Merge remote-tracking branch 'upstream/master' into Baer

105 files changed, 49870 insertions, 437 deletions

diff --git a/buch/chapters/10-vektorenmatrizen/linear.tex b/buch/chapters/10-vektorenmatrizen/linear.tex
index e368364..3ad51f1 100644..100755
--- a/buch/chapters/10-vektorenmatrizen/linear.tex
+++ b/buch/chapters/10-vektorenmatrizen/linear.tex
@@ -33,7 +33,7 @@ aber mit Punkten kann man trotzdem noch nicht rechnen.
 Ein Vektor fasst die Koordinaten eines Punktes in einem Objekt zusammen,
 mit dem man auch rechnen und zum Beispiel Parallelverschiebungen
 algebraisieren kann.
-Um auch Streckungen ausdrücken zu können, wird auch eine Menge von 
+Um auch Streckungen ausdrücken zu können, wird auch eine Menge von
 Streckungsfaktoren benötigt, mit denen alle Komponenten eines Vektors
 multipliziert werden können.
 Sie heissen auch {\em Skalare} und liegen in $\Bbbk$.
@@ -73,7 +73,7 @@ a+b
 =
 \begin{pmatrix}\lambda a_1\\\vdots\\\lambda a_n\end{pmatrix}.
 \]
-Die üblichen Rechenregeln sind erfüllt, nämlich 
+Die üblichen Rechenregeln sind erfüllt, nämlich
 \begin{equation}
 \begin{aligned}
 &\text{Kommutativität:}
@@ -149,7 +149,7 @@ kann als (abstrakter) Vektor betrachtet werden.
 \begin{definition}
 Eine Menge $V$ von Objekten, auf der zwei Operationen definiert,
 nämlich die Addition, geschrieben $a+b$ für $a,b\in V$ und die
-Multiplikation mit Skalaren, geschrieben $\lambda a$ für $a\in V$ und 
+Multiplikation mit Skalaren, geschrieben $\lambda a$ für $a\in V$ und
 $\lambda\in \Bbbk$, heisst ein {\em $\Bbbk$-Vektorraum} oder {\em Vektorraum
 über $\Bbbk$} (oder
 einfach nur {\em Vektorraum}, wenn $\Bbbk$ aus dem Kontext klar sind),
@@ -172,7 +172,7 @@ $\mathbb{C}$ ein Vektorraum über $\mathbb{R}$.
 \end{beispiel}
 
 \begin{beispiel}
-Die Menge $C([a,b])$ der stetigen Funktionen $[a,b]\to\mathbb{Re}$ 
+Die Menge $C([a,b])$ der stetigen Funktionen $[a,b]\to\mathbb{Re}$
 bildet ein Vektorraum.
 Funktionen können addiert und mit reellen Zahlen multipliziert werden:
 \[
@@ -188,7 +188,7 @@ Die Vektorraum-Rechenregeln
 \end{beispiel}
 
 Die Beispiele zeigen, dass der Begriff des Vektorraums die algebraischen
-Eigenschaften eine grosse Zahl sehr verschiedenartiger mathematischer 
+Eigenschaften eine grosse Zahl sehr verschiedenartiger mathematischer
 Objekte beschreiben kann.
 Alle Erkenntnisse, die man ausschliesslich aus Vekotorraumeigenschaften
 gewonnen hat, sind auf alle diese Objekte übertragbar.
@@ -300,7 +300,7 @@ folgt, dass alle $\lambda_1,\dots,\lambda_n=0$ sind.
 Lineare Abhängigkeit der Vektoren $a_1,\dots,a_n$ bedeutet auch, dass
 man einzelne der Vektoren durch andere ausdrücken kann.
 Hat man nämlich eine
-Linearkombination~\eqref{buch:vektoren-und-matrizen:eqn:linabhdef} und 
+Linearkombination~\eqref{buch:vektoren-und-matrizen:eqn:linabhdef} und
 ist der Koeffizient $\lambda_k\ne 0$, dann kann man nach $a_k$ auflösen:
 \[
 a_k = -\frac{1}{\lambda_k}(\lambda_1a_1+\dots+\widehat{\lambda_ka_k}+\dots+\lambda_na_n).
@@ -323,7 +323,7 @@ offenbar eine besondere Bedeutung.
 Eine linear unabhängig Menge von Vektoren
 $\mathcal{B}=\{a_1,\dots,a_n\}\subset V$
 heisst {\em Basis} von $V$.
-Die maximale Anzahl linear unabhängiger Vektoren in $V$ heisst 
+Die maximale Anzahl linear unabhängiger Vektoren in $V$ heisst
 {\em Dimension} von $V$.
 \end{definition}
 
@@ -331,7 +331,7 @@ Die Standardbasisvektoren bilden eine Basis von $V=\Bbbk^n$.
 
 \subsubsection{Unterräume}
 Die Mengen $\langle a_1,\dots,a_n\rangle$ sind Teilmengen
-von $V$, in denen die Addition von Vektoren und die Multiplikation mit 
+von $V$, in denen die Addition von Vektoren und die Multiplikation mit
 Skalaren immer noch möglich ist.
 
 \begin{definition}
@@ -352,7 +352,7 @@ gilt.
 %
 \subsection{Matrizen
 \label{buch:grundlagen:subsection:matrizen}}
-Die Koeffizienten eines linearen Gleichungssystems finden in einem 
+Die Koeffizienten eines linearen Gleichungssystems finden in einem
 Zeilen- oder Spaltenvektor nicht Platz.
 Wir erweitern das Konzept daher in einer Art, dass Zeilen- und
 Spaltenvektoren Spezialfälle sind.
@@ -378,14 +378,14 @@ M_{m\times n}(\Bbbk) = \{ A\;|\; \text{$A$ ist eine $m\times n$-Matrix}\}.
 \]
 Falls $m=n$ gilt, heisst die Matrix $A$ auch {\em quadratisch}
 \index{quadratische Matrix}%
-Man kürzt die Menge der quadratischen Matrizen als 
+Man kürzt die Menge der quadratischen Matrizen als
 $M_n(\Bbbk) = M_{n\times n}(\Bbbk)$ ab.
 \end{definition}
 
