdf 100644
--- a/buch/papers/erdbeben/Gausskurve2.pdf
+++ b/buch/papers/erdbeben/Gausskurve2.pdf
diff --git a/buch/papers/erdbeben/Gausskurve2.tex b/buch/papers/erdbeben/Gausskurve2.tex
index 44319c3..2441766 100644
--- a/buch/papers/erdbeben/Gausskurve2.tex
+++ b/buch/papers/erdbeben/Gausskurve2.tex
@@ -1,13 +1,12 @@
 \documentclass{standalone}
  
 \usepackage{pgfplots}
- 
+\usepackage{txfonts}
 \pgfplotsset{compat = newest}
  
 \begin{document}
 
-
-\begin{tikzpicture}
+\begin{tikzpicture}[>=latex,thick]
 
 
 \begin{axis}[
diff --git a/buch/papers/erdbeben/Gausskurve3.pdf b/buch/papers/erdbeben/Gausskurve3.pdf
index e86a403..b86023f 100644
--- a/buch/papers/erdbeben/Gausskurve3.pdf
+++ b/buch/papers/erdbeben/Gausskurve3.pdf
diff --git a/buch/papers/erdbeben/Gausskurve3.tex b/buch/papers/erdbeben/Gausskurve3.tex
index 85455ef..032d6de 100644
--- a/buch/papers/erdbeben/Gausskurve3.tex
+++ b/buch/papers/erdbeben/Gausskurve3.tex
@@ -1,13 +1,12 @@
 \documentclass{standalone}
  
 \usepackage{pgfplots}
- 
+\usepackage{txfonts}
 \pgfplotsset{compat = newest}
  
 \begin{document}
 
-
-\begin{tikzpicture}
+\begin{tikzpicture}[>=latex,thick]
 
 
 \begin{axis}[
diff --git a/buch/papers/erdbeben/main.tex b/buch/papers/erdbeben/main.tex
index 95f1f4b..4167475 100644
--- a/buch/papers/erdbeben/main.tex
+++ b/buch/papers/erdbeben/main.tex
@@ -4,7 +4,7 @@
 % (c) 2020 Hochschule Rapperswil
 %
 \chapter{Erdbebenmessung\label{chapter:erdbeben}}
-\lhead{Thema}
+\lhead{Erdbeben}
 \begin{refsection}
 \chapterauthor{Lukas Zogg und
 Fabio Veicelli}
diff --git a/buch/papers/erdbeben/references.bib b/buch/papers/erdbeben/references.bib
index 56ca24b..444c82d 100644
--- a/buch/papers/erdbeben/references.bib
+++ b/buch/papers/erdbeben/references.bib
@@ -1,22 +1,22 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% https://bibdesk.sourceforge.io/
 
-%% Created for lukas zogg at 2021-07-17 16:48:19 +0200 
+%% Created for lukas zogg at 2021-07-27 17:56:45 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
-@article{aragher_understanding_2012,
+@article{erdbeben:aragher_understanding_2012,
 	author = {Faragher, Ramsey},
 	date-added = {2021-07-17 16:44:00 +0200},
 	date-modified = {2021-07-17 16:45:54 +0200},
-	journal = { Signal Processing Magazine},
+	journal = {Signal Processing Magazine},
 	month = {09},
 	number = {5},
 	pages = {128--132},
-	title = {Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation },
+	title = {Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation},
 	volume = {29},
 	year = {2012},
 	Bdsk-File-1 = {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}}
diff --git a/buch/papers/erdbeben/teil0.tex b/buch/papers/erdbeben/teil0.tex
index 8ce8ff2..c099340 100644
--- a/buch/papers/erdbeben/teil0.tex
+++ b/buch/papers/erdbeben/teil0.tex
@@ -23,6 +23,7 @@ Die Masse schwing jedoch in seiner Eigendynamik weiter.
 Relativbewegung des Bodens kann damit als Auslenkung 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 mit eine Gehäuse, an dem zwei Federn und eine Masse befestigt sind. 
+Der Seismograph ist in Abbildung ~\ref{erdbeben:Seismograph} ersichtlich.
 Ein Sensor unter der Masse misst die Position, bzw. die Auslenkung der Feder und der Masse.
 Dies bedeutet, unser Seismograph kann nur in eine Dimension Messwerte aufnehmen. 
 
@@ -30,52 +31,52 @@ Dies bedeutet, unser Seismograph kann nur in eine Dimension Messwerte aufnehmen.
  \begin{center}
  \includegraphics[width=5cm]{papers/erdbeben/Apperatur}
  \caption{Aufbau des Seismographen mit Gehäuse, Masse, Federn und Sensor}
+ \label{erdbeben:Seismograph}
  \end{center}
 \end{figure}
 
 \subsection{Ziel}
 Unser Seismograph misst nur die Position der Masse über die Zeit. 
-Wir wollen jedoch die Beschleunigung $a(t)$ des Boden bzw. die Kraft $f(t)$ welche auf das Gehäuse wirkt bestimmten.  
-Anhand dieser Beschleunigung bzw. der Krafteinwirkung durch die Bodenbewegung wird später das Bauwerk bemessen.
+Wir wollen jedoch die Beschleunigung $a(t)$ des Boden, bzw. die Kraft $f(t)$, welche auf das Gehäuse wirkt, bestimmten.  
+Anhand dieser Beschleunigung, bzw. der Krafteinwirkung durch die Bodenbewegung, wird später das Bauwerk bemessen.
 Dies bedeutet, die für uns interessante Grösse $f(t)$ wird nicht durch einen Sensor erfasst. 
 Jedoch können wir durch zweifaches ableiten der Positionsmessung $s(t)$ die Beschleunigung der Masse berechnen. 
 Das heisst: Die Messung ist zweifach Integriert die Kraft $f(t)$ inklusive der Eigendynamik der Masse.
-Um die Bewegung der Masse zu berechnen, müssen wir Gleichungen für unser System finden.
+Um die Krafteinwirkung der Masse zu berechnen, müssen wir Gleichungen für unser System finden.
 
 \subsection{Systemgleichung}
-Im Fall unseres Seismographen, kann die Differentialgleichung zweiter Ordnung einer gedämpften Schwingung am harmonischen Oszillator verwendet werden. 
-Diese lautet:
+Im Paper~\cite{erdbeben:mendezmueller} wurde das System gleich definiert und vorgegangen. 
+Im Fall unseres Seismographen, handelt es sich um ein Feder-Masse-Pendel.
+Dieser kann durch die Differentialgleichung zweiter Ordnung einer gedämpften Schwingung am harmonischen Oszillator beschrieben werden. 
+Die Gleichung lautet:
 \begin{equation}
-m\ddot s + 2k \dot s + Ds = f
+m\ddot s + 2k \dot s + Ds = f.
 \end{equation}
-mit den Konstanten $m$ = Masse, $k$ = Dämpfungskonstante und $D$  = Federkonstante.
-Da die DGL linear ist, kann sie in die kompaktere und einfachere Matrix-Form umgewandelt werden. Dazu wird die Differentialgleichung zweiter Ordnung substituiert:
-\[ {s_1}=s \qquad
-{s_2}=\dot s,  \qquad\]
-Somit entstehen die Gleichungen für die Position $s(t)$ der Masse :
+wobei $m$ die Masse, $k$ die Dämpfungskonstante und $D$ die Federkonstante bezeichnet.
+Da die Differentialgleichung linear ist, kann sie in die kompaktere und einfachere Matrix-Form umgewandelt werden.
+Dazu verwenden wir die Subsitution:
+\[ 
+s_1 = s 
+\qquad \text{und} \qquad
+s_2 = \dot s
+.
+\]
+Somit entstehen die Gleichungen für die Position $ \dot s_1(t)$ der Masse :
 \[ \dot {s_1} = {s_2}\] 
 und
-\[ \dot s_2 = -\frac{D}{m} {s_1} -\frac{2k}{m} {s_2} + \frac{f} {m} \] für die Beschleunigung $a(t)$ der Masse.
-
+\[ \dot s_2 = -\frac{D}{m} {s_1} -\frac{2k}{m} {s_2} + \frac{f} {m} \] 
+für die Beschleunigung $\dot s_2(t)$ der Masse.
 Diese können wir nun in der Form
-\[ {s_3}=-\frac{D}{m} {s_1} -\frac{2k}{m} {s_2} + \frac{f} {m} \]
+\[ f =-\frac{D}{m} {s_1} -\frac{2k}{m} {s_2} + \frac{f} {m} \]
 auch als Matrix-Vektor-Gleichung darstellen.
 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, innere Beschleunigung, als auch eine Beschleunigung der ganzen Apparatur, äussere Beschleunigung. 
-In unserem Fall wird die äusseren Beschleunigung gesucht, da diese der Erdbebenanregung gleich kommt. 
-\begin{equation}
-\frac{d}{dt} \left(\begin{array}{c} {s_1} \\ {s_2}  \end{array}\right) = \left(
- \begin{array}{ccc} 	
-0 & 1& 0 \\ 
-- \frac{D}{m} &-\frac{2k}{m} & \frac{1} {m}\\
-\end{array}\right)  \left(\begin{array}{c} {s_1} \\ {s_2} \\ {s_3} \end{array}\right).
-\end{equation}
-
-Durch Rücksubstituion ergibt sich:
+Die für uns relevanten Zustände sind die Position der Masse, die Geschwindigkeit der Masse und die äussere Beschleunigung des ganzen Systems. 
+
+Dabei muss unterschieden werden, um welche Beschleunigung es sich handelt.
+Das System beinhaltet sowohl eine Beschleunigung der Masse (innere Beschleunigung) als auch eine Beschleunigung der ganzen Apparatur (äussere Beschleunigung). 
+In unserem Fall wird die äusseren Beschleunigung gesucht, da diese der Erdbebenanregung gleich kommt.
+Dazu wird ein Zustandsvektor definiert:
+\[ 
+ \left(\begin{array}{c} {s_1} \\ {s_2} \\ {f} \end{array}\right).
+ \] 
+Durch Rücksubstituion ergibt sich uns folgende Systemgleichung in Matrix schreibweise, , wobei $\sot {s_1}= v$ ist:
 \begin{equation}
-\frac{d}{dt} \left(\begin{array}{c} s(t) \\ v(t) \end{array}\right) = \left(
+\frac{d}{dt} \left(\begin{array}{c} s(t) \\ v(t) \\ f(t) \end{array}\right) = \left(
  \begin{array}{ccc} 	
 0 & 1& 0 \\ 
 - \frac{D}{m} &-\frac{2k}{m} & \frac{1} {m}\\
diff --git a/buch/papers/erdbeben/teil1.tex b/buch/papers/erdbeben/teil1.tex
index e07800f..6c334bf 100644
--- a/buch/papers/erdbeben/teil1.tex
+++ b/buch/papers/erdbeben/teil1.tex
@@ -14,6 +14,8 @@
 \rhead{Kalman-Filter}
 
 \section{Kalman-Filter}
+Interessante Grösse ist also Integral von Überlagerung zweier Kräfte. 
+Wir brauchen also dir zweite Ableitung von der Messung , ohne deren Eigendynamik.
 Da wir die äussere Kraft nicht direkt messen können, benötigen wir ein Werkzeug, welches aus der gemessenen Position, die Krafteinwirkung auf unsere System schätzt. 
 Dies ist eine typische Anwendung für das Kalman-Filter.
 Unser Ziel ist es, anhand der Messung die eigentlich interessante Grösse $f$ zu bestimmen. 
@@ -23,8 +25,8 @@ Die Idee dahinter ist, dass das Kalman-Filter die nicht-deterministische Grösse
 Für mehrere Dimensionen (x,y,z) 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 auf den linearen Fall, da dadurch die wesentlichen Punkte bereits aufgezeigt werden. 
+Für ein nicht-lineares System werden Extended Kalman-Filter benötigt, bei denen die System-Matrix (A) durch die Jacobi-Matrix des System ersetzt wird.
 
 \subsection{Geschichte}
 Das Kalman-Filter wurde 1960 von Rudolf Emil Kalman entdeckt und direkt von der NASA für die Appollo Mission benutzt.
@@ -35,57 +37,60 @@ Das Filter schätzt den Zustand eines Systems anhand von Messungen und kann den
 Das Kalman-Filter schätzt den wahrscheinlichsten Wert zwischen Normalverteilungen.
 Dies bedeutet, das Filter schätzt nicht nur den Mittelwert, sondern auch die Standartabweichung.
 Da Normalverteilungen dadurch vollständig definiert sind, schätzt ein Kalman-Filter die gesamte Verteilungsfunktion des Zustandes.
+In der Abbildung~\ref{erdbeben: Zwei Normalverteilungen} sind zwei Funktionen dargestellt. 
 Die eine Funktion zeigt die errechnete Vorhersage des Zustands, bzw. deren Normalverteilung. 
 Die andere Funktion zeigt die verrauschte Messung des nächsten Zustand, bzw. deren Normalverteilung. 
-Wie man am Beispiel der Gauss-Verteilungen unten sehen kann, ist sowohl der geschätzte Zustand als auch der gemessene Zustand normalverteilt und haben dementsprechend unterschiedliche Standardabweichungen $\sigma$ und Erwartungswerte $\mu$.
-
+Wie man am Beispiel der Gauss-Verteilungen in Abblidung~\ref{erdbeben: Zwei Normalverteilungen} sehen kann, ist sowohl der geschätzte Zustand als auch der gemessene Zustand normalverteilt und haben dementsprechend unterschiedliche Standardabweichungen $\sigma$ und Erwartungswerte $\mu$. Dies wird in~\cite{erdbeben:aragher_understanding_2012}beschrieben.
 \begin{figure}
  \begin{center}
  \includegraphics[width=5cm]{papers/erdbeben/Gausskurve2.pdf}
  \caption{Zwei Normalerteilungen; Die eine Funktion zeigt die Vorhersage, die andere die Messung}
+    \label{erdbeben: Zwei Normalverteilungen}
  \end{center}
 \end{figure}
-
-
+Wir haben eine Vorhersage aus der Systemdynamik und eine Messung des Zustandes.
+Diese widersprechen sich im Allgemeinen. 
+Jedoch wissen wir die Wahrscheinlichkeiten der beiden Aussagen. 
 Um eine genauere Schätzung des Zustandes zu machen, wird nun ein Wert zwischen den beiden Verteilungen berechnet. 
 Nun wird eine Eigenschaft der Normalverteilung ausgenutzt. Durch das Multiplizieren zweier Normalverteilungen entsteht eine neue Normalverteilung. 
 Wir haben eine Normalverteilung der Vorhersage:
-
-\[ {y_1}(x;{\mu_1},{\sigma_1})=\frac{1}{\sqrt{2\pi\sigma_1^2}}\quad e^{-\frac{(x-{\mu_1})^2}{2{\sigma_1}^2}} \]
+\[ 
+{y_1}(x;{\mu_1},{\sigma_1})=\frac{1}{\sqrt{2\pi\sigma_1^2}}\quad e^{-\frac{(x-{\mu_1})^2}{2{\sigma_1}^2}} 
+\]
 und der Messung:
-\[ {y_2}(x;{\mu_2},{\sigma_2})=\frac{1}{\sqrt{2\pi\sigma_2^2}}\quad e^{-\frac{(x-{\mu_2})^2}{2{\sigma_2}^2}}. \]
-
-
-
-Diesen werden nun Multipliziert und durch deren Fläche geteilt um sie wieder zu Normieren:
-\[
-{y_f}(x;{\mu_f},{\sigma_f})=\frac{ \frac{1}{\sqrt{2\pi\sigma_1^2}}e^{-\frac{(x-{\mu_1})^2}{2{\sigma_1}^2}} \cdot \frac{1}{\sqrt{2\pi\sigma_2^2}}e^{-\frac{(x-{\mu_2})^2}{2{\sigma_2}^2}}}{\int {y_1}\cdot{y_2} dx\,}
- \] 
-
+\[ 
+{y_2}(x;{\mu_2},{\sigma_2})=\frac{1}{\sqrt{2\pi\sigma_2^2}}\quad e^{-\frac{(x-{\mu_2})^2}{2{\sigma_2}^2}}.
+\]
+Diesen werden nun multipliziert und durch deren Fläche geteilt um sie wieder zu normieren, $\odot$ beschreibt dabei die Multiplikation und die Normierung auf den Flächeninhalt eins :
+\begin{align*}
+	{y_f}(x; {\mu_f}, {\sigma_f}) = {y_1}(x;{ \mu_1},{ \sigma_1}) \odot {y_2}(x; {\mu_2}, {\sigma_2})
+	&=
+	\frac{1}{\sqrt{2\pi\sigma_1^2}}\quad e^{-\frac{(x-{\mu_1})^2}{2{\sigma_1}^2}} \odot \frac{1}{\sqrt{2\pi\sigma_2^2}}\quad e^{-\frac{(x-{\mu_2})^2}{2{\sigma_2}^2}}
+	\\
+	&=
+	\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{align*}
 Diese Kombination der beiden Verteilungen resultiert wiederum in einer Normalverteilung
-\[ {y_f}(x; {\mu_f}, {\sigma_f}) = {y_1}(x;{ \mu_1},{ \sigma_1}) {\cdot y_2}(x; {\mu_2}, {\sigma_2}), \]
 mit Erwartungswert
 \[ \mu_f = \frac{\mu_1\sigma_2^2 + \mu_2 \sigma_1^2}{\sigma_1^2 + \sigma_2^2} \]
 und Varianz
-\[ \sigma_f^2 = \frac{\sigma_1^2 \sigma_2^2}{\sigma_1^2 + \sigma_2^2}. \]
-
+\[
+\sigma_f^2 = \frac{\sigma_1^2 \sigma_2^2}{\sigma_1^2 + \sigma_2^2}.
+\]
 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. 
-
-
+Somit ist $\mu_f$ ist das gewichtete Mittel der beiden $\mu_{1,2}$, und die Varianzen sind die Gewichte!
+Die neue Funktion ist die best mögliche Schätzung für zwei Verteilungen, welche den selben Zustand beschreiben. 
+Dies ist in der Abbildung~\ref{erdbeben:Gauss3} anhand der rote Funktion ersichtlich. 
 \begin{figure}
  \begin{center}
  \includegraphics[width=5cm]{papers/erdbeben/Gausskurve3.pdf}
  \caption{Durch das Multiplizieren der blauen und der orangen Verteilung entsteht die die rote, optimale Funktion}
+ \label{erdbeben:Gauss3}
  \end{center}
 \end{figure}
-
-
 Was in zwei Dimensionen erklärt wurde, funktioniert auch in mehreren Dimensionen. 
 Dieses Prinzip mach sich das Kalman Filter zu nutze, und wird von uns für die Erdbeben Berechnung genutzt. 
 
 \section{Filter-Matrizen}
+Da wir nun ein Werkzeug besitzen, dass die Beschleunigung, welche auf das Gehäuse wirkt, ermitteln kann, wird dieses nun Schritt für Schritt erklärt. 
 Um den Kalman Filter zu starten, müssen gewisse Bedingungen definiert werden. 
 In diesem Abschnitt werden die einzelnen Parameter und Matrizen erklärt und erläutert, wofür sie nützlich sind. 
 
@@ -94,8 +99,6 @@ In diesem Abschnitt werden die einzelnen Parameter und Matrizen erklärt und erl
 Das Filter benötigt eine Anfangsbedingung. 
 In unserem Fall ist es die Ruhelage, die Masse bewegt sich nicht. 
 Zudem erfährt die Apparatur keine äussere Kraft.
-
-
 \[ {x_0 }= \left( \begin{array}{c} {s_0}\\ {v_0}\\{f_0}\end{array}\right) = \left( \begin{array}{c} 0\\ 0\\ 0\end{array}\right) \]
 
 \subsubsection*{Anfangsfehler / Kovarianzmatrix $P$}
@@ -108,7 +111,6 @@ Kovarianz: Cov(x, y) und Varianz: 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. 
 Als Initialwert für die Kovarianzmatrix ergibt sich
-
 \[ 
 {P_0 }=
 \left(
@@ -145,9 +147,9 @@ Die Matrix $\Phi$ beschreibt die Übergänge zwischen zeitlich aufeinanderfolgen
 
 \subsubsection*{Prozessrauschkovarianzmatrix $Q$}
 Die Prozessrauschmatrix teilt dem Filter mit, wie sich der Prozess verändert. 
-Kalman-Filter berücksichtigen sowohl Unsicherheiten wie Messfehler und -rauschen. 
-In der Matrix $Q$ geht es jedoch im die Unsicherheit die der Prozess mit sich bringt. 
-Bei unserem Modell könnte das beispielsweise ein Windstoss an die Masse sein. 
+Kalman-Filter berücksichtigen Unsicherheiten wie Messfehler und -rauschen. 
+In der Matrix $Q$ geht es jedoch um die Unsicherheit, die der Prozess mit sich bringt. 
+Bei unserem Modell könnte das beispielsweise ein Windstoss an die Masse sein oder auch die Ungenauigkeiten im Modell, wie die Annahme das dich die Kraft nicht ändert.
 Für uns wäre dies:
 \[ 
 Q = \left(
@@ -157,7 +159,6 @@ Q = \left(
 0 & 0& {\sigma_f }^2\\
 \end{array}\right)  
  \]
-
 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 nicht die der Messung.
 
@@ -165,13 +166,15 @@ Das Bedeutet wiederum dass $Q$ die Unsicherheit des Prozesses beschreibt und nic
 Die Messmatrix gibt an, welche Parameter gemessen werden. 
 $H$ ist die Gleichung die für die Vorhersage der Messung.
 In unserem Falle ist es die Position der Massen. 
-
-\[ H = (1, 0, 0) \]
+\[ 
+H = (1, 0, 0) 
+\]
 
 \subsubsection*{Messrauschkovarianz $R$}
 Die Messrauschkovarianzmatrix beinhaltet, wie der Name schon sagt, das Rauschen der Messung. 
 In unserem Fall wird nur die Position der Masse gemessen. Da wir keine anderen Sensoren haben ist $R$ lediglich:
-\[ R= ({\sigma_{sensor}}^2).
+\[ 
+R= ({\sigma_\mathrm{sensor}}^2).
  \] 
 Diese Messrauchen wird meistens vom Sensorhersteller angegeben. 
 Für unsere theoretische Apparatur wird hier ein kleiner Fehler eingesetzt da heutige Sensoren sehr genau messen können. 
@@ -182,19 +185,25 @@ Zuerst wird der nächste Zustand der Masse vorhergesagt, danach wird die Messung
 Das Filter berechnet aufgrund der aktuellen Schätzung eine Vorhersage. 
 Diese wird, sobald verfügbar, mit der Messung verglichen. 
 Aus dieser Differenz und den Unsicherheiten des Prozesses ($Q$) und der Messung ($R$) wird der wahrscheinlichste, neue Zustand geschätzt.
+Dabei muss genau auf den Index geachtet werden. Nach dem Artikel~\cite{erdbeben:wikipedia} ist die Indexierung so genormt:
+Der Zeitschritt wird mit $k$ definiert, $k-1$ ist somit ein Zeitschritt vor $k$.
+Auf der linken Seite von | wird der aktuelle Zustand verlangt, bzw. ausgegeben, auf der rechten Seiten den bisherigen Zustand.
+Dies bedeutet, dass die Notation $x_{n|m}$ die Schätzung von $x$ zum Zeitpunkt $n$ bis und mit zur Zeitpunkt $m \leq \ n$ präsentiert. 
 
 \subsubsection*{Vorhersage}
 Im Filterschritt Vorhersage wird der nächste Zustand anhand des Anfangszustand und der Systemmatrix berechnet. 
 Dies funktioniert mit dem Rechenschritt:
-\[ 
-{x_{k-1}}=\Phi \cdot {x_{k-1}}= \exp(A\Delta t)\cdot{x_{k-1}}.
- \] 
-
-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. 
+\[
+{x_{k|k-1}}=\Phi{x_{k-1|k-1}}= \exp(A\Delta t){x_{k-1|k-1}}.
+\] 
+Die Kovarianz $P_{k|k-1}$ wird ebenfalls neu berechnet. Zudem 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
-\[ {P_{k-1}} = {\Phi_k}  {P_{k-1}} {\Phi_k} ^T + {Q_{k-1}} .\] 
-Es vergeht genau $t$ Zeit, und dieser Vorgang wird wiederholt. 
+\[
+{P_{k|k-1}}=\Phi {P_{k-1|k-1}} {\Phi _{k}}^T + {Q_{k-1}}.
+\] 
+Es vergeht genau $\Delta t$ Zeit, und dieser Vorgang wird wiederholt. 
+Das hochgestellte T bezeichnet die transponierte Matrix.
 Dabei wird in den späteren Schritten überprüft, 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.
@@ -202,74 +211,83 @@ Das Filter passt sich selber an und korrigiert sich bei grosser Abweichung.
 \subsubsection*{Messen}
 Der Sensor wurde noch nicht benutz, doch genau der liefert Werte für das Filter. 
 Die aktuellen Messwerte $z$ werden die Innovation $w$ mit dem Zustandsvektor $x$ und der Messmatrix $H$ zusammengerechnet.
-Hier bei wird lediglich die Messung mit dem Fehler behaftet, und die Messmatrix $H$ mit der Vorhersage multipliziert
-
-\[{w_{k}}={z_{k}}-{H}\cdot{x_{k-1}}.\] 
-
+Hier bei wird lediglich die Messung mit dem Fehler behaftet, und die Messmatrix $H$ mit der Vorhersage multipliziert.
+\[
+{w_{k}}={z_{k}}-{H}{x_{k|k-1}}.
+\] 
 Die Innovation ist der Teil der Messung, die nicht durch die Systemdynamik erklärt werden kann. 
 Die Hilfsgröße Innovation beschreibt, wie genau die Vorhersage den aktuellen Messwert mittels der Systemmatrix $\Phi$ 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.
+Innovation = Messung - Vorhersage. Dies leuchtet ein, 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. 
 \[ 
-{S_{k}}={H}{P_{k-1}}{H}^T+{R_{k}}
- \] 
+{S_{k}}={H}{P_{k|k-1}}{H}^T+{R_{k}}
+\] 
 
 \subsubsection*{Aktualisieren}
 Im nächsten Schritt kommt nun die Wahrscheinlichkeit dazu. 
-\[ 
-{K_{k}}= {{P_{k-1}} \cdot {H_{k}^T}}\cdot {S_{k}}^{-1} 
- \] 
+\[{K_{k}}= {P_{k|k-1}} {H^T}{S_{k}^{-1}}\] 
 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, wenn der Messwert dem vorhergesagten Systemzustand entspricht. 
-Sind die Messwerte komplett anders als die Vorhersage, werden die Elemente in der Matrix $K$ grösser.
-Anhand der Informationen aus dem Kalman-Gain $K$ wird das System aktualisiert.
+Das Kalman-Gain gibt dem Zustand die Gewichtung, bzw. wie die Vorhersage auf den Zustand passt.
+Vereinfacht gesagt: Es wird das das Verhältnis zwischen der Unsicherheit der Vorhersage $P_k$ zu der zugehörigen Messunsicherheit $R_k$ gebildet. 
+In unserem Fall wird werden die Elemente der Kalman-Matrix vorweg berechnet, da das Kalman-Gain ohne Messungen auskommt. 
 
-\[ 
-{x_{k|k}}={x_{k-1}}+({K_{k}}\cdot {w_{k}}) 
- \] 
+Anhand der Informationen aus dem Kalman-Gain $K$ wird das System aktualisiert.
+\[
+{x_{k|k}}={x_{k|k-1}}+{K_{k}}{w_{k}}
+\] 
+Dabei wird der Unterschied zwischen dem erwarteten, errechneten, Zustand und dem gemessenen Zustand berechnet.
 
