diff options
| author | Ayexor <9105454+Ayexor@users.noreply.github.com> | 2021-08-27 18:09:54 +0200 |
|---|---|---|
| committer | GitHub <noreply@github.com> | 2021-08-27 18:09:54 +0200 |
| commit | 2b2c5daa139aec08d091b658ad6191d6e57024ef (patch) | |
| tree | 2c8f3fc7017394746d8e4f92a358e2a11015e072 /buch/papers | |
| parent | Anpassungen nach Mail (diff) | |
| parent | new image: tetraeder (diff) | |
| download | SeminarMatrizen-2b2c5daa139aec08d091b658ad6191d6e57024ef.tar.gz SeminarMatrizen-2b2c5daa139aec08d091b658ad6191d6e57024ef.zip | |
Merge branch 'master' into master
Diffstat (limited to 'buch/papers')
279 files changed, 57915 insertions, 3126 deletions
diff --git a/buch/papers/clifford/0_ElevatorPitch.tex b/buch/papers/clifford/0_ElevatorPitch.tex index 0db5617..ad9bcc2 100644 --- a/buch/papers/clifford/0_ElevatorPitch.tex +++ b/buch/papers/clifford/0_ElevatorPitch.tex @@ -1,2 +1,6 @@ -TODO... -GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains.
\ No newline at end of file + +Der Nutzen, welche die Clifford Algebra hat, lässt sich am besten mit den Worten des modernen Begründers dieser erläutern. + +"GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains." \cite{clifford:hestenes_GA} + +Im folgenden hoffen wir den Leser von der Nützlichkeit und der geometrischen Schönheit der Clifford Algebra zu überzeugen.
\ No newline at end of file diff --git a/buch/papers/clifford/10_Quaternionen.tex b/buch/papers/clifford/10_Quaternionen.tex index 375c6e7..d04ea38 100644 --- a/buch/papers/clifford/10_Quaternionen.tex +++ b/buch/papers/clifford/10_Quaternionen.tex @@ -6,222 +6,172 @@ \section{Quaternionen} \rhead{Quaternionen} -Wie die komplexen Zahlen eine Erweiterung der reellen Zahlen sind, sind die Quaternionen eine Erweiterung der komplexen Zahlen für den dreidimensionalen Raum. Sie haben, wie die komplexen Zahlen, eine dreh-streckende Eigenschaft. -Sie finden beispielsweise in der Computergraphik und in der Robotik Anwendung. +Wie die komplexen Zahlen eine Erweiterung der reellen Zahlen sind, sind die Quaternionen eine Erweiterung der komplexen Zahlen für den dreidimensionalen Raum. Sie haben, wie die komplexen Zahlen, eine drehstreckende Eigenschaft. +Sie finden beispielsweise in der Computergrafik und Robotik Anwendung. Die Quaternionen \begin{align} - q = w + xi + yj + zk \quad w,x,y,z \in \mathbb{R}\enspace q \in \mathbb{H} +q = w + xi + yj + zk \quad w,x,y,z \in \mathbb{R}\enspace q \in \mathbb{H} \end{align} -können dabei eine Drehstreckung mit dieser Formel erreichen +können dabei eine Drehstreckung mit \begin{align} \label{QuatRot} - \begin{split} - &v'' = qvq^{-1};\quad q,v,q^{-1} \in \mathbb{H}\\ - &\operatorname{Re}(q) = \operatorname{Re}(q^{-1})\quad \operatorname{Im}(q) = -\operatorname{Im}(q^{-1}) - \end{split} +\begin{split} +v \mapsto v'' = qvq^{-1} +\end{split} \end{align} -Auffallend ist hier schon die Ähnlichkeit zu dem Kapitel Rotation. Man könnte sich nun fragen wieso es drei imaginäre Einheiten $i,j,k$ gibt und nicht zwei, was doch näherliegender wäre. Der Grund liegt darin, weil es in der dritten Dimension drei Drehachsen gibt, anstatt nur eine. Wie im Kapitel Rotation beschrieben können wir auch hier die drei Drehungen durch Linearkombinationen von drei Bivektoren beschreiben. In der geometrischen Algebra ist es leicht herauszufinden wie viele Imaginärteile für jede weitere Dimension existieren. Dabei muss man nur die Anzahl der unabhängigen Bivektoren ermitteln. In der vierten Dimension würden es beispielsweise durch alle Vektorkombinationen von $\mathbf{e}_1, \mathbf{e}_2,\mathbf{e}_3, \mathbf{e}_4$ insgesamt 8 Bivektoren existieren (Nicht 16, da $\mathbf{e}_{ij} = -\mathbf{e}_{ji}$ nicht unabhängig voneinander sind). +erreichen, falls $q,v,q^{-1} \in \mathbb{H}$ und die Zusammenhänge +\begin{align} +\operatorname{Re}(q) = \operatorname{Re}(q^{-1})\quad\text{und}\quad \operatorname{Im}(q) = -\operatorname{Im}(q^{-1}) +\end{align} +gelten. Auffallend ist bei der abbildenden Funktion \eqref{QuatRot} schon die Ähnlichkeit zur Funktion \eqref{rotGA} im Abschnitt Drehung. Man könnte sich nun fragen wieso es drei imaginäre Einheiten $i,j,k$ gibt und nicht zwei, was doch näherliegender wäre. Der Grund liegt darin, weil es in drei Dimensionen drei Drehachsen gibt, anstatt nur eine. Wie im Abschnitt Drehung beschrieben, können wir auch hier die drei Drehungen durch Linearkombinationen von drei Bivektoren beschreiben. In der geometrischen Algebra ist es leicht herauszufinden, wie viele Imaginärteile für jede weitere Dimension existieren. Dabei muss man nur die Anzahl der unabhängigen Bivektoren ermitteln. In vier Dimensionen würden es beispielsweise durch alle Vektorkombinationen von $\mathbf{e}_1, \mathbf{e}_2,\mathbf{e}_3, \mathbf{e}_4$ insgesamt 8 Bivektoren existieren (Nicht 16, da $\mathbf{e}_{ij} = -\mathbf{e}_{ji}$ nicht unabhängig voneinander sind). -Ohne die geometrische Algebra, haben wir jetzt aber leider ein kleines Problem. Für die Darstellung der Quaternionen bräuchten wir insgesamt vier Achsen. Drei für die imaginären Einheiten und eine für die reelle Einheit. Ein weiterer Nachteil in visueller Hinsicht entsteht beim Anwenden eines Quaternion auf einen Vektor. Sie befinden sich nicht im gleichen Raum und müssen zuerst ineinander umgewandelt werden, um damit zu rechnen, wie man bei $v \in \mathbb{H}$ in der Formel (\ref{QuatRot}) sieht. +Ohne die geometrische Algebra, haben wir jetzt aber leider ein kleines Problem. Für die Darstellung der Quaternionen bräuchten wir insgesamt vier Achsen. Drei für die imaginären Einheiten und eine für die reelle Einheit. Ein weiterer Nachteil in visueller Hinsicht entsteht beim Anwenden einer Quaternion auf einen Vektor. Sie befinden sich nicht im gleichen Raum und müssen zuerst durch +\begin{align} +\mathbf{v} = x\mathbf{\hat{x}} + y\mathbf{\hat{y}} + z \mathbf{\hat{z}} \in \mathbb{R}^3 \enspace\mapsto\enspace v = 0 + xi + yj + zk \in \mathbb{H} +\end{align} +ineinander umgewandelt werden, um damit zu rechnen. \subsection{Geometrische Algebra} -Die geometrische Algebra besitzt die Fähigkeit beide Probleme zu lösen. Die Quaternionen können, wie schon im 2 dimensionalen Fall durch die gerade Grade $G_3^+(\mathbb{R}) \cong \mathbb{H}$ dargestellt werden. Da wir uns jetzt aber in $G_3(\mathbb{R})$ befinden haben wir drei Basisvektoren $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ und können somit drei Bivektoren bilden $\mathbf{e}_{12}, \mathbf{e}_{23}, \mathbf{e}_{31}$. +Die geometrische Algebra kann beide Probleme beheben. Die Quaternionen können, wie schon im zweidimensionalen Fall durch die gerade Grade $G_3^+(\mathbb{R}) \cong \mathbb{H}$ dargestellt werden. Da wir uns jetzt aber in $G_3(\mathbb{R})$ befinden haben wir drei Basisvektoren $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ und können somit drei Bivektoren $\mathbf{e}_{12}, \mathbf{e}_{23}, \mathbf{e}_{31}$ bilden. \begin{definition} - Multivektoren mit Drehstreckenden Eigenschaften in $G_3(\mathbb{R})$ (gleichbedeutend zu Quaternionen) + Die Multivektoren mit drehstreckenden Eigenschaften in $G_3(\mathbb{R})$ sind \begin{align} - \mathbf{q} = w + x\mathbf{e}_{12} + y\mathbf{e_{23}} + z\mathbf{e_{31}}; \quad w,x,y,z \in \mathbb{R};\enspace \mathbf{q} \in \mathbb{G}_3^+ + \mathbf{q} = w + x\mathbf{e}_{12} + y\mathbf{e}_{23} + z\mathbf{e}_{31} \quad w,x,y,z \in \mathbb{R}\enspace \mathbf{q} \in \mathbb{G}_3^+. \end{align} \end{definition} -Die Probleme werden dadurch gelöst, da wir die Bivektoren im Raum nicht durch einzelne Achsen darstellen müssen, sondern sie als eine orientiere Fläche darstellen können. Anstatt die Vektoren in Quaternionen umzurechnen, können wir jetzt die Vektoren separat im gleichen Raum darstellen. +Die Probleme werden dadurch gelöst, da wir die Bivektoren im Raum nicht durch einzelne Achsen darstellen müssen, sondern sie als eine orientiere Fläche darstellen können. Anstatt die Vektoren in Quaternionen umzurechnen, können wir jetzt die Vektoren separat im gleichen Raum, wie in Abbildung \ref{BildQuaternionen} gezeigt, darstellen. \begin{figure} \centering - \begin{tikzpicture} - % Koordinatensystem - \draw[thin,gray!40] (-3,-2) grid (3,3); - \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; - \draw[<->] (0,-2)--(0,3) node[above]{$a_2$}; - \draw[<->] (3,3)--(-2,-2) node[left]{$a_3$}; - - % v Vektor - \draw[line width=2pt,black,-stealth](0,0)--(2,-1) node[anchor=north]{$\boldsymbol{v}$}; - - % q Quaternion - \draw[line width=0,fill=blue!40] (0,0)--(0.75,0)--(0.75,0.75)--(0,0.75) - node[xshift=0.375cm, yshift=-0.5cm, blue]{$x\boldsymbol{e_{12}}$}; - \draw[->] (0.7,0.55) arc (0:310:0.15); - - \draw[line width=0,fill=blue!40] (0,0)--(-1,-1)--(-1,0.71)--(0,1.71) - node[xshift=-0.5cm, yshift=-1.5cm, blue]{$y\boldsymbol{e_{23}}$}; - \draw[->] (-0.1,1.1) arc (0:310:0.15); - - \draw[line width=0,fill=blue!40] (0,0)--(-0.71,-0.71)--(0.29,-0.71)--(1,0) - node[xshift=-0.7cm, yshift=-0.2cm, blue]{$z\boldsymbol{e_{31}}$}; - \draw[->] (0,-0.5) arc (0:310:0.15); - - % Basisvektoren - \draw[line width=1.5pt,gray,-stealth](0,0)--(1,0) node[anchor=south west]{$\boldsymbol{e_1}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(0,1) node[anchor=north west, yshift=0.2cm]{$\boldsymbol{e_2}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(-0.71,-0.71) node[anchor=south, yshift=0.2cm]{$\boldsymbol{e_3}$}; - \end{tikzpicture} + \includegraphics{papers/clifford/3d/dq.pdf} \caption{Darstellung eines Quaternion $\mathbf{q}$ und eines Vektors $\mathbf{v}$ im selben Raum} \label{BildQuaternionen} \end{figure} -Wie schon im 2 dimensionalen Fall beschreibt ein Bivektor, um wie viel der um 90 grad gedrehte orginale Vektor gestreckt wird. Dabei dreht jeder Bivektor den Vektor um eine andere Achse. -\\BILD?\\ -In der Computergraphik und Robotik macht eine Drehstreckung aber nicht viel Sinn. Wieso sollte ein Objekt bei einer Drehung zusätzlich noch grösser werden? Darum verwendet man sogenannte Einheitsquaternionen, welche den Betrag $|q|=1$ haben. Sie rotieren die Objekte bzw. Vektoren lediglich. + +Betrachten wir nun das Produkt +\begin{align} +\mathbf{qv} &= (w + x\mathbf{e}_{12} + y\mathbf{e}_{23} + z\mathbf{e}_{31})(a\mathbf{e}_1+b\mathbf{e}_2+c\mathbf{e}_3)\\ +&= \underbrace{w(a\mathbf{e}_1+b\mathbf{e}_2+c\mathbf{e}_3)}_{\displaystyle{w\mathbf{v}}} + \underbrace{x(-a\mathbf{e}_2+b\mathbf{e}_1}_{\displaystyle{x\mathbf{v}_{\angle 90^\circ, \parallel \mathbf{e}_{12}}}}+c\mathbf{e}_{123}) + \underbrace{y(-b\mathbf{e}_3+c\mathbf{e}_2}_{\displaystyle{y\mathbf{v}_{\angle 90^\circ, \parallel \mathbf{e}_{23}}}}+a\mathbf{e}_{123}) + \underbrace{z(a\mathbf{e}_3-c\mathbf{e}_1}_{\displaystyle{z\mathbf{v}_{\angle 90^\circ, \parallel \mathbf{e}_{31}}}}-b\mathbf{e}_{123}). +\end{align} +Wie schon im zweidimensionalen Fall \eqref{GAdrehstreck}, beschreibt im dreidimensionalen Fall mit drei Bivektoren jeder Bivektoranteil um wie viel der um 90° gedrehte zu der Ebene parallele Teil des Vektors gestreckt wird. Dabei dreht jeder Bivektor den Vektor um eine andere Achse und man sieht die drehstreckende Eigenschaft ähnlich zu den komplexen Zahlen. Der störende Trivektoranteil $(xc+ya+zb)\mathbf{e}_{123}$ bekommt man aber nur weg, indem man, wie in der Drehungsgleichung \eqref{QuatRot}, mit der Inversen Quaternion $\mathbf{q}^{-1}$ multipliziert, wobei die drehgestreckten parallelen Anteile nochmals drehgestreckt werden. Da nur so der Trivektoranteil wegfällt, sieht man, dass die Drehungsformel, der einzige vernünftige Weg ist, mit Quaternionen zu arbeiten. + +In der Computergraphik und Robotik macht eine Drehstreckung aber nicht viel Sinn. Wieso sollte ein Objekt bei einer Drehung zusätzlich noch grösser werden? Darum verwendet man sogenannte Einheitsquaternionen, welche den Betrag $|\mathbf{q}|=1$ haben und somit drehen sie die Objekte bzw. Vektoren lediglich. \begin{definition} - Einheitsquaternionen + Die Einheitsquaternionen sind definiert als \begin{align} - \mathbf{q} = \cos(\alpha) + sin(\alpha)(\tilde{x}\mathbf{e}_{12} + \tilde{y}\mathbf{e}_{23} + \tilde{z}\mathbf{e}_{31}) + \mathbf{q} = \cos(\alpha) + \sin(\alpha)(\tilde{x}\mathbf{e}_{12} + \tilde{y}\mathbf{e}_{23} + \tilde{z}\mathbf{e}_{31}) \end{align} \end{definition} -Dabei ist definiert, dass $\tilde{x}^2+\tilde{y}^2+\tilde{z}^2=1$. Somit beträgt der Betrag von $\mathbf{q}$ immer 1. +Zudem setzten wir $\tilde{x}^2+\tilde{y}^2+\tilde{z}^2=1$, damit \begin{align} - |\mathbf{q}| = \sqrt{cos(\alpha)^2 + sin(\alpha)^2(\tilde{x}^2+\tilde{y}^2+\tilde{z}^2) } = \sqrt{cos(\alpha)^2 + sin(\alpha)^2} = 1 +|\mathbf{q}| = \sqrt{\cos(\alpha)^2 + \sin(\alpha)^2(\tilde{x}^2+\tilde{y}^2+\tilde{z}^2) } = \sqrt{\cos(\alpha)^2 + \sin(\alpha)^2} = 1. \end{align} -Der Winkel $\alpha$ beschreibt dabei, wie im Bild (...) gezeigt den halben Winkel, um welchen der parallelen Anteil $\mathbf{v_{\perp}}$ des Vektors $\mathbf{v}$ zur kombinierten Bivektorebene $sin(\alpha)^2(\tilde{x}^2+\tilde{y}^2+\tilde{z}^2)$ gedreht wird. +Der Winkel $\alpha$ beschreibt dabei, wie im Bild \ref{BildQuaternionBeispiel2} gezeigt, den halben Winkel, um welchen der parallelen Anteil $\mathbf{v_{\parallel}}$ des Vektors $\mathbf{v}$ zur kombinierten Bivektorebene $sin(\alpha)(\tilde{x}\mathbf{e}_{12} + \tilde{y}\mathbf{e}_{23} + \tilde{z}\mathbf{e}_{31})$ gedreht wird. -Um einen Vektor zu drehen, verwendet man wieder die gleiche Formel, wie auch schon im zweidimensionalen Fall. +Um einen Vektor zu drehen, verwendet man die in Abschnitt 18.4 hergeleitete Formel \begin{align} \label{QuatRotGA} - \begin{split} - &\mathbf{v}'' = \mathbf{qvq}^{-1}\\ - &\operatorname{Re}(\mathbf{q}) = \operatorname{Re}(\mathbf{q}^{-1});\enspace \operatorname{Im}(\mathbf{q}) = -\operatorname{Im}(\mathbf{q}^-1) - \end{split} +\begin{split} +\mathbf{v}'' = \mathbf{qvq}^{-1}, +\end{split} +\end{align} +wobei wie auch schon bei den Quaternionen gelten muss, dass +\begin{align} \label{GAReIm} +\operatorname{Re}(\mathbf{q}) = \operatorname{Re}(\mathbf{q}^{-1}) \quad\text{und}\quad \operatorname{Im}(\mathbf{q}) = -\operatorname{Im}(\mathbf{q}^{-1}). +\end{align} +Der Grund für die Zusammenhänge \eqref{GAReIm} kann man durch die hergeleitete vereinfachte Drehungsgleichung \eqref{GAvereinfRot} sehen, weil durch den negierten Winkel $\theta$ der Reelle bzw. Grad 0 Anteil +\begin{align} +\operatorname{Re}(e^{-\theta \mathbf{e}_{12}}) = \operatorname{Re}(e^{\theta \mathbf{e}_{12}}) \end{align} -Es ist wichtig bei Quaternionen für eine reine Drehstreckung mit $q$ und $q^{-1}$ beidseitig zu multiplizieren, sonst werden die senkrechten Anteile zu den Bivektorebenen ebenfalls beeinflusst, wie man im Kapitel Rotation bei der Formel (\ref{RotAufPerpPar}) sehen kann. +und der imaginäre bzw. Grad 2 Anteil +\begin{align} +\operatorname{Im}(e^{-\theta \mathbf{e}_{12}}) = -\operatorname{Im}(e^{\theta \mathbf{e}_{12}}) +\end{align} +ist. Durch die geometrische Algebra sieht man nun, wieso es wichtig ist, bei Quaternionen für eine reine Drehstreckung mit $\mathbf{q}$ und $\mathbf{q}^{-1}$ beidseitig zu multiplizieren, sonst werden die senkrechten Anteile zu den Bivektorebenen ebenfalls beeinflusst, wie man im Abschnitt Drehung bei der Formel \eqref{RotAufPerpPar} sehen kann. \begin{beispiel} - Eine Drehung eines Vektors $\mathbf{v}= 1\mathbf{e}_2$ um 90 Grad um die $\mathbf{e}_1$-Achse und danach 90 Grad um die $\mathbf{e}_2$-Achse. Dafür nehmen wir zuerst einen Einheitsquaternion welcher um die Orientierte Ebene $\mathbf{e}_{23}$ um 90 Grad dreht + Eine Drehung eines Vektors $\mathbf{v}= 1\mathbf{e}_2$ um 90 Grad um die $\mathbf{e}_1$-Achse und danach 90 Grad um die $\mathbf{e}_2$-Achse. Dafür nehmen wir zuerst die Einheitsquaternion \begin{align} - \mathbf{q}_{23} &= \cos(\pi/4) + sin(\pi/4)(1\mathbf{e}_{23}) = e^{(\pi/4)\mathbf{e}_{23}} &= 0.71 + 0.71\mathbf{e}_{23}\\ - \mathbf{q}_{23}^{-1} &&= 0.71 - 0.71\mathbf{e}_{23} + \mathbf{q}_{23} &= \cos(\pi/4) + \sin(\pi/4)(1\mathbf{e}_{23}) = e^{(\pi/4)\mathbf{e}_{23}} &= \textstyle{\frac{\sqrt{2}}{2}}(1 + \mathbf{e}_{23})\\ + \mathbf{q}_{23}^{-1} &&= \textstyle{\frac{\sqrt{2}}{2}} (1- \mathbf{e}_{23}) \end{align} - und danach Einheitsquaternion welcher um die Orientierte Ebene $\mathbf{e}_{31}$ um 90 Grad dreht + welche um die $\mathbf{e}_{2}$-$\mathbf{e}_{3}$-Ebene um 90 Grad dreht und danach die Einheitsquaternion \begin{align} - \mathbf{q}_{31} &= \cos(\pi/4) + sin(\pi/4)(1\mathbf{e}_{31}) = e^{(\pi/4)\mathbf{e}_{31}} &= 0.71 + 0.71\mathbf{e}_{31}\\ - \mathbf{q}_{31}^{-1} &&= 0.71 - 0.71\mathbf{e}_{31} + \mathbf{q}_{31} &= \cos(\pi/4) + \sin(\pi/4)(1\mathbf{e}_{31}) = e^{(\pi/4)\mathbf{e}_{31}} &= \textstyle{\frac{\sqrt{2}}{2}}(1 + \mathbf{e}_{31})\\ + \mathbf{q}_{31}^{-1} &&= \textstyle{\frac{\sqrt{2}}{2}}(1 - \mathbf{e}_{31}), \end{align} - Um die vollständige Rotation zu beschreiben können die Einheitsquaternion multipliziert werden, wobei die Reihenfolge der Ausführung beachtet werden muss + welche um die $\mathbf{e}_{3}$-$\mathbf{e}_{1}$-Ebene um 90 Grad dreht. Um die vollständige Drehung zu beschreiben, können die Einheitsquaternionen multipliziert werden, wobei die Reihenfolge der Ausführung beachtet werden muss. Somit ist \begin{align} \label{FormelBeispielQuaternion} - \mathbf{q} &= \mathbf{q}_{31}\mathbf{q}_{23} = (0.71 + 0.71\mathbf{e}_{31})(0.71 + 0.71\mathbf{e}_{23}) &= 0.5 + 0.5\mathbf{e}_{31} + 0.5 \mathbf{e}_{23} + 0.5 \mathbf{e}_{12}\\ - \mathbf{q}^{-1} &= \mathbf{q}_{23}^{-1}\mathbf{q}_{31}^{-1} = (0.71 - 0.71\mathbf{e}_{23})(0.71 - 0.71\mathbf{e}_{31}) &= 0.5 - 0.5\mathbf{e}_{31} - 0.5 \mathbf{e}_{23} - 0.5 \mathbf{e}_{12} + \mathbf{q} &= \mathbf{q}_{31}\mathbf{q}_{23} = \textstyle{\frac{\sqrt{2}}{2}}(1 + \mathbf{e}_{31})\textstyle{\frac{\sqrt{2}}{2}}(1 + \mathbf{e}_{23}) &= \textstyle{\frac{1}{2}}(1 + \mathbf{e}_{31} + \mathbf{e}_{23} + \mathbf{e}_{12})\\ + \mathbf{q}^{-1} &= \mathbf{q}_{23}^{-1}\mathbf{q}_{31}^{-1} = \textstyle{\frac{\sqrt{2}}{2}} (1- \mathbf{e}_{23})\textstyle{\frac{\sqrt{2}}{2}}(1 -\mathbf{e}_{31}) &= \textstyle{\frac{1}{2}}(1 - \mathbf{e}_{31} - \mathbf{e}_{23} - \mathbf{e}_{12}). \end{align} - Wenn wir nun den Quaternion $\mathbf{q}$ auf den Vektor $\mathbf{v}$ anwenden + Wenn wir nun die Quaternion $\mathbf{q}$ auf den Vektor $\mathbf{v}$ anwenden, erhalten wir \begin{align} - \mathbf{v}'' = \mathbf{qvq}^{-1} &= (0.5 + 0.5\mathbf{e}_{31} + 0.5 \mathbf{e}_{23} + 0.5 \mathbf{e}_{12})(1\mathbf{e}_2)(0.5 - 0.5\mathbf{e}_{31} - 0.5 \mathbf{e}_{23} - 0.5 \mathbf{e}_{12})\\ - &= (0.5\mathbf{e}_2 + 0.5 \mathbf{e}_{123} - 0.5 \mathbf{e}_3 + 0.5 \mathbf{e}_1)(0.5 - 0.5\mathbf{e}_{31} - 0.5 \mathbf{e}_{23} - 0.5 \mathbf{e}_{12})\\ - &= (0.25 + 0.25 + 0.25 + 0.25)\mathbf{e}_1 + (0.25 + 0.25 - 0.25 - 0.25)\mathbf{e}_2 +\\ &(-0.25 + 0.25 - 0.25 + 0.25)\mathbf{e}_3 + (0.25 - 0.25 - 0.25 + 0.25)\mathbf{e}_{123}\\ - &= 1e_1 + \mathbf{v}'' = \mathbf{qvq}^{-1} &= \textstyle{\frac{1}{2}}(1 + \mathbf{e}_{31} + \mathbf{e}_{23} + \mathbf{e}_{12})(1\mathbf{e}_2)\textstyle{\frac{1}{2}}(1 - \mathbf{e}_{31} - \mathbf{e}_{23} - \mathbf{e}_{12})\\ + &= \textstyle{\frac{1}{4}}(\mathbf{e}_2 + \mathbf{e}_{123} - \mathbf{e}_3 + \mathbf{e}_1)(1 - \mathbf{e}_{31} - \mathbf{e}_{23} - \mathbf{e}_{12})\\ + &= (\textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}})\mathbf{e}_1 + (\textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}} - \textstyle{\frac{1}{4}} - \textstyle{\frac{1}{4}})\mathbf{e}_2 +\\ &\qquad(-\textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}} - \textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}})\mathbf{e}_3 + (\textstyle{\frac{1}{4}} - \textstyle{\frac{1}{4}} - \textstyle{\frac{1}{4}} + \textstyle{\frac{1}{4}})\mathbf{e}_{123}\\ + &= 1e_1. \end{align} Anders betrachtet könnte man von der Formel \eqref{FormelBeispielQuaternion} sehen, dass der Drehwinkel \begin{align} - \alpha = \arccos(w) = \arccos(0.5) = 60° + \alpha = \arccos(w) = \arccos(\textstyle{\frac{1}{2}}) = 60^\circ \end{align} und die Ebene der kombinierten Bivektoren wie in Abbildung \ref{BildQuaternionBeispiel2} aussieht. - Somit kann man sich ebenfalls Vorstellen, wie der parallele Anteil zur Ebene insgesamt um 120° rotiert wird während der senkrechte Anteil unverändert bleibt + Somit kann man sich ebenfalls vorstellen, wie der parallele Anteil zur Ebene insgesamt um 120° gedreht wird, während der senkrechte Anteil unverändert bleibt. \end{beispiel} \begin{figure} \centering - \begin{tikzpicture} - % Koordinatensystem - \draw[thin,gray!40] (-3,-2) grid (3,3); - \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; - \draw[<->] (0,-2)--(0,3) node[above]{$a_2$}; - \draw[<->] (3,3)--(-2,-2) node[left]{$a_3$}; - - % q Quaternion - \draw[line width=0,fill=blue!40] (0,0)--(1.41,0)--(1.41,1.41)--(0,1.41) - node[xshift=0.375cm, yshift=-0.5cm, blue]{$x\boldsymbol{e_{12}}$}; - \draw[->] (1.35, 1.2) arc (0:310:0.15); - - \draw[line width=0,fill=blue!40] (0,0)--(-1,-1)--(-1,0.41)--(0,1.41) - node[xshift=-0.5cm, yshift=-1.5cm, blue]{$y\boldsymbol{e_{23}}$}; - \draw[->] (-0.65,-0.5) arc (0:310:0.15); - - \draw[line width=0,fill=blue!40] (0,0)--(-1,-1)--(0.41,-1)--(1.41,0) - node[xshift=-0.7cm, yshift=-0.2cm, blue]{$z\boldsymbol{e_{31}}$}; - \draw[->] (0.4,-0.8) arc (0:310:0.15); - - % Basisvektoren - \draw[line width=1.5pt,gray,-stealth](0,0)--(2,0) node[anchor=south west]{$\boldsymbol{e_1}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(0,2) node[anchor=north west, yshift=0.2cm]{$\boldsymbol{e_2}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(-1.41,-1.41) node[anchor=south, yshift=0.2cm]{$\boldsymbol{e_3}$}; - - % v Vektor - \draw[line width=2pt,black,-stealth](-0.05,0)--(-0.05,2) node[anchor=east]{$\boldsymbol{v}$}; - % v'' Vektor - \draw[line width=2pt,black,-stealth](0,0.05)--(2,0.05) node[anchor=north]{$\boldsymbol{v}''$}; - \end{tikzpicture} + \includegraphics{papers/clifford/3d/qq.pdf} + \caption{Beispiel für Drehung um 90 Grad je um die $\mathbf{e}_1$- und $\mathbf{e}_2$-Achse.} \label{BildQuaternionBeispiel} \end{figure} \begin{figure} \centering - \begin{tikzpicture} - % q Quaternion - \draw[line width=0,fill=blue!40] (-0.75,-1)--(1.5,-0.5)--(0.55,1.35)--(-1.5,1) - node[xshift=0.375cm, yshift=-0.5cm, blue]{$\boldsymbol{q}$}; - \draw[->] (-0.7, 0.5) arc (310:0:0.15); - - % Koordinatensystem - \draw[thin,gray!40] (-3,-2) grid (3,3); - \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; - \draw[<->] (0,-2)--(0,3) node[above]{$a_2$}; - \draw[<->] (3,3)--(-2,-2) node[left]{$a_3$}; - - % Basisvektoren - \draw[line width=1.5pt,gray,-stealth](0,0)--(2,0) node[anchor=south west]{$\boldsymbol{e_1}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(0,2) node[anchor=north west, yshift=0.2cm]{$\boldsymbol{e_2}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(-1.41,-1.41) node[anchor=south, yshift=0.2cm]{$\boldsymbol{e_3}$}; - - % v Vektor - \draw[line width=2pt,black,-stealth](-0.05,0)--(-0.05,2) node[anchor=east]{$\boldsymbol{v}$}; - % vpar Vektor - \draw[line width=2pt,red,-stealth](0,0)--(-0.33,1.25) node[anchor=east]{$\boldsymbol{v_{\parallel}}$}; - % vperp Vektor - \draw[line width=2pt,green,-stealth](-0.33,1.25)--(0,2) node[anchor=east, xshift = -0.05, yshift = -0.3cm]{$\boldsymbol{v_{\perp}}$}; - % v'' Vektor - \draw[line width=2pt,black,-stealth](0,0.05)--(2,0.05) node[anchor=north, xshift = 0.25cm]{$\boldsymbol{v}''$}; - % vpar'' Vektor - \draw[line width=2pt,red,-stealth](0,0)--(1.66,-0.75) node[anchor=east, yshift = -0.2cm, xshift = -0.1cm]{$\boldsymbol{v_{\parallel}''}$}; - % vperp'' Vektor - \draw[line width=2pt,green,-stealth](1.66,-0.75)--(2,0) node[anchor=east, xshift = 0.5cm, yshift = -0.65cm]{$\boldsymbol{v_{\perp}''}$}; - - \coordinate (A) at (0,0); - \coordinate (B) at (-0.33,1.25); - \coordinate (C) at (1.66,-0.75); - \tikzset{anglestyle/.style={angle eccentricity=2, draw, thick, angle radius=0.75cm, purple}} - \draw pic ["120° $=2\alpha$", anglestyle] {angle = C--A--B}; - \end{tikzpicture} + \includegraphics{papers/clifford/3d/drehung.pdf} \caption{Beim Beispiel wird der parallele Anteil um 120° gedreht während der senkrechte Anteil zur kombinierten Ebene (Bivektoraddition) gleich bleibt} \label{BildQuaternionBeispiel2} \end{figure} \subsection{Interpolation} -In der Computergrafik wird Interpolation verwendet, um eine flüssige Drehbewegung zu erreichen. Dabei wird die gewünschte Drehbewegungen des Objektes in kleinere aufgeteilt. Man kann dabei mit zwei verschiedenen Systemen arbeiten. +In der Computergrafik wird Interpolation verwendet, um eine flüssige Drehbewegung zu erreichen. Dabei wird die ganze gewünschte Drehbewegungen des Objektes in kleinere Drehbewegungen aufgeteilt, wobei diese zeitlich nacheinander auf das Objekt angewendet werden. Als Vergleich könnte man sagen, dass ein Film auch nur Bilder sind, welche zeitlich nacheinander gezeigt werden. Man kann dabei mit zwei verschiedenen Systemen arbeiten. \begin{itemize} - \item Mit den Eulerschen Winkeln, welche für die Meisten zwar intuitiver sind, aber dafür Nachteile haben, worauf ich in diesem Abschnitt eingehen werde. Dabei kann eine ganze Drehbewegung $\mathbf{v}'' = R\mathbf{v}$ durch die Drehmatrix $R$ dargestellt werden. - \begin{align} - \begin{split} - &R = R_z(\gamma) R_y(\beta) R_x(\alpha)\\ - &R = - \begin{pmatrix} - \cos(\gamma) & -\sin(\gamma) & 0\\ \sin(\gamma) & \cos(\gamma) & 0 \\ 0 & 0 & 1 - \end{pmatrix} - \begin{pmatrix} - \cos(\beta) & 0 & \sin(\beta)\\ 0 & 1 & 0 \\ -\sin(\beta) & 0 & \cos(\beta) - \end{pmatrix} - \begin{pmatrix} - 1 & 0 & 0 \\ 0 & \cos(\alpha) & -\sin(\alpha)\\ 0 & \sin(\alpha) & \cos(\alpha) - \end{pmatrix} - \end{split} + \item Mit den Eulerschen Winkeln, welche für die Meisten zwar intuitiver sind, aber dafür Nachteile haben, worauf ich in diesem Abschnitt eingehen werde. Dabei kann eine ganze Drehbewegung $\mathbf{v}'' = R\mathbf{v}$ durch die Drehmatrix + \begin{align} \label{GADrehmatrix} + R = + \underbrace{ + \begin{pmatrix} + \cos(\gamma) & -\sin(\gamma) & 0\\ \sin(\gamma) & \cos(\gamma) & 0 \\ 0 & 0 & 1 + \end{pmatrix} + }_{\displaystyle{R_z(\gamma)}} + \underbrace{ + \begin{pmatrix} + \cos(\beta) & 0 & \sin(\beta)\\ 0 & 1 & 0 \\ -\sin(\beta) & 0 & \cos(\beta) + \end{pmatrix} + }_{\displaystyle{R_y(\beta)}} + \underbrace{ + \begin{pmatrix} + 1 & 0 & 0 \\ 0 & \cos(\alpha) & -\sin(\alpha)\\ 0 & \sin(\alpha) & \cos(\alpha) + \end{pmatrix} + }_{\displaystyle{R_x(\alpha)}} \end{align} - Wichtig dabei zu sehen ist, dass die Drehbewegungen durch die einzelnen Matrizen nacheinander ausgeführt werden. Das bedeutet, wenn man die Reihenfolge vertauscht, bekommt man eine völlig andere Drehung. Man kann die Auswirkungen der Reihenfolge gut bei einem Gimbal (REF zu BILD) sehen. Die Matrix ganz links ist die, welche als letztes Angewendet wird. Somit bildet sie die Drehung des äusseren Rings, welche auch die zwei inneren Ringe und das Objekt mitdreht. Die Matrix ganz rechts hingegen bildet nur die Drehung des inneren Rings, welche nur das Objekt selber dreht. Man kann dabei erkennen, dass vorgehen dabei sehr intuitiv ist, aber es kompliziert sein kann eine gewünschte Drehbewegung auszuführen, da sich beim Drehen der äusseren Achse, sich auch die Inneren drehen. Das bedeutet, wenn man sich eine Drehbewegung um die anfängliche x Achse mit $R_x(\alpha_2)$ wünscht, und vorher eine beliebige Drehung $R = R_z(\gamma_1) R_y(\beta_1) R_x(\alpha_1)$ ausgeführt hat, bekommt man nicht das richtige Ergebnis, da die anfängliche x-Achse durch die Drehmatrizen $R_z(\gamma_1)$ und $R_y(\beta_1)$ zu einer neuen, lokalen x-Achse wurde. - \item Andererseits mit den Quaternionen, welche die besondere Eigenschaft haben, dass eine Drehung immer um die globale Achsen ausgeführt wird, egal in welcher Rotationsposition sich das Objekt befindet. + dargestellt werden. Wichtig dabei zu sehen ist, dass die Drehbewegungen durch die einzelnen Matrizen nacheinander ausgeführt werden. Das bedeutet, wenn man die Reihenfolge vertauscht, bekommt man eine völlig andere Drehung. Man kann die Auswirkungen der Reihenfolge gut bei einem Gimbal im Bild \ref{BildReihenfolgeGimbal} sehen. Die Matrix ganz links in der Gleichung \eqref{GADrehmatrix} ist die, welche als letztes Angewendet wird. Somit bildet sie die Drehung des äusseren Rings, welche auch die zwei inneren Ringe und das Objekt mitdreht. Die Matrix ganz rechts hingegen bildet nur die Drehung des inneren Rings, welche nur das Objekt mitdreht. Man kann dabei erkennen, dass Vorgehen dabei sehr intuitiv ist, aber es kompliziert sein kann, eine gewünschte Drehbewegung auszuführen, da sich beim Drehen der äusseren Achse, sich auch die inneren drehen. Das bedeutet, wenn man sich eine Drehbewegung um die anfängliche x Achse mit $R_x(\alpha_2)$ wünscht, und vorher eine beliebige Drehung $R = R_z(\gamma_1) R_y(\beta_1) R_x(\alpha_1)$ ausgeführt hat, bekommt man nicht das richtige Ergebnis, da die anfängliche $x$-Achse durch die Drehmatrizen $R_z(\gamma_1)$ und $R_y(\beta_1)$ zu einer neuen, lokalen $x$-Achse wurde. + \item Mit den Quaternionen, welche die besondere Eigenschaft haben, dass eine Drehung immer um die globale Achsen ausgeführt wird, egal in welcher Drehungsposition sich das Objekt befindet. \end{itemize} Für Spielentwickler ist es darum meist sinnvoller Quaternionen für Drehbewegungen anzuwenden, als sich mit komplizierten Berechnungen mit Eulerschen Winkeln herumzuschlagen. + +\begin{figure} + \centering + \includegraphics[width=10cm]{papers/clifford/Bilder/ReihenfolgeGimbal.png} + \caption{Das Gimbal Lock tritt ein, wenn zwei Drehachsen in der gleichen Ebene liegen. Dies ist im rechten Bild bei der grünen und blauen Achse der Fall. Der rote Kreis würde sich an der oberen Halterung genau in die gleiche Richtung drehen, wie der grüne Kreis an der unteren Halterung. Man verliert somit eine Drehrichtung.} + \label{BildReihenfolgeGimbal} +\end{figure} + \subsection{Gimbal-Lock} -Ein weiterer Nachteil der Eulerschen Winkel ist das Gimbal-Lock. Es entsteht dann, wenn der äussere Ring Deckungsgleich über denn Inneren gedreht wird. Dabei verliert das Gimbal eine Drehrichtung, da der äussere und Innere Ring nun die gleiche Drehrichtung besitzen. Dies kann beispielsweise Probleme bei Spielen bei der Berechnung der Interpolation führen. Man hat das bei älteren Spielen dann gesehen, wenn plötzlich Gliedmassen bei den Spielermodellen in unnatürlichen Richtungen gesprungen sind. -\subsection{Fazit} -andere Darstellungsweise. Besser für Verständnis => komplexe Zahlen erscheinen ähnlicher zu Quaternionen? Eine Sprache für alle Geometrische Probleme
\ No newline at end of file +Ein weiterer Nachteil der Eulerschen Winkel ist das Gimbal-Lock. Es entsteht dann, wenn zwei Ringe Deckungsgleich übereinander gedreht werden, wie man im Bild \eqref{BildReihenfolgeGimbal} sieht. Dabei verliert das Gimbal eine Drehrichtung, da der äussere und Innere Ring nun die gleiche Drehrichtung besitzen. Dies kann beispielsweise Probleme bei Spielen bei der Berechnung der Interpolation führen. Man hat dies bei älteren Spielen wie im Bild \ref{BildGimbalLock} dann gesehen, wenn plötzlich Gliedmassen bei den Spielermodellen in unnatürliche Richtungen gesprungen sind. + +\begin{figure} + \centering + \includegraphics[width=10cm]{papers/clifford/Bilder/GimbalLock.png} + \caption{Interpolationsfehler durch Gimbal-Lock} + \label{BildGimbalLock} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/clifford/11_Fazit.tex b/buch/papers/clifford/11_Fazit.tex new file mode 100644 index 0000000..79a683d --- /dev/null +++ b/buch/papers/clifford/11_Fazit.tex @@ -0,0 +1,9 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Fazit} +\rhead{Fazit} + +Die geometrische Algebra ist dafür ausgelegt, geometrische Operationen, wie die Spiegelung oder Drehung, einfach zu beschreiben. Dadurch kann sie als gute Alternative zu der linearen Algebra angewendet werden, um grafische Probleme zu lösen. Sie kann zudem zum Verständnis der drehenden Eigenschaften der komplexen Zahlen und Quaternionen beitragen und die Zusammenhänge zwischen den komplexen Zahlen und den Quaternionen zeigen.
\ No newline at end of file diff --git a/buch/papers/clifford/1_Vektordarstellung.tex b/buch/papers/clifford/1_Vektordarstellung.tex index 88a5789..ac00a33 100644 --- a/buch/papers/clifford/1_Vektordarstellung.tex +++ b/buch/papers/clifford/1_Vektordarstellung.tex @@ -1,7 +1,7 @@ \section{Vektoroperationen\label{clifford:section:Vektoroperationen}} \rhead{Vektoroperationen} \subsection{Vektordarstellung\label{clifford:section:Vektordarstellung}} -Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus Basisvektoren dargestellt werden +Vektoren können neben der üblichen Spaltendarstellung, auch als Linearkombination aus Basisvektoren \begin{equation} \begin{split} \textbf{a} @@ -31,12 +31,14 @@ Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus \sum_{i=1}^{n} a_i \textbf{e}_i \qquad a_i \in \mathbb{R} - , \textbf{e}_i \in \mathbb{R}^n. + , \textbf{e}_i \in \mathbb{R}^n \end{split} \end{equation} -Diese Basisvektoren sollen orthonormal sein und um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, ...$ ersetzt. +dargestellt werden. +Diese Basisvektoren werden so gewählt, dass sie orthonormal sind. +Um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, \dots$ ersetzt. \begin{beispiel} -Linearkombination von Basisvektoren in $\mathbb{R}^4$ +Eine Linearkombination von Basisvektoren in $\mathbb{R}^4$ könnte wie folgt aussehen \begin{equation} \begin{pmatrix} 42 \\ 2 \\ 1291 \\ 4 @@ -65,7 +67,7 @@ Linearkombination von Basisvektoren in $\mathbb{R}^4$ + 1291\textbf{e}_3 + - 4\textbf{e}_4 + 4\textbf{e}_4. \end{equation} +Dieses Beispiel ist für einen vier dimensionalen Vektor, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden. \end{beispiel} -Wobei Beispiel für einen vier dimensionalen Vektor ist, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden.
\ No newline at end of file diff --git a/buch/papers/clifford/2_QuadratVektoren.tex b/buch/papers/clifford/2_QuadratVektoren.tex index cfb05d6..6c6fb7d 100644 --- a/buch/papers/clifford/2_QuadratVektoren.tex +++ b/buch/papers/clifford/2_QuadratVektoren.tex @@ -1,54 +1,71 @@ \subsection{Quadrat von Vektoren} -Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. +\subsubsection{Ziel der Multiplikation} +Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen wie in Abbildung \ref{figure:addition}, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. +\begin{figure}[htb] + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[blue,thick,->] (0,0)--(3.5,2) node[midway,above,sloped] {$\textbf{a}$}; + \draw[red,thick,->] (3.5,2)--(1.5,3.8) node[midway,above,sloped] {$\textbf{b}$}; + \draw[black,thick,->] (0,0)--(1.5,3.8)node[midway,above,sloped] {$\textbf{a} +\textbf{b} = \textbf{c} $}; + \end{tikzpicture} + \caption{Addition von zwei Vektoren\label{figure:addition}} +\end{figure} Was soll es schon heissen zwei Vektoren miteinander zu multiplizieren? -\newline Im Folgenden werden wir versuchen diese Operation ähnlich intuitiv darzustellen. -\newline -Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation ist. -Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen, auch so definiert ist. -\newline -Zusätzlich wollen wir auch das Assoziativgesetz und das Kommutativgesetz für Skalare beibehalten. Wobei das Kommutativgesetz leider, oder wie man sehen wird zum Glück, in der geometrischen Algebra im generellen nicht mehr gilt. Das heisst wir dürfen ausklammern \ref{eq:assoziativ} und die Position von Skalaren im Produkt ändern \ref{eq:kommSkalar}, allerdings nicht die Position der Vektoren \ref{eq:kommVector}. + +Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall sind diese Elemente Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation sein soll. + +Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen,so definiert ist. + +Zusätzlich soll auch das Assoziativgesetz für die Multiplikation von Vektoren gelten, dass heisst wir dürfen ausklammern \begin{equation} \label{eq:assoziativ} \textbf{e}_i(\textbf{e}_j + \textbf{e}_k) = - \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k + \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k. \end{equation} +Allerdings gilt das Kommutativgesetz leider, oder wie man sehen wird zum Glück, nur für skalare Elemente \begin{equation} \label{eq:kommSkalar} a\textbf{e}_ib\textbf{e}_j = - ab\textbf{e}_i\textbf{e}_j + ab\textbf{e}_i\textbf{e}_j \qquad a,b \in \mathbb{R} \end{equation} +und nicht für Vektoren \begin{equation} \label{eq:kommVector} \textbf{e}_i\textbf{e}_j \neq - \textbf{e}_j\textbf{e}_i + \textbf{e}_j\textbf{e}_i. +\end{equation} +\subsubsection{Quadrieren eines Vektors} +Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors. Zuerst werden die Vektoren als Linearkombinationen geschrieben +\begin{equation} + \textbf{a}^2 = + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \label{eq:quad_a_1}. +\end{equation} +Das Quadrat kann nun in zwei Summen aufgeteilt werden +\begin{equation} + \textbf{a}^2 = + \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} + + + \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j } + \label{eq:quad_a_2}, +\end{equation} +wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. Da $\textbf{e}_i^2 = 1$ gilt, weil das zuvor definierte Ziel des Quadrates eines Vektors dessen Länge ergibt und die Basisvektoren Länge 1 haben, wird dies nun eingesetzt +\begin{equation} + \textbf{a}^2 = \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j}. + \label{eq:quad_a_3} \end{equation} -Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors. -\begin{align} - \textbf{a}^2 &= - \left ( - \sum_{i=1}^{n} a_i \textbf{e}_i - \right ) - \left ( - \sum_{i=1}^{n} a_i \textbf{e}_i - \right ) - \label{eq:quad_a_1} - \\ - &= - \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} - + - \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j } - \label{eq:quad_a_2} - \\ - &= \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j}. - \label{eq:quad_a_3} -\end{align} - \begin{beispiel} -Quadrat eines Vektors in $\mathbb{R}^2$ +Das Quadrat des Vektor $a$ in $\mathbb{R}^2$ ist \begin{equation} \begin{split} \textbf{a}^2 @@ -56,22 +73,17 @@ Quadrat eines Vektors in $\mathbb{R}^2$ &= \textcolor{red}{a_1^2\textbf{e}_1^2 + a_2^2\textbf{e}_2^2} + \textcolor{blue}{a_1\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} \\\ & = \textcolor{cyan}{a_1^2 + a_2^2} + \textcolor{orange}{a_1b\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} - \end{split} + \end{split}. \end{equation} - \end{beispiel} -Der Vektor wird in \ref{eq:quad_a_1} als Linearkombination geschrieben. -Das Quadrat kann, wie in \ref{eq:quad_a_2} gezeigt, in zwei Summen aufteilen werden , wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. -\newline -Da $\textbf{e}_i^2 = 1$ gilt, da zuvor vorausgesetzt wurde, dass man mit orthonormalen Einheitsvektoren arbeitet, wird dies nun eingesetzt ergibt sich \ref{eq:quad_a_3} -\newline -Die hellblaue Teil ist nun bereits Länge im Quadrat eines Vektors, also das Ziel der Multiplikation. -Daher muss der restliche Teil dieser Gleichung null ergeben. -Aus dieser Erkenntnis leiten wir in \ref{eq:Mischterme_Null} weitere Eigenschaften für die Multiplikation her. + +Die hellblaue Teil ist nun bereits die Länge im Quadrat, also das zuvor definierte Ziel der Multiplikation. +Daraus lässt sich schliessen, dass der restliche Teil dieser Gleichung null ergeben muss \begin{equation} \label{eq:Mischterme_Null} - \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots = 0 + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots = 0. \end{equation} +Aus dieser Erkenntnis können weitere Eigenschaften für die Multiplikation hergeleitet werden. Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_2$ gleich null gesetzt. Somit fallen alle Terme bis auf den blauen weg. Wird dies weiter vereinfacht ergibt sich \begin{equation} \begin{split} @@ -81,15 +93,13 @@ Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_ \end{split} \end{equation} \begin{satz} - Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind. + Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind, es gilt also \begin{equation} - \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \qquad \textbf{e}_i \perp \textbf{e}_j + \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \quad \textrm{für} \quad \textbf{e}_i \perp \textbf{e}_j. \end{equation} \end{satz} -Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in \ref{tab:multip_vec} gemacht wurde. +Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in Tabelle \ref{tab:multip_vec} gemacht wurde. \begin{table} -\caption{Multiplikationstabelle für Vektoren} -\label{tab:multip_vec} \begin{center} \begin{tabular}{ |c|c|c|c|c|c| } \hline @@ -107,4 +117,6 @@ Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen \hline \end{tabular} \end{center} +\caption{Multiplikationstabelle für Vektoren} +\label{tab:multip_vec} \end{table}
\ No newline at end of file diff --git a/buch/papers/clifford/3_MultiplikationVektoren.tex b/buch/papers/clifford/3_MultiplikationVektoren.tex index 841dde4..0969b89 100644 --- a/buch/papers/clifford/3_MultiplikationVektoren.tex +++ b/buch/papers/clifford/3_MultiplikationVektoren.tex @@ -1,11 +1,14 @@ \subsection{Multiplikation von Vektoren} -Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipliziert werden? +Was geschieht nun wenn zwei beliebige Vektoren, $u$ und $v$ \begin{equation} \textbf{u} = \sum_{i=1}^{n} u_i \textbf{e}_i \qquad \textbf{v} = \sum_{i=1}^{n} v_i \textbf{e}_i \end{equation} + miteinander multipliziert werden? + + Wieder werden die Vektoren zuerst als Linearkombinationen darstellen und danach in zwei Summen aufgeteilt, eine Summe mit quadrierten Termen und eine Summe mit Mischtermen \begin{equation} \begin{split} \textbf{u}\textbf{v} @@ -18,12 +21,12 @@ Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipli \right) = \sum_{i=1}^n u_iv_i\underbrace{\textbf{e}_i^2}_{1} - + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j, \end{split} \end{equation} +wobei die Summe der quadrierten Termen bereits bekannt vorkommen könnte, es ist nämlich das Skalarprodukt von $u$ und $v$. Die Summe der Mischterme bilden etwas neues, dass wir das äussere Produkt von $u$ und $v$ nennen. \begin{beispiel} Multiplikation von Vektoren in $\mathbb{R}^2$ -\end{beispiel} \begin{equation} \begin{split} \textbf{u}\textbf{v} @@ -44,7 +47,7 @@ Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipli \underbrace{(u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2}_{\text{Äusseres Produkt}} \end{split} \end{equation} -Der linke Teil dieser Multiplikation ergibt das Skalarprodukt der zwei Vektoren, der rechte Term ergibt etwas neues das sich das äussere Produkt der zwei Vektoren nennt. +\end{beispiel} \subsubsection{Äusseres Produkt} Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt \begin{equation} @@ -53,123 +56,118 @@ Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j \end{equation} \begin{beispiel} -Äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$ -\end{beispiel} +Das äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$ ist \begin{equation} - \begin{split} - u \wedge v - &= - u_1v_2\textbf{e}_1\textbf{e}_2 - + - u_1v_3\textbf{e}_1\textbf{e}_3 - + - u_2v_2\textbf{e}_2\textbf{e}_3 - + - u_2v_1\textbf{e}_2\textbf{e}_1 - + - u_3v_1\textbf{e}_3\textbf{e}_1 - + - u_3v_2\textbf{e}_3\textbf{e}_2 \\\ - &= - (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 - + - (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 - + - (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3 - \end{split} + \begin{split} + u \wedge v + &= + u_1v_2\textbf{e}_1\textbf{e}_2 + + + u_1v_3\textbf{e}_1\textbf{e}_3 + + + u_2v_2\textbf{e}_2\textbf{e}_3 + + + u_2v_1\textbf{e}_2\textbf{e}_1 + + + u_3v_1\textbf{e}_3\textbf{e}_1 + + + u_3v_2\textbf{e}_3\textbf{e}_2 \\\ + &= + (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 + + + (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 + + + (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3. + \end{split} \end{equation} -Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \ref{eq:u_wedge_v}-\ref{eq:u_wedge_v_5} hergeleitet. +\end{beispiel} + +Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \eqref{eq:u_wedge_v}-\eqref{eq:u_wedge_v_5} hergeleitet. Die Summe, \begin{align} \textbf{u}\wedge \textbf{v} &= \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n - u_iv_j\textbf{e}_i\textbf{e}_j + u_iv_j\textbf{e}_i\textbf{e}_j, \label{eq:u_wedge_v} - \\ + \intertext{wird in zwei verschiedene Summen aufgeteilt. + Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat} \label{eq:u_wedge_v_1} &= \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + - \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j - \\ + \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j. + \intertext{Nun werden die Indexe der zweiten Summe vertauscht} \label{eq:u_wedge_v_2} &= \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + - \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i - \\ - \label{eq:u_wedge_v_3} + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i, + \intertext{und diese wird nun mit Hilfe der Antikommutativität umgeformt zu} &= \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j - - \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j - \\ + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j. + \intertext{Nun können die zwei Summen wieder zusammengefasst werden} \label{eq:u_wedge_v_4} &= - \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j - \\ - \label{eq:u_wedge_v_5} + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j. + \intertext{Der Term in der Summe könnte einem bereits bekannt vorkommen, es ist nämlich die Determinante einer Matrix mit $u$ und $v$ als ihre Spalten} &= + \label{eq:u_wedge_v_5} \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n \begin{vmatrix} u_i & v_i \\ u_j & v_j - \end{vmatrix}\textbf{e}_i\textbf{e}_j + \end{vmatrix}\textbf{e}_i\textbf{e}_j. \end{align} -Die Summe aus \ref{eq:u_wedge_v_1} wird in \ref{eq:u_wedge_v} in zwei verschiedene Summen aufgeteilt. -Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat. -\newline -Bei \ref{eq:u_wedge_v_2} werden die Indexe der zweiten Summe vertauscht, damit man nun bei beiden Teilen die gleiche Summe hat. -Danach werden in \ref{eq:u_wedge_v_3}, mit Hilfe der Antikommutativität, die Einheitsvektoren der zweiten Summe vertauscht. -\newline -Nun können die Summen, wie in \ref{eq:u_wedge_v_4} wieder in eine Summe zusammengefasst werden. -\newline -Der Term in der Klammer in \ref{eq:u_wedge_v_4} kann auch als Determinante einer 2x2 Matrix dargestellt werden, was in \ref{eq:u_wedge_v_5} gemacht wird. -\newline -Die Determinante einer Matrix beschreibt welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt. -\begin{figure} -\centering -\begin{tikzpicture} - \draw[thin,gray!40] (0,0) grid (4,4); - \draw[<->] (0,0)--(4,0) ; - \draw[<->] (0,0)--(0,4) ; - \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); - \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north - west]{$\boldsymbol{u}$}; - \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; - \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$\begin{vmatrix} - u_i & v_i \\ - u_j & v_j - \end{vmatrix} = u_iv_j - v_iu_j$}; -\end{tikzpicture} -\caption{Geometrische Interpretation der Determinante einer 2x2 Matrix\label{figure:det}} +Die Determinante einer Matrix beschreibt die Fläche, welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt. +\begin{figure}[htb] + \centering + \begin{minipage}[t]{.45\linewidth} + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) ; + \draw[<->] (0,0)--(0,4) ; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[black] (2,1.5)--(1.8,3.2) node[anchor = south]{$\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} = u_iv_j - v_iu_j$}; + \end{tikzpicture} + \caption{Geometrische Interpretation der Determinante einer $2 \times 2$ Matrix\label{figure:det}} + \end{minipage}% + \hfill% + \begin{minipage}[t]{.45\linewidth} + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) node[right]{$x$}; + \draw[<->] (0,0)--(0,4) node[above]{$y$}; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[->] (2.15,1.5) arc (0:310:0.3); + \draw[black] (2,1.5)--(2.5,3.2) node[anchor = south]{$u\wedge v = \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$}; + \end{tikzpicture} + \caption{Geometrische Interpretation des äusseren Produktes \label{figure:wedge}} + \end{minipage} \end{figure} -\newline Das äussere Produkt besteht nun also aus der Summe - $\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n$ + \(\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n\) von Flächen - $\begin{vmatrix} - u_i & v_i \\ - u_j & v_j - \end{vmatrix}$, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. + \(\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}\) +, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. Dieses Produkt $\textbf{e}_i\textbf{e}_j$ der Basisvektoren interpretiert man als Umlaufrichtung. Wobei die gebildete Fläche in Richtung des ersten Vektors umschritten wird. -Dies ist in \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. +Dies ist in Abbildung \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. Diese Fläche mit einer Richtung nennt man in der geometrischen Algebra einen Bivektor, da er eine Art zwei dimensionaler Vektor ist. -\begin{figure} -\centering -\begin{tikzpicture} - \draw[thin,gray!40] (0,0) grid (4,4); - \draw[<->] (0,0)--(4,0) node[right]{$x$}; - \draw[<->] (0,0)--(0,4) node[above]{$y$}; - \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); - \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north - west]{$\boldsymbol{u}$}; - \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; - \draw[->] (2.15,1.5) arc (0:310:0.3); - \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$u\wedge v = \begin{vmatrix} - u_i & v_i \\ - u_j & v_j - \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$}; -\end{tikzpicture} -\caption{Geometrische Interpretation des äusseren Produkt in $\mathbb{R}^2$\label{figure:wedge}} -\end{figure}
\ No newline at end of file diff --git a/buch/papers/clifford/3d/Makefile b/buch/papers/clifford/3d/Makefile new file mode 100644 index 0000000..147ca81 --- /dev/null +++ b/buch/papers/clifford/3d/Makefile @@ -0,0 +1,38 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: dq.jpg q23.jpg q31.jpg drehung.jpg dq.pdf qq.pdf drehung.pdf + +size="+W3840 +H2160" + +dq.png: dq.pov common.inc + povray +A0.1 $(size) -Odq.png dq.pov +dq.jpg: dq.png Makefile + convert -extract 1600x1400+1500+60 dq.png -density 300 -units PixelsPerInch dq.jpg +dq.pdf: dq.jpg dq.tex + pdflatex dq.tex + +extract="1200x1200+1450+350" + +q23.png: q23.pov common.inc + povray +A0.1 $(size) -Oq23.png q23.pov +q23.jpg: q23.png Makefile + convert -extract $(extract) q23.png -density 300 -units PixelsPerInch q23.jpg + +q31.png: q31.pov common.inc + povray +A0.1 $(size) -Oq31.png q31.pov +q31.jpg: q31.png Makefile + convert -extract $(extract) q31.png -density 300 -units PixelsPerInch q31.jpg + +qq.pdf: qq.tex q31.jpg q23.jpg + pdflatex qq.tex + +drehung.png: drehung.pov common.inc + povray +A0.1 $(size) -Odrehung.png drehung.pov +drehung.jpg: drehung.png Makefile + convert -extract 1600x1450+1400+50 drehung.png -density 300 -units PixelsPerInch drehung.jpg +drehung.pdf: drehung.tex drehung.jpg + pdflatex drehung.tex + diff --git a/buch/papers/clifford/3d/common.inc b/buch/papers/clifford/3d/common.inc new file mode 100644 index 0000000..55bf6e1 --- /dev/null +++ b/buch/papers/clifford/3d/common.inc @@ -0,0 +1,271 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.14; +#declare r = 0.02; +#declare thick = 0.040; + +camera { + location <40, 12, 15> + look_at <0, 0, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <40, 20, 20> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +// +// draw an arrow from <from> to <to> with thickness <arrowthickness> with +// color <c> +// +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + + +arrow(< -3, 0, 0 >, < 3, 0, 0 >, r, White) +arrow(< 0, -3, 0 >, < 0, 3, 0 >, r, White) +arrow(< 0, 0, -3 >, < 0, 0, 3 >, r, White) + +#macro circlearrow0(e1, e2, e3, r1, r2, h, winkel) + +mesh { + #declare N = 100; + #declare phi = 0; + #declare phimax = winkel - pi / 12; + #declare phistep = (phimax - phi) / N; + #while (phi < phimax - phistep/2) + triangle { + center + r1 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) - h * e3 + } + triangle { + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) - h * e3, + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r2 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) - h * e3 + } + triangle { + center + r1 * (cos(phi ) * e1 + sin(phi ) * e2) + h * e3, + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) + h * e3, + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3 + } + triangle { + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3, + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) + h * e3, + center + r2 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3 + } + triangle { + center + r1 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) - h * e3, + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3 + } + triangle { + center + r1 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r1 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3, + center + r1 * (cos(phi ) * e1 + sin(phi ) * e2) + h * e3 + } + triangle { + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r2 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) - h * e3, + center + r2 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3 + } + triangle { + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) - h * e3, + center + r2 * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + h * e3, + center + r2 * (cos(phi ) * e1 + sin(phi ) * e2) + h * e3 + } + #declare phi = phi + phistep; + #end + + triangle { + center + r1 * e1 - h * e3, + center + r1 * e1 + h * e3, + center + r2 * e1 + h * e3 + } + triangle { + center + r2 * e1 - h * e3, + center + r2 * e1 + h * e3, + center + r1 * e1 - h * e3 + } + triangle { + center + r1 * cos(phi) * e1 + r1 * sin(phi) * e2 - h * e3, + center + r2 * cos(phi) * e1 + r2 * sin(phi) * e2 - h * e3, + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) - h * e3 + } + triangle { + center + r1 * cos(phi) * e1 + r1 * sin(phi) * e2 + h * e3, + center + r2 * cos(phi) * e1 + r2 * sin(phi) * e2 + h * e3, + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) + h * e3 + } + triangle { + center + r1 * cos(phi) * e1 + r1 * sin(phi) * e2 - h * e3, + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) - h * e3 + center + r1 * cos(phi) * e1 + r1 * sin(phi) * e2 + h * e3 + } + triangle { + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) - h * e3 + center + r1 * cos(phi) * e1 + r1 * sin(phi) * e2 + h * e3, + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) + h * e3 + } + triangle { + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) - h * e3, + center + r2 * cos(phi) * e1 + r2 * sin(phi) * e2 - h * e3, + center + r2 * cos(phi) * e1 + r2 * sin(phi) * e2 + h * e3 + } + triangle { + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) - h * e3, + center + r2 * cos(phi) * e1 + r2 * sin(phi) * e2 + h * e3, + center + 0.5*(r1+r2) * (cos(phi + pi/12) * e1 + sin(phi + pi/12) * e2) + h * e3 + } + + pigment { + color rgb<1, 0.4, 0.4> + } +} + +#end + + +#macro circlearrow(fromdirection, axis, center, r, h, winkel, anzahl) + +#declare e1 = vnormalize(fromdirection); +#declare e2 = -vnormalize(vcross(axis, fromdirection)); +#declare e3 = vnormalize(axis); + +#declare r1 = 0.4 * r; +#declare r2 = r; + +#declare w = 0; +#while (w < anzahl) + #declare a = 2 * w * pi / anzahl; + circlearrow0(e1 * cos(a) - e2 * sin(a), e1 * sin(a) + e2 * cos(a), e3, r1, r2, 1.2 * h, winkel) + #declare w = w + 1; +#end + +mesh { + #declare vlu = center - r * e1 - r * e2 - h * e3; + #declare vlo = center - r * e1 - r * e2 + h * e3; + #declare vru = center - r * e1 + r * e2 - h * e3; + #declare vro = center - r * e1 + r * e2 + h * e3; + #declare hlu = center + r * e1 - r * e2 - h * e3; + #declare hlo = center + r * e1 - r * e2 + h * e3; + #declare hru = center + r * e1 + r * e2 - h * e3; + #declare hro = center + r * e1 + r * e2 + h * e3; + triangle { vlu, vru, vro } + triangle { vlu, vro, vlo } + + triangle { vru, hru, hro } + triangle { vru, hro, vro } + + triangle { hru, hlu, hlo } + triangle { hru, hlo, hro } + + triangle { hlu, vlu, vlo } + triangle { hlu, vlo, hlo } + + triangle { vlu, vru, hru } + triangle { vlu, hru, hlu } + + triangle { vlo, vro, hro } + triangle { vlo, hro, hlo } + + pigment { + color rgb<0.6,0.6,1> + } + finish { + specular 0.96 + metallic + } +} + +#if (vlength(axis) > 0.1) +cone { + center + 1.19 * h * e3, r, center + 2 * r * e3, 0 + pigment { + color rgbt<0.6,0.6,1,0.8> + } +} +#end + +cylinder { + center, center + 2 * r * e3, 0.04*0.2 + pigment { + color rgb<1.0,0.6,0.6> + } + finish { + specular 0.96 + metallic + } +} + +#end + +#macro bogen(v1, v2, center, winkelbogen, farbe) + +union { + #declare phi = 0; + #declare phimax = winkelbogen; + #declare phistep = (phimax - phi) / N; + #while (phi < phimax - phistep/2) + cylinder { + cos(phi ) * v1 + sin(phi ) * v2 + center, + cos(phi+phistep) * v1 + sin(phi+phistep) * v2 + center, + 0.01 + } + sphere { + cos(phi ) * v1 + sin(phi ) * v2 + center, + 0.01 + } + #declare phi = phi + phistep; + #end + pigment { + color farbe + } +} + +#end diff --git a/buch/papers/clifford/3d/dq.jpg b/buch/papers/clifford/3d/dq.jpg Binary files differnew file mode 100644 index 0000000..690cfdc --- /dev/null +++ b/buch/papers/clifford/3d/dq.jpg diff --git a/buch/papers/clifford/3d/dq.pdf b/buch/papers/clifford/3d/dq.pdf Binary files differnew file mode 100644 index 0000000..797a558 --- /dev/null +++ b/buch/papers/clifford/3d/dq.pdf diff --git a/buch/papers/clifford/3d/dq.pov b/buch/papers/clifford/3d/dq.pov new file mode 100644 index 0000000..762eee2 --- /dev/null +++ b/buch/papers/clifford/3d/dq.pov @@ -0,0 +1,30 @@ +// +// dq.pov -- Drehung und Quaternion +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +arrow(<0,0,0>, <1, sqrt(2), 2>, r, Red) + +#declare r = 0.2 * r; + +#declare drehwinkel = 0.95 * 2*pi/3 * 3; +#declare drehwinkel23 = drehwinkel; +#declare drehwinkel12 = drehwinkel / sqrt(2); +#declare drehwinkel13 = drehwinkel / 2; + +circlearrow(<1,0,0>, <0,0,1>, <1, sqrt(2), 0>, 1, thick, drehwinkel23, 1) +circlearrow(<1,0,0>, <0,1,0>, <1, 0, 2>, sqrt(2)/2, thick, drehwinkel12, 1) +circlearrow(<0,0,1>, <1,0,0>, <0, sqrt(2), 2>, 0.5, thick, drehwinkel13, 1) + +#declare l = 2.8; +#declare h = 0.0001; +union { + box { <-l,-l,-h>, <l,l,-h> } + box { <-l,-h,-l>, <l,-h,l> } + box { <-h,-l,-l>, <-h,l,l> } + pigment { + color rgbt<0.6,0.6,0.6,0.0> + } +} diff --git a/buch/papers/clifford/3d/dq.tex b/buch/papers/clifford/3d/dq.tex new file mode 100644 index 0000000..6b28452 --- /dev/null +++ b/buch/papers/clifford/3d/dq.tex @@ -0,0 +1,51 @@ +% +% dq.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\definecolor{darkred}{rgb}{0.7,0,0} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{6} +\def\hoehe{6} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=12cm]{dq.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (-2.8,-2.7) {$O$}; +\node at (4.7,-3.4) {$a_1$}; +\node at (-2.6,5.2) {$a_2$}; +\fill[color=white,opacity=0.7] ({-5.7-0.25},{-4.8-0.15}) rectangle ({-5.7+0.25},{-4.8+0.2}); +\node at (-5.7,-4.8) {$a_3$}; + +\node[color=blue] at (-3.6,0.8) {$y\mathbf{e}_{23}$}; +\node[color=blue] at (2.1,0.9) {$x\mathbf{e}_{12}$}; +\node[color=blue] at (1.3,-3.7) {$z\mathbf{e}_{13}$}; + +\node[color=darkred] at (1.3,0.4) {$\vec{q}$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/clifford/3d/drehung.jpg b/buch/papers/clifford/3d/drehung.jpg Binary files differnew file mode 100644 index 0000000..2347296 --- /dev/null +++ b/buch/papers/clifford/3d/drehung.jpg diff --git a/buch/papers/clifford/3d/drehung.pdf b/buch/papers/clifford/3d/drehung.pdf Binary files differnew file mode 100644 index 0000000..bc8036e --- /dev/null +++ b/buch/papers/clifford/3d/drehung.pdf diff --git a/buch/papers/clifford/3d/drehung.pov b/buch/papers/clifford/3d/drehung.pov new file mode 100644 index 0000000..b86a2c5 --- /dev/null +++ b/buch/papers/clifford/3d/drehung.pov @@ -0,0 +1,87 @@ +// +// drehung.pov -- Drehung um (1,1,1) +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +#declare n = vnormalize(<1,1,1>); +#declare V = <0,2.6,0>; +#declare W = <0,0,2.6>; + +#declare Vparallel = vdot(n, V) * n; +#declare Vperp = V - Vparallel; +#declare Wparallel = vdot(n, W) * n; +#declare Wperp = W - Wparallel; + +arrow(<0,0,0>, 2*n, thick, Red) + +arrow(<0,0,0>, V, thick, rgb<0.0,1.0,1.0>) +arrow(<0,0,0>, W, thick, rgb<0.0,1.0,1.0>) + +circlearrow(vnormalize(vcross(<-1,0,1>,n)), -0.01 * <1,1,1>, <0,0,0>, 1, 0.8*thick, 1.98*pi/3, 3) + +arrow(<0,0,0>, Vperp, 0.99*thick, Blue) +arrow(<0,0,0>, Wperp, 0.99*thick, Blue) + +arrow(Vperp, V, thick, Green) +arrow(Wperp, W, thick, Green) + +#declare l = 2.4; +intersection { + box { <-l,-l,-l>, <l,l,l> } + //cylinder { -n, n, 3 } + plane { n, 0.01 } + plane { -n, 0.01 } + pigment { + color rgbt<0.6,0.6,1.0,0.8> + } +} + +#declare e1 = vnormalize(Vperp); +#declare e3 = n; +#declare e2 = vnormalize(vcross(e3, e1)); +#declare r = vlength(Vperp); + +mesh { + #declare phi = 0; + #declare phimax = 2*pi/3; + #declare phistep = (phimax - phi) / N; + #while (phi < phimax - phistep/2) + triangle { + <0,0,0>, + r * (cos(phi ) * e1 + sin(phi ) * e2), + r * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + } + #declare phi = phi + phistep; + #end + pigment { + color rgbt<0.2,0.2,1.0,0.4> + } +} + +mesh { + #declare phi = 0; + #declare phimax = 2*pi/3; + #declare phistep = (phimax - phi) / N; + #while (phi < phimax - phistep/2) + triangle { + r * (cos(phi ) * e1 + sin(phi ) * e2), + r * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2), + r * (cos(phi ) * e1 + sin(phi ) * e2) + Vparallel + } + triangle { + r * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2), + r * (cos(phi ) * e1 + sin(phi ) * e2) + Vparallel, + r * (cos(phi+phistep) * e1 + sin(phi+phistep) * e2) + Vparallel + } + #declare phi = phi + phistep; + #end + pigment { + color rgbt<0.2,1,0.2,0.4> + } +} + +bogen(r * e1, r * e2, <0,0,0>, 2*pi/3, Blue) +bogen(r * e1, r * e2, Vparallel, 2*pi/3, Green) + diff --git a/buch/papers/clifford/3d/drehung.tex b/buch/papers/clifford/3d/drehung.tex new file mode 100644 index 0000000..2ed6789 --- /dev/null +++ b/buch/papers/clifford/3d/drehung.tex @@ -0,0 +1,56 @@ +% +% drehung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.6,0,0} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{7} +\def\hoehe{6} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\node at (0,0) {\includegraphics[width=13cm]{drehung.jpg}}; + +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\node at (6.1,-3.3) {$a_1$}; +\node at (-2.0,5.7) {$a_2$}; +\node at (-5.7,-4.9) {$a_3$}; + +\node[color=white] at (-1.9,4.4) {$\boldsymbol{v}$}; +\node[color=white] at (4.5,-2.7) {$\boldsymbol{v}''$}; + +\node[color=darkgreen] at (-3.3,4.4) {$\boldsymbol{v}_{\perp}$}; +\node[color=darkgreen] at (4.2,-4.3) {$\boldsymbol{v}''_{\perp}$}; + +\node[color=blue] at (-3.7,1.5) {$\boldsymbol{v}_{\|}$}; +\node[color=blue] at (1.9,-4.7) {$\boldsymbol{v}''_{\|}$}; + +\node[color=darkred] at (-1.6,-4.2) {$2\alpha=120^\circ$}; +\node[color=darkred] at (-4.9,-0.6) {$\boldsymbol{q}$}; + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/clifford/3d/q23.jpg b/buch/papers/clifford/3d/q23.jpg Binary files differnew file mode 100644 index 0000000..929ef90 --- /dev/null +++ b/buch/papers/clifford/3d/q23.jpg diff --git a/buch/papers/clifford/3d/q23.pov b/buch/papers/clifford/3d/q23.pov new file mode 100644 index 0000000..2e55c96 --- /dev/null +++ b/buch/papers/clifford/3d/q23.pov @@ -0,0 +1,14 @@ +// +// q23.pov -- Drehung und Quaternion +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +circlearrow(<1,0,0>, 0.01*<0,0,-1>, <0, 0, 0>, 1.0, thick, 0.98*pi/2, 4) + +bogen( <0,1.7,0>, <-1.7, 0, 0>, <0,0,0>, pi/2, Blue) + +arrow( <0,0,0>, <-2.0,0,0>, 0.99*thick, Blue) +arrow( <0,0,0>, <0,2.0,0>, 0.99*thick, Blue) +arrow( <0,0,0>, <0,0,2.0>, 0.99*thick, Red) diff --git a/buch/papers/clifford/3d/q31.jpg b/buch/papers/clifford/3d/q31.jpg Binary files differnew file mode 100644 index 0000000..c240b4f --- /dev/null +++ b/buch/papers/clifford/3d/q31.jpg diff --git a/buch/papers/clifford/3d/q31.pov b/buch/papers/clifford/3d/q31.pov new file mode 100644 index 0000000..4abe1ed --- /dev/null +++ b/buch/papers/clifford/3d/q31.pov @@ -0,0 +1,15 @@ +// +// q31.pov -- Drehung und Quaternion +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +circlearrow(<1,0,0>, 0.01*<0,-1,0>, <0, 0, 0>, 1.0, thick, 0.98*pi/2, 4) + +arrow( <0,0,0>, <-2.0,0,0>, 0.99*thick, Blue) +arrow( <0,0,0>, <0,2.0,0>, 0.99*thick, Red) +arrow( <0,0,0>, <0,0,2.0>, 0.99*thick, Blue) + +bogen( <0,0,1.7>, <-1.7, 0, 0>, <0,0,0>, pi/2, Blue) + diff --git a/buch/papers/clifford/3d/qq.pdf b/buch/papers/clifford/3d/qq.pdf Binary files differnew file mode 100644 index 0000000..fd7dbfa --- /dev/null +++ b/buch/papers/clifford/3d/qq.pdf diff --git a/buch/papers/clifford/3d/qq.tex b/buch/papers/clifford/3d/qq.tex new file mode 100644 index 0000000..9baa8bb --- /dev/null +++ b/buch/papers/clifford/3d/qq.tex @@ -0,0 +1,68 @@ +% +% qq.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{times} +\usepackage{amsmath} +\usepackage{txfonts} +\usepackage[utf8]{inputenc} +\usepackage{graphics} +\usetikzlibrary{arrows,intersections,math} +\usepackage{ifthen} +\begin{document} + +\definecolor{darkred}{rgb}{0.7,0,0} + +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{4} +\def\hoehe{4} + +\begin{tikzpicture}[>=latex,thick] + +% Povray Bild +\begin{scope}[xshift=-3.3cm] +\node at (0,0) {\includegraphics[width=6.3cm]{q23.jpg}}; +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} +\fill[color=white,opacity=0.5] ({-0.6-0.3},{-0.2-0.2}) rectangle ({-0.6+0.3},{-0.2+0.2}); +\node[color=darkred] at (-0.6,-0.2) {$\boldsymbol{q}_{23}$}; +\node[color=blue] at (-0.4,2.7) {$\boldsymbol{v}$}; +\node[color=blue] at (0.7,0.4) {$\boldsymbol{v}''_{23}$}; +\node at (3.1,-1.4) {$a_1$}; +\node at (-2.7,-2.4) {$a_3$}; +\node at (-0.7,3.4) {$a_2$}; +\end{scope} + +\setboolean{showgrid}{false} + +\begin{scope}[xshift=3.3cm] +\node at (0,0) {\includegraphics[width=6.3cm]{q31.jpg}}; +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} +\fill[color=white,opacity=0.5] ({-0.7-0.3},{-0.9-0.2}) rectangle ({-0.7+0.3},{-0.9+0.2}); +\node[color=darkred] at (-0.7,-0.9) {$\boldsymbol{q}_{13}$}; +\node[color=blue] at (0.7,0.4) {$\boldsymbol{v}''_{23}$}; +\node[color=blue] at (2.7,-0.7) {$\boldsymbol{v}''$}; +\node at (3.1,-1.4) {$a_1$}; +\node at (-2.7,-2.4) {$a_3$}; +\node at (-0.7,3.4) {$a_2$}; +\end{scope} + + +\end{tikzpicture} + +\end{document} + diff --git a/buch/papers/clifford/4_GeometrischesProdukt.tex b/buch/papers/clifford/4_GeometrischesProdukt.tex index a19e983..f18b90d 100644 --- a/buch/papers/clifford/4_GeometrischesProdukt.tex +++ b/buch/papers/clifford/4_GeometrischesProdukt.tex @@ -12,9 +12,9 @@ Ein Multivektor besteht aus den verschiedenen Bauteilen, wie zum Beispiel Vektor M = \sum \left ( \prod a_i\textbf{e}_j \right) \end{equation} \end{definition} -Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $\mathbb{G}^n$ beschrieben. +Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $G_n(\mathbb{R})$ beschrieben. Dies wird so geschrieben da man eine neue Algebrastruktur um die Vektoren einführt. \begin{beispiel} -Allgemeiner Multivektor in $\mathbb{G}^3$ +Allgemeiner Multivektor in $G_3(\mathbb{R})$ \begin{equation} M = a + @@ -26,34 +26,30 @@ Allgemeiner Multivektor in $\mathbb{G}^3$ \end{equation} \end{beispiel} \begin{definition} -Um das Produkt von Basisvektoren in Zukunft darzustellen wird folgende Notation definiert +Für das Produkt von Basisvektoren wird folgende Notation definiert \begin{equation} - e_ie_j = e_{ij} + e_ie_j = e_{ij}. \end{equation} \end{definition} -Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $\mathbb{G}^n$ erstellt werden. In \ref{tab:multip} ist dies für $\mathbb{G}^3$ gemacht. +Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $G_n(\mathbb{R})$ erstellt werden. In Tabelle \ref{tab:multip} ist dies für $G_3(\mathbb{R})$ gemacht. \begin{table} - \caption{Multiplikationstabelle für $\mathbb{G^3}$} \label{tab:multip} \begin{center} - \begin{tabular}{ |c|c|c|c|c|c|c|c| } + \begin{tabular}{ |c|ccc|ccc|c| } \hline 1 & $\textbf{e}_1$ & $\textbf{e}_2$ &$\textbf{e}_3$ & $\textbf{e}_{12}$ & $\textbf{e}_{13}$ & $\textbf{e}_{23}$ & $\textbf{e}_{123}$\\ \hline $\textbf{e}_1$ & 1 & $\textbf{e}_{12}$ & $\textbf{e}_{12}$ & $\textbf{e}_2$ & $\textbf{e}_3$ & $\textbf{e}_{123}$ & $\textbf{e}_{23}$\\ - \hline $\textbf{e}_2$ & $-\textbf{e}_{12}$ & 1 & $\textbf{e}_{23}$ & $-\textbf{e}_1$ & $-\textbf{e}_{123}$ & $\textbf{e}_3$ & $-\textbf{e}_{13}$\\ - \hline $\textbf{e}_3$ & $-\textbf{e}_{13}$ & $-\textbf{e}_{23}$ & 1 & $\textbf{e}_{123}$ & $-\textbf{e}_1$ & $-\textbf{e}_2$ & $\textbf{e}_{12}$\\ \hline $\textbf{e}_{12}$ & -$\textbf{e}_2$ & $\textbf{e}_1$& $\textbf{e}_{123}$ & -1 & $-\textbf{e}_{23}$ & $\textbf{e}_{13}$ & $-\textbf{e}_{3}$\\ - \hline $\textbf{e}_{13}$ & $-\textbf{e}_{3}$ & $-\textbf{e}_{123}$ & $\textbf{e}_{1}$ & $\textbf{e}_{23}$ & -1 & $-\textbf{e}_{12}$ & $\textbf{e}_{2}$\\ - \hline $\textbf{e}_{23}$ & $\textbf{e}_{123}$ & $-\textbf{e}_{3}$ & $\textbf{e}_{2}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & -1 & $-\textbf{e}_{1}$ \\ \hline $\textbf{e}_{123}$ & $\textbf{e}_{23}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & $-\textbf{e}_{3}$& $\textbf{e}_{2}$ & $-\textbf{e}_{1}$ & -1 \\ \hline \end{tabular} \end{center} + \caption{Multiplikationstabelle für $G_3(\mathbb{R})$} \end{table} diff --git a/buch/papers/clifford/6_PauliMatrizen.tex b/buch/papers/clifford/6_PauliMatrizen.tex index e41275a..4e82f28 100644 --- a/buch/papers/clifford/6_PauliMatrizen.tex +++ b/buch/papers/clifford/6_PauliMatrizen.tex @@ -10,33 +10,33 @@ Was ist der beste Weg um einen Computeralgorithmus für die Rechenoperationen in \begin{beispiel} Der Algorithmus weiss, dass er $a\mathbf{e}_1\cdot b\mathbf{e}_1$ zu $ab\cdot1$ vereinfachen kann. Dies ermöglicht zum Beispiel die Vereinfachung \begin{align} - 3\mathbf{e}_1 \cdot 2\mathbf{e}_1 + 3\mathbf{e}_2 \Rightarrow 6 + 3\mathbf{e}_2 + 3\mathbf{e}_1 \cdot 2\mathbf{e}_1 + 3\mathbf{e}_2 \Rightarrow 6 + 3\mathbf{e}_2 \end{align} \end{beispiel} Ein textueller Algorithmus ist aber sehr ineffizient. Die Pauli-Matrizen bilden eine elegante und schnellere Alternative, welche für die dreidimensionale Clifford-Algebra verwendet werden können und alle Operationen aus der Clifford-Algebra gleich wie die Matrixoperationen ausführen lassen. \begin{definition} \label{def:defPauli} Die Matrizen \begin{align} \label{Pauli} - \mathbf{e}_0 = E = - \begin{pmatrix} - 1 & 0 \\ - 0 & 1 - \end{pmatrix},\quad - \mathbf{e}_1 = - \begin{pmatrix} - 0 & 1 \\ - 1 & 0 - \end{pmatrix},\quad - \mathbf{e}_2 = - \begin{pmatrix} - 0 & -j \\ - j & 0 - \end{pmatrix},\quad - \mathbf{e}_3 = - \begin{pmatrix} - 1 & 0 \\ - 0 & -1 - \end{pmatrix} + \mathbf{e}_0 = E = + \begin{pmatrix} + 1 & 0 \\ + 0 & 1 + \end{pmatrix},\quad + \mathbf{e}_1 = + \begin{pmatrix} + 0 & 1 \\ + 1 & 0 + \end{pmatrix},\quad + \mathbf{e}_2 = + \begin{pmatrix} + 0 & -j \\ + j & 0 + \end{pmatrix},\quad + \mathbf{e}_3 = + \begin{pmatrix} + 1 & 0 \\ + 0 & -1 + \end{pmatrix} \end{align} heissen Pauli-Matrizen ($\mathbf{e}_0$ = Skalare) \end{definition} @@ -44,85 +44,85 @@ Die Matrix-Multiplikationen der Pauli-Matrizen führt auf die gleichen algebrais \begin{definition} \label{def:defPauli2} Die Bivektoren und Trivektoren hergeleitet aus den Pauli-Matrizen sind \begin{align} \label{Pauli2} - \mathbf{e}_{12} = - \begin{pmatrix} - j & 0 \\ - 0 & -j - \end{pmatrix}\quad - \mathbf{e}_{23} = - \begin{pmatrix} - 0 & j \\ - j & 0 - \end{pmatrix}\quad - \mathbf{e}_{31} = - \begin{pmatrix} - 0 & 1 \\ - -1 & 0 - \end{pmatrix}\enspace\text{und}\enspace - \mathbf{e}_{123} = - \begin{pmatrix} - j & 0 \\ - 0 & j - \end{pmatrix}. - \end{align} -\end{definition} -Dabei ist wichtig, dass sich die Matrizen gleich verhalten, wie es die Clifford-Algebra für die Basiselemente definiert hat. Zum Beispiel gilt in der Clifford-Algebra $\mathbf{e}_1^2=\mathbf{e}_0$ und $\mathbf{e}_{12}^2=-\mathbf{e}_0$, genau die selbe Relation gilt auch für die zugehörigen Matrizen, wie man durch die Matrizenrechnungen -\begin{align} - \mathbf{e}_1^2 &= + \mathbf{e}_{12} = \begin{pmatrix} - 0 & 1 \\ - 1 & 0 - \end{pmatrix}^2 = + j & 0 \\ + 0 & -j + \end{pmatrix}\quad + \mathbf{e}_{23} = \begin{pmatrix} - 1 & 0 \\ - 0 & 1 - \end{pmatrix}= \mathbf{e}_0 \quad\text{und}\\ - \mathbf{e}_{12}^2 &= + 0 & j \\ + j & 0 + \end{pmatrix}\quad + \mathbf{e}_{31} = \begin{pmatrix} - j & 0 \\ - 0 & -j - \end{pmatrix}^2 = + 0 & 1 \\ + -1 & 0 + \end{pmatrix}\enspace\text{und}\enspace + \mathbf{e}_{123} = \begin{pmatrix} - -1 & 0 \\ - 0 & -1 - \end{pmatrix} = -\mathbf{e}_0 + j & 0 \\ + 0 & j + \end{pmatrix}. + \end{align} +\end{definition} +Dabei ist wichtig, dass sich die Matrizen gleich verhalten, wie es die Clifford-Algebra für die Basiselemente definiert hat. Zum Beispiel gilt in der Clifford-Algebra $\mathbf{e}_1^2=\mathbf{e}_0$ und $\mathbf{e}_{12}^2=-\mathbf{e}_0$, genau die selbe Relation gilt auch für die zugehörigen Matrizen, wie man durch die Matrizenrechnungen +\begin{align} +\mathbf{e}_1^2 &= +\begin{pmatrix} +0 & 1 \\ +1 & 0 +\end{pmatrix}^2 = +\begin{pmatrix} +1 & 0 \\ +0 & 1 +\end{pmatrix}= \mathbf{e}_0 \quad\text{und}\\ +\mathbf{e}_{12}^2 &= +\begin{pmatrix} +j & 0 \\ +0 & -j +\end{pmatrix}^2 = +\begin{pmatrix} +-1 & 0 \\ +0 & -1 +\end{pmatrix} = -\mathbf{e}_0 \end{align} -bestätigt. Man kann bei den Definitionen \ref{def:defPauli} und \ref{def:defPauli2} sehen, dass alle Matrizen linear unabhängig voneinander sind. Das bedeutet, dass wenn man die Matrizen der Basiselemente normal addiert und zu einer Matrix zusammenfasst, kann man anschliessend die einzelnen Anteile der Basiselemente wieder herausgelesen. +bestätigt. Man kann bei den Definitionen \ref{def:defPauli} und \ref{def:defPauli2} sehen, dass alle Matrizen linear unabhängig voneinander sind. Das bedeutet, dass wenn man die Matrizen der Basiselemente normal addiert und zu einer Matrix zusammenfasst, kann man anschliessend die einzelnen Anteile der Basiselemente wieder herauslesen. \begin{hilfssatz} Ein beliebiger Multivektor \begin{align} \label{MultiVektorAllg} - M = a_0\mathbf{e}_0 + a_1\mathbf{e}_1 + a_2\mathbf{e}_3 + a_{12}\mathbf{e}_{12} + a_{23}\mathbf{e}_{23} + a_{31}\mathbf{e}_{31} + a_{123}\mathbf{e}_{123}\\ + M = a_0\mathbf{e}_0 + a_1\mathbf{e}_1 + a_2\mathbf{e}_3 + a_{12}\mathbf{e}_{12} + a_{23}\mathbf{e}_{23} + a_{31}\mathbf{e}_{31} + a_{123}\mathbf{e}_{123} \end{align} erhält durch das einsetzten der Formel Matrizen \eqref{Pauli} und \eqref{Pauli2} die Form \begin{align} - M = - \begin{pmatrix} - (a_0+a_3) + (a_{12}+a_{123})j & (a_1+a_{31})+(-a_2+a_{23})j \\ - (a_1-a_{31})+(a_2+a_{23})j & (a_0-a_3)+(-a_{12}+a_{123})j - \end{pmatrix}.\label{MultivektorMatirx} + M = + \begin{pmatrix} + (a_0+a_3) + (a_{12}+a_{123})j & (a_1+a_{31})+(-a_2+a_{23})j \\ + (a_1-a_{31})+(a_2+a_{23})j & (a_0-a_3)+(-a_{12}+a_{123})j + \end{pmatrix}.\label{MultivektorMatirx} \end{align} \end{hilfssatz} Die Anteile treten zudem immer paarweise auf und können somit immer je durch zwei Gleichungen bestimmt werden. \begin{beispiel} Die Matrix \begin{align} - M &= - \begin{pmatrix} - 1 & 0 \\ - 0 & -1j - \end{pmatrix} + M &= + \begin{pmatrix} + 1 & 0 \\ + 0 & -1j + \end{pmatrix} \end{align} soll als Multivektor in der Form \eqref{MultiVektorAllg} geschrieben werden. Dafür entnehmen wir aus \eqref{MultivektorMatirx} die Gleichungen \begin{align} - a_0 + a_3 = 1,\quad a_0 - a_3 = 0,\quad a_{12}+a_{123} = 0\enspace\text{und}\enspace -a_{12}+a_{123}=-1 + a_0 + a_3 = 1,\quad a_0 - a_3 = 0,\quad a_{12}+a_{123} = 0\enspace\text{und}\enspace -a_{12}+a_{123}=-1, \end{align} aus denen man auf \begin{align} - a_0 = \dfrac{1}{2},\quad a_3 = \dfrac{1}{2},\quad a_{12}=\dfrac{1}{2}\enspace\text{und}\enspace a_{123}=-\dfrac{1}{2} + a_0 = \dfrac{1}{2},\quad a_3 = \dfrac{1}{2},\quad a_{12}=\dfrac{1}{2}\enspace\text{und}\enspace a_{123}=-\dfrac{1}{2} \end{align} schliessen kann. Da die restlichen Realteile und Imaginärteile 0 sind, werden die anderen Anteile ebenfalls 0 sein. Daher ist \begin{align} - M = \dfrac{1}{2} \mathbf{e}_0+ \dfrac{1}{2} \mathbf{e}_3 + \dfrac{1}{2} \mathbf{e}_{12} - \dfrac{1}{2} \mathbf{e}_{123}. + M = \dfrac{1}{2} \mathbf{e}_0+ \dfrac{1}{2} \mathbf{e}_3 + \dfrac{1}{2} \mathbf{e}_{12} - \dfrac{1}{2} \mathbf{e}_{123}. \end{align} \end{beispiel} Die Clifford-Algebra ist bei der Darstellung durch Matrizen kein Ausnahmefall. Es lässt sich theoretisch jede algebraische Struktur durch Matrizen darstellen.
\ No newline at end of file diff --git a/buch/papers/clifford/7_Reflektion.tex b/buch/papers/clifford/7_Reflektion.tex index bdfb4e8..549848c 100644 --- a/buch/papers/clifford/7_Reflektion.tex +++ b/buch/papers/clifford/7_Reflektion.tex @@ -6,15 +6,15 @@ \section{Spiegelung} \rhead{Spiegelung} -Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere, wie beispielsweise die später beschriebene Rotation, ableiten kann. Da die geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Spiegelung auch eine einfache, praktische Formulierung besitzen. +Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere Operationen, wie beispielsweise die später beschriebene Rotation, ableiten kann. Da die geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Spiegelung auch eine einfache, praktische Formulierung besitzen. \begin{figure} \centering \begin{tikzpicture} \draw[thin,gray!40] (-3,-1) grid (3,3); \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; \draw[<->] (0,-1)--(0,3) node[above]{$a_2$}; + \draw[blue, line width=1.0pt] (0,3)--(0,-1) node[anchor=south east]{$\sigma_u$}; \draw[line width=2pt,black,-stealth](0,0)--(2,2) node[anchor=south east]{$\boldsymbol{v}$}; - \draw[line width=1.5pt,blue,-stealth](0,0)--(0,2.5) node[anchor=south east]{$\boldsymbol{u}$}; \draw[line width=2pt,black,-stealth](0,0)--(-2,2) node[anchor=south east]{$\boldsymbol{v'}$}; \draw[line width=1.5pt,gray,-stealth](0,0)--(1,0) node[anchor=north]{$\boldsymbol{e_1}$}; \draw[line width=1.5pt,gray,-stealth](0,0)--(0,1) node[anchor=north east]{$\boldsymbol{e_2}$}; @@ -22,62 +22,74 @@ Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man we 0.25cm]{$\boldsymbol{v_{\perp u}}$}; \draw[line width=1.5pt,red,-stealth](-2,2)--(0,2) node[xshift=-1cm, yshift= 0.25cm]{$\boldsymbol{v_{\perp u}}$}; - \draw[line width=1.5pt,purple,-stealth](0,1.5)--(1,1.5) node[xshift=-0.5cm, yshift=-0.25cm]{$\boldsymbol{\hat{n}}$}; + \draw[line width=1.5pt,blue,-stealth](0,0.05)--(1,0.05) node[xshift=-0.5cm, yshift=-0.25cm]{$\boldsymbol{\hat{u}}$}; \end{tikzpicture} - \caption{Spiegelung des Vektors \textbf{v} an Spiegelachse bzw. Vektor \textbf{u}} + \caption{Spiegelung des Vektors $\mathbf{v}$ an der Spiegelebene $\sigma_u$ mit dem Normalenvektor $\mathbf{\hat{u}}$} \label{BildSpiegelung} \end{figure} \subsection{Linearen Algebra} Aus der linearen Algebra ist bekannt, dass man eine Spiegelung an einer Ebene wie folgt beschreiben kann. \begin{definition} - Die Spiegelungsgleichung in der linearen Algebra mit dem Normalenvektor $\mathbf{\hat{n}}$ zur Spiegelebene ist + Die Abbildung der Spiegelung in der linearen Algebra mit dem Normalenvektor $\mathbf{\hat{u}}$ zur Spiegelebene ist \begin{equation} \label{RefLinAlg} - \mathbf{v^{'}} = \mathbf{v} - 2 \cdot \mathbf{v_{\parallel \hat{n}}} = \mathbf{v} - 2 \cdot \mathbf{v_{\perp u}}. + \mathbf{v} = \mathbf{v_{\perp u}} + \mathbf{v_{\parallel u}} \enspace\mapsto\enspace \mathbf{v'} = \mathbf{v_{\perp u}} - \mathbf{v_{\parallel u}} = \mathbf{v} - 2 \cdot \mathbf{v_{\parallel u}}. \end{equation} - Per Definition sind $\mathbf{v_{\parallel \hat{n}}} = \mathbf{v_{\perp u}}$. In der geometrischen Algebra verwenden wir aber in den Formeln Vektoren, welche Spiegelachsen, nicht Spiegelebenen, repräsentieren. \end{definition} -Es scheint für diese Formel aber umständlich zu sein, weitere Spiegelungen mit weiteren Spiegelebenen anzufügen. Man kann diese Abbildung aber auch als Matrix schreiben. Sei $\mathbf{\hat{n}}$ ein Normalenvektor auf die Spiegelungs-Achse bzw. -Ebene, also $\mathbf{\hat{n}}\perp \mathbf{u}$, und sei ausserdem normiert $|\mathbf{\hat{n}}| = 1$, dann kann man die Spiegelung durch die Matrix +Es scheint für diese Formel \eqref{RefLinAlg} aber umständlich zu sein, weitere Spiegelungen mit weiteren Spiegelebenen anzufügen. Weil man $\mathbf{v_{\parallel u}}$ auch als Skalarprodukt $\mathbf{v_{\parallel u}} = \mathbf{\hat{u}} \cdot \mathbf{v}$ schreiben kann, ist es leicht diese Abbildung auch als Matrix darzustellen. Sei $\mathbf{\hat{u}}$ ein Normalenvektor auf die Spiegelungsebene, also $\mathbf{\hat{u}}\perp \sigma_u$, und sei ausserdem normiert $|\mathbf{\hat{u}}| = 1$, dann kann man die Spiegelung durch die Matrix \begin{align} - S = E - 2\dfrac{1}{|\mathbf{n}|^2}\mathbf{nn}^t + S = E - 2\mathbf{\hat{u}\hat{u}}^t \end{align} beschrieben werden. In der zweiten und dritten Dimension ergibt die Berechnung \begin{align} \label{Spiegelmatrizen} S_2 = \begin{pmatrix} - 1-2n_1^2 & -2n_1n_2 \\ - -2n_1n_2 & 1-2n_2^2 - \end{pmatrix} \quad + 1-2u_1^2 & -2u_1u_2 \\ + -2u_1u_2 & 1-2u_2^2 + \end{pmatrix}\enspace\text{und}\enspace S_3 = \begin{pmatrix} - 1-2n_1^2 & -2n_1n_2 & -2n_1n_3\\ - -2n_1n_2 & 1-2n_2^2 & -2n_2n_3\\ - -2n_1n_3 & -2n_2n_3 & 1-2n_3^2\\ + 1-2u_1^2 & -2u_1u_2 & -2u_1u_3\\ + -2u_1u_2 & 1-2u_2^2 & -2u_2u_3\\ + -2u_1u_3 & -2u_2u_3 & 1-2u_3^2\\ \end{pmatrix}. \end{align} -Diese Spiegelmatrizen gehören der orthogonalen Matrizengruppe $S\in \text{O}(n)$ an. Die Matrizengruppe $\text{O}(n)$ haben die Eigenschaft $S^t S = E$, was bedeutet, dass die Länge und Winkel bei der Abbildung beibehalten bleiben. Zusätzlich sind die Spiegelmatrizen symmetrisch, es gilt $S^t = S$. Somit liefert zweimal dieselbe Spiegelung wieder die identische Abbildung, wie man aus +Diese Spiegelmatrizen gehören der orthogonalen Matrizengruppe $S_n\in \text{O}(n)$ an. Die Matrizengruppe $\text{O}(n)$ haben die Eigenschaft $S_n^t S_n = E$, was bedeutet, dass die Länge und Winkel bei der Abbildung beibehalten bleiben. Zusätzlich sind die Spiegelmatrizen symmetrisch, es gilt $S_n^t = S_n$. Somit liefert zweimal dieselbe Spiegelung wieder die identische Abbildung, wie man aus \begin{align} - S^t S = S^2 = E + S_n^t S_n = S_n^2 = E \end{align} schliessen kann. \subsection{Geometrische Algebra} -Um die folgenden Formeln zu verstehen, definieren wir zuerst die Inverse eines Vektors, welche in dieser Form nicht in der linearen Algebra nicht existiert. +Wir definieren zuerst die Inverse eines Vektors, welche in dieser Form nicht in der linearen Algebra nicht existiert. \begin{definition} Die Inverse eines Vektors wird definiert als - \begin{align} - \mathbf{u}^{-1} = \dfrac{\mathbf{u}}{|\mathbf{u}|^2} \Rightarrow \mathbf{uu}^{-1} = \dfrac{\mathbf{u}^2}{|\mathbf{u}|^2} = 1. + \begin{align} \label{InverseGA} + \mathbf{u}^{-1} = \dfrac{\mathbf{u}}{|\mathbf{u}|^2}. \end{align} - Wie schon aus anderen algebraischen Strukturen bekannt, ergibt ein Element, hier $\mathbf{u}$, multipliziert mit dessen Inversen, hier $\mathbf{u}^{-1}$, das neutrale Element der Struktur, hier 1. \end{definition} +Diese Definition ist sinnvoll, da wegen $\mathbf{u}^2 = |\mathbf{u}|^2$ folgt +\begin{align} + \mathbf{uu}^{-1} = \mathbf{u} \frac{\mathbf{u}}{|\mathbf{u}|^2} = \frac{\mathbf{u}^2}{|\mathbf{u}|^2} = \frac{|\mathbf{u}|^2}{|\mathbf{u}|^2} = 1. +\end{align} +Der Vektor $\mathbf{u}^{-1}$ in \eqref{InverseGA} ist also tatsächlich das inverse Element im Sinne des Produktes in der geometrischen Algebra. Die geometrische Algebra leitet aus der obigen Formel \eqref{RefLinAlg} für eine Spiegelung eine einfache und intuitive Form her, welche auch für weitere Operationen erweitert werden kann. \begin{definition} - Die Spiegelungsgleichung in der geometrischen Algebra mit der Spiegelachse $\mathbf{u}$ ist definiert als + Die Abbildung der Spiegelung in der geometrischen Algebra mit dem senkrechten Vektor $\mathbf{u}$ zur Spiegelungsebene $\sigma_u$ ist \begin{align}\label{RefGA} - \mathbf{v}' = \mathbf{uvu}^{-1} + \mathbf{v} \enspace\mapsto\enspace \mathbf{v}' = -\mathbf{uvu}^{-1} \end{align} \end{definition} +Diese Abbildung muss stimmen, weil man durch die Schlussfolgerungen \eqref{uperpv} und \eqref{uparallelv} die Zusammenhänge +\begin{align} + \mathbf{uv_{\perp u}} = -\mathbf{v_{\perp u}u} \enspace\text{und}\enspace \mathbf{uv_{\parallel u}}=\mathbf{v_{\parallel u}u} +\end{align} +der geometrischen Produkte findet und somit die Abbildung aus der geometrischen Algebra \eqref{RefGA} wegen +\begin{align} + \mathbf{v}' = -\mathbf{uvu}^{-1} = -\mathbf{uv_{\perp u}u}^{-1} - \mathbf{uv_{\parallel u}u}^{-1} = -(-\mathbf{v_{\perp u}}\underbrace{\mathbf{u})\mathbf{u}^{-1}}_{1} -(\mathbf{v_{\parallel u}}\underbrace{\mathbf{u})\mathbf{u}^{-1}}_{1} = \mathbf{v_{\perp u}} - \mathbf{v_{\parallel u}} +\end{align} +gleichbedeutend zu der Definition \eqref{RefLinAlg} der Spiegelung ist. -verwendet man für $\mathbf{u}$ nur einen Einheitsvektor $\mathbf{\hat{u}}$, welcher die Länge 1 besitzt, wird die Gleichung zu +Verwendet man für $\mathbf{u}$ nur einen Einheitsvektor $\mathbf{\hat{u}}$, welcher die Länge 1 besitzt, wird die Gleichung \eqref{RefGA} zu \begin{align} - \mathbf{v'} = \mathbf{\hat{u}v\hat{u}} + \mathbf{v'} = -\mathbf{\hat{u}v\hat{u}} \end{align} vereinfacht. Im Gegensatz zu den Abbildungen in der linearen Algebra, welche in jeder anderen Dimension, durch andere Matrizen \eqref{Spiegelmatrizen} beschrieben werden müssen, ist es in der geometrischen Algebra immer der gleiche Vorgehensweise. Zudem ist diese kompakte Schreibweise in der linearen Algebra nicht möglich, da bis auf das Vektorprodukt in der dritten Dimension keine Multiplikation von Vektoren definiert ist.
\ No newline at end of file diff --git a/buch/papers/clifford/7_Spiegelung.tex b/buch/papers/clifford/7_Spiegelung.tex new file mode 100644 index 0000000..c79d908 --- /dev/null +++ b/buch/papers/clifford/7_Spiegelung.tex @@ -0,0 +1,100 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Spiegelung} +\rhead{Spiegelung} + +Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere Operationen, wie beispielsweise die später beschriebene Drehung, ableiten kann. Da die geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Spiegelung auch eine einfache, praktische Formulierung besitzen. +\begin{figure} + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (-3,-1) grid (3,3); + \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; + \draw[<->] (0,-1)--(0,3) node[above]{$a_2$}; + \draw[blue, line width=1.0pt] (0,3)--(0,-1) node[anchor=south east]{$\sigma_u$}; + \draw[line width=2pt,black,-stealth](0,0)--(2,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[line width=2pt,black,-stealth](0,0)--(-2,2) node[anchor=south east]{$\boldsymbol{v'}$}; + \draw[line width=1.5pt,gray,-stealth](0,0)--(1,0) node[anchor=north]{$\boldsymbol{e_1}$}; + \draw[line width=1.5pt,gray,-stealth](0,0)--(0,1) node[anchor=north east]{$\boldsymbol{e_2}$}; + \draw[line width=1.5pt,red,-stealth](0,2)--(2,2) node[xshift=-1cm, yshift= + 0.25cm]{$\boldsymbol{v_{\parallel u}}$}; + \draw[line width=1.5pt,red,-stealth](-2,2)--(0,2) node[xshift=-1cm, yshift= + 0.25cm]{$\boldsymbol{v_{\parallel u}}$}; + \draw[line width=1.5pt,blue,-stealth](0,0.05)--(1,0.05) node[xshift=-0.5cm, yshift=-0.25cm]{$\boldsymbol{\hat{u}}$}; + \end{tikzpicture} + \caption{Spiegelung des Vektors $\mathbf{v}$ an der Spiegelebene $\sigma_u$ mit dem Normalenvektor $\mathbf{\hat{u}}$} + \label{BildSpiegelung} +\end{figure} + +\subsection{Linearen Algebra} +Aus der linearen Algebra ist bekannt, dass man eine Spiegelung an einer Ebene, wie in Abbildung \ref{BildSpiegelung} gezeigt, wie folgt beschreiben kann. +\begin{definition} + Die Abbildung der Spiegelung in der linearen Algebra mit dem Normalenvektor $\mathbf{\hat{u}}$ zur Spiegelebene ist + \begin{equation} \label{RefLinAlg} + \mathbf{v} = \mathbf{v_{\perp u}} + \mathbf{v_{\parallel u}} \enspace\mapsto\enspace \mathbf{v'} = \mathbf{v_{\perp u}} - \mathbf{v_{\parallel u}} = \mathbf{v} - 2 \cdot \mathbf{v_{\parallel u}}. + \end{equation} +\end{definition} +Es scheint für diese Formel \eqref{RefLinAlg} aber umständlich zu sein, weitere Spiegelungen mit weiteren Spiegelebenen anzufügen. Weil man $\mathbf{v_{\parallel u}}$ auch als Skalarprodukt $\mathbf{v_{\parallel u}} = \mathbf{\hat{u}} \cdot \mathbf{v}$ schreiben kann, ist es leicht diese Abbildung auch als Matrix darzustellen. Sei $\mathbf{\hat{u}}$ ein Normalenvektor auf die Spiegelungsebene, also $\mathbf{\hat{u}}\perp \sigma_u$, und sei ausserdem normiert $|\mathbf{\hat{u}}| = 1$, dann kann man die Spiegelung durch die Matrix +\begin{align} +S = E - 2\mathbf{\hat{u}\hat{u}}^t +\end{align} +beschrieben werden. In zwei und drei Dimensionen ergibt die Berechnung +\begin{align} \label{Spiegelmatrizen} +S_2 = \begin{pmatrix} +1-2u_1^2 & -2u_1u_2 \\ +-2u_1u_2 & 1-2u_2^2 +\end{pmatrix}\quad\text{und}\quad +S_3 = \begin{pmatrix} +1-2u_1^2 & -2u_1u_2 & -2u_1u_3\\ +-2u_1u_2 & 1-2u_2^2 & -2u_2u_3\\ +-2u_1u_3 & -2u_2u_3 & 1-2u_3^2\\ +\end{pmatrix}. +\end{align} +Diese Spiegelmatrizen gehören der orthogonalen Matrizengruppe $S_n\in \text{O}(n)$ an. Die Matrizengruppe $\text{O}(n)$ hat die Eigenschaft $S_n^t S_n = E$, was bedeutet, dass die Länge und Winkel bei der Abbildung beibehalten bleiben. Zusätzlich sind die Spiegelmatrizen symmetrisch, es gilt $S_n^t = S_n$. Somit liefert zweimal dieselbe Spiegelung wieder die identische Abbildung, wie man aus +\begin{align} +S_n^t S_n = S_n^2 = E +\end{align} +schliessen kann. + +\subsection{Geometrische Algebra} +Wir definieren zuerst die Inverse eines Vektors, welche in dieser Form nicht in der linearen Algebra nicht existiert. +\begin{definition} + Die Inverse eines Vektors wird definiert als + \begin{align} \label{InverseGA} + \mathbf{u}^{-1} = \dfrac{\mathbf{u}}{|\mathbf{u}|^2}. + \end{align} +\end{definition} +Diese Definition ist sinnvoll, da wegen $\mathbf{u}^2 = |\mathbf{u}|^2$ folgt +\begin{align} +\mathbf{uu}^{-1} = \mathbf{u} \frac{\mathbf{u}}{|\mathbf{u}|^2} = \frac{\mathbf{u}^2}{|\mathbf{u}|^2} = \frac{|\mathbf{u}|^2}{|\mathbf{u}|^2} = 1. +\end{align} +Der Vektor $\mathbf{u}^{-1}$ in \eqref{InverseGA} ist also tatsächlich das inverse Element im Sinne des Produktes in der geometrischen Algebra. +Die geometrische Algebra leitet aus der obigen Formel \eqref{RefLinAlg} für eine Spiegelung eine einfache und intuitive Form her, welche auch für weitere Operationen erweitert werden kann. +\begin{definition} + Die Abbildung der Spiegelung in der geometrischen Algebra mit dem senkrechten Vektor $\mathbf{u}$ zur Spiegelungsebene $\sigma_u$ ist + \begin{align}\label{RefGA} + \mathbf{v} \enspace\mapsto\enspace \mathbf{v}' = -\mathbf{uvu}^{-1} + \end{align} +\end{definition} +Um zu überprüfen, ob die Spiegelungsgleichung \eqref{RefGA} wirklich eine Spiegelung ist, setzen wir zuerst in diese Gleichung $\mathbf{v} = \mathbf{v_{\perp u}} + \mathbf{v_{\parallel u}}$ ein. Wir bekommen somit +\begin{align} +\mathbf{v}' = -\mathbf{uv_{\perp u}u}^{-1} - \mathbf{uv_{\parallel u}u}^{-1}. +\end{align} +Danach können wir mit Hilfe der aus der Schlussfolgerung \eqref{uperpv} und \eqref{uparallelv} hergeleiteten Zusammenhänge +\begin{align} +\mathbf{uv_{\perp u}} = -\mathbf{v_{\perp u}u} \quad\text{und}\quad \mathbf{uv_{\parallel u}}=\mathbf{v_{\parallel u}u}, +\end{align} +die Gleichung weiter umformen zu +\begin{align} +\mathbf{v}' = -(-\mathbf{v_{\perp u}}\underbrace{\mathbf{u})\mathbf{u}^{-1}}_{1} -(\mathbf{v_{\parallel u}}\underbrace{\mathbf{u})\mathbf{u}^{-1}}_{1} = \mathbf{v_{\perp u}} - \mathbf{v_{\parallel u}}. +\end{align} +Man sieht, dass das Resultat $\mathbf{v}' = \mathbf{v_{\perp u}} - \mathbf{v_{\parallel u}}$ +gleichbedeutend zu der Definition \eqref{RefLinAlg} der Spiegelung ist. + +Verwendet man für $\mathbf{u}$ nur einen Einheitsvektor $\mathbf{\hat{u}}$, welcher die Länge 1 besitzt, wird die Gleichung \eqref{RefGA} zu +\begin{align} +\mathbf{v'} = -\mathbf{\hat{u}v\hat{u}} +\end{align} +vereinfacht. Im Gegensatz zu den Abbildungen in der linearen Algebra, welche in jeder anderen Dimension, durch andere Matrizen \eqref{Spiegelmatrizen} beschrieben werden müssen, ist es in der geometrischen Algebra immer der gleiche Vorgehensweise. Zudem ist diese kompakte Schreibweise in der linearen Algebra nicht möglich, da bis auf das Vektorprodukt in drei Dimensionen keine Multiplikation von Vektoren definiert ist.
\ No newline at end of file diff --git a/buch/papers/clifford/8_Rotation.tex b/buch/papers/clifford/8_Rotation.tex index 6a3251a..43d8f8a 100644 --- a/buch/papers/clifford/8_Rotation.tex +++ b/buch/papers/clifford/8_Rotation.tex @@ -3,168 +3,196 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Rotation} -\rhead{Rotation} +\section{Drehung} +\rhead{Drehung} -Eine Rotation kann man aus zwei aufeinanderfolgenden Spiegelungen bilden. Das wird für einige zuerst eine verwirrende Aussage sein, da man aus den vorherig gezeigten Formeln annehmen könnte, dass die Spiegelung schon für eine Drehung ausreicht. Obwohl sich die Längen, Winkel und Volumen sich bei einer Spiegelung, wie bei einer Rotation, nicht ändert, sind sie doch verschieden, da die Orientierung bei der Spiegelung invertiert wird. Stellt man sich beispielsweise ein Objekt im Dreidimensionalen vor und spiegelt dieses an einer Fläche, dann ist es unmöglich nur durch eine Rotation (egal an welchem Punkt) das ursprüngliche Objekt deckungsgleich auf das Gespiegelte zu drehen. Hingegen ist es wiederum möglich ein zweifach gespiegeltes Objekt durch eine Drehung zu erreichen. Das liegt daran, da die Orientierung zweimal invertiert wurde. -\\(Hier wird noch ein Bild für das Verständnis eingefügt) +Eine Drehung kann man aus zwei aufeinanderfolgenden Spiegelungen bilden. Das kann vielleicht zuerst eine verwirrende Aussage sein, da man aus den vorherig gezeigten Formeln annehmen könnte, dass die Spiegelung schon für eine Drehung ausreicht. Obwohl sich die Längen, Winkel und Volumen sich bei einer Spiegelung, wie bei einer Drehung, nicht ändert, sind sie doch verschieden, da die Orientierung bei der Spiegelung invertiert wird. Stellt man sich, wie im Bild \ref{BildSpiegRot} dargestellt, beispielsweise ein Objekt vor und spiegelt dieses an einer Ebene, dann ist es unmöglich, nur durch eine Drehung (egal an welchem Punkt) das ursprüngliche Objekt deckungsgleich auf das Gespiegelte zu drehen. Hingegen ist es wiederum möglich ein zweifach gespiegeltes Objekt durch eine Drehung zu erreichen. Das liegt daran, da die Orientierung zweimal invertiert wurde. + +\begin{figure} + \centering + \includegraphics[width=10cm]{papers/clifford/images/spiegelung.pdf} + \caption{Der wesentliche Unterschied zwischen Spiegelung und Drehung ist die Umkehrung der Orientierung} + \label{BildSpiegRot} +\end{figure} \begin{figure} \centering \begin{tikzpicture} - \draw[thin,gray!40] (-3,-1) grid (3,3); - \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; - \draw[<->] (0,-1)--(0,3) node[above]{$a_2$}; - \draw[line width=2pt,black,-stealth](0,0)--(2,2) node[anchor=south east]{$\boldsymbol{v}$}; - \draw[line width=1.5pt,blue,-stealth](0,0)--(0,2.5) node[anchor=south east]{$\boldsymbol{u}$}; - \draw[line width=2pt,black,-stealth](0,0)--(-2,2) node[anchor=south east]{$\boldsymbol{v'}$}; - \draw[line width=1.5pt,red,-stealth](0,0)--(-2.31, 0.957) node[anchor=south east]{$\boldsymbol{w}$}; - \draw[line width=2pt,black,-stealth](0,0)--(-2.828,0) node[anchor=south east]{$\boldsymbol{v''}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(1,0) node[anchor=north]{$\boldsymbol{e_1}$}; - \draw[line width=1.5pt,gray,-stealth](0,0)--(0,1) node[anchor=north west]{$\boldsymbol{e_2}$}; - - \coordinate (A) at (0,0); - \coordinate (B) at (0,2.5); - \coordinate (C) at (-2.31, 0.957); - \tikzset{anglestyle/.style={angle eccentricity=1.25, draw, thick, angle radius=1.25cm}} - \draw pic ["$\theta$", anglestyle] {angle = B--A--C}; + \draw[thin,gray!40] (-3,-1) grid (3,3); + \draw[<->] (-3,0)--(3,0) node[right]{$a_1$}; + \draw[<->] (0,-1)--(0,3) node[above]{$a_2$}; + \draw[line width=1.0pt,green,-stealth](2,2)--(-2,2) node[anchor=south west]{$\boldsymbol{-2v_{\parallel u}}$}; + \draw[line width=1.0pt,green,-stealth](-2,2)--(-2.828,0) node[anchor=north west]{$\boldsymbol{-2v'_{\parallel w}}$}; + \draw[blue, line width=1.0pt] (0,3)--(0,-1) node[anchor=south east]{$\sigma_u$}; + \draw[red, line width=1.0pt] (-3,1.24)--(2.21,-1) node[anchor=south]{$\sigma_w$}; + \draw[line width=2pt,black,-stealth](0,0)--(2,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[line width=1.5pt,blue,-stealth](0,0)--(2.5, 0) node[anchor=south east]{$\boldsymbol{u}$}; + \draw[line width=2pt,black,-stealth](0,0)--(-2,2) node[anchor=south east]{$\boldsymbol{v'}$}; + \draw[line width=1.5pt,red,-stealth](0,0)--(0.957, 2.31) node[anchor=south east]{$\boldsymbol{w}$}; + \draw[line width=2pt,black,-stealth](0,0)--(-2.828,0) node[anchor=south east]{$\boldsymbol{v''}$}; + \draw[line width=1.5pt,gray,-stealth](0,0)--(1,0) node[anchor=north]{$\boldsymbol{e_1}$}; + \draw[line width=1.5pt,gray,-stealth](0,0)--(0,1) node[anchor=north east]{$\boldsymbol{e_2}$}; + + \coordinate (A) at (0,0); + \coordinate (B) at (2.5,0); + \coordinate (C) at (0.957, 2.31); + \tikzset{anglestyle/.style={angle eccentricity=1.25, purple, draw, thick, angle radius=1cm}} + \draw pic ["$\theta$", anglestyle] {angle = B--A--C}; + \coordinate (D) at (0,0); + \coordinate (E) at (1,1); + \coordinate (F) at (-1, 0); + \tikzset{anglestyle/.style={angle eccentricity=1.25, purple, draw, thick, angle radius=1.25cm}} + \draw pic ["$2\theta$", anglestyle] {angle = E--D--F}; \end{tikzpicture} - \caption{Rotation des Vektors $\textbf{v}$ um $2\theta$} - \label{BildRotation} + \caption{Drehung des Vektors $\textbf{v}$ um $2\theta$} + \label{BildDrehung} \end{figure} \subsection{Linearen Algebra} In der linearen Algebra haben wir Drehungen durch die Matrizen der Gruppe $\text{SO}(n)$ beschrieben. Beispielsweise besteht $\text{SO}(2)$ aus den Matrizen \begin{align} - D = - \begin{pmatrix} - \cos(\alpha) & \sin(\alpha) \\ - -\sin(\alpha) & \cos(\alpha) - \end{pmatrix},\quad - \alpha \in [0, 2\pi). -\end{align} -Diese Drehmatrizen gehören der speziellen orthogonalen Matrizengruppe $D\in \text{SO}(n) = \text{SL}_n(\mathbb{R})\enspace \cap \enspace \text{O}(n)$ an. $\text{SL}_n(\mathbb{R})$ beinhaltet die Matrizen mit scherenden Eigenschaften. Diese Drehmatrizen haben die Eigenschaft $D^t D = E \enspace \land \enspace \det(D)=1$. Da $\det(D) = 1$ und nicht $-1$ sein kann fallen alle Spiegelungen aus der Menge heraus. $\det(D) = -1$ bedeutet, dass eine Orientierungsinversion stattfindet. -\\(BILD Mengen Spezieller Matrizen von Herrn Müller Präsentation) +D = +\begin{pmatrix} +\cos(\alpha) & \sin(\alpha) \\ +-\sin(\alpha) & \cos(\alpha) +\end{pmatrix},\quad +\alpha \in [0, 2\pi). +\end{align} +Diese Drehmatrizen gehören der speziellen orthogonalen Matrizengruppe $D\in \text{SO}(n) = \text{SL}_n(\mathbb{R})\enspace \cap \enspace \text{O}(n)$ an. $\text{SL}_n(\mathbb{R})$ beinhaltet die Matrizen mit scherenden Eigenschaften. Die Drehmatrizen haben die Eigenschaft $D^t D = E \enspace \land \enspace \det(D)=1$. Da $\det(D) = 1$ und nicht $-1$ sein kann fallen alle Spiegelungen aus der Menge heraus. $\det(D) = -1$ bedeutet, dass eine Orientierungsinversion stattfindet. Eine übersichtliche Darstellung der beschriebenen Matrizengruppen sieht man in der Abbildung \ref{BildMatrizenGruppen} + +\begin{figure} + \centering + \includegraphics[width=10cm]{papers/clifford/Bilder/MatrizenGruppen.png} + \caption{Matrizengruppen} + \label{BildMatrizenGruppen} +\end{figure} \subsection{Geometrische Algebra} -Da wir jetzt aus der Geometrie wissen, dass eine Rotation durch zwei Spiegelungen gebildet werden kann, können wir die Rotation mit der Formel \eqref{RefGA} einfach herleiten. +Da wir jetzt aus der Geometrie wissen, dass eine Drehung durch zwei Spiegelungen gebildet werden kann, können wir die Drehung mit der Formel \eqref{RefGA} einfach herleiten. \begin{satz} - Eine Rotation + Durch zwei nacheinander auf einen Vektor $\mathbf{v}$ angewendete Spiegelungen lässt sich eine Drehung \begin{align} \label{rotGA} - \mathbf{v}'' = \mathbf{wv}'\mathbf{w}^{-1} = \mathbf{w}(\mathbf{uvu}^{-1})\mathbf{w}^{-1} = (\mathbf{wu})\mathbf{v}(\mathbf{u}^{-1}\mathbf{w}^{-1}) + \mathbf{v}'' = -\mathbf{wv}'\mathbf{w}^{-1} = -\mathbf{w}(-\mathbf{uvu}^{-1})\mathbf{w}^{-1} = (\mathbf{wu})\mathbf{v}(\mathbf{u}^{-1}\mathbf{w}^{-1}) \end{align} - lässt sich durch zwei nacheinander auf einen Vektor $\mathbf{v}$ angewendete Spiegelungen beschreiben. + beschreiben. \end{satz} Die Vektoren $\mathbf{w}$ und $\mathbf{u}$ bilden hier wiederum die Spiegelachsen. Diese Formel versuchen wir jetzt noch durch Umstrukturierung zu verbessern. \subsubsection{Exponentialform} -Dazu leiten wir zuerst die Exponentialform eines Vektors her. Es wird dabei zur Vereinfachung davon ausgegangen, dass alle Vektoren $\mathbf{w}, \mathbf{u}, \mathbf{v}$ in der $\mathbf{e}_{12}$ Ebene liegen. Weitere Drehungen können in höheren Dimensionen durch Linearkombinationen von Drehungen in den $\mathbf{e}_{ij}, i\not=j$ Ebenen erreicht werden. Für die Herleitung erweitern wir nun als erstes die Polarform +Dazu leiten wir zuerst die Exponentialform eines Vektors her. Es wird dabei zur Vereinfachung davon ausgegangen, dass alle Vektoren $\mathbf{w}, \mathbf{u}, \mathbf{v}$ in der $\mathbf{e}_{1}$-$\mathbf{e}_{2}$-Ebene liegen. Weitere Drehungen können in höheren Dimensionen durch Linearkombinationen von Drehungen in den $\mathbf{e}_{i}$-$\mathbf{e}_{j}$-Ebenen $(i\not=j)$ erreicht werden. Für die Herleitung ersetzen wir als erstes in der Polarform \begin{align} - \mathbf{w} = |\mathbf{w}| \left(\cos(\theta_w) \mathbf{e}_1 + \sin(\theta_w) \mathbf{e}_2\right) +\mathbf{w} = |\mathbf{w}| \left(\cos(\theta_w) \mathbf{e}_1 + \sin(\theta_w) \mathbf{e}_2\right) \end{align} -eines Vektors mit $\mathbf{e}_1^2 = 1$ beim Sinus +eines Vektors einen Faktor 1 durch $1=\mathbf{e}_1^2$ und erhalten beim Sinus \begin{align}\label{e1ausklammern} - \mathbf{w} &= |\mathbf{w}| \left(\cos(\theta_w) \mathbf{e}_1 + \sin(\theta_w) \mathbf{e}_1\mathbf{e}_1\mathbf{e}_2\right), +\mathbf{w} &= |\mathbf{w}| \left(\cos(\theta_w) \mathbf{e}_1 + \sin(\theta_w) \mathbf{e}_1\mathbf{e}_1\mathbf{e}_2\right). \end{align} -um dann $\mathbf{e}_1$ +In einem zweiten Schritt klammern wir $\mathbf{e}_1$ aus, dies ergibt \begin{align} - \mathbf{w} = |\mathbf{w}|\mathbf{e}_1\left(\cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_{12}\right) \label{ExponentialGA} +\mathbf{w} = |\mathbf{w}|\mathbf{e}_1\left(\cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_{12}\right). \label{ExponentialGA} \end{align} -ausklammern zu können. Die Ähnlichkeit des Klammerausdrucks zu der Eulerschen Formel bei den Komplexen Zahlen ist nun schon gut erkennbar. Versuchen wir nun mithilfe der Reihenentwicklungen +Die Ähnlichkeit des Klammerausdrucks in der Formel \eqref{ExponentialGA} zu der Eulerschen Formel bei den komplexen Zahlen ist nun schon gut erkennbar. Versuchen wir nun mithilfe der Reihenentwicklungen \begin{align} - \sin(\theta_w)\mathbf{e}_{12}&=\sum _{n=0}^{\infty }(-1)^{n}{\frac {\theta_w^{2n+1}}{(2n+1)!}}\mathbf{e}_{12} =\theta_w\mathbf{e}_{12}-{\frac {\theta_w^{3}}{3!}}\mathbf{e}_{12}+{\frac {\theta_w^{5}}{5!}}\mathbf{e}_{12}-\cdots \\ - \cos(\theta_w)&=\sum _{n=0}^{\infty }(-1)^{n}{\frac {\theta_w^{2n}}{(2n)!}} =1-{\frac {\theta_w^{2}}{2!}}+{\frac {\theta_w^{4}}{4!}}-\cdots +\sin(\theta_w)\mathbf{e}_{12}&=\sum _{n=0}^{\infty }(-1)^{n}{\frac {\theta_w^{2n+1}}{(2n+1)!}}\mathbf{e}_{12} =\theta_w\mathbf{e}_{12}-{\frac {\theta_w^{3}}{3!}}\mathbf{e}_{12}+{\frac {\theta_w^{5}}{5!}}\mathbf{e}_{12}-\cdots \\ +\cos(\theta_w)&=\sum _{n=0}^{\infty }(-1)^{n}{\frac {\theta_w^{2n}}{(2n)!}} =1-{\frac {\theta_w^{2}}{2!}}+{\frac {\theta_w^{4}}{4!}}-\cdots \end{align} -den Zusammenhang auch hier herzustellen. Verwenden wir jetzt noch die Eigenschaft, dass $\mathbf{e}_{12}^2=-1, \enspace\mathbf{e}_{12}^3=-\mathbf{e}_{12}, \dots$, bei dem Klammerausdruck in Formel \eqref{ExponentialGA} +diesen Zusammenhang auch hier herzustellen. Setzt man diese beiden Reihenentwicklungen in \eqref{ExponentialGA} ein, erhält man \begin{align} - \cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_{12} &= 1+\theta_w\mathbf{e}_{12}-{\frac {\theta_w^{2}}{2!}}-{\frac {\theta_w^{3}}{3!}}\mathbf{e}_{12}+{\frac {\theta_w^{4}}{4!}}+{\frac {\theta_w^{5}}{5!}}\mathbf{e}_{12}-\cdots\\ - &= 1 \mathbf{e}_{12}^0+\theta_w\mathbf{e}_{12}^1+{\frac {\theta_w^{2}}{2!}}\mathbf{e}_{12}^2+{\frac {\theta_w^{3}}{3!}}\mathbf{e}_{12}^3+{\frac {\theta_w^{4}}{4!}}\mathbf{e}_{12}^4+{\frac {\theta_w^{5}}{5!}}\mathbf{e}_{12}^5+\cdots - \label{ExponentialGA2} +\cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_{12} &= 1+\theta_w\mathbf{e}_{12}-{\frac {\theta_w^{2}}{2!}}-{\frac {\theta_w^{3}}{3!}}\mathbf{e}_{12}+{\frac {\theta_w^{4}}{4!}}+{\frac {\theta_w^{5}}{5!}}\mathbf{e}_{12}-\cdots \end{align} -dann sieht man die Übereinstimmung mit der Reihenentwicklung der Exponentialfunktion +Dies sieht noch nicht wie eine Exponentialreihe aus, da $\mathbf{e}_{12}$ nur in jedem zweiten Term auftritt. Da aber $\mathbf{e}_{12}=-1$ gibt, erhält man für \begin{align} - &e^{\theta_w\mathbf{e}_{12}}=\sum _{n=0}^{\infty }{\frac {(\theta_w\mathbf{e}_{12})^{n}}{n!}}={\frac {(\theta_w\mathbf{e}_{12})^{0}}{0!}}+{\frac {(\theta_w\mathbf{e}_{12})^{1}}{1!}}+{\frac {(\theta_w\mathbf{e}_{12})^{2}}{2!}}+{\frac {(\theta_w\mathbf{e}_{12})^{3}}{3!}}+\cdots\\ - &\Rightarrow \mathbf{w} = |w|\mathbf{e}_1 e^{\theta_w \mathbf{e}_{12}} = |w|\mathbf{e}_1\left(\cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_{12}\right). +e^{\theta_w\mathbf{e}_{12}} = 1 \mathbf{e}_{12}^0+\theta_w\mathbf{e}_{12}^1+{\frac {\theta_w^{2}}{2!}}\mathbf{e}_{12}^2+{\frac {\theta_w^{3}}{3!}}\mathbf{e}_{12}^3+{\frac {\theta_w^{4}}{4!}}\mathbf{e}_{12}^4+{\frac {\theta_w^{5}}{5!}}\mathbf{e}_{12}^5+\cdots +\label{ExponentialGA2} +\end{align} +Man sieht, dass die beiden Reihen übereinstimmen. Es folgt somit +\begin{align}\label{EulerGA} +e^{\theta_w \mathbf{e}_{12}} = \cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_{12}, +\end{align} +was zeigt, dass es eine Euler-Formel mit $\mathbf{e}_{12}$ anstelle der imaginären Einheit $j$ gibt. + +Wenn man jetzt den Vektor \eqref{ExponentialGA} durch die eulersche Schreibweise +\begin{align} +\mathbf{w} = |\mathbf{w}|\mathbf{e}_1e^{\theta_w\mathbf{e}_{12}} \end{align} -Man kann die Exponentialform des Vektors ähnlich wie die der komplexen Zahlen interpretieren. Der Einheitsvektor $\mathbf{e}_1$ wird um die Länge $|\mathbf{w}|$ gestreckt und um $\theta_w$ gedreht. -Bei den komplexen Zahlen würden man vom Punkt 1 anstatt $\mathbf{e}_1$ ausgehen. +ersetzt, kann die Exponentialform des Vektors ähnlich wie die der komplexen Zahlen interpretieren. Der Einheitsvektor $\mathbf{e}_1$ wird um die Länge $|\mathbf{w}|$ gestreckt und um $\theta_w$ gedreht. \subsubsection{Vektormultiplikation} -Nun werden wir das Produkt von zwei Vektoren $\mathbf{wu}$ -\begin{align} - \mathbf{wu} = |\mathbf{w}|\mathbf{e}_1 e^{\theta_w \mathbf{e}_{12}}|\mathbf{u}|\mathbf{e}_1 e^{\theta_u \mathbf{e}_{12}} +Nun werden wir das Vektorprodukt +\begin{align} \label{VektorproduktformelGA} +\mathbf{wu} = |\mathbf{w}|\mathbf{e}_1 e^{\theta_w \mathbf{e}_{12}}|\mathbf{u}|\mathbf{e}_1 e^{\theta_u \mathbf{e}_{12}} \end{align} -so umformen, dass wir eine bessere Darstellung erhalten. Wir tauschen dafür zuerst beim Vektor $\mathbf{w}$ die Reihenfolge von -$\mathbf{e}_1$ mit dem Exponentialterm $e^{\theta_w \mathbf{e}_{12}}$, indem wir bei der Gleichung \eqref{e1ausklammern}, anstatt mit $\mathbf{e}_1\mathbf{e}_1\mathbf{e}_2$ mit $\mathbf{e}_2\mathbf{e}_1\mathbf{e}_1$ erweitern +so umformen, dass wir die Drehung nur durch Exponentialterme beschreiben können. Wir tauschen dafür zuerst beim Vektor $\mathbf{w}$ die Reihenfolge von +$\mathbf{e}_1$ mit dem Exponentialterm $e^{\theta_w \mathbf{e}_{12}}$, indem wir bei der Gleichung \eqref{e1ausklammern} $1=\mathbf{e}_1^2$ an einer anderen Position einsetzten. Wir erhalten \begin{align} - \mathbf{w} &= |\mathbf{w}|\left(\cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_2\mathbf{e}_1\right)\mathbf{e}_1\\ - &= |\mathbf{w}|e^{\theta_w \mathbf{e}_{21}}\mathbf{e}_1\\ - &= |\mathbf{w}|e^{-\theta_w \mathbf{e}_{12}}\mathbf{e}_1 +\mathbf{w} &= |\mathbf{w}|\left(\cos(\theta_w)+ \sin(\theta_w) \mathbf{e}_2\mathbf{e}_1\right)\mathbf{e}_1. \end{align} -und umstrukturiert wieder in die Vektorproduktformel einsetzen +Mithilfe der Formel \eqref{EulerGA} und dem Wissen, dass $\mathbf{e}_{21}= -\mathbf{e}_{12}$ können wir die Umformung \begin{align} - \mathbf{wu} = |\mathbf{w}||\mathbf{u}|e^{-\theta_w \mathbf{e}_{12}}\mathbf{e}_1\mathbf{e}_1 e^{\theta_u \mathbf{e}_{12}}\\ - \mathbf{wu} = |\mathbf{w}||\mathbf{u}|e^{(\theta_u-\theta_w) \mathbf{e}_{12}}. +|\mathbf{w}|e^{-\theta_w \mathbf{e}_{12}}\mathbf{e}_1 \end{align} -Der Term $\mathbf{u}^{-1}\mathbf{w}^{-1}$ +ausführen. Diese wichtige Umstrukturierung können wir wieder in die Vektorproduktformel \eqref{VektorproduktformelGA} einsetzen un erhalten \begin{align} - \mathbf{u}^{-1}\mathbf{w}^{-1} = \dfrac{1}{|\mathbf{w}||\mathbf{u}|}e^{(\theta_w-\theta_u) \mathbf{e}_{12}} +\mathbf{wu} &= |\mathbf{w}|\,|\mathbf{u}|e^{-\theta_w \mathbf{e}_{12}}\mathbf{e}_1\mathbf{e}_1 e^{\theta_u \mathbf{e}_{12}}\\ +&= |\mathbf{w}|\,|\mathbf{u}|e^{(\theta_u-\theta_w) \mathbf{e}_{12}}. \end{align} -kann durch die selbe Methode zusammengefasst werden. -Wenn wir den Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ als $\theta = \theta_w - \theta_u$ definieren erhalten wir +Das inverse Vektorprodukt +\begin{align} +\mathbf{u}^{-1}\mathbf{w}^{-1} = \dfrac{1}{|\mathbf{w}|\,|\mathbf{u}|}e^{(\theta_w-\theta_u) \mathbf{e}_{12}} +\end{align} +kann durch die selbe Methode vereinfacht werden. +Wenn wir den Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ als $\theta = \theta_w - \theta_u$ definieren erhalten wir als endgültige Form der Vektorprodukte \begin{align}\label{wuExpo} - \mathbf{wu} = |\mathbf{w}||\mathbf{u}|e^{-\theta \mathbf{e}_{12}}\\ - \mathbf{u}^{-1}\mathbf{w}^{-1} = \dfrac{1}{|\mathbf{w}||\mathbf{u}|}e^{\theta \mathbf{e}_{12}} \label{wuExpoInv} +\mathbf{wu} &= |\mathbf{w}|\,|\mathbf{u}|e^{-\theta \mathbf{e}_{12}}\enspace\text{und}\\ +\mathbf{u}^{-1}\mathbf{w}^{-1} &= \dfrac{1}{|\mathbf{w}|\,|\mathbf{u}|}e^{\theta \mathbf{e}_{12}} \label{wuExpoInv}. \end{align} -die finale Form der Vektorprodukte. \subsubsection{Umstrukturierte Drehungsgleichung} -Setzten wir nun unsere neuen Erkenntnisse in die Gleichung \eqref{rotGA} ein +Setzten wir nun unsere neuen Erkenntnisse in die Gleichung \eqref{rotGA} ein, erhalten wir \begin{align} - \mathbf{v''} = (|\mathbf{w}||\mathbf{u}|e^{-\theta \mathbf{e}_{12}}) \mathbf{v}( \dfrac{1}{|\mathbf{w}||\mathbf{u}|}e^{\theta \mathbf{e}_{12}}), +\mathbf{v''} = (|\mathbf{w}|\,|\mathbf{u}|e^{-\theta \mathbf{e}_{12}})\mathbf{v}\biggl(\dfrac{1}{|\mathbf{w}|\,|\mathbf{u}|}e^{\theta \mathbf{e}_{12}}\biggr) \end{align} -erhalten wir durch die Kürzungen der Längen die vereinfachte Drehungsgleichung -\begin{align} - \mathbf{v''} = e^{-\theta \mathbf{e}_{12}} v e^{\theta \mathbf{e}_{12}}. +und können durch die Kürzungen der Längen die vereinfachte Drehungsgleichung +\begin{align} \label{GAvereinfRot} +\mathbf{v''} = e^{-\theta \mathbf{e}_{12}} v e^{\theta \mathbf{e}_{12}} \end{align} - -Wir wissen nun, dass das diese beidseitige Multiplikation die Länge von $\mathbf{v}$ nicht verändert, da sich die Längen von $\mathbf{w}$ und $\mathbf{u}$ kürzen. Betrachten wir nun den Effekt der Exponentialterme auf $\mathbf{v}$. Dabei Teilen wir den Vektor $\mathbf{v}$ auf in einen Anteil $\mathbf{v_\parallel}$, welcher auf der Ebene $\mathbf{e}_{12}$ liegt, und einen Anteil $\mathbf{v_\perp}$, welcher senkrecht zu der Ebene steht. Wir bekommen durch Einsetzten nun diese Form +bilden. Wir wissen nun, dass das diese beidseitige Multiplikation die Länge von $\mathbf{v}$ nicht verändert, da sich die Längen von $\mathbf{w}$ und $\mathbf{u}$ kürzen. Betrachten wir nun den Effekt der Exponentialterme auf $\mathbf{v}$. Dabei teilen wir den Vektor $\mathbf{v}$ auf in einen Anteil $\mathbf{v_\parallel}$, welcher auf der Ebene $\mathbf{e}_{12}$ liegt, und einen Anteil $\mathbf{v_\perp}$, welcher senkrecht zu der Ebene steht. Wir bekommen durch Einsetzten nun diese Form \begin{align} \label{RotAufPerpPar} - \mathbf{v}'' = e^{-\theta \mathbf{e}_{12}} (\mathbf{v_\perp + v_\parallel}) e^{\theta \mathbf{e}_{12}} = e^{-\theta \mathbf{e}_{12}} \mathbf{v_\perp} e^{\theta \mathbf{e}_{12}} + e^{-\theta \mathbf{e}_{12}} \mathbf{v_\parallel} e^{\theta \mathbf{e}_{12}}. +\mathbf{v}'' = e^{-\theta \mathbf{e}_{12}} (\mathbf{v_\perp + v_\parallel}) e^{\theta \mathbf{e}_{12}} = e^{-\theta \mathbf{e}_{12}} \mathbf{v_\perp} e^{\theta \mathbf{e}_{12}} + e^{-\theta \mathbf{e}_{12}} \mathbf{v_\parallel} e^{\theta \mathbf{e}_{12}}. \end{align} -Auf eine allgemeine Herleitung wird hier zwar verzichtet, aber man kann zeigen, dass die Reihenfolge so umstrukturiert werden kann +Auf eine allgemeine Herleitung wird hier zwar verzichtet, aber man kann zeigen, dass man die Reihenfolge der Vektoranteile $\mathbf{v_\perp}$ und $\mathbf{v_\parallel}$ mit dem Exponentialterm $e^{-\theta \mathbf{e}_{12}}$ so vertauschen kann, dass sich \begin{align} - \mathbf{v}'' = \mathbf{v_\perp} e^{-\theta \mathbf{e}_{12}} e^{\theta \mathbf{e}_{12}} + \mathbf{v_\parallel} e^{-(-\theta) \mathbf{e}_{12}} e^{\theta \mathbf{e}_{12}}, +\mathbf{v}'' = \mathbf{v_\perp} e^{-\theta \mathbf{e}_{12}} e^{\theta \mathbf{e}_{12}} + \mathbf{v_\parallel} e^{-(-\theta) \mathbf{e}_{12}} e^{\theta \mathbf{e}_{12}} \end{align} -dass der Winkel beim parallelen Anteil negiert wird. An der Zusammengefassten Gleichung +ergibt. Der Winkel wird beim parallelen Anteil negiert. An der Zusammengefassten Gleichung \begin{align}\label{RotParPerp} - \mathbf{v}'' = \mathbf{v_\perp} + \mathbf{v_\parallel} e^{2\theta \mathbf{e}_{12}} +\mathbf{v}'' = \mathbf{v_\perp} + \mathbf{v_\parallel} e^{2\theta \mathbf{e}_{12}} \end{align} kann man sehen, dass nur der parallele Anteil $\mathbf{v_\parallel}$ des Vektors $\mathbf{v}$ auf der Ebene $\mathbf{e}_{12}$ um $2\theta$ gedreht wird. Der senkrechte Anteil $\mathbf{v_\perp}$ bleibt gleich. Wichtig dabei zu sehen ist, dass nur der Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ von Bedeutung ist. Die Länge und Richtung der einzelnen Vektoren spielt keine Rolle. Zeigen wir nun diese Eigenschaften an einem Beispiel \begin{beispiel} - Gegeben sei ein Vektor $\mathbf{v} = 1\mathbf{e}_1 + 2\mathbf{e}_2 + 3\mathbf{e}_3$ mit zur $\mathbf{e}_{12}$-Ebene parallelen Anteil $\mathbf{v_\parallel} = 1\mathbf{e}_1 + 2\mathbf{e}_2$ und senkrechten Anteil $\mathbf{v_\perp} = 3\mathbf{e}_3$. Zusätzlich sind die Spiegelachsen $\mathbf{u} = \mathbf{e}_1$ und $\mathbf{w} = 2\mathbf{e}_2$ gegeben. Gesucht ist der rotierte Vektor $\mathbf{v}''$. Bestimmen wir als erstes das Vektorprodukt $\mathbf{wu}$ + Gegeben sei ein Vektor $\mathbf{v} = 1\mathbf{e}_1 + 2\mathbf{e}_2 + 3\mathbf{e}_3$ mit zur $\mathbf{e}_{12}$-Ebene parallelen Anteil $\mathbf{v_\parallel} = 1\mathbf{e}_1 + 2\mathbf{e}_2$ und senkrechten Anteil $\mathbf{v_\perp} = 3\mathbf{e}_3$. Zusätzlich sind die Spiegelachsen $\mathbf{u} = \mathbf{e}_1$ und $\mathbf{w} = 2\mathbf{e}_2$ gegeben. Gesucht ist der rotierte Vektor $\mathbf{v}''$. Bestimmen wir als erstes das Vektorprodukt \begin{align} - \mathbf{wu} = (2\mathbf{e}_2)(\mathbf{e}_1) = -2\mathbf{e}_{12} + \mathbf{wu} = (2\mathbf{e}_2)(\mathbf{e}_1) = -2\mathbf{e}_{12} \end{align} - und das Produkt der Inversen $\mathbf{u}^{-1}\mathbf{w}^{-1}$ + und das Produkt der Inversen \begin{align} - \mathbf{u}^{-1}\mathbf{w}^{-1} = (\dfrac{\mathbf{e}_1}{1^2})(\dfrac{2\mathbf{e}_2}{2^2}) = \dfrac{1}{2}\mathbf{e}_{12}. + \mathbf{u}^{-1}\mathbf{w}^{-1} = \biggl(\dfrac{\mathbf{e}_1}{1^2}\biggr) \left(\dfrac{2\mathbf{e}_2}{2^2}\right) = \dfrac{1}{2}\mathbf{e}_{12}. \end{align} - Der rotierte Vektor $\mathbf{v}''$ können wir nun durch das einsetzten und auflösen der Produkte in die Gleichung \eqref{rotGA} + Den gedrehten Vektor $\mathbf{v}''$ können wir nun durch Einsetzen und Auflösen der Produkte in die Gleichung \eqref{rotGA} bestimmen. Der Rechnenvorgang ist \begin{align} - \mathbf{v}'' = (\mathbf{wu})\mathbf{v}(\mathbf{u}^{-1}\mathbf{w}^{-1}) &= (-2e_{12})(1\mathbf{e}_1 + \mathbf{e}_2 + 1\mathbf{e}_3)(\dfrac{1}{2}\mathbf{e}_{12})\\ - &= (2\mathbf{e}_2-2\mathbf{e}_1-2\mathbf{e}_{123})(\dfrac{1}{2}\mathbf{e}_{12})\\ - &= -1\mathbf{e}_1 - 1\mathbf{e}_2 + 1\mathbf{e}_3 + \mathbf{v}'' = (\mathbf{wu})\mathbf{v}(\mathbf{u}^{-1}\mathbf{w}^{-1}) &= (-2e_{12})(1\mathbf{e}_1 + \mathbf{e}_2 + 1\mathbf{e}_3)(\textstyle{\frac{1}{2}}\mathbf{e}_{12})\\ + &= (2\mathbf{e}_2-2\mathbf{e}_1-2\mathbf{e}_{123})(\textstyle{\frac{1}{2}}\mathbf{e}_{12})\\ + &= -1\mathbf{e}_1 - 1\mathbf{e}_2 + 1\mathbf{e}_3. \end{align} - finden. Aus dem Resultat $\mathbf{v}''= -1\mathbf{e}_1 + 1\mathbf{e}_2 + 1\mathbf{e}_3$ können wir bestätigen, dass + Aus dem Resultat $\mathbf{v}''= -1\mathbf{e}_1 + 1\mathbf{e}_2 + 1\mathbf{e}_3$ können wir bestätigen, dass \begin{itemize} \item die Länge $|\mathbf{v}| = \sqrt{3}$ zur Länge $|\mathbf{v}''|=\sqrt{3}$ gleich blieb. \item sich der parallele Anteil $\mathbf{v_\parallel}'' = -1\mathbf{e}_1 - 1\mathbf{e}_2$ gedreht hat und der senkrechte Anteil $\mathbf{v_\perp}'' = 1\mathbf{e}_3$ unverändert blieb. \item der parallele Teil sich genau um $2\theta=180$° gedreht hat. $\theta$ kann übrigens durch die Umformung des Produkt $\mathbf{wu}$ in die Exponentialschreibweise \begin{align} - &\mathbf{wu} = -2\mathbf{e}_{12} = 2(0-1\mathbf{e}_{12})=2(\cos(\dfrac{-\pi}{2} + \sin(\dfrac{-\pi}{2})\mathbf{e}_{12})) = 2e^{(-\pi/2)\mathbf{e}_{12}} + &\mathbf{wu} = -2\mathbf{e}_{12} = 2(0-1\mathbf{e}_{12})=2(\cos\biggl(\dfrac{-\pi}{2}\biggr) + \sin\biggl(\dfrac{-\pi}{2}\biggr)\mathbf{e}_{12}) = 2e^{(-\pi/2)\mathbf{e}_{12}} \end{align} durch einen Vergleich mir der Formel \eqref{wuExpo} \begin{align} - \theta = -(\dfrac{-\pi}{2}) = \dfrac{\pi}{2} + \theta = -\biggl(\dfrac{-\pi}{2}\biggr) = \dfrac{\pi}{2} \end{align} - ausgelesen werden. + ausgelesen werden. \qedhere \end{itemize} -\end{beispiel}
\ No newline at end of file +\end{beispiel}
\ No newline at end of file diff --git a/buch/papers/clifford/9_KomplexeZahlen.tex b/buch/papers/clifford/9_KomplexeZahlen.tex index 70107da..12fa546 100644 --- a/buch/papers/clifford/9_KomplexeZahlen.tex +++ b/buch/papers/clifford/9_KomplexeZahlen.tex @@ -6,23 +6,34 @@ \section{Komplexe Zahlen} \rhead{Komplexe Zahlen} -Die komplexen Zahlen finden eine Vielzahl von Anwendungsgebiete in den Ingenieurwissenschaften. Das liegt daran, weil die komplexen Zahlen Rotationen und Schwingungen gut beschreiben können. Nach dem vorherigen Kapitel überrascht es wahrscheinlich nicht viele, dass es möglich ist komplexe Zahlen in der geometrischen Algebra darzustellen. Sie können durch die geraden Grade der 2 Dimensionalen geometrischen Algebra vollständig beschrieben werden: $\mathbf{g}_n \in G_2^+(\mathbb{R}) \cong \mathbb{C}$. Das bedeutet eine komplexe Zahl kann durch ein Skalar (Grad 0) und einem Bivektor (Grad 2) dargestellt werden +Die komplexen Zahlen finden eine Vielzahl von Anwendungsgebiete in den Ingenieurwissenschaften. Das liegt daran, weil die komplexen Zahlen Drehungen und Schwingungen gut beschreiben können. Nach dem vorherigen Abschnitt ist es nicht überraschend, dass es möglich ist, komplexe Zahlen in der geometrischen Algebra darzustellen. Sie können durch die geraden Grade der zweidimensionalen geometrischen Algebra vollständig beschrieben werden: $\mathbf{g}_n \in G_2^+(\mathbb{R}) \cong \mathbb{C}$. Das bedeutet eine komplexe Zahl \begin{align} - a_0 + a_1 j \cong a_0 + a_1 \mathbf{e}_{12} = \mathbf{g}_n\quad a_0, a_1 \in \mathbb{R}\\ - |r|e^{\theta j} \cong |r|e^{\theta \mathbf{e}_{12}} = \mathbf{g}_n; \quad r, \theta \in \mathbb{R} +a_0 + a_1 j \cong a_0 + a_1 \mathbf{e}_{12} = \mathbf{g}_n\quad a_0, a_1 \in \mathbb{R}\\ +|r|e^{\theta j} \cong |r|e^{\theta \mathbf{e}_{12}} = \mathbf{g}_n; \quad r, \theta \in \mathbb{R} \end{align} -weil $j$ und $\mathbf{e}_{12}$ beide die Eigenschaft besitzen quadriert $-1$ zu ergeben +kann durch ein Skalar (Grad 0) und einem Bivektor (Grad 2) dargestellt werden, weil $j$ und $\mathbf{e}_{12}$ beide die Eigenschaft \begin{align} - j^2 = -1\quad \mathbf{e}_{12}^2 = -1 +j^2 = -1\quad\text{und}\quad\mathbf{e}_{12}^2 = -1 \end{align} -Man beachte, dass wenn wir, wie bei den komplexen Zahlen, Elemente von $G_2^+(\mathbb{R})$ miteinander Multiplizieren, ist es nicht, wie im Kapitel Rotation bei der Formel (\ref{rotGA})beschrieben, eine Multiplikation von zwei $g_n$ mit einem Vektor. Im zweidimensionalen bewirken beide Multiplikationen grundsätzlich das Gleiche (eine Drehstreckung), aber die Multiplikation von mehreren $g_n$ ist kommutativ, wie wir es von den komplexen Zahlen kennen. +besitzen. Die Kommutativität \begin{align} - \mathbf{g}_1\mathbf{g}_2 = \mathbf{g}_2\mathbf{g}_1 \quad&\Leftrightarrow\quad (a + b \mathbf{e}_{12})(f + g \mathbf{e}_{12}) = (f + g \mathbf{e}_{12})(a + b \mathbf{e}_{12})\\ - \mathbf{g}_1\mathbf{v}\not= \mathbf{v}\mathbf{g}_1 \quad&\Leftrightarrow\quad(a + b \mathbf{e}_{12})(x\mathbf{e}_1+y\mathbf{e}_2)\not= (x\mathbf{e}_1+y\mathbf{e}_2)(a + b \mathbf{e}_{12}) +\begin{split} +\mathbf{g}_1\mathbf{g}_2 = \mathbf{g}_2\mathbf{g}_1 \enspace&\Leftrightarrow\enspace (a + b \mathbf{e}_{12})(f + g \mathbf{e}_{12}) = (f + g \mathbf{e}_{12})(a + b \mathbf{e}_{12})\\ &\Leftrightarrow\enspace |\mathbf{g}_1|\,|\mathbf{g}_2|e^{(\theta_{g_1} + \theta_{g_2})\mathbf{e}_{12}} = |\mathbf{g}_2|\,|\mathbf{g}_1|e^{(\theta_{g_2} + \theta_{g_1})\mathbf{e}_{12}}, +\end{split} \end{align} -Um später die Auswirkung der Quaternionen besser zu verstehen, möchte ich kurz darauf eingehen, was ein $g_n$ für eine Auswirkung auf einen Vektor hat. -Wir kennen diesen Effekt schon von den komplexen Zahlen. Wenn eine komplexe Zahl $c_1=a+bj$ mit einer zweiten $c_2=f+gj$ multipliziert wird, dann kann man diese so aufteilen. +welche wir schon von den komplexen Zahlen her kennen, ist dabei eine in der geometrischen Algebra nur selten anzutreffende Eigenschaft. Beispielsweise ist das geometrische Produkt von \begin{align} - c = c_1\cdot c_2 = (a + bj)(d + ej) = f\cdot(a+bj) + gj\cdot(a+bj) +\mathbf{g}_1\mathbf{v}\not= \mathbf{v}\mathbf{g}_1 \quad\Leftrightarrow\quad(a + b \mathbf{e}_{12})(x\mathbf{e}_1+y\mathbf{e}_2)\not= (x\mathbf{e}_1+y\mathbf{e}_2)(a + b \mathbf{e}_{12}) \end{align} -Dabei ist $f\cdot(a+bj)$ die jetzige komplexe Zahl $c_1$ um den Faktor $f$ steckt und $gj\cdot(a+bj)$ die um 90° im Gegenuhrzeigersinn gedrehte Zahl $c_2$ um den Faktor $g$ streckt. Diese Anteile addiert ergeben, dann den um $c_2$ dreh-gestreckten Vektor $c_1$. Die wirklichen Vorteile der geometrischen Algebra werden sich aber erst bei den Quaternionen zeigen.