-Die $m$-dimensionalen Spaltenvektoren $v\in \Bbbk^m$ sind $m\times 1$-Matrizen 
+Die $m$-dimensionalen Spaltenvektoren $v\in \Bbbk^m$ sind $m\times 1$-Matrizen
 $v\in M_{n\times 1}(\Bbbk)$, die $n$-dimensionalen Zeilenvetoren $u\in\Bbbk^n$
 sind $1\times n$-Matrizen $v\in M_{1\times n}(\Bbbk)$.
-Eine $m\times n$-Matrix $A$ mit den Koeffizienten $a_{ij}$ besteht aus 
+Eine $m\times n$-Matrix $A$ mit den Koeffizienten $a_{ij}$ besteht aus
 den $n$ Spaltenvektoren
 \[
 a_1 = \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix},\quad
@@ -435,7 +435,7 @@ werden kann.
 \begin{definition}
 Eine $m\times n$-Matrix $A\in M_{m\times n}(\Bbbk)$ und eine
 $n\times l$-Matrix $B\in M_{n\times l}(\Bbbk)$ haben als Produkt
-eine $n\times l$-Matrix $C=AB\in M_{n\times l}(\Bbbk)$ mit den
+eine $m\times l$-Matrix $C=AB\in M_{m\times l}(\Bbbk)$ mit den
 Koeffizienten
 \begin{equation}
 c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}.
@@ -483,7 +483,7 @@ I
 1     &0     &\dots &0     \\
 0     &1     &\dots &0     \\[-2pt]
 \vdots&\vdots&\ddots&\vdots\\
-0     &0     &\dots &1  
+0     &0     &\dots &1
 \end{pmatrix}.
 \]
 
@@ -521,10 +521,10 @@ Ein Gleichungssystem mit $0$ auf der rechten Seite ist also bereits
 ausreichend um zu entscheiden, ob die Lösung eindeutig ist.
 Ein Gleichungssystem mit rechter Seite $0$ heisst {\em homogen}.
 \index{homogenes Gleichungssystem}%
-Zu jedem {\em inhomogenen} Gleichungssystem $Ax=b$ mit $b\ne 0$ 
+Zu jedem {\em inhomogenen} Gleichungssystem $Ax=b$ mit $b\ne 0$
 ist $Ax=0$ das zugehörige homogene Gleichungssystem.
 
-Ein homogenes Gleichungssytem $Ax=0$ hat immer mindestens die 
+Ein homogenes Gleichungssytem $Ax=0$ hat immer mindestens die
 Lösung $x=0$, man nennt sie auch die {\em triviale} Lösung.
 Eine Lösung $x\ne 0$ heisst auch eine nichttriviale Lösung.
 Die Lösungen eines inhomgenen Gleichungssystem $Ax=b$ ist also nur dann
@@ -535,7 +535,7 @@ Lösung hat.
 Der Gauss-Algorithmus oder genauer Gausssche Eliminations-Algorithmus
 löst ein lineare Gleichungssystem der
 Form~\eqref{buch:vektoren-und-matrizen:eqn:vektorform}.
-Die Koeffizienten werden dazu in das Tableau 
+Die Koeffizienten werden dazu in das Tableau
 \[
 \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
 \hline
@@ -552,7 +552,7 @@ Der Algorithmus is so gestaltet, dass er nicht mehr Speicher als
 das Tableau benötigt, alle Schritte operieren direkt auf den Daten
 des Tableaus.
 
-In jedem Schritt des Algorithmus wird zunächst eine Zeile $i$ und 
+In jedem Schritt des Algorithmus wird zunächst eine Zeile $i$ und
 Spalte $j$ ausgewählt, das Elemente $a_{ij}$ heisst das Pivotelement.
 \index{Pivotelement}%
 Die {\em Pivotdivision}
@@ -646,7 +646,7 @@ In der Phase der {\em Vorwärtsreduktion} werden Pivotelemente von links
 nach rechts möglichst auf der Diagonale gewählt und mit Zeilensubtraktionen
 die darunterliegenden Spalten freigeräumt.
 \index{Vorwärtsreduktion}%
-Während des Rückwärtseinsetzens werden die gleichen Pivotelemente von 
+Während des Rückwärtseinsetzens werden die gleichen Pivotelemente von
 rechts nach links genutzt, um mit Zeilensubtraktionen auch die
 Spalten über den Pivotelemnten frei zu räumen.
 \index{Rückwärtseinsetzen}%
@@ -800,7 +800,7 @@ $x = b_1c_1+b_2c_2+\dots+b_nc_n$ konstruieren.
 Tatsächlich gilt
 \begin{align*}
 Ax
-&= 
+&=
 A( b_1c_1+b_2c_2+\dots+b_nc_n)
 \\
 &=
@@ -851,10 +851,10 @@ für eine Gleichungssystem mit quadratischer Koeffizientenmatrix $A$
 heisst die Determinante $\det(A)$ der Matrix $A$.
 \end{definition}
 
-Aus den Regeln für die Durchführung des Gauss-Algorithmus kann man die 
+Aus den Regeln für die Durchführung des Gauss-Algorithmus kann man die
 folgenden Regeln für die Determinante ableiten.
 Wir stellen die Eigenschaften hier nur zusammen, detaillierte Herleitungen
-kann man in jedem Kurs zur linearen Algebra finden, zum Beispiel im 
+kann man in jedem Kurs zur linearen Algebra finden, zum Beispiel im
 Kapitel~2 des Skripts \cite{buch:linalg}.
 \begin{enumerate}
 \item
@@ -877,11 +877,11 @@ wird auch der Wert der Determinanten mit $\lambda$ multipliziert.
 \item
 \label{buch:linear:determinante:asymetrisch}
 Die Determinante ist eine lineare Funktion der Zeilen von $A$.
-Zusammen mit der Eigeschaft~\ref{buch:linear:determinante:vorzeichen} 
+Zusammen mit der Eigeschaft~\ref{buch:linear:determinante:vorzeichen}
 folgt, dass die Determinante eine antisymmetrische lineare Funktion
 der Zeilen ist.
 \item
-Die Determinante ist durch die Eigenschaften 
+Die Determinante ist durch die Eigenschaften
 \ref{buch:linear:determinante:einheitsmatrix}
 und
 \ref{buch:linear:determinante:asymetrisch}
@@ -895,7 +895,7 @@ Die Determinante der $n\times n$-Matrix $A$ kann mit der Formel
 =
 \sum_{i=1}^n (-1)^{i+j} a_{ij} \cdot \det(A_{ij})
 \end{equation}
-wobei die $(n-1)\times(n-1)$-Matrix $A_{ij}$ die Matrix $A$ ist, aus der 
+wobei die $(n-1)\times(n-1)$-Matrix $A_{ij}$ die Matrix $A$ ist, aus der
 man Zeile $i$ und Spalte $j$ entfernt hat.
 $A_{ij}$ heisst ein {\em Minor} der Matrix $A$.
 \index{Minor einer Matrix}%
@@ -949,7 +949,7 @@ der rechten Seiten ersetzt worden ist.
 \end{satz}
 
 Die Cramersche Formel ist besonders nützlich, wenn die Abhängigkeit
-einer Lösungsvariablen von den Einträgen der Koeffizientenmatrix 
+einer Lösungsvariablen von den Einträgen der Koeffizientenmatrix
 untersucht werden soll.
 Für die Details der Herleitung sei wieder auf \cite{buch:linalg}
 verwiesen.
@@ -993,7 +993,7 @@ heisst die {\em Adjunkte} $\operatorname{adj}A$ von $A$.
 \end{satz}
 