 Dazu kommt  eine neue Kovarianz für den nächste Vorhersageschritt:
-
-\[ 
-{P_{k}}=(I-({K_{k}} \cdot {H})) \cdot {P_{k-1}}  
- \] 
-
+\[
+{P_{k|k}}=(I-{K_{k}}{H}){P_{k|k-1}} 
+\] 
 Der ganze Algorithmus und beginnt wieder mit der Vorhersage 
-
-\[ 
-{x_{k-1}}=\Phi \cdot {x_{k-1}}= \exp(A\Delta t)\cdot{x_{k-1}}.
- \] 
-
+\[
+{x_{k|k-1}}=\Phi{x_{k-1|k-1}}= \exp(A\Delta t){x_{k|k-1}}.
+\] 
 
 \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
-\[{x_{k-1}}={\Phi} \cdot {x_{k-1}}= \exp(A\Delta t)\cdot{x_{k-1}}.\] 
+\[
+{x_{k|k-1}}=\Phi{x_{k-1|k-1}}= \exp(A\Delta t){x_{k-1|k-1}}.
+\] 
 
 2. Nächste Fehlerkovarianz vorhersagen
-\[{P_{k-1}}={\Phi} {P_{k-1}} {\Phi _{k}}^T + {Q_{k-1}}.\] 
+\[
+{P_{k|k-1}}=\Phi {P_{k-1|k-1}} {\Phi _{k}}^T + {Q_{k-1}}.
+\] 
 
 3. Zustand wird gemessen
-\[{w_{k}}={z_{k}}-{H}\cdot{x_{k-1}}.\] 
+\[
+{w_{k}}={z_{k}}-{H}{x_{k|k-1}}.
+\] 
 
 4. Innovation (= Messung -  Vorhersage)
-\[ {S_{k}}={H}{P_{k-1}}{H}^T+{R_{k}}\] 
+\[ 
+{S_{k}}={H}{P_{k|k-1}}{H}^T+{R_{k}}
+\] 
 
 5. Das Kalman Filter anwenden
-\[{K_{k}}= {P_{k-1}} \cdot {H^T}\cdot {S_{k}^{-1}}\] 
+\[
+{K_{k}}= {P_{k|k-1}} {H^T}{S_{k}^{-1}}
+\] 
 
 6. Schätzung aktualisieren
-\[{x_{k}}={x_{k-1}}+({K_{k}}\cdot {w_{k}}) \] 
+\[
+{x_{k|k}}={x_{k|k-1}}+{K_{k}}{w_{k}}
+\] 
 
 7. Fehlerkovarianz aktualisieren
-\[{P_{k}}=(I-({K_{k}}\cdot {H})) \cdot {P_{k-1}} \] 
+\[
+{P_{k|k}}=(I-{K_{k}}{H}){P_{k|k-1}}
+\] 
 
 8. Die Outputs von $k$ werden die Inputs für ${k-1}$ und werden wieder im Schritt 1 verwendet
 
diff --git a/buch/papers/multiplikation/Makefile b/buch/papers/multiplikation/Makefile
index 8f04c2c..8f04c2c 100644..100755
--- a/buch/papers/multiplikation/Makefile
+++ b/buch/papers/multiplikation/Makefile
diff --git a/buch/papers/multiplikation/Makefile.inc b/buch/papers/multiplikation/Makefile.inc
index b78d67e..074020f 100644..100755
--- a/buch/papers/multiplikation/Makefile.inc
+++ b/buch/papers/multiplikation/Makefile.inc
@@ -7,8 +7,7 @@ dependencies-multiplikation =						\
 	papers/multiplikation/packages.tex					\
 	papers/multiplikation/main.tex					\
 	papers/multiplikation/references.bib				\
-	papers/multiplikation/teil0.tex					\
-	papers/multiplikation/teil1.tex					\
-	papers/multiplikation/teil2.tex					\
-	papers/multiplikation/teil3.tex
+	papers/multiplikation/einlteung.tex					\
+	papers/multiplikation/loesungsmethoden.tex					\
+	papers/multiplikation/problemstellung.tex					
 