\ No newline at end of file +und auch die im folgenden Kapitel behandelten Quaternionen sind nicht kommutativ. + +Um später die Auswirkung der Quaternionen auf Vektoren besser zu verstehen, möchten wir kurz darauf eingehen, was ein $\mathbf{g}_n$ für eine Auswirkung auf einen Vektor hat. +Wir kennen diesen Effekt schon von den komplexen Zahlen. Wenn eine komplexe Zahl $c_1=a+bj$ mit einer zweiten $c_2=f+gj$ multipliziert wird, dann kann man +\begin{align} +c = c_1\cdot c_2 = (a + bj)(d + ej) = \underbrace{a\cdot(d+ej)}_{\displaystyle{a\cdot c_2}} + \underbrace{bj\cdot(d+ej)}_{\displaystyle{b\cdot c_2 \cdot (1\angle 90^\circ)}} +\end{align} +so aufteilen. Dabei ist $a\cdot(d+ej)$ die komplexe Zahl $c_2$ um den Faktor $a$ steckt und $bj\cdot(d+ej)$ die um 90° im Gegenuhrzeigersinn gedrehte Zahl $c_2$ um den Faktor $b$ streckt. Diese Anteile addiert ergeben dann den um $c_1$ drehgestreckten Vektor $c_2$. Den gleichen Effekt hat +\begin{align}\label{GAdrehstreck} +\mathbf{v}' = \mathbf{g}\mathbf{v} = (a + b\mathbf{e}_{12})(d\mathbf{e}_{1} + e\mathbf{e}_{2}) = a(d\mathbf{e}_{1} + e\mathbf{e}_{2}) + b\mathbf{e}_{12}(d\mathbf{e}_{1} + e\mathbf{e}_{2}) +\end{align} +in der zweidimensionalen geometrischen Algebra. Im Falle der komplexen Zahlen macht es jetzt noch nicht wirklich Sinn in die geometrische Algebra zu wechseln. Die potenziellen Vorteile der geometrischen Algebra werden sich aber erst bei den Quaternionen zeigen.
\ No newline at end of file diff --git a/buch/papers/clifford/Bilder/ReihenfolgeGimbal.png b/buch/papers/clifford/Bilder/ReihenfolgeGimbal.png Binary files differnew file mode 100644 index 0000000..625757d --- /dev/null +++ b/buch/papers/clifford/Bilder/ReihenfolgeGimbal.png diff --git a/buch/papers/clifford/Bilder/test.png b/buch/papers/clifford/Bilder/test.png Binary files differdeleted file mode 100644 index 1633a2e..0000000 --- a/buch/papers/clifford/Bilder/test.png +++ /dev/null diff --git a/buch/papers/clifford/Makefile.inc b/buch/papers/clifford/Makefile.inc index e168ae8..fe32eba 100644 --- a/buch/papers/clifford/Makefile.inc +++ b/buch/papers/clifford/Makefile.inc @@ -14,7 +14,9 @@ dependencies-clifford = \ papers/clifford/4_GeometrischesProdukt.tex \ papers/clifford/5_PolareDarstellung.tex \ papers/clifford/6_PauliMatrizen.tex \ - papers/clifford/7_Reflektion.tex \ + papers/clifford/7_Spiegelung.tex \ papers/clifford/8_Rotation.tex \ papers/clifford/9_KomplexeZahlen.tex \ - papers/clifford/10_Quaternionen.tex + papers/clifford/10_Quaternionen.tex \ + papers/clifford/11_Fazit.tex + diff --git a/buch/papers/clifford/images/Makefile b/buch/papers/clifford/images/Makefile new file mode 100644 index 0000000..cc621fb --- /dev/null +++ b/buch/papers/clifford/images/Makefile @@ -0,0 +1,13 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: spiegelung.pdf + +spiegelung.pdf: spiegelung.tex punkte.tex + pdflatex spiegelung.tex + +punkte.tex: spiegelung.m + octave spiegelung.m + diff --git a/buch/papers/clifford/images/punkte.tex b/buch/papers/clifford/images/punkte.tex new file mode 100644 index 0000000..41d2247 --- /dev/null +++ b/buch/papers/clifford/images/punkte.tex @@ -0,0 +1,21 @@ +\coordinate (A) at (2.300,1.700); +\coordinate (B) at (4.300,2.500); +\coordinate (C) at (2.800,2.700); +\coordinate (S) at (3.133,2.300); +\coordinate (G1) at (0.489,0.873); +\coordinate (G1oben) at (4.886,8.725); +\coordinate (G1unten) at (-4.886,-8.725); +\coordinate (G2) at (0.336,-0.942); +\coordinate (G2oben) at (3.363,-9.417); +\coordinate (G2unten) at (-3.363,9.417); +\coordinate (A1) at (0.248,2.849); +\coordinate (B1) at (-0.115,4.973); +\coordinate (C1) at (0.839,3.798); +\coordinate (S1) at (0.324,3.873); +\coordinate (A2) at (-1.997,2.048); +\coordinate (B2) at (-3.061,3.921); +\coordinate (C2) at (-3.055,2.407); +\coordinate (S2) at (-2.704,2.792); +\def\winkela{60.7512} +\def\winkelb{48.9027} +\coordinate (G) at (0.489,0.873); diff --git a/buch/papers/clifford/images/spiegelung.m b/buch/papers/clifford/images/spiegelung.m new file mode 100644 index 0000000..a086cb5 --- /dev/null +++ b/buch/papers/clifford/images/spiegelung.m @@ -0,0 +1,66 @@ +# +# spiegelung.m +# +# +fn = fopen("punkte.tex", "w"); + + +a = [ 2.3; 1.7 ]; +b = [ 4.3; 2.5 ]; +c = [ 2.8; 2.7 ]; +s = (a + b + c)/3; + +fprintf(fn, "\\coordinate (A) at (%.3f,%.3f);\n", a(1, 1), a(2, 1)); +fprintf(fn, "\\coordinate (B) at (%.3f,%.3f);\n", b(1, 1), b(2, 1)); +fprintf(fn, "\\coordinate (C) at (%.3f,%.3f);\n", c(1, 1), c(2, 1)); +fprintf(fn, "\\coordinate (S) at (%.3f,%.3f);\n", s(1, 1), s(2, 1)); + +n1 = [ -2.5; 1.4 ]; +n1 = n1 / norm(n1); +S1 = eye(2) - 2 * (n1 * n1'); +g1 = [ n1(2,1); -n1(1,1) ]; + +fprintf(fn, "\\coordinate (G1) at (%.3f,%.3f);\n", g1(1,1), g1(2,1)); +fprintf(fn, "\\coordinate (G1oben) at (%.3f,%.3f);\n", 10*g1(1,1), 10*g1(2,1)); +fprintf(fn, "\\coordinate (G1unten) at (%.3f,%.3f);\n", -10*g1(1,1), -10*g1(2,1)); + +n2 = [ 1.4; 0.5 ]; +n2 = n2 / norm(n2); +S2 = eye(2) - 2 * (n2 * n2'); +g2 = [ n2(2,1); -n2(1,1) ]; + +fprintf(fn, "\\coordinate (G2) at (%.3f,%.3f);\n", g2(1,1), g2(2,1)); +fprintf(fn, "\\coordinate (G2oben) at (%.3f,%.3f);\n", 10*g2(1,1), 10*g2(2,1)); +fprintf(fn, "\\coordinate (G2unten) at (%.3f,%.3f);\n", -10*g2(1,1), -10*g2(2,1)); + +D = S2 * S1; + +a1 = S1 * a; +b1 = S1 * b; +c1 = S1 * c; +s1 = S1 * s; + +fprintf(fn, "\\coordinate (A1) at (%.3f,%.3f);\n", a1(1, 1), a1(2, 1)); +fprintf(fn, "\\coordinate (B1) at (%.3f,%.3f);\n", b1(1, 1), b1(2, 1)); +fprintf(fn, "\\coordinate (C1) at (%.3f,%.3f);\n", c1(1, 1), c1(2, 1)); +fprintf(fn, "\\coordinate (S1) at (%.3f,%.3f);\n", s1(1, 1), s1(2, 1)); + +a2 = D * a; +b2 = D * b; +c2 = D * c; +s2 = D * s; + +fprintf(fn, "\\coordinate (A2) at (%.3f,%.3f);\n", a2(1, 1), a2(2, 1)); +fprintf(fn, "\\coordinate (B2) at (%.3f,%.3f);\n", b2(1, 1), b2(2, 1)); +fprintf(fn, "\\coordinate (C2) at (%.3f,%.3f);\n", c2(1, 1), c2(2, 1)); +fprintf(fn, "\\coordinate (S2) at (%.3f,%.3f);\n", s2(1, 1), s2(2, 1)); + +winkel1 = atan2(g1(2,1), g1(1,1)) * (180 / pi); +winkel2 = acosd(g1' * g2); + +fprintf(fn, "\\def\\winkela{%.4f}\n", winkel1); +fprintf(fn, "\\def\\winkelb{%.4f}\n", 180 - winkel2); + +fprintf(fn, "\\coordinate (G) at (%.3f,%.3f);\n", g1(1,1), g1(2,1)); + +fclose(fn); diff --git a/buch/papers/clifford/images/spiegelung.pdf b/buch/papers/clifford/images/spiegelung.pdf Binary files differnew file mode 100644 index 0000000..a17d369 --- /dev/null +++ b/buch/papers/clifford/images/spiegelung.pdf diff --git a/buch/papers/clifford/images/spiegelung.tex b/buch/papers/clifford/images/spiegelung.tex new file mode 100644 index 0000000..0960456 --- /dev/null +++ b/buch/papers/clifford/images/spiegelung.tex @@ -0,0 +1,85 @@ +% +% spiegelung.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math,calc} +\begin{document} +\def\skala{1.1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\def\punkt#1{ +\fill #1 circle[radius=0.06]; +} + +\coordinate (M) at (0,0); + +\fill[color=blue] (M) circle[radius=0.06]; +\node[color=blue] at (M) [left] {$M$}; + +\input{punkte.tex} + +\fill[color=red!30] (A) -- (B) -- (C) -- cycle; +\draw[color=red] (A) -- (B) -- (C) -- cycle; +\node at (A) [below] {$A$}; +\node at (B) [above right] {$B$}; +\node at (C) [above] {$C$}; +\node at (S) {$\circlearrowleft$}; + +\fill[color=red!30] (A1) -- (B1) -- (C1) -- cycle; +\draw[color=red] (A1) -- (B1) -- (C1) -- cycle; +\node at (A1) [below] {$A'$}; +\node at (B1) [above] {$B'$}; +\node at (C1) [above right] {$C'$}; +\node at (S1) {$\circlearrowright$}; + +\fill[color=red!30] (A2) -- (B2) -- (C2) -- cycle; +\draw[color=red] (A2) -- (B2) -- (C2) -- cycle; +\node at (A2) [below] {$A''$}; +\node at (B2) [above] {$B''$}; +\node at (C2) [left] {$C''$}; +\node at (S2) {$\circlearrowleft$}; + +\draw[color=gray,dotted] (A) -- (A1); +\draw[color=gray,dotted] (B) -- (B1); +\draw[color=gray,dotted] (C) -- (C1); + +\draw[color=gray,dotted] (A1) -- (A2); +\draw[color=gray,dotted] (B1) -- (B2); +\draw[color=gray,dotted] (C1) -- (C2); + +\punkt{(A)} +\punkt{(B)} +\punkt{(C)} +\punkt{(A1)} +\punkt{(B1)} +\punkt{(C1)} +\punkt{(A2)} +\punkt{(B2)} +\punkt{(C2)} + +\fill[color=darkgreen!30] (M) -- (G1) arc ({\winkela}:{\winkela+\winkelb}:1) -- cycle; +\draw[color=darkgreen] (G1) arc ({\winkela}:{\winkela+\winkelb}:1); +\node[color=darkgreen] at ({\winkela+0.5*\winkelb}:0.7) {$\alpha$}; + +\node at ($6*(G1)$) [right] {$g\mathstrut$}; +\node at ($-5.6*(G2)$) [left] {$h\mathstrut$}; + +\clip (-3,-0.2) rectangle (4.5,5.5); + +\draw[line width=1pt] (G1oben) -- (G1unten); +\draw[line width=1pt] (G2oben) -- (G2unten); + +\fill[color=blue] (M) circle[radius=0.06]; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/clifford/main.tex b/buch/papers/clifford/main.tex index ec44963..3649b20 100644 --- a/buch/papers/clifford/main.tex +++ b/buch/papers/clifford/main.tex @@ -16,10 +16,11 @@ \input{papers/clifford/4_GeometrischesProdukt.tex} \input{papers/clifford/5_PolareDarstellung.tex} \input{papers/clifford/6_PauliMatrizen.tex} -\input{papers/clifford/7_Reflektion.tex} +\input{papers/clifford/7_Spiegelung.tex} \input{papers/clifford/8_Rotation.tex} \input{papers/clifford/9_KomplexeZahlen.tex} \input{papers/clifford/10_Quaternionen.tex} +\input{papers/clifford/11_Fazit.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/clifford/references.bib b/buch/papers/clifford/references.bib index ff829d6..9090005 100644 --- a/buch/papers/clifford/references.bib +++ b/buch/papers/clifford/references.bib @@ -4,32 +4,13 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{clifford:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{clifford:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{clifford:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@article{clifford:hestenes_GA, + author = { David Hestenes, Garret Eugene Sobczyk and James S. Marsh }, + title = { Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics }, + journal = { American Journal of Physics }, + year = 1985, + volume = 53, + pages = {24}, + url = {https://www.researchgate.net/publication/258944244_Clifford_Algebra_to_Geometric_Calculus_A_Unified_Language_for_Mathematics_and_Physics} } diff --git a/buch/papers/erdbeben/teil0.tex b/buch/papers/erdbeben/teil0.tex index afa1244..d32b316 100644 --- a/buch/papers/erdbeben/teil0.tex +++ b/buch/papers/erdbeben/teil0.tex @@ -77,7 +77,7 @@ Deshalb nehmen wir $f$ als dritte Grösse in den Zustandsvektor auf und definier \[ x = (s_1, s_2, f)^T. - \] +\] Für die Standard-Form $\dot x = Ax$ brauchen wir als nächstes die Ableitungen aller Elemente von $x$. Für $\dot s_1$ und $\dot s_2$ folgen diese direkt aus Gleichung (20.1), aber über $\dot f$ wissen wir nichts. Wir müssen also eine Annahme treffen: $\dot f = 0$. Diese Annahme ist im Allgemeinen falsch, aber etwas Besseres haben wir zurzeit nicht zur Verfügung. diff --git a/buch/papers/ifs/images/FIC.pdf b/buch/papers/ifs/images/FIC.pdf index 1c76dfe..525a857 100644 --- a/buch/papers/ifs/images/FIC.pdf +++ b/buch/papers/ifs/images/FIC.pdf @@ -1,7 +1,7 @@ %PDF-1.6
%
-1 0 obj
<</Metadata 2 0 R/OCProperties<</D<</ON[5 0 R 6 0 R 7 0 R 8 0 R 9 0 R]/Order 10 0 R/RBGroups[]>>/OCGs[5 0 R 6 0 R 7 0 R 8 0 R 9 0 R]>>/Pages 3 0 R/Type/Catalog>>
endobj
2 0 obj
<</Length 50391/Subtype/XML/Type/Metadata>>stream
+1 0 obj
<</Metadata 2 0 R/OCProperties<</D<</ON[5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 37 0 R 38 0 R 39 0 R 40 0 R 41 0 R]/Order 42 0 R/RBGroups[]>>/OCGs[5 0 R 6 0 R 7 0 R 8 0 R 9 0 R 37 0 R 38 0 R 39 0 R 40 0 R 41 0 R]>>/Pages 3 0 R/Type/Catalog>>
endobj
2 0 obj
<</Length 51947/Subtype/XML/Type/Metadata>>stream
<?xpacket begin="" id="W5M0MpCehiHzreSzNTczkc9d"?> -<x:xmpmeta xmlns:x="adobe:ns:meta/" x:xmptk="Adobe XMP Core 6.0-c004 79.164570, 2020/11/18-15:51:46 "> +<x:xmpmeta xmlns:x="adobe:ns:meta/" x:xmptk="Adobe XMP Core 6.0-c006 120.b669747, 2021/05/19-19:07:51 "> <rdf:RDF xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"> <rdf:Description rdf:about="" xmlns:dc="http://purl.org/dc/elements/1.1/" @@ -13,6 +13,7 @@ xmlns:illustrator="http://ns.adobe.com/illustrator/1.0/" xmlns:xmpTPg="http://ns.adobe.com/xap/1.0/t/pg/" xmlns:stDim="http://ns.adobe.com/xap/1.0/sType/Dimensions#" + xmlns:stFnt="http://ns.adobe.com/xap/1.0/sType/Font#" xmlns:xmpG="http://ns.adobe.com/xap/1.0/g/" xmlns:pdf="http://ns.adobe.com/pdf/1.3/"> <dc:format>application/pdf</dc:format> @@ -21,8 +22,8 @@ <rdf:li xml:lang="x-default">FIC</rdf:li> </rdf:Alt> </dc:title> - <xmp:MetadataDate>2021-06-20T20:23:48+02:00</xmp:MetadataDate> - <xmp:ModifyDate>2021-06-20T20:23:48+02:00</xmp:ModifyDate> + <xmp:MetadataDate>2021-07-16T16:31+02:00</xmp:MetadataDate> + <xmp:ModifyDate>2021-07-16T16:31+02:00</xmp:ModifyDate> <xmp:CreateDate>2021-06-20T20:23:48+02:00</xmp:CreateDate> <xmp:CreatorTool>Adobe Illustrator 25.2 (Windows)</xmp:CreatorTool> <xmp:Thumbnails> @@ -31,11 +32,11 @@ <xmpGImg:width>256</xmpGImg:width> <xmpGImg:height>128</xmpGImg:height> <xmpGImg:format>JPEG</xmpGImg:format> - <xmpGImg:image>/9j/4AAQSkZJRgABAgEASABIAAD/7QAsUGhvdG9zaG9wIDMuMAA4QklNA+0AAAAAABAASAAAAAEA
AQBIAAAAAQAB/+4ADkFkb2JlAGTAAAAAAf/bAIQABgQEBAUEBgUFBgkGBQYJCwgGBggLDAoKCwoK
DBAMDAwMDAwQDA4PEA8ODBMTFBQTExwbGxscHx8fHx8fHx8fHwEHBwcNDA0YEBAYGhURFRofHx8f
Hx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8fHx8f/8AAEQgAgAEAAwER
AAIRAQMRAf/EAaIAAAAHAQEBAQEAAAAAAAAAAAQFAwIGAQAHCAkKCwEAAgIDAQEBAQEAAAAAAAAA
AQACAwQFBgcICQoLEAACAQMDAgQCBgcDBAIGAnMBAgMRBAAFIRIxQVEGE2EicYEUMpGhBxWxQiPB
UtHhMxZi8CRygvElQzRTkqKyY3PCNUQnk6OzNhdUZHTD0uIIJoMJChgZhJRFRqS0VtNVKBry4/PE
1OT0ZXWFlaW1xdXl9WZ2hpamtsbW5vY3R1dnd4eXp7fH1+f3OEhYaHiImKi4yNjo+Ck5SVlpeYmZ
qbnJ2en5KjpKWmp6ipqqusra6voRAAICAQIDBQUEBQYECAMDbQEAAhEDBCESMUEFURNhIgZxgZEy
obHwFMHR4SNCFVJicvEzJDRDghaSUyWiY7LCB3PSNeJEgxdUkwgJChgZJjZFGidkdFU38qOzwygp
0+PzhJSktMTU5PRldYWVpbXF1eX1RlZmdoaWprbG1ub2R1dnd4eXp7fH1+f3OEhYaHiImKi4yNjo
+DlJWWl5iZmpucnZ6fkqOkpaanqKmqq6ytrq+v/aAAwDAQACEQMRAD8Ah+jaNo8mj2MkljbvI9vE
zu0SEklASSSMVRn6C0T/AKt9t/yJj/pil36C0T/q323/ACJj/pirv0Fon/Vvtv8AkTH/AExV36C0
T/q323/ImP8Apirv0Fon/Vvtv+RMf9MVd+gtE/6t9t/yJj/pirv0Fon/AFb7b/kTH/TFXDQ9EH/S
vtv+RMf/ADTiqfaJqOkaa6Cfy3omo267GK5020LU70kWNXr8ycUPYvJdj+TfmmErb+U9Ht9QjXlP
ZSWFoWA7sjen8a17/eBirKP+VY/lr/1Kejf9w+1/6p4q7/lWP5a/9Sno3/cPtf8Aqnirv+VY/lr/
ANSno3/cPtf+qeKu/wCVY/lr/wBSno3/AHD7X/qniqWWvljy1of5laP+hNJstL+saNq31j6lbxW/
qcLrTeHP0lXlx5GlelcVeV/n5Y2V1+ZSfWreKfho1nw9VFelbq9rTkDStMVeffoLRP8Aq323/ImP
+mKVG80TRltJ2WwtgwjYgiGMEEKfbFDrPRNGa0gZrC2LGNSSYYySSo9sVVv0Fon/AFb7b/kTH/TF
KEn0bRxqdqgsbcI0cxZfSShIKUqKdq4oRf6C0T/q323/ACJj/pil36C0T/q323/ImP8ApiqD0nRt
Ie2dnsbdmFxcqC0SHZbiRVG47AUGKE70xNN09gY9I0yda1ZLmwtLgH2rJGzD6DirM7XWfy21OLTr
G/8AKOjadfPqNgpuoLK2WGSM3SCVXqlUBStQSQR1xV7N/wAqx/LX/qU9G/7h9r/1TxV3/Ksfy1/6
lPRv+4fa/wDVPFUh0H8ufy9l8weZYpPLGkvFBd26wRtY2xVFaygchAUooLMTt3xVPv8AlWP5a/8A
Up6N/wBw+1/6p4qx78xfy6/L60/L7zPdWvljSbe6t9JvpYJ4rG2SSORLZ2V0ZUBVlIqCMVfPGhf8
cTT/APmGh/5NjFKOxV2KuxV2KuxV2KuxV2KuxVFaVql7pWo2+oWUhiurZw8bjxHUHxBGxHcYq+pP
Lmt2+uaHZ6rAKJdRhila8XHwulf8lgRihMcVdirsVY1qH/kytB/7Y2sf9RWl4q8f/PP/AMmUP+2N
Zf8AUVe4qwXFKhe/7xXH/GN/+InFXWX+8Vv/AMY0/wCIjFVfFUFcf8daz/4xT/rjxVG4q7FUDo/+
8kn/ADE3X/UTJiqOxVBar/cwf8xMH/J1cVfQ/wCTPm+XVtIk0m8k53umhfSdj8T252X58D8Pypih
6LirHfLv/KS+av8AmMtv+oCDFWRYqxr8zv8AyWvmz/tjah/1CyYq+V9C/wCOJp//ADDQ/wDJsYpR
2KuxV2KuxV2KuxV2KuxV2KuxV7x+RNzJJ5TuoXJKwXjiOvYNGjUH01P04oej4q7FXYqxrUP/ACZW
g/8AbG1j/qK0vFXjH59zTxfmUnpW7T10azrxZFpS6vf5yvXFXn/12+/6t8v/AAcP/NeKVG8vL02k
4NhIAY2qecW3wn/LxQ6zvL0WkAFhIQI1oecW/wAI/wAvFVb67ff9W+X/AIOH/mvFKDnu7z9J2h+o
yAiOai84qmpT/K7YoRn12+/6t8v/AAcP/NeKXfXb7/q3y/8ABw/814qg9Ku7xbVwtjIw+sXJqHiG
5uJCRu3bpihGfXb7/q3y/wDBw/8ANeKUHqd3eGKGtjItLiEgl4tyJBts3fFDPvyl17WLLzrbC20m
e4a5imheFJbdWZRGZOryKuxjB64q94/xF5l/6lW8/wCkmw/6r4qkOg695hXzD5mZfLN27Pd25dBc
WQKEWUAAJM4BqBXbFU+/xF5l/wCpVvP+kmw/6r4qx78xte8wS/l75njl8tXcET6TfLJO1xZMqKbZ
wXISdmIUb7CuKvnXQv8Ajiaf/wAw0P8AybGKUdirsVdirsVdirsVdirsVdirsVe9/kZZvD5QmncU
+s3cjx+6KiJX/glbFD0TFXYqp3MvpW8kg6opI+YG2KsMs7qSb8xNFSRi7JpGsHkdzRrrTP6Yq8x/
PP8A8mUP+2NZf9RV7irBcUqF7/vFcf8AGN/+InFXWX+8Vv8A8Y0/4iMVV8VQVx/x1rP/AIxT/rjx
VG4q7FUDo/8AvJJ/zE3X/UTJiqOxVBar/cwf8xMH/J1cVej/AJLWTXHnmCYdLOCaZvky+j/zNxQ+
hcVY75d/5SXzV/zGW3/UBBirIsVYD+Y1458n+cFLfA+j6lGBXb4bWSn6sVfOGhf8cTT/APmGh/5N
jFKOxV2KuxV2KuxV2KuxV2KuxVtEd3VEBZ2ICqNySegGKvqbyhon6E8s6dphFJLeEetQ1HquS8lD
/rscUJxirsVQesNx0+TxbiB94xVhum/+TK0n/tjar/1Fabirzj88/wDyZQ/7Y1l/1FXuKsFxSoXv
+8Vx/wAY3/4icVdZf7xW/wDxjT/iIxVXxVBXH/HWs/8AjFP+uPFUbirsVQOj/wC8kn/MTdf9RMmK
o7FUFqv9zB/zEwf8nVxV7v8AkPoTQ6ff61KlDdMLe2JG/CPdyPZmIH+xxQ9VxVjvl3/lJfNX/MZb
f9QEGKsiJAFT0GKvMPzDYnyH5mY9TpV9X6bZ8VeGWU8ps4CSCfTSpKqSfhHUkZi4sEDAEgcnc9pa
vLHU5YxkQBkkAL/pFX9eT/J/4Ff6ZZ+Xx/zQ4P5/P/Pl83evJ/k/8Cv9Mfy+P+aF/P5/58vmzr8u
vy5u/MxN9eubbR42481RRJMw6iMlSAB3b6PGj+Xx/wA0L+fz/wA+Xzeqj8qvIYiCfotSQKczJLyr
4/bx/L4/5oX8/n/ny+bAJ/yabUra5udHvRFJBdXVultcD4SsM7ov7xRUHiP5fuyOm+kj+kfvZ9oE
mYJ5mED/ALEMA13yvr+gzelqtlJb1NElI5Rt3+GRaofvzIcFKsUuxV2KvTfyc8jSX9+nmG+jIsbN
62asP72df2h/kxn/AIb5HFD3LFXYq7FUt140s1HjIP1HFWJ6b/5MrSf+2Nqv/UVpuKvMfz7tvX/M
pP3skXHRrP8Au241rdXvXFXn/wCjP+Xu5/5Gf2Yqo3mm0tJz9auDSNjQybfZPtiqZ+UfJmpeYbqz
07T57gzSorOxkokaADk7Gmyr/Z1xV7jpv/OP/lG2tkW6vL+7uKD1JmmCKT34oF2HzJ+eKpB5j/Ij
y5J5l0qy0/UL6za6tb2QyNIJgGhaDj8JCmh9Q1+LFWIeZfyY82aGHmL3F9ZJubm1kL0G+7x05rt1
NKDxxVh/6M/5e7n/AJGf2Yqg9K0/laufrM6/6Rcigeg2uJBXp3xVGfoz/l7uf+Rn9mKq9h5SutZ1
Cxsobm5CS3lrHNMW5LEstwkfM7DoW298VfTun/l/aafZQ2Vpq+qRW0ChI41uQAAPknfqcVRH+D/+
13q3/SV/zbiqQaD5U5+YfMyfpjU19O7txyW5oWrZQNVjx3O9PliqdXPlHhbSt+m9W+FGP+9XgP8A
VxV53568t+j5I8wy/pTUJPT0y8f05J+SNxt3NGHHcHvirypLC9tLOy+tQPCJ7eKaEupAeN0DK6k9
QRlWH6I+4Ow7W/xvL/wyf+6LkALqD0JANMOWXDEnuDrMs+GBl3AllHk7yXL5ov8A6vZpNHbR73N4
9PTjHhUDdj2X+GCsnePl+1an3j8fF71p+ma7p9lBZWcthFbW6COKMQTbKo/4zbnxORrL3x+R/Wip
94+X7UR6Xmj/AJabH/kRN/1WxrL3x+R/WtT7x8v2pR5aj8wG1vPQntAv1+8584ZCef1h+ZFJR8PL
plGnGSjvH6j0Pf73M1oycUbMfoh0P80eaY3en69eW72122nXFvIKPDLbSOjD3VpSMvrL3x+R/W4d
T7x8v2vLNb/KCe90yPV/L/ESyBjNphJC1Vip9B3JPb7Ln6e2TwzMogllCViywB/K/mVLj6u+lXgn
/wB9+hJyPyHHfLGbOPJX5Narfzx3fmBGsdPHxfVSaTy+xA/ux41+L274oe4Wttb2ttFbW0axW8Ki
OKJRRVVRQAD2xVUxV2KuxVJNfnBkjhB+wCzfM9MVY5pv/kytJ/7Y2q/9RWm4qwD87NPvp/zAnu4I
HltrXRtPFzKillj9S6v+PMjoDxO+KvPcUrJbee5glhgjaWZ43CxoCSaKT0GKvov8pfJlv5d8q2kr
0fUr+CKW6l/lBQFYl9lrv4n6MUM3xVjuqf8AKdaB/wAwepf8StcVZFirC/Of5WaD5hWS5gUafqp3
FzEvwOf+LUFAa/zDf59MVeBWnkrzPawTr+jp7iJL7UIRcW8byxs0F9NC9GUfzoetDiqfeXvyv836
zOq/Unsbao9S5u1MQA/yVYB3+gfTir1W58n6V5Y8vaZZ2K85X1fTGubpwPUlYXke5p0A/ZXt86nF
WfYq7FWO+Xf+Ul81f8xlt/1AQYqmmsziKyZa/FKeI+XU4q88/ML/AJQHzL/2yr7/AKhnxVlPlHS9
N1L8vPLlvqFrFdw/ouyPpzIrgH6um4qNj75Xh+iPuDsO1v8AG8v/AAyf+6KU+Y/yx8jWmhapfW+m
CO6t7WeaFxNPRXSJmU8TJx2I8MjqP7uX9U/c6jU/3Uv6p+5m9lY2VjbJbWUEdtbp9iGJQiiu52FM
ub1fFXYqkvlP/eO+/wC2lf8A/US+Y+n5H+tL73O1/wBUf+Fw/wByE6zIcFK/LH/HDtv9n/ycbKNP
9Aa8X0ppl7Y7FXYq7FXYqpzzpBC0rn4VFfn7YqxWeZ5pnlf7TmpxVL9N/wDJlaT/ANsbVf8AqK03
FVW78w6fpH5k6wLyG8lFxo2k8BZ2N5fU4XWpV5/VIZ+H2tudK706HFUu1GL8qdQlMtz5e1MSHqYd
D1yCvuRDbIMVbfVfIGj6FqUekaNqdrJLazIZToesciGjOzTSWtePzamKproPn/Qo9D06NrXWCyWs
KkromrstRGBsy2hBHuDiqP8A+Vh6B/yy6z/3AtZ/7JMVSHUvPeiN5y0ScW2rcIrTUFYHRtWDku1t
TihtebD4dyoNO/UYqn3/ACsPQP8All1n/uBaz/2SYq7/AJWHoH/LLrP/AHAtZ/7JMVY95F89aJb6
JcxyW2rMzatrMgMejatKvGXVrqRQWjtWAYBviXqpqrAMCMVZD/ysPQP+WXWf+4FrP/ZJiqQ+cPPe
iT2WnqltqwKapp8h56Nq0Yol1GxAL2qgtQbKNz0G+Kp9/wArD0D/AJZdZ/7gWs/9kmKu/wCVh6B/
yy6z/wBwLWf+yTFUt8ueZdOOteYrv0b5Yrm6gkiDaffLIFWzijPONoRJH8SGgdRUbjYg5Uc0Qa3+
RP6GqWaINb/In7gu1PzTZXVxVY7v0k2j/wBDu/pP913wePH+l/pZfqY/mI90v9LL9TGvOuow3nk3
XrO3huWnudOu4YVa1uEUu8DqoLvGqqKnqxoMTqIgWb/0sv1KdTACzxf6WX6meeQP+UE8t/8AbLsv
+odMlh+iPuDt+1v8by/8Mn/uiiPN3/KKa1/zAXP/ACZbI6j+7l/VP3Oo1P8AdS/qn7k2y5vdirsV
SXyn/vHff9tK/wD+ol8x9PyP9aX3udr/AKo/8Lh/uQnWZDgpX5Y/44dt/s/+TjZRp/oDXi+lNMvb
HYq7FXYqpzzxQRmSVuKj8T4DFWO6hqD3b/yxL9hP4n3xVCYqgtN/8mVpP/bG1X/qK03FU50//wAm
Vr3/AGxtH/6itUxVkuKpf5i/5R/U/wDmEn/5NtirvLv/ACj+mf8AMJB/ybXFUwxVjuqf8p1oH/MH
qX/ErXFWRYq7FWNfl5/xwLr/ALbOu/8AdZu8VZLirHfO/wDvBpv/AG19M/6jI8VZFirTOqKWYhVG
5J2AxVhMWoGTXdfWE0ilmtyT0JC2yD7sqx/VL3/oDXDmfx0VstbEv8w/8cDU/wDmEn/5NtlOo/u5
f1T9zRqf7qX9U/cyPyB/ygnlv/tl2X/UOmSw/RH3B2/a3+N5f+GT/wB0UR5u/wCUU1r/AJgLn/ky
2R1H93L+qfudRqf7qX9U/cm2XN7sVdiqS+U/9477/tpX/wD1EvmPp+R/rS+9ztf9Uf8AhcP9yE6z
IcFK/LH/ABw7b/Z/8nGyjT/QGvF9KaZe2OxV2KuxVjGo3bXNyzV/dqSIx2p4/TiqFxV2KoLTf/Jl
aT/2xtV/6itNxVOdP/8AJla9/wBsbR/+orVMVZLiqX+Yv+Uf1P8A5hJ/+TbYq7y7/wAo/pn/ADCQ
f8m1xVMMVY7qn/KdaB/zB6l/xK1xVkWKuxVjX5ef8cC6/wC2zrv/AHWbvFWS4qx3zv8A7wab/wBt
fTP+oyPFWRYqkmu3TGVbdT8CgM48SemKsV0z/jsax/xkh/5MLlWP6pe/9Aa4cz+OiaZa2Jf5h/44
Gp/8wk//ACbbKdR/dy/qn7mjU/3Uv6p+5Efl/oZvPIXlu7OpX0JuNKspTFFOVjTnbo3FAQSFFaDf
IDTf0pfN2+TtGU5GUoQMibJ4eqM8z+XfS8tatL+k7+T07O4b03n5I3GJjRhx3B75XmwVCR4pcj1c
TWay8MxwQ+k9PJM/8M/9rbUf+kj/AJtyz8v/AEpfNyfzv9DH/pXf4Z/7W2o/9JH/ADbj+X/pS+a/
nf6GP/Su/wAM/wDa21H/AKSP+bcfy/8ASl81/O/0Mf8ApUdpOlW+mWhtoHkkVpJJnkmbm7PK5dyW
92bLceMQFBx9RnOWXEa5AbeWyMyxpSvyx/xw7b/Z/wDJxso0/wBAa8X0ppl7Y7FXYq5q0NOvbFWH
Yq7FXYqgtN/8mVpP/bG1X/qK03FURPa67P8AmVrH6K1CGx46NpPretbG551utS40pLDxpv41xVNP
0X56/wCr/Z/9w1v+yrFUDr2medxoWol9es2QWsxZRpzAkem1RX60aYq7QdM87nQtOKa9ZqhtYSqn
TmJA9NaCv1oVxVHfovz1/wBX+z/7hrf9lWKpDqWm+cx5z0NW1y0MxtNQMcg09gFAa25Ar9Z3rt32
xVPv0X56/wCr/Z/9w1v+yrFXfovz1/1f7P8A7hrf9lWKse8iab5ybRLkwa3aRJ+ltZBVtPZyXGrX
Qdq/WV2Z6sB2rTfrirIf0X56/wCr/Z/9w1v+yrFUh846b5zWy08y65aSA6ppwULp7LRzdxhW/wB6
WqAd6d8VT79F+ev+r/Z/9w1v+yrFWP6tpnnQX8nPXLQk8dxp7D9kf8vOKpLo0OvSaprMTajEJ7ee
JJpVtvhkJt43UhDIeNA/HqfHKTjlZIPPyazE2SDzTf6j5g/6ukf/AEij/qph4Z94+X7VqXf9n7Uq
82DW7HytrN7LfxzxWtjczPAIAhdY4WYpz5tx5UpWhpkZ45yBBI38v2sZ45SiYk8/Jmv5Y/8AktfK
f/bG0/8A6hY8vbk28w2k97oGpWduA09zazwxKSAC8kbKoqem5yvNEygQOZBas0TKEgOZBUv0rq3/
AFZZ/wDkbbf9VMh4k/5p+Y/Wnjl/NP2frd+ldW/6ss//ACNtv+qmPiT/AJp+Y/WvHL+afs/W79K6
t/1ZZ/8Akbbf9VMfEn/NPzH6145fzT9n63fpXVv+rLP/AMjbb/qpj4k/5p+Y/WvHL+afs/W79K6t
/wBWWf8A5G23/VTHxJ/zT8x+teOX80/Z+tV0C2uLbSLeG4T05lDF0qGoWYtSoqO+SwRIgAeacYIj
uj8tZuxV2KuxVi+o2xt7t1p8LHknyOKobFXYqgtN/wDJlaT/ANsbVf8AqK03FU50/wD8mVr3/bG0
f/qK1TFWS4ql/mL/AJR/U/8AmEn/AOTbYq7y7/yj+mf8wkH/ACbXFUwxVjuqf8p1oH/MHqX/ABK1
xVkWKuxVjX5ef8cC6/7bOu/91m7xVkuKsd87/wC8Gm/9tfTP+oyPFWRYqk+v2x+C5UbD4H/gcVYT
oX/Hf8yf8xVv/wBQUOKp7iqQfmD/AMoD5l/7ZV9/1DPirK/yx/8AJa+U/wDtjaf/ANQseKslxV2K
uxV2KuxV2KuxV2KuxV2KuxVL9atvVtfUA+OLf/Ynr/XFWPYq7FUFpv8A5MrSf+2Nqv8A1FabiqYR
6jp9n+ZWt/XLqK29TRtI4etIqcqXWp1pyIrSuKp9/iLy/wD9XO0/5Hx/81Yql/mDzBoLaDqSrqVq
WNrOABPGSSY2/wArFXeX/MGgroOmq2pWoYWsAIM8YIIjX/KxVMP8ReX/APq52n/I+P8A5qxVj+p6
9oR876C41G1KLZ6iGYTR0BLWtKnl3pirIP8AEXl//q52n/I+P/mrFXf4i8v/APVztP8AkfH/AM1Y
qxvyBr2hx6FdLJqNqjHWNbYBpoweLavdsp3PQggjFWSf4i8v/wDVztP+R8f/ADViqQec9c0SWx08
RahbOV1TTnYLNGaIl3GzMaHooFScBIHNMYkmgnv+JvLf/V2s/wDpIi/5qyPix7w2eBk/mn5LJ/MP
liaF4m1Wz4uKH/SIv+asfFj3hfAyfzT8nnukappUOv8AmIyXtuqPdQmNzKnFwtpEpKmvxDkpG2SE
geTCUDHYik4/Tuif9XC2/wCR0f8AXCxSLz7rOjyeRfMccd9bvI+l3qoiyoSSbdwAAD1xVm/5Y/8A
ktfKf/bG0/8A6hY8VZLirsVdirsVdirsVdirsVdirsVdiqjeTRQ27tKfhIIp4kjpirFMVdiqC03/
AMmVpP8A2xtV/wCorTcVTCPTtPvPzK1v65axXPp6NpHD1o1fjW61OtOQNK0xVPv8O+X/APq2Wn/I
iP8A5pxVL/MHl/QV0HUmXTbUMLWcgiCMEERt/k4q7y/5f0FtB01m021LG1gJJgjJJMa/5OKph/h3
y/8A9Wy0/wCREf8AzTirH9T0HQh530FBp1qEaz1EsohjoSGtaVHHtXFWQf4d8v8A/VstP+REf/NO
Ku/w75f/AOrZaf8AIiP/AJpxVjfkDQdDk0K6aTTrV2Gsa2oLQxk8V1e7VRuOgAAGKsk/w75f/wCr
Zaf8iI/+acVY/wCc9C0SO10v09Ptk56rYI/GGMVVrhQVNBuD3GUagbD+sPvc3QkiUiP5k/8AclPv
8M+W/wDq02f/AEjxf805Z4Ue4OP4+T+cfm0/lvyyilm0uyVRuSbeID/iOPhR7gvj5P5x+bArbTNE
n8y+YOFhb+gk8AhT0U4qPq0deIptU75XiAEpAd4+4ORqpGWPGSb9J/3Ukx/QWif9W+2/5Ex/0y9w
ki8+6No8fkXzHJHY26SJpd6yOsSAgi3cgggdcVZv+WP/AJLXyn/2xtP/AOoWPFWS4q7FXYq7FXYq
7FXYq7FXYq7FXE0BPhirFLq6luZTJIf9VewHgMVUcVdiqC03/wAmVpP/AGxtV/6itNxVOdP/APJl
a9/2xtH/AOorVMVZLiqX+Yv+Uf1P/mEn/wCTbYq7y7/yj+mf8wkH/JtcVTDFWO6p/wAp1oH/ADB6
l/xK1xVkWKuxVjX5ef8AHAuv+2zrv/dZu8VZLirHfO/+8uk/9tfT/wDqIXKM/If1h97maL6pf1J/
7ksiy9w0g1u5le6aCtI46UXxJFan78VYno//AB3tf/4zW/8A1DR5Tj+uXvH3By9R/d4/6p/3Uk6y
5xEg/MH/AJQHzL/2yr7/AKhnxVOvKsXnzQ/K+j6K+iWU76XY21k0y6iyhzbxLEWCm1NOXGtMVTT9
Keev+rBZ/wDcSb/slxV36U89f9WCz/7iTf8AZLirv0p56/6sFn/3Em/7JcVd+lPPX/Vgs/8AuJN/
2S4q79Keev8AqwWf/cSb/slxV36U89f9WCz/AO4k3/ZLirv0p56/6sFn/wBxJv8AslxV36U89f8A
Vgs/+4k3/ZLirv0p56/6sFn/ANxJv+yXFXfpTz1/1YLP/uJN/wBkuKu/Snnn/qX7P/uJN/2S4qkr
2nnzm3DRLPjX4R+kW6f9I2Krfqnn/wD6sll/3EW/7JsVd9U8/wD/AFZLL/uIt/2TYqr+X9D80nzh
a6vqllbWdpaafeWgENybh3kup7SRdjFFQKtq3fviqOvbPzPZ+cL7WNMsLa+tb7T7G0IlumtnSS0n
vJG2EMwYMt2tN+xxVX/Snnr/AKsFn/3Em/7JcVQ+pXPnu8066tBoVkhuYZIg51FjTmpWtPqvauKu
025892enWtodCsnNtDHEXGosK8FC1p9V70xVEfpTz1/1YLP/ALiTf9kuKpbdp58n1/TtUGiWSrYw
3UJi/SLEt9ZMRBr9V24+j+OKpl+lPPX/AFYLP/uJN/2S4q79Keev+rBZ/wDcSb/slxVK/LkXnzR9
PltG0WymMt7f3vMagy0F9ezXYSn1Y/YE/GvelcVTT9Keev8AqwWf/cSb/slxVLtcTzzqcNon6EtI
vqt5bXm2oFi31eUPw3t0pyp1yrNEkbdCC5WkyRjI8WwMZDv5ikx/S/nf/qXbf/uIj/qhkePJ/NHz
/Yy8LB/Pl/pP+PJdff41upvVGg28ZIAYfXwa07/3Ix48n80fP9i+Fg/ny/0n/Hkps9C8929/qF2N
KtG+vSRv6bXxXj6cSx9RA9a8a4cUZWSdr/UjUzgRCMCTwjqK6k95Rv1Lz7/1ZrL/ALiDf9k2XOIl
3mPy75/1fy9qmkppVjC+oWk9qspv3YIZ4mjDEfVhWnKuKv8A/9k=</xmpGImg:image> + <xmpGImg:image>/9j/4AAQSkZJRgABAgEASABIAAD/7QAsUGhvdG9zaG9wIDMuMAA4QklNA+0AAAAAABAASAAAAAEA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==</xmpGImg:image> </rdf:li> </rdf:Alt> </xmp:Thumbnails> - <xmpMM:InstanceID>uuid:e882c6a4-b7dc-4cb1-95f3-adfed3b77c13</xmpMM:InstanceID> + <xmpMM:InstanceID>uuid:b9fcd98f-6fe2-4cbb-a19d-4367321d727e</xmpMM:InstanceID> <xmpMM:DocumentID>xmp.did:ffdebe24-c43c-ae47-828c-2b9d14439d09</xmpMM:DocumentID> <xmpMM:OriginalDocumentID>uuid:5D20892493BFDB11914A8590D31508C8</xmpMM:OriginalDocumentID> <xmpMM:RenditionClass>proof:pdf</xmpMM:RenditionClass> @@ -73,6 +74,19 @@ <stDim:h>60.000000</stDim:h> <stDim:unit>Millimeters</stDim:unit> </xmpTPg:MaxPageSize> + <xmpTPg:Fonts> + <rdf:Bag> + <rdf:li rdf:parseType="Resource"> + <stFnt:fontName>MyriadPro-It</stFnt:fontName> + <stFnt:fontFamily>Myriad Pro</stFnt:fontFamily> + <stFnt:fontFace>Italic</stFnt:fontFace> + <stFnt:fontType>Open Type</stFnt:fontType> + <stFnt:versionString>Version 2.106;PS 2.000;hotconv 1.0.70;makeotf.lib2.5.58329</stFnt:versionString> + <stFnt:composite>False</stFnt:composite> + <stFnt:fontFileName>MyriadPro-It.otf</stFnt:fontFileName> + </rdf:li> + </rdf:Bag> + </xmpTPg:Fonts> <xmpTPg:PlateNames> <rdf:Seq> <rdf:li>Cyan</rdf:li> @@ -648,20 +662,22 @@ <?xpacket end="w"?>
-endstream
endobj
3 0 obj
<</Count 1/Kids[11 0 R]/Type/Pages>>
endobj
11 0 obj
<</ArtBox[8.16666 14.1824 288.781 151.356]/BleedBox[0.0 0.0 297.638 170.079]/Contents 12 0 R/CropBox[0.0 0.0 297.638 170.079]/LastModified(D:20210620202348+02'00')/MediaBox[0.0 0.0 297.638 170.079]/Parent 3 0 R/PieceInfo<</Illustrator 13 0 R>>/Resources<</ExtGState<</GS0 14 0 R>>/Properties<</MC0 5 0 R/MC1 6 0 R/MC2 7 0 R/MC3 8 0 R/MC4 9 0 R>>>>/Thumb 15 0 R/TrimBox[0.0 0.0 297.638 170.079]/Type/Page>>
endobj
12 0 obj
<</Filter/FlateDecode/Length 1133>>stream
-HWˎT9 ߯T*<j!ԋ - {s_*L 6pYB<"-l}5kWB3IG}X>Aҍ㢯e -`F]Gy8_ -:fNY- ۊT} (7[ - Lg#d,0Ȃ#e 0;- } -%*J<ŞG\f+˭z -endstream
endobj
15 0 obj
<</BitsPerComponent 8/ColorSpace 16 0 R/Filter[/ASCII85Decode/FlateDecode]/Height 21/Length 280/Width 37>>stream
-8;XF2_%FR-$j6r9!TZZWr$[(j>QmjWcVM*4CF>9rfZ'><*tdEK_f@d@jPN9F"=9\7 -mE.?=]D[&B,p1C*,`Bo=p2A(oZF)fL1&.2T;DRH4&m'Z]X1,t[/tN:.$JL`sTHr!Z -jqp^HTfF<cOpUImc*bg#/h^+*HJ'tk&*6f-4d)(=0_!!Ldh./XTPZ)c763_kCEQ<M -(->f'!cI8XZ17^;kn'b7c+#G3D^VB>V[f8,`O/I@RsLVXd&^1Q]d_2P:nmfor-W@1 -YPnm#`-VJX.%=9~>
-endstream
endobj
16 0 obj
[/Indexed/DeviceRGB 255 17 0 R]
endobj
17 0 obj
<</Filter[/ASCII85Decode/FlateDecode]/Length 428>>stream
+endstream
endobj
3 0 obj
<</Count 1/Kids[11 0 R]/Type/Pages>>
endobj
11 0 obj
<</ArtBox[8.16666 14.1824 288.781 151.356]/BleedBox[0.0 0.0 297.638 170.079]/Contents 43 0 R/CropBox[0.0 0.0 297.638 170.079]/LastModified(D:20210716163100+02'00')/MediaBox[0.0 0.0 297.638 170.079]/Parent 3 0 R/PieceInfo<</Illustrator 44 0 R>>/Resources<</ExtGState<</GS0 45 0 R>>/Font<</T1_0 36 0 R>>/ProcSet[/PDF/Text]/Properties<</MC0 37 0 R/MC1 38 0 R/MC2 39 0 R/MC3 40 0 R/MC4 41 0 R>>>>/Thumb 46 0 R/TrimBox[0.0 0.0 297.638 170.079]/Type/Page>>
endobj
43 0 obj
<</Filter/FlateDecode/Length 1309>>stream
+HWM[7 _cz,R>
rAz!Mf +1,l#QfZQ )Q!قbqT8h-[,VihI\r$1NƒGd aW\C|+_*%,&!IZ豶N
M h%Z3?6bw %UHK#!H!5h
9^ +X[r .RJE9Xlɨi Fˈc"ؠ-Q2as"bF$
`*9No OB-`9$y|'0 7Myuf?c} k'7}me~X{. +$}E7+EC_͑W]w& {cfU`_DN# + r{"v/AnNp> G<x`L̑G~ y4%&l"Tb ! J p ºP (4@#'R%o@OXJ q_4\DcqZzFSa04z[\ tX + p80#! +6h9< L5Xs xb܀[ Y6
-G +#%\ D1Ha癑=ndCluHyH)^( Lf ' @eB ]&.00 Jal%VxiLօL2=<]tV²} ۛk
P/nSxip Vݐo`ʙ4||s'a% y 檐DžAlEY:7p\dё[9O +>AIi+GtLZ|;(i Y:3yZ
֞uJ/yPhDfA|X|p<xel~\vMA qi$J%4l_5xVaN9u?\%4LaĄZ*E˳~ؿ_*euO y.((y4 TrCq:I\WƆg
t(o1FJ [XzaAUuyv-yF8*5
wSGNv7@*hgf'peB=6T̴f=' 0 +endstream
endobj
46 0 obj
<</BitsPerComponent 8/ColorSpace 47 0 R/Filter[/ASCII85Decode/FlateDecode]/Height 21/Length 288/Width 37>>stream
+8;XEGYn=kr$lk'(P_p4>(k>>JJgJpWPkXrqncBfc,aV.98Pl)3lOa,HZg`SjjGT!I +"D!S;*Bca6V`Q;\MTT,KUc^rej^Smt+58A@"BD%u[dU$>!,M##-A>)&)'>2OH0` +.4bjmUD/P?'L;t&^E>U92,DTa"#,s"2I2#0GnfBccl`Q_qj^ud;,q!c62SKoS#-QA +@1t&W>2ek+*(4R(=%n"fQO1OWD?pc/idP=IBpt]QD0hh<fks!O)e`et:jZ\U7ps^+ +cgpb<q7HJVpr/g,!!a")bl~>
+endstream
endobj
47 0 obj
[/Indexed/DeviceRGB 255 48 0 R]
endobj
48 0 obj
<</Filter[/ASCII85Decode/FlateDecode]/Length 428>>stream
8;X]O>EqN@%''O_@%e@?J;%+8(9e>X=MR6S?i^YgA3=].HDXF.R$lIL@"pJ+EP(%0 b]6ajmNZn*!='OQZeQ^Y*,=]?C.B+\Ulg9dhD*"iC[;*=3`oP1[!S^)?1)IZ4dup` E1r!/,*0[*9.aFIR2&b-C#s<Xl5FH@[<=!#6V)uDBXnIr.F>oRZ7Dl%MLY\.?d>Mn @@ -669,25 +685,27 @@ E1r!/,*0[*9.aFIR2&b-C#s<Xl5FH@[<=!#6V)uDBXnIr.F>oRZ7Dl%MLY\.?d>Mn VNWFKf>nDZ4OTs0S!saG>GGKUlQ*Q?45:CI&4J'_2j<etJICj7e7nPMb=O6S7UOH< PO7r\I.Hu&e0d&E<.')fERr/l+*W,)q^D*ai5<uuLX.7g/>$XKrcYp0n+Xl_nU*O( l[$6Nn+Z_Nq0]s7hs]`XX1nZ8&94a\~>
-endstream
endobj
5 0 obj
<</Intent 18 0 R/Name(Ebene 4)/Type/OCG/Usage 19 0 R>>
endobj
6 0 obj
<</Intent 20 0 R/Name(Ebene 1)/Type/OCG/Usage 21 0 R>>
endobj
7 0 obj
<</Intent 22 0 R/Name(Ebene 2)/Type/OCG/Usage 23 0 R>>
endobj
8 0 obj
<</Intent 24 0 R/Name(Ebene 5)/Type/OCG/Usage 25 0 R>>
endobj
9 0 obj
<</Intent 26 0 R/Name(Ebene 3)/Type/OCG/Usage 27 0 R>>
endobj
26 0 obj
[/View/Design]
endobj
27 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
24 0 obj
[/View/Design]
endobj
25 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
22 0 obj
[/View/Design]
endobj
23 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
20 0 obj
[/View/Design]
endobj
21 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
18 0 obj
[/View/Design]
endobj
19 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
14 0 obj
<</AIS false/BM/Normal/CA 1.0/OP false/OPM 1/SA true/SMask/None/Type/ExtGState/ca 1.0/op false>>
endobj
13 0 obj
<</LastModified(D:20210620202348+02'00')/Private 28 0 R>>
endobj
28 0 obj
<</AIMetaData 29 0 R/AIPDFPrivateData1 30 0 R/AIPDFPrivateData2 31 0 R/AIPDFPrivateData3 32 0 R/AIPDFPrivateData4 33 0 R/AIPDFPrivateData5 34 0 R/ContainerVersion 12/CreatorVersion 25/NumBlock 5/RoundtripStreamType 2/RoundtripVersion 25>>
endobj
29 0 obj
<</Length 1223>>stream
+endstream
endobj
37 0 obj
<</Intent 49 0 R/Name(Ebene 4)/Type/OCG/Usage 50 0 R>>
endobj
38 0 obj
<</Intent 51 0 R/Name(Ebene 1)/Type/OCG/Usage 52 0 R>>
endobj
39 0 obj
<</Intent 53 0 R/Name(Ebene 2)/Type/OCG/Usage 54 0 R>>
endobj
40 0 obj
<</Intent 55 0 R/Name(Ebene 5)/Type/OCG/Usage 56 0 R>>
endobj
41 0 obj
<</Intent 57 0 R/Name(Ebene 3)/Type/OCG/Usage 58 0 R>>
endobj
57 0 obj
[/View/Design]
endobj
58 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.3)/Subtype/Artwork>>>>
endobj
55 0 obj
[/View/Design]
endobj
56 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.3)/Subtype/Artwork>>>>
endobj
53 0 obj
[/View/Design]
endobj
54 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.3)/Subtype/Artwork>>>>
endobj
51 0 obj
[/View/Design]
endobj
52 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.3)/Subtype/Artwork>>>>
endobj
49 0 obj
[/View/Design]
endobj
50 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.3)/Subtype/Artwork>>>>
endobj
36 0 obj
<</BaseFont/QGHLNM+MyriadPro-It/Encoding/WinAnsiEncoding/FirstChar 46/FontDescriptor 59 0 R/LastChar 106/Subtype/Type1/Type/Font/Widths[211 0 492 492 492 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 647 0 0 0 0 0 0 0 0 0 0 0 0 0 523 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 229 227]>>
endobj
59 0 obj
<</Ascent 953/CapHeight 674/CharSet(/period/zero/one/two/D/R/i/j)/Descent -250/Flags 96/FontBBox[-185 -250 1090 953]/FontFamily(Myriad Pro)/FontFile3 60 0 R/FontName/QGHLNM+MyriadPro-It/FontStretch/Normal/FontWeight 400/ItalicAngle -11/StemV 84/Type/FontDescriptor/XHeight 484>>
endobj
60 0 obj
<</Filter/FlateDecode/Length 928/Subtype/Type1C>>stream
+H|Pmlu pzsY_#:::L ºؽXTH[[vו R˶n ib:`Dtʐ9' H4wݿ /|˓</?$I7{GtXmMwɗ TtRjr-xpeaL<`rBBC7kÇؼ"t6 Y܈6o,=OpKi^lrlDADF{rHHp:YeC
DMp[E'f6PT!Ye#,>UcRSٖ6o':jegO6!`'#
$zX~h \ħ'erIqC} 3TқjJ20g9pAGV׳3l*ḡ3E($cK] pc >w{F&x6P$ȩe/SKQ?ipzЂ 8 |>X- 4ּ<T/ 4)^%Rz7 2u3lR&I:X0ECCڤ/ڲ$kF:6~nlV0j Z~ r`'¯+gIe*>ą~{I_o\Z퓫gw?ǴPpl ih4͖'҉!7ς5s0
t2sӗća~fdоי~ʹD¥-t~9Zmy''ã+8kod_ w(Jn)2L49.~ +endstream
endobj
45 0 obj
<</AIS false/BM/Normal/CA 1.0/OP false/OPM 1/SA true/SMask/None/Type/ExtGState/ca 1.0/op false>>
endobj
44 0 obj
<</LastModified(D:20210716163100+02'00')/Private 61 0 R>>
endobj
61 0 obj
<</AIMetaData 62 0 R/AIPDFPrivateData1 63 0 R/AIPDFPrivateData2 64 0 R/AIPDFPrivateData3 65 0 R/AIPDFPrivateData4 66 0 R/AIPDFPrivateData5 67 0 R/ContainerVersion 12/CreatorVersion 25/NumBlock 5/RoundtripStreamType 2/RoundtripVersion 25>>
endobj
62 0 obj
<</Length 1462>>stream
%!PS-Adobe-3.0
%%Creator: Adobe Illustrator(R) 24.0
-%%AI8_CreatorVersion: 25.2.3
+%%AI8_CreatorVersion: 25.3.1
%%For: (Alain) ()
-%%Title: (FIC.ai)
-%%CreationDate: 6/20/2021 8:23 PM
+%%Title: (FIC.pdf)
+%%CreationDate: 7/16/2021 4:31 PM
%%Canvassize: 16383
%%BoundingBox: 8 -156 289 -18
-%%HiResBoundingBox: 8.16666698455811 -155.896331787109 288.781496063013 -18.722782152231
+%%HiResBoundingBox: 8.16666698455811 -155.896331787109 288.781496063013 -18.722782152232
%%DocumentProcessColors: Cyan Magenta Yellow Black
%AI5_FileFormat 14.0
-%AI12_BuildNumber: 259
+%AI12_BuildNumber: 390
%AI3_ColorUsage: Color
%AI7_ImageSettings: 0
%%CMYKProcessColor: 1 1 1 1 ([Passermarken])
%AI3_Cropmarks: 0 -170.07874015748 297.63779527559 0
%AI3_TemplateBox: 149.5 -85.5 149.5 -85.5
-%AI3_TileBox: -260.125987616111 -370.677156943973 557.793995294045 200.522855263057
+%AI3_TileBox: -260.125987616111 -370.677156943975 557.793995294045 200.522855263056
%AI3_DocumentPreview: None
%AI5_ArtSize: 14400 14400
%AI5_RulerUnits: 1
@@ -696,1308 +714,1313 @@ endstream
endobj
5 0 obj
<</Intent 18 0 R/Name(Ebene 4)/Type/OCG/Usage 19 0 R>> %AI5_ArtFlags: 0 0 0 1 0 0 1 0 0
%AI5_TargetResolution: 800
%AI5_NumLayers: 5
-%AI9_OpenToView: -66.666666666667 34.666666666667 6 2256 1308 18 0 0 46 87 0 0 0 1 1 0 1 1 0 0
+%AI17_Begin_Content_if_version_gt:24 4
+%AI10_OpenToVie: -67 35 6 0 0 0 2256 1308 18 0 0 46 87 0 0 0 1 1 0 1 1 0 0
+%AI17_Alternate_Content
+%AI9_OpenToView: -67 35 6 2256 1308 18 0 0 46 87 0 0 0 1 1 0 1 1 0 0
+%AI17_End_Versioned_Content
%AI5_OpenViewLayers: 77777
+%AI17_Begin_Content_if_version_gt:24 4
+%AI17_Alternate_Content
+%AI17_End_Versioned_Content
%%PageOrigin:-157 -481
%AI7_GridSettings: 72 8 72 8 1 0 0.800000011920929 0.800000011920929 0.800000011920929 0.899999976158142 0.899999976158142 0.899999976158142
%AI9_Flatten: 1
%AI12_CMSettings: 00.MS
%%EndComments
-endstream
endobj
30 0 obj
<</Length 65536>>stream
-%AI24_ZStandard_Data(/ -# -DKt8Dν%θ!CLta#دX^By <#Wu:bCgi2;#[}ޖ*"r7D$Ƚ ,#yvF3}8Vn;wQ62EăGv7b&.L$tR݆Tܷ_.zA2bHES,+GI=bIJ.UD|4<^!Z7ҟ+v\""xIC.EF -GD`)n}0hl0kB644[K XlM^,ɓ.'@U%eeتYŲp˫Yr1]+#ڬAO2be)D3J]%|i&.-mf)mO;ʒ]0Kuw1 -6( t DHT N@LA$ "Q `
<4Lz,2te ٔJm#v%?f-椭 y D`,jU, PY -zac!C$"
d(Q"ETͬzVX`,`( ˍ"q x, 5xp4Dxql WTNYfˮ GP$^!H, GcH$"AFc@$8C$≈# @$~8xV> Taxl3,ġ:UjeR-|!0!Vl:em ZDMrRk
@:t)FeHf2XudSnq
<C$,ɁH$#"*"QF$˅UV =@$D("AW F \0 CR wsq{+DWVH'= E"}8 L](HʄH<TS^R*C~Xڋ١h( DĐ0"[Ģ8~@$ -`(ҎHd"".9HXX\/#< E$ 2!m@$̃H<C$qD"1p c /6f1 ,Dp4Wn46qq0 CP1 (X8BP,n>x781G5,,rŹ ԠQѐ*VS0F197543!v2JXPJ 饮2N7ʉ)Cem̶M@$p0XԢ> b)v4 p
ñ@$"HG$TDV."8"qccV:CK49P\@FHУQRAUh&{/CEfe:
zD}&"*CJs3eHe/C?s ŏ@$<H<2"a Dxt -A$ DP( -1.hE-lq \EcX<(fÒnTF:jmC$]4jA$ȶ
hP^n - Tgy_8 -
1 *LE# yH "H4 E$J.+"aÜ.D<N/t8>kU]hScV -S`0
`<h8CX0<p4BD$L bØ81ӖVWlUtQFuѡ) DbA j 4v|Fofa jQ^ZG [ZMa cX /笠1q{`XhxxT "@6ЁlR:+-/13\$#Gtnq E~#yłJ -DeyłP<CPLeybX,cD A$*(рd0 ؋XsYjrĊjW\Up*1)A$*hp
%x`A"X
Ё 2h, t)ID7e!:,ve,C*ka"2S2FYe "H0';Db
GcX(̎q<geF$lS jPĠ-HA -:L&""146(*YjTj;}cØ,F1 Mg@$ld7Ec`4Fp4 G8ꨃ 䰣 09qt# P8`8x8 ỵ>A?G=a{ ( x0 Gx8 QlN>Ё
dxLD<4,4 1nqz 0-La432أ (F-JQ,+*Ukafxa4!B"-j-CP4CP(W0E) Ex_\?~HG6T3Ҳ(RDbF:qs up4
FcP46ajA6`8
` d 0e p, b0 - XpBC C d@&4\@ $Dpp *l` 0h4A"84HЀB$,H0 Lx
JpB&L`.8lyhȀ:p -\h -H$0L !p @ (0HR
(t CC 4@P@*x - - 2p -
.0\@"Q -P -*<\@A &LȀÁD0 P)4\ TMO z;Yfp - -@"hpA,:cA3bgE20o2G$8X|@pA&4D -J,X XЀÅ
4h0!XР @
25 *<$8< XD` " AB -
04P` X`ᡂ - "Tx Dp -
(8d@a -XP B@IJC .04(Tp0X C 8@$*T@AbC #b - & - d 4lB\ -TT@9ʠ$0" -TL 8@$4X[U24+7uݎ-!4ٚ2H
,x6 mlLZaUq
bt]r ia݉]R93Ǵ"|U%h 'XjHxf|,3wJVI~StUD,j)+!gV3E{!Qr,Xg -ZItZzͽ\& 6C̤R,RxY]I,10m;1ѥYEeOZaJ'n3;3L*ZW 2WrLߘG/WEiU+Kw ,43L9DâՍI/"y*{֩L^< cH3wTJ|"4Rc$Օ_nʦ -Ϯ߱[!{!JeZM +(YxmFj ߂MɥS S}9cs.ddل(Tpp&l!bh@ ("bN)_.,CR*imS* -8uU/:+YB"X5Q=2NDY;Mt? 6E9~o%PR=+S=!|Fؾf tQE[[{|5o**vM2T/W:<{钙yScwB d9Q U}lU)=;{mb/K,֎ Z冎~ބiXǡw~Z7szU_
=^ETfu+u"zȔDW&/ "X',{Oա[u_P"bVZBt[\,$zC7K5:XoDzt-):I4::g6 IX {@.B{f OĨk s6j>jϑ<%~uoDTkCwe\FI#+䳷|2G/F5 ̨/gEidE[ZRTTS7aϢ5o&BMLf-zbNI
A4@ -S:vo -Y>7WS}ӊϹ*TVKʈ4X[;,-CUZ%<xUìLZZνJ&?c6P%c=2nFlȯegxrʔYi56ϼ|}nԾo}Wi -w8^2 \*lu>e> 瑽-l< Qsf9sΫfr /oGDMKBS8 gb-1C;gu3/~"Mb42xU6)2Zu6ħ!!NwT[3BIw9 ,}:SqyFs-TD|3[ОIΊbhSU>*6rrlWYuwZo=ď=
хz\,&"8C˲3r=ys/S,s.iHmL-G| d*|`tubgJKJ {c͌ ;ɬ/GSh]7MţJk?Lx|Y]ljŵqe!QeF
-~ly4[&Z9<XC$KP'irXӜdjQ_2vLicNfe0ГϺ3)3?.]պ[ LEs鬰]y'_Nޏ$ZX6rCCEisаHyF%9G^ J%UwSɝ!I7쨤܂K/a|.;+` *Ss ]R11_j=ˏ"x,T;]JfĺXN4 Uee
ǚ07TeuȦY !<2Z`N!_46sJrx$ZbT<bc -:nBݑ.t @w9vϔ A}<ʼU8E9*OVr5Sfҕ%̯OݜS ϶ZDUgg E3Y߭(sTi{VtSusTiyh Wl]W|ժ9|fNSC9VtyX5ci+"M i5-Tfk1*;yTIgXwH9'Wmk' -AD|!*ZeY9qvUyaoepu+ 9|732?]=0DO}3VFkOEDXrlF?*_FoEǖFWs+~< UG C^E*?**קz !% } 4IyRR$C"et!
j6ݙ5͚EC`%$äkzh&"E v=\3̙ùZ:le㹨sjuE%ODe1ȔjZ 31G 87tɕuSYԥ4V3Rzj(C"̾ܚȦ38tcz؝C4&7Sh\J0mYŖ8D磻Mϼ|CL#PKQ,]H)$?vYCԼc v"^k
͏[b$E ȡYt:t"U 愝DDg$̼j
针at<.lcU!3A"YWUijTc'3=S\5tÖ&ъtV˭&=<C$L$I̳2<o'uZvł mJ擨>f[7s28Uww"B~83*+_:6dBNxLB -ٜ;s<U?v|,vG J2f ja=kKR%ב4<s&2ƺޞ$;tX:NϪ\MUߣJ6"7:x>U8VV8Gƺ+DvΛ+'.z-M[ξ2_>;{19># fm(MSټ v6[ՈԼNVt]=T3;]0Bj U{S{hXʎm -by>͝9&M䱥<4ki>KU/5";ȫuҋHSi,6gp-cDBeO^nJ'CjЦ9Aw6יܣOoy+d2X+,T.+#N!Vmv9Y6t_!,C;ƥg\gjܚ_nSv spjݜˆXhj1yTzns<%S~WVmD khiN1">fŦIS3~[L|ܗJߣQXsL4JW<wF7'nl>JQ8/k;:!"11
'1cM^#T:VT|Ekck~eM53j -z愆"qtUXu. bӥ~d\WeV̳"/,vZ%fa4LB -- Yg]DDϣl/e5dVVTr.e֔2љ|u8vYW4W~a3>%S~5ƚfb "n2UfZUf\ -fUj._Uu6U>]"γ2YYX*"LcI*JU 3f>E̜VL̾ IS4c,THL՚b}4+55{NkNfO%
)HBS<[y̞V-t*lƯl>'=SĨ/CL4L[U2uU*sR뎞g)ʺk9T솙U'<\[ ,5*7EihC_VeR]y -Sy_]fG<˦LsN&*Qg]mg47;s?Ci-d4ِnj9 ʠcF%ϧ:ܮS|-*e.y FotJ2+u.z6'<r5^O*'M33ow0eT<gmѦ~'[G:t,iyr|]nyoeFOE',xT.x"ObOkJ[,>ˍD8&g'ϹJ dVHj7ұej].+%8tמ7x l90Ѫ%6ˉ8wsZƄ}W\v;,}=Lq]b7G}^Vkhf3*X&y3W%[j&*oBew)YUlXx%LZThE?WS4OԸeELefJnd=W.".=[Zww5휾yY|U^n"벝Y˶r6o>XƧ>᙮:enGt~~Aryn3Ws3 =эBWCŰ,O.vu/KWʷ ]!˻z%llqڐ\RtǃzAA$H -Z! -
2=Dͺ1$Y1K>PP@:Ɨ;M~
m.@}BH-?DѲWH^įAWoou@' -6:)yAAW88}Ӿ w$-w;QDq^ m (/( -@DߧOybي j< p@&Y)$O*CrwHuz}Krn!r%E7b"8$ÖI[ 5 VC!\1XV߇'A^Eă, 4U;яkgڎmV(G& E} v
)RlCBuH5} -6QN/;}=Y<m b< >c p 4ȩ~}*F+ Ex9 W͵֘#*j}\t着oцqy8 zCtBʉOoM{"4ZZz:M>Gg..)=hͶ$DXgFEd<y =~[\J</ak;ؓM% 9|_! =:e)qƥd Wv/,ֈZ
iDT.5)5kL |6Y/Ýq8\/THL2KV t qi\Mz{S8 5BK+B&ڒKn/e`MoH "7pNdj?1P/3
~NL
ʲ!}EKrMWj1{- -y_=L|S 7lyAe
( -UX[d!MHSԘaTq.O,qoң)2r
_W}F>m#yD(qԋiM$~ -&<Zb/ z>˴|w,5AY ݶ G'Y/F@Wq}ZV" -:h^Gstu_ 9. Ƿ fI r P7\
% ]sЦ`{ V%*nA*?Y1ep̩\cD)[dkcxRs)VP˒I+
֯{G5jzk}Ɲ6 oN]l܇{6c-Yuh ͽkc:<Zj -
s%'
[ungl˨c*N zU {QQ> -M?Tz&XZV>`i -,`e\2సAХk!:hm^"q(:'7N7Ā@!*(;t٩+5(ں|ƅY@q>X}b -!| !A@[r9jvm *}[:ydP3VO5D' hh v. Eua2`TsȫR X{ -6 =ڦGUA[6 ->5IKn=PIucL4ҍKVD;/64@}39mX4 %OdC -DŽd&<#D8c paF<]m:X: 8<] E..铲`fa0q] _#F锉_Ӣ!x=S -rh& < (< E 偅x*̂\2vO B<1Op^NJ
ťV ? -34av%>iAٮXJxyT.v -
5j(wI$;kΐ$Rzd!WU4:[ bIJLⓘQb?vv ؆EbA>2| }`K
sV -ǁ,tNipE pjP'yۓsaM '9 )c'I.CD! fܰLVrr.xcU<L 7 B犳HX-i#R)5AGe -fiylG, ,`
2`xr^n f0BH>!yAh07}MgR d6hGs7k1x45jH;]L( <#bs ]v5@nH=ZSBj EDs0H=bAB,$.E xlUDg;_V8Q:[p&T1D(DFd))_H%J -"'1*t}0UDb͂ "%>k"N hH '<X"#P" - -ZqlR MȤ6elKAvF~KRDldwnFquCTݒUd<s}Tb)+& -Cr.bi9w i -1Tꭲws]PE<sX^|a:(b)4ܮQHwtbj /Ƅ oMb^K?0h4ƈ3Mbu.=/2QEdZ- -.[!ENChp ~$7Yۿꚕ5gy$b%eRcOY$Tv Da!x" "-TidH+ SD0DoRe&Z"{(P1:! \e`/_>EJj -qt~Ua
O1J@]Aׁ -=ɾcHAl boC<>i2 aO(G]P&kUz`QeT<\XxVG;+:e |:6DT$%\g\V6 -yx)3uN!lB+{9"EiٷXJi{o>
:@Bd!45]A
M
e! eDJ4j4>rjO= # DZdK9g8,)ԉ )э@5'^"l)V!!1 =hzV} =OC w pR̼d%h -adP L"KO.@]ݱDڅ<*(]џTxa kHBt
-Ob|= yv R-W"Ѣ7k2 C9est
3dSάV=|WxS> *Jv -Xud -QOs>{3]is2Fp! 榹./ 0* Siw -`yg%(ȏם,?HqkiχHAڋ)x -v $P; h%MI}@6PI-TŤKKd@h[ C30 5#$6?@UP[#eZ^s HaHC?BT\ıbS -( -7-f20u%Ԯ4vc& QP7$I~qt5._ u 5鰝7\u(d3 l -Mq,U8j -1À:|Cd/xv7)5J }nUQ4=e{Ioj
P 髭[Uو>X@|[uF#Z`+lAO -_ -t+k̘ -|JQu|r>6|j͆/q&8-\ЍJ =ǾIe0&Gj -9(N9ιzQq,ى2JEsDea'vSfۉ^!aDۅkM'!uO=4Jz7͐W̨zP,Nj,@9(Ta '9 Ē, rb1g_'^=sO\xu9{dS(ٵ4iX?d=,pM7M}3=iӘ83Xo/Gk@$/5q- -<5!i\)M7.j"+~E;DaT}>O0 -]fMԥ֡gJɆ*kVE=y`A
U4ԧa -'j\CQP /dòUPEePCh"]gP)~ct~B0qձӔ7YT,K% /~abހ41Oҭqtkt*[N5`zA;a+уWj[V>'8F$7IT -Iрg_d>#F`Sq <B -21Zs"2?)j ˟Ƅ6&ÂQ ɘ!GG_C%'J CmTF: ZdͿ_@\8 n
pz= *AL Pika+4װ$yhk/@m}h&
4.c)hϡ -1<o͏QHqIJPI'Dj l43-RsV}x@3<īϝ'<U '8'XYKC.3 -A
A,z? Ӛ Ġ<b@H^Z#Aqʲ -tww )ڤْ8<"(^iDrdX|շH< *
9<FhE 9%!L܌ԊDROqaBHma=k4G[u+rZc1K a71 ƀvOTݜ\I3Mxf,r -ˬ9?@V1Ɵ TR0v?8L= XS/ $gAFlHظԼ%zg?, w8齔NV6$H>Fp
LW%H,S834X0tf?fdLSe( GDf6ډ-imKa˜"t-Ek],/+Wy<tHyTIJlE7X$ܿJa$sx)R/d,T?~O Ղlx|_%TZ⸥q^+n&/ -pY`%yK,DCvh=2E>L>d]ïnC;\{BQ}!"?.1 |"7qw@Ë>pa o4pB;I,y -+N < -e?W?[lɇ݂W0uHf| - 'ih56cS3C !p%EX<7{ :GT<Ă^+;,v
n E4t!TA- -^'# 1/[1Y -2QʬJLL,{\#H2uI~/ޗՂƃA5$QEVñǀ
Db|Ѳӽ#D d/L'ejL&üi'o{Lze'%zes
q563JMxK+5n& -?G`o OA5w CY_ - ! @L n a -qlLkbZn=WȚjR<fCxIKXmz啶0vA -/ݨ 4A7c5fhBk 2My=~86#zbJm% -'3 -D?d I)g=ID7ב$ -{EcKB$>$z(@vRM5{'>с؋ -5B Ö́ANqc9 Tje@Z4K쏛 '8+Fr01w5 vRwUjd;.F#fNqq 7cVY#vį wp\^u<N;.) -3L2wkY f1,:3Wi ^"\́{FZeLh}`m$]UM2ʵ{ZVd8JFqr̀ZDe\W,M8n7; ݞm[FP5nb]5K!6O[@jAk)H Il]oS |R*{ 3ڋ Ԍ+iݠ}2sWGph -Id1bDRýRL+"h&~ -P N9L ` Nh6g/)Xl>arЉy%E41lZUDފN_굦d N #ˤ*u2 Ptu-:1A;MT\p$e pv.OT:MJ>P-|_ -;/LOn5~C* -Ƴ8T#hl~WuH&>yʖP널3[PH?E"JQR;n: /_ATS7ChSꀫ0n - 2wiw^F$mFI1 "6vZ Zꘃ`֭NjmPpݰ(s%h/͑Uͪ ۷r\=*q( -*zpָ*1d}nomfɃf`V[e -.L|rZld AXj$aވ$mDU6"3EQ$f;|);D oF[ ˦!̤+C舐ioCgDtKd%M -6UXx}}Y an:/ƇC :A0sZ -mV h*i ", ->pƌC* ";Nk!A{88)~(1 1F*TV FDς3?_T:}~G3NƗO|VVm v)*t>od'#Sc qGv_<'&y
^vJvqv(OF@[-ؓF nSQ9)<4oYfIccnA P -3~`3 >yЙ> \ 3'Zɺd6=`iN\pepJ[nl#V%Rbc:s5U*1pb'Lŝ9_}<]J\1`[.$GzXiH&ʽ! [ }<OBYT`,/cX ?G}hm}#XT? fql8zMB543 - P"BB?zX}PX -)7YN}\qO`iBQ/d"Zĭ~wo0W{~ފ&yTDtw ) 31j Ď9E1Y[W+x;k
ϷnG Ae+o(GEh0)7kbmj, J&z\BPFF؍X\ۓE` +b 1 ]!K0jpn 6SAZ+6cgVNEW#? {<:]1xxQDMW5d2L^ITam -rpRTS=8nީ|u\Q
* M!qQ[[ -_^`p|h@eW`Tس -ړUv59%z>X $}hC7`#ޫ
ݼ;zu׃b /M,8ꦾ'Sm=·#5Pr=/$PQqtUvUjcJ[wJ؟ͽ:vf381QsA&D}cULѓ*|߄_ au2
&Nt."~OXQ g_AZԐMl'k}Tqw ' )o16Q>/ 5InP,wV:[-S+bxBO߂CD>Bܶ@AD䇙͉҂^U3) s"ET#i#I Τ#s Ly2EZRGĨqG2HMÂ3W
-x[ '88rԙ^(Z40ZęUB!YngVCX*'W C@.\XF9AGI0hߪٽ4q.JG jp(CMPtk<l B*<Ɗ& -bvzi[d}%ʳhC7CΙ":2H Hm~1{ uѬZFpIQdB 80\nd|뿳o7O9}-7H=I*CכKgzܘj6ȶVU!"7kو|>5v5v#v0nf9C4s2;Kof$_HE{J:_*|!@"Ek -rʑFh2-LJv fZTy`[yx/wpA
Nf Y߬<$h
-5b
J9Wک_,φxbX[/}ԫֹFZ&(Oh:1n,J ]ae,YyK0գc[% JwG7ڿX۩MdS𨖵ɏԌ3h!3\O,xB~ -$b{.ң? t<a[1O'# d Do\$Лعzi I ςaK4㖊6ow5 Co [:¾o! Y/?ļ -LU[72d+`\> "Ϊ.T㦇D\_/y` -q2H%ieWjVhw!+z}lUG:Y!0&(T$iIf4G6ly8O^éXҡ7xE{t"T>8c#:W`6 T\j=enJ
- -x8B!-71~2^4
%3 -%C^@߄cr(T:dsU+{ -ȨԡTb25!_^dFm ͤmdP")IźGDRXsa!F`!uM9m{ x0հ'\<e% w\Hoi7]`Iن 4F_2GLM(qhMT v@g0,sxh(ϊ1O`垧'sXz:n\!JeSdsM>G I|(%)@|)%/BҎ\(i&앢d)[Ed=}}_My ʤ %.q>!,]\Z#U -ѥ pdzü'.-= krp5>{ݕ~g ,Bue 8fC\w[HM5?Em`5| j,'<:;6ٳ>+m1 5{"/pҬrO %lwCXI,?Q `l&X8aXڐ䗵"*,nLj\. >Ϳi5F r >m9y..3F 6jBI\FԾ$,Fj!x[ >Mu -4-hF:hӀ: -LBbGjm!(ztpiDi, 1B {/6LcNg -0@c(H(6B:;K< ,ۑu. ̈́mo<"M'91 )TVF?mLLйSC&=m\(4Oj-tTCۯ@l)G 0 D &(8h/{+ -.(ԓ<*p *+q!$`/۵ZCWI\/נ "Nb#]ߥ}r&]Y[?9Wָb2 -< "Z*4;nk^:JSLB PfKZ ِ>I<ܠ,JM 52fJ"ځ y).^9VzXe`t2Xh4A>orgDOy_/J0XT@G01 t 3UR=\ vm VW۾mKcCwk ըﴤ7d@[q8;D}BO$2JI$ȋ@6]ED -x_Oíy fqky<ձY@%]skWr̭RWk:Js7;D(0&-pvD<hׂ)*8^7c2D$&\/>N'8; @9t -?`#qD:vmM -|D ,[Zq1TU -N^w -MC)siLL851- 3L+2b{%S㯓-,ZaIev { n,FYɞΑM6dWo1R뒲GuQ- -G%Zi\eeulMnK[7w I7յ <!0]h%=>C^^7ЧjV̙9b1܌@" b5>9=Kѵ}2 -0- @n rx[<:cvU
Y Nni
** 0kxr^yK -,oC#a 24Ä
w {MwFE.ۺqJĺa%^4'CqH d:D $%+e^Q $,>nRG62 tRF)h%u!FdzfkW$z!%4(P 2:ԓ, zd?'P, >)wg71_ OAo:ˠ`* $gr 71pMuEo<ӟ 8QػƵQ}8ݝk29 DI -p9 >OVHAΰfLuG\R_Q0ߛ<JYD-.x`jt lWq/+ P ~\ @)$ǞHc(OO!d<^dE O5ʮ7( P8&(eV
e$L))Q^ * lC ՚`g$8H8P ~hES)Q"VTA5⪢ S5.Q("a(9j]LCuAVZ jADt&!Dbx&3R=]2 5N- - pP:G> QN1Cˁ -` TP -Jp@ H -Dp XP8 -CPI"%:qu p lH T!m̲\E/suI3J(@D UScArDc) V%_:Ga,5ZGhR_/J5WT8i*ݐ6&T#VF*'~ v¡J8cw,cϯ.r dqp 4!cٌ=>4A"gtO
D+:DqMU) %MT Qŷq3y 5(^`*hF\1,J:îZM*Ѹ/2
T2w<Uh%t :G>ͣhkH< 0` ;D]y֝D`F1
x}h9$9I7!08 j3փn&"R#댋]MX݁ˬpfp/$CW [=8FQG)NYXȖ8M] D$!98*H# D2ܕSqQ<.V_UK$~ฎբ@#GNᔧksT$b3;"14 #)x$Hdƃ˅ B
%4-!$@GX*)xvI@^v;U?h'Df@#'DBE"_)N0<E!U?*
MO N-ܣq|bx 5.f 'Ft` >^#eho5!&xv % -~$_I S&\ i)O - ~.n*?g"e[IDQcl"Ui<CtL͝!U @4;H [j AZhQħ de>hB?UX^m5**66}cF]cnǖr3dkᰎImGY}^V
RDdbb DgϤk^( P*'"d<Ak LM$ Md"j0$+Ͼ~BާՓ9 >vO*P.f!yeE2!/yǩ[.Uj*X2 Ɂ4zQ_ҤB&l&',r{"(H PъKA3Ty$,(̱,2Q(9HCrFCl [}6GAQdeHT5'0eۜLRj<TM߉S)WYQQӻd $mR,Ę=R%@4=V {ӨѺ251.uRU -5߅H8 ͩ*"eSߞiЋXhGIw&DOoFȚwwNc>j;Fֲ l/C'aձi()R&AR~HXSe\9Q)ŸfK;Z" Ut="*_$M +Yi0IMYx#BD:vnjJx^%{P!OnZ 4%&lr$U(GZ4CDmj"5Dvt%3(ʸ-u1NiZjOIhsR}Pna4Bv
}v9&a!EMd^#Z1o.bO."FxUG9xBȐb0*}(;EAEv)У$v*ИBDbq$T%2e%L62"e)1U TE|ժ@b*'ܧjAa=bWb (oAm+.&"J#Nb Nx9X؋&X#
K]pSѨ#C |03JYаhFM?7 Qq,ijl8 cI żԊhkȊzX|d&^%ǼAR ӏDĉfV)87c
#kFszWu . ]q]QdYQzL *rPĈ=P(:DUԨLZ ZτsPDpA+C:|qM u)y=xȄ"ID؈g$ CDL& -),2\E 큜*G$Ji_"gM+#НP4kQg<|+AR#˃bV/DPM G㎐(LW\:; .^FJh>^G1 TLr,Xz| D])mfdB v$=4H ))WCA5>b55A2A$3DSB![FbHV4&:<37>.rld(Ӻbeģl<QQjQt ECQ"-)KFVmzqP5 L4NM!9oc9t`_4SG)_r?
EUL|N298 ԧBxNH9Fh81,}Nf8 !.dHm S7AC
[)kď?rt} +endstream
endobj
63 0 obj
<</Length 65536>>stream
+%AI24_ZStandard_Data(/ + P8@@0!8P$D0<@4LDhpLD, 8P$ +P@@P(H$ +pdPA&PHbQER!~G?'ͪT<n=v,;kW q\T`P@e"Q$E"E + EpX(2utvl4;*O, m-ȃXEbX|5ʌ~5 mPy?hcH CX+)ƙxC"i,HvU犹dQ$TCĊD("Ð"A#Ei0RTOAa
=qtS$eáHU$HG9QGp4f5ƣ`0HLJH">ΡHac Ⱥzs-h)騨HhCSmKJr.>[ *:k^FCE/=TffcVX(R$cc"1`," cC D<h< d(EB +QBr! 0
H8 zi!P6 ZI!HG6|njfb^ZV: % 2pEDB#! 1nq=<;: 氆1laR$c8cH2abA p,IcX,_B-lQ /h7^> ulcn."9Ŋɡc >HQ?q< Gp, wC @9Q;i<Fh
mdװF5ܠ`8
HH +HI")H`,T8D Pр x`BdBD "4LD0T@A + A&<T BCaBBCāQ2H@!R" HD&<D(P`PDC JDDCL0T0C "
`L<PP" *H + px@@Bd0PC$Рa$.hP@&8`A ` ET0CÄ +@X
0Q hHxx$x &<4H + +(x<cXx,&<4D,bA,.80dyXLD0T@baסB +" CHH .(,D&*D`D"2XhhPA (HTp`ALxhXx@aAB$"2!0!@d0aBKᡁB +.]"4X(
H +, ]$$"" D ""&" &(D2`Y88T`"
dˣ D *LXx`` +LDD4X@ +"D4DXp@BDd DAB " 0XH0\@Qᱰـ!bBD &<4HD& *l`H,H40`DD
" + ( 0D h@(?D&& ^Eu۴TBC ރ<4Lhh@dCTU <2pXX˒$VʂV4 "
xCD@ _`b
` + $"L`x * *8D$@,D&D0( +,Q&01 2ER&L` + @DT ! (HL<@@4Q8 +]U:L?nuf D?(">33=4 E~PUl
*@0 8P$.D> !*( Z&L08 $@Ӡ +]XwݽݕVv{{&aqb+XyGS̓^]xYQH.gU{3ں7wT$ɶc{EjGw/{>x4W]{zw.J[6-ll~eڸ6V<eD;LLlszz']iL?>jM}|%] +kS%,[zNh{6m_jgThWe-\IC|) e6Vy;h?Y PwWq71&)~x,)uFN]q/Cş7_ESyQצJ[.1$'"*j퓉].Y'W~6xID4*n^s*;z裪-rqm1r}sKI! tK7hfGrceܵ!óE]-sjhӜҗ{4eTtLnjyE,4ɺh! U !ZWSs꼄{5itڌ J]ëӨd#m>zIwH~ + ,D<*M n-Qn9w ƽS_q֣/Ż$ܬ- ]Ot* ] tjkMi^{4ltpm{xZT$4Y[G;`
Z1C[ЕY;v̒`M mEw[z 4ʒ:Tǹ9e(ҷElH=/V %i]K6%ro\cQؕ}EJSVfw=fŊk7kx{CVNY2*CW\{m1inv϶,MwG&)^4UܽQ ZUzjڭ֒f+ZlK\VlNtҊ߭}ԕ#}m}[GжiH [#Mˣ=^2O;rݺ5Ǖ%+'|{-,_9bź)/ y𦲘6uY7k8kYSXniWgUg4ҖsXRַ{Ghky[[eś;WusFj!MӾT3o mu[xU=d4[m:Uؖ"+[ %R\R7EjS/8,Z[Vn*6%Q1og,VJ[76lT_T[Rùk'LUͳڢ۱ʒ:bʴ=2<f)[.tlh/' # -wŅJg{~s,IDŴdfZ-5-\> Jki͢. TS8ԭUbGSZnDu}Nz[b"ݼ$սeӻI-W7YƲ]f1ץweoٖWǙϜk˭g8Y^o[,k-M˚b zQ*kZjKOW %4[LYۤrl4|K7W: +(D@` + +
p`8`ABDBD& PB{j?굜ҜC *vuimP |Q 7CbT;i{%=6X~P2ܴ(fBí '
AQA 9wAsR胒xck @pH&0aEb Ѓ"V^tqcԻs)"]N.wR(Uss%g=
-\Ԕlu1jiGʞ+o==j{ߕhWvLsR{}^)ͫUct9zeU_獢յŝ^QmiK5Ն[r{x+欺X uHgꮪʕ] ~<#]Z`e"~yǛv2EYUejO8kZRkҥҝ"WMnWRʅw-ʦ ~5.QSGtKoZDy,utQݝǺ B[V?Ido eNWϕںT9a#*']%ţYFԱlrT%U"E,j}lLU +u.Ji /qɨf n sd*>q&~xstEgRvݽqU٪2g&& MwI-Wpnx2͐vU1,cAr.M]̭Pkn)Â3,3>ZXPUT lsWDzz)3j [mr\6mݰphnQ\a9=˺-VRsHG6II2{"֬UY#/ " 9.\<ߚO{{hg{RJܼu9.Y*!
ѨhFCgs(V9.vz۟yOϩ/iҰשuIFeӴ46X( +Nzuhݻh0ʱ:nʘSWuˬܻt/ +օ7sѼ'L<t]rW.$VWu_]4 +zNҭAAӋy*R{SK{v/xUS곲m.shiGJoҭf{-<Ӣstu +C*;U5zzͅMTu`}S=v^*ԫÓD[mNY ZכluꝪIg}շE+LI4:xϭRBݫ**{ykJ{7² ~3M}mWz_*սs}jV-hwIxUsjY.c7cgg(~PVMQ_&Kb>(N;ܬoA<Ow{}U{So
Ͳ)g7b=v*N9eeCZxro(Wd'QIWu[٫«ףW>[םu5we[վWot=Z! +~]VhZlUG, HL0D48 +KtE@-!YytZw +Dk ]t/ݚn+sJ.ۻa ;{G**YBΩl
Jd)<ƭoMJ$n] +uޅYdVܺٵ_Du-)j" yW ץ 4&EC|ïک)t6U(w;=~y]!=-V)\6Ogm!u9WN B9h,]W 9?[fUjYzn?Sߜ1-;vv:_ +Uiw&r=IIy< g2ydt<.JL \>^EQZ4w>F<xt*mu['%mٞ͛7'˩x*[EuyB~tml+SbokO-<%ʹ./Cx+&u@S!Nnu_%+eJ5T"4u|o~TZE2햚"wK^2n?;<ٷu;WZ4;4B3ʴ+ g VZuZSi:iL/}3T|dO3ҝלּ Ѥy|Fyӝ \JW˧^ɯ DMJm|b.sK3
1iD~,1f^4Եҝ^b!ZHT˛j>jefOwR;_Io"dH|-ۚunR_FvF5],;=ԓZnIڏMmk[biJ[ 5G?OtT!S-uf/$ ʮKvϬUR3
ҪΜuqShG{Vvrv:xJzs39㽶AE6J~~ZFtՠd DTiMkfk,hSG3OhkdCä3T;tVkfW隖t<-tC*ҽjW"44U_5 ~LW_oN.2 Y,f.Z3mͦ5wMߺ7 `z^ڨt]]Tڹ]B5Oi[=^5K67꣧futҼq4VsjG{)8(aFV/,UWmmnӮnwvx姃y7Ꞟ{%zROMѕy;߿Kfݭ>=* ë3ܚD][\ ޮ$f^^QiԻy<4hȻny*nwnX6^ʣ=%*sgm.깫wUxWY;ޜӮpTuv/Rw]-W%ΝݺR-7G8GGk% ҕ/"]~StM7WN>~ͦcC"Kk4FW.Cujkx]
mrW53]f}Ѿsu6㗨<3Y:/ա?fGݽ;KU ; 7k$]/*~S-|;3:}-mE̢ozTt?D_+~MGd*Wa趖톉Ez]SY7:5L?J6
TKFuIDzR[#D5Uidi{- 4}5vhHYY
%mи³cHi1#2v!u}m~uҦe.4ъx[5M]úutWWl]u=ͩ%;mR{&
bp?BzUԫ+NV?~$*%Ru]evCB<;HfIY2ջC6dQLɶ\G2H7utK+*7Ϛ{RoK R -;<DcgJl]F+,{ <[<Sg=+D;?iӞ x[Vi'٧G4C2փfz=Y[~G #,o[sdF *-ۿgr5UKD㲽lNϼZe3LD??;S4$%vuH
Ҵ3
m k{zI:huż)U7NQ
<q_6w̛6NbV 6Sgs}7k|Sjn.c ks)pMu])!]Q[\lIDHڦ2L:^]'{ +"DX0
, DPW,Po휋mHskz +؎Igsޫ9t4:G,MT;cҢ bi=uwTd顴oe鈖hU-3E,[3;X:[e w^_;)uӱݖlO4Kh_N6M0l':g^_<Us bYzVWXyGM4u}xJyD )s"54ig:բ,[5fӼ-L3<$͆xV#ZL45{+/T|<,7ulCN(vWN?+}'('ј&uYJh^ <jy\)aP~v_hV!+ݯTCX n
Yՠ*W g]խXiVJ\4N@t;
9dætlV 59E2*%{7w%9ӏ*洲ƪɴm !+Mioh䡟LMZ?,Vң_}^!Y.#zkU>iI8h$E,1Sz 9AAA +zg4l Df^[Dyz<1YEq5QuECR{ 2gN/ +xQ
"|1hX{y$[$1 koy +O [ē2"OD 0kH EyBsn֧%ɜ Q2>MKIv! =t)On=_y"0 +<y +ةWC)K8Ɍd )h`˸F9:;%H2dA ZcF@JT1|E2pTUfh aբQ3QT +\ABB+PF2P{<oC!ەлBq"\)
+P*S/ʿ|ݵ=k!>+d.zhEQpD9%uavAA kn7(4lϷM <7jdK\0*ߎ@XP{R:'( b6p._I g((jQd'䭛sAWiG8~QA hB@/L}F
1ZM5,P;
ν04Q +k +wdXUY@ QRP_$%e$;x q-C + +|W${\LW,gh3 D]h M$Y/ӐsbuT6(, gW"#@dz,a1a1+~0Ux\^,>1N1sF&1PA+vWL.ҦpcqM<ZTj5$C`xZA;m:Yi?Iڏ,*RAvhc-[ `>FR +U$J?
Ta˺?(kXΕbv-r-G(l _kA+]b<emKܠ.9ȋ%%T -78*cb _ m;/F `7i +-@r8x2?t/3AԷ2 FրJ:_SMFd.m4bPmN ]2]SGg*7wN u53GOnU~G 7T [*djZ~hiͳuq/ !ĕ $_ރ 䄋|5i@ +*ۨO隟zYc# Ņ} H6ѰQ()HQny>ȌbDu:Uk"v-& ]5GDԔ>k
``aw +? NF',=ᅶ=M(#Op<is<mFc Ր>2}}J ¶ 9',*TF06wEI'P_ƆaJݳ3RXpn;/.FS`ů>m\5Fn['&*$ K穣0 + +rb^π4 9[Fᕗ5VLQB]_ntƌRA'ߒٸtn[<;T#2A9dSVFSf1UfVWV+ 2"8_=u NOؠ9˶ACSD~0crtXf%\G
:j&\s>9|bʊ,YdUYO+CĖ_*NZqJkɚ؇ieћSe*kVWIy +սxp˅Fkk:{%<^S-Rf`Q0^Sq(CԞ.Z%еy d$S +rHECNq[ +HdqQ4ofq4uM[f(-<GY!mrUA6P*XJRIp&5g0^Gy.s\Ѐ8K.NL tw Wഘ"wZT_LũRB%~)xL5,(^U\A5 ~} }psNe4 +3!-~-l M&FOީs~d*\5?"om@RX9 b088k,]PZ>Rb86 [۞6Hk1ia F(|O &2%)r^ V\k[d9 kԯ@T}P˟"eB%m ( +j#' ,ܧrpg +CoƘG54$R3TrEL+~?i'0\W!)ʎxIfDpc)fx ̉5 cW;-rgS401( LR79 BaK ,n9iQ
{beg ? ̹} |<TԻVMxRʃjJ ԾӘU";@rsUY\n)77*{Q J | +:f8MQ'ڻlq#p&ƻK֦
:f|ƴXt~ &xWyj 2dZl #w+1QzC(kaB{ox*=8<j츫 S@&ijόeBmQ딮GF;4n^]GW(djN|yL;H#9^K-+E!!/Cӣ"!Հ +/S@QY|ϭ}ǣشXR[u[! ]F8@ kUilh
@!lsW3emqzY̢Be}g۱<{KզLB*߲Uz>KƬaumT)m
D{{'mj۪_aDgۗ1{h2B)m`Ѷ8AED ym[?6~9 ŷ}QVk6 +Do~IB +ݧ gdN;N]b 0n,3[#9oZB.ch@j;ku]5X13b,Ч$(M=Wn*aж0x|_ Tsq*eB)c큸hfb9`S~i\R W:>P %!O;I maputnV9
R5.`lXDݲY]j+KWbgBu я2D JVn5&3E#<n1c*>KҮߝ>FdҡGYDž[KZ6HmE?M,,,=O8kBl0 871! V¼j
\[21Juq5XPj1O1@* awf\&`kZܗq+z|ynP9vtdMS4s vA$ EU
N5C4q{iPÌvdq"NzGq.[Uk^"x- Fq8s!$J6q.R&CMأa<} QB4 $k
2[ꦱ#H<ܡ/XuXp9ݲ|^dռu.$%MS#^(8ܯ+=-܉Ó"L\ᙛ ض +'oS}ָjU%.[lnήmo:;Թmuu%gC<~,qi@Jt7ѫ+,TiG*X~f_~$sTRi*|D,G68&wqe9225Ѝb7:h24nŜŕ,uvF8˶\q={/g)e85,=K D`mopԨj&Yj
:K8I";}'֦TN-{@+W|_iafXX= 2@Sx:YXIW<ha ikքRZy _xjIUvR!^~#2 +%ԣ3^ϟ;]1^ϲZ!
3xWEW/,$#1趪ER2o%CЋ7 rQ.^E%=0 {{An/I)VYQixo 7G6yؕ|Ws^*+S0`Gt2 ! u`Tl^mbpK 2
@{~.ڜk쾈cܰN/sȁyy1DĄ5l&1!9 yĎ#(Dszh@zMcǕ2 F(`6ǫnNh8|3%
Pau
#wE#F0Fɾ^ø*$5Rc|c[h 1wfWS +K˲= @)ߪ~+V,F|/D3&:y +~П e}ٳ-e*;FNs9mo;sXV/s}E + T葯"|4KUkbѾR"n1G(9- Pec~BSdRo2ܕ q1oƌ@h02kjBT^zԡ4x/\
u$2יӑg-
Q bR~kܺwePI
/
Uj,Ӊgm N3> +MhPo 4TS-X {RPݮI@tt!ˈcÞ^( [*e:7 Q*vt[ᨡd0Z)6VgYĖVp,]?&
qU^UD5< |RQ ? &Dd^>۴_yb6 ؼ + M PF4WvaHg6h۴ +c_ᨀD)UjD#+'VT;EFYq\p\(5c(=z\F4cI>G*$T}L@aw@Cܐ S8!]6Lt^)@Y`?&B +
) y܊jh +7;5
6V (d([% r3ILӗa6\ŝ`( J恐Tp +GA]n)'Bc}7(DEv#TOVpe |m[<x
G |T?'YNQ:6R*<û>OYOFd\Is|V́P/7o!-'L:_m[dO:3\Er$@xV#dHe- q!FHLPHy~@tfaAW$|ܳ
[Lyɺn>` +\И| XBf^J>02<t)X[g'$(+E.fzv +gnZ$GWt/7mu{ +]U\LMلwaOd2RP +-jLZM[._3:Sp QF )XBx
WW<O8} X{%Q-l& MT9\ ~UpPǼʠ,7Ϟn.(5M婄 ]حhEڞDu Vo6E ܶZGt> JCA>VQWN`L2y}GFOsƾ+X;"LK q HI23?I
]C,\.xr)Qh;kSH"\()c\04}Q.ʋ;Aӡ| + +H)yѾ$5OZx1ݿ>W}BHv6Ω<l{N`RaQ`+C0V2ǯ f:Rzaa|]+˙ V6KZf mI UZC $MDeA6E3~"մ;8^VH rHc_|z)
-447wD:Ď)i5&Ϥ1c|n\&%VBb"Σ.bnkf@Ȱ/ߏ% C"Ɗpjk'^>OV5;0=V<ξ/{dp4]_E$hjmҫCn%v~"ES>z6ek^/l-pOÉ|#5M,^}
ZZ?b;m0jH >?p%5<42s&L[!%}iD wB +g*К.,/(w$kl'#[녭`2%Q(k)^"u1>.>a`xdP lɅyxf + 8/ ح +e9r&x& iA3&xLxM ZXKg~z49AuP1_ΰ-s UjDP (MVv#2.Wx7LcwRѭz @!n+Uf!vo141f *jVkPf˦"e+˕
bJ7SY*W873f ݫ:}Gk%~qbAFȑ1[x"[+:*E(D#RM,vvu .,y71ދ;ĽLUds I]9%K+3H6zl Dl> +kŇ~:@ٜsH扪W'gj +Q";Aֿ=n5Q Y9$hʡaI
A[($T3 +{H 43'v: Af +j)K ++(__}*flw8 Ҫ"\L@ +j"{iyѥ](K2ӈ,;,U~!MC?r2F['
bq?'(ЏMxŅO{ +Ҟg2hЃS7\{Wϖ/ u=-|M;z#ùnr-X2?H5Z`1O|╼ <:;{gWس>mՉfrΪe֬j(3iCI zJOJ iNBd2L,Krd-%h +ˆ(gYoؕm%D*BF:a60G? $"JS>)Mkvsڱ]v ֩ ivmj~4H7rdi05T{F&8v r0.W! +F\h<~q`ؗC o ( r*s]p:Fnne49Z +&]I*\d;d4eR9$˗{"`C.K)O 7kiJ́Dߝ
&{Gf B"V
y<K <Nh.GFcRv\(O 'ew=:@鍐nhTH:mH +|&[ W av @ :< ^ 6^`]C3{$ wY˱39;Dɥpuv˝8| 胋df+x5r&_\y*h^rqNl&aל"v֨,8, GH,sQc4+}( +n|v~ +e`V"8Ӳ쏐{?2tIT0
һjߦڈoMQOQ8쵢[H~td̄*rl=ۅȝ`{-c0N ?y$%<YFpXp2jhsmZxyD#c߮}ӏ)T\r d+D@c%^eڹpgl nNMV2ju|]QcrI8*Ynnb{c5| bn.φ ZfG|䟩ˮJ7I%Q;{jKO^j6坃4kw `f*Ooi*9[b bHt%-d,G1 + bn@d(F@%t\EEԚϔLPy"$`2Et,U7I IĜMÂ$fx$ *ʿ"9U -`FI]'`K˅#d9Aw}$~NLە(GcK}si ]By%iq( C
#g +sB{W
(pvB s 5S{97bژUz'R!a؊B\Ԍ휣=8*a
$y@R)jKk ltHOi +k 6.-*KpqX.$ԣrw{Mf1y"~G;HVI{oꇡ + W@f@\ǁV<ώÚfNGTq +ދDiI[Л
M-&P!i7ZWSC.1sSgdhJY%XԔ _DS9#B(}mH:$
L= *1zY ب֛ n %O8&> 5jɫAiFCh08X,A(@v]JB!v0[~"i -|(\BAPUPՆ_Ys;aVJfZf?h)M3ade+8 +h:e %s2* +y.i;}ft]*L&q">$fz[/_) 6ݴ RP:U*j"mi@eBr ~~@H#gD aH%dcMen]QGigx06)#m-,pV_ n kCp@LD Q\.5Y"r/n/jLj7q }v$ +Q9s{8H+4S{iorT7-pMG( lϦ.VuﵸgfW3z[KT5#gcG5'騼ڵ:'V T"j
1gtg]eI S6$}>$;'t<1i̧+R^L1#o9P;5Qg! ^ + +Z֟R1&f;#v l@Vqsi +A*o +FQ.*5h#'u0\is?ZA*ߥƩv B]ǡm;
Ya8}[G) +Sx1|HQcQt8d OX> + Et;Q;d cc>{cv?;3x1Mv /4Ĥi32bM6f~2Y_)iZk&S +ԛ#q& >(h)ʓuoqv+{.JdQ2Bq, 25My SYF -!S YKnvxs;NUtTq]Vf" [!v@
?jU&11m3Y!=*{ &BLEt%#1%%T^y;a ^ؔQ4'9=UeE[MlS+[h;ت".cnkՉ$AgL;$ͥ/ά^.\9ᔋeBw&= ፄ+ +g˾IoFD t7Un)
!_I#V eR c٢86B
:VfMy-GODK(PRaS=)AݩeR6 % Ycq=q~ ?/&*:tpOsj[V4έWKݬH +;^U1j8eE5Yծ}y
'zI E[ +)os z sT .__Z,'zK2qa#(L/_qBҭ]_
'
gG.z5R| Mhȳ*,:ם:RNژ.O\$>lQZߣz2mH:MIh +a Հ[FqP+m=?֓8 9in[nVh= +}N bJ&\%"&ŨJU!CC/5C$ nߦ3Øn7
dJA66PR1=A -K`4T1 iz !I WhJ6EkΟP/!~Z*E]W7ɬBje̐W6[ ]`٩ql 9@3z\|~ 6 -0
I 9+m Ɇ~5L{Pz~)CqP> U|&97,PnšOaL#LpTt qSqi0/ S>wu̝RF\~R%#R83I{:f + +"@+xBD\;in.en6 !WR&mO &)%%0 +&dvh\ׇXSO*R*G#
HQ5$EFG0?S ^].<nNY : C܈<5ﶯ3$X#'&saCmxbz0-~DPpD7pcqb;y,9dhp'h01e0 ٝR:$$W\)zqR#[r*6Q`Ԙ ב:Zh)3~mjVLz&sG$fq:a$iBxpp7\XF% Hݖ=X#H`4X;9B>ȫn$p3ġ \UK: 睞(J>G WrكQکe{4XŌ߈^ k 7$c#X4Dozw ;) 9s0 ~ʄ:
<IOŭBAXL3p"(<cM$j7t3DjC BFvvpCxvBx~~ cR;2BC25H a M4L5MG٫^EFӾצo軫ws Rnz- #i-(2Co5˪ALLL2#Ȓ⌔- qsYCZݑXZ'(yS)ҵiLD\ƛD{|7_Լz"xAI +ӌ1$HF>d%8uŽWMKP#9F/KT EnOX 8Zw)V8h>ϐ%㢐9E&j %iH`m>Aڼ/(( Iʰ&֘ ,qI_B;Rs**{>k;=rzLpMG* $ gST=Cco5ZW<%NJ9¡JA0 5A9U_ElT"
zȿ!<8{(XYՕ.1ӉyQ^-cG6ԈZ-CPo$:v07%9\jD4H {jK1'*EUL:{=bAV[sJCDTye?=
9)VuBH6MM ϫd*p_ MDфtV +TfhM
ZN㽄|e[%6&);ZBt"t^K)\
rP-A^HѮO`2dt?,9qD+FEL1E1"OJ(<'O RBQ2b((rP.ezNZe PXb\,*\Ɇ]f\,t< +<B>V彵4"Bo~tvܕ~J$18nX,"+'K2Q7A3[vH%#;#SUUGL{d-HY8գ +${qS+ +MjD*R Mj]r3|4`% +Xv?2Z$ +l,CfPs -a5 rtuVpfIߙ{I9 [3:q U(pZa܅_/a5M{T1(]Upj4x?"ψ;1g'ITr)2 xS^+,R*sNX*qFIg\P3amGPSJ%n΄dz 2.:!1gN㠗EE +.ɾN.#233m]Y`|\J&*lLWw7hg +V.T,/.Ӧq/iỵf?Q̈_V6k9 $ Րaup!QZ1bfHnlbzE7"`H)( +~ ΰ{U;V GXղ)O|%᪢{P+\vEFЇeOE +m2(pbdX"[ bBfZ]4+#͈ا#T Hɦ+0jԦ*C/Njt|U2} UIKD8Du qJBDC$hB &svCD9UiCBB[xO"C(t9u jF K$6*$QKޙ`Ibhb#5 ϐ: jYD3L}$fcRLK-眑d3-Kb`W zoyTM뢉*"efJEa -
3Ob*|C iO $*{|'Hj)*,O
k=skjrGM
tHޕ<՚X
"&k"bJHѶ͍(t 2غ*V(u"},fT;29 N}ĆFG>!BD̐ejZ-)̪oC~ Iۋ:k`JA5
},}o }hIhp ┼Z( r4uVc:({FaFi>A X_7)E :7רK*yZ~J8r -# <|}h_5_2$ЬL&F *Z`{a/G ÉQfps P#\*-j 4qD:bF0`,*J)A`Zqdꯢb5]y;(&B;N %A(#dצ<AlND2Q<</4ggnDh#x
6 <n̄B\ՆxC(e$<E<gּ(lwrMJH#! "S2Q<a<x2 % 52NMy.f"C$ L4dO DTW{jte#D4 P )A Q#Ses ;HVIH憌.".ýY&8CU";t14bBL( ATHA !!o PԑGդ 4 [#萋P̑Lg =U=PJbBQçh.v3 R߃)LI1`&Tb!,|(.is`24G"#6Ǽk%,$[Z'L9}Ӥ
}EU43TX|ל¢պS^P -zI9 n Ec{ \ (*ZσyusOwUUs[^R _n(M -<فd "
<7A+萑*) ;j"p,A
5 XFJ rpÁ fcp@4}FB"Il6 fyf:aKCjQHVO ЏO0HRyFF9
-5A -/,"u^ZuF"EBe%3`w#WEe3aPFx5C^5<]!BrAj&#TWe}BX>F:qTi$gwF(Uġ<BTX=_GR"HI>3, au88HNlӢAgCI8F<(HJ
MEBPu>J<^3hJb㝩rgyXӃ&F$3 / t\UD2 R#PEE{,CVшe=K6DT$SD(!#VfD+r8:g0*H*hLT{PmE -.+ST['ج!&e=UQ gv 9yV,tn:sS,SKfq)PͩVZ[ Z!2k$~NL;ʑ暡T_q -z]ZtufF"G:g<-ϰIR⢪&,!ҷSM72f=hoԤӴl`iyzZЃHdJ&CT=TibuK1=]B1$?oEEOvtjUstW$~4 jBE#&BDƉ8ϠDYY<̚ &*x ]$zL1Ҵg<,EUK(Rxʊ!9h "#
nͦE}2.RɎ:Jb#ibCjihFge'%Y%((f ㏧H>ʕ*aՒ"~n(#!tƥ2rLWQu89hLAT4!g3%Ҙ:<ffA斯椩("QF֢̰wH1sKTׄ4'ql "!y*(<-$ӽ Ӕ5R45V(@)>'{&C¬mA|!pٖ/mWBg@'Ld܀XF"ZT+Du;lpʏh}!AhI+a%qLPtP9>ƥ#ċb!w NgU_x>ڸ\6;X)0*v|8!@ 5]j\S=j4Eҩt 2UC2u%t1aljcU icOULԸbEׄpH5H$Q -30SOPJ:q yHH+YY(RBN=)GYY: \9`u0;ܘB#?