 Der Satz~\ref{buch:linalg:inverse:adjoint} liefert eine algebraische
-Formel für die Elemente der inversen Matrix. 
+Formel für die Elemente der inversen Matrix.
 Für kleine Matrizen wie im nachfolgenden Beispiel ist die
 Formel~\eqref{buch:linalg:inverse:formel} oft einfachter anzuwenden.
 Besonders einfach wird die Formel für eine $2\times 2$-Matrix,
@@ -1035,7 +1035,7 @@ Die Adjunkte ist
 \begin{pmatrix*}[r]
  \det A_{11} & -\det A_{21} &  \det A_{31} \\
 -\det A_{12} &  \det A_{22} & -\det A_{32} \\
- \det A_{13} & -\det A_{23} &  \det A_{33} 
+ \det A_{13} & -\det A_{23} &  \det A_{33}
 \end{pmatrix*}
 \intertext{und damit ist die inverse Matrix}
 A^{-1}
@@ -1084,7 +1084,7 @@ A^{-1}
 \end{pmatrix}.
 \label{buch:vektoren-und-matrizen:abeispiel:eqn2}
 \end{equation}
-für die Inverse einer Matrix der Form 
+für die Inverse einer Matrix der Form
 \eqref{buch:vektoren-und-matrizen:abeispiel:eqn1}.
 \end{beispiel}
 
@@ -1118,7 +1118,7 @@ Eine Abbildung $f\colon V\to U$ zwischen Vektorräumen $V$ und $U$
 heisst linear, wenn
 \[
 \begin{aligned}
-f(v+w) &= f(v) + f(w)&&\forall v,w\in V 
+f(v+w) &= f(v) + f(w)&&\forall v,w\in V
 \\
 f(\lambda v) &= \lambda f(v) &&\forall v\in V,\lambda \in \Bbbk
 \end{aligned}
@@ -1129,16 +1129,16 @@ gilt.
 Lineare Abbildungen sind in der Mathematik sehr verbreitet.
 
 \begin{beispiel}
-Sie $V=C^1([a,b])$ die Menge der stetig differenzierbaren Funktionen 
+Sie $V=C^1([a,b])$ die Menge der stetig differenzierbaren Funktionen
 auf dem Intervall $[a,b]$ und $U=C([a,b])$ die Menge der
-stetigen Funktion aif $[a,b]$. 
+stetigen Funktion aif $[a,b]$.
 Die Ableitung $\frac{d}{dx}$ macht aus einer Funktion $f(x)$ die
 Ableitung $f'(x)$.
-Die Rechenregeln für die Ableitung stellen sicher, dass 
+Die Rechenregeln für die Ableitung stellen sicher, dass
 \[
 \frac{d}{dx}
 \colon
-C^1([a,b]) \to  C([a,b]) 
+C^1([a,b]) \to  C([a,b])
 :
 f \mapsto f'
 \]
@@ -1157,7 +1157,7 @@ eine lineare Abbildung.
 \end{beispiel}
 
 \subsubsection{Matrix}
-Um mit linearen Abbildungen rechnen zu können, ist eine Darstellung 
+Um mit linearen Abbildungen rechnen zu können, ist eine Darstellung
 mit Hilfe von Matrizen nötig.
 Sei also $\mathcal{B}=\{b_1,\dots,b_n\}$ eine Basis von $V$ und
 $\mathcal{C} = \{ c_1,\dots,c_m\}$ eine Basis von $U$.
@@ -1165,12 +1165,12 @@ Das Bild des Basisvektors $b_i$ kann als Linearkombination der
 Vektoren $c_1,\dots,c_m$ dargestellt werden.
 Wir verwenden die Bezeichnung
 \[
-f(b_i) 
+f(b_i)
 =
 a_{1i} c_1 + \dots + a_{mi} c_m.
 \]
 Die lineare Abbildung $f$ bildet den Vektor $x$ mit Koordinaten
-$x_1,\dots,x_n$ ab auf 
+$x_1,\dots,x_n$ ab auf
 \begin{align*}
 f(x)
 &=
@@ -1193,7 +1193,7 @@ x_n(a_{1n} c_1 + \dots + a_{mn} c_m)
 +
 ( a_{m1} x_1 + \dots + a_{mn} x_n ) c_m
 \end{align*}
-Die Koordinaten von $f(x)$ in der Basis $\mathcal{C}$ in $U$ sind 
+Die Koordinaten von $f(x)$ in der Basis $\mathcal{C}$ in $U$ sind
 also gegeben durch das Matrizenprodukt $Ax$, wenn $x$ der Spaltenvektor
 aus den Koordinaten in der Basis $\mathcal{B}$ in $V$ ist.
 
@@ -1231,7 +1231,7 @@ b_{m1}x_1&+& \dots &+&b_{mn}x_n&=&b_{m1}'x_1'&+& \dots &+&b_{mn}'x_n'
 \end{linsys}
 \]
 Dieses Gleichungssystem kann man mit Hilfe eines Gauss-Tableaus lösen.
-Wir schreiben die zugehörigen Variablen 
+Wir schreiben die zugehörigen Variablen
 \[
 \renewcommand{\arraystretch}{1.1}
 \begin{tabular}{|>{$}c<{$} >{$}c<{$} >{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}|}
@@ -1277,7 +1277,7 @@ Für zwei Vektoren $u$ und $w$ in $U$ gibt es daher Vektoren $a=g(u)$
 und $b=g(w)$ in $V$ derart, dass $f(a)=u$ und $f(b)=w$.
 Weil $f$ linear ist, folgt daraus $f(a+b)=u+w$ und $f(\lambda a)=\lambda a$
 für jedes $\lambda\in\Bbbk$.
-Damit kann man jetzt 
+Damit kann man jetzt
 \begin{align*}
 g(u+w)&=g(f(a)+f(b)) = g(f(a+b)) = a+b = g(u)+g(w)
 \\
@@ -1315,7 +1315,7 @@ Der Kern oder Nullraum der Matrix $A$ ist die Menge
 \]
 \end{definition}
 
-Der Kern ist ein Unterraum, denn für zwei Vektoren $u,w\in \ker f$ 
+Der Kern ist ein Unterraum, denn für zwei Vektoren $u,w\in \ker f$
 \[
 \begin{aligned}
 f(u+v)&=f(u) + f(v) = 0+0 = 0 &&\Rightarrow& u+v&\in\ker f\\
@@ -1331,7 +1331,7 @@ Wir definieren daher das Bild einer linearen Abbildung oder Matrix.
 
 \begin{definition}
 Ist $f\colon V\to U$ eine lineare Abbildung dann ist das Bild von $f$
-der Unterraum 
+der Unterraum
 \[
 \operatorname{im}f = \{ f(v)\;|\;v\in V\} \subset U
 \]
@@ -1375,7 +1375,7 @@ $\operatorname{def}A=\dim\ker A$.
 \end{definition}
 
 Da der Kern mit Hilfe des Gauss-Algorithmus bestimmt werden kann,
-können Rang und Defekt aus dem Schlusstableau 
+können Rang und Defekt aus dem Schlusstableau
 eines homogenen Gleichungssystems mit $A$ als Koeffizientenmatrix
 abgelesen werden.
 