diff --git a/buch/papers/multiplikation/code/Figure_1.png b/buch/papers/multiplikation/code/Figure_1.png
new file mode 100755
index 0000000..9def15a
--- /dev/null
+++ b/buch/papers/multiplikation/code/Figure_1.png
diff --git a/buch/papers/multiplikation/code/MM b/buch/papers/multiplikation/code/MM
new file mode 100755
index 0000000..f07985f
--- /dev/null
+++ b/buch/papers/multiplikation/code/MM
diff --git a/buch/papers/multiplikation/code/MM.c b/buch/papers/multiplikation/code/MM.c
new file mode 100755
index 0000000..04c4dab
--- /dev/null
+++ b/buch/papers/multiplikation/code/MM.c
@@ -0,0 +1,465 @@
+#include <stdio.h>
+#include <stdint.h>
+#include <stdlib.h>
+#include <time.h>
+#include <omp.h>
+#include "c_matrix.h"
+#include <gsl/gsl_cblas.h>
+#include <string.h>
+
+void MM(int *A, int *B, int *C, int n);
+void openMP_MM(int *A, int *B, int *C, int n);
+void winograd(int *A, int *B, int *C, int n);
+int winograd_inner(int *a, int *b, int n);
+void run_algo(void (*algo)(), char alog_name[], int print);
+void run_algo_cblas(int print);
+void MM_dc(int *A, int *B, int *C, int n);
+void strassen(int *A, int *B, int *C, int n);
+void printMatrix(int *C, int n);
+void printMatrix_double(double *C, int n);
+void split(int *in, int *out, int n, int col, int row);
+void join(int *in, int *out, int n, int col, int row);
+void add(int *A, int *B, int *C, int n);
+void sub(int *A, int *B, int *C, int n);
+void multiply(int *A, int *B, int *C, int n);
+
+int main() {
+	// omp_set_dynamic(0);
+	// omp_set_num_threads(4);
+//	run_algo(openMP_MM, "openMP_MM",0);
+	run_algo(MM_dc, "MM_dc",0);
+	run_algo(strassen, "strassen",0);
+
+	run_algo(MM, "MM", 0);
+  // run_algo(winograd, "winograd", 0);
+  run_algo_cblas(0);
+
+	return 0;
+}
+
+void MM(int *A, int *B, int *C, int n) {
+	for (int i = 0; i < n; ++i) {
+		for (int j = 0; j < n; ++j) {
+			int sum = 0;
+			for (int k = 0; k < n; ++k) {
+				sum += (*((A + i * n) + k)) * (*((B + k * n) + j));
+			}
+			*((C + i * n) + j) = sum;
+		}
+	}
+}
+
+int winograd_inner(int *a, int *b, int n){
+	int ab = 0;
+	if(n%2==0)
+		{
+			int xi = 0;
+			int etha = 0;
+			for(int i = 0; i<n/2;++i)
+			{
+				xi += a[2*i]*a[2*i+1];
+				etha += b[2*i]*b[2*i+1];
+				ab += (a[2*i]+b[2*i+1])*(a[2*i+1]+b[2*i]);
+			}
+			ab = ab-etha-xi;
+		}
+		return ab;
+	}
+
+	void winograd(int *A, int *B, int *C, int n) {
+
+		int xi_array[n];
+		int etha_array[n];
+		int xi = 0;
+		int etha = 0;
+		int ab = 0;
+
+		for (int i = 0; i < n; ++i) {
+			xi = 0;
+			etha = 0;
+			for(int k = 0;k<n/2;++k)
+			{
+				xi += (*((A + i * n) + 2*k))*(*((A + i * n) + (2*k+1)));
+				etha += (*((B + 2*k * n) + i))*(*((B + (2*k+1) * n) + i));
+			}
+			xi_array[i] = xi;
+			etha_array[i] = etha;
+    }
+
+		for (int i = 0; i < n; ++i) {
+			for (int j = 0; j < n; ++j) {
+				ab = 0;
+				for(int k = 0;k<n/2;++k)
+				{
+			  	ab += ((*((A + i * n) + 2*k))+(*((B + (2*k+1) * n) + j)))*((*((A + i * n) + (2*k+1)))+(*((B + 2*k * n) + j)));
+				}
+				*((C + i * n) + j) =  ab-etha_array[j]-xi_array[i];
+				}
+			}
+
+
+
+
+		// for (int i = 0; i < n; ++i) {
+		// 	int *a = (int*) malloc(n * sizeof(int));
+		// 	for(int k = 0; k<n; ++k)
+		// 	{
+		// 		a[k] = (*((A + i * n) + k));
+		// 	}
+		//
+		// 	for (int j = 0; j < n; ++j) {
+		// 		int *b = (int*) malloc(n * sizeof(int));
+		// 		for(int k = 0; k<n; ++k)
+		// 		{
+		// 			b[k] =(*((B + k * n) + j));
+		// 		}
+		// 		*((C + i * n) + j) = winograd_inner(a,b,n);
+		// 		}
+		// 	}
+		}
+
+
+void openMP_MM(int *A, int *B, int *C, int n) {
+
+		#pragma omp parallel for
+		for (int i = 0; i < n; ++i) {
+			for (int j = 0; j < n; ++j) {
+				int sum = 0;
+				for (int k = 0; k < n; ++k) {
+					sum += (*((A + i * n) + k)) * (*((B + k * n) + j));
+				}
+				*((C + i * n) + j) = sum;
+			}
+		}
+}
+
+void MM_dc(int *A, int *B, int *C, int n) {
+	if (n <= 2) {
+		MM((int*) A, (int*) B, (int*) C, n);
+	} else {
+		int *A11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *A12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *A21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *A22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		split((int*) A, (int*) A11, n / 2, 0, 0);
+		split((int*) A, (int*) A12, n / 2, 0, n / 2);
+		split((int*) A, (int*) A21, n / 2, n / 2, 0);
+		split((int*) A, (int*) A22, n / 2, n / 2, n / 2);
+		split((int*) B, (int*) B11, n / 2, 0, 0);
+		split((int*) B, (int*) B12, n / 2, 0, n / 2);
+		split((int*) B, (int*) B21, n / 2, n / 2, 0);
+		split((int*) B, (int*) B22, n / 2, n / 2, n / 2);
+
+		int *tmp1 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp2 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp3 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp4 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp5 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp6 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp7 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *tmp8 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		MM_dc((int*) A11, (int*) B11, (int*) tmp1, n / 2);
+		MM_dc((int*) A12, (int*) B21, (int*) tmp2, n / 2);
+		MM_dc((int*) A11, (int*) B12, (int*) tmp3, n / 2);
+		MM_dc((int*) A12, (int*) B22, (int*) tmp4, n / 2);
+		MM_dc((int*) A21, (int*) B11, (int*) tmp5, n / 2);
+		MM_dc((int*) A22, (int*) B21, (int*) tmp6, n / 2);
+		MM_dc((int*) A21, (int*) B12, (int*) tmp7, n / 2);
+		MM_dc((int*) A22, (int*) B22, (int*) tmp8, n / 2);
+
+		free(A11);
+		free(A12);
+		free(A21);
+		free(A22);
+		free(B11);
+		free(B12);
+		free(B21);
+		free(B22);
+
+		int *C11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *C12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *C21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *C22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		add((int*) tmp1, (int*) tmp2, (int*) C11, n / 2);
+		add((int*) tmp3, (int*) tmp4, (int*) C12, n / 2);
+		add((int*) tmp5, (int*) tmp6, (int*) C21, n / 2);
+		add((int*) tmp7, (int*) tmp8, (int*) C22, n / 2);
+
+		free(tmp1);
+		free(tmp2);
+		free(tmp3);
+		free(tmp4);
+		free(tmp5);
+		free(tmp6);
+		free(tmp7);
+		free(tmp8);
+
+		join((int*) C11, (int*) C, n / 2, 0, 0);
+		join((int*) C12, (int*) C, n / 2, 0, n / 2);
+		join((int*) C21, (int*) C, n / 2, n / 2, 0);
+		join((int*) C22, (int*) C, n / 2, n / 2, n / 2);
+
+		free(C11);
+		free(C12);
+		free(C21);
+		free(C22);
+
+	}
+}
+
+void strassen(int *A, int *B, int *C, int n) {
+	if (n <= 2) {
+
+		int P, Q, R, S, T, U, V;
+		P = ((*((A + 0 * n) + 0)) + (*((A + 1 * n) + 1)))
+				* ((*((B + 0 * n) + 0)) + (*((B + 1 * n) + 1)));
+		Q = ((*((A + 1 * n) + 0)) + (*((A + 1 * n) + 1)))
+				* ((*((B + 0 * n) + 0)));
+		R = ((*((A + 0 * n) + 0)))
+				* ((*((B + 0 * n) + 1)) - (*((B + 1 * n) + 1)));
+		S = ((*((A + 1 * n) + 1)))
+				* ((*((B + 1 * n) + 0)) - (*((B + 0 * n) + 0)));
+		T = ((*((A + 0 * n) + 0)) + (*((A + 0 * n) + 1)))
+				* ((*((B + 1 * n) + 1)));
+		U = ((*((A + 1 * n) + 0)) - (*((A + 0 * n) + 0)))
+				* ((*((B + 0 * n) + 0)) + (*((B + 0 * n) + 1)));
+		V = ((*((A + 0 * n) + 1)) - (*((A + 1 * n) + 1)))
+				* ((*((B + 1 * n) + 0)) + (*((B + 1 * n) + 1)));
+		(*((C + 0 * n) + 0)) = P + S - T + V;
+		(*((C + 0 * n) + 1)) = R + T;
+		(*((C + 1 * n) + 0)) = Q + S;
+		(*((C + 1 * n) + 1)) = P + R - Q + U;
+
+	} else {
+		int *A11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *A12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *A21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *A22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *B22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		split((int*) A, (int*) A11, n / 2, 0, 0);
+		split((int*) A, (int*) A12, n / 2, 0, n / 2);
+		split((int*) A, (int*) A21, n / 2, n / 2, 0);
+		split((int*) A, (int*) A22, n / 2, n / 2, n / 2);
+		split((int*) B, (int*) B11, n / 2, 0, 0);
+		split((int*) B, (int*) B12, n / 2, 0, n / 2);
+		split((int*) B, (int*) B21, n / 2, n / 2, 0);
+		split((int*) B, (int*) B22, n / 2, n / 2, n / 2);
+
+		int *P = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *Q = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *R = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *S = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *T = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *U = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *V = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		int *addA = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *addB = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+    add((int*) A11, (int*) A22, (int*) addA, n / 2);
+		add((int*) B11, (int*) B22, (int*) addB, n / 2);
+		strassen((int*) addA, (int*) addB, (int*) P, n / 2);
+
+		add((int*) A21, (int*) A22, (int*) addA, n / 2);
+		strassen((int*) addA, (int*) B11, (int*) Q, n / 2);
+
+		sub((int*) B12, (int*) B22, (int*) addB, n / 2);
+		strassen((int*) A11, (int*) addB, (int*) R, n / 2);
+
+		sub((int*) B21, (int*) B11, (int*) addB, n / 2);
+		strassen((int*) A22, (int*) addB, (int*) S, n / 2);
+
+		add((int*) A11, (int*) A12, (int*) addA, n / 2);
+		strassen((int*) addA, (int*) B22, (int*) T, n / 2);
+
+		sub((int*) A21, (int*) A11, (int*) addA, n / 2);
+		add((int*) B11, (int*) B12, (int*) addB, n / 2);
+		strassen((int*) addA, (int*) addB, (int*) U, n / 2);
+
+		sub((int*) A12, (int*) A22, (int*) addA, n / 2);
+		add((int*) B21, (int*) B22, (int*) addB, n / 2);
+		strassen((int*) addA, (int*) addB, (int*) V, n / 2);
+
+		free(A11);
+		free(A12);
+		free(A21);
+		free(A22);
+		free(B11);
+		free(B12);
+		free(B21);
+		free(B22);
+		free(addA);
+		free(addB);
+
+		int *C11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *C12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *C21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *C22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		int *resAdd1 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+		int *resAdd2 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+		add((int*) R, (int*) T, (int*) C12, n / 2);
+		add((int*) Q, (int*) S, (int*) C21, n / 2);
+
+		add((int*) P, (int*) S, (int*) resAdd1, n / 2);
+		add((int*) resAdd1, (int*) V, (int*) resAdd2, n / 2);
+		sub((int*) resAdd2, (int*) T, (int*) C11, n / 2);
+
+		add((int*) P, (int*) R, (int*) resAdd1, n / 2);
+		add((int*) resAdd1, (int*) U, (int*) resAdd2, n / 2);
+		sub((int*) resAdd2, (int*) Q, (int*) C22, n / 2);
+
+		free(P);
+		free(Q);
+		free(R);
+		free(S);
+		free(T);
+		free(U);
+		free(V);
+		free(resAdd1);
+		free(resAdd2);
+
+		join((int*) C11, (int*) C, n / 2, 0, 0);
+		join((int*) C12, (int*) C, n / 2, 0, n / 2);
+		join((int*) C21, (int*) C, n / 2, n / 2, 0);
+		join((int*) C22, (int*) C, n / 2, n / 2, n / 2);
+
+		free(C11);
+		free(C12);
+		free(C21);
+		free(C22);
+	}
+}
+
+void add(int *A, int *B, int *C, int n) {
+	for (int i = 0; i < n; i++) {
+		for (int j = 0; j < n; j++) {
+			*((C + i * n) + j) = *((A + i * n) + j) + *((B + i * n) + j);
+		}
+	}
+}
+
+void sub(int *A, int *B, int *C, int n) {
+	for (int i = 0; i < n; i++) {
+		for (int j = 0; j < n; j++) {
+			*((C + i * n) + j) = *((A + i * n) + j) - *((B + i * n) + j);
+		}
+	}
+}
+
+void multiply(int *A, int *B, int *C, int n) {
+	int mul;
+
+	for (int i = 0; i < n; ++i) {
+		for (int j = 0; j < n; ++j) {
+			mul = (*((A + i * n) + j)) * (*((B + i * n) + j));
+			*((C + i * n) + j) = mul;
+		}
+	}
+}
+
+void split(int *in, int *out, int n, int col, int row) {
+	for (int i1 = 0, i2 = col; i1 < n; i1++, i2++)
+		for (int j1 = 0, j2 = row; j1 < n; j1++, j2++) {
+			*((out + i1 * n) + j1) = *((in + i2 * n * 2) + j2);
+
+		}
+}
+
+void join(int *in, int *out, int n, int col, int row) {
+	for (int i1 = 0, i2 = col; i1 < n; i1++, i2++)
+		for (int j1 = 0, j2 = row; j1 < n; j1++, j2++)
+			*((out + i2 * n * 2) + j2) = *((in + i1 * n) + j1);
+}
+
+void printMatrix(int *C, int n) {
+	for (int i = 0; i < n; ++i) {
+		for (int j = 0; j < n; ++j) {
+			printf("%d ", *((C + i * n) + j));
+		}
+		printf("\n");
+	}
+}
+
+void printMatrix_double(double *C, int n) {
+	for (int i = 0; i < n; ++i) {
+		for (int j = 0; j < n; ++j) {
+			printf("%.0f ", *((C + i * n) + j));
+		}
+		printf("\n");
+	}
+}
+
+void run_algo(void (*algo)(), char alog_name[], int print)
+{
+	FILE *fptr;
+
+	char fileName[40] =  "meas/";
+	strcat(fileName, alog_name);
+	strcat(fileName, ".txt");
+	fptr = fopen(fileName, "w");
+
+
+	for(int i=0; i<n_arrays; ++i)
+	{
+		for(int j = 0; j<1; ++j)
+		{
+	    int *C = (int*) malloc(n[i] * n[i] * sizeof(int));
+    	double dtime = omp_get_wtime();
+      algo(Ap[i], Bp[i], (int*) C, n[i]);
+	    dtime = omp_get_wtime() - dtime;
+			// printf("The %s program took %f seconds to execute \n", alog_name, dtime);
+			fprintf(fptr, "%f,%d\n", dtime, n[i]);
+
+			if(print==1)
+			{
+				printMatrix((int*)C, n[i]);
+			}
+			free(C);
+  }
+	}
+	fclose(fptr);
+
+}
+
+void run_algo_cblas(int print)
+{
+
+	FILE *fptr;
+
+	fptr = fopen("meas/blas.txt", "w");
+	for(int i=0; i<n_arrays; ++i)
+	{
+		for(int j = 0; j<1; ++j)
+		{
+			double *dC = (double*) malloc(n[i] * n[i] * sizeof(double));
+			double dtime = omp_get_wtime();
+			cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, n[i], n[i], n[i], 1.0, dAp[i], n[i],
+			dBp[i], n[i], 0.0, dC, n[i]);
+			dtime = omp_get_wtime() - dtime;
+			// printf("The cblas program took %f seconds to execute \n", dtime);
+			fprintf(fptr, "%f,%d\n",dtime, n[i]);
+
+			if(print==1)
+			{
+				printMatrix_double( (double*)dC, n[i]);
+			}
+
+			free(dC);
+		}
+	}
+	fclose(fptr);
+
+}
diff --git a/buch/papers/multiplikation/code/MM.py b/buch/papers/multiplikation/code/MM.py
new file mode 100644
index 0000000..626b82d
--- /dev/null
+++ b/buch/papers/multiplikation/code/MM.py
@@ -0,0 +1,311 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Fri Mar 19 07:31:29 2021
+
+@author: nunigan
+"""
+import numpy as np
+import time
+import matplotlib.pyplot as plt
+from scipy.optimize import curve_fit
+import tikzplotlib
+def MM(A, B):
+    n = np.shape(A)[0]
+    C = np.zeros((n, n))
+    for i in range(n):
+        for j in range(n):
+            C[i, j] = 0
+            for k in range(n):
+                C[i, j] += A[i, k]*B[k, j]
+    return C
+
+
+def MM_dc(A, B):
+    n = np.shape(A)[0]
+    if(n <= 2):
+        C = np.zeros((n, n))
+        C[0, 0] = A[0, 0]*B[0, 0]+A[0, 1]*B[1, 0]
+        C[0, 1] = A[0, 0]*B[0, 1]+A[0, 1]*B[1, 1]
+        C[1, 0] = A[1, 0]*B[0, 0]+A[1, 1]*B[1, 0]
+        C[1, 1] = A[1, 0]*B[0, 1]+A[1, 1]*B[1, 1]
+        return C
+    else:
+        A11, A12, A21, A22 = A[:n//2, :n//2], A[:n//2, n//2:], A[n//2:, :n//2], A[n//2:, n//2:]
+        B11, B12, B21, B22 = B[:n//2, :n//2], B[:n//2, n//2:], B[n//2:, :n//2], B[n//2:, n//2:]
+        C11 = MM_dc(A11, B11) + MM_dc(A12, B21)
+        C12 = MM_dc(A11, B12) + MM_dc(A12, B22)
+        C21 = MM_dc(A21, B11) + MM_dc(A22, B21)
+        C22 = MM_dc(A21, B12) + MM_dc(A22, B22)
+        C = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22))))
+    return C
+
+
+def strassen(A, B):
+    n = np.shape(A)[0]
+    if(n <= 2):
+        C = np.zeros((n, n))
+        P = (A[0, 0]+A[1, 1])*(B[0, 0]+B[1, 1])
+        Q = (A[1, 0]+A[1, 1])*B[0, 0]
+        R = A[0, 0]*(B[0, 1]-B[1, 1])
+        S = A[1, 1]*(B[1, 0]-B[0, 0])
+        T = (A[0, 0]+A[0, 1])*B[1, 1]
+        U = (A[1, 0]-A[0, 0])*(B[0, 0]+B[0, 1])
+        V = (A[0, 1]-A[1, 1])*(B[1, 0]+B[1, 1])
+        C[0, 0] = P+S-T+V
+        C[0, 1] = R+T
+        C[1, 0] = Q+S
+        C[1, 1] = P+R-Q+U
+        return C
+    else:
+        m = n//2
+        A11, A12, A21, A22 = A[:m, :m], A[:m, m:], A[m:, :m], A[m:, m:]
+        B11, B12, B21, B22 = B[:m, :m], B[:m, m:], B[m:, :m], B[m:, m:]
+        P = strassen((A11+A22),(B11+B22))
+        Q = strassen((A21+A22),B11)
+        R = strassen(A11,(B12-B22))
+        S = strassen(A22,(B21-B11))
+        T = strassen((A11+A12),B22)
+        U = strassen((A21-A11),(B11+B12))
+        V = strassen((A12-A22),(B21+B22))
+        
+        C11 = P+S-T+V
+        C12 = R+T
+        C21 = Q+S
+        C22 = P+R-Q+U
+
+        C = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22))))
+    return C
+
+def winograd_inner(a, b):
+    n = np.shape(a)[0]
+    if n%2 == 0:
+        xi = np.sum(a[::2]*a[1::2])
+        etha = np.sum(b[::2]*b[1::2])
+        # print("xi = {}, etha = {}".format(xi, etha))
+        ab = np.sum((a[::2]+b[1::2])*(a[1::2]+b[::2]))-xi-etha
+    else:
+        xi = np.sum(a[0:-1:2]*a[1::2])
+        etha = np.sum(b[0:-1:2]*b[1::2])
+        ab = np.sum((a[0:-1:2]+b[1::2])*(a[1::2]+b[0:-1:2]))-xi-etha+a[-1]*b[-1]
+    return ab
+
+def winograd(A, B):
+    m,n = np.shape(A)
+    n2,p = np.shape(B)
+    C = np.zeros((m,p))
+    for i in range(np.shape(A)[0]):
+        for j in range(np.shape(B)[1]):
+            C[i,j] = winograd_inner(A[i,:], B[:,j])
+    return C
+
+def winograd2(A, B):
+    m,n = np.shape(A)
+    n2,p = np.shape(B)
+    C = np.zeros((m,p))
+    xi = np.zeros((m))
+    eta = np.zeros((p))
+    ab = 0
+    for i in range(m):
+        for j in range(n//2):
+            xi[i] += A[i,2*j]*A[i,2*j+1]
+
+    for i in range(p):
+        for j in range(n//2):
+           eta[i] += B[2*j,i]*B[2*j+1,i]
+
+    if n%2==0:
+        for i in range(m):
+            for j in range(p):
+                ab = 0
+                for k in range(n//2):
+                    ab += (A[i,2*k]+B[2*k+1,j])*(A[i,2*k+1]+B[2*k,j])
+                C[i,j] = ab-eta[j]-xi[i]
+    else:
+        for i in range(m):
+            for j in range(p):
+                ab = 0
+                for k in range(n//2):
+                    ab += (A[i,2*k]+B[2*k+1,j])*(A[i,2*k+1]+B[2*k,j])
+                C[i,j] = ab-eta[j]-xi[i]+A[i,-1]*B[-1,j]
+        
+    return C
+        
+def test_perfomance(n):
+    t_mm = []
+    t_mm_dc = []
+    t_mm_strassen = []
+    t_wino = []
+    t_np = []
+
+    for i in n:
+        A = np.random.randn(i, i)
+        B = np.random.randn(i, i)
+        # A = np.random.randint(-100, 100,(i, i))
+        # B = np.random.randint(-100, 100,(i, i))
+
+        start = time.time()
+        C3 = strassen(A, B)
+        t_mm_strassen.append(time.time() - start)
+
+        start = time.time()
+        C1 = MM(A, B)
+        t_mm.append(time.time() - start)
+
+        start = time.time()
+        C2 = MM_dc(A, B)
+        t_mm_dc.append(time.time() - start)
+
+        start = time.time()
+        C4 = winograd2(A, B)
+        t_wino.append(time.time() - start)
+
+        start = time.time()
+        C = A@B
+        t_np.append(time.time() - start)
+
+    plt.figure(figsize=(13,8))
+    plt.rcParams['font.family'] = 'STIXGeneral'
+    plt.rc('axes', labelsize=23)
+    plt.rc('xtick', labelsize=23)
+    plt.rc('ytick', labelsize=23)
+    plt.plot(n, t_mm, label='Standard', lw=5)
+    plt.plot(n, t_mm_dc, label='Divide and conquer', lw=5)
+    plt.plot(n, t_mm_strassen, label='Strassen', lw=5)
+    plt.plot(n, t_wino, label='Winograd', lw=5)
+    plt.plot(n, t_np, label='NumPy A@B', lw=5)
+    plt.legend()
+    plt.xlabel("n")
+    plt.ylabel("time (s)")
+    plt.grid(True)
+    plt.tight_layout()
+    # plt.yscale('log')
+    plt.legend(fontsize=19)
+    plt.savefig('meas_' + str(max(n))+ '.pdf')
+    arr = np.array([n, t_mm, t_mm_dc, t_mm_strassen, t_wino, t_np])
+    np.savetxt('meas_' + str(max(n))+ '.txt',arr)    
+    return arr
+
+
+def plot(num):
+    arr = np.loadtxt('meas_{}.txt'.format(num))
+    n, t_mm, t_mm_dc, t_mm_strassen, t_wino, t_np = arr
+    plt.figure(figsize=(13,8))
+    plt.rcParams['font.family'] = 'STIXGeneral'
+    plt.rc('axes', labelsize=23)
+    plt.rc('xtick', labelsize=23)
+    plt.rc('ytick', labelsize=23)
+    plt.plot(n, t_mm, label='3 For Loops', lw=5)
+    plt.plot(n, t_mm_dc, label='Divide and Conquer', lw=5)
+    plt.plot(n, t_mm_strassen, label='Strassen', lw=5)
+    # plt.plot(n, t_wino, label='Winograd', lw=5)
+    plt.plot(n, t_np, label='NumPy A@B', lw=5)
+    plt.legend()
+    plt.xlabel("n")
+    plt.ylabel("time (s)")
+    plt.grid(True)
+    plt.tight_layout()
+    # plt.yscale('log')
+    plt.legend(fontsize=19)
+    plt.savefig('meas_' + str(num)+ '.pdf')    
+    return arr
+
+def plot_c_res(ave, num):
+    MM = np.loadtxt("meas/MM.txt", delimiter=',')
+    # winograd = np.loadtxt("meas/winograd.txt", delimiter=',')
+    blas = np.loadtxt("meas/blas.txt", delimiter=',')
+    MM_dc = np.loadtxt("meas/MM_dc.txt", delimiter=',')
+    strassen = np.loadtxt("meas/strassen.txt", delimiter=',')
+
+    MM_t = MM[:,0] 
+    MM_n = MM[:,1] 
+    MM_t = np.mean(MM_t.reshape(-1,ave),axis=1)
+    MM_n = np.mean(MM_n.reshape(-1,ave),axis=1)
+
+    MM_dc_t = MM_dc[:,0] 
+    MM_dc_n = MM_dc[:,1] 
+    MM_dc_t = np.mean(MM_dc_t.reshape(-1,ave),axis=1)
+    MM_dc_n = np.mean(MM_dc_n.reshape(-1,ave),axis=1)
+
+    strassen_t = strassen[:,0] 
+    strassen_n = strassen[:,1] 
+    strassen_t = np.mean(strassen_t.reshape(-1,ave),axis=1)
+    strassen_n = np.mean(strassen_n.reshape(-1,ave),axis=1)
+    
+    # winograd_t = winograd[:,0] 
+    # winograd_n = winograd[:,1] 
+    # winograd_t = np.mean(winograd_t.reshape(-1,ave),axis=1)
+    # winograd_n = np.mean(winograd_n.reshape(-1,ave),axis=1)
+    
+    blas_t = blas[:,0] 
+    blas_n = blas[:,1] 
+    blas_t = np.mean(blas_t.reshape(-1,ave),axis=1)
+    blas_n = np.mean(blas_n.reshape(-1,ave),axis=1)
+
+    def func(x, a,b):
+        return b*x**a
+
+    # popt, pcov = curve_fit(func, blas_n, blas_t)
+    # popt1, pcov2 = curve_fit(func, blas_n, winograd_t)
+    # popt2, pcov2 = curve_fit(func, blas_n, MM_t)
+
+    plt.figure(figsize=(13,8))
+    plt.rcParams['font.family'] = 'STIXGeneral'
+    plt.rc('axes', labelsize=23)
+    plt.rc('xtick', labelsize=23)
+    plt.rc('ytick', labelsize=23)
+    plt.plot(MM_n, MM_t, label='3 For Loops', lw=5)
+    # plt.plot(winograd_n, winograd_t, label='Winograd MM', lw=5)
+    plt.plot(blas_n, blas_t, label='Blas', lw=5)
+    plt.plot(strassen_n, strassen_t, label='Strassen', lw=5)
+    plt.plot(MM_dc_n, MM_dc_t, label='Divide and Conquer', lw=5)
+    plt.xlabel("n")
+    plt.ylabel("time (s)")
+    plt.grid(True)
+    plt.tight_layout()
+    plt.legend(fontsize=19)
+    plt.savefig('c_meas_' + str(num)+ '.pdf')    
+
+    # plt.plot(blas_n, func(blas_n, *popt), 'r-', label='fit blas: a=%5.5f, b=%5.10f' % tuple(popt))
+    # plt.plot(blas_n, func(blas_n, *popt1), 'r-', label='fit winograd: a=%5.5f, b=%5.10f' % tuple(popt1))
+    # plt.plot(blas_n, func(blas_n, *popt2), 'r-', label='fit MM: a=%5.5f, b=%5.10f' % tuple(popt2))
+  
+    plt.legend()
+    
+
+# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+if __name__ == '__main__':
+    plot_c_res(1, 4096)
+
+ 
+    # plot(8)    
+    # n = np.logspace(1,10,10,base=2,dtype=(np.int))
+    # n = np.arange(1,50,2)  
+    A = np.random.randint(-10, 10, (5,3))
+    B = np.random.randint(-10, 10, (3,5))
+
+    C = winograd2(A, B)
+    C_test = A@B
+    print(C)
+    print(C_test)
+    # print(np.equal(C, C_test))
+
+    # t_np = test_perfomance(n)
+    # C = strassen(A, B)
+    # C_test = A@B
+    
+    
+    # plot_c_res()
+    # def func(x, a):
+    #     return x**a
+
+    # popt, pcov = curve_fit(func, n, t_np, bounds=(2, 3))
+
+
+    # plt.figure()
+    # plt.plot(n, t_np, 'b-', label='data')
+    # plt.plot(n, func(n, *popt), 'r-', label='fit: a=%5.3f' % tuple(popt))
+    # plt.xlabel('x')
+    # plt.ylabel('y')
+    # plt.legend()
+    
+\ No newline at end of file
diff --git a/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc b/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc
new file mode 100644
index 0000000..7768772
--- /dev/null
+++ b/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc
diff --git a/buch/papers/multiplikation/code/c_matrix.h b/buch/papers/multiplikation/code/c_matrix.h
new file mode 100644
index 0000000..13df55d
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_matrix.h
@@ -0,0 +1,101 @@
+/* Seminar Matrizen, autogenerated File, Michael Schmid, 30/05/2021, 22:00:57 */ 
+ 
+#include <stdint.h> 
+const int A0[][2] = 
+  {
+    {-15,68},
+    {49,86}
+  };
+const int B0[][2] = 
+  {
+    {33,73},
+    {38,-76}
+  };
+const double dB0[][2] = 
+  {
+    {33,73},
+    {38,-76}
+  };
+const double dA0[][2] = 
+  {
+    {-15,68},
+    {49,86}
+  };
+const int A1[][4] = 
+  {
+    {75,-38,-32,-65},
+    {37,74,-31,29},
+    {15,-62,-20,-20},
+    {-31,-35,-89,47}
+  };
+const int B1[][4] = 
+  {
+    {71,90,78,-98},
+    {4,63,12,-47},
+    {11,-44,75,-69},
+    {95,-15,64,23}
+  };
+const double dB1[][4] = 
+  {
+    {71,90,78,-98},
+    {4,63,12,-47},
+    {11,-44,75,-69},
+    {95,-15,64,23}
+  };
+const double dA1[][4] = 
+  {
+    {75,-38,-32,-65},
+    {37,74,-31,29},
+    {15,-62,-20,-20},
+    {-31,-35,-89,47}
+  };
+const int A2[][8] = 
+  {
+    {80,42,3,-16,6,55,87,16},
+    {-99,-14,21,-1,-94,-56,91,10},
+    {-47,-55,-59,62,12,-53,87,-65},
+    {-60,94,-67,23,-62,33,-63,-72},
+    {12,-75,16,21,22,-37,1,16},
+    {-100,-99,82,-66,2,64,-13,44},
+    {59,-100,-90,8,36,-24,18,88},
+    {73,-58,75,-100,-19,-29,85,-19}
+  };
+const int B2[][8] = 
+  {
+    {-61,88,69,49,-53,47,73,45},
+    {16,14,-88,-11,-67,-73,-20,43},
+    {-60,-63,26,32,-29,18,-44,-69},
+    {1,21,21,38,7,-100,-61,-76},
+    {-90,95,-99,88,49,-80,27,-36},
+    {24,-12,-47,-7,29,15,52,37},
+    {-98,-76,29,76,-41,-75,97,79},
+    {62,-90,-35,-14,-30,-42,-95,52}
+  };
+const double dB2[][8] = 
+  {
+    {-61,88,69,49,-53,47,73,45},
+    {16,14,-88,-11,-67,-73,-20,43},
+    {-60,-63,26,32,-29,18,-44,-69},
+    {1,21,21,38,7,-100,-61,-76},
+    {-90,95,-99,88,49,-80,27,-36},
+    {24,-12,-47,-7,29,15,52,37},
+    {-98,-76,29,76,-41,-75,97,79},
+    {62,-90,-35,-14,-30,-42,-95,52}
+  };
+const double dA2[][8] = 
+  {
+    {80,42,3,-16,6,55,87,16},
+    {-99,-14,21,-1,-94,-56,91,10},
+    {-47,-55,-59,62,12,-53,87,-65},
+    {-60,94,-67,23,-62,33,-63,-72},
+    {12,-75,16,21,22,-37,1,16},
+    {-100,-99,82,-66,2,64,-13,44},
+    {59,-100,-90,8,36,-24,18,88},
+    {73,-58,75,-100,-19,-29,85,-19}
+  };
+const int *Ap[3] = {(int*) A0,(int*) A1,(int*) A2}; 
+const int *Bp[3] = {(int*) B0,(int*) B1,(int*) B2}; 
+const double *dAp[3] = {(double*) dA0,(double*) dA1,(double*) dA2}; 
+const double *dBp[3] = {(double*) dB0,(double*) dB1,(double*) dB2}; 
+int n[3] = {2,4,8}; 
+int n_arrays = 3;
diff --git a/buch/papers/multiplikation/code/c_meas_1024.pdf b/buch/papers/multiplikation/code/c_meas_1024.pdf
new file mode 100644
index 0000000..95b68b5
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_1024.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_128.pdf b/buch/papers/multiplikation/code/c_meas_128.pdf
new file mode 100644
index 0000000..56b9200
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_128.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_16.pdf b/buch/papers/multiplikation/code/c_meas_16.pdf
new file mode 100644
index 0000000..2edc82d
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_16.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_2048.pdf b/buch/papers/multiplikation/code/c_meas_2048.pdf
new file mode 100644
index 0000000..caba698
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_2048.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_256.pdf b/buch/papers/multiplikation/code/c_meas_256.pdf
new file mode 100644
index 0000000..383ae86
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_256.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_32.pdf b/buch/papers/multiplikation/code/c_meas_32.pdf
new file mode 100644
index 0000000..180fd22
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_32.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_4096.pdf b/buch/papers/multiplikation/code/c_meas_4096.pdf
new file mode 100644
index 0000000..547d794
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_4096.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_512.pdf b/buch/papers/multiplikation/code/c_meas_512.pdf
new file mode 100644
index 0000000..5e8894e
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_512.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_64.pdf b/buch/papers/multiplikation/code/c_meas_64.pdf
new file mode 100644
index 0000000..8ff905c
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_64.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_8.pdf b/buch/papers/multiplikation/code/c_meas_8.pdf
new file mode 100644
index 0000000..9682aca
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_8.pdf
diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py
new file mode 100755
index 0000000..485fa76
--- /dev/null
+++ b/buch/papers/multiplikation/code/helper_class.py
@@ -0,0 +1,105 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Fri Mar 12 09:02:48 2021
+
+@author: nunigan
+"""
+
+from datetime import datetime
+import numpy as np
+
+class Helper():
+    def __init__(self):
+        pass
+
+    def write_c_matrix(self, n_array):
+  
+        with open('c_matrix.h', 'w') as file:
+            file.writelines('/* Seminar Matrizen, autogenerated File, Michael Schmid, {} */ \n \n'.format(datetime.now().strftime("%d/%m/%Y, %H:%M:%S")))
+
+            file.writelines('#include <stdint.h> \n')
+
+
+
+            for k, n in enumerate(n_array):
+                A = np.random.randint(-100,100,(n,n))
+                B = np.random.randint(-100,100,(n,n))
+                file.writelines('const int A{}[][{}] = \n'.format(k, n))
+                file.writelines('  {\n')
+                for i in range(n):
+                    file.writelines('    {')
+                    for j in range(n):
+                        if j == n-1:
+                            file.writelines('{}'.format(A[i,j]))
+                        else:
+                            file.writelines('{},'.format(A[i,j]))
+                    if i == n-1:
+                        file.writelines('}\n')
+                    else:
+                        file.writelines('},\n')
+    
+                file.writelines('  };\n')
+    
+                file.writelines('const int B{}[][{}] = \n'.format(k,n))
+                file.writelines('  {\n')
+                for i in range(n):
+                    file.writelines('    {')
+                    for j in range(n):
+                        if j == n-1:
+                            file.writelines('{}'.format(B[i,j]))
+                        else:
+                            file.writelines('{},'.format(B[i,j]))
+                    if i == n-1:
+                        file.writelines('}\n')
+                    else:
+                        file.writelines('},\n')
+    
+                file.writelines('  };\n')
+    
+                file.writelines('const double dB{}[][{}] = \n'.format(k,n))
+                file.writelines('  {\n')
+                for i in range(n):
+                    file.writelines('    {')
+                    for j in range(n):
+                        if j == n-1:
+                            file.writelines('{}'.format(B[i,j]))
+                        else:
+                            file.writelines('{},'.format(B[i,j]))
+                    if i == n-1:
+                        file.writelines('}\n')
+                    else:
+                        file.writelines('},\n')
+    
+                file.writelines('  };\n')
+    
+                file.writelines('const double dA{}[][{}] = \n'.format(k,n))
+                file.writelines('  {\n')
+                for i in range(n):
+                    file.writelines('    {')
+                    for j in range(n):
+                        if j == n-1:
+                            file.writelines('{}'.format(A[i,j]))
+                        else:
+                            file.writelines('{},'.format(A[i,j]))
+                    if i == n-1:
+                        file.writelines('}\n')
+                    else:
+                        file.writelines('},\n')
+    
+                file.writelines('  };\n')
+
+            file.writelines('const int *Ap[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(int*) A'+str(element) for element in np.arange(len(n_array))])))
+            file.writelines('const int *Bp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(int*) B'+str(element) for element in np.arange(len(n_array))])))
+            file.writelines('const double *dAp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(double*) dA'+str(element) for element in np.arange(len(n_array))])))
+            file.writelines('const double *dBp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(double*) dB'+str(element) for element in np.arange(len(n_array))])))
+            file.writelines('int n[{}] = {{{}}}; \n'.format(len(n_array),",".join([str(element) for element in n_array])))
+            file.writelines('int n_arrays = {};\n'.format(len(n_array)))
+
+# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+if __name__ == '__main__':
+
+    helper = Helper()
+    # n = np.arange(2,10)
+    n = np.logspace(1,3,3,base=2,dtype=(np.int))
+    C = helper.write_c_matrix(n)
diff --git a/buch/papers/multiplikation/code/meas/MM.txt b/buch/papers/multiplikation/code/meas/MM.txt
new file mode 100644
index 0000000..1a0cd5d
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/MM.txt
@@ -0,0 +1,12 @@
+0.000000,2
+0.000000,4
+0.000002,8
+0.000011,16
+0.000080,32
+0.000653,64
+0.005397,128
+0.045147,256
+0.487710,512
+3.964180,1024
+128.863544,2048
+996.370209,4096
diff --git a/buch/papers/multiplikation/code/meas/MM_dc.txt b/buch/papers/multiplikation/code/meas/MM_dc.txt
new file mode 100644
index 0000000..0d5580a
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/MM_dc.txt
@@ -0,0 +1,12 @@
+0.000006,2
+0.000007,4
+0.000035,8
+0.000228,16
+0.001310,32
+0.007204,64
+0.034338,128
+0.267511,256
+2.131212,512
+17.177403,1024
+146.112874,2048
+1156.777565,4096
diff --git a/buch/papers/multiplikation/code/meas/blas.txt b/buch/papers/multiplikation/code/meas/blas.txt
new file mode 100644
index 0000000..6b7cd0b
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/blas.txt
@@ -0,0 +1,12 @@
+0.000001,2
+0.000000,4
+0.000001,8
+0.000003,16
+0.000021,32
+0.000164,64
+0.001240,128
+0.009657,256
+0.072523,512
+0.735149,1024
+6.895747,2048
+56.812183,4096
diff --git a/buch/papers/multiplikation/code/meas/strassen.txt b/buch/papers/multiplikation/code/meas/strassen.txt
new file mode 100644
index 0000000..89cf41a
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/strassen.txt
@@ -0,0 +1,12 @@
+0.000000,2
+0.000003,4
+0.000010,8
+0.000086,16
+0.000476,32
+0.003366,64
+0.025547,128
+0.184593,256
+1.248713,512
+9.007700,1024
+61.079879,2048
+424.493037,4096
diff --git a/buch/papers/multiplikation/code/meas/test/4096/MM.txt b/buch/papers/multiplikation/code/meas/test/4096/MM.txt
new file mode 100644
index 0000000..25e40e1
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/4096/MM.txt
@@ -0,0 +1,12 @@
+0.000000,2
+0.000000,4
+0.000002,8
+0.000011,16
+0.000100,32
+0.000712,64
+0.005498,128
+0.046711,256
+0.489233,512
+4.006544,1024
+124.427496,2048
+993.405615,4096
diff --git a/buch/papers/multiplikation/code/meas/test/4096/strassen.txt b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt
new file mode 100644
index 0000000..eb2a496
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt
@@ -0,0 +1,12 @@
+0.000007,2
+0.000007,4
+0.000029,8
+0.000199,16
+0.001414,32
+0.007583,64
+0.028096,128
+0.171662,256
+1.198323,512
+8.421896,1024
+58.803644,2048
+415.115401,4096
diff --git a/buch/papers/multiplikation/code/meas/test/MM.txt b/buch/papers/multiplikation/code/meas/test/MM.txt
new file mode 100644
index 0000000..e0754ab
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/MM.txt
@@ -0,0 +1,14900 @@
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diff --git a/buch/papers/multiplikation/code/meas/test/blas.txt b/buch/papers/multiplikation/code/meas/test/blas.txt
new file mode 100644
index 0000000..7b0a9d1
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/blas.txt
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diff --git a/buch/papers/multiplikation/code/meas/test/winograd.txt b/buch/papers/multiplikation/code/meas/test/winograd.txt
new file mode 100644
index 0000000..d01fefd
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/winograd.txt
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+
+\end{tikzpicture}
diff --git a/buch/papers/multiplikation/einlteung.tex b/buch/papers/multiplikation/einlteung.tex
new file mode 100755
index 0000000..bc4bfcf
--- /dev/null
+++ b/buch/papers/multiplikation/einlteung.tex
@@ -0,0 +1,52 @@
+%
+% einleitung.tex -- Beispiel-File für die Einleitung
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Einleitung \label{multiplikation:section:einleitung}}
+\rhead{Einleitung}
+
+Die Multiplikation zweier Matrizen ist eine wichtige Operation die in verschiedensten Teilen der Mathematik Anwendung findet.
+Die Beschreibung der Multiplikation aus der Definition 2.10 (\textcolor{blue} {Kein Hyperlink zu einer Definition?)}:
+
+Eine $m\times n$-Matrix $\mathbf{A}\in M_{m\times n}(\Bbbk)$ und eine
+$n\times p$-Matrix $\mathbf{B}\in M_{n\times l}(\Bbbk)$ haben als Produkt
+eine $n\times l$-Matrix $\mathbf{C}=\mathbf{AB}\in M_{n\times l}(\Bbbk)$ mit den
+Koeffizienten
+\begin{equation}
+c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}.
+\label{multiplikation:eq:MM}
+\end{equation}
+Grafisch kann die Matrizenmultiplikation $AB=C$ wie in \ref{multiplikation:fig:mm_viz} visualisiert werden.
+\begin{figure}
+	\center
+	\includegraphics[]{papers/multiplikation/images/mm_visualisation}
+	\caption{Matrizen Multiplikation}
+	\label{multiplikation:fig:mm_viz}
+\end{figure}
+Im Fall einer Matrizengr\"osse von $2\times 2$
+\begin{equation}
+  \begin{bmatrix}
+A_{11} & A_{12}\\
+A_{21} & A_{22}
+\end{bmatrix}
+\begin{bmatrix}
+B_{11} & B_{12}\\
+B_{21} & B_{22}
+\end{bmatrix}
+=
+\begin{bmatrix}
+C_{11} & C_{12}\\
+C_{21} & C_{22}
+\end{bmatrix}
+\end{equation}
+kann die Gleichung der einzelnen Terme
+\begin{equation} \label{multiplikation:eq:MM_exp}
+\begin{split}
+C_{11} &= A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\
+C_{12} &= A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\
+C_{21} &= A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\
+C_{22} &= A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+\end{split}
+\end{equation}
+explizit geschrieben werden.
diff --git a/buch/papers/multiplikation/images/bigo.pdf b/buch/papers/multiplikation/images/bigo.pdf
new file mode 100644
index 0000000..dfa2ba4
--- /dev/null
+++ b/buch/papers/multiplikation/images/bigo.pdf
diff --git a/buch/papers/multiplikation/images/bigo.tex b/buch/papers/multiplikation/images/bigo.tex
new file mode 100644
index 0000000..e3293e4
--- /dev/null
+++ b/buch/papers/multiplikation/images/bigo.tex
@@ -0,0 +1,107 @@
+\documentclass[border=10pt,varwidth]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{pgfplots}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usetikzlibrary{decorations.text}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+\begin{tikzpicture}
+\begin{axis}[
+    axis lines = left,
+    xlabel = $n$ (Data Input),
+    ylabel = {$t$ (time)},
+    legend pos=north east,
+    very thick,
+    ymax = 500,
+    yticklabels=\empty,
+    xticklabels=\empty,
+    scale only axis=true,
+  width=12cm, height=6cm,
+    ]
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}(n^2)$}
+\addplot [
+    domain= 1:10,
+    samples=100,
+    color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}(n^3)$}
+\addplot [
+    domain= 1:10,
+    samples=100,
+    color=black,
+]
+{exp(x)};
+\addlegendentry{$\mathcal{O}(e^n)$}
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=orange,
+]
+{log2(x)};
+\addlegendentry{$\mathcal{O}(\log n)$}
+
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=gray,
+]
+{x*log2(x)};
+\addlegendentry{$\mathcal{O}(n \log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/images/mm_visualisation.pdf b/buch/papers/multiplikation/images/mm_visualisation.pdf
new file mode 100644
index 0000000..9309df1
--- /dev/null
+++ b/buch/papers/multiplikation/images/mm_visualisation.pdf
diff --git a/buch/papers/multiplikation/images/mm_visualisation.tex b/buch/papers/multiplikation/images/mm_visualisation.tex
new file mode 100644
index 0000000..6e8f789
--- /dev/null
+++ b/buch/papers/multiplikation/images/mm_visualisation.tex
@@ -0,0 +1,45 @@
+
+  \begin{tikzpicture}[ampersand replacement=\&]
+
+    \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (0,0)
+     {
+     A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\
+     \vdots \&  \& \vdots \&  \& \vdots \\
+     A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\
+     \vdots \&  \& \vdots \& \& \vdots \\
+     A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\
+     };
+ 
+  	 \node [right=0.1 of A] (mul) {$\cdot$};
+
+	 
+     \matrix (B)[right=0.1 of mul, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
+      {
+      B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\
+      \vdots \&  \& \vdots \&  \& \vdots \\
+      B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\
+      \vdots \&  \& \vdots \& \& \vdots \\
+      B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\
+      };
+  
+  	 \node [right=0.1 of B] (eq) {$=$};
+
+     \matrix (C)[right=0.1 of eq, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
+      {
+      C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\
+      \vdots \&  \& \vdots \&  \& \vdots \\
+      C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\
+      \vdots \&  \& \vdots \& \& \vdots \\
+      C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\
+      };
+
+
+       \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {};
+       \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {};
+       \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {};
+
+
+   \end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/multiplikation/images/strassen.pdf b/buch/papers/multiplikation/images/strassen.pdf
new file mode 100644
index 0000000..9899dcb
--- /dev/null
+++ b/buch/papers/multiplikation/images/strassen.pdf
diff --git a/buch/papers/multiplikation/images/strassen.tex b/buch/papers/multiplikation/images/strassen.tex
new file mode 100644
index 0000000..797772b
--- /dev/null
+++ b/buch/papers/multiplikation/images/strassen.tex
@@ -0,0 +1,140 @@
+\documentclass[border=10pt]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{pgfplots}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usetikzlibrary{decorations.text}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+\begin{tikzpicture}[ampersand replacement=\&]
+
+\foreach \i in {1,...,4}
+{
+	\small{
+		\matrix (X\i)[matrix of math nodes,nodes in empty cells,
+		nodes = {draw, minimum size=10mm,
+			anchor=center,
+			inner sep=0pt, outer sep=0pt},
+		column sep=-\pgflinewidth,
+		row sep=-\pgflinewidth,
+		] at (0,-\i*5)
+		{
+			A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\
+			A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\
+			A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\
+			A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\
+		};}
+	
+	\foreach \j in {1,...,7}
+	{
+		\matrix(M\i\j)[matrix of math nodes,nodes in empty cells,
+		nodes = {draw, minimum size=10mm,
+			anchor=center,
+			inner sep=0pt, outer sep=0pt},
+		column sep=-\pgflinewidth,
+		row sep=-\pgflinewidth,
+		] at (\j*5,-\i*5)
+		{
+			\& \& \& \\
+			\& \& \& \\
+			\& \& \& \\
+			\& \& \& \\
+		};
+	}
+}
+
+\huge{
+	\node at (-3,-20)  {$C_{22}=$};
+	\node at (-3,-15) {$C_{21}=$} ;
+	\node at (-3,-10) {$C_{12}=$} ;
+	\node at (-3,-5) {$C_{11}=$} ;
+	
+	\node at (5,-2)  {I};
+	\node at (10,-2)  {II};
+	\node at (15,-2)  {III};
+	\node at (20,-2)  {IV};
+	\node at (25,-2)  {V};
+	\node at (30,-2)  {VI};
+	\node at (35,-2)  {VII};
+}
+
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-2-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-3-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-4-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-2-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-3-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-4-4)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M14-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M14-2-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-2-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-2-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-4-2)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M23-3-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M23-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-2)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M34-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M34-2-4)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex
new file mode 100755
index 0000000..83be814
--- /dev/null
+++ b/buch/papers/multiplikation/loesungsmethoden.tex
@@ -0,0 +1,309 @@
+%
+% teil2.tex -- Beispiel-File für teil2
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\section{L\"osungsmethoden}
+\rhead{L\"osungsmethoden}
+
+In diesem Abschnitt werden mehrere Algorithmen zur Berechnung der Matrizenmultiplikation vorgestellt, auch werden Libraries zur automatisierten Verwendung von vordefinierten Algorithmen gezeigt.
+
+\subsection{Standard Algorithmus}
+
+Der Standard Methode kann im Algorithmus \ref{multiplikation:alg:smm} entnommen werden.
+Hierf\"ur wurde die Gleichung \eqref{multiplikation:eq:MM} direkt implementiert.
+Die \texttt{For i} Schleife iteriert \"uber alle Zeilen der $\mathbf{A}$ Matrix, die \texttt{For j} Schleife iteriert \"uber alle Spalten der $\mathbf{B}$ Matrix und die \texttt{For k} Schleife iteriert \"uber alle Eintr\"age dieser Zeilen bzw. Spalten.
+
+\begin{algorithm}\caption{Matrix Multiplication}
+	\label{multiplikation:alg:smm}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}[1]
+		\Function{MM}{$\textbf{A}, \textbf{B}$}
+		\State $sum \gets 0$
+		\State $n \gets columns(\textbf{A}) == rows(\textbf{B})$
+		\State $m \gets rows(\textbf{A})$
+		\State $p \gets columns(\textbf{B})$
+		\State $\textbf{C} \gets zeros(m,p)$
+		\For{$i = 0,1,2 \dots,m-1$}
+		\For{$j = 0,1,2 \dots,p-1$}
+		\State $sum \gets 0$
+		\For{$k = 0,1,2 \dots,n-1$}
+		\State $sum \gets  sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+		\EndFor
+		\State $\textbf{C}[i][j] \gets  sum $
+		\EndFor
+		\EndFor
+		\State \textbf{return} $\textbf{C}$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+Die Laufzeit dieser Struktur mit drei \texttt{For} Schleifen ist $\mathcal{O}(n^3)$
+
+\subsubsection{Divide and Conquer Methode}
+
+F\"ur gewisse Algorithmen f\"uhren \textit{Divide and Conquer} Ans\"atze zu markant besseren Laufzeiten.
+Das bekannteste Beispiel ist wohl die \textit{Fast Fourier Transform} wobei die Laufzeit von $\mathcal{O}(n^2)$ zu $\mathcal{O}(n \log n)$ verbessert werden kann.
+
+Die Matrizenmultiplikation kann ebenfalls mit solch einem Ansatz berechnet werden.
+Zur vereinfachten Veranschaulichung kann die Situation, mit $\mathbf{A}$ und $\mathbf{B}$ der gr\"osse $2^n \times 2^n$ verwendet werden.
+Die Matrizen $\mathbf{A}$ und $\mathbf{B}$ werden in jeweils vier Blockmatrizen der gr\"osse $2^{n-1} \times 2^{n-1}$
+\begin{equation}
+\mathbf{A}\mathbf{B}=
+\begin{bmatrix}
+\mathbf{A}_{11} & \mathbf{A}_{12}\\
+\mathbf{A}_{21} & \mathbf{A}_{22}
+\end{bmatrix}
+\begin{bmatrix}
+\mathbf{B}_{11} & \mathbf{B}_{12}\\
+\mathbf{B}_{21} & \mathbf{B}_{22}
+\end{bmatrix}
+=
+\begin{bmatrix}
+\mathbf{C}_{11} & \mathbf{C}_{12}\\
+\mathbf{C}_{21} & \mathbf{C}_{22}
+\end{bmatrix}
+\end{equation}
+aufgeteilt.
+Die Berechnung
+\begin{equation}
+\mathbf{C}_{ij} = \sum_{k=1}^n \mathbf{A}_{ik} \mathbf{B}_{kj}
+\label{multiplikation:eq:MM_block}
+\end{equation}
+ist identisch zu der Gleichung \eqref{multiplikation:eq:MM}, wobei hier f\"ur die Multiplikation die Matrizenmultiplikation verwendet wird.
+
+Der Algorithmus \ref{multiplikation:alg:devide_mm} zeigt den \textit{Divide and Conquer} Ansatz,
+Der Grundstruktur dieser Methode besteht aus dem rekursiven Aufruf der Funktion mit den erzeugten Blockmatrizen.
+Der rekursive Aufruf wird bis zu der Gr\"osse der Matrizen von $N = 2 \times 2$ durchgef\"uhrt.
+\begin{algorithm}\caption{Divide and Conquer Matrix Multiplication}
+	\setlength{\lineskip}{7pt}
+	\label{multiplikation:alg:devide_mm}
+	\begin{algorithmic}
+		\Function{MM}{$\textbf{A}, \textbf{B}, n$}
+		\If{$n = 2$}
+		\State  $ \mathbf{C} \gets zeros(n, n)$
+		\State  $C[0, 0] \gets  A[0][0]\cdot B[0][0]+A[0][1]\cdot B[1][0]$
+		\State  $C[0, 1] \gets  A[0][0]\cdot B[0][1]+A[0][1]\cdot B[1][1]$
+		\State  $C[1, 0] \gets  A[1][0]\cdot B[0][0]+A[1][1]\cdot B[1][0]$
+		\State  $C[1, 1] \gets  A[1][0]\cdot B[0][1]+A[1][1]\cdot B[1][1]$
+		\Else
+		\State  $ m \gets n/2$
+		\State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+		\State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+		\State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11},n) + \text{MM}(\mathbf{A12}, \mathbf{B21},n)$
+		\State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12},n) + \text{MM}(\mathbf{A12}, \mathbf{B22},n)$
+		\State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11},n) + \text{MM}(\mathbf{A22}, \mathbf{B21},n)$
+		\State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12},n) + \text{MM}(\mathbf{A22}, \mathbf{B22},n)$
+		\State $  C \gets vstack(hstack(C11, C12), hstack(C21, C22))$
+
+		\EndIf
+		\State \textbf{return} $\textbf{C}$
+
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+Die Laufzeit dieser rekursiven Funktion kann mit dem \textit{Master Theorem} berechnet werden.
+Ohne auf diesen vertieft einzugehen, bestimmt die Anzahl rekursiver Aufrufe der Funktion die Laufzeit.
+In diesem Fall wird die Funktion pro Durchlauf acht mal rekursiv aufgerufen, dies f\"uhrt
+\begin{equation} \label{multiplikation:eq:laufzeitdac}
+	\mathcal{T}(n) =
+	\begin{cases}
+	1  & \text{if }  n \leq 2\\
+	8 \cdot \mathcal{T}(\frac{n}{2}) + n^2  & \text{if }  n > 2
+	\end{cases} = \mathcal{O}(n^{\log_2 8}) = \mathcal{O}(n^{3})
+\end{equation}
+zu einer kubischen Laufzeit.
+Die Addition zweier Matrizen $\mathbf{A} + \mathbf{B} = \mathbf{C}$ hat eine Laufzeit von $\mathcal{O}(n^{2})$ und kann neben dem dominierendem Anteil von $\mathcal{O}(n^{3})$ ignoriert werden.
+In diesem Fall hat der \textit{Divide and Conquer} Ansatz zu keiner Verbesserung gef\"uhrt.
+
+
+\subsection{Strassen's Algorithmus}
+
+Strassen's Algorithmus \cite{multiplikation:strassen_1969} beschreibt die Matrizenmultiplikation mit einer Vielzahl von Additionen, Subtraktionen und Multiplikationen.
+Die Grundlegenden Terme
+\begin{equation} \label{multiplikation:eq:strassen}
+\begin{split}
+\text{\textbf{P}}   &= (\mathbf{A}_{11} + \mathbf{A}_{22}) \cdot (\mathbf{B}_{11} + \mathbf{B}_{22}) \\
+\text{\textbf{Q}}  &= (\mathbf{A}_{21} + \mathbf{A}_{22}) \cdot \mathbf{B}_{11} \\
+\text{\textbf{R}} &= \mathbf{A}_{11} \cdot (\mathbf{B}_{12}-\mathbf{B}_{22}) \\
+\text{\textbf{S}}  &= \mathbf{A}_{22} \cdot (-\mathbf{B}_{11}+\mathbf{B}_{21}) \\
+\text{\textbf{T}}   &= (\mathbf{A}_{11} + \mathbf{A}_{12}) \cdot \mathbf{B}_{22} \\
+\text{\textbf{U}}  &= (-\mathbf{A}_{11} + \mathbf{A}_{21}) \cdot (\mathbf{B}_{11} + \mathbf{B}_{12}) \\
+\text{\textbf{V}} &= (\mathbf{A}_{12} - \mathbf{A}_{22}) \cdot (\mathbf{B}_{21} + \mathbf{B}_{22})
+\end{split}
+\end{equation}
+aus $\mathbf{A}$ und $\mathbf{B}$, werden f\"ur die Berechnung der Matrix $\mathbf{C}$
+\begin{equation} \label{multiplikation:eq:strassen2}
+\begin{split}
+\mathbf{C}_{11} &= \text{\textbf{P}} + \text{\textbf{S}} - \text{\textbf{T}} + \text{\textbf{V}} \\
+\mathbf{C}_{21} &= \text{\textbf{R}} + \text{\textbf{T}} \\
+\mathbf{C}_{12} &= \text{\textbf{Q}} + \text{\textbf{S}}\\
+\mathbf{C}_{22} &= \text{\textbf{P}} + \text{\textbf{R}} - \text{\textbf{Q}} + \text{\textbf{U}}
+\end{split}
+\end{equation}
+gebraucht.
+\begin{algorithm}\caption{Strassen Matrix Multiplication}
+	\label{multiplikation:alg:strassen}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}
+		\Function{strassen}{$\textbf{A}, \textbf{B}, n$}
+		\If{$n = 2$}
+		\State  $ \mathbf{C} \gets zeros((n, n))$
+		\State $P  \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$
+		\State   $Q  \gets (A[1][0]+A[1][1])\cdot B[0][0]$
+		\State   $R  \gets A[0][0]\cdot (B[0][1]-B[1][1])$
+		\State   $S  \gets A[1][1]\cdot (B[1][0]-B[0][0])$
+		\State   $T  \gets (A[0][0]+A[0][1])\cdot B[1][1]$
+		\State   $U  \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$
+		\State   $V  \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$
+		\State   $C[0][0]  \gets P+S-T+V$
+		\State   $C[0][1]  \gets R+T$
+		\State   $C[1][0]  \gets Q+S$
+		\State   $C[1][1]  \gets P+R-Q+U$
+		\Else
+		\State  $ m \gets n/2$
+		\State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+		\State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+		\State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$
+		\State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$
+		\State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}-  \mathbf{B22}),m)$
+		\State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}-  \mathbf{B11}),m)$
+		\State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$
+		\State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$
+		\State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$
+
+
+
+		\State   $\mathbf{C11}  \gets \mathbf{P+S-T+V}$
+		\State   $\mathbf{C12}  \gets \mathbf{R+T}$
+		\State   $\mathbf{C21}  \gets \mathbf{Q+S}$
+		\State   $\mathbf{C22}  \gets \mathbf{P+R-Q+U}$
+		\State $  C \gets vstack(hstack(C11, C12), hstack(C21, C22))$
+
+		\EndIf
+		\State \textbf{return} $\textbf{C}$
+
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+Strassens's Methode wird in der Abbildung \ref{multiplikation:fig:strassen} grafisch dargestellt.
+\begin{figure}
+	\center
+	\includegraphics[width=\linewidth]{papers/multiplikation/images/strassen.pdf}
+	\caption{Strassen's Algorithmus}
+	\label{multiplikation:fig:strassen}
+\end{figure}
+
+Die Funktion wird sieben mal rekursiv aufgerufen.
+Dies f\"uhrt zu einer Laufzeit von
+\begin{equation} \label{multiplikation:eq:laufzeitstrassen}
+\mathcal{T}(n) =
+\begin{cases}
+1  & \text{if }  n \leq 2\\
+7 \cdot \mathcal{T}(\frac{n}{2}) + n^2  & \text{if }  n > 2
+\end{cases} = \mathcal{O}(n^{\log_2 7}) = \mathcal{O}(n^{2.8074})
+\end{equation}
+und ist somit schneller als die Standard Methode.
+
+\subsection{Winograd's Algorithmus}
+
+Ein weiterer Ansatz lieferte Shmuel Winograd im Jahre 1968 \cite{multiplikation:winograd_1968}.
+Er zeigte einen neuen Algorithmus f\"ur das
+\begin{equation}
+	\langle x,y \rangle = \sum_{i=1}^{n}x_i y_i
+\end{equation}
+Skalarprodukt.
+F\"ur jeden Vektor berechne
+\begin{equation}
+	\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j}
+\end{equation}
+und
+\begin{equation}
+	\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}.
+\end{equation}
+Das Skalarprodukt ist nun geben mit
+\begin{equation}
+	\langle x,y \rangle =
+	\begin{cases}
+	 \displaystyle \quad \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if  $n$ is even}\\
+	\displaystyle  \quad \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if  $n$ is odd}.
+	\end{cases}
+\end{equation}
+
+Angenommen man hat $N$ Vektoren mit welchen man $T$ Skalarprodukte berechnen m\"ochte.
+Daf\"ur werden $N\lfloor n/2 \rfloor + T\lfloor (n+1)/2 \rfloor $ Multiplikationen ben\"otigt.
+Eine Matrizenmultiplikation mit $\mathbf{A}$ einer $m \times n$ und $\mathbf{B}$ einer $n \times p$ Matrix, entspricht $N=m+p$ Vektoren mit welchen man $T=mp$ Skalarprodukte berechnet.
+Dies f\"uhrt zu
+\begin{equation}
+		(m+p) \left \lfloor \frac{n}{2} \right \rfloor + mp \left \lfloor \frac{n+1}{2} \right \rfloor = \frac{mn}{2} + \frac{pn}{2} + \frac{mpn}{2} + \frac{mp}{2}
+\end{equation}
+Multiplikationen.
+Wenn $m,p,n$ gross werden, dominiert der Term $\frac{mpn}{2}$ und es werden $\frac{mpn}{2}$ Multiplikationen ben\"otigt.
+Was im Vergleich zu den $mpn$ Multiplikation der Standard Methode nur die H\"alfte ist.
+Die Implementation kann im Algorithmus \ref{multiplikation:alg:winograd} entnommen werden.
+
+\begin{algorithm}\caption{Winograd Matrix Multiplication}
+	\setlength{\lineskip}{7pt}
+	\label{multiplikation:alg:winograd}
+	\begin{algorithmic}
+		\Function{Winograd}{$\textbf{A}, \textbf{B}, n$}
+		\State  $ m \gets rows(\mathbf{A})$
+		\State  $ n \gets columns(\mathbf{A}) == rows(\mathbf{B})$
+		\State  $ p \gets columns(\mathbf{B})$
+		\State  $ \mathbf{\xi} \gets zeros(m)$
+		\State  $ \mathbf{\eta} \gets zeros(p)$
+
+
+		\For{$i = 0,1,2 \dots,m-1$}
+		\For{$j = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+		\State $\xi[i] \gets \xi[i]+A[i,2 j]A[i,2 j+1]$
+		\EndFor
+		\EndFor
+
+		\For{$i = 0,1,2 \dots,p-1$}
+		\For{$j = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+		\State $\eta[i] \gets   \eta[i]+B[2 j,i]B[2 j+1,i]$
+		\EndFor
+		\EndFor
+
+		\If{$n \% 2 == 0$}
+		\For{$i = 0,1,2 \dots,m-1$}
+		\For{$j = 0,1,2 \dots,p-1$}
+		\State $ab \gets 0$
+		\For{$k = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+		\State $ab \gets ab + (A[i,2k]+B[2k+1,j])(A[i,2k+1]+B[2k,j])$
+		\EndFor
+		\State $C[i,j] \gets ab-\eta[j]-\xi[i]$
+		\EndFor
+		\EndFor
+    \Else
+		\For{$i = 0,1,2 \dots,n-1$}
+		\For{$j = 0,1,2 \dots,n-1$}
+		\State $ab \gets 0$
+		\For{$k = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+		\State $ab \gets ab + (A[i,2k]+B[2k+1,j])(A[i,2k+1]+B[2k,j])$
+		\EndFor
+		\State $C[i,j] \gets ab-\eta[j]-\xi[i]+A[i,-1]B[-1,j]$
+		\EndFor
+		\EndFor
+		\EndIf
+		\State \textbf{return} $\textbf{C}$
+
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+\subsection{Weitere Algorithmen}
+
+\textcolor{red}{TODO: BLAS}
+
+\section{Implementation}
+\rhead{Implementation}
+\textcolor{red}{TODO: messresultate}
+
+\section{Fazit}
+\rhead{Fazit}
diff --git a/buch/papers/multiplikation/main.tex b/buch/papers/multiplikation/main.tex
index 42f2768..8d0a8df 100644..100755
--- a/buch/papers/multiplikation/main.tex
+++ b/buch/papers/multiplikation/main.tex
@@ -1,36 +1,18 @@
+% !TEX root = ../../buch.tex
 %
 % main.tex -- Paper zum Thema <multiplikation>
 %
-% (c) 2020 Hochschule Rapperswil
+% (c) 2021 Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:multiplikation}}
-\lhead{Thema}
+\chapter{Schnelle Matrizen Multiplikation\label{chapter:multiplikation}}
+\lhead{FMM}
 \begin{refsection}
-\chapterauthor{Hans Muster}
+\chapterauthor{Michael Schmid}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht. 
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile. 
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt 
-anzuwenden.
-\item 
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
 