L?* {(A.pas@^@8fLյJt@N'w4>D< ja>3.UEj,]:ES4N9R*#f I!2N7cȬBߔ* JPyi+{I(4OG^Y&<k2 K3mdהMHɣ9< )&rmڌr7DxyYj>C,Ik S_\xvDhN:CY1 9f$4e}j,~$ rH)\aMA[ǹ Uqa6'd$Mym}<TQL% -3 $٬bE(UZT5`"HʍYh$eָ#)s%Aq_u}P+>*^bzJ$NȤS|TqCTk*N2UEh -ʛ>u~Mu^-<jh&&ZTx"sjD"q.cǽۢ*Q3fVS? V9Lv͑gJta& [|aQ\$i"%q)r/QI:OǸOS#*Hh : -g?C.c,K!I%T%Z#\=</"GNONۭA:mADH.vS:.iUQQIZ }1Q MD.h*B ֘ʎ6\7Dejtʺ&JAcSL [D' Elx|@RU1Thaf6G"y&UH=;R'^܃ a
1CaPŹ)' :^$ / uT +# <|}h_5_2$ЬL&F *Z`{a/G ÉQfps P#\*-j 4qD:bF0`,*J)A`Zqdꯢb5]y;(&B;N %A(#dצ<AlND2Q<</4ggnDh#x
6 <n̄B\ՆxC(e$<E<gּ(lwrMJH#! "S2Q<a<x2 % 52NMy.f"C$ L4dO DTW{jte#D4 P )A Q#Ses ;HVIH憌.".ýY&8CU";t14bBL( ATHA !!o PԑGդ 4 [#萋P̑Lg =] EĻ*,+(5|b 0s0 ` =Ri˔dqf8HE,\ވ:V*C#y$2bj̻6[Bb +5qL*). 7Mz ]"̪az`ov*,jkNajQ)/Ht\$ Fch"1={B.}YJksPQiAx<غ'*9-)ф/7x@S LJA2EdžRBtHB 58]Rz#af`9zh8doLWm$-!iv/Nobkgf4V8/iYNzphȺ0o t +$5WmdRd0 "RU9\k$R$T\B1 vG:l@|UTH8-߮a`\3:U8!*G< klX8BuUG.ԌENn@eFyg(iȋRE z#xoHquʪ +En.ԜdO3yY +V͌4Ħ8-Z dz6că[dQ$UG5$<ޙ*wVU:=hbDK؆5ՉH\qW0"2q0Њ2SQQE 5"j.UEAj\NVQDQrDLh ꂬ6G1ajM(eC:/Q)Ŗ ,1M<%yg&{.d>j L[@?[F 9t}cy +\` DB}$@ $!J
8!@l +A5 cة",N<!4|gnAMTJn +fQ`мĩ f"{ F(ъmل[t̡ZKM+ZR9 ] ,xlTaH> *I\5 X APAbچ,U2GPZD8ڋBp ^J$P_95$gKT8`U_sRu\,E[Tq |EULV醴1 4R? v+U+Hd 3}~uA { f y8{Yh$ZU|(^!GoJ(am84*E>~
gRM983Q fm3̘
Ü Ն-϶<x1h:j?a $ 22עZ!aVnV~D FKZ -ycgS5Eݘz1. !^# t:8:0 媰2HQ Q&̰xf E5:TV)N$T祎qZ7C-Q
cS|
M {мf5.&;EqEz0$&PzjVBՉsfCҬ6EB_BrI92&1QEƬY.~&hd-6wP Q9C>p[uV-Z0eU +r*<+ "o(V Cq9(:Us@`)pʑjP1#OgH:Ut +_ Cf]V(VrGK$O[܃HBy:2Q\ L8YXB,i#MlRFJ ́PO1kOf̗V!zReIZCML*ES'XDsz"/nΊL1$Q/#TSf#ȐCJrkBJ<4K8!Pe'Ghk㡚gb,gLmAD f+B< +Lİ(FARn̊(XD#) I+ OZi T#{P"q"'F&E\ +_SqJ-E6UהWkjQkES75QԢDS+% ctQ% <U6 fsm<S 4y⫈]+"1XNud)KTK~Ny*<ƍh~PD}F&ƟQ!E"D#P>)Vrcy^ +IT,r/ l)ye?rڵ4|tnM?iFOH.vS:.iUQQIZ }1Q MD.h*B ֘ʎ6\7Dejtʺ&JAcSL [D' Elx|@RU1Thaf6G"y&UH=;R'^܃ a
1CaPŹ)' :^$ / uT 7MNȡD1.a b& ?B`2śM6'BYlV=pY :Փn*bk^t3(; IBE:eNH~: "<P:zJ8o[c otprE <[A(4* HQ$ N}B&<xRm*{B Rd ԃ`FӒ4l2Q$_b4+ ~7I
> 5CDr"@aϜ^^~EcoqHS<85xY㯰 =g9%$5a.IjNfr8 }N9&U"]ʮ͢RP(I"{˫~*^P4 1*ۋ$:y -zIB
Ջt b&fCN #nˈPvaݝE_Pl ndQ D(]N7ݎS\^"K bY3tYeѣ*> q ]!UWd 9Z,E1h!씸?]eֈ<4(%rJrVisռ^_ Iq<rVgQ̐jHeQ" We[HrI}ĢEӊPLli !I92y J[3V\HK5 GWUnE{}TOh {E վoC3bGo}5!ku=% _4*J< 2<!ŸJވm@ MEFC 7LlbXH0.ب#o<*p bcgi+l4T,ҎlJZ
9( oixӤ{F"u9x(2+>dxb;WP_7B<E+,N=F 骚nJՄ_ZJIQaיA;Nj'o%vקBף~h]Gmf4Do ,}p^2ᆅp酣Z/49ʹ
#e$>,ɤ 9L0"BgNIV=;zӬxҾ(wE%O48O3)[ǬU> G!yX~_ydޚwȓ64h(־L <ke*Z-Ō/Zg"ۦ~kޫ6Jp&1^:"?Z[6p+IqXW7r^\DcE-,B7Lu YR6ʭbr9]qEv$rZת-7oq-RpJlKE+QHlDң#BD(.EX 8b&},QS̒M(8%~de|ħ8OejTxM_Lx'D. -)g9kJ -O>ɧW1b^}$VAs[4{8Ś|OPY9۸Ƅ">5όbE[ys~r[ wWT3Q =ET }HHOGQIo}T<uϔO
KNo?+!YGSG$kB zբXNоoioQS*µGUx",CȒ?^E?:nՇ1X"òp=[ o^61~W wz>Z6^$YE&¯/¯qR7UƛVnf) ŏtj:Әt0=qeEHDOS7<.#V\h2!J mDQ(Ƌ"? xX>V/ -U3_PFy,2E)
#j,H $׆`*L4;X?q2xࢊ$J" -ҬWlʼn=l_z*9)E>CoOy Y'e - 7,)o r-" 8)yKrtn!d\
*ZZH^9 YdaJ1HVimd̋lP*Ϥ WEl)Ҟ0'SDrNC= b AAN6_PcVN mfR鄠UH[ۤ 76G!KvWyJ:rӕG J[ׯ.R2(郻p喤N #Y\>hGm m`i;>+dƑ$ ɠ W:A$&K%w/q.-HI8$%Dw&J &ɜ#ߐqW絼1MY<L,T2ײ]҉'2"2d&y$R1|(]6NKiU c3T;11dieubO rj/Bbh 6֗ luy D` F\ -2 -k}O%ckHz`4&FfbAUuh:SerIĦp8U;UuIETeN
.|g?ʧ -Q T+2'd8 kr~"0Ur=p<"A>a[ljT0Ra]B&<m@\(SCQ.{S[ s]-H,$-?(pXQo0hæL -CbAlP -hzk3|h|0TW`/AD+A{U@ʇ[x`$Ӏfn+psn- lbW(RbwݜceTc}|1?(c ɩܴK8tݥ cxNjcWc - lc%I9OJ|VX* ^ 1x8r RSkFXQޓ^_MaS -İ* #Z1`L cSa + hwPGtjt -ƒSԟ!Ɗ>{k>65@0 X h¿$=<zcAvBRac-Rd{HŎ 怱hmt[ ٜXE{l˴V.m9Z?c1tb9V)_!$ cda3ƲI$lℱ]%cƺ =0of:ʕ@ V0X7Y"yi&8PcAm[{-@ % -0l@eJXwkmf0֔q"/n_E`m@Nj5 t Hleыӟ'
5k Zź81P]O:2w$?C+ d O" Ib/VDM X a+}p`rC&(6Ef/ hz /IuփkM=؋dWCS{pS.(/vPykYɋUHGh =NU~I^In.bGr J|b-,g1Bbaݒ+/V]A_R^XXn0m:/âSC_bRÿ{Gm &˒3>f+
?WYh7"r`
Z/I ,a}B@Lvj~>PLrb-gKi^,} Q -E/'J[P PջX -(?DqkdkZV.H~Lc(*qX0N1ZcP1
NX1.jH U cx[$ƍQDXDg!yn?:!XЏfyc:r}ґ k-ƻNܷ3qfp48GED6^)ocy mc)~{Q 13.ҚRw&A!2nO6胱i0d u$Qs߁ dP, 0]mHZ&}S0[cu1na"Hʊ `,,WLg#%G͈ ML%QobGT!5p\l/tEQ07E _ X<DX4 -wc=`\aF ciJcH zncEE~kL hH҉wښX -bc, - 7rkPv8{p0bX3%#L'X3{Rvgio5X
=imطK %Xa-RKݵcmt`]kXH]73wX<c9I̘Z:3~ O83bAR$܅2>ŧp-X˩^mx5usM cRo1=33vX O9]~lUmx dK+[bYMe8Rc).r?Q$1֠Sk/1FcYM czfck>-$ jb,#}{o';X^Ն]_ xRB |llg{q5z g<2*rp!|K Y#5AR5^p=]02t(GJȎÞӟ#c]GSXUR;2ֳ^'+&蘥1Ϸ|hb /ĥ *yY\DV
]aꊓhy"=X0oAc%04
C0ɣ*J` -x1灺dP^w#ȟ!]jѧ[zGXs8`H&RRcA(GVc -ВEhFfl,h\Oi-fx2tAUM40LhcKuc X6J¨7Vr*.+De8{)ĉaL ,U raBčŸ {qꍕUZ
|6ƪ,ݍ:fS#p#~KXns)Kt jj:C -e{vkr¨5VT rKbA]cη,$l",/r 8 ǟb<#Қ -, C F8cb6֮}:9ЃGmc-̿ un,e )^aI5QrcYٍRl\Tʁn*⻱0o -u94EcRNܜ1opK, qiϕܱ,AcW eNi,cj,sG_%wť2 -X]Q)o/ob -Uf -]xc)c=cW
!Ɣn6 -ce>$ c!86p,7
<p` VXt "m| -ƺ_,J\lP -<tl,K0! -:Ȝ[qe#h~H痟zGuk҆W7(C@+ZK7"mQ@JDa'U%(7VBTjb ȢavqW9Jw ֍5`B DBܴE^I[7ַ:Juc3rXlgf Տ1i@@XN% PӍ/S-˼M{t)dLs0ޕrd[u5aE}4!brW~ #C\c?RadGʼnG?UIx"˂c+ST0[k8Z5H0)jzB-mrmı܌ Yc83 p *Hwٍc8~*x[ 2E)Zz$FX ֡Nq) OՖ`4]ks7sMS:ˈ>Xci= /R|y.?%I?pxM -YX<?M,ydW*Naa8ER~NJM2Ufʯ4 c;7V -6?W 1]-L4Jx=Q`V=@ `x/7`G)ZOjV~<q5ȥx*
D,a CgaDJ=h. 3frz. Kͤf~&bFˏ,.P
m~ m6qi:ڧPt_ %YÄanuZ ]S)Z:Oح"͒Amhd# M0ġ`'N`sGqu> "[xOV qBIf¹v9-F&-WC05$ `;tޏJ
[p7 (
D% B[ 06ǷV{5a;w6ގ2 -@Wga , )(hW - t -|\RVH6-CXg&̀ ww<c&-ؘ:'bCOf4?)~JXzSw90-~^12 NV$Ӽ 28KCaٴ'VB!Ry m'%BU~@|2ЪD9<GrO,25m7%":x6i>yΫ& CلčQ NTmi(]4슶w|i" /ޱ_XPb 4h}X*2g|de&Ig&=*ga;,TzQDx:ު _v0-Fƌ#[C:}"q*
$s@[ys>KւwA;^ "J{67
Ij{m܍q/;="C@֕^\]hR`X}>nږ%;O
9"#U51~f{C_MxӋZnE?&Nև -!Ό'+nHƠ ';4&C*]ef id.~xUB\+g,% .okE* -} ^46|Yਂf1*!jm@%GXwطѣҿjjI3t
[S*CˬS'Cy_&ywy b^
q - Q~У%XO,uQ!5块xAV*ɧCCJ7Jgɉlk(\SvݶJҠE~=.5T5d]09I]l" 2"/X.lD,ώ_ -@hX
F,nlLQ!D\ ȟ){$X%L^M|ZDROvtRFzJIjE嗇S-Q& EE/skB7tId8y,ɵiS䚇8P畠 svBPl
sD\9}"6.kº=sELѲ9t[JIWS lVł;o HRdHAX-;nWoN!URY{-xݞH$[eN뎕ЮaliscAQ'qYa'1M2 -)JXϑ-:@[V[7 Tۏu9 -u¨T :"$49O-O " -I5 G;| l6/==O T --kn -2[#͂ySeTz
(dTߪyQjP>jy?/{pIx=3[Cث/N 6%(
ǚa -͘7 ]ZEDU.!Uឳm1t>H , -Y"UMu< ݬ5LAhy !MsYevI¬ͩk4U}*jjWQPWlH:P mnTБPyK' =ʉs.r:od|Eя\k HV?]|b7ё m2p_֊@( -Z<
fnl@ #*2 ܣŃr\ m
aY&YA!bVɃw xĵ -vE&9KR$E[WFk2wBS9FB+w'k ]!<:b2V9.ӱFH->5*%s lKW ;ӢuKb( V:U'& -{ X]J%d \/OlF 7kϏM=ΠPsI
I"5; P XnM[Ѷth4 -~ wUUX+ T7Ω K[,6NlnGYݜSa>|3((![y -^ rn%dN{q"هI QJC|w'̻ͷGy,0"m}\]G!`(Ec5$EfHK2.$k oZdU9W -p,n@}kAIVm@CU/<L)HDp0 -w(k#-3% :;k|[VT(
y_c0jvwTijyA& ):n% [6Ab!eug\QU4)?sɨ]x. ~|(=ns"EQًK骐 -88͎x}WAZ"Ծ`NqMgQ'D`.hGSO`3WDMLlb !=g;ID9a痠 -gwIM,)#Wh | UhhnL@+-| a~PF ޒݰ lQpp@@ g8vLDr]FGxiFzJa߁)pp bI3zXp)dus{"=' ])d_<wuvTz)ӵa -GC?>A)P
f9"(+ =jj:"!z9Gy::/">pO:e+uyi:_FH,EV2!{`3նZ\Q1Ԃ?%:loF -Ʒ j- M\\AZTǀCU+o>Iv?) fCpRgʱ -^3$ lϱ/q - ģ< -J˺1N?#¿NmM _y^`,pdHX`E`"> O/ Gx.cF.眊$R<(xbtZ1<usw*1=$.&Ɨ (%-ʛj!,٭:{*jQVlU%9Qu"]ݠ>J>Ng ) -xX@bZ͚9B Wu剳\7J'Du ,ʉYx풁# J>??EtZf -^<闰.1Aɀfِ7ZW>{xL9rt3 {PԢC}9z#S4 VG= -['O8 "mEC9I.V)̉ .t -_ sYQԇBw Ì0|5hoбe |){mթp@䓥BIot4v -%_Aىh g+ -szR )0m'J hRCجcQ@:"]˺ :` {'@Z*B fLhJZ}?YiчB^x=CY"1I -Ȝ<ݷwBlR28 z s(A^֚ bF?3 [949hA-ۼT`i_x"T<N~f%v#Vw}< - z% !\ 눲zHF<?x<q ^l!Ĵ}%Ot:@H?7'qۗn;8x>0EI{х҂> -Ϊ{.Q>%=1G٠Ci\y=v JY~ -u ݗTGij`VXg_懝PRcYƵ\:%⺴ -{qZ ( A)NzuN +'V - -OB!2+'Ɨot6$x -~ucG?1q LgY <dr]Nqae$KxHεe ް,#jMslɶSRXdT1~n9hOO|$<mZ3_
D#:T[*DH7jNT8$?يeOLJuJ̌Ê1ƧW 'U(p ';MBVu$q4/lhfp](([>N^g -Fқ[
ց -"'3TY:((
.Ϲ -n`Zf* (%pOs6JfvQ{)flX Dތj{9g.ccTc(>'$@Q_NM3lBk#
¤ҖL4'Z0'TDAdY-zGCa)͒T~v{%Dmh' 0K}Y /# .Ľie o|ٲb WDV@2 ؑ =\:i`8'N hU)OŤ զ -%K%9x^5roU --# Pe(v.K8lɫ%7ڷDoK5D#h9;:y!xĄ"TF2
6,ҩA
"HH_Qtr|78) A$K/Q'LV., qbfEgt$ѣ W⧏1F !xSJ:5DLJR v;1p6
- /p#>q J) H@v Qૡmt@Z 9"Z e].ᒡ\j'; -[h+*!kH5##%}hh( -r@j|~YX0|ƴP`k) NG@eXWcxn'=(ˍy -Vôߟ,3,Guv-v0il' F@/g;B=eZXJ'9`E$f1e5Y7F ~L v/?D[CdCYrw<B^ --gCT w -²S7e,kDqJ# h aԘp'̶h"kΕX4F$pBD٫* #5i@
%n7 -,ԟ7/}2`^*ubfCvM9&fn|K3 -]3=T,H#>GYNǘ%' ]Wh(i [u|9+:.mٲSa -nXr&QBLs;<ldHj!~5Vl ֖#f0({b/}8X}ڴD:&;mFDr' v84O;ڎbքwgPၠ/$ @DXBկ+vA\#! x$h](4GO+ -cr]&A5 Rý U,FVZxqU~p<N2x-'<\uȌp-rl (x!m'V'zxM'pj< -˞Z7:Ȼp"bRw4 فQ:%DLЀOB^TZOzJ˿Nt딅vy { {WDŽŏn7TU>FQ=Ƅ`XPX D<vVv)ӚPq3=yb
S -PR<C5=r -bkL: -,>Q -F_a/,+U - - h8i_BB ^'[X?m*.6pa|N - `0]3eÂl4/Ј:* -ل"ˡ9ؾ85] !@UdI8'dJlr A}_1q
h.d c -+⅃8ahnBg+Ǭ2-TjAWC~@e - f -A_'(sA;B ^C|, -k2aֿvDSf Ba|7IN@ B'l ފ@: ۓ+'8@W{#hGO&twߢ͛
/?* Na<vBjTbGy:x{O7 1Tj;fN|p쉔0ӊ'?~ az#sw/;u+ w(aNJX'&yˊkk<-Õ ` -I= -0k7)CF> -D"YUW/FON*h'_g&"U}ITׯq+MS
_bD +=BI})t]/|[qvQx0Wxo)AUd_JރyCqteT~ ^ G"p }Qy*hWwb>( z7{)Rbp&u&*xz'K|PG9T05 -+Ix! ^78k~/7k!1]ĹWf
"jOu>ድ{vQ;sٵ7Ԁ߲+0`fQ -3 W JOB\&./Jmͅy37>sF! }c~O i?K`Tz] ]GD9=~(ciӸ~Йz<RV\,S" %<=P \
-endstream
endobj
31 0 obj
<</Length 65536>>stream
-s"ݕ3KﶜЪvN5^ žy=V'trֿ'Mo-- -Hw4Ǻ֡
-Vj;L5zQֺࠠ!D7bASdNZ6'iiKV)k1~Z5(njb fo`5M, a g'UjQ$5%(He9D/& uEɒl< `ڽeӹB?b#UïPRo@ 0$IPÃ+y rҫ Mhe
jFpLx垦wT7HTa`g]R'tV׃o $fEj d -RZ &tE0>{A]jȐDepyBF,fQ' {poА4 -ׇ -f
F4)c|@%M/ڧ5pGi^tk Ў@Gݸ4 - `56L&;bt2X]?7( ݞe{fi/]Dz,CbÑ`:˜ŃF4u"sg̭Zv] VL5=:|8mB^@]o}o~2/F<y>c!OGp:qo擩Iq[ .̺s0,YFwO'n>s:ώ媻q=Q\>Kq& 8"ۍDu>}QN4 niKcgM9I
Ī3:T -~F -UE2^ZU]p ̦ә1}aKgc4*
m1ZgApQJ -&r c@ _;r3z(%BT: Z]jv].6e`B8L4:QfNhp?RoCD^Ő5"Q -wȂ(E}W_Kf ; `.,Et
R4; -NFn]Uﳼ)DPvXW˅V'Pݨ02 a˔m k(7ͷ.^F>A`gʚ"+ʤȓb+$=ŗ>ʼn AFA;|Hv{4'`|c xeWRpy;(`z"1Y 'qLӕ9^" O -[KIPV.nxȄSɱoJx .ƌ;0 wYZp5z;,q t&^E]3BZ;#1^ ǖ^WK:I,+8
A!wM϶?8 -\U OpQ/2IO ]8T"C>L$^.ތ!XGԠ -E\P@/ns %Dw^{d2b9֤`wW0D,\x9/M%/P!FR X-z&TM1Z64 R?mE=o 4u, {vV] -B>-,lbBh%&FE_<2 - aq]" -z('k$J{~A(n o$Ѷk69Z?lyq|<sI;ܠۧf* S3@S! -?5@rZ L <Q{G7'o~s -08XBT!bd'0[d~x>}F:w TE1?R7Qrޢ&Z1 -. N&ED̃W^ ߡlQ~"md HesG97صCM\Jj\DG1V& %\..w]D;hRsK -%wI|0`rR
t@+PHyBAߔ0<7
-ɻįkAu2Na! F, J Hb `^,l,*S]Z#`>DeƎsR$56d"% NŴ/- ?cK
`s=!Ob
60=b9a ;< ȹ0+ h#v= -4 -B~Ek-x$.vJZ
܍TO\hj[YAA^4)5.eC(CB;4R]?7%Q
`~ -C:Ѫ.Iٰt5q m uhXR> %S`F -YQFTa-OG2U4`(D -ˑHkZw&=@v4y;a' (p(6 1gwPc;=[7C{h\މ7 ?oV$ T3 A%}E!&*ds
rSOQW-Arw[C0Z -Hw-xf~ 8t SڕH:*Ve7&(n]jy1A Eub=C]X|: f2qwF -\ zQF>X{ YM| -TkOJݿq+ފR [p&\4`mPXJտ$vT^=壨NrT_ȡUA6K-B &2dg2Y-fXA]nw&5nayt~02(sZط, ƻu:lv9ME-oXq"T:gN"TUY*I8Ч8Ea
K9I`2L4q0%֑b# -x<+7d8țAKq4+oq3ɜnF|Op -(_;#|@&V :Ja0p -䜪H^nP\EVB3$GM,GƌLcVE$]\CGL-vt͎f`]SQ{)y+
ZL$tzf<%JAJ~d7.aOeVkd|5"? -36cJVu)gA!N0Z9qɔ` l-=ZzhGW>$"}nB1Yʩ4>R|/Ń:4
=NR Uk2ŕp?9+)n,G!~=rkR9]97k -y d(rN$AVMN IE94?JD4?J (Zpb/PHS/DEj.= %&m%3!HO/du 9^FY2R2 %Lff SM2>s&γSOqӸA~-ԬJyFgL2N=Zٽx=t$^j)VDhi*#RIZqWUPS"{f.(iw启Gă攻HHCSěȟ'eUZV8D'̊:ďb83 `qu_"qT=* \OWEG'e&s<4}_37<|8ڞxTy 1P̿0zD^e\lhj9{%xjIH~GՓ٪nj~\IwdQS 1fVWߺ[ѡΦKmtT9 [x<\;"|,%#LSǨU$cO#84bBӠ
Yc}x&QQg(>ty<F?RHT8 r=)/NO!ǛNTF#hH4dRgK\+FU|_ !?ξ;ވxY+ 4\sьUW\D8h*f i%es咢ٝ= -fFH'="hF ( k4iU[ghd z RC]7^ b}ps]f$x#ct`/#&WA/ZtcDªg$*F1.)1-C!1RH -'ugQ-$ӵ$C qޣϲY 9 .g)FD -[2)!($5\h5#9k&tLu 5Y(j&4i@>D\& /:YU IµF>7w<:?;jFu*cȝ砗"FV,]CǚBUSBU˨H䰴CT, -0䉖K<HŢ:iVS "ths
<SW-L 443Ox<peVP KRsƍÉ}clN#b}Ǽ21ѹ-FzFZl/Nl a6DU"H #y MĴ Fb7y ͍"Zsd]t]ԑ<85y?ʐx$!|!綋CPY.wA -Ј= iL=!E -։4!QHLȏ &ׂF$YN[ EY4ZiT)O:D`''gg&NnF J<' }q -$ЙPȅND(>;;E "HlL:WHDm҈^m -95U,NT1ǥBLNqWsn r.17t]1$! 4,Iz˖֖. ]uj -E,B>?%}LE格D\Z1]%bN*I , )(qa?؉ =N T.?H: "B*hj!GS(Mh4Ct&ZAN8d!L1(O2J2bT*F)ҍ4Jڠ>=h*6q0dJwn!vN8{s%cKk]B#li"q }AK㣖bnB%!
Z 6ӈ&F G"Xr cr& nbOWnn]D/j7GE)\/NJFP9BtB7X.r3,+W)^-!+[èENC
"B9hrFC9; -̈!e()\mbϸ5[o&՛"T\3[]-fL|`b.]sޢw OȨUkeER -VdHXt-(-e ]d)JUE`8"2ŠlFJ i~B$6IQB3;MI&JgB-d~ 1Y? :BQ'Eϒ25D iTEu -.gΏC+^L[5UG91nf)( -18h(ZygsR( m h7҈J
m(zmCDF;ǭ5k9[+9̵ʍˍ1UID>1♜{T
P(,IXJG J;J^R}-D\>ZN&&,=ZXJ)j=&K.zF.YvaS]^~x\02=R+"cX6O[wDYĨ9:'Ŧk'P,.udMsQ%#Bbtoetz~D^^D\٢+7[czEf79QT&BEjUTg JgUm|V2#"+m2tϹ!7jAHi$iEqua((BF^^d%" 9PGB gT -].W*\iV?"]\. K\VD杮:HG$'fzg".MHF*."O3(O!ݨ[.XJJ dDN*V%& 0 -[S) D/М%3ԓ9%4mS"]8W֥䈐2呐PeI) <AF4NbŒ8!<B@4V"аʥ;2A1묛;T2DkyX3|upqxkȎFH6*n3.tB`_ʰHF9ǒV?1$-LZokj
|-ir8a 0 #Q&Q 3Uն`{1*(P k#o 贂FԊ15!~F+Quƫ&dJm460US8U"V>B#YH#5[Eh _4"#KN$JV{,LBB!S|1 -ʌU4.LY-6?6CIdL͊ Ejnl - -\L+[gB -'| -vXBU8UP WDlP -}*jETv*h? -NJ²\TH * -AP!ms z<La^3)H/sPRT -B;R -?pO-" hC<5Qi l@b0U( -Ln -d Ym+:P { -(. U:?a>! '8'\vs! +Bc~'Wo'tn_A&ZiK:!\ -d'& -tpK!%s [Zz {8+@PW m2S«A)!w-J@.A < o 4 RaBЗ% - P -Id8 &a _) LdK =+A dn⁡$}$Cv$P(N{ `/EC D~_H9HhhG] @P; 5 O=B|G8^8`ZY!EhY4B IfVoC3DtoɃz +xN -h& D1 -cAp#n x ocv
O88}AP`03 7A](Ysb mcƝM81=?c{A #P@f -AFw8Ls #;Xy)AD _ 3x -A$ # n:W-۾O.B -J--|vjJ,#C0@CF -`w Hby(" {{y.$QA!g~>0/rAT4fBPC͆t~hZM45g(*6[BeNJ
"!X -> :GA؋, -= $}l=(g -͓N7,wN<CVAxf<j? ATd tnrYAD E; -]ӁD:T\!7с?HBՔZ1 Azf0Iq`;q -Z1p\ (
̿ԠVw
jM8
a%
\,C"4p,3 <g7g^3 Og'p1e`622 ~V_A{dH\!c 1H~2oA
Cp\k%1 - DCV -UuU@DRTONiR+*( -،x -bM!dLA.H].V)(dI}H6$D60,2B -{.MxR@3D#R!H&K0( E -B8!XI{D - NQz2 -5EAq;+@4D(eE| -AC7"?P -VUI\bKLbC9y(O `ć}?DUDAavQ*#5C})t -$ C;u(@,C+P7o(Y:%|?8OAB שp<Cl -,0CdDz(Ov(Cu -hA wck -jP G
l^DF%* `Ew3 6 Z9v=!Et#@U0M p{RR u:NEH{jfFMWDW?ٲS\%6p?nIU,c6꽲u7U2j V4M?e
7Ԁl32l_g@G/qUڇ_-J6^нu R.BDxwKoNb)'[P ` Ycݞ;%j!х]%dL -NH}< ȣrMUL7b%UIb\σ*$A`KAxRMF4DI0 )g,\hy#1c$=?t.r {X GTl#PDֲej4dhKvBkk@5&D\yt$0l--ߗjj% -q-7>ߣ hM"(@-"(orw -DpUZ@] bW !X* -x(CS)q!pphi -F -4d?+Zܩ
Xry]C`V4=`]`2Fp|u#e|Q$YA2~d;
X>ysa*Jk5ׁ֭=TY {sL ^Am: B7D9`[p\XK덱@ v<_#\ -
m`\Aȋ6s<["V=@8ĮFqD3M5 -*
IAE/LCN-3 覘p8rFmhz
T PI - It Wu_t p:KLȲD[Y>RP!<83xFl?whbbL!G xj`u - Bq@daBҬI8Uly.fhtlNJXd(*@)/z -*I^W a Bi/ ĊG !0Q 2R@*^-P-_"Z@ vLKKx
b#4%uY \آUŬ@ -XFVGx tu-hZFZ]_iz .{{- --5eIU?>W @255>_ LU$# -Ҳb֯k`7nP;ܐ!ʩG(K* +V81GBYPGQz4DW'{TB%xح<X pSK (w 4/F$ TA+𐍄 ٨ cX:b q:bsx(f6 o}]I qgb T :!C,P &h\?! HQ d>aڐ T0 ^33mu H5Z=7rX@'"@6Nk 26hU` ipK[UW @r2m] :ޙ|X - @| Ĵ>oK}G&o/cފ,1c) Pv⻊:Щ}10ñ -d] @μ|T,2ݞ:*/(=AbxS&Oכ裤r@]*' -s -_P+ h%+@Gg<U -4D[#`E}yb vPxnL
xD";B^dմ0CړYi -Pq W2<+D@Y! -P_&E -bؑAL ¨ -@} -SP &˙I * -o)I - =Ǿ[%l(d -#a G'R(u$ ->A*! -5c Bqh%j0F+G9
B]Gn
:ᛅii uCia@?R7&GXg/At(ڮ|
= Ki -K쾚hU'eŴ <F~+i5З]84 - bѨ5VMmԶҰNH¥Z7⇹/{PD%DŽThӍ' ~||ٶ~-u cؿ?=(dΉo&]oLpx - _o?P*$<X"=H'9=NXtHo^Zy -gdHSIi U -oncu_./B} -:p,b%6QFo"Nu(.(pޗQDf@re^U -R[_bMbH3~o_)+,P/3#r_g7n% yx|wc?`~ݩDhH>'3sVi件Ly罨I0?Z23/RC)8*U'!VM 7u9r1D_T0/ŗ3a~'k
U U襖BG&2 ga`'fu^AY -$[Qz#K~Dz1?0.QaE -|**^~WKܶjb,![*nO5D%t}f Fp - -"(X/JY
OR0Џ kjM}C;_onK!ucfBNՈ v; !wR$ }.~OH< -zM s+ b@o
g
UJdY[{fQk8Z Wt IJ-lb ڶFo(u*'ѽ"k8<J?×SH#?~&z -{?D=i ǍMv~ K,?Q}pT"='ym -Dvߑ.OD!̺Ztwooy -8.ou-obP}]=Ÿjr/EAp_gjyh a{ Ow+"c F㵿";ӹ?\f<)h Nb+TqgWM痸CaN U*pe#9O !a?; u6!!m3z;L;jg(YM'k^}^%"'uAr)z
d*J \0Pm% -VIŁs&X)^7q6%npqnf,ҏp#IZős0RRŦw A3 4\[o,֚6GK_}b?-4i}m3A6|h'6J;PKoCh7"0W?,g\j[gEf)U_t8_0 -[FD`UQDn -lBf~`~
]rjpLp |;o (6S|FHh\@} :]#je -!\?x 7][G'L{hVYVIK:*U5 D/DL/;F8hev -o_S ~kdTK=s%~!F*O4@tg8oF#EqO}vI͉Sb' Ӌ
~V -@rWÎmw4)ݿw7wU1j3w^yܥѷjFWamCحe _F p -d h}:'g=:
fPKFZ_,3 ce3Y-ρGI)IB[ i]e*u <h}8Gzk!A~c -_=>~RWimuIq<9\2iHI0PS;+͏T(v25Z=]OWA04r8vׁ03L[Q"_=qŷ#eg7E7L %Wooآ{BI9>N]_=W߱Z~T>>uuiZW?{퉣g3 Fyګ~ScǔnIUp"#S9^2^<Dtso -פ}2 Y2pWHٕU%I|_,N.Fs s$8$Mՙa2瘱{.g]O;>+Z!s` nTv!ca"M\N圁VQ-,~A o,1`Q쫤?Q0
mgIus"Ẕ]%ϱN7#*ͳ%e}g,5҆5g~l19ϡCt -}Nzͱ{T& %MSH'Iю'_NԃW'I^t$2Ih/`=BI>
9KcJr$}~ٓyM -ÚUy>2g|A - /*ɫ1q?sH A'`ѩΡ
sEyΙRhh`dzrZXÀKJfusz -Sa%^8E&SWeRsVƜcfXs @a)Hg$'taĠJmM=j Y9;=]DF93w:0DƈcI dtcKN4@%qs{4$}Ѥ<|AMoK:*kXrɒܤB9%˦ gsv>"Cb9HAD9ݶ"QeI YUoνߴ9 n -d t gMsl1ӵ)w?M\,/|ޠ%)`XIpN~Q 09
B YIPF+h Kzb1c 85-b`I`"T]$Z֡ (=&-"PQ^ Ɩdt&At! r<li~t ºKc>&O I{:>(=}}R+oagGEP:gav,EhGzAjVv%ၓ75c-xv}-T: g) ¨9PA<')ոC -<ڂu8VҧIjP! -N.dk=D8䎭R 6 huzB:!g
ǙIg+' ieMr_%P8_ vKx #&hLV[y<5:OwL"Trcn<uG1D3=t(bIbW=."ZO2}мq eFh©)'rgl=e|yKh}̗cGHB=Fy9C{Dtg] --iK^"x :g)q*T> GAdoD#b>`|t9)`#x)(|\Td!agf2p R)C]'Ֆ-K_"43S)d~F@7~ -S6_cGAoKyzz Tsb2yY?tRV=
)4e -4pGl-
$ew3ӣ@!ZrM 4LEo%u ֥1tZwۻ@ - cfT><gmePi;)U. K:(斩9e$#e@c -Ν*- kW&5X EcV&n4 N2*d'A ՓEKhPmrzi QuP?DU + A '5XlM2^( 1äm!|JAUʼn ^^HagF 1M+PKr2͌ gR2mQ++9d,ɿ@MbΝ + X/zm>["Ob >VQ&8\UH^)]:0 \@]6< b~d -t>a@ -C8B -0&U5:r {h(E0 G>(@ZHhRLNIMCJ -{ ˓ugR= {&Q~ ,d&Wu7%qxi,۸v;ʱ+dQz -Mms',=vQ Y4 .(u$ -" -"&~bh$yD - ݒ(48Tu8t88T Rk @aq% 1W*zHAL -ʑ$FM\IӘ苂>-I-@ QMZA&-t
0Q'͈@ST`47h9F40:s)KԲ_HK/ -v$Vu=DAGhтD2Z5-yULMf8U+<>08[a,a(d6ש'F FtLx6 g -0L -$ /4<;=:9ntXA&# -FH6L(Kg-3A+ tW 2Av -\ -L34>|0U傸'0Aa&u$}AFIO> MjLLA &4h35zܽgjGvP,;P89e2pe@3%GF;h$EwLtX A3fJ$c}.$8! R.fKcuu}gOZ(]ʟ(VP# o {R<}:^A)خSh
9X&I
} D> < ڂ R+Bw#5W*~ߤ8)©j,*Xw -&PK{Y
5[TF[-!i D$H5;Htf1BMBt O$
4G`GdidA%x֧AS~uTAXV9aރ~̷7JG[5}$u:)\v)L/66cb xL/lJ B 6
Xz3X
C-dz -=f m
{8<I2tKTVDQ8(Y4/2sM3DL!*NUVbCS ~ @`FC>ma5w 94)!!T,N# @r-
OGP -{**]ƥ~]FN<$Dй^ h(&*'#hjİ' -o*)cQtE6i ou(k?O xR7<Y\ O^ *)i@q|pE'sÈ' TO[{QjބцuKfXYܗI|)PӣijF lΧx/ -M$$85?I#|H*M -%Er?,Beddą2]K:nXF2R7Z'H]5b ;(HqO$!1Q#S'nkITK>99Ӱ o&4iI%P2
%Nt%o&d0=dOShG|N&hN,mϴ+ em*/OexCtam0#Eiԥ̨:)h/ot -\QFJ -)Tٽ~CT$F= (N櫕5IXHbI -IDƦ݆@"Og N+W6p6$uV`ﮨ#(dīCSQ(W;dcH5amz!|. F%uL
q5U*.Lr2D5JMuhģ0 -1l(5@rk\D??sI$ aU {K KG W̥ZONAc-K%J1:EJgƛ(Su e qRƱ.tXmjiRJ|ec)SZ{kaU[BtZRR-}I-+-rT)чۈF,C-] -+dj>
)ItK&3 8;3+R ӅnG{vYTR=όR&-ɺJXڒHdHA]0<0+|jn)HZc-츦 U9'-UXK֥81k*HÕQ"!pHGåO -!\q -c
cjrL3UqV]NBtP W2SRl;[2STnFRgpT]6 7 -&AVM ):6B8-nj$SSm6+~8 $XG3lm{[jEIiM6[ٷy$_n0h*z$(Mnzd0ƽڔoFyZp h -cK([8Xe|e RݱdAS5> MmtӀ@JҐu od~i}cI41܈yլJo:$wY^Sd# -u ̛rqu5eR1%<oL }[^S6u#dY#yNo:AuwIӛ,s?A bty5EӦ
l -IF M6mXwOK og '0SgA>ԽM5;ONXk`
v?}V}ob|s'nc.h) $-Byܔ ->k -F Yb -s8 oRvB)̬9M$['*/GYʣ|-CX&JF$\G0+ۤP_B!ɠv"2v tH(}֪61 -αu0uKx@-}>[ tdYiVNkY7+22, -Z I xMӁ\3k P6
*b,|>g8 8jRObf}1g2SؿXPBRcW[u{"@140y-S8e*<N7XTjJ1 c?Ba)Sq=&« w+X 3Iӭ<tVޤe +zC1T)TSm2Wѱ4z
4 1G&2]DC|ևC.Ժ!pv !jLVw*IVW14exk. sݜܱ0q2nw*F;;Z3
Uעm< - ˷X(;`LC%FR h9U3b9ֲrV25+*0wz(y?=| ʬBQ@Rc=TEs}cyz)DoL*rPv k?zKa\վA~6|,^z%֘^QE@k=U_W#oyLO#bۚw<b -XR!pB`U -/[}<E\ Kt2 \y"J9*WVP7 -2kx}-a3h!ۛőəjw`0`eB'+o*Vƫ Y2*#ٰXE45Ӫ uCjKV{[+oWt -4I3JeHV5Q;&,PɝvBXJtOH p\ui8mXB\b RjcZ~ R?3 ,S fKgw,r@/ ץ>XAy,6;MMu0;pp|EX7X-p% u?EhM \$z.( H<A|%4L FX tF xݜ?ƴ( - cQ?qΕF> s3ZJiouJo5,m7k(c1 wLܮt)Yj%BcU͔& cz>!_t0 Jw*Ha4Ǣ7S،{>VΗ@H*¦C N>H -$H fDG&Ue ikʨO_b_%(ƽ su&-XO48{~t-]?V$Mۭ{K K,\<$> b%QRơ_'+;++P!dǢޔߏo~oz gk[Gtqc6C)B. -^;XWq2 cN8D\R¬Ę +N|Zcm8b^ 9.eLd19
k`:b_omv4 -{xU X2K^mv:F?mC c9RNXc 1
"jV^)c_\d;&(!^TOy݄|,a=V~U=j F[=U3[CRtd˄GȅlQơa@>Uz6=֚BiG{,.Ч8 -VK z+Hjm #X{'vo,xRi^%WHIX h҅cIT:/}cIKH- A@R@\ 8~rm:קX1e
FE) ;zc55EuoYV`+Unk }|y pD\x~n,l~D-kʻSX
43IAXIƢI~xo,n%84 -78I\mXat2j7[8+wצA ۉ̄x1 GpjfoM0ؗ@\<I |No,Xn8|%+;$ c5H&.ʋ7VU:Qj7V6KQz M%0~j}c1\4qXb]QX,3dX/E+R
SXH8q{DX3Z+MIt ?$.Mvc# n-5'o,O>$@Ed8E h=o [ 7& o]Rv`ocI4xﱩ:UNrE͕p,Ǵ+ XxcU]68kBc*3F){يp,±0vL" +*,*|'`M8}xHQ$z+HK8A -nhc!58i!(sgX ˸p7Ӹ[zGu~j c'~qOEFPZUFQ\Xd-_."?TZo<3|bn#Z"z! k[^,r`h wX@`c3jciSџcr;.ee x݄*$ ܁V>)k3vH_C+TW݅X2v' EO/ DIKHyJeQ0P .`,*uށ̝w4 hZ)ZilH0ǴW*71SR)"#d'އdS%T$X: -R \Xfa@z!1Цcxd%WɈf:[ͪ -Jk1V=O܅1 'Ҋ Y^-
IfsVnXg"X4|u],0C,*M? Xԙ0ja,/Z:O QRX_l&Xl|r&jkǎ;U a׀#{evYG
,ZG y=AQa@0ֶ QEi畬7`,A; QI{ub -
I!/[T!p"MW #+N):x ra0\0 OЕP s# /T(:uk7ӊM3գ0f, - t&) aH Rظذk<dJ>Y.;vq?wFFcBy] dF|I!H#EGZk8>&-eo&(D$tP *bݢi360QNk$ On3 )XhTDE b3#D3nEaOeeAx >{g<9)" ?n} , Mx9Kw\+% U0g#daC? #X{bs@0hо@dX]NL<$I(0M d}^hͬ\/,kd.\~nprXv ywdIHq1!N)Kg.EM! "a* #RЇ˰,[I H I֑M f
W!}Ɓį:ae++#ʜؼٝͲh6
6֘m9߹Hcv#< hx>]Lt34Ӟa皞R%%*QRD"@JsU4'{k.hT^c<I4aAB^1<v W8E:J0UQ0OOz+@"/O _S#Pli<9 aLihX1>02CPBNuy 꾜$D>/YAZr0̸0 !aD^?±9|G7$FGp %"0X_eWX4b&)ƙ6-Dٛzf)wp)8a!$D¶P/\ĈMXѭ3ZW&D.AAZ*!:}} ^џ:2gἿF7i 9?%#|L -GMg1T?;t|a31p>/Ql
` $5/d$ؠ2K "@B -q&&C3 B;jSRVjf*a+% Jr L=?С)>U">UtXh5/`<|_L1C~F_Oz"lixr9vBP<OqFa -R'*xoՅ2vb!!tCx<<Ҵ®}XKpy cلpm!5a$/a`bgbdRH>/4&A_٩
@Q|j"p L(Bwp -+j#HO VDBdW6x# M I7R%KT^HD̵ fBc1$42 -wz]Da$1!a~Й>;MqoმxrBHl~pEKaI&bXE_4Nd CaFƐK& =#f~_= "ea.?"S2="h֘T&EQj %K@ND,݊ňt'7v|zpx -V\fS|6 t_ys2\ Fe(WPf/ s1^R"
7q2Y T}9OqoXQ$
w@["a6L1G!*HnC"Byĉg m.Q0Ee-ޞO
᥉x̃t 2O?D)M| & ')ha AQ(f6:/TxxOb 6]8z4yW3p - -F^0 -Ȍo 2iV"3QEf <S2V`CSCq<1{
z0U5>f,Xe?!N[Jis!h_u 3.njy!d&5`)-ݐt-U!a8׆I3 2"Efs |*+h w -1CVWJ 3^ I} -Ev_5X Qnj״)3OF0@*$N] lT*$!l̨=pc >"rqBktB{%:7f6(-5y aMt c IIl L;fVdnjaNLTcR D<T0Tj]L>eaC#&aæ&Ňw :f;1nT$P!A^fcF ᆤfYl`2eĖjNWk 38\ρӵ.;D. fߝooG*M QcFd SRGm48]6OCabq{89_;uS>anjKA> I`1 ;0W*h.cFz=fv̨0njD@ wt$1T"mN<H22) :f}NץB棰[(qғ By8U' 3R gX?9T=֡ܜv̹졂4 3 - \ n1`K1;u -43FS<N" -bIj
dL(1 Hю/ɓr-lpCF;bD2#r 5fj#K$UvMȌ- LiIP
2CHɶ/g\qY?QǭHQy O7)bÀq@~_ĸKx, !3r%34g -GW?$DW EtBnl> -q9jJqEf䦴#zlEfؠ Hʀ
kEK2ǵ4Ʌnd]4dssV`!3F Qܢ \y' Nk -F/3p.NZ&-R+Xnҧ-1ku2
tH8p -~1:|p%`S iv/Xyp~ . 7" GD̹[eFr -Q sR_~zJ9SXIeFv(Qe|Z"֓"AqVG{Ϋ2
Td(HYeF f$p[J "JJL3JȬq(xIΔf!4rhTʌ:! ʌj a2A5¸Љn>1]eF V:h]EVI zEHzkA*3^CU'o,ϒ0T24OFƪ$ ReFk],A+3x>,3G2#M/2ätoq> k@T0yQՙS -PM8ca "_YcG հ :"n=x8ۜF\fp[} a_Gh8gsҵ 3zacM̽[K` Q.@-:5- Dh8 - -9ֹj0C $EҙŁ1xaBcNDC\;ˌ;Yfq $fOa=7
44#= 5.?ߥe,=,L0݈;?X6mT=Чޮ%X 6J2bn#לX :+ -' I0\P>o -f(ÈQ٭,F濄3 h[f )l!1XjjecG8Y jw6Xf^Y 0%Sa`MW/U",vHY<P: %k#n/XPOlYfP'^YF]; rϰYfkY Ĺi
7LY ,m\ 1_rjЅ p&怆,:{+hQ9ʌh -B2?Fأ*S&q8ڢ6`ʛ }*(@ {le/>
*(Oh-6v74J
%јõoT<AA-k%1s&8:k0..eFyMN2 ~Jʌ g#ڑ+}nO2/`&qבm$bv(*d8Cb?As` }K5X
!hfF eԢ<NaJdҖ,$!夕PFArtI!+ -w8$;N+%Ñn Bp\tz-wW2*ӴC'mzrfsٵ dFҴ(
jQf`эVꡛ_(3jh j -M*W+3 idE4Yaca.mI/Vvj'BUP2S -KpJk4 .7Tjzy%fxe q_$s{re˾]he! LD)+W:7'*SA\al?P#:eP#1VWf0/ _qd+pE -9%CASڰ*3ZK | Wm# Eb5-cw*3ʏ/mMhQ/X0
l6, - )QHV?|r;y]Њz<ʌJa)cm2heFmsV2JUR+E ffTy⣆Ci*R-6y 5h81| 8oYfC{ Y ,mR5 -nPgOCt*f9嵵4|VfSRҧ7 - -9 Mofg &3̓qjcdv$ js<j2tLb1B<P:t4ܵևuua7_Gi')u92sOftbA̠ޓ#hX SE@GIJFMfx(RMfe5rLG<1PAUXז hHm6 -t4 -T̘%8 dिDUL7ɰJc@1ѫ h -| MĜ#4]O`ݎ7:s ͥ8ױ<14MrQ ]pi Oni2< ߉ FwqsNJi$i u>DaH\7Q&3X3R$oos#(љD4
vU lM*%AU7 U-yFdF7~:~2&jo=*ʌU<ʌA@/D2"aq-F Oz91,) D
,Pfњ<'BA\MΚԀG3'3ɑ9Ofry" Of$Ѿi
`< [ Pf@X+7y@ /WM' Ld6iB$/aS3NbH ou; cJe{љ +̣@ ˺7adH?'AA}T:bΓO -j2Hw%3ȨwN"]F4m!OSp {G %paqi{]4FuP8C<x)-Q烿#0 \!0Jf$O s;C2#i@dFmͪxWQ2c ]C|8 C49- Po4 \|uʒH{h![0jަ9u])%aM΅à̂ P7SΗs@4C!<BW ð{4En#KfdaaNd A[p&mJaN rqG䃎 a&ELV#/Jt Az]+ɆnPc̨Bz6 ' ;lʼtU
j-e't +j*053c'wt =fr 3>u~d>q)O66 0#0`P}ZF1ˎ s_d%Xꌉot2Sl|7dLc~!3{qiN8\O? -tX͞sS2c-
SƋ̀WMK -c nxldF rs ` UҖɌg[؎] W0p@o -]f?oLx݉=rpkڈvdF큱t3Df>Yӿ -'{^!3B
J - m~HF* -x**R G"32g Bs, =dV{~={ -]ֆjOYDȽ> O @:lO&F2ê& ؊Hcb :{LAjƕ/hbqDgܤLf y -/ -6 ()#E.yfwZ[M [NR`Ɍ]Ob3?<U76}/\)DBa{"%3v/q_?/jU_(bɌ"j5%3Z'#TY ~
يa?P/$3}ZF/e@hS`ܪbHZX- -&=D -
#3̡@աmwqE҃T->/x&7e{Ǧ%\JͽV`; .@f~␇1Bˌy xNEf\q|VTa*;1bed <z 3*X: C ,/Ū_U1c4bTm4cHaU
{* cg>fLe Hufc@ I)Q -=.Hj*ִg^ҭ2h+0^_(uZnr/ x g!%n9Ӟ ԛjʶ?nos{zHH{j+_cO <P#62$x: LDTד'ppZ9E̔ۅ{dchEf ԸPP:{+|W~
3+a{S8-{AX { -.=gw](:){Po_C|lzZ2{(`n*t:S ciK*(T1 BNz 3' )j:TAf䙜(6֗(u 3fq{я6wc
L G!L15H(ڛr l>ÆpWrYVD2@jP|ɇ/!c{ʎjz8mm~q\Bf -}1` {V y$4dSilU~(kA[rНZ UZ _-Ⴟ>Pɀ*J}ؾ.OktT `7_? -ԄT|l}R1cp$3q 4 $T1ci( "x bT|E{XhTHljr ͪ$1`Rޗ~Y7&bF̍5.Ajd-6J RM@/ä9ĸ ZA -Pzm <1Uyh/-⇢G -~qosǐqQeΚWn87uߒ~(4mP *'Z1
;z"ZΑV3 ;؟.Ō :ĔGO3 { ڲ -w PܱeH31 e - - -Mi @ѥ> (kAwKy >jpc=T7fؑL2[K -bei_blXƌXU㘑s48 ڸD!0-|f&?Ck4DAr{OtrLNWq'vB$SyO kw^ah0Oy̨pnj`҇#*Q@ v1c $ kO3Dg5 X;f Feh+X$qTړQ% -<!QOJVZ"z2zJs(+l> ?6 3`{BcCן0pC>m̨^T] {3J9@2$#a\ _1c -k|-b?jY{ E@)AR -~;q0 T~a.00(xQ E DƇL @0>!nb ^`bk00`<gqNbX (V`p*1[F,i`x7 RbDRe k <pTK71jɉ/M%\,Z/rZr/*un8Hŀ>nx6Y*$ J/ΪZBߋͽ@g1Q{a +^d}F/ =/$h^"Q/|/pbB ] 6Ng(7.bB|څ徒|PAKH`1[مB -Tn I-IbFR nbX/,
c{rj+RƺolcNtl%Xiٞ=}@]+A;k$I9Kf -CLk(O|Cj'l(vgʄjaik2L}y[Yoe -tANp06sѠE mJsB! %'.4H`.Mi -K, -%HeAhʂޕ&d!dҴD - -wڍ*+rxe!+8<MXOV6v@-"6 -S<JB?XAG{=
V-PK D
߽QX` %jbOZP V"j[$&= -fFPЯªsqM"Q{ļ0U<7jXuru mv8@o!Bq - `cUfHWE٪z=bTq -*D<[$XN0<Zw*5W Q a<Qh J3Vz*"CED֖3G -T|T0 {ܽvTe QӨZʇ`gT -#Kmũa7 -jj@T{ڟKcfsJfJ* R)8*x~5 -pX'*,d !kB*i -њbAcRZWm+?(oSHu\_Aǵ)g{pNQ| o;9 >YQϦoUS篡 -*6KhlZ -ǦR0X)ʓ?lQ -Pxjl0 -OgWR,TR0C@H -`(e\)D) QGAbZGڂ8 -^5e(ʳM 6ߋ
(06* -:oEQ -Xׄ?܈$NuȻninv`nR -[( -zfV@ b -J -LCBW&t Bwf=&P YLWxB& 4JBL: :Rte}]*@0?kL\D -& :Vswq"Z_%Ke%lm -sf/5K8j.akOnG?%tt葝%gtG3;)W܂%J7:HR.)ݫ3Bt -t'-ԅw>u[F%X.nY tK<O, p`(%%IVݱDqc*] -kUf.V,`5D@ s} 8 -aS+B,
`$4!WB$c$D! -JyAlAH=,@DAsV Wz k awcUF=}a= -O` ? ث?)߲e= F -ټx' aO*ao!e`ZP F@z ,=}qpW&BOz` hk _|xTz=&f<hqd nyn*WpXäGA}x2l02aLvqw -}
.7K
pXzA -CʄAF
f ! A -@P/xw&\' YvAe)0 q""҅tszpoqA
\ -Gouw Z4 ^ -+8Bgt땃B cWzSnST_~@&d<X X} Ap'
مo}'3@P5+<.%(p鷯Jg^a +D - -ebbfa0q䞓++4#jIh!$`BV 4AQ81a9GzPG,kgsv<ef01&0b `Ak\PTn`ZnLgrFĂp10vh` 1ؐvE6ܐ:bݳ \FQ5%Fe<e~cK'/$[$Z!r pCd'ZZ"{˄Ϩ0!녛!j]S#
N0' 4"UGw/)~VCU
S
90~l) -8 -^qR#NKؐ4*W[;c^CIb7>0lp
!$^j j-1c%f2b# U]=]D!hl4B`:vCK"!0uCJH0 KH"86:8
`,!> `Fcb#2<s za[ wiv5\42Ls=WBMWdv]ɵK%CbuK{e!۷viW[٥]Z`Op t/smhڥCj.9]z\HܺL"[=`إi:$"B +ćL"D߹sеk` yz/ ϭ]ڇhJBn`d(`gW]y~o;U>__}i vGjt.\ǭ[k$}[?~i:ۺ㶆Gb? r]kO3 >]Wnkiו[viV???~)4pcvwᙦḅ_irgXؿr_p m_ne__oLǟ]xvam]mg~_i_=\}a˲ #4n ׯ}Ұ[ îu {xk{_p˭i?1c/#1 ?]<i{X -ޕeV,L#3at+˱wVaϯg ؿsҿni . m p r p, -o=0]m
#aa_ {xGza9neX_@z{o_~~W4:~4=K/,ïeK_$var]{ڷ4=M{W-1=yvu^a_8?_c};q Ҳ+11]nku+_nM_ W[ׅnZkX_Zk}c_Zzi}uivc=g=,~n㹦g?/[n´뗦ߘGܖ~gnvis-nuMom.ҴKÞ[[mkڭcwr.\uw_Edۺ W״-\oݺo î6_q]q<u~Vv u~vk[sq,îҰֵҴҵ;-]E\ޅ]yKm]-tz}]yn<߭aauiq vw~k8vc=3<p,q~unzB<kح[
-endstream
endobj
32 0 obj
<</Length 65536>>stream
-n ۾K{<qzO/u_?o?Lo=>m_ӭӭ[.6m n4z9Op3} ˵w_][ оrg} n\ӯw =<˟n:~[oggzawr {W[ߥ[rL q]߾nڻeBݺ ~*u=v_ϭʭ/ a_ V?]3
muߺV˴뾲춳_ٕi~ճ,2ӱۘvEVoeڵ~s;ϲ,皎}v o v_k_5 +m}]!o~=e[k8nLm
s[s[qLӱ|O1,/4 "q~#[~:ne XzkyZdz9nv[ynkyvenٷ\mv~kߺpݙܺ.
{֥k]Ǵ Ls[ӳ֥XiX~oӮog[O ~[ u[m
_4=-MLgY;rzڅ+ܾ4 ۗ~_/u=> uo[4=ۺ[2=WױXt4M4,߾ߗvו~㶖߶gmYvim-ߕznZ? ~nvߦ[ycwnZVv]wwc:pK_ݷk~ݖڷم+4,?uz.Mank.䷎gWdvᶆes<O[ K~g[v -n;Ϟn+ȯ/纥tLKtZ15 oi]_ocm>6m6]B LZm<"]W~ >aV]_~gZZp[?p7oΰ ##}k 9L Rt?\2m7]8BF ]BdW``
!M r,! a4 dU2~?"˝=zE6pzEDk˜qV7q-"CVES,( -:56$nc[333S45$n"nHI\HA,xN0밈k^ NH?4$^CjC9f`E嗙jHAH*r2*ySf 2' -hEؘ8P0zA vI Lijv̔jg];B+VQ`HnS -R=0ch s#$BO(q8N숖2B;butIl
̎6m̘mk<եFܒxˬH*z fi)2zP 9|M,TMh5Sq_Y%@ -= *[dAL=}p
M=JdBN+ -CC$U -jHh%MvN[xzGK4 BC)dN
/ /2&1<<ƝC+ >(hĎXAPf8*
Q1P -zη]fEY*)AaLjlFuJ iA-V (5o|vj-16u%[4 #DYjC>`jGN tM rjmfP, s'a⭜#d& -QbZ>4G+3c:殂yE\² - ʸOpS77bhesh$FR#S#] EĆә^ٚ9n`9ՁET@u!GJF1x2x!U]㞓;g|Xb;R.H6Vխx̫"Y -`FQ($(x Q_c#l^}vWx^b1(`Ia!'2#\:8'nvY $!mOiؔU*CK q1]Ҟ閹upH7A!!N̨@]@lLY{ &i♯lF ,P`{<R秄
' -@.2BؑFtc -'Y~B(%v3!E O Bc ]t56$kB!3 ! xP,P)R6綴ij4U!I 浶;5̍'Q1 d9(vk.cTKqfƴpIWvJS]TM-yXI6sS輦Đ-K6A/)-uXyPD@sKăupiKȒ-0g{\tbs@# qy6c憔vLq3V,sL,wW̖T R/A3)2T-'P2+&xSAN
(,YO-1#)1Udcf<u tT,u3$3CZ@'EVȘy|MNW h`xadp \r'B Q)7 P^)~=Wẗ6kQfE<#C.p\Ə
jhtG ILMƠ!EFHNqᆈRCcpGˇ䌬5#kQ{ BÇso)ES#T7QǀCäFSIMGnȩ -9ba -h̢%˖-3ljlj`')BWJV-o Zw=g z+zINcgv -h -EcK @wh3CI9w /)$ 2o ]W鈷g]9bnA 1q!YO`ŜDľnVD mԗ0Zy5w홍閻، \DQ2#n5"XF=q߀S0GvQp+1omnz+MzN=23 = -MbHSYНk_SWA%)a/N
vJ`o k"tzerHМ9w mhJ} ")s*EbP9p/^2G!ZD*mv=&ȖMrНvSp82}AuWιnif$|^wf-?3g-oVٞך.թE pR JȡAԤCRIӕ*%m<Ď58R֎ 5 ]bFەcbP)wy1s B:HfW7]E@;MO4ǑA3LQI+t HؼЎM ,1SOI*kbU78 irXϨxˬhxN)M&?HBX12|ЊG\CP#\FkYR7y 9BN5pII`GRِ2+!V} W!GBx -hz+*=-FqKV]bJLubQ0 1$t"<w/xK{Зt픮ҵ[zKW/zѽ{^IӖٱ)ѪK%/vr˽EWd[cbc!8ݐcȜN/Gv R.3'i sw :%%a$,xݦfխp;1u3s rP#\(W1ȩevz[S(o@GȢדc9+ #K^_`lĜO y5m`F 3G:Hm:%3 |EvD*j)mMzlG[ F WA59 45e>)~bCh唦VXL,7C 97>V`%T5ՈMmA#) Z64 HX(ڑAs]ihxU`DPq
45lJq̢ uUxU1w]/56:<^T_1)ewl[9f[)1s [)y풰@%/̅'b]9e5vI:=Ix{?YڸԕӮ=Z.v~:8Aeă0#[Z댬;:DJИn$%EytQ ȗ#KG?:)J; (nlőZ<u `twq)8$]Ȕ./ )oر -= aS9 M\Eȧcq"piup@Χb!jZ5䐢^lHoyDRWRғH_*H)r0SjKA\ xl;&rJ-TנR6^3 .R,_YU ҔM4#c%`L兼sl\ggud)#dAjvk:g6C"X5<x)9o6dp;[-.'Vf5;!LnH H-z[Ҕ-:5K -R#K5QBsnG+֑ET%q]ωUيJzf!HjAK&8&)v[9UuO45u 926\9F۔z8`Et j ^&J+E^s'qAu)jJ
~ .wWP<" -W\G152 Ae cKĔ !sj#8'(ĆM<@fv/2JRU5sbUD<_S_ 4 k#\zUQc.SFw aAwΙn?IR'Q X| ])] EPiLY8ݺ1zLD*
m8 Sݒ,ź1^c;f.-IԞ7U $aKI®Iځ= 7U jՔ2/!K35+zKij%A0]Vo=&Kǜڿ^`ƨinK=e_b<[r-ϹzEU1$GS/TGi-SS`Gb&,z9Pȴz\%T%w2- -!ł'Hsag|r9J i %l $l lIGx|gyzI[k@;\3_(ZoiuU -UeCJ?\fE̾rĺ>h.- ~W /3C 疔|O;kFBq|`)on4G}CfulIR/\4 6sc'T'vWXIѓ ]3fU5T Ʊ5oWiYZO$7-w@DĄ ybY EҖxPl$'gYէhzDLi];UXƦ,X&pp4U Ӟ`3Ӂ2Ș㍡dF"yhaIPq;<kb]J4-I|wxE'4*x:BÇ"TZf3!#XfTEUB%z˝[4+OP-..VuTWUhETR~aUDf$==،B *YO|=[8RlfLfBR -9TlDP%lGT3b+x\#h
EĈ6JbEQ]!Uu,V]5'g[jG
V3> -T9 --~t?μ{^u\ -ޱJA>S Bs`QZM<sPy57',AԒTY 20N$#@ h@GFaECLjȇȴ2# {ҹ-**7 +<6XӴCy s!$dGSZ2咤|IR?BN HRc`/ND 8AQL@ T((x={^ihƶYcC,xR/ݐOӂCM%OvI={fU:c3N_=T[:;`jê)M~͝w$@%XoB -fE &fLΈsQ Mg-dUI?@ -W3 5ĒvbV*) -)ܒ~a0LkB8 RC̐ʌ8 5P -aT** :' 7ɬH%qXQZ? -a[ -TeglHQ)&ָ7!]vWL3I"A6{$/q -~p? -2jFvV7BY5.l5YK) _'a?Rªש)inDS egG:̒b0]GRxlcӯuϙ
7-eSJΦkf
+F{YXFhH_C9e*>ʉt+G!* tTB7PĐ >Jρ -@8L_V; b[~RfuRm$Ez5c cɜ _>1 cB~Kx+vW ͋:XNhg& m.n.1r -5ȝĤVD%2&7[M
??Ps A sx -Vˤ^HH)˞uW: z\'?a:hZ 3ZunS'-f M"Õ픴d9Rt%"|t>Nl}"qA.b) q)(YẂ nb&|IwMZ=*#Ҷ-`w: -: kI=78]I鯄vx(wz&SC߿80V-]̚P
J4vU0P$0sT(Y41;QﵚF"{)WgP4lU= -ݪ1K" aBG}K?ECd+E![;lPt:tOaY=NG{P -6>e -_
ݐ/Wѽ:6|m+-)3\'?N -iJiQO\tdk۬FbY A+iCt7 bp{J8U/]+y 0zj P0.LU. fUBt𤣠=c9::5'\{*(6ѫ~5Q ;'wZt{&o+hP v|c]pg/ t9
ErЛbY%
_ ,Bv=BBA'R F 9n־A?ר[D Zn?DB )P&sՕ(qh_zJ Hfѭ̦cnؙp,_LC]şU }^F8fu=J th **m0/fSueL I6.*@rVU@GDՍչV=2an~"n0Ϸghun[L0[X!;yf~:=s(c|.lD M<&}N<JAa$ ^o%xEVD oc c - yKW}.:ҹj،Qd&@ԣʭ -PN v:N$'^|b?]Ey bוVD.^F?79F:z] 'aΐ0+oaole+@o<x,I3
7PT cgԉ*%qQ"#\/\)!"3SDoYvnn% wa2,inɓ,x!I' sBynqWZ@ -( 3 8MFql@8?# -&TB"3z:1\L*_r W V gYXJz.ZI'V~V x?v:X#] ^ܘ=ۧ]H4pMA:"Pv4pnQ!LqEK^ -n%baI - -tWAtv:88R^X`'/2N6 -0Cs£Z! cѵr_lEvh(}7!}=X 2GK 4)I`EЂ>=0T/@Eb'Ŭ -hn kK/(*uKnH97َB˚pn {&#fe q^ドL%Rj|O#Z5l\>BUD:&@ų 1:mT\
< n0>0}h -Lmq z0BhXbj -UbyKKb ypbM^ D'k⧕4:$ -O`]nPYQ=4 M8-K>(. ' A~tlD: Tbm0Qvf-o]ʤ M{I0(xVāAK)P= uZn L\9 .9˽)[sl~霺ãKt4vJHkws^SFzWp ء{X
=Bs|eɟCV:niP`i<Z4ƣ}6TB0O) 1 - B^vӴ" YXL" 7 ]w[t+41PR -@fPo[ \l3\0Ɍ LDu7>צ"IRt -%crEϬ|K؟l骄/3-DqzE;43 5>m*W^JR^ZKܾeA26v"i7|K*G9pPj -l\A(D|Yomͷj9 #F -@|7&ŠH^=T ;Vg[dIHGLwJʙ_q>__KʒEC {`eF1< + "U
>x ̇VNE
dsLc(΄_VHEt~ẇvN\ -gf{_s͊qB *Bq~T -ZAq7}JIߜn=8ҽX;?.
n&>i|1-n3KR-W $29h\~*|'[ lXslTVsk s[e 2 i7Ѓ:gzk j T+ڪ -@* -XY]V1tQf]۱n(L3$z`\wgܒl= -4I(K*\
zT @:Qp7e;/]Z Nj3݃ou2$l^@ oUSA%CI -KE]w_HU)L9pZͮt]:Ga_&i*ђ4&̲w -O C"ɵ -bt oQ6G<-cBԑ@=Fa?:e^#H}8vkǣ ZA'7cfzFmኋ aڔYm1-^GU!aVμ]lnekJ*cX|5 J[gkpolOא60r~!s$8<&[dzmP{[~Moi?wf|u~ȑLEPqOKk?hb<I0đt"dp/j"/eZdY+`#%ӢU,~y_Qa;ɸ"VLi;Y-q.]i;k,,f*$ЀQ C%+!Ik
Ȃ%]%V|m̵]E>
(=cq=o0oӖ#18Tv79lm -cqA; v0Ap8@ ,S~u4ĩ)kM 2x{
5JVC -'ob
2q1ʟd{ gϐf (X \H#DX/$f;?a -υP MYDyW@>oAWXq?$\GNs -RڔtѥmoM\6T -^kvwYZڲ TߙVK -pFbyBd *15VJ5Ɲ.guHBVQz g> -:`(ᘩfܹAU(`Oŷ~Rz}+{ds c,8D1eJ"7jvG}?ѻX/ ?zϭH1q1Y t uԨ ^ʧz$-~%pQ PR/|0 ( m J:m8*R4ϰuMyYCt=HO(_i2xcc oT9jA];LQlm}P$bWIi>9Uq*7D`L,t8ϫ7 SenI^#q vRwYpcVޢW_q[k[ׄf( 7.otԇƦdMKkcZ)`9 J둯,q{?Z aQ$nElq?
.knc"&n u}Pi)q6g$QQ[c|;%A -eDDx9 -#Ws -i<UlEsD `Lo @+ԡ~"S4Ux\͖纆]dg(p! -z쇡{\ Xs &^0F a0!5 8{0ŁC puT(FmCSU@Zp) î駠 Ϭ$_ (6eދUq
W?+unlݣ)Ttw cr2kBlìwYPK^GJҾ(dpY d@0Qw -dᏡD
A, hҪeh jm-( -ΩP?1`oEN5Ccff=<05Kڍz4{!?C"OoH/˱lq%][ATZ>*Pa* - tn?J?,:Svc
W臂{ Óۨ4H - ScvdWLCյ|?mhOR:~nI5
éN5Hp/L(GXMf(~ ɓҔ =!( *ACɚ '5C+4 -c)qFAV^ϊ -X9.-3Wҳo 82
1 ˊaPFD o' g$/- -->,"Ƞe' -ƖӉcu),k7gq`''C^rD -X
Gtyy4JL_(y;ö - yiV#I(-U?V3)g,(q< WZ#;;?Б|<;eG1Ο1 L%c7O:77'+s1_)Iv-^ rYEa],qQ5-~/<|F0zucZv(\ M&qWۿ |l%dTt=҉ w1℄h wW[6$U f (68<`g-ZU:~.Ң>%n PV| Ph~~Jl -OJT XSTY -CL\F}ڮ͖quC {3S-5<l1jkWq*}2衯Q
>~7~KX,F0;RUFO*蒎n1I"֭y0ci#'ŦXn+G]/I.6.d:GU -qBK߷T9 /{ؑ ~(s9ڟ(&[r"SgzU2P>H7٫< -js4D/d/]6ϝ
䜐;paP5w_OmsjCqd:.6{:_˙3 % DҤT?Bܗ~l\Hl$K7TG\4z{dc;^r'G3B䙜6 -ʐK& >0f*`0`:71i{1j; )9XVby -7=^kyQȅkXZ&mb? -iab,ɘ -ndXb,3c3i21ÅRl.ݞC cyq[**DJإf{'ؘܘ -R?wcezE#L5cx%)l#B -3[fٷ V3 kV3f ,Z5i`
1Pzf HQL}ʙ휉 ><sYMφbͰ{:>˄óP?CH -
fpΕxY]E#
6~5 - U6 :#j ř/!jkF
O|q8i] -|i߲"Ӕ{mՌAFN-eۮAj$_j9
v)˧fHYznFXl/5h7M yjF|4OL4@SijE5oӪLQ[-z4fNi i6N1.~U3y Ռ!߲a:mǓi
i̾m aK/%ELZilZ_8mG/~i!qZvu'kQ:K9ră8MַpӔH0 ""npO9Y_(D`q5 -g?ެZtvhhO3 iV-jӌIڲt5 -Lk[&5ӌߟ* ZK+ZuLZH\˹JiZ5CǛE=D
5B|M~͠lTzAĆ -j^fDI/3`F
6{ X6(<lf)j|mkDoԌ[ S(qlrCɖv o-¤n۹u&Uͨ P-Dtn~7
#ț{՛ioԌ(q|fD7L}C1=5ct pf m su|ɤf N), Gh(.)3"n5>8cŽRy8qjF ry@H-p0I͘gF=?α "(\ݕ 87''ܙSj\:iN\^n&jIۡfФ1a鉚QjaĘS\ ʜ
5m2>͐5%&Cp$/Y'<sA3W3wi.C/ݜiF@ssNO3|璫[C~5# f20I3::ylE7G7jn&(DS:qKV1G+ThtKá4MK3o^nPbq;ܦNgj
SvL3w:SfpNfHYKflM5]q5KtF4cN3 -ƕi ,.4 DSxA ֘fL[:f"&f]RRHעtåf`Wӌ4ì_if4d:FH+5cw͇tQ̨
if0]6md] R'S/= # Y'\Zwۺ4 Jkn^?_WC~3NM3|4CK3f1ͨ#{⠬.]d_툀톗fdUO;`6{qv1~qx;'wi,irPidl~v0pw$B. iR -A-_L?~Bh i2JQfhO3 pw/'KѠ t8wR<P1bO3 x˸4t"RgӀWRޱC CiF? -/ xxBƥ/xNxI3&ݘ,Y;Y_\zA`,.o4üyʫyǤ7o9/GyfF瑿4t iFj,eH8jҌalsޘ_8 -` -yky357c!=+>\as}>o4|9#Gn8,dy/G+&L3,x<!DI~㧠ךQ7=]lAo qR!hhD342"' x>hAkhh zOvЌ a=8fL̐)MU3c2B}fһoXyό|IO36R蕨!KG̨hs YcǗkό3{fT>3J*^gl%b谞'Jz[zD FWѷό)PO9M=#Ќ0:`}$>'h_0eO.KwŴg_{I,:ܓ!͐{c 4y?/hN{v gnhoh uk4$[|5A5(Z>=||NH3OAn-4#E9W3Q !D@جʔrz}G3h(W<c_V¸!`0^y}}rɾLB S% -?G>M4~nE~ _HPF'Cf$>DJ3mcYg4~G3p*fV4#_z_? y1!_ -:QY td -A!,'0fD_S@R. j B RR`n -w@ -ŀ$o@E -^ Tp4F1
3T?فY=?PahF-,͐ޒ=1GFP<;ߑUGP&c -G1HXmPN>7@ /1[|i=f\2hnjah1zʣP||<>Hp{ g*1ˎg%xcͨ&`|"cF$h/Ll4![VY;EL*٨uK,tqt4x1#3`C'\E8LR0Yp4 Y,^ڌ%R4ɌfGQEe*3:9eF^ f,CH0 i5Bˊe A>w$xy eƌbBR
T] -a -UPZAU\{2²p+3dDA` ˕xSyec$1eFtAPڪǯۄ(3T2#q:=ǟzO1 -0 -h=GM_ 6S+3 N_AE)"-6M o88:KgR"`Ư2Apŕg'[/W08 AнkAeAl[Jgs +3LR㠻pb/=\`e=-3BA2?Dzd+3v2` s|;ޖqy<ic -k;` FˌV?|"aZ~XfnvrP4Yf̄@P^' X/ˌݮ B"xq(+3z*t5@0c.ˌ٠SsYAz'u>eF5+3Azkʌ۰@ЏtxZ9I:AjE6,NcA3pA2CFDꟁ[xy2#ٗ&3&H+oOmRH &1 2 -WAPT$z4m<`\$qӸwh 8̠<:2Pd!oAma1N. @\
̗̐ v$q$bFdS%3hy-8dw+ hbO\26.q| x-Wh}@2[?=!n#Qf9eF\e @oWf p&m~)3._`2~0e{*S!d]2C`@0/8/8JpP& Γ28+eʌPPl":>qSf 'FQrOfH7XT
~Ɍ@Pn A<aɌ(QP-`ww [2# Z\2Í>aAp1 -Of0l'@!Kfw:7Wu@pz ҹ\!3z$ji eY@ͅTTTmL32|@p'"8hPZ|ۏ-iXA$*5#34jɌ&O%3|^VPD@ˢɌ{ῸɌ7tHf+&_@" :`՟+= Ne"3*Ȍ -BAA2A~dƫ&%nEfHnv!X닑ez8S} jM@" %{adFH]D-aluA;` wZ>'3YrF!i8 "Ϝ!iZR2X7 4EvJ2 ȌևJEK -!3ȌȌ!~9D)Fs.'%"3b*<dJD)aXχhۛBg\p-NNEN@1}w}8WJ)5" - -$QSDi ۈ UzrkcJB륔|(# r-E˧57jq)M0=ůˍ -VIo?ĉW ͒&J4m+qێ4mzt_{ ^s8S4_hUYc
癦)[ -
p'՞q$KsLt^%uɬNJ*믖~3<?4A.&9OL}>2ws h,;ϓਗ਼ir|Uu |tTCd2ϱcGIlH GtvWS E8"K
v%P{μ4&编<~G`k n2{ %o9rO"#̺Z5[T1r}q Bq?x[y!N.6HfېVY2Q**s2j2\ -"lMteݸ\ ToQDt9S} Hiᔄr[Hrm/rf=~C-ׇ@bp.)Yŷma$`n>V iHtL Un~Q\Ԯ:Rt:ڮaM io:elI"Η^"՚chTPg4t( DK7g7ߵ$y;Zk&1'5Dܒ6>h/VE l- #s+i~8qۏkD~#"Rs6Hͷ<.l]pZĢUHn:q(^]a B:}ҘE`k!/3Rw)lDU&R;jDAi=vH슥%f緬ᖐ|wi~Ҷ?ʖa!$;v>.fؙ+UF#%!h" ћ.-vjܳ~=߉c6p8%XlyIr+G;B3<dDK '7?h-ն8UOvn:yEqfK\tלFyGW&1KʏSXLω\2ԻLp{g:ns*z[j
P&G Dv*}YQ~)1..vM*%H4BYvB{ќ`STH얳Rqn$ԎS\W>;J 12n:)|*Pz: 8Kl;D:e>CX[%3 irZDrf" GAј+7פֿ}sUE] -&r`evT"Xbj'פXĐi\ 3{mUK*pPH8"7>Vq0=ǻe3ª`&٩G' J
ӗMS ,vK$Ȣ[v.C(U&S
ceonm:D> "a Yv\3FKg iOH|tFDQ*A{ 5 4Jc*oܺm%22ޙu$fxd$QJ x( }Eޫ-wdU2cR9jW;N3::۽p^we+: T+nr(ԫ51)Ε2 *[FGP"|9"|݂1r'
NoV(mm؎*QR%.5mTVZ )աJdbTM#2Q8-kD^[Z2}n|2 6<d \-4 EG %x$ - -H:V x"F,x"0/ff@ ͥ LRg5 Ihl .6Z~mpP|/I - B -.lؠ" -7,r@J͒#n GfB|)G;0:Nl@lؘ#@8&Aǩ!jԎ A}M)Ib!eI]s'YUf:EZj%Te*,3%J 38' ~(? M 2)H$X*aNeqX\Xe%,(&DN 15 LM -E VQXq\
@|AZ8dQʧY(Uȗ+UkD%ŖPF9$E 0lDaB8aب@a &%Db - -].\FTnӤ^6\yNXj -"BDlh -4x@Ć-*P#FqDh!hDVy |Sg lW*>j -Ad#&l@ -Kԅ$6 BDaf0pb8S݂ - ؘ h QBc= l@)ԡ/{e22I;X1`ƆHX -8p
,6B|Qc dMfU^A-B`a4*Q1<Y0 /I( ->Z -0A`b4 IP9[uS4eu}\03Vߙ*Pu$A
3 -6 -yXˮjSQz[9FbE#4&9ۍ 8Lpa -6 H@AB .P60`p"ˆM/.E|<W -(ԋDJCɆ7&JJt - -( - 5ہ~|iNBi}jFg3.#`lD -;$ZE?atrXD=Wg:گ5O,P5Rp -4LC,XWhean$N.ů5$xM)fIk:)H9-D -8y."h - -pDQ{fqCi DB06c*ך !Ի'-ۍS~IYuxCj@j@
!L`a `#O=G -$k^B J 9[@ O*xV߯+TY4D@وd
D -0& -t r-@ BDxq.`t`RYRm֚GSl9e滆TFe-E 5B%4|8b -eFzWcxU-]ƉMNVg
(@ P *q!Ë$d -p2:K(Պ%[E:k^s( U=I$[*rQ%2XWk )hNPc 7O 6GhWZZ/ ch5aݲHgxA[\ (& <yE{ѢIPf!N-e -4ꬄ Cbk《}D*3 @abUeϚ=뎲Բ bRgˡ/ItdB;tEbfw-AF -qH\{/ +|Ǜci]!}oQ}.Ԛ-ѯ&9 %wW尌sS*ODOX Z[xhW~xe YQl!V.SUHlu6}2 -4"[b_kOqv,7Q8[g+NuHU , ( tS аC3@)f&zζM:$-2Y^v%'=9Q(CJBh 5gzF_|(.m7>' f)= OD9%5 C;AF<O8=|Zg'.U[4~x'M:Ugt 8B`ao r hz,p=h.%(*AJ}§:`+YOOYdZlEB
;K~瓦/l^#v+[l*I -rL,MUrs7L>Jfx!Fi 9^f ܄ֵϔ7Ajqb`nF.΄Kj[,,o*M0{ >M*ŏ nB -faz'E,Ju=.>]yDPKN6|apǬ:O(E17=2/; ?< @xl=MnҼCÛ)X.U1 -:^Պl$~sJeY4r@* !nb;B$4rJLzG ٮ܊Qb{0S c8^%= 4Nc'xJ`qψ˟(Hz)Ь1OnBrHz]o7- - -)Ϗd eYr8x*Ԟ>k8cb! <]
f ^jtBkd*^ٟHYzEk:˲ Y5Vbsr`ᶼ~}7] oytA]z%x(z
:Qd&SfmiaR p!Du~R滚~q^K>mˎ[I(q/2~D&T71BYzވ섩U'!Rcy[r:DKxo{=Y)r; - -烊T -0Bg8_I0' +$Mքײ~kw"rI`jYse_~OW`o>l8_ fV(n !r 61^5 X腱L!M糚Z]p)ȫ)Xp%ϧ)Q\T`H`l0S5lPz/܊Y{ -kA.L}ء*CAs꺝85ϥ(&HcRhp$ -Mr$ J~9̪6 F^Xb'Rg%~Nc5[WH-'B;z4Faq*Zj,]j t|`
`^ -fh]0Xj Ƭ5=)s !I.Ouy!zІV3_i{pms_y܊,'bKmM;$y'Ohx +rW sn"鎎&-\
vM)U'hm ś4Oqz7()XɊeZ*^p,j-EɬEi%ֆa덀vȪ%)ס.ZmVlRx!drTR']p(|GTN5 XwL] EsAjh.ͬ);[n*zXj -f\IS>GDw`e5HeB8UցgibxƈAhTTBՏ?`ۦaܲ -PF3ح6=PhALj!11 & -;d1T >:vͩV, %pɞ- K\lz9ѯ7v}ǡ? /Rk/|`ߑ7 ͑/[6@ƨ6/\!],Bj8[tOރηQ\HgYDI$*K1_D vyH@u <Hc\ -jiG-` [K!2@rg? Ԟ(|W -dxp~&| Lye'A.YP,ɚ! -U[1f}'7A :F5Anb0HEZ+B]Z8R!j@EfV
A!iCYuv_p ݔ W/0rz0
8"kYk^ -1Ir=Wj!@1Lyvʼn7~*.5"J(>Mh ïer i PI~ -Xc 8H2Cq% i1<6@ LZ)qU[+a`\ -כ B,?G&'j-ZNx!U,oA >BzC)>KfhZb&Ъ3й噅 _ъM JgQdԪI\p+ʥ9 0 3@6 ,:P[ď -2_#f}fX>1 -7Get%ɯ X -mwŖEPl'zq2c@˃*7<Zp E,7J] ϶ lځv1>8Y.),<0aԠD.4 # T 5Co =DG& tʑ -sIKV, -W8ƂXǩ̃ q -emXd3Rd,I6JR}g$?]eV[ $BHeF~ -c0+`
V -dY<2e< -ZXltF>[m=-[U,?r :!y(6 -6Nj&z6Mv]%qj,F0
DYZ ߔ - I'^vJn :F]Y*{ d! kyC
6[!31ʼnbb{%A -eAk.yN9B pbA^ 7Cr VF_T#-<GD@3x&NexlfE -ϐ[֬Rρ5YzI@za%?,Xl|)Ti3S}dyGi>aG "D ajJv r-K -@N>\:+уeCVjKV%հ6Fq1Bn =.ĎF\N_-_w 1I1Do0_.V144 yI.bj -< X -A ( --L 5[&p,Vr P{M<.3
˗٪
R`n@LA݀J4HM8Et & -Rؘ)R8s%z( /4:`>5>SHl;/f!*z1f? 1=GAl=$ -m`'L0H 7Ep[PZ桔Kᷞ,ww
Gʬ
X/\ϓZ_J3,.*0TBAQO EK -w F ۞ &bC9:\pC9?zV!zÐdv"xeaԂqʯ4r1Xw>'D+7E7A
=Kf$~Ui$08*ZN8)d % ?D]zY 0 ) -zmd_NR(3.{=gb v*Vjrh!z@b,e8~t 3'{(˱^iWj7ɒ!eMX--
/CϔZ V #!S3*7uDq?@Zov zRgr\^ Vahyd gU::8qQ*Db*hm 9 گl<e/Թ&$i,s7N8z<r%/ "Gs*mĉ Qks!캯,ؾD)wy!8Oer7p :Wv \d*2B= x ѣuOa+M0<ɒ
qHQl.1,G~"rA,Z -fb , N.\eЉB{jHj0$!r#46Kv<Aw-+y織&X@jBy,%Ni+Rl+Qk+Ph(*U !A,C/<>F NZDr.(WIv Rr(u8My^ r=B$sn]wZ;1R -!Œ - Ca=^m(Ls+ =:
v;3U<4<Id:g=LQaYELqGnj~ھ뱞bd -W\C .J,WkعrŨ sC'
W\ /r]>KS)UƢf$ԶP[ c ->f>Tl#Si#Th(Zn$Hr?kyv#z(K1+_w,[8o qj:W\uʳkͤ;"Syfc`Ȳ+ЊMp # ?QZ I.L'3AB?<0)"4U|> 8TϹ<zbpH"ZzOneX7 4]Ǽkdh8Z2Hλy<]h!M/'9ߒlۍt(HZhcIVl ˪5(eW qQLNS~ǬaM -2MWr<<x% 7= <wbhF8v.J6p,'o+ˁ u0KݯSa zPc&M0Xdө r+KtEE0tۭ(t-\A*4=#\ו,(Vm%J0\MHn@{e `%If-=t=G$=X#,}Z A~iQgA䲣,{*7@xOr40wCZY-YTK;Nx$EmEЫm '\ nhX*S9[9!zQ O;IDj\4;V5#
U_yߥ(e2X^
Ԍ錄t*M6 :huwoԪ6 -0 I<u-2 HmQf54,f4x Er(7I:P7 γ}Ǵixd`j4Zb}L3ͯY8~HQ߉W_MԲJa
7Uc`#G2 ɖiPjQ=QQ`a EIi^鹒XO=j)7~ݾٴFjN2r;e#Vy=n |ceOz 9!F?Ei{q\B^vį[qHRh0u'.y]sNSn4t9N ZVz=V#lN|
"1jP:tZvtK@d3jtp;z V['Ś-cw>P MFUPڌ2
DžDsU-Υ(91< t>dfC皂zv
d2ڱ<%Z
-V1jSv8qBi\Nz Yi3۴N?j4jTZ] -9ft2Jw̒=}dh 4xhcol8wh}mDWFv -^:#*F]vQ<Rtn z;Xʅy\8sqxu㿈l`ZP;^#Rl\-}Nk*h黖WZ|Hn j '[ÙZ %K-tka z d40x@k7Է 4;snBHe7
$7jt Ê9Z/7$U%N, .Uȧ9u8VMbц=<PКG-[BWh2ӯ6(M4m=LN焸exL_,8T}wC,p'<_w7Th x(|.%)Ue#f Ԍ0zq<Zl:r~H w܆"F MQ캛0Z:`n.J 'L/:tR|H]n ~Mc
Pa\-aDy -n6\ufg+7ԪLOUˇUWz{ȱKZV-JK`GM ,˾O7ӱ[Rl0qF?u&
KSrs6^E֠c䮠[vC&$"8DE) Ajn(r0qIQj.$skcnqJ{Ir|jx̩1i"@kO/) r<z q*BZmFi0-8
RYz[OvsVEcw_.ZcdpT~Yϼ0Њ^+ |Km$ǨL<HCE7 }8RɧY?(e],W[jG5˫ZQ @VqalέU]P`>Vh j 4 b!8TaBksJu;qJCN#;P|\rPZY1_EW 2
[ --& RN -j=9.Pu$?d - [1'@ej؍Xdvo˓Hg* r;8I^yKSe"J*4#t|2anq:={ѶږJtGj\y}t6;"Lm1N5}D 'r}+&h,Z`6i1;DŽDZ z8Z.D5l3,f@ -L.DЩ.Sbܰ l:#Zu,0̑g)<Ot݉2,/߭?i9_Y =\w1Ri#rHP -4Fe\<!*+V`+jǫ@rMй& -~@ -.D-|Qbg8{3 Z̓si ;ϸ :r暀Kdx?[DϪ,k]Uq=KrҺޫ n'vb 1Od -8Hc zX7O9 -
Ֆ
Jt -6\ K-C C _8eCH f ݪ'ˮ;&۰-@!7e@|0[KA~ lbyzdU 7@ (:mF GpkX Tlﳢ/[N(^ڷ Pݮ0s%l0\/D 0A#]ewO)x 81b$ InE2_b02`J'P%Oj
g"XVzIZARwL`:&u.G
o|feT*9c|I2 & -R=p..HFAJ 8 Yb*8̺w!7#؞1mxdhV 2]t۶+NP -rQ句dpr2?r;Gt_) 4Nt;Nc n.zhb $:IQj-E @ +Mw
Jv
`pCX\缗i BSr;I 0;ҵHn[NwCi\N?;s##Չ#vkV :F7vd.F3QRT,َ; -EDZeAu+ 3Eس^0JTLDNAADWqJc*]r>N%3)=F L!.ٍM]7?iq~+ϟ~uFzz<TKGaU`=
kX3g傣D -&zAT -hƬƋJDFA˖ \j -@:W)ãv әU[ -ˎs
_{P0]3Q7u:?XeߥZ:]u;)Վt/2ݒie6mv!/Y) -Z8B `-G+Kb/<a[ S-Y/ -0Y b~"8<CUOQЩl:ƁƺeAb - mه)bMd$ɪ izJju@hߢ̞kĘ s:2?ją&'(NaHQr]r * -4Pc)K/ ,@d Cf 4/9Y̮ 0`Xe%L ?=ёٮu{98 $+Sv{' ۈ'>xQKdeбDŽ noyBS)G,7R[ EߴC8ȌYH(>^Mu?I73n]h<VpڒP}H(b߃2Ԯ[vi E KBh!0j.w -)-aV$@t\d
;LdPV\R%Cf ]m\hnZWܦ[5>m~}fֽ~-Z *MƎ| (ӪĞ4QM|gy|gpjΐXeeeGD -mi; -ˉ)崊$|0XP`4btN4fsZOT fi"/l%](?' -F\&JmH(}Dtj,Cr~Eo -eHA 8xS?6gTN]9fau 4
jxt~w2Gxц -/د 5B RXHO 7Dn 1J}F*Q`pkY==뇄)=ܴ 3z~tL, U_dFUxRkXu/PO$ǩй"[Hd7enYMkEdbml+|We!Dp[OF ` -Cf'9Nd|0{Tq.P8dvMAj[As$00/,q چ_/}&1zCz۰lעu uKdzrwl ߳\]s3 UDt/7UwlS bij)B3>x!XQlz@]X6,Kc ~L%nNم&2E/ ГTWi~뼎쾐_o`q&|Rc'GI&#j5@xw%*
Kғƀ -(W%\%jq18]J2!% {6)QJpI{CUfJ$ bm)Ƭ3D*e[N WԞߪߔ.j q{~rjfnۡKJM ^[[j B)3<IL|l{85
8gnT3 Tl~~}=EbA |5OujGk>Ҋ*_zX%y mJD~-
BY⬢kA2㙢B@q>ܟeT -"jZxhu盄sTCdwG5h _R)ry$HV'Tj˷˃̗<A464gQRY`Ybm8 --=
:Ab$U2 [4 '$=OJȘK+]sl牺[oE4FE*lIo EЫͤqVg
NJ'zCaWB܂V > DN 4,. -&Gm) -`#tEvALb $/&GN 76̊q$6<G`;q*fٵU[]u007dپuV7Ҍ_qیZt^t`p:pn:Yd0Yg(Ϯ5hWZo J -auD&R# 2?lz *
鵦R}7hN85pKzy@_ ).Br~e9@_l'L6&r;nmYR@$X Z5**5ZzK4L]-@B&Ȃqno>U[)b;.[&>jd8Ps, ou= p`O$+Y$Gqs|0xTm<(Gr{Oxvc @EN -$O5M H\{NA26%caE+4CĬ5&v/Rr/r;nݍ3GZ+R{YE?c42p!xT"؊uis2PVgqRE d者 -bUX bG - w*jpKPq0Vi:R1I]g" AT
gT=/q>:OT\vO5-W-O_>C -PT הAiE 뀄vXbry i4qiT+ aԺs_ -7P)U b M`RLqЭD(E PӜ (Xr 2dn^3}u: G0|GrSb+b1y]z $u m%gEÕ V|Aڅ0j Xevʴf.k ,Kn@LN^o(O.C( s I1NeuwA*á~~wKCeV 0Ȱܯ8Θ%[^l,Yh?_8 oj
j,! /:Z!xH. C 2=,ʳݡe;`٢! =_oD]Ħ1AOo{1R9 [d[n~3_n{kͦFˑ Fp - [`y)6&]x=ϝ -y\Xv PU 20;[<\m@,/ҜZ#r<f= -4Ldl4<ɁV5INH.cզ kqbVN<OH;4Ia! -Y,߳6H+g\+k@킉-p{*Vݳ '`#<U̪%J/7>^k:yJqV-Gܒ(R3vܶDax8RfLc
6Of\\%- ~AGE`۳rY.4g-k<4Ki/K}i <t}?YCfuA:_$c9于)Z\ί@zV[s n&:![s aN8uq"Бs8r5y0x~= 1 -ˠin\c\Jp~ږ{ 3@DAZ{: >? -#TYCB:d8)Qc'Qu`@",Y<ӱԾrvAYΦ -oS̟@"SC&,fρym:w=85f:b<Oehhtjp$K5v@R̄oJmÍ}/l*
fravJTۥ-9e>bLp1Q T|7-f 5Fq Vb[ -}@(:AЂ@l!G NB8yvj[qb#l'U -#9 K˭==G5?lw=B"rZ^z PEq +JBCU8q,~Xn vPT8Uv?&2 -R~!rLb<Mgjr:T3?E덎pZ"} Nwf|+MV kkQjn4fǼ#I.3oǁ+J1$Zϲ<!]o'|2-DKaH)YIavj>La 3be K_w>4gW\ Fis!AAj=,- Fe$; -} |7a18(#8iR#C@RCXg E-y?Vns V /QC;QNbDyu1 x;2ރ UYp23Zy^k6"̲ 3:Zj<T3'/r^*s1ox9w - jׯ]L}I1Vͧ*{y:Ch
$ M^ς\[2>M^Ë:Fp DC5 sC1~c;Ch59+j2 ?ƈ.&ML<A6 A>V0,Pq TR-AK^h2εYM1c!$Ԗ=_m'Md(MZ &sr (>+xڅ'Yf)繚jX>v;˖2Bq4& T.s#QjT2<-̮\8u q6CSIQFrNAgxNBdv_wHoIV9S4[&9s3wZj} cevG_;g9EѪ&*rܲa:kQrp3Goyxam2Яg[u/ zT1{LTlwaV~)hNvDWäC#Kl-%5UvfJƣ4jb&8A zF=p -rxLAc42( -f
F9#9zt8NXUu}KU(V<vm(H2Amۦ Z{&}W3f}s 3E ۑ-c{foa6abi0q[_UI2| VtϬ /m:b=\br0ZwA/;Ow"[ :Y\hv!K/ Dg漏yZgzW%OGubG mxޣ s(ti=v3N( -nT -][:ޖKvCM5qZz~#W
4n_?Qߵ ~ zz%5^"Bkr_B f݇Sv^*!45@mЁ@UF&*ԮZn.[&C.+~fM0cԎa[gr[i! R0P^l'Vt.4M
(1/ $4g^>Rsb 35VKVQY
a,,藽Jֿ b4y -咆vRF: jRK-f -nSbc rہ4]
XH9 !s0V/M^8+)VgN?u;+3"H#$݅9{i6ѳ]O9 Uwcs -H7W+6Dv,mjKSgi- -quuU
3]>_d9k).ڵEwOVˍSZ#&9Zp 0?e( (W{& -jQ*aӸ uzl:s[{a2ۊqa< ,AKU (r0Ju_hU]6 vs$cmYSZ1bv;`li:/NIfs1ʲ݄(v8 lω9X5ﺔ D[OV=g阈vZƹ<qlۿVevvHBpB}x mMEe q,8}<v;N8vKDvs3M0 qϵ-2^$vӡz8*J5BQu>NgO"ǫC-&+jw!?H>gb44T0|r(I3iq=X.Ax)YlTzO6M[ZdpLUk`P!Ns;7i Ъf'dR[I'B;I~}
ᔼj2\2#'phHy9P]I2ObS1R3a#M.;L
-endstream
endobj
33 0 obj
<</Length 65536>>stream
-- <WU ë,*nd˱F`Fm6jc a p1O[S MC0]mY݈-\ʲ2Or\KQ$`n`#.}nEÝ-JpZ 7 㽊Lr+4$cRu2H2eo"k2~ӵ'텍U -rO^tF*mۍvbjSEsP5EIC]-Q qd|헁y z.4c6bgKmjU~tWDduB@p?P|.I.G|
o~W[232}z *iHrXQ 8Ef=:.,Y?X|7U]8 {NGI5rGk?%ˣ -Z@=m0{|r U~ C/B0;T] -q^b8 ڑK%5U.c ̺=1\0;g{oB`6$j_~#"u #gF)6/zM Bp'J x5י5Id9.ہzCNY&2 ;GsV[QkRjf:X΄/
.)ޗ4|U\3,#BxŶ\-Lu;fNcxt1~s Ⴄn4y5IjOX asLzArZ&#ya]hk E&IaD=2+Vu,|IJ-d-Y*#ImPAc - 6 6Kt#]F*QbːcT0v@~1^De !:4xv oeV=35O5a,Mu< 3Qf VZX4xUY6MtDݳ9̷\G0 g-<t>ְ} hbΈP N|/1|X~ p @у M$rJN=W - )3$wM7 cĦBn`tZ -(̐L{cۃ |x,Û - ̓(52Hhxߡ> ď1 5XApQW9V}p9 h^F:W J -_ns
V -5W/(ډ 9e Ϋ2JnǬ3 ,N;RKDaTu_a%I#6 ִ\ -v+O:Kx (1 {riq}[ .$pDQR* -ݠ{[ bC$q( -U EF - %z2:Yn"^.0Cd0 D. R î6A -Xxn= -
'9g]+ -_U}'ʞqJd=HJK̲ t~*B7Sh X|Hx t.PUi* AN ]opfDXyrȪn @D**J4?R, JL h1QPK2
Ĩ{{Y$tpQ ->D1hN CҘ![qMDAZQfy -7G0J>aq 9Xh2Ny<r<Msf'A~9pTK(K\u x?Z4| ቕ}1Iesm f A5"5nyk 9;ქTklJrGѫBk { 0I\Ѫ[k
7Oc-tt=\x8Sk <Zo&fPa~ LRL4˱ؑJ뉞u+Doweq삻@]K|p3UTEdg8u4 79^n/|_mJ*݂S=MT 5y"F0C$ؒ fq1^5Yv X c֯0SHeӱ - | -<C
T zFT0s'Krjl>*sb3\q\kƳ4o4M1ycQT M!D2Mc==7@0-"e7I?`w Z1kFPs4]H7H 탸u}b -HR %l u M:Z|< -4I -D 4HqA*dEybz®7xO<Ub=/8 0UkTTp 9 5=gHAނGŖ,} nB\qބX~%%H݀$hHx/:
2I)P-E>Vj zDdb)G x! "{Z -3Jf UH_w!XɲKBR{P>a B)Vi&0N6IK6*-eD'?rb롂QoĞ&Iq~ )54[o8M7 q"&k>!3 -7&Zysi]X]n˭8#`+Uc IZ3,_ܺP ^^HG0Ԭg~TوNɮiS6ݳ -c^Vm%!泤$RmF5Yv}e7m"듓ZF>zTh];Ւ Ny=DNڅ5Xi60Qw -Ȟ:FS@iv|څ: .699 mNhK&aT 1Mq6Ru?,3 7s:0$IԒYsTvAcDd6r { -2feaiwVh,{@ .H쓢o5Hp[GQ̭G !:;YjPژ8˥qIчVr 4v'?/0I%W 먌wlLsg1ߌ]ub$mLb9h<v omYGKM3*#Ѽ.$ .Si0xԄS}hFD~cn|D,(l.@BQ!K-g]jUvd2@yg9#eT_5i(+3k %Y -cUY eYK->%w8fיlfUyb]( +'9.)q<n7+P04硦:5DI1jt+s)DFi7v|CM l?wq)ȳHuH:in4Vbp!0;^Z>վsGLa< &{Fp:(3 a t^lM-f{6|' f
eGKꒂY]e[S߮/W1ɋ}IaxLl* e4% ^d&VJnm'_
ݱYk "4?]玨5n*]`F]xC-y -ĪqEVy TI -VOS7*pEg Yܡ+"&,"*1<b+]Q贰(O .Ivu8| S-JŨq^$0Ht j0DSb G[BVGZ8?]+<:$d"Ev2kiYo0| Soӡ3(% C2mJ\PRVSPK BE2.k~6Yrz| jVlE㏈#h6\<EZNWΗSJQՋ", %rp) du!b3G .9M[ -fWKhz[SOC/+)(*%-VQWUk s yꌺk2ٳ(7w&ŧNL֕SK - -C7EǬ finj~:FvY{H:BV@I
GM䇨E[VZjZbmiih-Pi -+ͥIӁ隟s8M}`@ \QgU t R)m5ЇlQ [EjiV!rg1 FtY*m%3S^O @1*<I: ΔX:r"?lM a Ȧ:/qyDbhHjcTEt9P<iaOŇކ &4P@vZ, +u.>.ζHPՇ
tUc-uua })nccvխXbl;)&5zbxh[q*4DLDD> ",`u -( a-6"k&TSe"d5bD003ט|dZ <PZ] XC?-[ex;t!"/pmX2lkP0pp&K\q!U^+{_% IaQ=YQ<OI(
E<'jD -s)s7!B#$6z , -g% -NуVY.QHī+nyVfœB)YD০ŀI
'"P%Q -%Q -A5bE(*agFI +zIB
Ջt b&fCN #nˈPvaݝE_PC̼Rc%4,ᤡQS
v| =m&{RB17",b$ z1oPT#a#gga.qU%XÈќE E$CxaBW}g8~ /~wVԣ 1wO)Qp*|hU&5jE#V.Vf3a ie" %tPʐ(Ey\(CgJ^ 2!$A6A:%) ;Qo{0mE@ +~ 9'#Č8 pH!)k0LeT
Et6knUӧr!? u4ŚRCU[n RyH5UU%8G/ -O~8Y]ʕN+9@.E{nKu&:x)e1V";G9q kTʼnH%)-|FG$"9>K$ +QEbߗsd +a\L\8GQL1K6%Nr)n<1ko%5jN\Rr)4|JϓO)(c)R]Ih q5 r%q E}UkƋ+γ|qV)&f6 =z@. '~.x)i j!5VB232 }(Hxׄ:TE9}|ޢ4Uk DX1\g%#X
~j {t6cfDh#e#z˷>j1z_m|c"j :"5&}^mI_ZM_m5__m5n7 R u11Ra".zvˢR};nx]Fdz)C +"?PE~4>}Ms_3 +f&XdRFTYH
T0 NyiwHEű~M"d0]EIDEY؊{*Ur&R|rԟ*N D&rZ oXR<4ZD8qR 4B yOǹTdUys:^É,Ôcl: %Ӝ!Ȝ٠TLI'I9٠S=aOd{tEc=ѽ="݃mh'ٿƬo"?6|ͤ A_s.I?nl\C t*5FU+y8y_E]dQw-I <.A8G =}(>ANv +ҽ}4W4:'Ȁ{#I>AAtIr!MJNcE-( /_(-\ZΑʗpHJ M\8M9G!4kyc-AxXXd8eԥ#!NexEdxHM$&Ib~PllI06, "g:`g4{vlcncZ4Ğ8 +&C#_څ:l/?d1=X;@ ) 79#f\>Hs~XU5ZLrTΉ\])D{Z^8 Y84'5&ipH3HDUX>K 1 6(Go 4TS<D*tEXdK\VITiU]SRԽFZaMd,r
I,$LW9MgL=@:Ty.h*BU%l;_TA] 4*ajE[愬Cqm"Tv^D;?xU_B. 'vѴT>('lMmURF!KȄ!V
4Uk<ej(7e/b*` S}ka<k<e= +3 / t\UD2 R#PEE{,CVшe=K6DT$SD(!#VfD+r8:g0*H*hLT{PmE +.+ST['ج!&e=UQ gv 9yV,tn:sS,SKfq)PͩVZ[ Z!2k$~NL;ʑ暡T_q +z]ZtufF"G:g<-ϰIR⢪&,!ҷSM72f=hoԤӴl`iyzZЃHdJ&CT=TibuK1=]B1$?oEEOvtjUstW$~4 jBE#&BDƉ8ϠDYY<̚ &*x ]$zL1Ҵg<,E#UK(Rxʊ!9h |h(2lJQtч(+hCG$8r&v; 6jitVYvb\"Ub&1x +\ѩP-y)2'6JQM1RNg\*C.wyQ3SD%)Or6S)CcflnjN"ed- {3'4AE~MHSq6] r9*bӢA21MYs(LSs(ki*/f˦ +%{ʛ (Jc[LU˿KI0ZXR
19J`P .t &RJUVɲ0E\d `,nQeUuIW7uDt$qJx;~>XcP<M0ֵ@M +(v`<E-|}c;YC +Nʯ QUX;OƢ>-.חf<! gwb 1?ocQlU&cMdBo.$cGU xnlX *8rI1bNnc,6+f"0enYVe-V@d.+./VG+x
X(>TزTs8 !OY5&
!z4CWu7`ily/Vi,Ʋу(lDQJj5Vaxa_c5GxrPHkEl,nrOHHs(q6@GRIX!=1/ &!ƒc_c6+VUNXۉqm.:Jg℃.a?nF6H6V>n-n qCƲ^AƲLj +H?Abr*!n,ɇRU"zQhKscex:,}e3,܍u# WaOG{K=ZQ@S)nŒBҭ, 1hn0X\XqgT$M ~93lccڪ ؍u &oÍjHMdmQucy0(>`5lX9sPrscSԴ X
8eM4&$5)BԜHk;e}&Gy+R^c=Mu si`LޱM+Zc5/ YYbcaȁO>BX1Z\iR!n ^ԗ0( +Ɗ_ sՍ ^z_1QTJp~C$=߲JmWP vmN_幢 "@lԤY=:(
e!ѰMI2HpʬZ8
SKIƺ,h"2+r5fXG3Y7 $Ĩ tS˨61˛n kj-nF<Σvc */KVՍu{3Yqt;+G7!l9MX;Pیm7V&ݑ/D 8=nccPbn, Kv.0ÍU>6VecͼhhX5 +ʫz l +S&S,keݬkylXϢGt1@XbB2 Ǜ~DC*ZˌJ^DE|9 .6 {D kX[(q4Agi +kX!/:bj,z2HupќƂk Ɨ[X,YT.J~⊑k'k-wíeXkզ&
)z_[TsX' +O֕N,o X:;I@cٺ͒qĤN+2n +(5xJƲƆ벳 9;rh ,?%fϸ2;O9L,7XgQcA5:*k(C"N3" 4e7V1FL鉓k /݂,SXC\7V%w17*_6nqOE(|Y3(:X&B]X"[nǷ qg(=L!|q`c!o%0ȓ}p%ԉk)tЦ+ w,c" ՠ5mA$AC?
^%pazßÚ]cqɶ*/l^cRĽ0xTDBX8Tlk]vAP| wRDK Zca?L 囊EXNQA]ߤ RKE<<;`c+X)ztQ\'~ ',K7-q4JƂ3\A+Yߧ՟ ӆ&,m! +ìJoPb +X!B?Xc-0u'1`;`Xi]߰2肮<HXHW MI =Dl ,lSˆ&89B$ZkJpÛ\8`йL
Dz-6~X3kJW^xOSX7)S58\dbiy0[x
n|m/B<پ)V:5!R`Q+J__K8q]+KVDzy Ӌj c?B%mLv!>9&o,Nc̦Xtn,m jcU# +UPcULۘBoUq#e{17NUwSWY,
$cgbSD),QY5/3[)rBamS 2(< @8YFD24:"; + +! nȿ&GFe}^uEK-"(FSD/81:74~ceϫ7¢J\
~5dX#]s%(Ig,U \@5" gL +8lCJ\~8&~iC +mm\>d5x-s +!:n KkiOPqHӶNQ:xB_F̦8C9VtD3kse(A01EU<c#͛oU
^ Iųіxdj?ϸeZ32Y_*/l@J^)lQV7vS^q2ٽl G:m;N4 *y0IzO/*k
WÎ!vj=3RM +dThFF#E;)h/֓N\D9{&Ta=9OO_4WHr^b2Z | +6>5[+1Z@00$b}>#JYn_Ε
<;ɧlK- 6&bKRo@P:L5 13eXpOi*۠$k UrvaSDlDdXz՝`*4ad@O ,O^Cp- 0 jOpcfܑ< >u:рRc ;{niyo`7.7
aѹeCnψ(\1 mt4"T5މdČZdN4iw,tɿ/Eء$vM:,(OzOz+aUg _'0OF}
lKT~7pUw n ϗi +*YO'{w2+ajFxtN+β}z{y)@bJf6ސM} ܍zZ1&аf.Ɗen,3OMm&Vz`mW_M3'ǵU'}LNc +/նC3:C?ѱk eDNWmTWj MEM,QI.)H7Z͞!WzQEQHy7mbur S}6ma#I?7卶(ڄWϵM_`fctn\æ"w04l +|8Q5yB& /Qx$g JR kcW-#CY EB A&͋5[S .^һQw(̍{hvE>ṟ39+>$YlM HtK&e2xKRҡ0J:ŵ#)gQM"]owsg1AcZyK +\ ABhlJ +E.daGO4=P=|t.% *(\=D * t)huRc:(d
c7M{ +cSGK-p;%텲 A~Sr?ߞ](}_kȦ/, 2 +DZ쎈"g,PX
%ć栩)& +G"R[䩱tiGc!$О{B7UG u>#Kb>9SHPNPvdz^j鎢XC 0+*?2OǤsN/@ _DdKZ& +4rlEJ6=/c~5+풕(c/!(PXtb ȸ@})YrRAV>ًM<()MbbϲS dص\`_q|fQ*x YG˫ A?1̕kEFw^NOf4{ۆ;0q9aEi'<
Q#- ~tڐǙ(Sʩ 2JHC82.ƢB _9`-RMy`ۻ\NU(D; ÐᚋOe_( g}MoRn'4O4(ݔo3ݲ
d*iyK\97ͼ(}mZqt_8XJ!p"my&_ /490FL"]H`YTEZ6 2
gk҉w:B. /I+8M5"l>|FYvIѽ\$-ZS2)R&
;)&edQ3w&78I +d +jRqő{VPY)p g DfN;X״:<3轲зH
?O pJFq(8ԸQȰYC<,le8 iY xF>)喘Ջ,GU"c∬0fh^_ K(-M2K|S9X֝D4-!Ò} ڶT of#>A4Ol;&ʛZ3]J ʁZ U7?@U
LlEDy/D%{%[NF3J 9ڐ:/,"P"$mQ6@}>^i<>6@h'TorI>{TD1`mt %$w 7 {&ȷ:Fe;rP)Y8 +siNBہ $8 3dx?q !S2ȯ$ +aymPP&L%W 9IӧZS!dD|3:!/ j%389Has]a9!fN(3ͮ4:ʄҰE* #$@615<\q@˛N3?sYeK3g5T.VD)x9&$JTQXR
y6h$tN~sp>.Pwf<IE73 Lk="[_-ywV׀gdա~N4{pm0Vm gJ}B i7ol c>$ꎸ㵤mVv|Rdԋp62`t0G j}QGvߙZP- Eqd%{
lBe,XV7'W<gb$\%& ~#|k!:vr LAqG攳 V %d jZD51&=ގS=: | +qD?* +CJxtwH뵯)2ovsW@3A#X87kfl8d`jkb +]߶Qt?*ey9a +!#9KHBUg~7υ8n9}EvaR_
!mF'D$r*jn6 + &ΑiD_z[O쒨lo? +7'? zij? E9i4#Hдm- +FcJhk*AKE_d@70-ѭOVsD i)o9D
s
+b17PЗǧ&15yJMY)b<:#S9JO[O8)<E}d t;wo~x'!^DFcXƘ43~ @2T tZ20"+ٟ + d& X,@c|2v 5tZiH廬ڡokM-|ó61|Z&u%D9PzƵaf8v 9<l3~IQ߆PX mmDeM1jxxuB*7Y[Md"'?)#-l0|w-腧-w/`Q#>ݸ;4~x)lys - cР,$ՔldH
Fj^KUoB̉i,r B4 +AoV4-\Vdn$+"WÚ 1 +W +dHɀ\Uzd o.j ǠB)K#4܁\gP,ф+ !/8D3/z꒔'&R[D #R{o^,N +/40s~l]*D[bHFD`9}H W@;dsbqGQQ)F*9^:Za֔^$~++ˤygz@ D[{ L x\AT~*9_33z +F^&DAB0g +pϷHKuT(/ yv<CK|bN + +?̝,[{ FJ p! |oxx*TGY +cz,8YZ0DC"#Aqe݅&,mJQ7EX 8Z J:{()HY.(23>>@}í `8HY/ll!)xZy4c(\o(4fb%h!;\3 +o^/v|cx@3Fһ6{0<~-nQoiQ}sbbPҚP-gi1i2'?/c]Sre~[Z+uq7w=3=Q ?Q?P4dWfNm@Y vb&[0
-H +*Ѻf,C%)D : D)WwDX.⚲ 0Acfn&X%ڌy, <
e;ZOPӊG 9z fElPIZna:SR(VɄWJNl6x +#v 8<Ϭ&"N.GnZ{ bNM蘌 vUdGғgufn +aA?y *^u79CH&6$_#y S3NXf3Q5=hzx8@{X~v1*ԺF՟mur >Pq`bWf3pu E#Yi>^6di|1/JV"ji]YvS +:.o#쮜X j _rH|(@T+ UqvY>/z I~ +Cɗ bafP_IAŜO`CMG`*1/y7d Fkw> ]9RLg7Z8$ @?[ż;{> y 07zQ G&Q~u
Vm Ɠߒr BV/3!AVr
878e_3]0ZfpvSXsB^ +7OO6Ph>FS+FZ-C8AvahFs"^ssh:D{ ѝJw<W"BECs@z bZȐCr#^\jS)2
qVzkOD$,U )Z|3rp>Ko; 2PGf,P.sTHк 5Hwe pR,0kqI>t}Ҁ>z +trQXsM pHyK A%p#1&ScwL.H #9+ݝ=D)FXo2JU{)Op̮8y ,lk +pt@Р#LX UMQB,;F[M:ɏ ߭aӐXh@L=]o]8!Z.OxHyԭl\[&w#AIX5'Wy⢍=+>MMt #{9DuLm p + +p kDQ2V" $j(}7վ<~D=vl,bp_F6߇0qK<po7ut!\g"]!bV9OHBh\l%- =ݔ-ŃCx& ;tr<F d F9k$ 3`OqGBp +i/'{˩waܲ@uwt-F|㲋'wt !mfSҏf98:5-O:\C f6V]!u 2 +md!>+ETI ^>L@"fGc;:+jTfkgV^j<'tzϫCtu*.4 ,WY*.EۛB|H'~W0ղ]R#U@XYrJdRC,;0<t9HNf yjusmVC7ͺg@-,`CS7+Ay;ڬ!1y 4 +$ ++2hŜ\,2 jIh C٤y&blA3;j{Lwfqu
=39g.г/_v +y +9}T 1 C9הѩhU B_x5h-VWFCR%9Sa< )J
A|Q[ chf3usdn3#4242aTUT"H HTvV?. F +*HӰrx̰|6 DXssasݰY2WY dsXe<4v֗|Uf=;3,9@ ~"RӠ\]&N + her #TI +Em4;pP NA*IOܐ T'H&j:\JvO Ψki@_slD0t$c&?
C!Q +CKH
v j^(\=ن $Bixq +m{-QMgbb +:̅.&
63hpv5v[pV+MJfZL[S5Go:ؐچ Ppt
c,E{
: h IxFW 트:-yOIj{ȚA it3]Z@sP]:_ڰ + + D8BvoȆGNm +AMV$B 20>JO`<A? 0؟} +hJ;LFβNR`**NG +}jR1<5(Oyr:ziMǔuP@â騱 +}\t34r3Y toD$#ٸeOL orn n:l /1XMgu;FshdyW89]Z"-eCWP(iF
?0\. c0GC
TCB21K+ g#b>@5 +nc\ t` ؈z AULv́)=l:- 08S ө N yh=HqWjH` +fNKW_K']Ѕ +΅XaDla?kOyا-ݧ) ~~G-]FNT/r + u̜ +F!BZ m +kDpމYrJ@ZB?a9Fny2G܃k +\ <P + +~p5nW1){ܩy %Qnra(U" Fന( \P<:(izpS +p3GcT/Zc[]GX)E |=3{DR}q^\B$?}|H _
+fXZ*(#4R٫~n<EvHnf]ਝnQNJ:ARF 1skKбr ei#X5d@hCv+2 l%)+@sJ"2Jc&0@mPqþ<٧gtݲԚY +)+*ѭ_&oJO4!c/0 3+LjFw O>by1D=MYDf3/$qd[Vҕ =/HPQ]9ЪG`HP`Mt-␑#feNlfd.+$z.V[ G *4USKry5ɬd&.OZUƚ"C2c~ PD/++bv?{zj%Ɍߥqi4Rx 9 s N
iw|pd,S%bgZ2va"Ҍ&^]⥋#gi2ؿ* +;.Zm.@ +W4|&f_<6H {Nf >ڏ<BXx +@`k0n'q cSl*gV'O CL|E
+}( B_r}( +}cyX>B?7w5l9<H0x& RS {K&0@ ,6-+[-Y#lrNӠ&5ĖgI&e-KEԍ`z4M~>mx0MM R 6æ]p;8zX [)4JJ> +Z-*Z";@'ም uTS'Dے,=ϋE\"%[t+#.[h]ĈHZVE| +̠26aj*G H
{*tmIYieB{ ѕ&/
2QNRf1tҺȫ=.i#19t. =.=f1%⡠t
+OɁҽ`DuƥSFy gx ]/Kn ү +S Ii<V(߄װX<cvGd7H^-yÿE @Q([xnǐU A"cר +t" +#Wˢ_@)纲;rz;6}7gr5ELSBo5P#0X3*d`b>t˷ܽh˴$na+>eP sAQMٯ,A(wx3s LnwN2zWʹm j6<'tyG3qo?e;SK{ڳ>vsK"- mc ڢ?~vvf×Ue6[겙6=dCۛٱ2s˙b@F DIA$ 9H q$1 #q)"/j@"Bƾ@- +]" +endstream
endobj
64 0 obj
<</Length 65536>>stream
+*4'۞Kt+Yq;}#QuQE: J$Y<־nZ ̨ENX1Š\b;hZo\2:kW~*DeIUq/DK Cu .)y0 E?<7 +M
/]L hRb`3YqW鱪 +{BeDt3Bos$A]ē|[ s\& 2(4Y +yo{kǹ 0r
+ԡ3) s$1Jax\ɫKHΛ|܆?\! Gbx);vWZen~x\wVV%AsF$o5s:a<~0 ɹIp6 x-yg< w\<O_Uߏv #'ui'3GŽd6(:& ,٩j!=Z7
l Bzr[/jn +M)d0V/R+NiD %}g0%ꡀ!ό2pYc>^ GpŶ(*=SQʨl2; RY ),$t2!WsRn)'@TiT< N{lFFM<.zb aP#` +]g+ 9`(ڸ #,ه:gU""8) wTx y<d`zo각xx+2}6<^rˉvԽĂF 0R7` jO9},4ųȃ[m*ȲQޝz9tGF=z6;d9&~=K # +"Q6W- 9>}ȁK1-|74 f#qĝd6` ;<n})<$Աo+|7j9x}r}ӍiLUy$]*;*솛5Z վ}a,֬RkOV08F؉ _77\Q8l;ḅvMS .֖琇a閬B$>8 O3rA}T4hBwޓR |$?Z L@x Fq 4c,>˵(M7䔦ChJ*Y +)
&-:Y)YT{o(E >z$F)1*p:<Rs(vu m2g.N1ًC+ڢb[Q'ɴ0=Mb^b66N1gphdC̝X1VD&sN_Ѳnۂ-y\/zU#ɔCe70%@R h9俛& iA% =K1'e a@kl j Ofݑ9X?«< Wp m[pӫНTHqͲ9~mmh![Qez~k.kj\+JK?}/4<$ں5eGc.X%\8eR'D:[0qlp5fФB#0\諟_-x0X<}ߒ`Ëu^CD+ 6{gL/"|p;:/Rpu_s\4מ1C[.V#Q<>}f1 6>Nf` - SsiY 1d!pA`2#?xEֺM/ NӢa *hRtW'm +_ Haȁ$qBS;f28}EV!`-X+A-d6˨HrbOW3e^dO e:3q\ '><CI1@1<˫ zKG[a6Ƌ̭b!'pmN" ICv"h;p+(-Ma`@])"QoH
Ih/.+ž<nߑIeKgg +Ny%+YeSBDWZ6Z)_?̲;j$ ' <zN¥9] }@%@m_!D')<[h>S% 2_R[ )N!,&"Megs4@d11(Dh~-+Jk<)Oj F%{_X0Ǎ]'j:GYY m3]66<۰?n̓5=\wJ$)LC o"? +Ze Goضq :-/6Z?3c$#,"SUZ GnEEg1<$qKTsTB^flٗId}raջ=ަ2b|;KL)ŷ!W1: '6[ӟ> 3ue`'x" +Gk1 ?ݴ'9pRg ^qŷw22sP$Id +?']y k:-x#g8"x
2 dE[~0 espva}L\. kUZ +0(,ٛ+݁4@s_+ŘQ'`!gO^IREږ$\I!P\ R3=c{?0$"|N8S7\l7Co:5+ۿLrH `M#t>ir i;y4<5L~J5 |:V=P5<BBeP:hVe U&<>qJ_ 4!3<uZu^cVVi4zr f!W ɦqfZ?B'3@5Hic,O .Qqcթv$]f/5 . #08+~3!C.
B6$N9p4)p)'di +9 +
XP n0N݅G욊szˉu_
Fdo] CN2hSjA}V?^ 1⤔.SO~ + p+Ep(Yc4z8 l!nrIJMt)y* E.r$AB inrK |!O)=؊ +9츰0|q22%&sJNBcLf5#xzn?f&!A&L9lQh8%
[z롺-AAݨZm1UB7R;Z{L "<l&3A0~bN`Q;'J!K7#T! ѹ6qUXc;#Hm
I'~2c' y4{gpF ?/wGVPN1WG0
ŅWL]}
+!<Oj#P!lգHq%SUUK#
5){UH=u\XĵšY<!MHM䢈a4HRe` "SD"H\uiOx=qSo R* +?Bbr[RjSBkBLV +P$s0C. 7P&["
˄VPhwi&d7ALJeTxp8Ff2,W埈ԉi3USy* +$$"8h4{L+єu] _xOE +E#+I+Ax塺*9B|$C& +g(
"P|Ef4#pGpD?!+ FQQ!De &>+;B kq! +$
> mgTҌM8K4n5 ~!llRt}&N\&Z s/NDU"#}Q&EaDq_#Q$!MjmD)F$0ь-mxƥbVP`JQ4 +3npt~>1Yΐ,g0"p`W ;H(* 9+tbEjW?!d0G)DKC| + LP=T0 sښbr<i&>-t \"ejjN/ +i )")w()PQX Bpx#G`"x
G%]F . +諗+b +>b7k!Qv_ZE&a)Υä :c5ׁ~Fa!m4?We'43cY$q"я Q&(4աξXp0QI"DZ-"WĪI.p d<B8v}s5f jwpy5E?j_lqr)Tk@.J $hf| FbF Jh_1CB7:BCg.O!l5By3;1X"az3K!e2]F 9\! . XL +b1ku0
6SF@(s@p@@dp>L(YGa!F $d#{|DxS[ !L<Eq0A M]1DzߜH!.B;E4wT؉ s ɿN37q6Bv#p"^am1^["'B=#ٴ?N"DEQQ"j6~]dIRL +QmPTSU(Rj@ 8~r 0sqZZS^6BrG=+9qbl5 ONH&]z1&z. [6Ep + Q!d>N$OkCNәT)Ρ)مre|8wF3 ShYYG2?ps:sL߈uyh),4¬*V?-l"@*ݺUfF,p
lMD3p7|ǓV&D! 7ω|E<$aBM1g +B̺3rCZM!;afY@ 1Gg!o zhJL\a@O !ϬRF+rG +,L\.e"S2
|t[Ү(jc aX"NTG蠠A!TM4A2'j եW<X7AJicw##iFa`h{̂,+TQx +E)hٓ t1 .qCCC2B藭K^lQc\ySHXQA 0ETpLQ? $H(>cy BC8ေ,Фp>|ݳNÙ 8$B-DFrG|khk%;Π ;TFP^H(* [ȎH#/ a~@u*})j,2= fj&q (JD$O8,Ep lp;BeZ/19'( "b *Vrg0_a$~><Λ!Ixb-8/O^p頵n!z&[89VՆ% wpS6VjffO[ +maJ"
S:H7D$.`+L?JH-uxD-㡕"o
f \Ѻ>HD&q>6T#V0ya d< {茰 a2::wHD r5Ȕ5aU 1f.` 8 +CtBtYL M8tR<s
۽҉y]TQB KSuB%Ou¢*0
6"E9 " +^eq4R
:(DRy`(cXt1*ar ⫐CK!t +¹2S 6Od~.d[ O^:U^0#g2^HCx777·[*g0 7HxF7fq|Ø]Yb4SK A)^NڃCTHKqS- y7HOaWMhC 6J&f!J),,p?:6b͟k.,UL L hI nłxD@It2j}tA63Hb? +cx&ḇV
3 8 'ð Bld#J"Jx˧ua0c IK1 b&FY1#yXmD>M&ppBa$FfϹ B6nCPvŌ5 oeT&+w ڰ*rAC@:u)P5+ @0uK;1{"J
@)ml1/lmA a0]e>X ;1e!.4bh"tChþIb G)"1 cA& b(Ȃ풸bA#HF 582}B5Ɛl8A61+lSD6 wXwy C3K|0#@7Fv63"['. f8&8b@3VfTB +MJ"9eݰ4b/aߩAe dS` g8V,!a颏ib/zylh
m-n=a mt/AJxaBY5Kn[蜜 Lt$|D5pg#u 9uL0zsaRNtl֕ŪM &jW"BA &&b,&4^^gei a20uk(F";1r͍DN.M1&))dhMZ 2DCLXP0 w62R̈́4 r",u_m4 #Rz+gmBHj"a"
"hpD7gDC(qx΅ぜ ))MA6Ø Ø} +zaAP1D/KZџ)";[^ E
Q [S|LSƴ֜H] <q0CtWd5 +EzG6X 'TnefEpWA"5WF!"C,@`AuE(`B0
A AK$! TFC@)c<߬) YZ;)7!T/i+l>;"{CypF(ȳ߇AC6p-UW|ˣ_'TBW6#'Ti!c +!"R6P"f!HR5?"H~")>(<PR$/+Q%S ؗ*xu!\V "@'gePP?!Џ~~j98nZ^FbOfCsu nJi,o7 BCp B!xBC@|!ӧ7wݥOvwK!80J [BB<Up^~A.pA@W 0G ȅ B
B BxK
sB T 5 e h=v}p?@~;+??@7*>ҴBTs>C. + +/pxlZ@h)[[=WƈW +r +0dq +HJ$$D45GGqnHތCA;s0)P@z(PpA +T7 \P 10=x AY6P][WX +(،KͱKZԈ,/PP8Moqn>wa y.]LsbԉwժN(
\2,;Ů<{t t
a
VM0$heg8EvH5NyDS @ !5`CYE4% ^^dC%]!%FӸ zG}8 J ; +:JZ{F0Zi } ov ,[38J-h
R=q9ILRRFeèki32JEBTd;;&'ULR#A|55ʐBG@܆N^̡ I9L1+"x
>V@
PE[D. 9PHTQP72hFO
wS<Xr)ǭVVB %EuWu $, $W4=A +ý$N0 G[TaS* \B~>> :В$|`iiʝZU@'YNaPO4t @Jy?|3!<s\JB*C sDD$
@4>\#}}w=@ 9o[5"n: )uCFi:"^y9Ft +P78H nk7@G>
Z݃7P6-ٝ;5 ?g
$zxWـB1 +c mI( ~k?~c)j*`g)*jsӉ)Sd< m
?X}nimƕ/X<Y\kC_q?D%gQNe+U@r2u=T<qMJX| 0x.Up ,L0;u `.v,:&?@Q_l ELz[ 3hH +#o +F H$ ة6Z)JDtW$i @?$Ī'݉_ҕ +JvF%^m.?="NWuZJAUcY]J s"B?Tg2
k@* l1$"@
[ YE#6;3@J_y`;R#[lrݯhdnY) +T +4zecbn ; :%lგ,K +lC@יT0evk䄏Fb0ܳDWakmP&Få_ #'(y,Y!" {Ijf[SWN^%D3dkй +\n~Q0ѻ $; !(X`A3 z<ڒu0`8," H)@FeYU)ζ<,ޗ ~x=̴}:,s<rna"_drD hl;KJ,Nx܅wժx^9|%w %Ր: `Bo?J^u;X4l93~/5\yhSDu}4Kp&)tɸnAu<7MXv ]6H,0)K$7P,09uAjsN
Wuv;+0錴j]L˭dN {QD+ Tj +?(v{(nD(-P, + J>4 +H0 +A-$ +n +8A2r K=
XKM\8qJ#K%K +`GyB qzw_4 +^_,R[o0hܜq +" + YH+%p0qYԱ5:Dy0Gخi)$cԔҗ&EE ++ eBx?E`g,v:j\T +P +b9EvnUtoa[JMU rfńOX`.U +D*@ zA|x +UJK0~D!C;^ +<B2ۯh_ +s1୭K%Fn +eVWȾ +:Ǚ[\ R]0[)'Q]E5Ax@) Ŵ)SW`Y6 +Faˈ9c:K[?xIsG! # oNNt&^Ti&>gm +Rs7 ӂ(``^M>AcS6e +?&<%Ki``ň[p? ?j&umIMGm/ IJ9m,hr Lg$ RK?FnJRe,'PnTs)ȟKQHj +.o*jS8or^q0h OnOOoLx6, IA +p}rc 0a~ !H_Oy ̠(.Bp. }y34`H +=0?pęw]@X٢ Y"H:4;˭ +tP n9 E/r,ȸ;%C:ܷUFq(-C`=ʏ:ۮ7vha3b: 6ᷟ] dZ0hc= @d~ +\oVI 7W|_Neʀ1(מ['Y&/ySY&0{FؗcDV?`9 CaοEz)ҠդC7M х%+v~c=^:ڝ6< 2[Gvst3PBI}#.ǖuُBt'@ZGC [+ zAD+f!s%vBk- E7OA k qu=pN7S8::Ϡ^q۠{} ;]ίP_!+9ʘEQƁm*S#is92C[ 7dAdyd ^- x+̝ <mVR w\Ұ!;ۣxW~Vs;2GiE ra#v~F(vE \˻cnh7H@:ApU;Lko.$S xzhĔ;ZgGKT' w~ +K +[MV?wȟצkadecߐ ;JG6 & +dtkdpB07+'J<g D~iK@_}o#\nۍ8D l@* ~Һߥj(2~Hf5!r읎M?~?Jzh<t=6AF 4.~e+xwE:U[bkaC~J!^vs#l 4^hÕrQ} Ga9$ue\
2kJ)Za +ߜ[ ܐ.z©_PK M堨ήQC?w^A~Y#`-e42I") y/}hїT"/%ȁ* 15 2K!}x!mhkhr'*0ED* }塟eyqsNq[
~D O}x~ (?[[*6 + f <nw~Dp!'-7>|C)kc"a1.- }O:%]| + |H
s$|oYZn,d; lc@ B&Z'xFJ)n ^
t(%
4-D[Z|
"@4/kk<v(f{7ب"J2 K-DRCQ-7,|E pG?|V"ԽL50}Z"c)o%dQz{~D) =ex$#{ɚ=h:ebD=ܽfifj[1XWF
`,cG>L +֫ +dQKh") +i=+ȞNn+o2ǧAv^pڀ.s`LdqIR@6 40|:Xl$]pEPdCoj(" ? dnV0oDEpU~K| :~he#p
Z$X_)]K6C'@DԵ'!z*F'ݕdt9E4EAԯ {;̘t7
1m3;i: %POغe1~STj@z%F]ݫgLߑ8՛ߡB-@[0-sDOb1f'&w]lWNzo}H +k~u!1-asK.4 rƔ{>qQ '
rf#a -K/ +mD^s.YՇ}r-"o,y>ŤzZFW_iv
:Tf 쫯$RcQ䫗 + "VY|sV=xYA9 bUF Ltrd͋M9贯$NG1g I<[(Nkp$ ':+awNO*3Y|^R=jzЩ]՝ptʇŌ7x@[qMBnqs@zצšqc*_09X[{!{s xf$PwxbgD
YY sXi
-W6ڶ Jmuj@9q*;o6s1 +R=r~2G( K jXT
U:1|w*<jƐSZ-WffUҎ'Ri +9ZO5q֜Bb磭C' +6csJ{6 !PTH>eM6f(Q ݝ fUf=&8O: sv/$y(?TZ5gX<!k Vfԍ~sn_Y-tlNT*5:{*HeHTАor{:x{9'NzIm%Y;S]q8t 9G y~9 d:c)$E7{ȜM,z %8[鸂ƪ&HuK`AJʨ$GJN9s +s*> +][9]既_{G0K +B_9L6ōs +scJncn#O8mN:xXbV=tZ7>D "4)Nsgl=7'>ʥB +f#B=FI{>t\oKSAK8^[xq* :eR.菏G|2=H 1 d|4^ϑqS>Jp gAI>)_2O8"I1?+Þj p
f?v O03?z+a:.=O +l/,
D2MjꇕhJEGUi
D^=ބ)vL) +JP}"h3^-H8Z$LY 8]lu|ժDI3 2 j +G~&3j"KQ[@(pMMtQaO*Sr2YhuVЏɻ|`! kmL X/1cA+tྍ7 ۠$D uȻ- կK+*E J) + ԽP
ۖ#(&(E\rá=dcBќ?=- xHie-b -`a90 +}yBxl7%JASXSCl , q8@eF>9{Wە@ z^ J ,BTX +]>ew + P-T աI^qH_`X=42 @{i \)jU@. +D +~eLO@ihoUh? I#'p_ # v3l *!~AؐT@~m]a1 -t?:==.8 + ۚmC .'UO tDL=Gī\ ւ0Q7v +E/m"݁}Y(nƅ;2Y܁ܓHA +)fMs~L PI@vQ%"cϑ'!cZ\fƾ+Iw Uc LʾVS܁r<x
FC 9 ,K@ > Djrr_2gILY xc#dL]"g?=% B"JlDw+urz ^* 0 PL]S1 ;_m?PG< nߛ'CYA")䷯c- D05ADVX́QώlTrmip4D gK{3ڎ +iTJ{.v'L"UR"& +*J7J &VVnN0; +qret@T{#M5{c!뙍}4]_a \kR\6E, + +{űN>lC$ +Ee#
V)ӃQ ~j(/ Rv
6hZ + sfTO`ܹRi$1gzl6RlfoEI + + a5U ݇((*WJUQ9 +U%LQ lQ*!Pd^wJ;Z_ Ze&4!p6![Z.
1
O &(%ĔQ =0&ƒc ]^ $*yMbr7xY Pk&UL 9ٞL2oG"J );l&(eWW8^
$0 L֒D
Z63
3,DݙB[u9.ugsOm)Cd(ELvLD Ayv(߂ncH L6 ЂWBBgm9~X -&(C?PL/{$RØ $=iLdur-a[uI jFWX=2إ J#s +!) E +WL4K1 -2A 1 ɚRm5AN 1e>TP&X&s TyB$~WWUС(i2㧄|iAKF (M^Ù96TtW m.vߟI +aU ɖX|oh~D2$= l~AtۘEHIhZ_tнUi-<bU IU$OR4k5XImB
SO Y' __ +&i\fM4C=" ~2OGF
hiPvbxiNOu7=Ij K7c?VKEH˕k='x&l":M~l2*GS_G(C@jS{-P4nj\[8DMhMJ +1Or/C/Q?SMHAˆ@*0rV=2JC8{= ;KU)0C:m44aNm;:wM +qMX6 95CF5Þ`U +`/j˒ ?K +Z" <ݒpIk./~*-]A 5_.)CҸ|i0}xF +suM>6] P̑䋧K~N9yek1 N(0R]$XS#%$ˊ%$IKs Thņ;RʌG%HēP'}Fa,טK+!)Rћc.}? +;_cgK;_)^LcK-rC%T9\N[՜^M)R:zȬ@dpլlQR>j#%-}<2_d[2O|j!UjK:{%*-mKKhͷ4<qOWdԉK' +"Ra/VG:hei@r=ѹ +u5Ng QC_4%
6m7K_wR`ۏ . c؏z#p7Xk
*9@f8$r,}Z +lJ%$tÑH&N=f8B~T|P-e*;YtW[Zp -=mbeS-jHͶ)4\T:s0z!0 ݒhw. F)]-kͭ$vI$2l @z^ b Y/0t~&LR3;4 +{Z&pqQuK<.[Pk9g,FIdxHw$GH[h!c;8i8pi +8\n7?u(%lm)-E^tb +]kX]bKhlam2]VdAk"VwM5Of3M6]ms(զkA0@SCV#t*; +)Z{h7WG 9 ~pZ`j~p'7&?2YNi)˩mDE*R$( PnjD)PCpB} +tSW1A@>';R7_i@T_/_AW!9gk0N +VO {F c(]NC ScLRD\NGuV 5
< ])vV<Lw<T6Ω"S(I)ijvz TnSPY?T!-+'(nZAE ړ=퐹oOةFDOD4k!~" +Ap\&Jp AGÚ"tMmP8yҠ6Txjf:f'}^)B}%U꛰H4v@}8[gVPx;N1R 3 +n*"ć7(U*{x;{NhɫTDU.KK,(ss")A;^e #:X. , 5^鄲W,8zfV^#($ uI uˑbuRgTP28De2z6M2q#yC E;b ? LU +W/zSew#tRO* !<\AJ00) WP='U#ORL]U7*\ !Xΐ~i2-J( +uݻd
UX j7CLaDVf#̵I+1P>I$w Qhlqe/+ъc:TSeSi ء$UL%ihՒ"IUa@HHP;po@膅j VUD\?*G_X,Tr/z_AbH{!} (^o @+S(|hB/ +J4iwO;+z|;ɲfm" +❕HBq]]k;Cu4CQ+0 yN >2F%qVVI8-*𑋖ToU*&rmm5W*Xw<7H*h2&"Vѧ(˫&si I¤*XP ^le WJ*lX 8LU5#ɿ' W]7\; +ʽ+}U-]͖NUgL(vP. ޯdU +',,Bz}UDxk,NUyzb ~Zd_E d1U_9<wbkOIB!u*7a1q'478Q b CtJ,M%b5q14+LJgDPKpji谮E ̬G_Ay p3Mܱ +^HGᓠ +#ٸcGZ7)Ns?BXo>^n!DH.KXfT tڔoکRX鰑F~=>+2<4csm#q +)'?lL2Q#( k?_VQXɾQX4$s$wcq7bӝ8H~8Ie?*@cQĶǚVX"ZxHB`e 6% +`MN`:1n1C͇ {S 0/ gz
K^>V:쮁_oXCUK YU_h> +[c!IFk^`/XAX;SXӊehsy]KV#ʬ%+c1jHmh8-X iCX`K9#b5܀
*i5Eҿ ;:'=:
J1? >s~ +h ; öcVq-=V]y4+ ԼG 52E(?O,y6 hU@|!1cj4p Z[g.=) o2YB u] +51 ?c&C\ˇY6 +A /~\?yApvc`7VZ4#$'dJe#.֘vn`V[%- + DK!+VbW[ƪX_ ,uU\}i⢊z;+ƚ0~#.|%$፵2o%,WS7W/PFRZYZɼVXlm%^{r|"JG}1~Z*\b)(Vn XS2o,KhŦVAabMvSt~+yN[}|59< A# H7qbn`]S[$~2CƊ1LcVxo,D{Dq%/^Xabozձx,c?yu±rEW8{Z:Ap,bOv]i5c& 6 k"I63tJ&' +'cW;c
>5%!9ԌGaHJ'輤cY :_͜nLc ;M΄` .|bu1Dxq2I7 \;|4^Ɨ~*nɱ М(ھ9HUc^8W{ oy"2 4?5*0k\[]m8rr, jdX%Mȱ01ʱY kВP n +nv,H&-|њ;B,SgR tiH:זM咂p== XqF\j< e=CmT 5;v*TZk@6 + c~W,(2[ +$o9^'{X AӿX$1Z-Pq6XhH^YM +cVL4U6um# > +X h0V-zǔEIj[0>XzԿef7#5?9* m] f0V|Xhh bڽ^a>Z* 0JgWK#HEv Dpo2c%&4^/EL`i[|q2[LE,:#|W0V,z;X*Y0 ^0o:ɔl#cgdTA/<_$Bb,67sӷ"P'?ɪ0@aƋf@حq4LޕnO/aѷZH !EC" @/nβ=st^Xk#BVƽin^~'H`,"qiyXϚfb[-L^C0e cOAScEsBd.# ƊÙw!M7l ͇V=~^|\(p_e+c=(}Eqc,\% X at0 +
:F+."i%21bAJ B IymOPe=e, [Xh
0 8&cՇ7XQwj7~*_ ba,.Kt1<; Vc1mzM +jL^\ES~&sB<&2`_,#)J&b 7XpJL1}`j_Qq7sP!5I^,^]@9vf/V ʛ iDX +bALϊ#4*h źp^, GJďo bf1KGașBzʱ<H ŢZC 6Y ;d i/. w*0
0ggKɧ + +|D-zhXv!D|x{475/_Y5C f=KHoU_Zv(f?^ZK./Xtz4)| + `&)qgN RBKcK`, <"Xv !+r^òx#Q?1--چ 9) |Pk-(?>1A[D`9 V-))ZA,w
;90_Ib)@Pc su40P^U}Mk]p0VsLSr氄-gYCXX̎xGa̷ mjX*/ +`\_ 3|nl[3\ie2j/V0ybz2J|lÅb2^Nvޘ"dyS_D@mğXm=@S~]+k=5Jϰ_|o/V_,c?5ŰP bM
6Ӄ)~M7"ݹ E:*bŹ؉ ㋅c닅awXq e{b'A `\BFgr=& Kb⛰ d[PJXFMxۆSX +pgװ'!5vuW0L0YGkP X<qID0P юN4_&c/Tʂ&>cYSXtQ{ˉNHsW2G0F _. c9_ cq<tc +K4a|;=c҇?X +݃a0$ )lBadlµ? 2% B, 8 +;p#}ONO1Zټ.{L 2 + EXJ7ĈVI
w ]BPGQcx:B(Co\n4y +(Fj}'ܵ]|'ovw|`_4B*"QѢj\JD "'L2EBjyy_ {=3 jVq Nl} &| %;*T|30u ڎCJ,q\=c 4h_
v +|2. AA&
Z pm A&pEA 2Ǿ/ffp C52O.qyN7~8 ,gmEü;e$JW踘_ԌQN\LJuR̃a~3&ʐ0at +Eߎӑ?)eX᭤IcI$ H|&pGmaWS2tpIeNVAlNfYJ4_ԆNk挶~ʜ\1Cґx EJC<.D&:_ϙ_iϰsMO|yHBD)"~PC %9*ۊOFSX=5g4UEU1$!/K;Vr+"D%ZG*(Cߧ mG[hm/)?(} 044,UUF!(!:X~D<}u_ _S"Za~\, FqDf\Py0"Z/
^ؿ >Rܣ##d_d,/2P +Kؚ6S$38Ӧ({^,EԽ U VUPN= ',VC +4u W +dUH?q%9HyB%D3@8 1SG@?&9!5'1d)Y葲,rߒ!lfo23ER"9dﰿ4 5^t)bC"YA1ndC(_vJJ[zL%l$CIN ɳ:!ŧJUħ*⠐N G?`)f( I^0
R.>B |<#( +QA^DﭺSTB.Y,<]4bxϘXVkI?5!|,-&% LL@J:Fs8 ";(*xST-)ENaEMx驣ÊH +WFoD i`a4UdjTR hlTh,&FFSNK(:&;,oC8:g)64
BQ>9|pONivM;bI2l6ID &[ I,WcH>wЄz<agĬ?+"C5°:>9\ ?]GTd^GM1ʤ-R((_ +r"b5V,F;ԽĴ\׃Sش2AXgr40.C.2{!g؟sԕn1ENJ-
4~{Ê"ia dz9QADp#N<h gu|J)/*kg|hX/Mă'f cy|a%JlS_6NH<!OA ((
zGB1(q`xqFC|nM<z4yWp +w.\ + D`0 +4aRpwhqL`#2 3~EF5dy6fЪ"~y` +{cN}G]pгܘ̔! 3ёq *\aQOP Slގ(K{'6uNŲW*ZQ-Γv"t@TUt + 3rZssڠi&<]E@f Sw&
RB p#1(ǀ9AfƤzuT6ui''xրRԗ"6[ +M? e@n Apf^@fp<n<EtF`7Ȍ3ݤVc 2 +ϡ.mZtWK+2#7 2̨DAД+
e2A|goadn4d;PsV`|!. }訸
r $ sX`eB֕*EW~OfC
8b۶h@-X(AvɌ7@,GOy-"̸%9: +h R` R~Nf@A`UNʌlR4a6 5Q}$ Sf 3P[a@̈"n
e, >RXʌ3 54ʌ{fN&? *mdP(B@ȁ0Qk2ڙfu'3U{@8!P4}k:NS vRJ3V@k.[
u n-O SnX^Jxi55<^ǵdF]R2 m|
\iX>X)!88-mis22 "ᛡs̰bg/qPf$UHMMR)E8:ikk '16ᚊ2)Z.eF̉RE,aza#V嘟.Q]&BUfa(bF J|w9X{i(_qU1X8QF*3eZ߃@ 5.o vM9RQ+IC-i%*BFJ|SڕtMkcO*3⬨ʌ9VRg }sI^nM*3)1BYK&S +3 jK #l*3d%BQ.dUf$;6<Bok BW \@EOVVQ젡qÒ24)q&AűTfR} W88=Y@eƗif-|rLC5.We*L?M]Nȕ?ix ,3!8gewO9SeF 2IPފa`BeXUb֪S +*p8c ` "`cіlN-Bְ wyo= +]k R%mNh .3(8ʜNQp ^k sZrcNҴǭqJ"dCrSr%5# j8Kqj+33+]O&̰,u>De2:TvC<JiM +2C4U!VFرY@yufKQ1+Q! iw;[YxX1!xRj2"9eH;ĎqJ8p.4kl 8gJ\&Y К,"G"P|bAjd1NF۳F)v:БYf3J9_gq fji
4L$gqpM8JB/墈C#^8 +sR}96 + eFPBMU*L +888H+tlY ʃNd%EIJhg@vqz_ $GI$Xq^e)%y +IL z+E+׳3CR~ '3HVzPWdhAAvO,b B+E'qV$0 t1S^<U/ʌ>S884ieF +zZ +01 +Vpdb_/2F@|ͻm0 f]lKH`5Žj̮܆BC)QZzeT +*&Upo!NiSW)-*3-slb3
Tl^/肽 P
Z8_/Tf Qۚ^xd^3>^(8 +=e$`ՐVq[6&%Sy uYUf< %XR3VP=|N+y]؊$z BG(4ʌmbWze\ x+~fA! $cq3-G
ȕQZ +(~)|;VEZ,38oX6>6:<xbZ5 bK.|,3i=
eǤkU> -CZbyKhc {rryo!'
1b^qCRmЗP +)QaYfp2#q e4b̳HfZNOF =6#f)G
hbuYfPV ˌ"d`U2cݏOXe<jhszJ1n QWRhkVk]ʌOZ +3d]
Wv2`[
b%ފIYf|1+ŪYfORʌXEzm%I`Al+3
U c ,x x>
w+z̘FUO- +e) )fe\c boG4 8+2XLRѻ :kR1/
>eUi)¿8SJ;(Bo LCiD9Ӱ}=ʌ%̇Å +䘌xv ۥ^GB˻p=x`!-Y5QYud2P`c C
+|qGfx!L#3
{*w
d|v4 opsΑO +&72C-#F)@HfT:;rTx:~m2.BYAjαB JFAd]$3a؉P4 XmUGR41 $p%d Vֻ-]D5=QwY <? 6R'h +<C2:aXj5d /7Q/W&m'~ v<YtZn4)?C"/ddO~NPx> #1 憟5 S# oӪ`$4bR,qs-!]v2qʆf,6&3^MV+ 4̢?Yfs!dB"DUph5$O &3a Ȇ5ACI:膟&3PH[;r-|O3 m2#Ձ̈́kS`6 wH̀>#k2cV fWeXI$"h2qFU֘ +ēR6< +Sot;~cL(3"l&Qj- WKVh5*3Fw[e>$P@`
2 #U U(ge@[~`(O"^`H1T# Mpծ g ȑDVgʹlBQnIpeFx-I !hb4Ыda#Ø" 9'2-8yd ^}X@3ZNdՍh(3B(2
xmiyXTJu5r$l`28c8. .sW`z8cH`.-b cɌ)n +&3"WaY19yG>d|'&3s߆J)%3v2{&3ga^o
0MfּJA QkZ#W1\̲U 6 )UȺ0b.2u$$Pe}/(ξw+'fQA(3~^!2"3F G?v9?) (^Pf<'AX]:ΚDdG'3@ȴg93?'3f-(40\e18Eqfmsڬ~8gZ7\meƞq + +`oVM=r8$3'*( ZCx$3O3o_"zW$ (Rm8 ;P_,K\SB )M}.+uJf( 2pa {o_!wÀ G87Ԓ tXUFT 3OVdY;0qDʮ3@T 0\aH&ven:f j4ȉ 1Cep@G1W!ui#czAY =-k=Pg >Nk?fR"RUXO07o!31uUpWc!3&!2-N`j}0kxQ;ɧG E(o:7'!3BD4LwU@ 2cjwE)5@2At7&J-,2cqyՌM'EfXz;;,#ìh#zmȌv; `+d
0, _H%$և%xEgt8;P U6Hf IB E8tYɃxoHd?3C~Eu0H12X.g.UtaH[ $3vBq\ +5dxOEj" ZiPreƨ/Za1 /l jq[^/D]EqI$g%L 3XT[.%&2c\=eddbne:8#3Ba>DNz5 k֑Jø0NPrS +DG,@[ Qk|!r "CFfĹ +N~` HExUp՚U&a;'edWBB}ildC\4x]zIz +]XB].2N%)32b9XzۃQ@g <Sh6NeI` <f^DMlaC6Bw"|x
D[j |ð?nj-YVDh2Pj͇/sh{gWO|5H1>H6 + WD pߙ["Q KLK;qv网(#LT܆x7|-赂w!AXej%Y2,Cf 9ec R P 6 %O{ Bl^%g ja >w%TJnNWQdnj>nj<WNǣ@|m N\eAnPEnj">h 3.(8(f.Q>p+ Nea- 3b1W3;|r̘ѐm1f:4_9Ôt~1mר$BȔB R-f&e6Sv:d@Ō+"qAǫ]xfŌ )3~@c/R\<ww[ufcT# U>3<ߛbF¾)o37F@Ol,`otnD] l(2`(ZhFZ +)fDFa=U ++r>uNcǣXQ("(goz67s+ zȎM +=8:["8T +VD3bd8.%f͝|/ +A?:I1PMBsoU$*M~Č~<tS5} f{{ +t~~7s.ңؾaeEfVX7BM((W *A?]1#e x@u=FU3 ;(zbm61ebFRg{
f"h +IGBA24B1W Ji8)H8)*w` +hu.STSPcO_ )D`2E@ +w$/\å|ɟ-ÊvoNXJ9 +yƽFa(0
!ztzS2-~Z r磆7fSKO-GotcF͢>Sbx WuP7쐓wQp$6W(l iPvQF?d8fxmܦ.Z֘A*Nn=2w2wBg~gصpC`Kt>ƷBUM(7ft{1Þ^ +!M] xQ?y{TÆQˋB +cF7Pj 3J +y \ЃqJĪF1ڞXFp($?<P 3(t;1C(Xi}ሥJD~`]ew̨ +HguOH"CTd8fz-A=w-3*/GZƌ *PLu'1.q?5W ĊySE =bmc>1}O?16 3 +ךg +F7=#M2q 8n h\>!00`p%7x+-9m +$fzX/K
c<nd*gQt7;c*:6v,^1 +A Kq 3,#('CNȼ"+p -dg4$<vQ%kz$&sdP )C]Ld< { o+ *KÖ428tENBp[+ٶ +)Wl5fq2[.vs3E'>=Ҍ eO #PZ] j̝o%2\8.DAu𛱢J cCB <
Q/6 uvgnp~ϼlG3gMEY +GZ=4ʨ Uu?ԅD ,Z[ gy͔FuH%ݠ4"PjQvFwxPdWB]awiEubZUi¸.igêi !TE jXyP^ U+֤ѣ.ɓ9
FsriR8 D]0TӾALH Q#RE] ˛jإ.K;SKDkA.&E:}(K9͂ b!LJ(]^.1<֛f.tu@rLi˗\\ +pもquv<\|{ . PX婴Ǭ-,[Xڟx u_uXfnaҪE& +" i-rve)[@[+b_\b ^U]47iaHU ZjѳZe> .LI \ZhQh`iBXZEς +,NI--Nd + FpǢ<;\Ƃ L-XHLbb<0-Ȃic\X$REĥ-,. :m@ IBj +TH(6:&23Di>e~[^Q˴GsW
vvrМv;W+]ap{]V4r+q]؊j|biEq<+fViNCVi
cE8v'ـ( +P +8ytF*FMjn_^ŔFCWήݨU ʎptUAV7i*Q`FڭEjE_+vg R2j[ +tRE*zUtTG;TxpSB +IAB#jR j8Q{*RxS4U6u?]BM0G<W;cqJB +5aG
USa5 +ToJjjՈ\* +lG%hh +I@.FSmb`~OP(uVQ(MFmŶ^nmТUWQyEqmW&݁ +(<m2D]q}(Dǁ'Qni
E[#Btn,1{ք,$]n +v;LPx}@o;yP#yB +6bMjs+ _ٹmB& >}*е㠳 ]bl%Ef]6Q!SPUMyM5QzR5N +5Dϣ :hә(Keċb&Y&B&] e"&ıDŽo ZL@`a2Dh0A +Fׅ:0F]`LTw$İFċu0: +)ta`bXzir:}$tɁ ېkkMP82E30ADLXc\ b-%^ⅼD2{x 9 +-[ҕ8I()04m: @ D|2sԥK] :F`b(<g-I%ujpŅ(@[@.3#.!NbfG{ +_ZYumWǵ w *Ti!X`~u\ctp?Kh.Z ޮKW VCȖ /H-J5oDYBKZbݵ"XqJ %*
|1\gw ,a _E/wmCuktoוn Ӯf =( +uQ%ҵЗ:S&{lJ
^ ޢij;JlxE IR-ItM$oǓ ''$ kI!HM@m:}QDcuO@;/-ϽP5%qR$( 0& He"~I(Hp%( *JHH!7`HDW;&记G!7OSH[ +JsW w GԠ <u}ώyS#L}G(@pGy潎G|{n'ǚnZ Kj+y9"qČi 7/4+,H.5BbFp ԑ6Y2;xmSbZB%nH;o 7<-3tҼ( qE + QE>EԶyk,EvypBDy2M{}&"i+$"lD $] y!hh%D(clU ~^YŊ<id ^w +B),3!lN8Đ
ѱrPC`L!d!s / ̅`B ϣZw $+iBWB9U!z*v=t!Y TA8<c{)z "čWz9 "ЋJ) A T8;DoGq#I+2EP1"b1*1ɰ2;Np Tp] ^ ۑ?b{yJ{[sϡ0XLCD3?@ބp;HiAC}#F{
W>ެW:ug>x|~>Hy͇3L>,x> -| +v[ҭy2+L_䨇7FP!JFP<աoNJ{Gz CNvCp< MԖ\[C Jx0+( !, b 0% ;0ּʨ;YçT"\g-'x?{ ;uV렔a +e[ \1@k) ' -hPq@ ޑf`({*A -ږC]{mLZ8ŕ5 -Ñ]2%W 3(PhlaFÄ,B!X>5aň @r9a % -nV6ǎ7K-
: rYR I* -!2aHOxt\ -m2mn -WX 3Q)+d5jE`YrђF1k
6R4X`D
# - -jB4 ؞xp!Fذ0QE
68.xP -蒴(@b'Ć1^ -``= xqy`# 0` أb~efPNO8R $%OUI'J: L+9R+
)̇ ! D($Q -jMLT7%b))Zz`q Kݦ;bmAHe,W6qI38Wl iQ-(dX 3C -TVВna|]0 !ve-HKlrݗsPVRU} 0!uY*B) -FKb -^i1hGZ%ꈉ53[YѝK\.