@@ -1391,8 +1391,3 @@ n-\operatorname{def}A.
 
 \subsubsection{Quotient}
 TODO: $\operatorname{im} A \simeq \Bbbk^m/\ker A$
-
-
-
-
-
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},
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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
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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
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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
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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
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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 @@
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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 @@
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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
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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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diff --git a/buch/papers/multiplikation/code/meas/winograd.txt b/buch/papers/multiplikation/code/meas/winograd.txt
new file mode 100644
index 0000000..3a4d88b
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/winograd.txt
@@ -0,0 +1,11 @@
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diff --git a/buch/papers/multiplikation/code/meas_1024.pdf b/buch/papers/multiplikation/code/meas_1024.pdf
new file mode 100644
index 0000000..fd0a108
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_1024.pdf
diff --git a/buch/papers/multiplikation/code/meas_1024.txt b/buch/papers/multiplikation/code/meas_1024.txt
new file mode 100644
index 0000000..c5ce619
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_1024.txt
@@ -0,0 +1,6 @@
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new file mode 100644
index 0000000..ed1ec63
--- /dev/null
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diff --git a/buch/papers/multiplikation/code/meas_128.txt b/buch/papers/multiplikation/code/meas_128.txt
new file mode 100644
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index 0000000..4d8f04b
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new file mode 100644
index 0000000..3a90949
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diff --git a/buch/papers/multiplikation/code/meas_8.pdf b/buch/papers/multiplikation/code/meas_8.pdf
new file mode 100644
index 0000000..16d177d
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_8.pdf
diff --git a/buch/papers/multiplikation/code/meas_8.txt b/buch/papers/multiplikation/code/meas_8.txt
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diff --git a/buch/papers/multiplikation/code/test.tex b/buch/papers/multiplikation/code/test.tex
new file mode 100644
index 0000000..40ea239
--- /dev/null
+++ b/buch/papers/multiplikation/code/test.tex
@@ -0,0 +1,92 @@
+% This file was created by tikzplotlib v0.9.8.
+\begin{tikzpicture}
+
+\definecolor{color0}{rgb}{0.886274509803922,0.290196078431373,0.2}
+\definecolor{color1}{rgb}{0.203921568627451,0.541176470588235,0.741176470588235}
+\definecolor{color2}{rgb}{0.596078431372549,0.556862745098039,0.835294117647059}
+\definecolor{color3}{rgb}{0.984313725490196,0.756862745098039,0.368627450980392}
+
+\begin{axis}[
+axis background/.style={fill=white!89.8039215686275!black},
+axis line style={white},
+legend cell align={left},
+legend style={
+  fill opacity=0.8,
+  draw opacity=1,
+  text opacity=1,
+  at={(0.03,0.97)},
+  anchor=north west,
+  draw=white!80!black,
+  fill=white!89.8039215686275!black
+},
+tick align=outside,
+tick pos=left,
+x grid style={white},
+xlabel={n},
+xmajorgrids,
+xmin=-4.3, xmax=134.3,
+xtick style={color=white!33.3333333333333!black},
+y grid style={white},
+ylabel={time (s)},
+ymajorgrids,
+ymin=-0.0834965705871582, ymax=1.75356960296631,
+ytick style={color=white!33.3333333333333!black}
+]
+\addplot [line width=2pt, color0]
+table {%
+2 1.57356262207031e-05
+4 5.96046447753906e-05
+8 0.000428915023803711
+16 0.00276041030883789
+32 0.0217020511627197
+64 0.160412073135376
+128 1.3419406414032
+};
+\addlegendentry{Standard MM}
+\addplot [line width=2pt, color1]
+table {%
+2 6.43730163574219e-06
+4 6.69956207275391e-05
+8 0.00048065185546875
+16 0.00336766242980957
+32 0.0257236957550049
+64 0.231612205505371
+128 1.67006659507751
+};
+\addlegendentry{Divide and conquer MM}
+\addplot [line width=2pt, color2]
+table {%
+2 2.90870666503906e-05
+4 0.000133275985717773
+8 0.000703096389770508
+16 0.00453472137451172
+32 0.0282893180847168
+64 0.181003332138062
+128 1.40816903114319
+};
+\addlegendentry{Strassen MM}
+\addplot [line width=2pt, white!46.6666666666667!black]
+table {%
+2 2.19345092773438e-05
+4 9.01222229003906e-05
+8 0.000406503677368164
+16 0.00258469581604004
+32 0.0171687602996826
+64 0.126588344573975
+128 1.02698183059692
+};
+\addlegendentry{Winograd MM}
+\addplot [line width=2pt, color3]
+table {%
+2 1.45435333251953e-05
+4 1.1444091796875e-05
+8 7.39097595214844e-06
+16 1.28746032714844e-05
+32 2.83718109130859e-05
+64 0.000111103057861328
+128 0.000159025192260742
+};
+\addlegendentry{np MM}
+\end{axis}
+
+\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)
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+     \vdots \&  \& \vdots \& \& \vdots \\
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+     };
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+  	 \node [right=0.1 of A] (mul) {$\cdot$};
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+	 
+     \matrix (B)[right=0.1 of mul, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
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+  	 \node [right=0.1 of B] (eq) {$=$};
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+      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)] {};
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+       \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}
+
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+\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}}
+\headcommand {\beamer@framepages {5}{6}}
+\headcommand {\slideentry {1}{0}{3}{7/8}{}{0}}
+\headcommand {\beamer@framepages {7}{8}}
+\headcommand {\slideentry {1}{0}{4}{9/10}{}{0}}
+\headcommand {\beamer@framepages {9}{10}}
+\headcommand {\slideentry {1}{0}{5}{11/12}{}{0}}
+\headcommand {\beamer@framepages {11}{12}}
+\headcommand {\slideentry {1}{0}{6}{13/13}{}{0}}
+\headcommand {\beamer@framepages {13}{13}}
+\headcommand {\slideentry {1}{0}{7}{14/14}{}{0}}
+\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}}
+\headcommand {\slideentry {2}{0}{1}{15/15}{}{0}}
+\headcommand {\beamer@framepages {15}{15}}
+\headcommand {\slideentry {2}{0}{2}{16/18}{}{0}}
+\headcommand {\beamer@framepages {16}{18}}
+\headcommand {\slideentry {2}{0}{3}{19/19}{}{0}}
+\headcommand {\beamer@framepages {19}{19}}
+\headcommand {\slideentry {2}{0}{4}{20/20}{}{0}}
+\headcommand {\beamer@framepages {20}{20}}
+\headcommand {\slideentry {2}{0}{5}{21/23}{}{0}}
+\headcommand {\beamer@framepages {21}{23}}
+\headcommand {\slideentry {2}{0}{6}{24/24}{}{0}}
+\headcommand {\beamer@framepages {24}{24}}
+\headcommand {\slideentry {2}{0}{7}{25/25}{}{0}}
+\headcommand {\beamer@framepages {25}{25}}
+\headcommand {\slideentry {2}{0}{8}{26/26}{}{0}}
+\headcommand {\beamer@framepages {26}{26}}
+\headcommand {\slideentry {2}{0}{9}{27/29}{}{0}}
+\headcommand {\beamer@framepages {27}{29}}
+\headcommand {\slideentry {2}{0}{10}{30/32}{}{0}}
+\headcommand {\beamer@framepages {30}{32}}
+\headcommand {\beamer@sectionpages {15}{32}}
+\headcommand {\beamer@subsectionpages {15}{32}}
+\headcommand {\sectionentry {3}{Measurements}{33}{Measurements}{0}}
+\headcommand {\slideentry {3}{0}{1}{33/40}{}{0}}
+\headcommand {\beamer@framepages {33}{40}}
+\headcommand {\slideentry {3}{0}{2}{41/49}{}{0}}
+\headcommand {\beamer@framepages {41}{49}}
+\headcommand {\beamer@sectionpages {33}{49}}
+\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}}
+\headcommand {\beamer@subsectionpages {50}{50}}
+\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
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+\usepackage{textcomp}
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+\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
@@ -0,0 +1,245 @@
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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
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf
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
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.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{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}
+
+\begin{tikzpicture}[ampersand replacement=\&]
+
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+  {
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+     \& \& \& \\
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+
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+ \node at (-3,-5) {$C_{11}=$} ;
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+ \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};
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diff --git a/buch/papers/verkehr/main.tex b/buch/papers/verkehr/main.tex
index 6348993..98d0581 100644
--- a/buch/papers/verkehr/main.tex
+++ b/buch/papers/verkehr/main.tex
@@ -3,8 +3,7 @@
 %
 % (c) 2020 Hochschule Rapperswil
 %
-\chapter{Thema\label{chapter:verkehr}}
-\lhead{Verkehrsfluss und Verkehrsnetze}
+\chapter{Verkehrsfluss und Verkehrsnetze\label{chapter:verkehr}}
 \begin{refsection}
 \chapterauthor{Pascal Andreas Schmid und Robine Luchsinger}
 
diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex
index d96d450..416e311 100644
--- a/buch/papers/verkehr/section1.tex
+++ b/buch/papers/verkehr/section1.tex
@@ -1,99 +1,75 @@
-\section{Einführung}
 \label{section:verkehr/einfuehrung}
 
-\subsection{Verkehrsnetze}
 Das Verkehrsnetz besteht aus allen Anlagen, auf oder unter der Erdoberfläche, auf denen eine räumliche Fortbewegung von Personen oder auch Gütern stattfindet. Verkehrsnetze sind ein Bestandteil der Verkehrsinfrastruktur, die auf topografischen Karten festgehalten werden. Sie umfassen den Schienenverkehr, alle Strassen und Wege, wie auch Flugplätze und alle dazugehörigen Bauwerke.
 Aus verkehrsgeografischer Sicht besteht das Verkehrsnetz aus Kanten, Knotenpunkten und dem Hinterland. Die Knotenpunkte werden auch hier durch die Kanten verbunden, die den Verkehrsstrom aufnehmen, wobei das Hinterland durch einzelne Knoten versorgt wird. Die Aufteilung in Kanten und Knotenpunkte ermöglicht eine Vereinfachung komplexer Verkehrsnetze, damit sie mittels der Graphentheorie untersucht werden können.
 Grundsätzlich können kurze Wege zwischen den Knotenpunkten das Ziel beim Aufbau eines Verkehrsnetzes sein. Es kann aber auch versucht werden, die Bau- und Unterhaltskosten des Verkehrsnetzes in einem gewissen Rahmen zu halten. Aus diesen Vorgaben ergibt sich dann, je nach dem was gewünscht wird, eine grob- oder feinmaschige Struktur des Netzes.
 Ziel ist aber ein möglichst wirtschaftliches und optimales Verkehrsnetz.
 
-\subsection{Suchalgorithmen}
+\section{Suchalgorithmen}
+Inbesondere bei Graphen in Form von Verkehrsnetzen ist das Finden eines kürzesten Weges von Interesse. Mathematisch betrachtet handelt es sich hierbei um ein Optimierungsproblem, bei dem die Summe der Kantengewichte zwischen zwei Knoten minimiert werden soll. Zu diesem Zweck existieren verschiedene Suchalgorithmen. In den folgenden Abschnitten wird auf eine Auswahl davon eingegangen. Zuvor ist es jedoch notwendig, einige Begriffe und Eigenschaften von Suchalgorithmen zu definieren.
 
-\subsubsection{Dijkstra-Algorithmus}
-Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Infomratikprofessor Edsger Dijkstra. Den Algorithmus hat er im Jahr 1959 erfunden.
-Der Algorithmus von Dijkstra ist ein Greedy-Algorithmus (gieriger Algorithmus), der schrittweise einen Folgezustand auswählt, damit beim Zeitpunkt der Wahl der grösste Gewinn bzw. das beste Ergebnis erzielt werden kann.
-Trotz der Schnelligkeit der Greedy-Algorithmen, können viele Probleme nicht optimal gelöst werden.
-Vereinfacht wird beim Dijkstra-Algorithmus, ausgehend von einem Startknoten so lange dem kürzesten Pfad gefolgt, bis der Zielknoten erreicht wird. Dabei muss für jeden besuchten Knoten die Kostenfunktion als auch der Pfad dahin (vorheriger Knoten) gespeichert werden.
-Dadurch wird hingegen garantiert, dass, wenn der Zielknoten erreicht wird, auch der kürzeste Pfad gefunden wurde.
-Grundlegende Voraussetzung für den Dijkstra-Algorithmus ist die strikte Positivität der Kantengewichte. Andernfalls würde ein wiederholtes Ablaufen einer Kante mit negativem Gewicht zu einer stetigen Reduktion der Kostenfunktion führen, was zu einer unendlichen Schlaufe führen würde.
+Einerseits wird zwischen optimalen und nicht-optimalen Algorithmen unterschieden. Ein Suchalgorithmus gilt als optimal, falls er einen günstigsten Pfad zwischen zwei Knoten findet. Es gilt zu beachten, dass im Falle des Vorhandenseins von mehrerern Pfaden mit identischer, minimaler Summe der Kantengewichte zwischen zwei Knoten, mindestens einer dieser Pfade gefunden wird.
 