-\input{papers/multiplikation/teil0.tex}
-\input{papers/multiplikation/teil1.tex}
-\input{papers/multiplikation/teil2.tex}
-\input{papers/multiplikation/teil3.tex}
+\input{papers/multiplikation/einlteung.tex}
+\input{papers/multiplikation/problemstellung.tex}
+\input{papers/multiplikation/loesungsmethoden.tex}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/multiplikation/packages.tex b/buch/papers/multiplikation/packages.tex
index e4173c0..e4173c0 100644..100755
--- a/buch/papers/multiplikation/packages.tex
+++ b/buch/papers/multiplikation/packages.tex
diff --git a/buch/papers/multiplikation/papers/Strassen_GPU.pdf b/buch/papers/multiplikation/papers/Strassen_GPU.pdf
new file mode 100755
index 0000000..4ce7625
--- /dev/null
+++ b/buch/papers/multiplikation/papers/Strassen_GPU.pdf
diff --git a/buch/papers/multiplikation/papers/Strassen_original_1969.pdf b/buch/papers/multiplikation/papers/Strassen_original_1969.pdf
new file mode 100755
index 0000000..b647fc0
--- /dev/null
+++ b/buch/papers/multiplikation/papers/Strassen_original_1969.pdf
diff --git a/buch/papers/multiplikation/papers/assay_fast_MM.pdf b/buch/papers/multiplikation/papers/assay_fast_MM.pdf
new file mode 100755
index 0000000..3cd6b63
--- /dev/null
+++ b/buch/papers/multiplikation/papers/assay_fast_MM.pdf
diff --git a/buch/papers/multiplikation/papers/strassen_video.txt b/buch/papers/multiplikation/papers/strassen_video.txt
new file mode 100755
index 0000000..f84122c
--- /dev/null
+++ b/buch/papers/multiplikation/papers/strassen_video.txt
@@ -0,0 +1 @@
+https://www.youtube.com/watch?v=0oJyNmEbS4w
diff --git a/buch/papers/multiplikation/papers/winograd_original.pdf b/buch/papers/multiplikation/papers/winograd_original.pdf
new file mode 100755
index 0000000..a7aba36
--- /dev/null
+++ b/buch/papers/multiplikation/papers/winograd_original.pdf
diff --git a/buch/papers/multiplikation/presentation/common.tex b/buch/papers/multiplikation/presentation/common.tex
new file mode 100644
index 0000000..200d244
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/common.tex
@@ -0,0 +1,79 @@
+%
+% common.tex -- gemeinsame Definitionen
+%
+% (c) 2021 Michael Schmid, OST Campus Rapperswil
+%
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{epic}
+\usepackage{color}
+\usepackage{array}
+\usepackage{algorithm}
+\usepackage{ifthen}
+\usepackage{adjustbox}
+\usepackage[noend]{algpseudocode}
+\usepackage{neuralnetwork}
+\usepackage{amsmath}
+\usepackage{lmodern}
+\usepackage{tikz}
+\usetikzlibrary{decorations.text}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{svg}
+
+\definecolor{codegreen}{rgb}{0,0.6,0}
+\definecolor{codegray}{rgb}{0.5,0.5,0.5}
+\definecolor{codepurple}{rgb}{0.58,0,0.82}
+\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
+\definecolor{ost}{rgb}{164,0,136}
+
+\lstdefinestyle{mystyle}{
+	backgroundcolor=\color{backcolour},
+	commentstyle=\color{codegreen},
+	keywordstyle=\color{magenta},
+	numberstyle=\tiny\color{codegray},
+	stringstyle=\color{codepurple},
+	basicstyle=\footnotesize,
+	breakatwhitespace=false,
+	breaklines=true,
+	captionpos=b,
+	keepspaces=true,
+	numbers=left,
+	numbersep=2pt,
+	showspaces=false,
+	showstringspaces=false,
+	showtabs=false,
+	tabsize=2
+}
+
+\usetikzlibrary{fit}
+\tikzset{%
+  highlight/.style={rectangle,rounded corners,fill=red!15,draw,fill opacity=0.5,inner sep=0pt}
+}
+\newcommand{\tikzmark}[2]{\tikz[overlay,remember picture,baseline=(#1.base)] \node (#1) {#2};}
+%
+\newcommand{\Highlight}[1][submatrix]{%
+    \tikz[overlay,remember picture]{
+    \node[highlight,fit=(left.north west) (right.south east)] (#1) {};}
+}
+
+
+\lstset{style=mystyle}
+\lstdefinestyle{mystyle}{
+	morekeywords={cwt,contourf,datetick}
+}
+
+
+\usetikzlibrary{shapes.geometric}
+\mode<beamer>{%
+\usetheme[]{Frankfurt}}
+\beamertemplatenavigationsymbolsempty
+\title[]{Fast Matrix Multiplication}
+\author[]{Michael Schmid}
+\usecolortheme[named=ost]{structure}
+
+\date[]{31.05.2021}
+\newboolean{presentation}
diff --git a/buch/papers/multiplikation/presentation/presentation.nav b/buch/papers/multiplikation/presentation/presentation.nav
new file mode 100644
index 0000000..2a01568
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.nav
@@ -0,0 +1,59 @@
+\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}
+\headcommand {\beamer@framepages {1}{1}}
+\headcommand {\beamer@sectionpages {1}{1}}
+\headcommand {\beamer@subsectionpages {1}{1}}
+\headcommand {\sectionentry {1}{Big $\mathcal {O}$}{2}{Big $\mathcal {O}$}{0}}
+\headcommand {\slideentry {1}{0}{1}{2/4}{}{0}}
+\headcommand {\beamer@framepages {2}{4}}
+\headcommand {\slideentry {1}{0}{2}{5/6}{}{0}}
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+\headcommand {\beamer@framepages {14}{14}}
+\headcommand {\beamer@sectionpages {2}{14}}
+\headcommand {\beamer@subsectionpages {2}{14}}
+\headcommand {\sectionentry {2}{Strassen's Algorithm}{15}{Strassen's Algorithm}{0}}
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+\headcommand {\beamer@framepages {15}{15}}
+\headcommand {\slideentry {2}{0}{2}{16/18}{}{0}}
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+\headcommand {\beamer@framepages {30}{32}}
+\headcommand {\beamer@sectionpages {15}{32}}
+\headcommand {\beamer@subsectionpages {15}{32}}
+\headcommand {\sectionentry {3}{Measurements}{33}{Measurements}{0}}
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+\headcommand {\slideentry {3}{0}{2}{41/49}{}{0}}
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+\headcommand {\beamer@subsectionpages {33}{49}}
+\headcommand {\sectionentry {4}{How To Matrix Multiply}{50}{How To Matrix Multiply}{0}}
+\headcommand {\slideentry {4}{0}{1}{50/50}{}{0}}
+\headcommand {\beamer@framepages {50}{50}}
+\headcommand {\beamer@partpages {1}{50}}
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+\headcommand {\beamer@sectionpages {50}{50}}
+\headcommand {\beamer@documentpages {50}}
+\headcommand {\gdef \inserttotalframenumber {21}}
diff --git a/buch/papers/multiplikation/presentation/presentation.pdf b/buch/papers/multiplikation/presentation/presentation.pdf
new file mode 100644
index 0000000..842e68c
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.pdf
diff --git a/buch/papers/multiplikation/presentation/presentation.snm b/buch/papers/multiplikation/presentation/presentation.snm
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.snm
diff --git a/buch/papers/multiplikation/presentation/presentation.tex b/buch/papers/multiplikation/presentation/presentation.tex
new file mode 100644
index 0000000..2a4af45
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.tex
@@ -0,0 +1,12 @@
+%
+% MathSem-yyy-xxx.tex -- Präsentation
+%
+% (c) 2021 Michael Schmid, OST campus Rapperswil
+%
+
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+%\setboolean{presentation}{true}
+\begin{document}
+\input{slides/slides.tex}
+\end{document}
diff --git a/buch/papers/multiplikation/presentation/slides/algo.tex b/buch/papers/multiplikation/presentation/slides/algo.tex
new file mode 100644
index 0000000..0c3d130
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/algo.tex
@@ -0,0 +1,111 @@
+\begin{frame}
+  \frametitle{Algorithm}
+  \begin{columns}
+    \begin{column}{0.6\textwidth}
+  \begin{algorithm}[H]\caption{Square Matrix Multiplication}
+    \setlength{\lineskip}{7pt}
+    \begin{algorithmic}[1]
+      \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}$}
+      \State $sum \gets 0$
+      \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$
+      \State $m \gets rows(\textbf{A})$
+      \State $p \gets columns(\textbf{B})$
+
+        \For{$i = 0,1,2 \dots,m-1$}
+        \For{$j = 0,1,2 \dots,p-1$}
+        \State $sum \gets 0$
+        \For{$k = 0,1,2 \dots,n-1$}
+          \State $sum \gets  sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+        \EndFor
+        \State $\textbf{C}[i][j] \gets  sum $
+        \EndFor
+        \EndFor
+        \State \textbf{return} $\textbf{C}$
+     \EndFunction
+    \end{algorithmic}
+  \end{algorithm}
+\end{column}
+\begin{column}{0.4\textwidth}
+  \scalebox{0.6}{\parbox{\linewidth}{
+
+  \begin{tikzpicture}[ampersand replacement=\&,remember picture,overlay]
+
+    \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (2,-2.8)
+     {
+     A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\
+     \vdots \&  \& \vdots \&  \& \vdots \\
+     A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\
+     \vdots \&  \& \vdots \& \& \vdots \\
+     A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\
+     };
+
+     \matrix (B)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (7.5,1.2)
+      {
+      B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\
+      \vdots \&  \& \vdots \&  \& \vdots \\
+      B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\
+      \vdots \&  \& \vdots \& \& \vdots \\
+      B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\
+      };
+
+     \matrix (C)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (7.5,-2.8)
+      {
+      C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\
+      \vdots \&  \& \vdots \&  \& \vdots \\
+      C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\
+      \vdots \&  \& \vdots \& \& \vdots \\
+      C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\
+      };
+
+
+               \begin{scope}[on background layer]
+               \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {};
+               \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {};
+               \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {};
+
+               \end{scope}
+
+
+
+
+   \end{tikzpicture}
+   }}
+ \end{column}
+\end{columns}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{Algorithm}
+
+\begin{columns}
+  \begin{column}{0.6\textwidth}
+\begin{algorithm}[H]\caption{Square Matrix Multiplication}
+  \setlength{\lineskip}{7pt}
+  \begin{algorithmic}[1]
+    \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}$}
+    \State $sum \gets 0$
+    \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$
+    \State $m \gets rows(\textbf{A})$
+    \State $p \gets columns(\textbf{B})$
+
+      \For{$i = 0,1,2 \dots,m-1$}
+      \For{$j = 0,1,2 \dots,p-1$}
+      \State $sum \gets 0$
+      \For{$k = 0,1,2 \dots,n-1$}
+        \State $sum \gets  sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+      \EndFor
+      \State $\textbf{C}[i][j] \gets  sum $
+      \EndFor
+      \EndFor
+      \State \textbf{return} $\textbf{C}$
+   \EndFunction
+  \end{algorithmic}
+\end{algorithm}
+\end{column}
+\begin{column}{0.4\textwidth}
+\Huge$\mathcal{O}(n^3)$
+\end{column}
+\end{columns}
+
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/bigO.tex b/buch/papers/multiplikation/presentation/slides/bigO.tex
new file mode 100644
index 0000000..d425da8
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/bigO.tex
@@ -0,0 +1,251 @@
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+\begin{itemize}
+  \item <1-> Time complexity of an algorithm
+  \item <2-> How many multiplications in a function
+  \item <3-> Drop Constants
+\end{itemize}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+  \onslide<1->{
+
+  \begin{algorithm}[H]\caption{Foo 1}
+    \setlength{\lineskip}{7pt}
+    \begin{algorithmic}[1]
+      \Function{foo}{$a, b$}
+        \State \textbf{return} $a+b$
+     \EndFunction
+    \end{algorithmic}
+  \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(1)$
+  }
+\end{frame}
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+  \onslide<1->{
+
+  \begin{algorithm}[H]\caption{Foo 2}
+    \setlength{\lineskip}{7pt}
+    \begin{algorithmic}[1]
+      \Function{foo}{$a, b$}
+        \State $ x \gets a+b $
+        \State $ y \gets a \cdot b $
+        \State \textbf{return} $x+y$
+     \EndFunction
+    \end{algorithmic}
+  \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(1) + \mathcal{O}(1) = 2\mathcal{O}(1) = \mathcal{O}(1) $
+  }
+\end{frame}
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+  \onslide<1->{
+
+  \begin{algorithm}[H]\caption{Foo 3}
+    \setlength{\lineskip}{7pt}
+    \begin{algorithmic}[1]
+      \Function{foo}{$\mathbf{A}, \mathbf{B}$,n}
+      \State $ sum \gets 0$
+      \For{$i = 0,1,2 \dots,n$}
+        \State $ sum \gets sum + A[i] \cdot B[i] $
+        \EndFor
+
+        \State \textbf{return} $sum$
+
+     \EndFunction
+    \end{algorithmic}
+  \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(n)$
+  }
+\end{frame}
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+  \onslide<1->{
+
+  \begin{algorithm}[H]\caption{Foo 4}
+    \setlength{\lineskip}{7pt}
+    \begin{algorithmic}[1]
+      \Function{foo}{$\mathbf{A}, \mathbf{B}$,n}
+      \State $ sum \gets 0$
+      \For{$i = 0,1,2 \dots,n$}
+      \For{$j = 0,1,2 \dots,n$}
+      \State $ sum \gets sum + A[i] \cdot B[j] $
+      \EndFor
+      \EndFor
+        \State \textbf{return} $sum$
+     \EndFunction
+    \end{algorithmic}
+  \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(n^2)$
+  }
+\end{frame}
+
+% \begin{frame}
+%   \frametitle{Big $\mathcal{O}$ notation}
+%   \onslide<1->{
+%
+%   \begin{algorithm}[H]\caption{Fibonacci}
+%     \setlength{\lineskip}{7pt}
+%     \begin{algorithmic}[1]
+%       \Function{fib}{$n$}
+%       \If{$n <= 1$}
+%       \State \textbf{return} $1$
+%       \Else
+%         \State \textbf{return} fib($n-1$) + fib($n-2$)
+%         \EndIf
+%
+%      \EndFunction
+%     \end{algorithmic}
+%   \end{algorithm}
+% }
+% \onslide<2->{
+% \[
+% \langle x,y \rangle =
+% \begin{cases}
+%  \displaystyle $\mathcal{O}(1)$  & \text{if  $n \leq 2$}\\
+% \displaystyle $ 2 \mathcal{T}(\frac{n}{2})$  & \text{if  $n > 2$}
+% \end{cases}
+% \]  }
+% \end{frame}
+
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+\begin{tikzpicture}
+\begin{axis}[
+    axis lines = left,
+    xlabel = $n$ (Data Input),
+    ylabel = {$t$ (time)},
+    legend pos=north east,
+    very thick,
+    ymax = 20,
+    yticklabels=\empty,
+    xticklabels=\empty,
+    scale only axis=true,
+  width=12cm, height=6cm,
+    ]
+%Below the red parabola is defined
+\addplot [
+    domain= 1:6,
+    samples=100,
+    color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+%Here the blue parabloa is defined
+\addplot [
+    domain= 1:6,
+    samples=100,
+    color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+    domain= 1:6,
+    samples=100,
+    color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}(n^2)$}
+\addplot [
+    domain= 1:6,
+    samples=100,
+    color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}(n^3)$}
+\addplot [
+    domain= 1:3,
+    samples=100,
+    color=black,
+]
+{exp(x)};
+\addlegendentry{$\mathcal{O}(e^n)$}
+\addplot [
+    domain= 1:6,
+    samples=100,
+    color=orange,
+]
+{log2(x)};
+\addlegendentry{$\mathcal{O}(\log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{frame}
+
+\begin{frame}
+  \frametitle{Big $\mathcal{O}$ notation}
+\begin{tikzpicture}
+\begin{axis}[
+    axis lines = left,
+    xlabel = $n$ (Data Input),
+    ylabel = {$t$ (time)},
+    legend pos=north east,
+    very thick,
+    ymax = 500,
+    yticklabels=\empty,
+    xticklabels=\empty,
+    scale only axis=true,
+  width=12cm, height=6cm,
+    ]
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}(n^2)$}
+\addplot [
+    domain= 1:10,
+    samples=100,
+    color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}(n^3)$}
+\addplot [
+    domain= 1:10,
+    samples=100,
+    color=black,
+]
+{exp(x)};
+\addlegendentry{$\mathcal{O}(e^n)$}
+\addplot [
+    domain= 1:20,
+    samples=100,
+    color=orange,
+]
+{log2(x)};
+\addlegendentry{$\mathcal{O}(\log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/blas.tex b/buch/papers/multiplikation/presentation/slides/blas.tex
new file mode 100644
index 0000000..ed498a3
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/blas.tex
@@ -0,0 +1,18 @@
+\begin{frame}
+\frametitle{BLAS, LAPACK}
+\begin{itemize}
+  \item Basic Linear Algebra Subprograms
+  \begin{itemize}
+    \item $\mathbf{y} = \alpha \mathbf{x}+\mathbf{y}$
+    \item $\mathbf{y} = \alpha \mathbf{A}\mathbf{x}+ \beta \mathbf{y}$
+    \item $\mathbf{C} = \alpha \mathbf{A}\mathbf{B}+ \beta \mathbf{C}$
+
+  \end{itemize}
+  \item Linear Algebra Package
+  \begin{itemize}
+    \item QR decomposition
+    \item Singular value decomposition
+    \item Eigenvalues
+  \end{itemize}
+\end{itemize}
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/conclusuion.tex b/buch/papers/multiplikation/presentation/slides/conclusuion.tex
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/conclusuion.tex
diff --git a/buch/papers/multiplikation/presentation/slides/logo.pdf b/buch/papers/multiplikation/presentation/slides/logo.pdf
new file mode 100644
index 0000000..d78ca88
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/logo.pdf
diff --git a/buch/papers/multiplikation/presentation/slides/meas.tex b/buch/papers/multiplikation/presentation/slides/meas.tex
new file mode 100644
index 0000000..489c010
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/meas.tex
@@ -0,0 +1,42 @@
+\begin{frame}
+  \frametitle{Measurements Python}
+  \only<1>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_8.pdf}}
+  \only<2>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_16.pdf}}
+  \only<3>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_32.pdf}}
+  \only<4>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_64.pdf}}
+  \only<5>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_128.pdf}}
+  \only<6>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_256.pdf}}
+  \only<7>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_512.pdf}}
+  \only<8>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_1024.pdf}}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{Measurements C}
+  \only<1>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_8.pdf}}
+  \only<2>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_16.pdf}}
+  \only<3>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_32.pdf}}
+  \only<4>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_64.pdf}}
+  \only<5>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_128.pdf}}
+  \only<6>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_256.pdf}}
+  \only<7>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_512.pdf}}
+  \only<8>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_1024.pdf}}
+  \only<9>{
+  \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_2048.pdf}}
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/nn.tex b/buch/papers/multiplikation/presentation/slides/nn.tex
new file mode 100644
index 0000000..e74e970
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/nn.tex
@@ -0,0 +1,97 @@
+
+\begin{frame}
+  \frametitle{Neural Network}
+  \centering
+\newcommand{\inputnum}{4}
+
+% Hidden layer neurons'number
+\newcommand{\hiddennumA}{5}
+\newcommand{\hiddennumB}{6}
+
+% Output layer neurons'number
+\newcommand{\outputnum}{4}
+
+\begin{tikzpicture}
+
+
+% Input Layer
+\foreach \i in {1,...,\inputnum}
+{
+	\node[circle,
+		minimum size = 6mm,
+		fill=blue!30] (Input-\i) at (0,-\i) {};
+}
+
+% Hidden Layer1
+\foreach \i in {1,...,\hiddennumA}
+{
+	\node[circle,
+		minimum size = 6mm,
+		fill=red!50,
+		yshift=(\hiddennumA-\inputnum)*5 mm
+	] (Hidden1-\i) at (2.5,-\i) {};
+}
+
+% Hidden Layer2
+\foreach \i in {1,...,\hiddennumB}
+{
+	\node[circle,
+		minimum size = 6mm,
+		fill=red!50,
+		yshift=(\hiddennumB-\inputnum)*5 mm
+	] (Hidden2-\i) at (5,-\i) {};
+}
+
+% Output Layer
+\foreach \i in {1,...,\outputnum}
+{
+	\node[circle,
+		minimum size = 6mm,
+		fill=green!50,
+		yshift=(\outputnum-\inputnum)*5 mm
+	] (Output-\i) at (7.5,-\i) {};
+}
+
+% Connect neurons In-Hidden
+\foreach \i in {1,...,\inputnum}
+{
+	\foreach \j in {1,...,\hiddennumA}
+	{
+		\draw[->, shorten >=1pt] (Input-\i) -- (Hidden1-\j);
+	}
+}
+
+% Connect neurons In-Hidden
+\foreach \i in {1,...,\hiddennumA}
+{
+	\foreach \j in {1,...,\hiddennumB}
+	{
+		\draw[->, shorten >=1pt] (Hidden1-\i) -- (Hidden2-\j);
+	}
+}
+
+% Connect neurons Hidden-Out
+\foreach \i in {1,...,\hiddennumB}
+{
+	\foreach \j in {1,...,\outputnum}
+	{
+		\draw[->, shorten >=1pt] (Hidden2-\i) -- (Output-\j);
+	}
+}
+
+% Inputs
+\foreach \i in {1,...,\inputnum}
+{
+	\draw[<-, shorten <=1pt] (Input-\i) -- ++(-1,0)
+		node[left]{\LARGE{$x_{\i}$}};
+}
+
+% Outputs
+\foreach \i in {1,...,\outputnum}
+{
+	\draw[->, shorten <=1pt] (Output-\i) -- ++(1,0)
+		node[right]{\LARGE{$y_{\i}$}};
+}
+
+\end{tikzpicture}
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/parcomp.tex b/buch/papers/multiplikation/presentation/slides/parcomp.tex
new file mode 100644
index 0000000..1ba39ee
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/parcomp.tex
@@ -0,0 +1,66 @@
+% !TEX root = presentation.tex
+
+\begin{frame}
+  \frametitle{Vector-Matrix Multiplication}
+\center{
+  \begin{tikzpicture}[ampersand replacement=\&]
+
+    \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
+     {
+     A_{1,1} \& A_{1,2} \& A_{1,3} \& A_{1,4} \\
+     };
+
+    \matrix (B)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (5,-0.95)
+     {
+     B_{1,1} \& B_{1,2} \& B_{1,3} \& B_{1,4} \& B_{1,5} \\
+     B_{2,1} \& B_{2,2} \& B_{2,3} \& B_{2,4} \& B_{2,5} \\
+     B_{3,1} \& B_{3,2} \& B_{3,3} \& B_{3,4} \& B_{3,5} \\
+     B_{4,1} \& B_{4,2} \& B_{4,3} \& B_{4,4} \& B_{4,5} \\
+     };
+
+     \matrix (C)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (5,-3)
+      {
+      C_{1,1} \& C_{1,2} \& C_{1,3} \& C_{1,4} \& C_{1,5}\\
+      };
+
+      \foreach \i in {1,...,4}
+      {
+      \pgfmathtruncatemacro{\ii}{\i+1}
+          \onslide<\ii>{
+
+       \foreach \j in {1,...,5}
+       {
+        \draw[thick] (A-1-\i.south) to [out=-90,in=135]node[visible on=<\i->, anchor=north]{} (B-\i-\j.center);
+
+          }
+        }
+        }
+
+
+     \end{tikzpicture}
+}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{DSP Architecture}
+\scalebox{2}{
+  \begin{tikzpicture}
+    \node (mul) at (0,0) [circle,draw=black,inner sep=0pt,minimum size=0.5cm] {X};
+    \node (mac) at (2,0) [circle,draw=black,inner sep=0pt,minimum size=0.5cm] {\textbf{+}};
+
+    \node at (-2,0.3) {$A[n]$};
+    \node at (0.4,2) {$B[n]$};
+    \node at (4,0.3) {$C[n]$};
+
+    \draw[thick, ->] (-2,0) --++ (mul);
+    \draw[thick, ->] (0,2) --++ (mul);
+    \draw[thick, ->] (mul) -- (mac);
+    \draw[thick] (mac) --++ (1,0) node (i) {};
+    \draw[thick, ->] (i.center) --++ (0,1) --++ (-1,0) -- (mac);
+    \draw[thick, ->] (i.center) --++ (1,0);
+
+
+  \end{tikzpicture}
+  }
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/slides.tex b/buch/papers/multiplikation/presentation/slides/slides.tex
new file mode 100644
index 0000000..64edb86
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/slides.tex
@@ -0,0 +1,15 @@
+% !TEX root = presentation.tex
+\begin{frame}
+\titlepage
+\end{frame}
+%
+\section{Big $\mathcal{O}$}
+\input{slides/BigO.tex}
+\section{Strassen's Algorithm}
+\input{slides/strassen.tex}
+% \input{slides/nn.tex}
+\section{Measurements}
+\input{slides/meas.tex}
+% \input{slides/parcomp.tex}
+\section{How To Matrix Multiply}
+\input{slides/blas.tex}
diff --git a/buch/papers/multiplikation/presentation/slides/strassen.tex b/buch/papers/multiplikation/presentation/slides/strassen.tex
new file mode 100644
index 0000000..c3398d5
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/strassen.tex
@@ -0,0 +1,429 @@
+\begin{frame}
+  \frametitle{Strassen's Algorithm}
+            \includegraphics[page=1,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}
+      \includegraphics[page=2,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}      \includegraphics[page=3,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}
+          \end{frame}
+
+\begin{frame}
+  \frametitle{Strassen's Algorithm}
+  \centering
+  \large
+\onslide<1->{
+  $
+  \mathbf{A B = C}
+  $
+}
+
+\onslide<2->{
+
+
+\medskip
+  $
+  \begin{bmatrix}
+  A_{11} & A_{12}\\
+  A_{21} & A_{22}
+  \end{bmatrix}
+  \begin{bmatrix}
+  B_{11} & B_{12}\\
+  B_{21} & B_{22}
+  \end{bmatrix}
+  =
+  \begin{bmatrix}
+  C_{11} & C_{12}\\
+  C_{21} & C_{22}
+  \end{bmatrix}
+  $
+  }
+
+
+  \onslide<3->{
+
+\medskip
+$
+C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}
+$
+
+$
+C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}
+$
+
+$
+C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}
+$
+
+$
+C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+$
+}
+\end{frame}
+
+\input{slides/algo.tex}
+
+
+
+\begin{frame}
+  \frametitle{Strassen's Algorithm}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \onslide<1->{
+      \large
+      \begin{math}
+      \begin{aligned}
+      \text{I}   &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+      \text{II}  &= (A_{21} + A_{22}) \cdot B_{11} \\
+      \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+      \text{IV}  &= A_{22} \cdot (-B_{11}+B_{21}) \\
+      \text{V}   &= (A_{11} + A_{12}) \cdot B_{22} \\
+      \text{VI}  &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\
+      \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+      \end{aligned}
+      \end{math}
+      }
+    \end{column}
+
+    \begin{column}{0.5\textwidth}
+        \onslide<2->{
+        \large
+        \begin{math}
+        \begin{aligned}
+        C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+        C_{21} &= \text{II} + \text{IV} \\
+        C_{12} &= \text{III} + \text{V}\\
+        C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+        \end{aligned}
+        \end{math}
+        }
+    \end{column}
+\end{columns}
+
+\onslide<3->{
+
+\bigskip
+\centering
+\tiny
+\begin{math}
+\begin{aligned}
+  C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+  C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
+  C_{11} &= A_{11}B_{11} + A_{12}B_{21}
+\end{aligned}
+\end{math}
+}
+
+\end{frame}
+
+
+\begin{frame}
+\begin{adjustbox}{width=\textwidth}
+\begin{tikzpicture}[ampersand replacement=\&]
+
+  \foreach \i in {1,...,4}
+  {
+  \small{
+    \matrix (X\i)[matrix of math nodes,nodes in empty cells,
+              nodes = {draw, minimum size=10mm,
+                       anchor=center,
+                       inner sep=0pt, outer sep=0pt},
+              column sep=-\pgflinewidth,
+              row sep=-\pgflinewidth,
+              ] at (0,-\i*5)
+   {
+   A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\
+   A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\
+   A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\
+   A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\
+   };}
+
+    \foreach \j in {1,...,7}
+    {
+    \matrix(M\i\j)[matrix of math nodes,nodes in empty cells,
+                nodes = {draw, minimum size=10mm,
+                         anchor=center,
+                         inner sep=0pt, outer sep=0pt},
+                column sep=-\pgflinewidth,
+                row sep=-\pgflinewidth,
+                ] at (\j*5,-\i*5)
+     {
+     \& \& \& \\
+     \& \& \& \\
+     \& \& \& \\
+     \& \& \& \\
+     };
+   }
+ }
+
+\huge{
+ \node at (-3,-20)  {$C_{22}=$};
+ \node at (-3,-15) {$C_{21}=$} ;
+ \node at (-3,-10) {$C_{12}=$} ;
+ \node at (-3,-5) {$C_{11}=$} ;
+
+ \node at (5,-2)  {I};
+ \node at (10,-2)  {II};
+ \node at (15,-2)  {III};
+ \node at (20,-2)  {IV};
+ \node at (25,-2)  {V};
+ \node at (30,-2)  {VI};
+ \node at (35,-2)  {VII};
+ }
+
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-4-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M14-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M14-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M23-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M23-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M34-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M34-2-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\end{tikzpicture}
+\end{adjustbox}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{Strassen's Algorithm}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \large
+      \begin{math}
+      \begin{aligned}
+      \text{I}   &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+      \text{II}  &= (A_{21} + A_{22}) \cdot B_{11} \\
+      \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+      \text{IV}  &= A_{22} \cdot (-B_{11}+B_{21}) \\
+      \text{V}   &= (A_{11} + A_{12}) \cdot B_{22} \\
+      \text{VI}  &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\
+      \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+      \end{aligned}
+      \end{math}
+
+    \end{column}
+
+    \begin{column}{0.5\textwidth}
+        \large
+        \begin{math}
+        \begin{aligned}
+        C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+        C_{21} &= \text{II} + \text{IV} \\
+        C_{12} &= \text{III} + \text{V}\\
+        C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+        \end{aligned}
+        \end{math}
+
+    \end{column}
+\end{columns}
+\end{frame}
+
+
+
+\begin{frame}
+  \frametitle{Strassen's Algorithm}
+
+\begin{columns}
+  \begin{column}{0.5\textwidth}
+\large
+\begin{math}
+\begin{aligned}
+\text{\textbf{I}}   &= (\mathbf{A_{11}} + \mathbf{A_{22}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{22}}) \\
+\text{\textbf{II}}  &= (\mathbf{A_{21}} + \mathbf{A_{22}}) \cdot \mathbf{B_{11}} \\
+\text{\textbf{III}} &= \mathbf{A_{11}} \cdot (\mathbf{B_{12}}-\mathbf{B_{22}}) \\
+\text{\textbf{IV}}  &= \mathbf{A_{22}} \cdot (-\mathbf{B_{11}}+\mathbf{B_{21}}) \\
+\text{\textbf{V}}   &= (\mathbf{A_{11}} + \mathbf{A_{12}}) \cdot \mathbf{B_{22}} \\
+\text{\textbf{VI}}  &= (-\mathbf{A_{11}} + \mathbf{A_{21}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{12}}) \\
+\text{\textbf{VII}} &= (\mathbf{A_{12}} - \mathbf{A_{22}}) \cdot (\mathbf{B_{21}} + \mathbf{B_{22}}) \\
+\end{aligned}
+\end{math}
+
+\end{column}
+
+\begin{column}{0.5\textwidth}
+  \large
+  \begin{math}
+  \begin{aligned}
+  \mathbf{C_{11}} &= \text{\textbf{I}} + \text{\textbf{IV}} - \text{\textbf{V}} + \text{\textbf{VII}} \\
+  \mathbf{C_{21}} &= \text{\textbf{II}} + \text{\textbf{IV}} \\
+  \mathbf{C_{12}} &= \text{\textbf{III}} + \text{\textbf{V}}\\
+  \mathbf{C_{22}} &= \text{\textbf{I}} + \text{\textbf{III}} - \text{\textbf{II}} + \text{\textbf{VI}} \\
+  \end{aligned}
+  \end{math}
+
+\end{column}
+\end{columns}
+
+\end{frame}
+
+\begin{frame}
+  \frametitle{Algorithm}
+  \onslide<1->{
+
+  \scalebox{0.45}{\parbox{\linewidth}{
+    \begin{algorithm}[H]\caption{Strassen Matrix Multiplication}
+      \setlength{\lineskip}{7pt}
+      \begin{algorithmic}[1]
+        \Function{strassen}{$\textbf{A}, \textbf{B}, n$}
+        \If{$n = 2$}
+        \State  $ \mathbf{C} \gets zeros((n, n))$
+        \State $P  \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$
+        \State   $Q  \gets (A[1][0]+A[1][1])\cdot B[0][0]$
+        \State   $R  \gets A[0][0]\cdot (B[0][1]-B[1][1])$
+        \State   $S  \gets A[1][1]\cdot (B[1][0]-B[0][0])$
+        \State   $T  \gets (A[0][0]+A[0][1])\cdot B[1][1]$
+        \State   $U  \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$
+        \State   $V  \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$
+        \State   $C[0][0]  \gets P+S-T+V$
+        \State   $C[0][1]  \gets R+T$
+        \State   $C[1][0]  \gets Q+S$
+        \State   $C[1][1]  \gets P+R-Q+U$
+        \Else
+      \State  $ m \gets n/2$
+  \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+  \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+  \State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$
+  \State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$
+  \State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}-  \mathbf{B22}),m)$
+  \State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}-  \mathbf{B11}),m)$
+  \State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$
+  \State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$
+  \State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$
+
+
+
+  \State   $\mathbf{C11}  \gets \mathbf{P+S-T+V}$
+  \State   $\mathbf{C12}  \gets \mathbf{R+T}$
+  \State   $\mathbf{C21}  \gets \mathbf{Q+S}$
+  \State   $\mathbf{C22}  \gets \mathbf{P+R-Q+U}$
+  \State $  C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$
+
+        \EndIf
+        \State \textbf{return} $\textbf{C}$
+
+       \EndFunction
+      \end{algorithmic}
+  \end{algorithm}
+    }}}
+%     \[
+%   \mathcal{T}(n) = \left\{\begin{array}{lr}
+%       1, & \text{if}  n \leq 2\\
+%       7 \mathcal{T}(\frac{n}{2}) + n^2, & \text{if} n > 2\\
+%       \end{array}\right\}
+% \]
+\only<2>{
+    $
+    \mathcal{T}(n) =
+    \begin{cases}
+      1  & \text{if }  n \leq 2\\
+      7 \cdot \mathcal{T}(\frac{n}{2}) + n^2  & \text{if }  n > 2
+    \end{cases} = \mathcal{O}(n^{\log_2 7})$
+
+}
+\only<3>{
+    $
+    \mathcal{T}(n) =
+    \begin{cases}
+      1  & \text{if }  n \leq 2\\
+      7 \cdot \mathcal{T}(\frac{n}{2}) + n^2  & \text{if }  n > 2
+    \end{cases} = \mathcal{O}(n^{2.81})$
+
+}
+
+\end{frame}
+
+\begin{frame}
+  \frametitle{Algorithm}
+  \onslide<1->{
+
+  \scalebox{0.45}{\parbox{\linewidth}{
+    \begin{algorithm}[H]\caption{Strassen Matrix Multiplication}
+      \setlength{\lineskip}{7pt}
+      \begin{algorithmic}[1]
+        \Function{MM}{$\textbf{A}, \textbf{B}, n$}
+        \If{$n = 2$}
+        \State  $ \mathbf{C} \gets zeros((n, n))$
+                \State  $C[0, 0] \gets  A[0][0]*B[0][0]+A[0][1]*B[1][0]$
+                \State  $C[0, 1] \gets  A[0][0]*B[0][1]+A[0][1]*B[1][1]$
+                \State  $C[1, 0] \gets  A[1][0]*B[0][0]+A[1][1]*B[1][0]$
+                \State  $C[1, 1] \gets  A[1][0]*B[0][1]+A[1][1]*B[1][1]$
+        \Else
+      \State  $ m \gets n/2$
+  \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+  \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+    \State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11}) + \text{MM}(\mathbf{A12}, \mathbf{B21})$
+    \State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12}) + \text{MM}(\mathbf{A12},\mathbf{B22})$
+    \State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11}) + \text{MM}(\mathbf{A22}, \mathbf{B21})$
+    \State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12}) + \text{MM}(\mathbf{A22}, \mathbf{B22})$
+    \State $  C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$
+
+        \EndIf
+        \State \textbf{return} $\textbf{C}$
+
+       \EndFunction
+      \end{algorithmic}
+  \end{algorithm}
+  \bigskip
+  \bigskip
+  \bigskip
+  \bigskip
+  \bigskip
+    }}}
+
+\only<2>{
+
+
+    $
+    \mathcal{T}(n) =
+    \begin{cases}
+      1  & \text{if }  n \leq 2\\
+      8 \cdot \mathcal{T}(\frac{n}{2}) + n^2  & \text{if }  n > 2
+    \end{cases} = \mathcal{O}(n^{\log_2 8})$
+
+}
+\only<3>{
+    $
+    \mathcal{T}(n) =
+    \begin{cases}
+      1  & \text{if }  n \leq 2\\
+      8 \cdot \mathcal{T}(\frac{n}{2}) + n^2  & \text{if }  n > 2
+    \end{cases} = \mathcal{O}(n^{3})$
+
+}
+
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/tikz/algo.pdf b/buch/papers/multiplikation/presentation/tikz/algo.pdf
new file mode 100644
index 0000000..752f42e
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/tikz/algo.pdf
diff --git a/buch/papers/multiplikation/presentation/tikz/algo.tex b/buch/papers/multiplikation/presentation/tikz/algo.tex
new file mode 100644
index 0000000..0b2c567
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/tikz/algo.tex
@@ -0,0 +1,52 @@
+\documentclass[border=10pt]{article}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{algorithm}
+\usepackage[noend]{algpseudocode}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+\begin{algorithm}[H]\caption{Square Matrix Multiplication}
+  \setlength{\lineskip}{7pt}
+  \begin{algorithmic}[1]
+    \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}, n$}
+    \State $sum \gets 0$
+      \For{$i = 0,1,2 \dots,n-1$}
+      \For{$j = 0,1,2 \dots,n-1$}
+      \State $sum \gets 0$
+      \For{$k = 0,1,2 \dots,n-1$}
+        \State $sum \gets  sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+      \EndFor
+      \State $\textbf{C}[i][j] \gets  sum $
+      \EndFor
+      \EndFor
+    \EndFunction
+  \end{algorithmic}
+\end{algorithm}
+\end{document}
diff --git a/buch/papers/multiplikation/problemstellung.tex b/buch/papers/multiplikation/problemstellung.tex
new file mode 100755
index 0000000..b20a791
--- /dev/null
+++ b/buch/papers/multiplikation/problemstellung.tex
@@ -0,0 +1,104 @@
+%
+% teil1.tex -- Beispiel-File für das Paper
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Problemstellung}
+\rhead{Problemstellung}
+Dank der breiten Anwendung der Matrizenmultiplikation ist eine effiziente L\"osung dieser Operation von grosser Bedeutung.
+Das Ziel dieses Papers ist verschiedenen Algorithmen der Matrizenmultiplikation vorzustellen.
+Wobei gezielt auf Algorithmen, welche das Problem schneller als der Standard Algorithmus L\"osen eingegangen wird.
+
+\subsection{Big $\mathcal{O}$ Notation}
+Die Big $\mathcal{O}$ Notation beschreibt die Laufzeitkomplexit\"at eines Algorithmus \cite{multiplikation:bigo}.
+$f(x) \in \mathcal{O}(g(x))$ besagt das die Funktion $f$ nicht wesentlich schneller w\"achst als $g$ wenn $x \rightarrow \infty$.
+Vereinfacht werden f\"ur Algorithmen die folgende Notation verwendet:
+\begin{itemize}
+	\item $f \in \mathcal{O}(1) \rightarrow f$ ist beschr\"ankt
+	\item $f \in \mathcal{O}(n) \rightarrow f$ w\"achst linear
+	\item $f \in \mathcal{O}(n^2) \rightarrow f$ w\"achst quadratisch
+	\item $f \in \mathcal{O}(\log n) \rightarrow f$ w\"achst logarithmisch
+	\item $f \in \mathcal{O}(n \log n) \rightarrow f$ hat super-lineares Wachstum
+	\item $f \in \mathcal{O}(e^n) \rightarrow f$ w\"achst exponentiell
+	\item usw.
+\end{itemize}
+
+In der Abbildung \ref{multiplikation:fig:bigo} k\"onnen die Verschiedenen Laufzeiten miteinander verglichen werden. 
+
+\begin{figure}
+	\center
+	\includegraphics[]{papers/multiplikation/images/bigo}
+	\caption{Verschiedene Laufzeiten}
+	\label{multiplikation:fig:bigo}
+\end{figure}
+
+\subsubsection{Beispiel Algorithmen}
+\paragraph{Beschr\"ankter Algorithmus}
+
+Ein Beispiel eines Beschr\"ankter Verhalten $\mathcal{O}(1)$, kann im Algorithmus \ref{multiplikation:alg:b1} entnommen werden.
+
+\begin{algorithm}\caption{}
+	\label{multiplikation:alg:b1}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}
+		\Function{B1}{$a, b$}
+		\State \textbf{return} $a+b$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+Wobei Konstanten nicht beachtet werden, der Algorithmus \ref{multiplikation:alg:b2} f\"uhrt ebenso zu  $\mathcal{O}(1)$ und nicht zu $\mathcal{O}(2)$.
+
+\begin{algorithm}\caption{}
+	\label{multiplikation:alg:b2}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}
+		\Function{B2}{$a, b$}
+		\State $ x \gets a+b $
+		\State $ y \gets a \cdot b $
+		\State \textbf{return} $x+y$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+\paragraph{Linearer Algorithmus}
+
+Folgender Algorithmus \ref{multiplikation:alg:l1} hat ein lineares $\mathcal{O}(n)$ Verhalten.
+
+\begin{algorithm}\caption{}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}
+		\label{multiplikation:alg:l1}
+		\Function{L}{$\mathbf{A}, \mathbf{B}$,n}
+		\State $ sum \gets 0$
+		\For{$i = 0,1,2 \dots,n$}
+		\State $ sum \gets sum + A[i] \cdot B[i] $
+		\EndFor
+		
+		\State \textbf{return} $sum$
+		
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+\paragraph{Quadratischer Algorithmus}
+
+Folgender Algorithmus \ref{multiplikation:alg:q1} hat ein quadratisches $\mathcal{O}(n^2)$ Verhalten.
+
+\begin{algorithm}[H]\caption{}
+	\label{multiplikation:alg:q1}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}
+		\Function{Q}{$\mathbf{A}, \mathbf{B}$,n}
+		\State $ sum \gets 0$
+		\For{$i = 0,1,2 \dots,n$}
+		\For{$j = 0,1,2 \dots,n$}
+		\State $ sum \gets sum + A[i] \cdot B[j] $
+		\EndFor
+		\EndFor
+		\State \textbf{return} $sum$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+
diff --git a/buch/papers/multiplikation/references.bib b/buch/papers/multiplikation/references.bib
index 7149fb1..9d76e8e 100644..100755
--- a/buch/papers/multiplikation/references.bib
+++ b/buch/papers/multiplikation/references.bib
@@ -33,3 +33,33 @@
         url = {https://doi.org/10.1016/j.acha.2017.11.004}
 }
 