ߦJ, paɢU70|W^($L xPUT>"Q^QPJR Ӫ1<o%SRՓkjwՑ}ZfYu-]CKi& ^"xSBguD@mZ5emSF[9E@W-BSx7-8)0 .:"7Fk1Z rCC:`%&,$WWPX
Hд?f, AZ
ZU
8l|lKK -AP5e ҅%AæT_hX|EC=uk@J:PUI ESDƁ$RETUt -%tdDFAT=Ғ
LWKke^BmC1$&õsزMZd,Cܯ cR:uJ%χ )au*0Q*!$t%2ӻeTEpKRͅ+튕YUJm"&q7Lɴbdi a4JILURIˇ ˅Eu)kۆwn
ێWKfE@ -H((@&$1,.ȊZW!|h %JE19OR3UikD5VF)6>+ -fzlsLّɁ!FR8`E\-(4Me$"\X*ZX"|;BUY,)YZ,B j%8eVK)~|9 LK)H L55 -<z5Io -*,,]2ϙ@籣]O:2Hpf5/fZklQXN- -hDOa:RL߽~JȬ*mNEk2[yپ-rjUn'άw5L rCAXP姕Du6o_4%DlB_rvrq^G~Ln)ﱚ˘3V"):b܉NsCAh
UrX P=R`^k*Y˫
:\jɭMuߙZ =%4k:f&%"~G1J[&0SxE%{z:Qo2TFXN-+pIK}+ci~ٱb'Ɗ?bg 3WH G3*kqJSPȡ1&{ev2eRhC
H}*SΡ/()f)hf8U#Y]!3(l$;b$E! b
G\o#7}סJ^c:[LQSJɜSDARSB?،W~ B9)y%RO- %J jpbեAIJZh-v3iUb"z;RMaH)2 uc=ᨚq(, -v:뽒r&I3g yhKK=5d# RJ|Y*,3SďKj nrIOW4+s}|OE4ߦZ=&GŎճ t ÆѰݸ^IHBְ3iW=$:Dvj G$O MH嶜V#
٨naHu<yi47HY]
(Q,5lB9ջfi$hX- VBۧz#z !k6 ^(Rz秘wmԜGda:i*i/Eսn\Rf(|J+
jh7VYgHe'9 NEf-k 5˒B`+f+uSYR،D@>{ .;b揹u9PDr? -S CXgڅF7а 4J~ ߎiE$5QS7i<%C+2Bߗ[MԼYoo
'Q*x u Ԓَ[ Mߛ]+Bcֆ]uK.Ԏ|T4gUYs
+fOoӑԮ)'*фڶL`pP,5"J\a2fBPl\EyʠD5+:y}R oS
(D&VQջ7UuҰ?m=[y#>Zې+`Frt.ܪ7Mn
:1rybkuFJJeb8$as|gځLT(UWZɉ0ZRnωrSHn90t6?+z-F*6KniPEpTѼOg5f~%vĦߴ1p zb6Uf5 ZpIjF{vn*6ကy+IL ڒP9Vy!3h,( × 6{qK:[kyz)6ї QaɏB`eȢ/<ϴ@[vz!ԦyoP%)5+ mm!"6 -FPhC
ih5ծ3i -nf
k} -W;Uq܋`WZ 9N$ւ6ۙ[4^ 8btgU[69)$FTNjBum 141u=7tXv^hX ug=S8'Rq$Z n2sc͒V@bk LA@a؎d5p;&v>GnpF@uZr~KzkUNus_j)(wQޢ(DAP9N5۱CnTbl#ϭzKkMUj8ݯ*WC*N[8Npkb4` vIx,TIUOFE\ni ݆ŔG!whc!S
Xo)uggrۻH5"7&P*E$x\c [F$.;CMEV
ZAXf*Lbu c:- B_h:?Y +9 TB"2;qRٝVQY/@#Ӫwۇv;Nx9 jiFC$= 7̆r uvEPk -.!G ݰ7\sVjdMm^C2*r 9h.E,~3
céCX<*b5_cCJ峮
3ñ~d^Ze<[04sh8T݄NT6JlmX5q8YVmN빒\ඛ' c
/M^sR^
ufoHx?V_l#Ps j:3ug< oAU\Ί iEqFL㳧#7 ޟ -Ϛ@ai=p0kfcj L]8(XGX qN߱<Q#R.QŇEYe>Rf8(9_nޓP "c2{4@!7Ti ND~s -2"UlAlymHJۅ\1 -JCx縘'n k9VI颏Z40dp[ov=jDZO# Ie8h `6.S 3 =colּJFnTLTe:Z0&؛PİHMAcdHҫ*ֶsaoد;r㑌w7q]c
` fvjے7
5X%O"caJ]m/Vf{g4p=зH)rnr +̕jDV8bl' ΑLFMS fqmԲ$z@b t]?lD)%MPi jkVX>Z
Ke&3Xp^nNKF7CFg-~D^k
b߯]]{W1 ÒDJq<F`d1ͳ v1xPoQo~n\sƳ |4Pd!~M)4B$&F$
:٩I~gMDoJA P;ڕ}*;ڒ@N:SF LVaڏ#h>ޏiu n3j WaU307|im-N윬X}of<b)H_[*Q#.҃ -! /K鵆$7C` -RC0G`k.z* -bUم 9zq -srKtHȢ@h7len3ѯ%XUoJDaVu Q[ - -5T ukMGV$B3dbXNY3@j'.2$VSko|=Њ7<_zqG~B։0E6:Hm%h}X +>ճ yEٮ_4 ]# '2Wa<y5)_c瑪Z`4 -BJd6Hv
T7ذvjvKnx]e S=ORz)5pg9NLC8cDm 8xo.̴YΛ(8Wi'G-50,(ɦ a&+Mse c0N"+Sۆ,C.v?Y2U-?1 GZm'~w >Dj J00䉲B2% āf*-U* X_#" +t4W\eH8,Qb;RbvMAfwOAd2^x( ACE#UV/ -ܦ3 X ZX^ O7]Ml >OcQ fù8%I NZ -x^^a Z;ajm n CọR1L")@ry8\2ث1K25jLb`5
:5L
H݇v| *9Vb4*cvKan8N {δ7s&!finIa X#+L1JVaǫ~tHj] 7?b.
<ߛX\W>GeO!Xt-̂oZO&/ Z W̟q\Sib*XVIYGQo9QZl>]/Vȼ߹j\)Km - lP7=}j -;Rw>h|n88hPogQ*C)(\( 8` %ZbDnq"YThXK -:۪ ;ER8!|`T/bSuk r]E̓E l*{!4T H1_~
@FIM;_!E'T$.J`q%Ŋ/X\X/L"R$!@*ZXA I@tqE | x`%*66) -!L +RN!ic+TP"x )`HBPE ).RXQ|IaX2lpOt [7(_hΒU:Y?(e{)Jܷ qh 03z=qO٦-i^\˽33ls&@/ -k
Yn:7:]v=bhL -V`1ZeP@i5Z_8 Y \ эUD6 $v_lb0ӷ JSGl# Ez|Zm7Nȅ̶CA]"(= mFS~;{^w_ٌV-.*0JSߩ'L'Rwݎ0Zv>o }U[Qk@{&{EӁzj YNFJޟ4p<Q7^YXFZR-w|%<-bnSI,=)W4MkjS -#GPDJ4w?κfՀY\,P:(z `=sRQ R nZjnXy0ny1,8`o[l}2 -}FS4qٲ_B+1k]F{ - Xۈ^Ha% - UY) \;@yv%=t}DzIBam.`; 1?B0b ӌHFͲHӉU
:$]GӸ]FF -Y+Js}5P^u0ރ(w@d6AE~bgU8Ҽfuf]7zbkM[=דPPnʫn˭Ϡvy)~QhX`pgG,5ם'Z -oR0^ -309Qbyƭ3hڑkާD>\l"x@xq9l,.ڮMV[1=Grg͓-g8Nf8u -CE1dl[1MeD4LIaio;"x!GJ/'zC5
j+CԘ'ts3̸ZgUU sZ0iebs0E[y |g-<r%|Wm,|`o8Kz^ɖtfb,y i|H[p xUϘv@t*zXc*U]R GRLW 3ʪnGZ{$V|ΒƏyVCkUQ͈A=жJT\p:QT)za0J{9"q\
MSFû ˋQ °0{ -pyvo2ȱ^EvM%ZH\5 -+Qg\@ uЪYOkҬrvUBh5 2]itn;{Ĉ0kLݟQ,x</tߚ.dhh9M Id@5H9ฟbTt(v̮NaABX
g}N3r<aIo>Su$0hF_0Rj*hM@%I0*-w\Q}rc-2 y~
5KJer+jV-wIJF< AMT8 r@~ݕyjQ:!DaUcˣe@(ɈVfXd3 4MӮzu8U -]ō$t7P0 :&B(䶐Vbx* 9֏Bsa.E;h CaqH4yߙ 4NȾa_25v'Gvh $ZZ'
/Sj,I jÉZ繚y'N#Y9j dx8~lhoE-Z߲L}xo -e - ]9a9mS/] -FTHbq9PVu7R4D9jF\ r5p.O9rpUT gI1]W U8Hi5PO]gFj i l&~Qq«z2ڷ iOX s9k:zUhh&.6 Z?b,}2cK~0B$i:8oC q=WmVc-Deq5Jx *{BVAд8I ?'Sǭ }!/=I~hk? WiHrĪOĖ Լc,D0-H4LsW/Fu2^i9@Nx&^sQ7P>h]9 y}f@g3P [ u)ʬ5 ~cy ]X?݉ -
eSe<OsEtD<oEN,bX p\ʖ Z۳?'j(#9b(88IdD}b5.7;@z[@;(=\q zJs$ӫ͆J3f|X SJS DX0y>wދ 0IwKr= E[n(xH8iL+r{' -F;濁~sHBp<wIJ >\n `8`0}-Z1bx<Zl15]>C)- ˈY,1l
oUxm -q7NvHAu<&gD<s D|*{%\5 O# y=Au.6]o\SxsWbx63hui9?C-zUI~]e0tnGڮEvOƩ /> Dxu< tmG jV1l?ZyUm ^ci8A(M3. 5YV / -Rff8r(|_m D1,IWhVf<7m睆~eUهʬE ~s Q6 mn`8[ $X KCS1?91<#U^g~d@eyA]3_n=Oyޛ{aYLm潐ʞk2` sڱTkY=\Iь'Iߎߋh
zqfk-4^d:&zazCr+JrQZEPj棚x
ĹvNa -%efTϹ$|e3e~k/]
8Zr2N^f{#0wOmu}trwLBu^-dᄩ#]1m; >'jn+1d(Y N_ R)/~a|!xq`ZXo: 4 %Qe%db/>YS-)璄4Ҹ] gg)|]q3kV-F-:O9R #%:1=5V3 wM{Q7]RMB,c{%K^wP..Yq,>RLHuM7# ]gN=gUS8ڮyzM,wĺ~]L_Y60Zj&.9$bAkIu=QvQY`k>-g.
=B& - %[5 -~N.CI-e Iw5ε J$oᣥƢ#rߔ/V|(M3!0Tg!L| -pH4<8 dߜ岊rmY}쾊Pg#fZr" v^4رB]H?dy^EfxC.$s&[iacx\mY^1̾ ۽^ܔ8y$r\Ob{ߖ繖;!v9x"0Z0 x0(Heq"0Dv Jqjy(Wb ߓ5 H&Q|r#!Hz~[ -Se(I ˔Y1q_~~#3 #I-WBJCHԪtf V H8t/p>E_K< ?ˍMZ
GWp?Dk65K̺\by -1Jq`؛y/2vȜTc7eIr'|>Dq=YNն:el;3 %b$Gl | *=%k RsLF
]"#4%U]T] _ )# !6=gֳ Q&tx8eM'aвUK?iNbSx##x@t6ѴG -Ru9G`VR\4x7M ʶ;B˔ -%Y9XR2=w$=^kVT dB͒Ì|XSdx] ,OH6ci1 - #&r{yU<, $Fws9cOv)ewj [A1#UfT9 3=ǩ0o4MIjhDc2Bxyk[Y3lݎMr; U, -Q+4漆Q:*!Nt܅0
6-!F-MX qmli,:-w,%BESۍS Hmy+9b䙮ZHiN=]IVXEƓȑ2`N7Ag ľ|O4˖~x)qMڈ {=|:&觬 4wqKmn%լM]$sY>7\Sڶ3j(Vv3s_9j4[w <F*#5U ߖ - p.;(Z8ŭ9,ǭ4q5R<nwjzevWUGa~Hp?+{ߣ$0 -Sݺ]<oe)2c!RIzohWjѳJca/βe7B9Vb8 ,B[t^GKt<oNx]4q(L1Y${iyԚȯ8=e~
4Rv@L@e8r/N EiΟ 3YlbP |bmz
J_sw-C^n%Vg%2Sz4 - -ȍ -_XJAxP!-R@jxcw)\_B<?~ߖ۳On:&zoy h;O(s(B?B˂"emϔ(2Oru6My@s4ε JzDa;]k-u_)=!
-ZGm0/K
D8$x7O]O頻4Rq^%e'fr n -z;jnp<XTl?!*;qTv nu=J:^J8ōZ [}ˍ\c{6 gL -3G,_ζܹ s-"6 -ZȰGA=Q%N.Œɀv -S̟"xE#F~!h}T cӆ- 1?ҜoYE˞ y_vcW7$d^f
-.Cj0I頴ݯq%xXg -SJu.DqDeCA 8Q=Xve,5~ĺs_zqMĪ t@2Izw#Jw(t~$z=UseĻv -;2:^v<]m vxe}[@h 2V ՃM=IrdX0k\Z N -4CRh&:4̞Dv[~f<c3>罉ޗۇ 2 2JsB,5+P1 y\by"Xۍs([1R $Z
-6W" Qd0 /6oezBƁ&Er_yXY -xVe,8r2B0I"WI]:Er FS4$94Y]lJ-# --E)~kqn4~ٰ'z>XnZ<h?b|2/Vr6P^ub}' u7rc1rˁ9ѶƉ$'YUϝ4 +&Zy)K7$Wu} Z*3$e@k%˛Ns۫ /C{qZ<Hl q5kj[" - 6aj|*vs/z((s: e
a.NZ $7ѩnQZ5l9#-^ۏ T\U -qYbav9Ph'zY.=~{PYUAX8XNugY)Un-[gվb$J]l<R5' a=<̷b$PM -Xoܢ:ӯ2+ )ѧh;ʷ'jηRa& 67gRw86^ / --C
;Rj G+jyB\'uh9fYN.ق Q*ͦV~T -}Ķ>y)zgJL rMlQ8YoT&R&1m~Ms'JlPJ':7Pj jDg~9ϻnxخg I4OMY\T4&(?@8y^P -F~ >ag7˖AL0킑*P;1pbNԊB )
A'(mj#Nx>eh;!R_qd^ >&6KR-v+j<|Hbr2p -_$ƬA*3 7Is+Ep ,Oe(tCZ
k@ -u 0*f9ᦩ-!fhaEh!FL0Rr@C qP 9I L ,F푣xOTg -nX(U !$rVHi89"D, -LU\PL%z/Xn+l:`yjq8zCZM |%DE'_vf^v@qD -N`x(J,ɉ]wL˥r$o(J6 5\,Y!yv9&MV\⬢78e<'u<G gDwy:=bG-ϖێT=G 8zAHE!zUH^`|,7&kM -л?:?(*/P!?HDX !,#8P6B<#x +Ó# 85,88!G$*4\'_piӴ// + @p/^wwۅ[J " *Pve)` +I +\1Fƾ^?`<F`b!"y6@pخW +o +G}U 6MYU\ 4U*X*dTU)d@*2P!^C +ۧP!O0DSqM!3uQRLKfd)R`#RXQEZQXQQ.B \ xXKނEDh-3P@TaP0jw:PB?!V >A2ʞ 7P y%@;!N@(ĀB넛& 87'<@ВUb&nBY7&<6քf&54 e33.L@ :`̒//x s %U-"e 5 a ,Õ +duIQ@P. dnRmm'} a>4eCA>N~[8C)ːPv-R2TzEE? {B}wA[4[NsUWn{_xUsu #dw1UW]n23+|*= AjrTw-G!;fخwɼ'(; |euQ]L L`29Ey;3 L=_g.ba aJ=.;UF#-{丟O"\G?[]2tTS~cC܄nk wÔ}Uű; 9 Ɨc~썏ٝ?O{[e#v1~y=W7Uc5Tu魩Х>?ot鹯N=Bm7tw.Fn;w[ +s;FaPߩgکFֻݺ] ʝ._uq3] ٿ߅}!Tv컞 >pEλ.Ə >]t壃1>̌ϜϟdعΙ߫a:|ucyc^ndݷT^u.bl
Uk/:O~vq970Q v+]l\йus_ 'Ggo +~vg8=}c^ur {*)pu{N:۵k5f ~?/0;ٮ>^[Eitq]'(/'()]Ƚ.-i5*瘳]'(vX>AʝD +gfK?lwa°) >OнgBמ~2 8~~ 74;s).7>
U.tͰ]?fO2sv^Vv~ u yo/mBUӽؿuk%֎ޮ4SvA.:}ϕ>߿ wcB} #lvֆ]Beu2:Tͭ
A4APM ?g`gvcׯ ~v=u9察T<y뻋Pďso˷i} ;em3 EoY=*;טo_܆ +YBǛw FNy 5uuRQy{]BknUNMYAWV`tS%uLݗwA|/Bugݪ s?7| 2;~ݧ}
ft|ݮƘ)P; 5ߣFfe{>wt9a|\_wV5{~~;dɝv}{ן._g{ KV~&ٮ}3~svfzgC1:R293y ,l0>M_KtPE拎;;3 >W?~|PyŎs풕a~j^tﲝT)꺅1s2*\t%cnuϗ_=>آ̟N%Ww^^'G +>euO zvUƆ +endstream
endobj
65 0 obj
<</Length 65536>>stream
+ h +@ OS F:x T)^H0FW'`M:P4(ZLk$ .=Նl-, #bt c%1MbdW-9OCD +͏6FW-;KY1;M7ۊ
&88сdf"'fBΏ*6>^@GY%jt\u`1,-T`F6P1<00cGA J8A
Xmi S=5|^$&hU V0 K<x!x3:dW:@OY5?I3:FW.;,"A-.J*
kFvv4Kya%GX`q-=\$]qH Ep`qe<^ S f +n(Qi@*X|T1}a!f|xI J GhVtN Y +RsTub >ou`xh<[--^NPFhX!!ťENPDX pa1:h]z-_; oꏮwߔ˜? Jkkza"L}n?f1= Xt ÏvzJS >N{:JO<Av'NPT:ݽG + n̴'("Z6>G-4Qs𣮷^tڊhͨ +^ c&xUP_ooǸn)*\+T TzV<Vyn s;gO_tܾ1?FaMR ڊ hi3T+#dc#jqtCVnn z--Рr#<:t ++0)8MW@BOP,YIZ#&xqaƉɋҕ 6L3
P$k\ +NLXr8 KD8B8aPe{#4P$QĢn ^I,KW Xu +E 'm6n(*¤4^DEn*(r#A#1=lxDb@Q6DDl)VCT-G6{-b|dW + !G60 +PCJdʈH + dHaK,@#&(2 %fv .cH" Rmņ 5Jfp Y"*z j*^8.RDUS!%[3+0B(8f) `8nFZbMX7?PL|5dxd +# + S,tnf1P&<AO9MY-pl!ff+pf2?M GhV$h3@KX1b,1I@0x(4kb p +V4f# + +b + O/ "5 =EL.!NNa &|f- p a +]3XErB$Ȇ~T,y)#R<{ D@H+Z`uiT )f,Wl5狀]ŰN[h E~i3Cw;`ؖ{U +5_ sAht`Y~[ kjXS湩Y8"@"ux& ;F:?%5d + IG܄L2ukӮ[Y<Q^ +a A.5̻y:mB7C+T=S5HȚ S6)29l9q?:Ѡd +bu +9S,.@>8
h^akp
۪@TKSp?J9Q}*;+ЩLB? +J +[(9qv00dxZTbVaףQ) MNN1<9M$W n WghՅ"s3#SkC6.# 0B6x˓clxE1^jg$癐hȞcDl MO!%}>;J@0%2eJ3 8beP#o{[MR +E|TBѠ?0zVK{a*\Xl&_esT| +[wʌ:;g.Q%#n MM0|-[)e#4hpࢷ=@|@\r$8^@ .Ţ٭BcqUi *' pgqe\ح2\ΞC'?Ʀjn" In&r>%s̽"`y<_(SyF'<EOQ¸S잇i
;'?r 'y.:&~*rsHٯ'ٹA^E'U}2,6sYp˕7[{s Y3!Ovfli6tp;zTwinx}O nM YC T,ŋv."2bq.zJ9 =C[?[2q *)>䅙D섄٦ij 0nDQƉy-& f!Ώpp7)RI +fෝ/ 5ïlHS@3 tlHl*9d3 VLxK/qx[X^ +3 V)YDLcy|"SI!fEy(gR
6Z_`2^N:xYb Vc@rR=Y=pX0)RP6ЭOwє.vh+K ~@rEv`E
^P6#D pƄf70$zf
G#NxX%7Lv=v9ZE N)vYf@=)' 7awoVX]cfR +|fÅ<L9 t5O?L n㷎Jk{\.O]3tXzD_*vD XF_0dGHf%o1[; o0(.i3g7ԝ1 = gK X@ǼBs"EЦ VaR/x#|9#%o:G1Rdhi + 9KAK[sDΖe
(%'>.fx#)b
*}S@ti_u<
ڊm}t!栾5[E ߩO/](TMN}At3YakލJ`njԕZ'dhqU +MV-
6
k8Tqs +5x_ +!m }[ خRZgv]2d |˟9%푯_p|t5 +r$b&:t-b t/=;f/ 6 +V,I invx +"ȇ.]e ĺKbfC V:5ew[lw)8 S2օK"-VhzSQf$Rj)8pc~Nk(|XTL{y&/ v&j ֎B
V+Y'koSF0T[&j;)E&6k/` 9$̚*=>)s6KʥA5͎XB^Ӡ W/Њ ${[Cde<gM$UM #~ ֹ2ܟWuYRÁnc짨SIj $_k J37Z } o+NN٧^3 +Q{QUy13#AI +M,ɟH]CGrtOPٳoLyьOGbE2.Lщ +/_h'JF//{ 6 + "Sm#NggO5O5%A2 =Uٟ ⋓. I +{rea+Z#mxM2$t
6-hoby{ZG mJ/Woڀ|'.x8F7n?K7~+\EG| +ev!yY7p +xJ
?Se 0bXkgKȖ+Ձɴ;˦ +&vim'˲Q̹]8\8[&
ʅI)TY'TޘCrI?Ѥx/yvwe4o' +x=Gg,6ٹ}heS ㄓG㤶E.dE©bmHR|;Ӥs7ŖGemȑ_Z 1Rt)X/aWYC2_%a4@ 7 tma H@ A>4Lπ(`ahO5#nj}/:G%0o-JPFP/$hg>nX@BCAH2"#4⟊{j v)jNFw 姜K,Q;&YcP4m
$ ^Qd(?+"Fiչ^Vx7* ID!" +fHNڨ::NLһ[/|ReD=x5-KR:y%=B=Ί>ycQ)~C/ Q`/bb×YVL|<SA5"v +?JZ?_L*0.AR ү.G<[5%\H] h7.P64 WDߵ+'x Bg?/\i+(e͎,ƦsŌtLD3icz_M|S͕QIL-+\` +T:pN Vo9?/yN{%h'e jB#%jgޤ@"?.<9[Ro \L$BIt tƲ4 +zcCM@ynR.O"hr-RBZ +FHVao} k|.?.Gf{'+j%HQ=nrl@d!)$v 3*VQͮMR'
ѭg~pcSC4T·Ռb!jnONE& +9}l +P6zy~lLdOf Ũ"M3Dt v:t5uоčt&#Ng6t>Z,CI " Z3 #bU74Ōmz On<7Dmm<`6m" eG g FGa Rq_Du /`3v)bVӚCUZsHf`)k^xNP`WYȠh;$`:7`Q4LMN5%EeϨ:.4*BEi} -8ѭYĢy|, L=h}WDi闘zqkkIQ?͓&5b/.-cܤڱk E +XYQ^&>$E[!4Y_}r [,,ęAcQ⺌gs_N@ɈNQ+)0 dO_&~Wqu#s9AYqi|鄕("9iߞsFn] 3;)/-gQ*I Ai*G .u[\
Υzmg,"_Ee@j>'5k.KHdܱh;: q^@m[5x +14hTb+T .+Z bC{fl +긎F1d2Fs fB,ˑBVsJb̂1ۈ>d Gpb,B +J i,K1CscH)r"3u xV"z$1؏Q& l'1 #@G&C2di,5@dBdثf1(s[̀(!yOSO2ZTW6uXvR WʒٷeJ!\F3%j#l
C
]Y3Ԃ=\̾2+,X;JN4ShjJ5c|ͼ66U3 yC 1^4bXQ2!9{tt7,ufÏg rV=={ k{sMqЬ*d$848a +P]pO |TQݿjP{8lQAiJtq}id;
u8l J iVL#5%Nj-7}fj6MԷjQT6%z' b}(U3̟jF!Ԝ:xiu/ɒHvjiamp٦NF AgQx+Z2iiŢ-N+qi\4 TEhv9r"NaGz N 4|! ǑV(l?zi~jbCf +!` ʴZ:ة]KTSPּK=gSkwj V4/[ZܩiFfjRij9S+Ev*#njl?՚Vɲjӌ m$V
!W5V5i,h-=UU XM+OnmaI^fwsf3 ]S9z)YSM 1 ڵZH +C1Q35=V\& 66<lG1R ixB6}-DT6ep6y6mX
J}j&%]^Km#mwK'ppNhՌ &7lfs}#$M6߭ d7AՌvt=OX0&gDo\}E{Jހ^m mj֛ +%;f[oSԌC/5G͐s+pQ3*#RZ\r[Hspٴ-.'Qu w8qԌn"q2i4s!+u ZsKD@Wi 'Kfaf=.?NF9;S%ԌÕ܆Ԍ{O SΥ]rNL9!;5"̹f,o̙uX3,Q36s,@CPf`MƧ9s5b ^Tgnd9i=9#-2Ьe{;" 5c,>g}Ԍ:#`yfUa6Mn% *TH3O閸tۤV1+*=Sۦ 勷MW dSaqyK߿=4w`4C>P7q C7A 6ttiUN3&,`:tiӌ60 5nSa:ڬ!XO3HEE`;K,<P:ΨVt ԌE$ 4;͈lt)ҐgmtPXM3= <45 2ywJQJ#< y4X4N34\:iƵ/+VM3wdAn|Ȓ1+QJ!3OL3 4mEf,Lwt@4NfDes}K!eP5 SMg9"n6 }E×:J cSLiY0թ|ՙ{`::֩f4r]GziF:#d0a1cm0;\-X&s4lԈ1"E;:(
"]A" v>4v{΄?pqiq4c*ٹ3?
OQK3aA^= +:H3QoCAf,p9oߋs ɩ
yzW<C0G+tVBgf|^;'̲/42|<%=5LfA/7W̠?fhRf0 ͘hKf,m +\2A$4㳂flAof8 `AO@3꽲A3p6 -z#)gDШgkH =dңD< wUT3O +bzpdu +}9Ҍ꙾~Q _k=IG3طl4cp)W@ϲs 2_~d:]oQ +Opό_]_("? V~~QAٓf +x +PBx~`Ҍ9 =˝y~Qu#fhi+`hFE36:Ә+$?U MO + @6v^ +1Q]y9fP1 F@X=ȫɟ0`c 8f.xnjA(ˠ +\$I\B Q DN=ҟ 3- A| y8-$$`:9 ,, +!3ݙuF } F72c=8(h~ +V@W> Wܪ7kOΥW!; ^C-uGP:K5Gsf=#X#'Zf< >%FoL_8|qp]Ӥt>:eAR_B#dQmCaQx2c̨+8$8Ae +tB&b Ȁ 8R5L5A" +=@0̨"}Xfl#Fw9n=eZ( `AzA)ze,3&^Y n[f,|[fTnO,2o0; /LF ++3
,3 k S mZXg
e,(_E HLP%'3곇>ʌ TdƜf&3h H+ #>(<Hfi D4x иFfH+vis '\" Zb/dIieQɌ( *$3l +Z9 +L!G-Û ~*T9}chQ!q2bDf@/"yj!3 + +cwlB̾T <fdY:/~2j`1B*C{ Z El!!PXnj1#°YQ9!iՙq$3?bz,*͍@ `>JKsD0,@#@SU o_y&×Afl.;#D~u1K*7QQ>*]oI 3 +vA&nOWm'|Wk ރ
U/EiԘ164v ق-]q + <=ye5^plx SBP:+/P+ ْ= IZlbX-O+5)^bdVBDO!\8Ъݳ]/J7 k|MWB{F?$E1FbgHA|H1Qmh^d -1"uK-EYe~A橬U 2ë4dVҊ CNH^4psFJ #P#6ALH9LpbKmEY V9d ,6q-
@be{ɅFUu܊ ՚,> +1D6Z \ Ά sZn&,
.Uf5ɳy%=`c]a~ f$pdk搳T /kk`EV
C +mBLAJwlXazI|18Z*z]l-rg9w+:OhtL[/F-2 2_5|zh{q ċCM8kLG4:Ug-Ot6?i JW*xM
V hP@8u)bn0(n\Wh"r.P jsy2B4W~y
ۇ`jUb%9%@s %] +S'O|HXl K3 9Bhj +*X 1<&* Z8eZ$. +(h@FE5auAJK72۰2+mҩS RDO QL*Uza[!Z 'I/| Yc2'X'ή>3Ri +p$g1ZSZ k"{q2nق +1 Itm +X*zZjXr%@*. +H܂ְH '82C 5AoZ( Nu_ +r ++!JDn0&FH0? 9@DvH +
)Q+X + /b]P.4,cU', ;EB'xkwY7PQ`n)Z>D]БUK"~ vCkFar/0`>FuQk9r >LvZlNa\'s b o8Uh,̭6F4?(:(~S Nm1rNVy1640 0(A&A
D+Hb 2H:Sg#zRg0Vi!Ǭ#M) +W +,Obd5 D1S1%t D+4
1W2 +Vr&'SPtAR + "稽E 2|UqYw=yJ;CbT;h!j|!~Mt +6Gs-Dt
Xt\ /@]: ?dE.[DPZ(:AO%Z*MR,smNR|&U - N^ ƫ7C<4-l"J 7A1װ![iBۄIuA'5lTCs%MQ{)j(jU)zo9z +mEr)ނ$@ A݃Y^Z +!Z dZHjC0I^RfQ`1zC5D'[ >4JT:;ؑ~S^i
bzr* yؑRc1zz"M2Et c]nE +J:PkZ $"THXji +\n:D(h<aTY)X ? d Zc Vb
d zRg'L/ S˭X-__*v*] Y1 Uۇѫ0~I^|9Foՙ ܂h")DV 06asN`&b"w&Y0>k4[,1 yԴ F Tf +4J)C6īk cHaG^"(Bh&'v!EQjӑOk{ND$AO: +{MΔ + Z`ۋVۊ~,JOy 3A 8Lb K,UXR@T]u(Al
^ zQf)WiEW ' +W^@3B-hhp t>T}ЙJàDpB !\l )@yW5jAםBMQ ELY3Okn&Nb-Sf+.*o*f F +Di : v_UZKmFuwqzM A +\g V 2Dn
9Oe1Vnd +p@" 74EZ7δ )vM&v h6FI\m̈́مwq!HḮ,Î= Í| qfLaCh[ j8a=ǒ<E +̗Y):n ۃ xPfp"Lb'2%Y}kfI`w"URPCeA+k.zSjdH*~`w<{ +6DJcJ4!~ALY0^<w_uraw9Ti )LYi[QBsaBIz9^?4:J T H_ 3Ps=*G-fW v$J:HRh49Dvtvdv46":qJuP( GvBeajki1 Zs %ƬA@c?lˡ|`fS^~`TنҜODU_a~'Jj} +CNr E0 +&1JkA.4b8oBdfH1&.;ѫ c4gDE[`3G$aRˑK,O|OEI8t$D#6 `߉>j?
ӷ,?5 c +bTTY'1+A- fA݃1bHuFb;Y~9]o#rFKׁL;jAq+-9hy/vl9XA :SMŦA5Vށ4~݈Y0aay!
3Ya-8[oh8`{aTm7f*!4+DoMMUZ56Xn)|^7O^< a'M8q.Ym*Ui(Vh!ǰ;/}+)<roЁ:{"#J(ih9?H! >{HnA\vC,6+5 rg~qM 3 3,Dնaj3$c`VHHVk ,?3+Fѫ sH1n_o)L07ob筗&[xIUfmYHY_3O?I}Yc Ns H:Pza/Yc J-:2bD=Hil=, +8=鏦ԙ d b1 -%9,SGjmT.HK6` + ؿC=]o)H1H9dcZ
G +
]P_cWM9ޯ$|enW*C^Q kRh!gʌ {yA,b(z %>2WT\y}Px܇0uv§v֢4THeU:1yQW,+Fpic:`=Yz K'-6 [W9( qA4f %J04hٍ8Mmj=tZ6#T+|xtg_t r9x JVǽXJsP+ ,e|g>*FGԥusB]9 ^vG(NDsv$n f+>uCLzrZN(e(s/L3']1q;땬18'Unsj8'n`A-Fw\gi!<QꆻqRf^.t
o8
Wt8Mط1~b4kn9n|Η>^猺tɨ +j# 9v3ܳJCNp:5';6CF3NmXDo7͗^aۏ ^_Ǒ7ӛ<^
lѶZ-#:rD9 45r*KF-{)){1hJйMrt$,Atm~rM+I ia be/Rd$q세t
s\Q +=t @s(vJj\.}ָ
w6̶ =lR7\Z6 +͎ y`|xq?ݷg ;&vggt9^xfu>ز"O/n3x"cB䂴btʩoĥy`F>O}ztm|Nu}du@X3 lY];QO<[QM)mf qdtbsF[2]H[&c)
pgl +Rdd|bsO<H;Mm9 lKkL-74a!szx}bnHJuz+J&^M@n88 W/w ЫEPe9>Q78Pw/N>'KZ {8gU^ff8eL[gyյ\tlWq~8| +bAp9
WU nnĕpv2s^IuUN8]S}zuuV:V6;3N&UW#SA!k<.jhv:N6 m)Uljfl + +;Nq=Y2] +F)6N'idd:5.Z9\^NЄߪU1 1;% 6;c!!bk)='+46W3fW qZ ?/\H1q47 S +[w ݮC)[Ŏ̀Rf0p77\MrBqjOͲoN>TԸݒK1~I^4KVEYvZ W}p,ʰWwcA%A&)KFf +C>2L(+VNͪH+xr+pF|Xn
ٿbT]c>rru.zu8MÝi-<aqsjyX
'sį si:qˣnskN}-n9)ul +Q ݾ[En~Z)yÉk9HQ[mMr:%-~csR?brN?+VNV^g0/UwRuX\g;^qB=~PW 7S7@+{JݾZBbrR?V(rg ^af%jsK=sŨ;fwm; M !RG CSy]2.g$sTLVLw>̪LAV۟%D2CZ1ZsKuR\ e$`&zH"/2 Yݐ?i}zSJs re>: zRr FъLy-۱qT,~.9Retw_}sKCLS/1M@jv +h>Cqv<\/Dzadv,Y.ĥ%uꀸq*fjn +[Ns7ys=x0 ͓8UX1<Zz6 ^pFiϢ4QiME4^3mV%3J| {8]rJ E vlo#fjɒ)YX=Uj? zcX1;#\NKh>wDi(F4"՜y1~j1njd@ĥ e(h9Z[bKS JpJmp[FmլQwiv r 7z4y,ugU\
3p%04a]HZߙ9ϲ 9mLi*&E較\ۍ4\ W0M4S @ a(DYOU: U|H +z=j99ava:O;Zy+6958v usR*Fs=:6CZ{%JZ4,juxqk b}
SNqϬ..' Ŋ-n9TN͊ ʆCRp[ΈK691j<9luSmNiUQp9# O qsG<.#wS{94-wLcg=vr UMy@Vu0_wym4Qt9\ \pZ;u0 ;N}M]GŲ^Jp!o:=ׂ4M bd?ޙ=Ho` + W>2 ,Zl!C2_HMT܉ -ޔz l6PB mf8 )ĚvoJY 2nBG+mZ1߁Y,1 /i ,5eө-8d<מt1 ̮<?u>W;fqɪUC<)*R6\6NVruڇۆxs\Bb2xvlȩjlچZt]KVǣ uu< YeX̎Ǥbt]l(1; :P\z˽(z5 Wԥ 8b=wl
n--KNp1|c{\jq:tE4~$jrF=8oQ<eQe3>s9̸s>Ì
s: Q|YU_f6t}Q'F ':slL
_<:^U9 ˆnt)\Ft"Y *fG$s%뒺bvE1^Ml~ 4D͙Rlx ԏ z n>*GjR71B(3CV®3㹮V1`c(Jq yM!5Zg.vm.HsEkYdNsQ&IoݚUwc=}& 3FvK{(]@|dTApcZ P j pHU8^4?Vr Eߢ'ޟIxa*84s(xVjMd8q,q܈ rA^n-Lq^g_g# + ډYN:ln:#7d49\j)x6 Z[^Pwc +q#E=(Ɨ wh/描rUEj>J8 neOc _NqI[3: ܒOu߳ Jp.K~ [Js+}g;br,\
mdpf2y:VmztGݱ:LFZr]-}.um9`>Lv4OA p}y'8%r<Qܳ +Y;L J-9gt)-q\9
9-〸r Y<F$!TNOHpe24,$Z?LW'>ԍa:)L +Ft%Ĵ x+9v]Ħ~3\MZ2zϱ_D]p/
$ 5'ϼP +2ݿi\h "fULiN| +RWA$Fr7?N@ Ǥӽ p&4(s:L6 m|qV^{0eւGΟ~ |8Ԍnjϥp.Hw\2a D8mVAp?{4s
?ay_[ +MB + ~eZao+/Q`fXLF +@~eO6S bS +г۞Ix)Ϳ +5зF"9!$gi:NFSs9bB -ݮKQ:д>Z!m`%$S
:$s{5-;)T +yR/R,_k w#_pg: ]qʆCNr;v R'FqL]JfGĥy~s?<9f-GU:JfbŎɠbsgm\RJ9:Tgk>Uێ y!xk6NP̲-!4S$n7
L + +/IcD +6L 01 C* + SfہߒT8'==~}ΜiWIv۱n@C+9
i%a뷟<XT^(De<GVڊ
K ]Y7 $h +"UZSkxgO .Ya̦ +&Fܒ绐uƟR FzeYc7S/# 8?ȇ9(8x51rMkQ||旚_NJ{jWZ|to]Бr1^1Bw
IR +BJ)'|3Mz k 2J:6#̢@]w>iM߯ S <V|'ibHJ{Uڋ+vjӉ/zO +>L!l~fsY݆z{0*hacXTAe47iޯY5Ni]X (Y"0!M1)v h:^)Q4``NG E%4 +Rl #!^ѫtǝRre9~aD9tݭMVq]% V +Rw` +?`ً!Du#SBo4aj.Nd +TM''7r F{uȱ)Љ>y^$>p" .p&@IF *\9/nZO# q BvDIR kAk;. _&r {Y[J˭h5_N-Ucd) [Ѐ\-/I,bT~\Ā]^kܚݢw4[Z5+h<[ p~ ly)KۉslEiu +*8! ;TٔQj){+oݲ\I ĚJ 1LqVI;i]ý=X|@BJmAEsz )E.Yj+Xj&DoEYCM
%{x4~s# 1IU7E 6#tlFs +Œ̢O-նGj#XEOEтESձ:.wݦQ:qۈf3RT)! +[^^ <if4v_jw+2 P +/O$Zɵ! mg +M&Ŧ*q,<0v8mwQ\N '=]^ET8|m8fBHEAJOMzI&)Ȇ% #E+O.5%VZHr M{еurI%VgAu u /0mcaCUa[ +BbJ?f3IKaTIV_(0!`R CE ѫ3^us kI]lⶐ M^=ݟbUͷr}Lf֫1ViT d+v$I1Q8/#̦4Ec
1>b +R]D <mkCeʀVF)K-mָj#e;/PۿJ1lp o&/u\:o[=q*9`3ˬęE*H|\8"No"V9gf1"Ij]yo:SKJZ[
G5S]GQ:_&ȱ W ֙H m$uV~bWer406#WB˒'3G+3 U\EgWkn[ +.T",P\(-ӛL"~|cQ:@n<yhV
ӳ?Q~JO? 9 OϬiũϼMZ:q= Ɨ )Xg .'9bIh)R & +m$zHRK
s<F4OM['N&ۇS_uԷKHMc͒-G o>Du jzoNKd5M'
t:U^烢ϫ;Ή]a>L)p+XOe_U`4FBH%EĆm_n,Rd5O7yi q"JmT˲_f~ )yE[jC +ҫ 6~~_\1YdIazyMUrݤU3
[Sbw4jA~mqU A䊕=BA @a*YXt$xMc<8ZUym?h>"'
Wۆ
e]긠 ܋ MF2ZQ <$ +Hg:PcŲZt0a =)5
; %c'kM3XbhTb͢QnVZ&?m Bz`~z|dt,(N8P""C!>})o-g #JzSAT#Z2X1)P"WVp!G4ĩ2*Uݗi7M8% F+olD{xrF}{B ˂QEZ!Knݪ:|IBF ɣos*
/]
)RX&Lsxcjð#.l5`Ѵ +MCL< dCF`K,օ6u c
ۋz|n7}WfnX.l'I/KHM5^7|xһŢ'|Ts" lIb?VwyzV^D2eI0Į #uAK>
7
L +s f!pR^3v+2^ 4;:~rj}&\iհޒ<CZ1CiufZ#Q~y 8Qt Oxa:Ӡew冒_=8|Ԝqaj?u j0)~+L:}q^,l[gĩ9GPH*6^l^i +8>&O֜d +&G: ;E,n(xTi l$~Mg&Oh-Qj"xHr'1 H4O9'}M47fv!lGy.S6hUBC9ۘT>g7'IjaĢokL3\79Gfz`uf,.8K"Kl/>r?&0cZodR6_%HE 1~H@,R,|DfQS0H\ePBP/(0$YZvcZӬ2,&E6JEtPZN6[o&ȱ>Qb1cߘU_
GզH>?<Φ ~Pp5 +DptND>]o#)X'`5ٞEQi)I L\?&v Zz Eَ@_5aBA-z@F$~IhVY ![q#]p& ;Kd9T/Z;f|iWŦ#mbs{HJ_lRڥioiif4eT wB
JeXch%v 1asJ=bqJ>?GiQ^i'xOmY(5_595cَ\+aV2N0Z$¦ 2?k6FkWM e:b# 6 ' +$ni~w
`PpR0m['Rjᷠ(``v bך`w#X5CQYq67me+V#Vn} D9Ã[9j%(QP^j)OO[$s{)M#J,5e
a߶ IJ=* v~Ք.5dz FEC_l@iϴܵ m3-S`{C+6>*-ibInS2 *9Zg"I/ slwQ:ϰ>I$x UUt3a< (16Vc A}hyd-߫XםUEKMKCՆ =
kM2Q!T<L(Hacң gLP[CNr#)NE`a8 )%Q2c%,w{VSN|UtJT6K/ٖ˖Q"g +M͓MMaٍ
UJƐJU8fma*+Qn[^y^5l!y&~H6"֓@j9/>?fרr)g,~~2DUx
g=P 5h:f=fp: td7eU}'BpۭZ/5mds<[Sf;kEP+C lOj F
=eX߁
B
| %':T0\f |}z)>`\ȭwMYU6S+!tJRl?b)Qd=GRlu6(6AI[2 & z Ź櫂͐|$ n!0n4p 9]|ꮂv NU'-i}EE c/YhU X==_j D/ޔۆ~w*6#'\۳vͭ3^Ew n o QV<ۇ~z% 8EϠ(HNI~Ta{SlVx! t^H2qal-Ygĭ3A/=h@ ǧ&զWZӃD<]e(̯mMpUFjQz?{ +vry,<Yb,.4 >ä"~\$8ءJIn@^$<Yn,P/SM*ub} d`|̲WQ'ʯJ6?ٹ Qj'|PBR4a2k4Ѭ(`bU H4:Kd# ejs;8Xac6 +
U7nތ{$\GojLĸU%aKf-fqYuYzs-G<Á%xUg&Ue!I/6%ڊT+h+(R0v zivG*iMe3U' +~SсFd} +%4Ҿ@MdhF9z$\7 EjJ?m~2T&y)?ff:Sjֻȩ:[#EfRڍu]{NCf+{ +rG#NԼv#CY9HM[\m/Or hPzg*`sU ۅԳG-\v +HY|"^gٚj +Ɋ=㸄z!.~ -M?i}6"h>!|8wg^gqfb<p~f| h94GhW.c6XmY4g8v`4! sPڍ)HE{~ϸ~xnct
G ,}jLoˌE㝊z) +b)vKZՌMo9oe۞se9^g#E4|B@|/Zm+N(v( B-tdhj}j $F˳&vC*x&xuV&#rb7% +Z ^V ӫ#KL?n"[01Gag[ dEyծ[wˣo $6G-gcoZΟ$o2ͱJ(HYh zG<'w an8 |:*ףv&}ȬGʼnYa(/K]i37"v|V%c:(4؎զ].# 2j-lnP=bp0o"<alZFӓ懏%75bz2Nt ذ ,C
V jPΔ4mp c7C7OB|{-"z_pJU{g8
/Us 6Iǩ J.P.3\Vyj{vϑך(mF>Qզ¬3e`8l UFv)@_Wtl/N6tʫ9v[R} ". vrtDZό@^t.i +ǠcPId2%X- H}nkK-ͧC1 lB7g fߪx07 ' +VÆ0X`i ʭ1XE +۸Bak@@e{R+vS 2uzP|ԲX ۣ}o/(mXzDvYd .3={U=2yR;Se"Mg[lxӌZ026۠#"G
(xOyT }CsydVLK,Eւx5y iHJRｔxbVl^l@><J@f`#ZKMم̒G7 oOK:kq~ٞ<4 Ӑ:]J㻂^N F7-5S0Um&$NqMZa85{6Uj + xٰF
IŶ&p*Ș[q[OQXh+#5 .y$=!eaaR'V-<U 0DVc+Oj6{xhYe +Q-{m.MgG 0L£Edro8n2yZLTUgӸTyrPc` Y/ 5<̊Hg٭]zw+~Yߥpy%a= +-&UjJ|4EEU Х DEWqDzg +b祌mLDh7[ޓV_6'DQEZDzV Y" C1HJ[1IIX <FW+DT+ka̒;|eFB4ovX֊Z'LGD#5J2r:WHBT`_ BB^#' +S8 +.2^}4+Ӫ|rz%6Ob-P>TߖEWb,=<`D$ p|@ R'C]-gTx;uU12](IJQ}fHLhH }V/6)cBl~$:eY'-zTּ76=Lf!fsa6LL߳"Vv +߰> ڑUݢ@`lO#kS0iI^aA*D:HJ!yt*fvt>^oFkOzц]6[UY@eh@ (Xq +!0 +Ki*`RVqQ[sm 5ϕ$db@ dЀ` 4*x +a +"QaB
x +v u`*B!)I`lhLA# $hp
@*и +@@h0@ xE ++ar[<UkV&6` +C#6Ѐ +46T 4` :h +'46x +8Р 4"XpAB@142Ё +!c ji^t nσ%3F{1 4A +7-\|5,xA# и H A h
4(Ppa0蠱G0>XQJ쉰c$LČ۰.G
{5v")T +.thHB +4hX`#^ 8"R +*8Q"EŋTR P9h:VX@K E r:֯ 4 C +8A<h@` -4J!E 04(pQN +h 1uh\}hNVǫN4]q t +bF@ +014, Fp1 Bphx@& +& Xu,%1IL `0 + (^(HA< +
OdϹVyߘh K[g--4֜7^tF[`:@A2=a5`64$xA#Bs +-4.`(@ +\`y*@Z@' +h.F$׃}K055$G !vXuds,b @Ug&&ä7KTL
s Yve .)9GZl#H+
9=g^I q>m(ƈݬ7nwCJԚ# BHmYȱ2bCV}eՌn{goQ|& vf~'e텔 ͞.B(VH<lך, zLe0Vf>Ԓ,߳^^3w%IWlbq 5LҌ>X&}$b +VL/ɬ! w
Q[k#v nN %S Y GHJWo Mm^pj9>6qL*Z
Fh.DGQhh r6o_)MzS1BZl6
G0 .{롲d6AСe͒mL n*H+FfC" +c($Dvl u+a-CKIr +: b6/6h-׃r}DjbgKo[h9 ` +ҫmy "Iy 35S -
Onmsb1^xfA(نOL *ŶMC_7Tf rH8C[j+Ԇ,Po hX[J{(ɂQP!{y(S_;qXFf)_o0Ѭ3%V$8-D?)(WNkPrJX8u"ɽˏYBLZ/Gۯv%tU棆ߨg
E[j&-#6'Ae&NRBC*9zI_?}fm?":Ӂ긿3
QYm$)4
H^0F`L(Al %dGk +ˬC`/C)Vl#E:
a0
w +C]cbNAPQRM4J!lR57`ђ3(J> iXf'ҫ)4w"yLÌ܄moL fae]j(~Vf#vn cݲ{%|1H%/ Cn&D͔ڄ
*{R J^%fN6@);A=0qP( +0V#Cʬj"U"X(ٲ<jCuR Ǭc,` <ZPrG齀 ;Ssex M0|,|ņDEt Azy!8Ii . _l: +6H .a9jB ۟;H@3 +Z;V4D1z[YJr$ dkpSV÷bM@CvBCZ +0`*~Tk0P_iUrK]o%K.
73j-GC + ! +V}F + *@oh"B͒Y._d%|Qc$-hW=O@)OE
9Jd pVJDB4izwl)8_x2Yn"vbF[o!~Od.Ю{ ~3DQ Dh}!j0˭JeU4A
a;0c`&Lr(JΖ
E x +Jsd~ <|u
\ `X?-b ke|
rݿi@
7u`APCә *aMלB]2x?Xf&^\ @e0BvzcI*I/kC>r(hhԼ'쁎*. :9Uk9M s@ W"N*@QcuZCCʂIPjcuOJ2VR +%>2Oc N~l<"ٚ7LQx24(>Yk %4yH* j} dfrd<o%1^( B@ HZ9vZl)*4C%ך +'FaDPRmaz뿅3 ,9JbXTZ'Zt q @j T Ūtǫ&BsBԎB
;ыBŎ҉ RY&¦](nH"x:k2qj3r{4+PfT~ ;\.H2MG)@Fy<v + jl7T0;_Z{a%I EV[(3%Nc&t~2'JAeYzqQ XZ)_k71Zipҳ~C{<1c9+?"+Z4N:Hiqk<*P.K E{cՖUم w7Tk!r>DIHgE=qOnǁ~Պұ]ILRlOVuj]T?cuE<*^!jdWo0U8O_o,<.ޯ F 9s&é1T +Vf`h"Т$6 +-C +ƿ~Pv]0ͯ
ՙ T!ԲފVr^EYg*Xc>8\-;_o*vW4K>9eG`B^d1R(~m +
FQkE1{/\?!q+bmn^)Y$qk*\k1px xb
d5v%J@dW e8zjggכ Ȩ +Ra zPc G0|ѫ1b+{"If,jXgeH+*;scIkbD@enWb AAYeh'zNoʬO ^'E"j9O (ECYe.Wd,zXi dJh gβ { jDԜ[rWC N4\(qXn5|@ H|(rV=ROQrU'̬4$[/nTN/Td ˩2fW $yݲgPyjM8DHYojjg27B}tORy
++nbEE'"t3݁̂DiWG+筄 U&#fUNp1 +(Q0Td1ï6!m<nGmr _tELQ`!bOH!:#"15z,+ +u;RE5 ӚiEaFKRpkV2Mqe <{ Γ.J0~$9E_VW %Q*1!VY"K3 +vPf0\Yɥ3 +8Naan 4zs&J<xcX sĶXF;9mˉ~г~eVOYIb0#);X^(&L6b8Re-z[lЫD!}QbSX
+endstream
endobj
66 0 obj
<</Length 65536>>stream
+M6,Í Eq^C:S\y(\FTa 38Iةٲ?<o5(y'{zrIMYY[X8FT["ZfFcQ`"<f{Q5gmi_p?nS*&I*DS?,1S뾉^b 7UFժߺqH9 'Eo*sQLi.R/蕬Ԧժ@9SX~c_2hXYZ^KnlT OMgm}>L3QQ9Ԛ45L=c.լ=1O +ze"Z]0RњLJ?ɑȁ +l r #|p,rI칏
mڔŎVIyqH +B熩ėuA<QbriJ\.٫˝4&j*\0YrC:K`Pa-טJrBV2 $},fYe*ㅳX vVK.<:9j_az$wn G|6
"4yE1K 3)"QBjZMI iiy9D[!#/.9?bn";yŔP:3ЀGSp&rJ DU%5d%'H +;Iuɢ5J8*~x!b4#0ʨ8ҡ +TABPKQTS+l6 R@n9f6$Wl+hc( zn,= DF/BsرTer*ʠA`!]iBv1LJFlFX%EX8KYV/&xJe i %ŔDB5btDtUڢ:i}WYk-/5ZY6~Ym 5MVV.USJ4^$-Oчb <!UG|:ht\pKG;YON'HS6=Ziϗ5&:C!zH4NH`+] P<4/
$AX`lQBBN(\ D0A\5(g3cGئ[*Hhħi +JvO7lvF݀<T pS$bR C h#p` $ ZBȆx`%U~Hn7OYW>V]`NX]&/hWaUݦM^c,)qԇR !$LO%IL| `%1D1=!UZ[/=PL&rslE ki;P3 Qz:Zr.%H! F|%#|3&!G謀}RORX[6] TзD ܒݜeQ]Xiҗ̙Rb[ QD U+(ϵ;>"bjbfUQ
qey ai@`c(-G( hP$<4p.lBA@t +h +1lGsfꡖE_nϙ͆ECS?fT`d+J ;PlрH0/_(hᘠ&l` I'A +LQrJJb +!r^IXg}eŌHYfR*O7^1'0#jL0<b0D=!PS +--u&|WUТ~zYh?9\I\EnD t Q@qÄ $ՄLR 8s4@qᰅy1YԊ.s" -%l:$ʡ~j[0h~PF +8078ȠS!
+( B7ha^-)sNr*Fe !s$AF_20pVzXC 4z 2z(FK@ChCL zx H%UǞ3
hBYaUd ψYH[*aZk @2EI*
a(w$Ve/Q Tz#yp3<:F%ԄVѡ +B ` + aD
D
VV&8*T7TUb)dZ6xŠPX W$. ; *h@T0<Q tS: \mDTmE\PL:KPWϟ/U\q@" G.(@ +a3
+r0cZPը8QȆ&j )+OT3s\XV_!9ɩ- | +4)2htƉ0 |N6p
a
Ja+ dL.DdړU"c{AF`-bс, +8aF4 6ЈPF +4(Иp + + %*X@! +r8 +Շ$'YMS: +j#fi ˦*Y ߖY0 "pY\U%4y`&F ! H+. D&VO,-vB\AaX(ƪ(H + 1{r$ M&<02 +L +4"0(" E13txK +[F-SP!Lq-*xB:_Qi}RR`ۄ(r@F( HFEdV 0 +5.pMZLc61 +DuQT,}8Wr $O` ?H( KZ(<[nT@9+[>C!HQ ؟CS p=41qi-m!Hpϕ3QdZTZRQי5J~`J,Z9*B$٩ +s5^bZm籠(#t[[%(& +<P
LIV4Fyngmm.V+s5
z\@K_o"#ؕfn' 猎iVx#7>J]h'vMNYVi]meVyb3%(%A(= Au1!qۨH]Ӱٶ<Fzix̚͘=3\'h +E9
>&Hf&鵅 ĥe^ 1Y\fĖў4[fYeFlS&[vg"01cSC8)jKn-}avJPc#%o JGdUrO^f< .nu~F! & m}n/Ъ9ҩMeb(|oZg1)rJ_4"2Ls$ЄB}Xl`AEd/QAےh '%k}gK/K X}\KPkЗqK4Nj_e Kx
D֑2 +DEunp 1 %ՏӊĕHq&z!ꃖHilRY0&'Z0]>08jQ-۠n$|* +KJWXi)O9
jm*ZC>b .+UU1"pV(*z ;*Bz%5ǥP9im߃P=g}۵=55V%G:O*48կm7iuߴK,BjY
*<8jQ
g4~K"J!`2|%N" H?7%)f 3ۈAmX<;5(J^cml4v'#VCzn+ZgF=o^e?!bMnϮh)܁3%4uuZ"a9ǎ!]?NxkzŬ[5[UE4/ۨmϞ26YmXO&n.|'km}Bs-Hať0P^u\`g<L
Q6% w '\bW@ƢL+M
EM܂
׆ȄDVm +5$4fCbi1ΰّ[lTZ#~qBQWo5쌈 5d66ƫURDU!03Bj߳7-}ְLLEz56̊K귥>y5u~WJR+5iB"q5NK(-viV9L k%ҷH]&ЗY +M$RO:=T'OXeF#2#9ry!"L5@Qj *Eܴ۫}=וwN.?UZDTNF_Zi Imo3L L`"dDEd赦RYԘYr=cƟ,|Zz=+x U ~e6ߋ?,_g./tVMf,O/݈ 1ߞy.9aU|ݒudFx+ ofm9SpZ^ѣrJzmƫ- cjiKM +6ͮxx*_7D:jD4SرZVپS?tm*(ԳnF|{n{rl۞Z0PZc(BWUy}D y1h r3ybW 3q Njы]!v}C^[n̎[Y˔Ԣϰ=h s/5FLTȌU0,h{ը]qMcVnnMmޛ~V,TuR\^$v~F\d1WmzU0OL&Jih %WluT Uˮiܦ5V)ySqO*h<~Q*K1|DBQ3ӪjfQS<_j3^i]s5 R>crGy
I嶣5n\B ,x +!K )uEjZݎڰZ'
ق[3 řgnkRn`m0QybT^|P(F_
ßjV~{2CDe +8I, J?SIc4*$ZQ'
WmD _e6S 6ߐjZ/`묣.q4*{0l ;3IA1a &H
aS{X%V:nEn՛+9B<<?${pVAs gYͤFc@:q~3q㙄uL&^*Z5 ݀/3N& D̤=㴕d:͕ U6%\+ZMU! Khofo?6~ Ƚrwa5Mu6#`dBO\&7З݊ŴB^:QEng[d*U
k'H 2=ai?*xEFl*9
(
Q JCf!Y5,E& {~t}
;vfJ?o{ +fKQtz8ja;z<o12 UM[g L(M]GpR$IܿEP8Y(Wli(ݖ"=6
zᄉ=b
cc^RcQm$E3:Ss8# 2"^@6S^eE,̞EkFUכYϺ3ficko:VVn=,ڒv `ℙA{:洒o
e]zόZqZRNahlYs^͞>_W-ËS 6Ktl噤@D !9BBal:R0f cPzEMPUiQyj1 [ꅶsR[yf<ΒWtw͚T+WTLőB.V`ʄ\!St\~Yz~ d +n+if}07U/,5F%C,L
"B<*R,(ԼU[3!A +v[blūF( Й 0CF\rPQsȥ/eS*Sn@i:Sq6 p?ͮ)x@a8,FծĪCkézT8B`0 I.9R$?^j&|S9Vל[t[K^k^yY/K~;^ xIȧ1?!CjO6X-m2,<hiwik>Du"EyLHd "qZ[*3YvmxѲh At2y6P1,
5ۯ*WP"J]l;W/Z +ҙƪٔtKo$Dm^WS;u•g7 <AyLњKQCuAPy]ѶLF4
H\=]EY7e3CV[ohK2#SQdD;`$[ +6Fn 7Kp!tWJ+Sb<NqrƳ~8DXW63e +m"t+X)I)8a$f(}RӀZl*4.-*͵+M +F2Sa~ي45<k߫zZUf KFx _Em^c:. >{z@0RyMkP}tf1drC(IFA T +g
K[&̒(R7װv 91JʖAzFzF5q}E]r:Nvg
d7 #9%_ngT!-@ǐj-_њc9_vvnr<{U>Fag ,w%KZe]b̫NȪ;T^ĎNxB#37 bҀe!fM4bfnC-ө: fk%Kz.%Iֿ0cZ ѪʳÈ1ap")JZ_?j2ErBkB +'zs+T`xq[*`5 +PRNU1?IZ]b?jɡ#a|jU>4yLPT c}VyXךG\rk3Ӯ2 t-gan0Ud!IFI yc8U +w
@o"t" '$_ d]j! JOyŰi ҄*+1T`-0*OTtj.Xkr ljm'"5CvHB;U %ۃTq=ĸJ <X.Q0%ө2,[ LGv mVe)T)RB +CyWir^m7U+Y 刔nPOfZ`h?Ne X`D\:Q LN1Rq
<V tlF_n-ɱ~uYr9Oa⤂ŋgŶ5,
8CZLblvS +>^xXLDz \L
@FjdjwP3tRC!ɍzMgjq0Ҥ*[!)ذ{ ":H ٴ6eZl3S/7c֚ing 2DR -Us+1WZ̄0"31[p$BXxW0!">Ib^%Vӊ g +f1z8mOA>ՐCe{1+v]j,-]G r!ůZʱkDP,NӘ ،5 Cabp|F 9I9=(ƤTJDqD +irz)KpISK
+@~*=}m}6<g˪:Īm۟ծf1Dn4ϰ5gGfҋ
!J{!0βy ֻ8V__j-Ѭ9 kNR[BtT¼J+Z;Sp QC)E'Eۥ'q̀>+`R +3J
EWg.OtL%J5"{xZj5Z*3䏖3,p,iI
'V1MH
$Q]iЫ;nǹn&7K&z _@a !>A W+؆J-WFYOPc8SAѣ*q~WR 7v,A)x &ԌKMD͑{/
4Of,M &ņD(YFHMj̢{a: +H_x ruAr 'QݰH&3p*`BGǩV&X(R`dz#SYFPh`j.ھ$w{xgmy!溵@<n?HJPk,ZeY.Bk +4G&t)SYgS +1>9/>h d +;<¡v]A!mSBùrHư]=][v;K lm JSj_`{V?b 3$0_n~w8؊Y2"h5'!X7% 4-i
z?i>J!IDYU!f+MyevVL+[_M{Ğ9P =%酶 [{Rm=>2[~~v:U+EjNȒ*BTDq=2}ʤEZg94\1yQ=jc95Bqr8W
ElOv 0J[vkj5iP߷MfM ؆3k#iA +$/bvfPˬŎًY\1r#b:\h$BZK!r 'Y_ߩ]5]`h}6H Pa#@ X\_ j~0.Ƨf.Ϯ|=l X`$,Pz%%M +ٞzt Fmfd7.oB@۪Y6?Iaz!;J[ +"Tjm}s^OJlzUFC\\ HK-E1 +>#:[bgٮBH5 AzБ +|(hqjQRV
XNc6т-p,Oc+N- )X
E.b < h;?q EQ!q -۩wʇ6~敌W:#9~>z\kxL +; +}ӓ~ b`$ GCMʇג"p#[j8z&z|({_?*!\R
@XZf/M06#[*ǽR3y& 8jSrvE2iA;H鉠#i!=sJ 2g0c*aVq4xI;@咠D§ofanܥu NW[ 1D0~cW6=o$[``q"@slEjlhuv_u +A +RL81 +6)p ]ז|gy0Qg \~ t r\ , i^V
&&&Բ(L?'\H 1UāL6AGӫ +sPa>pR,uD/B@,58?ͦwϖnWb$ە '^3Jn'p ++X4vslkQs Qӵ:)~
8Z0!d#Ux%V|jTQ٦}"lx6 !bh<>=zn= 1;,R.6 '=#;</Kf S 9
N{ +B [LJLc*[qs7A WVZh r>%g\iPmy̞`,01`rA~՛ZsS7Ė",6 5I B_7^k;г~C6 +1, V1j5O0Rj O <$@~J v` ?
u̴ +vߙRyӳ[#Z_\CĄ?`ePR[X1{ JVjlPyZAef kF>W@hx0NcX]!xAh2;Ηacqe'uZ0J9]DVh2k}VBS~@UcCY4kFpKXoQ~π4o5'>Ө + 4tLFro +)d^'zD
TZVY .DPI
&aGeR|sc fOo}z&t &)egqz${O =Xf,Rw-k7!JW&F0\_l/Noȱ uݘ)M|!A r/=R`9Mbns;ӓC~sm[ߘ6ZEjVD[R<[ET}0ÃS:b@*V +*w x< x< x< x< x< x< x< xw; wǻxw; wǻxw; wǻxw; wǻxw; wǻxwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww +L\1Vs& 2=G$p&bF 㬂tU 0ǤsHKUGhr[ \tGσ-j}Aa1h +˖TctEݱVrygRpܬ [6q:%lNכҋ"I06-MW Nh9 b<p+Nz}۽4噟rT4r9&*<"Jg:g Ba:`se,ej8#-7*^h%Fqi}πf2t>[0VhnN_xSMǁ;N6\m34 /bt\+ +G'62U1(uF +PgF`u+gW186Xg n8ha-8k. +Hk9X/<Oz *YY +w 'v܄t\G *Io9g=Gf)UjDNH侒sZt/4M7Q 뎁Ʃ́.CVr|݀ղ>e ;#j|${NYL4Eq˩\=Av<ae × ;S0ؙ݇BCõ6|g34b3P ׅR߃Z !D0s?([Zj2d xR8$Q}'U.wԥ la23A4H /)fwIbt,Er>OF| 9L|K|η +q| +.5~Dm깖^rbÁG[D2w &MbR8YeSJJa{6Gn8:ao/t<uFɆi/I5c8$vu>W<]d<.5/+j ;>0 N{Eg-g4v B0E'8YW_I +C,t3J Q̯R~Do3R/ #XWGtEϩ' S v1rI轌Ҭ'ն*
?7 DRPV*#=z{u6Xk+|\:(n n9(>#tϵ {fnE+VL$xUsfS:Rt+.مv?# +.,Yj+Fs__>y'uP )~ tROs^rN,.IƓ uۈ.o GQz=o5S7 N]2<}c0 qӅTm-gR )E0b{Tvb{0܌SGA00ˮ3Jt;K +Q>"
ׂw8&(Hr}ɖAtMj|F!g!U-Ǔ cN +7@B(|wQ.AH
T\ɐ`[RlI_]9!24 9^'t9OzaNdY@. EI^9[g;k&uÕu2^v2Pg0VgL+z 2#nmYRw +1|&. XBřņ ] +^Dy$tT2I\Kၹ R:y Cpۆ(0 @m9^ c4mGUdv.8ηs0ǵ\Q/5:`k*to&QNWֿyLxy[.
g/D~Bu7>(MxfȡnDƟ>K ;~[n<T[{2%4&+Bq!@Creoj/ofϤA9]S*F72,C ޗ !ɱl'ƴm\.@#
wBTc9_vE˳~Q;Nu?m˝js +>47]5Jt+Ƴ $ q0t% ^a(|\i=od5tHbcq$7BqP<huj\/4t|"u\v;V
+Qcd`<vlF 3 s>'|)Οo0It^|16β>YlPm +KFr1ʱoԪ[Rb$9jL^߷ 8ʹ?We+05d
4Uv4i.M1>)J+au(^wg$iiKRs=oDq?kY~ϛ'Px\*&zBv'D՛( 6Td /5-'eHo;]'R4Gh~=Ų?gYBNJ ZIZt6ڋ [ǑNrcIһEו~zȨqrZK3`@F}W[ Z"[0ʖ+V2TyͶeP0OQw>הz9n.WC >R$c/z'e Uqt>'ٞ >_kD*Zdפ3pUTߢ 7Yk$p҄Y-I -hxcYk"˫=+ 2Yj H8~/ciuvba)0 ׁ<<lTS?bsiz߲ [f o%c^6Jw9;|q8^w"H;\m5O?iEo`vWx-s律z 9XZ +ۉPaP>I5-q=HEi}7̧ixorq8t X1~f=/O1F=ñ0`=w- Yj{R}緞kh~LBq;MI<p1XMh8g(p@|F2J +_I7~HMQXyDU5D@e/Wxzk=p0K~(w`Y`Yjб*;9}=߅4iF^DZ%!l@'X/Kt%8߳ϵz% +{+/J{9_+QԀ1q8f E8pd~ +Vv>Z QW.DF~%Yb,}~9]x\: v$Զ6 -YW!"B{+5 y/2{JKA"EB~)N,4¤vf0` V!D/4A>u֗(Ҕ5 l2 @[K{ 1zT l7rvZ[O!>DT&1x-ka Y 4ae90XbV-W?Pa*sZyZ_զ/ίYS6ִ? Vňݢ̢ܠq] v [ߴ7hEy~&9 SnXv3m62 R F<XkPm8Ұ63a`j?o
DR|vI hObzSY~hc< [H1Ci;9tD|@̀ s@Az%E/6%MD&0r9K݊0>q}8z4M4V"4e8e pŰa}> +WL4wZ-{ZV y*0 "=c +%>ddޣmmԬ7zD`)P 4Mbxih86_{ `?َ&t5Pf\XzR G:b(pfG&rQV`
y5 ($E`m2ΰI 0K Un}liLYh5Kr X
{UȀ+ m fv3a7g
u $ 3Mbh!Jꗨ7ֆ3
Ya 1M +DBfNFFw
I}P6w@4<hng?3>C_l+JF*0 +KFf p5F]Nb\m2〥 祖{;Pu X拠R +3&T, nGJVG#ɭ27=qcK$]kby-g9DQ:Zj2ϱY}";IzT`{ᖚԘfTެ>c;< Ts܈m$};Q!= +zuSr8$Qg;CXVs*} x .Ug8I +s B@nuۭ[ +(VjLaPYC*/_aⷵv!&HX]?yM__n2ү w"v%`M4̭'Ek= 4Ϋߖ[2 s`Δ/^x^OFJJmAn$T[i50iXTkO HOQn*Iq^|C@6M +E~|JfGJ|S zN6\0'
C3kJR\_A.Js F9~P4D)zא<.kn
,4ϴ<\ OUbTFyfW*Rz?l;= T:9FkJrn
U +3%Y[o-KQå2O}s_Zs1ϰ;M[^|!<HEKXJ + dǫֲe2yUxuYN Ә 6&Z +!ռHb~ &OREdy 0}0%m}ˡ/x\k)vn"pv~:kj=Dp|S:!tZѬ CxyݘU7],O@-ע =^A9 `e)?m:4mgV-rbQZуˑ .70Qd/xUh7P|Nܧi/|{79[n p^Z^
#֚[oYiJޫ<N 1Ue/Q6*3 8L>+z\7f^J56XNc Z^iXG%yQs/FuX>g3gupRX9Sgp2U/ K ne3: +͢b5bmQ!h\ j)F0<ievq~
9#:8w yAby9u8~6 ݿF-<ȖY"o 5bODy9-CUFrZq!?I0 ?LBSY+7Rt ܆G#Bs0ݟp> +&!VLV + |TZl8vwxO6 q|!~mbr.{b6;y)0'锸ev;L/,AVpn| `X
GRԱh=(2 +`\$+8N5 G:^essvy͵\W 9rJa
Ӽ7Bp5fj g߁uk%XtE + `õ`DxF͏Zp[qCǜG U[ ֛q+- 妴 f@Q
[ jH( .Lp-6۳KK=b:v9ʹ~ gY mJVS/3~tPEoJ^mBbIcُl1&o|`L7HE8gYg4D"LQuj44.4;Wb>?a{Mr\Rp;M]t;bT)8BGUFS~p.(5maIқ7asFehmBZqzf&H
ǬV; 5:Vj 5}Ot f9ƇB;":`m|)Ao=䗥F#2gc5"8mcj)#ld5GVy6C JS-O`lPUA K*$El:`
G+C#
?Zjtll0 \$xUkC/c{)<¦jm 7t +_˭-W]m|@fKMEMN1lYA"l'gYd ,z45[lR j@dey*~Rqn뿆3 T$ PB)jw~pʕ(E6G&Bg Q{XuBnʆ@Yc8,ᖚ-79Sk pZ VWi1U4 LF"7zVkؖ@a2dz=b;zދy]Nj8,E%C wЙ(v-g8]M X PЋ K{1;I?o8si/4Uv &Y~KOh
a s78n "7:;F}>V + wơU7q#buWYr3Aб2yJc^ZD+dRͳܯy$rlx ˬ55Nb]؆)۹ z$ZLEЊJ-0x/RbD%l(vTT^{_ +X5!o8Xp d +9%&zTB-+W
B-hz=@Qs)?WeQXMTk-'efb~Bgdbsa9OGK1%|NgLX)z\l/`vcrhLWc2;ӵ& ەY3^EW^kWkwOdTj߳uL{\ql p;'
۫ݰ:ɒE9fR :*1:m/Ft
W2{Jɉ?Yt ]J(
71 -'r4βf~9~׀Ҳ )=^)EyQ^ >,͆ˌp?&z.ZHxy@ sIz4Ԋxp#I. N+Fǜ6z
4N-Fn?"5AKxZ)߶T-w S`!|i;1 %6aFR ]U DLUϋ-鬒D +Ҍ^s(r6@%2z +B^ 2Pt<.؍8Z_j6Oq( +BTft'ʬwz`D; K`$4"Fr,>Oq +_:TiP4Xmo(HP4·<LWC
U + VMq0˱}sageR
rCu6 l
FlEyX(Zf\r9(TRqhx$lX Q: 8 qZdRP`YdNޣBZh d*vDap S{5n~ct ~>Lu@M!Ү Ԭa=[
gQz'ʯIkyzߊ~|-'Ⴢ ҞTw4{ߴIx-< ++[-|9~c +1
4Jo%/zQ4 EOJ +9LRC|j)Fcc4>>rV R_ E &N)zj^"*EĩdD'R !8 +6\*q&|Yh+z_j0D3dۭeQTb tj~,CLqq] igZ^ EXd+zTfh*Zc#J04^(z0_u"='C']XTT`c=Qul7 <J,H.#U
g˓Bz7 r\RmT혿"')љ u'??MX 4(Wb{D,A,5"݃xVbQ
VU dNiW٥ilNܷZܳg}& ԛf6Sb~'H{ _TZߕ o^+2dͳܿP2CRehA :yP"zRe ƮeX>NڊՙI|gI D;L 1@z4O(}G\qj 2Ī4ބ,DND|2QJ eLZ[Ŷ W[w(W^w ֮[5\Gkl 4?,6(2NR R ef{nyT 6
T'zUe ~JNBc +r,~Ӭ7)Xbe2$ڇE:UĹ9RkPoi}S<mP o.J -ծu2 +9, /x +hKP~ j}0ߠ$6"8E#ZBz`C"iL;S'
1?D Kܥh~DbA?\j1q F}i &Tϰ7# Tfʬ'0rT;M_=$ߓ[۹s Z0 +191
`'M!(AILFe9Rm036ij<ufնqrc1b0z߮s\ $ / 0F@W/÷ jJ-O\j1ϰ%& ߖGRrCAÑ3Ѱ3-SيӫL|CKڂ |7ֳ +-<a
D\HRmp" z/ɲMd$hB0w ىDت)0 W݂e(-5O?/|(>1RcX4\,8?PItzB)Jr{.M-5=)16 ha8Os D\8>9jd8mW '$K@ߵ#ZlxkX
Y@%ۃdVrnG-/A; V+Y;$gr>:Wi62iL Ʃ1}[Fnj R
QEʌ +M魯( /B
0EcVʼnu(Fy*P#֠E;Cơ<Yjh"t(7Yl-Xm"GG
E8հ ԛ (1 7I 6CXVD`~SFA\m;dH7ufR4'<Sd$pI'HLMt>'iH!*XiUQB ߁3*6C* qXb 551zTr7Km+A >iPjck!SY**na-MۇB !8*6[z[k? +[`sAgY_\ w&Bp 1Pj'F/uk-kpIzO$qj#c"kуK E֛>9H}N89 J151\PqcWfG)rY1s*=k +: BW 5y%1 TX$İDA Љ SI5Cv6<=igr4-Ae d9a:93- -`DŽ +aՙ$aHSwu >0(ID_v(xo50^$Ŧ%WkOZ &YoD,We;NZψQ.z\i>gI.|_m"r8e
Sי~!V ޕX96cD vn:Iם@ b$p, ~@E"T.ƨC1FbƉ"(aE+EK"3Vn 2[\eЙb+#ŦEh bU$->y >!3<`8In>3@%z +s"ɧ JV'K k?[[AzWц7=˲MQH6\֙#?Vd+(aKxz(R8 +4Do .>j>2 37Pْ1%=iZi&H@]+v0K/2OqS ?A$I%$8AOxJ$xu· w %x/! +/ԕ#T|N [r1"DHd/@u0#48>]z`|G ++xB j!G+)P PR0Wb%p^ $xA4B̐ng(@ ք#t ^S/?q X vVd/Udh+8bSB @"F1t*rՑ2W UI56$'r +BQJfC>0J3;,cL;1.$788xߔ/1SnDq0p @o}}_hdP<n9EO`m|u.'Q9|^nd3r= +~[Eˁ- L'ܦ#$AiK1`: +Πc +F&q? +HT&.6/G h .7(A*2K4MdF_ 0*ޓQk.4fEJm#y$z$%Jҫ~z2t +V}#gQRJ+A~!pj~ ʫ +6?n +Va V /p xǒʬi&!C&Od +Amܦ(8Q.NMA)@YlF(e><q0dϐI)!d4M_%90Jrc卡I7%]_& Uy1 AQ0T*jD4Z.muBw " ghGl?-xní:}xoBK;xi
Gv-Jzo j 2 nCMCn[qlï頧>{oF>b;isnl{ZzHw%"IܗJ^q/q+yOD䝄wEj%Iܗ|'q/q+Ľ:w"Kn%I|'q/V Q4Pu/L'm h
-WQ48 -/XMLNode1 /Int (xmlnode-nodet2;attribute(-28 1D -yB4 ƭG-` փZF5 < -Xa Nt !aa<4KJD%jq&1Ha B -diM DҤG0.x
8 P|R -.#J4" $@'D0@B - - c -&Ā@ -gC7L^kKm8`#YeUdm6|CL\r;(Tp5Rޭӛ0*"4!$!XC1J%Z7BrI_rY f&H>s"-e eoG G)MYb+Z=+$%[C%pB_Dz?PTKω -vn7,6UY}xV(:m?' T1PL4 g $ddA $Gy骺'Ǽk<Sqp4)]G.7A5?-1U (`g\4W]0iIӽE 16F?N)Rz9pH ȫ@>b -~ oi'7 , --qΧReI >cr5><QT$ /Ҹzk T m3#a^xCg30״~EJ/nrPh9i$I#HU^n4Fj9~|iC]l!ř_bOkvnPNwH(-.K.}SFS<?ق^ PI -YІw\ 353 Mny{; /w6 -{O-:YCm4ae\`\ '><x DiʩFZr,7Tx
(KPpM }HNf -tB8wɐvsAad&i4-%hIY._ ">`sU<e !@?#Kʙ #R&0Vn6j\@ -O2Y$.wd\x'Zi]Xc7 Z5cNNjFUs j) G G ˻pkk%kh դc)A^&mܟA(FTeޓpdߺ bL=& u#9d B^fahَf2 ~0kjFN$ I.һR,z;R3]NBebM_'c%1qSDْ=:Xrkn)lˌ j9=i ވسiMza}xA8{q:
ǹ='"h+b1BJ 3D#MߙFlg+lln[s)ҋ'50达*F$Mi~ܢU|˱S{cZs -vR7X-EPب{}=CJ -i+ H"q6JGl V W! / ʨ$A.Xr7Vʻq -jהoj՝b. gFŀBf_G MoJknO@ \KMg !ŊT$R;@At8#dgNȧ;=1=L+"ܱ$^b"QHcUs2m2q,ۛ|Ή6o -ضPUp ZZCoYr>Ci"r8t=4L3* <X} =sK3{$+O"iFLdW8 uߡKѮn eHlT^ õI"J !KrjD;x:C ][d~U -W~hV)yH8'B
b=i
v^?VĐA
mst!- , ?\i9."`%(KZ
A63 $3 eKZ<fՉ0n% Kq 2oZx '+K$,jG.C䐡4C{y}Z|+8S -k;%1 >^Zޔ ^wz* 9lE߾CQ.|ď"qŮ)k=pz/p:~ĿplV뺘x*< ӑoVHe(8v; -me +^_
a"彛tX9^UbDϝnyyDStN\<jXt! %Y,,-My -KYkDRFFNIo/ZKx!Ŗ0ov4u9>q'wEOxRQiPUu +1X}8}TCJB5Xbo)++`HPO8cY;z?s=`!|j Jk*l,bhÑ#%|݁,z^3~8˟\; -8 E <EŔhYx}F?^dQӧ+jU »'T_".K4J Udmj>+y ÆpAt<(Ʌ#-Fv]KM[o,lSY +-ݍ9/, -Z" -{LB9c?^R$ǟğ.S3c_Sߦ0tF6Ăpr.<уPK"}l|7pd(<H _s" 7 -mĭEVR%% -F\o;ӗ.&a3( t９Gֵk3 v -6Ah'u[oXu(O{ -Z~& -[G#Z/4ш;)1Fkdo4WqQ NJsFmiڬE]kwMC4yǣ)@@=/ؠPduO?M$nI<jq\ʢJ! -_?ՀOȳ?~f -{UB` |>~09jl "i,8~&6֫ӳ aCȩQ y9P˕f^t8p~#B"< A`aDx5 -dQ uĖUE&nӦcJvI'q_Z9}ZJkc%K&UL]*}H
JS8EVgR?.LC$1-Ljh}
mid' ~0[y4Vuେh! %4%Ll6,6d }*̙ hn^DsiYf iTuG+ҿH -t* -4{Їs07/v+J-/ cn1~6!.lw#) b -}_k-s5wz ^{{!(Bo1 (RI(F2:g.6V^{)n6}9ѝR5z{Vŕ vvO.BJ
sE]Il rE CRon7O]sjN*T&+7l͜[}[dQ} /PT BfR'_FމdU)5~+@
Yluݐ:|efұoA֯/C - dmn5>Fu љKnСFA5Ω7tAEH&}=zc<yF }kͺ[>o{! l -}7skϢ/ٓbdE2Z7nc[F "ٱel=jKY:Ȳ=$7
26[[BwZ冝S*oJkY&Jw^!;\=oRֲlS*~lNl(b~R<J®;;}IJ#+S*CkNk*] zWwn춼ۖ{JQʦTB>zs(neOCn7F˴G\ ?lp7>E6)ua7+pwe{ Fp -_:lP6DzU{CՖ+ٷg +;^X!vL3PL!qbXt~C46kϗ {ѵdƥ`?,2ga$[>(3gO6g~y=0 9F\֭cp/R*9Je1lN ca-vzٙ>\H|d.Fj?
* )k-asdon]sU+tK\nioͅtl̺9ǔ5cord_P#l#䧚jfT6K)| ?M_ø_~wkJssua|0BJF ^ 1~!mG76l+U.7 J| #|=}B_)_>Fno;vt
9rgȐ 2B(#1l5)]s_mmb˶~˕g{Z| >ݷokr9?`7+dȱ];ccl9J~ י{23쐛A{ {1lIwo+Ydݎ=zte毻+۶͘wor)l )7e+#Kbno_R?gg -fgJes.{csCaCJ5:t >it Sve>ϲ|]Pv>Vwu0v?P B^٤3ػח?O>{RHϤsCw9l{KcvR*g -R컏l]Gaǰw{);篿4J~R3L| _|ŗo{-)r {T.P 9n-\~zdw ; !s>ee82Ǝ ;k,at{ e*?(c1x"l/. /pu6%UQ DUI"K(;L #d~~ݻ _^:hϽskOBw^Z}cbȾ ^suJO9疁4?[s̹[c'AfҾ+JB~G5n sNjC2Gn[q:kKs۽ۯ5|zw}~_{ﹿ}y>cK^x~zߓ߾w3??33s3;3333;3333?;~~gg톚6K~
cvwG^R,a^>R_Y|$_r{ 7谛f 9:!L߷\wݛ㋱cZGݱW"? 9J,Y:2G'a'_LJ<zt2zͣѣ{Lgͥ.]7.%tKoҥ3퐙2fݱuݑWr}؟u_s~){2S?|>vw͛;COK=ݼ ×$|0:^f#|a3uBaknȐt/Cݬ=6{d0]7;J:~f <2l7 '<y@.+¨ɴ˒LhâEy7,JL -h$Q5j -Bz)&Ù )ÉLHhtMБaӉ4E9tLϜ"x\e= }#O7a㶘"
#ֆ )q6`C]V;M J sQ S<QXpkB -IϸF&f8\\#+0,S$] jD^Ք02L k -h +"8L :Oiʃ:ݰ@cQЦE M$=)UHMdi59G%UաL j2RFHiJɡròe] ҫù: *7,m&H4"ᘩtU |X<t@(42( @<p8! -h\*pq -PeP*E}@:#1kJf8$z4s\p옯Dƭ -rIV3Qr= %2ݣV5CT5po*L%AVÁ* 'iP&գyJ -x`;DִL'FĽJ 'U皨v?)X՜*.e tSMhrTL sa;rB9]{p"asiJ ȅph`T"̦3iXDCú<DB pn6fzC$dgE2-D\TeႦ42)LhĔ" DOTTE:NTd*$)tLbZ$)¸f.WxL(@/(J (:$&Tu@T@"e"+ -ХT 䂡:$ <6@fA(i5=T)u@"]k3?2CP* ~թSpt2V_U kkw8^'__RTCD3uw Փ0SٰoU AD4JiMB' (v@HYI LTB= Z<tB( %PPPT=Ʒ2TFLC_j·NhD55UV -:UEt$;|RQJf)
J~ Io,۲Ƕ5T.y:_ʅvz.Ya?_) Cnٱ/{Pc+]ݭ #T A - @8`uFXM -)04n oh$ a -q -40@xT4 k* -.ъP]XlDH`S -.0 - <J -qɘBbrE{sXp#
9bQqIh " 9$&\_%
$&8$&D&H<h(hZ|ظX&TR*)aw`EOI#SSq ISq"ii*EPJZ%A4#+2MSA s[ǏRZ<+ʴ $y6Q84-C)P -HSCm4 #]tYG*sװ?Lj4$f!JjfYIX@H IEU$EWT%mJO#ꙢL8p dN͢q<%V8$q8PPGáCp84uF69V
*сy=b^S64'q8 j U8 +m Nj ܬJSsHő4F\F*ˤz6광J3i&kN#\jjGHIeL"I a\Pd!IiMGnX oX>@lx*ሐL 1@@. -Ƥz86Q%fP<)"ͅEY+!D2 -u5pRbzLm. SĆS" ! -DemPw xT`b8[ m"*&3XӖ`+3p|++ƓM4-_ >z8>&ȢK?HUJ؏Eĥ|Bkjbw)1Ox6 A/82xB&'VJ*Mu~܈(_yetc+4(Ue"o *3ܚT.jVl 147 -D2zQ x%_2?A<glw40_:aPK-[UapUD`ͭf!M\D-EoT>qQA -M2sMRD$N>3w>!*M7 Qd"%EyF -@,7 gs˞Eˢ$lJQZWݬlhH1*-B9 śld^h&h%$9ݧTU1=Ah= ӧ|r)ͱlo.&ArMDPX׆dBXK->0Z]I`z5k[?MJ&(Иa"~ e8|/ϗyN8ZW -jq aop>vmpfE -4ix` e}uV3M1+=C/F08Чn
-ݞQxYtz<PE^hMcPHhMa蚁 >>S0| -eBF+zj -*dIqpAEFJ\5NϘRAh -
(=TTܩ xxGZ -tS9Ybc|髯;LU˫ $NFަ;:Rg%}C@~0 Wxgk~;Yx<ˏ &%Aa /
! צ0f0{h+#v0|ty -dxM"[&c^ tOcd Y vZf8=YNm~9~`fE_RuedPv,Eq=}ӂ< ˢtsPMCt,HT߇ԃ@z /hrt赨wXϺ\fc3 - Q}䖺Àb6rJ+y#~<jP ?@ - ,?Z2@^vUMe'0"@*}2.DQ @ɐ Eh*))vd'bBXʢ0#tª -1R .SeƪM4vS譥Qt@
E*0s} q7ז@7 -s6hBA xՅkr5 /Yb?\3Qa{Kdf/5l/zV|5ʴ2ʉq7RݟRp}ƹ¤S(z -^T<\j^Oӏ q<-NBY^dZYC]D8l^恿ῼwETrp -S+E 9oҲ'5!尽3e.Ubp"P6;*IG3w9?wQ^<6ata bCbzd{)sN`
f0 -gz&'4fCR75DZ_ -n^btKQHE)ߛvU>Ea3R#k/ܥ^5eĖB\p tپ -gǞS, /wڇA i? X1MZeiUWצ:]E],ҊIVzE \_
LVR'tj~
Q@yI'ײ?+f`2E #V]EA<αt+ ڴrNev@\|@ԹS,D@1J -R#E4x8 p͂RB|B3 -^ nb -Y8z"g`p pז|ItOxig$"i *yVdC`dcơ L h'3Y" k:25n^BCRWUB ()B6ףƎcT'x* =o,fJ#BwOU dPjE9U4\ p|^lXD[vƈwxm -E/ )ۍIF1;lQ)"`0.ĺXqg? {Mh+pԛ%V&<V} Vu75.DXFc?a 8"7q%ٌ6/Tny4 |qvhXkg`& Y Ƿa۴+χӢ\S S -n~I8q!+'~X^PI&]_$W3Vwb>
GqtnDИ6zqnfd#`#r$E9W W4ӻr)pABo~,M
Os8kI tߜʣ.ıli:X^ -O!"i3 ЯL6:%qB<kAĎ]M= LyD
+U_~aզin&>C51qQkub-\ <] Hh}5
p@D x=Ktg_,IDZ4=EY 03#\ -XF{d7\wE #_jwn
&pbح_o))37(>[B@L*z%;M'K0L`}}bڝZ sLIVQFԯ7rk"k$'Mݴ`Eva " -"\ldy6kt+9JLY!S8H[x6V\ZgSahhladF x{m'KbP -VH]ĦDe+Qe߄ -!vgͭZ.B)d>X߅l1 [n_xCPCVl*T<{G羘 -PIkӼȺVOg}F!5 4,
!t=S)|ӝ7RYfB 1۴xP_(d0Fen =tj&UOF]71-k^OFGR_a\z0{<H[AE 5dsR6[ք[)KC3c Nf֑~7W{* \Sk RjYiCcBTbqRE%G0x_XR)'+| -֚45B(.z -ΆXn pU δ4PD&pFC7ZߑDVB<pNjCV|DqыD/p8qںhҷ`b+*gp}`dG`fx. d[(M_F -> AB - -vX*A/q $dX7"B.^a@Yn9 t\tӴL\ZgBE`j%C]z>AGsy2]iUwkQrt%ŝc~K<2'ӹKaЉA x5tf hU@[=U=lιjNG צ[k9 1ڂ -l"bx*KbS3k-AsZEz XCA _>o/^"~;\Dۆ[_pBkhr)_-&@?8J ED -^^*bl u붋JnAZ;=^@ 5
qCKvAk,ǜ/ugY}5S56}As?#wyaz P:e%s ܶ7CN Q ǽ?Z/ \_@WEcB[ݪ *Ӥ_1Ts%$lp$ 8ڈtFy)R1%-f~xIg;
o}w#Aq!
ۂFvN#* 5jD@v,º8$mJZf?-Cxr }(a.o@c P״0Mc -3ݩv $#aCԜM 9SW-d_~ 7H#ؐ[$RZla (xw Gtnd$¸x't2 l7$Z$Q4š-cmO *Zb 6Ʊ JjߌH¿/U^q A%zw1+<>PX=6 ]C16-gjpDZ}BpDۊF)>.Jڶ^wF [8
䞅 ~vV=۷_xC "N|2Xz pkߖQY|s{){H `ySgJgO!XPt۫zetMD& Ok}U.aCyH-
x0Iq\xd -grf)nxqщZ:akiUǜ2FΑ38c6dvp&dZ; -@hr|ªvqHZyƆMVfX -S(L~I jhL4qVc@D>7G -1"cͯV i9.~)cC*9vRV i t_\DUև9`hwT2%0ˣi5 -`U -:;Js(Iaޜz
;'bAӫnLuGI)joƨm@H:1@ĝ<ex1$jb;p[t ?W"~#AT^'`4c]xs9E%d, /c-. [^qdFm[w Z|SV\ְ +{AT'JX3PA0)g2l1t:&]2 -,';Ȝ MGseGj,GNYaHR51#kQj 3t- a[ -VfW< 4V-GuW[0Lpd{ RuHLd(Ǘ9ū֦7c|(-a1FFk1(EGP-7MWwk}yIcN8[!1E,) NU5kF Y!¼9݊ VFDp{,+O$>F,JOu\`Gޫ7:ж{ E%#2xc _gͥ@*c(#uM$D0iQw$OK~nwxPC1Yg"]Z ٠d$LT~\]ovo'e"rFN0,AFVYʊwgDߦCiZGf-+m mư/R\[Uɕ? n&˪65@}'=un"R|mB`D\FScK
ZFൂ4v/XG$MA@ /JZUi@}XVc[jL贤Ӵ(HjIRKAifKrUe;fpm0+5bKv
~yt.'k$@ї1\@s2a ٷ9tՈtVrV -GWt/c|3$ GP$S9P(C.@d *RC:q/n5k@"cE ?Di[|.;S?S;0 J(䋯65(P !g{0fonC2H٣nSɟp[g[$dO%.|%eR-M*2L@'-gNeZ&Ap*v}Y/ X]=#詴 b7 f/(QɾgR1 NC/=GW۰\g4`kq@dB}-"
-endstream
endobj
34 0 obj
<</Length 33421>>stream
-jLW#by`:Z<l4L=1q 3(0 ({&K=ϧa忥F̃{v af-7q52mօ I+۷҂5 ,zoLWμ0An?˦X5),[tĴʏ/ ^w]*:N^ b_U~-Zǩd@ -xrXL.s̺,!5YW^t^JKb F<t%q_Ƴ/FR8G𘫌
]9 $= S8|:y.(^C ->6yM9 -:SW\KJC(w\ln^)GG
1eǶe"y6np Xa2`Ҳs&1BL6*Qt9ە~K j̅g<-bioZtrKGsԭGLv*m>p\Z3h -|z/ -u\`Ǎ|Z;y\6^$H籎sL=e5 \ XkeVvr# p|P#`H]FPaUBSLءJ? K)nƞE<ތ@S}:3 (҉B7 :7.eGm ۟3(ok4lJ!f c!X >pJ7dڭ -K"cCnܸ<IᶰEbJssn@"es%dMȘ=zy¸2 q77TC/$|FL -e<H@>n%_ #^hX|g ,gWsG=6x`0M;D!twG:(
6 -^ xpzͼq ֍iF
*!2\8K^RaEmeA
Df%:'`ʜus FU<R\XtuX6:K}x靾a -ݾ2Jw#j@Pm ʬ mC$ /n_}h ^dI< ԯڟƸiBkB-Dԟ @Y7A -DK}q o`ej -VRԢfփo ;oejDL^ -Ҷ#зy5N4FP"ȧ { Ҥ._;:{]&̤ KNHqN(`} [=Ni}Bdw.:Kz>` --ڥt}I ra`@r80\Eh]I%%uf Đ*wijJ\ΦN Q oMVr* -+ssq2N9[BEEb8UmR K..>Ag3TjgZ:5vcՕHY>38us -оˇDW [i'"E!2TvGq+lm;J.S_"4c3 懜e -6q*n㛵̓*lB;Rj盲lw WtRmb;>0T'LYZw -*n@&UZZ͖"fF[~7ײWx"?h q,| ;WD//3 -QNRTR<
E -_}ٌLPF'U3y"XN77]C?p B;Z.
7&Ic|dXSg1HJCW;ds 6pEK#A:h<чU =1f3Qٺ#Ts<l]|%E2Uv0#;
x8Y^&]%bߔ k0LiYc8}g۫|&drY_/k_ؐh!KI -&Zu[(C6Fe6({Z&%(Q;I 7qy` oZo=;AB8.
n[v*e-2#s%;\s_Pвm~ Ap \9u%}-% -g!IR$,Lnx.# Ȼ>eLf)Z
t K` -9%ʙ& *MlTޣpv,g2ܔpdC6HAntic -.)IN hexuXf쳞7sudo.]trz3<w0_uMgm P( x% @ #SfQZ H⑉Xg g&Ѷ̋ʚ,3]KVv1%9ڷ\ktx,`y!H W(4F5[B( -еђL>] \;*FjUč%InC֏ELynX1/ѐf B FF Z%|k5Lm8-#ВL~eOœ2k٠^TiO<wo ꩕| -on o[ՀOy4dk g[린t$£s*m̹,@j*i[FZnj= - SE}ʰ FKҧ߾ GU ˑ
R68:/81 uQ |5{+gZP./[XĪdUU-yЕvth9 -ybTV! "!_ ĺVP١oLMУu -{qb~6lPKBa`JuKwp D{Cj髊QmpLH]L' `7^uRʌ@j &|]w -33`!SIΎ]?C'lp -?y2uSTܴSz֒*n!g; bhn^rW˪v#߬*Aj HiH)aԪQcN?&j ƯA%W)XZڧf{5U _;э3L0 ZK<[S YGvX<FBu1WAWYF`n *~]7
dVBWq2˷(;d <+:Кe_: t45 +~
||[?DfM"~v}FֶQ}^HR!dIzjܬZ6+FZKFy-/m-rF嶥/aY<~R-Kq> VAS'3ԑ4fhN <"0 -l 8u0b]Q;SNgzA i>S -UzBdE/Ǔ{ Fl h3<lJy8)T -<A1D;pf!6x1w4H} :iM# :^Fã)_/)?S-M<|4j얿I=U#>!258p>%0> 9=EZˍ0pHNK{ -qG0SHoJU<;B6ʼ,`xi|fez2bkdBL+ɭ{<k&H1Nm|j -@'n a:' X,7vswwD9.]:5\d*jJd\4Vp -|ڏ>,nlA;-GpT - z' YA$g}lPhWRÍ~%-nP0);UoxxI\Fr [,UK) oA)qժ -.̿>S sɮg7e zP & 9asY.ICGGՒ`!]Mm -P$b`J>.mzsNpvM:,VF }-0{"TIc@bip!Ҙ\N{2)Zs5gE?H X $^O%- -XޅW=bmbH 8 P!سr 4"K@8衄T8cs"Wxh$<1xx Kiح#>2 -M ) -B o/."ݙ\WHgS tb ]5֍z DԳmuK|Uk' Ñ&Ovi+ -PrI00x3#c$q(%NK@l&VGb R53]gȸBF84ĔB jx e-+U7-Uq(y;&8_muo[U ]$hȈH(PZ_I'i H -LU l_h(]=E͉~PJjk+@Kh CB?iD6:)2} W$H -YIkzR\|6uDRgBMc:# eT-wldp]%2]D <> ד <dgS<l0`}iv}HF[ -SR XBځ"nm0בB$`
gxetGy+>fnމБҋܖ #Ҭ-܅%t\,%
GG -ZEI0ڴ@c8>B%Hn:Ԙ%Ir8sόNX&qFysEjݲo 2~oX6#>yyYw6&z v(+RclNyjN
5
4
=gzĂ~%:9' -ěn&c*m T8;U7\Ș]U^8e 2G xPX#< }sA]f -Ȯh1Ў O\ *)$^Iʊ@w8|'B@r@ā3U/
lHҐp@}w1 M^ npcK!!A<X{i3(5E_ZևbW;hRYQ~JO`Zn$Ζyl4 $ǺSE_! p\)'p10\bqE_ - py2'}UXa& JV@A PAњbӰ>Y2 UQ8&V"tw|aAUAz^) b&wnK[h{YU)t2mF hw`ԥ`zn)B8k`JP)>PJ2C|x |rhKcZlF#)%:ꩠ\3%ꛝPBDk NF+q܁Ncl&b %L8Y#1TỦD 0S(Q/ yt3 itH?a8$B\UjQ4 -L{ ,{P$K;'ykoi
Iau\WGbl^K -IFVdV\* =hk~{+UɃqӒqq6`s``Y#f&*P0mTP1I{a iCh@ˣr34k_Cʸ8JoKn]ȫR q G'1!xHݥ]JSRb ̈>E3PB -!C2\G[aTx=Fm[zQ>&+TlGA8:
eY Х6 xO,D;@ňL+q5(k5DDbV# -0)
F =zx,Mթ4M0h2:V awڰ912S_>`IƦL/}{h.aWuCJy) -͖> G:ɅyyvƲ95vB E7Ls*Jva 5iSghG4>:AJ*/bU:OE/A98BqƊfzwp -EћN XhȲWP3r$S9Md&64=C`̈́w*S`/2X:DMv|fLfbww2hrˬw:6skwl)8D Rք4 :E/82U96pPq!* =CYoK8z6ͺus1;u׳Ԏ /LJ2uR dlVsٝ$y;6gKgd]*$\C&/%"0:t* zI.n87S%jVD{%[8 6nq -TI~^#xK'Y>uυ3R݅N -QNLq0d&H{@"DɧQrdh -ͣARt n(|Ef~ -:uo~ߒԓ(]cv}Kڽ~el*ևɰYlhYR K!#&f~a%wR( -Y*S%0uɣ_HO"W{uquLNiCrPfا\a@0 pr|th
-a<j)bQh6pޥ;W MI?%{ػX ֭1EGቱyyFi Ci?[rBB8ӼOXhҡ xRڬKݔф% vz }Ěo}Fg&]&+-pB)m1V0!; - I_ !'+'ɒ C+tj㝪) ,>LJ4(Uy346 JP3Gr8F[ٟ:9NcLyL Cb E#.{fIʤi\ ޯ=)6l|jGЙT- WJCbk U.(07) XS)h."2vԎW
5*8r?JOW )χ.ܠcܨԀ$oQv,WԞ~$} *̙H + ITIR| RǟeUjA2aׅ5J: G!t*~( 3N-N$.yl9FhuhTN u -J£[tl &0P:T wR&;^|`Ҕ u7l -dO>SŠRK $)^|[0?/Y^%xplCt'ns!#,𱀤 YeǏ>
e\ -"qƿj_;{GxTѱxKhC"- EQ=vՂaREnbݞ t/{ o4HC|pɰGW@!!՟W=!C艒\:P~n+Ebt!Ѫоxya p͵Q>"!f`:f* - -y̨TH*A <^XE. }dO ^heX.@1'L;Z Gwq ˁht -ݕu2BdeBԞ6H,[r FˠB yB224D -Ђ -:R5 -*h#ڈ6݇\"Rڈ62Kpfp5U]
p5jB}L4@qY- $uY- J媲2d}̒=I "I^KW,͵ŽK! fр,FM7q}J%&9IMÖl%4R(|z4$+OZq@w@ЊTʐphe! W\ Xks!h4|V X BfJLGeNx4T oHqW̄+!3` sv L6=bmiTZHb8RĿ*`$F2K yke8R D".3Y̩z"r(M 1~\[H<FaE: F35aK]1r5c<L,w@\PӐ ("<)[; . -f/A# FVRZ FS>rzY?"YVqdAh 1a < -'(@f 3".S -QIMc-Yg/kKgRr -@Sml4@ BEn7 A@^BL*H2+DĢ*PTaHi6%+k@ @ *m7DsFzRܢRsKr),[&P*ZJ4 ՛"%YM]ZZ!I# $d 8 L6 -j&4q5ĕwP
q)hS rR<)Bm BbLQ%#;S0L<,R!LkNڲk_m#]ÑucS0Wt.Ӂ"]¸YAq2n)M.MλŠp.Cy,*!KZ! p(':8%6:R ʡj' -9+ - XˈKp"Z9⦬AV .7ZM!RD@ < \!3`+_8x -TլjB@PMFm1pG%ھ%p54P*W%F8Y_1#mK\X{,&s:*C'U <~qq-Rr7u:8 -L TXljj}Z5jE20cTVB*TvrYO)S'F #'H V @v ڣT=:hLC `6;Dm$'9Q|H Lj cqd,5R2dΈ; - -fZW
ȫyU] -P6 I< 'ʉJ>_#H5R 0oOp wf` fAn:E*WE\)kNNV21戈$SspI:D6/m^q^r7
%XjӐ80
,5YJMĝAX=w)liv>&8YKc|J1.}Eq)i ˔t!D"ᓛ -D]-882TZ&0 -sBR bNq1'LA SHKPH(
Bybܭ$\VNԀm@}ܛK -V*R8 %xādLA|)C*%^C!VȂYo M1_EGCVC\'d5&(:j̓$1O_PHMi':5[ء-TB%J(X%hs+ -7[1k -EByJiF*cLaEHh"bh V4l@K8̀T",2R!u4"Y<\DQҧs0Ӥ@Tm%; N6:,ս)<I#alzkj3KX0a e ,Aw2NES0a (Q ÒHnB=2'\td bcBDr >e `$lpd -2fy9N.~y$P H.!RYS4ٱ!kN|2qlUOKf'G`4D{A.b[jnw3eGpRz)M
+5BςL0LOI.,5YѦ"k:QV3Ww@4l&[d*d阪E f$hRp$ hY/ïV'˛
IkF<GH; D`'Gcpܧ8f4·RjU̷ݛC+H_; ƯZ(\Q%OBD`_!(jH㨠 w9f1ĩnTvy ; -n$ۨ $FT2Ѫ7̓b.7n7Zv̼VQN<IY 1vÅ0Y ,VDYX\6r8 '%X -D - T ^3ޑ͵/mͻc3v{1wvݒ.lwj-轻v]oދ;b@uLuS}0wMp3wko-U@ŀci2 ƢTUA( - -isc,x,ZHkٱ,j?)Lb A<ۏN]M484689O>âY`M] (i)*,:eԑT @
- 윚X -w^Wc
6QPi"4YC":
+B1e<xlM x3(ˌ
#; 3l;O8oTXov*b#Aء j 1}֛5QŤ0||JӂMa:wܯۈ)IvGcvrY - ݚQw;1ri6+9 ,]봙0a+0 H!b,nIr >c!M^ި }f-jF Br >n@x¤baM?XJ{H)ƾH3W43_%Z?>|nm57gq r7 V4]W0rbnNdkai,r{TA-ޘmA%F2En{$q<'վ=!NW ,ӳU
E4h@9+&T5g5 O7Aw?;,A-DB7a - VK} - -f= $IC]P͚ACa
%;HʦVEO> -&BxհmǭNq X+Q={8=Y^r*Ny38A1 r[XA -P$7~/eF n6@ -Z5E` Qtp?]鶣8m&{IKm&$·ݖ+9:>h6e_AS \PsSɄ? ]>O {[J8dw /Z(zԙ< Z
o%8bior;4=3k\kΊLMGOWpv{ J
`ʲ5S4`2
?aEڢ]'FuʊuJ"j.DïB TAImiNj^ti`:j9J6OޞmZ:BPIG} Tr-`Ac*@WX 2.7,]ޙS?%Z~m.^lzMFP{c5|yaUzM}#"eea#C8mjarAu>y4̦jt̲}Sf%ϰĶach5
rq*8]"WǰS݊ c\E^Sft|6_*l'G |%]xa9IY uո^FiA+-P·COTC|&.A
W:%T.{YgQN#4x*hIk lЦ(D+hRmȠfOs.6Q{|)c%놝 B[igF u`|ҋ9l#U CROS [)0q ?/5xg%[ T3\WAZ97#OkȀ4_.
DrN2zR ,u'-B9dVgLp\K̨z mieG_'K^nΖ -瑜7/]̲P|L
C3v<S]ȩ)U#&#sGͭkv㧅?/zjMߞ1eJ?d;M+7{\ݑ ݀ފpV+y -O 1N .P+h+,b;rmD S
/h[h<?Q -ò=c0:T.Ղ.P.n S -G}D֎0{ Ld9e> 0#~;Q ' -Q!Z"&ܘzF -#&Z ܇ߐ2@! ڀBANP:1B|݈K8jc D| nSU.cP<dOCb?;P .a{>Ps]}kW/!O?GnM4h:( -HYW8a o?{B0",駧yr8>PpI6y՟f5d&pW"^@ -̥- 8xZs -? -AY胄?nDpYe'] Ԫ(KS$]{N=E'u -K ^2Q ዄw'#ZEoVA?.V' CeK~x" "I}U"t S2)" ->ޢ(Ok'c_I8_ϯTb"f9A - j uҶWn3GG;EDQtڠ -w97ɑ3?&3? >jPiBc#$klSőO 8fƅUY;89 b1àTUtcb{P;V kE}amU\LdxC~>|%FY%BC,c Td~I<n ǒ%j*6bf쯉^Rdȡ!9{3mkL35PTMQH7_FZ?M>&zUyNՇ?TaE 8A;Nʌֶx הՆlr #8m {Tg'#>v :=X+pq`JHby\#.Uaɥ*BCg_|w-O -Jzx@e+`α97
72 -=' zjZb)S% 4g:/gMpUwUXkpNTsV+5r% -Z9Bvuׅd ZdJs" TsI@5s\mjG -6ONJd9ʧ>ۤzt87&;ΡƼaVEՒ>] -Rn* =MX
ĺ> -9B*~قzx`8p_ }kCf6Ƨ6_BjZ.&y |^-yuK`&_ Kݦ F '=}
!N8W88 /MoN=?5!^Dva<By{(/
u! ʨ?~|hFKBø-ȧZYƹqگR֭iۣ tK6 w a -i2{;әx - -P& gNSAOH@-j|3CQ|#|bLDx02B<CLMzCJ Jӵ.? b4?73i b- C1~7㈦T;/P:5hR` qg2 3nYi9='UQs B|jsR Ia<680'jO +77 6⣭z %([@`xw{ RO/l5xS&27@v!<NLnb}}i:ʬY j_RBPPu*fv TQ Z=/c\A#,( |:}|UZ:$:u^ـؽwxT &!". <Vް
kr)CM@dR.^+~-s@68) +<N>˓o(kӡID9~fRyi .b7ʺt[U,.|kH6;SAЯ R}IL@t(k;5F 3W -'øLU,]^XzVaq8_6A[2qTkTIn6E^ -R3:R}Zvo?gP%_laPRܳ v@x1n֚Er+/?e -PS`;Br "g]ֹD,g)bbH.L@&I0AZ~܃CB mWuJ^'exFAkC#@J!DBu}vmpzNMFMlBtʊ(|Zd?_c)Q29$ɰFRJw6yX55ױbTA׆u1lʚ S)d f# -Eܱo/\#e/7Ap -I$ -,?^?lphHE
Gw7_Z-J~* 0f^> K7Aop9ATk" )*W{ۄ
^8
w`XL%$ lpK}[!Jcs ePsj,Kg)BoH]Z@Ě.%!lGvJ<sEGaE͇>r {;X=@x&qJrum¦3=sHeݫ =y3#Ȁ
ŗjg8j6K*(L8(~TІ
16/y?Q} PҢJiU~2IG,NT/."8FҬl#W8缐Q~N: -38[rv] fEd3
YťZfA"OʿF8ng]Gj &RSwT{()!ڞdT#`PoZT ˰?.[Q|E3 -6 uԣ&M0IE6 /i% os:() HGD>/<6]'Uvh+>e; ʶgSƚ3
R|)/mq<G!:TVٳp!. P_s#.2XP9Gu\3љQ
x8
Fv>gӫnsr˚[S_M?lIb/R -G ^ N;= N#i& l<c!8c.X^pX\_`UW-8|ϊjގSlzv8AsV hA ۋUS BA|: C72`a\+KUdoWQ j] % 9'mf~BTVʳc dyB8Q"˟St}5aWys5ҩ,9"oKeY2.ʼn|;Si<̵Ob~y@Qħ"8:"6vQ KELc%~A <ؤD_ 2fa=y% }) Bpd\JliG 0 -ʹSp=wտR]* "DÀr)ҢzFaSi[5kG B>ύzօ -K$ |\!o)Yz7Bs*dD{=c*YnEa "F8|' - 7[>ב *n -CL|qYstڥss3Kh(8m G4X G4)4|'0L>y3H6*7P)kכgLp -)ϥְ%ae -Y - 2k
F 26o.fY:B)Y= >ӌ bL(V::IԒt0 - -Kp^= [&S>)'z %I C4T4h: ڼTlȈB.䫝*~JA{=j#%54(t ;S-װ#u- ~Tpe|x,͘7Ve%ry 8PSӻfn$:5aiӼ&^md\d =>-uC<3*ʖpB ^Ke=J:~]Ӡ-9 -endstream
endobj
10 0 obj
[9 0 R 8 0 R 7 0 R 6 0 R 5 0 R]
endobj
35 0 obj
<</CreationDate(D:20210620202348+02'00')/Creator(Adobe Illustrator 25.2 \(Windows\))/ModDate(D:20210620202348+02'00')/Producer(Adobe PDF library 15.00)/Title(FIC)>>
endobj
xref
-0 36
-0000000000 65535 f
+WQ48 +jbc!9d T+iG@99< +w:܉ +"E$+H(a +%FU +#z X +TL=$Y)1 / *N (7;!Dv0#?eTB .?eT$($tEA!
3>=' TT "= + +j9$1b 2JD4YRH` 3*A>K`@"ĈUhc.)8zH pD +K,H`й$,@AD|VD 'W 83+(F*ȠY:2!F@,B{+R0A +%Q.JPTHɠ.F#F0bD (HPQ +$HP)"OeQ* ^t"EYPP~$~ߊH<)AAs0NaQ.(ɥStJ"Ed{!IdH P sY5i +`@@hDB>fRsJ`Ԙ
,Iw(į+@6G:qcQeԍ8:ooU7Dž۠ G~._xDa Hh$4LCXh)4xtt mXF#
^yОd @ Op{,SUvZUG<Sn" szsϽ(Q8`>* D[ DE0I|'=_Vᣊw2Z,2?r.IXWo"ddԉ9DvعwfZL3[yHƩCߘNlr[3'Q$b;@Nl9 b3ՊX] ڔd +th^sZ
`sK穌?$T |w zS%
yFS 4Oguc/\d&- uaT3YC {=^_CR''W16*vD:J]8Ł!f/шi˷U<r5\X/tN +,U靸<(VQcPu͋%djWO<
3[v@.Q\ܤH:g&J5q^ ˹M*zۚgyu@[={T5kEO +K3bHfJ)ۧ>OZuoF{P +f9k7gp8±kkS˲+\r9F>mtVNQēktk>a磾E"
l*KA$/7{hv@4 ,$ &MB;RZS
LI߱
w16n@'jg/i GMǹ y1f(e* RvңdOЏlU} v oP"W0.|iѺ.dƺ! ?\υď ShDRj%jq%AЬrՃdbkr )1ibz%zik)gmd?Q^&FY [K}a 0"#DhK2̀ +#-^e| g` ̊ +U5D-X- +!t heUnθ`jՠZ=|a=GJV
l=WyZJDr G7ڿZP
F2E07 pHI{`_an
I) )Vm +j +NSXĐU +V5)s p ZiL2Q&dEBV[ 4+ 9=jȘBP/c7I Sy[Gߨ{ +L; { b@5z][L&#`;xMinZӈByW]Ho]{EViZ'وwL*%l ^cҙ (I2_թ
Sٵq~ȑ +h)bscFLLj.i)+uBˤ[zd,M7: yYI7!J('D(=6>P #2zE;ObȵQG=Bw[t4a'[TN9A1Djμb4 -IK+~T@d{v47EM *J<EE 3 ?>EQ}l^nWh<8ٕ&;ҽԕ ϮRm%]2R;T;T(o"Eը" ?DxnHrq$.N +q?ʹQ/&jfm >'j6(2c^?zJ3w:M۔-sQ5&GO^H13u["Fٗ>5if'|Բ]dhi>3*GC+}g7Z;Bz?V~?,rKrLvٟ '~jh4zQ}Dֈt~7U 5rěe5&گDKg]YYɷo<):ZBh4 1k33 ?!G坨aDƎo۷19v%h4gmj=!#>v;,ƼUOgvM2ڢ9"Tȳf_:;FcU\Yikζΰt4y7hSCmLҼD=)*oDC~ -%2D[kdƨ2 zLFqKNh4gh +Ӑ јp57^KG ?"\cT]yսKQb#vcz#.t6EŘFSIϼmqWrT<CTݎ*O]uuӴT w#GG +9)*ODv4Qaq7DTxo 9'{?5E4FJom2m;ϰUf=V䦧Qb9NFc &Td `T&ʄ&y`T@< 0FSȩjJ1
*' /j&Qoo|Fl5 Zd4R,n{UEmFxk7De؎1B<00OEb + $髿frK~1NٸyY٘acfnVOmMjmEŋp7YWC4_'SD\oeN/M~mw sbNDiS'۴ nڤl6!'ߴnڴ~Y33 m&SO|v;br{:no&~^w"扅>Ļ6ɋm$OnWNfKLޫꚙq]} z:t|'zh짶oY_Us4Q)szƿ)_S͝@!B%6&>̄؈vFu@PAF@Ѐd$#"1H `v`4 {HEȸ1qJ{V<<=k=dnLj^̯n芜5 > L؋k8ƹ`0NKĢ瑀P.sX.0R8P"` ++c<mG;@
xՍ/%"[:Lx ={"[S '{)qQݾ1t|iW jQ%BxHu NP'Bq2`\ßmz,&TE~Wzd'&d},zZbZC|Pc hCi"!EkX-Tw+B+˴
tSᝨ-R^Nud祏7WNлϮ|i\5Mc[M4p)*6ͧ9oF֦iߔDMhFQjYXeq)2#3k:.w
ׯj= [h8QAL.p4Fh8R#&Fc2¼lR)[*I& +(K)QԵTOHj*6VlwQTaD)JF}cmɞ :zRۭF:W&Lز i.#ZGZՈ~ܳ47Mͫj~H7 /*e扷< ]js8BMB%Fo4ܦHմo13nh4xƪhQԊp7E 5bh\8!6EͶ6NTѸx8R BT&E.#3irKTt}vk~=FBy~>V.Xc4((ٶn!zGGujDFɖߴϘxhZؽd1W>mFьFM
XYZ +/qq!Ѡ@DD( |. y2!Cuao P`!SL$A500(S` EP(eB5<'4(TSQqK ājx* EeC54`-T204X&BqP,$TC)8 8 +AU`,.( +'>^cT`*Bk>~iS27&1JYƔx
sڒp<#W.ZEEne4 t iSr/g4 ynG-Dry*WUN=l[}gmjGVU[VFqշ]5'[w_fkuoO)aڙ[nwe]]ˋ)a7!W{~UOmNOLD RB6<4D^Q1*CM01R%7nϾ sU?Kێ\3|=]R7KI)3dʏ6QbfTqv u]X[ ^Ri
GRT錆)*KLh8<u/2\N~\h<&5=]px}zrH~o#ęԜ zl/T.u=N߽JZ{țvIM_}vP1YZdlm97_ow'nh8R3J{gx]C9%m]E89μwFc,'@>FxyQuW.^.FF)nb4 ˨'Fh[
իflϩѮ51ިS73Nƾnt<ctϽ|ċ~ .ZvT(yL,%:vjۮƍ +Q%)2 )2Ť\* JE\2 #Bq&.NH.
PP$ḦUelcVD tLw[wZwrk1.~c0"p*4=wL6twBoO5 ;̾G_Ө ;**؆ 2>ݤv?;dS {#|0S4Ք3U U5O?3efFn.9{ļ]\D>TEmSoUlE]]EfnEݻɌ8;ύSwbwjk+v'fUE^FdŔY9aBLNLLĜəə)5S]i
H2dlbP$τIDcP0D@rIq0\.f8Jc \, Y.Td*6i(KH24L "0D XrBXD.A( a,bFA.HۉC* 4T ՙy X,4X<P&e(a,P*J1bT| c<Y.8&Mb,\X,uq$$eBU4 FqXvy\" ơd τa,vX0LP HT P*\8 #aX8SiLtP <%P z@ay4b8 ?T2s.<C +GBa) s c<
&`0#i4 + I8]*ADB¡@ɥ('y* " EAP R<+XT,4S8*caX@A%*)s2we. L&LX8sqLF 2#8CU2H H\ł@,&DN#0DcX*9p(H%`\ S@`ũh$$8D8pq `< AHd.IFD`ʄǑRPĂy"@lFH*sP8!3<&L"iY>s_`<&0/P ' Ƣd 2`@Drq\$ MY +3=wDa.b@"h $O"#GBhs4\*ʅDY*w +S@ !Y͝ @4!ay(P8 D
,8p@i !I@ + + +ʅHQ*2@,8 + eA+e, A P 0d,8 B(((\ERq % sq(Ce"Tű0PNƉP,.E\`@ lW
y +?ME( !40AX4@X +@@2D(H +8 +2L8 +DHp2LRL +EerH&4sq$#iP +,1(\*T +Ѩxxx I4H,"#sDh(Fc4
+TŢdg9d D"G AyBqLLF8 ϳ*
Xܨ.UԨt4$-C4" +J=TI[YWIpAfkߗny'};I]]Y9)@^w-pEQMO+0w0*P[bAA=րmOl-tяf&%rpW}9zTܱ=*xBMW쥔7W4;cFEi$} +ϷAPz|3&'| !)Y; +;`EJBz> o'm Xǻ-am"cL;YFTNcIhmэ)=6-(9i/n]qI<sgJo@HwKP
voo99!Anfiڜ c+ 1#
jk c~oXBC&P'P|
XNҠAKaA\O{ͬdYM6ſ(]?Gɒ(߸w]@_91-:?pM#^Tl2^FjtnEO0H43Y͂E]C~g 힑'"r)p;iz2 +̒:QuPX?Cy
pH_ +J Un3',|CX0V @nMõQvvԗdw+f'+`CA0&9ic )D\@nQXCxЯSeBg/)z'f +Wסli-8u;@i{*5 }e > S ]# +.PzBH>/D߽֞)SzHKA [*눧xºKG/媮<i9(OA讒v#h +[vpCOs-DcOam_*./T<!+ 36q BNu"-bJDny u %C)pa
מ7OEk +T\^RzRO~x" /L=g#&@zؖ6# bt=_1`ο"WM~\/m/܈9R= b8 +$lfCgspp/o:"iUk~+0~ujsC7a[f
@~^FVM +
k',,T2 IAٽx%%MtA TKu2xt(Úmf%V4[t8gZ
=6ǴZr2-TV88iS:mb1L2Oci/E<w0kx-uc!1$~ȓ.% l)-H5Ʉ[V ]BNbԫy~+ +Sٮjd:i^)ۮKh+f]>Ȅr Z)=bޏ0;) C + mRl#7F>܌NϹら!!._{P2gb OPB$hZkʈ"g+:a{z
h!n I sꕫu KA[ +F8V2U@)P\!=3@ +RR{sxm ; hxBp]v BAd?fڋ6fU
KֈT *Zz Z"t &e1X v]zyos~M# H7/h$ +1/ +d-Ǘ+ vՃ.>[Q3W^&\ +>0P +潾"8В8J RS|No%(mf4ikXE +eLYj"|c^ӻ D4&J#)ڴI^yLLö)@ .O-MFAAzRPo P77aˬ8fXǨ +,r}$2) ܚx4>3wLspai<ó|-vR5-ztjjN :ws' +X_o~ S6%Cԇ6z:_7]1Fa\nyP{(N&;h-h"~ϵ)IHA4E,A9&^>! 5DsYh|<f2 wp2?l')mK( +MW-D"EnMx^#An(Fma-<)$
| d6
h=+ +bʳC S +Mu!+ܺXm~܁b '<&& '8P W$`Lx3*j@ -2 nMٜT6*鋱'C a1b Z5l)v x mkw喭80Э!˷B|lOtfҌk* +(%d%,xXw51Y0R}^bK
"7x.JQ{ALL G+D
FHU˭pѪyg6T-_ĉbRTڏ[fg^诖:_ȅ +Es4zhQ ixq;5+?u^[:g q 6yWG[e߄ٲHL +e/0Z,,Qi'J-̓)E*"`pjz[0Mpd|_.+nH?)Y0M\Z +p% 6qN +,TF/^0 @4Tdd'>hxB6,?k1pz2 ڶي䕅x=B1^gAJoCfG{tPsE@أKdj Ɉڼr8ȶx I錯Ru=DT!ܰ!v&>9o@h1oY':ғb(fF @=mK%։tW.{R4\|\kK,"P&_|TjD<b]CS,uZAHHl/-.ܪ.0^+K,ZU˶ $ˬ^ Aw(+q-Ґko\ }).4TAK(<ݚnT(9 "iu||~D4K~(m!5_12B|h>* + ~ [E1l PuCDiH֘ˉ$s:C6&ꂴ +SWD\BSn⼆&ŏRFBۯKTb~{5 hv,?-j,J }!.\5\̲KxMGAD.k"P[0M; 1uE]@d?&ScLLpۃ|A%b'qNg27γ +.ý N $CzicdO6BFD2K82 ?-UWs(d[čA ILG%WIUy8Ֆ'@K%e3' -V!n*h<<֠i9[ pDCeD<~ +Yފ@ ps} +P t[Ie C^Le@S܌ +j %1dnƧpV6g +~f&lTw̦ C#08r g ILVZJB}@2 sHUA5p +nB,u~l#B:oZL%O&48xM&f?o=e10U7D"*~E({յǟ:> +~XQ I؉M (
:Ay(D' 1F6LB/ٽP1IEy1ə?E_3uyΐ,vEZ-5ɷZRL2]q
5pp N֖}8Q*Jp6Y" i<s+U Nȫߓ5/>mXDD)nи64a}N2 =k]! +q_ݿ +%Iߜ
)W~L 7J m%J>FW +0nOmz,̛ѼӦt(pX#?$ >k=I2`By
.%G X0ؙ7T +IU!YC|!iI$6+U?." 5IiE^sj$4e5LB]QFǎccaD?!@48PB&Δc<6ۺPxCi} cGnCCЙ稑aJX^O; +耘ÛUP lp۫KZR3.k۸HHfXk1RGCeAo`0 B ²&2<"E:ѶZ%A, 7V*U99]w* V F%eT3 aQze]H.şgmh^ kU} ޟw`b`MM/S P$ k2j
F]j'iX@+w@^
wI^j>ϾˎF $1'S۪;Y,~ jSD_Va줔v6l1"4xRƧ<eõfGq +Iuޣր0x!J<* ˄.L %#UӴހ?`mk/HtnFji, `)npfd=u,ZEBeY)"8u ,)8"=@#%JuEMܼ;JZ?d0Xݖj *N(f+LDѢ1Ɩ1FA(7 O nJm:+ݲik_s`rJDև i5L
z(4}ZgcĴ>$V1ma[Q ydfj(9˱[X7Z`i +n@Q}8Zۦ<\F(<07,9sЭJ Fҵ' ڭB-Rcc-S-?ao｠<[Kwbh)8 B>ƾWBM \Sgtd.Ư⽞zhJ]Gp1ju]#Zp2Ӄ^ +آ8 5Q_']W1%ſ3?d=^NhuydXZB]rFX,gvSξ+J l@x6kj˴'cdܝ8.gg3l=R4% U! LXwݔoC> LNcvA2x0HbDrolmu"K{eW]o?,}NV-t spVPP{*:t
'0 +5Fkz&0ȉe;/{- *,T;~pGZf8"bK753 z~e@5~bt^p,@"#B"bڒf W)6jz 467pap4JG&noK =dq
1Y% DW?HjVLzr_MЮv͡SH4:.)w9%s,\FX9G +Ue8 d\纉t-ȁܲ"c;GjgSmtk1o_hHC╥aS?ݼ +b*S*{DW Вk*A.4@!*i}A"jH +p di_q6t7l(5 l
n PZ|2!-As#ܷ }X5[c&t1
zv[^0m-ud6
p{2f@ rb߯ɸXCFd$nbL*Y| +VV 5 xSN!,ݞ%IJ)FтDaE-JN{^>8AѸɖZHq/6 ѵi%+RwN6b b 槣B9Vv:,n+$K$·ΰ]254/pK:_a$}ؿmK9Y&<ݺ3%$Uoi-~M` 8# 6{Qi
sNi$@X oM!GPW&-g<bV +֏
C[% +Gs"Y6!GV5= '5X˝BN<sxg9woYj +pW') A}Fv +er[esQ +q/(oᓸ`e {!&D:g`UzH]գ +Dk.tI7;Wa2pC5-_X%OW2QvK C- f;$8n2!G;;Nh L0Q $UƧWN6
ʣF0 < Ŷhfcvn)s\Vʉ
-
ЈslҖqP*Rט +rv35w6I:d 1=^ Q# +^~2eWq mu~ +݆^ҹY_#h@ Lx3^[7Vdw@A `րf_Gpl!U +m#%X[u1] [B'Wv!A'x4muhz@DhXj%}S+ +Y%̵%S2yĮ`b ސ垾Y\A!36,R@f;؇ґWq(_(PN՝tϮ:#zNV;kn6QT=]XP"7y+Y QW1 Aќzƌ tG`~㾮{ 0/6 ʖbYJYߋ ˖)s`!"l zd<Y"P
AdL2T #I-HqˊfNZ +]p# Xάg5ۗ7 gKegUӟbԷ'0EZ "bmiM+gش?Kw٦aʒx\vCao;|=,<6qa$ٕ1 +&كgEֳ y\SBb!+] 뽆")K/ +dk$gqp muLLrꯪP)|Ju' AW">i8kaVQ +C O +%Xr68Gi $| ]E,\)Z.
7aOBO :M-pxOTi~:%&HTnL1?K%Q
bU bQoDH:/{|dUM֢?gu/eg3g[څ]Wo y<BUPyuzT
zP"o
UcWC4\!J0 juH3T/C:3zZk,
.tslƽKdUog(ŬR u1L&!P{p8s[UNF ÝֿW^g9-ͣJ '=o10~j6-$xEՙU{ T4;,"uS'jQ}0sF_(*~D<{TI_nTD so ?~_S _sj1<EԦ01ՙqa x~ +tgr7Pl]@srS;U +Vm#5Ő]bw4R[.>hZv2D՝FPEg.[F׆g++; n\'RfDxk5VQۊZVL4pb0 ? Ľ zfx`OYd9KfQ$jN'Vdk\ \D; + 8pqc yZwVsB +n8saG\/]a*m\&JeVytI@TK7߷a+dq1^ k!=`KߵE!W K4/6;@ /DEM[drj|PըmoڲҴA\[&QkI"p3 i ʨV ?!kĺX$ڡ '>kRq1
>߲R+n8h;
;h8l0$folHs!+X, b)v ak4/(}+ 7h1DtQů1gt@|lZ"12a ht
m|Zb\+ qzH:_Њ ,0jDo7(cZ93ĬjʹfR%9 +eaHm#Jx~,/o#R +_&hvI^1},c5 +-3O\Xo*话F9os؆1>
+endstream
endobj
67 0 obj
<</Length 38184>>stream
+:̖" ʹ-XwLMH6_!ŌCDVמ{a`nVS2 q`K'ѯē!\š3
wUu.z sʃT&`N%mR٨S$&##=_ǧ + <"z}&ut9Yk[Bցp!uYgp40hb |բWFn[TUᢾnlA]N<Ց eG5PʜnLĒP>e ZY~E-_O}g + ,qnxFʒSw t$eFx -f0l*>)X*uu8. xo~2ymmߕSDy1E摵"YƓez^kt><M(m_?w {邪S%08 M +#99!-Lne]Ǹn}tk#@桞$uٗwmGU{e0
zX ؕiY~=oHݑIż9,@'MoXԚ'aa^Kb9Q/+s&I,_8RDZ8y ׀"iqK;+ (cqq>9[Rb +381Þz? eSZ& +5;H2v*GD&WY2.t<B U<yae-Ap31Yrzt)ks^xH pR X ia8pDzSt"
}
TG:j(4RlC%&[ vm̰
%c+ˉu5M6Gk5X +% Se?0Jq p3r Lw-~uT* C A&o n5 v! Z +qFfn,\Zl]H%6O`O4ȍA&q L
3|Vdn +oTTԳo+.S9+?>C :(9F +l"dډci~lv| +*{z%.T18
E +wIc !NyoF\(ԂldLbW买P4Va m (8?S1 l8r8i3 J+G"6-8 =_ u} .WLXX
fe{)ipUtL-nph]}+1dC +nq #P:5+(BR]c@fuJfqm
6@ >8H tkCq +5a/b`RuLM3V1WG8>.bX"uo ])A
$D +Ha" a|ߧcw}^ 3 ĈL+)&hߏDͣJޭ&j`duH~E +=C0І +RFU"2tLfjmG:#3:2
pLƣ`N)[,Rl2DM%)t7ۍi_R]̪ y +P7Jn [jT(g-SO 9j -矇2JYClt"@N <# CE*S/ꐼUD^m1lZRK#ɀ,I}=ڮ)љ0V;;ΖϨT`-;CQh/%JHM >%¤zf*zDD%\A + f^kBnZVDdM{N -JWe"iߎ'$|^Ÿ6) {K4"`5d%FD(KC$%Mo@v'BYa. I`!t>QGѡu,֏@ _`_76'T4U$;A~A)6-x,}}cׇɰ7ڐahYgIaGDie&܂B7ްc.?PR;UeTa J" $$89L Xh4|:Ŷ}Z~3 G}1% +_ + pш<ehwSZ?Ji +)KF2n<m@$wbpWw4DZ*w%=) =+X7 CvзOVEf ~]@rT,ȅLRmš:Y9d>rrLPE +)2SPچI}D(g'O/2,6I,}t# $V +1$aa:@N +ҕXZ FڂE=3 +jR S]i4C +䩼5\6C&Kow81Pr4] 1$ J(Mig* [@9\@ +bk g?oAdeaM $o%:{~hBR'RFX~w ԿljMT(<)K GzMH+%jJCcwM冋& T +yqpThbl!an_x\S=MY+]pz=#z,i#M19QH (7#$%+ +QJKX9=zWuj:2 \/K} DŽiJQ<
UwmnRyNl JIN1 h
{k5L pJ[(sG5->`dUoy_!vjbEwm0Ib\6j\@Be =YOiZ@BF3^`$*}ztFvX0cj.xǑȽ.}y`~/ݾd"s@d~ +_\ D֜3tG
b)y`HRc"1L(LB6, KHUSKYx#E3:JNٽgQJI6YơzB%4;
JF
frP}cCed', x>w D5U
h{ٖ AD d ވI'ni띻"V
-Rb"H5o,vtWLipB$IbN #QTkg'dӭQt;B*K?⭩FlLl +0d#QGfzoL 5 +ThJa);6-#D V0:)%]eZAGR>Kmb1~2 +Ġ(ê@$®S6jdڥJNidLz= + CcHzlNQ~UP,'FSFҰ"5䤖W'1HlMMSZ N ` P* +GO Qͻb4aN=RcO||3 Qs1`u|'kK依,7g-ۋ 0^qhuxd;iT qt{|/ZW
X&'i7Sd)" )u.?p#=ȇ60dHR
/2^ñ;Vg5@ѽY986ks1 +^6 + + +"g +9 aH4`lLt7敺TK9EvTD>,=,pQEӅ( +ADP#, SS1Xz.h$?KûW}1wbxaHԒڔ dDhKj"PH +BRd,%XW&Úvi4IB@]a:
b+؊J' )c;^1݀(WZieVi;Ňr* Չ Hm>mqTjW Gp*} }0 PD7v@m +Rl(_< G D30Sd$~6Rd +nvfA5<>a6*d +ƕ%/A´x*V _xi$xv0j[Ng#>Wi{-dXGLLӦE-F
(7v)2"OjEQ +Fd"ڈia&jSP`.