-Gegeben sei ein Netzwerk mit $n$ Knoten und dem Startknoten $a$.
-Alle Kanten sind mit $k(i, j)$ bewertet.
-Gesucht wird der kürzeste Pfad zwischen dem Startknoten und allen übrigen Knoten im Netz.
-$D(i)$ ist die kürzeste Distanz vom Startknoten $a$ zum Knoten $i, V(i)$ ist der unmittelbare Vorgängerknoten vom Knoten $i$ auf dem kürzesten Weg vom Startknoten $a$ zum Konten $i$ und die Menge $M$ ist die Menge einer bestimmten Auswahl an Knoten.
+Weiter wird zwischen informierten und uninformierten Algorithmen differenziert. Während uninformierte Suchalgorithmen den Suchraum schematisch auf Basis der Eigenschaften des Graphen absuchen, bis eine günstigste Lösung gefunden wurde, verwenden informierte Suchalgorithmen eine Heuristik zur Abschätzung der Suchrichtung. Oftmals wird bei informierten Algorithmen ein Verlust der Optimalität zugunsten einer verbesserten Rechenzeit in Kauf genommen. Es exisitieren jedoch auch Heurstiken, die eine optimale Lösung gewährleisten.
 
-Dabei gilt
-\begin{equation}M={a}\end{equation}
-\begin{equation}D(a)=0\end{equation} wobei
-\begin{equation}D(i)=\infty\end{equation} und
-\begin{equation}i \neq a \end{equation}
-Ausserdem gilt \begin{equation}V(i)=(-) \text{für alle Knoten $i$}\end{equation}\\
+Eine besondere Art von Suchalgorithmen stellen die sogenannten Greedy-Algorithmen, zu deutsch gierige Algorithmen, dar. Sie zeichnen sich dadurch aus, dass sie stets den zurzeit günstigsten Folgezustand auswählen. Dadurch sind sie in der Regel äusserst effizient, garantieren bei vielen Problemstellungen jedoch keine optimale Lösung.
 
-%THEORIE...
-Iteration
+\subsection{Dijkstra-Algorithmus}
+Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Informatikprofessor Edsger Dijkstra. Er gehört zur Klasse der uninformierten Greedy-Algorithmen. Zudem ist die Optimalität bei strikt positiven Kantengewichten gewährleistet.
+Vorteilhaft ist die einfache Implementierung. Abhängig von der Programmiersprache sind zwischen 30 und 40 Zeilen an Code ausreichend, damit er den kürzesten Pfad zwischen einem Startknoten $a$ und Zielknoten $b$ finden kann. 
 
-1. Auswahl eines Knotens \begin{equation} K\in M \text{mit} D(K)=D(i);i\in M\end{equation}
+Die für dieses Paper verwendete programmierte Funktion (MATLAB) verwendet eine abgewandelte Form der gewichteten Adjazenz-Matrix $A$, für welche gilt:
+Der Matrix-Eintrag $A_{i,j}$ enthält das Kantengewicht der Kante von Knoten $j$ nach $i$ auf. Falls keine Kante zwischen $j$ und $i$ vorhanden ist, beträgt der Eintrag $\infty$. Dies vereinfacht die Implementierung zur Bestimmung des nächst-günstigsten Pfades.
+Zudem werden zwei Hilfs-Vektoren $\vec{d}$ und $\vec{b}$ der Länge $n$ eingeführt, wobei $n$ die Anzahl Knoten des Graphen ist. Im Vektoreintrag $\vec{d}(i)$ wird das kummulierte Kantengewicht zur Erreichung von Knoten $i$ vom Startknoten $a$ gespeichert. Der Eintrag $\vec{d}(a)$ beträgt somit $0$. Im Vektor $\vec{b}$ wird zudem vermerkt, falls ein Knoten bereits als Ziel eines kürzesten Pfads gefunden wurde und somit für die weitere Suche nicht mehr berücksichtigt werden muss ($\vec{b}(i)=1$, sonst $\vec{b}(i)=0$).
 
-2. Für alle Nachfolger $N(j)$ vom Knoten $K$ gilt:
-\begin{equation}D(K) + k_Kj < D(j)\end{equation} dann wird \begin{equation}D(j) = D(K) + k_Kj, V(j) = K\end{equation} gesetzt und somit wird der Knoten $j$ in die Menge $M$ aufgenommen.
+Ausgehend vom Startknoten $a$ wird nun anhand der Matrix $A$ in der Spalte $a$ nach dem kleinsten Eintrag gesucht. Somit wird der Folgeknoten $c$ gefunden. Dieser Vorgang wird nun wiederholt, wobei jedoch sämtliche von Knoten $a$ und $c$ erreichbaren Knoten berücksichtigt werden, die noch nicht besucht wurden. In anderen Worten alle nicht verschwindenden Einträge $i$ der Spalten $a$ und $c$ der Matrix $A$, für welche gilt $\vec{b}(i)=0$. Ausschlaggebend für die folgende Auswahl ist die Summe der kummulierten Kantengewichte und des Kantengewichts des nächsten Knotens. Als Beispiel zur Erreichung von Knoten $k$ über Knoten $j$:
+\begin{equation}
+\vec{d}(k)=\vec{d}(j)+A(k,j)
+\end{equation}
+Diese Iteration wird solange durchgeführt, bis der Folgeknoten dem Zielknoten entspricht.
 
-3. Der ausgewählte Knoten \begin{equation}K\in M\text{wird aus der Menge herausgelöscht}\end{equation}\\
-Diese drei Schritte werden so lange wiederholt bis gilt
-\begin{equation}M=\{\}\end{equation}
+\subsection{A*-Algorithmus}
+Der A*-Algorithmus basiert auf dem Dijkstra-Algorithmus, verwendet jedoch eine Heuristik zur Abschätzung der günstigsten Suchrichtung. Somit handelt es sich um einen informierten Greedy-Algorithmus, der abhängig von der verwendeten Heuristik auch optimal sein kann. Er wurde von Peter Hart, Nils Nilsson und Bertram Raphael entwickelt.
 