+@article{multiplikation:winograd_1968,
+ 				title={A New Algorithm for Inner Product},
+				volume={C-17},
+  			DOI={10.1109/tc.1968.227420},
+  			number={7},
+  			journal={IEEE Transactions on Computers},
+  			author={Winograd, S.},
+ 				year={1968},
+   		pages={693–694}
+}
+
+@article{multiplikation:strassen_1969,
+	      title={Gaussian elimination is not optimal},
+        volume={13},
+        DOI={10.1007/bf02165411},
+        number={4},
+        journal={Numerische Mathematik},
+        author={Strassen, Volker},
+        year={1969},
+        pages={354–356}
+}
+
+@online{multiplikation:bigo,
+	title = {Big O notation},
+	url = {https://en.wikipedia.org/wiki/Big_O_notation},
+	date = {2021-07-27},
+	year = {2021},
+	month = {7},
+	day = {27}
+}
diff --git a/buch/papers/multiplikation/teil0.tex b/buch/papers/multiplikation/teil0.tex
deleted file mode 100644
index 082b7f5..0000000
--- a/buch/papers/multiplikation/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{multiplikation: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{multiplikation:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum.  Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
-
-
diff --git a/buch/papers/multiplikation/teil1.tex b/buch/papers/multiplikation/teil1.tex
deleted file mode 100644
index 0a6903a..0000000
--- a/buch/papers/multiplikation/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{multiplikation: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{multiplikation: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{multiplikation: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{multiplikation: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{multiplikation:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
-
-
diff --git a/buch/papers/multiplikation/teil2.tex b/buch/papers/multiplikation/teil2.tex
deleted file mode 100644
index efbf31a..0000000
--- a/buch/papers/multiplikation/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{multiplikation: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{multiplikation:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/multiplikation/teil3.tex b/buch/papers/multiplikation/teil3.tex
deleted file mode 100644
index f58508b..0000000
--- a/buch/papers/multiplikation/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{multiplikation:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{multiplikation:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk
new file mode 100644
index 0000000..5f14129
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk
@@ -0,0 +1,254 @@
+# Fdb version 3
+["pdflatex"] 1620305767 "algo.tex" "algo.pdf" "algo" 1621586452
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+INPUT /usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty
+INPUT algo.out
+INPUT algo.out
+INPUT algo.out
+INPUT algo.out
+OUTPUT algo.pdf
+INPUT ./algo.out
+INPUT ./algo.out
+OUTPUT algo.out
+INPUT /usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty
+INPUT /usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty
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+INPUT algo.aux
+INPUT ./algo.out
+INPUT ./algo.out
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo.pdf b/buch/papers/multiplikation/tikz_formulas/algo.pdf
new file mode 100644
index 0000000..f711224
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.pdf
diff --git a/buch/papers/multiplikation/tikz_formulas/algo.tex b/buch/papers/multiplikation/tikz_formulas/algo.tex
new file mode 100755
index 0000000..1e437c2
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.tex
@@ -0,0 +1,131 @@
+\documentclass[border=10pt,varwidth]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+$
+A=
+\begin{bmatrix}
+A_{11} & A_{12}\\
+A_{21} & A_{22}
+\end{bmatrix},
+B=
+\begin{bmatrix}
+B_{11} & B_{12}\\
+B_{21} & B_{22}
+\end{bmatrix},
+C=
+\begin{bmatrix}
+C_{11} & C_{12}\\
+C_{21} & C_{22}
+\end{bmatrix}
+$
+
+\medskip
+$
+A \cdot B = C
+$
+
+\medskip
+$
+C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\
+C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\
+C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\
+C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+$
+
+\medskip
+\begin{math}
+\begin{aligned}
+\text{I}   &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+\text{II}  &= (A_{21} + A_{22}) \cdot B_{11} \\
+\text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+\text{IV}  &= A_{22} \cdot (-B_{11}+B_{21}) \\
+\text{V}   &= (A_{11} + A_{12}) \cdot B_{22} \\
+\text{VI}  &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12})) \\
+\text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+\end{aligned}
+\end{math}
+
+
+\medskip
+\begin{math}
+\begin{aligned}
+C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+C_{21} &= \text{II} + \text{IV} \\
+C_{12} &= \text{III} + \text{V}\\
+C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+\end{aligned}
+\end{math}
+
+
+\medskip
+\begin{math}
+\begin{aligned}
+C_{11} &= \text{II} + \text{IV} \\
+C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22})C_{21} \\
+C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
+C_{11} &= A_{11}B_{11} + A_{12}B_{21}
+\end{aligned}
+\end{math}
+
+\section{Winograd}
+
+$
+x_1 y_1 + x_2 y_2 = (x_1 +y_2)(y_1 + x_2)-x_1 x_2 - y_1 y_2
+$
+
+$
+x = (x_1, \cdots, x_n), y=(y_1, \cdots, y_n)
+$
+
+\[
+\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j}
+\]
+
+\[
+\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}
+\]
+
+\[
+\langle x,y \rangle =
+\begin{cases}
+ \displaystyle  \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if  $n$ is even}\\
+\displaystyle  \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if  $n$ is odd}
+\end{cases}
+\]
+
+\end{document}
diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk
new file mode 100644
index 0000000..ddfa880
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf
new file mode 100755
index 0000000..7f5a984
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.tex b/buch/papers/multiplikation/tikz_formulas/algo_graph.tex
new file mode 100755
index 0000000..ad4228b
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+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.tex
@@ -0,0 +1,140 @@
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+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
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+\usepackage{slashed}
+\usepackage{pdfpages}
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+\usepackage{url}
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diff --git a/buch/papers/munkres/figures/Matrixdarstellung.png b/buch/papers/munkres/figures/Matrixdarstellung.png
new file mode 100644
index 0000000..91a376d
--- /dev/null
+++ b/buch/papers/munkres/figures/Matrixdarstellung.png
diff --git a/buch/papers/munkres/main.tex b/buch/papers/munkres/main.tex
index 8915a3d..e5282dc 100644
--- a/buch/papers/munkres/main.tex
+++ b/buch/papers/munkres/main.tex
@@ -3,8 +3,8 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Munkres-Algorithmus\label{chapter:munkres}}
-\lhead{Munkres-Algorithmus}
+\chapter{Das Zuordnungsproblem und der Munkres-Algorithmus\label{chapter:munkres}}
+\lhead{Das Zuordnungsproblem und der Munkres-Algorithmus}
 \begin{refsection}
 \chapterauthor{Marc Kühne}
 