-\subsubsection{A*-Algorithmus}
-Suchalgorithmen werden nach einfachen (uninformierte) und heuristischen (informierten) Algorithmen unterschieden. Während einfache Algorithmen den Suchraum intuitiv durchsuchen, beziehen heuristische Algorithmen Wissen über den Suchraum mit ein.
-Der A*-Algorithmus geht auf seine Erfinder Peter Hart, Nils Nilsson und Bertram Raphael zurück, die den Algorithmus erstmals im Jahr 1968 beschrieben.
-Der A*-Algorithmus ist ein heuristischer Suchalgorithmus, der den kürzesten Pfad zwischen zwei Knoten in einem Graphen mit positiven Kantengewichten berechnet.
-Im Gegensatz zu einfachen Suchalgorithmen, wird beim A*-Algorithmus eine Schätzfunktion, die sogenannte Heuristik, verwendet. Dies ermöglicht ein zielgerichtetes Suchen und gleichzeitig wird die Laufzeit verringert.
-Ausserdem findet der A*-Algorithmus immer eine optimale Lösung, sofern eine vorhanden ist.
-Der A*-Algorithmus wird als Verallgemeinerung gehandhabt und gilt als Erweiterung des Dijkstra-Algorithmus.
+\subsection{Anwendung A*-Algorithmus}
+Wie oben erwähnt basiert der A*-Algorithmus auf dem Shortest-Path-Algorithmus von Dijkstra. Gemäss dem Algorihtmus von Dijkstra werden von einem Startknoten aus die jeweiligen Nachbarknoten, die Nachbarknoten der Nachbarknoten usw. verarbeitet. Die Kantengewichte werden dabei aufsummiert und die Priorität wird auf die Kante gelegt, die das geringste Gewicht aufweist. Mit diesem Verfahren wird sichergestellt, dass die erste gefundene Lösung auch eine optimale Lösung darstellt.\\
 
-\subsubsection{Anwendung A*-Algorithmus}
-Wie oben erwähnt basiert der A*-Algorithmus auf dem Shortest-Path-Algorithmus von Dijkstra. Gemäss dem Algorihtmus von Dijkstra werden von einem Startknoten aus die jeweiligen Nachbarknoten, die Nachbarknoten der Nachbarknoten usw. verarbeitet. Die Kantengewichte werden dabei aufsummiert und die Priorität wird auf die Kante gelegt, die das geringste Gewicht aufweist. Mit diesem Verfahren wird sichergestellt, dass die erste gefundene Lösung auch die optimalste Lösung darstellt.\\
+Der A*-Algorithmus unterscheidet sich vom Dijkstra-Algorithmus dahingehend, dass bei der Auswahl des Folgeknotens, nicht nur die Summe der Kantengewichte $\vec{d}(j)+A(k,j)$, sondern zusätzlich die für jeden Knoten definierte Abschätzfunktion $f(k)$ hinzuaddiert wird. Dies passiert jedoch nur bei der \emph{Auswahl} des Folgeknotens. Der Wert von $f(k)$ wird nicht im Eintrag $\vec{d}(k)$ gespeichert. Somit wird gewährleistet, dass der gefundene Pfad, der Summe der Kantengewichte entspricht. Ein Beispiel dafür, wie eine Abschätzfunktion gebildet werden kann findet sich in Abschnitt \ref{sec:verkehr/euklidische}
 
-Die Kantengewichte werden für jeden Knoten in Form einer Funktion dargestellt
-\begin{equation}f(n)=g(n)\end{equation} mit
-\begin{equation}g(n)=\text{Summe aller Kantengewichte vom Startknoten bis n}\end{equation}\\
-Der A*-Algorithmus erweitert die Vorgehensweise des Algorithmus von Dijkstra um die Heuristik $h(n)$, die für jeden Knoten $n$ die geschätzte Entfernung zum Zielknoten beschreibt.
-Somit gilt:
-\begin{equation}f(n)=g(n)+h(n)\end{equation}\\
-Wie auch der Algorithmus von Dijkstra findet der A*-Algorithmus die optimalste Lösung.
+\subsection{Euklidische Heuristik}
+\label{sec:verkehr/euklidische}
+Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar. Unter Verwendung dieser Heuristik gilt der A*-Algorithmus als optimal.
 
-\subsubsection{Floyd-Warshall-Algorithmus}
-Der Floyd-Warshall-Algorithmus, auch Tripel-Algorithmus genannt, wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt.
-Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die kürzesten , beziehungsweise die optimalsten Wege zwischen allen Paaren von Knoten berechnet. Der Floyd-Warhshall-Algrithmus kann ausserdem mit negativen Kantengewichten umgehen, sofern der Graph aber keinen negativen Kreis (Zyklus) aufweist. Ist dies der Fall, führt der Algorithmus zu einem falschen Ergebnis.
-Ein Kreis (Zyklus) in einem Graphen ist ein Weg, bei dem Start- und Endpunkt den gleichen Knoten aufweisen. Dieser wird negativ, wenn die Summe der gewichteten Kanten kleiner als Null wird.\\
-Der Floyd-Warshall-Algorithmus besteht grundsätzlich aus Floyd's Berechnung der kürzesten Distanzen zwischen zwei Knoten und Warshall's Konstruktion der kürzesten Wege. Werden diese beiden Teilgebiete zusammengefügt, ergibt sich der Floyd-Warshall-Algorithmus.
+Bei der euklidischen Heuristik wird die Abschätzfunktion $f(k)$ für jeden Knoten $k$ durch euklidische Distanz zum Zielknoten $b$ gebildet.
+\begin{equation}
+f(k)=\sqrt{(x_k-x_b)^2+(y_k-y_b)^2}
+\end{equation}
+
+Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant. Hier können hingegen andere Eigenschaften des Netzwerks verwendet werden, auf welche in diesem Paper nicht weiter eingegangen wird.
 
-\subsubsection{Anwendung Floyd-Warshall-Algorithmus}
+\subsection{Floyd-Warshall-Algorithmus}
+Der Floyd-Warshall-Algorithmus, auch Tripel-Algorithmus genannt, wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt.
+Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die günstigsten Wege zwischen allen Paaren von Knoten berechnet. Der Floyd-Warhshall-Algrithmus kann ausserdem mit negativen Kantengewichten umgehen, sofern der Graph keinen negativen Kreis (Zyklus) aufweist. Ein Kreis, sprich ein Weg mit identischem Start- und Zielknoten, ist negativ, falls die Summe der Kantengewichte des Weges kleiner als null ist. Ist dies der Fall, führt der Algorithmus zu einem falschen Ergebnis.
 
-Wie oben erwähnt, besteht der Floyd-Warshall-Algorithmus aus dem Teil von Floyd zur Berechnung der kürzesten Pfade und dem Teil von Warshall zur Konstruktion der kürzesten Pfade.
+\subsection{Anwendung Floyd-Warshall-Algorithmus}
 
 %THEORIE...
-Als erstes wird eine Gewichtsmatrix $W$ mit den Matrixeinträgen $W[i, j]$ erstellt.
+In einem ersten Schritt wird eine Gewichtsmatrix $W$ mit den Matrixeinträgen $W[i, j]$ erstellt.
 Der Algorithmus berechnet danach in einer Hauptschleife alle Knoten $k$ von 1 bis $n$.
 Dabei versucht er in jeder Iteration alle Wege von $i$ nach $j$ durch die Wege $(i, k)$ und $(k, j)$ zu verbessern.
-Falls dieser mögliche Umweg zu einer Verbesserung führt, wird der Algorithmus aktualisiert.
+Falls dieser mögliche Umweg zu einer Verbesserung führt, wird der entsprechende Eintrag aktualisiert.
 
 Die aktuelle Gewichtung der Pfade wird mit
-\begin{equation}d[i, j]=min[d[i,j], d[i,k] + d[k,i]]\end{equation}
+\begin{equation}d[i, j]=\min[d[i,j], d[i,k] + d[k,i]]\end{equation}
 ermittelt.
 
-\subsubsection{Euklidische Heuristik}
-Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar.
-Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant.
 
-\subsection{PageRank-Algorithmus}
-Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc..
-Beim PageRank-Algorithmus handelt es sich um den Algorithmus von Google, aus dem die Google-Matrix abgeleitet wird.
-Die Google-Matrix ist eine immens grosse Matrix mit Millionen Zeilen und Spalten, die für die schnelle und vor allem exakte Bestimmung der PageRanks (Gewichtung) eine grosse Bedeutung hat.
-Der PageRank-Algorithmus analysiert und gewichtet beispielsweise die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur.
-Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\
-Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche gilt.
 
-%THEORIE...
-Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen.
+\section{PageRank-Algorithmus}
+Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc.
+Beim PageRank-Algorithmus handelt es sich nicht um einen Suchalgorithmus, stattdessen werden Knoten aufgrund der Vernetzung des vorliegenden Graphen bewertet.
+Verwendet wird er beispielsweise um die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur zu bewerten und relevante Suchergebnisse zu ermittteln. Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\
+Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche folgendes gilt:
 
 \begin{equation}
 A_{i,j}=\left\{ \begin{matrix}
@@ -103,16 +79,20 @@ A_{i,j}=\left\{ \begin{matrix}
 \label{verkehr:Adja}
 \end{equation}
 
+%THEORIE...
+Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen.
+
+
 
-Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1...n\right\}\end{equation}
+Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1\dot n\right\}\end{equation}
 Beim PageRank-Algorithmus wird eine abgewandelte Form der Adjazenz-Matrix verwendet.
 Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt.
-\begin{equation} P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \end{equation}
+\[ P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \]
 Anschliessend multipliziert man diese Matrix $P$ mit einem Spaltenvektor $\Vec{r_0}$ mit $n$ Einträgen, für welchen gilt:
-\begin{equation} \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1...n\right\} \end{equation}
+\[ \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1\dot n\right\} \]
 Dieser Vektor stellt ein neutrales Ranking dar. Alle Knoten werden gleich gewichtet.
-Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das "erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt.
-\begin{equation} \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\end{equation}
+Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das ``erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt.
+\[ \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\]
 somit
 \begin{equation} \Vec{r_i} = P^i\cdot\Vec{r_0}\end{equation}
-Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ und stellt das abschliessende Ranking dar.
+Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ der das abschliessende Ranking darstellt.
diff --git a/buch/papers/verkehr/section2.tex b/buch/papers/verkehr/section2.tex
index 638d9dd..527885e 100644
--- a/buch/papers/verkehr/section2.tex
+++ b/buch/papers/verkehr/section2.tex
@@ -1,12 +1,12 @@
 \section{Versuchsreihe}
 \label{section:verkehr/versuchsreihe}
 
-Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt.
-Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert.
-Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt.
-Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen.
+Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der Dijkstra- und der A*-Algorithmus auf das Netzwerk angewandt.
+Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal wiederholt.
+Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(|E|\log{}|V|)$ auf, wobei $E$ die Menge der Kanten (engl. \emph{edges}) und $V$ die Menge der Knoten (engl. \emph{vertices}) des Graphen $G$ darstellt.
+Für den A*-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine definitive Angabe zur Zeitkomplexität machen.
 
-Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind.
+Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind.
 
 \subsection{Einfluss der Knotenzahl auf die Rechenzeit}
 \label{verkehr:Knotenzahl}
@@ -19,9 +19,9 @@ Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start-
 \label{verkehr:Vr1}
 \end{figure}
 
-In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt.
+In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen Dijkstra und A* erst ab einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt.
 Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist.
-Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt.
+Abbildung \ref{verkehr:pathDifference} illustriert dies, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt.
 
 \begin{figure}
 \centering
@@ -36,13 +36,13 @@ Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei di
 
 \begin{figure}
 \centering
-\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}\\
+\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}
 \caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.}
 \label{verkehr:Vr2}
 \end{figure}
 
-Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\
-Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen  \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären.
+Zum Vergleich der Resultate in Abschnitt \ref{verkehr:Knotenzahl} zeigt Abbildung \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.
+Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen Dijkstra und A* deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des A*-Algorithmus erklären.
 
 \begin{figure}
 \centering
@@ -52,4 +52,4 @@ Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwis
 \label{verkehr:Comparison}
 \end{figure}
 
-In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde.
+In Abbildung \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde.
diff --git a/buch/papers/verkehr/section3.tex b/buch/papers/verkehr/section3.tex
index 99a0d92..9aa8ae4 100644
--- a/buch/papers/verkehr/section3.tex
+++ b/buch/papers/verkehr/section3.tex
@@ -1,8 +1,9 @@
 \section{Ausblick}
 \subsection{Optimierungsprobleme bei Graphen}
-Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Grafen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.\\
-Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem \emph{Dijkstra}, oder dem \emph{A*}-Algorithmus anstelle der Graph-Matrix (mit Kantengewichten als Einträgen) die Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten.  
+Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Graphen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.
+
+Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem Dijkstra- oder dem A*-Algorithmus anstelle der gewichteten Adjazenz-Matrix (mit Kantengewichten als Einträgen) die ungewichtet Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten.  
 
 \subsection{Wahl der Heuristik}
-Ein grundlegendes Problem bei der Anwendung des \emph{A*} oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.\\
+Ein grundlegendes Problem bei der Anwendung des A* oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.
 In nicht-physischen Netzwerken stellt sich jedoch eine zweite Problematik. Da eine physische Distanz entweder nicht ermittelt werden kann, oder aber nicht ausschlaggebend ist, sind andere Netzwerk-Eigenschaften zur Beurteilung beizuziehen. Die Zuverlässigkeit ist dabei aber in den meisten Fällen nicht vergleichbar hoch, wie bei der euklidischen Heuristik. Oftmals werden deshalb bei derartigen Problem auch Algorithmen angewendet, die eine deutlich optimierte Zeitkomplexität aufweisen, dafür aber nicht mit Sicherheit den effizienstesten Pfad finden.