diff --git a/buch/papers/munkres/teil0.tex b/buch/papers/munkres/teil0.tex
index 1ef0538..0578429 100644
--- a/buch/papers/munkres/teil0.tex
+++ b/buch/papers/munkres/teil0.tex
@@ -3,19 +3,8 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Geschichte\label{munkres:section:teil0}}
-\rhead{Geschichte}
-Die Ungarische Methode wurde 1955 von Harold Kuhn entwickelt und veröffentlicht.
-Der Name ``Ungarische Methode'' ergab sich, weil der Algorithmus
-weitestgehend auf den früheren Arbeiten zweier ungarischer Mathematiker
-basierte: Dénes Kőnig und Jenő Egerváry.
-James Munkres überprüfte den Algorithmus im Jahr 1957 und stellte fest,
-dass der Algorithmus (stark) polynomiell ist.
-Seitdem ist der Algorithmus auch als Kuhn-Munkres oder
-Munkres-Zuordnungsalgorithmus bekannt.
-Die Zeitkomplexität des ursprünglichen Algorithmus war $O(n^4)$,
-später wurde zudem festgestellt, dass er modifiziert werden kann,
-um eine  $O(n^3)$-Laufzeit zu erreichen.
-
-
+\section{Einleitung\label{munkres:section:teil0}}
+\rhead{Einleitung}
 
+Im Bereich der Unternehmensplanung (Operations Research) gibt es verschiedene Fragestellungen. Eine davon ist das sogenannte Transportproblem. Zum Transport einheitlicher Objekte von mehreren Angebots- zu mehreren Nachfrageorten ist ein optimaler, d. h. kostenminimaler Plan zu finden, wobei die vorhandenen und zu liefernden Mengen an den einzelnen Standorten gegeben sowie die jeweiligen Transportkosten pro Einheit zwischen allen Standorten bekannt sind.
+Nun gibt es im Bereich des klassischen Transportproblems Sonderfälle. Ein Sonderfall ist z.B. das Zuordnungsproblem.
diff --git a/buch/papers/munkres/teil1.tex b/buch/papers/munkres/teil1.tex
index 7cbbbfd..c13732c 100644
--- a/buch/papers/munkres/teil1.tex
+++ b/buch/papers/munkres/teil1.tex
@@ -3,19 +3,56 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Was ist die ungarische Methode?
+\section{Beschrieb des Zuordnungsproblems
 \label{munkres:section:teil1}}
 \rhead{Problemstellung}
-Es ist ein kombinatorischer Optimierungsalgorithmus, der das Zuordnungsproblem
-in polynomieller Zeit löst.
-\begin{itemize}
-\item
-Polynom = vielgliedrig
-\end{itemize}
-Der Begriff polynomielle Laufzeit bedeutet, dass die Laufzeit des Programms
-wie $n^2$, $n^3$, $n^4$, etc.~wächst und vernünftig skaliert.
-Mit der ungarischen Methode können also lineare Optimierungsprobleme gelöst
-werden, die bei gewichteten Zuordnungen in bipartiten Graphen entstehen.
-Mit ihr kann die eindeutige Zuordnung von Objekten aus zwei Gruppen so
-optimiert werden, dass die Gesamtkosten minimiert werden bzw.~der
-Gesamtgewinn maximiert werden kann. 
+
+Das spezielle an einem Zuordnungsproblem ist, dass es an jedem Ort nur eine Einheit angeboten bzw. nachgefragt wird. Es werden hier nicht Mengen möglichst kostenminimal von einem zum anderen
+Ort transportiert, sondern es geht um die kostenminimale Zuordnung von z.B. Personen, oder Bau-Materialien auf bestimmte Orte, Stellen oder Aufgaben.
+Um dieses Problem in einer einfachen, händischen Art und Weise zu lösen wurde der Munkres-Algorithmus, auch die Ungarische Methode genannt, entwickelt. Diese Methode ist ein weiteres Hauptthema dieses Kapitels.
+
+\subsection{Zuordnungsproblem an einem konkreten Beispiel
+\label{munkres:subsection:bonorum}}
+
+\subsection{Zuordnungsproblem abstrakt
+\label{munkres:subsection:bonorum}}
+
+Es sind alle Angebots- und Bedarfsmengen gleich 1 
+\begin{equation}
+a_{i}=b_{j}=1
+\end{equation}
+
+\subsection{alternative Darstellungen des Zuordnungsproblems
+\label{munkres:subsection:bonorum}}
+\begin{equation}
+Netzwerk
+\end{equation}
+\begin{equation}
+Matrix
+\end{equation}
+\begin{equation}
+Bitpartiter Graph
+\end{equation}
+Ein bipartiter Graph ist ein mathematisches Modell für Beziehungen
+zwischen den Elementen zweier Mengen.
+Es eignet sich sehr gut zur Untersuchung von Zuordnungsproblemen» 
+\begin{figure}
+\centering
+\includegraphics[width=5cm]{papers/munkres/figures/Netzwerkdarstellung}
+\caption{Typische Netzwerkdarstellung eines Zuordnungsproblems.}
+\label{munkres:Vr2}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=5cm]{papers/munkres/figures/Matrixdarstellung}
+\caption{Typische 4x4 Matrixdarstellung eines Zuordnungsproblems.}
+\label{munkres:Vr2}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics[width=5cm]{papers/munkres/figures/bipartiter_graph}
+\caption{$K_{3,3}$ vollständig bipartiter Graph mit 3 Knoten pro Teilmenge.}
+\label{munkres:Vr2}
+\end{figure}
diff --git a/buch/papers/munkres/teil2.tex b/buch/papers/munkres/teil2.tex
index 29db8d7..9a44cd4 100644
--- a/buch/papers/munkres/teil2.tex
+++ b/buch/papers/munkres/teil2.tex
@@ -3,86 +3,11 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Das Zuordnungsproblem
+\section{Schwierigkeit der Lösung (Permutationen)
 \label{munkres:section:teil2}}
-\rhead{Das Zuordnungsproblem}
-Das (lineare) Zuordnungsproblem ist ein diskretes Optimierungsproblem aus
-der Graphentheorie.
-Es handelt sich um einen Spezialfall eines maximalen Matchings
-minimalen Gewichtes in einem bipartiten, gewichteten Graphen
+\rhead{Schwierigkeit der Lösung (Permutationen)}
 
-Vereinfacht gesagt sind Zuordnungsprobleme spezielle Transportprobleme.
-Der Unterschied zu klassischen Transportproblemen liegen darin,
-dass hier nicht Mengen möglichst kostenminimal von einem zum anderen
-Ort transportiert werden sollen, sondern es geht um die kostenminimale
-Zuordnung von z.~B.~Personen, oder Bau-Materialien auf bestimmte
-Orte, Stellen oder Aufgaben.
-Dabei sind alle Angebots- und Bedarfsmenge gleich 1 
-\begin{equation}
-a_{i}=b_{j}=1
-\end{equation}
+Eine Permutation ist eine Anordnung von Objekten in einer bestimmten Reihenfolge oder eine Umordnung von Objekten aus einer vorgegebenen Reihung. Ist eine maximale Zuordnung (maximales Matching) gefunden, so steht in jeder Zeile und jeder Spalte der Matrix genau ein Element, das zur optimalen Lösung gehört, eine solche Gruppe von Positionen wird auch als Transversale der Matrix bezeichnet. 
 
-\subsection{Zuordnungsproblem in Netzwerkdarstellung
-\label{munkres:subsection:bonorum}}
-
-\begin{figure}
-\centering
-\includegraphics[width=5cm]{papers/munkres/figures/Netzwerkdarstellung}
-\caption{Typische Netzwerkdarstellung eines Zuordnungsproblems.}
-\label{munkres:Vr2}
-\end{figure}
-
-\subsection{Matrix Formulierung
-\label{munkres:subsection:bonorum}}
-In der Matrixformulierung ist eine nicht-negative $n\times n$-Matrix
-gegeben, wobei das Element in der $i$-ten Zeile und $j$-ten Spalte
-die Kosten für die Zuweisung des $j$-ten Jobs an den $i$-ten Arbeiter
-darstellt.
-Wir müssen eine Zuordnung der Jobs zu den Arbeitern finden, so dass
-jeder Job einem Arbeiter zugewiesen wird und jeder Arbeiter einen
-Job zugewiesen bekommt, so dass die Gesamtkosten der Zuordnung
-minimal sind.
-Dies kann als Permutation der Zeilen und Spalten einer Kostenmatrix
-$C$ ausgedrückt werden, um die Spur einer Matrix zu minimieren:
-\begin{equation}
-\min(L,R)Tr (LCR)
-\end{equation}
-wobei $L$ und $R$ Permutationsmatrizen sind.
-Wenn das Ziel ist, die Zuordnung zu finden, die die maximalen Kosten
-ergibt, kann das Problem durch Negieren der Kostenmatrix $C$ gelöst
-werden.
-
-\subsection{Suche der optimalen Lösung
-\label{munkres:subsection:bonorum}}
-Ist eine maximale Zuordnung (maximales Matching) gefunden, so steht
-in jeder Zeile und jeder Spalte der Matrix genau ein Element, das
-zur optimalen Lösung gehört, eine solche Gruppe von Positionen wird
-auch als Transversale der Matrix bezeichnet.
-Deshalb kann die Problemstellung auch anders formuliert werden: Man
-ordne die Zeilen- oder die Spaltenvektoren so um, dass die Summe
-der Elemente in der Hauptdiagonale maximal wird.
-Hieraus wird sofort ersichtlich, dass es in einer 
-$n\times n$-Matrix genau so viele Möglichkeiten gibt, die Zeilen-
-bzw.~Spaltenvektoren zu ordnen, wie es Permutationen von $n$ Elementen
-gibt, also $n!$.
-Außer bei kleinen Matrizen ist es nahezu aussichtslos, die optimale
-Lösung durch Berechnung aller Möglichkeiten zu finden.
-Schon bei einer $10\times 10$-Matrix gibt es nahezu 3,63 Millionen (3.628.800)
-zu berücksichtigender Permutationen.
-
-\subsection{Formulierung Bipartiter Graph
-\label{munkres:subsection:bonorum}}
-Der Algorithmus ist einfacher zu beschreiben, wenn wir das Problem
-anhand eines bipartiten Graphen formulieren.
-Wir haben einen vollständigen zweistufigen Graphen $G=(S,T;E)$ mit
-$n$ Arbeiter-Eckpunkten ($S$) und $n$ Job-Scheitelpunkte ($T$), und
-jede Kante hat einen nichtnegativen Preis $c(i,j)$.
-Wir wollen ein perfektes Matching mit minimalen Gesamtkosten finden.
-
-\begin{figure}
-\centering
-\includegraphics[width=5cm]{papers/munkres/figures/bipartiter_graph}
-\caption{$K_{3,3}$ vollständig bipartiter Graph mit 3 Knoten pro Teilmenge.}
-\label{munkres:Vr2}
-\end{figure}
+Die Problemstellung kann auch so formuliert werden, dass man die Zeilen- oder die Spaltenvektoren so umordnet soll, dass die Summe der Elemente in der Hauptdiagonale maximal wird. Hieraus wird sofort ersichtlich, dass es in einer n×n-Matrix genau so viele Möglichkeiten gibt, die Zeilen- bzw. Spaltenvektoren zu ordnen, wie es Permutationen von n Elementen gibt, also n!. Außer bei kleinen Matrizen ist es nahezu aussichtslos, die optimale Lösung durch Berechnung aller Möglichkeiten zu finden. Schon bei einer 10×10-Matrix gibt es nahezu 3,63 Millionen (3.628.800) zu berücksichtigender Permutationen.
 
diff --git a/buch/papers/munkres/teil3.tex b/buch/papers/munkres/teil3.tex
index 806cd83..cd47c92 100644
--- a/buch/papers/munkres/teil3.tex
+++ b/buch/papers/munkres/teil3.tex
@@ -3,102 +3,44 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Der Algorithmus in Form von bipartiten Graphen
+\section{Der Munkres-Algorithmus (Ungarische Methode)
 \label{munkres:section:teil3}}
-\rhead{Der Algorithmus in Form von bipartiten Graphen}
-Mit der ungarischen Methode können also lineare Optimierungsprobleme
-gelöst werden, die bei gewichteten Zuordnungen in bipartiten Graphen
-entstehen.
+\rhead{Der Munkres-Algorithmus (Ungarische Methode)}
 
-Mit ihr kann die eindeutige Zuordnung von Objekten aus zwei Gruppen
-so optimiert werden, dass die Gesamtkosten minimiert werden bzw.~der
-Gesamtgewinn maximiert werden kann.
+Mit der ungarischen Methode können also lineare Optimierungsprobleme gelöst
+werden, die bei gewichteten Zuordnungen in bipartiten Graphen entstehen.
+Mit ihr kann die eindeutige Zuordnung von Objekten aus zwei Gruppen so
+optimiert werden, dass die Gesamtkosten minimiert werden bzw.~der
+Gesamtgewinn maximiert werden kann. 
 
-Ein bipartiter Graph ist ein mathematisches Modell für Beziehungen
-zwischen den Elementen zweier Mengen.
-Es eignet sich sehr gut zur Untersuchung von Zuordnungsproblemen» 
-
-\subsection{Beweis, dass der Algorithmus Fortschritte macht
+\subsection{Geschichte
 \label{munkres:subsection:malorum}}
-Wir müssen zeigen, dass der Algorithmus, solange das Matching nicht
-die maximal mögliche Größe hat, immer in der Lage ist, Fortschritte
-zu machen --- das heißt, entweder die Anzahl der übereinstimmenden
-Kanten zu erhöhen oder mindestens eine Kante zu straffen.
-Es genügt zu zeigen, dass bei jedem Schritt mindestens eine der
-folgenden Bedingungen erfüllt ist:
-
-\begin{itemize}
-\item
-$M$ die maximal mögliche Größe.
-\item
-$Gy$ enthält einen Erweiterungspfad.
-\item
-$G$ enthält einen losen Pfad: einen Pfad von einem Knoten in $Rs$
-zu einem Knoten in $T$ / $Z$ die aus einer beliebigen Anzahl von
-festen Kanten, gefolgt von einer einzelnen losen Kante, besteht.
-Die freie Kante einer freien Bahn ist also $Z$ (beinhaltet $T$),
-so garantiert es, dass Delta gut definiert ist.
-\end{itemize}
-Wenn $M$ die maximal mögliche Größe hat, sind wir natürlich fertig.
-Andernfalls muss es nach Berges Lemma im zugrundeliegenden Graphen
-$G$ einen Augmentierungspfad $P$ in Bezug auf $M$ geben.
-Dieser Pfad darf jedoch nicht in $G_y$ existieren: Obwohl jede
-geradzahlige Kante in $P$ durch die Definition von $M$ fest ist,
-können ungeradzahlige Kanten lose sein und in $G_y$ fehlen.
-Ein Endpunkt von $P$ liegt in $R_{S}$, der andere in $R_T$; w.l.o.g.,
-nehmen Sie an, es beginnt in $R_{S}$.
-Wenn jede Kante von $P$ dicht ist, dann bleibt sie ein augmentierender
-Pfad in $G_y$ und wir sind fertig.
-Andernfalls sei $uv$ die erste lose Kante auf $P$.
-Wenn $v$ kein Element von $Z$ ist, dann haben wir einen losen Pfad
-gefunden und sind fertig.
-Andernfalls ist $v$ von irgendeinem anderen Pfad $Q$ aus festen
-Kanten von einem Knoten in $R_{S}$ erreichbar.
-Sei $P_{v}$ der Teilpfad von $P$, der bei $v$ beginnt und bis zum
-Ende reicht, und sei $P'$ der Pfad, der gebildet wird, indem man
-entlang $Q$ gebildet wird, bis ein Scheitelpunkt auf $P_{v}$ erreicht
-wird, und dann weiter bis zum Ende von $P_{v}$.
-Beachten Sie, dass $P'$ ein erweiternder Pfad in $G$ mit mindestens
-einer losen Kante weniger als $P$ ist.
-$P$ kann durch $P'$ ersetzt und dieser Argumentationsprozess iteriert
-werden (formal, unter Verwendung von Induktion auf die Anzahl der
-losen Kanten), bis entweder ein erweiternder Pfad in $G_y$ oder ein
-losender Pfad in $G$ gefunden wird.
+Die Ungarische Methode wurde 1955 von Harold Kuhn entwickelt und veröffentlicht.
+Der Name ``Ungarische Methode'' ergab sich, weil der Algorithmus
+weitestgehend auf den früheren Arbeiten zweier ungarischer Mathematiker
+basierte: Dénes Kőnig und Jenő Egerváry.
+James Munkres überprüfte den Algorithmus im Jahr 1957 und stellte fest,
+dass der Algorithmus (stark) polynomiell ist.
+Seitdem ist der Algorithmus auch als Kuhn-Munkres oder
+Munkres-Zuordnungsalgorithmus bekannt.
+Die Zeitkomplexität des ursprünglichen Algorithmus war $O(n^4)$,
+später wurde zudem festgestellt, dass er modifiziert werden kann,
+um eine  $O(n^3)$-Laufzeit zu erreichen.
 
-\subsection{Beweis, dass die Anpassung des Potentials $y$ $M$ unverändert lässt
+\subsection{Besondere Leistung der Ungarischen Methode
 \label{munkres:subsection:malorum}}
-Um zu zeigen, dass jede Kante in $M$ nach der Anpassung von $y$
-erhalten bleibt, genügt es zu zeigen, dass für eine beliebige Kante
-in $M$ entweder beide Endpunkte oder keiner von ihnen in $Z$ liegen.
-Zu diesem Zweck sei $vu$ eine Kante in $M$ von $T$ nach $S$.
-Es ist leicht zu sehen, dass wenn $v$ in $Z$ ist, dann muss auch
-$u$ in $Z$ sein, da jede Kante in $M$ dicht ist.
-Nehmen wir nun an, dass $u$ kein Element von $Z$ und auch $v$ kein
-Element von $Z$ ist.
-$u$ selbst kann nicht in $R_{S}$ sein, da es der Endpunkt einer
-angepassten Kante ist, also muss es einen gerichteten Pfad von engen
-Kanten von einem Knoten in $R_{S}$ zu $u$ geben.
-Dieser Pfad muss $v$ vermeiden, da es per Annahme nicht in $Z$ ist,
-also ist der Knoten, der $u$ in diesem Pfad unmittelbar vorausgeht,
-ein anderer Knoten $v$ (ein Element von $T$) und $v$ ein Element
-von $u$ ist eine enge Kante von $T$ nach $S$ und ist somit in $M$.
-Aber dann enthält $M$ zwei Kanten, die den Knoten $u$ teilen, was
-der Tatsache widerspricht, dass $M$ ein Matching ist.
-Jede Kante in $M$ hat also entweder beide Endpunkte oder keinen
-Endpunkt in $Z$.
+Es ist ein kombinatorischer Optimierungsalgorithmus, der das Zuordnungsproblem
+in polynomieller Zeit löst.
+Der Begriff polynomielle Laufzeit bedeutet, dass die Laufzeit des Programms
+wie $n^2$, $n^3$, $n^4$, etc.~wächst und vernünftig skaliert.
+
 
-\subsection{Beweis, dass $y$ ein Potential bleibt
+\subsection{Beispiel eines händischen Verfahrens
 \label{munkres:subsection:malorum}}
-Um zu zeigen, dass y nach der Anpassung ein Potenzial bleibt, genügt
-es zu zeigen, dass keine Kante ihr Gesamtpotenzial über ihre Kosten
-hinaus erhöht.
-Dies ist für Kanten in $M$ bereits durch den vorangegangenen Absatz
-bewiesen.
-Man betrachtet also eine beliebige Kante $uv$ von $S$ nach $T$.
-Wenn $y(u)$ erhöht wird um $\Delta$, dann wird entweder $v\in
-\mathbb{Z}_n$ in diesem Fall wird $y(v)$ verringert um $\Delta$,
-wobei das Gesamtpotenzial der Kante unverändert bleibt, oder $v\in
-T\setminus Z$, wobei die Definition von $\Delta$ garantiert, dass
-$y(u)+y(v)+\Delta \le c(u,v)$
-Also $y$ bleibt ein Potential.
 
+\begin{figure}
+\centering
+\includegraphics[width=14cm]{papers/munkres/figures/beispiel_munkres}
+\caption{Händisches Beispiel des Munkres Algorithmus.}
+\label{munkres:Vr2}
+\end{figure}
diff --git a/buch/papers/munkres/teil4.tex b/buch/papers/munkres/teil4.tex
index 3d76743..9a27227 100644
--- a/buch/papers/munkres/teil4.tex
+++ b/buch/papers/munkres/teil4.tex
@@ -3,34 +3,7 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Matrix-Interpretation
+\section{-
 \label{munkres:section:teil4}}
-\rhead{Matrix-Interpretation}
-Gegeben ist die quadratische Matrix $C=(c_{ij})$ der Grösse $n\times n$.
-Ohne Beschränkung der Allgemeinheit werden eine Zuordnung $j
-\rightarrow  s_j$, $j = 1, \dots, n$ mit minimaler Gesamtsumme
-$\sum_{j=1}^{n}c_{s_j,j}$ gesucht, wobei die $s_j$ eine Permutation
-von $\{1,\ldots ,n\}$ sind.
-Soll die Summe maximiert werden, dann kann $C$ durch $-C$ ersetzt werden.
-Die Grundlage dieses Verfahrens ist, dass sich die optimale Zuordnung
-unter bestimmten Änderungen der Matrix nicht ändert, sondern nur
-der Optimalwert.
-Diese Änderungen sind durch Knotenpotentiale bzw.~duale Variablen 
-\begin{equation}
-u_1 u_2,{\dots}, u_n
-\end{equation}
+\rhead{-}
 
-für die Zeilen und 
-
-\begin{equation}v_1,v_2,\dots,v_n \end{equation} fuer die Spalten angegeben.
-Die modifizierte Matrix hat dann die Komponenten $\tilde{c}_{i,j}
-= c_{ij} - u_j - v_j$.
-
-In der Summe über jede kantenmaximale Zuordnung kommt jedes
-Knotenpotential genau einmal vor, so dass die Änderung der Zielfunktion
-eine Konstante ist.
-Sind die Einträge von $C$ nichtnegativ, und sind alle Knotenpotentiale
-ebenfalls nichtnegativ, so nennt man die modifizierte Matrix \~{C}
-auch eine Reduktion.
-Ziel ist, in der reduzierten Matrix möglichst viele Komponenten auf
-den Wert Null zu bringen und unter diesen die Zuordnung zu konstruieren.
diff --git a/buch/papers/munkres/teil5.tex b/buch/papers/munkres/teil5.tex
index f8138f4..b938c50 100644
--- a/buch/papers/munkres/teil5.tex
+++ b/buch/papers/munkres/teil5.tex
@@ -3,12 +3,6 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Ungarische Methode anhand eines Beispiels
+\section{-
 \label{munkres:section:teil5}}
-\rhead{Ungarische Methode anhand eines Beispiels}
-\begin{figure}
-\centering
-\includegraphics[width=14cm]{papers/munkres/figures/beispiel_munkres}
-\caption{Händisches Beispiel des Munkres Algorithmus.}
-\label{munkres:Vr2}
-\end{figure}
+\rhead{-}
diff --git a/buch/papers/reedsolomon/Makefile b/buch/papers/reedsolomon/Makefile
index 9c96e88..25fd98b 100644
--- a/buch/papers/reedsolomon/Makefile
+++ b/buch/papers/reedsolomon/Makefile
@@ -4,6 +4,52 @@
 # (c) 2020 Prof Dr Andreas Mueller
 #
 
-images:
-	@echo "no images to be created in reedsolomon"
+SOURCES := \
+	anwendungen.tex \
+	codebsp.tex \
+	decmitfehler.tex \
+	decohnefehler.tex \
+	dtf.tex \
+	einleitung.tex \
+	endlichekoerper.tex \
+	hilfstabellen.tex \
+	idee.tex \
+	main.tex \
+	packages.tex \
+	rekonstruktion.tex \
+	restetabelle1.tex \
+	restetabelle2.tex \
+	standalone.tex \
+	zusammenfassung.tex
+
+TIKZFIGURES := \
+	tikz/polynom2.tex \
+	tikz/plotfft.tex
+
+FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES))
+
+
+all: images standalone
+
+
+.PHONY: images
+images: $(FIGURES)
+
+figures/%.pdf: tikz/%.tex
+	mkdir -p figures
+	pdflatex --output-directory=figures $<
+
+.PHONY: standalone
+standalone: standalone.tex $(SOURCES) $(FIGURES)
+	mkdir -p standalone
+	cd ../..; \
+		pdflatex \
+			--halt-on-error \
+			--shell-escape \
+			--output-directory=papers/reedsolomon/standalone \
+			papers/reedsolomon/standalone.tex;
+	cd standalone; \
+		bibtex standalone; \
+		makeindex standalone;
+
 
diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex
index a111527..e9aacfb 100644
--- a/buch/papers/reedsolomon/dtf.tex
+++ b/buch/papers/reedsolomon/dtf.tex
@@ -3,52 +3,71 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Diskrete Fourier Transformation
+\section{Übertragung mit hilfe der Diskrete Fourier Transformation
 \label{reedsolomon:section:dtf}}
 \rhead{Umwandlung mit DTF}
 Um die Polynominterpolation zu umgehen, gehen wir nun über in die Fourientransformation.
-Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauch
-für den Reed-Solomon-Code. Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert.
+Dies wird weder eine Erklärung der Forientransorfmation, noch ein genauer gebrauch für den Reed-Solomon-Code. 
+Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert.
 wobei sie dann bei späteren Berchnungen ganz nützlich ist.
 
-\subsection{Diskrete Fourientransformation Zusamenhang
+\subsection{Diskrete Fourietransformation Zusamenhang
 \label{reedsolomon:subsection:dtfzusamenhang}}
-Die Diskrete Fourientransformation ist definiert als
-	\[
-	\label{ft_discrete}
+Mit hilfe der Fourietransformation werden die \textcolor{blue}{blauen Datenpunkte} transformiert,
+zu den \textcolor{darkgreen}{grünen Übertragungspunkten}. 
+Durch eine Rücktransformation könnnen die \textcolor{blue}{blauen Datenpunkte} wieder rekonstruiert werden.
+Nun zur definition der Diskrete Fourietransformation, diese ist definiert als
+\begin{equation}
 	\hat{c}_{k} 
 	= \frac{1}{N} \sum_{n=0}^{N-1}
 	{f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn}
-	\]
-, wenn man nun 
-	\[
-	w = e^{-\frac{2\pi j}{N} k}
-	\]
+	,\label{reedsolomon:DFT}
+\end{equation}
+wenn man nun 
+\begin{equation}
+	w =
+	e^{-\frac{2\pi j}{N} k}
+	\label{reedsolomon:DFT_summand}
+\end{equation}
 ersetzte, und $N$ konstantbleibt, erhält man
-	\[
-	\hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N)
-	\]
+\begin{equation}
+	\hat{c}_{k}=
+	\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N)
+	\label{reedsolomon:DFT_polynom}
+\end{equation}
 was überaust ähnlich zu unserem Polynomidee ist.
-\subsection{Übertragungsabfolge
+
+\subsection{Beispiel
 \label{reedsolomon:subsection:Übertragungsabfolge}}
+Der Auftrag ist nun 64 Daten zu übertragen und nach 32 Fehler abzusicheren,
+16 Fehler erkennen und rekonstruieren.
 
-\begin{enumerate}[1)]
+Dieser Auftrag soll mittels Fouriertransformation bewerkstelligt werden. 
+In der Abbildung \ref{reedsolomon:subsection:Übertragungsabfolge} sieht man dies Schritt für schritt,
+und hier werden die einzelne Schritte erklärt:
+\begin{enumerate}[(1)]
 \item Das Signal hat 64 die Daten, Zahlen welche übertragen werden sollen. 
 Dabei zusätzlich nach 16 Fehler abgesichert, macht insgesamt 96 Übertragungszahlen.
-\item Nun wurde mittels der schnellen diskreten Fourientransformation diese 96 codiert.
-Das heisst alle information ist in alle Zahlenvorhanden.
-\item Nun kommen drei Fehler dazu an den Übertragungsstellen 7, 21 und 75.
-\item Dieses wird nun Empfangen und mittels inversen diskreten Fourientransormation, wieder rücktransformiert.
-\item Nun sieht man den Fehler im Decodieren in den Übertragungsstellen 64 bis 96.
-\item Nimmt man nun nur diese Stellen 64 bis 96, auch Syndrom genannt, und Transformiert diese.
-\item Bekommt man die Fehlerstellen im Locator wieder, zwar nichtso genau, dennoch erkkent man wo die Fehler stattgefunden haben.
+(siehe Abschnitt \externaldocument{papers/reedsolomon/idee}\ref{reedsolomon:section:Fehlerkorrekturstellen})
+Die 32 Fehlerkorrekturstellen werden als Null Übertragen
+\item Nun wurde mittels der diskreten Fourientransformation diese 96 codiert.
+Das heisst alle Informationen ist in alle Zahlenvorhanden. (Auch die Fehlerkorrekturstellen Null)
+\item Nun kommen drei Fehler dazu an den Übertragungsstellen 7, 21 und 75.(die Skala ist Rechts)
+Die Fehler können auf den ganzen 96 Übertragungswerten liegen, wie die 75 zeigt.
+\item Dieses wird nun Empfangen und mittels inversen diskreten Fourientransormation, wieder rücktransformiert.(Iklusive der Fehler)
+\item Nun sieht man den Fehler im Decodieren in den Übertragungsstellen 64 bis 96, da es dort nicht mehr Null ist.
+\item Nimmt man nun nur diese Stellen 64 bis 96, dies definieren wir als Syndrom, und transformiert nur dieses Syndrom.
+\item Bekommt man die Fehlerstellen wieder, zwar nichtso genau, dennoch erkennt man wo die Fehler stattgefunden haben. 
+Dies definieren wir als Locator.
 \end{enumerate}
+Nun haben wir mit Hilfe der Fourietransformation die 3 Fehlerstellen durch das Syndrom lokalisiert, 
+jetzt gilt es nur noch diese zu korrigieren und wir haben unser originales Signal wieder.
 
 \begin{figure}
 	\centering
-	\resizebox{0.9\textwidth}{!}{
-	%\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/plot.pdf}
-    \input{papers/reedsolomon/images/plotfft.tex}
+	\resizebox{\textwidth}{!}{
+	\includegraphics[width=\textwidth]{papers/reedsolomon/figures/plotfft}
+    %\input{papers/reedsolomon/images/plotfft.tex}
 	}
 	\caption{Übertragungsabfolge \ref{reedsolomon:subsection:Übertragungsabfolge}}
 	\label{fig:sendorder}
diff --git a/buch/papers/reedsolomon/einleitung.tex b/buch/papers/reedsolomon/einleitung.tex
index 2b1d878..074df05 100644
--- a/buch/papers/reedsolomon/einleitung.tex
+++ b/buch/papers/reedsolomon/einleitung.tex
@@ -7,13 +7,11 @@
 \label{reedsolomon:section:einleitung}}
 \rhead{Einleitung}
 Der Reed-Solomon-Code ist entstanden um,
-das Problem der Fehler, bei der Datenübertragung, zu lösen.
-In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt.
+das Problem der Fehler bei der Datenübertragung, zu lösen.
+In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge, 
+Funktion oder Algorithmus des Reed-Solomon-Code erklärt.
 Es wird jedoch nicht auf die technische Umsetzung oder Implementierung eingegangen.
-Um beim Datenübertragen Fehler zu erkennen, könnte man die Daten jeweils doppelt senden,
-und so jeweilige Fehler zu erkennen.
-Doch nur schon um weinige Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach gesendet.
-Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise.
+
 
 
 
diff --git a/buch/papers/reedsolomon/experiments/plot.tex b/buch/papers/reedsolomon/experiments/plot.tex
index 2196c82..4b156bb 100644
--- a/buch/papers/reedsolomon/experiments/plot.tex
+++ b/buch/papers/reedsolomon/experiments/plot.tex
@@ -90,7 +90,7 @@
 	\draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north);
 	
 	%item
-	\node[circle, draw, fill =lightgray] at (signal.north west)+(1,0) {1};
+	\node[circle, draw, fill =lightgray] at (signal.north west) {1};
 	\node[circle, draw, fill =lightgray] at (codiert.north west) {2};
 	\node[circle, draw, fill =lightgray] at (fehler.north west) {3};
 	\node[circle, draw, fill =lightgray] at (empfangen.north west) {4};
diff --git a/buch/papers/reedsolomon/figures/plotfft.pdf b/buch/papers/reedsolomon/figures/plotfft.pdf
new file mode 100644
index 0000000..c5e21e3
--- /dev/null
+++ b/buch/papers/reedsolomon/figures/plotfft.pdf
diff --git a/buch/papers/reedsolomon/figures/polynom2.pdf b/buch/papers/reedsolomon/figures/polynom2.pdf
new file mode 100644
index 0000000..55a50ac
--- /dev/null
+++ b/buch/papers/reedsolomon/figures/polynom2.pdf
diff --git a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex
index 39adbbf..8ad3d27 100644
--- a/buch/papers/reedsolomon/idee.tex
+++ b/buch/papers/reedsolomon/idee.tex
@@ -1,21 +1,32 @@
 %
-% teil1.tex -- Beispiel-File für das Paper
+% idee.tex -- Polynom Idee
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
 \section{Idee
 \label{reedsolomon:section:idee}}
 \rhead{Problemstellung}
+Um beim Datenübertragen Fehler zu erkennen, könnte man die Daten jeweils doppelt senden,
+und so jeweilige Fehler zu erkennen.
+Doch nur schon um Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach gesendet.
+Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise.
 Das Problem liegt darin Informationen, Zahlen, 
 zu Übertragen und Fehler zu erkennen.
 Beim Reed-Solomon-Code kann man nicht nur Fehler erkennen, 
 man kann sogar einige Fehler korrigieren.
+Der unterschied des Fehler erkennen und korrigiren, ist das beim Erkennen nur die Frage beantwortet wird mit: Ist die Übertragung fehlerhaft oder nicht?
+Beim Korrigieren werden Fehler erkennt und dann zusätzlich noch den original Wert rekonstruieren.
+Auch eine Variante wäre es die Daten nach einem Fehler nachdem Fehlerhaften senden, nochmals versenden(auch hier wieder doppelt und dreifach Sendung), 
+was bei Reed-Solomon-Code-Anwendungen nicht immer sinnvoll ist. 
+\externaldocument{papers/reedsolomon/anwendungen}
+\ref{reedsolomon:section:anwendung}
 
+\subsection{Polynom-Ansatz
+\label{reedsolomon:section:polynomansatz}}
 \rhead{Polynom-Ansatz}
-Eine Idee ist aus den Daten 
-ein Polynom zu bilden.
+Eine Idee ist aus den Daten ein Polynom zu bilden.
 Diese Polynomfunktion bei bestimmten Werten, ausrechnet und diese Punkte dann überträgt.
-Nehmen wir als beisbiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5},
+\begin{beispiel} Nehmen wir die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5},
 welche uns dann das Polynom 
 \begin{equation}
 p(x)
@@ -24,49 +35,63 @@ p(x)
 \label{reedsolomon:equation1}
 \end{equation}
 ergeben.
-Übertragen werden nun die Werte an den stellen 1, 2, 3\dots 7 dieses Polynomes.
+Übertragen werden nun die \textcolor{darkgreen}{grünen Werte} 
+dieses \textcolor{blue}{blauen Polynomes} an den Stellen 1, 2, 3\dots 7 dieses Polynomes.
 Grafisch sieht man dies dann in Abbildung \ref{fig:polynom}, 
-mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{green}{8}, 
-\textcolor{green}{15}, \textcolor{green}{26},
-\textcolor{green}{41}, \textcolor{green}{60}, 
-\textcolor{green}{83}, \textcolor{green}{110})$
-Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren.
+mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{darkgreen}{8}, 
+\textcolor{darkgreen}{15}, \textcolor{darkgreen}{26},
+\textcolor{darkgreen}{41}, \textcolor{darkgreen}{60}, 
+\textcolor{darkgreen}{83}, \textcolor{darkgreen}{110})$
+Wenn ein Fehler sich in die Übertragung eingeschlichen hat, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren.
 Der Leser/Empfänger weiss, den Grad des Polynoms und dessen Werte übermittelt wurden. 
+Die Farbe blau brauchen wir für die \textcolor{blue}{Daten} welche wir mit der Farbe grün \textcolor{darkgreen}{Übermitteln}.
+\end{beispiel}
 
-\subsection{Beispiel}
-Für das Beispeil aus der Gleichung \eqref{reedsolomon:equation1},
+\begin{beispiel}
+Aus der Gleichung \eqref{reedsolomon:equation1},
 ist ein Polynome zweiten Grades durch drei Punkte eindeutig bestimmbar.
 Hat es Fehler in der Übertragunge gegeben,(Bei Abbildung \ref{fig:polynom}\textcolor{red}{roten Punkte}) kann man diese erkennen,
 da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen. 
-(Bei Abbildung \ref{fig:polynom}\textcolor{green}{grünen Punkte})
+(Bei Abbildung \ref{fig:polynom}\textcolor{darkgreen}{grünen Punkte})
 Ab wie vielen Fehler ist das Polynom nicht mehr erkennbar beim Übertragen von 7 Punkten?
 Bei 2 Fehlern kann man noch eindeutig bestimmen, dass das Polynom mit 4 Punkten,
 gegenüber dem mit 5 Punkten falsch liegt.\ref{fig:polynom}
 Werden es mehr Fehler kann nur erkennt werden, dass das Polynom nicht stimmt.
 Das orginale Polynom kann aber nicht mehr gefunden werden.
-Dafür sind mehr übertragene Werte nötig.
+Da das Konkurenzpolynom, grau gestrichelt in Abbildung \ref{fig:polynom}, das orginal fehlleited.
+Um das Konkurenzpolynom auszuschliessen, währen mehr \textcolor{darkgreen}{Übertragungspunkte} nötig.
+\end{beispiel}
 
 \begin{figure}
 	\centering
-	%\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/polynom2}
-    \input{papers/reedsolomon/images/polynom2.tex}
-	\caption{Polynom $p(x)$ \eqref{reedsolomon:equation1}}
+	\includegraphics[width=\textwidth]{papers/reedsolomon/figures/polynom2}
+    %\input{papers/reedsolomon/tikz/polynom2.tex}
+	\caption{Polynom $p(x)$ von der Gleichung\eqref{reedsolomon:equation1}}
 	\label{fig:polynom}
 \end{figure}
 
-\section{Fehlerbestimmung
-\label{reedsolomon:section:Fehlerbestimmmung}}
-So wird ein Muster indentifiziert, welches genau vorherbestimmen kann,
-wie gross das Polynom sein muss und wie viele Übertragungspunkte gegeben werden müssen.
-Um zu bestimmen wie viel Fehler erkennt und korriegiert werden können.
-Die Anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast),
-die Entschlüsselt werden sollen, brauchen die gleiche Anzahl an  Polynomgraden, beginnend bei Grad 0. ( \( k-1 \) )
-Für die Anzahl an Übertragungspunkte, muss bestimmt werden wieviel Fehler erkennt und korrigiert werden sollen.
-Mit Hilfe der Tabelle, sieht man das es bei $t$ Fehlern und $k$ Nutzlast Zahlen,
-$k+2t$ Punkte übertragen werden müssen.
+\section{Fehlerkorekturstellen bestimmen
+\label{reedsolomon:section:Fehlerkorrekturstellen}}
+Um zu bestimmen wieviel zusätzliche \textcolor{darkgreen}{Übertragungspunkte} notwendig sind, die dann Fehler korrigieren,
+muss man zuerst Wissen wieviel \textcolor{blue}{Daten} gesendet und wieviel \textcolor{red}{Fehler} erkennt werden sollen. 
+Die Anzahl \textcolor{blue}{Daten} (ab hier verwenden wir das Wort Nutzlast), die als Polynomkoeffizente $k$ übergeben werden, 
+brauchen die gleiche Anzahl an Polynomgraden, beginnend bei Grad 0 somit ergibt sich der Polynomgrad mit $k-1$.
+Für die Anzahl der Fehler $t$, welche korrigiert werden können, gehen wir zum Beispiel.
+\begin{beispiel} von den Polynom \ref{reedsolomon:equation1} in, welchem wir 7 \textcolor{darkgreen}{Übertragungspunkte} senden.
+Durch 3 Punkte wird das Polyom eindeutig bestimmt, nun haben wir mehrere Konkurenzpolynome, doch mit maximal 2 Fehler liegen auf einem Konkurenzpolynom,
+maximal 4 Punkte und auf unserem orginal 5 Punkte. Ansonsten hatt es mehr Fehler oder unser Konkurenzpolynom ist das gleiche wie das Original. 
+Somit können wir nun bestimmen, dass von den \textcolor{darkgreen}{7 Übertragungspunkten$u$} bis zu 2 Fehler korrigiert werden können und 4 Übertragungspunkte zusätzlich gesendet werden müssen.
+\end{beispiel}
+Durch das erkennen des Schemas in der Tabelle\ref{tabel:fehlerkorrekturstellen}
+\begin{equation}
+    \frac{\textcolor{darkgreen}{u}-\textcolor{blue}{k}}{\textcolor{red}{t}}
+    =2
+    \label{reedsolomon:equation2}
+\end{equation}
+zeigt sich das es $k+2t$ Übertragungspunkte braucht.
 
 \begin{center}
-    \begin{tabular}{ c c c } 
+    \begin{tabular}{ c c | c} 
         \hline
         Nutzlas & Fehler & Übertragen \\
         \hline 
@@ -77,12 +102,11 @@ $k+2t$ Punkte übertragen werden müssen.
         $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ 
         \hline
     \end{tabular}
+    Fehlerkorrekturstellen Bestimmung TODO: Tabellenreferenz
+	\label{tabel:fehlerkorrekturstellen}
 \end{center}
 
-Ein toller Nebeneffekt ist das dadurch auch $2t$ Fehler erkannt werden. 
-Um zurück auf unser Beispiel zu kommen, 
-können von den 7 Übertragungspunkten bis zu $2t = 2\cdot2 = 4 $ Punkten falsch liegen 
-und es wird kein eindeutiges Polynom zweiten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert.
+Ein Nebeneffekt ist das dadurch auch $2t$ Fehler erkannt werden können, nicht aber korrigiert.
 Um aus den Übertragenen Zahlen wieder die Nutzlastzahlen zu bekommen könnte man eine Polynominterpolation anwenden,
 doch die Punkte mit Polynominterpolation zu einem Polynom zu rekonstruieren ist schwierig und Fehleranfällig.
 
diff --git a/buch/papers/reedsolomon/images/codiert.txt b/buch/papers/reedsolomon/images/codiert.txt
deleted file mode 100644
index 4a481d8..0000000
--- a/buch/papers/reedsolomon/images/codiert.txt
+++ /dev/null
@@ -1,96 +0,0 @@
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diff --git a/buch/papers/reedsolomon/images/decodiert.txt b/buch/papers/reedsolomon/images/decodiert.txt
deleted file mode 100644
index f6221e6..0000000
--- a/buch/papers/reedsolomon/images/decodiert.txt
+++ /dev/null
@@ -1,96 +0,0 @@
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diff --git a/buch/papers/reedsolomon/images/empfangen.txt b/buch/papers/reedsolomon/images/empfangen.txt
deleted file mode 100644
index 38c13b0..0000000
--- a/buch/papers/reedsolomon/images/empfangen.txt
+++ /dev/null
@@ -1,96 +0,0 @@
-0,284
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diff --git a/buch/papers/reedsolomon/images/fehler.txt b/buch/papers/reedsolomon/images/fehler.txt
deleted file mode 100644
index 23f1a83..0000000
--- a/buch/papers/reedsolomon/images/fehler.txt
+++ /dev/null
@@ -1,96 +0,0 @@
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diff --git a/buch/papers/reedsolomon/images/locator.txt b/buch/papers/reedsolomon/images/locator.txt
deleted file mode 100644
index b28988c..0000000
--- a/buch/papers/reedsolomon/images/locator.txt
+++ /dev/null
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diff --git a/buch/papers/reedsolomon/images/plotfft.tex b/buch/papers/reedsolomon/images/plotfft.tex
deleted file mode 100644
index 83a89eb..0000000
--- a/buch/papers/reedsolomon/images/plotfft.tex
+++ /dev/null
@@ -1,89 +0,0 @@
-%
-% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT
-%
-\begin{tikzpicture}[]
-
-%---------------------------------------------------------------
-	%Knote
-\matrix[draw = none, column sep=25mm, row sep=2mm]{
-	\node(signal)  []    {
-	\begin{tikzpicture}
-		\begin{axis}
-			[title = {\Large {Signal}}, 
-			xlabel={Anzahl Übertragene Zahlen},
-			xtick={0,20,40,64,80,98},]
-			\addplot[blue] table[col sep=comma] {papers/reedsolomon/images/signal.txt};
-		\end{axis}
-	\end{tikzpicture}}; &
-	
-	\node(codiert) []    {
-	\begin{tikzpicture}
-		\begin{axis}[title = {\Large {Codiert}}]
-			\addplot[] table[col sep=comma] {papers/reedsolomon/images/codiert.txt};
-		\end{axis}
-	\end{tikzpicture}}; \\
-		
-	&\node(fehler) []    {
-	\begin{tikzpicture}
-		\begin{axis}[scale=0.6, title = {\Large {Fehler}},
-			xtick={7,21,75}]
-			\addplot[red] table[col sep=comma] {papers/reedsolomon/images/fehler.txt};
-		\end{axis}
-	\end{tikzpicture}};\\
-	
-	\node(decodiert) []    {
-	\begin{tikzpicture}
-		\begin{axis}[title = {\Large {Decodiert}}]
-			\addplot[blue] table[col sep=comma] {papers/reedsolomon/images/decodiert.txt};
-		\end{axis}
-	\end{tikzpicture}}; &
-
-	\node(empfangen) []    {
-	\begin{tikzpicture}
-		\begin{axis}[title = {\Large {Empfangen}}]
-			\addplot[] table[col sep=comma] {papers/reedsolomon/images/empfangen.txt};
-		\end{axis}
-	\end{tikzpicture}};\\
-
-	\node(syndrom) []   {
-	\begin{tikzpicture}
-		\begin{axis}[title = {\Large {Syndrom}}]
-			\addplot[blue] table[col sep=comma] {papers/reedsolomon/images/syndrom.txt};
-		\end{axis}
-	\end{tikzpicture}}; &
-
-	\node(locator) []    {
-	\begin{tikzpicture}
-		\begin{axis}[title = {\Large {Locator}}]
-			\addplot[] table[col sep=comma] {papers/reedsolomon/images/locator.txt};
-		\end{axis}
-	\end{tikzpicture}};\\
-};
-%-------------------------------------------------------------
-	%FFT & IFFT deskription
-
-\draw[thin,gray,dashed] (0,12) to (0,-12);
-\node(IFFT)  [scale=0.7]  at (0,12.3)   {IFFT};
-\draw[<-](IFFT.south west)--(IFFT.south east);
-\node(FFT)  [scale=0.7, above of=IFFT]    {FFT};
-\draw[->](FFT.north west)--(FFT.north east);
-
-\draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5);
-%Arrows
-\draw[ultra thick, ->] (signal.east) to (codiert.west);
-\draw[ultra thick, ->] (codiert.south) to (fehler.north);
-\draw[ultra thick, ->] (fehler.south) to (empfangen.north);
-\draw[ultra thick, ->] (empfangen.west) to (decodiert.east);
-\draw[ultra thick, ->] (syndrom.east) to (locator.west);
-\draw(decodiert.south east)++(-1.8,1)  ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ;
-\draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north);
-
-%item
-\node[circle, draw, fill =lightgray] at (signal.north west) {1};
-\node[circle, draw, fill =lightgray] at (codiert.north west) {2};
-\node[circle, draw, fill =lightgray] at (fehler.north west) {3};
-\node[circle, draw, fill =lightgray] at (empfangen.north west) {4};
-\node[circle, draw, fill =lightgray] at (decodiert.north west) {5};
-\node[circle, draw, fill =lightgray] at (syndrom.north west) {6};
-\node[circle, draw, fill =lightgray] at (locator.north west) {7};
-\end{tikzpicture}
-\ No newline at end of file
diff --git a/buch/papers/reedsolomon/images/signal.txt b/buch/papers/reedsolomon/images/signal.txt
deleted file mode 100644
index c4fa5f8..0000000
--- a/buch/papers/reedsolomon/images/signal.txt
+++ /dev/null
@@ -1,96 +0,0 @@
-0,6
-1,6
-2,0
-3,6
-4,4
-5,0
-6,5
-7,2
-8,1
-9,2
-10,1
-11,2
-12,0
-13,6
-14,3
-15,5
-16,7
-17,5
-18,5
-19,4
-20,1
-21,5
-22,9
-23,9
-24,3
-25,2
-26,6
-27,6
-28,4
-29,2
-30,9
-31,1
-32,1
-33,1
-34,2
-35,6
-36,6
-37,1
-38,9
-39,7
-40,7
-41,1
-42,9
-43,9
-44,10
-45,9
-46,8
-47,5
-48,2
-49,4
-50,1
-51,0
-52,9
-53,3
-54,3
-55,3
-56,5
-57,6
-58,0
-59,8
-60,6
-61,9
-62,3
-63,4
-64,0
-65,0
-66,0
-67,0
-68,0
-69,0
-70,0
-71,0
-72,0
-73,0
-74,0
-75,0
-76,0
-77,0
-78,0
-79,0
-80,0
-81,0
-82,0
-83,0
-84,0
-85,0
-86,0
-87,0
-88,0
-89,0
-90,0
-91,0
-92,0
-93,0
-94,0
-95,0
diff --git a/buch/papers/reedsolomon/images/syndrom.txt b/buch/papers/reedsolomon/images/syndrom.txt
deleted file mode 100644
index 8ca9eed..0000000
--- a/buch/papers/reedsolomon/images/syndrom.txt
+++ /dev/null
@@ -1,96 +0,0 @@
-0,0
-1,0
-2,0
-3,0
-4,0
-5,0
-6,0
-7,0
-8,0
-9,0
-10,0
-11,0
-12,0
-13,0
-14,0
-15,0
-16,0
-17,0
-18,0
-19,0
-20,0
-21,0
-22,0
-23,0
-24,0
-25,0
-26,0
-27,0
-28,0
-29,0
-30,0
-31,0
-32,0
-33,0
-34,0
-35,0
-36,0
-37,0
-38,0
-39,0
-40,0
-41,0
-42,0
-43,0
-44,0
-45,0
-46,0
-47,0
-48,0
-49,0
-50,0
-51,0
-52,0
-53,0
-54,0
-55,0
-56,0
-57,0
-58,0
-59,0
-60,0
-61,0
-62,0
-63,0
-64,0.0275599094902563
-65,0.0115837187254191
-66,0.025877761014238
-67,0.0224618032819697
-68,0.04410594689944
-69,0.0474504002669341
-70,0.0227694695500626
-71,0.0271436638090525
-72,0.0104166666666667
-73,0.0271436638090523
-74,0.0227694695500608
-75,0.0474504002669343
-76,0.0441059468994397
-77,0.0224618032819701
-78,0.0258777610142379
-79,0.0115837187254183
-80,0.027559909490256
-81,0.0245124379481793
-82,0.0499782237195209
-83,0.0401432022864265
-84,0.0232923747656228
-85,0.0237974288564099
-86,0.0143895905726624
-87,0.0271745729691685
-88,0.0275599094902567
-89,0.0515501672184983
-90,0.0358255004834542
-91,0.024700508366373
-92,0.0210194725405171
-93,0.0177592928994296
-94,0.0261327016093158
-95,0.0314909067039411
diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex
index ab4e4be..017fe94 100644
--- a/buch/papers/reedsolomon/main.tex
+++ b/buch/papers/reedsolomon/main.tex
@@ -8,29 +8,9 @@
 \begin{refsection}
 \chapterauthor{Joshua Bär und Michael Steiner}
 
-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}
-
 % Joshua
 \input{papers/reedsolomon/einleitung.tex}
 \input{papers/reedsolomon/idee.tex}
-%\input{papers/reedsolomon/teil2.tex}
 \input{papers/reedsolomon/dtf.tex}
 
 % Michael
diff --git a/buch/papers/reedsolomon/packages.tex b/buch/papers/reedsolomon/packages.tex
index b84e228..40c6ea3 100644
--- a/buch/papers/reedsolomon/packages.tex
+++ b/buch/papers/reedsolomon/packages.tex
@@ -10,3 +10,5 @@
 
 \usepackage{pgfplots}
 \usepackage{filecontents}
+\usepackage{xr}
+
diff --git a/buch/papers/reedsolomon/standalone.tex b/buch/papers/reedsolomon/standalone.tex
new file mode 100644
index 0000000..c850d1f
--- /dev/null
+++ b/buch/papers/reedsolomon/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/reedsolomon/main.tex}
+\end{document}
diff --git a/buch/papers/reedsolomon/standalone/standalone.pdf b/buch/papers/reedsolomon/standalone/standalone.pdf
new file mode 100644
index 0000000..1f2f0b9
--- /dev/null
+++ b/buch/papers/reedsolomon/standalone/standalone.pdf
diff --git a/buch/papers/reedsolomon/experiments/codiert.txt b/buch/papers/reedsolomon/tikz/codiert.txt
index 4a481d8..4a481d8 100644
--- a/buch/papers/reedsolomon/experiments/codiert.txt
+++ b/buch/papers/reedsolomon/tikz/codiert.txt
diff --git a/buch/papers/reedsolomon/experiments/decodiert.txt b/buch/papers/reedsolomon/tikz/decodiert.txt
index f6221e6..f6221e6 100644
--- a/buch/papers/reedsolomon/experiments/decodiert.txt
+++ b/buch/papers/reedsolomon/tikz/decodiert.txt
diff --git a/buch/papers/reedsolomon/experiments/empfangen.txt b/buch/papers/reedsolomon/tikz/empfangen.txt
index 38c13b0..38c13b0 100644
--- a/buch/papers/reedsolomon/experiments/empfangen.txt
+++ b/buch/papers/reedsolomon/tikz/empfangen.txt
diff --git a/buch/papers/reedsolomon/experiments/fehler.txt b/buch/papers/reedsolomon/tikz/fehler.txt
index 23f1a83..23f1a83 100644
--- a/buch/papers/reedsolomon/experiments/fehler.txt
+++ b/buch/papers/reedsolomon/tikz/fehler.txt
diff --git a/buch/papers/reedsolomon/experiments/locator.txt b/buch/papers/reedsolomon/tikz/locator.txt
index b28988c..b28988c 100644
--- a/buch/papers/reedsolomon/experiments/locator.txt
+++ b/buch/papers/reedsolomon/tikz/locator.txt
diff --git a/buch/papers/reedsolomon/tikz/plotfft.tex b/buch/papers/reedsolomon/tikz/plotfft.tex
new file mode 100644
index 0000000..14af683
--- /dev/null
+++ b/buch/papers/reedsolomon/tikz/plotfft.tex
@@ -0,0 +1,94 @@
+%
+% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{pgfplots}
+\usepackage{pgfplotstable}
+\usepackage{csvsimple}
+\usepackage{filecontents}
+
+
+\begin{document}
+\begin{tikzpicture}[]
+
+	%---------------------------------------------------------------
+	%Knote
+	\matrix(m) [draw = none, column sep=25mm, row sep=2mm]{
+
+		\node(signal)  []    {
+		\begin{tikzpicture}
+			\begin{axis}
+				[title = {\Large {Signal}}, 
+				xtick={0,20,40,64,80,98}]
+				\addplot[blue] table[col sep=comma] {tikz/signal.txt};
+			\end{axis}
+		\end{tikzpicture}}; &
+		
+		\node(codiert) []    {
+		\begin{tikzpicture}[]
+			\begin{axis}[ title = {\Large {Codiert \space + \space Fehler}},
+				xtick={0,40,60,100}, axis y line*=left]
+				\addplot[green] table[col sep=comma] {tikz/codiert.txt};
+			\end{axis}
+			\begin{axis}[xtick={7,21,75}, axis y line*=right]
+					\addplot[red] table[col sep=comma] {tikz/fehler.txt};
+			\end{axis}
+		\end{tikzpicture}}; \\
+		
+		\node(decodiert) []    {
+		\begin{tikzpicture}
+			\begin{axis}[title = {\Large {Decodiert}}]
+				\addplot[blue] table[col sep=comma] {tikz/decodiert.txt};
+			\end{axis}
+		\end{tikzpicture}}; &
+	
+		\node(empfangen) []    {
+		\begin{tikzpicture}
+			\begin{axis}[title = {\Large {Empfangen}}]
+				\addplot[green] table[col sep=comma] {tikz/empfangen.txt};
+			\end{axis}
+		\end{tikzpicture}};\\
+	
+		\node(syndrom) []   {
+		\begin{tikzpicture}
+			\begin{axis}[title = {\Large {Syndrom}}]
+				\addplot[black] table[col sep=comma] {tikz/syndrom.txt};
+			\end{axis}
+		\end{tikzpicture}}; &
+	
+		\node(locator) []    {
+		\begin{tikzpicture}
+			\begin{axis}[title = {\Large {Locator}}]
+				\addplot[gray] table[col sep=comma] {tikz/locator.txt};
+			\end{axis}
+		\end{tikzpicture}};\\
+	};
+	%-------------------------------------------------------------
+		%FFT & IFFT deskription
+	
+	\draw[thin,gray,dashed] (0,9) to (0,-9);
+	\node(IFFT)  [scale=0.8]  at (0,9.3)   {IFFT};
+	\draw[stealth-](IFFT.south west)--(IFFT.south east);
+	\node(FFT)  [scale=0.8, above of=IFFT]    {FFT};
+	\draw[-stealth](FFT.north west)--(FFT.north east);
+	
+	\draw[thick, ->,] (codiert)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5);
+	%Arrows
+	\draw[thick, ->] (signal.east) to (codiert.west);
+	\draw[thick, ->] (codiert.south) to (empfangen.north);
+	\draw[thick, ->] (empfangen.west) to (decodiert.east);
+	\draw[thick, ->] (syndrom.east) to (locator.west);
+	\draw[thick](decodiert.south east)++(-1.8,1)  ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ;
+	\draw[thick, ->] (zoom) to[out=180, in=90] (syndrom.north);
+	
+	%item
+	\node[circle, draw, fill =lightgray] at (signal.north west) {1};
+	\node[circle, draw, fill =lightgray] at (codiert.north west) {2+3};
+	\node[circle, draw, fill =lightgray] at (empfangen.north west) {4};
+	\node[circle, draw, fill =lightgray] at (decodiert.north west) {5};
+	\node[circle, draw, fill =lightgray] at (syndrom.north west) {6};
+	\node[circle, draw, fill =lightgray] at (locator.north west) {7};
+\end{tikzpicture}	
+\end{document}
+\ No newline at end of file
diff --git a/buch/papers/reedsolomon/images/polynom2.tex b/buch/papers/reedsolomon/tikz/polynom2.tex
index 288b51c..47dc679 100644
--- a/buch/papers/reedsolomon/images/polynom2.tex
+++ b/buch/papers/reedsolomon/tikz/polynom2.tex
@@ -1,5 +1,13 @@
 % polynome
 %-------------------
+
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{pgfplots}
+
+
+\begin{document}
 % Teiler für das Skalieren der Grafik /40
 \newcommand{\teiler}{40}
 
@@ -21,9 +29,14 @@
 
 	\def\hellpunkt#1{
 		\fill[color=lightgray] #1 circle[radius=0.08];
-		\draw #1 circle[radius=0.07];
+		\draw[gray] #1 circle[ radius=0.07];
 	}
 	
+	\draw[color=gray,line width=1pt,dashed] 
+	plot[domain=0.5:7, samples=100]
+	({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler});
+
+
 	\punkt{(1,8/\teiler)}
 	\hellpunkt{(2,15/\teiler)}
 	\hellpunkt{(3,26/\teiler)}
@@ -32,9 +45,7 @@
 	\punkt{(6,83/\teiler)}
 	\punkt{(7,110/\teiler)}
 	
-	\draw[color=gray,line width=1pt,dashed] 
-	plot[domain=0.5:7, samples=100]
-	({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler});
+
 	
 	\def\erpunkt#1{
 		\fill[color=red] #1 circle[radius=0.08];
@@ -46,4 +57,4 @@
 	\draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}];
 	\draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}];		
 \end{tikzpicture}
-%\end{document}
+\end{document}
diff --git a/buch/papers/reedsolomon/experiments/signal.txt b/buch/papers/reedsolomon/tikz/signal.txt
index c4fa5f8..c4fa5f8 100644
--- a/buch/papers/reedsolomon/experiments/signal.txt
+++ b/buch/papers/reedsolomon/tikz/signal.txt
diff --git a/buch/papers/reedsolomon/experiments/syndrom.txt b/buch/papers/reedsolomon/tikz/syndrom.txt
index 8ca9eed..8ca9eed 100644
--- a/buch/papers/reedsolomon/experiments/syndrom.txt
+++ b/buch/papers/reedsolomon/tikz/syndrom.txt
diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex
index b1588ff..8e0d36d 100644
--- a/buch/papers/spannung/Einleitung.tex
+++ b/buch/papers/spannung/Einleitung.tex
@@ -1,17 +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.
+In diesem Kapitel geht es darum, das Hook'sche Gesetz im Dreidimensionalen zu beschreiben.
 Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen.
 In jedem erdenklichen Punkt im Dreidimensionalen herrscht daher ein entsprechender individueller Spannungszustand.
 Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus.
-Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen.
+Darum werden Vektoren, Matrizen und Tensoren zu Hilfe gezogen.
 Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden.
 Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch.
 
-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.
-In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein,
+Um die mathematischen und physikalischen Berechnungen anwenden zu können,
+müssen vorerst ein paar spezifische Bedingungen vorausgesetzt und Annahmen getroffen werden.
+Ebenfalls gilt es, ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen.
+In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannungen, 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}}
@@ -29,7 +30,7 @@ Belastet man den Boden mit einer Spannung
 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.
+Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{fig:Bild4} breitet sich die Zusatzspannung zwiebelartig aus.
 
 \begin{figure}
 	\centering
@@ -38,11 +39,11 @@ Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bi
 	\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.
+Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{fig:Bild5}).
+Wie diese Geometrie der Ausbreitung aussieht, 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.
+Das können beispielsweise verschiedene Bodenkennwerte oder auch der Wassergehalt sein.
 
 \begin{figure}
 	\centering
@@ -72,18 +73,18 @@ berechnet werden mit:
 	          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.
+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~\ref{spannung:Bild3}).
+Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{fig: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 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.
+Generell wird im Bauingenieurwesen oder auch im Maschinenbau versucht, manche Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können.
 
 \begin{figure}
 	\centering
 	\includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png}
-	\caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat}
+	\caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welche kompliziertere Spannungsausbreitung zur Folge hat}
 	\label{fig:Bild3}
 \end{figure}
diff --git a/buch/papers/spannung/main.tex b/buch/papers/spannung/main.tex
index bbdf730..d2aeda9 100644
--- a/buch/papers/spannung/main.tex
+++ b/buch/papers/spannung/main.tex
@@ -3,7 +3,7 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:spannung}}
+\chapter{Dreidimensionaler Spannungszustand\label{chapter:spannung}}
 \lhead{Dreiachsiger Spannungszustand}
 \begin{refsection}
 \chapterauthor{Adrian Schuler und Thomas Reichlin}
diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex
index 7647252..089c28e 100644
--- a/buch/papers/spannung/teil0.tex
+++ b/buch/papers/spannung/teil0.tex
@@ -1,9 +1,10 @@
 \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}).
+Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{fig: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.
+Um alle Spannungen eines Punktes darstellen zu können,
+stellt man sich modellhaft ein infinitesimales Bodenelement in Form eines Würfels vor.
+Man spricht auch von einem Elementarwürfel.
 
 \begin{figure}
 	\centering
@@ -15,19 +16,19 @@ Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist.
 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 $1$, $2$, $3$.
+Das infinitesimale Bodenteilchen hat die Koordinatenachsen $1$, $2$, $3$.
 Veränderungen der Normalspannungen können durch Schubspannungen kompensiert werden und umgekehrt.
-So sind insgesamt neun verschiedene Spannungen möglich, wobei drei Normal- und sechs Schubspannungen sind.
+So sind insgesamt neun verschiedene Spannungen möglich, konkret sind dies drei Normal- und sechs Schubspannungen.
 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.
+Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetzes berechnet werden.
 Daher gibt es auch den entsprechenden Dehnungszustand.
 
 
 \section{Spannungszustand\label{spannung:section:Spannungsustand}}
 \rhead{Spannungszustand}
 
-Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{spannung:Bild1}).
+Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{fig:Bild1}).
 Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand.
 Nach Hooke gilt:
 \[
@@ -59,7 +60,7 @@ mit
 	  A &= \text{Fläche [\si{\meter\squared}].}
 \end{align*}
 Diese Beziehung gilt bei linear-elastischen Materialien, welche reversible Verformungen zulassen.
-Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$.
+Es ist praktisch, die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$.
 \begin{figure}
 	\centering
 	\includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png}
@@ -73,10 +74,10 @@ Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich
 E\cdot\varepsilon
 \]
 beschreiben.
-Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch
+Im Falle, dass $E$ nicht konstant ist, wird dieser durch
 \[
 E
 =
-\frac{\Delta\sigma}{\Delta\varepsilon}
+\frac{\text{d}\sigma}{\text{d}\varepsilon}
 \]
-ausgedrückt werden.
-\ No newline at end of file
+ausgedrückt.
+\ No newline at end of file
diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex
index 74516c1..647b452 100644
--- a/buch/papers/spannung/teil1.tex
+++ b/buch/papers/spannung/teil1.tex
@@ -1,8 +1,8 @@
 \section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}}
 \rhead{Skalare, Vektoren, Matrizen und Tensoren}
-Der Begriff Tensor kann als Überbegriff, der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden.
+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.
+Skalare, Vektoren oder Matrizen sind daher auch Tensoren.
 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.
@@ -14,11 +14,10 @@ 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.
+In den 40er Jahren vom 19. Jahrhundert wurde der Begriff Tensor 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 6326eab..8620afe 100644
--- a/buch/papers/spannung/teil2.tex
+++ b/buch/papers/spannung/teil2.tex
@@ -3,7 +3,7 @@
 Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen.
 \begin{figure}
 	\centering
-	\includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png}
+	\includegraphics[width=0.30\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png}
 	\caption{Beispiel eines Spannungszustandes; Vergrösserung eines infinitesimalen Bodenteilchen}
 	\label{fig:infinitesimalerWuerfel}
 \end{figure}
@@ -49,7 +49,7 @@ Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\
 dargestellt werden und beschreibt den gesamten Dehnungszustand.
 
 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.
+Man darf Zeile um Zeile in eine Spalte notieren, sodass es einen Spaltenvektor ergibt.
 So ergibt sich der Spannungsvektor
 
 \[
@@ -79,7 +79,7 @@ So ergibt sich der Spannungsvektor
 	\sigma_{33}
 \end{pmatrix}
 \]
-und Dehnungsvektor
+und der Dehnungsvektor
 \[
 \overline{\varepsilon}
 =
@@ -140,14 +140,6 @@ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{
 \end{pmatrix}
 \]
 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:
 \[
 \vec\sigma
@@ -155,8 +147,7 @@ Die allgemeine Spannungsgleichung lautet nun:
 \overline{\overline{C}}\cdot\vec{\varepsilon}
 .
 \]
-
-Als Indexnotation
+Sie kann ebenfalls als Indexnotation
 \[
 \sigma_{ij}
 =
@@ -164,7 +155,15 @@ Als Indexnotation
 \sum_{l=1}^3
 C_{ijkl}\cdot\varepsilon_{kl}
 \]
-kann dies ebenfalls geschrieben werden.
+geschrieben werden.
+Der Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein.
+Folglich gilt:
+\[
+\overline{\overline{C}}
+=
+\overline{\overline{C}}~^{T}
+.
+\]
 
 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,
@@ -221,7 +220,7 @@ definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich
 \end{pmatrix}
 .
 \]
-Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als
+Die Normalspannung $\sigma_{22}$ lässt sich zum Beispiel als
 \[
 \sigma_{22}
 =
@@ -229,11 +228,13 @@ Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als
 \]
 berechnen.
 
+Reduzierte Spannungs- und Dehnungsgleichungen
+
 Man betrachte nun die Eigenschaften des Elastizitätstensors.
 Dieser ist quadratisch und symmetrisch, die verschiedenen Einträge wechseln sich aber miteinander ab.
 Es ergeben sich keine Blöcke mit einheitlichen Einträgen.
 
-Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch Elastizitätstensor symmetrisch sind.
+Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch der Elastizitätstensor symmetrisch sind.
 Wäre dem nicht so, würde sich das Material je nach Richtung unterschiedlich elastisch verhalten.
 Diese Symmetrie setzt daher voraus, dass
 \[
@@ -399,7 +400,7 @@ Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit
 \]
 beschreiben.
 Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert.
-Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist:
+Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist:
 \begin{equation}
 \begin{pmatrix}
 	\sigma_{11}\\
@@ -433,7 +434,7 @@ Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist:
 
 Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also,
 dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben.
-Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben.
+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.
 
 Man betrachte als Beispiel die Berechnung von $\sigma_{33}$.
@@ -441,8 +442,8 @@ Es ist ersichtlich, dass die Schubdehnungen keinen Einfluss auf $\sigma_{33}$ ha
 Der Einfluss der zu $\sigma_{33}$ äquivalenten Dehnung $\varepsilon_{33}$ hat den grössten Einfluss.
 Die anderen Normalspannungen $\sigma_{11}$ und $\sigma_{22}$ haben einen unter anderem mit $\nu$ korrigierten Einfluss.
 
-Von  $\overline{\overline{C}}$ bildet man noch die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$ um die Gleichung umstellen zu können.
-Dadurch erhält man die Dehnungsgleichung:
+Von  $\overline{\overline{C}}$ bildet man die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$, mithilfe des Gauss - Jordan Algorithmus, um die Gleichung umstellen zu können.
+Durch einige Berechnungsschritte erhält man die Dehnungsgleichung:
 
 \[
 \vec{\varepsilon}
diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex
index 3e456c3..a9080ea 100644
--- a/buch/papers/spannung/teil3.tex
+++ b/buch/papers/spannung/teil3.tex
@@ -30,7 +30,7 @@ q
 \label{spannung:Invariante_q}
 .
 \end{equation}
-Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt.
+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
 \[
 p
@@ -38,28 +38,28 @@ p
 \frac{\sigma_{11}+2\sigma_{33}}{3}
 \]
 vereinfacht werden.
-Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q}als
+Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q} als
 \[
 q
 =
 \sigma_{11}-\sigma_{33}
 \]
-vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten.
+vereinfacht. Man kann $p$ als Druck und $q$ als Schub betrachten.
 
-Die Invarianten können mit der Spannungsformel \eqref{spannung:Spannungsgleichung} berechnet werden.
+Die Invarianten $p$ und $q$ können mit der Spannungsgleichung \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
 \[
-\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p}
+\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{\displaystyle{p}}
 =
-\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{v}}
+\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\displaystyle{{\varepsilon_{v}}}}
 \]
 und
 \[
-\overbrace{\sigma_{11}-\sigma_{33}}^{q}
+\overbrace{\sigma_{11}-\sigma_{33}}^{\displaystyle{q}}
 =
-\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{s}}
+\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\displaystyle{\varepsilon_{s}}}
 .
 \]
 Die Faktoren mit den Dehnungskomponenten können so mit
@@ -79,8 +79,8 @@ eingeführt werden, mit
 	\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.
+Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression und
+die deviatorische Dehnung $\varepsilon_{s}$  mit einer Verzerrung verglichen werden.
 
 Diese zwei Gleichungen kann man durch die Matrixschreibweise
 \begin{equation}
@@ -90,8 +90,8 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise
 \end{pmatrix}
 =
 \begin{pmatrix}
-	\frac{3E}{2(1+\nu)} &                   0 \\
-	                  0 & \frac{E}{3(1-2\nu)}
+	\displaystyle{\frac{3E}{2(1+\nu)}} &                                  0 \\
+	                                 0 & \displaystyle{\frac{E}{3(1-2\nu)}}
 \end{pmatrix}
 \begin{pmatrix}
 	\varepsilon_{s}\\
@@ -100,9 +100,11 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise
 \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.
 
+Änderungen des Spannungszustandes können mit diesen Gleichungen vollumfänglich erfasst werden.
+Diese Spannungsgleichung mit den zwei Einträgen ($p$ und $q$) ist gleichwertig
+wie die ursprüngliche Spannungsgleichung mit den neun Einträgen
+($\sigma_{11}$, $\sigma_{12}$, $\sigma_{13}$, $\sigma_{21}$, $\sigma_{22}$, $\sigma_{23}$, $\sigma_{31}$, $\sigma_{32}$, $\sigma_{33}$).
 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.
+Ein solcher Versuch, der 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 2f2e4ce..00b2d4f 100644
--- a/buch/papers/spannung/teil4.tex
+++ b/buch/papers/spannung/teil4.tex
@@ -1,6 +1,6 @@
-\section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}}
-\rhead{Oedometer-Versuch}
-Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden.
+\section{Oedometrischer Elastizitätsmodul\label{spannung:section:Oedometrischer Elastizitätsmodul}}
+\rhead{Oedometrischer Elastizitätsmodul}
+Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{\text{OED}}$ 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.
@@ -43,8 +43,8 @@ Diese lautet nun:
 \end{pmatrix}
 =
 \begin{pmatrix}
-	\frac{E_{OED}}{(1+\nu)} &                             0 \\
-                          0 & \frac{E_{OED}}{3(1-2\nu)}
+	\displaystyle{\frac{E_{\text{OED}}}{(1+\nu)}} &                                               0 \\
+                                                0 & \displaystyle{\frac{E_{\text{OED}}}{3(1-2\nu)}}
 \end{pmatrix}
 \begin{pmatrix}
 	\varepsilon_{11}\\
@@ -52,28 +52,28 @@ Diese lautet nun:
 \end{pmatrix}
 .
 \]
-Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen
+Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{\text{OED}}$ und die seitlichen Spannungen $\sigma_{33}$ mit den zwei Gleichungen
 \[
 \sigma_{11}-\sigma_{33}
 =
-\frac{E_{OED}}{(1+\nu)}\cdot\varepsilon_{11}
+\frac{E_{\text{OED}}}{(1+\nu)}\cdot\varepsilon_{11}
 \]
 und
 \[
 \sigma_{11}+2\sigma_{33}
 =
-\frac{E_{OED}}{3(1-2\nu)}\cdot\varepsilon_{11}
+\frac{E_{\text{OED}}}{3(1-2\nu)}\cdot\varepsilon_{11}
 \]
 berechnen.
-Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen.
+Mit diesen Gleichungen hat man das Gleichungssystem um $E_{\text{OED}}$ 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}).
+Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{fig: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.
+Mit diesem ermittelten $E_{\text{OED}}$ kann man nun weitere Berechnungen für die Geotechnik durchführen.
 
 \begin{figure}
 	\centering
-	\includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png}
+	\includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png}
 	\caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material}
 	\label{fig:DiagrammOedometer-Versuch}
 \end{figure}
 \ No newline at end of file