& H Ȏ]$dUy
.aISpsiRd-n+Co\hФ + ƋOyt<I
DpZ7pMeYFa!yF8l$gl 0s[<&\&1tBȓnp(92yqn1?@B= 2B^^Pn &WCHi@p} eD?J +zܡXuo .loD݀ {P0n@(J`5,1 DGa|-+
+,fz+ŗS`wEӢF^#Ev7 pO9 N5 TW JI$. ̖ĀѨDD>92bEus݀]
p9(6Bb3sS$?@w{+ر.SaIo RIG<mD +D-MC$%L 7xݖn@J aM7m7 lHRLbEJڟloϋeqm +NK d]g[iF3n@ +LCkafduZ.2PB UusAAR@fo+|ru8a_*.*A[¼rq*mK.4fJ&/r>2 xL#5OBw(Ve׀{ E/c% !4Z +uL0r` \,Nָgp硶fă:N<^~,u*,mp.EY,d836_@R +p +2xWU +S<3XvXQj +r-e 1<{Y\d/8U-gy kUX^rA_l-1_H~2e$ƿ8{8,~/Wm Z7EU*ߴ +^ل6]XlXy&mb-M4-cUF%3h,,.mFİ +6K>YKpF) qٜ VHS<dJ^l2+$V]p jo艴</jw+S][)D:qu +buEʳrMa0 shi E]5F4c¾n@(1/bjviҚ0,Ἦ尜)1 ,>36\ ք;P"QWnvYl/fvCd +芇oW% $RaIjJޏ
1ʊL ؙRRSQI}MZZR8(iIM{"۴p_P -*0?]%3C ^%)5mV(DW*`|X1(L61{a~RJc譴
*(w+NRF@fA"F *JqHi JiV6KFm4& /{( x?lI.'9`Ɍ=BF4}Z'LEu +_6]"UBI6.|~ +~ԥ:X)e +
#]' +6OAE)$OArݰ\$)2 ,n=LGmOAY"/ĭQwa(Un@tkTX KQ1< + \
P> R67?ωCOUr SPOl-6?;pR{sZR%8\
B 'k+E%%4*Vb"xg4TT~ + |>%& + D@PQ(8S04l0'(WˤS,"IB!=օ,h" &SnjZvUN!>!JxȕgS%
pMRZpk&cf+nA -3TL oxڅT"j7lC<n3ws.LG^$E +Bw`Ca#!>18a` { Id 9 ˢJF,8rBF!<r['/:\TeF\y<ơnb
He<O:UaH``.v4 $&p"4[pQ `7^bbq<s<t4۷XFc2滠t"QzfL&1JZ G0tl r
zM<헎vnd}L,ZD7hME &D"{y@G` +B!1X:$H6A(exڝ ^
B,)&jn@ښMAX0Ya
x + cLk15I D%D\Z7uC +/*XF: &q@5 Bt%5< v(%uXO[p/FQpZKѲxHv! y݀@ѬLs"l!E6Srji(4Ƨd#\4lV'8w6RKj D"E`%X:k||<CG]p k kU| + +֕Z8Jt7 XeSdnC7ʦ>@Pdٍ s-,=p +A+'k Ai^i c %].. 7Kl] E%E"&Q1E))-X-`l$Pd;3$[V{G,=8.q"1#2?8q(,1
-) +d[ f:QH He7 #{7xuÆ eTc+<\!'B)݀A)"gEL;mf"ג[ci$'Ő7U; `ԇ8culi%ʴpr\Y上B_UO17S#Ҥa4h,]Dm +2HshK@
@mYouWY.a:}i;m O/e66 aCF
Ee;(vPdCe;n5ev(~ خͥ/9\)9霂nAcN'Lq[r:&ֶN;k$8vEq)v'+P"y%pm +Jz-bȮAELPT[IY(ҋi7O49<Mq2-~2O;s>dV7)O
&S Hܡ<PZrЭMHXw(9nI[ 1j 1 + r 6@e3\j39K Ϭ5÷^ʅ]HǬA`$âݼ>(2*:"@: 1(H t 㻪%#I-P-@W7#SɸFGLǰ6Sñ}t +IZY +,P:u꜉ +2Z)RM)E]"bbIm J'fRp)jzģ֙ݨ^G(Բ[ ֹ@CY"rhH ~uɌpٟƋ\H4>- ]ZJwk 槠fڍM-5X]8>#n@(TaDKJ:.AxrkoLӤ 4D
LJSS O2_tdna/2M eCqP`G1I_;a6uGøg:@<;#~@<*f7 % +RDqaN 'N-^*ǹ>`HєCf*w݀Ze70cqfT@T8mC v:J],n]Hӷ=NZRيo^^71_<˖Cr.SjO> W'8|t'^e䓾HPjx 2>~DPprZ 7 +B2 )мȉV +.8^jߴϑ&vС}aZD.zq -
O/-rV vYb9Y)Mt }xwH{=rԎbHNHT͕J[td T{Ė}HHyȌu9x*z7 OYQsau2]髮sOrO|Oj8zfzUq7(5kNnb +JB 9 </>Sa?ꍫ0zeN\6~ +FZ
@rqbrR]Y-41*gWE\tL [E* Îݘ+ܠր٫0ݧ.4(SuR[+:m(5*83)DZ4u~DS#Ȧd#qt
7|LOjΊM>&SݍM +'ɨ`]lOeѡuF +ޅWKl5l L(
Dn@VدufG]O9fDIH Sݸ"3<՞eF-; .M&vHK0Un)|=\ Z-TkAqqw,Шs݀jq6 ^nTI{#`m9;C!Gk \+uN!s6
q, APx )ڑI1dsT;⋈i% "η RB|R"SGJrwWp(h(d`] n^jfX3,ٵ}^`@j_?.1X:B P{!RW
Mn>n怓E/SD}L,} + I}>^ g D3,_,SV)Ѣo=v ++06aY(JA#Qa Ew[BZїZE"+\d7pHA)X 4 /< n Ax7ej-1QԵ?۴) vHI]0r㾉Q>"PpEp݀}j Zbb p@>{m AD袎 R 0d'<X4Չg<T^h<biQZɐXporq0)} FS
+jrLI
a +))9&LA!$麮[# pLe8n[ +wm}-sMoM 7??K>(7-Zdj4Ի6uRS,hGʎj7~5 E +DV%SOZjOIO F19@R5n~7?]Kmp1?k5W5\T5*''u0>%urO\fp#hS(ZƆN*N</OjQ=2j;&Zv7 4DM*5l)5gn29m)]-M>Ȩ۷FSnd)-\ 0`H6Hm=.X J=A݀ +Hȥ3yi*r( ");=f }v Cb"Ԟvo,+%F JxDù0bfFd +h>*NDL*2DHƁ aA!RdU +_z!iu 2+Tg? l;e86k4xcO +ߤFZc7D:*uik~j s7`FhH\4r&R;U6j(Ӻ2 +pOm?8
hv㑧cZQ ûyFЊЅ06Own +;/E/x&60Ռ)E{UmR~4ڍL gb~Z&3\q_L>=CFU_IP-˞[ˠq {SF"8D
\(
q\x1ޭg $@<"ѺU`_ɩ`~ڠAH%%>lۥ{9m'yF1{MQ9LT L7 v1jhz*m1U̧tGԈK<TI ĕVI-iӏtp`9-/q +~/w9 [gM ֽ"pHտU`?KF1-+@dVjhl R;1C Zš*b !D1
Ixqc +-˃*W_l6 +Y½n'aLBLBqNZfl8E&Qp6 .͡Kw7Z_D:^?I`~]0f4aQr0k5-"m#3W Gx +y~*Vq3 9ԵK5}s>NLcj-盟"b7*l0]U)ȸ!opǰB_js5c#sXS8Nص|{Fx#3*sYiAXUek!.$l4,+j$ZѪ52[݉VeSLZ]]c(3N'LEyVQI̚ TӸCGw'yQYG7^Hk̀"veB" s'%g2dwNZ-ԏ [ :(imx/mI ~C5=Uz5nx.3/X"<D`}1o~JI;Ozq8;8 g|26B9G6TPcfz\^6њS'qߝ'+Ҙ si9yښ# E:/uy3=ivqAb/ʦfGp4 +]BCˊ22lG=@a8 9h4T[8>5CAi#sbDk4-qѨC+Za4ĶS/O\8nǷJ[k2(i&JWa@H!65Ce,h7-â?r yȗUmaŹ]wo${mw:`s" ,fԤ4ܩ.v$=wZ\oW+s`Djh䝮ceC{8t
֪'O+֮k"/Ea}֗?
-n@0XID.qB:dU, goZoe۴\lT]*n^~
+ɚ/. vݎ"0τ|`` ]rY=dse *tWzNRHWt(38ڄB27( *61KMvkL䚷c)@i
kA2OVČKx9!:!,4pB |st"u0taz0)ZÔm-g0'y ^ډ`XL V9=vD $Ʃ6G`X~U + <IJdQ oF0nX0!v- {Wr# +_DΣ~4[] Ӫrǖq{d& Cdm-GSo;0Gby~q|ykɇBrq LsWO2օ璍q<qT'MJJSpѣoѐ0DDM`j +a3- D4%<C1;Іb Y$ }"j[9`cb=ͼS@y0K'*twPk, +v!}V?ԲeS"Y?E(rɎ~* M
;w,(Qp ɫ߯+n2t J-]ϊ^s(5 Z9*
)}elc`31ȶ ,tza1T'h RkMDmzs%GI#V?U%x +J!| +k < vHfA":خE !| ah/R;ړB܋\GTn(^2l!I{.(ҖRbfKi<(A5e@iRKbl +ee]!eGfff1Ρ"/dR =>!Z^C~z_cuLgx3jF?<% Խ=x2<@ΡY hQl2)R1S?G`
ЂP`KT&sC|\Qh B[D:9b:`qK"k + FoDjw/g%!UvKw\YMN 9+T\`$+YT@yI!3g#OpxP%>Kؑ.J!NAobY C\T>E 3`9~$x6Hu8:8ob-x18ѧtu4]S٪1\o4d֦>f /PИrRJ&%5l +"FmĕV+Ĩ69Q!RlQa!DŽUyIvĨr9FVU0jjtY3jQ\* 9IQ:)ضĨKdq3fERDrmj[}u49\+~*t&hJ8#!d.ޤA|ArSuebda>B.{~a.hkdiH՚AM6V\iȕm5vdt+7ڗ+wVLޘ4aie@eNJSF,Oy Fhݼ0Јr M$*.$gW;%{pfAn8_2ljƯG]%rp| 8r ȖoY|Gz͝J, +BcR 7D\!zr% ^lF;IX$Gi-S
e\a ؟΄_icefs +mARЃ+)%u/|O >( > +a+E&hz lj.vR=?^4 +qZ09LNLc䥣]MۢҠ gv#|)ylo)_G3 3 a|o9 ]9o| Io<ڦ> kۙoey,
YQVn }n]|n|hJ̸V- МBj+}h%]bF],^2L "{ >_O(` EYRQG,V@䅼$>8p)b("hԈqn-jX*y1 /o
ajwom!$6 b]"#jSD6?Oh [EKKOv@4f0DzilX?L t+Q.sn*]Tr`iK'7_n+[YiktJ +P5+7dWj$P&\;10œ.m? +*0z*;HM+ߤmQUĩ93V&cJL9T@lD=2.O@
3tdžnM-c1pb=nVeԆbD6honm}ўk=@IXi7T~ s(xڻj#~ޓ/*?8 sG!s P6Lh' +A-taufYtzxNz[SiCX]S!q-P*}8ql'8t~7,bTKkKg +5U'xaVx]ճUd+7Jɸ Z2V [ԭiNQV֝M/:>BT30Ǧݧ ATTH ٥Ί |4àYUloۄ*.AKY5΅w=?r K;,wtW2zV ȡF@%K:t5)a%P[ 4H]a@Jxo XZiVHO(p;$ +-y D /[u$G[97 +OK0w5l02ɵ]^(cc;-o$ȡ5
6l4Tij~aP%\]պLI5,hVb;uWؚ_,WԵhҕ (9n%k+<SAӦD ZBuP;x^q6=mՑHV&x][?*ϟ%OI]gWaKpbSrtmC>w+b9 +o + c9d ;u@Z^A8T!\⑬Sr +`.UH-L.> +ф +%=k%)[u&~uDOqEN9eIQ`RTI +oge!$sY駈ݞX'iD{B/O0BSmQn{Q47_pDtX#\B`&lf:%]nkR8(x.alg:hq]>a; gumA`Ђ+q fDA`RuYVt.+%荺TE'r72]KہϿ1Ģ& N @Dɣ]y2) +GWewƣFR Bbt(/S&+|APP24܂J_ {ۨoA}]H_ʡ6!..3\C +QD [a3 .. +-TT:E2X0cif ,pX +cѵCVaw$rCgGbz O>gD{L(ܙIZ"R| L ڕ= _+1!-%s, +` 45<vRFD҅u*9u0M$L8%j8+d ]alfk!MҒ)@[';-B+b" Nq=Wb_z}ש~{.H|CYIP\eD{<t}\iѝ۶1 jR|^fUQ{0aRחr% UKwL Ђ8HH ڥNrAH KI!2K3I\1$VG Z"+A0J@GM +܀ Օv7{ѵcg,@m%Y^_<0!/l +Ah¦ rŔt- X ۉxwbh5`֠7' +aZam4Sd}jm;曋2uBny⧜MH$/-~JU5^^gy;CB}% o NزmsG Kׄ kSYYjhq9TN {`!?<%HwV
2q +w]]a +ϬlMZN{CJ^Vᤉ)~B^快d}C{.И +g;"j#0\i{!o7$Ȥrh +DhffP͝ +Uh!z2Dszu~!s +E_=ˋ+Xm^iaP$pJM3 X9ԩ'em{5<zb!)Ҍ$ +PP) + +ǟ^w Jԑ(-d;'?di/+ >aa1y{`XR i5@"%YlB B E +z$YO`Dh'pb:cI@Nٵsá&/)ItvX$$s<7ڇ7wd
8 ZA<N4h(a,K=i m[ŷ&p5V +<܋[}~.z@?OԺp +GA{"ٲ +,Lȣ̜ʣEbAÕ X[4חQO1 8O ĞyhsŎ?C>O9= S4% *Y9O33.ݜo@M4Zpɉꥴ䌦ꨐ4쮃GCMnRT^dz + ӂ1<Bxg0M?F +E <FeAIogj%ԤVhF݁ Bc^bQyT&O{֯"_)u[3Z"DV*ؕ1z ၮ{Lt\ƾtRmWmGX + 2rڢs1 s'RVñmrOB=htV.mN+UY1\]F/cPǻK3-c k_"zj"x@ +Dؔ + 2jꗁ /,$^tbW: Nhw &|P!LN]xИ{;fz +ʘ; 2|<T.m +6zhW> 5_& 7ݪ wP*?O67A$ $`&uVD)&|>4A9܇8G I@ ^{7* uS@na\cDQP̲dMT +z_%Wx +f< [ +d)ΙB*{Ρr*tێ8K2iYz&[@쫁C?6F_8`Yk.`b<] ]Kf}\F1y i~ +w8
+endstream
endobj
5 0 obj
<</Intent 18 0 R/Name(Ebene 4)/Type/OCG/Usage 19 0 R>>
endobj
6 0 obj
<</Intent 20 0 R/Name(Ebene 1)/Type/OCG/Usage 21 0 R>>
endobj
7 0 obj
<</Intent 22 0 R/Name(Ebene 2)/Type/OCG/Usage 23 0 R>>
endobj
8 0 obj
<</Intent 24 0 R/Name(Ebene 5)/Type/OCG/Usage 25 0 R>>
endobj
9 0 obj
<</Intent 26 0 R/Name(Ebene 3)/Type/OCG/Usage 27 0 R>>
endobj
26 0 obj
[/View/Design]
endobj
27 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
24 0 obj
[/View/Design]
endobj
25 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
22 0 obj
[/View/Design]
endobj
23 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
20 0 obj
[/View/Design]
endobj
21 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
18 0 obj
[/View/Design]
endobj
19 0 obj
<</CreatorInfo<</Creator(Adobe Illustrator 25.2)/Subtype/Artwork>>>>
endobj
42 0 obj
[41 0 R 40 0 R 39 0 R 38 0 R 37 0 R]
endobj
68 0 obj
<</CreationDate(D:20210620202348+02'00')/Creator(Adobe Illustrator 25.2 \(Windows\))/ModDate(D:20210716163100+02'00')/Producer(Adobe PDF library 15.00)/Title(FIC)>>
endobj
xref
+0 69
+0000000004 65535 f
0000000016 00000 n
-0000000193 00000 n
-0000050662 00000 n
+0000000263 00000 n
+0000052288 00000 n
+0000000000 00000 f
+0000360264 00000 n
+0000360334 00000 n
+0000360404 00000 n
+0000360474 00000 n
+0000360544 00000 n
0000000000 00000 f
-0000053324 00000 n
-0000053394 00000 n
-0000053464 00000 n
-0000053534 00000 n
-0000053604 00000 n
-0000351801 00000 n
-0000050714 00000 n
-0000051136 00000 n
-0000054367 00000 n
-0000054254 00000 n
0000052340 00000 n
-0000052762 00000 n
-0000052810 00000 n
-0000054138 00000 n
-0000054169 00000 n
-0000054022 00000 n
-0000054053 00000 n
-0000053906 00000 n
-0000053937 00000 n
-0000053790 00000 n
-0000053821 00000 n
-0000053674 00000 n
-0000053705 00000 n
-0000054441 00000 n
-0000054696 00000 n
-0000055971 00000 n
-0000121560 00000 n
-0000187149 00000 n
-0000252738 00000 n
-0000318327 00000 n
-0000351849 00000 n
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000361078 00000 n
+0000361109 00000 n
+0000360962 00000 n
+0000360993 00000 n
+0000360846 00000 n
+0000360877 00000 n
+0000360730 00000 n
+0000360761 00000 n
+0000360614 00000 n
+0000360645 00000 n
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000000000 00000 f
+0000056114 00000 n
+0000055179 00000 n
+0000055250 00000 n
+0000055321 00000 n
+0000055392 00000 n
+0000055463 00000 n
+0000361194 00000 n
+0000052807 00000 n
+0000057828 00000 n
+0000057715 00000 n
+0000054187 00000 n
+0000054617 00000 n
+0000054665 00000 n
+0000055998 00000 n
+0000056029 00000 n
+0000055882 00000 n
+0000055913 00000 n
+0000055766 00000 n
+0000055797 00000 n
+0000055650 00000 n
+0000055681 00000 n
+0000055534 00000 n
+0000055565 00000 n
+0000056407 00000 n
+0000056702 00000 n
+0000057902 00000 n
+0000058157 00000 n
+0000059671 00000 n
+0000125260 00000 n
+0000190849 00000 n
+0000256438 00000 n
+0000322027 00000 n
+0000361247 00000 n
trailer
-<</Size 36/Root 1 0 R/Info 35 0 R/ID[<8F914AC017E7274DA4FF36C5815F2B8A><ADB0D5143910BA4C92D0FEA7476B1AE2>]>>
+<</Size 69/Root 1 0 R/Info 68 0 R/ID[<8F914AC017E7274DA4FF36C5815F2B8A><196BBE183DA3F8489DF09FFAFB9C2A6A>]>>
startxref
-352030
+361428
%%EOF
diff --git a/buch/papers/ifs/images/Makefile b/buch/papers/ifs/images/Makefile new file mode 100644 index 0000000..c6d3fb5 --- /dev/null +++ b/buch/papers/ifs/images/Makefile @@ -0,0 +1,9 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chaosspiel.pdf: chaosspiel.tex \ + farnnotweight-eps-converted-to.pdf \ + farnrightwight-eps-converted-to.pdf + pdflatex chaosspiel.tex diff --git a/buch/papers/ifs/images/chaosspiel.pdf b/buch/papers/ifs/images/chaosspiel.pdf Binary files differnew file mode 100644 index 0000000..23f0dd2 --- /dev/null +++ b/buch/papers/ifs/images/chaosspiel.pdf diff --git a/buch/papers/ifs/images/chaosspiel.tex b/buch/papers/ifs/images/chaosspiel.tex new file mode 100644 index 0000000..7c69ad3 --- /dev/null +++ b/buch/papers/ifs/images/chaosspiel.tex @@ -0,0 +1,37 @@ +% +% tikztemplate.tex -- template for standalon tikz images +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\begin{document} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +% add image content here + +\begin{scope}[xshift=-3.6cm] +%\clip (-3.3,-3) rectangle (3.3,3); +\node at (0,0) { +\includegraphics[width=6.8cm]{farnnotweight-eps-converted-to.pdf} +}; +\node at (0.2,-5.7) {(a)}; +\end{scope} + +\begin{scope}[xshift=3.6cm] +%\clip (-3.3,-3) rectangle (3.3,3); +\node at (0,0) { +\includegraphics[width=6.8cm]{farnrightwight-eps-converted-to.pdf} +}; +\node at (0.2,-5.7) {(b)}; +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/ifs/images/farnnotweight-eps-converted-to.pdf b/buch/papers/ifs/images/farnnotweight-eps-converted-to.pdf Binary files differindex 35bff32..f5e4093 100644 --- a/buch/papers/ifs/images/farnnotweight-eps-converted-to.pdf +++ b/buch/papers/ifs/images/farnnotweight-eps-converted-to.pdf diff --git a/buch/papers/ifs/images/farnrightwight-eps-converted-to.pdf b/buch/papers/ifs/images/farnrightwight-eps-converted-to.pdf Binary files differindex 3652e8f..fa69d77 100644 --- a/buch/papers/ifs/images/farnrightwight-eps-converted-to.pdf +++ b/buch/papers/ifs/images/farnrightwight-eps-converted-to.pdf diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex index 833748c..af2105e 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -5,7 +5,7 @@ % \section{Einleitung \label{ifs:section:teil0}} \rhead{Was ist ein Iteriertes Funktionsschema} -Mit der Hilfe von Iterierten Funktionsschemata (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden. +Mit der Hilfe von Iterierten Funktionsschemata (IFS) können mit nur wenigen affinen Funktionen komplexe Bilder beschrieben werden. In der Regel sind diese Bilder Fraktale. Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir in diesem Kapitel untersuchen. diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index a75b529..caba120 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -7,29 +7,27 @@ \label{ifs:section:teil1}} \rhead{Problemstellung} Bevor wir die IFS ansehen, schauen wir uns Fraktale genauer an. - - Über die genaue Definition von Fraktalen sind sich die Mathematiker nicht einig. -In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry \cite{ifs:fractal-geometry} beschreibt. +In diesem Kapitel orientieren wir uns an den Eigenschaften, welche Kenneth Falconer in seinem Buch {\em Fractal Geometry} \cite{ifs:fractal-geometry} beschreibt. Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \begin{enumerate} \item $F$ hat eine unendlich feine Struktur \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. \item Oftmals hat $F$ eine Form von Selbstähnlichkeit. - \item Die 'fraktale Dimension' ist grösser als die topologische Dimension + Man spricht von einer selbstähnlichen Menge, wenn sich diese Menge überdecken lässt mit echten Teilmengen, die zur ganzen Menge ähnlich sind. + \item Die `fraktale Dimension' ist grösser als die topologische Dimension. \item Viele Fraktale lassen sich auf eine simple Art definieren. Es genügen zum Beispiel nur wenige Funktionen, welche rekursiv ausgeführt werden, um ein Fraktal zu definieren. \end{enumerate} \subsection{Koch Kurve \label{ifs:subsection:lilkoch}} Diese Eigenschaften möchten wir nun am Beispiel der Koch Kurve näher anschauen. -In Abbildung \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Sie besteht aus lauter kleineren Kopien von sich selber. -Den Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt. +In Abbildung \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Sie besteht aus lauter kleineren Kopien von sich selbst. +Der Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt. Gestartet wird mit einer einzelnen Strecke der Länge $a$. Diese wird in ersten Schritt durch vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt. Die Kurve besteht also aus vier kleineren Kopien der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. -Man spricht von einer selbstähnlichen Menge, wenn sich diese Menge überdecken lässt mit echten Teilmengen, die zur ganzen Menge ähnlich sind. \begin{figure} @@ -66,7 +64,7 @@ berechnen. In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Daraus resultiert, dass die Länge gegen $\infty$ divergiert. -Die Fläche unter der Kurve lässt sich folgendermassen berechnen +Die Fläche zwischen der Strecke von $O$ nach $(1,0)$ und der Kurve lässt sich folgendermassen berechnen \begin{align*} A_0 &= 0 \\ A_1 &= \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ @@ -88,22 +86,22 @@ Wie wir sehen ist die Koch-Kurve ein Objekt mit endlicher Fläche, aber unendlic Zu guter Letzt bestimmen wir die Dimension der Kurve. Es gibt viele verschiedene Methoden die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur unterschiedliche Arten. -In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. +Da die Kochsche Kurve selbstähnlich ist, ist die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry} die angemessene Messzahl für die Dimension. Die Ähnlichkeits-Dimension $D$ ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$ \begin{align*} D = - \frac{\log N}{\log \epsilon }. \end{align*} -Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen. -Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge $l$, Abbildung \ref{ifs:trinagle}. +Die Ähnlichkeits-Dimension stimmt für viele gewöhnliche Geometrische Objekte mit der intuitiven Vorstellung von Dimension überein. +Zum Beispiel besteht ein Dreieck aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge $l$, Abbildung \ref{ifs:trinagle}. Somit hat das Dreieck die Dimension $D = 2$. Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon =l \cdot 1/3$. Ihre Ähnlichkeits-Dimension ist somit \begin{align*} D = - \frac{\log N }{\log \epsilon } = - \frac{\log 4 }{\log 1/3 } \approx 1.2619. \end{align*} -Wie wir nun sehen besitzt die Koch-Kurve alle oben beschriebenen Eigenschaften von Fraktalen. -Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten. +Wie wir nun sehen, besitzt die Koch-Kurve alle oben beschriebenen Eigenschaften von Fraktalen. +Dies muss jedoch nicht bei allen Fraktalen der Fall sein. Sonst wäre die Frage nach einer `richtigen' Definition einfach zu beantworten. \begin{figure} \centering \begin{tikzpicture} diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index fd10634..d0110ed 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -6,8 +6,9 @@ \section{Fraktale mit IFS \label{ifs:section:teil2}} \rhead{Teil 2} -Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale machen kann. -Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. +Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale erzeugen kann. +Im Beispiel auf Seite \pageref{ifs:trinagle} haben wir ein Dreieck aus 4 skalierten Kopien zusammengefügt. +Lässt man die Kopie im Zentrum des Dreiecks weg, entsteht die Grundlage des sogenannten Sierpinski-Dreieck in Abbildung \ref{ifs:sierpinski10}. \begin{figure} \centering \includegraphics[width=0.5\textwidth]{papers/ifs/images/sierpinski} @@ -92,21 +93,22 @@ Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktion \label{ifs:sierpconst} \end{figure} Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird. -Der Abstand zum Original wird immer kleiner, und konvergiert gegen null. +Der `Abstand' zum Original wird immer kleiner, und konvergiert gegen null. \subsection{Iterierte Funktionensysteme \label{ifs:subsection:IteratedFunktionensysteme}} In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. -$S_1,\dots,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt +$S_1,\dots,S_n$ sind Kontraktionen auf einer Menge $D \subset \mathbb{R}^n$. Es gilt \begin{align} |S_i(x) - S_i(y)| \leq c_i|x - y| \end{align} für jedes i mit einem $c_i < 1$. -Der Banachsche Fixpunktsatz besagt, dass für solche Kontraktionen ein Eindeutiges $A$ existiert, für das $S(A) = A$ gilt. +Man kann zeigen, dass für solche Kontraktionen ein eindeutiges $A$ existiert, für das $S_i(A) = A$ gilt. Den Beweis kann man in \cite{ifs:Rousseau2012} nachlesen. -Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeutige kompakte Menge $F$ für die gilt + +Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeutige kompakte Menge $F$, für die gilt \begin{equation} F = \bigcup\limits_{i = 1}^{m} S_i(F). \end{equation} @@ -115,17 +117,17 @@ Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leer S(E) = \bigcup\limits_{i = 1}^m S_i(E). \label{ifs:transformation} \end{equation} -Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, gilt +Wird diese Transformation iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, gilt \begin{equation} F = \bigcap\limits_{k = 1}^{\infty} S^k(E). \label{ifs:ifsForm} \end{equation} -In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. +In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt gegen eine eindeutige Menge konvergiert. Diese Menge ist auch als Attraktor eines IFS bekannt. Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. \subsection{Beispiel: Barnsley-Farn} -Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann. +Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktals, welches mit einem IFS generiert werden kann. Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben. Die vier affinen Transformationen \begin{align} @@ -153,7 +155,7 @@ Die vier affinen Transformationen \begin{pmatrix} 0 \\ 1.6 - \end{pmatrix}\\ + \end{pmatrix},\\ & {S_3(x,y)} = \begin{pmatrix} @@ -183,25 +185,25 @@ Die vier affinen Transformationen \begin{pmatrix} 0 \\ 0.44 - \end{pmatrix}\\ + \end{pmatrix},\\ \label{ifs:farnFormel} \end{align} -, welche für die konstruktion des Farns benötigt werden sind in der Abbildung \ref{ifs:farncolor} farblich dargestellt. +welche für die Konstruktion des Farns benötigt werden, sind in der Abbildung \ref{ifs:farncolor} farblich dargestellt. Das gesamte Farnblatt ist in der schwarzen Box. -Auf diese werden die Transformationen angewendet +Auf diese werden die Transformationen angewendet. $S_1$ erstellt den Stiel des Farnblattes (rot). -Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab. +Die Transformation bildet das gesamte Blatt auf die $y$-Achse ab. $S_2$ (grün) erstellt den Hauptteil des Farnes. Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$. -$S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. +$S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten links ab. $S_4$ spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. \subsection{Erzeugung eines Bildes zu einem IFS} -Es gibt zwei verschiedene Methoden um das Bild zu einem IFS zu erzeugen. +Es gibt zwei verschiedene Methoden, um das Bild zu einem IFS zu erzeugen. Die erste Methode ist wahrscheinlich die intuitivste. -Wir beginnen mit einm Startbild, zum Beispiel ein Schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab. -Das neue Bild, dass entsteht, ist die nächste Iterierte. +Wir beginnen mit einem Startbild, zum Beispiel ein schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab. +Das neue Bild, das entsteht, ist die nächste Iterierte. Dieses wird wieder mit den Transformationen abgebildet. -Wir wiederholen den letzten schritt, bis wir zufrieden mit der neusten Iterierten sind. +Wir wiederholen den letzten Schritt, bis wir zufrieden mit der neusten Iterierten sind. Diesen Vorgang haben wir beim Sierpinski-Dreieck in Abbildung \ref{ifs:sierpconst} gebraucht. In Abbildung \ref{ifs:sierpinski10} ist die zehnte Iterierte zu sehen. @@ -213,11 +215,12 @@ Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Men Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist die erste Methode ziemlich rechenintensiv. Beim Chaosspiel werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt. Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein. -Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet. +Es wird bei jedem Iterationsschritt nur eine Transformation $S_i$, welche zufällig gewählt wurde, angewendet. + Da, wie wir beim Barnsley-Farn gut sehen, nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet. -Je mehr eine Transformation kontrahiert, desto weniger Punkte braucht es um die resultierende Teilabbildung darzustellen. -Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. -Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grüne Hauptteil des Blattes, $S_2$. +Je mehr eine Transformation kontrahiert, desto weniger Punkte braucht es, um die resultierende Teilabbildung darzustellen. +Im Fall des Barnsley-Farns wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3$ und $S_4$ in $7\%$ der Iterationen ausgeführt. +Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, viel weniger Punkte braucht als der grüne Hauptteil des Blattes, $S_2$. In Abbildung \ref{ifs:farnNoWeight} wurden die vier gleich stark gewichtet. Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn nicht so gut abgebildet wird. @@ -245,12 +248,13 @@ In jeder Kopie des ganzen Farns fehlen die Punkte für dieses rechte untere Teil \begin{figure} \centering - \subfigure[]{ - \label{ifs:farnNoWeight} - \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnnotweight}} - \subfigure[]{ - \label{ifs:farnrightWeight} - \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnrightwight}} + \includegraphics{papers/ifs/images/chaosspiel.pdf} + %\subfigure[]{ + % \label{ifs:farnNoWeight} + % \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnnotweight}} + %\subfigure[]{ + % \label{ifs:farnrightWeight} + % \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnrightwight}} \caption{(a) Chaosspiel ohne Gewichtung (b) $S_4$ zu wenig gewichtet} \label{ifs:farnweight} \end{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 78fb935..cebb664 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -6,32 +6,31 @@ \section{Fraktale Bildkomprimierung \label{ifs:section:teil3}} \rhead{Fraktale Bildkomprimierung} -Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren. -Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat. -Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. +Mit dem Prinzip dieser IFS ist es auch möglich, Bilder zu komprimieren. +Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch {\em Fractals Everywhere} einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat. +Das Ziel ist, ein IFS zu finden, welches das Bild als Attraktor hat. In diesem Unterkapitel wollen wir eine Methode dafür anschauen, wie sie in \cite{ifs:Rousseau2012} beschrieben ist. Es ist wohl nicht falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. Ein IFS, wie wir es in \ref{ifs:subsection:IteratedFunktionensysteme} definiert haben, wird uns also nicht weiter helfen. -Die Lösung dazu sind Partitionierte IFS (PIFS) \cite{ifs:pifs}. +Anders sieht es mit partitionierten IFS (PIFS) \cite{ifs:pifs} aus. + In \ref{ifs:transformation} wurde definiert, dass die Kontraktionen $S_i$ bei IFS auf die gesamte Menge $E$ angewendet werden. Bei einem PIFS wird der Attraktor in disjunkte Teilmengen aufgeteilt. Für jede dieser Teilmengen $R_i$ braucht es dann eine grössere Teilmenge, welche mit einer affinen Transformation eine zu $R_i$ ähnliche Menge bildet. -Wir müssen nicht mehr Ähnlichkeiten zum ganzen Bild finden, sondern zwischen Teilen des Bildes. +Wir müssen nicht mehr Ähnlichkeiten zum ganzen Bild finden, sondern nur zwischen Teilen des Bildes. Doch wie finden wir das PIFS, welches das Bild als Attraktor hat? -\subsection{das Kompressionsverfahren +\subsection{Das Kompressionsverfahren \label{ifs:subsection:malorum}} Wir beschränken das Verfahren für Graustufenbilder. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. -Ein Graustufenbild kann man als Pixelraster mit einer x und y Achse verstehen. +Ein Graustufenbild kann man als Pixelraster mit einer $x$ und $y$ Achse verstehen. Jedem dieser Pixel wird ein Grauwert zugeordnet. -Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert $z$ zuweist -\begin{align*} - z = f(x,y). -\end{align*} +Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert \(z = f(x,y)\) zuweist. + +Wir suchen ein PIFS, welches das zu komprimierende Bild als Attraktor hat. +In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$. Diese sind als Raster im rechten Bild der Abbildung \ref{ifs:FIC} dargestellt. -Wir suchen ein PIFS welches das zu komprimierende Bild als Attraktor hat. -In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$ Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist. Zwei Beispiele wie solche Domain-, und Range-Block Paare aussehen können, sehen wir in Abbildung \ref{ifs:FIC} @@ -57,8 +56,10 @@ Zuerst brauchen wir die Transformation g_i \end{pmatrix} \end{align*} -um ein Element aus $D$ auf ein Element von $R$ Abzubilden. -Wenn wir die Grauwerte ausser acht lassen, haben wir die affine Abbildung +um ein Element aus $D$ auf ein Element von $R$ abzubilden. +Das bestimmen der besten Transformation kann man in drei Schritte aufteilen. + +\textbf{Schritt 1: }Wenn wir die Grauwerte ausser acht lassen, haben wir die affine Abbildung \begin{align} t_i(x,y) = \begin{pmatrix} @@ -83,39 +84,47 @@ Wir sind auf folgende acht Abbildungen beschränkt: \item Drehung um 90, 180 oder 270 Grad. \item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen. \end{itemize} -Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $G_j$ um $1/2$ skalieren. -Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. +Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $D_j$ um $1/2$ skalieren. +Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ Pixel Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. - -Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Grautöne auf die richtige Helligkeit, sie bilden die lineare Funktion +\textbf{Schritt 2: }Es muss nicht nur eine geometrische Abbildung, sondern auch eine Abbildung für die Grautöne gewählt werden. Letztere lässt sich mit den Parametern $s_i$ und $g_i$ beschrieben. +Wir suchen einen linearen Zusammenhang zwischen den Grautönen des Domain-, und Range-Block. $s_i$ verändert den Kontrast und $g_i$ verschiebt die Grautöne auf die richtige Helligkeit, sie bilden die lineare Funktion \begin{align*} z' = s_i z + g_i. \end{align*} Für die Bestimmung dieser Parameter führen wir zuerst die Bildfunktionen $f_{R_i}$ und $\tilde{f_{R_i}}$ ein. -$f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst Skalierten und dann mit \ref{ifs:affTrans} transformierten Domain-Blocks $D_j$. +$f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst skalierten und dann mit \eqref{ifs:affTrans} transformierten Domain-Blocks $D_j$. -Wir suchen $s_i$ und $g_i$ so das +Wir suchen $s_i$ und $g_i$ so das der quadratische Abstand zwischen \begin{align*} - f_{R_i} = s_i \tilde{f_{R_i}} + g_i = \bar{f_{R_i}}. + \bar{f_{R_i}} = s_i \tilde{f_{R_i}} + g_i \end{align*} -Die Parameter lassen sich mit +und $f_{R_i}$ am kleinsten ist. +Dies ist ein klassisches Problem der linearen Regression. Die Parameter lassen sich mit \begin{align*} - s = \frac{\operatorname{cov}(f_{R_i}), f(\tilde{f_{R_i}}))}{\operatorname{var}(\tilde{f_{R_i}})} \\ - g = E(f_{R_i}) - s E(f(\tilde{f_{R_i}})) + s_i = \frac{\operatorname{cov}(f_{R_i}, \tilde{f_{R_i}})}{\operatorname{var}(\tilde{f_{R_i}})} \\ + g_i = E(f_{R_i}) - s E(\tilde{f_{R_i}}) \end{align*} berechnen. +Die Varianz und Kovarianz erstrecken sich über die Grauwerte der Pixel der Blöcke. Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt. -Um zu beurteilen wie ähnlich der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block ist, berechnet man den quadratischen Abstand + +Um zu beurteilen wie ähnlich der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block ist, berechnet man den quadratischen Fehler \begin{align*} e = d(f_{R_i}, \bar{f_{R_i}}). \end{align*} -Dieser Abstand sollte so klein wie möglich sein. +$e$ sollte so klein wie möglich sein. + +\textbf{Schritt 3: } +Somit haben wir die zwei Schritte um eine Transformation $T_i$ zu finden. +Wir führen den zweiten Schritt für jede der acht möglichen affinen Abbildungen vom ersten Schritt aus, und bestimmen den jeweilig resultierenden Fehler $e$. +Es resultieren acht $T_j$ mit ihren jeweiligen Fehlern. -Wir bestimmen die Parameter $s$ und $g$ für jede der acht möglichen affinen Abbildungen und das mit jedem Domain-Block. -Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist die beste. +Um den besten Domain-Block zu finden, führen wir die drei Schritte für jeden Domain-Block aus. +Der Domain-Block $D_j$, welcher die Transformation $T_j$ mit dem kleinsten Fehler $e$ hat, ist der ähnlichste. -Diese Schritte führen wir für jeden Range-Block $R_i$ aus. -Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und Transformation gefunden. +Wir suchen nun für jeden Range-Block $R_i$ den ähnlichsten Domain-Block. +Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und die dazugehörige Transformation gefunden. \begin{figure} \centering @@ -128,7 +137,7 @@ Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain Mit den gefundenen Abbildungen lässt sich das Bild generieren. Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge. In unserem Fall ist dieses ein Bild $f_0$ derselben Grösse. -Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$. +Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(D_j)$. Dies wird verkürzt als Operator $W$ geschrieben. So erhalten wir ein neues Bild $f_1 = W(f_0)$. Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen Unterschied feststellen. Die Iteration hat nun ihren Attraktor, das Bild, erreicht. @@ -140,22 +149,21 @@ Teilt man ein Bild in die drei Farbkanäle auf, das heisst, es wird nur noch ein Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt nach der Rekonstruktion die Kanäle wieder zusammen. \subsubsection{Performance des Verfahren} -Dieser Grundalgorithmus der fraktalen Bildkompression ist recht langsam und skaliert auch schlecht für grössere Bilder. -Dies resultiert aus eigenen Experimenten. +Experimentelle Beobachtungen haben gezeigt, dass dieser Grundalgorithmus der fraktalen Bildkompression recht langsam ist und auch schlecht für grössere Bilder skaliert. Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nicht so schnell wie zum Beispiel das JPEG-Verfahren. Es wurden bessere Algorithmen der fraktalen Bildkompression entwickelt, doch auch diese können, vor allem in der Laufzeit, noch nicht mit herkömmlichen Komprimierungsverfahren mithalten. \subsection{Beispiel} -Wir Verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert. +Wir verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert. Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren. -Die Range-Blöcke wurden $4\times4$ gewählt und die Dommain dementsprechend $8\times8$. +Die Range-Blöcke wurden $4\times4$ gewählt und die Domain dementsprechend $8\times8$. Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Blöcke gebraucht. -Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}. -Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}. -Es kann jedoch ein beliebiges Startbild +Als erstes Beispiel wählen wir das 360$\times$360 Pixel Bild von Rapperswil in Abbildung \ref{ifs:original}. +Das Startbild ist ein mittelgraues 360$\times$360 Pixel Bild, Abbildung \ref{ifs:bild0}. +Es kann jedoch ein beliebiges Startbild sein. Nun lassen wir das PIFS laufen. Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar. -Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. +Nach der fünften Iteration, Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. \begin{figure} \centering \includegraphics[width=0.4\textwidth]{papers/ifs/images/original} diff --git a/buch/papers/mceliece/Makefile.inc b/buch/papers/mceliece/Makefile.inc index ed1affa..53ecf7a 100644 --- a/buch/papers/mceliece/Makefile.inc +++ b/buch/papers/mceliece/Makefile.inc @@ -7,8 +7,8 @@ dependencies-mceliece = \ papers/mceliece/packages.tex \ papers/mceliece/main.tex \ papers/mceliece/references.bib \ - papers/mceliece/teil0.tex \ - papers/mceliece/teil1.tex \ - papers/mceliece/teil2.tex \ - papers/mceliece/teil3.tex + papers/mceliece/einleitung.tex \ + papers/mceliece/aufbau.tex \ + papers/mceliece/funktionsweise.tex \ + papers/mceliece/fazit.tex diff --git a/buch/papers/mceliece/aufbau.tex b/buch/papers/mceliece/aufbau.tex new file mode 100644 index 0000000..200cb7b --- /dev/null +++ b/buch/papers/mceliece/aufbau.tex @@ -0,0 +1,161 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Aufbau\label{mceliece:section:Aufbau}} +\rhead{Aufbau} +Das McEliece-Kryptosystem besteht aus folgenden Elementen: + +\subsection{Datenvektor $d_k$ +\label{mceliece:subsection:d_k}} +In diesem Vektor der Länge $k$ sind die zu verschlüsselnden Daten enthalten. + +Beispiel: +\[d_4= +\begin{pmatrix} + 1\\ + 1\\ + 1\\ + 0 +\end{pmatrix} +\] + +\subsection{Binäre Zufallsmatrix $S_k$ +\label{mceliece:subsection:s_k}} +$S_k$ ist eine Binäre Zufallsmatrix der Grösse $k \times k$. +Auch muss diese Matrix in $\mathbb{F}_2$ invertierbar sein. +Für kleine Matrizen kann durchaus jedes Matrizenelement zufällig generiert werden, +wobei danach mithilfe des Gauss-Algorithmus deren Inverse bestimmt werden kann. +Da eine solche Matrix möglicherweise singulär ist, muss in diesem Fall eine neue Zufallsmatrix erzeugt werden. +Für grössere Matrizen existieren bessere Methoden, auf welche hier nicht weiter eingegangen wird \cite{mceliece:GenerationRandMatrix}. + +Beispiel: +\[S_4= + \begin{pmatrix} + 0 & 0 & 1 & 1\\ + 0 & 0 & 0 & 1\\ + 0 & 1 & 0 & 1\\ + 1 & 0 & 0 & 1 + \end{pmatrix} +\] + +\[ + S_4^{-1}= + \begin{pmatrix} + 0 & 1 & 0 & 1\\ + 0 & 1 & 1 & 0\\ + 1 & 1 & 0 & 0\\ + 0 & 1 & 0 & 0\\ + \end{pmatrix} +\] + +\subsection{Linear-Code-Generatormatrix $G_{n,k}$ +\label{mceliece:subsection:g_nk}} +Das wichtigste Element des McEliece-Systems ist ein fehlerkorrigierender Code, +der in der Lage ist, $t$ Fehler zu korrigieren. +Im Zusammenhang mit McEliece werden dabei meist binäre Goppa-Codes \cite{mceliece:goppa} verwendet, +es können prinzipiell auch andere Codes wie beispielsweise Reed-Solomon verwendet werden, +jedoch besitzen einige (unter anderem auch Reed-Solomon) Codes Schwachstellen \cite{mceliece:lorenz}. +Das Codieren mit diesem linearen Code kann mithilfe dessen Generatormatrix $G_{n,k}$ erfolgen. +Da es sich um einen fehlerkorrigierenden Code handelt, +wird das Codewort länger als das Datenwort, +es wird also Redundanz hinzugefügt, +um die Fehlerkorrektur möglich zu machen. + +Beispiel +\[ + G_{7,4}= + \begin{pmatrix} + 1 & 0 & 0 & 0\\ + 1 & 1 & 0 & 0\\ + 0 & 1 & 1 & 0\\ + 1 & 0 & 1 & 1\\ + 0 & 1 & 0 & 1\\ + 0 & 0 & 1 & 0\\ + 0 & 0 & 0 & 1 + \end{pmatrix} +\] + +\subsection{Permutations-Matrix $P_n$ +\label{mceliece:subsection:p_n}} +Mit der zufällig generierten Permutationsmatrix $P_n$ wird die Reihenfolge der Bits geändert. +Mit der Inversen $P_n^{-1}$ kann die Bitvertauschung rückgängig gemacht werden. + +Beispiel +\[ + P_7= + \begin{pmatrix} + 0 & 1 & 0 & 0 & 0 & 0 & 0\\ + 0 & 0 & 0 & 0 & 0 & 0 & 1\\ + 0 & 0 & 0 & 0 & 0 & 1 & 0\\ + 0 & 0 & 1 & 0 & 0 & 0 & 0\\ + 0 & 0 & 0 & 1 & 0 & 0 & 0\\ + 1 & 0 & 0 & 0 & 0 & 0 & 0\\ + 0 & 0 & 0 & 0 & 1 & 0 & 0 + \end{pmatrix} +\] +, +\[ + P_7^{-1}=P_7^t= + \begin{pmatrix} + 0 & 0 & 0 & 0 & 0 & 1 & 0\\ + 1 & 0 & 0 & 0 & 0 & 0 & 0\\ + 0 & 0 & 0 & 1 & 0 & 0 & 0\\ + 0 & 0 & 0 & 0 & 1 & 0 & 0\\ + 0 & 0 & 0 & 0 & 0 & 0 & 1\\ + 0 & 0 & 1 & 0 & 0 & 0 & 0\\ + 0 & 1 & 0 & 0 & 0 & 0 & 0 + \end{pmatrix} +\] + +\subsection{Public-Key $K_{n,k}$ +\label{mceliece:subsection:k_nk}} +Der öffentliche Schlüssel, welcher zum Verschlüsseln verwendet wird, +berechnet sich aus den bereits bekannten Matrizen wiefolgt: +\[ + K_{n,k}=P_{n}\cdot G_{n,k}\cdot S_{k}\,. +\] + +Beispiel +\[ + K_{7,4}= + \begin{pmatrix} + 0 & 0 & 1 & 0\\ + 1 & 0 & 0 & 1\\ + 0 & 0 & 1 & 1\\ + 1 & 1 & 1 & 1\\ + 0 & 1 & 0 & 1\\ + 0 & 1 & 0 & 0\\ + 1 & 0 & 0 & 0 + \end{pmatrix} +\] + +\subsection{Fehler-Vektor $e_n$ +\label{mceliece:subsection:e_n}} +Dieser Vektor der Länge $n$ besteht aus $t$ Einsen, welche zufällig innerhalb des Vektors angeordnet sind, +alle anderen Einträge sind Null. +Dieser Fehlervektor besitzt also gleich viele Einer, +wie die Anzahl Fehler, die der Linearcode der Generatormatrix $G_{n,k}$ zu korrigieren vermag. + +Beispiel +\[ + E_7= + \begin{pmatrix} + 0\\ + 0\\ + 1\\ + 0\\ + 0\\ + 0\\ + 0 + \end{pmatrix} +\] + +\subsection{Daten-Vektor $d_k$ +\label{mceliece:subsection:d_k}} +In diesem Vektor der länge $k$ ist die Nachricht (oder einen Teil davon) enthalten. + +\subsection{Code-Vektor $c_n$ +\label{mceliece:subsection:c_n}} +In diesem Vektor der länge $n$ ist die verschlüsselte Nachricht (oder einen Teil davon) enthalten.
\ No newline at end of file diff --git a/buch/papers/mceliece/einleitung.tex b/buch/papers/mceliece/einleitung.tex new file mode 100644 index 0000000..cebb8ed --- /dev/null +++ b/buch/papers/mceliece/einleitung.tex @@ -0,0 +1,16 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Einleitung +\label{mceliece:section:einleitung}} +\rhead{Einleitung} +Beim McEliece-Kryptosystem handelt es sich um ein asymetrisches Verschlüsselungsverfahren, welches erlaubt, +Daten verschlüsselt über ein Netzwerk zu übermitteln, ohne dass vorab ein gemeinsamer, +geheimer Schlüssel unter den Teilnehmern ausgetauscht werden müsste. +Eine andere, bereits erläuterte Variante einer asymetrischen Verschlüsselung ist das Diffie-Hellman-Verfahren \ref{buch:subsection:diffie-hellman}. +Im Gegensatz zu Diffie-Hellman gilt das McEliece-System als Quantencomputerresistent +und das Verschlüsseln/Entschlüsseln von Nachrichten wird hauptsächlich mit Matrizenoperationen durchgeführt. + + diff --git a/buch/papers/mceliece/fazit.tex b/buch/papers/mceliece/fazit.tex new file mode 100644 index 0000000..186708b --- /dev/null +++ b/buch/papers/mceliece/fazit.tex @@ -0,0 +1,57 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Fazit +\label{mceliece:section:fazit}} +\rhead{Fazit} +Ein kurzer Vergleich des McEliece-Systems +mit dem oft verwendeten RSA-System soll zeigen, wo dessen Vor- und Nachteile liegen. + +\subsection{Resourcen} +Eine Eigenheit des McEliece-Systems ist das hinzufügen von Rauschen (mit Fehlervektor $e_n$). +Damit diese mit dem Lienarcode-Decoder wieder entfernt werden können, +wird Redundanz benötigt, +weshalb dessen Kanalefizienz (Nutzbits/Übertragungsbits) sinkt. +Die Schlüsselgrösse des McEliece-Systems ist deshalb so riesig, weil es sich um eine zweidimensionale Matrix handelt, währenddem RSA mit nur zwei Skalaren auskommt. +Das McEliece-System benötigt dafür weniger Rechenaufwand beim Verschlüsseln/Entschlüsseln, da die meisten Operationen mit Matrixmultiplikationen ausgeführt werden können (Aufwand ist in binären Operationen pro Informationsbit)\cite{mceliece:CodeBasedCrypto}. +Beim Rechenaufwand sei noch erwähnt, +dass asymetrische Verschlüsselungen meist nur dazu verwendet werden, +um einen Schlüssel für eine symetrische Verschlüsselung auszutauschen. +\begin{center} +\begin{tabular}{c|c|c} + &McEliece (n=2048, k=1718, t = 30) &RSA (2048, e = 216 + 1)\\ + \hline + Schlüssegrösse: (Public) &429.5 KByte &0.5 KByte \\ + Kanaleffizienz: &83.9 \% &100 \% \\ + Verschlüsselungsaufwand: &1025 &40555 \\ + Entschlüsselungsaufwand: &2311 &6557176, 5 +\end{tabular} +\end{center} + +\subsection{Sicherheit} +Grosse unterschiede zwischen den beiden Kryptosystemen gibt es jedoch bei der Sicherheit. +Der Kern der RSA-Verschlüsselung beruht auf dem Problem, eine grosse Zahl in ihre beiden Primfaktoren zu zerlegen. +Bei genügend grossen Zahlen ist diese Zerlegung auch mit den heute besten verfügbaren Computern kaum innerhalb vernünftiger Zeit zu lösen. +Weiter ist aber bekannt, +dass mithilfe des sogenannten Shor-Algorithmus \cite{mceliece:shor} und einem Quantencomputer auch diese Zerlegung zügig realisiert werden könnte, +was zur Folge hätte, dass die Verschlüsselung von RSA unwirksam würde. +Zurzeit sind die Quantencomputer jedoch noch bei weitem nicht in der Lage, grosse Zahlen mithilfe dieses Algorithmuses zu zerlegen. +Das McEliece-System hingegen beruht auf dem Problem des ``Syndrome decoding'' (Korrektur von Bitfehlern eines Codewortes, das mit einem entsprechenden Linearcode codiert wurde). +Für das ``Syndrome decoding'' sind bis heute keine Methoden bekannt, +welche nennenswerte Vorteile gegenüber dem Durchprobieren (brute-force) bringen, +auch nicht mithilfe eines Quantencomputers. +\begin{center} +\begin{tabular}{c|c|c} + &McEliece &RSA \\ +\hline + Grundlage Verschlüsselung &Syndrome decoding &Integer factoring\\ + Aufwand (gewöhnliche CPU) &exponential &< exponential \\ + Aufwand (Quantencomputer) &> polynomial &$\mathcal{O}(\log(N)^3)$ +\end{tabular} +\end{center} +Die Verbreitung des McEliece-Kryptosystems ist zurzeit äusserst gering. +Das liegt einerseits an der immensen Grösse des öffentlichen Schlüssels, +andererseits wird aber auch in naher Zukunft nicht mit einem genügend starken Quantencomputer gerechnet, +welcher andere asymetrische Verschlüsselungen gefährden würde. diff --git a/buch/papers/mceliece/funktionsweise.tex b/buch/papers/mceliece/funktionsweise.tex new file mode 100644 index 0000000..7c69b13 --- /dev/null +++ b/buch/papers/mceliece/funktionsweise.tex @@ -0,0 +1,83 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Funktionsweise +\label{mceliece:section:funktionsweise}} +\rhead{Funktionsweise} +Um den Ablauf des Datenaustausches mittels McEliece-Verschlüsselung zu erläutern, +wird ein Szenario verwendet, +bei dem Bob an Alice eine verschlüsselte Nachticht über ein öffentliches Netzwerk zukommen lässt. + +\subsection{Vorbereitung +\label{mceliece:section:vorbereitung}} +Damit der Nachrichtenaustausch stattfinden kann, muss Alice (Empfängerin) +zuerst ein Schlüsselpaar definieren. +Dazu erstellt sie die einzelnen Matrizen $S_k$, $G_{n,k}$ und $P_n$. +Diese drei einzelnen Matrizen bilden den privaten Schlüssel von Alice +und sollen geheim bleiben. +Der öffentliche Schlüssel $K_{n,k}$ hingegen berechnet sich +aus der Multiplikation der privaten Matrizen (Abschnitt \ref{mceliece:subsection:k_nk}) +und wird anschliessend Bob zugestellt. + +\subsection{Verschlüsselung +\label{mceliece:section:verschl}} +Bob berechnet nun die verschlüsselte Nachricht $c_n$, indem er seine Daten $d_k$ +mit dem öffentlichen Schlüssel $K_{n,k}$ von Alice multipliziert +und anschliessend durch eine Addition mit einem Fehlervektor $e_n$ einige Bitfehler hinzufügt. +\[ + c_n\,=\,K_{n,k}\cdot d_k + e_n\,. +\] +Dabei wird für jede Nachricht (oder für jedes Nachrichtenfragment) +einen neuen, zufälligen Fehlervektor generiert. +Die verschlüsselte Nachricht $c_n$ wird anschliessend Alice zugestellt. + +\subsection{Entschlüsselung +\label{mceliece:section:entschl}} +Alice entschlüsselt die erhaltene Nachricht in mehreren einzelnen Schritten. +Um etwas Transparenz in diese Prozedur zu bringen, wird der öffentliche Schlüssel $K_{n,k}$ mit seinen Ursprungsmatrizen dargestellt. +\begin{align*} + c_n\,&=\,K_{n,k}\cdot d_k + e_n \\ + &= P_{n}\cdot G_{n,k}\cdot S_{k}\cdot d_k + e_n +\end{align*} +Zuerst wird der Effekt der Permutationsmatrix rückgängig gemacht, +indem das Codewort mit dessen Inversen $P_n^{-1}$ multipliziert wird. +\begin{align*} + c_{n}''\,=\,P_n^{-1}\cdot c_n\,&= P_n^{-1}\cdot P_{n}\cdot G_{n,k}\cdot S_{k}\cdot d_k + P_n^{-1}\cdot e_n \\ + &= G_{n,k}\cdot S_{k}\cdot d_k + P_n^{-1}\cdot e_n \\ +\end{align*} +Eine weitere Vereinfachung ist nun möglich, +weil $P_n^{-1}$ einerseits auch eine gewöhnliche Permutationsmatrix ist +und andererseits ein zufälliger Fehlervektor $e_n$ multipliziert mit einer Permutationsmatrix +wiederum einen gleichwertigen, zufälligen Fehlervektor $e_n'$ ergibt. +\begin{align*} + c_{n}''\,&=\,G_{n,k}\cdot S_{k}\cdot d_k + P_n^{-1}\cdot e_n \\ + &=\,G_{n,k}\cdot S_{k}\cdot d_k + e'_n\quad \quad \quad | \, + e'_n\,=\,P_n^{-1}\cdot e_n +\end{align*} +Dank des fehlerkorrigierenden Codes, der durch die implizite Multiplikation mittels $G_{n,k}$ auf die Daten angewendet wurde, +können nun die Bitfehler, verursacht durch den Fehlervektor $e'_n$, +entfernt werden. +Da es sich bei diesem Schritt nicht um eine einfache Matrixmultiplikation handelt, +wird die Operation durch eine Funktion dargestellt. +Wie dieser Decoder genau aufgebaut ist, +hängt vom verwendeten Linearcode ab. +\begin{align*} + c_{k}'\,&=\text{Linear-Code-Decoder($c''_n$)}\\ + &=\text{Linear-Code-Decoder($G_{n,k}\cdot S_{k}\cdot d_k + e'_n$)}\\ + &=S_{k}\cdot d_k +\end{align*} +Zum Schluss wird das inzwischen fast entschlüsselte Codewort $c'_k$ mit der inversen der zufälligen Binärmatrix $S^{-1}$ multipliziert, +womit der Inhalt der ursprünglichen Nachricht nun wiederhergestellt wurde. +\begin{align*} + c_{k}'\,&=S_{k}\cdot d_k \quad | \cdot S_k^{-1}\\ + d'_{k}\,=\,S_{k}^{-1} \cdot c'_k&=S_{k}^{-1} \cdot S_{k}\cdot d_k\\ + &=d_k +\end{align*} + +\subsection{Beispiel} + +TODO: +-alle Beispielmatrizen- und Vektoren hierhin zügeln, numerisches Beispiel kreieren\\ +-erläutern des 7/4-codes (ja/nein)?
\ No newline at end of file diff --git a/buch/papers/mceliece/main.tex b/buch/papers/mceliece/main.tex index dbbaaac..352a6be 100644 --- a/buch/papers/mceliece/main.tex +++ b/buch/papers/mceliece/main.tex @@ -8,29 +8,10 @@ \begin{refsection} \chapterauthor{Reto Fritsche} -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/mceliece/teil0.tex} -\input{papers/mceliece/teil1.tex} -\input{papers/mceliece/teil2.tex} -\input{papers/mceliece/teil3.tex} +\input{papers/mceliece/einleitung.tex} +\input{papers/mceliece/aufbau.tex} +\input{papers/mceliece/funktionsweise.tex} +\input{papers/mceliece/fazit.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/mceliece/references.bib b/buch/papers/mceliece/references.bib index 47798d3..0388ff4 100644 --- a/buch/papers/mceliece/references.bib +++ b/buch/papers/mceliece/references.bib @@ -4,32 +4,45 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{mceliece:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@online{mceliece:GenerationRandMatrix, + title = {Efficient Generation of Random Nonsingular Matrices}, + url = {https://www.researchgate.net/publication/2729950_Efficient_Generation_of_Random_Nonsingular_Matrices}, + date = {Januar 1993}, + year = {2021}, + month = {7}, + day = {29} } -@book{mceliece:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} +@online{mceliece:lorenz, + title = {Cryptography based on error correcting codes}, + url = {https://algo.epfl.ch/_media/en/projects/lorenz_thesis.pdf}, + date = {2007-07-27}, + year = {2021}, + month = {7}, + day = {29} } -@article{mceliece:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@online{mceliece:shor, + title = {Shor's algorithm}, + url = {https://en.wikipedia.org/wiki/Shor%27s_algorithm}, + year = {2021}, + month = {8}, + day = {9} } +@online{mceliece:CodeBasedCrypto, + title = {Code based cryptography and steganography}, + url = {https://www.researchgate.net/publication/268009418_Code_Based_Cryptography_and_Steganography}, + date = {2013-05-30}, + year = {2021}, + month = {8}, + day = {9} +} + +@online{mceliece:goppa, + title = {Binary Goppa code}, + url = {https://en.m.wikipedia.org/wiki/Binary_Goppa_code}, + year = {2021}, + month = {8}, + day = {10} +}
\ No newline at end of file diff --git a/buch/papers/mceliece/teil0.tex b/buch/papers/mceliece/teil0.tex deleted file mode 100644 index b98f8be..0000000 --- a/buch/papers/mceliece/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{mceliece: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{mceliece: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/mceliece/teil1.tex b/buch/papers/mceliece/teil1.tex deleted file mode 100644 index 06035a6..0000000 --- a/buch/papers/mceliece/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{mceliece: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{mceliece: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{mceliece: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{mceliece: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{mceliece: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/mceliece/teil2.tex b/buch/papers/mceliece/teil2.tex deleted file mode 100644 index fd247c7..0000000 --- a/buch/papers/mceliece/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{mceliece: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{mceliece: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/mceliece/teil3.tex b/buch/papers/mceliece/teil3.tex deleted file mode 100644 index 421b331..0000000 --- a/buch/papers/mceliece/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{mceliece: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{mceliece: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/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 Binary files differnew file mode 100755 index 0000000..9def15a --- /dev/null +++ b/buch/papers/multiplikation/code/Figure_1.png diff --git a/buch/papers/multiplikation/code/MM.c b/buch/papers/multiplikation/code/MM.c new file mode 100755 index 0000000..2588262 --- /dev/null +++ b/buch/papers/multiplikation/code/MM.c @@ -0,0 +1,466 @@ +#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<10; ++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<10; ++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..8057850 --- /dev/null +++ b/buch/papers/multiplikation/code/MM.py @@ -0,0 +1,339 @@ +#!/usr/bin/env python3 +# -*- coding: utf-8 -*- +""" +Created on Fri Mar 19 07:31:29 2021 + +@author: nunigan +""" +import scipy.stats +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.xscale('log', base=2) + plt.legend() + plt.xlabel("n") + plt.ylabel("time (s)") + plt.grid(True, which="both", ls="-") + 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 t_np + + +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.yscale('log', base=10) + 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.loglog(MM_n, MM_t, '.', label='3 For Loops', lw=5) + plt.loglog(winograd_n, winograd_t, '.', label='Winograd MM', lw=5) + plt.loglog(blas_n, blas_t, '.', label='Blas', lw=5) + plt.loglog(strassen_n, strassen_t, '.', label='Strassen', lw=5) + plt.loglog(MM_dc_n, MM_dc_t, '.', label='Divide and Conquer', lw=5) + plt.xlabel("n") + # plt.yscale('log', base=10) + # plt.xscale('log', base=2) + plt.ylabel("time (s)") + plt.grid(True, which="both", ls="-") + 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() + # return [MM_n,winograd_n,blas_n,strassen_n,MM_dc_n] + + + return [MM_t,winograd_t,blas_t,strassen_t,MM_dc_t] + + +def mean_confidence_interval(data, confidence=0.95): + a = 1.0 * np.array(data) + n = len(a) + m, se = np.mean(a), scipy.stats.sem(a) + h = se * scipy.stats.t.ppf((1 + confidence) / 2., n-1) + return m, h + +# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +if __name__ == '__main__': + # A = plot_c_res(10, 4096) + # name = ['MM', 'Wino', 'blas', 'strassen', 'dc'] + # for i in range(5): + # ci_inner = [] + # print(name[i]) + # for j in range(11): + # m,h=mean_confidence_interval(A[i][j*10:(j+1)*10]) + # print("({},{})".format(2**(j+1),m)) + # np.savetxt('meas/ci/' + name[i]+'.txt',ci_inner) + + arr = plot(4096) + # n = np.logspace(1,12,12,base=2,dtype=(np.int)) + # n=[2048,4096] + # n = np.arange(1,50,2) + # A = np.random.randint(-10, 6, (5,3)) + # B = np.random.randint(-10, 6, (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 Binary files differnew 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..63d5390 --- /dev/null +++ b/buch/papers/multiplikation/code/c_matrix.h @@ -0,0 +1,177 @@ +/* Seminar Matrizen, autogenerated File, Michael Schmid, 10/08/2021, 05:46:32 */ + +#include <stdint.h> +const int A0[][2] = + { + {60,-84}, + {-66,-1} + }; +const int B0[][2] = + { + {-45,87}, + {-38,-73} + }; +const double dB0[][2] = + { + {-45,87}, + {-38,-73} + }; +const double dA0[][2] = + { + {60,-84}, + {-66,-1} + }; +const int A1[][4] = + { + {-72,-19,-91,62}, + {-36,-74,-44,-47}, + {-39,-31,50,-93}, + {-81,2,-17,-86} + }; +const int B1[][4] = + { + {-66,39,-23,52}, + {-88,-13,13,-13}, + {-45,-70,28,-20}, + {96,5,88,96} + }; +const double dB1[][4] = + { + {-66,39,-23,52}, + {-88,-13,13,-13}, + {-45,-70,28,-20}, + {96,5,88,96} + }; +const double dA1[][4] = + { + {-72,-19,-91,62}, + {-36,-74,-44,-47}, + {-39,-31,50,-93}, + {-81,2,-17,-86} + }; +const int A2[][8] = + { + {-36,-2,-58,-32,34,-89,49,-55}, + {-68,-73,52,-3,-51,-37,-31,70}, + {73,-90,-21,-79,-15,96,-99,12}, + {68,-25,38,-73,-60,35,-99,72}, + {-43,-87,48,-84,-100,37,80,53}, + {-27,88,-5,-82,-57,-27,20,10}, + {-91,-47,54,-90,-99,-76,50,-18}, + {69,-36,76,5,-67,-38,-95,91} + }; +const int B2[][8] = + { + {-84,22,-13,-66,-42,51,66,0}, + {37,-65,66,-85,-10,-23,77,5}, + {1,41,-79,0,63,-37,-10,29}, + {72,66,-99,92,-28,65,25,-40}, + {69,-49,65,-18,64,-97,-47,30}, + {36,86,66,-12,-17,89,1,-37}, + {-100,11,27,23,-75,-23,96,-9}, + {68,90,-87,-99,-70,-28,98,-76} + }; +const double dB2[][8] = + { + {-84,22,-13,-66,-42,51,66,0}, + {37,-65,66,-85,-10,-23,77,5}, + {1,41,-79,0,63,-37,-10,29}, + {72,66,-99,92,-28,65,25,-40}, + {69,-49,65,-18,64,-97,-47,30}, + {36,86,66,-12,-17,89,1,-37}, + {-100,11,27,23,-75,-23,96,-9}, + {68,90,-87,-99,-70,-28,98,-76} + }; +const double dA2[][8] = + { + {-36,-2,-58,-32,34,-89,49,-55}, + {-68,-73,52,-3,-51,-37,-31,70}, + {73,-90,-21,-79,-15,96,-99,12}, + {68,-25,38,-73,-60,35,-99,72}, + {-43,-87,48,-84,-100,37,80,53}, + {-27,88,-5,-82,-57,-27,20,10}, + {-91,-47,54,-90,-99,-76,50,-18}, + {69,-36,76,5,-67,-38,-95,91} + }; +const int A3[][16] = + { + {-24,65,21,19,94,70,-90,-81,53,-41,-23,-1,58,-80,-54,59}, + {-42,76,-19,98,29,-56,92,14,45,11,82,83,48,-13,81,66}, + {43,-57,-67,95,5,72,11,0,-47,55,-24,36,84,54,-31,-54}, + {-39,-40,19,97,-82,-56,27,95,81,-21,-50,-74,-35,-87,-28,-26}, + {-74,-98,79,92,-24,-48,99,94,55,-83,70,98,-24,18,-67,14}, + {20,76,11,-23,-56,21,0,42,64,86,-74,44,93,-76,-30,97}, + {13,20,-73,-11,-30,80,53,-8,60,21,17,-42,82,-72,-6,-80}, + {36,-93,-64,-21,20,-85,15,24,99,81,-52,64,71,-56,52,63}, + {32,9,-2,-85,17,62,-98,-35,75,-58,-44,-20,-47,89,-95,52}, + {93,-43,86,68,-6,-25,90,57,60,-10,65,-97,43,46,-60,-41}, + {43,-33,0,50,-100,26,-60,95,39,-70,-61,-81,9,-23,-99,-4}, + {20,61,15,43,-96,93,-55,38,-29,-1,-10,26,-87,18,64,6}, + {-98,-84,51,16,-14,86,52,59,44,-39,-2,10,82,-66,54,19}, + {89,-49,-37,-6,-53,40,-11,46,-51,-56,86,34,11,13,-20,-49}, + {-90,14,28,-45,-25,-56,-51,-61,28,-8,51,91,95,-10,-85,58}, + {8,-44,88,-71,-27,11,89,37,86,-78,-44,-56,-87,0,-42,-61} + }; +const int B3[][16] = + { + {62,-30,62,92,29,-93,-95,44,-33,-88,-29,9,-88,-42,-90,-70}, + {60,37,-44,-93,-87,6,-53,2,-29,53,-49,59,6,83,-15,50}, + {-19,85,-49,-14,84,-4,12,88,-83,-81,-24,-16,-12,-42,-63,-71}, + {-42,-78,-58,-61,-29,67,-28,-46,64,7,6,-13,88,-42,95,-24}, + {-90,-56,8,-30,-89,70,37,-29,24,-8,-10,-2,-25,-63,-95,-91}, + {10,-81,42,-28,-13,-68,-72,-20,-22,5,-79,-50,-88,62,57,69}, + {-67,24,-71,-43,11,48,33,-93,-82,-65,-4,5,-15,25,-54,-45}, + {-49,19,-29,90,-97,-87,78,-39,-75,-85,-79,-35,54,3,-73,7}, + {-7,39,70,-42,32,-100,56,4,-24,-57,38,-49,-50,-44,79,-42}, + {37,-65,-55,22,-97,-42,-76,95,97,-27,38,11,0,-81,-23,35}, + {26,-70,10,-29,47,-70,-52,29,-13,-18,5,34,18,32,87,91}, + {-84,41,-19,96,-51,-19,81,75,81,92,2,-40,-42,-69,-10,-61}, + {-30,98,71,-51,91,-59,58,86,86,-22,-84,7,66,-55,-52,23}, + {-71,-44,-9,90,26,18,26,-10,-85,64,-47,3,72,81,74,-8}, + {52,-59,-91,22,8,-63,84,9,-11,-54,-78,-71,-98,42,96,57}, + {18,-39,34,-50,-62,-96,-2,-78,52,94,-33,2,-19,-9,-86,-75} + }; +const double dB3[][16] = + { + {62,-30,62,92,29,-93,-95,44,-33,-88,-29,9,-88,-42,-90,-70}, + {60,37,-44,-93,-87,6,-53,2,-29,53,-49,59,6,83,-15,50}, + {-19,85,-49,-14,84,-4,12,88,-83,-81,-24,-16,-12,-42,-63,-71}, + {-42,-78,-58,-61,-29,67,-28,-46,64,7,6,-13,88,-42,95,-24}, + {-90,-56,8,-30,-89,70,37,-29,24,-8,-10,-2,-25,-63,-95,-91}, + {10,-81,42,-28,-13,-68,-72,-20,-22,5,-79,-50,-88,62,57,69}, + {-67,24,-71,-43,11,48,33,-93,-82,-65,-4,5,-15,25,-54,-45}, + {-49,19,-29,90,-97,-87,78,-39,-75,-85,-79,-35,54,3,-73,7}, + {-7,39,70,-42,32,-100,56,4,-24,-57,38,-49,-50,-44,79,-42}, + {37,-65,-55,22,-97,-42,-76,95,97,-27,38,11,0,-81,-23,35}, + {26,-70,10,-29,47,-70,-52,29,-13,-18,5,34,18,32,87,91}, + {-84,41,-19,96,-51,-19,81,75,81,92,2,-40,-42,-69,-10,-61}, + {-30,98,71,-51,91,-59,58,86,86,-22,-84,7,66,-55,-52,23}, + {-71,-44,-9,90,26,18,26,-10,-85,64,-47,3,72,81,74,-8}, + {52,-59,-91,22,8,-63,84,9,-11,-54,-78,-71,-98,42,96,57}, + {18,-39,34,-50,-62,-96,-2,-78,52,94,-33,2,-19,-9,-86,-75} + }; +const double dA3[][16] = + { + {-24,65,21,19,94,70,-90,-81,53,-41,-23,-1,58,-80,-54,59}, + {-42,76,-19,98,29,-56,92,14,45,11,82,83,48,-13,81,66}, + {43,-57,-67,95,5,72,11,0,-47,55,-24,36,84,54,-31,-54}, + {-39,-40,19,97,-82,-56,27,95,81,-21,-50,-74,-35,-87,-28,-26}, + {-74,-98,79,92,-24,-48,99,94,55,-83,70,98,-24,18,-67,14}, + {20,76,11,-23,-56,21,0,42,64,86,-74,44,93,-76,-30,97}, + {13,20,-73,-11,-30,80,53,-8,60,21,17,-42,82,-72,-6,-80}, + {36,-93,-64,-21,20,-85,15,24,99,81,-52,64,71,-56,52,63}, + {32,9,-2,-85,17,62,-98,-35,75,-58,-44,-20,-47,89,-95,52}, + {93,-43,86,68,-6,-25,90,57,60,-10,65,-97,43,46,-60,-41}, + {43,-33,0,50,-100,26,-60,95,39,-70,-61,-81,9,-23,-99,-4}, + {20,61,15,43,-96,93,-55,38,-29,-1,-10,26,-87,18,64,6}, + {-98,-84,51,16,-14,86,52,59,44,-39,-2,10,82,-66,54,19}, + {89,-49,-37,-6,-53,40,-11,46,-51,-56,86,34,11,13,-20,-49}, + {-90,14,28,-45,-25,-56,-51,-61,28,-8,51,91,95,-10,-85,58}, + {8,-44,88,-71,-27,11,89,37,86,-78,-44,-56,-87,0,-42,-61} + }; +const int *Ap[4] = {(int*) A0,(int*) A1,(int*) A2,(int*) A3}; +const int *Bp[4] = {(int*) B0,(int*) B1,(int*) B2,(int*) B3}; +const double *dAp[4] = {(double*) dA0,(double*) dA1,(double*) dA2,(double*) dA3}; +const double *dBp[4] = {(double*) dB0,(double*) dB1,(double*) dB2,(double*) dB3}; +int n[4] = {2,4,8,16}; +int n_arrays = 4; diff --git a/buch/papers/multiplikation/code/c_meas_1024.pdf b/buch/papers/multiplikation/code/c_meas_1024.pdf Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew file mode 100644 index 0000000..f637ae4 --- /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 Binary files differnew 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 Binary files differnew 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 Binary files differnew file mode 100644 index 0000000..9682aca --- /dev/null +++ b/buch/papers/multiplikation/code/c_meas_8.pdf diff --git a/buch/papers/multiplikation/code/ci.txt b/buch/papers/multiplikation/code/ci.txt new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/code/ci.txt diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py new file mode 100755 index 0000000..3b74f67 --- /dev/null +++ b/buch/papers/multiplikation/code/helper_class.py @@ -0,0 +1,106 @@ +#!/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(1,11,11,base=2,dtype=(np.int)) + # n=[8192] + # 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..7bffb6e --- /dev/null +++ b/buch/papers/multiplikation/code/meas/MM.txt @@ -0,0 +1,110 @@ +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000001,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000001,4 +0.000001,4 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000021,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000090,32 +0.000093,32 +0.000083,32 +0.000082,32 +0.000090,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000089,32 +0.000126,32 +0.000771,64 +0.000651,64 +0.000651,64 +0.000651,64 +0.000731,64 +0.000673,64 +0.000745,64 +0.000672,64 +0.000671,64 +0.000707,64 +0.005642,128 +0.005579,128 +0.005768,128 +0.005745,128 +0.005518,128 +0.005877,128 +0.005513,128 +0.005850,128 +0.005769,128 +0.005581,128 +0.052188,256 +0.051988,256 +0.051888,256 +0.051518,256 +0.051709,256 +0.051543,256 +0.051707,256 +0.051845,256 +0.051495,256 +0.051834,256 +0.507020,512 +0.504111,512 +0.502049,512 +0.529743,512 +0.501028,512 +0.502097,512 +0.503490,512 +0.502079,512 +0.506688,512 +0.504163,512 +4.538722,1024 +4.291473,1024 +4.516302,1024 +4.374630,1024 +4.719557,1024 +4.438999,1024 +4.641680,1024 +4.407959,1024 +4.441451,1024 +4.677313,1024 +129.433279,2048 +129.277802,2048 +129.284817,2048 +129.086884,2048 +129.197444,2048 +129.350999,2048 +129.264250,2048 +129.295723,2048 +129.402601,2048 +129.300820,2048 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..b78b925 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/MM_dc.txt @@ -0,0 +1,110 @@ +0.000003,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000002,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000008,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000007,8 +0.000007,8 +0.000007,8 +0.000007,8 +0.000018,8 +0.000008,8 +0.000075,16 +0.000063,16 +0.000088,16 +0.000062,16 +0.000086,16 +0.000092,16 +0.000081,16 +0.000080,16 +0.000070,16 +0.000085,16 +0.000581,32 +0.000659,32 +0.000584,32 +0.000714,32 +0.000666,32 +0.000574,32 +0.000616,32 +0.000534,32 +0.000506,32 +0.000506,32 +0.004567,64 +0.004502,64 +0.004332,64 +0.004578,64 +0.004543,64 +0.004426,64 +0.004497,64 +0.004329,64 +0.004288,64 +0.004277,64 +0.036456,128 +0.034901,128 +0.034545,128 +0.034283,128 +0.035150,128 +0.034663,128 +0.034901,128 +0.034022,128 +0.034368,128 +0.035154,128 +0.296292,256 +0.297592,256 +0.302464,256 +0.299557,256 +0.299367,256 +0.306394,256 +0.287616,256 +0.292630,256 +0.289542,256 +0.277019,256 +2.331956,512 +2.224501,512 +2.203910,512 +2.198937,512 +2.206083,512 +2.199477,512 +2.199847,512 +2.225379,512 +2.202491,512 +2.235926,512 +17.649432,1024 +17.636769,1024 +17.639024,1024 +17.625402,1024 +17.722286,1024 +17.611777,1024 +17.653120,1024 +17.748270,1024 +17.691817,1024 +17.614448,1024 +141.943689,2048 +141.580812,2048 +141.882050,2048 +141.516253,2048 +141.351237,2048 +141.641167,2048 +141.596407,2048 +141.607048,2048 +141.469723,2048 +141.515550,2048 diff --git a/buch/papers/multiplikation/code/meas/blas.txt b/buch/papers/multiplikation/code/meas/blas.txt new file mode 100644 index 0000000..9414d8f --- /dev/null +++ b/buch/papers/multiplikation/code/meas/blas.txt @@ -0,0 +1,110 @@ +0.000001,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000001,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000012,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000021,32 +0.000019,32 +0.000030,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000180,64 +0.000192,64 +0.000163,64 +0.000153,64 +0.000153,64 +0.000197,64 +0.000163,64 +0.000267,64 +0.000226,64 +0.000164,64 +0.001216,128 +0.001233,128 +0.001364,128 +0.001278,128 +0.001211,128 +0.001295,128 +0.001206,128 +0.001371,128 +0.001225,128 +0.001250,128 +0.009733,256 +0.009497,256 +0.009586,256 +0.009600,256 +0.009768,256 +0.009566,256 +0.009731,256 +0.009550,256 +0.009664,256 +0.009794,256 +0.077453,512 +0.076616,512 +0.088812,512 +0.075990,512 +0.076925,512 +0.076303,512 +0.075915,512 +0.075600,512 +0.075122,512 +0.075029,512 +0.769186,1024 +0.775780,1024 +0.753906,1024 +0.757834,1024 +0.772001,1024 +0.770950,1024 +0.791317,1024 +0.753319,1024 +0.747228,1024 +0.752347,1024 +7.625205,2048 +7.652278,2048 +7.640682,2048 +7.649428,2048 +7.632806,2048 +7.579347,2048 +7.612317,2048 +7.676742,2048 +7.632979,2048 +7.619210,2048 diff --git a/buch/papers/multiplikation/code/meas/ci/MM.txt b/buch/papers/multiplikation/code/meas/ci/MM.txt new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/ci/MM.txt diff --git a/buch/papers/multiplikation/code/meas/ci/Wino.txt b/buch/papers/multiplikation/code/meas/ci/Wino.txt new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/ci/Wino.txt diff --git a/buch/papers/multiplikation/code/meas/ci/blas.txt b/buch/papers/multiplikation/code/meas/ci/blas.txt new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/ci/blas.txt diff --git a/buch/papers/multiplikation/code/meas/ci/dc.txt b/buch/papers/multiplikation/code/meas/ci/dc.txt new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/ci/dc.txt diff --git a/buch/papers/multiplikation/code/meas/ci/strassen.txt b/buch/papers/multiplikation/code/meas/ci/strassen.txt new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/ci/strassen.txt diff --git a/buch/papers/multiplikation/code/meas/old/8196/MM.txt b/buch/papers/multiplikation/code/meas/old/8196/MM.txt new file mode 100644 index 0000000..0edf9f6 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/MM.txt @@ -0,0 +1 @@ +9376.173434,8192 diff --git a/buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt b/buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt new file mode 100644 index 0000000..36f6ff0 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt @@ -0,0 +1 @@ +9606.402522,8192 diff --git a/buch/papers/multiplikation/code/meas/old/8196/blas.txt b/buch/papers/multiplikation/code/meas/old/8196/blas.txt new file mode 100644 index 0000000..b5989fb --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/blas.txt @@ -0,0 +1 @@ +478.429957,8192 diff --git a/buch/papers/multiplikation/code/meas/old/8196/strassen.txt b/buch/papers/multiplikation/code/meas/old/8196/strassen.txt new file mode 100644 index 0000000..ca06e97 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/strassen.txt @@ -0,0 +1 @@ +3014.235467,8192 diff --git a/buch/papers/multiplikation/code/meas/old/8196/winograd.txt b/buch/papers/multiplikation/code/meas/old/8196/winograd.txt new file mode 100644 index 0000000..2a529c4 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/winograd.txt @@ -0,0 +1 @@ +10071.512655,8192 diff --git a/buch/papers/multiplikation/code/meas/old/MM.txt b/buch/papers/multiplikation/code/meas/old/MM.txt new file mode 100644 index 0000000..e296dd7 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/MM.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000001,4 +0.000001,8 +0.000010,16 +0.000081,32 +0.000654,64 +0.005556,128 +0.054253,256 +0.487317,512 +4.162845,1024 +125.909034,2048 +1111.312696,4096 diff --git a/buch/papers/multiplikation/code/meas/old/MM_dc.txt b/buch/papers/multiplikation/code/meas/old/MM_dc.txt new file mode 100644 index 0000000..f6be928 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/MM_dc.txt @@ -0,0 +1,12 @@ +0.000003,2 +0.000002,4 +0.000010,8 +0.000068,16 +0.000594,32 +0.004264,64 +0.036289,128 +0.324645,256 +2.612010,512 +19.928951,1024 +159.333884,2048 +1147.106865,4096 diff --git a/buch/papers/multiplikation/code/meas/old/blas.txt b/buch/papers/multiplikation/code/meas/old/blas.txt new file mode 100644 index 0000000..92a61b9 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/blas.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000001,4 +0.000001,8 +0.000003,16 +0.000022,32 +0.000179,64 +0.001278,128 +0.010165,256 +0.074739,512 +0.704748,1024 +6.845095,2048 +55.845038,4096 diff --git a/buch/papers/multiplikation/code/meas/old/strassen.txt b/buch/papers/multiplikation/code/meas/old/strassen.txt new file mode 100644 index 0000000..fdfbf2b --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/strassen.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000003,4 +0.000010,8 +0.000066,16 +0.000470,32 +0.003368,64 +0.024232,128 +0.172000,256 +1.209262,512 +8.457472,1024 +59.267256,2048 +414.648901,4096 diff --git a/buch/papers/multiplikation/code/meas/old/winograd.txt b/buch/papers/multiplikation/code/meas/old/winograd.txt new file mode 100644 index 0000000..d185906 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/winograd.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000001,4 +0.000002,8 +0.000011,16 +0.000100,32 +0.000654,64 +0.005229,128 +0.057440,256 +0.517850,512 +4.539413,1024 +130.627663,2048 +1179.261048,4096 diff --git a/buch/papers/multiplikation/code/meas/strassen.txt b/buch/papers/multiplikation/code/meas/strassen.txt new file mode 100644 index 0000000..d6e040e --- /dev/null +++ b/buch/papers/multiplikation/code/meas/strassen.txt @@ -0,0 +1,110 @@ +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000004,4 +0.000002,4 +0.000002,4 +0.000002,4 +0.000002,4 +0.000002,4 +0.000002,4 +0.000002,4 +0.000002,4 +0.000001,4 +0.000020,8 +0.000018,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000019,8 +0.000080,16 +0.000075,16 +0.000078,16 +0.000085,16 +0.000065,16 +0.000065,16 +0.000065,16 +0.000064,16 +0.000065,16 +0.000065,16 +0.000546,32 +0.000480,32 +0.000563,32 +0.000551,32 +0.000502,32 +0.000504,32 +0.000463,32 +0.000462,32 +0.000508,32 +0.000462,32 +0.003675,64 +0.003665,64 +0.003493,64 +0.003708,64 +0.003465,64 +0.003502,64 +0.003710,64 +0.003537,64 +0.003637,64 +0.003568,64 +0.025342,128 +0.025179,128 +0.026475,128 +0.025758,128 +0.025333,128 +0.024988,128 +0.025727,128 +0.025298,128 +0.025283,128 +0.025098,128 +0.181311,256 +0.178432,256 +0.177075,256 +0.177474,256 +0.177025,256 +0.177805,256 +0.177944,256 +0.178151,256 +0.177858,256 +0.178742,256 +1.283374,512 +1.246682,512 +1.245898,512 +1.251547,512 +1.250288,512 +1.250495,512 +1.257037,512 +1.255247,512 +1.255382,512 +1.259050,512 +8.784102,1024 +8.845725,1024 +8.771100,1024 +8.770184,1024 +8.955977,1024 +8.849161,1024 +8.806902,1024 +8.808937,1024 +8.848900,1024 +8.861383,1024 +61.787123,2048 +61.972599,2048 +61.822434,2048 +62.051331,2048 +61.946171,2048 +61.911404,2048 +61.872671,2048 +61.791260,2048 +61.818110,2048 +62.045588,2048 diff --git a/buch/papers/multiplikation/code/meas/test/4096/MM.txt b/buch/papers/multiplikation/code/meas/test/4096/MM.txt new file mode 100644 index 0000000..25e40e1 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/test/4096/MM.txt @@ -0,0 +1,12 @@ +0.000000,2 +0.000000,4 +0.000002,8 +0.000011,16 +0.000100,32 +0.000712,64 +0.005498,128 +0.046711,256 +0.489233,512 +4.006544,1024 +124.427496,2048 +993.405615,4096 diff --git a/buch/papers/multiplikation/code/meas/test/4096/strassen.txt b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt new file mode 100644 index 0000000..eb2a496 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt @@ -0,0 +1,12 @@ +0.000007,2 +0.000007,4 +0.000029,8 +0.000199,16 +0.001414,32 +0.007583,64 +0.028096,128 +0.171662,256 +1.198323,512 +8.421896,1024 +58.803644,2048 +415.115401,4096 diff --git a/buch/papers/multiplikation/code/meas/test/MM.txt b/buch/papers/multiplikation/code/meas/test/MM.txt new file mode 100644 index 0000000..e0754ab --- /dev/null +++ b/buch/papers/multiplikation/code/meas/test/MM.txt @@ -0,0 +1,14900 @@ +0.000004,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 +0.000001,8 +0.000002,8 +0.000002,8 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000006,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000013,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000008,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000016,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000007,14 +0.000011,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000025,16 +0.000011,16 +0.000020,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000010,16 +0.000016,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000015,18 +0.000014,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000015,18 +0.000014,18 +0.000021,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000030,20 +0.000029,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000030,20 +0.000030,20 +0.000029,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000048,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000020,20 +0.000027,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000033,22 +0.000040,22 +0.000045,22 +0.000046,22 +0.000041,22 +0.000040,22 +0.000040,22 +0.000040,22 +0.000042,22 +0.000040,22 +0.000043,22 +0.000030,22 +0.000036,22 +0.000026,22 +0.000037,22 +0.000049,22 +0.000036,22 +0.000046,22 +0.000047,22 +0.000049,22 +0.000037,22 +0.000035,22 +0.000037,22 +0.000050,22 +0.000055,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000036,22 +0.000036,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000026,22 +0.000036,22 +0.000046,22 +0.000062,22 +0.000047,22 +0.000036,22 +0.000047,22 +0.000041,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000050,24 +0.000053,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000055,24 +0.000058,26 +0.000055,26 +0.000077,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000052,26 +0.000043,26 +0.000043,26 +0.000066,26 +0.000061,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000054,28 +0.000054,28 +0.000053,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000065,28 +0.000066,28 +0.000058,28 +0.000097,28 +0.000084,28 +0.000073,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000073,28 +0.000054,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000053,28 +0.000073,28 +0.000054,28 +0.000064,28 +0.000063,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000073,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000082,28 +0.000063,28 +0.000083,28 +0.000063,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000069,30 +0.000066,30 +0.000066,30 +0.000074,30 +0.000103,30 +0.000108,30 +0.000107,30 +0.000112,30 +0.000111,30 +0.000087,30 +0.000105,30 +0.000076,30 +0.000066,30 +0.000107,30 +0.000119,30 +0.000105,30 +0.000117,30 +0.000077,30 +0.000077,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000079,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000069,30 +0.000077,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000096,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000085,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000089,30 +0.000066,30 +0.000066,30 +0.000066,30 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000079,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000102,32 +0.000091,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000090,32 +0.000119,32 +0.000129,32 +0.000134,32 +0.000095,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000100,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000102,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000100,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000100,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000114,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000080,32 +0.000098,34 +0.000096,34 +0.000106,34 +0.000124,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000134,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000131,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000119,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000154,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000116,36 +0.000153,36 +0.000133,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000123,36 +0.000142,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000150,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000113,36 +0.000143,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000143,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000143,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000145,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000161,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000180,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000141,38 +0.000143,38 +0.000168,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000168,40 +0.000164,40 +0.000165,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000166,40 +0.000164,40 +0.000268,40 +0.000164,40 +0.000164,40 +0.000165,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000188,40 +0.000183,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000174,40 +0.000293,40 +0.000184,40 +0.000164,40 +0.000164,40 +0.000170,40 +0.000234,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000195,40 +0.000174,40 +0.000164,40 +0.000214,40 +0.000234,40 +0.000203,40 +0.000164,40 +0.000183,40 +0.000183,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000186,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000164,40 +0.000190,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000200,42 +0.000198,42 +0.000215,42 +0.000258,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000231,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000201,42 +0.000252,42 +0.000189,42 +0.000189,42 +0.000347,42 +0.000296,42 +0.000208,42 +0.000194,42 +0.000195,42 +0.000213,42 +0.000215,42 +0.000323,42 +0.000235,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000199,42 +0.000220,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000199,42 +0.000240,42 +0.000189,42 +0.000222,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000209,42 +0.000199,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000194,42 +0.000202,42 +0.000223,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000189,42 +0.000222,44 +0.000216,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000288,44 +0.000228,44 +0.000216,44 +0.000217,44 +0.000254,44 +0.000216,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000216,44 +0.000268,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000256,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000254,44 +0.000255,44 +0.000217,44 +0.000216,44 +0.000216,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000240,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000245,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000216,44 +0.000217,44 +0.000217,44 +0.000217,44 +0.000250,46 +0.000246,46 +0.000246,46 +0.000249,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000257,46 +0.000275,46 +0.000303,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000285,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000250,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000252,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000253,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000253,46 +0.000257,46 +0.000277,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000285,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000246,46 +0.000250,46 +0.000286,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000286,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000279,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000284,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000284,48 +0.000280,48 +0.000280,48 +0.000290,48 +0.000311,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000318,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000281,48 +0.000279,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000283,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000281,48 +0.000321,48 +0.000280,48 +0.000332,48 +0.000316,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000279,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000280,48 +0.000334,48 +0.000343,48 +0.000319,50 +0.000338,50 +0.000315,50 +0.000431,50 +0.000315,50 +0.000335,50 +0.000315,50 +0.000446,50 +0.000315,50 +0.000315,50 +0.000351,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000359,50 +0.000315,50 +0.000343,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000355,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000352,50 +0.000315,50 +0.000315,50 +0.000325,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000326,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000354,50 +0.000339,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000343,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000334,50 +0.000376,50 +0.000317,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000315,50 +0.000319,50 +0.000315,50 +0.000359,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000362,52 +0.000353,52 +0.000354,52 +0.000356,52 +0.000392,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000354,52 +0.000354,52 +0.000358,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000354,52 +0.000355,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000357,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000354,52 +0.000362,52 +0.000356,52 +0.000354,52 +0.000353,52 +0.000392,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000358,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000354,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000355,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000358,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000353,52 +0.000355,52 +0.000409,54 +0.000395,54 +0.000395,54 +0.000405,54 +0.000423,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000400,54 +0.000394,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000396,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000394,54 +0.000395,54 +0.000395,54 +0.000396,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000394,54 +0.000395,54 +0.000395,54 +0.000398,54 +0.000395,54 +0.000403,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000434,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000397,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000399,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000395,54 +0.000421,54 +0.000395,54 +0.000395,54 +0.000473,54 +0.000404,54 +0.000419,54 +0.000415,54 +0.000419,54 +0.000408,54 +0.000443,54 +0.000419,54 +0.000395,54 +0.000419,54 +0.000434,54 +0.000409,54 +0.000467,54 +0.000462,54 +0.000429,54 +0.000395,54 +0.000440,54 +0.000415,54 +0.000395,54 +0.000497,54 +0.000415,54 +0.000395,54 +0.000436,54 +0.000395,54 +0.000395,54 +0.000431,54 +0.000395,54 +0.000444,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000469,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000463,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000448,56 +0.000439,56 +0.000439,56 +0.000523,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000472,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000535,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000461,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000450,56 +0.000439,56 +0.000468,56 +0.000478,56 +0.000439,56 +0.000439,56 +0.000440,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000441,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000461,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000457,56 +0.000451,56 +0.000451,56 +0.000448,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000439,56 +0.000470,56 +0.000439,56 +0.000439,56 +0.000537,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000512,58 +0.000500,58 +0.000497,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000529,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000491,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000511,58 +0.000496,58 +0.000487,58 +0.000487,58 +0.000526,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000492,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000489,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000489,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000491,58 +0.000487,58 +0.000495,58 +0.000487,58 +0.000487,58 +0.000526,58 +0.000487,58 +0.000487,58 +0.000489,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000521,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000487,58 +0.000545,58 +0.000521,58 +0.000511,58 +0.000557,58 +0.000544,58 +0.000531,58 +0.000500,58 +0.000498,58 +0.000539,58 +0.000521,58 +0.000517,58 +0.000549,58 +0.000508,58 +0.000576,60 +0.000609,60 +0.000601,60 +0.000538,60 +0.000538,60 +0.000582,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000543,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000540,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000542,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000546,60 +0.000538,60 +0.000579,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000569,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000570,60 +0.000567,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000542,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000546,60 +0.000538,60 +0.000541,60 +0.000577,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000543,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000540,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000542,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000548,60 +0.000609,60 +0.000538,60 +0.000570,60 +0.000538,60 +0.000558,60 +0.000558,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000538,60 +0.000542,60 +0.000538,60 +0.000597,62 +0.000593,62 +0.000593,62 +0.000595,62 +0.000593,62 +0.000594,62 +0.000593,62 +0.000592,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000597,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000633,62 +0.000595,62 +0.000601,62 +0.000593,62 +0.000632,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000598,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000595,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000594,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000597,62 +0.000593,62 +0.000601,62 +0.000593,62 +0.000632,62 +0.000593,62 +0.000595,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000597,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000606,62 +0.000668,62 +0.000617,62 +0.000617,62 +0.000637,62 +0.000607,62 +0.000634,62 +0.000625,62 +0.000608,62 +0.000667,62 +0.000634,62 +0.000653,62 +0.000683,62 +0.000625,62 +0.000593,62 +0.000593,62 +0.000635,62 +0.000593,62 +0.000593,62 +0.000633,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000593,62 +0.000613,62 +0.000677,62 +0.000746,62 +0.000613,62 +0.000749,62 +0.000623,62 +0.000612,62 +0.000593,62 +0.000632,62 +0.000593,62 +0.000612,62 +0.000658,64 +0.000681,64 +0.000651,64 +0.000697,64 +0.000650,64 +0.000650,64 +0.000671,64 +0.000650,64 +0.000650,64 +0.000680,64 +0.000650,64 +0.000650,64 +0.000651,64 +0.000650,64 +0.000651,64 +0.000673,64 +0.000732,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000654,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000659,64 +0.000653,64 +0.000690,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000655,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000652,64 +0.000651,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000654,64 +0.000650,64 +0.000670,64 +0.000670,64 +0.000650,64 +0.000709,64 +0.000663,64 +0.000689,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000655,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000652,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000654,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000652,64 +0.000658,64 +0.000650,64 +0.000689,64 +0.000650,64 +0.000650,64 +0.000655,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000652,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000650,64 +0.000654,64 +0.000650,64 +0.000650,64 +0.000651,64 +0.000651,64 +0.000650,64 +0.000725,66 +0.000713,66 +0.000722,66 +0.000713,66 +0.000752,66 +0.000713,66 +0.000718,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000759,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000717,66 +0.000713,66 +0.000812,66 +0.000736,66 +0.000740,66 +0.000776,66 +0.000755,66 +0.000738,66 +0.000766,66 +0.000775,66 +0.000797,66 +0.000776,66 +0.000829,66 +0.000722,66 +0.000713,66 +0.000713,66 +0.000736,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000859,66 +0.000825,66 +0.000713,66 +0.000713,66 +0.000733,66 +0.000757,66 +0.000713,66 +0.000733,66 +0.000765,66 +0.000772,66 +0.000894,66 +0.000713,66 +0.000713,66 +0.000752,66 +0.000731,66 +0.000754,66 +0.000723,66 +0.000713,66 +0.000734,66 +0.000713,66 +0.000713,66 +0.000749,66 +0.000793,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000840,66 +0.000768,66 +0.000752,66 +0.000756,66 +0.000713,66 +0.000724,66 +0.000781,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000736,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000736,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000744,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000713,66 +0.000712,66 +0.000745,66 +0.000713,66 +0.000752,66 +0.000713,66 +0.000713,66 +0.000719,66 +0.000713,66 +0.000789,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000785,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000783,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000780,68 +0.000787,68 +0.000817,68 +0.000778,68 +0.000778,68 +0.000783,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000780,68 +0.000779,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000780,68 +0.000779,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000782,68 +0.000778,68 +0.000787,68 +0.000778,68 +0.000817,68 +0.000781,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000783,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000780,68 +0.000778,68 +0.000845,68 +0.000822,68 +0.000813,68 +0.000832,68 +0.000826,68 +0.000902,68 +0.000890,68 +0.000835,68 +0.000799,68 +0.000868,68 +0.000778,68 +0.000778,68 +0.000816,68 +0.000778,68 +0.000779,68 +0.000778,68 +0.000778,68 +0.000813,68 +0.000798,68 +0.000778,68 +0.000778,68 +0.000798,68 +0.000820,68 +0.000778,68 +0.000779,68 +0.000778,68 +0.000787,68 +0.000841,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000783,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000780,68 +0.000846,68 +0.000778,68 +0.000778,68 +0.000778,68 +0.000786,68 +0.000779,68 +0.000778,68 +0.000779,68 +0.000790,68 +0.000864,70 +0.000887,70 +0.000848,70 +0.000895,70 +0.000856,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000850,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000849,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000863,70 +0.000848,70 +0.000887,70 +0.000848,70 +0.000848,70 +0.000852,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000854,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000850,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000852,70 +0.000857,70 +0.000848,70 +0.000887,70 +0.000848,70 +0.000850,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000896,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000850,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000848,70 +0.000852,70 +0.000848,70 +0.000857,70 +0.000887,70 +0.000848,70 +0.000851,70 +0.001152,70 +0.000848,70 +0.000848,70 +0.000880,70 +0.001018,70 +0.000848,70 +0.001016,70 +0.000885,70 +0.000848,70 +0.000935,70 +0.000894,70 +0.000883,70 +0.000921,70 +0.000926,70 +0.000922,70 +0.001052,70 +0.000956,70 +0.000883,70 +0.001083,70 +0.000970,70 +0.001244,70 +0.000980,70 +0.000928,70 +0.000927,70 +0.000914,70 +0.000951,70 +0.000948,70 +0.000969,70 +0.000974,70 +0.000970,70 +0.001081,70 +0.001080,70 +0.000962,70 +0.000948,70 +0.000948,70 +0.000946,70 +0.000934,70 +0.000946,70 +0.001066,72 +0.001042,72 +0.001018,72 +0.001084,72 +0.001068,72 +0.001038,72 +0.001056,72 +0.001073,72 +0.001119,72 +0.001100,72 +0.001092,72 +0.001032,72 +0.001029,72 +0.001016,72 +0.001057,72 +0.001057,72 +0.001057,72 +0.001068,72 +0.001039,72 +0.001038,72 +0.001066,72 +0.001032,72 +0.001037,72 +0.001037,72 +0.001135,72 +0.001046,72 +0.001085,72 +0.001089,72 +0.001054,72 +0.001079,72 +0.001024,72 +0.001033,72 +0.001036,72 +0.001050,72 +0.000921,72 +0.001033,72 +0.001062,72 +0.000960,72 +0.000921,72 +0.001050,72 +0.001081,72 +0.001111,72 +0.001031,72 +0.001009,72 +0.001014,72 +0.000946,72 +0.001031,72 +0.000950,72 +0.001043,72 +0.000921,72 +0.000921,72 +0.001055,72 +0.000958,72 +0.001054,72 +0.001056,72 +0.000921,72 +0.000980,72 +0.001135,72 +0.001088,72 +0.001116,72 +0.000955,72 +0.000921,72 +0.001174,72 +0.000941,72 +0.000967,72 +0.001058,72 +0.000921,72 +0.000921,72 +0.000947,72 +0.001054,72 +0.000941,72 +0.001007,72 +0.001018,72 +0.001027,72 +0.000987,72 +0.001003,72 +0.001095,72 +0.000942,72 +0.001094,72 +0.000921,72 +0.000971,72 +0.000921,72 +0.000921,72 +0.000942,72 +0.000965,72 +0.000921,72 +0.000921,72 +0.000921,72 +0.000921,72 +0.000966,72 +0.000941,72 +0.000921,72 +0.000941,72 +0.000996,72 +0.000921,72 +0.000921,72 +0.000921,72 +0.000967,72 +0.000921,72 +0.000921,72 +0.001012,74 +0.001031,74 +0.000999,74 +0.001078,74 +0.000998,74 +0.001047,74 +0.000999,74 +0.000999,74 +0.000999,74 +0.001067,74 +0.001024,74 +0.000998,74 +0.000999,74 +0.001002,74 +0.000998,74 +0.000998,74 +0.000998,74 +0.001007,74 +0.000998,74 +0.000998,74 +0.000998,74 +0.001001,74 +0.000998,74 +0.000999,74 +0.001053,74 +0.001033,74 +0.001023,74 +0.000999,74 +0.000998,74 +0.001003,74 +0.000999,74 +0.000999,74 +0.000998,74 +0.001000,74 +0.000999,74 +0.000998,74 +0.000998,74 +0.001005,74 +0.000999,74 +0.000998,74 +0.000998,74 +0.001000,74 +0.001022,74 +0.001024,74 +0.000999,74 +0.001007,74 +0.000999,74 +0.000999,74 +0.000998,74 +0.001000,74 +0.000998,74 +0.000998,74 +0.000999,74 +0.001000,74 +0.000999,74 +0.000999,74 +0.000999,74 +0.001002,74 +0.001021,74 +0.000998,74 +0.001023,74 +0.001002,74 +0.000998,74 +0.000998,74 +0.000998,74 +0.001005,74 +0.000999,74 +0.000999,74 +0.000998,74 +0.001000,74 +0.000998,74 +0.000999,74 +0.001075,74 +0.001077,74 +0.001043,74 +0.001087,74 +0.001126,74 +0.001090,74 +0.001029,74 +0.001098,74 +0.001032,74 +0.000999,74 +0.000998,74 +0.001048,74 +0.001004,74 +0.000999,74 +0.000999,74 +0.000999,74 +0.001006,74 +0.000999,74 +0.000999,74 +0.001023,74 +0.001028,74 +0.000999,74 +0.000998,74 +0.000999,74 +0.001007,74 +0.000998,74 +0.000998,74 +0.000998,74 +0.001095,76 +0.001081,76 +0.001153,76 +0.001081,76 +0.001095,76 +0.001081,76 +0.001081,76 +0.001111,76 +0.001116,76 +0.001082,76 +0.001080,76 +0.001091,76 +0.001080,76 +0.001081,76 +0.001081,76 +0.001083,76 +0.001081,76 +0.001081,76 +0.001082,76 +0.001277,76 +0.001102,76 +0.001101,76 +0.001132,76 +0.001106,76 +0.001081,76 +0.001105,76 +0.001081,76 +0.001081,76 +0.001081,76 +0.001104,76 +0.001081,76 +0.001080,76 +0.001081,76 +0.001109,76 +0.001081,76 +0.001081,76 +0.001109,76 +0.001103,76 +0.001105,76 +0.001081,76 +0.001089,76 +0.001081,76 +0.001081,76 +0.001080,76 +0.001082,76 +0.001081,76 +0.001081,76 +0.001102,76 +0.001081,76 +0.001081,76 +0.001081,76 +0.001085,76 +0.001080,76 +0.001103,76 +0.001104,76 +0.001107,76 +0.001081,76 +0.001081,76 +0.001089,76 +0.001174,76 +0.001081,76 +0.001081,76 +0.001330,76 +0.001080,76 +0.001081,76 +0.001148,76 +0.001121,76 +0.001161,76 +0.001117,76 +0.001203,76 +0.001255,76 +0.001144,76 +0.001592,76 +0.002119,76 +0.002177,76 +0.001425,76 +0.001322,76 +0.001309,76 +0.001290,76 +0.001266,76 +0.001303,76 +0.001343,76 +0.001229,76 +0.001141,76 +0.001222,76 +0.001349,76 +0.001260,76 +0.001221,76 +0.001172,76 +0.001276,76 +0.001166,76 +0.001110,76 +0.001237,76 +0.001164,76 +0.001207,76 +0.001205,76 +0.001145,76 +0.001226,76 +0.001109,76 +0.001230,76 +0.001250,78 +0.001198,78 +0.001318,78 +0.001202,78 +0.001205,78 +0.001384,78 +0.001220,78 +0.001353,78 +0.001168,78 +0.001194,78 +0.001167,78 +0.001167,78 +0.001167,78 +0.001215,78 +0.001167,78 +0.001167,78 +0.001201,78 +0.001167,78 +0.001167,78 +0.001239,78 +0.001190,78 +0.001191,78 +0.001168,78 +0.001204,78 +0.001167,78 +0.001167,78 +0.001198,78 +0.001167,78 +0.001177,78 +0.001198,78 +0.001167,78 +0.001167,78 +0.001167,78 +0.001221,78 +0.001190,78 +0.001194,78 +0.001362,78 +0.001167,78 +0.001167,78 +0.001210,78 +0.001167,78 +0.001167,78 +0.001167,78 +0.001204,78 +0.001167,78 +0.001167,78 +0.001283,78 +0.001278,78 +0.001323,78 +0.001268,78 +0.001204,78 +0.001262,78 +0.001289,78 +0.001167,78 +0.001167,78 +0.001167,78 +0.001202,78 +0.001309,78 +0.001167,78 +0.001206,78 +0.001167,78 +0.001167,78 +0.001423,78 +0.001194,78 +0.001167,78 +0.001167,78 +0.001211,78 +0.001168,78 +0.001168,78 +0.001192,78 +0.001251,78 +0.001167,78 +0.001332,78 +0.001199,78 +0.001190,78 +0.001189,78 +0.001204,78 +0.001191,78 +0.001167,78 +0.001212,78 +0.001167,78 +0.001167,78 +0.001207,78 +0.001167,78 +0.001167,78 +0.001167,78 +0.001201,78 +0.001167,78 +0.001167,78 +0.001239,78 +0.001167,78 +0.001192,78 +0.001184,78 +0.001186,78 +0.001168,78 +0.001167,78 +0.001173,78 +0.001168,78 +0.001167,78 +0.001169,78 +0.001268,80 +0.001257,80 +0.001261,80 +0.001279,80 +0.001282,80 +0.001257,80 +0.001261,80 +0.001257,80 +0.001257,80 +0.001263,80 +0.001257,80 +0.001257,80 +0.001259,80 +0.001257,80 +0.001287,80 +0.001268,80 +0.001257,80 +0.001297,80 +0.001284,80 +0.001257,80 +0.001257,80 +0.001259,80 +0.001257,80 +0.001257,80 +0.001256,80 +0.001263,80 +0.001257,80 +0.001257,80 +0.001259,80 +0.001353,80 +0.001333,80 +0.001366,80 +0.001305,80 +0.001357,80 +0.001332,80 +0.001365,80 +0.001257,80 +0.001291,80 +0.001257,80 +0.001257,80 +0.001282,80 +0.001257,80 +0.001278,80 +0.001294,80 +0.001280,80 +0.001257,80 +0.001286,80 +0.001257,80 +0.001257,80 +0.001280,80 +0.001257,80 +0.001330,80 +0.001257,80 +0.001302,80 +0.001257,80 +0.001257,80 +0.001283,80 +0.001281,80 +0.001257,80 +0.001261,80 +0.001257,80 +0.001257,80 +0.001297,80 +0.001306,80 +0.001257,80 +0.001300,80 +0.001267,80 +0.001257,80 +0.001323,80 +0.001278,80 +0.001280,80 +0.001281,80 +0.001257,80 +0.001257,80 +0.001274,80 +0.001266,80 +0.001257,80 +0.001257,80 +0.001259,80 +0.001257,80 +0.001257,80 +0.001260,80 +0.001278,80 +0.001353,80 +0.001274,80 +0.001259,80 +0.001257,80 +0.001263,80 +0.001257,80 +0.001257,80 +0.001259,80 +0.001257,80 +0.001257,80 +0.001261,80 +0.001434,80 +0.001276,80 +0.001527,80 +0.001383,80 +0.001498,80 +0.001345,80 +0.001604,82 +0.001498,82 +0.001578,82 +0.001498,82 +0.001565,82 +0.001591,82 +0.001523,82 +0.001503,82 +0.001502,82 +0.001531,82 +0.001463,82 +0.001638,82 +0.001432,82 +0.001530,82 +0.001492,82 +0.001578,82 +0.001501,82 +0.001977,82 +0.001577,82 +0.001540,82 +0.001526,82 +0.001540,82 +0.001564,82 +0.001492,82 +0.001627,82 +0.001528,82 +0.001529,82 +0.001539,82 +0.001479,82 +0.001453,82 +0.001450,82 +0.001442,82 +0.001445,82 +0.001457,82 +0.001457,82 +0.001469,82 +0.001572,82 +0.001430,82 +0.001776,82 +0.001605,82 +0.001509,82 +0.001503,82 +0.001509,82 +0.001536,82 +0.001509,82 +0.001493,82 +0.001517,82 +0.001511,82 +0.001508,82 +0.001513,82 +0.001536,82 +0.001484,82 +0.001468,82 +0.001473,82 +0.001476,82 +0.001487,82 +0.001474,82 +0.001489,82 +0.001553,82 +0.001523,82 +0.001489,82 +0.001544,82 +0.001542,82 +0.001464,82 +0.001463,82 +0.001469,82 +0.001482,82 +0.001511,82 +0.001423,82 +0.001497,82 +0.001488,82 +0.001635,82 +0.001456,82 +0.001527,82 +0.001452,82 +0.001566,82 +0.001485,82 +0.001559,82 +0.001428,82 +0.001697,82 +0.001431,82 +0.001431,82 +0.001410,82 +0.001421,82 +0.001736,82 +0.001462,82 +0.001439,82 +0.001382,82 +0.001366,82 +0.001766,82 +0.001578,82 +0.001466,82 +0.001366,82 +0.001556,82 +0.001485,82 +0.001559,82 +0.001420,82 +0.001509,82 +0.001448,82 +0.001433,82 +0.001505,84 +0.001708,84 +0.001583,84 +0.001532,84 +0.001867,84 +0.001624,84 +0.001513,84 +0.001615,84 +0.001491,84 +0.001673,84 +0.001537,84 +0.001602,84 +0.001517,84 +0.001776,84 +0.001573,84 +0.001652,84 +0.001731,84 +0.001606,84 +0.001568,84 +0.001583,84 +0.001534,84 +0.001584,84 +0.001581,84 +0.001996,84 +0.001828,84 +0.001970,84 +0.001907,84 +0.001872,84 +0.001870,84 +0.001966,84 +0.001880,84 +0.001653,84 +0.001818,84 +0.001933,84 +0.001632,84 +0.001582,84 +0.001560,84 +0.001675,84 +0.001603,84 +0.001697,84 +0.001657,84 +0.001648,84 +0.001725,84 +0.001654,84 +0.001623,84 +0.001516,84 +0.001491,84 +0.001517,84 +0.001491,84 +0.001679,84 +0.001637,84 +0.001610,84 +0.001630,84 +0.001572,84 +0.001564,84 +0.001569,84 +0.001545,84 +0.001588,84 +0.001658,84 +0.001571,84 +0.001573,84 +0.001709,84 +0.001576,84 +0.001554,84 +0.001481,84 +0.001796,84 +0.001491,84 +0.001621,84 +0.001545,84 +0.001453,84 +0.001494,84 +0.001452,84 +0.001563,84 +0.001522,84 +0.001584,84 +0.001637,84 +0.001725,84 +0.001619,84 +0.001604,84 +0.001670,84 +0.001588,84 +0.001708,84 +0.001615,84 +0.001452,84 +0.001606,84 +0.001490,84 +0.001604,84 +0.001579,84 +0.001527,84 +0.001452,84 +0.001499,84 +0.001452,84 +0.001465,84 +0.001486,84 +0.001555,84 +0.001452,84 +0.001526,84 +0.001675,84 +0.001599,84 +0.001499,84 +0.001638,86 +0.001737,86 +0.001600,86 +0.001737,86 +0.001559,86 +0.001559,86 +0.001749,86 +0.001691,86 +0.001658,86 +0.001558,86 +0.001590,86 +0.001558,86 +0.001559,86 +0.001587,86 +0.001558,86 +0.001680,86 +0.001558,86 +0.001579,86 +0.001627,86 +0.001597,86 +0.001789,86 +0.001558,86 +0.001558,86 +0.001584,86 +0.001559,86 +0.001583,86 +0.001558,86 +0.001703,86 +0.001776,86 +0.001597,86 +0.001583,86 +0.001558,86 +0.001559,86 +0.001690,86 +0.001600,86 +0.001592,86 +0.001559,86 +0.001559,86 +0.001683,86 +0.001761,86 +0.001623,86 +0.001558,86 +0.001558,86 +0.001584,86 +0.001558,86 +0.001581,86 +0.001796,86 +0.001659,86 +0.001857,86 +0.001810,86 +0.001826,86 +0.001671,86 +0.001702,86 +0.001696,86 +0.001702,86 +0.001628,86 +0.001734,86 +0.001704,86 +0.001711,86 +0.001851,86 +0.001669,86 +0.001721,86 +0.001756,86 +0.001707,86 +0.001834,86 +0.001838,86 +0.001635,86 +0.001871,86 +0.001859,86 +0.001666,86 +0.001666,86 +0.001657,86 +0.001668,86 +0.001654,86 +0.001675,86 +0.001891,86 +0.001799,86 +0.001804,86 +0.001750,86 +0.001787,86 +0.001738,86 +0.001770,86 +0.001625,86 +0.001771,86 +0.001757,86 +0.001690,86 +0.001711,86 +0.001751,86 +0.001820,86 +0.001612,86 +0.001733,86 +0.001706,86 +0.001678,86 +0.001699,86 +0.001700,86 +0.001734,86 +0.001750,86 +0.001779,86 +0.001700,86 +0.001779,86 +0.001813,88 +0.001813,88 +0.001888,88 +0.001836,88 +0.001887,88 +0.001911,88 +0.001892,88 +0.001787,88 +0.001867,88 +0.001848,88 +0.001905,88 +0.001843,88 +0.001897,88 +0.002130,88 +0.004302,88 +0.001848,88 +0.001766,88 +0.001722,88 +0.001804,88 +0.001802,88 +0.001815,88 +0.001818,88 +0.001847,88 +0.001865,88 +0.001767,88 +0.001776,88 +0.001751,88 +0.001773,88 +0.001744,88 +0.001782,88 +0.001919,88 +0.002034,88 +0.001918,88 +0.001979,88 +0.001885,88 +0.001895,88 +0.001865,88 +0.001859,88 +0.001773,88 +0.001917,88 +0.001851,88 +0.001844,88 +0.001803,88 +0.001832,88 +0.001774,88 +0.001863,88 +0.001940,88 +0.001916,88 +0.001999,88 +0.001922,88 +0.001866,88 +0.001984,88 +0.001864,88 +0.001809,88 +0.001784,88 +0.001841,88 +0.001970,88 +0.001865,88 +0.001875,88 +0.001867,88 +0.001998,88 +0.001879,88 +0.001818,88 +0.001863,88 +0.001943,88 +0.002102,88 +0.001872,88 +0.001922,88 +0.001922,88 +0.001738,88 +0.001803,88 +0.001777,88 +0.001788,88 +0.002044,88 +0.002014,88 +0.001786,88 +0.001942,88 +0.001800,88 +0.001764,88 +0.001819,88 +0.001800,88 +0.001826,88 +0.001763,88 +0.001832,88 +0.001832,88 +0.001867,88 +0.001826,88 +0.001783,88 +0.001833,88 +0.001755,88 +0.001767,88 +0.001849,88 +0.001800,88 +0.001838,88 +0.002000,88 +0.001914,88 +0.001889,88 +0.001822,88 +0.001900,88 +0.001868,88 +0.002045,90 +0.002083,90 +0.001996,90 +0.002012,90 +0.001956,90 +0.001960,90 +0.001994,90 +0.001975,90 +0.002229,90 +0.001996,90 +0.001926,90 +0.002033,90 +0.001940,90 +0.001917,90 +0.001934,90 +0.001913,90 +0.001920,90 +0.001983,90 +0.001978,90 +0.002018,90 +0.001928,90 +0.001949,90 +0.001948,90 +0.001969,90 +0.002095,90 +0.001965,90 +0.002043,90 +0.001930,90 +0.001987,90 +0.001946,90 +0.001935,90 +0.001928,90 +0.001970,90 +0.001996,90 +0.001952,90 +0.002076,90 +0.002171,90 +0.002512,90 +0.002524,90 +0.002317,90 +0.002413,90 +0.002428,90 +0.002173,90 +0.002052,90 +0.002012,90 +0.001995,90 +0.001990,90 +0.002001,90 +0.001979,90 +0.001961,90 +0.002068,90 +0.001985,90 +0.001988,90 +0.002031,90 +0.002024,90 +0.002012,90 +0.001985,90 +0.001977,90 +0.002043,90 +0.002001,90 +0.002052,90 +0.002040,90 +0.002011,90 +0.002026,90 +0.001964,90 +0.002007,90 +0.001987,90 +0.001951,90 +0.001966,90 +0.002065,90 +0.001977,90 +0.001908,90 +0.001914,90 +0.002070,90 +0.002188,90 +0.002242,90 +0.002210,90 +0.002093,90 +0.002013,90 +0.001985,90 +0.001969,90 +0.001980,90 +0.001984,90 +0.001978,90 +0.001962,90 +0.001997,90 +0.001951,90 +0.001964,90 +0.001948,90 +0.001950,90 +0.001944,90 +0.002016,90 +0.001914,90 +0.001957,90 +0.001954,90 +0.001952,90 +0.001933,90 +0.001929,90 +0.001929,90 +0.001955,90 +0.002176,92 +0.002076,92 +0.002074,92 +0.002073,92 +0.002086,92 +0.002072,92 +0.002070,92 +0.002071,92 +0.002150,92 +0.002102,92 +0.002085,92 +0.002072,92 +0.002090,92 +0.002103,92 +0.002081,92 +0.002080,92 +0.002178,92 +0.002117,92 +0.002096,92 +0.002074,92 +0.002081,92 +0.002049,92 +0.001983,92 +0.001978,92 +0.001988,92 +0.002048,92 +0.002083,92 +0.001966,92 +0.001939,92 +0.001905,92 +0.001938,92 +0.001905,92 +0.001998,92 +0.001988,92 +0.001990,92 +0.001956,92 +0.001944,92 +0.001905,92 +0.001940,92 +0.001905,92 +0.001931,92 +0.002051,92 +0.001932,92 +0.001916,92 +0.001945,92 +0.001934,92 +0.001905,92 +0.001934,92 +0.001905,92 +0.001959,92 +0.001905,92 +0.001956,92 +0.002195,92 +0.002039,92 +0.002183,92 +0.001987,92 +0.002163,92 +0.002143,92 +0.002184,92 +0.002212,92 +0.002241,92 +0.002146,92 +0.002183,92 +0.002171,92 +0.002169,92 +0.002408,92 +0.002204,92 +0.002163,92 +0.002097,92 +0.002111,92 +0.002057,92 +0.002114,92 +0.002097,92 +0.002273,92 +0.002191,92 +0.002150,92 +0.002058,92 +0.002047,92 +0.002057,92 +0.002034,92 +0.001942,92 +0.002168,92 +0.002163,92 +0.003191,92 +0.002162,92 +0.002106,92 +0.002028,92 +0.002057,92 +0.002162,92 +0.002098,92 +0.002227,92 +0.002239,92 +0.002172,92 +0.002044,92 +0.002031,92 +0.002021,92 +0.002299,92 +0.002519,92 +0.002628,92 +0.002104,92 +0.002270,94 +0.002232,94 +0.002239,94 +0.002805,94 +0.002718,94 +0.002506,94 +0.002880,94 +0.002652,94 +0.002438,94 +0.002516,94 +0.002958,94 +0.002180,94 +0.002139,94 +0.002145,94 +0.002162,94 +0.002192,94 +0.002308,94 +0.002193,94 +0.003367,94 +0.002772,94 +0.002902,94 +0.002238,94 +0.002120,94 +0.002267,94 +0.002291,94 +0.002955,94 +0.002548,94 +0.002764,94 +0.002785,94 +0.002638,94 +0.002414,94 +0.002271,94 +0.002242,94 +0.002232,94 +0.002199,94 +0.002193,94 +0.002248,94 +0.002287,94 +0.002269,94 +0.002258,94 +0.002242,94 +0.002174,94 +0.002188,94 +0.002180,94 +0.002209,94 +0.002219,94 +0.002284,94 +0.002228,94 +0.002259,94 +0.002227,94 +0.002286,94 +0.002191,94 +0.002205,94 +0.002281,94 +0.002241,94 +0.002187,94 +0.002224,94 +0.002170,94 +0.002151,94 +0.002234,94 +0.002278,94 +0.002294,94 +0.002267,94 +0.002179,94 +0.002405,94 +0.002226,94 +0.002247,94 +0.002268,94 +0.002221,94 +0.002960,94 +0.003003,94 +0.002788,94 +0.002929,94 +0.002821,94 +0.003428,94 +0.002702,94 +0.002226,94 +0.002220,94 +0.002311,94 +0.002193,94 +0.002503,94 +0.002530,94 +0.002313,94 +0.002327,94 +0.002228,94 +0.002315,94 +0.002801,94 +0.003093,94 +0.002355,94 +0.002123,94 +0.002102,94 +0.002036,94 +0.002268,94 +0.002291,94 +0.002212,94 +0.002548,94 +0.002096,94 +0.002070,94 +0.002223,94 +0.003631,94 +0.003131,96 +0.002345,96 +0.002444,96 +0.002200,96 +0.002164,96 +0.002201,96 +0.002165,96 +0.002459,96 +0.002256,96 +0.002386,96 +0.002270,96 +0.002165,96 +0.002200,96 +0.002165,96 +0.002303,96 +0.002474,96 +0.002570,96 +0.002492,96 +0.002222,96 +0.002253,96 +0.002445,96 +0.002405,96 +0.002374,96 +0.002388,96 +0.002320,96 +0.002348,96 +0.002412,96 +0.002335,96 +0.002826,96 +0.002458,96 +0.002369,96 +0.002287,96 +0.002271,96 +0.002221,96 +0.002232,96 +0.002600,96 +0.002674,96 +0.002477,96 +0.002204,96 +0.002200,96 +0.002249,96 +0.002207,96 +0.002188,96 +0.003085,96 +0.002353,96 +0.002224,96 +0.002199,96 +0.002165,96 +0.002200,96 +0.002524,96 +0.002687,96 +0.002478,96 +0.002204,96 +0.002224,96 +0.002204,96 +0.002476,96 +0.002659,96 +0.002713,96 +0.002460,96 +0.002290,96 +0.002394,96 +0.002247,96 +0.002490,96 +0.002873,96 +0.002604,96 +0.002406,96 +0.002254,96 +0.002250,96 +0.002389,96 +0.002253,96 +0.002941,96 +0.002625,96 +0.002418,96 +0.002181,96 +0.002247,96 +0.002221,96 +0.002249,96 +0.002854,96 +0.002349,96 +0.002397,96 +0.002164,96 +0.002199,96 +0.002287,96 +0.002282,96 +0.002452,96 +0.002915,96 +0.002363,96 +0.002283,96 +0.002221,96 +0.002247,96 +0.002222,96 +0.002580,96 +0.002344,96 +0.002385,96 +0.002205,96 +0.002164,96 +0.002200,96 +0.002164,96 +0.002477,96 +0.002228,96 +0.002862,98 +0.002740,98 +0.002681,98 +0.002558,98 +0.003056,98 +0.003342,98 +0.002864,98 +0.003272,98 +0.003514,98 +0.002586,98 +0.002846,98 +0.003309,98 +0.002700,98 +0.002704,98 +0.002443,98 +0.002605,98 +0.003502,98 +0.003117,98 +0.002557,98 +0.002528,98 +0.002538,98 +0.002366,98 +0.002391,98 +0.003854,98 +0.003106,98 +0.002539,98 +0.002463,98 +0.002305,98 +0.003020,98 +0.002789,98 +0.002425,98 +0.002469,98 +0.002396,98 +0.002366,98 +0.002765,98 +0.002778,98 +0.002635,98 +0.002610,98 +0.002480,98 +0.002392,98 +0.002396,98 +0.002714,98 +0.002756,98 +0.002841,98 +0.004154,98 +0.002943,98 +0.002630,98 +0.002467,98 +0.003615,98 +0.003165,98 +0.002790,98 +0.002379,98 +0.003336,98 +0.002897,98 +0.003019,98 +0.002448,98 +0.002841,98 +0.002532,98 +0.002560,98 +0.002856,98 +0.003478,98 +0.003149,98 +0.002553,98 +0.002413,98 +0.002631,98 +0.003139,98 +0.002597,98 +0.002427,98 +0.002502,98 +0.002393,98 +0.002618,98 +0.002441,98 +0.002753,98 +0.003863,98 +0.003689,98 +0.002348,98 +0.002846,98 +0.002731,98 +0.002495,98 +0.002508,98 +0.002396,98 +0.002401,98 +0.002703,98 +0.002545,98 +0.002685,98 +0.002514,98 +0.002346,98 +0.002623,98 +0.002410,98 +0.002921,98 +0.003268,98 +0.002623,98 +0.002346,98 +0.002697,98 +0.002615,98 +0.002521,98 +0.003498,98 +0.003344,98 +0.003301,98 +0.002581,98 +0.002742,100 +0.002698,100 +0.003454,100 +0.002797,100 +0.002585,100 +0.002507,100 +0.002585,100 +0.002588,100 +0.003374,100 +0.002645,100 +0.002527,100 +0.002447,100 +0.002799,100 +0.002665,100 +0.003417,100 +0.002694,100 +0.002484,100 +0.002548,100 +0.002658,100 +0.002574,100 +0.003506,100 +0.002723,100 +0.002596,100 +0.002688,100 +0.002624,100 +0.002649,100 +0.003460,100 +0.002747,100 +0.002662,100 +0.002631,100 +0.002617,100 +0.002584,100 +0.002649,100 +0.002590,100 +0.002796,100 +0.002484,100 +0.002778,100 +0.002643,100 +0.002649,100 +0.002754,100 +0.002555,100 +0.002564,100 +0.002531,100 +0.004300,100 +0.002859,100 +0.002562,100 +0.002571,100 +0.002561,100 +0.002549,100 +0.002654,100 +0.002806,100 +0.002635,100 +0.002647,100 +0.002842,100 +0.002732,100 +0.002691,100 +0.004549,100 +0.003644,100 +0.003237,100 +0.003062,100 +0.003510,100 +0.002940,100 +0.002762,100 +0.002677,100 +0.002640,100 +0.002693,100 +0.002635,100 +0.002778,100 +0.004505,100 +0.002998,100 +0.003058,100 +0.002824,100 +0.002817,100 +0.002728,100 +0.002627,100 +0.002649,100 +0.002588,100 +0.003822,100 +0.003904,100 +0.003254,100 +0.003074,100 +0.003190,100 +0.003889,100 +0.003142,100 +0.002844,100 +0.002776,100 +0.002816,100 +0.003645,100 +0.003645,100 +0.002665,100 +0.002592,100 +0.002575,100 +0.002658,100 +0.003486,100 +0.002725,100 +0.002622,100 +0.002620,100 +0.002645,100 +0.002613,100 +0.002580,100 +0.002724,102 +0.003303,102 +0.003048,102 +0.002780,102 +0.002951,102 +0.004601,102 +0.003367,102 +0.003190,102 +0.003475,102 +0.003924,102 +0.003630,102 +0.002814,102 +0.002834,102 +0.002798,102 +0.002829,102 +0.003342,102 +0.003120,102 +0.002870,102 +0.002729,102 +0.002820,102 +0.003945,102 +0.003139,102 +0.003297,102 +0.002713,102 +0.002807,102 +0.002903,102 +0.003155,102 +0.003056,102 +0.002928,102 +0.002823,102 +0.002892,102 +0.003179,102 +0.002805,102 +0.002963,102 +0.003007,102 +0.002791,102 +0.002833,102 +0.003276,102 +0.003000,102 +0.002734,102 +0.002885,102 +0.002941,102 +0.003071,102 +0.003235,102 +0.003131,102 +0.002980,102 +0.003010,102 +0.002969,102 +0.002932,102 +0.003002,102 +0.002888,102 +0.002757,102 +0.002811,102 +0.002874,102 +0.002963,102 +0.002898,102 +0.002741,102 +0.002809,102 +0.002712,102 +0.003060,102 +0.003273,102 +0.003245,102 +0.003031,102 +0.002997,102 +0.003172,102 +0.002861,102 +0.002823,102 +0.002939,102 +0.002998,102 +0.002820,102 +0.002794,102 +0.003012,102 +0.002913,102 +0.002679,102 +0.002762,102 +0.002738,102 +0.002825,102 +0.002687,102 +0.002860,102 +0.002803,102 +0.002834,102 +0.002665,102 +0.002883,102 +0.002636,102 +0.002978,102 +0.002732,102 +0.002656,102 +0.002633,102 +0.002697,102 +0.002831,102 +0.002744,102 +0.002656,102 +0.002900,102 +0.002691,102 +0.002800,102 +0.002862,102 +0.003009,102 +0.002916,102 +0.002882,102 +0.002770,102 +0.003104,104 +0.004950,104 +0.005103,104 +0.004573,104 +0.004026,104 +0.003457,104 +0.003062,104 +0.003048,104 +0.002974,104 +0.003241,104 +0.003673,104 +0.002939,104 +0.002939,104 +0.002946,104 +0.003087,104 +0.003103,104 +0.003060,104 +0.003008,104 +0.003019,104 +0.003130,104 +0.003015,104 +0.003053,104 +0.002991,104 +0.003000,104 +0.003002,104 +0.002993,104 +0.003105,104 +0.003042,104 +0.003033,104 +0.003031,104 +0.003215,104 +0.002948,104 +0.002988,104 +0.003126,104 +0.003376,104 +0.003281,104 +0.003323,104 +0.003151,104 +0.003025,104 +0.002992,104 +0.002998,104 +0.003124,104 +0.003087,104 +0.003148,104 +0.003070,104 +0.003095,104 +0.003224,104 +0.003072,104 +0.002990,104 +0.003106,104 +0.003007,104 +0.003000,104 +0.003089,104 +0.002909,104 +0.002937,104 +0.003134,104 +0.003013,104 +0.003071,104 +0.003501,104 +0.003272,104 +0.002931,104 +0.002971,104 +0.003224,104 +0.003673,104 +0.004977,104 +0.005140,104 +0.004520,104 +0.003520,104 +0.003634,104 +0.003674,104 +0.003072,104 +0.004450,104 +0.005067,104 +0.003972,104 +0.003040,104 +0.004208,104 +0.003163,104 +0.003146,104 +0.003114,104 +0.003184,104 +0.004320,104 +0.004138,104 +0.003283,104 +0.003101,104 +0.003353,104 +0.003452,104 +0.003375,104 +0.003198,104 +0.002984,104 +0.002996,104 +0.003103,104 +0.003065,104 +0.002990,104 +0.003533,104 +0.002977,104 +0.003177,104 +0.003234,104 +0.003008,104 +0.002749,104 +0.002798,104 +0.003047,106 +0.003276,106 +0.003251,106 +0.002976,106 +0.002943,106 +0.002945,106 +0.003244,106 +0.003172,106 +0.003055,106 +0.002941,106 +0.002908,106 +0.003125,106 +0.003282,106 +0.003298,106 +0.002979,106 +0.002907,106 +0.002947,106 +0.003287,106 +0.003200,106 +0.003031,106 +0.002942,106 +0.002940,106 +0.003263,106 +0.003190,106 +0.003135,106 +0.003207,106 +0.003102,106 +0.003048,106 +0.003238,106 +0.004084,106 +0.003262,106 +0.002951,106 +0.003074,106 +0.003253,106 +0.003276,106 +0.002987,106 +0.002948,106 +0.002943,106 +0.003273,106 +0.003223,106 +0.003069,106 +0.002946,106 +0.002907,106 +0.003239,106 +0.003170,106 +0.003175,106 +0.002966,106 +0.002907,106 +0.002944,106 +0.003637,106 +0.003381,106 +0.003021,106 +0.002907,106 +0.002945,106 +0.003403,106 +0.003164,106 +0.003231,106 +0.003006,106 +0.003251,106 +0.003278,106 +0.003057,106 +0.003331,106 +0.003197,106 +0.003204,106 +0.003135,106 +0.003388,106 +0.003199,106 +0.003293,106 +0.003074,106 +0.003239,106 +0.003718,106 +0.003302,106 +0.003199,106 +0.003224,106 +0.003473,106 +0.004491,106 +0.003305,106 +0.002990,106 +0.003136,106 +0.003739,106 +0.003203,106 +0.003375,106 +0.003028,106 +0.003151,106 +0.003945,106 +0.003031,106 +0.003347,106 +0.003095,106 +0.003152,106 +0.003090,106 +0.003259,106 +0.003189,106 +0.002980,106 +0.002939,106 +0.002946,106 +0.003175,106 +0.003065,106 +0.003058,106 +0.002943,106 +0.002907,106 +0.003187,108 +0.003114,108 +0.003144,108 +0.003167,108 +0.003105,108 +0.003137,108 +0.003077,108 +0.003091,108 +0.003151,108 +0.003082,108 +0.003082,108 +0.003137,108 +0.003076,108 +0.003128,108 +0.003080,108 +0.003082,108 +0.003117,108 +0.003076,108 +0.003171,108 +0.003267,108 +0.003295,108 +0.003273,108 +0.003076,108 +0.003203,108 +0.003136,108 +0.003176,108 +0.003111,108 +0.003154,108 +0.003122,108 +0.003130,108 +0.003084,108 +0.003076,108 +0.003118,108 +0.003448,108 +0.003253,108 +0.003176,108 +0.003076,108 +0.003272,108 +0.003213,108 +0.003440,108 +0.003461,108 +0.003272,108 +0.003390,108 +0.003595,108 +0.003716,108 +0.003763,108 +0.003569,108 +0.003491,108 +0.004074,108 +0.004008,108 +0.003675,108 +0.003863,108 +0.004063,108 +0.004119,108 +0.003293,108 +0.003363,108 +0.003859,108 +0.004069,108 +0.003520,108 +0.003406,108 +0.003918,108 +0.003379,108 +0.003471,108 +0.003419,108 +0.003413,108 +0.003456,108 +0.003645,108 +0.003479,108 +0.003432,108 +0.003446,108 +0.003743,108 +0.003456,108 +0.003485,108 +0.003335,108 +0.003717,108 +0.003818,108 +0.004702,108 +0.003204,108 +0.003619,108 +0.004511,108 +0.003444,108 +0.003417,108 +0.003457,108 +0.003391,108 +0.004317,108 +0.003866,108 +0.003940,108 +0.004001,108 +0.004094,108 +0.003464,108 +0.003306,108 +0.003358,108 +0.004539,108 +0.003901,108 +0.003670,108 +0.003260,108 +0.003190,108 +0.003597,108 +0.003501,108 +0.003408,108 +0.003429,110 +0.003429,110 +0.003797,110 +0.004116,110 +0.004120,110 +0.003768,110 +0.004135,110 +0.003665,110 +0.003478,110 +0.003370,110 +0.003650,110 +0.003790,110 +0.003530,110 +0.003283,110 +0.003287,110 +0.003979,110 +0.003637,110 +0.003364,110 +0.003284,110 +0.003276,110 +0.003956,110 +0.003654,110 +0.003303,110 +0.003277,110 +0.003401,110 +0.003906,110 +0.003552,110 +0.003310,110 +0.003291,110 +0.003906,110 +0.003494,110 +0.004124,110 +0.004018,110 +0.003607,110 +0.004405,110 +0.003534,110 +0.003367,110 +0.003302,110 +0.004091,110 +0.003674,110 +0.003433,110 +0.003281,110 +0.003282,110 +0.004209,110 +0.004076,110 +0.003343,110 +0.003248,110 +0.003524,110 +0.004683,110 +0.003419,110 +0.003310,110 +0.003274,110 +0.003638,110 +0.004112,110 +0.003305,110 +0.003293,110 +0.003420,110 +0.004203,110 +0.003878,110 +0.003977,110 +0.003768,110 +0.004266,110 +0.003622,110 +0.003306,110 +0.003291,110 +0.003711,110 +0.004202,110 +0.003488,110 +0.003287,110 +0.003288,110 +0.004122,110 +0.003463,110 +0.003373,110 +0.003309,110 +0.003280,110 +0.003765,110 +0.003592,110 +0.003307,110 +0.003268,110 +0.003380,110 +0.003534,110 +0.003634,110 +0.003308,110 +0.003291,110 +0.003403,110 +0.003537,110 +0.003463,110 +0.004221,110 +0.003886,110 +0.003534,110 +0.003454,110 +0.003349,110 +0.003297,110 +0.003337,110 +0.003307,110 +0.003330,110 +0.003333,110 +0.003292,110 +0.003253,110 +0.003314,110 +0.003456,112 +0.003488,112 +0.003424,112 +0.003424,112 +0.003485,112 +0.003424,112 +0.003509,112 +0.003465,112 +0.003487,112 +0.003426,112 +0.003655,112 +0.003506,112 +0.003448,112 +0.003691,112 +0.003444,112 +0.003525,112 +0.004253,112 +0.004014,112 +0.003595,112 +0.003568,112 +0.003467,112 +0.003682,112 +0.003663,112 +0.003452,112 +0.003849,112 +0.003829,112 +0.003802,112 +0.004047,112 +0.003946,112 +0.003842,112 +0.003872,112 +0.003809,112 +0.003732,112 +0.003750,112 +0.003766,112 +0.003737,112 +0.003740,112 +0.003876,112 +0.003732,112 +0.003700,112 +0.004181,112 +0.004431,112 +0.004791,112 +0.004435,112 +0.004282,112 +0.004833,112 +0.003635,112 +0.003641,112 +0.004416,112 +0.004212,112 +0.003849,112 +0.003800,112 +0.004405,112 +0.004508,112 +0.003790,112 +0.003760,112 +0.004236,112 +0.004309,112 +0.004036,112 +0.004067,112 +0.004138,112 +0.004338,112 +0.003998,112 +0.003950,112 +0.004470,112 +0.004087,112 +0.003750,112 +0.003674,112 +0.004195,112 +0.003909,112 +0.003730,112 +0.003789,112 +0.003871,112 +0.004001,112 +0.003885,112 +0.003800,112 +0.003805,112 +0.003705,112 +0.003966,112 +0.003826,112 +0.003758,112 +0.004308,112 +0.004099,112 +0.003878,112 +0.003669,112 +0.004048,112 +0.003747,112 +0.004013,112 +0.003836,112 +0.004167,112 +0.004281,112 +0.003775,112 +0.003679,112 +0.003618,112 +0.004103,112 +0.004505,112 +0.003802,112 +0.003920,112 +0.004098,112 +0.003758,112 +0.004096,114 +0.004092,114 +0.004524,114 +0.003986,114 +0.004240,114 +0.004373,114 +0.004719,114 +0.004651,114 +0.004453,114 +0.004648,114 +0.004415,114 +0.004713,114 +0.004440,114 +0.004585,114 +0.004241,114 +0.003937,114 +0.003874,114 +0.004223,114 +0.004237,114 +0.004248,114 +0.003954,114 +0.004599,114 +0.003933,114 +0.004173,114 +0.003968,114 +0.004382,114 +0.004115,114 +0.004127,114 +0.004023,114 +0.004088,114 +0.004349,114 +0.004448,114 +0.004103,114 +0.004327,114 +0.004140,114 +0.004051,114 +0.004106,114 +0.004270,114 +0.004178,114 +0.004092,114 +0.004039,114 +0.003899,114 +0.004260,114 +0.004283,114 +0.004056,114 +0.004069,114 +0.004379,114 +0.004036,114 +0.004061,114 +0.004247,114 +0.003940,114 +0.004357,114 +0.004266,114 +0.004872,114 +0.004120,114 +0.004111,114 +0.004133,114 +0.004764,114 +0.004693,114 +0.004351,114 +0.004444,114 +0.004386,114 +0.004131,114 +0.004052,114 +0.003993,114 +0.004490,114 +0.004194,114 +0.004035,114 +0.004154,114 +0.004109,114 +0.004098,114 +0.004010,114 +0.004149,114 +0.004626,114 +0.004505,114 +0.003911,114 +0.004349,114 +0.004280,114 +0.004227,114 +0.003865,114 +0.004203,114 +0.004726,114 +0.004246,114 +0.004057,114 +0.004151,114 +0.004506,114 +0.004091,114 +0.004129,114 +0.003975,114 +0.004005,114 +0.004205,114 +0.004115,114 +0.004046,114 +0.004115,114 +0.004059,114 +0.003898,114 +0.003900,114 +0.004033,114 +0.003874,114 +0.003878,114 +0.004165,116 +0.004023,116 +0.003963,116 +0.003937,116 +0.003927,116 +0.003950,116 +0.004177,116 +0.003951,116 +0.003834,116 +0.003977,116 +0.003862,116 +0.003893,116 +0.003824,116 +0.003883,116 +0.004006,116 +0.004065,116 +0.003836,116 +0.003913,116 +0.003836,116 +0.003851,116 +0.003851,116 +0.003833,116 +0.003878,116 +0.003807,116 +0.003856,116 +0.003801,116 +0.003897,116 +0.003805,116 +0.003903,116 +0.003825,116 +0.003925,116 +0.003808,116 +0.003824,116 +0.003827,116 +0.003812,116 +0.003879,116 +0.003935,116 +0.003904,116 +0.003808,116 +0.003932,116 +0.003906,116 +0.004212,116 +0.003857,116 +0.004047,116 +0.003861,116 +0.003930,116 +0.003859,116 +0.003830,116 +0.003823,116 +0.003885,116 +0.003844,116 +0.003812,116 +0.003804,116 +0.003832,116 +0.003877,116 +0.003803,116 +0.003807,116 +0.003801,116 +0.003831,116 +0.003943,116 +0.003805,116 +0.003826,116 +0.003802,116 +0.003859,116 +0.003803,116 +0.003913,116 +0.003920,116 +0.004088,116 +0.003964,116 +0.003828,116 +0.003826,116 +0.003879,116 +0.003828,116 +0.003809,116 +0.003812,116 +0.003832,116 +0.003845,116 +0.003829,116 +0.003806,116 +0.003802,116 +0.003875,116 +0.003803,116 +0.003804,116 +0.003800,116 +0.003830,116 +0.003849,116 +0.003802,116 +0.003805,116 +0.003802,116 +0.003875,116 +0.003802,116 +0.003932,116 +0.003888,116 +0.004108,116 +0.003871,116 +0.003825,116 +0.003979,116 +0.003924,116 +0.003832,116 +0.003967,116 +0.004099,118 +0.004075,118 +0.004074,118 +0.004004,118 +0.004010,118 +0.004033,118 +0.004051,118 +0.004004,118 +0.004004,118 +0.004007,118 +0.004094,118 +0.004050,118 +0.004003,118 +0.004006,118 +0.004077,118 +0.004006,118 +0.004087,118 +0.004210,118 +0.004358,118 +0.004105,118 +0.004024,118 +0.004011,118 +0.004085,118 +0.004140,118 +0.004008,118 +0.004016,118 +0.004140,118 +0.004010,118 +0.004005,118 +0.004004,118 +0.004075,118 +0.004006,118 +0.004006,118 +0.004004,118 +0.004124,118 +0.004004,118 +0.004149,118 +0.004055,118 +0.004022,118 +0.004042,118 +0.004037,118 +0.004253,118 +0.004472,118 +0.004140,118 +0.004036,118 +0.004067,118 +0.004177,118 +0.004095,118 +0.004128,118 +0.004145,118 +0.004078,118 +0.004070,118 +0.004007,118 +0.004009,118 +0.004034,118 +0.004069,118 +0.004007,118 +0.004009,118 +0.004007,118 +0.004071,118 +0.004006,118 +0.004003,118 +0.004006,118 +0.004093,118 +0.004039,118 +0.004071,118 +0.004192,118 +0.004376,118 +0.004032,118 +0.004024,118 +0.004029,118 +0.004094,118 +0.004011,118 +0.004007,118 +0.004009,118 +0.004075,118 +0.004004,118 +0.004006,118 +0.004004,118 +0.004078,118 +0.004024,118 +0.004006,118 +0.004003,118 +0.004033,118 +0.004112,118 +0.004038,118 +0.004023,118 +0.004035,118 +0.004048,118 +0.004004,118 +0.004159,118 +0.004318,118 +0.004226,118 +0.004037,118 +0.004078,118 +0.004100,118 +0.004078,118 +0.004041,118 +0.004005,118 +0.004013,118 +0.004287,120 +0.004207,120 +0.004205,120 +0.004205,120 +0.004275,120 +0.004208,120 +0.004204,120 +0.004206,120 +0.004272,120 +0.004207,120 +0.004204,120 +0.004236,120 +0.004244,120 +0.004211,120 +0.004343,120 +0.004570,120 +0.004448,120 +0.004240,120 +0.004242,120 +0.004260,120 +0.004269,120 +0.004209,120 +0.004207,120 +0.004272,120 +0.004207,120 +0.004203,120 +0.004206,120 +0.004277,120 +0.004204,120 +0.004205,120 +0.004203,120 +0.004275,120 +0.004203,120 +0.004205,120 +0.004223,120 +0.004274,120 +0.004203,120 +0.004205,120 +0.004403,120 +0.004593,120 +0.004256,120 +0.004269,120 +0.004335,120 +0.004336,120 +0.004315,120 +0.004269,120 +0.004230,120 +0.004365,120 +0.004229,120 +0.004224,120 +0.004224,120 +0.004274,120 +0.004204,120 +0.004206,120 +0.004233,120 +0.004246,120 +0.004391,120 +0.004207,120 +0.004236,120 +0.004243,120 +0.004206,120 +0.004343,120 +0.004532,120 +0.004434,120 +0.004247,120 +0.004307,120 +0.004299,120 +0.004225,120 +0.004233,120 +0.004225,120 +0.004294,120 +0.004319,120 +0.004244,120 +0.004232,120 +0.004317,120 +0.004256,120 +0.004245,120 +0.004250,120 +0.004315,120 +0.004206,120 +0.004204,120 +0.004205,120 +0.004273,120 +0.004206,120 +0.004238,120 +0.004449,120 +0.004641,120 +0.004204,120 +0.004205,120 +0.004224,120 +0.004247,120 +0.004204,120 +0.004205,120 +0.004233,120 +0.004265,120 +0.004204,120 +0.004205,120 +0.004231,120 +0.004290,120 +0.004203,120 +0.004448,122 +0.004493,122 +0.004564,122 +0.004424,122 +0.004419,122 +0.004477,122 +0.004420,122 +0.004419,122 +0.004614,122 +0.004852,122 +0.004420,122 +0.004643,122 +0.004469,122 +0.004523,122 +0.004598,122 +0.004456,122 +0.004554,122 +0.004446,122 +0.004423,122 +0.004419,122 +0.004492,122 +0.004420,122 +0.004420,122 +0.004452,122 +0.004460,122 +0.004424,122 +0.004421,122 +0.004471,122 +0.004488,122 +0.004445,122 +0.004588,122 +0.004816,122 +0.004674,122 +0.004440,122 +0.004448,122 +0.004523,122 +0.004466,122 +0.004424,122 +0.004470,122 +0.004481,122 +0.004423,122 +0.004464,122 +0.004488,122 +0.004422,122 +0.004427,122 +0.004422,122 +0.004530,122 +0.004423,122 +0.004420,122 +0.004422,122 +0.004537,122 +0.004419,122 +0.004470,122 +0.004943,122 +0.004728,122 +0.004450,122 +0.004420,122 +0.004705,122 +0.004813,122 +0.004763,122 +0.004560,122 +0.004446,122 +0.004525,122 +0.004445,122 +0.004512,122 +0.004572,122 +0.004654,122 +0.004420,122 +0.004476,122 +0.004561,122 +0.004623,122 +0.004454,122 +0.004489,122 +0.004460,122 +0.004494,122 +0.004738,122 +0.004682,122 +0.004465,122 +0.004475,122 +0.004489,122 +0.004479,122 +0.004420,122 +0.004420,122 +0.004477,122 +0.004420,122 +0.004442,122 +0.004431,122 +0.004486,122 +0.004441,122 +0.004464,122 +0.004468,122 +0.004420,122 +0.004422,122 +0.004419,122 +0.004512,122 +0.004420,122 +0.004425,122 +0.004646,122 +0.004711,122 +0.004576,122 +0.004711,124 +0.004750,124 +0.004739,124 +0.004677,124 +0.004678,124 +0.004712,124 +0.004638,124 +0.004635,124 +0.004690,124 +0.004638,124 +0.004636,124 +0.004637,124 +0.004684,124 +0.004695,124 +0.004646,124 +0.004648,124 +0.004681,124 +0.004635,124 +0.004683,124 +0.004996,124 +0.004826,124 +0.004724,124 +0.004688,124 +0.004707,124 +0.004640,124 +0.004642,124 +0.004685,124 +0.004681,124 +0.004666,124 +0.004638,124 +0.004685,124 +0.004640,124 +0.004692,124 +0.004689,124 +0.004635,124 +0.004641,124 +0.004636,124 +0.004689,124 +0.004661,124 +0.004636,124 +0.004882,124 +0.004919,124 +0.004701,124 +0.004638,124 +0.004705,124 +0.004638,124 +0.004636,124 +0.004647,124 +0.004680,124 +0.004636,124 +0.005736,124 +0.004788,124 +0.004636,124 +0.004643,124 +0.004678,124 +0.004667,124 +0.004638,124 +0.004635,124 +0.004687,124 +0.004641,124 +0.004636,124 +0.004925,124 +0.005010,124 +0.004684,124 +0.004679,124 +0.004749,124 +0.004758,124 +0.004657,124 +0.004758,124 +0.004704,124 +0.004636,124 +0.004638,124 +0.004706,124 +0.004639,124 +0.004636,124 +0.004667,124 +0.004699,124 +0.004636,124 +0.004670,124 +0.004706,124 +0.004659,124 +0.004635,124 +0.004787,124 +0.005047,124 +0.004857,124 +0.004656,124 +0.004749,124 +0.004659,124 +0.004636,124 +0.004662,124 +0.004766,124 +0.004687,124 +0.004664,124 +0.004730,124 +0.004661,124 +0.004636,124 +0.004642,124 +0.004704,124 +0.004640,124 +0.004638,124 +0.004915,126 +0.004906,126 +0.004864,126 +0.004917,126 +0.005215,126 +0.005071,126 +0.004884,126 +0.005222,126 +0.004990,126 +0.005259,126 +0.005390,126 +0.005463,126 +0.005577,126 +0.005841,126 +0.005507,126 +0.005470,126 +0.005750,126 +0.007375,126 +0.006174,126 +0.006265,126 +0.005507,126 +0.005868,126 +0.009030,126 +0.007537,126 +0.007350,126 +0.005591,126 +0.005574,126 +0.007123,126 +0.005460,126 +0.005456,126 +0.007157,126 +0.005286,126 +0.005380,126 +0.006827,126 +0.005305,126 +0.005246,126 +0.006802,126 +0.005030,126 +0.008349,126 +0.008952,126 +0.004990,126 +0.004984,126 +0.004962,126 +0.004979,126 +0.005130,126 +0.004950,126 +0.005002,126 +0.004897,126 +0.004938,126 +0.004991,126 +0.004948,126 +0.005013,126 +0.004889,126 +0.004946,126 +0.004927,126 +0.004868,126 +0.006621,126 +0.008387,126 +0.004923,126 +0.004954,126 +0.004954,126 +0.004868,126 +0.005026,126 +0.004867,126 +0.004864,126 +0.004906,126 +0.004905,126 +0.004864,126 +0.004866,126 +0.004947,126 +0.004884,126 +0.004866,126 +0.004942,126 +0.004866,126 +0.004867,126 +0.005186,126 +0.008795,126 +0.005915,126 +0.004976,126 +0.004869,126 +0.004885,126 +0.004933,126 +0.004925,126 +0.004887,126 +0.004868,126 +0.004964,126 +0.004888,126 +0.004887,126 +0.004947,126 +0.004867,126 +0.004949,126 +0.004907,126 +0.004930,126 +0.004866,126 +0.004866,126 +0.007738,126 +0.007211,126 +0.004975,126 +0.004872,126 +0.004864,126 +0.005386,128 +0.005457,128 +0.005374,128 +0.005376,128 +0.005440,128 +0.005372,128 +0.005418,128 +0.005458,128 +0.005413,128 +0.005375,128 +0.005435,128 +0.005372,128 +0.005376,128 +0.007662,128 +0.008165,128 +0.005499,128 +0.005405,128 +0.005429,128 +0.005458,128 +0.005401,128 +0.005376,128 +0.005431,128 +0.005375,128 +0.005376,128 +0.005431,128 +0.005375,128 +0.005376,128 +0.005392,128 +0.005415,128 +0.005377,128 +0.005393,128 +0.008756,128 +0.007322,128 +0.005482,128 +0.005442,128 +0.005407,128 +0.005487,128 +0.005377,128 +0.005396,128 +0.005436,128 +0.005373,128 +0.005375,128 +0.005435,128 +0.005373,128 +0.005418,128 +0.005435,128 +0.005373,128 +0.005376,128 +0.006360,128 +0.009397,128 +0.005654,128 +0.005469,128 +0.005642,128 +0.005554,128 +0.005407,128 +0.005405,128 +0.005435,128 +0.005401,128 +0.005411,128 +0.005434,128 +0.005408,128 +0.005464,128 +0.005396,128 +0.005417,128 +0.005377,128 +0.005395,128 +0.008415,128 +0.007413,128 +0.005487,128 +0.005378,128 +0.005376,128 +0.005459,128 +0.005373,128 +0.005375,128 +0.005466,128 +0.005374,128 +0.005660,128 +0.006978,128 +0.005549,128 +0.005383,128 +0.005587,128 +0.006194,128 +0.005526,128 +0.008246,128 +0.008427,128 +0.005702,128 +0.005406,128 +0.005692,128 +0.005657,128 +0.005604,128 +0.005617,128 +0.005458,128 +0.005567,128 +0.005881,128 +0.005510,128 +0.005733,128 +0.005493,128 +0.005481,128 +0.005583,128 +0.005448,128 +0.007634,130 +0.008371,130 +0.005475,130 +0.005438,130 +0.005381,130 +0.005505,130 +0.005396,130 +0.005344,130 +0.005421,130 +0.005375,130 +0.005337,130 +0.005420,130 +0.005397,130 +0.005339,130 +0.005418,130 +0.005390,130 +0.005344,130 +0.005521,130 +0.007434,130 +0.008883,130 +0.005438,130 +0.005415,130 +0.005448,130 +0.005405,130 +0.005400,130 +0.005437,130 +0.005424,130 +0.005340,130 +0.005439,130 +0.005372,130 +0.005340,130 +0.005378,130 +0.005383,130 +0.005380,130 +0.005378,130 +0.006334,130 +0.009437,130 +0.005541,130 +0.005377,130 +0.005616,130 +0.005437,130 +0.005339,130 +0.005600,130 +0.005439,130 +0.005337,130 +0.005339,130 +0.005459,130 +0.005337,130 +0.005339,130 +0.005439,130 +0.005337,130 +0.005341,130 +0.005442,130 +0.007503,130 +0.008373,130 +0.005348,130 +0.005343,130 +0.005455,130 +0.005339,130 +0.005364,130 +0.005397,130 +0.005379,130 +0.005341,130 +0.005378,130 +0.005380,130 +0.005342,130 +0.005357,130 +0.005421,130 +0.005341,130 +0.005337,130 +0.005517,130 +0.008094,130 +0.007734,130 +0.005471,130 +0.006000,130 +0.006253,130 +0.005450,130 +0.005444,130 +0.005474,130 +0.005337,130 +0.005420,130 +0.005439,130 +0.005398,130 +0.005341,130 +0.005421,130 +0.005338,130 +0.005342,130 +0.005425,130 +0.006189,130 +0.009636,130 +0.005515,130 +0.005400,130 +0.005429,130 +0.005374,130 +0.005482,130 +0.005847,130 +0.005396,130 +0.005360,130 +0.005418,130 +0.005342,130 +0.005615,132 +0.005666,132 +0.005921,132 +0.005611,132 +0.005735,132 +0.005710,132 +0.010125,132 +0.006354,132 +0.005653,132 +0.005672,132 +0.005607,132 +0.005646,132 +0.005715,132 +0.006172,132 +0.006019,132 +0.005687,132 +0.005589,132 +0.005627,132 +0.005667,132 +0.005585,132 +0.005590,132 +0.005691,132 +0.007254,132 +0.009219,132 +0.005691,132 +0.005624,132 +0.005865,132 +0.005591,132 +0.005606,132 +0.005692,132 +0.005588,132 +0.005589,132 +0.005665,132 +0.005587,132 +0.005611,132 +0.005685,132 +0.005606,132 +0.005592,132 +0.005745,132 +0.007300,132 +0.009084,132 +0.005619,132 +0.005597,132 +0.005708,132 +0.005624,132 +0.005675,132 +0.005747,132 +0.005725,132 +0.005683,132 +0.005650,132 +0.005598,132 +0.005613,132 +0.005687,132 +0.005588,132 +0.005605,132 +0.005759,132 +0.007564,132 +0.008772,132 +0.005623,132 +0.005671,132 +0.005660,132 +0.005645,132 +0.005625,132 +0.005628,132 +0.005589,132 +0.005606,132 +0.005635,132 +0.005744,132 +0.005645,132 +0.005590,132 +0.005586,132 +0.005669,132 +0.005738,132 +0.007667,132 +0.008944,132 +0.005686,132 +0.005690,132 +0.006929,132 +0.010806,132 +0.010826,132 +0.010964,132 +0.010989,132 +0.010404,132 +0.008319,132 +0.005731,132 +0.008331,132 +0.008752,132 +0.005674,132 +0.005879,132 +0.005639,132 +0.005672,132 +0.005971,132 +0.005622,132 +0.005645,132 +0.005624,132 +0.005586,132 +0.005622,132 +0.005624,132 +0.005585,132 +0.005644,132 +0.005968,134 +0.006600,134 +0.010243,134 +0.005955,134 +0.006033,134 +0.005905,134 +0.005864,134 +0.005923,134 +0.005841,134 +0.005846,134 +0.005925,134 +0.005844,134 +0.005881,134 +0.005923,134 +0.005842,134 +0.005863,134 +0.005983,134 +0.006247,134 +0.010562,134 +0.006022,134 +0.005924,134 +0.005888,134 +0.005843,134 +0.005924,134 +0.005841,134 +0.005843,134 +0.005926,134 +0.005840,134 +0.005844,134 +0.005922,134 +0.005846,134 +0.005840,134 +0.005944,134 +0.005923,134 +0.009654,134 +0.007257,134 +0.005909,134 +0.005901,134 +0.005865,134 +0.005921,134 +0.005845,134 +0.005841,134 +0.005926,134 +0.005883,134 +0.005865,134 +0.005927,134 +0.005846,134 +0.005843,134 +0.005923,134 +0.005908,134 +0.008324,134 +0.008524,134 +0.005931,134 +0.005912,134 +0.005887,134 +0.006011,134 +0.005851,134 +0.005841,134 +0.005926,134 +0.005842,134 +0.005844,134 +0.005920,134 +0.005845,134 +0.005842,134 +0.005939,134 +0.005904,134 +0.007211,134 +0.009723,134 +0.005917,134 +0.005949,134 +0.005866,134 +0.005883,134 +0.005884,134 +0.005840,134 +0.005925,134 +0.005842,134 +0.005845,134 +0.005928,134 +0.005843,134 +0.005844,134 +0.005949,134 +0.005903,134 +0.005978,134 +0.010587,134 +0.006265,134 +0.005974,134 +0.005847,134 +0.005888,134 +0.005879,134 +0.005845,134 +0.005881,134 +0.005883,134 +0.005842,134 +0.005926,134 +0.005844,134 +0.005844,134 +0.005926,134 +0.005949,134 +0.005843,134 +0.009580,134 +0.007544,136 +0.006259,136 +0.006137,136 +0.006177,136 +0.006148,136 +0.006099,136 +0.006257,136 +0.006103,136 +0.006102,136 +0.006180,136 +0.006124,136 +0.006101,136 +0.006234,136 +0.006189,136 +0.007276,136 +0.010162,136 +0.006208,136 +0.006129,136 +0.006098,136 +0.006184,136 +0.006101,136 +0.006133,136 +0.006182,136 +0.006138,136 +0.006145,136 +0.006140,136 +0.006122,136 +0.006238,136 +0.006165,136 +0.006100,136 +0.009424,136 +0.007949,136 +0.006220,136 +0.006165,136 +0.006179,136 +0.006205,136 +0.006099,136 +0.006204,136 +0.006101,136 +0.006102,136 +0.006177,136 +0.006145,136 +0.006163,136 +0.006159,136 +0.006162,136 +0.006208,136 +0.010491,136 +0.007331,136 +0.006136,136 +0.006138,136 +0.006207,136 +0.006110,136 +0.006183,136 +0.006164,136 +0.006098,136 +0.006185,136 +0.006101,136 +0.006102,136 +0.006207,136 +0.006182,136 +0.006102,136 +0.009914,136 +0.007608,136 +0.006170,136 +0.006119,136 +0.006205,136 +0.006101,136 +0.006102,136 +0.006182,136 +0.006102,136 +0.006121,136 +0.006210,136 +0.006097,136 +0.006181,136 +0.006171,136 +0.006104,136 +0.007201,136 +0.010121,136 +0.006260,136 +0.006129,136 +0.006142,136 +0.006253,136 +0.006098,136 +0.006221,136 +0.006100,136 +0.006102,136 +0.006218,136 +0.006102,136 +0.006113,136 +0.006188,136 +0.006159,136 +0.006146,136 +0.009205,136 +0.008398,136 +0.006163,136 +0.006129,136 +0.006238,136 +0.006110,136 +0.006101,136 +0.006184,136 +0.006475,138 +0.006483,138 +0.006375,138 +0.006380,138 +0.006462,138 +0.006437,138 +0.006422,138 +0.008880,138 +0.009047,138 +0.006436,138 +0.006377,138 +0.006463,138 +0.006380,138 +0.006415,138 +0.006424,138 +0.006379,138 +0.006465,138 +0.006380,138 +0.006375,138 +0.006458,138 +0.006482,138 +0.006478,138 +0.009612,138 +0.008407,138 +0.006416,138 +0.006408,138 +0.006474,138 +0.006380,138 +0.006463,138 +0.006381,138 +0.006463,138 +0.006476,138 +0.006379,138 +0.006420,138 +0.006414,138 +0.006450,138 +0.006463,138 +0.010245,138 +0.007760,138 +0.006435,138 +0.006458,138 +0.006458,138 +0.006388,138 +0.006456,138 +0.006402,138 +0.006378,138 +0.006481,138 +0.006399,138 +0.006458,138 +0.006480,138 +0.006398,138 +0.006459,138 +0.010950,138 +0.006932,138 +0.006438,138 +0.006532,138 +0.006417,138 +0.006415,138 +0.006459,138 +0.006376,138 +0.006423,138 +0.006416,138 +0.006381,138 +0.006499,138 +0.006475,138 +0.006380,138 +0.006794,138 +0.011207,138 +0.006485,138 +0.006396,138 +0.006503,138 +0.006445,138 +0.006410,138 +0.006490,138 +0.006439,138 +0.006463,138 +0.006386,138 +0.006378,138 +0.006462,138 +0.006463,138 +0.006458,138 +0.007488,138 +0.010416,138 +0.006409,138 +0.006401,138 +0.006456,138 +0.006436,138 +0.006480,138 +0.006379,138 +0.006380,138 +0.006474,138 +0.006375,138 +0.006423,138 +0.006417,138 +0.006450,138 +0.006461,138 +0.007113,138 +0.011225,138 +0.006471,138 +0.006478,138 +0.006769,140 +0.006656,140 +0.006753,140 +0.006655,140 +0.006738,140 +0.006691,140 +0.006655,140 +0.006737,140 +0.006713,140 +0.006737,140 +0.006694,140 +0.011793,140 +0.006709,140 +0.006807,140 +0.006790,140 +0.006687,140 +0.006729,140 +0.006655,140 +0.006696,140 +0.006692,140 +0.006655,140 +0.006736,140 +0.006710,140 +0.006765,140 +0.006694,140 +0.010488,140 +0.008057,140 +0.006740,140 +0.006800,140 +0.006695,140 +0.006780,140 +0.006676,140 +0.006678,140 +0.006751,140 +0.006655,140 +0.006747,140 +0.006761,140 +0.006656,140 +0.006732,140 +0.009127,140 +0.009353,140 +0.006724,140 +0.006806,140 +0.006677,140 +0.006760,140 +0.006654,140 +0.006715,140 +0.006734,140 +0.006651,140 +0.006733,140 +0.006684,140 +0.006766,140 +0.006730,140 +0.007838,140 +0.010583,140 +0.006669,140 +0.006790,140 +0.006676,140 +0.006704,140 +0.006691,140 +0.006655,140 +0.006734,140 +0.006676,140 +0.006805,140 +0.006760,140 +0.006711,140 +0.006734,140 +0.006655,140 +0.011768,140 +0.006691,140 +0.006817,140 +0.006717,140 +0.006691,140 +0.006778,140 +0.006655,140 +0.006731,140 +0.006655,140 +0.006719,140 +0.006782,140 +0.006765,140 +0.006796,140 +0.006700,140 +0.010922,140 +0.007698,140 +0.006781,140 +0.006674,140 +0.006694,140 +0.006740,140 +0.006651,140 +0.006738,140 +0.006656,140 +0.006655,140 +0.006736,140 +0.006822,140 +0.006776,140 +0.006673,140 +0.009524,140 +0.008979,140 +0.006754,140 +0.006715,140 +0.007035,142 +0.007048,142 +0.006946,142 +0.007020,142 +0.007048,142 +0.007000,142 +0.007080,142 +0.007065,142 +0.007026,142 +0.006946,142 +0.011171,142 +0.007917,142 +0.007063,142 +0.007218,142 +0.007036,142 +0.006964,142 +0.007092,142 +0.007063,142 +0.008122,142 +0.007109,142 +0.007142,142 +0.007097,142 +0.006951,142 +0.008729,142 +0.010398,142 +0.007144,142 +0.007088,142 +0.006942,142 +0.007028,142 +0.006949,142 +0.007069,142 +0.007492,142 +0.007120,142 +0.007258,142 +0.007069,142 +0.007050,142 +0.006944,142 +0.010432,142 +0.009191,142 +0.007110,142 +0.007057,142 +0.007094,142 +0.006976,142 +0.007043,142 +0.006947,142 +0.006946,142 +0.007023,142 +0.007052,142 +0.007024,142 +0.006946,142 +0.008870,142 +0.010285,142 +0.007081,142 +0.006970,142 +0.007068,142 +0.007010,142 +0.007010,142 +0.007087,142 +0.006966,142 +0.007024,142 +0.006985,142 +0.007014,142 +0.007026,142 +0.006945,142 +0.010603,142 +0.008532,142 +0.007087,142 +0.006987,142 +0.007045,142 +0.006968,142 +0.007052,142 +0.007006,142 +0.006958,142 +0.007029,142 +0.007006,142 +0.007024,142 +0.006943,142 +0.006946,142 +0.012083,142 +0.007096,142 +0.006994,142 +0.006994,142 +0.007034,142 +0.007008,142 +0.007026,142 +0.006946,142 +0.006985,142 +0.006985,142 +0.007024,142 +0.007025,142 +0.006942,142 +0.008762,142 +0.010367,142 +0.007151,142 +0.006977,142 +0.007046,142 +0.006955,142 +0.006988,142 +0.006987,142 +0.006944,142 +0.007327,144 +0.007334,144 +0.007310,144 +0.007232,144 +0.007268,144 +0.012184,144 +0.007532,144 +0.007281,144 +0.007333,144 +0.007233,144 +0.007237,144 +0.007310,144 +0.007274,144 +0.007310,144 +0.007298,144 +0.007331,144 +0.007454,144 +0.007391,144 +0.011581,144 +0.007960,144 +0.007244,144 +0.007270,144 +0.007314,144 +0.007273,144 +0.007328,144 +0.007248,144 +0.007312,144 +0.007335,144 +0.007310,144 +0.007232,144 +0.007269,144 +0.010878,144 +0.008735,144 +0.007287,144 +0.007312,144 +0.007293,144 +0.007232,144 +0.007358,144 +0.007243,144 +0.007335,144 +0.007291,144 +0.007310,144 +0.007231,144 +0.007227,144 +0.010332,144 +0.009383,144 +0.007242,144 +0.007305,144 +0.007304,144 +0.007293,144 +0.007309,144 +0.007232,144 +0.007310,144 +0.007312,144 +0.007339,144 +0.007228,144 +0.007231,144 +0.009600,144 +0.010036,144 +0.007243,144 +0.007316,144 +0.007283,144 +0.007262,144 +0.007314,144 +0.007254,144 +0.007333,144 +0.007283,144 +0.007355,144 +0.007228,144 +0.007274,144 +0.007931,144 +0.012148,144 +0.007279,144 +0.007336,144 +0.007293,144 +0.007232,144 +0.007309,144 +0.007231,144 +0.007310,144 +0.007292,144 +0.007352,144 +0.007228,144 +0.007375,144 +0.009157,144 +0.010462,144 +0.007231,144 +0.007276,144 +0.007271,144 +0.007230,144 +0.007311,144 +0.007231,144 +0.007312,144 +0.007291,144 +0.007270,144 +0.007267,144 +0.007231,144 +0.008402,144 +0.011334,144 +0.007284,144 +0.007240,144 +0.007692,146 +0.007635,146 +0.007671,146 +0.007604,146 +0.007675,146 +0.007674,146 +0.007691,146 +0.007592,146 +0.008196,146 +0.011100,146 +0.009205,146 +0.007645,146 +0.007745,146 +0.007741,146 +0.007693,146 +0.007592,146 +0.007713,146 +0.007721,146 +0.007635,146 +0.007631,146 +0.007589,146 +0.008456,146 +0.011895,146 +0.007599,146 +0.007672,146 +0.007591,146 +0.007850,146 +0.007654,146 +0.007612,146 +0.007675,146 +0.007654,146 +0.007694,146 +0.007590,146 +0.007653,146 +0.011486,146 +0.008898,146 +0.007667,146 +0.007743,146 +0.007588,146 +0.007674,146 +0.007592,146 +0.007675,146 +0.007653,146 +0.007671,146 +0.007592,146 +0.007812,146 +0.008585,146 +0.011660,146 +0.007667,146 +0.007763,146 +0.007664,146 +0.007691,146 +0.007589,146 +0.007630,146 +0.007656,146 +0.007675,146 +0.007671,146 +0.007592,146 +0.007714,146 +0.011731,146 +0.008435,146 +0.007684,146 +0.007714,146 +0.007672,146 +0.007723,146 +0.007733,146 +0.007689,146 +0.007660,146 +0.007674,146 +0.007592,146 +0.007674,146 +0.008910,146 +0.011245,146 +0.007650,146 +0.007695,146 +0.007592,146 +0.007752,146 +0.007666,146 +0.007710,146 +0.007593,146 +0.007711,146 +0.007716,146 +0.007592,146 +0.007674,146 +0.012258,146 +0.007952,146 +0.007695,146 +0.007613,146 +0.007652,146 +0.007758,146 +0.007596,146 +0.007690,146 +0.007652,146 +0.007679,146 +0.007635,146 +0.007674,146 +0.008160,146 +0.012631,146 +0.008593,146 +0.007761,146 +0.007959,148 +0.007968,148 +0.007851,148 +0.007933,148 +0.007973,148 +0.007929,148 +0.008064,148 +0.007951,148 +0.009425,148 +0.011397,148 +0.007870,148 +0.007970,148 +0.007905,148 +0.007930,148 +0.007850,148 +0.007932,148 +0.007918,148 +0.007932,148 +0.007850,148 +0.007933,148 +0.008677,148 +0.012120,148 +0.007928,148 +0.007977,148 +0.007935,148 +0.007924,148 +0.007869,148 +0.007936,148 +0.007914,148 +0.007929,148 +0.007850,148 +0.007940,148 +0.008258,148 +0.012461,148 +0.007890,148 +0.007965,148 +0.007929,148 +0.007910,148 +0.007851,148 +0.007914,148 +0.007934,148 +0.007977,148 +0.007850,148 +0.007929,148 +0.007850,148 +0.012867,148 +0.007892,148 +0.007950,148 +0.007874,148 +0.007932,148 +0.007872,148 +0.007944,148 +0.007951,148 +0.007930,148 +0.007850,148 +0.007911,148 +0.007851,148 +0.012936,148 +0.007927,148 +0.007969,148 +0.007873,148 +0.007961,148 +0.007850,148 +0.007933,148 +0.007970,148 +0.007910,148 +0.007927,148 +0.007972,148 +0.007849,148 +0.012914,148 +0.007896,148 +0.007971,148 +0.007897,148 +0.007961,148 +0.007914,148 +0.007989,148 +0.007942,148 +0.007972,148 +0.007896,148 +0.007957,148 +0.007873,148 +0.012887,148 +0.007898,148 +0.007967,148 +0.007851,148 +0.007972,148 +0.007890,148 +0.007907,148 +0.007930,148 +0.007909,148 +0.007850,148 +0.007909,148 +0.007851,148 +0.012450,148 +0.008353,148 +0.007964,148 +0.007935,148 +0.007967,148 +0.007860,148 +0.008012,148 +0.008326,150 +0.008283,150 +0.008174,150 +0.008234,150 +0.008176,150 +0.013231,150 +0.008333,150 +0.008238,150 +0.008228,150 +0.008234,150 +0.008227,150 +0.008236,150 +0.008307,150 +0.008221,150 +0.008196,150 +0.008215,150 +0.009105,150 +0.013067,150 +0.009257,150 +0.008500,150 +0.008297,150 +0.008421,150 +0.008322,150 +0.008274,150 +0.008333,150 +0.008374,150 +0.008256,150 +0.008175,150 +0.013310,150 +0.008296,150 +0.008247,150 +0.008317,150 +0.008242,150 +0.008293,150 +0.008203,150 +0.008344,150 +0.008177,150 +0.008256,150 +0.008198,150 +0.010603,150 +0.010949,150 +0.008287,150 +0.008198,150 +0.008256,150 +0.008192,150 +0.008240,150 +0.008258,150 +0.008269,150 +0.008235,150 +0.008241,150 +0.008175,150 +0.013210,150 +0.008325,150 +0.008201,150 +0.008306,150 +0.008176,150 +0.008239,150 +0.008211,150 +0.008391,150 +0.008218,150 +0.008248,150 +0.008181,150 +0.009482,150 +0.011948,150 +0.008287,150 +0.008195,150 +0.008310,150 +0.008197,150 +0.008256,150 +0.008283,150 +0.008265,150 +0.008176,150 +0.008239,150 +0.008176,150 +0.012253,150 +0.009329,150 +0.008246,150 +0.008255,150 +0.008195,150 +0.008235,150 +0.008176,150 +0.008328,150 +0.008177,150 +0.008353,150 +0.008176,150 +0.008301,150 +0.013030,150 +0.008253,150 +0.008203,150 +0.008289,150 +0.008261,150 +0.008237,150 +0.008196,150 +0.008282,150 +0.008176,150 +0.008215,150 +0.008199,150 +0.011124,150 +0.010348,150 +0.008221,150 +0.008590,152 +0.008528,152 +0.008581,152 +0.008496,152 +0.008618,152 +0.008500,152 +0.008539,152 +0.008495,152 +0.009897,152 +0.012220,152 +0.008549,152 +0.008580,152 +0.008496,152 +0.008536,152 +0.008536,152 +0.008594,152 +0.008504,152 +0.008537,152 +0.008508,152 +0.008783,152 +0.013258,152 +0.008581,152 +0.008518,152 +0.008561,152 +0.008531,152 +0.008556,152 +0.008678,152 +0.008503,152 +0.008537,152 +0.008538,152 +0.008536,152 +0.013405,152 +0.009086,152 +0.008519,152 +0.008583,152 +0.008531,152 +0.008603,152 +0.008575,152 +0.008543,152 +0.008708,152 +0.008499,152 +0.008728,152 +0.013398,152 +0.008614,152 +0.008541,152 +0.008572,152 +0.008527,152 +0.008578,152 +0.008571,152 +0.008540,152 +0.008496,152 +0.008633,152 +0.008537,152 +0.012668,152 +0.009393,152 +0.008538,152 +0.008580,152 +0.008513,152 +0.008534,152 +0.008556,152 +0.008553,152 +0.008498,152 +0.008536,152 +0.008495,152 +0.011647,152 +0.010350,152 +0.008567,152 +0.008598,152 +0.008497,152 +0.008541,152 +0.008564,152 +0.008630,152 +0.008497,152 +0.008536,152 +0.008499,152 +0.010562,152 +0.011428,152 +0.008526,152 +0.008558,152 +0.008550,152 +0.008571,152 +0.008993,152 +0.009109,152 +0.008602,152 +0.008796,152 +0.008519,152 +0.010517,152 +0.011600,152 +0.008552,152 +0.008619,152 +0.008497,152 +0.008579,152 +0.008496,152 +0.008652,152 +0.008569,152 +0.008620,152 +0.008498,152 +0.009709,152 +0.012485,152 +0.008564,152 +0.009016,154 +0.008902,154 +0.008946,154 +0.008847,154 +0.008991,154 +0.008844,154 +0.008929,154 +0.008883,154 +0.011722,154 +0.011108,154 +0.008916,154 +0.008952,154 +0.008897,154 +0.008927,154 +0.008921,154 +0.008936,154 +0.008879,154 +0.008908,154 +0.008925,154 +0.013578,154 +0.009007,154 +0.008936,154 +0.008911,154 +0.008889,154 +0.008881,154 +0.009014,154 +0.008844,154 +0.008920,154 +0.008842,154 +0.008925,154 +0.013867,154 +0.008872,154 +0.008942,154 +0.008897,154 +0.008921,154 +0.008842,154 +0.008996,154 +0.009369,154 +0.009038,154 +0.008919,154 +0.009989,154 +0.013399,154 +0.008900,154 +0.008974,154 +0.008902,154 +0.008949,154 +0.008947,154 +0.008939,154 +0.009119,154 +0.008847,154 +0.008954,154 +0.013207,154 +0.009464,154 +0.008922,154 +0.008902,154 +0.008945,154 +0.008842,154 +0.009057,154 +0.008846,154 +0.008920,154 +0.008889,154 +0.008921,154 +0.013838,154 +0.008916,154 +0.008943,154 +0.008860,154 +0.008921,154 +0.008842,154 +0.009052,154 +0.008841,154 +0.008921,154 +0.008842,154 +0.010556,154 +0.012137,154 +0.008893,154 +0.009020,154 +0.008856,154 +0.008920,154 +0.008948,154 +0.009503,154 +0.008969,154 +0.008888,154 +0.008961,154 +0.012439,154 +0.010243,154 +0.008968,154 +0.008888,154 +0.008947,154 +0.008843,154 +0.008980,154 +0.008842,154 +0.008925,154 +0.008843,154 +0.008920,154 +0.013791,154 +0.008942,154 +0.008941,154 +0.008903,154 +0.008926,154 +0.008848,154 +0.009499,156 +0.009334,156 +0.009283,156 +0.009266,156 +0.011629,156 +0.011676,156 +0.009271,156 +0.009246,156 +0.009345,156 +0.009240,156 +0.009374,156 +0.009283,156 +0.009286,156 +0.009184,156 +0.009266,156 +0.014065,156 +0.009249,156 +0.009320,156 +0.009289,156 +0.009245,156 +0.009339,156 +0.009187,156 +0.009283,156 +0.009188,156 +0.009263,156 +0.013484,156 +0.009916,156 +0.009344,156 +0.009206,156 +0.009293,156 +0.009184,156 +0.009405,156 +0.009270,156 +0.009185,156 +0.009287,156 +0.011293,156 +0.012023,156 +0.009330,156 +0.009187,156 +0.009264,156 +0.009192,156 +0.009396,156 +0.009184,156 +0.009284,156 +0.009228,156 +0.009224,156 +0.014616,156 +0.009291,156 +0.009270,156 +0.009321,156 +0.009187,156 +0.009374,156 +0.009282,156 +0.009344,156 +0.009185,156 +0.009271,156 +0.013772,156 +0.009505,156 +0.009288,156 +0.009205,156 +0.009265,156 +0.009332,156 +0.009187,156 +0.009263,156 +0.009286,156 +0.009381,156 +0.010971,156 +0.012399,156 +0.009319,156 +0.009214,156 +0.009270,156 +0.009185,156 +0.009331,156 +0.009207,156 +0.009204,156 +0.009225,156 +0.009189,156 +0.014033,156 +0.009291,156 +0.009208,156 +0.009224,156 +0.009228,156 +0.009336,156 +0.009186,156 +0.009225,156 +0.009186,156 +0.009245,156 +0.012972,156 +0.010474,156 +0.009281,156 +0.009432,156 +0.009240,156 +0.009350,156 +0.009186,156 +0.009249,156 +0.009185,156 +0.009224,156 +0.010326,156 +0.013002,156 +0.009279,156 +0.009701,158 +0.009625,158 +0.009621,158 +0.009726,158 +0.009628,158 +0.009543,158 +0.009582,158 +0.010835,158 +0.013186,158 +0.009713,158 +0.009620,158 +0.009625,158 +0.009666,158 +0.009545,158 +0.009610,158 +0.009585,158 +0.009679,158 +0.010873,158 +0.013045,158 +0.009651,158 +0.009623,158 +0.009568,158 +0.009643,158 +0.009545,158 +0.009583,158 +0.009542,158 +0.009586,158 +0.010793,158 +0.013279,158 +0.009682,158 +0.009608,158 +0.009561,158 +0.009687,158 +0.009548,158 +0.009582,158 +0.009545,158 +0.009612,158 +0.010642,158 +0.013318,158 +0.009618,158 +0.009640,158 +0.009560,158 +0.009687,158 +0.009544,158 +0.009582,158 +0.009546,158 +0.009582,158 +0.009657,158 +0.014768,158 +0.009654,158 +0.009671,158 +0.009584,158 +0.009713,158 +0.009808,158 +0.009650,158 +0.009599,158 +0.009547,158 +0.009723,158 +0.009852,158 +0.009656,158 +0.009584,158 +0.009627,158 +0.009685,158 +0.009587,158 +0.009604,158 +0.009544,158 +0.009607,158 +0.009562,158 +0.009986,158 +0.009774,158 +0.009682,158 +0.011187,158 +0.009877,158 +0.009714,158 +0.009629,158 +0.009544,158 +0.009622,158 +0.009546,158 +0.009776,158 +0.009973,158 +0.009588,158 +0.010187,158 +0.009548,158 +0.009694,158 +0.009594,158 +0.009582,158 +0.009623,158 +0.009543,158 +0.009730,158 +0.009879,158 +0.009693,158 +0.009872,158 +0.009792,158 +0.009710,158 +0.009545,158 +0.009621,158 +0.009547,158 +0.009622,158 +0.009627,158 +0.010054,158 +0.010054,160 +0.009929,160 +0.010009,160 +0.009988,160 +0.010043,160 +0.010026,160 +0.009913,160 +0.009992,160 +0.009988,160 +0.010275,160 +0.010065,160 +0.009904,160 +0.010016,160 +0.009985,160 +0.009989,160 +0.009988,160 +0.009997,160 +0.010034,160 +0.010070,160 +0.010303,160 +0.010059,160 +0.009946,160 +0.010008,160 +0.009986,160 +0.009979,160 +0.009984,160 +0.009905,160 +0.009988,160 +0.009993,160 +0.010278,160 +0.010234,160 +0.009945,160 +0.010079,160 +0.009987,160 +0.009990,160 +0.012101,160 +0.010456,160 +0.009979,160 +0.010182,160 +0.010254,160 +0.010008,160 +0.009989,160 +0.009967,160 +0.010090,160 +0.009973,160 +0.010025,160 +0.009985,160 +0.009903,160 +0.010008,160 +0.010262,160 +0.010137,160 +0.010036,160 +0.009937,160 +0.010067,160 +0.010004,160 +0.010024,160 +0.009989,160 +0.009907,160 +0.009991,160 +0.010275,160 +0.010042,160 +0.009989,160 +0.009911,160 +0.010050,160 +0.009907,160 +0.010129,160 +0.010013,160 +0.009909,160 +0.009995,160 +0.010495,160 +0.010028,160 +0.010029,160 +0.009951,160 +0.010063,160 +0.010007,160 +0.009944,160 +0.009986,160 +0.009904,160 +0.010006,160 +0.010279,160 +0.010067,160 +0.010054,160 +0.009934,160 +0.010086,160 +0.009944,160 +0.009944,160 +0.009987,160 +0.009904,160 +0.009985,160 +0.010425,160 +0.009962,160 +0.009988,160 +0.009906,160 +0.010069,160 +0.009944,160 +0.009949,160 +0.009985,160 +0.009904,160 +0.010007,160 +0.010422,160 +0.010407,162 +0.010363,162 +0.010280,162 +0.010447,162 +0.010445,162 +0.010283,162 +0.010360,162 +0.010319,162 +0.010479,162 +0.010749,162 +0.010282,162 +0.010404,162 +0.010358,162 +0.010366,162 +0.010361,162 +0.010360,162 +0.010364,162 +0.010358,162 +0.010496,162 +0.010656,162 +0.010365,162 +0.010284,162 +0.010423,162 +0.010320,162 +0.010323,162 +0.010378,162 +0.010281,162 +0.010360,162 +0.010843,162 +0.010325,162 +0.010400,162 +0.010281,162 +0.010441,162 +0.010361,162 +0.010284,162 +0.010378,162 +0.010372,162 +0.010364,162 +0.010847,162 +0.010389,162 +0.010406,162 +0.010388,162 +0.010346,162 +0.010407,162 +0.010365,162 +0.010317,162 +0.010364,162 +0.010498,162 +0.010612,162 +0.010410,162 +0.010280,162 +0.010466,162 +0.010487,162 +0.010487,162 +0.010363,162 +0.010280,162 +0.010450,162 +0.010746,162 +0.010334,162 +0.010491,162 +0.010358,162 +0.010363,162 +0.010387,162 +0.010300,162 +0.010363,162 +0.010362,162 +0.010541,162 +0.010729,162 +0.010459,162 +0.010328,162 +0.010466,162 +0.010283,162 +0.010364,162 +0.010362,162 +0.010282,162 +0.010362,162 +0.010745,162 +0.010464,162 +0.010403,162 +0.010316,162 +0.010428,162 +0.010365,162 +0.010284,162 +0.010382,162 +0.010364,162 +0.010383,162 +0.010771,162 +0.010351,162 +0.010378,162 +0.010404,162 +0.010364,162 +0.011207,162 +0.010435,162 +0.010319,162 +0.010426,162 +0.010802,162 +0.010531,162 +0.010431,162 +0.010284,162 +0.010522,162 +0.010784,164 +0.010711,164 +0.010748,164 +0.010809,164 +0.010786,164 +0.011112,164 +0.010752,164 +0.010789,164 +0.010813,164 +0.010773,164 +0.010780,164 +0.011292,164 +0.011594,164 +0.011430,164 +0.011138,164 +0.010801,164 +0.010771,164 +0.010786,164 +0.010763,164 +0.010774,164 +0.010746,164 +0.010671,164 +0.010771,164 +0.011398,164 +0.010791,164 +0.010771,164 +0.010810,164 +0.010777,164 +0.010796,164 +0.010822,164 +0.011003,164 +0.010760,164 +0.010943,164 +0.011030,164 +0.010857,164 +0.010801,164 +0.010826,164 +0.010754,164 +0.010670,164 +0.010748,164 +0.010750,164 +0.010836,164 +0.011151,164 +0.010840,164 +0.010732,164 +0.010915,164 +0.010751,164 +0.010709,164 +0.010746,164 +0.010754,164 +0.010714,164 +0.011173,164 +0.010763,164 +0.010850,164 +0.010793,164 +0.010779,164 +0.010792,164 +0.010793,164 +0.010669,164 +0.010780,164 +0.011376,164 +0.010779,164 +0.010790,164 +0.010751,164 +0.010751,164 +0.010753,164 +0.010751,164 +0.010669,164 +0.010745,164 +0.010846,164 +0.012370,164 +0.010926,164 +0.011042,164 +0.010808,164 +0.010782,164 +0.010733,164 +0.010707,164 +0.010750,164 +0.010891,164 +0.011047,164 +0.011147,164 +0.010669,164 +0.010834,164 +0.010770,164 +0.010670,164 +0.010747,164 +0.010746,164 +0.010755,164 +0.011173,164 +0.011209,164 +0.010669,164 +0.011009,164 +0.010817,164 +0.010712,164 +0.010766,164 +0.010734,164 +0.010828,164 +0.011192,164 +0.010784,164 +0.010732,164 +0.011191,166 +0.011185,166 +0.011145,166 +0.011888,166 +0.012399,166 +0.012089,166 +0.011909,166 +0.011759,166 +0.011249,166 +0.011758,166 +0.011934,166 +0.011491,166 +0.011819,166 +0.011546,166 +0.011587,166 +0.011658,166 +0.012568,166 +0.011181,166 +0.011217,166 +0.011118,166 +0.011121,166 +0.011059,166 +0.011164,166 +0.011387,166 +0.011276,166 +0.011226,166 +0.011204,166 +0.011228,166 +0.011131,166 +0.011142,166 +0.011100,166 +0.011137,166 +0.011355,166 +0.011357,166 +0.011142,166 +0.011250,166 +0.011281,166 +0.011159,166 +0.011142,166 +0.011099,166 +0.011100,166 +0.011505,166 +0.011219,166 +0.011137,166 +0.011220,166 +0.011239,166 +0.011102,166 +0.011155,166 +0.011142,166 +0.011062,166 +0.011673,166 +0.011270,166 +0.011083,166 +0.011158,166 +0.011245,166 +0.011062,166 +0.011140,166 +0.011156,166 +0.011061,166 +0.011578,166 +0.011292,166 +0.011134,166 +0.011167,166 +0.011235,166 +0.011059,166 +0.011144,166 +0.011139,166 +0.011159,166 +0.011497,166 +0.011209,166 +0.011156,166 +0.011138,166 +0.011201,166 +0.011061,166 +0.011138,166 +0.011169,166 +0.011059,166 +0.011572,166 +0.011204,166 +0.011161,166 +0.011254,166 +0.011203,166 +0.011098,166 +0.011138,166 +0.011144,166 +0.011083,166 +0.011473,166 +0.011272,166 +0.011144,166 +0.011296,166 +0.011161,166 +0.011103,166 +0.011360,166 +0.011138,166 +0.011134,166 +0.011465,166 +0.011145,166 +0.011102,166 +0.011164,166 +0.011182,166 +0.011523,168 +0.011490,168 +0.011535,168 +0.011532,168 +0.011907,168 +0.011672,168 +0.011655,168 +0.011612,168 +0.011542,168 +0.011551,168 +0.011531,168 +0.011524,168 +0.011651,168 +0.011834,168 +0.011450,168 +0.011632,168 +0.011648,168 +0.011473,168 +0.011532,168 +0.011528,168 +0.011452,168 +0.011698,168 +0.011850,168 +0.011598,168 +0.011687,168 +0.011610,168 +0.011528,168 +0.011452,168 +0.011555,168 +0.011512,168 +0.011784,168 +0.011572,168 +0.011512,168 +0.011580,168 +0.011532,168 +0.011554,168 +0.011490,168 +0.011495,168 +0.011668,168 +0.011871,168 +0.011485,168 +0.011530,168 +0.011591,168 +0.011449,168 +0.011529,168 +0.011532,168 +0.011516,168 +0.011658,168 +0.011794,168 +0.011571,168 +0.011533,168 +0.011528,168 +0.011533,168 +0.011450,168 +0.011528,168 +0.011552,168 +0.011742,168 +0.011668,168 +0.011667,168 +0.011600,168 +0.011559,168 +0.011602,168 +0.011623,168 +0.011590,168 +0.011655,168 +0.012261,168 +0.011516,168 +0.011774,168 +0.011784,168 +0.011491,168 +0.011541,168 +0.011557,168 +0.011546,168 +0.011727,168 +0.011933,168 +0.011591,168 +0.011511,168 +0.011567,168 +0.011619,168 +0.011493,168 +0.011491,168 +0.011567,168 +0.012135,168 +0.011495,168 +0.011536,168 +0.011575,168 +0.011452,168 +0.011510,168 +0.011513,168 +0.011452,168 +0.011755,168 +0.012195,168 +0.011600,168 +0.011611,168 +0.011585,168 +0.011533,168 +0.011453,168 +0.011526,168 +0.011563,168 +0.011778,168 +0.012368,170 +0.011991,170 +0.012023,170 +0.011874,170 +0.011995,170 +0.011975,170 +0.011873,170 +0.012019,170 +0.012605,170 +0.011974,170 +0.012064,170 +0.012011,170 +0.011956,170 +0.011913,170 +0.011915,170 +0.011957,170 +0.012581,170 +0.011992,170 +0.011972,170 +0.012224,170 +0.011968,170 +0.011880,170 +0.011963,170 +0.011961,170 +0.012367,170 +0.012179,170 +0.012012,170 +0.012050,170 +0.011872,170 +0.011946,170 +0.011983,170 +0.011910,170 +0.012138,170 +0.012355,170 +0.012069,170 +0.011953,170 +0.011972,170 +0.011951,170 +0.011969,170 +0.011873,170 +0.011954,170 +0.012545,170 +0.011952,170 +0.012034,170 +0.011974,170 +0.011952,170 +0.011891,170 +0.011952,170 +0.011949,170 +0.012361,170 +0.012149,170 +0.012212,170 +0.012514,170 +0.012752,170 +0.012681,170 +0.012633,170 +0.012701,170 +0.012961,170 +0.012976,170 +0.013305,170 +0.013122,170 +0.013128,170 +0.013197,170 +0.013457,170 +0.012856,170 +0.014079,170 +0.012390,170 +0.012251,170 +0.012068,170 +0.011998,170 +0.012010,170 +0.011881,170 +0.012047,170 +0.013430,170 +0.012022,170 +0.012064,170 +0.012012,170 +0.011969,170 +0.011879,170 +0.011970,170 +0.012012,170 +0.013293,170 +0.012159,170 +0.012003,170 +0.012090,170 +0.011921,170 +0.011919,170 +0.011975,170 +0.011976,170 +0.013089,170 +0.013243,170 +0.012087,170 +0.012064,170 +0.012142,170 +0.011978,170 +0.012004,170 +0.011963,170 +0.011878,170 +0.013426,170 +0.011999,170 +0.012542,172 +0.012343,172 +0.012399,172 +0.012399,172 +0.012399,172 +0.012335,172 +0.013855,172 +0.012485,172 +0.012488,172 +0.012322,172 +0.012376,172 +0.012417,172 +0.012376,172 +0.012359,172 +0.013822,172 +0.012456,172 +0.012528,172 +0.012299,172 +0.012400,172 +0.012376,172 +0.012380,172 +0.012348,172 +0.013919,172 +0.012396,172 +0.012518,172 +0.012300,172 +0.012436,172 +0.012373,172 +0.012377,172 +0.012768,172 +0.013555,172 +0.012493,172 +0.012480,172 +0.012365,172 +0.012372,172 +0.012416,172 +0.012470,172 +0.012911,172 +0.013230,172 +0.012400,172 +0.012480,172 +0.012401,172 +0.012358,172 +0.012374,172 +0.012399,172 +0.012936,172 +0.013161,172 +0.012432,172 +0.012491,172 +0.012339,172 +0.012351,172 +0.012387,172 +0.012380,172 +0.012970,172 +0.013251,172 +0.012388,172 +0.012437,172 +0.012403,172 +0.012297,172 +0.012398,172 +0.012422,172 +0.012949,172 +0.013123,172 +0.012379,172 +0.012475,172 +0.012376,172 +0.012297,172 +0.012430,172 +0.012380,172 +0.013706,172 +0.013553,172 +0.012487,172 +0.012477,172 +0.012439,172 +0.012298,172 +0.012377,172 +0.012376,172 +0.013300,172 +0.012890,172 +0.012415,172 +0.012543,172 +0.012378,172 +0.012305,172 +0.012375,172 +0.012374,172 +0.013355,172 +0.012861,172 +0.012377,172 +0.012459,172 +0.012386,172 +0.012342,172 +0.012377,172 +0.012387,172 +0.013479,172 +0.012737,172 +0.012374,172 +0.012439,172 +0.012419,172 +0.012339,172 +0.012335,172 +0.012924,174 +0.014114,174 +0.013398,174 +0.012870,174 +0.012829,174 +0.012810,174 +0.012898,174 +0.012773,174 +0.012770,174 +0.014461,174 +0.012943,174 +0.012952,174 +0.012893,174 +0.012770,174 +0.012819,174 +0.012815,174 +0.013198,174 +0.013806,174 +0.012834,174 +0.012930,174 +0.012905,174 +0.013421,174 +0.012810,174 +0.012833,174 +0.014129,174 +0.013014,174 +0.012818,174 +0.012832,174 +0.012812,174 +0.012827,174 +0.012774,174 +0.012827,174 +0.014350,174 +0.012876,174 +0.012939,174 +0.012771,174 +0.012778,174 +0.012826,174 +0.012832,174 +0.012847,174 +0.014217,174 +0.012893,174 +0.012886,174 +0.012816,174 +0.012773,174 +0.012781,174 +0.012813,174 +0.014284,174 +0.013887,174 +0.013176,174 +0.012836,174 +0.012812,174 +0.012850,174 +0.012854,174 +0.012751,174 +0.014305,174 +0.012886,174 +0.012893,174 +0.012796,174 +0.012766,174 +0.012814,174 +0.012808,174 +0.012820,174 +0.014250,174 +0.012876,174 +0.012902,174 +0.012811,174 +0.012777,174 +0.012769,174 +0.012870,174 +0.013717,174 +0.013426,174 +0.012750,174 +0.012822,174 +0.012850,174 +0.012808,174 +0.012984,174 +0.012814,174 +0.013902,174 +0.012985,174 +0.012943,174 +0.012767,174 +0.012812,174 +0.012810,174 +0.012809,174 +0.012775,174 +0.014276,174 +0.012971,174 +0.012871,174 +0.012878,174 +0.012763,174 +0.012850,174 +0.012817,174 +0.012812,174 +0.014245,174 +0.012864,174 +0.013058,174 +0.012952,174 +0.012890,174 +0.012775,174 +0.013229,176 +0.014257,176 +0.013616,176 +0.013359,176 +0.013247,176 +0.013200,176 +0.013245,176 +0.013236,176 +0.013294,176 +0.014650,176 +0.013181,176 +0.013312,176 +0.013240,176 +0.013241,176 +0.013264,176 +0.013196,176 +0.014077,176 +0.013740,176 +0.013320,176 +0.013239,176 +0.013155,176 +0.013252,176 +0.013269,176 +0.013234,176 +0.015676,176 +0.013342,176 +0.013354,176 +0.013246,176 +0.013241,176 +0.013235,176 +0.013202,176 +0.014308,176 +0.013464,176 +0.013379,176 +0.013273,176 +0.013302,176 +0.013184,176 +0.013255,176 +0.013260,176 +0.014662,176 +0.013348,176 +0.013158,176 +0.013240,176 +0.013244,176 +0.013241,176 +0.013302,176 +0.014299,176 +0.013475,176 +0.013319,176 +0.013235,176 +0.013244,176 +0.013223,176 +0.013300,176 +0.013448,176 +0.014596,176 +0.013353,176 +0.013236,176 +0.013191,176 +0.013241,176 +0.013890,176 +0.013450,176 +0.014640,176 +0.013330,176 +0.013404,176 +0.013243,176 +0.013239,176 +0.013238,176 +0.013216,176 +0.013558,176 +0.014477,176 +0.013404,176 +0.013368,176 +0.013281,176 +0.013190,176 +0.013259,176 +0.013234,176 +0.014619,176 +0.013257,176 +0.013273,176 +0.013255,176 +0.013242,176 +0.013230,176 +0.013262,176 +0.013705,176 +0.014162,176 +0.013396,176 +0.013258,176 +0.013253,176 +0.013167,176 +0.013313,176 +0.013258,176 +0.014743,176 +0.013368,176 +0.013229,176 +0.013240,176 +0.013265,176 +0.013312,176 +0.013221,176 +0.014368,176 +0.014610,176 +0.014034,178 +0.013804,178 +0.013749,178 +0.013730,178 +0.013697,178 +0.013688,178 +0.015298,178 +0.013925,178 +0.013806,178 +0.013768,178 +0.013693,178 +0.013685,178 +0.013730,178 +0.015227,178 +0.013905,178 +0.013775,178 +0.013737,178 +0.013648,178 +0.013803,178 +0.013741,178 +0.015209,178 +0.013851,178 +0.013757,178 +0.013646,178 +0.013730,178 +0.013816,178 +0.013727,178 +0.014983,178 +0.013994,178 +0.013671,178 +0.013728,178 +0.013737,178 +0.013749,178 +0.013727,178 +0.014320,178 +0.014624,178 +0.013803,178 +0.013733,178 +0.013728,178 +0.013793,178 +0.013732,178 +0.013658,178 +0.015135,178 +0.013893,178 +0.013753,178 +0.013730,178 +0.013949,178 +0.014305,178 +0.013707,178 +0.015145,178 +0.013894,178 +0.013792,178 +0.013752,178 +0.013750,178 +0.013733,178 +0.013735,178 +0.015082,178 +0.013848,178 +0.013724,178 +0.013745,178 +0.013650,178 +0.013775,178 +0.013733,178 +0.014791,178 +0.014185,178 +0.013790,178 +0.013675,178 +0.013743,178 +0.013723,178 +0.013748,178 +0.013917,178 +0.016409,178 +0.013868,178 +0.013665,178 +0.013760,178 +0.013723,178 +0.013813,178 +0.013727,178 +0.015177,178 +0.013749,178 +0.013743,178 +0.013724,178 +0.013724,178 +0.013728,178 +0.013685,178 +0.015080,178 +0.013865,178 +0.013746,178 +0.013721,178 +0.013739,178 +0.013731,178 +0.013676,178 +0.015119,178 +0.013960,178 +0.013744,178 +0.013728,178 +0.013740,178 +0.013644,178 +0.013724,178 +0.014658,178 +0.014728,180 +0.014195,180 +0.014154,180 +0.014114,180 +0.014113,180 +0.014158,180 +0.015102,180 +0.014781,180 +0.014233,180 +0.014154,180 +0.014149,180 +0.014076,180 +0.014217,180 +0.014995,180 +0.014969,180 +0.014297,180 +0.014176,180 +0.014168,180 +0.014086,180 +0.014151,180 +0.014921,180 +0.014900,180 +0.014164,180 +0.014154,180 +0.014149,180 +0.014153,180 +0.014112,180 +0.014883,180 +0.014967,180 +0.014175,180 +0.014164,180 +0.014196,180 +0.014109,180 +0.014114,180 +0.014899,180 +0.015049,180 +0.014223,180 +0.014156,180 +0.014178,180 +0.014150,180 +0.014075,180 +0.015177,180 +0.015930,180 +0.014194,180 +0.014163,180 +0.014190,180 +0.014157,180 +0.014110,180 +0.014913,180 +0.014957,180 +0.014214,180 +0.014177,180 +0.014151,180 +0.014155,180 +0.014152,180 +0.014882,180 +0.014835,180 +0.014170,180 +0.014159,180 +0.014152,180 +0.014232,180 +0.014157,180 +0.015053,180 +0.014741,180 +0.014195,180 +0.014153,180 +0.014174,180 +0.014151,180 +0.014157,180 +0.015240,180 +0.014557,180 +0.014173,180 +0.014154,180 +0.014157,180 +0.014151,180 +0.014394,180 +0.015120,180 +0.014651,180 +0.014242,180 +0.014196,180 +0.014211,180 +0.014216,180 +0.014153,180 +0.015174,180 +0.014765,180 +0.014133,180 +0.014154,180 +0.014189,180 +0.014172,180 +0.014150,180 +0.015249,180 +0.014679,180 +0.014261,180 +0.014184,180 +0.014153,180 +0.014189,180 +0.014164,180 +0.015344,180 +0.014498,180 +0.014108,180 +0.014750,182 +0.014647,182 +0.014627,182 +0.014627,182 +0.016130,182 +0.014691,182 +0.014653,182 +0.014672,182 +0.014551,182 +0.014638,182 +0.014671,182 +0.017304,182 +0.014720,182 +0.014631,182 +0.014627,182 +0.014625,182 +0.014627,182 +0.015040,182 +0.015694,182 +0.014651,182 +0.014626,182 +0.014629,182 +0.014627,182 +0.014628,182 +0.015926,182 +0.014910,182 +0.014648,182 +0.014550,182 +0.014717,182 +0.014645,182 +0.014633,182 +0.016201,182 +0.014751,182 +0.014878,182 +0.014713,182 +0.014713,182 +0.014655,182 +0.014610,182 +0.016184,182 +0.014636,182 +0.014632,182 +0.014627,182 +0.014632,182 +0.014793,182 +0.015369,182 +0.015352,182 +0.014551,182 +0.014627,182 +0.014633,182 +0.014628,182 +0.014625,182 +0.016009,182 +0.014969,182 +0.014744,182 +0.014692,182 +0.014556,182 +0.014629,182 +0.014630,182 +0.016128,182 +0.015961,182 +0.014838,182 +0.014701,182 +0.014754,182 +0.014700,182 +0.014594,182 +0.016209,182 +0.014634,182 +0.014625,182 +0.014630,182 +0.014634,182 +0.014626,182 +0.015189,182 +0.015702,182 +0.014674,182 +0.014607,182 +0.014588,182 +0.014629,182 +0.014722,182 +0.016583,182 +0.015458,182 +0.014648,182 +0.014629,182 +0.014654,182 +0.014622,182 +0.014551,182 +0.016172,182 +0.014692,182 +0.014670,182 +0.014647,182 +0.014626,182 +0.014631,182 +0.014628,182 +0.016231,182 +0.014702,182 +0.014695,182 +0.014647,182 +0.014689,182 +0.014630,182 +0.015389,182 +0.015426,182 +0.015261,184 +0.015180,184 +0.015138,184 +0.015059,184 +0.015058,184 +0.016465,184 +0.015138,184 +0.015098,184 +0.015097,184 +0.015119,184 +0.015141,184 +0.015094,184 +0.016480,184 +0.015096,184 +0.015165,184 +0.015160,184 +0.015066,184 +0.015055,184 +0.016511,184 +0.015245,184 +0.015117,184 +0.015112,184 +0.015097,184 +0.015093,184 +0.015099,184 +0.016688,184 +0.015242,184 +0.015145,184 +0.015171,184 +0.015055,184 +0.015094,184 +0.015913,184 +0.016243,184 +0.016548,184 +0.016859,184 +0.016615,184 +0.016799,184 +0.017313,184 +0.016958,184 +0.016048,184 +0.016098,184 +0.015850,184 +0.015472,184 +0.015232,184 +0.015827,184 +0.015177,184 +0.015074,184 +0.016331,184 +0.017115,184 +0.016935,184 +0.016747,184 +0.016620,184 +0.016549,184 +0.016444,184 +0.016073,184 +0.016638,184 +0.016661,184 +0.017202,184 +0.016771,184 +0.016886,184 +0.016380,184 +0.016665,184 +0.016209,184 +0.016098,184 +0.015419,184 +0.015109,184 +0.015642,184 +0.017083,184 +0.016561,184 +0.015799,184 +0.015359,184 +0.015220,184 +0.015253,184 +0.016316,184 +0.017189,184 +0.016395,184 +0.016625,184 +0.016639,184 +0.016613,184 +0.016478,184 +0.016285,184 +0.016545,184 +0.016739,184 +0.017193,184 +0.018658,184 +0.016758,184 +0.016925,184 +0.016327,184 +0.015753,184 +0.016161,184 +0.017930,184 +0.016808,184 +0.016986,184 +0.016812,184 +0.018616,184 +0.017193,184 +0.017303,184 +0.019176,184 +0.018340,184 +0.019973,184 +0.020155,186 +0.020461,186 +0.018463,186 +0.019075,186 +0.024476,186 +0.022738,186 +0.020556,186 +0.019866,186 +0.017708,186 +0.018356,186 +0.017593,186 +0.018223,186 +0.019948,186 +0.020820,186 +0.022140,186 +0.017918,186 +0.019225,186 +0.019098,186 +0.019009,186 +0.022953,186 +0.017083,186 +0.018396,186 +0.017968,186 +0.017585,186 +0.016829,186 +0.016879,186 +0.016319,186 +0.016866,186 +0.018337,186 +0.018310,186 +0.023732,186 +0.018696,186 +0.019207,186 +0.019455,186 +0.018706,186 +0.017146,186 +0.016986,186 +0.017028,186 +0.019344,186 +0.019544,186 +0.017739,186 +0.017609,186 +0.017587,186 +0.017680,186 +0.017810,186 +0.017584,186 +0.017399,186 +0.018720,186 +0.017195,186 +0.017573,186 +0.017484,186 +0.018048,186 +0.021768,186 +0.016621,186 +0.017351,186 +0.016885,186 +0.016948,186 +0.017927,186 +0.017122,186 +0.017224,186 +0.017469,186 +0.017162,186 +0.017409,186 +0.017101,186 +0.017373,186 +0.016490,186 +0.017339,186 +0.017151,186 +0.017822,186 +0.017549,186 +0.017109,186 +0.016668,186 +0.016529,186 +0.016398,186 +0.017232,186 +0.019138,186 +0.017349,186 +0.016798,186 +0.016739,186 +0.017289,186 +0.016649,186 +0.017188,186 +0.016960,186 +0.016722,186 +0.017282,186 +0.016667,186 +0.016988,186 +0.016480,186 +0.016038,186 +0.015908,186 +0.015761,186 +0.015867,186 +0.019385,186 +0.017620,186 +0.018975,186 +0.017683,186 +0.016649,186 +0.017682,186 +0.017443,186 +0.017369,186 +0.017585,188 +0.017031,188 +0.016650,188 +0.016580,188 +0.016551,188 +0.016911,188 +0.017369,188 +0.017895,188 +0.020415,188 +0.021828,188 +0.020295,188 +0.020074,188 +0.020325,188 +0.018933,188 +0.016831,188 +0.016935,188 +0.016587,188 +0.016250,188 +0.016128,188 +0.016203,188 +0.016281,188 +0.016457,188 +0.016501,188 +0.016292,188 +0.016134,188 +0.017463,188 +0.016336,188 +0.016530,188 +0.016388,188 +0.016234,188 +0.016892,188 +0.021228,188 +0.018101,188 +0.018236,188 +0.019020,188 +0.022148,188 +0.028301,188 +0.030740,188 +0.028133,188 +0.030402,188 +0.030221,188 +0.021442,188 +0.026085,188 +0.021689,188 +0.021782,188 +0.019309,188 +0.018858,188 +0.016389,188 +0.016632,188 +0.019297,188 +0.017420,188 +0.018507,188 +0.016734,188 +0.017010,188 +0.017164,188 +0.017296,188 +0.019475,188 +0.018051,188 +0.016476,188 +0.018042,188 +0.016878,188 +0.016812,188 +0.018170,188 +0.017259,188 +0.016538,188 +0.016762,188 +0.017282,188 +0.017397,188 +0.022841,188 +0.018055,188 +0.016597,188 +0.016511,188 +0.016567,188 +0.016492,188 +0.018150,188 +0.017119,188 +0.016564,188 +0.017333,188 +0.016671,188 +0.016647,188 +0.016544,188 +0.016785,188 +0.016593,188 +0.017043,188 +0.016483,188 +0.016506,188 +0.020677,188 +0.016481,188 +0.016269,188 +0.016441,188 +0.016390,188 +0.016472,188 +0.020768,188 +0.016408,188 +0.016359,188 +0.016669,188 +0.016571,188 +0.017968,188 +0.019100,188 +0.016422,188 +0.019606,190 +0.018196,190 +0.017098,190 +0.017581,190 +0.017235,190 +0.016891,190 +0.017145,190 +0.017270,190 +0.017056,190 +0.021680,190 +0.020330,190 +0.019509,190 +0.017363,190 +0.017317,190 +0.017935,190 +0.017530,190 +0.017335,190 +0.018212,190 +0.017494,190 +0.017188,190 +0.017696,190 +0.016806,190 +0.016855,190 +0.017064,190 +0.017031,190 +0.017257,190 +0.017479,190 +0.016840,190 +0.016886,190 +0.017175,190 +0.016892,190 +0.016751,190 +0.017574,190 +0.016842,190 +0.016957,190 +0.017035,190 +0.016909,190 +0.017014,190 +0.017247,190 +0.016834,190 +0.016835,190 +0.017247,190 +0.017006,190 +0.017128,190 +0.018882,190 +0.018347,190 +0.017874,190 +0.017716,190 +0.016836,190 +0.017167,190 +0.016848,190 +0.016718,190 +0.016687,190 +0.016791,190 +0.016808,190 +0.017179,190 +0.016809,190 +0.016723,190 +0.016876,190 +0.017000,190 +0.016726,190 +0.017112,190 +0.016764,190 +0.016678,190 +0.016707,190 +0.016774,190 +0.016693,190 +0.017114,190 +0.016812,190 +0.016817,190 +0.016675,190 +0.016843,190 +0.016779,190 +0.017179,190 +0.016748,190 +0.016754,190 +0.017375,190 +0.017169,190 +0.018405,190 +0.018924,190 +0.018512,190 +0.018088,190 +0.017260,190 +0.017003,190 +0.016994,190 +0.018513,190 +0.017271,190 +0.017029,190 +0.017007,190 +0.016876,190 +0.016957,190 +0.019317,190 +0.019256,190 +0.018045,190 +0.017711,190 +0.017179,190 +0.021522,190 +0.019157,190 +0.022006,190 +0.022502,190 +0.024940,192 +0.020805,192 +0.018494,192 +0.018604,192 +0.018872,192 +0.020463,192 +0.019122,192 +0.018341,192 +0.018278,192 +0.018342,192 +0.021521,192 +0.021664,192 +0.023983,192 +0.022446,192 +0.020787,192 +0.020216,192 +0.021117,192 +0.021752,192 +0.020148,192 +0.020033,192 +0.019823,192 +0.019258,192 +0.018466,192 +0.018968,192 +0.019993,192 +0.024043,192 +0.020247,192 +0.020460,192 +0.022285,192 +0.032353,192 +0.020860,192 +0.019693,192 +0.019178,192 +0.019545,192 +0.019595,192 +0.022936,192 +0.019735,192 +0.020366,192 +0.020715,192 +0.020817,192 +0.019938,192 +0.019200,192 +0.019328,192 +0.019081,192 +0.019019,192 +0.024591,192 +0.020063,192 +0.023090,192 +0.020689,192 +0.025426,192 +0.024649,192 +0.022197,192 +0.023562,192 +0.025272,192 +0.021982,192 +0.019063,192 +0.020815,192 +0.018930,192 +0.019309,192 +0.021683,192 +0.018894,192 +0.018693,192 +0.018402,192 +0.018643,192 +0.018803,192 +0.018865,192 +0.018530,192 +0.018566,192 +0.018457,192 +0.019478,192 +0.019840,192 +0.019749,192 +0.021034,192 +0.020868,192 +0.023697,192 +0.019251,192 +0.019106,192 +0.021241,192 +0.019936,192 +0.020166,192 +0.019412,192 +0.019460,192 +0.019504,192 +0.019592,192 +0.019792,192 +0.019798,192 +0.019799,192 +0.019798,192 +0.021687,192 +0.020310,192 +0.020672,192 +0.020906,192 +0.020747,192 +0.020806,192 +0.020942,192 +0.022014,192 +0.020865,192 +0.020481,192 +0.021178,192 +0.020042,192 +0.019345,194 +0.020383,194 +0.034984,194 +0.031715,194 +0.019095,194 +0.019188,194 +0.019084,194 +0.019325,194 +0.018859,194 +0.019776,194 +0.018427,194 +0.018625,194 +0.018061,194 +0.018269,194 +0.018030,194 +0.018468,194 +0.017982,194 +0.018246,194 +0.018547,194 +0.019330,194 +0.019578,194 +0.019471,194 +0.020268,194 +0.019597,194 +0.019489,194 +0.019100,194 +0.019014,194 +0.019099,194 +0.018724,194 +0.018987,194 +0.018608,194 +0.018548,194 +0.018599,194 +0.018297,194 +0.018905,194 +0.018515,194 +0.018691,194 +0.018296,194 +0.018482,194 +0.018275,194 +0.018193,194 +0.017980,194 +0.018071,194 +0.018040,194 +0.017936,194 +0.019145,194 +0.019417,194 +0.019837,194 +0.020615,194 +0.020887,194 +0.019485,194 +0.018711,194 +0.018365,194 +0.019080,194 +0.018409,194 +0.018458,194 +0.018273,194 +0.017884,194 +0.017895,194 +0.018663,194 +0.019215,194 +0.019486,194 +0.019641,194 +0.019856,194 +0.019866,194 +0.019767,194 +0.020457,194 +0.019443,194 +0.019975,194 +0.019074,194 +0.018775,194 +0.019472,194 +0.020400,194 +0.020998,194 +0.020603,194 +0.020092,194 +0.019683,194 +0.020105,194 +0.018864,194 +0.018451,194 +0.017971,194 +0.018226,194 +0.017998,194 +0.017893,194 +0.018022,194 +0.017815,194 +0.017903,194 +0.018246,194 +0.019260,194 +0.019605,194 +0.019167,194 +0.019089,194 +0.019311,194 +0.019635,194 +0.019365,194 +0.019919,194 +0.020112,194 +0.019885,194 +0.019468,194 +0.019149,194 +0.020764,196 +0.021087,196 +0.020729,196 +0.023838,196 +0.024036,196 +0.019970,196 +0.019897,196 +0.020047,196 +0.019956,196 +0.019867,196 +0.019909,196 +0.021403,196 +0.021023,196 +0.021649,196 +0.020910,196 +0.020714,196 +0.021551,196 +0.021570,196 +0.021105,196 +0.019902,196 +0.019433,196 +0.021446,196 +0.021803,196 +0.020211,196 +0.019600,196 +0.018518,196 +0.019530,196 +0.020621,196 +0.021180,196 +0.020092,196 +0.019216,196 +0.019500,196 +0.021280,196 +0.021502,196 +0.020392,196 +0.020218,196 +0.019788,196 +0.019832,196 +0.019178,196 +0.019315,196 +0.018625,196 +0.018539,196 +0.019664,196 +0.019096,196 +0.019631,196 +0.019488,196 +0.019109,196 +0.019395,196 +0.019174,196 +0.019152,196 +0.018386,196 +0.018477,196 +0.018987,196 +0.018399,196 +0.019330,196 +0.019047,196 +0.018537,196 +0.018668,196 +0.018769,196 +0.018420,196 +0.018592,196 +0.018385,196 +0.018368,196 +0.018968,196 +0.018484,196 +0.018433,196 +0.018916,196 +0.018328,196 +0.018830,196 +0.018451,196 +0.018356,196 +0.018438,196 +0.018313,196 +0.018338,196 +0.018935,196 +0.018504,196 +0.018554,196 +0.018682,196 +0.018337,196 +0.018847,196 +0.019009,196 +0.018384,196 +0.019396,196 +0.018847,196 +0.018546,196 +0.018771,196 +0.018494,196 +0.018434,196 +0.018456,196 +0.018361,196 +0.019148,196 +0.021199,196 +0.019170,196 +0.018582,196 +0.018312,196 +0.018855,196 +0.023224,196 +0.019320,196 +0.023507,196 +0.021675,196 +0.020363,198 +0.021072,198 +0.020478,198 +0.019864,198 +0.022227,198 +0.020976,198 +0.019821,198 +0.019979,198 +0.019839,198 +0.021022,198 +0.020894,198 +0.019946,198 +0.020093,198 +0.024282,198 +0.022166,198 +0.020987,198 +0.021312,198 +0.020292,198 +0.019523,198 +0.019733,198 +0.019085,198 +0.019108,198 +0.018861,198 +0.020641,198 +0.022971,198 +0.023156,198 +0.022558,198 +0.023455,198 +0.021469,198 +0.020916,198 +0.021697,198 +0.021540,198 +0.021498,198 +0.022028,198 +0.023256,198 +0.021457,198 +0.021465,198 +0.020630,198 +0.021839,198 +0.020525,198 +0.020847,198 +0.020563,198 +0.020045,198 +0.020640,198 +0.020792,198 +0.020395,198 +0.019915,198 +0.019510,198 +0.019546,198 +0.018955,198 +0.019296,198 +0.018956,198 +0.018959,198 +0.019419,198 +0.021557,198 +0.024834,198 +0.020640,198 +0.019464,198 +0.019883,198 +0.020924,198 +0.020797,198 +0.019979,198 +0.020777,198 +0.020949,198 +0.020096,198 +0.020566,198 +0.020725,198 +0.020589,198 +0.020503,198 +0.021146,198 +0.020854,198 +0.020709,198 +0.020673,198 +0.021265,198 +0.020957,198 +0.020087,198 +0.020293,198 +0.020254,198 +0.020485,198 +0.020327,198 +0.019938,198 +0.019418,198 +0.020064,198 +0.019959,198 +0.020235,198 +0.019863,198 +0.019784,198 +0.019847,198 +0.019759,198 +0.019440,198 +0.019175,198 +0.019037,198 +0.019055,198 +0.019500,198 +0.018947,198 +0.018935,198 +0.018942,198 +0.019043,198 +0.019386,198 +0.018894,198 +0.019847,200 +0.019664,200 +0.019531,200 +0.019984,200 +0.019496,200 +0.019535,200 +0.019523,200 +0.019396,200 +0.019900,200 +0.019467,200 +0.019590,200 +0.019588,200 +0.019606,200 +0.019987,200 +0.020046,200 +0.019704,200 +0.019461,200 +0.020299,200 +0.020112,200 +0.019736,200 +0.019486,200 +0.019468,200 +0.019677,200 +0.019764,200 +0.020011,200 +0.019543,200 +0.019450,200 +0.019493,200 +0.019740,200 +0.019989,200 +0.019563,200 +0.019551,200 +0.019500,200 +0.019565,200 +0.020163,200 +0.019851,200 +0.019784,200 +0.019433,200 +0.019716,200 +0.020197,200 +0.019575,200 +0.019520,200 +0.019471,200 +0.019425,200 +0.020289,200 +0.019476,200 +0.019521,200 +0.019553,200 +0.019613,200 +0.020180,200 +0.019593,200 +0.019575,200 +0.019641,200 +0.019589,200 +0.020003,200 +0.019793,200 +0.019558,200 +0.019446,200 +0.019542,200 +0.019870,200 +0.019855,200 +0.020083,200 +0.019634,200 +0.019726,200 +0.020650,200 +0.019880,200 +0.019689,200 +0.020130,200 +0.019559,200 +0.019814,200 +0.019751,200 +0.019683,200 +0.019536,200 +0.019511,200 +0.019739,200 +0.019761,200 +0.019844,200 +0.019614,200 +0.019510,200 +0.019615,200 +0.019933,200 +0.019595,200 +0.019627,200 +0.019514,200 +0.019771,200 +0.019989,200 +0.019759,200 +0.019605,200 +0.019446,200 +0.019555,200 +0.020018,200 +0.019672,200 +0.019454,200 +0.019554,200 +0.019475,200 +0.019982,200 +0.019757,200 +0.019448,200 +0.019456,200 +0.019556,200 +0.020752,202 +0.020315,202 +0.020130,202 +0.020352,202 +0.020185,202 +0.020527,202 +0.020467,202 +0.020032,202 +0.020118,202 +0.020047,202 +0.020468,202 +0.020382,202 +0.020043,202 +0.020195,202 +0.020701,202 +0.022772,202 +0.020800,202 +0.020752,202 +0.020241,202 +0.020193,202 +0.020553,202 +0.021345,202 +0.021025,202 +0.021084,202 +0.021334,202 +0.021501,202 +0.021238,202 +0.020361,202 +0.020826,202 +0.020559,202 +0.021365,202 +0.021705,202 +0.021215,202 +0.020839,202 +0.021010,202 +0.024230,202 +0.020649,202 +0.020775,202 +0.020427,202 +0.021548,202 +0.020331,202 +0.020407,202 +0.020270,202 +0.020106,202 +0.020695,202 +0.022353,202 +0.020113,202 +0.020133,202 +0.020351,202 +0.020511,202 +0.020268,202 +0.020175,202 +0.020279,202 +0.020470,202 +0.020585,202 +0.020178,202 +0.020050,202 +0.021290,202 +0.020567,202 +0.020655,202 +0.020220,202 +0.020202,202 +0.020527,202 +0.022670,202 +0.021468,202 +0.020876,202 +0.022385,202 +0.032710,202 +0.039798,202 +0.023425,202 +0.021605,202 +0.023522,202 +0.021182,202 +0.020169,202 +0.020112,202 +0.020041,202 +0.021666,202 +0.020232,202 +0.020173,202 +0.021271,202 +0.022316,202 +0.022846,202 +0.021547,202 +0.023117,202 +0.022271,202 +0.023128,202 +0.024963,202 +0.022957,202 +0.025744,202 +0.023609,202 +0.021734,202 +0.021805,202 +0.021457,202 +0.021720,202 +0.024516,202 +0.022754,202 +0.023032,202 +0.021576,202 +0.022953,202 +0.020970,202 +0.021015,204 +0.020775,204 +0.020687,204 +0.025122,204 +0.020993,204 +0.020820,204 +0.020792,204 +0.020859,204 +0.030238,204 +0.025164,204 +0.026316,204 +0.023593,204 +0.025047,204 +0.023127,204 +0.022618,204 +0.022424,204 +0.022037,204 +0.022084,204 +0.021993,204 +0.022581,204 +0.021904,204 +0.021925,204 +0.022160,204 +0.022226,204 +0.022588,204 +0.028373,204 +0.022144,204 +0.023337,204 +0.023259,204 +0.022549,204 +0.023086,204 +0.023397,204 +0.022853,204 +0.028375,204 +0.023074,204 +0.022656,204 +0.022478,204 +0.022516,204 +0.025986,204 +0.020992,204 +0.020852,204 +0.020867,204 +0.025466,204 +0.021012,204 +0.020711,204 +0.020913,204 +0.021363,204 +0.027080,204 +0.022212,204 +0.021433,204 +0.020768,204 +0.025439,204 +0.021289,204 +0.023418,204 +0.026657,204 +0.022670,204 +0.023941,204 +0.023481,204 +0.025152,204 +0.022343,204 +0.023359,204 +0.023745,204 +0.023200,204 +0.022828,204 +0.023002,204 +0.022384,204 +0.022177,204 +0.022080,204 +0.021990,204 +0.022732,204 +0.022659,204 +0.023318,204 +0.022884,204 +0.022393,204 +0.023143,204 +0.023760,204 +0.024075,204 +0.022287,204 +0.022691,204 +0.021653,204 +0.021537,204 +0.020961,204 +0.022480,204 +0.020901,204 +0.022064,204 +0.024598,204 +0.028897,204 +0.024398,204 +0.022677,204 +0.022657,204 +0.022089,204 +0.021633,204 +0.022691,204 +0.024345,204 +0.022917,204 +0.024167,204 +0.022165,204 +0.022049,204 +0.022787,204 +0.024235,204 +0.023591,206 +0.021978,206 +0.021865,206 +0.021844,206 +0.022636,206 +0.022349,206 +0.021367,206 +0.021330,206 +0.022050,206 +0.021592,206 +0.021232,206 +0.021594,206 +0.022438,206 +0.022745,206 +0.022449,206 +0.021736,206 +0.021858,206 +0.022040,206 +0.021889,206 +0.021391,206 +0.021359,206 +0.021393,206 +0.021920,206 +0.021935,206 +0.022188,206 +0.021793,206 +0.021650,206 +0.022413,206 +0.021509,206 +0.021635,206 +0.022036,206 +0.023080,206 +0.021413,206 +0.021489,206 +0.021287,206 +0.021372,206 +0.022243,206 +0.022011,206 +0.021568,206 +0.022526,206 +0.022047,206 +0.021898,206 +0.021383,206 +0.021404,206 +0.021395,206 +0.021860,206 +0.021563,206 +0.021460,206 +0.021246,206 +0.021360,206 +0.022058,206 +0.021339,206 +0.022831,206 +0.021602,206 +0.021552,206 +0.021860,206 +0.021494,206 +0.021493,206 +0.021341,206 +0.022150,206 +0.023089,206 +0.023404,206 +0.023811,206 +0.023137,206 +0.023687,206 +0.023525,206 +0.022122,206 +0.021608,206 +0.022198,206 +0.021359,206 +0.022185,206 +0.021756,206 +0.021619,206 +0.021804,206 +0.021404,206 +0.021511,206 +0.021431,206 +0.022140,206 +0.021525,206 +0.021277,206 +0.021323,206 +0.021513,206 +0.022128,206 +0.021489,206 +0.023350,206 +0.021501,206 +0.022080,206 +0.021529,206 +0.021403,206 +0.021514,206 +0.021426,206 +0.022244,206 +0.021401,206 +0.021319,206 +0.022172,206 +0.022706,206 +0.022246,206 +0.021965,206 +0.021504,206 +0.022424,206 +0.023821,208 +0.023756,208 +0.022642,208 +0.025208,208 +0.023727,208 +0.022863,208 +0.022806,208 +0.022508,208 +0.022684,208 +0.022997,208 +0.023056,208 +0.023113,208 +0.022604,208 +0.023785,208 +0.029033,208 +0.024733,208 +0.024317,208 +0.024554,208 +0.024225,208 +0.025887,208 +0.024642,208 +0.025190,208 +0.024142,208 +0.024546,208 +0.022975,208 +0.023746,208 +0.023763,208 +0.026540,208 +0.024444,208 +0.023780,208 +0.023109,208 +0.022683,208 +0.022263,208 +0.021896,208 +0.022531,208 +0.025044,208 +0.023229,208 +0.022732,208 +0.023278,208 +0.022880,208 +0.024107,208 +0.024268,208 +0.024868,208 +0.025449,208 +0.024925,208 +0.024397,208 +0.025436,208 +0.025071,208 +0.025322,208 +0.024356,208 +0.026775,208 +0.024093,208 +0.026201,208 +0.025142,208 +0.027131,208 +0.024924,208 +0.025702,208 +0.027040,208 +0.024313,208 +0.024575,208 +0.024335,208 +0.024293,208 +0.023861,208 +0.023234,208 +0.022936,208 +0.022000,208 +0.022155,208 +0.023588,208 +0.024183,208 +0.022588,208 +0.021907,208 +0.022326,208 +0.022018,208 +0.021848,208 +0.021839,208 +0.022088,208 +0.022409,208 +0.021985,208 +0.021948,208 +0.021927,208 +0.022259,208 +0.021915,208 +0.022055,208 +0.021779,208 +0.021774,208 +0.022601,208 +0.022210,208 +0.021924,208 +0.022733,208 +0.023556,208 +0.025291,208 +0.024384,208 +0.024610,208 +0.028181,208 +0.024314,208 +0.024542,208 +0.024557,208 +0.023488,208 +0.023457,208 +0.023533,208 +0.024409,210 +0.024628,210 +0.023856,210 +0.023852,210 +0.023106,210 +0.023128,210 +0.023243,210 +0.023334,210 +0.022987,210 +0.022623,210 +0.022790,210 +0.022955,210 +0.022740,210 +0.022705,210 +0.022813,210 +0.022882,210 +0.023087,210 +0.022539,210 +0.022487,210 +0.023094,210 +0.022589,210 +0.022518,210 +0.022693,210 +0.022863,210 +0.022825,210 +0.022861,210 +0.022547,210 +0.022669,210 +0.022950,210 +0.022901,210 +0.023057,210 +0.022649,210 +0.023103,210 +0.022778,210 +0.022605,210 +0.023652,210 +0.024703,210 +0.025658,210 +0.024492,210 +0.023719,210 +0.024246,210 +0.024166,210 +0.024462,210 +0.023157,210 +0.023018,210 +0.024428,210 +0.026391,210 +0.024282,210 +0.023907,210 +0.025583,210 +0.023684,210 +0.023131,210 +0.022876,210 +0.024564,210 +0.022791,210 +0.022825,210 +0.022587,210 +0.024037,210 +0.023186,210 +0.022641,210 +0.022504,210 +0.022791,210 +0.024149,210 +0.022694,210 +0.022492,210 +0.022470,210 +0.024329,210 +0.023516,210 +0.022771,210 +0.022512,210 +0.023819,210 +0.023292,210 +0.022586,210 +0.022538,210 +0.022760,210 +0.023383,210 +0.023399,210 +0.022960,210 +0.022517,210 +0.023320,210 +0.022742,210 +0.022517,210 +0.022731,210 +0.022745,210 +0.023166,210 +0.023166,210 +0.022732,210 +0.023988,210 +0.024082,210 +0.024657,210 +0.024366,210 +0.027884,210 +0.024597,210 +0.025377,210 +0.023830,210 +0.023895,210 +0.024427,210 +0.023678,210 +0.025564,210 +0.024984,210 +0.028780,212 +0.029890,212 +0.027592,212 +0.026942,212 +0.030883,212 +0.027178,212 +0.028512,212 +0.039651,212 +0.028242,212 +0.028991,212 +0.026668,212 +0.027360,212 +0.026401,212 +0.029501,212 +0.026908,212 +0.027813,212 +0.027291,212 +0.026234,212 +0.027539,212 +0.028791,212 +0.027450,212 +0.025350,212 +0.023689,212 +0.023496,212 +0.023258,212 +0.023522,212 +0.024155,212 +0.024813,212 +0.026071,212 +0.027582,212 +0.025773,212 +0.024817,212 +0.023433,212 +0.023553,212 +0.023926,212 +0.024485,212 +0.024293,212 +0.024646,212 +0.029259,212 +0.025543,212 +0.024065,212 +0.024499,212 +0.030856,212 +0.027562,212 +0.024638,212 +0.023682,212 +0.023831,212 +0.023276,212 +0.023219,212 +0.023363,212 +0.023826,212 +0.023578,212 +0.023316,212 +0.023402,212 +0.023999,212 +0.023216,212 +0.023572,212 +0.023125,212 +0.023700,212 +0.025048,212 +0.024970,212 +0.024854,212 +0.024401,212 +0.025642,212 +0.025771,212 +0.025708,212 +0.024839,212 +0.025416,212 +0.023692,212 +0.023449,212 +0.024185,212 +0.024160,212 +0.023241,212 +0.025328,212 +0.023427,212 +0.024180,212 +0.023727,212 +0.023739,212 +0.023470,212 +0.023785,212 +0.023785,212 +0.026845,212 +0.026689,212 +0.032715,212 +0.027248,212 +0.026124,212 +0.025759,212 +0.025625,212 +0.024113,212 +0.023480,212 +0.023333,212 +0.023605,212 +0.023243,212 +0.023336,212 +0.023319,212 +0.023747,212 +0.023551,212 +0.023361,212 +0.024556,212 +0.024892,212 +0.026700,214 +0.025213,214 +0.026438,214 +0.025136,214 +0.026425,214 +0.025937,214 +0.025493,214 +0.024801,214 +0.024645,214 +0.025036,214 +0.024143,214 +0.024138,214 +0.024073,214 +0.023911,214 +0.024143,214 +0.024270,214 +0.024084,214 +0.024074,214 +0.023962,214 +0.024107,214 +0.024537,214 +0.024517,214 +0.024748,214 +0.024388,214 +0.024160,214 +0.024152,214 +0.023880,214 +0.024127,214 +0.024769,214 +0.024106,214 +0.023967,214 +0.024425,214 +0.025422,214 +0.024347,214 +0.024800,214 +0.023930,214 +0.024677,214 +0.023916,214 +0.024202,214 +0.023872,214 +0.024607,214 +0.023927,214 +0.023970,214 +0.024261,214 +0.025395,214 +0.024604,214 +0.024098,214 +0.024544,214 +0.024544,214 +0.024166,214 +0.023802,214 +0.023982,214 +0.024028,214 +0.024306,214 +0.024041,214 +0.024064,214 +0.024016,214 +0.024493,214 +0.024140,214 +0.023797,214 +0.023974,214 +0.024329,214 +0.023998,214 +0.023922,214 +0.023908,214 +0.024607,214 +0.023883,214 +0.024193,214 +0.024173,214 +0.024611,214 +0.023901,214 +0.023930,214 +0.023932,214 +0.024221,214 +0.024181,214 +0.023771,214 +0.024193,214 +0.023987,214 +0.024391,214 +0.023819,214 +0.023993,214 +0.023855,214 +0.024511,214 +0.023991,214 +0.023807,214 +0.024126,214 +0.024321,214 +0.024102,214 +0.023873,214 +0.023921,214 +0.024535,214 +0.024155,214 +0.023932,214 +0.024392,214 +0.024515,214 +0.023947,214 +0.024139,214 +0.023944,214 +0.027535,214 +0.025248,214 +0.025484,216 +0.025123,216 +0.030448,216 +0.025789,216 +0.024894,216 +0.025904,216 +0.026197,216 +0.024432,216 +0.024402,216 +0.024708,216 +0.026282,216 +0.024615,216 +0.024535,216 +0.024753,216 +0.026081,216 +0.024561,216 +0.024475,216 +0.024744,216 +0.025945,216 +0.024761,216 +0.024542,216 +0.024724,216 +0.025159,216 +0.024840,216 +0.024601,216 +0.024741,216 +0.025042,216 +0.024965,216 +0.024499,216 +0.024812,216 +0.024926,216 +0.024801,216 +0.024435,216 +0.024676,216 +0.024981,216 +0.025004,216 +0.024671,216 +0.024859,216 +0.025062,216 +0.024994,216 +0.024513,216 +0.024685,216 +0.024627,216 +0.025029,216 +0.024553,216 +0.024694,216 +0.024752,216 +0.024954,216 +0.024574,216 +0.024582,216 +0.024523,216 +0.025109,216 +0.024530,216 +0.024503,216 +0.024665,216 +0.024862,216 +0.024782,216 +0.024510,216 +0.024871,216 +0.024840,216 +0.024585,216 +0.024626,216 +0.024589,216 +0.025097,216 +0.024512,216 +0.024515,216 +0.024539,216 +0.025020,216 +0.024724,216 +0.025950,216 +0.025361,216 +0.024966,216 +0.024595,216 +0.024620,216 +0.024548,216 +0.024962,216 +0.024853,216 +0.024712,216 +0.025908,216 +0.024977,216 +0.024504,216 +0.024641,216 +0.024624,216 +0.024878,216 +0.024699,216 +0.024645,216 +0.024614,216 +0.024994,216 +0.024638,216 +0.024640,216 +0.024676,216 +0.024891,216 +0.024802,216 +0.024641,216 +0.024596,216 +0.025361,216 +0.024752,216 +0.024759,216 +0.024553,216 +0.025133,216 +0.026054,218 +0.025362,218 +0.025420,218 +0.025677,218 +0.025630,218 +0.025347,218 +0.025488,218 +0.025886,218 +0.025560,218 +0.025564,218 +0.025548,218 +0.025869,218 +0.025400,218 +0.025229,218 +0.025425,218 +0.025760,218 +0.025523,218 +0.025417,218 +0.025613,218 +0.025950,218 +0.025742,218 +0.026214,218 +0.025615,218 +0.025978,218 +0.025597,218 +0.025436,218 +0.025336,218 +0.025929,218 +0.025384,218 +0.025450,218 +0.025279,218 +0.025758,218 +0.026162,218 +0.026038,218 +0.025452,218 +0.025823,218 +0.025876,218 +0.025534,218 +0.025492,218 +0.025908,218 +0.025268,218 +0.025463,218 +0.025701,218 +0.025900,218 +0.025401,218 +0.025399,218 +0.025506,218 +0.026239,218 +0.025726,218 +0.025523,218 +0.025512,218 +0.025693,218 +0.025337,218 +0.025391,218 +0.025715,218 +0.025980,218 +0.025722,218 +0.025406,218 +0.025888,218 +0.025402,218 +0.025444,218 +0.025225,218 +0.025808,218 +0.025480,218 +0.025621,218 +0.025296,218 +0.025849,218 +0.025335,218 +0.025513,218 +0.025507,218 +0.025734,218 +0.025532,218 +0.025320,218 +0.025351,218 +0.026005,218 +0.025512,218 +0.025464,218 +0.025527,218 +0.025952,218 +0.025491,218 +0.025561,218 +0.025331,218 +0.026120,218 +0.025413,218 +0.025770,218 +0.025433,218 +0.026001,218 +0.025613,218 +0.030177,218 +0.025411,218 +0.025951,218 +0.025443,218 +0.025476,218 +0.025370,218 +0.025936,218 +0.025675,218 +0.025550,218 +0.025630,218 +0.025618,218 +0.025579,218 +0.026034,220 +0.026495,220 +0.026471,220 +0.026135,220 +0.026024,220 +0.026502,220 +0.026192,220 +0.026208,220 +0.025793,220 +0.026485,220 +0.026102,220 +0.026000,220 +0.025925,220 +0.026730,220 +0.026285,220 +0.026114,220 +0.026022,220 +0.026625,220 +0.026283,220 +0.025947,220 +0.026193,220 +0.026647,220 +0.026127,220 +0.026263,220 +0.026584,220 +0.026220,220 +0.026227,220 +0.025974,220 +0.026622,220 +0.026000,220 +0.026058,220 +0.025875,220 +0.026576,220 +0.026332,220 +0.026333,220 +0.026040,220 +0.026581,220 +0.026081,220 +0.025933,220 +0.025890,220 +0.026702,220 +0.026128,220 +0.026023,220 +0.026261,220 +0.026445,220 +0.026108,220 +0.025919,220 +0.026601,220 +0.026105,220 +0.026200,220 +0.025909,220 +0.026559,220 +0.026029,220 +0.026357,220 +0.025926,220 +0.026565,220 +0.026243,220 +0.026013,220 +0.026061,220 +0.026618,220 +0.026076,220 +0.026005,220 +0.026074,220 +0.026654,220 +0.027963,220 +0.026178,220 +0.026506,220 +0.026037,220 +0.026183,220 +0.025942,220 +0.026730,220 +0.026153,220 +0.026348,220 +0.025951,220 +0.026421,220 +0.026054,220 +0.026047,220 +0.025945,220 +0.026649,220 +0.026028,220 +0.026105,220 +0.025931,220 +0.026659,220 +0.026037,220 +0.025900,220 +0.026170,220 +0.026410,220 +0.027071,220 +0.029181,220 +0.029791,220 +0.032931,220 +0.029965,220 +0.028955,220 +0.028804,220 +0.028369,220 +0.033877,220 +0.035271,220 +0.029037,220 +0.030304,220 +0.028764,220 +0.032144,222 +0.030095,222 +0.029826,222 +0.030025,222 +0.030439,222 +0.029537,222 +0.030335,222 +0.030394,222 +0.031119,222 +0.030428,222 +0.029719,222 +0.030357,222 +0.030121,222 +0.030463,222 +0.030936,222 +0.029935,222 +0.030507,222 +0.030769,222 +0.030753,222 +0.031559,222 +0.030556,222 +0.028491,222 +0.038898,222 +0.031032,222 +0.032999,222 +0.030822,222 +0.031671,222 +0.027785,222 +0.028481,222 +0.029536,222 +0.028762,222 +0.029051,222 +0.028534,222 +0.028510,222 +0.028585,222 +0.028222,222 +0.027511,222 +0.026921,222 +0.027167,222 +0.028879,222 +0.029678,222 +0.029246,222 +0.028604,222 +0.027927,222 +0.027112,222 +0.026937,222 +0.027582,222 +0.027922,222 +0.028084,222 +0.027279,222 +0.032112,222 +0.028924,222 +0.028764,222 +0.028468,222 +0.028902,222 +0.029415,222 +0.028632,222 +0.028597,222 +0.028101,222 +0.027938,222 +0.027737,222 +0.027592,222 +0.027508,222 +0.026710,222 +0.027587,222 +0.027275,222 +0.026938,222 +0.026820,222 +0.027447,222 +0.026860,222 +0.027257,222 +0.027210,222 +0.027288,222 +0.026897,222 +0.026723,222 +0.027392,222 +0.027271,222 +0.026700,222 +0.027218,222 +0.027228,222 +0.026802,222 +0.026738,222 +0.026919,222 +0.027479,222 +0.026767,222 +0.027356,222 +0.028218,222 +0.027190,222 +0.026627,222 +0.026783,222 +0.029107,222 +0.027437,222 +0.027101,222 +0.027245,222 +0.027957,222 +0.027147,222 +0.026625,222 +0.027619,222 +0.026972,222 +0.026861,222 +0.030481,224 +0.031907,224 +0.030276,224 +0.030199,224 +0.030781,224 +0.030660,224 +0.029803,224 +0.029641,224 +0.030029,224 +0.030143,224 +0.030204,224 +0.029160,224 +0.028092,224 +0.027552,224 +0.028139,224 +0.027974,224 +0.027323,224 +0.027517,224 +0.028085,224 +0.030571,224 +0.030403,224 +0.029997,224 +0.028308,224 +0.027460,224 +0.028766,224 +0.029718,224 +0.028819,224 +0.029245,224 +0.031629,224 +0.032029,224 +0.030914,224 +0.030517,224 +0.030474,224 +0.030768,224 +0.031882,224 +0.031387,224 +0.030420,224 +0.031337,224 +0.030044,224 +0.034327,224 +0.030854,224 +0.030328,224 +0.033406,224 +0.031817,224 +0.030631,224 +0.032101,224 +0.033730,224 +0.029826,224 +0.031530,224 +0.031753,224 +0.033048,224 +0.032152,224 +0.030505,224 +0.029881,224 +0.031639,224 +0.029774,224 +0.028924,224 +0.028632,224 +0.027792,224 +0.029400,224 +0.030336,224 +0.030603,224 +0.028470,224 +0.028010,224 +0.027968,224 +0.027848,224 +0.034007,224 +0.034008,224 +0.032426,224 +0.031746,224 +0.038141,224 +0.032608,224 +0.031625,224 +0.034657,224 +0.038820,224 +0.033090,224 +0.033139,224 +0.031559,224 +0.032271,224 +0.030568,224 +0.032382,224 +0.030899,224 +0.030876,224 +0.031431,224 +0.032091,224 +0.030831,224 +0.033164,224 +0.031260,224 +0.032525,224 +0.030558,224 +0.031484,224 +0.032766,224 +0.040871,224 +0.034256,224 +0.029736,224 +0.030002,224 +0.030190,224 +0.031873,224 +0.035653,224 +0.030355,224 +0.038359,226 +0.032737,226 +0.032086,226 +0.031910,226 +0.033470,226 +0.035968,226 +0.035538,226 +0.031602,226 +0.031934,226 +0.033544,226 +0.031603,226 +0.032140,226 +0.030537,226 +0.029187,226 +0.033465,226 +0.032267,226 +0.028616,226 +0.029906,226 +0.030404,226 +0.031123,226 +0.030498,226 +0.029185,226 +0.029922,226 +0.029873,226 +0.029993,226 +0.029503,226 +0.029013,226 +0.029772,226 +0.029598,226 +0.029569,226 +0.029438,226 +0.029786,226 +0.029251,226 +0.029219,226 +0.029285,226 +0.028921,226 +0.029585,226 +0.029455,226 +0.030155,226 +0.037745,226 +0.048594,226 +0.035934,226 +0.029589,226 +0.030131,226 +0.031797,226 +0.032539,226 +0.037528,226 +0.030358,226 +0.030639,226 +0.033117,226 +0.030044,226 +0.032242,226 +0.055458,226 +0.031261,226 +0.030087,226 +0.030046,226 +0.030232,226 +0.029933,226 +0.033404,226 +0.031345,226 +0.029815,226 +0.034788,226 +0.030248,226 +0.031458,226 +0.030393,226 +0.029576,226 +0.034623,226 +0.031584,226 +0.030099,226 +0.032214,226 +0.038271,226 +0.053570,226 +0.032201,226 +0.037901,226 +0.032488,226 +0.040133,226 +0.047461,226 +0.054163,226 +0.034901,226 +0.029900,226 +0.029762,226 +0.033021,226 +0.033003,226 +0.038823,226 +0.037308,226 +0.032131,226 +0.034172,226 +0.033802,226 +0.030919,226 +0.031852,226 +0.033025,226 +0.031556,226 +0.032073,226 +0.033406,226 +0.033584,226 +0.032928,226 +0.038199,226 +0.033780,226 +0.038584,226 +0.055719,226 +0.058413,228 +0.035285,228 +0.035209,228 +0.037280,228 +0.033502,228 +0.032071,228 +0.035548,228 +0.034351,228 +0.036300,228 +0.033368,228 +0.032850,228 +0.034856,228 +0.031111,228 +0.031256,228 +0.032246,228 +0.032399,228 +0.030678,228 +0.032127,228 +0.034021,228 +0.033264,228 +0.035448,228 +0.033460,228 +0.032239,228 +0.034237,228 +0.033503,228 +0.035946,228 +0.038068,228 +0.037110,228 +0.032372,228 +0.034622,228 +0.043915,228 +0.038023,228 +0.032146,228 +0.033678,228 +0.035175,228 +0.039960,228 +0.038647,228 +0.032987,228 +0.034119,228 +0.031473,228 +0.034447,228 +0.037187,228 +0.034195,228 +0.053888,228 +0.057387,228 +0.041434,228 +0.035605,228 +0.037209,228 +0.038886,228 +0.038520,228 +0.037813,228 +0.038613,228 +0.037488,228 +0.035828,228 +0.055271,228 +0.044974,228 +0.036878,228 +0.033910,228 +0.040586,228 +0.034270,228 +0.035660,228 +0.037039,228 +0.035461,228 +0.033405,228 +0.034598,228 +0.033075,228 +0.031569,228 +0.029700,228 +0.028999,228 +0.028899,228 +0.030214,228 +0.029377,228 +0.029040,228 +0.028972,228 +0.030507,228 +0.030097,228 +0.029211,228 +0.031429,228 +0.031037,228 +0.030696,228 +0.032641,228 +0.030338,228 +0.030568,228 +0.035138,228 +0.033369,228 +0.033579,228 +0.032370,228 +0.033078,228 +0.033028,228 +0.036455,228 +0.031926,228 +0.037841,228 +0.043416,228 +0.034424,228 +0.035560,228 +0.035623,228 +0.033878,228 +0.034861,228 +0.040343,228 +0.031714,228 +0.031521,230 +0.032262,230 +0.031759,230 +0.031238,230 +0.040024,230 +0.030795,230 +0.030423,230 +0.037331,230 +0.034220,230 +0.038911,230 +0.033386,230 +0.030560,230 +0.032715,230 +0.032480,230 +0.032481,230 +0.032610,230 +0.031889,230 +0.029772,230 +0.031411,230 +0.036283,230 +0.035074,230 +0.031736,230 +0.032945,230 +0.033298,230 +0.035190,230 +0.035525,230 +0.033094,230 +0.030818,230 +0.030266,230 +0.030043,230 +0.033915,230 +0.033520,230 +0.043449,230 +0.031992,230 +0.032614,230 +0.030324,230 +0.029697,230 +0.031615,230 +0.029970,230 +0.029960,230 +0.030578,230 +0.030815,230 +0.029851,230 +0.029909,230 +0.031756,230 +0.030288,230 +0.029887,230 +0.031984,230 +0.030395,230 +0.030012,230 +0.031443,230 +0.030140,230 +0.029813,230 +0.029760,230 +0.030528,230 +0.031063,230 +0.030089,230 +0.030302,230 +0.031332,230 +0.030670,230 +0.030580,230 +0.030771,230 +0.029953,230 +0.029818,230 +0.030290,230 +0.029702,230 +0.029878,230 +0.030488,230 +0.029758,230 +0.029761,230 +0.030089,230 +0.029880,230 +0.029779,230 +0.029914,230 +0.030102,230 +0.030568,230 +0.029937,230 +0.040401,230 +0.031918,230 +0.033614,230 +0.033388,230 +0.031643,230 +0.033828,230 +0.033263,230 +0.032154,230 +0.037540,230 +0.031902,230 +0.031859,230 +0.031949,230 +0.032026,230 +0.033004,230 +0.033935,230 +0.032172,230 +0.030865,230 +0.034507,230 +0.031688,230 +0.030815,230 +0.031229,230 +0.032014,230 +0.031126,230 +0.032550,232 +0.031897,232 +0.031908,232 +0.035073,232 +0.038418,232 +0.041864,232 +0.035232,232 +0.033638,232 +0.037859,232 +0.036304,232 +0.041679,232 +0.037868,232 +0.035952,232 +0.034724,232 +0.035198,232 +0.037757,232 +0.048921,232 +0.045694,232 +0.034858,232 +0.034677,232 +0.037713,232 +0.035391,232 +0.034319,232 +0.035481,232 +0.031987,232 +0.032051,232 +0.036022,232 +0.032238,232 +0.032866,232 +0.036448,232 +0.037238,232 +0.036668,232 +0.034088,232 +0.036922,232 +0.035981,232 +0.035254,232 +0.040008,232 +0.036387,232 +0.036067,232 +0.035025,232 +0.040029,232 +0.035266,232 +0.035036,232 +0.033987,232 +0.033387,232 +0.033578,232 +0.035668,232 +0.033234,232 +0.032961,232 +0.044879,232 +0.034288,232 +0.036545,232 +0.037218,232 +0.060757,232 +0.052725,232 +0.059088,232 +0.046550,232 +0.035530,232 +0.037320,232 +0.035810,232 +0.037426,232 +0.038875,232 +0.039483,232 +0.036936,232 +0.039523,232 +0.035185,232 +0.036760,232 +0.039975,232 +0.033352,232 +0.034196,232 +0.038098,232 +0.035772,232 +0.034154,232 +0.046065,232 +0.036488,232 +0.036674,232 +0.032738,232 +0.040295,232 +0.034094,232 +0.032441,232 +0.034736,232 +0.039548,232 +0.043287,232 +0.035976,232 +0.034000,232 +0.039252,232 +0.036005,232 +0.035834,232 +0.038083,232 +0.042782,232 +0.034514,232 +0.036371,232 +0.037417,232 +0.036355,232 +0.035273,232 +0.034276,232 +0.034102,232 +0.034029,232 +0.032980,232 +0.033573,232 +0.038926,234 +0.040618,234 +0.040820,234 +0.037968,234 +0.038272,234 +0.042271,234 +0.049123,234 +0.034518,234 +0.038881,234 +0.036649,234 +0.034415,234 +0.036345,234 +0.042409,234 +0.046327,234 +0.040199,234 +0.035942,234 +0.033875,234 +0.034114,234 +0.032950,234 +0.034286,234 +0.033797,234 +0.033200,234 +0.034005,234 +0.034584,234 +0.033112,234 +0.035728,234 +0.034922,234 +0.034962,234 +0.033979,234 +0.037586,234 +0.032872,234 +0.032678,234 +0.032931,234 +0.031773,234 +0.031569,234 +0.032165,234 +0.031644,234 +0.031421,234 +0.031764,234 +0.032788,234 +0.031787,234 +0.032033,234 +0.032698,234 +0.034101,234 +0.032329,234 +0.031548,234 +0.031429,234 +0.031976,234 +0.031711,234 +0.031761,234 +0.031984,234 +0.032145,234 +0.032423,234 +0.031948,234 +0.032846,234 +0.032110,234 +0.031985,234 +0.032124,234 +0.032101,234 +0.032569,234 +0.031868,234 +0.031619,234 +0.031396,234 +0.031854,234 +0.031510,234 +0.031874,234 +0.032488,234 +0.031960,234 +0.032351,234 +0.032967,234 +0.031426,234 +0.031421,234 +0.032175,234 +0.032020,234 +0.031694,234 +0.031361,234 +0.032057,234 +0.031503,234 +0.031530,234 +0.031835,234 +0.031568,234 +0.032580,234 +0.033096,234 +0.031693,234 +0.031557,234 +0.031821,234 +0.031663,234 +0.031552,234 +0.031997,234 +0.031551,234 +0.032328,234 +0.038241,234 +0.033073,234 +0.034025,234 +0.041936,234 +0.034898,234 +0.034670,234 +0.033454,234 +0.032831,234 +0.032554,234 +0.034014,236 +0.042267,236 +0.033598,236 +0.037150,236 +0.037716,236 +0.042123,236 +0.035850,236 +0.033674,236 +0.034933,236 +0.046021,236 +0.036961,236 +0.034046,236 +0.034640,236 +0.038851,236 +0.038601,236 +0.037612,236 +0.038283,236 +0.050662,236 +0.042437,236 +0.035757,236 +0.042565,236 +0.042066,236 +0.036041,236 +0.045430,236 +0.046189,236 +0.035335,236 +0.035933,236 +0.037539,236 +0.035753,236 +0.038120,236 +0.038759,236 +0.035861,236 +0.037570,236 +0.037263,236 +0.037237,236 +0.034930,236 +0.036549,236 +0.038395,236 +0.035150,236 +0.034880,236 +0.056546,236 +0.065572,236 +0.064669,236 +0.043473,236 +0.038197,236 +0.036195,236 +0.037004,236 +0.037704,236 +0.039867,236 +0.034735,236 +0.035096,236 +0.036566,236 +0.036038,236 +0.035488,236 +0.036002,236 +0.034937,236 +0.037259,236 +0.037143,236 +0.038535,236 +0.037414,236 +0.037420,236 +0.035069,236 +0.044932,236 +0.042434,236 +0.038816,236 +0.037132,236 +0.039159,236 +0.043116,236 +0.053508,236 +0.036529,236 +0.042496,236 +0.041511,236 +0.042414,236 +0.036079,236 +0.035603,236 +0.038234,236 +0.036367,236 +0.035904,236 +0.043415,236 +0.038683,236 +0.049272,236 +0.037985,236 +0.040943,236 +0.036566,236 +0.036520,236 +0.036361,236 +0.040316,236 +0.038872,236 +0.046996,236 +0.038305,236 +0.037079,236 +0.037051,236 +0.036403,236 +0.052353,236 +0.065731,236 +0.038483,236 +0.039731,236 +0.040667,236 +0.039422,236 +0.036323,236 +0.039984,238 +0.039154,238 +0.038000,238 +0.040795,238 +0.040740,238 +0.037750,238 +0.039052,238 +0.037504,238 +0.039023,238 +0.039670,238 +0.039332,238 +0.038281,238 +0.037550,238 +0.039648,238 +0.042368,238 +0.040971,238 +0.046427,238 +0.041052,238 +0.038373,238 +0.040200,238 +0.043690,238 +0.037832,238 +0.038493,238 +0.038403,238 +0.037142,238 +0.034706,238 +0.033901,238 +0.040440,238 +0.035300,238 +0.041074,238 +0.037595,238 +0.035440,238 +0.037362,238 +0.035476,238 +0.035796,238 +0.037895,238 +0.038695,238 +0.037677,238 +0.041286,238 +0.037201,238 +0.033769,238 +0.036268,238 +0.035800,238 +0.036867,238 +0.035524,238 +0.036707,238 +0.033865,238 +0.034250,238 +0.033525,238 +0.034638,238 +0.034202,238 +0.034417,238 +0.036239,238 +0.035166,238 +0.033821,238 +0.033657,238 +0.033593,238 +0.034345,238 +0.033903,238 +0.033662,238 +0.034174,238 +0.034770,238 +0.033359,238 +0.033261,238 +0.033956,238 +0.033516,238 +0.034037,238 +0.033512,238 +0.035305,238 +0.043315,238 +0.036623,238 +0.036925,238 +0.035807,238 +0.035129,238 +0.035297,238 +0.035685,238 +0.034809,238 +0.062845,238 +0.063883,238 +0.037364,238 +0.034707,238 +0.036881,238 +0.036212,238 +0.034828,238 +0.039247,238 +0.037213,238 +0.035516,238 +0.039169,238 +0.034531,238 +0.034429,238 +0.058864,238 +0.034223,238 +0.036844,238 +0.034868,238 +0.039806,238 +0.039903,238 +0.036842,238 +0.045442,238 +0.061311,238 +0.068015,238 +0.061888,240 +0.037788,240 +0.038708,240 +0.040065,240 +0.037985,240 +0.038610,240 +0.038260,240 +0.035348,240 +0.043080,240 +0.065015,240 +0.065295,240 +0.047075,240 +0.039273,240 +0.039100,240 +0.041221,240 +0.036884,240 +0.037328,240 +0.038838,240 +0.035996,240 +0.035691,240 +0.038985,240 +0.040828,240 +0.063071,240 +0.040215,240 +0.040135,240 +0.044682,240 +0.058434,240 +0.035911,240 +0.035433,240 +0.036398,240 +0.038814,240 +0.039300,240 +0.040816,240 +0.039196,240 +0.039069,240 +0.036121,240 +0.037252,240 +0.040057,240 +0.037747,240 +0.049648,240 +0.044225,240 +0.037219,240 +0.043359,240 +0.039678,240 +0.044457,240 +0.043541,240 +0.037499,240 +0.036685,240 +0.035518,240 +0.036836,240 +0.036822,240 +0.038609,240 +0.042677,240 +0.042368,240 +0.041357,240 +0.036577,240 +0.037395,240 +0.044514,240 +0.041369,240 +0.040472,240 +0.043148,240 +0.035156,240 +0.036309,240 +0.037413,240 +0.041998,240 +0.039106,240 +0.037416,240 +0.049919,240 +0.058601,240 +0.063010,240 +0.039447,240 +0.041990,240 +0.036290,240 +0.035762,240 +0.036162,240 +0.036034,240 +0.038136,240 +0.035628,240 +0.037692,240 +0.037752,240 +0.041216,240 +0.043586,240 +0.036112,240 +0.038512,240 +0.037177,240 +0.035050,240 +0.034959,240 +0.035005,240 +0.034586,240 +0.040625,240 +0.036259,240 +0.035390,240 +0.047543,240 +0.046283,240 +0.035670,240 +0.037121,240 +0.036174,240 +0.036019,240 +0.035681,240 +0.035161,240 +0.037298,242 +0.036379,242 +0.037063,242 +0.039403,242 +0.035977,242 +0.035764,242 +0.038621,242 +0.040435,242 +0.040444,242 +0.046900,242 +0.044130,242 +0.043335,242 +0.041598,242 +0.043373,242 +0.051978,242 +0.047757,242 +0.043976,242 +0.044748,242 +0.044415,242 +0.045642,242 +0.043595,242 +0.040760,242 +0.037238,242 +0.037202,242 +0.039095,242 +0.037753,242 +0.044098,242 +0.043148,242 +0.037186,242 +0.037201,242 +0.037794,242 +0.037017,242 +0.046732,242 +0.042538,242 +0.040082,242 +0.037564,242 +0.036788,242 +0.039120,242 +0.037258,242 +0.039025,242 +0.038419,242 +0.037153,242 +0.039335,242 +0.036763,242 +0.037089,242 +0.038095,242 +0.036415,242 +0.035928,242 +0.037595,242 +0.035858,242 +0.035469,242 +0.035986,242 +0.035906,242 +0.036469,242 +0.036442,242 +0.036042,242 +0.036749,242 +0.036919,242 +0.036366,242 +0.042866,242 +0.036113,242 +0.037040,242 +0.039419,242 +0.037763,242 +0.036457,242 +0.035895,242 +0.036282,242 +0.037048,242 +0.035670,242 +0.036967,242 +0.036571,242 +0.035999,242 +0.038157,242 +0.036348,242 +0.036195,242 +0.037875,242 +0.038158,242 +0.037128,242 +0.037198,242 +0.036315,242 +0.039334,242 +0.040416,242 +0.040462,242 +0.040766,242 +0.036621,242 +0.041799,242 +0.041783,242 +0.041681,242 +0.039947,242 +0.045828,242 +0.051787,242 +0.055497,242 +0.062481,242 +0.054816,242 +0.051234,242 +0.053183,242 +0.049677,242 +0.043943,242 +0.049858,242 +0.048604,242 +0.042604,244 +0.037067,244 +0.037008,244 +0.038071,244 +0.037895,244 +0.037857,244 +0.039293,244 +0.038633,244 +0.038398,244 +0.037568,244 +0.043805,244 +0.037950,244 +0.037571,244 +0.038194,244 +0.040907,244 +0.049489,244 +0.045770,244 +0.041992,244 +0.049068,244 +0.042707,244 +0.038987,244 +0.040554,244 +0.045714,244 +0.046163,244 +0.045404,244 +0.039588,244 +0.038019,244 +0.039299,244 +0.044092,244 +0.039754,244 +0.037165,244 +0.037098,244 +0.039174,244 +0.043341,244 +0.038992,244 +0.037748,244 +0.047128,244 +0.046255,244 +0.042033,244 +0.043816,244 +0.040439,244 +0.046624,244 +0.047540,244 +0.054191,244 +0.051626,244 +0.045808,244 +0.042352,244 +0.045372,244 +0.038918,244 +0.038240,244 +0.038005,244 +0.037630,244 +0.042605,244 +0.039113,244 +0.038081,244 +0.040873,244 +0.039740,244 +0.039959,244 +0.043360,244 +0.041199,244 +0.039120,244 +0.037091,244 +0.042271,244 +0.042701,244 +0.045315,244 +0.049741,244 +0.039499,244 +0.039563,244 +0.057227,244 +0.044174,244 +0.044178,244 +0.040710,244 +0.040150,244 +0.041164,244 +0.039595,244 +0.038822,244 +0.037958,244 +0.045212,244 +0.041920,244 +0.041130,244 +0.042547,244 +0.046054,244 +0.041133,244 +0.037234,244 +0.038638,244 +0.042100,244 +0.040171,244 +0.041663,244 +0.040491,244 +0.043518,244 +0.044038,244 +0.038680,244 +0.042164,244 +0.061726,244 +0.074289,244 +0.042699,244 +0.042803,244 +0.049122,244 +0.070119,244 +0.049622,244 +0.038539,246 +0.038196,246 +0.039307,246 +0.037428,246 +0.038324,246 +0.052023,246 +0.039487,246 +0.037586,246 +0.039020,246 +0.039549,246 +0.037492,246 +0.040484,246 +0.039250,246 +0.046656,246 +0.038669,246 +0.039248,246 +0.040040,246 +0.037132,246 +0.038674,246 +0.038312,246 +0.037191,246 +0.039565,246 +0.060067,246 +0.053942,246 +0.044020,246 +0.055706,246 +0.040239,246 +0.041632,246 +0.046325,246 +0.046631,246 +0.050694,246 +0.042951,246 +0.042646,246 +0.042530,246 +0.048909,246 +0.040221,246 +0.044465,246 +0.048841,246 +0.049324,246 +0.043825,246 +0.042679,246 +0.042340,246 +0.042111,246 +0.045769,246 +0.054803,246 +0.048428,246 +0.043964,246 +0.052360,246 +0.041956,246 +0.042329,246 +0.043834,246 +0.047139,246 +0.060399,246 +0.052413,246 +0.042464,246 +0.042927,246 +0.049934,246 +0.041210,246 +0.043826,246 +0.045054,246 +0.048279,246 +0.041517,246 +0.039144,246 +0.053363,246 +0.039289,246 +0.040568,246 +0.039130,246 +0.041380,246 +0.039996,246 +0.038805,246 +0.039796,246 +0.037353,246 +0.048057,246 +0.042571,246 +0.041057,246 +0.042104,246 +0.043330,246 +0.049639,246 +0.041997,246 +0.040691,246 +0.042721,246 +0.048252,246 +0.039384,246 +0.038983,246 +0.040735,246 +0.039003,246 +0.038468,246 +0.043975,246 +0.039244,246 +0.039227,246 +0.038976,246 +0.038386,246 +0.042636,246 +0.038402,246 +0.038908,246 +0.038447,246 +0.038969,246 +0.043644,246 +0.038643,246 +0.039654,246 +0.040417,248 +0.039043,248 +0.044185,248 +0.040981,248 +0.041167,248 +0.039010,248 +0.046503,248 +0.039729,248 +0.038554,248 +0.044928,248 +0.038168,248 +0.041322,248 +0.041021,248 +0.037554,248 +0.045301,248 +0.039310,248 +0.044383,248 +0.038072,248 +0.037636,248 +0.045089,248 +0.037994,248 +0.045009,248 +0.037771,248 +0.037889,248 +0.044825,248 +0.038956,248 +0.046335,248 +0.038490,248 +0.039092,248 +0.044719,248 +0.038244,248 +0.038259,248 +0.040093,248 +0.059074,248 +0.069364,248 +0.072384,248 +0.070536,248 +0.070802,248 +0.043605,248 +0.043621,248 +0.048433,248 +0.050848,248 +0.045514,248 +0.043148,248 +0.051421,248 +0.042470,248 +0.041860,248 +0.044064,248 +0.042480,248 +0.039761,248 +0.042519,248 +0.046595,248 +0.043080,248 +0.041553,248 +0.040925,248 +0.039039,248 +0.041334,248 +0.041843,248 +0.043456,248 +0.052848,248 +0.054206,248 +0.043880,248 +0.039692,248 +0.049379,248 +0.044408,248 +0.044133,248 +0.042658,248 +0.040781,248 +0.039622,248 +0.040741,248 +0.040740,248 +0.040613,248 +0.042183,248 +0.047676,248 +0.045665,248 +0.040499,248 +0.039285,248 +0.040557,248 +0.040458,248 +0.039474,248 +0.041121,248 +0.041969,248 +0.048307,248 +0.043854,248 +0.046862,248 +0.039862,248 +0.044932,248 +0.040158,248 +0.044907,248 +0.039039,248 +0.038666,248 +0.040718,248 +0.039401,248 +0.041912,248 +0.039699,248 +0.038408,248 +0.039801,248 +0.037801,248 +0.038018,248 +0.041077,248 +0.041118,250 +0.041625,250 +0.042281,250 +0.041250,250 +0.047224,250 +0.052398,250 +0.042484,250 +0.041450,250 +0.044243,250 +0.043093,250 +0.043305,250 +0.044385,250 +0.040806,250 +0.042173,250 +0.043244,250 +0.043859,250 +0.041508,250 +0.040480,250 +0.041561,250 +0.052677,250 +0.049975,250 +0.043095,250 +0.042289,250 +0.040271,250 +0.042032,250 +0.041356,250 +0.040589,250 +0.042898,250 +0.040365,250 +0.042170,250 +0.041341,250 +0.047620,250 +0.040095,250 +0.042915,250 +0.042191,250 +0.040765,250 +0.041442,250 +0.040396,250 +0.038723,250 +0.039274,250 +0.038511,250 +0.039335,250 +0.039447,250 +0.038701,250 +0.041427,250 +0.039387,250 +0.041807,250 +0.039079,250 +0.038644,250 +0.039729,250 +0.038817,250 +0.039518,250 +0.039032,250 +0.038899,250 +0.041545,250 +0.040623,250 +0.038601,250 +0.039554,250 +0.038782,250 +0.038914,250 +0.038213,250 +0.038458,250 +0.038830,250 +0.038276,250 +0.038921,250 +0.038286,250 +0.038049,250 +0.039573,250 +0.038401,250 +0.038713,250 +0.038589,250 +0.038117,250 +0.038831,250 +0.038402,250 +0.038166,250 +0.038875,250 +0.038836,250 +0.039089,250 +0.038327,250 +0.038419,250 +0.039411,250 +0.042661,250 +0.041217,250 +0.039954,250 +0.040938,250 +0.040428,250 +0.040414,250 +0.046576,250 +0.040224,250 +0.041112,250 +0.045884,250 +0.039604,250 +0.040516,250 +0.039564,250 +0.040219,250 +0.040064,250 +0.039664,250 +0.047404,250 +0.040092,250 +0.047241,250 +0.041014,252 +0.041330,252 +0.044109,252 +0.041699,252 +0.041691,252 +0.043507,252 +0.049178,252 +0.039971,252 +0.039065,252 +0.047117,252 +0.039674,252 +0.048511,252 +0.040083,252 +0.040037,252 +0.039856,252 +0.040304,252 +0.043659,252 +0.040051,252 +0.039996,252 +0.039877,252 +0.041357,252 +0.041120,252 +0.040048,252 +0.040099,252 +0.039636,252 +0.039485,252 +0.040500,252 +0.039753,252 +0.040429,252 +0.039844,252 +0.039803,252 +0.040322,252 +0.041272,252 +0.040620,252 +0.039722,252 +0.039304,252 +0.040193,252 +0.039097,252 +0.039661,252 +0.039098,252 +0.038839,252 +0.040025,252 +0.039618,252 +0.040018,252 +0.039926,252 +0.039517,252 +0.040126,252 +0.040258,252 +0.040113,252 +0.040086,252 +0.039453,252 +0.041034,252 +0.039821,252 +0.040005,252 +0.039831,252 +0.039674,252 +0.040629,252 +0.041473,252 +0.039830,252 +0.040176,252 +0.039588,252 +0.040798,252 +0.039369,252 +0.039816,252 +0.040228,252 +0.039705,252 +0.040574,252 +0.039849,252 +0.043661,252 +0.039510,252 +0.039278,252 +0.040141,252 +0.039366,252 +0.039836,252 +0.039810,252 +0.039245,252 +0.040290,252 +0.039499,252 +0.039695,252 +0.040437,252 +0.040627,252 +0.040927,252 +0.041921,252 +0.040642,252 +0.039550,252 +0.039489,252 +0.040235,252 +0.039898,252 +0.039762,252 +0.040119,252 +0.040233,252 +0.040398,252 +0.039181,252 +0.043033,252 +0.044545,252 +0.044985,252 +0.047034,252 +0.044280,252 +0.054477,252 +0.050319,252 +0.057066,254 +0.045915,254 +0.053807,254 +0.045821,254 +0.046224,254 +0.053504,254 +0.046008,254 +0.042736,254 +0.042936,254 +0.042031,254 +0.049899,254 +0.045839,254 +0.043794,254 +0.045294,254 +0.042477,254 +0.042991,254 +0.042728,254 +0.050819,254 +0.042482,254 +0.048271,254 +0.040985,254 +0.040710,254 +0.047932,254 +0.040411,254 +0.048790,254 +0.040995,254 +0.048399,254 +0.041008,254 +0.046948,254 +0.043205,254 +0.041398,254 +0.049819,254 +0.064929,254 +0.056505,254 +0.043840,254 +0.045254,254 +0.046367,254 +0.048988,254 +0.048867,254 +0.047641,254 +0.046717,254 +0.041652,254 +0.040698,254 +0.041314,254 +0.041291,254 +0.041492,254 +0.041668,254 +0.041032,254 +0.043226,254 +0.040972,254 +0.041882,254 +0.042040,254 +0.041288,254 +0.043481,254 +0.041075,254 +0.042863,254 +0.042624,254 +0.041805,254 +0.045990,254 +0.042427,254 +0.047742,254 +0.042355,254 +0.045689,254 +0.046819,254 +0.047558,254 +0.045499,254 +0.044920,254 +0.042057,254 +0.041470,254 +0.042317,254 +0.041304,254 +0.042961,254 +0.041028,254 +0.042244,254 +0.041813,254 +0.043338,254 +0.043398,254 +0.041723,254 +0.042069,254 +0.042506,254 +0.040893,254 +0.041919,254 +0.041103,254 +0.041800,254 +0.041996,254 +0.041740,254 +0.042997,254 +0.040920,254 +0.041795,254 +0.041676,254 +0.041098,254 +0.041497,254 +0.041171,254 +0.042121,254 +0.041001,254 +0.041654,254 +0.045172,254 +0.040708,254 +0.041422,254 +0.040343,254 +0.046370,256 +0.045655,256 +0.048878,256 +0.045398,256 +0.045694,256 +0.046966,256 +0.045426,256 +0.045866,256 +0.045093,256 +0.046127,256 +0.045129,256 +0.046048,256 +0.045077,256 +0.048681,256 +0.045522,256 +0.052923,256 +0.048435,256 +0.046260,256 +0.045517,256 +0.046115,256 +0.046228,256 +0.045122,256 +0.046320,256 +0.046687,256 +0.046604,256 +0.045214,256 +0.046405,256 +0.045484,256 +0.046130,256 +0.045156,256 +0.046180,256 +0.045839,256 +0.045490,256 +0.046040,256 +0.045252,256 +0.046244,256 +0.045412,256 +0.046311,256 +0.045113,256 +0.049488,256 +0.045643,256 +0.046900,256 +0.046661,256 +0.045409,256 +0.046062,256 +0.045338,256 +0.045944,256 +0.045376,256 +0.046158,256 +0.045406,256 +0.046151,256 +0.045319,256 +0.045680,256 +0.045820,256 +0.045328,256 +0.045860,256 +0.045192,256 +0.046020,256 +0.045257,256 +0.046302,256 +0.045438,256 +0.058704,256 +0.047125,256 +0.050324,256 +0.045177,256 +0.046064,256 +0.045252,256 +0.045498,256 +0.045675,256 +0.045127,256 +0.046056,256 +0.045348,256 +0.046808,256 +0.045555,256 +0.045877,256 +0.045550,256 +0.046480,256 +0.045690,256 +0.045546,256 +0.045840,256 +0.045534,256 +0.052565,256 +0.045433,256 +0.046270,256 +0.045094,256 +0.045598,256 +0.045110,256 +0.045658,256 +0.045319,256 +0.045527,256 +0.045396,256 +0.045103,256 +0.045891,256 +0.045382,256 +0.046379,256 +0.045006,256 +0.045845,256 +0.045230,256 +0.045804,256 +0.045221,256 +0.043128,258 +0.043097,258 +0.042982,258 +0.043606,258 +0.042851,258 +0.044028,258 +0.042502,258 +0.043378,258 +0.043095,258 +0.042678,258 +0.043282,258 +0.042649,258 +0.043546,258 +0.042667,258 +0.042888,258 +0.043258,258 +0.042892,258 +0.043451,258 +0.042961,258 +0.043597,258 +0.042769,258 +0.043045,258 +0.043404,258 +0.042599,258 +0.044687,258 +0.042961,258 +0.046317,258 +0.042790,258 +0.043971,258 +0.043444,258 +0.042786,258 +0.044754,258 +0.042692,258 +0.044956,258 +0.042697,258 +0.045201,258 +0.043495,258 +0.043298,258 +0.047522,258 +0.047937,258 +0.047038,258 +0.042625,258 +0.045165,258 +0.042676,258 +0.045141,258 +0.043566,258 +0.046159,258 +0.049445,258 +0.051943,258 +0.048570,258 +0.050922,258 +0.077064,258 +0.054108,258 +0.075734,258 +0.049988,258 +0.055326,258 +0.054183,258 +0.052520,258 +0.050749,258 +0.049838,258 +0.050774,258 +0.048713,258 +0.046663,258 +0.051461,258 +0.050249,258 +0.050005,258 +0.056744,258 +0.050853,258 +0.048940,258 +0.049486,258 +0.049413,258 +0.048170,258 +0.043561,258 +0.043528,258 +0.046977,258 +0.047641,258 +0.045539,258 +0.046903,258 +0.049043,258 +0.051012,258 +0.050541,258 +0.051858,258 +0.050450,258 +0.043314,258 +0.045638,258 +0.051711,258 +0.048072,258 +0.049489,258 +0.049104,258 +0.046936,258 +0.045119,258 +0.044753,258 +0.044203,258 +0.044305,258 +0.046464,258 +0.047061,258 +0.045169,258 +0.045110,258 +0.047880,258 +0.046840,258 +0.050664,260 +0.047291,260 +0.050600,260 +0.045224,260 +0.044480,260 +0.046777,260 +0.047652,260 +0.048565,260 +0.048601,260 +0.047835,260 +0.054151,260 +0.043818,260 +0.051183,260 +0.044731,260 +0.052353,260 +0.044937,260 +0.045090,260 +0.044738,260 +0.044539,260 +0.044189,260 +0.044068,260 +0.044165,260 +0.044131,260 +0.044706,260 +0.043923,260 +0.044485,260 +0.044039,260 +0.044352,260 +0.043917,260 +0.044257,260 +0.044521,260 +0.044229,260 +0.044743,260 +0.044407,260 +0.045388,260 +0.046720,260 +0.046403,260 +0.044388,260 +0.047111,260 +0.048361,260 +0.049818,260 +0.050172,260 +0.049104,260 +0.051559,260 +0.049807,260 +0.053649,260 +0.051756,260 +0.051425,260 +0.049529,260 +0.050346,260 +0.049346,260 +0.048547,260 +0.048216,260 +0.049552,260 +0.049367,260 +0.046374,260 +0.048519,260 +0.047924,260 +0.052343,260 +0.054300,260 +0.046514,260 +0.047875,260 +0.047481,260 +0.048849,260 +0.046673,260 +0.046245,260 +0.044684,260 +0.048325,260 +0.048049,260 +0.050329,260 +0.047061,260 +0.048243,260 +0.050512,260 +0.052190,260 +0.048308,260 +0.049647,260 +0.047268,260 +0.048502,260 +0.044475,260 +0.045791,260 +0.044151,260 +0.045445,260 +0.049574,260 +0.048230,260 +0.049027,260 +0.054493,260 +0.053909,260 +0.050246,260 +0.050147,260 +0.047211,260 +0.047581,260 +0.044123,260 +0.049601,260 +0.049980,260 +0.058061,260 +0.060819,260 +0.065946,260 +0.056675,260 +0.071167,260 +0.054238,260 +0.068391,262 +0.054044,262 +0.063854,262 +0.055572,262 +0.055039,262 +0.053090,262 +0.059861,262 +0.075603,262 +0.060356,262 +0.053670,262 +0.053980,262 +0.057140,262 +0.060211,262 +0.057499,262 +0.052951,262 +0.051282,262 +0.051657,262 +0.048438,262 +0.050060,262 +0.050079,262 +0.056088,262 +0.050441,262 +0.054786,262 +0.054944,262 +0.061158,262 +0.063592,262 +0.049756,262 +0.049989,262 +0.047498,262 +0.048026,262 +0.048181,262 +0.051458,262 +0.053053,262 +0.052789,262 +0.054084,262 +0.058270,262 +0.048062,262 +0.056312,262 +0.048521,262 +0.061651,262 +0.056249,262 +0.055687,262 +0.052286,262 +0.055558,262 +0.053551,262 +0.052897,262 +0.055583,262 +0.052118,262 +0.055595,262 +0.052332,262 +0.052024,262 +0.050782,262 +0.050073,262 +0.047154,262 +0.050550,262 +0.046832,262 +0.051025,262 +0.051053,262 +0.054520,262 +0.051870,262 +0.050002,262 +0.046938,262 +0.049972,262 +0.046934,262 +0.049755,262 +0.049122,262 +0.054521,262 +0.053756,262 +0.052077,262 +0.049274,262 +0.049231,262 +0.047972,262 +0.048660,262 +0.049238,262 +0.053002,262 +0.050456,262 +0.051622,262 +0.050363,262 +0.050675,262 +0.057281,262 +0.056032,262 +0.057048,262 +0.051817,262 +0.049484,262 +0.048476,262 +0.048065,262 +0.057571,262 +0.053732,262 +0.050675,262 +0.048509,262 +0.049214,262 +0.047276,262 +0.053407,262 +0.056771,262 +0.063283,262 +0.055163,262 +0.054900,262 +0.054134,262 +0.054612,262 +0.053658,262 +0.056285,264 +0.052211,264 +0.050864,264 +0.049217,264 +0.050019,264 +0.050148,264 +0.050645,264 +0.050201,264 +0.049772,264 +0.049309,264 +0.047067,264 +0.049003,264 +0.047070,264 +0.047391,264 +0.046381,264 +0.047166,264 +0.048396,264 +0.047551,264 +0.046881,264 +0.047159,264 +0.047023,264 +0.047268,264 +0.046398,264 +0.046534,264 +0.046563,264 +0.046307,264 +0.047036,264 +0.046239,264 +0.046973,264 +0.046144,264 +0.047086,264 +0.046353,264 +0.047711,264 +0.046246,264 +0.047048,264 +0.046189,264 +0.048609,264 +0.046199,264 +0.046267,264 +0.047407,264 +0.046089,264 +0.047786,264 +0.046747,264 +0.046999,264 +0.046237,264 +0.046987,264 +0.046402,264 +0.048332,264 +0.047240,264 +0.047567,264 +0.047951,264 +0.048270,264 +0.046454,264 +0.048006,264 +0.046843,264 +0.046841,264 +0.047899,264 +0.046480,264 +0.048260,264 +0.046080,264 +0.048094,264 +0.046167,264 +0.048128,264 +0.051792,264 +0.050430,264 +0.049137,264 +0.048897,264 +0.048034,264 +0.047286,264 +0.046188,264 +0.046870,264 +0.046286,264 +0.046630,264 +0.046781,264 +0.046262,264 +0.049960,264 +0.051342,264 +0.049265,264 +0.046303,264 +0.047130,264 +0.046226,264 +0.047704,264 +0.046374,264 +0.046911,264 +0.046082,264 +0.047771,264 +0.046475,264 +0.049328,264 +0.046435,264 +0.046622,264 +0.046266,264 +0.046550,264 +0.048710,264 +0.049719,264 +0.048527,264 +0.048581,264 +0.050060,264 +0.049059,264 +0.047369,264 +0.046636,264 +0.051725,266 +0.051314,266 +0.053410,266 +0.055430,266 +0.054625,266 +0.053391,266 +0.051943,266 +0.051466,266 +0.050024,266 +0.050617,266 +0.049899,266 +0.050297,266 +0.050409,266 +0.050722,266 +0.050274,266 +0.050730,266 +0.054686,266 +0.053915,266 +0.051247,266 +0.051398,266 +0.050113,266 +0.050549,266 +0.054031,266 +0.051283,266 +0.049816,266 +0.050561,266 +0.050195,266 +0.050661,266 +0.049893,266 +0.050586,266 +0.050073,266 +0.050903,266 +0.050330,266 +0.050627,266 +0.049771,266 +0.050922,266 +0.051920,266 +0.053144,266 +0.053605,266 +0.053413,266 +0.052102,266 +0.052975,266 +0.052856,266 +0.054334,266 +0.053598,266 +0.052921,266 +0.052794,266 +0.052685,266 +0.052804,266 +0.051450,266 +0.051631,266 +0.050223,266 +0.050900,266 +0.058479,266 +0.056501,266 +0.055018,266 +0.054116,266 +0.052089,266 +0.052082,266 +0.050745,266 +0.051203,266 +0.050035,266 +0.050797,266 +0.050062,266 +0.051296,266 +0.052480,266 +0.050936,266 +0.057681,266 +0.058276,266 +0.055856,266 +0.053685,266 +0.053777,266 +0.053911,266 +0.052941,266 +0.051741,266 +0.050494,266 +0.055502,266 +0.059932,266 +0.054018,266 +0.052270,266 +0.050435,266 +0.077285,266 +0.054011,266 +0.050785,266 +0.054649,266 +0.052420,266 +0.052819,266 +0.054659,266 +0.054409,266 +0.054260,266 +0.053533,266 +0.053754,266 +0.052733,266 +0.052416,266 +0.051040,266 +0.052033,266 +0.052009,266 +0.052529,266 +0.052487,266 +0.052117,266 +0.052258,268 +0.053511,268 +0.064842,268 +0.059756,268 +0.059964,268 +0.056388,268 +0.060106,268 +0.053551,268 +0.053074,268 +0.052905,268 +0.059101,268 +0.052455,268 +0.053274,268 +0.054025,268 +0.062822,268 +0.056024,268 +0.055818,268 +0.053038,268 +0.052923,268 +0.052858,268 +0.052785,268 +0.052192,268 +0.050187,268 +0.050496,268 +0.052006,268 +0.052511,268 +0.050390,268 +0.050760,268 +0.050301,268 +0.050783,268 +0.050071,268 +0.050577,268 +0.050098,268 +0.050878,268 +0.050046,268 +0.050771,268 +0.050167,268 +0.051274,268 +0.051997,268 +0.051412,268 +0.050190,268 +0.050241,268 +0.051350,268 +0.052092,268 +0.050312,268 +0.050354,268 +0.050081,268 +0.050403,268 +0.050523,268 +0.052009,268 +0.050330,268 +0.050170,268 +0.050514,268 +0.050418,268 +0.050490,268 +0.050214,268 +0.050398,268 +0.050309,268 +0.050384,268 +0.050148,268 +0.050418,268 +0.050590,268 +0.054156,268 +0.053955,268 +0.054889,268 +0.061761,268 +0.066656,268 +0.053520,268 +0.051113,268 +0.050269,268 +0.050530,268 +0.051181,268 +0.050927,268 +0.050556,268 +0.050343,268 +0.050250,268 +0.050333,268 +0.050440,268 +0.050843,268 +0.050310,268 +0.050533,268 +0.050466,268 +0.050440,268 +0.050083,268 +0.050723,268 +0.050275,268 +0.050505,268 +0.050215,268 +0.050782,268 +0.050139,268 +0.050525,268 +0.050406,268 +0.050330,268 +0.050213,268 +0.050382,268 +0.051692,268 +0.051835,268 +0.051324,268 +0.052542,268 +0.050617,268 +0.053960,270 +0.054397,270 +0.053719,270 +0.054373,270 +0.053741,270 +0.054123,270 +0.054112,270 +0.054448,270 +0.053833,270 +0.054301,270 +0.054003,270 +0.054069,270 +0.054124,270 +0.053797,270 +0.054256,270 +0.054018,270 +0.054296,270 +0.053865,270 +0.055765,270 +0.053850,270 +0.054618,270 +0.053893,270 +0.055188,270 +0.058444,270 +0.057940,270 +0.056482,270 +0.058558,270 +0.056220,270 +0.055996,270 +0.058367,270 +0.056417,270 +0.058341,270 +0.062754,270 +0.056253,270 +0.061714,270 +0.057928,270 +0.056349,270 +0.058408,270 +0.058224,270 +0.057903,270 +0.057873,270 +0.059330,270 +0.060017,270 +0.058020,270 +0.057583,270 +0.056950,270 +0.056838,270 +0.057761,270 +0.057302,270 +0.057690,270 +0.059148,270 +0.057890,270 +0.057142,270 +0.057580,270 +0.057715,270 +0.058074,270 +0.057601,270 +0.058007,270 +0.057046,270 +0.056160,270 +0.056786,270 +0.065296,270 +0.060188,270 +0.070213,270 +0.060761,270 +0.060641,270 +0.056608,270 +0.054322,270 +0.056702,270 +0.068307,270 +0.061488,270 +0.059540,270 +0.059322,270 +0.062787,270 +0.056925,270 +0.062799,270 +0.058443,270 +0.057363,270 +0.057748,270 +0.060106,270 +0.064343,270 +0.061347,270 +0.062039,270 +0.063974,270 +0.064517,270 +0.060165,270 +0.062633,270 +0.059059,270 +0.056930,270 +0.054548,270 +0.061077,270 +0.053595,270 +0.059761,270 +0.053469,270 +0.059381,270 +0.059112,270 +0.053274,270 +0.059535,270 +0.053350,270 +0.058998,270 +0.051533,272 +0.056155,272 +0.056381,272 +0.050555,272 +0.056389,272 +0.050467,272 +0.056186,272 +0.050446,272 +0.056900,272 +0.052880,272 +0.056857,272 +0.050859,272 +0.056792,272 +0.050541,272 +0.056909,272 +0.054983,272 +0.051940,272 +0.056142,272 +0.050491,272 +0.057792,272 +0.058770,272 +0.056321,272 +0.057960,272 +0.056229,272 +0.058982,272 +0.056232,272 +0.056269,272 +0.056030,272 +0.057696,272 +0.054746,272 +0.053061,272 +0.051260,272 +0.051097,272 +0.050458,272 +0.050925,272 +0.051690,272 +0.051156,272 +0.051179,272 +0.052154,272 +0.050605,272 +0.051061,272 +0.050549,272 +0.051130,272 +0.051958,272 +0.053018,272 +0.053542,272 +0.051333,272 +0.054512,272 +0.055631,272 +0.056081,272 +0.056137,272 +0.056946,272 +0.055434,272 +0.060452,272 +0.052961,272 +0.055519,272 +0.055727,272 +0.056935,272 +0.052991,272 +0.050753,272 +0.051385,272 +0.050604,272 +0.052218,272 +0.052902,272 +0.053580,272 +0.051193,272 +0.052471,272 +0.050534,272 +0.052734,272 +0.050638,272 +0.052118,272 +0.050569,272 +0.052254,272 +0.050403,272 +0.052877,272 +0.050546,272 +0.051199,272 +0.050424,272 +0.051033,272 +0.050499,272 +0.050719,272 +0.050657,272 +0.051270,272 +0.050571,272 +0.051184,272 +0.050471,272 +0.051083,272 +0.050534,272 +0.051226,272 +0.050497,272 +0.050930,272 +0.050515,272 +0.050652,272 +0.050458,272 +0.050849,272 +0.050960,272 +0.050717,272 +0.050660,272 +0.050604,272 +0.050627,272 +0.055355,274 +0.055286,274 +0.055058,274 +0.077759,274 +0.062612,274 +0.084260,274 +0.104903,274 +0.106103,274 +0.077543,274 +0.085944,274 +0.080646,274 +0.056136,274 +0.056251,274 +0.056378,274 +0.056039,274 +0.056316,274 +0.056571,274 +0.068323,274 +0.065931,274 +0.059785,274 +0.058998,274 +0.056701,274 +0.061016,274 +0.062724,274 +0.056139,274 +0.057446,274 +0.057002,274 +0.059934,274 +0.060444,274 +0.060331,274 +0.070187,274 +0.102343,274 +0.101983,274 +0.085944,274 +0.057725,274 +0.060507,274 +0.078247,274 +0.073117,274 +0.066163,274 +0.060576,274 +0.059118,274 +0.064332,274 +0.068698,274 +0.063827,274 +0.068858,274 +0.062736,274 +0.068263,274 +0.061657,274 +0.059037,274 +0.062543,274 +0.062379,274 +0.065076,274 +0.061394,274 +0.065859,274 +0.059628,274 +0.060438,274 +0.058751,274 +0.060736,274 +0.056983,274 +0.059483,274 +0.058811,274 +0.060062,274 +0.059005,274 +0.058105,274 +0.059569,274 +0.061596,274 +0.067420,274 +0.065639,274 +0.065058,274 +0.064804,274 +0.064150,274 +0.061249,274 +0.060533,274 +0.060707,274 +0.060454,274 +0.063287,274 +0.059066,274 +0.058902,274 +0.062817,274 +0.057138,274 +0.058739,274 +0.056870,274 +0.060131,274 +0.058255,274 +0.057846,274 +0.056016,274 +0.059232,274 +0.059195,274 +0.058514,274 +0.066063,274 +0.068293,274 +0.066232,274 +0.064580,274 +0.059091,274 +0.060501,274 +0.060111,274 +0.060989,274 +0.059421,274 +0.065569,274 +0.061068,274 +0.060733,276 +0.055370,276 +0.077650,276 +0.057427,276 +0.057217,276 +0.057550,276 +0.057177,276 +0.056801,276 +0.062281,276 +0.056746,276 +0.061197,276 +0.057997,276 +0.058909,276 +0.058194,276 +0.059997,276 +0.058988,276 +0.058429,276 +0.064624,276 +0.057617,276 +0.060444,276 +0.060571,276 +0.058261,276 +0.060075,276 +0.059849,276 +0.060039,276 +0.057928,276 +0.058232,276 +0.060680,276 +0.062127,276 +0.060752,276 +0.062138,276 +0.058386,276 +0.058208,276 +0.057690,276 +0.055231,276 +0.055463,276 +0.056137,276 +0.066303,276 +0.060803,276 +0.061295,276 +0.060621,276 +0.061494,276 +0.065527,276 +0.065753,276 +0.069152,276 +0.060415,276 +0.060932,276 +0.060823,276 +0.060634,276 +0.068160,276 +0.071643,276 +0.067290,276 +0.066832,276 +0.066297,276 +0.069288,276 +0.066156,276 +0.061526,276 +0.059984,276 +0.065361,276 +0.059491,276 +0.056447,276 +0.059148,276 +0.059728,276 +0.057638,276 +0.060436,276 +0.074956,276 +0.064729,276 +0.071866,276 +0.078464,276 +0.065329,276 +0.082083,276 +0.060430,276 +0.061510,276 +0.069063,276 +0.064846,276 +0.058104,276 +0.056249,276 +0.059217,276 +0.063967,276 +0.062880,276 +0.055427,276 +0.061798,276 +0.057072,276 +0.065297,276 +0.064682,276 +0.062539,276 +0.060433,276 +0.061693,276 +0.064777,276 +0.062460,276 +0.061834,276 +0.057039,276 +0.063286,276 +0.058734,276 +0.062493,276 +0.061756,276 +0.059587,276 +0.058489,276 +0.058649,276 +0.055728,276 +0.058655,278 +0.058376,278 +0.057495,278 +0.058710,278 +0.058263,278 +0.066848,278 +0.065266,278 +0.058434,278 +0.066578,278 +0.066915,278 +0.069207,278 +0.070718,278 +0.063471,278 +0.061312,278 +0.059197,278 +0.059363,278 +0.058063,278 +0.058907,278 +0.060127,278 +0.058821,278 +0.058679,278 +0.058371,278 +0.060246,278 +0.062742,278 +0.058916,278 +0.059371,278 +0.059822,278 +0.060244,278 +0.062157,278 +0.063462,278 +0.066566,278 +0.068093,278 +0.071604,278 +0.073728,278 +0.069347,278 +0.070758,278 +0.070580,278 +0.062849,278 +0.065264,278 +0.058941,278 +0.058012,278 +0.063789,278 +0.066193,278 +0.072972,278 +0.069772,278 +0.065957,278 +0.070013,278 +0.067595,278 +0.073879,278 +0.063638,278 +0.067512,278 +0.068782,278 +0.070986,278 +0.076221,278 +0.074177,278 +0.066303,278 +0.062857,278 +0.063683,278 +0.061570,278 +0.062001,278 +0.060927,278 +0.062514,278 +0.059205,278 +0.058478,278 +0.059849,278 +0.063694,278 +0.065171,278 +0.072492,278 +0.065300,278 +0.064288,278 +0.064443,278 +0.063192,278 +0.060911,278 +0.063925,278 +0.059304,278 +0.059943,278 +0.059610,278 +0.057801,278 +0.059888,278 +0.058579,278 +0.059326,278 +0.059327,278 +0.058302,278 +0.063137,278 +0.061572,278 +0.060818,278 +0.059413,278 +0.057830,278 +0.063835,278 +0.057556,278 +0.064423,278 +0.063405,278 +0.058073,278 +0.072706,278 +0.068393,278 +0.068173,278 +0.067488,278 +0.064341,278 +0.065489,278 +0.061618,278 +0.063219,280 +0.060395,280 +0.061169,280 +0.060930,280 +0.057626,280 +0.055814,280 +0.056872,280 +0.056238,280 +0.057810,280 +0.056633,280 +0.056554,280 +0.057295,280 +0.056093,280 +0.056544,280 +0.056032,280 +0.056596,280 +0.056284,280 +0.055857,280 +0.056353,280 +0.055744,280 +0.056309,280 +0.055758,280 +0.056570,280 +0.056137,280 +0.056888,280 +0.056344,280 +0.055935,280 +0.056404,280 +0.055796,280 +0.056614,280 +0.055893,280 +0.056381,280 +0.056788,280 +0.056024,280 +0.056255,280 +0.056147,280 +0.056318,280 +0.055886,280 +0.056246,280 +0.055850,280 +0.056286,280 +0.056400,280 +0.055959,280 +0.056493,280 +0.055887,280 +0.056421,280 +0.056425,280 +0.069027,280 +0.063965,280 +0.062629,280 +0.063214,280 +0.066614,280 +0.061499,280 +0.061985,280 +0.063162,280 +0.065707,280 +0.060343,280 +0.060719,280 +0.063382,280 +0.060085,280 +0.063920,280 +0.062268,280 +0.060334,280 +0.057546,280 +0.066008,280 +0.056917,280 +0.059449,280 +0.061266,280 +0.061639,280 +0.061490,280 +0.061053,280 +0.062394,280 +0.061806,280 +0.066203,280 +0.065063,280 +0.068803,280 +0.073897,280 +0.071415,280 +0.061651,280 +0.062140,280 +0.068908,280 +0.069307,280 +0.067386,280 +0.062752,280 +0.069230,280 +0.070114,280 +0.064809,280 +0.060479,280 +0.061138,280 +0.063538,280 +0.061519,280 +0.058772,280 +0.059980,280 +0.061173,280 +0.061562,280 +0.058449,280 +0.059663,280 +0.071151,280 +0.060882,280 +0.066724,280 +0.072016,282 +0.074860,282 +0.070270,282 +0.073070,282 +0.069344,282 +0.076691,282 +0.086220,282 +0.078314,282 +0.068736,282 +0.066885,282 +0.068434,282 +0.066834,282 +0.066249,282 +0.072824,282 +0.075319,282 +0.071495,282 +0.069390,282 +0.068081,282 +0.067893,282 +0.068946,282 +0.067668,282 +0.068085,282 +0.067531,282 +0.069401,282 +0.076482,282 +0.074109,282 +0.068777,282 +0.070797,282 +0.071662,282 +0.070121,282 +0.097231,282 +0.079167,282 +0.069282,282 +0.069333,282 +0.073691,282 +0.068289,282 +0.072362,282 +0.068569,282 +0.068555,282 +0.069764,282 +0.072091,282 +0.065155,282 +0.064922,282 +0.070280,282 +0.064525,282 +0.066633,282 +0.062893,282 +0.064499,282 +0.062042,282 +0.061602,282 +0.061429,282 +0.061333,282 +0.061341,282 +0.060931,282 +0.061759,282 +0.061270,282 +0.060762,282 +0.061671,282 +0.060706,282 +0.061936,282 +0.065297,282 +0.061463,282 +0.062186,282 +0.061000,282 +0.061329,282 +0.061405,282 +0.062654,282 +0.064562,282 +0.064346,282 +0.064517,282 +0.062018,282 +0.062551,282 +0.063137,282 +0.061378,282 +0.061309,282 +0.061441,282 +0.062240,282 +0.063585,282 +0.070333,282 +0.064389,282 +0.062171,282 +0.065352,282 +0.066117,282 +0.067361,282 +0.064329,282 +0.061419,282 +0.061788,282 +0.061899,282 +0.060952,282 +0.062098,282 +0.064113,282 +0.064734,282 +0.065608,282 +0.062708,282 +0.062502,282 +0.061444,282 +0.062500,282 +0.063233,282 +0.062965,282 +0.062939,282 +0.061046,284 +0.060496,284 +0.060836,284 +0.062635,284 +0.061098,284 +0.062422,284 +0.060454,284 +0.061306,284 +0.060970,284 +0.059322,284 +0.060996,284 +0.059282,284 +0.062343,284 +0.061150,284 +0.059339,284 +0.062270,284 +0.059404,284 +0.060982,284 +0.061201,284 +0.059487,284 +0.061295,284 +0.059402,284 +0.061040,284 +0.061154,284 +0.059433,284 +0.061046,284 +0.059347,284 +0.061015,284 +0.062313,284 +0.060296,284 +0.061069,284 +0.059390,284 +0.061376,284 +0.061045,284 +0.059254,284 +0.061072,284 +0.059390,284 +0.061039,284 +0.060863,284 +0.060137,284 +0.061132,284 +0.059248,284 +0.061549,284 +0.061096,284 +0.059492,284 +0.062208,284 +0.059378,284 +0.064014,284 +0.061461,284 +0.059358,284 +0.062296,284 +0.074642,284 +0.071049,284 +0.066325,284 +0.069245,284 +0.068743,284 +0.070516,284 +0.071255,284 +0.067085,284 +0.065974,284 +0.066825,284 +0.067573,284 +0.068968,284 +0.064459,284 +0.061483,284 +0.075049,284 +0.065255,284 +0.077691,284 +0.090725,284 +0.067188,284 +0.065421,284 +0.070782,284 +0.066749,284 +0.067901,284 +0.065804,284 +0.064306,284 +0.061525,284 +0.060358,284 +0.062015,284 +0.060604,284 +0.060064,284 +0.061275,284 +0.059862,284 +0.061431,284 +0.062032,284 +0.061327,284 +0.061746,284 +0.059494,284 +0.060965,284 +0.060872,284 +0.059400,284 +0.062127,284 +0.059301,284 +0.061337,284 +0.061756,284 +0.059648,284 +0.060987,284 +0.059261,284 +0.061118,284 +0.061140,284 +0.063425,286 +0.064737,286 +0.064355,286 +0.063657,286 +0.064667,286 +0.063042,286 +0.064707,286 +0.065789,286 +0.063100,286 +0.065326,286 +0.066364,286 +0.063675,286 +0.064873,286 +0.064375,286 +0.063434,286 +0.063620,286 +0.063581,286 +0.064280,286 +0.063903,286 +0.063200,286 +0.063643,286 +0.063740,286 +0.063307,286 +0.063884,286 +0.063368,286 +0.063508,286 +0.063826,286 +0.063174,286 +0.063968,286 +0.064779,286 +0.063249,286 +0.065129,286 +0.065503,286 +0.066801,286 +0.065673,286 +0.066431,286 +0.066196,286 +0.065085,286 +0.064430,286 +0.064809,286 +0.063816,286 +0.063359,286 +0.063642,286 +0.063762,286 +0.063233,286 +0.063836,286 +0.063836,286 +0.063249,286 +0.064058,286 +0.063400,286 +0.063894,286 +0.063750,286 +0.063127,286 +0.063740,286 +0.064056,286 +0.063366,286 +0.063874,286 +0.063578,286 +0.063819,286 +0.063598,286 +0.063287,286 +0.063787,286 +0.063828,286 +0.063347,286 +0.064371,286 +0.063440,286 +0.063083,286 +0.063631,286 +0.063373,286 +0.063355,286 +0.063701,286 +0.063124,286 +0.063970,286 +0.063441,286 +0.062995,286 +0.063621,286 +0.063674,286 +0.065323,286 +0.064494,286 +0.067087,286 +0.064575,286 +0.064647,286 +0.063564,286 +0.064806,286 +0.064180,286 +0.063215,286 +0.063833,286 +0.063644,286 +0.063111,286 +0.063867,286 +0.063236,286 +0.063614,286 +0.063930,286 +0.062981,286 +0.063888,286 +0.063725,286 +0.063463,286 +0.063839,286 +0.063619,286 +0.063223,286 +0.064376,288 +0.063707,288 +0.064364,288 +0.064462,288 +0.066459,288 +0.064684,288 +0.065017,288 +0.063947,288 +0.064447,288 +0.064137,288 +0.063646,288 +0.065175,288 +0.070360,288 +0.071598,288 +0.071383,288 +0.070829,288 +0.069477,288 +0.069394,288 +0.068970,288 +0.069672,288 +0.069716,288 +0.071996,288 +0.074610,288 +0.068118,288 +0.065774,288 +0.069773,288 +0.074340,288 +0.070091,288 +0.070158,288 +0.068244,288 +0.067423,288 +0.068650,288 +0.069801,288 +0.070965,288 +0.071539,288 +0.071692,288 +0.070481,288 +0.078654,288 +0.072991,288 +0.067273,288 +0.064580,288 +0.065642,288 +0.067566,288 +0.066009,288 +0.066068,288 +0.068610,288 +0.069114,288 +0.068622,288 +0.065788,288 +0.064485,288 +0.065000,288 +0.066887,288 +0.074228,288 +0.069965,288 +0.077213,288 +0.066733,288 +0.071594,288 +0.069941,288 +0.064325,288 +0.070085,288 +0.065749,288 +0.065430,288 +0.083415,288 +0.071016,288 +0.076565,288 +0.072772,288 +0.069500,288 +0.069351,288 +0.066924,288 +0.066275,288 +0.064787,288 +0.066885,288 +0.064586,288 +0.064665,288 +0.067937,288 +0.069406,288 +0.079028,288 +0.082235,288 +0.071772,288 +0.065477,288 +0.068255,288 +0.070726,288 +0.070777,288 +0.070487,288 +0.066012,288 +0.067938,288 +0.068077,288 +0.073226,288 +0.066543,288 +0.066800,288 +0.065631,288 +0.085718,288 +0.089092,288 +0.087413,288 +0.070510,288 +0.066891,288 +0.066570,288 +0.064278,288 +0.064291,288 +0.066164,288 +0.066856,290 +0.066542,290 +0.066700,290 +0.066331,290 +0.066329,290 +0.066563,290 +0.066186,290 +0.066713,290 +0.066740,290 +0.066146,290 +0.066181,290 +0.066832,290 +0.066045,290 +0.066206,290 +0.066781,290 +0.066178,290 +0.066023,290 +0.066379,290 +0.074327,290 +0.067359,290 +0.066890,290 +0.066038,290 +0.066210,290 +0.066857,290 +0.065996,290 +0.066408,290 +0.066721,290 +0.066083,290 +0.066572,290 +0.066601,290 +0.065704,290 +0.066854,290 +0.066663,290 +0.065723,290 +0.066815,290 +0.066464,290 +0.065716,290 +0.066601,290 +0.066705,290 +0.065860,290 +0.066663,290 +0.066405,290 +0.065973,290 +0.066908,290 +0.066657,290 +0.065943,290 +0.066938,290 +0.066389,290 +0.065999,290 +0.066745,290 +0.066303,290 +0.066258,290 +0.067397,290 +0.066411,290 +0.066299,290 +0.066642,290 +0.066156,290 +0.074817,290 +0.070316,290 +0.068684,290 +0.066628,290 +0.067040,290 +0.066562,290 +0.066173,290 +0.067013,290 +0.066810,290 +0.065973,290 +0.066960,290 +0.066530,290 +0.066101,290 +0.066809,290 +0.066522,290 +0.066021,290 +0.066746,290 +0.066434,290 +0.066097,290 +0.066708,290 +0.066651,290 +0.066335,290 +0.066961,290 +0.066217,290 +0.066568,290 +0.066737,290 +0.065997,290 +0.067041,290 +0.067766,290 +0.065829,290 +0.066719,290 +0.066944,290 +0.065817,290 +0.066878,290 +0.066380,290 +0.066074,290 +0.067010,290 +0.067114,290 +0.065857,290 +0.066773,290 +0.066490,290 +0.065858,290 +0.066554,290 +0.066309,292 +0.065868,292 +0.066050,292 +0.066186,292 +0.067014,292 +0.070358,292 +0.066037,292 +0.066376,292 +0.066464,292 +0.065910,292 +0.067182,292 +0.066370,292 +0.066138,292 +0.066665,292 +0.072453,292 +0.065768,292 +0.065263,292 +0.067144,292 +0.065889,292 +0.065232,292 +0.065898,292 +0.065375,292 +0.065452,292 +0.065864,292 +0.065250,292 +0.065853,292 +0.065723,292 +0.064962,292 +0.066235,292 +0.065645,292 +0.065551,292 +0.066315,292 +0.066393,292 +0.066390,292 +0.066038,292 +0.065460,292 +0.066657,292 +0.066426,292 +0.065633,292 +0.067215,292 +0.066436,292 +0.065902,292 +0.065923,292 +0.066517,292 +0.065837,292 +0.065519,292 +0.066377,292 +0.065731,292 +0.065825,292 +0.066306,292 +0.065418,292 +0.065969,292 +0.066479,292 +0.065072,292 +0.066340,292 +0.065952,292 +0.065452,292 +0.066103,292 +0.066080,292 +0.065526,292 +0.066222,292 +0.065902,292 +0.065540,292 +0.066321,292 +0.065830,292 +0.065732,292 +0.067067,292 +0.066852,292 +0.065767,292 +0.066326,292 +0.066414,292 +0.065622,292 +0.066265,292 +0.066090,292 +0.065993,292 +0.066376,292 +0.065403,292 +0.066079,292 +0.066214,292 +0.065233,292 +0.066158,292 +0.066097,292 +0.065168,292 +0.066237,292 +0.065953,292 +0.065484,292 +0.066720,292 +0.072731,292 +0.076935,292 +0.080018,292 +0.082379,292 +0.080917,292 +0.074605,292 +0.068439,292 +0.069450,292 +0.068890,292 +0.068080,292 +0.068419,292 +0.068819,292 +0.069171,292 +0.076822,294 +0.076086,294 +0.078940,294 +0.074760,294 +0.074426,294 +0.073137,294 +0.072081,294 +0.070823,294 +0.072792,294 +0.072130,294 +0.072409,294 +0.072391,294 +0.072992,294 +0.079275,294 +0.086944,294 +0.083957,294 +0.076999,294 +0.076847,294 +0.078900,294 +0.071525,294 +0.071786,294 +0.072034,294 +0.072026,294 +0.074384,294 +0.074108,294 +0.071156,294 +0.070194,294 +0.070710,294 +0.070922,294 +0.070716,294 +0.070229,294 +0.070987,294 +0.070825,294 +0.069836,294 +0.071088,294 +0.070921,294 +0.070434,294 +0.070411,294 +0.071038,294 +0.070787,294 +0.070330,294 +0.071036,294 +0.070979,294 +0.070156,294 +0.070996,294 +0.070858,294 +0.070398,294 +0.070803,294 +0.071150,294 +0.070646,294 +0.071168,294 +0.071539,294 +0.070709,294 +0.069758,294 +0.071038,294 +0.071165,294 +0.070349,294 +0.073770,294 +0.070953,294 +0.072926,294 +0.069917,294 +0.071281,294 +0.071955,294 +0.071535,294 +0.077098,294 +0.074606,294 +0.074303,294 +0.071320,294 +0.070532,294 +0.070896,294 +0.070341,294 +0.069827,294 +0.070714,294 +0.070326,294 +0.072954,294 +0.074486,294 +0.073188,294 +0.070619,294 +0.070657,294 +0.070815,294 +0.071126,294 +0.074661,294 +0.073268,294 +0.076067,294 +0.083637,294 +0.074073,294 +0.082686,294 +0.077119,294 +0.075453,294 +0.076002,294 +0.071392,294 +0.072592,294 +0.074008,294 +0.069877,294 +0.071344,294 +0.070870,294 +0.071150,294 +0.071804,294 +0.071079,294 +0.071553,294 +0.066974,296 +0.068582,296 +0.068594,296 +0.067236,296 +0.068186,296 +0.068300,296 +0.066251,296 +0.068456,296 +0.068182,296 +0.067526,296 +0.068491,296 +0.069412,296 +0.066596,296 +0.068758,296 +0.068368,296 +0.066651,296 +0.068386,296 +0.068434,296 +0.066997,296 +0.068163,296 +0.068334,296 +0.067285,296 +0.071464,296 +0.072527,296 +0.069374,296 +0.066762,296 +0.069384,296 +0.068125,296 +0.066689,296 +0.072023,296 +0.074263,296 +0.066657,296 +0.068572,296 +0.068314,296 +0.066261,296 +0.068553,296 +0.067026,296 +0.066611,296 +0.067615,296 +0.067246,296 +0.066506,296 +0.067668,296 +0.067284,296 +0.066467,296 +0.067471,296 +0.067280,296 +0.067456,296 +0.068320,296 +0.068635,296 +0.066139,296 +0.068106,296 +0.068149,296 +0.066075,296 +0.068784,296 +0.068268,296 +0.066233,296 +0.069456,296 +0.067947,296 +0.066422,296 +0.067812,296 +0.068153,296 +0.066867,296 +0.067462,296 +0.068132,296 +0.067164,296 +0.067326,296 +0.068220,296 +0.067258,296 +0.067231,296 +0.068053,296 +0.068183,296 +0.067775,296 +0.068161,296 +0.067675,296 +0.067005,296 +0.068616,296 +0.067729,296 +0.066907,296 +0.068243,296 +0.067927,296 +0.066670,296 +0.068298,296 +0.067857,296 +0.067380,296 +0.068326,296 +0.068842,296 +0.067039,296 +0.068302,296 +0.068213,296 +0.066634,296 +0.068386,296 +0.067673,296 +0.066894,296 +0.068327,296 +0.066926,296 +0.066740,296 +0.067472,296 +0.066996,296 +0.066620,296 +0.067563,296 +0.074082,298 +0.072820,298 +0.073513,298 +0.073754,298 +0.074084,298 +0.073198,298 +0.073715,298 +0.073480,298 +0.073005,298 +0.073204,298 +0.073714,298 +0.073746,298 +0.072662,298 +0.073693,298 +0.073421,298 +0.073221,298 +0.072732,298 +0.073666,298 +0.073410,298 +0.072768,298 +0.073171,298 +0.073607,298 +0.072946,298 +0.072686,298 +0.073423,298 +0.073634,298 +0.074629,298 +0.072765,298 +0.073651,298 +0.073395,298 +0.072872,298 +0.073192,298 +0.073592,298 +0.073454,298 +0.072587,298 +0.073284,298 +0.073709,298 +0.073488,298 +0.072771,298 +0.073605,298 +0.073534,298 +0.073027,298 +0.073515,298 +0.073689,298 +0.073708,298 +0.072601,298 +0.073632,298 +0.073809,298 +0.073004,298 +0.073058,298 +0.073719,298 +0.073876,298 +0.073224,298 +0.073554,298 +0.073837,298 +0.073483,298 +0.073045,298 +0.073656,298 +0.073569,298 +0.073026,298 +0.075732,298 +0.079497,298 +0.077026,298 +0.076725,298 +0.075296,298 +0.074288,298 +0.075321,298 +0.077556,298 +0.077182,298 +0.080477,298 +0.079437,298 +0.079661,298 +0.079812,298 +0.075677,298 +0.074235,298 +0.073422,298 +0.073126,298 +0.072768,298 +0.073769,298 +0.074062,298 +0.073833,298 +0.074786,298 +0.074297,298 +0.073377,298 +0.072905,298 +0.076307,298 +0.078593,298 +0.074184,298 +0.073057,298 +0.073524,298 +0.073492,298 +0.073582,298 +0.072937,298 +0.073603,298 +0.073564,298 +0.072902,298 +0.073522,298 +0.079887,298 +0.083117,298 +0.075722,298 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 @@ -0,0 +1,14900 @@ +0.000001,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000000,6 +0.000001,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000001,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000010,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000010,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000010,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000010,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000010,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000001,10 +0.000002,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000001,12 +0.000003,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000013,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000002,14 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000005,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000006,18 +0.000007,18 +0.000007,18 +0.000007,18 +0.000006,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000013,18 +0.000004,18 +0.000013,18 +0.000004,18 +0.000013,18 +0.000004,18 +0.000013,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000013,18 +0.000004,18 +0.000013,18 +0.000004,18 +0.000014,18 +0.000004,18 +0.000014,18 +0.000004,18 +0.000014,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000004,18 +0.000007,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000005,20 +0.000008,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000011,22 +0.000012,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000016,22 +0.000007,22 +0.000016,22 +0.000007,22 +0.000016,22 +0.000016,22 +0.000016,22 +0.000007,22 +0.000016,22 +0.000016,22 +0.000016,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000016,22 +0.000007,22 +0.000016,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000006,22 +0.000010,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000010,24 +0.000018,24 +0.000018,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000008,24 +0.000013,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000020,26 +0.000011,26 +0.000020,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000020,26 +0.000020,26 +0.000021,26 +0.000031,26 +0.000011,26 +0.000019,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000020,26 +0.000020,26 +0.000020,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000010,26 +0.000015,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000022,28 +0.000022,28 +0.000021,28 +0.000022,28 +0.000021,28 +0.000023,28 +0.000022,28 +0.000023,28 +0.000022,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000024,28 +0.000024,28 +0.000025,28 +0.000035,28 +0.000048,28 +0.000055,28 +0.000045,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000026,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000023,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000020,28 +0.000025,28 +0.000025,28 +0.000025,28 +0.000024,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000013,28 +0.000025,28 +0.000032,30 +0.000031,30 +0.000030,30 +0.000031,30 +0.000030,30 +0.000030,30 +0.000073,30 +0.000030,30 +0.000030,30 +0.000031,30 +0.000017,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000019,30 +0.000030,30 +0.000030,30 +0.000030,30 +0.000030,30 +0.000030,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000026,30 +0.000029,30 +0.000040,30 +0.000041,30 +0.000041,30 +0.000040,30 +0.000038,30 +0.000042,30 +0.000023,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000027,30 +0.000030,30 +0.000030,30 +0.000040,30 +0.000040,30 +0.000031,30 +0.000028,30 +0.000028,30 +0.000024,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000015,30 +0.000021,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000021,32 +0.000025,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000028,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000029,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000028,32 +0.000039,32 +0.000043,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000022,32 +0.000028,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000040,32 +0.000038,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000019,32 +0.000043,32 +0.000047,32 +0.000031,32 +0.000047,34 +0.000039,34 +0.000035,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000022,34 +0.000027,34 +0.000050,34 +0.000023,34 +0.000023,34 +0.000023,34 +0.000023,34 +0.000023,34 +0.000033,34 +0.000023,34 +0.000042,36 +0.000028,36 +0.000037,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000037,36 +0.000036,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000043,36 +0.000050,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000027,36 +0.000037,36 +0.000036,36 +0.000037,36 +0.000059,36 +0.000049,36 +0.000027,36 +0.000037,36 +0.000057,36 +0.000048,36 +0.000046,36 +0.000047,36 +0.000027,36 +0.000046,36 +0.000027,36 +0.000027,36 +0.000035,38 +0.000036,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000051,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000042,38 +0.000041,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000051,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000031,38 +0.000044,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000086,40 +0.000090,40 +0.000051,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000058,40 +0.000062,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000042,40 +0.000053,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000059,42 +0.000077,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000068,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000068,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000064,42 +0.000058,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000048,42 +0.000058,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000078,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000078,44 +0.000095,44 +0.000073,44 +0.000096,44 +0.000097,44 +0.000085,44 +0.000064,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000065,44 +0.000064,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000075,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000054,44 +0.000068,46 +0.000062,46 +0.000078,46 +0.000083,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000085,46 +0.000072,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000081,46 +0.000133,46 +0.000080,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000097,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000061,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000062,46 +0.000073,48 +0.000069,48 +0.000108,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000090,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000089,48 +0.000069,48 +0.000069,48 +0.000095,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000088,48 +0.000079,48 +0.000099,48 +0.000141,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000091,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000069,48 +0.000085,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000092,50 +0.000138,50 +0.000097,50 +0.000151,50 +0.000117,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000098,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000099,50 +0.000077,50 +0.000078,50 +0.000078,50 +0.000078,50 +0.000117,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000150,50 +0.000111,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000078,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000100,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000077,50 +0.000089,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000131,52 +0.000106,52 +0.000147,52 +0.000130,52 +0.000105,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000123,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000106,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000110,52 +0.000128,52 +0.000153,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000110,52 +0.000095,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000086,52 +0.000099,54 +0.000116,54 +0.000161,54 +0.000169,54 +0.000144,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000153,54 +0.000096,54 +0.000096,54 +0.000115,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000114,54 +0.000115,54 +0.000170,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000106,54 +0.000196,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000131,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000139,54 +0.000096,54 +0.000130,54 +0.000096,54 +0.000096,54 +0.000120,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000096,54 +0.000110,56 +0.000125,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000133,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000136,56 +0.000159,56 +0.000159,56 +0.000143,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000126,56 +0.000106,56 +0.000106,56 +0.000125,56 +0.000106,56 +0.000130,56 +0.000135,56 +0.000183,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000172,56 +0.000149,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000106,56 +0.000119,56 +0.000115,56 +0.000123,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000129,58 +0.000127,58 +0.000117,58 +0.000117,58 +0.000156,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000136,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000136,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000152,58 +0.000138,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000117,58 +0.000134,58 +0.000153,60 +0.000178,60 +0.000226,60 +0.000188,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000148,60 +0.000128,60 +0.000128,60 +0.000149,60 +0.000142,60 +0.000132,60 +0.000132,60 +0.000157,60 +0.000190,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000153,60 +0.000132,60 +0.000161,60 +0.000132,60 +0.000132,60 +0.000132,60 +0.000142,60 +0.000180,60 +0.000141,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000212,60 +0.000390,60 +0.000275,60 +0.000261,60 +0.000269,60 +0.000234,60 +0.000272,60 +0.000241,60 +0.000238,60 +0.000326,60 +0.000245,60 +0.000182,60 +0.000150,60 +0.000167,60 +0.000153,60 +0.000138,60 +0.000128,60 +0.000159,60 +0.000249,60 +0.000157,60 +0.000128,60 +0.000164,60 +0.000165,60 +0.000128,60 +0.000128,60 +0.000230,60 +0.000176,60 +0.000244,60 +0.000238,60 +0.000162,60 +0.000128,60 +0.000128,60 +0.000170,60 +0.000148,60 +0.000129,60 +0.000142,60 +0.000128,60 +0.000128,60 +0.000128,60 +0.000180,60 +0.000212,60 +0.000189,60 +0.000191,60 +0.000161,60 +0.000143,60 +0.000166,60 +0.000135,60 +0.000135,60 +0.000135,60 +0.000135,60 +0.000135,60 +0.000135,60 +0.000135,60 +0.000142,60 +0.000169,60 +0.000128,60 +0.000128,60 +0.000144,62 +0.000141,62 +0.000141,62 +0.000188,62 +0.000215,62 +0.000213,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000161,62 +0.000141,62 +0.000161,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000167,62 +0.000145,62 +0.000184,62 +0.000145,62 +0.000223,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000165,62 +0.000165,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000178,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000165,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000178,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000145,62 +0.000150,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000141,62 +0.000156,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000198,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000248,64 +0.000255,64 +0.000163,64 +0.000154,64 +0.000154,64 +0.000174,64 +0.000154,64 +0.000174,64 +0.000153,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000190,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000195,64 +0.000192,64 +0.000154,64 +0.000164,64 +0.000216,64 +0.000164,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000168,64 +0.000173,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000173,64 +0.000173,64 +0.000154,64 +0.000154,64 +0.000173,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000173,64 +0.000189,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000173,64 +0.000174,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000154,64 +0.000172,66 +0.000169,66 +0.000169,66 +0.000194,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000240,66 +0.000283,66 +0.000188,66 +0.000169,66 +0.000169,66 +0.000188,66 +0.000169,66 +0.000189,66 +0.000169,66 +0.000196,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000179,66 +0.000227,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000191,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000179,66 +0.000198,66 +0.000169,66 +0.000169,66 +0.000189,66 +0.000207,66 +0.000169,66 +0.000169,66 +0.000188,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000191,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000169,66 +0.000186,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000198,68 +0.000307,68 +0.000251,68 +0.000183,68 +0.000183,68 +0.000209,68 +0.000223,68 +0.000345,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000303,68 +0.000331,68 +0.000331,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000255,68 +0.000283,68 +0.000183,68 +0.000201,68 +0.000203,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000215,68 +0.000223,68 +0.000183,68 +0.000203,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000203,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000355,68 +0.000196,68 +0.000183,68 +0.000221,68 +0.000183,68 +0.000256,68 +0.000184,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000183,68 +0.000246,68 +0.000312,68 +0.000193,68 +0.000183,68 +0.000183,68 +0.000203,68 +0.000203,68 +0.000218,70 +0.000355,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000237,70 +0.000342,70 +0.000334,70 +0.000339,70 +0.000351,70 +0.000373,70 +0.000355,70 +0.000281,70 +0.000363,70 +0.000341,70 +0.000323,70 +0.000200,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000233,70 +0.000348,70 +0.000199,70 +0.000242,70 +0.000199,70 +0.000199,70 +0.000238,70 +0.000199,70 +0.000210,70 +0.000208,70 +0.000199,70 +0.000199,70 +0.000219,70 +0.000199,70 +0.000378,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000326,70 +0.000209,70 +0.000199,70 +0.000239,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000200,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000341,70 +0.000229,70 +0.000199,70 +0.000291,70 +0.000325,70 +0.000303,70 +0.000199,70 +0.000214,70 +0.000210,70 +0.000210,70 +0.000210,70 +0.000210,70 +0.000214,70 +0.000205,70 +0.000205,70 +0.000210,70 +0.000199,70 +0.000375,70 +0.000347,70 +0.000410,70 +0.000361,70 +0.000362,70 +0.000371,70 +0.000203,70 +0.000199,70 +0.000199,70 +0.000199,70 +0.000382,70 +0.000200,70 +0.000210,70 +0.000345,70 +0.000490,70 +0.000247,70 +0.000255,70 +0.000199,70 +0.000199,70 +0.000359,70 +0.000230,70 +0.000281,70 +0.000210,70 +0.000251,70 +0.000311,72 +0.000259,72 +0.000226,72 +0.000215,72 +0.000235,72 +0.000215,72 +0.000235,72 +0.000367,72 +0.000451,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000250,72 +0.000215,72 +0.000215,72 +0.000393,72 +0.000215,72 +0.000215,72 +0.000293,72 +0.000309,72 +0.000215,72 +0.000304,72 +0.000325,72 +0.000221,72 +0.000221,72 +0.000221,72 +0.000374,72 +0.000227,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000244,72 +0.000356,72 +0.000545,72 +0.000385,72 +0.000391,72 +0.000216,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000420,72 +0.000388,72 +0.000393,72 +0.000223,72 +0.000215,72 +0.000233,72 +0.000270,72 +0.000348,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000254,72 +0.000220,72 +0.000390,72 +0.000215,72 +0.000215,72 +0.000257,72 +0.000308,72 +0.000215,72 +0.000235,72 +0.000301,72 +0.000279,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000215,72 +0.000230,72 +0.000351,72 +0.000215,72 +0.000257,72 +0.000314,72 +0.000254,72 +0.000215,72 +0.000215,72 +0.000281,72 +0.000317,72 +0.000390,72 +0.000292,72 +0.000227,72 +0.000234,72 +0.000221,72 +0.000247,72 +0.000250,72 +0.000227,72 +0.000215,72 +0.000344,74 +0.000473,74 +0.000451,74 +0.000574,74 +0.000248,74 +0.000357,74 +0.000234,74 +0.000233,74 +0.000233,74 +0.000350,74 +0.000234,74 +0.000233,74 +0.000312,74 +0.000311,74 +0.000233,74 +0.000233,74 +0.000314,74 +0.000265,74 +0.000233,74 +0.000234,74 +0.000346,74 +0.000273,74 +0.000234,74 +0.000287,74 +0.000351,74 +0.000234,74 +0.000293,74 +0.000363,74 +0.000233,74 +0.000233,74 +0.000239,74 +0.000348,74 +0.000234,74 +0.000234,74 +0.000302,74 +0.000282,74 +0.000234,74 +0.000234,74 +0.000338,74 +0.000243,74 +0.000233,74 +0.000260,74 +0.000344,74 +0.000234,74 +0.000233,74 +0.000238,74 +0.000346,74 +0.000233,74 +0.000233,74 +0.000299,74 +0.000465,74 +0.000260,74 +0.000274,74 +0.000246,74 +0.000246,74 +0.000287,74 +0.000351,74 +0.000233,74 +0.000233,74 +0.000233,74 +0.000460,74 +0.000435,74 +0.000513,74 +0.000318,74 +0.000234,74 +0.000354,74 +0.000234,74 +0.000233,74 +0.000299,74 +0.000326,74 +0.000233,74 +0.000233,74 +0.000316,74 +0.000256,74 +0.000233,74 +0.000234,74 +0.000351,74 +0.000234,74 +0.000233,74 +0.000234,74 +0.000376,74 +0.000233,74 +0.000253,74 +0.000364,74 +0.000276,74 +0.000234,74 +0.000234,74 +0.000362,74 +0.000234,74 +0.000233,74 +0.000238,74 +0.000346,74 +0.000233,74 +0.000233,74 +0.000313,74 +0.000270,74 +0.000233,74 +0.000269,74 +0.000347,74 +0.000233,74 +0.000260,76 +0.000270,76 +0.000352,76 +0.000252,76 +0.000251,76 +0.000368,76 +0.000251,76 +0.000286,76 +0.000487,76 +0.000252,76 +0.000315,76 +0.000396,76 +0.000262,76 +0.000251,76 +0.000269,76 +0.000351,76 +0.000251,76 +0.000251,76 +0.000444,76 +0.000423,76 +0.000509,76 +0.000443,76 +0.000575,76 +0.000445,76 +0.000520,76 +0.000373,76 +0.000252,76 +0.000372,76 +0.000251,76 +0.000251,76 +0.000279,76 +0.000341,76 +0.000251,76 +0.000312,76 +0.000423,76 +0.000252,76 +0.000251,76 +0.000366,76 +0.000252,76 +0.000251,76 +0.000256,76 +0.000356,76 +0.000252,76 +0.000251,76 +0.000361,76 +0.000251,76 +0.000251,76 +0.000316,76 +0.000334,76 +0.000252,76 +0.000251,76 +0.000362,76 +0.000252,76 +0.000251,76 +0.000268,76 +0.000343,76 +0.000252,76 +0.000251,76 +0.000360,76 +0.000346,76 +0.000320,76 +0.000341,76 +0.000279,76 +0.000296,76 +0.000336,76 +0.000273,76 +0.000251,76 +0.000251,76 +0.000470,76 +0.000439,76 +0.000464,76 +0.000459,76 +0.000586,76 +0.000261,76 +0.000258,76 +0.000345,76 +0.000258,76 +0.000258,76 +0.000283,76 +0.000318,76 +0.000258,76 +0.000258,76 +0.000258,76 +0.000258,76 +0.000258,76 +0.000295,76 +0.000420,76 +0.000291,76 +0.000251,76 +0.000284,76 +0.000251,76 +0.000251,76 +0.000276,76 +0.000405,76 +0.000251,76 +0.000251,76 +0.000251,76 +0.000251,76 +0.000251,76 +0.000287,76 +0.000432,78 +0.000271,78 +0.000271,78 +0.000406,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000420,78 +0.000271,78 +0.000313,78 +0.000500,78 +0.000325,78 +0.000278,78 +0.000298,78 +0.000308,78 +0.000271,78 +0.000427,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000382,78 +0.000539,78 +0.000500,78 +0.000324,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000400,78 +0.000311,78 +0.000305,78 +0.000360,78 +0.000283,78 +0.000292,78 +0.000281,78 +0.000313,78 +0.000316,78 +0.000280,78 +0.000294,78 +0.000292,78 +0.000295,78 +0.000284,78 +0.000282,78 +0.000271,78 +0.000320,78 +0.000271,78 +0.000347,78 +0.000315,78 +0.000271,78 +0.000271,78 +0.000308,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000306,78 +0.000271,78 +0.000271,78 +0.000341,78 +0.000384,78 +0.000271,78 +0.000291,78 +0.000291,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000391,78 +0.000474,78 +0.000490,78 +0.000437,78 +0.000272,78 +0.000271,78 +0.000271,78 +0.000282,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000309,78 +0.000272,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000271,78 +0.000310,78 +0.000271,78 +0.000315,78 +0.000293,80 +0.000310,80 +0.000290,80 +0.000290,80 +0.000323,80 +0.000290,80 +0.000291,80 +0.000290,80 +0.000291,80 +0.000290,80 +0.000290,80 +0.000291,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000330,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000474,80 +0.000291,80 +0.000310,80 +0.000311,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000328,80 +0.000290,80 +0.000290,80 +0.000450,80 +0.000537,80 +0.000440,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000332,80 +0.000291,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000330,80 +0.000290,80 +0.000313,80 +0.000290,80 +0.000314,80 +0.000290,80 +0.000290,80 +0.000291,80 +0.000291,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000301,80 +0.000333,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000310,80 +0.000454,80 +0.000291,80 +0.000448,80 +0.000310,80 +0.000335,80 +0.000291,80 +0.000290,80 +0.000291,80 +0.000291,80 +0.000337,80 +0.000592,80 +0.000518,80 +0.000362,80 +0.000290,80 +0.000290,80 +0.000290,80 +0.000317,80 +0.000291,80 +0.000290,80 +0.000290,80 +0.000291,80 +0.000291,80 +0.000290,80 +0.000324,82 +0.000313,82 +0.000313,82 +0.000473,82 +0.000313,82 +0.000366,82 +0.000313,82 +0.000352,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000469,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000353,82 +0.000464,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000472,82 +0.000333,82 +0.000323,82 +0.000526,82 +0.000313,82 +0.000312,82 +0.000312,82 +0.000476,82 +0.000555,82 +0.000576,82 +0.000616,82 +0.000313,82 +0.000356,82 +0.000313,82 +0.000386,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000393,82 +0.000392,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000343,82 +0.000332,82 +0.000336,82 +0.000313,82 +0.000503,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000352,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000468,82 +0.000313,82 +0.000313,82 +0.000424,82 +0.000313,82 +0.000314,82 +0.000421,82 +0.000461,82 +0.000313,82 +0.000332,82 +0.000354,82 +0.000313,82 +0.000313,82 +0.000313,82 +0.000484,82 +0.000571,82 +0.000426,82 +0.000313,82 +0.000353,82 +0.000476,82 +0.000312,82 +0.000367,82 +0.000362,82 +0.000312,82 +0.000312,82 +0.000312,82 +0.000312,82 +0.000353,82 +0.000410,82 +0.000312,82 +0.000348,82 +0.000314,82 +0.000352,82 +0.000454,82 +0.000326,82 +0.000377,84 +0.000335,84 +0.000336,84 +0.000340,84 +0.000470,84 +0.000336,84 +0.000376,84 +0.000335,84 +0.000336,84 +0.000492,84 +0.000336,84 +0.000336,84 +0.000336,84 +0.000336,84 +0.000359,84 +0.000449,84 +0.000375,84 +0.000438,84 +0.000415,84 +0.000430,84 +0.000406,84 +0.000336,84 +0.000335,84 +0.000336,84 +0.000343,84 +0.000608,84 +0.000601,84 +0.000591,84 +0.000681,84 +0.000572,84 +0.000592,84 +0.000335,84 +0.000335,84 +0.000335,84 +0.000389,84 +0.000549,84 +0.000385,84 +0.000390,84 +0.000335,84 +0.000375,84 +0.000360,84 +0.000399,84 +0.000399,84 +0.000348,84 +0.000383,84 +0.000377,84 +0.000349,84 +0.000488,84 +0.000384,84 +0.000335,84 +0.000442,84 +0.000337,84 +0.000336,84 +0.000336,84 +0.000336,84 +0.000451,84 +0.000413,84 +0.000463,84 +0.000488,84 +0.000388,84 +0.000385,84 +0.000336,84 +0.000335,84 +0.000624,84 +0.000716,84 +0.000602,84 +0.000619,84 +0.000336,84 +0.000479,84 +0.000336,84 +0.000336,84 +0.000335,84 +0.000336,84 +0.000445,84 +0.000424,84 +0.000336,84 +0.000446,84 +0.000336,84 +0.000373,84 +0.000336,84 +0.000356,84 +0.000335,84 +0.000372,84 +0.000336,84 +0.000522,84 +0.000336,84 +0.000341,84 +0.000438,84 +0.000336,84 +0.000444,84 +0.000336,84 +0.000336,84 +0.000336,84 +0.000335,84 +0.000375,84 +0.000489,84 +0.000336,84 +0.000335,84 +0.000359,84 +0.000494,84 +0.000477,86 +0.000420,86 +0.000358,86 +0.000358,86 +0.000406,86 +0.000458,86 +0.000697,86 +0.000648,86 +0.000407,86 +0.000369,86 +0.000358,86 +0.000499,86 +0.000358,86 +0.000526,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000395,86 +0.000379,86 +0.000358,86 +0.000434,86 +0.000358,86 +0.000358,86 +0.000489,86 +0.000388,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000387,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000532,86 +0.000368,86 +0.000378,86 +0.000379,86 +0.000393,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000472,86 +0.000666,86 +0.000555,86 +0.000358,86 +0.000358,86 +0.000391,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000393,86 +0.000407,86 +0.000358,86 +0.000392,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000384,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000401,86 +0.000502,86 +0.000358,86 +0.000434,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000610,86 +0.000596,86 +0.000358,86 +0.000444,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000358,86 +0.000393,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000452,88 +0.000402,88 +0.000382,88 +0.000416,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000381,88 +0.000412,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000434,88 +0.000480,88 +0.000382,88 +0.000421,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000426,88 +0.000730,88 +0.000542,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000417,88 +0.000382,88 +0.000416,88 +0.000382,88 +0.000402,88 +0.000382,88 +0.000415,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000431,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000381,88 +0.000382,88 +0.000382,88 +0.000381,88 +0.000382,88 +0.000456,88 +0.000544,88 +0.000403,88 +0.000414,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000381,88 +0.000442,88 +0.000729,88 +0.000707,88 +0.000537,88 +0.000382,88 +0.000382,88 +0.000381,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000628,88 +0.000746,88 +0.000518,88 +0.000409,88 +0.000569,88 +0.000488,88 +0.000509,88 +0.000535,88 +0.000407,88 +0.000392,88 +0.000466,88 +0.000425,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000382,88 +0.000458,90 +0.000408,90 +0.000408,90 +0.000590,90 +0.000408,90 +0.000488,90 +0.000408,90 +0.000428,90 +0.000408,90 +0.000447,90 +0.000408,90 +0.000447,90 +0.000510,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000450,90 +0.000408,90 +0.000409,90 +0.000408,90 +0.000408,90 +0.000448,90 +0.000428,90 +0.000408,90 +0.000444,90 +0.000446,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000446,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000582,90 +0.000417,90 +0.000428,90 +0.000428,90 +0.000408,90 +0.000440,90 +0.000408,90 +0.000408,90 +0.000438,90 +0.000569,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000444,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000442,90 +0.000418,90 +0.000420,90 +0.000475,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000446,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000534,90 +0.000467,90 +0.000427,90 +0.000429,90 +0.000440,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000418,90 +0.000500,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000441,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000408,90 +0.000445,92 +0.000435,92 +0.000466,92 +0.000452,92 +0.000476,92 +0.000467,92 +0.000435,92 +0.000435,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000470,92 +0.000444,92 +0.000435,92 +0.000434,92 +0.000435,92 +0.000463,92 +0.000532,92 +0.000593,92 +0.000465,92 +0.000486,92 +0.000435,92 +0.000435,92 +0.000435,92 +0.000435,92 +0.000484,92 +0.000526,92 +0.000434,92 +0.000434,92 +0.000457,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000435,92 +0.000435,92 +0.000434,92 +0.000537,92 +0.000475,92 +0.000434,92 +0.000503,92 +0.000435,92 +0.000435,92 +0.000434,92 +0.000434,92 +0.000465,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000579,92 +0.000434,92 +0.000474,92 +0.000434,92 +0.000435,92 +0.000434,92 +0.000434,92 +0.000435,92 +0.000537,92 +0.000470,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000454,92 +0.000434,92 +0.000470,92 +0.000434,92 +0.000468,92 +0.000478,92 +0.000435,92 +0.000466,92 +0.000434,92 +0.000454,92 +0.000434,92 +0.000473,92 +0.000446,92 +0.000451,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000482,92 +0.000619,92 +0.000445,92 +0.000463,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000434,92 +0.000484,94 +0.000565,94 +0.000493,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000507,94 +0.000740,94 +0.000755,94 +0.000503,94 +0.000684,94 +0.000541,94 +0.000739,94 +0.000463,94 +0.000472,94 +0.000596,94 +0.000465,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000496,94 +0.000463,94 +0.000580,94 +0.000588,94 +0.000511,94 +0.000507,94 +0.000462,94 +0.000482,94 +0.000495,94 +0.000502,94 +0.000566,94 +0.000463,94 +0.000478,94 +0.000475,94 +0.000477,94 +0.000462,94 +0.000487,94 +0.000463,94 +0.000462,94 +0.000462,94 +0.000463,94 +0.000530,94 +0.000508,94 +0.000463,94 +0.000538,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000463,94 +0.000496,94 +0.000475,94 +0.000478,94 +0.000526,94 +0.000581,94 +0.000483,94 +0.000487,94 +0.000462,94 +0.000494,94 +0.000463,94 +0.000463,94 +0.000565,94 +0.000606,94 +0.000495,94 +0.000463,94 +0.000563,94 +0.000500,94 +0.000657,94 +0.000624,94 +0.000506,94 +0.000488,94 +0.000515,94 +0.000580,94 +0.000657,94 +0.000534,94 +0.000529,94 +0.000533,94 +0.000540,94 +0.000562,94 +0.000494,94 +0.000537,94 +0.000565,94 +0.000878,94 +0.000931,94 +0.000894,94 +0.000959,94 +0.000763,94 +0.000537,94 +0.000584,94 +0.000978,94 +0.001019,94 +0.001073,94 +0.000778,94 +0.000477,94 +0.000535,96 +0.000582,96 +0.000641,96 +0.000605,96 +0.000597,96 +0.000584,96 +0.000516,96 +0.000534,96 +0.000892,96 +0.000842,96 +0.000522,96 +0.000502,96 +0.000502,96 +0.000609,96 +0.000636,96 +0.000555,96 +0.000502,96 +0.000502,96 +0.000502,96 +0.000502,96 +0.000514,96 +0.000489,96 +0.000504,96 +0.000564,96 +0.000502,96 +0.000707,96 +0.000916,96 +0.000739,96 +0.000601,96 +0.000697,96 +0.000612,96 +0.000552,96 +0.000490,96 +0.000489,96 +0.000489,96 +0.000489,96 +0.000530,96 +0.000490,96 +0.000490,96 +0.000490,96 +0.000490,96 +0.000490,96 +0.000489,96 +0.000560,96 +0.000598,96 +0.000510,96 +0.000529,96 +0.000489,96 +0.000490,96 +0.000490,96 +0.000498,96 +0.000504,96 +0.000568,96 +0.000502,96 +0.000509,96 +0.000977,96 +0.000834,96 +0.000513,96 +0.000615,96 +0.000572,96 +0.000549,96 +0.000577,96 +0.000508,96 +0.000490,96 +0.000490,96 +0.000527,96 +0.000489,96 +0.000498,96 +0.000502,96 +0.000510,96 +0.000490,96 +0.000489,96 +0.000490,96 +0.000549,96 +0.000661,96 +0.000490,96 +0.000534,96 +0.000490,96 +0.000490,96 +0.000490,96 +0.000498,96 +0.000546,96 +0.000549,96 +0.000550,96 +0.000503,96 +0.000772,96 +0.000877,96 +0.000647,96 +0.000529,96 +0.001050,96 +0.000608,96 +0.000727,96 +0.000565,96 +0.000638,96 +0.000607,96 +0.000500,96 +0.000668,96 +0.000520,96 +0.000490,96 +0.000490,96 +0.000576,98 +0.000523,98 +0.000704,98 +0.000562,98 +0.000563,98 +0.000523,98 +0.000523,98 +0.000564,98 +0.000581,98 +0.000601,98 +0.000634,98 +0.000532,98 +0.000543,98 +0.000626,98 +0.000597,98 +0.000598,98 +0.000631,98 +0.000684,98 +0.000586,98 +0.000643,98 +0.000603,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000555,98 +0.000523,98 +0.000523,98 +0.000663,98 +0.000564,98 +0.000582,98 +0.000538,98 +0.000563,98 +0.000523,98 +0.000563,98 +0.000601,98 +0.000588,98 +0.000581,98 +0.000543,98 +0.000586,98 +0.000556,98 +0.000582,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000725,98 +0.000686,98 +0.000551,98 +0.000551,98 +0.000562,98 +0.000550,98 +0.000536,98 +0.000536,98 +0.000567,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000533,98 +0.000722,98 +0.000543,98 +0.000594,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000562,98 +0.000543,98 +0.000605,98 +0.000593,98 +0.000553,98 +0.000562,98 +0.000701,98 +0.000523,98 +0.000763,98 +0.000631,98 +0.000759,98 +0.000582,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000565,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000523,98 +0.000702,98 +0.000568,98 +0.000563,98 +0.000523,98 +0.000523,98 +0.000552,98 +0.000552,98 +0.000722,98 +0.000579,98 +0.000602,98 +0.000600,100 +0.000567,100 +0.000610,100 +0.000552,100 +0.000695,100 +0.000765,100 +0.000756,100 +0.000605,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000591,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000663,100 +0.000571,100 +0.000591,100 +0.000552,100 +0.000552,100 +0.000592,100 +0.000620,100 +0.000613,100 +0.000572,100 +0.000716,100 +0.000567,100 +0.000675,100 +0.000552,100 +0.000552,100 +0.000766,100 +0.000738,100 +0.000699,100 +0.000611,100 +0.000552,100 +0.000552,100 +0.000598,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000643,100 +0.000704,100 +0.000612,100 +0.000552,100 +0.000552,100 +0.000552,100 +0.000591,100 +0.000704,100 +0.000632,100 +0.000552,100 +0.000711,100 +0.000567,100 +0.000650,100 +0.000635,100 +0.000670,100 +0.001211,100 +0.000648,100 +0.000753,100 +0.000768,100 +0.000696,100 +0.000578,100 +0.000625,100 +0.000567,100 +0.000567,100 +0.000603,100 +0.000567,100 +0.000627,100 +0.000567,100 +0.000593,100 +0.000567,100 +0.000567,100 +0.000594,100 +0.000573,100 +0.000749,100 +0.000572,100 +0.000587,100 +0.000678,100 +0.000579,100 +0.000580,100 +0.000552,100 +0.000642,100 +0.000743,100 +0.000706,100 +0.000611,100 +0.000592,100 +0.000553,100 +0.000552,100 +0.000552,100 +0.000553,100 +0.000552,100 +0.000553,100 +0.000595,100 +0.000553,100 +0.000553,100 +0.000752,100 +0.000619,102 +0.000624,102 +0.000599,102 +0.000604,102 +0.000623,102 +0.000663,102 +0.000725,102 +0.000620,102 +0.000668,102 +0.000805,102 +0.000604,102 +0.000584,102 +0.000627,102 +0.000769,102 +0.000735,102 +0.000616,102 +0.000610,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000622,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000741,102 +0.000642,102 +0.000647,102 +0.000604,102 +0.000584,102 +0.000623,102 +0.000662,102 +0.000776,102 +0.000644,102 +0.000625,102 +0.000584,102 +0.000585,102 +0.000584,102 +0.000584,102 +0.000739,102 +0.000766,102 +0.000677,102 +0.000624,102 +0.000585,102 +0.000584,102 +0.000584,102 +0.000620,102 +0.000585,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000606,102 +0.000776,102 +0.000668,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000624,102 +0.000783,102 +0.000739,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000660,102 +0.000688,102 +0.000643,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000618,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000584,102 +0.000674,102 +0.000626,102 +0.000731,102 +0.000665,102 +0.000584,102 +0.000623,102 +0.000663,102 +0.000700,102 +0.000696,102 +0.000584,102 +0.000584,102 +0.000585,102 +0.000595,102 +0.000630,102 +0.000584,102 +0.000623,102 +0.000604,102 +0.000623,102 +0.000585,102 +0.000584,102 +0.000627,102 +0.000584,102 +0.000631,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000663,104 +0.000839,104 +0.000650,104 +0.000641,104 +0.000617,104 +0.000617,104 +0.000691,104 +0.000753,104 +0.000980,104 +0.000890,104 +0.000617,104 +0.000650,104 +0.000626,104 +0.000617,104 +0.001098,104 +0.000670,104 +0.000933,104 +0.000782,104 +0.000709,104 +0.000728,104 +0.000733,104 +0.000617,104 +0.000644,104 +0.000617,104 +0.000784,104 +0.000660,104 +0.000646,104 +0.000617,104 +0.000648,104 +0.000784,104 +0.000789,104 +0.000865,104 +0.000654,104 +0.000633,104 +0.000655,104 +0.000617,104 +0.000618,104 +0.000638,104 +0.000656,104 +0.000617,104 +0.000698,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000653,104 +0.000617,104 +0.000617,104 +0.000808,104 +0.000636,104 +0.000657,104 +0.000654,104 +0.000617,104 +0.000794,104 +0.000804,104 +0.000651,104 +0.000640,104 +0.000643,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000656,104 +0.000681,104 +0.000656,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000642,104 +0.000617,104 +0.000617,104 +0.000680,104 +0.000785,104 +0.000661,104 +0.000648,104 +0.000617,104 +0.000656,104 +0.000754,104 +0.000617,104 +0.000617,104 +0.000644,104 +0.000617,104 +0.000633,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000685,104 +0.000637,104 +0.000656,104 +0.000617,104 +0.000617,104 +0.000617,104 +0.000630,104 +0.000679,106 +0.000652,106 +0.000652,106 +0.000651,106 +0.000836,106 +0.000702,106 +0.000674,106 +0.000691,106 +0.000803,106 +0.000652,106 +0.000652,106 +0.000696,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000681,106 +0.000691,106 +0.000671,106 +0.000690,106 +0.000651,106 +0.000653,106 +0.000679,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000744,106 +0.000651,106 +0.000652,106 +0.000651,106 +0.000652,106 +0.000652,106 +0.000656,106 +0.000654,106 +0.000652,106 +0.000651,106 +0.000652,106 +0.000652,106 +0.000654,106 +0.000652,106 +0.000691,106 +0.000652,106 +0.000691,106 +0.000651,106 +0.000656,106 +0.000652,106 +0.000651,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000654,106 +0.000711,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000656,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000656,106 +0.000654,106 +0.000652,106 +0.000652,106 +0.001086,106 +0.000704,106 +0.000987,106 +0.000778,106 +0.000766,106 +0.000692,106 +0.000831,106 +0.000699,106 +0.000692,106 +0.000752,106 +0.000652,106 +0.000651,106 +0.000652,106 +0.000652,106 +0.000685,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000652,106 +0.000690,106 +0.000652,106 +0.000652,106 +0.000651,106 +0.000691,106 +0.000652,106 +0.000696,106 +0.000652,106 +0.000652,106 +0.000651,106 +0.000652,106 +0.000652,106 +0.000730,108 +0.000690,108 +0.000752,108 +0.000689,108 +0.000689,108 +0.000726,108 +0.000690,108 +0.000689,108 +0.000690,108 +0.000690,108 +0.000769,108 +0.000729,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000728,108 +0.000711,108 +0.000728,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000694,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000753,108 +0.000709,108 +0.000715,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000696,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000700,108 +0.000721,108 +0.000728,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000694,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000752,108 +0.000692,108 +0.000690,108 +0.000690,108 +0.000689,108 +0.000690,108 +0.000689,108 +0.000694,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000690,108 +0.000689,108 +0.000731,108 +0.000690,108 +0.000729,108 +0.000690,108 +0.000690,108 +0.000694,108 +0.000689,108 +0.000689,108 +0.000690,108 +0.000689,108 +0.000690,108 +0.000711,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000693,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000691,108 +0.000721,108 +0.000698,108 +0.000728,108 +0.000689,108 +0.000694,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000689,108 +0.000691,108 +0.000689,108 +0.000689,108 +0.000739,110 +0.000726,110 +0.000726,110 +0.000731,110 +0.000726,110 +0.000785,110 +0.000726,110 +0.000726,110 +0.000729,110 +0.000727,110 +0.000726,110 +0.000765,110 +0.000726,110 +0.000767,110 +0.000882,110 +0.000767,110 +0.000752,110 +0.000763,110 +0.000796,110 +0.000783,110 +0.000819,110 +0.000881,110 +0.000726,110 +0.000726,110 +0.000764,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000761,110 +0.000780,110 +0.000726,110 +0.000748,110 +0.000757,110 +0.000757,110 +0.000772,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000764,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000772,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000729,110 +0.000788,110 +0.000890,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000769,110 +0.000754,110 +0.000767,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000749,110 +0.000726,110 +0.000728,110 +0.000737,110 +0.000922,110 +0.000800,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000749,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000766,110 +0.000746,110 +0.000765,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000763,110 +0.000729,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000765,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000749,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000726,110 +0.000751,110 +0.000817,112 +0.000763,112 +0.000803,112 +0.000764,112 +0.000766,112 +0.000763,112 +0.000764,112 +0.000763,112 +0.000763,112 +0.000809,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000765,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000768,112 +0.000763,112 +0.000802,112 +0.000802,112 +0.000763,112 +0.000766,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000767,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000767,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000767,112 +0.000763,112 +0.000802,112 +0.000763,112 +0.000895,112 +0.000818,112 +0.000801,112 +0.000821,112 +0.000797,112 +0.000868,112 +0.000816,112 +0.000763,112 +0.000783,112 +0.000763,112 +0.000763,112 +0.000786,112 +0.000764,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000788,112 +0.000764,112 +0.000763,112 +0.000802,112 +0.000763,112 +0.000825,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000857,112 +0.000764,112 +0.000764,112 +0.000763,112 +0.000763,112 +0.000788,112 +0.000764,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000801,112 +0.000822,112 +0.000946,112 +0.000824,112 +0.000814,112 +0.000841,112 +0.000804,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000800,112 +0.000763,112 +0.000763,112 +0.000837,112 +0.000916,112 +0.000820,112 +0.000763,112 +0.000763,112 +0.000763,112 +0.000827,114 +0.000849,114 +0.000812,114 +0.000811,114 +0.000812,114 +0.000811,114 +0.000885,114 +0.000831,114 +0.000851,114 +0.000812,114 +0.000812,114 +0.000850,114 +0.000812,114 +0.000812,114 +0.000812,114 +0.000812,114 +0.000836,114 +0.000812,114 +0.000812,114 +0.000812,114 +0.000812,114 +0.000847,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000874,114 +0.000811,114 +0.000851,114 +0.000812,114 +0.000839,114 +0.000811,114 +0.000812,114 +0.000811,114 +0.000811,114 +0.000834,114 +0.000812,114 +0.000811,114 +0.000811,114 +0.000862,114 +0.000836,114 +0.000812,114 +0.000812,114 +0.000812,114 +0.000812,114 +0.000839,114 +0.000812,114 +0.000851,114 +0.000851,114 +0.000812,114 +0.000837,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000839,114 +0.000812,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000816,114 +0.000812,114 +0.000812,114 +0.000811,114 +0.000864,114 +0.000812,114 +0.000812,114 +0.000850,114 +0.000812,114 +0.000858,114 +0.000978,114 +0.000837,114 +0.000856,114 +0.000860,114 +0.000867,114 +0.000872,114 +0.000832,114 +0.000812,114 +0.000812,114 +0.000847,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000835,114 +0.000811,114 +0.000812,114 +0.000811,114 +0.000851,114 +0.000876,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000845,114 +0.000814,114 +0.000811,114 +0.000811,114 +0.000811,114 +0.000816,114 +0.000811,114 +0.000870,116 +0.000850,116 +0.000850,116 +0.000932,116 +0.000851,116 +0.000850,116 +0.000850,116 +0.000925,116 +0.000890,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000852,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000854,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000931,116 +0.000851,116 +0.000851,116 +0.000850,116 +0.000850,116 +0.000920,116 +0.000850,116 +0.000889,116 +0.000850,116 +0.000850,116 +0.000873,116 +0.000851,116 +0.000850,116 +0.000851,116 +0.000855,116 +0.000851,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000853,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000855,116 +0.000890,116 +0.000889,116 +0.000850,116 +0.000852,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000854,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000852,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000855,116 +0.000850,116 +0.000890,116 +0.000850,116 +0.000889,116 +0.000853,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000856,116 +0.000851,116 +0.000850,116 +0.000865,116 +0.000855,116 +0.000851,116 +0.000850,116 +0.000851,116 +0.000850,116 +0.000857,116 +0.000850,116 +0.000870,116 +0.000875,116 +0.000889,116 +0.000852,116 +0.000897,116 +0.000883,116 +0.000907,116 +0.000925,116 +0.000870,116 +0.000907,116 +0.000859,116 +0.000850,116 +0.000873,116 +0.000850,116 +0.000850,116 +0.000850,116 +0.000932,118 +0.000889,118 +0.000890,118 +0.000928,118 +0.000889,118 +0.000951,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000925,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000891,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000919,118 +0.000971,118 +0.000889,118 +0.000889,118 +0.000950,118 +0.000928,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000894,118 +0.000889,118 +0.000890,118 +0.000889,118 +0.000891,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000894,118 +0.000889,118 +0.000890,118 +0.000889,118 +0.000951,118 +0.000889,118 +0.000929,118 +0.000889,118 +0.000889,118 +0.000893,118 +0.000889,118 +0.000913,118 +0.000889,118 +0.000897,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000897,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000895,118 +0.000928,118 +0.000889,118 +0.000928,118 +0.000895,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000895,118 +0.000889,118 +0.000889,118 +0.000906,118 +0.000898,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000900,118 +0.000889,118 +0.000928,118 +0.000928,118 +0.000891,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000893,118 +0.000889,118 +0.000888,118 +0.000889,118 +0.000891,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000889,118 +0.000894,118 +0.000889,118 +0.000933,118 +0.000889,118 +0.000930,118 +0.000889,118 +0.000995,120 +0.000992,120 +0.001008,120 +0.000973,120 +0.000965,120 +0.001004,120 +0.000957,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000964,120 +0.000935,120 +0.000937,120 +0.000974,120 +0.001011,120 +0.000935,120 +0.000937,120 +0.000935,120 +0.000958,120 +0.000935,120 +0.000936,120 +0.000935,120 +0.000940,120 +0.000936,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000973,120 +0.001014,120 +0.000936,120 +0.000975,120 +0.000969,120 +0.000974,120 +0.000936,120 +0.000935,120 +0.000969,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000940,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000958,120 +0.000935,120 +0.000935,120 +0.000965,120 +0.000935,120 +0.001004,120 +0.000975,120 +0.000935,120 +0.000935,120 +0.000997,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000941,120 +0.000935,120 +0.000937,120 +0.000935,120 +0.000937,120 +0.000935,120 +0.000937,120 +0.000935,120 +0.000935,120 +0.000937,120 +0.000974,120 +0.000974,120 +0.000935,120 +0.000940,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000937,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000940,120 +0.000935,120 +0.000935,120 +0.000935,120 +0.000934,120 +0.000937,120 +0.000974,120 +0.000935,120 +0.000974,120 +0.000940,120 +0.000935,120 +0.000961,120 +0.000936,120 +0.000941,120 +0.000938,120 +0.000935,120 +0.000938,120 +0.000941,120 +0.000936,120 +0.000936,120 +0.000995,120 +0.000941,120 +0.001005,122 +0.000983,122 +0.001022,122 +0.001049,122 +0.000983,122 +0.000983,122 +0.001040,122 +0.001136,122 +0.001051,122 +0.001031,122 +0.001070,122 +0.001012,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.001006,122 +0.000983,122 +0.000984,122 +0.001022,122 +0.001019,122 +0.001012,122 +0.000983,122 +0.000983,122 +0.001006,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000988,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.001006,122 +0.001090,122 +0.001003,122 +0.000983,122 +0.001051,122 +0.001022,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000988,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.001027,122 +0.001022,122 +0.000983,122 +0.000987,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000987,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.000983,122 +0.001022,122 +0.001022,122 +0.000988,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000990,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.001009,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000986,122 +0.001022,122 +0.000983,122 +0.001022,122 +0.000989,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.000985,122 +0.000983,122 +0.000983,122 +0.000983,122 +0.001059,124 +0.001033,124 +0.001073,124 +0.001073,124 +0.001036,124 +0.001033,124 +0.001033,124 +0.001158,124 +0.001078,124 +0.001099,124 +0.001071,124 +0.001124,124 +0.001057,124 +0.001033,124 +0.001033,124 +0.001057,124 +0.001034,124 +0.001034,124 +0.001072,124 +0.001098,124 +0.001034,124 +0.001034,124 +0.001057,124 +0.001034,124 +0.001034,124 +0.001034,124 +0.001038,124 +0.001034,124 +0.001034,124 +0.001034,124 +0.001068,124 +0.001092,124 +0.001033,124 +0.001033,124 +0.001109,124 +0.001073,124 +0.001033,124 +0.001033,124 +0.001036,124 +0.001033,124 +0.001033,124 +0.001033,124 +0.001038,124 +0.001033,124 +0.001033,124 +0.001036,124 +0.001034,124 +0.001033,124 +0.001033,124 +0.001038,124 +0.001072,124 +0.001072,124 +0.001034,124 +0.001036,124 +0.001034,124 +0.001034,124 +0.001034,124 +0.001038,124 +0.001034,124 +0.001034,124 +0.001034,124 +0.001036,124 +0.001033,124 +0.001034,124 +0.001034,124 +0.001038,124 +0.001073,124 +0.001073,124 +0.001036,124 +0.001034,124 +0.001034,124 +0.001034,124 +0.001061,124 +0.001033,124 +0.001033,124 +0.001033,124 +0.001036,124 +0.001033,124 +0.001033,124 +0.001033,124 +0.001059,124 +0.001033,124 +0.001073,124 +0.001072,124 +0.001067,124 +0.001033,124 +0.001033,124 +0.001033,124 +0.001038,124 +0.001033,124 +0.001033,124 +0.001036,124 +0.001033,124 +0.001033,124 +0.001033,124 +0.001038,124 +0.001033,124 +0.001033,124 +0.001073,124 +0.001075,124 +0.001105,126 +0.001084,126 +0.001175,126 +0.001258,126 +0.001130,126 +0.001150,126 +0.001208,126 +0.001215,126 +0.001107,126 +0.001083,126 +0.001121,126 +0.001168,126 +0.001122,126 +0.001185,126 +0.001123,126 +0.001111,126 +0.001149,126 +0.001120,126 +0.001168,126 +0.001139,126 +0.001147,126 +0.001169,126 +0.001084,126 +0.001149,126 +0.001201,126 +0.001249,126 +0.001189,126 +0.001217,126 +0.001123,126 +0.001168,126 +0.001228,126 +0.001204,126 +0.001151,126 +0.001149,126 +0.001206,126 +0.001173,126 +0.001381,126 +0.001185,126 +0.001208,126 +0.001172,126 +0.001250,126 +0.001249,126 +0.001187,126 +0.001148,126 +0.001207,126 +0.001239,126 +0.001200,126 +0.001223,126 +0.001148,126 +0.001199,126 +0.001234,126 +0.001242,126 +0.001171,126 +0.001173,126 +0.001230,126 +0.001272,126 +0.001191,126 +0.001209,126 +0.001169,126 +0.001210,126 +0.001245,126 +0.001253,126 +0.001148,126 +0.001197,126 +0.001235,126 +0.001149,126 +0.001150,126 +0.001207,126 +0.001209,126 +0.001245,126 +0.001266,126 +0.001189,126 +0.001210,126 +0.001300,126 +0.001375,126 +0.001252,126 +0.001221,126 +0.001355,126 +0.001453,126 +0.001288,126 +0.001281,126 +0.001253,126 +0.001228,126 +0.001283,126 +0.001296,126 +0.001215,126 +0.001245,126 +0.001442,126 +0.001202,126 +0.001205,126 +0.001170,126 +0.001228,126 +0.001194,126 +0.001167,126 +0.001169,126 +0.001238,126 +0.001135,126 +0.001112,126 +0.001131,126 +0.001218,126 +0.001382,128 +0.001331,128 +0.001270,128 +0.001313,128 +0.001256,128 +0.001326,128 +0.001293,128 +0.001140,128 +0.001395,128 +0.001199,128 +0.001249,128 +0.001276,128 +0.001238,128 +0.001316,128 +0.001350,128 +0.001339,128 +0.001203,128 +0.001262,128 +0.001302,128 +0.001210,128 +0.001162,128 +0.001297,128 +0.001271,128 +0.001223,128 +0.001257,128 +0.001159,128 +0.001138,128 +0.001228,128 +0.001189,128 +0.001330,128 +0.001185,128 +0.001196,128 +0.001140,128 +0.001139,128 +0.001293,128 +0.001178,128 +0.001178,128 +0.001178,128 +0.001139,128 +0.001138,128 +0.001139,128 +0.001262,128 +0.001179,128 +0.001139,128 +0.001172,128 +0.001140,128 +0.001139,128 +0.001139,128 +0.001178,128 +0.001179,128 +0.001158,128 +0.001211,128 +0.001139,128 +0.001139,128 +0.001139,128 +0.001175,128 +0.001139,128 +0.001140,128 +0.001161,128 +0.001139,128 +0.001139,128 +0.001138,128 +0.001183,128 +0.001139,128 +0.001178,128 +0.001173,128 +0.001139,128 +0.001240,128 +0.001419,128 +0.001814,128 +0.001751,128 +0.001291,128 +0.001630,128 +0.001826,128 +0.001257,128 +0.001138,128 +0.001217,128 +0.001177,128 +0.001138,128 +0.001137,128 +0.001177,128 +0.001278,128 +0.001591,128 +0.001900,128 +0.001170,128 +0.001295,128 +0.001161,128 +0.001207,128 +0.001140,128 +0.001270,128 +0.001356,128 +0.001140,128 +0.001186,128 +0.001140,128 +0.001188,128 +0.001189,128 +0.001161,128 +0.001139,128 +0.001160,128 +0.001239,128 +0.001271,130 +0.001282,130 +0.001341,130 +0.001218,130 +0.001237,130 +0.001244,130 +0.001288,130 +0.001208,130 +0.001301,130 +0.001274,130 +0.001237,130 +0.001239,130 +0.001224,130 +0.001258,130 +0.001198,130 +0.001244,130 +0.001263,130 +0.001237,130 +0.001243,130 +0.001323,130 +0.001227,130 +0.001264,130 +0.001210,130 +0.001232,130 +0.001414,130 +0.001217,130 +0.001218,130 +0.001241,130 +0.001198,130 +0.001239,130 +0.001235,130 +0.001243,130 +0.001323,130 +0.001197,130 +0.001231,130 +0.001198,130 +0.001264,130 +0.001457,130 +0.001198,130 +0.001198,130 +0.001387,130 +0.001198,130 +0.001232,130 +0.001277,130 +0.001321,130 +0.001276,130 +0.001322,130 +0.001198,130 +0.001198,130 +0.001263,130 +0.001405,130 +0.001230,130 +0.001230,130 +0.001298,130 +0.001347,130 +0.001305,130 +0.001444,130 +0.001218,130 +0.001217,130 +0.001365,130 +0.001294,130 +0.001370,130 +0.001236,130 +0.001350,130 +0.001473,130 +0.001258,130 +0.001375,130 +0.001332,130 +0.001422,130 +0.001261,130 +0.001310,130 +0.001338,130 +0.001298,130 +0.001352,130 +0.001393,130 +0.001251,130 +0.001344,130 +0.001407,130 +0.001295,130 +0.001275,130 +0.001400,130 +0.001198,130 +0.001439,130 +0.001305,130 +0.001198,130 +0.001288,130 +0.001342,130 +0.001197,130 +0.001290,130 +0.001473,130 +0.001198,130 +0.001307,130 +0.001305,130 +0.001238,130 +0.001395,130 +0.001226,130 +0.001198,130 +0.001313,130 +0.001309,130 +0.001307,130 +0.001281,132 +0.001541,132 +0.001270,132 +0.001248,132 +0.001384,132 +0.001249,132 +0.001442,132 +0.001331,132 +0.001250,132 +0.001361,132 +0.001277,132 +0.001374,132 +0.001268,132 +0.001442,132 +0.001249,132 +0.001249,132 +0.001248,132 +0.001292,132 +0.001249,132 +0.001328,132 +0.001322,132 +0.001249,132 +0.001248,132 +0.001282,132 +0.001376,132 +0.001268,132 +0.001433,132 +0.001352,132 +0.001316,132 +0.001407,132 +0.001313,132 +0.001351,132 +0.001462,132 +0.001270,132 +0.001269,132 +0.001285,132 +0.001357,132 +0.001288,132 +0.001282,132 +0.001390,132 +0.001249,132 +0.001288,132 +0.001248,132 +0.001332,132 +0.001302,132 +0.001329,132 +0.001249,132 +0.001287,132 +0.001249,132 +0.001353,132 +0.001323,132 +0.001249,132 +0.001250,132 +0.001285,132 +0.001249,132 +0.001249,132 +0.001282,132 +0.001268,132 +0.001332,132 +0.001249,132 +0.001284,132 +0.001248,132 +0.001249,132 +0.001282,132 +0.001249,132 +0.001250,132 +0.001276,132 +0.001249,132 +0.001249,132 +0.001272,132 +0.001268,132 +0.001288,132 +0.001274,132 +0.001249,132 +0.001249,132 +0.001271,132 +0.001249,132 +0.001248,132 +0.001249,132 +0.001254,132 +0.001248,132 +0.001249,132 +0.001272,132 +0.001249,132 +0.001268,132 +0.001338,132 +0.001249,132 +0.001248,132 +0.001272,132 +0.001249,132 +0.001249,132 +0.001254,132 +0.001249,132 +0.001249,132 +0.001251,132 +0.001249,132 +0.001249,132 +0.001268,132 +0.001313,132 +0.001249,132 +0.001338,134 +0.001311,134 +0.001308,134 +0.001308,134 +0.001524,134 +0.001376,134 +0.001356,134 +0.001469,134 +0.001308,134 +0.001395,134 +0.001349,134 +0.001330,134 +0.001510,134 +0.001382,134 +0.001308,134 +0.001352,134 +0.001347,134 +0.001410,134 +0.001334,134 +0.001391,134 +0.001308,134 +0.001364,134 +0.001347,134 +0.001308,134 +0.001433,134 +0.001308,134 +0.001411,134 +0.001380,134 +0.001308,134 +0.001411,134 +0.001345,134 +0.001361,134 +0.001307,134 +0.001344,134 +0.001372,134 +0.001346,134 +0.001334,134 +0.001307,134 +0.001413,134 +0.001371,134 +0.001308,134 +0.001404,134 +0.001348,134 +0.001328,134 +0.001308,134 +0.001344,134 +0.001359,134 +0.001355,134 +0.001334,134 +0.001307,134 +0.001307,134 +0.001416,134 +0.001328,134 +0.001409,134 +0.001343,134 +0.001365,134 +0.001308,134 +0.001346,134 +0.001327,134 +0.001388,134 +0.001335,134 +0.001308,134 +0.001308,134 +0.001449,134 +0.001348,134 +0.001340,134 +0.001449,134 +0.001307,134 +0.001322,134 +0.001437,134 +0.001322,134 +0.001406,134 +0.001347,134 +0.001307,134 +0.001464,134 +0.001407,134 +0.001348,134 +0.001368,134 +0.001487,134 +0.001438,134 +0.001416,134 +0.001460,134 +0.001642,134 +0.001396,134 +0.001350,134 +0.001414,134 +0.001494,134 +0.001531,134 +0.001366,134 +0.001430,134 +0.001381,134 +0.001490,134 +0.001450,134 +0.001369,134 +0.001503,134 +0.001601,134 +0.001517,134 +0.001411,134 +0.001652,134 +0.001479,134 +0.001538,136 +0.001451,136 +0.001496,136 +0.001529,136 +0.001574,136 +0.001464,136 +0.001494,136 +0.001499,136 +0.001571,136 +0.001559,136 +0.001518,136 +0.001445,136 +0.001431,136 +0.001540,136 +0.001456,136 +0.001468,136 +0.001405,136 +0.001430,136 +0.001429,136 +0.001392,136 +0.001369,136 +0.001417,136 +0.001369,136 +0.001514,136 +0.001561,136 +0.001406,136 +0.001490,136 +0.001528,136 +0.001512,136 +0.001735,136 +0.001451,136 +0.001446,136 +0.001588,136 +0.001405,136 +0.001478,136 +0.001445,136 +0.001426,136 +0.001501,136 +0.001555,136 +0.001631,136 +0.001559,136 +0.001444,136 +0.001439,136 +0.001472,136 +0.001405,136 +0.002002,136 +0.001684,136 +0.001495,136 +0.001602,136 +0.001464,136 +0.001550,136 +0.001497,136 +0.001465,136 +0.001462,136 +0.001777,136 +0.002520,136 +0.002108,136 +0.001403,136 +0.001556,136 +0.001532,136 +0.001485,136 +0.001530,136 +0.001473,136 +0.001464,136 +0.001452,136 +0.001369,136 +0.002016,136 +0.001433,136 +0.001533,136 +0.001503,136 +0.001523,136 +0.001506,136 +0.001492,136 +0.001368,136 +0.001444,136 +0.001471,136 +0.001368,136 +0.001980,136 +0.001405,136 +0.001533,136 +0.001496,136 +0.001527,136 +0.001513,136 +0.001481,136 +0.001369,136 +0.001475,136 +0.001472,136 +0.001369,136 +0.002042,136 +0.001405,136 +0.001531,136 +0.001522,136 +0.001476,136 +0.001575,136 +0.001431,136 +0.001369,136 +0.001482,136 +0.001431,136 +0.001368,136 +0.002021,136 +0.001514,138 +0.001650,138 +0.001551,138 +0.001637,138 +0.001606,138 +0.001429,138 +0.001473,138 +0.001531,138 +0.001468,138 +0.001802,138 +0.001543,138 +0.001567,138 +0.001600,138 +0.001570,138 +0.001761,138 +0.001498,138 +0.001508,138 +0.001505,138 +0.001777,138 +0.002262,138 +0.001464,138 +0.001616,138 +0.001670,138 +0.001590,138 +0.001705,138 +0.001457,138 +0.001458,138 +0.001426,138 +0.001721,138 +0.001468,138 +0.001848,138 +0.001652,138 +0.001651,138 +0.001635,138 +0.001545,138 +0.001519,138 +0.001486,138 +0.001428,138 +0.001640,138 +0.001514,138 +0.001677,138 +0.001697,138 +0.001626,138 +0.001468,138 +0.001502,138 +0.001592,138 +0.001525,138 +0.001428,138 +0.001428,138 +0.001586,138 +0.001448,138 +0.001775,138 +0.001582,138 +0.001572,138 +0.001489,138 +0.001526,138 +0.001542,138 +0.001500,138 +0.001429,138 +0.001462,138 +0.001708,138 +0.001428,138 +0.001802,138 +0.001744,138 +0.001468,138 +0.001454,138 +0.001551,138 +0.001566,138 +0.001428,138 +0.001448,138 +0.001684,138 +0.001488,138 +0.001535,138 +0.001780,138 +0.001649,138 +0.001484,138 +0.001455,138 +0.001677,138 +0.001542,138 +0.001429,138 +0.001722,138 +0.001618,138 +0.001505,138 +0.001809,138 +0.001637,138 +0.001648,138 +0.001510,138 +0.001488,138 +0.001589,138 +0.001429,138 +0.001472,138 +0.001452,138 +0.001738,138 +0.001428,138 +0.001784,138 +0.001685,138 +0.001426,138 +0.001426,138 +0.001519,138 +0.001512,138 +0.001563,140 +0.001492,140 +0.001726,140 +0.001577,140 +0.001791,140 +0.001657,140 +0.001654,140 +0.001508,140 +0.001534,140 +0.001586,140 +0.001572,140 +0.001492,140 +0.001492,140 +0.001694,140 +0.001491,140 +0.001866,140 +0.001774,140 +0.001491,140 +0.001550,140 +0.001545,140 +0.001579,140 +0.001511,140 +0.001491,140 +0.001779,140 +0.001515,140 +0.001853,140 +0.001671,140 +0.001677,140 +0.001491,140 +0.001491,140 +0.001635,140 +0.001491,140 +0.001491,140 +0.001702,140 +0.001624,140 +0.001690,140 +0.001754,140 +0.001662,140 +0.001551,140 +0.001553,140 +0.001698,140 +0.001552,140 +0.001530,140 +0.001492,140 +0.001767,140 +0.001529,140 +0.001832,140 +0.001804,140 +0.001491,140 +0.001536,140 +0.001573,140 +0.001594,140 +0.001522,140 +0.001492,140 +0.001757,140 +0.001588,140 +0.001795,140 +0.001798,140 +0.001491,140 +0.001553,140 +0.001522,140 +0.001627,140 +0.001520,140 +0.001492,140 +0.001721,140 +0.001573,140 +0.001679,140 +0.001630,140 +0.001491,140 +0.001491,140 +0.001515,140 +0.001663,140 +0.001605,140 +0.001488,140 +0.001488,140 +0.001664,140 +0.001777,140 +0.001624,140 +0.001492,140 +0.001653,140 +0.001681,140 +0.001615,140 +0.001893,140 +0.001503,140 +0.001758,140 +0.001751,140 +0.001726,140 +0.001625,140 +0.001786,140 +0.001747,140 +0.001649,140 +0.001676,140 +0.001830,140 +0.001711,140 +0.001648,140 +0.001767,140 +0.001934,140 +0.002862,140 +0.002059,140 +0.001580,140 +0.001746,142 +0.001996,142 +0.001840,142 +0.001968,142 +0.002185,142 +0.002980,142 +0.001803,142 +0.001790,142 +0.001804,142 +0.001690,142 +0.001770,142 +0.001839,142 +0.001808,142 +0.002382,142 +0.002844,142 +0.001718,142 +0.001732,142 +0.001741,142 +0.001729,142 +0.001723,142 +0.001685,142 +0.001658,142 +0.002594,142 +0.002747,142 +0.001830,142 +0.001847,142 +0.001883,142 +0.001670,142 +0.001867,142 +0.001723,142 +0.002894,142 +0.002569,142 +0.001765,142 +0.001787,142 +0.001901,142 +0.001879,142 +0.002013,142 +0.001895,142 +0.001922,142 +0.002034,142 +0.001923,142 +0.001942,142 +0.001998,142 +0.002005,142 +0.002019,142 +0.001986,142 +0.001897,142 +0.001892,142 +0.001736,142 +0.001796,142 +0.001669,142 +0.001709,142 +0.001752,142 +0.001864,142 +0.001761,142 +0.002032,142 +0.001679,142 +0.001722,142 +0.001687,142 +0.001759,142 +0.001791,142 +0.001981,142 +0.002008,142 +0.002015,142 +0.002084,142 +0.002184,142 +0.002038,142 +0.002001,142 +0.002046,142 +0.002055,142 +0.002038,142 +0.001952,142 +0.001788,142 +0.002505,142 +0.001885,142 +0.001688,142 +0.001738,142 +0.001726,142 +0.001853,142 +0.001762,142 +0.001856,142 +0.001809,142 +0.002486,142 +0.001766,142 +0.001758,142 +0.001725,142 +0.001789,142 +0.001824,142 +0.001703,142 +0.001824,142 +0.001721,142 +0.002387,142 +0.001682,142 +0.001762,142 +0.001794,142 +0.001795,142 +0.001745,142 +0.001700,142 +0.001879,142 +0.001720,142 +0.001862,144 +0.001783,144 +0.001818,144 +0.001788,144 +0.001835,144 +0.001743,144 +0.001714,144 +0.001984,144 +0.001873,144 +0.002296,144 +0.001929,144 +0.001804,144 +0.001860,144 +0.001873,144 +0.002026,144 +0.001922,144 +0.001990,144 +0.001850,144 +0.001841,144 +0.001818,144 +0.001828,144 +0.001869,144 +0.001881,144 +0.001881,144 +0.001819,144 +0.001855,144 +0.001772,144 +0.001820,144 +0.001894,144 +0.001840,144 +0.001848,144 +0.001992,144 +0.001875,144 +0.001749,144 +0.001878,144 +0.001840,144 +0.001868,144 +0.001814,144 +0.001870,144 +0.001820,144 +0.001867,144 +0.001851,144 +0.001843,144 +0.001905,144 +0.001859,144 +0.001851,144 +0.001825,144 +0.001843,144 +0.001842,144 +0.001934,144 +0.001805,144 +0.001837,144 +0.001922,144 +0.001871,144 +0.001777,144 +0.001845,144 +0.001885,144 +0.001735,144 +0.001769,144 +0.001764,144 +0.001813,144 +0.001879,144 +0.001738,144 +0.001768,144 +0.001809,144 +0.001720,144 +0.001778,144 +0.001797,144 +0.001730,144 +0.001760,144 +0.001766,144 +0.001806,144 +0.001822,144 +0.001879,144 +0.001857,144 +0.001858,144 +0.001913,144 +0.002014,144 +0.001948,144 +0.001844,144 +0.001948,144 +0.001865,144 +0.001881,144 +0.001819,144 +0.001908,144 +0.001920,144 +0.002060,144 +0.001885,144 +0.002011,144 +0.001915,144 +0.001939,144 +0.001870,144 +0.001824,144 +0.001755,144 +0.001816,144 +0.001959,144 +0.001760,144 +0.001881,144 +0.001774,144 +0.001748,144 +0.001905,146 +0.001885,146 +0.001791,146 +0.001961,146 +0.001898,146 +0.001862,146 +0.001960,146 +0.001887,146 +0.001832,146 +0.001839,146 +0.001875,146 +0.001855,146 +0.001900,146 +0.001870,146 +0.001909,146 +0.001862,146 +0.001840,146 +0.001859,146 +0.001834,146 +0.001819,146 +0.001862,146 +0.002148,146 +0.002027,146 +0.001807,146 +0.001939,146 +0.001817,146 +0.001863,146 +0.001835,146 +0.001857,146 +0.001835,146 +0.001915,146 +0.001869,146 +0.001875,146 +0.001886,146 +0.001841,146 +0.001889,146 +0.001834,146 +0.001888,146 +0.001857,146 +0.001886,146 +0.001835,146 +0.001885,146 +0.001837,146 +0.001852,146 +0.001886,146 +0.001918,146 +0.001821,146 +0.001919,146 +0.001934,146 +0.001846,146 +0.001958,146 +0.001918,146 +0.001878,146 +0.001852,146 +0.001881,146 +0.001852,146 +0.001906,146 +0.001880,146 +0.001850,146 +0.001868,146 +0.001860,146 +0.001854,146 +0.001806,146 +0.001794,146 +0.001779,146 +0.001919,146 +0.001855,146 +0.001830,146 +0.002055,146 +0.001954,146 +0.002077,146 +0.001872,146 +0.001821,146 +0.001880,146 +0.001840,146 +0.001889,146 +0.001818,146 +0.001952,146 +0.001826,146 +0.001766,146 +0.001839,146 +0.001794,146 +0.001780,146 +0.001905,146 +0.002034,146 +0.001896,146 +0.001952,146 +0.001833,146 +0.001926,146 +0.002057,146 +0.001924,146 +0.001887,146 +0.001996,146 +0.001829,146 +0.001916,146 +0.001874,146 +0.001916,146 +0.001858,146 +0.001760,146 +0.001866,146 +0.001911,148 +0.001937,148 +0.001896,148 +0.001991,148 +0.001954,148 +0.001922,148 +0.001847,148 +0.001874,148 +0.001895,148 +0.001870,148 +0.001988,148 +0.002018,148 +0.001926,148 +0.001982,148 +0.001928,148 +0.001948,148 +0.001908,148 +0.001924,148 +0.001959,148 +0.001879,148 +0.001966,148 +0.002189,148 +0.001985,148 +0.001938,148 +0.001946,148 +0.001902,148 +0.002000,148 +0.001996,148 +0.001961,148 +0.002127,148 +0.002068,148 +0.001933,148 +0.001979,148 +0.001960,148 +0.002116,148 +0.001997,148 +0.001877,148 +0.002011,148 +0.001897,148 +0.001895,148 +0.001912,148 +0.001844,148 +0.001869,148 +0.001946,148 +0.002006,148 +0.001886,148 +0.001975,148 +0.001852,148 +0.001875,148 +0.001893,148 +0.001875,148 +0.001826,148 +0.001950,148 +0.001973,148 +0.001922,148 +0.001887,148 +0.001872,148 +0.001906,148 +0.001862,148 +0.001892,148 +0.001886,148 +0.002109,148 +0.002104,148 +0.002171,148 +0.002114,148 +0.002102,148 +0.002161,148 +0.002132,148 +0.001975,148 +0.001950,148 +0.001905,148 +0.001915,148 +0.002150,148 +0.001955,148 +0.001946,148 +0.001897,148 +0.001895,148 +0.001987,148 +0.001925,148 +0.001853,148 +0.001973,148 +0.001946,148 +0.001916,148 +0.001968,148 +0.001909,148 +0.001935,148 +0.002010,148 +0.002004,148 +0.001991,148 +0.002028,148 +0.001887,148 +0.001935,148 +0.001936,148 +0.001911,148 +0.001951,148 +0.002074,148 +0.001929,148 +0.001922,148 +0.001936,148 +0.001940,148 +0.002027,150 +0.002057,150 +0.001981,150 +0.002110,150 +0.002039,150 +0.002057,150 +0.002001,150 +0.002028,150 +0.002001,150 +0.002032,150 +0.001992,150 +0.002110,150 +0.001971,150 +0.002019,150 +0.001990,150 +0.002015,150 +0.001988,150 +0.002021,150 +0.001989,150 +0.002062,150 +0.002077,150 +0.001940,150 +0.001878,150 +0.002521,150 +0.002403,150 +0.002418,150 +0.002415,150 +0.002567,150 +0.002108,150 +0.002078,150 +0.002005,150 +0.001971,150 +0.002000,150 +0.001971,150 +0.001997,150 +0.002000,150 +0.002029,150 +0.002118,150 +0.002058,150 +0.002057,150 +0.002034,150 +0.002066,150 +0.002027,150 +0.002163,150 +0.002074,150 +0.002055,150 +0.002094,150 +0.002064,150 +0.002016,150 +0.002077,150 +0.002030,150 +0.002068,150 +0.002005,150 +0.002058,150 +0.002059,150 +0.002117,150 +0.001977,150 +0.001992,150 +0.001878,150 +0.002084,150 +0.001979,150 +0.001900,150 +0.001888,150 +0.002066,150 +0.001897,150 +0.001901,150 +0.001878,150 +0.002055,150 +0.002006,150 +0.001984,150 +0.001986,150 +0.001984,150 +0.002203,150 +0.002178,150 +0.002021,150 +0.001956,150 +0.002305,150 +0.002120,150 +0.002166,150 +0.002037,150 +0.001950,150 +0.002068,150 +0.002040,150 +0.002059,150 +0.002266,150 +0.002048,150 +0.002175,150 +0.002061,150 +0.002094,150 +0.002019,150 +0.002019,150 +0.001992,150 +0.002071,150 +0.001998,150 +0.002128,150 +0.001998,150 +0.002015,150 +0.001990,150 +0.002015,150 +0.001970,150 +0.002200,152 +0.002076,152 +0.002105,152 +0.002039,152 +0.002095,152 +0.002090,152 +0.002091,152 +0.002069,152 +0.002131,152 +0.002110,152 +0.002183,152 +0.002061,152 +0.002054,152 +0.002089,152 +0.002094,152 +0.002085,152 +0.002159,152 +0.002121,152 +0.002076,152 +0.002097,152 +0.002164,152 +0.002036,152 +0.002078,152 +0.002105,152 +0.002139,152 +0.002058,152 +0.002102,152 +0.002118,152 +0.002106,152 +0.002109,152 +0.002084,152 +0.002113,152 +0.002190,152 +0.002055,152 +0.002096,152 +0.002005,152 +0.002036,152 +0.002107,152 +0.002181,152 +0.002116,152 +0.002285,152 +0.002039,152 +0.002262,152 +0.002090,152 +0.002231,152 +0.002144,152 +0.002184,152 +0.002273,152 +0.002246,152 +0.002231,152 +0.002192,152 +0.002114,152 +0.002050,152 +0.002106,152 +0.002111,152 +0.002076,152 +0.002286,152 +0.002248,152 +0.002134,152 +0.002184,152 +0.002176,152 +0.002189,152 +0.002162,152 +0.002302,152 +0.002142,152 +0.002109,152 +0.002065,152 +0.002287,152 +0.002049,152 +0.002108,152 +0.002177,152 +0.002280,152 +0.002104,152 +0.002209,152 +0.002108,152 +0.002147,152 +0.002118,152 +0.002184,152 +0.002196,152 +0.002164,152 +0.002194,152 +0.002148,152 +0.002143,152 +0.002141,152 +0.002147,152 +0.002101,152 +0.002256,152 +0.002117,152 +0.002181,152 +0.002088,152 +0.002180,152 +0.002101,152 +0.002134,152 +0.002111,152 +0.002216,152 +0.002035,152 +0.002099,152 +0.002077,152 +0.002105,152 +0.002024,152 +0.002205,154 +0.002190,154 +0.002186,154 +0.002065,154 +0.002048,154 +0.002059,154 +0.002026,154 +0.002009,154 +0.001975,154 +0.002009,154 +0.002163,154 +0.002015,154 +0.002106,154 +0.002094,154 +0.001976,154 +0.002104,154 +0.002224,154 +0.002124,154 +0.002059,154 +0.002012,154 +0.002226,154 +0.001976,154 +0.001998,154 +0.001976,154 +0.001989,154 +0.001985,154 +0.002102,154 +0.001976,154 +0.002004,154 +0.001975,154 +0.001980,154 +0.001975,154 +0.001976,154 +0.001998,154 +0.001995,154 +0.002019,154 +0.002007,154 +0.002143,154 +0.001982,154 +0.001975,154 +0.001977,154 +0.001975,154 +0.002017,154 +0.002015,154 +0.001980,154 +0.001975,154 +0.001978,154 +0.001976,154 +0.001975,154 +0.001980,154 +0.001976,154 +0.002051,154 +0.002339,154 +0.002092,154 +0.002002,154 +0.002347,154 +0.002136,154 +0.001975,154 +0.002238,154 +0.002441,154 +0.002197,154 +0.002141,154 +0.002148,154 +0.002398,154 +0.002253,154 +0.002308,154 +0.002461,154 +0.002718,154 +0.002731,154 +0.002189,154 +0.002179,154 +0.002139,154 +0.002173,154 +0.002120,154 +0.002183,154 +0.002250,154 +0.002336,154 +0.002298,154 +0.002206,154 +0.002207,154 +0.002210,154 +0.002277,154 +0.002222,154 +0.002231,154 +0.002267,154 +0.002106,154 +0.002195,154 +0.002223,154 +0.002300,154 +0.002120,154 +0.002334,154 +0.002182,154 +0.002288,154 +0.002189,154 +0.002181,154 +0.002210,154 +0.002194,154 +0.002154,154 +0.002150,154 +0.002142,154 +0.002376,156 +0.002262,156 +0.002311,156 +0.002263,156 +0.002213,156 +0.002254,156 +0.002220,156 +0.002106,156 +0.002391,156 +0.002241,156 +0.002224,156 +0.002339,156 +0.002249,156 +0.002301,156 +0.002222,156 +0.002229,156 +0.002243,156 +0.002346,156 +0.002434,156 +0.002223,156 +0.002353,156 +0.002201,156 +0.002275,156 +0.002206,156 +0.002239,156 +0.002356,156 +0.002366,156 +0.002641,156 +0.002381,156 +0.002473,156 +0.002343,156 +0.002298,156 +0.002497,156 +0.002293,156 +0.002359,156 +0.002295,156 +0.002125,156 +0.002263,156 +0.002106,156 +0.002339,156 +0.002133,156 +0.002147,156 +0.002106,156 +0.002135,156 +0.002081,156 +0.002091,156 +0.002085,156 +0.002130,156 +0.002126,156 +0.002204,156 +0.002221,156 +0.002126,156 +0.002280,156 +0.002198,156 +0.002085,156 +0.002255,156 +0.002091,156 +0.002327,156 +0.002051,156 +0.002094,156 +0.002051,156 +0.002077,156 +0.002052,156 +0.002180,156 +0.002051,156 +0.002089,156 +0.002052,156 +0.002074,156 +0.002052,156 +0.002077,156 +0.002051,156 +0.002152,156 +0.002051,156 +0.002076,156 +0.002104,156 +0.002080,156 +0.002051,156 +0.002227,156 +0.002463,156 +0.002595,156 +0.002532,156 +0.003101,156 +0.002759,156 +0.002161,156 +0.002417,156 +0.002332,156 +0.002852,156 +0.002693,156 +0.002307,156 +0.002412,156 +0.002374,156 +0.002192,156 +0.002388,156 +0.002278,156 +0.003178,156 +0.002101,156 +0.002126,156 +0.002054,156 +0.002092,156 +0.002539,156 +0.002303,158 +0.002466,158 +0.002339,158 +0.002641,158 +0.002272,158 +0.002744,158 +0.003238,158 +0.002335,158 +0.002136,158 +0.002323,158 +0.002201,158 +0.002193,158 +0.002135,158 +0.002461,158 +0.002361,158 +0.002260,158 +0.003240,158 +0.002305,158 +0.002172,158 +0.002209,158 +0.002263,158 +0.002313,158 +0.002137,158 +0.002317,158 +0.002133,158 +0.002175,158 +0.002191,158 +0.002377,158 +0.002175,158 +0.002267,158 +0.002179,158 +0.002133,158 +0.002176,158 +0.002133,158 +0.002209,158 +0.002266,158 +0.002210,158 +0.002161,158 +0.002142,158 +0.002178,158 +0.002133,158 +0.002160,158 +0.002271,158 +0.002217,158 +0.002173,158 +0.002162,158 +0.002133,158 +0.002167,158 +0.002149,158 +0.002142,158 +0.002248,158 +0.002133,158 +0.002205,158 +0.002133,158 +0.002195,158 +0.002154,158 +0.002170,158 +0.002215,158 +0.002360,158 +0.002375,158 +0.002395,158 +0.002297,158 +0.002173,158 +0.002307,158 +0.002133,158 +0.002237,158 +0.002158,158 +0.002175,158 +0.002201,158 +0.002134,158 +0.002160,158 +0.002134,158 +0.002178,158 +0.002211,158 +0.002206,158 +0.002133,158 +0.002575,158 +0.002161,158 +0.002392,158 +0.002265,158 +0.002695,158 +0.002399,158 +0.002348,158 +0.002369,158 +0.002395,158 +0.002360,158 +0.002311,158 +0.002464,158 +0.002301,158 +0.002392,158 +0.002272,158 +0.002316,158 +0.002278,158 +0.002309,158 +0.002581,158 +0.002331,158 +0.002421,158 +0.002307,158 +0.002320,158 +0.002327,158 +0.002463,160 +0.002505,160 +0.002359,160 +0.002524,160 +0.002461,160 +0.002479,160 +0.002418,160 +0.002422,160 +0.002452,160 +0.002601,160 +0.002474,160 +0.002475,160 +0.002478,160 +0.002396,160 +0.002523,160 +0.002472,160 +0.002613,160 +0.002480,160 +0.002518,160 +0.002428,160 +0.002499,160 +0.002458,160 +0.002499,160 +0.002613,160 +0.002459,160 +0.002426,160 +0.002437,160 +0.002506,160 +0.002578,160 +0.002446,160 +0.002623,160 +0.002540,160 +0.002366,160 +0.002348,160 +0.002400,160 +0.002920,160 +0.004547,160 +0.004423,160 +0.003569,160 +0.003106,160 +0.003154,160 +0.003011,160 +0.003072,160 +0.002556,160 +0.002439,160 +0.002655,160 +0.002619,160 +0.002688,160 +0.002485,160 +0.002435,160 +0.002441,160 +0.002513,160 +0.002632,160 +0.002688,160 +0.003292,160 +0.002384,160 +0.002356,160 +0.002372,160 +0.002708,160 +0.002480,160 +0.002963,160 +0.002525,160 +0.002414,160 +0.002482,160 +0.002506,160 +0.003104,160 +0.002652,160 +0.002795,160 +0.002947,160 +0.002402,160 +0.003490,160 +0.004557,160 +0.004375,160 +0.002576,160 +0.002360,160 +0.002570,160 +0.002459,160 +0.002533,160 +0.002415,160 +0.002338,160 +0.002298,160 +0.002332,160 +0.002609,160 +0.002467,160 +0.002602,160 +0.002552,160 +0.002393,160 +0.002369,160 +0.002540,160 +0.002677,160 +0.002618,160 +0.002612,160 +0.002328,160 +0.002444,160 +0.002353,160 +0.002467,160 +0.002377,160 +0.002676,160 +0.002682,160 +0.002423,160 +0.002458,162 +0.002465,162 +0.002565,162 +0.002438,162 +0.002644,162 +0.002481,162 +0.002587,162 +0.002390,162 +0.002430,162 +0.002528,162 +0.002642,162 +0.002521,162 +0.002361,162 +0.002344,162 +0.002378,162 +0.002534,162 +0.002318,162 +0.003067,162 +0.002505,162 +0.002380,162 +0.002328,162 +0.002320,162 +0.002513,162 +0.002474,162 +0.002453,162 +0.002954,162 +0.002338,162 +0.002295,162 +0.002423,162 +0.002627,162 +0.002587,162 +0.002435,162 +0.002342,162 +0.002335,162 +0.002330,162 +0.002354,162 +0.002589,162 +0.002554,162 +0.002455,162 +0.002357,162 +0.002449,162 +0.002420,162 +0.002573,162 +0.002431,162 +0.002588,162 +0.002398,162 +0.002329,162 +0.002333,162 +0.002294,162 +0.002598,162 +0.002323,162 +0.002690,162 +0.002464,162 +0.002294,162 +0.002771,162 +0.002616,162 +0.002931,162 +0.003634,162 +0.002846,162 +0.002458,162 +0.002573,162 +0.002642,162 +0.002514,162 +0.003737,162 +0.003512,162 +0.003035,162 +0.003066,162 +0.002736,162 +0.002737,162 +0.002603,162 +0.002509,162 +0.002611,162 +0.002552,162 +0.002661,162 +0.002711,162 +0.002889,162 +0.003110,162 +0.002515,162 +0.002469,162 +0.002588,162 +0.002489,162 +0.002732,162 +0.002544,162 +0.002478,162 +0.002470,162 +0.002426,162 +0.002709,162 +0.002556,162 +0.003016,162 +0.002833,162 +0.002612,162 +0.002586,162 +0.003024,162 +0.003874,162 +0.003492,162 +0.002413,162 +0.002456,162 +0.002437,162 +0.002531,162 +0.002685,162 +0.002841,164 +0.002593,164 +0.002665,164 +0.002604,164 +0.002649,164 +0.002738,164 +0.003488,164 +0.002777,164 +0.002709,164 +0.002529,164 +0.002603,164 +0.002617,164 +0.003292,164 +0.002670,164 +0.002666,164 +0.002505,164 +0.002738,164 +0.002864,164 +0.003250,164 +0.002514,164 +0.002488,164 +0.002493,164 +0.002708,164 +0.002615,164 +0.002986,164 +0.002846,164 +0.002503,164 +0.002438,164 +0.002594,164 +0.002479,164 +0.002857,164 +0.002583,164 +0.002393,164 +0.002433,164 +0.002428,164 +0.002610,164 +0.002759,164 +0.002656,164 +0.002452,164 +0.002440,164 +0.002392,164 +0.002609,164 +0.002540,164 +0.002820,164 +0.002591,164 +0.002432,164 +0.002392,164 +0.002440,164 +0.002621,164 +0.002605,164 +0.002709,164 +0.002562,164 +0.002392,164 +0.002479,164 +0.002531,164 +0.002717,164 +0.002825,164 +0.002509,164 +0.002443,164 +0.002438,164 +0.002413,164 +0.002588,164 +0.002580,164 +0.002837,164 +0.002538,164 +0.002570,164 +0.002403,164 +0.002497,164 +0.002969,164 +0.004485,164 +0.003401,164 +0.002712,164 +0.002849,164 +0.002621,164 +0.003173,164 +0.002910,164 +0.002450,164 +0.002395,164 +0.002477,164 +0.002771,164 +0.002645,164 +0.003699,164 +0.003640,164 +0.002623,164 +0.002579,164 +0.002465,164 +0.002700,164 +0.003043,164 +0.002441,164 +0.002434,164 +0.002557,164 +0.002601,164 +0.002718,164 +0.002494,164 +0.002504,164 +0.002392,164 +0.002434,164 +0.002742,164 +0.002516,164 +0.002553,164 +0.002738,166 +0.002476,166 +0.002514,166 +0.002703,166 +0.002657,166 +0.002698,166 +0.003604,166 +0.002634,166 +0.002706,166 +0.002551,166 +0.002656,166 +0.002929,166 +0.002677,166 +0.002507,166 +0.002471,166 +0.002623,166 +0.002824,166 +0.002874,166 +0.002632,166 +0.002579,166 +0.002471,166 +0.002507,166 +0.002705,166 +0.002622,166 +0.002675,166 +0.002678,166 +0.002513,166 +0.002505,166 +0.002617,166 +0.002702,166 +0.002998,166 +0.002612,166 +0.002503,166 +0.002504,166 +0.002470,166 +0.002719,166 +0.002926,166 +0.002678,166 +0.002527,166 +0.002514,166 +0.002471,166 +0.002740,166 +0.002619,166 +0.002670,166 +0.002584,166 +0.002524,166 +0.002471,166 +0.002555,166 +0.002610,166 +0.002841,166 +0.002664,166 +0.002538,166 +0.002503,166 +0.002506,166 +0.002670,166 +0.002847,166 +0.002579,166 +0.002567,166 +0.002508,166 +0.002510,166 +0.002541,166 +0.002613,166 +0.002932,166 +0.002606,166 +0.002507,166 +0.002521,166 +0.002587,166 +0.002757,166 +0.002953,166 +0.002857,166 +0.002527,166 +0.002512,166 +0.002471,166 +0.002696,166 +0.002884,166 +0.002576,166 +0.002886,166 +0.002597,166 +0.002490,166 +0.002935,166 +0.003359,166 +0.003268,166 +0.002690,166 +0.002614,166 +0.002847,166 +0.002813,166 +0.002894,166 +0.002733,166 +0.002573,166 +0.002665,166 +0.002583,166 +0.002776,166 +0.002903,166 +0.002703,166 +0.002711,166 +0.002610,166 +0.002714,166 +0.002732,166 +0.002941,166 +0.002962,166 +0.002736,168 +0.002825,168 +0.002639,168 +0.002875,168 +0.002828,168 +0.002788,168 +0.002643,168 +0.002624,168 +0.002544,168 +0.002716,168 +0.002653,168 +0.002544,168 +0.002551,168 +0.002544,168 +0.002881,168 +0.002690,168 +0.002870,168 +0.002913,168 +0.002768,168 +0.002799,168 +0.002712,168 +0.002900,168 +0.002896,168 +0.002720,168 +0.002800,168 +0.002811,168 +0.002858,168 +0.002821,168 +0.002969,168 +0.002816,168 +0.002808,168 +0.002847,168 +0.002900,168 +0.002948,168 +0.002776,168 +0.002771,168 +0.003002,168 +0.003304,168 +0.002708,168 +0.002777,168 +0.002802,168 +0.002781,168 +0.002704,168 +0.002751,168 +0.002722,168 +0.003114,168 +0.002848,168 +0.002933,168 +0.002750,168 +0.002655,168 +0.002786,168 +0.002758,168 +0.002819,168 +0.002703,168 +0.002546,168 +0.002857,168 +0.002661,168 +0.003118,168 +0.002598,168 +0.002592,168 +0.002546,168 +0.002840,168 +0.002623,168 +0.002816,168 +0.002566,168 +0.002572,168 +0.002546,168 +0.002573,168 +0.002557,168 +0.002585,168 +0.002585,168 +0.002764,168 +0.002634,168 +0.002619,168 +0.002578,168 +0.002627,168 +0.002832,168 +0.002795,168 +0.002843,168 +0.002642,168 +0.002729,168 +0.002546,168 +0.002848,168 +0.002600,168 +0.002840,168 +0.002853,168 +0.002670,168 +0.002753,168 +0.003022,168 +0.002839,168 +0.002700,168 +0.002935,168 +0.002957,168 +0.002801,168 +0.002843,168 +0.002698,168 +0.002752,168 +0.002640,168 +0.002613,168 +0.002647,168 +0.003061,170 +0.002651,170 +0.002658,170 +0.002652,170 +0.002647,170 +0.002654,170 +0.002785,170 +0.002689,170 +0.002653,170 +0.002655,170 +0.002646,170 +0.002654,170 +0.002727,170 +0.002850,170 +0.002827,170 +0.002848,170 +0.002685,170 +0.002646,170 +0.002706,170 +0.002687,170 +0.002646,170 +0.002678,170 +0.002681,170 +0.002695,170 +0.002732,170 +0.002667,170 +0.002646,170 +0.002651,170 +0.002661,170 +0.002646,170 +0.002715,170 +0.002683,170 +0.002646,170 +0.002788,170 +0.002699,170 +0.002647,170 +0.002671,170 +0.002750,170 +0.002646,170 +0.002657,170 +0.003080,170 +0.002791,170 +0.002717,170 +0.003166,170 +0.002987,170 +0.002883,170 +0.002937,170 +0.002889,170 +0.002878,170 +0.002980,170 +0.003034,170 +0.003052,170 +0.002931,170 +0.002900,170 +0.003430,170 +0.004351,170 +0.003347,170 +0.002914,170 +0.002895,170 +0.002961,170 +0.003166,170 +0.002951,170 +0.002805,170 +0.002811,170 +0.002858,170 +0.002897,170 +0.003924,170 +0.003026,170 +0.003066,170 +0.003075,170 +0.003396,170 +0.002965,170 +0.002926,170 +0.002870,170 +0.002842,170 +0.002817,170 +0.002862,170 +0.002790,170 +0.002856,170 +0.002864,170 +0.002897,170 +0.002913,170 +0.002899,170 +0.002927,170 +0.002776,170 +0.002737,170 +0.002863,170 +0.003196,170 +0.003169,170 +0.003174,170 +0.002972,170 +0.002853,170 +0.002820,170 +0.002859,170 +0.003361,170 +0.003052,170 +0.002939,170 +0.002847,170 +0.002919,170 +0.003103,170 +0.003902,172 +0.003090,172 +0.003185,172 +0.003100,172 +0.003481,172 +0.003511,172 +0.002919,172 +0.002885,172 +0.002846,172 +0.002990,172 +0.003973,172 +0.003140,172 +0.003148,172 +0.003481,172 +0.004020,172 +0.003712,172 +0.003309,172 +0.004044,172 +0.003654,172 +0.003374,172 +0.003559,172 +0.003092,172 +0.003741,172 +0.003325,172 +0.003688,172 +0.003125,172 +0.002883,172 +0.002962,172 +0.003089,172 +0.003673,172 +0.003146,172 +0.002918,172 +0.002874,172 +0.003277,172 +0.003582,172 +0.003203,172 +0.002840,172 +0.002826,172 +0.002969,172 +0.002873,172 +0.003742,172 +0.002978,172 +0.002747,172 +0.002790,172 +0.002956,172 +0.003446,172 +0.003046,172 +0.003076,172 +0.002815,172 +0.002822,172 +0.003267,172 +0.003442,172 +0.003169,172 +0.002880,172 +0.003011,172 +0.002947,172 +0.003491,172 +0.003078,172 +0.003057,172 +0.002833,172 +0.002914,172 +0.003242,172 +0.003967,172 +0.003268,172 +0.003196,172 +0.003737,172 +0.003576,172 +0.003445,172 +0.003383,172 +0.003298,172 +0.003517,172 +0.003472,172 +0.003764,172 +0.003244,172 +0.003295,172 +0.003393,172 +0.004356,172 +0.002991,172 +0.003097,172 +0.002999,172 +0.003230,172 +0.003457,172 +0.003304,172 +0.003088,172 +0.003121,172 +0.003199,172 +0.003288,172 +0.003158,172 +0.003051,172 +0.003034,172 +0.003097,172 +0.003490,172 +0.003527,172 +0.003157,172 +0.003086,172 +0.003062,172 +0.003758,172 +0.003198,172 +0.003024,172 +0.003026,172 +0.003241,174 +0.003326,174 +0.003523,174 +0.003166,174 +0.003049,174 +0.002951,174 +0.003015,174 +0.003492,174 +0.003125,174 +0.003071,174 +0.002934,174 +0.003152,174 +0.003262,174 +0.003171,174 +0.002934,174 +0.002902,174 +0.002950,174 +0.003223,174 +0.003100,174 +0.003020,174 +0.002920,174 +0.002898,174 +0.003007,174 +0.003383,174 +0.003026,174 +0.002891,174 +0.002854,174 +0.002931,174 +0.003208,174 +0.003114,174 +0.003021,174 +0.002915,174 +0.002917,174 +0.003070,174 +0.003445,174 +0.003060,174 +0.002881,174 +0.002894,174 +0.002915,174 +0.003362,174 +0.003250,174 +0.003025,174 +0.003080,174 +0.002871,174 +0.003250,174 +0.003490,174 +0.003080,174 +0.002888,174 +0.002854,174 +0.003048,174 +0.003322,174 +0.003161,174 +0.002920,174 +0.002912,174 +0.002892,174 +0.003085,174 +0.003400,174 +0.003015,174 +0.002890,174 +0.002886,174 +0.003020,174 +0.003468,174 +0.003093,174 +0.002877,174 +0.002899,174 +0.002854,174 +0.003218,174 +0.002946,174 +0.003106,174 +0.002854,174 +0.002877,174 +0.002981,174 +0.003005,174 +0.003079,174 +0.003046,174 +0.003034,174 +0.002933,174 +0.003010,174 +0.003004,174 +0.002890,174 +0.002854,174 +0.002896,174 +0.002927,174 +0.003026,174 +0.002854,174 +0.002879,174 +0.002907,174 +0.002924,174 +0.002894,174 +0.002987,174 +0.002896,174 +0.002855,174 +0.002856,174 +0.002856,174 +0.002995,174 +0.002946,174 +0.004948,174 +0.002848,174 +0.002884,174 +0.002914,174 +0.003203,176 +0.003099,176 +0.002981,176 +0.003032,176 +0.002989,176 +0.003070,176 +0.003019,176 +0.003216,176 +0.003142,176 +0.002963,176 +0.003031,176 +0.003067,176 +0.002967,176 +0.002943,176 +0.002967,176 +0.002990,176 +0.003060,176 +0.002943,176 +0.002949,176 +0.002953,176 +0.002950,176 +0.002994,176 +0.003031,176 +0.002969,176 +0.002945,176 +0.002943,176 +0.002947,176 +0.003173,176 +0.002968,176 +0.002944,176 +0.002946,176 +0.002950,176 +0.002985,176 +0.003058,176 +0.002969,176 +0.002945,176 +0.002946,176 +0.002943,176 +0.002997,176 +0.003061,176 +0.003302,176 +0.003073,176 +0.003229,176 +0.003052,176 +0.003085,176 +0.002981,176 +0.002943,176 +0.002989,176 +0.002990,176 +0.003089,176 +0.002975,176 +0.002985,176 +0.002972,176 +0.002969,176 +0.002983,176 +0.003100,176 +0.002970,176 +0.002974,176 +0.002973,176 +0.002974,176 +0.003029,176 +0.003057,176 +0.002944,176 +0.002946,176 +0.002946,176 +0.002983,176 +0.003064,176 +0.002967,176 +0.002949,176 +0.002943,176 +0.002946,176 +0.003116,176 +0.003328,176 +0.003176,176 +0.003190,176 +0.003030,176 +0.003068,176 +0.003127,176 +0.002944,176 +0.002978,176 +0.002968,176 +0.002979,176 +0.002988,176 +0.003028,176 +0.002983,176 +0.002976,176 +0.002943,176 +0.003021,176 +0.003105,176 +0.002970,176 +0.003105,176 +0.002968,176 +0.002970,176 +0.003030,176 +0.003230,176 +0.002944,176 +0.002966,176 +0.002977,176 +0.002982,176 +0.003142,176 +0.003227,178 +0.003164,178 +0.003159,178 +0.003164,178 +0.003305,178 +0.003147,178 +0.003396,178 +0.003352,178 +0.003144,178 +0.003265,178 +0.003232,178 +0.003124,178 +0.003167,178 +0.003149,178 +0.003172,178 +0.003259,178 +0.003167,178 +0.003154,178 +0.003137,178 +0.003170,178 +0.003277,178 +0.003146,178 +0.003123,178 +0.003121,178 +0.003156,178 +0.003296,178 +0.003127,178 +0.003123,178 +0.003121,178 +0.003126,178 +0.003172,178 +0.003277,178 +0.003134,178 +0.003121,178 +0.003148,178 +0.003178,178 +0.003252,178 +0.003293,178 +0.003244,178 +0.003292,178 +0.003203,178 +0.003303,178 +0.003163,178 +0.003120,178 +0.003144,178 +0.003205,178 +0.003268,178 +0.003182,178 +0.003121,178 +0.003144,178 +0.003146,178 +0.003319,178 +0.003147,178 +0.003180,178 +0.003282,178 +0.003147,178 +0.003277,178 +0.003407,178 +0.003121,178 +0.003155,178 +0.003144,178 +0.003228,178 +0.003333,178 +0.003121,178 +0.003161,178 +0.003155,178 +0.003195,178 +0.003336,178 +0.003121,178 +0.003438,178 +0.003315,178 +0.003203,178 +0.003288,178 +0.003128,178 +0.003120,178 +0.003143,178 +0.003150,178 +0.003320,178 +0.003161,178 +0.003120,178 +0.003125,178 +0.003127,178 +0.003172,178 +0.003242,178 +0.003126,178 +0.003123,178 +0.003132,178 +0.003182,178 +0.003343,178 +0.003130,178 +0.003122,178 +0.003124,178 +0.003168,178 +0.003281,178 +0.003143,178 +0.003134,178 +0.003125,178 +0.003134,178 +0.003321,178 +0.003504,178 +0.003532,180 +0.003527,180 +0.003495,180 +0.004005,180 +0.003884,180 +0.003513,180 +0.003805,180 +0.004142,180 +0.003700,180 +0.003624,180 +0.003559,180 +0.003563,180 +0.004337,180 +0.003487,180 +0.003685,180 +0.003598,180 +0.003867,180 +0.003445,180 +0.003491,180 +0.003522,180 +0.003481,180 +0.003945,180 +0.003953,180 +0.004146,180 +0.004174,180 +0.004212,180 +0.004009,180 +0.003485,180 +0.003538,180 +0.003646,180 +0.003549,180 +0.003461,180 +0.003395,180 +0.003463,180 +0.003542,180 +0.003426,180 +0.003518,180 +0.003413,180 +0.003602,180 +0.003509,180 +0.003408,180 +0.003412,180 +0.003377,180 +0.003336,180 +0.003671,180 +0.003293,180 +0.003453,180 +0.003358,180 +0.003344,180 +0.003603,180 +0.003481,180 +0.003429,180 +0.003489,180 +0.003376,180 +0.003597,180 +0.003199,180 +0.006071,180 +0.004131,180 +0.003369,180 +0.003196,180 +0.003210,180 +0.003194,180 +0.003199,180 +0.003336,180 +0.003196,180 +0.003194,180 +0.003187,180 +0.003171,180 +0.003301,180 +0.003201,180 +0.003163,180 +0.003162,180 +0.003171,180 +0.003267,180 +0.003166,180 +0.003167,180 +0.003166,180 +0.003159,180 +0.003270,180 +0.003200,180 +0.003164,180 +0.003162,180 +0.003159,180 +0.003192,180 +0.003267,180 +0.003193,180 +0.005630,180 +0.004376,180 +0.003275,180 +0.003530,180 +0.003186,180 +0.003178,180 +0.003557,180 +0.003336,180 +0.003194,180 +0.003186,180 +0.003168,180 +0.003224,180 +0.003370,180 +0.003219,180 +0.003419,182 +0.003360,182 +0.003445,182 +0.003421,182 +0.003299,182 +0.003305,182 +0.003301,182 +0.003343,182 +0.003424,182 +0.003325,182 +0.003298,182 +0.003322,182 +0.003373,182 +0.003489,182 +0.003304,182 +0.005269,182 +0.005044,182 +0.003425,182 +0.003322,182 +0.003903,182 +0.003365,182 +0.003537,182 +0.003340,182 +0.003298,182 +0.003336,182 +0.003341,182 +0.003955,182 +0.003324,182 +0.003321,182 +0.003531,182 +0.003343,182 +0.003573,182 +0.003330,182 +0.003325,182 +0.003301,182 +0.003342,182 +0.003532,182 +0.003298,182 +0.003303,182 +0.003549,182 +0.003413,182 +0.003438,182 +0.003696,182 +0.003524,182 +0.003702,182 +0.003760,182 +0.003347,182 +0.003355,182 +0.003342,182 +0.003384,182 +0.003610,182 +0.003367,182 +0.003298,182 +0.003330,182 +0.003324,182 +0.003480,182 +0.003303,182 +0.003301,182 +0.003298,182 +0.003307,182 +0.003458,182 +0.003303,182 +0.003301,182 +0.003303,182 +0.003298,182 +0.003439,182 +0.003310,182 +0.003300,182 +0.003303,182 +0.003300,182 +0.003435,182 +0.003303,182 +0.003391,182 +0.003518,182 +0.003521,182 +0.003498,182 +0.003325,182 +0.003298,182 +0.003301,182 +0.003323,182 +0.003512,182 +0.003333,182 +0.003322,182 +0.003591,182 +0.003476,182 +0.004173,182 +0.003645,182 +0.003635,182 +0.003644,182 +0.003882,182 +0.003617,182 +0.003687,182 +0.003340,182 +0.003958,182 +0.003804,182 +0.003327,182 +0.003628,182 +0.003332,182 +0.003860,182 +0.003440,182 +0.003478,184 +0.003872,184 +0.003574,184 +0.003834,184 +0.003615,184 +0.003471,184 +0.003375,184 +0.003846,184 +0.003612,184 +0.003411,184 +0.003410,184 +0.003546,184 +0.003638,184 +0.003450,184 +0.003373,184 +0.003526,184 +0.003471,184 +0.003868,184 +0.003409,184 +0.003431,184 +0.003712,184 +0.003757,184 +0.003919,184 +0.003743,184 +0.003877,184 +0.004446,184 +0.003781,184 +0.003494,184 +0.003574,184 +0.003515,184 +0.003843,184 +0.003487,184 +0.003726,184 +0.003475,184 +0.003640,184 +0.003964,184 +0.003477,184 +0.003568,184 +0.003411,184 +0.003638,184 +0.003699,184 +0.003571,184 +0.003672,184 +0.003516,184 +0.004054,184 +0.003407,184 +0.003674,184 +0.003867,184 +0.003679,184 +0.003599,184 +0.003693,184 +0.003736,184 +0.003553,184 +0.003677,184 +0.003643,184 +0.003409,184 +0.003831,184 +0.003547,184 +0.003797,184 +0.003437,184 +0.003663,184 +0.003375,184 +0.003873,184 +0.003873,184 +0.003577,184 +0.003461,184 +0.003798,184 +0.003677,184 +0.003596,184 +0.003566,184 +0.003559,184 +0.003469,184 +0.003993,184 +0.003479,184 +0.003687,184 +0.003701,184 +0.003874,184 +0.003523,184 +0.003456,184 +0.003757,184 +0.003408,184 +0.003917,184 +0.003623,184 +0.003624,184 +0.003523,184 +0.003855,184 +0.003613,184 +0.003494,184 +0.003663,184 +0.003490,184 +0.003807,184 +0.003417,184 +0.003416,184 +0.003683,184 +0.003648,184 +0.003582,184 +0.003410,184 +0.003647,184 +0.003441,184 +0.003741,184 +0.003845,186 +0.003610,186 +0.003840,186 +0.003755,186 +0.003996,186 +0.003626,186 +0.003662,186 +0.003616,186 +0.003837,186 +0.003807,186 +0.003575,186 +0.003698,186 +0.003855,186 +0.004090,186 +0.003746,186 +0.003654,186 +0.003662,186 +0.003753,186 +0.003679,186 +0.003729,186 +0.003555,186 +0.004084,186 +0.003842,186 +0.004042,186 +0.003867,186 +0.003646,186 +0.004074,186 +0.003782,186 +0.003805,186 +0.003689,186 +0.004057,186 +0.003623,186 +0.003747,186 +0.003948,186 +0.003816,186 +0.003689,186 +0.003701,186 +0.004021,186 +0.003745,186 +0.004078,186 +0.003749,186 +0.003831,186 +0.003884,186 +0.003938,186 +0.003788,186 +0.003805,186 +0.003843,186 +0.003815,186 +0.004775,186 +0.003963,186 +0.003874,186 +0.003864,186 +0.003969,186 +0.003972,186 +0.003923,186 +0.003896,186 +0.003969,186 +0.003879,186 +0.003893,186 +0.003916,186 +0.004147,186 +0.003865,186 +0.004189,186 +0.004018,186 +0.003830,186 +0.004140,186 +0.003803,186 +0.003855,186 +0.003847,186 +0.003834,186 +0.003831,186 +0.003836,186 +0.003793,186 +0.003835,186 +0.003768,186 +0.003816,186 +0.003799,186 +0.003921,186 +0.003870,186 +0.004070,186 +0.003696,186 +0.003704,186 +0.003752,186 +0.003701,186 +0.003696,186 +0.003698,186 +0.003802,186 +0.003829,186 +0.003843,186 +0.003912,186 +0.003834,186 +0.003880,186 +0.003977,186 +0.003990,186 +0.003975,186 +0.004196,186 +0.003918,186 +0.003972,186 +0.003786,186 +0.003989,186 +0.004024,188 +0.003897,188 +0.004031,188 +0.004214,188 +0.004064,188 +0.004241,188 +0.004052,188 +0.004212,188 +0.003884,188 +0.003797,188 +0.003846,188 +0.004174,188 +0.003773,188 +0.003844,188 +0.003915,188 +0.003933,188 +0.003868,188 +0.004128,188 +0.003906,188 +0.003799,188 +0.004139,188 +0.003761,188 +0.003768,188 +0.003649,188 +0.004111,188 +0.003630,188 +0.003623,188 +0.003623,188 +0.004186,188 +0.003697,188 +0.003625,188 +0.003631,188 +0.003623,188 +0.003940,188 +0.003590,188 +0.003624,188 +0.003612,188 +0.004065,188 +0.003658,188 +0.003619,188 +0.004117,188 +0.003771,188 +0.003812,188 +0.004109,188 +0.004308,188 +0.004025,188 +0.004218,188 +0.004310,188 +0.004067,188 +0.004055,188 +0.004184,188 +0.003942,188 +0.003944,188 +0.003933,188 +0.004197,188 +0.003830,188 +0.003792,188 +0.003786,188 +0.003824,188 +0.003799,188 +0.004284,188 +0.004693,188 +0.004667,188 +0.004699,188 +0.004437,188 +0.004234,188 +0.004512,188 +0.004764,188 +0.003925,188 +0.003896,188 +0.003828,188 +0.003627,188 +0.003622,188 +0.004032,188 +0.003930,188 +0.003808,188 +0.003687,188 +0.003625,188 +0.004004,188 +0.004522,188 +0.003834,188 +0.003986,188 +0.003676,188 +0.004082,188 +0.003910,188 +0.003707,188 +0.004000,188 +0.003937,188 +0.003630,188 +0.003981,188 +0.004197,188 +0.004261,188 +0.004293,188 +0.003882,188 +0.003632,188 +0.003609,188 +0.003895,188 +0.003628,188 +0.003638,188 +0.003625,188 +0.004031,190 +0.003732,190 +0.003713,190 +0.004069,190 +0.003967,190 +0.003814,190 +0.003748,190 +0.003742,190 +0.003794,190 +0.003921,190 +0.003743,190 +0.003948,190 +0.003822,190 +0.003917,190 +0.004039,190 +0.003837,190 +0.003886,190 +0.004334,190 +0.004670,190 +0.003918,190 +0.003767,190 +0.003926,190 +0.004045,190 +0.003812,190 +0.003883,190 +0.004148,190 +0.003789,190 +0.003733,190 +0.003982,190 +0.003831,190 +0.004218,190 +0.003887,190 +0.004065,190 +0.003829,190 +0.004073,190 +0.003854,190 +0.003864,190 +0.004040,190 +0.003908,190 +0.003708,190 +0.003750,190 +0.003847,190 +0.003919,190 +0.004953,190 +0.003872,190 +0.003915,190 +0.003805,190 +0.003935,190 +0.003760,190 +0.003736,190 +0.003736,190 +0.003952,190 +0.003751,190 +0.003966,190 +0.003744,190 +0.004105,190 +0.004057,190 +0.003848,190 +0.004103,190 +0.004007,190 +0.003991,190 +0.003958,190 +0.003858,190 +0.003898,190 +0.004309,190 +0.003779,190 +0.004400,190 +0.003842,190 +0.004690,190 +0.004212,190 +0.003952,190 +0.003825,190 +0.004132,190 +0.003949,190 +0.003970,190 +0.003901,190 +0.004336,190 +0.004022,190 +0.004203,190 +0.004286,190 +0.004137,190 +0.003975,190 +0.004159,190 +0.004018,190 +0.004249,190 +0.003971,190 +0.004063,190 +0.004069,190 +0.004161,190 +0.004036,190 +0.004065,190 +0.004090,190 +0.004051,190 +0.005002,190 +0.004184,190 +0.004001,190 +0.004155,190 +0.004062,190 +0.004114,190 +0.003998,190 +0.004504,192 +0.004226,192 +0.004231,192 +0.004154,192 +0.004391,192 +0.004256,192 +0.004164,192 +0.004621,192 +0.004542,192 +0.004177,192 +0.004269,192 +0.004290,192 +0.004861,192 +0.004628,192 +0.004849,192 +0.004740,192 +0.005128,192 +0.004511,192 +0.004285,192 +0.004344,192 +0.004368,192 +0.004324,192 +0.004270,192 +0.004247,192 +0.004214,192 +0.004177,192 +0.004196,192 +0.004271,192 +0.004291,192 +0.004231,192 +0.004307,192 +0.004301,192 +0.004323,192 +0.004325,192 +0.004395,192 +0.004233,192 +0.004210,192 +0.004187,192 +0.004353,192 +0.004866,192 +0.004385,192 +0.004047,192 +0.004525,192 +0.003945,192 +0.004156,192 +0.004179,192 +0.004386,192 +0.003930,192 +0.004117,192 +0.004327,192 +0.004368,192 +0.004014,192 +0.003910,192 +0.003904,192 +0.004190,192 +0.003906,192 +0.003925,192 +0.004126,192 +0.004216,192 +0.003886,192 +0.004132,192 +0.003998,192 +0.004251,192 +0.004098,192 +0.004112,192 +0.004504,192 +0.004272,192 +0.003909,192 +0.004141,192 +0.003941,192 +0.004337,192 +0.003940,192 +0.004117,192 +0.003910,192 +0.004424,192 +0.003919,192 +0.003898,192 +0.003902,192 +0.004022,192 +0.004072,192 +0.004431,192 +0.003967,192 +0.003941,192 +0.004591,192 +0.003937,192 +0.003917,192 +0.004209,192 +0.004130,192 +0.004293,192 +0.004051,192 +0.003910,192 +0.004307,192 +0.004201,192 +0.004005,192 +0.003908,192 +0.004203,192 +0.003900,192 +0.003901,192 +0.003943,192 +0.004501,192 +0.004094,194 +0.003974,194 +0.003975,194 +0.004121,194 +0.003976,194 +0.003980,194 +0.004015,194 +0.004034,194 +0.003976,194 +0.003973,194 +0.004004,194 +0.004086,194 +0.004075,194 +0.004420,194 +0.004343,194 +0.004312,194 +0.004001,194 +0.004387,194 +0.004344,194 +0.004544,194 +0.004207,194 +0.004339,194 +0.004008,194 +0.004578,194 +0.004317,194 +0.004009,194 +0.004012,194 +0.004546,194 +0.004054,194 +0.004310,194 +0.004004,194 +0.004506,194 +0.004113,194 +0.004008,194 +0.004088,194 +0.004520,194 +0.004227,194 +0.004167,194 +0.004084,194 +0.004735,194 +0.004056,194 +0.004024,194 +0.004008,194 +0.004922,194 +0.004059,194 +0.004162,194 +0.004479,194 +0.004472,194 +0.004063,194 +0.004017,194 +0.004182,194 +0.004476,194 +0.004032,194 +0.004005,194 +0.004184,194 +0.004252,194 +0.004021,194 +0.004021,194 +0.004307,194 +0.004103,194 +0.004144,194 +0.004259,194 +0.004192,194 +0.004050,194 +0.003998,194 +0.004037,194 +0.004022,194 +0.004178,194 +0.003980,194 +0.003993,194 +0.003995,194 +0.004106,194 +0.004053,194 +0.003975,194 +0.003972,194 +0.004192,194 +0.003993,194 +0.004004,194 +0.003973,194 +0.004068,194 +0.003973,194 +0.003975,194 +0.003972,194 +0.004128,194 +0.003974,194 +0.004098,194 +0.004575,194 +0.004296,194 +0.004130,194 +0.004183,194 +0.004120,194 +0.004470,194 +0.004030,194 +0.004261,194 +0.004005,194 +0.004232,194 +0.004353,194 +0.004020,194 +0.004014,194 +0.004370,194 +0.004545,196 +0.004109,196 +0.004114,196 +0.004780,196 +0.004173,196 +0.004120,196 +0.004174,196 +0.004525,196 +0.004153,196 +0.004320,196 +0.004240,196 +0.004641,196 +0.004273,196 +0.004153,196 +0.004230,196 +0.005091,196 +0.004129,196 +0.004116,196 +0.004529,196 +0.004654,196 +0.004161,196 +0.004103,196 +0.004477,196 +0.004167,196 +0.004105,196 +0.004087,196 +0.004363,196 +0.004107,196 +0.004086,196 +0.004084,196 +0.004302,196 +0.004159,196 +0.004212,196 +0.004335,196 +0.004361,196 +0.004112,196 +0.004103,196 +0.004086,196 +0.004496,196 +0.004112,196 +0.004112,196 +0.004083,196 +0.004287,196 +0.004234,196 +0.004107,196 +0.004129,196 +0.004353,196 +0.004115,196 +0.004112,196 +0.004107,196 +0.004223,196 +0.004106,196 +0.004081,196 +0.004080,196 +0.004291,196 +0.004178,196 +0.004245,196 +0.004458,196 +0.004203,196 +0.004119,196 +0.004101,196 +0.004108,196 +0.004204,196 +0.004112,196 +0.004143,196 +0.004096,196 +0.004264,196 +0.004084,196 +0.004080,196 +0.004083,196 +0.004164,196 +0.004152,196 +0.004085,196 +0.004126,196 +0.004152,196 +0.004085,196 +0.004088,196 +0.004084,196 +0.004290,196 +0.004087,196 +0.004122,196 +0.004348,196 +0.004431,196 +0.004117,196 +0.004116,196 +0.004105,196 +0.004325,196 +0.004107,196 +0.004109,196 +0.004101,196 +0.004195,196 +0.004197,196 +0.004107,196 +0.004083,196 +0.004143,196 +0.004084,196 +0.004081,196 +0.004082,196 +0.004170,196 +0.004112,196 +0.004318,198 +0.004213,198 +0.004337,198 +0.004214,198 +0.004221,198 +0.004428,198 +0.004661,198 +0.004246,198 +0.004222,198 +0.004220,198 +0.004276,198 +0.004216,198 +0.004216,198 +0.004216,198 +0.004276,198 +0.004217,198 +0.004213,198 +0.004215,198 +0.004296,198 +0.004217,198 +0.004212,198 +0.004212,198 +0.004280,198 +0.004213,198 +0.004212,198 +0.004426,198 +0.004250,198 +0.004213,198 +0.004275,198 +0.004559,198 +0.004351,198 +0.004217,198 +0.004215,198 +0.004253,198 +0.004269,198 +0.004281,198 +0.004236,198 +0.004277,198 +0.004218,198 +0.004223,198 +0.004214,198 +0.004282,198 +0.004261,198 +0.004236,198 +0.004212,198 +0.004287,198 +0.004214,198 +0.004212,198 +0.004216,198 +0.004355,198 +0.004310,198 +0.004226,198 +0.004401,198 +0.004622,198 +0.004228,198 +0.004243,198 +0.004255,198 +0.004280,198 +0.004212,198 +0.004213,198 +0.004215,198 +0.004380,198 +0.004283,198 +0.004212,198 +0.004255,198 +0.004258,198 +0.004219,198 +0.004212,198 +0.004250,198 +0.004269,198 +0.004219,198 +0.004212,198 +0.004329,198 +0.004275,198 +0.004237,198 +0.004275,198 +0.004463,198 +0.004471,198 +0.004213,198 +0.004217,198 +0.004276,198 +0.004217,198 +0.004242,198 +0.004213,198 +0.004380,198 +0.004281,198 +0.004233,198 +0.004236,198 +0.004297,198 +0.004251,198 +0.004215,198 +0.004216,198 +0.004283,198 +0.004217,198 +0.004213,198 +0.004256,198 +0.004317,198 +0.004218,198 +0.004213,198 +0.004384,198 +0.004875,200 +0.004366,200 +0.004361,200 +0.004399,200 +0.004415,200 +0.004386,200 +0.004407,200 +0.004432,200 +0.004415,200 +0.004395,200 +0.004359,200 +0.004429,200 +0.004360,200 +0.004360,200 +0.004363,200 +0.004422,200 +0.004364,200 +0.004360,200 +0.004365,200 +0.004492,200 +0.004363,200 +0.004359,200 +0.004645,200 +0.004694,200 +0.004384,200 +0.004401,200 +0.004458,200 +0.004390,200 +0.004361,200 +0.004362,200 +0.004420,200 +0.004363,200 +0.004360,200 +0.004362,200 +0.004421,200 +0.004365,200 +0.004360,200 +0.004395,200 +0.004394,200 +0.004364,200 +0.004419,200 +0.004473,200 +0.004394,200 +0.004363,200 +0.004360,200 +0.004659,200 +0.004696,200 +0.004363,200 +0.004363,200 +0.004435,200 +0.004360,200 +0.004365,200 +0.004362,200 +0.004423,200 +0.004365,200 +0.004360,200 +0.004391,200 +0.004401,200 +0.004360,200 +0.004360,200 +0.004427,200 +0.004361,200 +0.004364,200 +0.004360,200 +0.004512,200 +0.004462,200 +0.004412,200 +0.004441,200 +0.004719,200 +0.004610,200 +0.004381,200 +0.004391,200 +0.004394,200 +0.004364,200 +0.004359,200 +0.004427,200 +0.004363,200 +0.004368,200 +0.004362,200 +0.004423,200 +0.004364,200 +0.004360,200 +0.004360,200 +0.004427,200 +0.004361,200 +0.004359,200 +0.004365,200 +0.004485,200 +0.004360,200 +0.004359,200 +0.004434,200 +0.004607,200 +0.004493,200 +0.004360,200 +0.004424,200 +0.004388,200 +0.004379,200 +0.004362,200 +0.004463,200 +0.004364,200 +0.004613,202 +0.004510,202 +0.004569,202 +0.004569,202 +0.004556,202 +0.004572,202 +0.004568,202 +0.004506,202 +0.004510,202 +0.004647,202 +0.004511,202 +0.004512,202 +0.004506,202 +0.004837,202 +0.004759,202 +0.004527,202 +0.004569,202 +0.004513,202 +0.004506,202 +0.004507,202 +0.004567,202 +0.004573,202 +0.004506,202 +0.004505,202 +0.004572,202 +0.004508,202 +0.004508,202 +0.004534,202 +0.004607,202 +0.004505,202 +0.004505,202 +0.004670,202 +0.004512,202 +0.004506,202 +0.004507,202 +0.004731,202 +0.004904,202 +0.004541,202 +0.004570,202 +0.004554,202 +0.004550,202 +0.004508,202 +0.004588,202 +0.004510,202 +0.004506,202 +0.004534,202 +0.004548,202 +0.004505,202 +0.004509,202 +0.004566,202 +0.004510,202 +0.004506,202 +0.004505,202 +0.004651,202 +0.004515,202 +0.004506,202 +0.004507,202 +0.004745,202 +0.004779,202 +0.004541,202 +0.005156,202 +0.004703,202 +0.004594,202 +0.004549,202 +0.004622,202 +0.004536,202 +0.004507,202 +0.004540,202 +0.004541,202 +0.004506,202 +0.004577,202 +0.004567,202 +0.004510,202 +0.004508,202 +0.004508,202 +0.004650,202 +0.004569,202 +0.004506,202 +0.004534,202 +0.004766,202 +0.004821,202 +0.004531,202 +0.004595,202 +0.004530,202 +0.004506,202 +0.004537,202 +0.004541,202 +0.004552,202 +0.004505,202 +0.004571,202 +0.004506,202 +0.004541,202 +0.004512,202 +0.004737,202 +0.004575,202 +0.005103,202 +0.004676,202 +0.004671,202 +0.004511,202 +0.004506,202 +0.005009,204 +0.004842,204 +0.004876,204 +0.004731,204 +0.004710,204 +0.004678,204 +0.004683,204 +0.004779,204 +0.004634,204 +0.004630,204 +0.004762,204 +0.004729,204 +0.004662,204 +0.004660,204 +0.004727,204 +0.004658,204 +0.004705,204 +0.004630,204 +0.004776,204 +0.004630,204 +0.004630,204 +0.004693,204 +0.004734,204 +0.004899,204 +0.004656,204 +0.004711,204 +0.004653,204 +0.004658,204 +0.004669,204 +0.004662,204 +0.004720,204 +0.004629,204 +0.004696,204 +0.004633,204 +0.004635,204 +0.004657,204 +0.004666,204 +0.004630,204 +0.004630,204 +0.004703,204 +0.004710,204 +0.004629,204 +0.004632,204 +0.004732,204 +0.004913,204 +0.004954,204 +0.004754,204 +0.004660,204 +0.004650,204 +0.004663,204 +0.004727,204 +0.004678,204 +0.004629,204 +0.004697,204 +0.004631,204 +0.004630,204 +0.004633,204 +0.004696,204 +0.004629,204 +0.004630,204 +0.004660,204 +0.004747,204 +0.004632,204 +0.004634,204 +0.004871,204 +0.004808,204 +0.004954,204 +0.004905,204 +0.004718,204 +0.004630,204 +0.004633,204 +0.004883,204 +0.004657,204 +0.004652,204 +0.004680,204 +0.004664,204 +0.004670,204 +0.004635,204 +0.004735,204 +0.004636,204 +0.004629,204 +0.004642,204 +0.004760,204 +0.004714,204 +0.004629,204 +0.004696,204 +0.004633,204 +0.004911,204 +0.004861,204 +0.004721,204 +0.004634,204 +0.004653,204 +0.004802,204 +0.004630,204 +0.004630,204 +0.004633,204 +0.004692,204 +0.004636,204 +0.004629,204 +0.004660,204 +0.004924,206 +0.004774,206 +0.004782,206 +0.004954,206 +0.004774,206 +0.004768,206 +0.004833,206 +0.004776,206 +0.005023,206 +0.005055,206 +0.004881,206 +0.004827,206 +0.004799,206 +0.004937,206 +0.004790,206 +0.004772,206 +0.004799,206 +0.004842,206 +0.004774,206 +0.004841,206 +0.004854,206 +0.004783,206 +0.004769,206 +0.004809,206 +0.004883,206 +0.004772,206 +0.004780,206 +0.004834,206 +0.004769,206 +0.005048,206 +0.005067,206 +0.004833,206 +0.004809,206 +0.004795,206 +0.004867,206 +0.004885,206 +0.004772,206 +0.004829,206 +0.004773,206 +0.004769,206 +0.004774,206 +0.004830,206 +0.004772,206 +0.004769,206 +0.004840,206 +0.004875,206 +0.004769,206 +0.004771,206 +0.004831,206 +0.004773,206 +0.005113,206 +0.007995,206 +0.008452,206 +0.008079,206 +0.004899,206 +0.004804,206 +0.004890,206 +0.004867,206 +0.004817,206 +0.004849,206 +0.004793,206 +0.004769,206 +0.004773,206 +0.004853,206 +0.004848,206 +0.004771,206 +0.004855,206 +0.004816,206 +0.004872,206 +0.005138,206 +0.004869,206 +0.004817,206 +0.004909,206 +0.004867,206 +0.004769,206 +0.004772,206 +0.004822,206 +0.004817,206 +0.004771,206 +0.004769,206 +0.004853,206 +0.004775,206 +0.004769,206 +0.004810,206 +0.004889,206 +0.004830,206 +0.004773,206 +0.004847,206 +0.004771,206 +0.004953,206 +0.005091,206 +0.004828,206 +0.004772,206 +0.004789,206 +0.004878,206 +0.004794,206 +0.004769,206 +0.004853,206 +0.004769,206 +0.004771,206 +0.005036,208 +0.005039,208 +0.004925,208 +0.004919,208 +0.005021,208 +0.005021,208 +0.004919,208 +0.004978,208 +0.004968,208 +0.004918,208 +0.005219,208 +0.005136,208 +0.004977,208 +0.005095,208 +0.005023,208 +0.004945,208 +0.004943,208 +0.005017,208 +0.004997,208 +0.004923,208 +0.004930,208 +0.005022,208 +0.004922,208 +0.004961,208 +0.004998,208 +0.004979,208 +0.004918,208 +0.004960,208 +0.004986,208 +0.004919,208 +0.005191,208 +0.005119,208 +0.004947,208 +0.004944,208 +0.004981,208 +0.004919,208 +0.005006,208 +0.004949,208 +0.004951,208 +0.004922,208 +0.004920,208 +0.004983,208 +0.004919,208 +0.004918,208 +0.004981,208 +0.005008,208 +0.004927,208 +0.004949,208 +0.004953,208 +0.004924,208 +0.005130,208 +0.005277,208 +0.004958,208 +0.004919,208 +0.005014,208 +0.004946,208 +0.004918,208 +0.004952,208 +0.004955,208 +0.004919,208 +0.004920,208 +0.004986,208 +0.004918,208 +0.004918,208 +0.004971,208 +0.004995,208 +0.004938,208 +0.005102,208 +0.005016,208 +0.004951,208 +0.005049,208 +0.005315,208 +0.004940,208 +0.004971,208 +0.005001,208 +0.004947,208 +0.004945,208 +0.004969,208 +0.005006,208 +0.004923,208 +0.004919,208 +0.004984,208 +0.004922,208 +0.004919,208 +0.004948,208 +0.005040,208 +0.005027,208 +0.004921,208 +0.004980,208 +0.004925,208 +0.004918,208 +0.005315,208 +0.005033,208 +0.004925,208 +0.004946,208 +0.004956,208 +0.004919,208 +0.004922,208 +0.004980,208 +0.004922,208 +0.005214,210 +0.005162,210 +0.005099,210 +0.005105,210 +0.005186,210 +0.005162,210 +0.005276,210 +0.005236,210 +0.005176,210 +0.005095,210 +0.005244,210 +0.005476,210 +0.005100,210 +0.005196,210 +0.005180,210 +0.005102,210 +0.005095,210 +0.005179,210 +0.005141,210 +0.005170,210 +0.005148,210 +0.005152,210 +0.005102,210 +0.005099,210 +0.005162,210 +0.005164,210 +0.005137,210 +0.005179,210 +0.005100,210 +0.005098,210 +0.005395,210 +0.005238,210 +0.005161,210 +0.005177,210 +0.005125,210 +0.005229,210 +0.005146,210 +0.005129,210 +0.005099,210 +0.005098,210 +0.005160,210 +0.005096,210 +0.005100,210 +0.005157,210 +0.005181,210 +0.005095,210 +0.005163,210 +0.005100,210 +0.005097,210 +0.005175,210 +0.005415,210 +0.005359,210 +0.005202,210 +0.005224,210 +0.005125,210 +0.005177,210 +0.005315,210 +0.005119,210 +0.005096,210 +0.005162,210 +0.005102,210 +0.005095,210 +0.005125,210 +0.005134,210 +0.005175,210 +0.005097,210 +0.005196,210 +0.005098,210 +0.005097,210 +0.005353,210 +0.005342,210 +0.005119,210 +0.005177,210 +0.005102,210 +0.005146,210 +0.005160,210 +0.005101,210 +0.005107,210 +0.005126,210 +0.005128,210 +0.005100,210 +0.005098,210 +0.005166,210 +0.005230,210 +0.005131,210 +0.005158,210 +0.005099,210 +0.005097,210 +0.005200,210 +0.005421,210 +0.005113,210 +0.005148,210 +0.005129,210 +0.005099,210 +0.005100,210 +0.005155,210 +0.005172,210 +0.005104,210 +0.005160,210 +0.005100,210 +0.005338,212 +0.005299,212 +0.005295,212 +0.005210,212 +0.005338,212 +0.005213,212 +0.005208,212 +0.005261,212 +0.005489,212 +0.005501,212 +0.005398,212 +0.005309,212 +0.005229,212 +0.005254,212 +0.005347,212 +0.005216,212 +0.005218,212 +0.005280,212 +0.005212,212 +0.005255,212 +0.005271,212 +0.005316,212 +0.005210,212 +0.005294,212 +0.005210,212 +0.005210,212 +0.005296,212 +0.005481,212 +0.005432,212 +0.005325,212 +0.005243,212 +0.005295,212 +0.005212,212 +0.005296,212 +0.005209,212 +0.005213,212 +0.005270,212 +0.005212,212 +0.005210,212 +0.005275,212 +0.005297,212 +0.005215,212 +0.005271,212 +0.005213,212 +0.005209,212 +0.005241,212 +0.005435,212 +0.005482,212 +0.005306,212 +0.005241,212 +0.005210,212 +0.005216,212 +0.005293,212 +0.005209,212 +0.005214,212 +0.005270,212 +0.005212,212 +0.005214,212 +0.005270,212 +0.005257,212 +0.005216,212 +0.005388,212 +0.005245,212 +0.005221,212 +0.005250,212 +0.005424,212 +0.005478,212 +0.005264,212 +0.005265,212 +0.005257,212 +0.005250,212 +0.005272,212 +0.005208,212 +0.005213,212 +0.005271,212 +0.005214,212 +0.005270,212 +0.005321,212 +0.005319,212 +0.005250,212 +0.005280,212 +0.005211,212 +0.005216,212 +0.005238,212 +0.005358,212 +0.005522,212 +0.005261,212 +0.005268,212 +0.005213,212 +0.005252,212 +0.005271,212 +0.005213,212 +0.005221,212 +0.005293,212 +0.005210,212 +0.005214,212 +0.005291,212 +0.005254,212 +0.005251,212 +0.005415,212 +0.005511,214 +0.005377,214 +0.005431,214 +0.005497,214 +0.005634,214 +0.005582,214 +0.005380,214 +0.005369,214 +0.005434,214 +0.005454,214 +0.005416,214 +0.005450,214 +0.005415,214 +0.005371,214 +0.005402,214 +0.005410,214 +0.005471,214 +0.005407,214 +0.005401,214 +0.005369,214 +0.005376,214 +0.005434,214 +0.005608,214 +0.005504,214 +0.005458,214 +0.005459,214 +0.005456,214 +0.005460,214 +0.005475,214 +0.005412,214 +0.005440,214 +0.005371,214 +0.005371,214 +0.005431,214 +0.005435,214 +0.005369,214 +0.005473,214 +0.005393,214 +0.005372,214 +0.005430,214 +0.005473,214 +0.005797,214 +0.005643,214 +0.005410,214 +0.005420,214 +0.005513,214 +0.005400,214 +0.005405,214 +0.005477,214 +0.005369,214 +0.005369,214 +0.005436,214 +0.005374,214 +0.005494,214 +0.005534,214 +0.005442,214 +0.005402,214 +0.005645,214 +0.005529,214 +0.005776,214 +0.005501,214 +0.005429,214 +0.005412,214 +0.005443,214 +0.005445,214 +0.005392,214 +0.005399,214 +0.005408,214 +0.005370,214 +0.005399,214 +0.005412,214 +0.005515,214 +0.005474,214 +0.005406,214 +0.005369,214 +0.005374,214 +0.005431,214 +0.005568,214 +0.005725,214 +0.005470,214 +0.005413,214 +0.005427,214 +0.005471,214 +0.005412,214 +0.005399,214 +0.005446,214 +0.005378,214 +0.005411,214 +0.005457,214 +0.005401,214 +0.005452,214 +0.005452,214 +0.005376,214 +0.005369,214 +0.005473,214 +0.005397,214 +0.005785,214 +0.005732,214 +0.005391,214 +0.005625,214 +0.006151,216 +0.005788,216 +0.006171,216 +0.006296,216 +0.006094,216 +0.006567,216 +0.008187,216 +0.006341,216 +0.006257,216 +0.006222,216 +0.006249,216 +0.006635,216 +0.007265,216 +0.006322,216 +0.006406,216 +0.006345,216 +0.006161,216 +0.006079,216 +0.007007,216 +0.006613,216 +0.006141,216 +0.006247,216 +0.006074,216 +0.007351,216 +0.005990,216 +0.006103,216 +0.007244,216 +0.007006,216 +0.008089,216 +0.006477,216 +0.007051,216 +0.007433,216 +0.007063,216 +0.006501,216 +0.006232,216 +0.006415,216 +0.006916,216 +0.007117,216 +0.007403,216 +0.006350,216 +0.007760,216 +0.006184,216 +0.005968,216 +0.007188,216 +0.006074,216 +0.007077,216 +0.005790,216 +0.006115,216 +0.008246,216 +0.007490,216 +0.007457,216 +0.005645,216 +0.006889,216 +0.006281,216 +0.006042,216 +0.006582,216 +0.006166,216 +0.006802,216 +0.006558,216 +0.006041,216 +0.006568,216 +0.006179,216 +0.006179,216 +0.006787,216 +0.007046,216 +0.008044,216 +0.006161,216 +0.006380,216 +0.006220,216 +0.005972,216 +0.007988,216 +0.006318,216 +0.007219,216 +0.007577,216 +0.007820,216 +0.006773,216 +0.006201,216 +0.006750,216 +0.006430,216 +0.006074,216 +0.006347,216 +0.005806,216 +0.006344,216 +0.006285,216 +0.006094,216 +0.006686,216 +0.005977,216 +0.005836,216 +0.006300,216 +0.007047,216 +0.006276,216 +0.006208,216 +0.005685,216 +0.006487,216 +0.006019,216 +0.005781,216 +0.007648,216 +0.005997,216 +0.007177,216 +0.006766,216 +0.006210,218 +0.007256,218 +0.005898,218 +0.006264,218 +0.009736,218 +0.005934,218 +0.006642,218 +0.005758,218 +0.006376,218 +0.006204,218 +0.005705,218 +0.006358,218 +0.005827,218 +0.005790,218 +0.006412,218 +0.005781,218 +0.006085,218 +0.006048,218 +0.005698,218 +0.006003,218 +0.006033,218 +0.005719,218 +0.005811,218 +0.005729,218 +0.005720,218 +0.005784,218 +0.005733,218 +0.005670,218 +0.005820,218 +0.005687,218 +0.005769,218 +0.005788,218 +0.005663,218 +0.005668,218 +0.005807,218 +0.005668,218 +0.005742,218 +0.006086,218 +0.005812,218 +0.005815,218 +0.005765,218 +0.005701,218 +0.005771,218 +0.005702,218 +0.005669,218 +0.005714,218 +0.005747,218 +0.005663,218 +0.005804,218 +0.005729,218 +0.005664,218 +0.005859,218 +0.005708,218 +0.005690,218 +0.005978,218 +0.006048,218 +0.005702,218 +0.005724,218 +0.005713,218 +0.005784,218 +0.006000,218 +0.005901,218 +0.005784,218 +0.006205,218 +0.005706,218 +0.005842,218 +0.005730,218 +0.005667,218 +0.005869,218 +0.005789,218 +0.005679,218 +0.005993,218 +0.006072,218 +0.005793,218 +0.005816,218 +0.006076,218 +0.005689,218 +0.005872,218 +0.006006,218 +0.005817,218 +0.005734,218 +0.005828,218 +0.005804,218 +0.005768,218 +0.005742,218 +0.005726,218 +0.005716,218 +0.005692,218 +0.005666,218 +0.006121,218 +0.005762,218 +0.005669,218 +0.005726,218 +0.005705,218 +0.005708,218 +0.005705,218 +0.005665,218 +0.005668,218 +0.005781,218 +0.005691,218 +0.006165,220 +0.005818,220 +0.005839,220 +0.005879,220 +0.005838,220 +0.005813,220 +0.006274,220 +0.005964,220 +0.005843,220 +0.005911,220 +0.005859,220 +0.005875,220 +0.005888,220 +0.005807,220 +0.005817,220 +0.005982,220 +0.005889,220 +0.006092,220 +0.005858,220 +0.005828,220 +0.005898,220 +0.005814,220 +0.005815,220 +0.006025,220 +0.005997,220 +0.005881,220 +0.005914,220 +0.005823,220 +0.005818,220 +0.005881,220 +0.005815,220 +0.005816,220 +0.005895,220 +0.005814,220 +0.005960,220 +0.005851,220 +0.005817,220 +0.005895,220 +0.005836,220 +0.005965,220 +0.006084,220 +0.005981,220 +0.005879,220 +0.005960,220 +0.005945,220 +0.005819,220 +0.005910,220 +0.005815,220 +0.006064,220 +0.005851,220 +0.005837,220 +0.006360,220 +0.005851,220 +0.005961,220 +0.006022,220 +0.005815,220 +0.005884,220 +0.006152,220 +0.006042,220 +0.005933,220 +0.005918,220 +0.005819,220 +0.005834,220 +0.005856,220 +0.005812,220 +0.005853,220 +0.005851,220 +0.005817,220 +0.005985,220 +0.005819,220 +0.005813,220 +0.005885,220 +0.005815,220 +0.005817,220 +0.006085,220 +0.006280,220 +0.005859,220 +0.005925,220 +0.005816,220 +0.005828,220 +0.005856,220 +0.005813,220 +0.005826,220 +0.005852,220 +0.005817,220 +0.005921,220 +0.005824,220 +0.005839,220 +0.005873,220 +0.005821,220 +0.005856,220 +0.006110,220 +0.006147,220 +0.005837,220 +0.005921,220 +0.005834,220 +0.005838,220 +0.005865,220 +0.005813,220 +0.005827,220 +0.006199,222 +0.006047,222 +0.006113,222 +0.005991,222 +0.005998,222 +0.006044,222 +0.005992,222 +0.005992,222 +0.006332,222 +0.006240,222 +0.006050,222 +0.006050,222 +0.006035,222 +0.006060,222 +0.006039,222 +0.006035,222 +0.006087,222 +0.005991,222 +0.006057,222 +0.006044,222 +0.005994,222 +0.006002,222 +0.006035,222 +0.006061,222 +0.006153,222 +0.006487,222 +0.006039,222 +0.006085,222 +0.006020,222 +0.006097,222 +0.006120,222 +0.006067,222 +0.006035,222 +0.006069,222 +0.005991,222 +0.006187,222 +0.005994,222 +0.005995,222 +0.006040,222 +0.005995,222 +0.005993,222 +0.006250,222 +0.006297,222 +0.006125,222 +0.006036,222 +0.006073,222 +0.006072,222 +0.005997,222 +0.005991,222 +0.006043,222 +0.005993,222 +0.006054,222 +0.006039,222 +0.005995,222 +0.006002,222 +0.006036,222 +0.005991,222 +0.006044,222 +0.006302,222 +0.006241,222 +0.006090,222 +0.006012,222 +0.006042,222 +0.006062,222 +0.005993,222 +0.006003,222 +0.006059,222 +0.005991,222 +0.006520,222 +0.005994,222 +0.006099,222 +0.006051,222 +0.006056,222 +0.006015,222 +0.006136,222 +0.006390,222 +0.006093,222 +0.006011,222 +0.006018,222 +0.006080,222 +0.006039,222 +0.006015,222 +0.006043,222 +0.005996,222 +0.006123,222 +0.006034,222 +0.005996,222 +0.006089,222 +0.005991,222 +0.005995,222 +0.006039,222 +0.006226,222 +0.006255,222 +0.006094,222 +0.006012,222 +0.006004,222 +0.006030,222 +0.005996,222 +0.006041,222 +0.005995,222 +0.006544,224 +0.006214,224 +0.006159,224 +0.006201,224 +0.006209,224 +0.006269,224 +0.006196,224 +0.006284,224 +0.006576,224 +0.006228,224 +0.006259,224 +0.006181,224 +0.006204,224 +0.006160,224 +0.006206,224 +0.006182,224 +0.006215,224 +0.006209,224 +0.006159,224 +0.006160,224 +0.006209,224 +0.006219,224 +0.006184,224 +0.006200,224 +0.006813,224 +0.006231,224 +0.006438,224 +0.006208,224 +0.006394,224 +0.006190,224 +0.006206,224 +0.006190,224 +0.006220,224 +0.006259,224 +0.006164,224 +0.006159,224 +0.006209,224 +0.006155,224 +0.006170,224 +0.006197,224 +0.006456,224 +0.006614,224 +0.006198,224 +0.006202,224 +0.006229,224 +0.006181,224 +0.006217,224 +0.006183,224 +0.006216,224 +0.006229,224 +0.006287,224 +0.006185,224 +0.006209,224 +0.006493,224 +0.006197,224 +0.006170,224 +0.006525,224 +0.006446,224 +0.006177,224 +0.006217,224 +0.006214,224 +0.006162,224 +0.006206,224 +0.006165,224 +0.006261,224 +0.006237,224 +0.006162,224 +0.006220,224 +0.006216,224 +0.006155,224 +0.006170,224 +0.006198,224 +0.006455,224 +0.006439,224 +0.006188,224 +0.006182,224 +0.006247,224 +0.006160,224 +0.006166,224 +0.006200,224 +0.006179,224 +0.006269,224 +0.006159,224 +0.006158,224 +0.006210,224 +0.006179,224 +0.006262,224 +0.006195,224 +0.006400,224 +0.006445,224 +0.006241,224 +0.006192,224 +0.006266,224 +0.006244,224 +0.006168,224 +0.006225,224 +0.006156,224 +0.006312,224 +0.006159,224 +0.006161,224 +0.006549,226 +0.006319,226 +0.006334,226 +0.006360,226 +0.006602,226 +0.006872,226 +0.006318,226 +0.006371,226 +0.006378,226 +0.006344,226 +0.006397,226 +0.006370,226 +0.006379,226 +0.006397,226 +0.006319,226 +0.006400,226 +0.006359,226 +0.006319,226 +0.006389,226 +0.006320,226 +0.006569,226 +0.006630,226 +0.006340,226 +0.006420,226 +0.006365,226 +0.006322,226 +0.006366,226 +0.006323,226 +0.006906,226 +0.006387,226 +0.006637,226 +0.006371,226 +0.006345,226 +0.006332,226 +0.006357,226 +0.006324,226 +0.006945,226 +0.006377,226 +0.006349,226 +0.006376,226 +0.006559,226 +0.006351,226 +0.006320,226 +0.006376,226 +0.006434,226 +0.006335,226 +0.006330,226 +0.006359,226 +0.006322,226 +0.006369,226 +0.006323,226 +0.006419,226 +0.006789,226 +0.006450,226 +0.006401,226 +0.006351,226 +0.006318,226 +0.006375,226 +0.006323,226 +0.006379,226 +0.006371,226 +0.006317,226 +0.006370,226 +0.006323,226 +0.006361,226 +0.006372,226 +0.006318,226 +0.006571,226 +0.006741,226 +0.006366,226 +0.006431,226 +0.006340,226 +0.006364,226 +0.006402,226 +0.006318,226 +0.006461,226 +0.006320,226 +0.006319,226 +0.006371,226 +0.006320,226 +0.006330,226 +0.006366,226 +0.006392,226 +0.006721,226 +0.006562,226 +0.006344,226 +0.006403,226 +0.006320,226 +0.006370,226 +0.006322,226 +0.006342,226 +0.006452,226 +0.006320,226 +0.006332,226 +0.006359,226 +0.006359,226 +0.006367,226 +0.006324,226 +0.006318,226 +0.006649,226 +0.007106,228 +0.006580,228 +0.006485,228 +0.006484,228 +0.006534,228 +0.006483,228 +0.006636,228 +0.006547,228 +0.006541,228 +0.006579,228 +0.006543,228 +0.006496,228 +0.006526,228 +0.006482,228 +0.006787,228 +0.006789,228 +0.006515,228 +0.006587,228 +0.006502,228 +0.006544,228 +0.006491,228 +0.006656,228 +0.006526,228 +0.006488,228 +0.006556,228 +0.006483,228 +0.006483,228 +0.006536,228 +0.006488,228 +0.006665,228 +0.006857,228 +0.006545,228 +0.006555,228 +0.006531,228 +0.006544,228 +0.006483,228 +0.006527,228 +0.006643,228 +0.006481,228 +0.006529,228 +0.006496,228 +0.006636,228 +0.006586,228 +0.006502,228 +0.006495,228 +0.007019,228 +0.006552,228 +0.006599,228 +0.006554,228 +0.006495,228 +0.006529,228 +0.006485,228 +0.006610,228 +0.006486,228 +0.006483,228 +0.006535,228 +0.006484,228 +0.006531,228 +0.006488,228 +0.006585,228 +0.006792,228 +0.006762,228 +0.006576,228 +0.006481,228 +0.006486,228 +0.006535,228 +0.006481,228 +0.006615,228 +0.006484,228 +0.006499,228 +0.006538,228 +0.006495,228 +0.006501,228 +0.006530,228 +0.006481,228 +0.006686,228 +0.006840,228 +0.006519,228 +0.006612,228 +0.006503,228 +0.006565,228 +0.006530,228 +0.006877,228 +0.006551,228 +0.006489,228 +0.006533,228 +0.006620,228 +0.006507,228 +0.006530,228 +0.006486,228 +0.006534,228 +0.006951,228 +0.006530,228 +0.006585,228 +0.006482,228 +0.006564,228 +0.006488,228 +0.006579,228 +0.006645,228 +0.006481,228 +0.006906,230 +0.006664,230 +0.006654,230 +0.006708,230 +0.006732,230 +0.007061,230 +0.007030,230 +0.007347,230 +0.007360,230 +0.007459,230 +0.007354,230 +0.007573,230 +0.007686,230 +0.007985,230 +0.007109,230 +0.007216,230 +0.007353,230 +0.008097,230 +0.009631,230 +0.007771,230 +0.009133,230 +0.007343,230 +0.008509,230 +0.007966,230 +0.007210,230 +0.008854,230 +0.008231,230 +0.007544,230 +0.008028,230 +0.007625,230 +0.007927,230 +0.008986,230 +0.008489,230 +0.009163,230 +0.011969,230 +0.011834,230 +0.008479,230 +0.007426,230 +0.007346,230 +0.008399,230 +0.008797,230 +0.007532,230 +0.007414,230 +0.006996,230 +0.007756,230 +0.006975,230 +0.007077,230 +0.007386,230 +0.007467,230 +0.007776,230 +0.007609,230 +0.007568,230 +0.007361,230 +0.007952,230 +0.007071,230 +0.008761,230 +0.007205,230 +0.008618,230 +0.007297,230 +0.007950,230 +0.007024,230 +0.008121,230 +0.007081,230 +0.007821,230 +0.007059,230 +0.006998,230 +0.007775,230 +0.007005,230 +0.008090,230 +0.007026,230 +0.007915,230 +0.007400,230 +0.007968,230 +0.007076,230 +0.006918,230 +0.007954,230 +0.006799,230 +0.008414,230 +0.006951,230 +0.008150,230 +0.007105,230 +0.008124,230 +0.006950,230 +0.006990,230 +0.008277,230 +0.006900,230 +0.007570,230 +0.007040,230 +0.007710,230 +0.007099,230 +0.006936,230 +0.007400,230 +0.006832,230 +0.006759,230 +0.006929,230 +0.006787,230 +0.006863,230 +0.006921,230 +0.007279,230 +0.008303,230 +0.007663,232 +0.008658,232 +0.006971,232 +0.007331,232 +0.007265,232 +0.007715,232 +0.008007,232 +0.007884,232 +0.007827,232 +0.008317,232 +0.007851,232 +0.008945,232 +0.008069,232 +0.008027,232 +0.008219,232 +0.010375,232 +0.009157,232 +0.007510,232 +0.008506,232 +0.007659,232 +0.007794,232 +0.008050,232 +0.007744,232 +0.008130,232 +0.008074,232 +0.009084,232 +0.007656,232 +0.007832,232 +0.007544,232 +0.007925,232 +0.007558,232 +0.008052,232 +0.007596,232 +0.007857,232 +0.007764,232 +0.008353,232 +0.007662,232 +0.007970,232 +0.007488,232 +0.007856,232 +0.007517,232 +0.007807,232 +0.007618,232 +0.007640,232 +0.007767,232 +0.007442,232 +0.008778,232 +0.008466,232 +0.009341,232 +0.007269,232 +0.007878,232 +0.007136,232 +0.007985,232 +0.007516,232 +0.008181,232 +0.007519,232 +0.007891,232 +0.007465,232 +0.008925,232 +0.007472,232 +0.007641,232 +0.008206,232 +0.007339,232 +0.008165,232 +0.007036,232 +0.008310,232 +0.007241,232 +0.007607,232 +0.007111,232 +0.007456,232 +0.007123,232 +0.007018,232 +0.007113,232 +0.006921,232 +0.007212,232 +0.007295,232 +0.006936,232 +0.006939,232 +0.007001,232 +0.006917,232 +0.006998,232 +0.006892,232 +0.006963,232 +0.006877,232 +0.006901,232 +0.006871,232 +0.006835,232 +0.006900,232 +0.006971,232 +0.007192,232 +0.006898,232 +0.006905,232 +0.006866,232 +0.006936,232 +0.007135,232 +0.006877,232 +0.006894,232 +0.006838,232 +0.006897,232 +0.006900,232 +0.007237,234 +0.007229,234 +0.007226,234 +0.007570,234 +0.007112,234 +0.007043,234 +0.007135,234 +0.007101,234 +0.007239,234 +0.007078,234 +0.007109,234 +0.007069,234 +0.007125,234 +0.007131,234 +0.007047,234 +0.007088,234 +0.007028,234 +0.007460,234 +0.007143,234 +0.007030,234 +0.007112,234 +0.007027,234 +0.007208,234 +0.007024,234 +0.007050,234 +0.007111,234 +0.007026,234 +0.007117,234 +0.007020,234 +0.007249,234 +0.007210,234 +0.007394,234 +0.007278,234 +0.007165,234 +0.007221,234 +0.007963,234 +0.009203,234 +0.008000,234 +0.007206,234 +0.007376,234 +0.007202,234 +0.007165,234 +0.007109,234 +0.007178,234 +0.007352,234 +0.007577,234 +0.007112,234 +0.007174,234 +0.007093,234 +0.007694,234 +0.007190,234 +0.007050,234 +0.007141,234 +0.007084,234 +0.007142,234 +0.007064,234 +0.007727,234 +0.007223,234 +0.007325,234 +0.007454,234 +0.007197,234 +0.007191,234 +0.007104,234 +0.007222,234 +0.007170,234 +0.007125,234 +0.007088,234 +0.007044,234 +0.007124,234 +0.007086,234 +0.007128,234 +0.007025,234 +0.007278,234 +0.007411,234 +0.007062,234 +0.007125,234 +0.007030,234 +0.007122,234 +0.007101,234 +0.007102,234 +0.007911,234 +0.007742,234 +0.008329,234 +0.010380,234 +0.008026,234 +0.008500,234 +0.008256,234 +0.008138,234 +0.008184,234 +0.008851,234 +0.008628,234 +0.009289,234 +0.008313,234 +0.007911,234 +0.007843,234 +0.008029,234 +0.007784,234 +0.007991,234 +0.008368,234 +0.008967,234 +0.008834,236 +0.008266,236 +0.008228,236 +0.008136,236 +0.007759,236 +0.007821,236 +0.007737,236 +0.008175,236 +0.007579,236 +0.007759,236 +0.007853,236 +0.008212,236 +0.007521,236 +0.007879,236 +0.007829,236 +0.008431,236 +0.007612,236 +0.007583,236 +0.007709,236 +0.007579,236 +0.007473,236 +0.007549,236 +0.007520,236 +0.007821,236 +0.007461,236 +0.007356,236 +0.007291,236 +0.007332,236 +0.007349,236 +0.007236,236 +0.007218,236 +0.007238,236 +0.007243,236 +0.007231,236 +0.007609,236 +0.007350,236 +0.007396,236 +0.007753,236 +0.007261,236 +0.007650,236 +0.007419,236 +0.007379,236 +0.007389,236 +0.007298,236 +0.007704,236 +0.007321,236 +0.007505,236 +0.007313,236 +0.007608,236 +0.007351,236 +0.007657,236 +0.007536,236 +0.007321,236 +0.008096,236 +0.007385,236 +0.007427,236 +0.007215,236 +0.007849,236 +0.007234,236 +0.007367,236 +0.007348,236 +0.007299,236 +0.007194,236 +0.007443,236 +0.007902,236 +0.007590,236 +0.007554,236 +0.007431,236 +0.007748,236 +0.007674,236 +0.007683,236 +0.007500,236 +0.007520,236 +0.007672,236 +0.007701,236 +0.007689,236 +0.007781,236 +0.007872,236 +0.007630,236 +0.007828,236 +0.007667,236 +0.008095,236 +0.008005,236 +0.007886,236 +0.011243,236 +0.008679,236 +0.007297,236 +0.007512,236 +0.007207,236 +0.007647,236 +0.007748,236 +0.007369,236 +0.007605,236 +0.007382,236 +0.008036,236 +0.007378,236 +0.007765,236 +0.007261,236 +0.008214,236 +0.007343,236 +0.008624,238 +0.007429,238 +0.008038,238 +0.007670,238 +0.007491,238 +0.007370,238 +0.007469,238 +0.007570,238 +0.007449,238 +0.007586,238 +0.007496,238 +0.007464,238 +0.007847,238 +0.007520,238 +0.007370,238 +0.007655,238 +0.007782,238 +0.007511,238 +0.007451,238 +0.007444,238 +0.007455,238 +0.007374,238 +0.007449,238 +0.007378,238 +0.007449,238 +0.007370,238 +0.007508,238 +0.007370,238 +0.007410,238 +0.007805,238 +0.007561,238 +0.007425,238 +0.007431,238 +0.007451,238 +0.007467,238 +0.007449,238 +0.007414,238 +0.007432,238 +0.007401,238 +0.007409,238 +0.007371,238 +0.007368,238 +0.007521,238 +0.007874,238 +0.007490,238 +0.007442,238 +0.007476,238 +0.007785,238 +0.007454,238 +0.007751,238 +0.007405,238 +0.007598,238 +0.007370,238 +0.007409,238 +0.007369,238 +0.007427,238 +0.008021,238 +0.007421,238 +0.007365,238 +0.007567,238 +0.008946,238 +0.008478,238 +0.010571,238 +0.009349,238 +0.011127,238 +0.009461,238 +0.008439,238 +0.008230,238 +0.008482,238 +0.009090,238 +0.008313,238 +0.008916,238 +0.009600,238 +0.009170,238 +0.008555,238 +0.008817,238 +0.008428,238 +0.008248,238 +0.008378,238 +0.008190,238 +0.008983,238 +0.010074,238 +0.009864,238 +0.009125,238 +0.008735,238 +0.007859,238 +0.007874,238 +0.007669,238 +0.007687,238 +0.007573,238 +0.008210,238 +0.008295,238 +0.008358,238 +0.007898,238 +0.008410,238 +0.007990,238 +0.007957,238 +0.008097,238 +0.008056,238 +0.007972,238 +0.008653,240 +0.008411,240 +0.009443,240 +0.010387,240 +0.009186,240 +0.009200,240 +0.008355,240 +0.008307,240 +0.008691,240 +0.009048,240 +0.009448,240 +0.008462,240 +0.010544,240 +0.009193,240 +0.010076,240 +0.008931,240 +0.008495,240 +0.008309,240 +0.008280,240 +0.008282,240 +0.008086,240 +0.008305,240 +0.008257,240 +0.008537,240 +0.007775,240 +0.008803,240 +0.008156,240 +0.008703,240 +0.007829,240 +0.009971,240 +0.010236,240 +0.011200,240 +0.008115,240 +0.009301,240 +0.007973,240 +0.008482,240 +0.007937,240 +0.008784,240 +0.013206,240 +0.008126,240 +0.008587,240 +0.008093,240 +0.008492,240 +0.007942,240 +0.008638,240 +0.007867,240 +0.008002,240 +0.007985,240 +0.008097,240 +0.008768,240 +0.008668,240 +0.007959,240 +0.008045,240 +0.007915,240 +0.007964,240 +0.009017,240 +0.010591,240 +0.010042,240 +0.011167,240 +0.010816,240 +0.008555,240 +0.010858,240 +0.009223,240 +0.010121,240 +0.008837,240 +0.008203,240 +0.008618,240 +0.008413,240 +0.008477,240 +0.008723,240 +0.008598,240 +0.008431,240 +0.008652,240 +0.008653,240 +0.008365,240 +0.008711,240 +0.008385,240 +0.008926,240 +0.008361,240 +0.008319,240 +0.008091,240 +0.008054,240 +0.008121,240 +0.008434,240 +0.010000,240 +0.011078,240 +0.009180,240 +0.008554,240 +0.008533,240 +0.008538,240 +0.008345,240 +0.008361,240 +0.009772,240 +0.009039,240 +0.008347,240 +0.010383,240 +0.011302,240 +0.009205,240 +0.012062,240 +0.009639,240 +0.010181,242 +0.009269,242 +0.009006,242 +0.009717,242 +0.008504,242 +0.010585,242 +0.008997,242 +0.011064,242 +0.010893,242 +0.009848,242 +0.010563,242 +0.009299,242 +0.010176,242 +0.009776,242 +0.013347,242 +0.009719,242 +0.009009,242 +0.009491,242 +0.009756,242 +0.008917,242 +0.009300,242 +0.008440,242 +0.009133,242 +0.008241,242 +0.008374,242 +0.008364,242 +0.008505,242 +0.008798,242 +0.008924,242 +0.009366,242 +0.009250,242 +0.008463,242 +0.008386,242 +0.009386,242 +0.008959,242 +0.008696,242 +0.008675,242 +0.009764,242 +0.008470,242 +0.008959,242 +0.008126,242 +0.008964,242 +0.007957,242 +0.008368,242 +0.007936,242 +0.008271,242 +0.007890,242 +0.008236,242 +0.008135,242 +0.008964,242 +0.007986,242 +0.008580,242 +0.008036,242 +0.008817,242 +0.007993,242 +0.008811,242 +0.008000,242 +0.008834,242 +0.007999,242 +0.008670,242 +0.008321,242 +0.008742,242 +0.008018,242 +0.008544,242 +0.007957,242 +0.008447,242 +0.008035,242 +0.008698,242 +0.007998,242 +0.009916,242 +0.008943,242 +0.008820,242 +0.008849,242 +0.008685,242 +0.009261,242 +0.010206,242 +0.009261,242 +0.008096,242 +0.008463,242 +0.007979,242 +0.008195,242 +0.007949,242 +0.008773,242 +0.008035,242 +0.009100,242 +0.008042,242 +0.008596,242 +0.008102,242 +0.008360,242 +0.008007,242 +0.008242,242 +0.007991,242 +0.008185,242 +0.007931,242 +0.008300,242 +0.007946,242 +0.009118,242 +0.008265,242 +0.008771,242 +0.008246,242 +0.008730,244 +0.008402,244 +0.009118,244 +0.008417,244 +0.009298,244 +0.008481,244 +0.009313,244 +0.008668,244 +0.008875,244 +0.008600,244 +0.008772,244 +0.009428,244 +0.008755,244 +0.009235,244 +0.009010,244 +0.010082,244 +0.008771,244 +0.010093,244 +0.008906,244 +0.009613,244 +0.009461,244 +0.009210,244 +0.010194,244 +0.009208,244 +0.011270,244 +0.009555,244 +0.009779,244 +0.010409,244 +0.009468,244 +0.009655,244 +0.008584,244 +0.008506,244 +0.008732,244 +0.008071,244 +0.008374,244 +0.008024,244 +0.008208,244 +0.008216,244 +0.008539,244 +0.008170,244 +0.008311,244 +0.009333,244 +0.010456,244 +0.009038,244 +0.009488,244 +0.008646,244 +0.008772,244 +0.008282,244 +0.008996,244 +0.013515,244 +0.011108,244 +0.009122,244 +0.009141,244 +0.010592,244 +0.012973,244 +0.010900,244 +0.008776,244 +0.008599,244 +0.008271,244 +0.008535,244 +0.008380,244 +0.008547,244 +0.008541,244 +0.009122,244 +0.009425,244 +0.009985,244 +0.008869,244 +0.009671,244 +0.008826,244 +0.009167,244 +0.009475,244 +0.010463,244 +0.008845,244 +0.009068,244 +0.009923,244 +0.008686,244 +0.009351,244 +0.008841,244 +0.009443,244 +0.009230,244 +0.009117,244 +0.009444,244 +0.008598,244 +0.011855,244 +0.013583,244 +0.009072,244 +0.009092,244 +0.009185,244 +0.014677,244 +0.011672,244 +0.009429,244 +0.008713,244 +0.008714,244 +0.009273,244 +0.010116,244 +0.009393,244 +0.010491,244 +0.008408,244 +0.009318,244 +0.008508,244 +0.009858,246 +0.009395,246 +0.008938,246 +0.011050,246 +0.008977,246 +0.009164,246 +0.009213,246 +0.009397,246 +0.009276,246 +0.009221,246 +0.009539,246 +0.008873,246 +0.009992,246 +0.009103,246 +0.009691,246 +0.010062,246 +0.009052,246 +0.009601,246 +0.009269,246 +0.009436,246 +0.008698,246 +0.008780,246 +0.008658,246 +0.008635,246 +0.009383,246 +0.008811,246 +0.009192,246 +0.008760,246 +0.009154,246 +0.008547,246 +0.009660,246 +0.010609,246 +0.009080,246 +0.009131,246 +0.009144,246 +0.009341,246 +0.013315,246 +0.009230,246 +0.009309,246 +0.009871,246 +0.008865,246 +0.008572,246 +0.010704,246 +0.008672,246 +0.009686,246 +0.011265,246 +0.009018,246 +0.009531,246 +0.008655,246 +0.010740,246 +0.008935,246 +0.008863,246 +0.008870,246 +0.009304,246 +0.009020,246 +0.008978,246 +0.009087,246 +0.008751,246 +0.009580,246 +0.008960,246 +0.009217,246 +0.008830,246 +0.009173,246 +0.008646,246 +0.009105,246 +0.009314,246 +0.008384,246 +0.009238,246 +0.008834,246 +0.009322,246 +0.008750,246 +0.009035,246 +0.008676,246 +0.009955,246 +0.008648,246 +0.009108,246 +0.008685,246 +0.008831,246 +0.009185,246 +0.009365,246 +0.009642,246 +0.008500,246 +0.009377,246 +0.008589,246 +0.008932,246 +0.008653,246 +0.008815,246 +0.008795,246 +0.009015,246 +0.008711,246 +0.009558,246 +0.009287,246 +0.008655,246 +0.008782,246 +0.008563,246 +0.010632,246 +0.008627,246 +0.009575,246 +0.008599,246 +0.009215,246 +0.009130,248 +0.009244,248 +0.009282,248 +0.008789,248 +0.009308,248 +0.008727,248 +0.009762,248 +0.008719,248 +0.008883,248 +0.008873,248 +0.008978,248 +0.008867,248 +0.010259,248 +0.009395,248 +0.008633,248 +0.009338,248 +0.008758,248 +0.009328,248 +0.009498,248 +0.008909,248 +0.009063,248 +0.009817,248 +0.009740,248 +0.009207,248 +0.009438,248 +0.012479,248 +0.010154,248 +0.009249,248 +0.009299,248 +0.010087,248 +0.009723,248 +0.009695,248 +0.010362,248 +0.010546,248 +0.010629,248 +0.010528,248 +0.015329,248 +0.012274,248 +0.011152,248 +0.009396,248 +0.009280,248 +0.009034,248 +0.012744,248 +0.015582,248 +0.015658,248 +0.015078,248 +0.010024,248 +0.009239,248 +0.009943,248 +0.009773,248 +0.014537,248 +0.017530,248 +0.015920,248 +0.011957,248 +0.009105,248 +0.009083,248 +0.009202,248 +0.009940,248 +0.010099,248 +0.009476,248 +0.009598,248 +0.010707,248 +0.009195,248 +0.009424,248 +0.008605,248 +0.009276,248 +0.008892,248 +0.009204,248 +0.008877,248 +0.009213,248 +0.011625,248 +0.009214,248 +0.009382,248 +0.009043,248 +0.009214,248 +0.008909,248 +0.010248,248 +0.009246,248 +0.009236,248 +0.009438,248 +0.008784,248 +0.011077,248 +0.010114,248 +0.009304,248 +0.009221,248 +0.008766,248 +0.009307,248 +0.008864,248 +0.009873,248 +0.008852,248 +0.009311,248 +0.008867,248 +0.009239,248 +0.009388,248 +0.009156,248 +0.009197,248 +0.008510,248 +0.009130,248 +0.008669,248 +0.009244,248 +0.009025,250 +0.009350,250 +0.008872,250 +0.009456,250 +0.009317,250 +0.008936,250 +0.009422,250 +0.008982,250 +0.009744,250 +0.008988,250 +0.009696,250 +0.009359,250 +0.009167,250 +0.012349,250 +0.009517,250 +0.009812,250 +0.009119,250 +0.009061,250 +0.009260,250 +0.009456,250 +0.009952,250 +0.008818,250 +0.009474,250 +0.008798,250 +0.009091,250 +0.009133,250 +0.009540,250 +0.009433,250 +0.009249,250 +0.010214,250 +0.008713,250 +0.010148,250 +0.008962,250 +0.009849,250 +0.009456,250 +0.009602,250 +0.009677,250 +0.008792,250 +0.009677,250 +0.008841,250 +0.009685,250 +0.009485,250 +0.008924,250 +0.009592,250 +0.008825,250 +0.009603,250 +0.009132,250 +0.009784,250 +0.009045,250 +0.009499,250 +0.009518,250 +0.009109,250 +0.009879,250 +0.008919,250 +0.009706,250 +0.008915,250 +0.009780,250 +0.009812,250 +0.009038,250 +0.009642,250 +0.009030,250 +0.009746,250 +0.008882,250 +0.009732,250 +0.009126,250 +0.009544,250 +0.009785,250 +0.009096,250 +0.009853,250 +0.008885,250 +0.009660,250 +0.009002,250 +0.009631,250 +0.009679,250 +0.008955,250 +0.009704,250 +0.008887,250 +0.013027,250 +0.010948,250 +0.008987,250 +0.009630,250 +0.008881,250 +0.009727,250 +0.009298,250 +0.009833,250 +0.009567,250 +0.009455,250 +0.010065,250 +0.009454,250 +0.010091,250 +0.009412,250 +0.009077,250 +0.009636,250 +0.009303,250 +0.009526,250 +0.008877,250 +0.009605,250 +0.009191,250 +0.010011,250 +0.009709,250 +0.010231,252 +0.012990,252 +0.009854,252 +0.010859,252 +0.014675,252 +0.015726,252 +0.012810,252 +0.014456,252 +0.009736,252 +0.010464,252 +0.010532,252 +0.012710,252 +0.010361,252 +0.009999,252 +0.009827,252 +0.013812,252 +0.010463,252 +0.010301,252 +0.009966,252 +0.009810,252 +0.009776,252 +0.009550,252 +0.009727,252 +0.009798,252 +0.009014,252 +0.011330,252 +0.010802,252 +0.009600,252 +0.010121,252 +0.011557,252 +0.009966,252 +0.010390,252 +0.009430,252 +0.009848,252 +0.010062,252 +0.012363,252 +0.012036,252 +0.009869,252 +0.009844,252 +0.010969,252 +0.011726,252 +0.011246,252 +0.011567,252 +0.009282,252 +0.010693,252 +0.010142,252 +0.009920,252 +0.012229,252 +0.010075,252 +0.009732,252 +0.010298,252 +0.009082,252 +0.010190,252 +0.009417,252 +0.009715,252 +0.015873,252 +0.016372,252 +0.016583,252 +0.015830,252 +0.016000,252 +0.011421,252 +0.010158,252 +0.010739,252 +0.009456,252 +0.012964,252 +0.012191,252 +0.012941,252 +0.011451,252 +0.011859,252 +0.013485,252 +0.011426,252 +0.011688,252 +0.016568,252 +0.016182,252 +0.016828,252 +0.016195,252 +0.016573,252 +0.016100,252 +0.010000,252 +0.009507,252 +0.010536,252 +0.010492,252 +0.009154,252 +0.013518,252 +0.010619,252 +0.012216,252 +0.010740,252 +0.009764,252 +0.009365,252 +0.009776,252 +0.010528,252 +0.009679,252 +0.009761,252 +0.009547,252 +0.010305,252 +0.012599,252 +0.011673,252 +0.009893,252 +0.010159,252 +0.009324,252 +0.010555,254 +0.009937,254 +0.010597,254 +0.010326,254 +0.009800,254 +0.010106,254 +0.009499,254 +0.010620,254 +0.010066,254 +0.009924,254 +0.011003,254 +0.009747,254 +0.009786,254 +0.009392,254 +0.010399,254 +0.011080,254 +0.009258,254 +0.010474,254 +0.009610,254 +0.010054,254 +0.011508,254 +0.009480,254 +0.009833,254 +0.009961,254 +0.009436,254 +0.012692,254 +0.009641,254 +0.010464,254 +0.011081,254 +0.009649,254 +0.010144,254 +0.010091,254 +0.009405,254 +0.009991,254 +0.010172,254 +0.010449,254 +0.010108,254 +0.010207,254 +0.010032,254 +0.009369,254 +0.010225,254 +0.009678,254 +0.016870,254 +0.016472,254 +0.010452,254 +0.011093,254 +0.010473,254 +0.009599,254 +0.014770,254 +0.016677,254 +0.017813,254 +0.017380,254 +0.016409,254 +0.017041,254 +0.016572,254 +0.016370,254 +0.016605,254 +0.016964,254 +0.017444,254 +0.017205,254 +0.016628,254 +0.013205,254 +0.010322,254 +0.009270,254 +0.009769,254 +0.009492,254 +0.009941,254 +0.010107,254 +0.010101,254 +0.010084,254 +0.010374,254 +0.012121,254 +0.012037,254 +0.009778,254 +0.010571,254 +0.009853,254 +0.009496,254 +0.009743,254 +0.009096,254 +0.009745,254 +0.009021,254 +0.010155,254 +0.010309,254 +0.009889,254 +0.011350,254 +0.010828,254 +0.010240,254 +0.009932,254 +0.009917,254 +0.010543,254 +0.010089,254 +0.013366,254 +0.016995,254 +0.017477,254 +0.016690,254 +0.016894,254 +0.018866,254 +0.014139,254 +0.010406,254 +0.009661,254 +0.009779,256 +0.010680,256 +0.010421,256 +0.010054,256 +0.010939,256 +0.009451,256 +0.013862,256 +0.010243,256 +0.009519,256 +0.010374,256 +0.009982,256 +0.010124,256 +0.011400,256 +0.010499,256 +0.009763,256 +0.010973,256 +0.009887,256 +0.010968,256 +0.011389,256 +0.013785,256 +0.009945,256 +0.010088,256 +0.010515,256 +0.010346,256 +0.010209,256 +0.009981,256 +0.010826,256 +0.009649,256 +0.011957,256 +0.010635,256 +0.009541,256 +0.010679,256 +0.010541,256 +0.009604,256 +0.010616,256 +0.009487,256 +0.010491,256 +0.010836,256 +0.013541,256 +0.017591,256 +0.014169,256 +0.012726,256 +0.013956,256 +0.011220,256 +0.010190,256 +0.011644,256 +0.010571,256 +0.010247,256 +0.010495,256 +0.010289,256 +0.010695,256 +0.010168,256 +0.009805,256 +0.011205,256 +0.010342,256 +0.010553,256 +0.010728,256 +0.009985,256 +0.010386,256 +0.010392,256 +0.010682,256 +0.011283,256 +0.011353,256 +0.010878,256 +0.015319,256 +0.015399,256 +0.010297,256 +0.010587,256 +0.017157,256 +0.017762,256 +0.017091,256 +0.017641,256 +0.013289,256 +0.011448,256 +0.011872,256 +0.011224,256 +0.010420,256 +0.011012,256 +0.011475,256 +0.015462,256 +0.012049,256 +0.011826,256 +0.011497,256 +0.011092,256 +0.011921,256 +0.011552,256 +0.011640,256 +0.010822,256 +0.011117,256 +0.010751,256 +0.010827,256 +0.010403,256 +0.009689,256 +0.011013,256 +0.012598,256 +0.009945,256 +0.010082,256 +0.014993,256 +0.009717,256 +0.009882,256 +0.010107,258 +0.009770,258 +0.009796,258 +0.009472,258 +0.009749,258 +0.009416,258 +0.010430,258 +0.009918,258 +0.009482,258 +0.009579,258 +0.009374,258 +0.009683,258 +0.009468,258 +0.009405,258 +0.009450,258 +0.009404,258 +0.009560,258 +0.009866,258 +0.009767,258 +0.009467,258 +0.009433,258 +0.009574,258 +0.009596,258 +0.009561,258 +0.009546,258 +0.009753,258 +0.009619,258 +0.009744,258 +0.010196,258 +0.010095,258 +0.010383,258 +0.010210,258 +0.010262,258 +0.010389,258 +0.010256,258 +0.009983,258 +0.009761,258 +0.009582,258 +0.010053,258 +0.009413,258 +0.009532,258 +0.009446,258 +0.009426,258 +0.009423,258 +0.009346,258 +0.009422,258 +0.009347,258 +0.009394,258 +0.009674,258 +0.009704,258 +0.009396,258 +0.009351,258 +0.009391,258 +0.009346,258 +0.009475,258 +0.009598,258 +0.010080,258 +0.010452,258 +0.010871,258 +0.010423,258 +0.010513,258 +0.010116,258 +0.010210,258 +0.009958,258 +0.009968,258 +0.010047,258 +0.010058,258 +0.010037,258 +0.010153,258 +0.011575,258 +0.012614,258 +0.010947,258 +0.009669,258 +0.009714,258 +0.009688,258 +0.010026,258 +0.009516,258 +0.009797,258 +0.009955,258 +0.010176,258 +0.009922,258 +0.010324,258 +0.010263,258 +0.009942,258 +0.010075,258 +0.010010,258 +0.010313,258 +0.010602,258 +0.010260,258 +0.010234,258 +0.010394,258 +0.010284,258 +0.013337,258 +0.010676,258 +0.010417,258 +0.010114,258 +0.010350,258 +0.012891,258 +0.013548,258 +0.010766,258 +0.011532,260 +0.010968,260 +0.010756,260 +0.010761,260 +0.010170,260 +0.010459,260 +0.013812,260 +0.011390,260 +0.009759,260 +0.010005,260 +0.009683,260 +0.009912,260 +0.009788,260 +0.009833,260 +0.012760,260 +0.019698,260 +0.018735,260 +0.018137,260 +0.018051,260 +0.018126,260 +0.016454,260 +0.010452,260 +0.009917,260 +0.009712,260 +0.010646,260 +0.010046,260 +0.010721,260 +0.010848,260 +0.010831,260 +0.011017,260 +0.010318,260 +0.010154,260 +0.009890,260 +0.009848,260 +0.010152,260 +0.009949,260 +0.009642,260 +0.009848,260 +0.010376,260 +0.010758,260 +0.011059,260 +0.010853,260 +0.012435,260 +0.010276,260 +0.010300,260 +0.012000,260 +0.013294,260 +0.012111,260 +0.011801,260 +0.010755,260 +0.011994,260 +0.011919,260 +0.010789,260 +0.012048,260 +0.011977,260 +0.012300,260 +0.012509,260 +0.012142,260 +0.010451,260 +0.010635,260 +0.010448,260 +0.010917,260 +0.010891,260 +0.010783,260 +0.010708,260 +0.010749,260 +0.010585,260 +0.010855,260 +0.011117,260 +0.010729,260 +0.010794,260 +0.010692,260 +0.010542,260 +0.010503,260 +0.010549,260 +0.010460,260 +0.010503,260 +0.010593,260 +0.010516,260 +0.010728,260 +0.010585,260 +0.010632,260 +0.010700,260 +0.013832,260 +0.011336,260 +0.010597,260 +0.010557,260 +0.010810,260 +0.011117,260 +0.011006,260 +0.011358,260 +0.010412,260 +0.010276,260 +0.011562,260 +0.012203,260 +0.015970,260 +0.009911,260 +0.011294,260 +0.011579,260 +0.012378,260 +0.011885,262 +0.011688,262 +0.012098,262 +0.017249,262 +0.012328,262 +0.017765,262 +0.022648,262 +0.018523,262 +0.011370,262 +0.012250,262 +0.011140,262 +0.010837,262 +0.010156,262 +0.010665,262 +0.010573,262 +0.012050,262 +0.011725,262 +0.011434,262 +0.011467,262 +0.011308,262 +0.012422,262 +0.010747,262 +0.011455,262 +0.012212,262 +0.010871,262 +0.011883,262 +0.011414,262 +0.011060,262 +0.011012,262 +0.011690,262 +0.013613,262 +0.010994,262 +0.011202,262 +0.013261,262 +0.012546,262 +0.013007,262 +0.011315,262 +0.011240,262 +0.010581,262 +0.012079,262 +0.011860,262 +0.011337,262 +0.011544,262 +0.013034,262 +0.011081,262 +0.010664,262 +0.010906,262 +0.012053,262 +0.010731,262 +0.014121,262 +0.011711,262 +0.013684,262 +0.012036,262 +0.012383,262 +0.011715,262 +0.010776,262 +0.011244,262 +0.011810,262 +0.010253,262 +0.010740,262 +0.010743,262 +0.011492,262 +0.011037,262 +0.011238,262 +0.010825,262 +0.011052,262 +0.011090,262 +0.010108,262 +0.011135,262 +0.011111,262 +0.011944,262 +0.011301,262 +0.011062,262 +0.010266,262 +0.012399,262 +0.011177,262 +0.009959,262 +0.010554,262 +0.010458,262 +0.010451,262 +0.010572,262 +0.010284,262 +0.011125,262 +0.011215,262 +0.010490,262 +0.011294,262 +0.010950,262 +0.010181,262 +0.010554,262 +0.010798,262 +0.010588,262 +0.010938,262 +0.011195,262 +0.010304,262 +0.016095,262 +0.013226,262 +0.010853,262 +0.010454,262 +0.010427,262 +0.010101,262 +0.010794,264 +0.011579,264 +0.011372,264 +0.013671,264 +0.012693,264 +0.012298,264 +0.011387,264 +0.010932,264 +0.012185,264 +0.012093,264 +0.010862,264 +0.013355,264 +0.013758,264 +0.012014,264 +0.010772,264 +0.013846,264 +0.010442,264 +0.010146,264 +0.010189,264 +0.010453,264 +0.010165,264 +0.010202,264 +0.010154,264 +0.010184,264 +0.016234,264 +0.010232,264 +0.010653,264 +0.010957,264 +0.010317,264 +0.010160,264 +0.010207,264 +0.010154,264 +0.010074,264 +0.015739,264 +0.010218,264 +0.010167,264 +0.010198,264 +0.010039,264 +0.011495,264 +0.010973,264 +0.010209,264 +0.010213,264 +0.010392,264 +0.010206,264 +0.010219,264 +0.010150,264 +0.010204,264 +0.010077,264 +0.010029,264 +0.010081,264 +0.010087,264 +0.010062,264 +0.010284,264 +0.010231,264 +0.010156,264 +0.010122,264 +0.010094,264 +0.010179,264 +0.009974,264 +0.010217,264 +0.010331,264 +0.010127,264 +0.010511,264 +0.010212,264 +0.010175,264 +0.010099,264 +0.010354,264 +0.010094,264 +0.010020,264 +0.010053,264 +0.010128,264 +0.010016,264 +0.010424,264 +0.010017,264 +0.010142,264 +0.010322,264 +0.009975,264 +0.011610,264 +0.010608,264 +0.010153,264 +0.010441,264 +0.010130,264 +0.010610,264 +0.010203,264 +0.010101,264 +0.010052,264 +0.010070,264 +0.010221,264 +0.010127,264 +0.010209,264 +0.010113,264 +0.010248,264 +0.010656,264 +0.010227,264 +0.010537,264 +0.010463,264 +0.010206,264 +0.009976,264 +0.010978,264 +0.012841,264 +0.011457,266 +0.011325,266 +0.012660,266 +0.011298,266 +0.010768,266 +0.011054,266 +0.010783,266 +0.010563,266 +0.010470,266 +0.010377,266 +0.010594,266 +0.010372,266 +0.010286,266 +0.010322,266 +0.010281,266 +0.010319,266 +0.010380,266 +0.010277,266 +0.010316,266 +0.010538,266 +0.010632,266 +0.010314,266 +0.010246,266 +0.010298,266 +0.010313,266 +0.010222,266 +0.010392,266 +0.010303,266 +0.010258,266 +0.011083,266 +0.011373,266 +0.011394,266 +0.011423,266 +0.011494,266 +0.011362,266 +0.011225,266 +0.011327,266 +0.011162,266 +0.011533,266 +0.012106,266 +0.011551,266 +0.011496,266 +0.012078,266 +0.010990,266 +0.011217,266 +0.011571,266 +0.011147,266 +0.010625,266 +0.011689,266 +0.012289,266 +0.011349,266 +0.012935,266 +0.012327,266 +0.011804,266 +0.011409,266 +0.011983,266 +0.011368,266 +0.011151,266 +0.011225,266 +0.010653,266 +0.010704,266 +0.011275,266 +0.011230,266 +0.011698,266 +0.013166,266 +0.015349,266 +0.018412,266 +0.017065,266 +0.013121,266 +0.011527,266 +0.010822,266 +0.011982,266 +0.011018,266 +0.010638,266 +0.011363,266 +0.012734,266 +0.010482,266 +0.010728,266 +0.010993,266 +0.011482,266 +0.011666,266 +0.011318,266 +0.011667,266 +0.011474,266 +0.012265,266 +0.012166,266 +0.011736,266 +0.012148,266 +0.012070,266 +0.011598,266 +0.013895,266 +0.012129,266 +0.014653,266 +0.012137,266 +0.014748,266 +0.015828,266 +0.013458,266 +0.014819,266 +0.012846,266 +0.016506,266 +0.012439,268 +0.011157,268 +0.012181,268 +0.012792,268 +0.018302,268 +0.012486,268 +0.011053,268 +0.011819,268 +0.012515,268 +0.014427,268 +0.018408,268 +0.021454,268 +0.012570,268 +0.011423,268 +0.011239,268 +0.011432,268 +0.011399,268 +0.011265,268 +0.012256,268 +0.011688,268 +0.011787,268 +0.011798,268 +0.011951,268 +0.011667,268 +0.011537,268 +0.012007,268 +0.010827,268 +0.011254,268 +0.011534,268 +0.011232,268 +0.010853,268 +0.011462,268 +0.011062,268 +0.010949,268 +0.011541,268 +0.011058,268 +0.011035,268 +0.011518,268 +0.011622,268 +0.010679,268 +0.011386,268 +0.012035,268 +0.014004,268 +0.011193,268 +0.012148,268 +0.011543,268 +0.012519,268 +0.011213,268 +0.014361,268 +0.013364,268 +0.013924,268 +0.012259,268 +0.013628,268 +0.012660,268 +0.012574,268 +0.012317,268 +0.012326,268 +0.011779,268 +0.012059,268 +0.011431,268 +0.011946,268 +0.013517,268 +0.015301,268 +0.014987,268 +0.012013,268 +0.012194,268 +0.011377,268 +0.011686,268 +0.011386,268 +0.012465,268 +0.015420,268 +0.020418,268 +0.015497,268 +0.012483,268 +0.011771,268 +0.013726,268 +0.012621,268 +0.012090,268 +0.010986,268 +0.012368,268 +0.011458,268 +0.011811,268 +0.011334,268 +0.017564,268 +0.014490,268 +0.016207,268 +0.013307,268 +0.012526,268 +0.011412,268 +0.012655,268 +0.011794,268 +0.011151,268 +0.012182,268 +0.012015,268 +0.011588,268 +0.011319,268 +0.011843,268 +0.011518,268 +0.013115,268 +0.011957,268 +0.013827,270 +0.011480,270 +0.011624,270 +0.011960,270 +0.013117,270 +0.013719,270 +0.012665,270 +0.014312,270 +0.018070,270 +0.012840,270 +0.013579,270 +0.013749,270 +0.011697,270 +0.011351,270 +0.012223,270 +0.017580,270 +0.011414,270 +0.011318,270 +0.011599,270 +0.011700,270 +0.012627,270 +0.011451,270 +0.011609,270 +0.011333,270 +0.011392,270 +0.010944,270 +0.010892,270 +0.011558,270 +0.011820,270 +0.011908,270 +0.011294,270 +0.011460,270 +0.013014,270 +0.014360,270 +0.011905,270 +0.011698,270 +0.011326,270 +0.011469,270 +0.011072,270 +0.011061,270 +0.012363,270 +0.011825,270 +0.012339,270 +0.011823,270 +0.014907,270 +0.011328,270 +0.011481,270 +0.011953,270 +0.014402,270 +0.014933,270 +0.011434,270 +0.011730,270 +0.011142,270 +0.011680,270 +0.011440,270 +0.011634,270 +0.011048,270 +0.012270,270 +0.011313,270 +0.011724,270 +0.011649,270 +0.014341,270 +0.017158,270 +0.013464,270 +0.011567,270 +0.011611,270 +0.011913,270 +0.013445,270 +0.011918,270 +0.015169,270 +0.014472,270 +0.011883,270 +0.012696,270 +0.012013,270 +0.011942,270 +0.011916,270 +0.011145,270 +0.013960,270 +0.012014,270 +0.011429,270 +0.011623,270 +0.013834,270 +0.011994,270 +0.011362,270 +0.010976,270 +0.011128,270 +0.010985,270 +0.011327,270 +0.011608,270 +0.013648,270 +0.011881,270 +0.011458,270 +0.011348,270 +0.011615,270 +0.012125,270 +0.011188,270 +0.013129,270 +0.011223,270 +0.011644,270 +0.011201,270 +0.011853,272 +0.012187,272 +0.012305,272 +0.013455,272 +0.012277,272 +0.012017,272 +0.015585,272 +0.012174,272 +0.011242,272 +0.011910,272 +0.011257,272 +0.011207,272 +0.011918,272 +0.011065,272 +0.011464,272 +0.012213,272 +0.011118,272 +0.011124,272 +0.011316,272 +0.011282,272 +0.011464,272 +0.011202,272 +0.011479,272 +0.011615,272 +0.011943,272 +0.011892,272 +0.011300,272 +0.013015,272 +0.011799,272 +0.012453,272 +0.011830,272 +0.015279,272 +0.013138,272 +0.011969,272 +0.012412,272 +0.013635,272 +0.011874,272 +0.012776,272 +0.012803,272 +0.012065,272 +0.012143,272 +0.011485,272 +0.011841,272 +0.012257,272 +0.012161,272 +0.011664,272 +0.012727,272 +0.012390,272 +0.011519,272 +0.012903,272 +0.012968,272 +0.011564,272 +0.011474,272 +0.012503,272 +0.013555,272 +0.011861,272 +0.017892,272 +0.012202,272 +0.011796,272 +0.011900,272 +0.011608,272 +0.011639,272 +0.012110,272 +0.011517,272 +0.011523,272 +0.011379,272 +0.011210,272 +0.011205,272 +0.011195,272 +0.011060,272 +0.011284,272 +0.011000,272 +0.011011,272 +0.011319,272 +0.010966,272 +0.011012,272 +0.011026,272 +0.010966,272 +0.010985,272 +0.011140,272 +0.010966,272 +0.011092,272 +0.011223,272 +0.010995,272 +0.011014,272 +0.010989,272 +0.010997,272 +0.010986,272 +0.011621,272 +0.010976,272 +0.011770,272 +0.011894,272 +0.012049,272 +0.012264,272 +0.012022,272 +0.011747,272 +0.012237,272 +0.012840,272 +0.012496,272 +0.018836,272 +0.012484,274 +0.011528,274 +0.011521,274 +0.011412,274 +0.011490,274 +0.011426,274 +0.011435,274 +0.017460,274 +0.011924,274 +0.011255,274 +0.011323,274 +0.011351,274 +0.011352,274 +0.011390,274 +0.011604,274 +0.014364,274 +0.014529,274 +0.011399,274 +0.011264,274 +0.011219,274 +0.011388,274 +0.011373,274 +0.011226,274 +0.011304,274 +0.017376,274 +0.011361,274 +0.011385,274 +0.011204,274 +0.011264,274 +0.011887,274 +0.011308,274 +0.011339,274 +0.017406,274 +0.013205,274 +0.012187,274 +0.011445,274 +0.011474,274 +0.011793,274 +0.011532,274 +0.011725,274 +0.017909,274 +0.011546,274 +0.012766,274 +0.011466,274 +0.011593,274 +0.011637,274 +0.011641,274 +0.013013,274 +0.018135,274 +0.013629,274 +0.011539,274 +0.012274,274 +0.012491,274 +0.011918,274 +0.012186,274 +0.011985,274 +0.017766,274 +0.011750,274 +0.011517,274 +0.011799,274 +0.011707,274 +0.011869,274 +0.012377,274 +0.012414,274 +0.011907,274 +0.012177,274 +0.011493,274 +0.011761,274 +0.012167,274 +0.012437,274 +0.011576,274 +0.011602,274 +0.012961,274 +0.011365,274 +0.011398,274 +0.011277,274 +0.011328,274 +0.011243,274 +0.011415,274 +0.011370,274 +0.011262,274 +0.013047,274 +0.011328,274 +0.011282,274 +0.011300,274 +0.011189,274 +0.011350,274 +0.011451,274 +0.011256,274 +0.011985,274 +0.012333,274 +0.011306,274 +0.011182,274 +0.011293,274 +0.011283,274 +0.011312,274 +0.011405,274 +0.011258,274 +0.012624,274 +0.011732,274 +0.011626,276 +0.011830,276 +0.011472,276 +0.011444,276 +0.012230,276 +0.011908,276 +0.011449,276 +0.013251,276 +0.011545,276 +0.011906,276 +0.011556,276 +0.011654,276 +0.011653,276 +0.011543,276 +0.011620,276 +0.014764,276 +0.012550,276 +0.011524,276 +0.011551,276 +0.011626,276 +0.011757,276 +0.012318,276 +0.011735,276 +0.012132,276 +0.012866,276 +0.011526,276 +0.011551,276 +0.011645,276 +0.011461,276 +0.011700,276 +0.011646,276 +0.011464,276 +0.013195,276 +0.011691,276 +0.011567,276 +0.011444,276 +0.011548,276 +0.011577,276 +0.011507,276 +0.011613,276 +0.011945,276 +0.013265,276 +0.011492,276 +0.011523,276 +0.011529,276 +0.011488,276 +0.011626,276 +0.011577,276 +0.011430,276 +0.013222,276 +0.011690,276 +0.011585,276 +0.011442,276 +0.011534,276 +0.011572,276 +0.011527,276 +0.011698,276 +0.011460,276 +0.013457,276 +0.011480,276 +0.011521,276 +0.011593,276 +0.011477,276 +0.011625,276 +0.011618,276 +0.011504,276 +0.013268,276 +0.011676,276 +0.011595,276 +0.011449,276 +0.011510,276 +0.011694,276 +0.011560,276 +0.011675,276 +0.011486,276 +0.013421,276 +0.011482,276 +0.011500,276 +0.011549,276 +0.011423,276 +0.011618,276 +0.011554,276 +0.011484,276 +0.013115,276 +0.011791,276 +0.011593,276 +0.011446,276 +0.011472,276 +0.011623,276 +0.011769,276 +0.011809,276 +0.011456,276 +0.013300,276 +0.011511,276 +0.011486,276 +0.011532,276 +0.011444,276 +0.011601,276 +0.011573,276 +0.011466,276 +0.014016,278 +0.012821,278 +0.011816,278 +0.011791,278 +0.011814,278 +0.011808,278 +0.011876,278 +0.011704,278 +0.012423,278 +0.013013,278 +0.011796,278 +0.011863,278 +0.011816,278 +0.012011,278 +0.011901,278 +0.011876,278 +0.011801,278 +0.013614,278 +0.012472,278 +0.011975,278 +0.011879,278 +0.011779,278 +0.011925,278 +0.011799,278 +0.011921,278 +0.013394,278 +0.012003,278 +0.011716,278 +0.011783,278 +0.011709,278 +0.011817,278 +0.011915,278 +0.011706,278 +0.012678,278 +0.012789,278 +0.011801,278 +0.011737,278 +0.011810,278 +0.011883,278 +0.011786,278 +0.011824,278 +0.011753,278 +0.013593,278 +0.011803,278 +0.011789,278 +0.011834,278 +0.011888,278 +0.011825,278 +0.011847,278 +0.011877,278 +0.013396,278 +0.011984,278 +0.011737,278 +0.011921,278 +0.011703,278 +0.011848,278 +0.012008,278 +0.012461,278 +0.013065,278 +0.015012,278 +0.011799,278 +0.011843,278 +0.011725,278 +0.011822,278 +0.011867,278 +0.011685,278 +0.012549,278 +0.017619,278 +0.012041,278 +0.011872,278 +0.011734,278 +0.011846,278 +0.011907,278 +0.011726,278 +0.013490,278 +0.016518,278 +0.012111,278 +0.011972,278 +0.011709,278 +0.011893,278 +0.011847,278 +0.011849,278 +0.013321,278 +0.017125,278 +0.011849,278 +0.012183,278 +0.011838,278 +0.011931,278 +0.011931,278 +0.011880,278 +0.016327,278 +0.013564,278 +0.011901,278 +0.011785,278 +0.012227,278 +0.012016,278 +0.015415,278 +0.012437,278 +0.018601,278 +0.011967,278 +0.012358,280 +0.012528,280 +0.012497,280 +0.012089,280 +0.012103,280 +0.013682,280 +0.016582,280 +0.011985,280 +0.012008,280 +0.012124,280 +0.012076,280 +0.012043,280 +0.012039,280 +0.015314,280 +0.015128,280 +0.011979,280 +0.012000,280 +0.012176,280 +0.012002,280 +0.012048,280 +0.011957,280 +0.017007,280 +0.013269,280 +0.011976,280 +0.012094,280 +0.012067,280 +0.012082,280 +0.012120,280 +0.011987,280 +0.018353,280 +0.012012,280 +0.012067,280 +0.011959,280 +0.012076,280 +0.012149,280 +0.011906,280 +0.012002,280 +0.018332,280 +0.012014,280 +0.012062,280 +0.011926,280 +0.012084,280 +0.012065,280 +0.012078,280 +0.011952,280 +0.018367,280 +0.012072,280 +0.011984,280 +0.012042,280 +0.012131,280 +0.012037,280 +0.012020,280 +0.011885,280 +0.018517,280 +0.012033,280 +0.011994,280 +0.012136,280 +0.012023,280 +0.012108,280 +0.012008,280 +0.012312,280 +0.019490,280 +0.012129,280 +0.012053,280 +0.012023,280 +0.012058,280 +0.011982,280 +0.012002,280 +0.015990,280 +0.014953,280 +0.012328,280 +0.012194,280 +0.012157,280 +0.012081,280 +0.012064,280 +0.011945,280 +0.018334,280 +0.012104,280 +0.011964,280 +0.012029,280 +0.012046,280 +0.012069,280 +0.012067,280 +0.012037,280 +0.018357,280 +0.011993,280 +0.012012,280 +0.012035,280 +0.012176,280 +0.011944,280 +0.011945,280 +0.012023,280 +0.018378,280 +0.012070,280 +0.011964,280 +0.012046,280 +0.012120,280 +0.011925,280 +0.011982,280 +0.012008,280 +0.018777,282 +0.012455,282 +0.012229,282 +0.012423,282 +0.012639,282 +0.012323,282 +0.012197,282 +0.017554,282 +0.013383,282 +0.012235,282 +0.012296,282 +0.012469,282 +0.012245,282 +0.012271,282 +0.012269,282 +0.018619,282 +0.012279,282 +0.012240,282 +0.012295,282 +0.012357,282 +0.012275,282 +0.012292,282 +0.012448,282 +0.018508,282 +0.012233,282 +0.012294,282 +0.012277,282 +0.012265,282 +0.012279,282 +0.012284,282 +0.016910,282 +0.014048,282 +0.012333,282 +0.012298,282 +0.012427,282 +0.012202,282 +0.012237,282 +0.012336,282 +0.019885,282 +0.012457,282 +0.012339,282 +0.012278,282 +0.012321,282 +0.012412,282 +0.012197,282 +0.015114,282 +0.016011,282 +0.012308,282 +0.012356,282 +0.012507,282 +0.012255,282 +0.012235,282 +0.012698,282 +0.019801,282 +0.013296,282 +0.013066,282 +0.012764,282 +0.012640,282 +0.012618,282 +0.012736,282 +0.016990,282 +0.014150,282 +0.012346,282 +0.012283,282 +0.012471,282 +0.012238,282 +0.012256,282 +0.012278,282 +0.018771,282 +0.012374,282 +0.012408,282 +0.012287,282 +0.012941,282 +0.012385,282 +0.012293,282 +0.014701,282 +0.016269,282 +0.012355,282 +0.012353,282 +0.012427,282 +0.012444,282 +0.012636,282 +0.013100,282 +0.020059,282 +0.013754,282 +0.015306,282 +0.013966,282 +0.013118,282 +0.013573,282 +0.013209,282 +0.013211,282 +0.013682,282 +0.013451,282 +0.013830,282 +0.016563,282 +0.014275,282 +0.013191,282 +0.014931,282 +0.013361,282 +0.013060,282 +0.013946,284 +0.013994,284 +0.013735,284 +0.013689,284 +0.015005,284 +0.013579,284 +0.013733,284 +0.013731,284 +0.014022,284 +0.013994,284 +0.013784,284 +0.013775,284 +0.013650,284 +0.013812,284 +0.013409,284 +0.013660,284 +0.013485,284 +0.013261,284 +0.012729,284 +0.013414,284 +0.012799,284 +0.012514,284 +0.012703,284 +0.012598,284 +0.012822,284 +0.012455,284 +0.012624,284 +0.012976,284 +0.012607,284 +0.012557,284 +0.012518,284 +0.012584,284 +0.012541,284 +0.013689,284 +0.013699,284 +0.013100,284 +0.013475,284 +0.012920,284 +0.013383,284 +0.014202,284 +0.014666,284 +0.014012,284 +0.013489,284 +0.013736,284 +0.013426,284 +0.013414,284 +0.013479,284 +0.013610,284 +0.013781,284 +0.013853,284 +0.014394,284 +0.013401,284 +0.013368,284 +0.013851,284 +0.013561,284 +0.013997,284 +0.013764,284 +0.015446,284 +0.013561,284 +0.013659,284 +0.013920,284 +0.013185,284 +0.013524,284 +0.013365,284 +0.015054,284 +0.014309,284 +0.013730,284 +0.013936,284 +0.013663,284 +0.013726,284 +0.013492,284 +0.014247,284 +0.013913,284 +0.013512,284 +0.013265,284 +0.013152,284 +0.012955,284 +0.012899,284 +0.012563,284 +0.012663,284 +0.012529,284 +0.012587,284 +0.012622,284 +0.012544,284 +0.012493,284 +0.012484,284 +0.012470,284 +0.012798,284 +0.012542,284 +0.012523,284 +0.012628,284 +0.012525,284 +0.012440,284 +0.012487,284 +0.012466,284 +0.012660,284 +0.012507,284 +0.012504,284 +0.012633,284 +0.012492,284 +0.012885,286 +0.012740,286 +0.012743,286 +0.012959,286 +0.012753,286 +0.012818,286 +0.012940,286 +0.012742,286 +0.012681,286 +0.012706,286 +0.012818,286 +0.013008,286 +0.012674,286 +0.012738,286 +0.013173,286 +0.012881,286 +0.012797,286 +0.013203,286 +0.013836,286 +0.013470,286 +0.013405,286 +0.013145,286 +0.013560,286 +0.013173,286 +0.012748,286 +0.012981,286 +0.013443,286 +0.013051,286 +0.012730,286 +0.012914,286 +0.012685,286 +0.012631,286 +0.012703,286 +0.012712,286 +0.013049,286 +0.012640,286 +0.012690,286 +0.012926,286 +0.012703,286 +0.012614,286 +0.012685,286 +0.012711,286 +0.013000,286 +0.012728,286 +0.012750,286 +0.012863,286 +0.012725,286 +0.012820,286 +0.012615,286 +0.012768,286 +0.012977,286 +0.012775,286 +0.012779,286 +0.012877,286 +0.012749,286 +0.012727,286 +0.012663,286 +0.012842,286 +0.012997,286 +0.012824,286 +0.012818,286 +0.012674,286 +0.012758,286 +0.012853,286 +0.012863,286 +0.012864,286 +0.012988,286 +0.012823,286 +0.012866,286 +0.012704,286 +0.012687,286 +0.012745,286 +0.012806,286 +0.013069,286 +0.012840,286 +0.012767,286 +0.012968,286 +0.012784,286 +0.012612,286 +0.012711,286 +0.012723,286 +0.013276,286 +0.012728,286 +0.012752,286 +0.012881,286 +0.012702,286 +0.012743,286 +0.012612,286 +0.012753,286 +0.013214,286 +0.012757,286 +0.012776,286 +0.012821,286 +0.012775,286 +0.012750,286 +0.012712,286 +0.012857,286 +0.013142,286 +0.012788,286 +0.012879,286 +0.013201,288 +0.012976,288 +0.013002,288 +0.012988,288 +0.013214,288 +0.013187,288 +0.013072,288 +0.013409,288 +0.013022,288 +0.012897,288 +0.013063,288 +0.012993,288 +0.013472,288 +0.013100,288 +0.012897,288 +0.013178,288 +0.013025,288 +0.013024,288 +0.013205,288 +0.013200,288 +0.013378,288 +0.013053,288 +0.013097,288 +0.014063,288 +0.013531,288 +0.013111,288 +0.012972,288 +0.013326,288 +0.013193,288 +0.013007,288 +0.013126,288 +0.013079,288 +0.013020,288 +0.013074,288 +0.013201,288 +0.013680,288 +0.013155,288 +0.013164,288 +0.013608,288 +0.013087,288 +0.013348,288 +0.013029,288 +0.013178,288 +0.013346,288 +0.013070,288 +0.013138,288 +0.013156,288 +0.013135,288 +0.013054,288 +0.013051,288 +0.013585,288 +0.014019,288 +0.013054,288 +0.013111,288 +0.013063,288 +0.013061,288 +0.013035,288 +0.013075,288 +0.013358,288 +0.013098,288 +0.013045,288 +0.013083,288 +0.012938,288 +0.012982,288 +0.013000,288 +0.013198,288 +0.013349,288 +0.013032,288 +0.013157,288 +0.013035,288 +0.013016,288 +0.012915,288 +0.013047,288 +0.013452,288 +0.013033,288 +0.012995,288 +0.012971,288 +0.013044,288 +0.013024,288 +0.012997,288 +0.013056,288 +0.013446,288 +0.013038,288 +0.013153,288 +0.013140,288 +0.012877,288 +0.012974,288 +0.013012,288 +0.013184,288 +0.013406,288 +0.012904,288 +0.013190,288 +0.013045,288 +0.013018,288 +0.012908,288 +0.012955,288 +0.013446,288 +0.013057,288 +0.013114,288 +0.012941,288 +0.013481,290 +0.013334,290 +0.013320,290 +0.013323,290 +0.013687,290 +0.013270,290 +0.013456,290 +0.013301,290 +0.013286,290 +0.013170,290 +0.013281,290 +0.013732,290 +0.013544,290 +0.013473,290 +0.014077,290 +0.013593,290 +0.013379,290 +0.013369,290 +0.013526,290 +0.013858,290 +0.013238,290 +0.013551,290 +0.013339,290 +0.013329,290 +0.013351,290 +0.013180,290 +0.013894,290 +0.013400,290 +0.013488,290 +0.013460,290 +0.013149,290 +0.013432,290 +0.014937,290 +0.013927,290 +0.013685,290 +0.013380,290 +0.013409,290 +0.013391,290 +0.013376,290 +0.013751,290 +0.013292,290 +0.013819,290 +0.013339,290 +0.013473,290 +0.013314,290 +0.013291,290 +0.013173,290 +0.013790,290 +0.013684,290 +0.013551,290 +0.013448,290 +0.013325,290 +0.013243,290 +0.013352,290 +0.013321,290 +0.013306,290 +0.013955,290 +0.013242,290 +0.013509,290 +0.013295,290 +0.013302,290 +0.013305,290 +0.013144,290 +0.013688,290 +0.013629,290 +0.013456,290 +0.013424,290 +0.013146,290 +0.013349,290 +0.013292,290 +0.013466,290 +0.013820,290 +0.013269,290 +0.013694,290 +0.013465,290 +0.013425,290 +0.013293,290 +0.013317,290 +0.013721,290 +0.013529,290 +0.013479,290 +0.013367,290 +0.013148,290 +0.013268,290 +0.013541,290 +0.013400,290 +0.013686,290 +0.013212,290 +0.013478,290 +0.013329,290 +0.013299,290 +0.013286,290 +0.013170,290 +0.013637,290 +0.013704,290 +0.013489,290 +0.013450,290 +0.013332,290 +0.013141,290 +0.013324,290 +0.014173,292 +0.013964,292 +0.013706,292 +0.013608,292 +0.013528,292 +0.013538,292 +0.013556,292 +0.013539,292 +0.014056,292 +0.013518,292 +0.013774,292 +0.013569,292 +0.013601,292 +0.013535,292 +0.013463,292 +0.013851,292 +0.013804,292 +0.013714,292 +0.013679,292 +0.013573,292 +0.013422,292 +0.013563,292 +0.013673,292 +0.013962,292 +0.013572,292 +0.013598,292 +0.013531,292 +0.013587,292 +0.013540,292 +0.013535,292 +0.013977,292 +0.013441,292 +0.013751,292 +0.013548,292 +0.013621,292 +0.013551,292 +0.013401,292 +0.013880,292 +0.013904,292 +0.013742,292 +0.013665,292 +0.013593,292 +0.013414,292 +0.013546,292 +0.013682,292 +0.014659,292 +0.014596,292 +0.014658,292 +0.013872,292 +0.014497,292 +0.014095,292 +0.014564,292 +0.014594,292 +0.015230,292 +0.014673,292 +0.014149,292 +0.014151,292 +0.014572,292 +0.014287,292 +0.014656,292 +0.014418,292 +0.014275,292 +0.013836,292 +0.013588,292 +0.013943,292 +0.014227,292 +0.014491,292 +0.014003,292 +0.013849,292 +0.013481,292 +0.014047,292 +0.013839,292 +0.013693,292 +0.014099,292 +0.015242,292 +0.014441,292 +0.014667,292 +0.015529,292 +0.016512,292 +0.016460,292 +0.015420,292 +0.016760,292 +0.014867,292 +0.014562,292 +0.014597,292 +0.014859,292 +0.014662,292 +0.014812,292 +0.015283,292 +0.015188,292 +0.014670,292 +0.014191,292 +0.014242,292 +0.014280,292 +0.014270,292 +0.014309,292 +0.013729,292 +0.013819,292 +0.014126,292 +0.013795,292 +0.014795,294 +0.014027,294 +0.013913,294 +0.013824,294 +0.015135,294 +0.016026,294 +0.015641,294 +0.015718,294 +0.015860,294 +0.015509,294 +0.015591,294 +0.014985,294 +0.015005,294 +0.016038,294 +0.016855,294 +0.016140,294 +0.015329,294 +0.015619,294 +0.015057,294 +0.014654,294 +0.015085,294 +0.015368,294 +0.015454,294 +0.015917,294 +0.015275,294 +0.015016,294 +0.015117,294 +0.015881,294 +0.017002,294 +0.016115,294 +0.016410,294 +0.015987,294 +0.014855,294 +0.015974,294 +0.014814,294 +0.015446,294 +0.015848,294 +0.015758,294 +0.014875,294 +0.014658,294 +0.014860,294 +0.015297,294 +0.016768,294 +0.015180,294 +0.014770,294 +0.015168,294 +0.015061,294 +0.015951,294 +0.017409,294 +0.017436,294 +0.017312,294 +0.015754,294 +0.016206,294 +0.016827,294 +0.016502,294 +0.016044,294 +0.015842,294 +0.018897,294 +0.019509,294 +0.018065,294 +0.019213,294 +0.015811,294 +0.015241,294 +0.014296,294 +0.015615,294 +0.017037,294 +0.017171,294 +0.015242,294 +0.015142,294 +0.015733,294 +0.016041,294 +0.015273,294 +0.015189,294 +0.015082,294 +0.015422,294 +0.016060,294 +0.016130,294 +0.016446,294 +0.017918,294 +0.016325,294 +0.016780,294 +0.015085,294 +0.014598,294 +0.015548,294 +0.018661,294 +0.018239,294 +0.015832,294 +0.015122,294 +0.018574,294 +0.017859,294 +0.016707,294 +0.014995,294 +0.015211,294 +0.014929,294 +0.015035,294 +0.014774,294 +0.014655,294 +0.015027,294 +0.015038,294 +0.015198,294 +0.014681,296 +0.014288,296 +0.015142,296 +0.015681,296 +0.015230,296 +0.015222,296 +0.015204,296 +0.015051,296 +0.015135,296 +0.015832,296 +0.016320,296 +0.015317,296 +0.015174,296 +0.014283,296 +0.016614,296 +0.016972,296 +0.014858,296 +0.014459,296 +0.014602,296 +0.014277,296 +0.015539,296 +0.015612,296 +0.016112,296 +0.015737,296 +0.015104,296 +0.014696,296 +0.015561,296 +0.015715,296 +0.015489,296 +0.015198,296 +0.015239,296 +0.015045,296 +0.015285,296 +0.015164,296 +0.014412,296 +0.014795,296 +0.014220,296 +0.014082,296 +0.013997,296 +0.014116,296 +0.014535,296 +0.014673,296 +0.015166,296 +0.015054,296 +0.014801,296 +0.016423,296 +0.019121,296 +0.016707,296 +0.017321,296 +0.015645,296 +0.015656,296 +0.014643,296 +0.014348,296 +0.014928,296 +0.015420,296 +0.015165,296 +0.015154,296 +0.014837,296 +0.015405,296 +0.015458,296 +0.023828,296 +0.024815,296 +0.016970,296 +0.017982,296 +0.015732,296 +0.016239,296 +0.015299,296 +0.014810,296 +0.014520,296 +0.014830,296 +0.015901,296 +0.015074,296 +0.015274,296 +0.016604,296 +0.015285,296 +0.014779,296 +0.015344,296 +0.015064,296 +0.014944,296 +0.016107,296 +0.015013,296 +0.014853,296 +0.014481,296 +0.015386,296 +0.014975,296 +0.015724,296 +0.014866,296 +0.015657,296 +0.014461,296 +0.014291,296 +0.014056,296 +0.014826,296 +0.014428,296 +0.015374,296 +0.014666,296 +0.015026,296 +0.014345,296 +0.014619,296 +0.015102,296 +0.014787,296 +0.015666,298 +0.014854,298 +0.015187,298 +0.015992,298 +0.016170,298 +0.015382,298 +0.016049,298 +0.015758,298 +0.016153,298 +0.015938,298 +0.016403,298 +0.016760,298 +0.016727,298 +0.017336,298 +0.015007,298 +0.015812,298 +0.016600,298 +0.017088,298 +0.021346,298 +0.016721,298 +0.018078,298 +0.016002,298 +0.015634,298 +0.016869,298 +0.015550,298 +0.015720,298 +0.015011,298 +0.015664,298 +0.015481,298 +0.015510,298 +0.018328,298 +0.020421,298 +0.015091,298 +0.014703,298 +0.014954,298 +0.014855,298 +0.016252,298 +0.017516,298 +0.018290,298 +0.017556,298 +0.016717,298 +0.019328,298 +0.018002,298 +0.017207,298 +0.017260,298 +0.016040,298 +0.015982,298 +0.015778,298 +0.016230,298 +0.017668,298 +0.016499,298 +0.017272,298 +0.015800,298 +0.015261,298 +0.015103,298 +0.015956,298 +0.016004,298 +0.016785,298 +0.016497,298 +0.015962,298 +0.015830,298 +0.017555,298 +0.016003,298 +0.016073,298 +0.016233,298 +0.016343,298 +0.015528,298 +0.017081,298 +0.015666,298 +0.015200,298 +0.015575,298 +0.014971,298 +0.015492,298 +0.016195,298 +0.016266,298 +0.015414,298 +0.015653,298 +0.015342,298 +0.015384,298 +0.015154,298 +0.017177,298 +0.016570,298 +0.017330,298 +0.016724,298 +0.018397,298 +0.021554,298 +0.025023,298 +0.016175,298 +0.017001,298 +0.016864,298 +0.016334,298 +0.016042,298 +0.015923,298 +0.017484,298 +0.016283,298 +0.014971,298 +0.015364,298 +0.016135,298 +0.016538,298 +0.015348,298 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 @@ -0,0 +1,14900 @@ +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000010,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000010,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000010,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000011,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000010,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000001,6 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000012,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000012,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000014,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000015,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000013,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000012,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000013,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000005,10 +0.000005,10 +0.000005,10 +0.000014,10 +0.000004,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000003,10 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000015,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000014,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000005,12 +0.000008,14 +0.000008,14 +0.000018,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000017,14 +0.000012,14 +0.000013,14 +0.000009,14 +0.000011,14 +0.000014,14 +0.000010,14 +0.000008,14 +0.000008,14 +0.000014,14 +0.000013,14 +0.000010,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000015,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000012,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000009,14 +0.000010,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000017,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000008,14 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000022,16 +0.000011,16 +0.000011,16 +0.000020,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000021,16 +0.000011,16 +0.000011,16 +0.000020,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000016,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000027,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000015,18 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000026,20 +0.000028,20 +0.000031,20 +0.000031,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000031,20 +0.000052,20 +0.000021,20 +0.000021,20 +0.000030,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000028,20 +0.000025,20 +0.000032,20 +0.000036,20 +0.000031,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000021,20 +0.000031,20 +0.000021,20 +0.000030,20 +0.000031,20 +0.000048,22 +0.000038,22 +0.000028,22 +0.000037,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000035,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000035,22 +0.000037,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000029,22 +0.000042,22 +0.000042,22 +0.000036,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000027,22 +0.000036,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000052,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000047,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000048,24 +0.000070,24 +0.000045,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000046,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000035,24 +0.000043,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000034,24 +0.000044,26 +0.000043,26 +0.000052,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000044,26 +0.000057,26 +0.000045,26 +0.000044,26 +0.000044,26 +0.000052,26 +0.000043,26 +0.000043,26 +0.000084,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000061,26 +0.000062,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000053,26 +0.000053,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000067,26 +0.000073,26 +0.000044,26 +0.000072,26 +0.000074,26 +0.000053,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000061,26 +0.000057,26 +0.000053,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000043,26 +0.000065,26 +0.000091,26 +0.000047,26 +0.000044,26 +0.000044,26 +0.000048,26 +0.000044,26 +0.000044,26 +0.000049,26 +0.000048,26 +0.000053,26 +0.000043,26 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000084,28 +0.000084,28 +0.000098,28 +0.000063,28 +0.000054,28 +0.000054,28 +0.000064,28 +0.000084,28 +0.000064,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000081,28 +0.000063,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000064,28 +0.000074,28 +0.000085,28 +0.000095,28 +0.000063,28 +0.000054,28 +0.000064,28 +0.000073,28 +0.000085,28 +0.000064,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000053,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000062,28 +0.000092,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000054,28 +0.000067,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000066,30 +0.000066,30 +0.000065,30 +0.000098,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000066,30 +0.000065,30 +0.000066,30 +0.000066,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000104,30 +0.000077,30 +0.000127,30 +0.000075,30 +0.000065,30 +0.000066,30 +0.000095,30 +0.000086,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000076,30 +0.000087,30 +0.000140,30 +0.000075,30 +0.000066,30 +0.000085,30 +0.000106,30 +0.000076,30 +0.000066,30 +0.000065,30 +0.000066,30 +0.000101,30 +0.000065,30 +0.000065,30 +0.000066,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000066,30 +0.000065,30 +0.000066,30 +0.000066,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000074,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000067,30 +0.000098,30 +0.000108,30 +0.000075,30 +0.000065,30 +0.000085,30 +0.000106,30 +0.000076,30 +0.000066,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000065,30 +0.000066,30 +0.000066,30 +0.000065,30 +0.000065,30 +0.000066,30 +0.000065,30 +0.000065,30 +0.000076,30 +0.000081,30 +0.000103,30 +0.000096,30 +0.000069,30 +0.000091,30 +0.000080,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000090,32 +0.000089,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000110,32 +0.000138,32 +0.000099,32 +0.000079,32 +0.000099,32 +0.000120,32 +0.000089,32 +0.000079,32 +0.000119,32 +0.000121,32 +0.000081,32 +0.000085,32 +0.000086,32 +0.000093,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000114,32 +0.000110,32 +0.000155,32 +0.000089,32 +0.000079,32 +0.000090,32 +0.000120,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000109,32 +0.000112,32 +0.000132,32 +0.000079,32 +0.000079,32 +0.000121,32 +0.000089,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000105,32 +0.000089,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000103,32 +0.000089,32 +0.000096,34 +0.000094,34 +0.000094,34 +0.000125,34 +0.000139,34 +0.000115,34 +0.000114,34 +0.000136,34 +0.000104,34 +0.000094,34 +0.000104,34 +0.000119,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000126,34 +0.000168,34 +0.000104,34 +0.000115,34 +0.000136,34 +0.000105,34 +0.000095,34 +0.000139,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000106,34 +0.000159,34 +0.000134,34 +0.000095,34 +0.000125,34 +0.000116,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000131,34 +0.000095,34 +0.000095,34 +0.000125,34 +0.000165,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000095,34 +0.000125,34 +0.000143,34 +0.000149,34 +0.000095,34 +0.000115,34 +0.000136,34 +0.000104,34 +0.000095,34 +0.000105,34 +0.000133,34 +0.000152,34 +0.000095,34 +0.000105,34 +0.000135,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000094,34 +0.000095,34 +0.000112,36 +0.000123,36 +0.000175,36 +0.000120,36 +0.000146,36 +0.000125,36 +0.000115,36 +0.000115,36 +0.000131,36 +0.000124,36 +0.000124,36 +0.000124,36 +0.000124,36 +0.000140,36 +0.000129,36 +0.000124,36 +0.000159,36 +0.000135,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000114,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000115,36 +0.000150,36 +0.000122,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000133,36 +0.000151,36 +0.000163,36 +0.000153,36 +0.000152,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000145,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000152,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000123,36 +0.000148,36 +0.000127,36 +0.000123,36 +0.000163,36 +0.000121,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000132,36 +0.000111,36 +0.000112,36 +0.000112,36 +0.000112,36 +0.000133,38 +0.000131,38 +0.000131,38 +0.000173,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000141,38 +0.000189,38 +0.000178,38 +0.000140,38 +0.000182,38 +0.000141,38 +0.000131,38 +0.000138,38 +0.000181,38 +0.000175,38 +0.000181,38 +0.000162,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000146,38 +0.000174,38 +0.000230,38 +0.000194,38 +0.000195,38 +0.000222,38 +0.000131,38 +0.000134,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000132,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000132,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000132,38 +0.000131,38 +0.000175,38 +0.000197,38 +0.000144,38 +0.000176,38 +0.000175,38 +0.000134,38 +0.000135,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000182,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000152,38 +0.000131,38 +0.000141,38 +0.000189,38 +0.000179,38 +0.000172,38 +0.000171,38 +0.000131,38 +0.000169,38 +0.000131,38 +0.000141,38 +0.000140,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000131,38 +0.000177,38 +0.000163,38 +0.000161,38 +0.000182,38 +0.000131,38 +0.000131,38 +0.000149,38 +0.000209,40 +0.000172,40 +0.000213,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000192,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000153,40 +0.000217,40 +0.000162,40 +0.000203,40 +0.000209,40 +0.000152,40 +0.000152,40 +0.000152,40 +0.000174,40 +0.000152,40 +0.000152,40 +0.000163,40 +0.000179,40 +0.000210,40 +0.000172,40 +0.000225,40 +0.000160,40 +0.000200,40 +0.000195,40 +0.000184,40 +0.000167,40 +0.000154,40 +0.000174,40 +0.000181,40 +0.000166,40 +0.000162,40 +0.000153,40 +0.000165,40 +0.000165,40 +0.000153,40 +0.000164,40 +0.000157,40 +0.000157,40 +0.000157,40 +0.000157,40 +0.000157,40 +0.000163,40 +0.000153,40 +0.000153,40 +0.000153,40 +0.000164,40 +0.000195,40 +0.000158,40 +0.000153,40 +0.000185,40 +0.000165,40 +0.000153,40 +0.000153,40 +0.000163,40 +0.000165,40 +0.000152,40 +0.000153,40 +0.000153,40 +0.000153,40 +0.000169,40 +0.000272,40 +0.000177,40 +0.000195,40 +0.000196,40 +0.000162,40 +0.000210,40 +0.000177,40 +0.000166,40 +0.000176,40 +0.000258,40 +0.000241,40 +0.000176,40 +0.000166,40 +0.000156,40 +0.000156,40 +0.000156,40 +0.000156,40 +0.000156,40 +0.000156,40 +0.000156,40 +0.000156,40 +0.000184,40 +0.000184,40 +0.000198,40 +0.000158,40 +0.000158,40 +0.000158,40 +0.000158,40 +0.000184,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000238,42 +0.000302,42 +0.000215,42 +0.000192,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000206,42 +0.000199,42 +0.000188,42 +0.000203,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000206,42 +0.000309,42 +0.000182,42 +0.000222,42 +0.000182,42 +0.000219,42 +0.000280,42 +0.000212,42 +0.000225,42 +0.000187,42 +0.000187,42 +0.000187,42 +0.000191,42 +0.000187,42 +0.000187,42 +0.000305,42 +0.000270,42 +0.000180,42 +0.000204,42 +0.000298,42 +0.000289,42 +0.000190,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000201,42 +0.000314,42 +0.000307,42 +0.000314,42 +0.000338,42 +0.000333,42 +0.000327,42 +0.000317,42 +0.000329,42 +0.000328,42 +0.000363,42 +0.000331,42 +0.000323,42 +0.000311,42 +0.000269,42 +0.000233,42 +0.000212,42 +0.000217,42 +0.000272,42 +0.000322,42 +0.000341,42 +0.000225,42 +0.000195,42 +0.000182,42 +0.000182,42 +0.000207,42 +0.000273,42 +0.000187,42 +0.000187,42 +0.000187,42 +0.000187,42 +0.000193,42 +0.000182,42 +0.000182,42 +0.000182,42 +0.000215,42 +0.000220,42 +0.000202,42 +0.000191,42 +0.000208,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000180,42 +0.000209,44 +0.000207,44 +0.000207,44 +0.000285,44 +0.000239,44 +0.000239,44 +0.000247,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000253,44 +0.000235,44 +0.000246,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000221,44 +0.000212,44 +0.000212,44 +0.000263,44 +0.000250,44 +0.000227,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000226,44 +0.000207,44 +0.000207,44 +0.000257,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000246,44 +0.000249,44 +0.000217,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000244,44 +0.000239,44 +0.000207,44 +0.000207,44 +0.000231,44 +0.000240,44 +0.000247,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000273,44 +0.000237,44 +0.000216,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000224,44 +0.000245,44 +0.000262,44 +0.000227,44 +0.000207,44 +0.000207,44 +0.000212,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000226,44 +0.000271,44 +0.000272,44 +0.000212,44 +0.000217,44 +0.000212,44 +0.000212,44 +0.000212,44 +0.000212,44 +0.000212,44 +0.000245,44 +0.000227,44 +0.000207,44 +0.000207,44 +0.000210,44 +0.000249,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000207,44 +0.000237,46 +0.000261,46 +0.000249,46 +0.000236,46 +0.000236,46 +0.000255,46 +0.000240,46 +0.000281,46 +0.000263,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000235,46 +0.000236,46 +0.000274,46 +0.000241,46 +0.000274,46 +0.000230,46 +0.000229,46 +0.000230,46 +0.000259,46 +0.000357,46 +0.000296,46 +0.000256,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000255,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000255,46 +0.000248,46 +0.000269,46 +0.000253,46 +0.000229,46 +0.000324,46 +0.000286,46 +0.000245,46 +0.000240,46 +0.000282,46 +0.000265,46 +0.000230,46 +0.000230,46 +0.000230,46 +0.000230,46 +0.000283,46 +0.000302,46 +0.000239,46 +0.000230,46 +0.000230,46 +0.000230,46 +0.000229,46 +0.000230,46 +0.000229,46 +0.000230,46 +0.000229,46 +0.000230,46 +0.000229,46 +0.000230,46 +0.000230,46 +0.000230,46 +0.000229,46 +0.000230,46 +0.000280,46 +0.000230,46 +0.000230,46 +0.000229,46 +0.000229,46 +0.000322,46 +0.000386,46 +0.000279,46 +0.000242,46 +0.000242,46 +0.000242,46 +0.000267,46 +0.000236,46 +0.000236,46 +0.000235,46 +0.000236,46 +0.000260,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000235,46 +0.000236,46 +0.000236,46 +0.000236,46 +0.000242,46 +0.000265,46 +0.000372,46 +0.000339,48 +0.000281,48 +0.000438,48 +0.000286,48 +0.000281,48 +0.000260,48 +0.000276,48 +0.000333,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000293,48 +0.000363,48 +0.000319,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000299,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000371,48 +0.000476,48 +0.000272,48 +0.000263,48 +0.000304,48 +0.000263,48 +0.000309,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000300,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000352,48 +0.000466,48 +0.000337,48 +0.000263,48 +0.000263,48 +0.000263,48 +0.000378,48 +0.000471,48 +0.000335,48 +0.000260,48 +0.000260,48 +0.000260,48 +0.000358,48 +0.000337,48 +0.000293,48 +0.000260,48 +0.000274,48 +0.000271,48 +0.000274,48 +0.000271,48 +0.000273,48 +0.000271,48 +0.000282,48 +0.000283,48 +0.000260,48 +0.000260,48 +0.000285,48 +0.000260,48 +0.000284,48 +0.000260,48 +0.000272,48 +0.000272,48 +0.000260,48 +0.000331,48 +0.000461,48 +0.000328,48 +0.000291,48 +0.000325,48 +0.000402,48 +0.000333,48 +0.000284,48 +0.000300,50 +0.000316,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000533,50 +0.000402,50 +0.000340,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000400,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000301,50 +0.000506,50 +0.000445,50 +0.000294,50 +0.000309,50 +0.000424,50 +0.000302,50 +0.000304,50 +0.000304,50 +0.000310,50 +0.000294,50 +0.000294,50 +0.000344,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000355,50 +0.000308,50 +0.000333,50 +0.000294,50 +0.000333,50 +0.000465,50 +0.000440,50 +0.000349,50 +0.000508,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000341,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000313,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000374,50 +0.000299,50 +0.000294,50 +0.000344,50 +0.000294,50 +0.000300,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000572,50 +0.000427,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000332,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000297,50 +0.000461,50 +0.000454,50 +0.000376,50 +0.000313,50 +0.000294,50 +0.000294,50 +0.000294,50 +0.000354,50 +0.000294,50 +0.000588,52 +0.000370,52 +0.000333,52 +0.000333,52 +0.000378,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000415,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000655,52 +0.000353,52 +0.000591,52 +0.000396,52 +0.000367,52 +0.000341,52 +0.000375,52 +0.000330,52 +0.000330,52 +0.000378,52 +0.000591,52 +0.000334,52 +0.000333,52 +0.000333,52 +0.000333,52 +0.000333,52 +0.000333,52 +0.000375,52 +0.000330,52 +0.000369,52 +0.000330,52 +0.000330,52 +0.000336,52 +0.000493,52 +0.000479,52 +0.000333,52 +0.000333,52 +0.000333,52 +0.000378,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000494,52 +0.000468,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000376,52 +0.000408,52 +0.000376,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000575,52 +0.000368,52 +0.000330,52 +0.000340,52 +0.000405,52 +0.000378,52 +0.000339,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000600,52 +0.000340,52 +0.000330,52 +0.000330,52 +0.000345,52 +0.000349,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000390,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000382,52 +0.000589,52 +0.000369,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000329,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000330,52 +0.000372,54 +0.000369,54 +0.000369,54 +0.000579,54 +0.000500,54 +0.000471,54 +0.000378,54 +0.000369,54 +0.000379,54 +0.000378,54 +0.000443,54 +0.000369,54 +0.000410,54 +0.000515,54 +0.000437,54 +0.000546,54 +0.000378,54 +0.000377,54 +0.000369,54 +0.000380,54 +0.000588,54 +0.000405,54 +0.000368,54 +0.000369,54 +0.000369,54 +0.000369,54 +0.000369,54 +0.000431,54 +0.000369,54 +0.000369,54 +0.000369,54 +0.000410,54 +0.000413,54 +0.000369,54 +0.000369,54 +0.000651,54 +0.000382,54 +0.000373,54 +0.000373,54 +0.000373,54 +0.000373,54 +0.000373,54 +0.000416,54 +0.000369,54 +0.000369,54 +0.000369,54 +0.000400,54 +0.000459,54 +0.000369,54 +0.000369,54 +0.000381,54 +0.000378,54 +0.000402,54 +0.000379,54 +0.000411,54 +0.000527,54 +0.000369,54 +0.000379,54 +0.000408,54 +0.000369,54 +0.000369,54 +0.000369,54 +0.000404,54 +0.000369,54 +0.000409,54 +0.000369,54 +0.000369,54 +0.000389,54 +0.000369,54 +0.000459,54 +0.000376,54 +0.000438,54 +0.000442,54 +0.000378,54 +0.000369,54 +0.000369,54 +0.000541,54 +0.000588,54 +0.000392,54 +0.000389,54 +0.000389,54 +0.000403,54 +0.000378,54 +0.000378,54 +0.000378,54 +0.000379,54 +0.000378,54 +0.000378,54 +0.000434,54 +0.000369,54 +0.000369,54 +0.000369,54 +0.000406,54 +0.000368,54 +0.000369,54 +0.000369,54 +0.000454,54 +0.000443,54 +0.000395,54 +0.000369,54 +0.000412,56 +0.000410,56 +0.000452,56 +0.000410,56 +0.000411,56 +0.000410,56 +0.000411,56 +0.000411,56 +0.000410,56 +0.000411,56 +0.000451,56 +0.000455,56 +0.000490,56 +0.000466,56 +0.000411,56 +0.000410,56 +0.000411,56 +0.000411,56 +0.000410,56 +0.000410,56 +0.000411,56 +0.000447,56 +0.000410,56 +0.000410,56 +0.000411,56 +0.000411,56 +0.000411,56 +0.000430,56 +0.000486,56 +0.000467,56 +0.000448,56 +0.000411,56 +0.000410,56 +0.000498,56 +0.000450,56 +0.000491,56 +0.000471,56 +0.000411,56 +0.000410,56 +0.000443,56 +0.000410,56 +0.000410,56 +0.000411,56 +0.000411,56 +0.000410,56 +0.000410,56 +0.000411,56 +0.000410,56 +0.000412,56 +0.000528,56 +0.000454,56 +0.000411,56 +0.000451,56 +0.000411,56 +0.000410,56 +0.000410,56 +0.000517,56 +0.000489,56 +0.000410,56 +0.000410,56 +0.000498,56 +0.000450,56 +0.000410,56 +0.000410,56 +0.000411,56 +0.000431,56 +0.000437,56 +0.000479,56 +0.000411,56 +0.000411,56 +0.000430,56 +0.000442,56 +0.000662,56 +0.000483,56 +0.000445,56 +0.000454,56 +0.000410,56 +0.000410,56 +0.000410,56 +0.000410,56 +0.000597,56 +0.000603,56 +0.000460,56 +0.000458,56 +0.000709,56 +0.000501,56 +0.000462,56 +0.000453,56 +0.000433,56 +0.000421,56 +0.000421,56 +0.000421,56 +0.000449,56 +0.000411,56 +0.000411,56 +0.000679,56 +0.000441,56 +0.000421,56 +0.000421,56 +0.000421,56 +0.000469,58 +0.000832,58 +0.000624,58 +0.000507,58 +0.000506,58 +0.000455,58 +0.000455,58 +0.000538,58 +0.000739,58 +0.000536,58 +0.000467,58 +0.000467,58 +0.000468,58 +0.000487,58 +0.000554,58 +0.000495,58 +0.000467,58 +0.000467,58 +0.000559,58 +0.000508,58 +0.000480,58 +0.000480,58 +0.000582,58 +0.000483,58 +0.000480,58 +0.000480,58 +0.000480,58 +0.000480,58 +0.000506,58 +0.000477,58 +0.000467,58 +0.000467,58 +0.000499,58 +0.000455,58 +0.000697,58 +0.000636,58 +0.000502,58 +0.000472,58 +0.000514,58 +0.000506,58 +0.000566,58 +0.000465,58 +0.000455,58 +0.000515,58 +0.000707,58 +0.000478,58 +0.000487,58 +0.000519,58 +0.000490,58 +0.000467,58 +0.000467,58 +0.000522,58 +0.000483,58 +0.000467,58 +0.000488,58 +0.000477,58 +0.000467,58 +0.000467,58 +0.000467,58 +0.000478,58 +0.000642,58 +0.000498,58 +0.000483,58 +0.000472,58 +0.000467,58 +0.000467,58 +0.000467,58 +0.000502,58 +0.000533,58 +0.000473,58 +0.000467,58 +0.000480,58 +0.000455,58 +0.000498,58 +0.000544,58 +0.000455,58 +0.000632,58 +0.000488,58 +0.000500,58 +0.000507,58 +0.000468,58 +0.000486,58 +0.000499,58 +0.000455,58 +0.000480,58 +0.000493,58 +0.000467,58 +0.000530,58 +0.000557,58 +0.000495,58 +0.000518,58 +0.000539,58 +0.000572,58 +0.000487,58 +0.000467,58 +0.000495,58 +0.000467,58 +0.000498,58 +0.000613,58 +0.000513,58 +0.000605,60 +0.000517,60 +0.000526,60 +0.000540,60 +0.000503,60 +0.000504,60 +0.000648,60 +0.000701,60 +0.000610,60 +0.000554,60 +0.000532,60 +0.000517,60 +0.000645,60 +0.000517,60 +0.000516,60 +0.000517,60 +0.000531,60 +0.000638,60 +0.000557,60 +0.000517,60 +0.000517,60 +0.000517,60 +0.000517,60 +0.000517,60 +0.000563,60 +0.000653,60 +0.000517,60 +0.000517,60 +0.000565,60 +0.000560,60 +0.000555,60 +0.000548,60 +0.000517,60 +0.000525,60 +0.000650,60 +0.000613,60 +0.000596,60 +0.000517,60 +0.000547,60 +0.000614,60 +0.000526,60 +0.000517,60 +0.000556,60 +0.000551,60 +0.000517,60 +0.000517,60 +0.000670,60 +0.000537,60 +0.000517,60 +0.000531,60 +0.000517,60 +0.000517,60 +0.000613,60 +0.000547,60 +0.000517,60 +0.000585,60 +0.000543,60 +0.000517,60 +0.000517,60 +0.000552,60 +0.000550,60 +0.000553,60 +0.000528,60 +0.000559,60 +0.000517,60 +0.000517,60 +0.000684,60 +0.000583,60 +0.000526,60 +0.000517,60 +0.000604,60 +0.000517,60 +0.000517,60 +0.000517,60 +0.000517,60 +0.000588,60 +0.000538,60 +0.000610,60 +0.000534,60 +0.000622,60 +0.000517,60 +0.000517,60 +0.000596,60 +0.000558,60 +0.000544,60 +0.000564,60 +0.000545,60 +0.000537,60 +0.000517,60 +0.000542,60 +0.000586,60 +0.000582,60 +0.000579,60 +0.000522,60 +0.000517,60 +0.000554,60 +0.000558,60 +0.000604,60 +0.000526,60 +0.000563,60 +0.000599,62 +0.000611,62 +0.000569,62 +0.000601,62 +0.000648,62 +0.000579,62 +0.000605,62 +0.000607,62 +0.000613,62 +0.000570,62 +0.000654,62 +0.000610,62 +0.000637,62 +0.000569,62 +0.000569,62 +0.000596,62 +0.000611,62 +0.000664,62 +0.000569,62 +0.000598,62 +0.000570,62 +0.000575,62 +0.000639,62 +0.000654,62 +0.000661,62 +0.000570,62 +0.000569,62 +0.000671,62 +0.000625,62 +0.000630,62 +0.000595,62 +0.000580,62 +0.000695,62 +0.000584,62 +0.000569,62 +0.000719,62 +0.000616,62 +0.000569,62 +0.000644,62 +0.000610,62 +0.000615,62 +0.000605,62 +0.000626,62 +0.000580,62 +0.000580,62 +0.000659,62 +0.000601,62 +0.000570,62 +0.000675,62 +0.000597,62 +0.000570,62 +0.000735,62 +0.000596,62 +0.000570,62 +0.000569,62 +0.000609,62 +0.000749,62 +0.000793,62 +0.000687,62 +0.000570,62 +0.000785,62 +0.000754,62 +0.000608,62 +0.000626,62 +0.000628,62 +0.000734,62 +0.000564,62 +0.000593,62 +0.000564,62 +0.000598,62 +0.000848,62 +0.000569,62 +0.000602,62 +0.000569,62 +0.000781,62 +0.000628,62 +0.000621,62 +0.000570,62 +0.000569,62 +0.000689,62 +0.000595,62 +0.000598,62 +0.000650,62 +0.000576,62 +0.000569,62 +0.000627,62 +0.000617,62 +0.000649,62 +0.000590,62 +0.000579,62 +0.000569,62 +0.000569,62 +0.000569,62 +0.000587,62 +0.000582,62 +0.000569,62 +0.000570,62 +0.000570,62 +0.000555,62 +0.000779,62 +0.000785,64 +0.000667,64 +0.000626,64 +0.000672,64 +0.000706,64 +0.000632,64 +0.000670,64 +0.000610,64 +0.000609,64 +0.000704,64 +0.000675,64 +0.000625,64 +0.000799,64 +0.000684,64 +0.000636,64 +0.000636,64 +0.000610,64 +0.000610,64 +0.000652,64 +0.000610,64 +0.000609,64 +0.000621,64 +0.000832,64 +0.000626,64 +0.000740,64 +0.000707,64 +0.000636,64 +0.000626,64 +0.000626,64 +0.000677,64 +0.000675,64 +0.000680,64 +0.000849,64 +0.000617,64 +0.000616,64 +0.000617,64 +0.000655,64 +0.000617,64 +0.000894,64 +0.000675,64 +0.000660,64 +0.000877,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000630,64 +0.000630,64 +0.000610,64 +0.000822,64 +0.000718,64 +0.000626,64 +0.000816,64 +0.000610,64 +0.000732,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000803,64 +0.000667,64 +0.000646,64 +0.000626,64 +0.000640,64 +0.000626,64 +0.000650,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000646,64 +0.000610,64 +0.000610,64 +0.000755,64 +0.000610,64 +0.000610,64 +0.000630,64 +0.000767,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000653,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000650,64 +0.000610,64 +0.000658,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000610,64 +0.000643,64 +0.000610,64 +0.000701,66 +0.000736,66 +0.000701,66 +0.000695,66 +0.000757,66 +0.000892,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000794,66 +0.000719,66 +0.000730,66 +0.000720,66 +0.000768,66 +0.000752,66 +0.000803,66 +0.000695,66 +0.000803,66 +0.000817,66 +0.000695,66 +0.000730,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000730,66 +0.000695,66 +0.000695,66 +0.000714,66 +0.000695,66 +0.000695,66 +0.000733,66 +0.000735,66 +0.000877,66 +0.000695,66 +0.000695,66 +0.000788,66 +0.000696,66 +0.000707,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000736,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000696,66 +0.000695,66 +0.000730,66 +0.000830,66 +0.001033,66 +0.000793,66 +0.000776,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000695,66 +0.000734,66 +0.000734,66 +0.000695,66 +0.000705,66 +0.000695,66 +0.000695,66 +0.000723,66 +0.000735,66 +0.000887,66 +0.000839,66 +0.001128,66 +0.000973,66 +0.000763,66 +0.000749,66 +0.000970,66 +0.000838,66 +0.000749,66 +0.000845,66 +0.000848,66 +0.000736,66 +0.001027,66 +0.001168,66 +0.001270,66 +0.000751,66 +0.000812,66 +0.000766,66 +0.000714,66 +0.000796,66 +0.000723,66 +0.000749,66 +0.000777,66 +0.000744,66 +0.000703,66 +0.000795,66 +0.000938,66 +0.000951,66 +0.001383,66 +0.000888,66 +0.000774,66 +0.000845,66 +0.001012,66 +0.000972,66 +0.000907,68 +0.001312,68 +0.001344,68 +0.001450,68 +0.001564,68 +0.001454,68 +0.001484,68 +0.001321,68 +0.001172,68 +0.000795,68 +0.000780,68 +0.000896,68 +0.000932,68 +0.000758,68 +0.000793,68 +0.000760,68 +0.000781,68 +0.000796,68 +0.000750,68 +0.000798,68 +0.000770,68 +0.000750,68 +0.000815,68 +0.000750,68 +0.000768,68 +0.000752,68 +0.000890,68 +0.000791,68 +0.000766,68 +0.000730,68 +0.000794,68 +0.000783,68 +0.000771,68 +0.000731,68 +0.000795,68 +0.000825,68 +0.000740,68 +0.000731,68 +0.000730,68 +0.000730,68 +0.000773,68 +0.000730,68 +0.000749,68 +0.000730,68 +0.000829,68 +0.000801,68 +0.000730,68 +0.000872,68 +0.000770,68 +0.000843,68 +0.000796,68 +0.000978,68 +0.000765,68 +0.000857,68 +0.000780,68 +0.000829,68 +0.000744,68 +0.000778,68 +0.000767,68 +0.000777,68 +0.000798,68 +0.000741,68 +0.000730,68 +0.000799,68 +0.000816,68 +0.000816,68 +0.000731,68 +0.000730,68 +0.000800,68 +0.000808,68 +0.000837,68 +0.000731,68 +0.000770,68 +0.000770,68 +0.000832,68 +0.000751,68 +0.000831,68 +0.000730,68 +0.000792,68 +0.000730,68 +0.000751,68 +0.000774,68 +0.000730,68 +0.000730,68 +0.000730,68 +0.000730,68 +0.000830,68 +0.000773,68 +0.000881,68 +0.000797,68 +0.000875,68 +0.000819,68 +0.000830,68 +0.000844,68 +0.000806,68 +0.000879,68 +0.000793,68 +0.000781,68 +0.000763,68 +0.000952,68 +0.000825,70 +0.000867,70 +0.000921,70 +0.000877,70 +0.000817,70 +0.000908,70 +0.000816,70 +0.000816,70 +0.000816,70 +0.000817,70 +0.000928,70 +0.000842,70 +0.000823,70 +0.000836,70 +0.000795,70 +0.000836,70 +0.000795,70 +0.000834,70 +0.000795,70 +0.000835,70 +0.000830,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000841,70 +0.000815,70 +0.000795,70 +0.000795,70 +0.000889,70 +0.000954,70 +0.000795,70 +0.000795,70 +0.000820,70 +0.000839,70 +0.000795,70 +0.000795,70 +0.000835,70 +0.000795,70 +0.000842,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000818,70 +0.000795,70 +0.000795,70 +0.000796,70 +0.000795,70 +0.000822,70 +0.000835,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000840,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000835,70 +0.000804,70 +0.000835,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000797,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000797,70 +0.000795,70 +0.000835,70 +0.000795,70 +0.000795,70 +0.000842,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000819,70 +0.000835,70 +0.000795,70 +0.000795,70 +0.000795,70 +0.000801,70 +0.000795,70 +0.000819,70 +0.000814,70 +0.000795,70 +0.000797,70 +0.000984,70 +0.000963,70 +0.000832,70 +0.000852,70 +0.000813,70 +0.000854,70 +0.000875,70 +0.000834,70 +0.001028,70 +0.000839,70 +0.000928,72 +0.000914,72 +0.000922,72 +0.000943,72 +0.001004,72 +0.000911,72 +0.000916,72 +0.000949,72 +0.000974,72 +0.001009,72 +0.001021,72 +0.000943,72 +0.000939,72 +0.001050,72 +0.001369,72 +0.000906,72 +0.000931,72 +0.000939,72 +0.001310,72 +0.001418,72 +0.001016,72 +0.000865,72 +0.000924,72 +0.000993,72 +0.000865,72 +0.001087,72 +0.001341,72 +0.000912,72 +0.000969,72 +0.000912,72 +0.000910,72 +0.000887,72 +0.001002,72 +0.000927,72 +0.000911,72 +0.000972,72 +0.001009,72 +0.000945,72 +0.000917,72 +0.000887,72 +0.000948,72 +0.001074,72 +0.000913,72 +0.001173,72 +0.000888,72 +0.001074,72 +0.000951,72 +0.001027,72 +0.000887,72 +0.000908,72 +0.000988,72 +0.000917,72 +0.000887,72 +0.000888,72 +0.000888,72 +0.000912,72 +0.001002,72 +0.000969,72 +0.000928,72 +0.000915,72 +0.001005,72 +0.001022,72 +0.000888,72 +0.000914,72 +0.001134,72 +0.000888,72 +0.000888,72 +0.000908,72 +0.000944,72 +0.001175,72 +0.000911,72 +0.000897,72 +0.000915,72 +0.000887,72 +0.000887,72 +0.000951,72 +0.000912,72 +0.000887,72 +0.000907,72 +0.000887,72 +0.001062,72 +0.000899,72 +0.000865,72 +0.000865,72 +0.000865,72 +0.000925,72 +0.000875,72 +0.000894,72 +0.000865,72 +0.000900,72 +0.000865,72 +0.001257,72 +0.000972,72 +0.000991,72 +0.001001,72 +0.000906,72 +0.000978,72 +0.000865,72 +0.000907,72 +0.000898,72 +0.000985,74 +0.001021,74 +0.000988,74 +0.001000,74 +0.001000,74 +0.001176,74 +0.000992,74 +0.000938,74 +0.000938,74 +0.000938,74 +0.001051,74 +0.000978,74 +0.001045,74 +0.000988,74 +0.001011,74 +0.001138,74 +0.001130,74 +0.001041,74 +0.000939,74 +0.001171,74 +0.001081,74 +0.001145,74 +0.001003,74 +0.000959,74 +0.000938,74 +0.000974,74 +0.001038,74 +0.001237,74 +0.001027,74 +0.001068,74 +0.000938,74 +0.000938,74 +0.000938,74 +0.000975,74 +0.000938,74 +0.000978,74 +0.000965,74 +0.001030,74 +0.000938,74 +0.000938,74 +0.001065,74 +0.001153,74 +0.000963,74 +0.000998,74 +0.000958,74 +0.001040,74 +0.000952,74 +0.000938,74 +0.000999,74 +0.001004,74 +0.000938,74 +0.000938,74 +0.000977,74 +0.001011,74 +0.000938,74 +0.000978,74 +0.001126,74 +0.001096,74 +0.001035,74 +0.000964,74 +0.000978,74 +0.000970,74 +0.001084,74 +0.000958,74 +0.000958,74 +0.001020,74 +0.000938,74 +0.000938,74 +0.000939,74 +0.001004,74 +0.000978,74 +0.000938,74 +0.000938,74 +0.000938,74 +0.000961,74 +0.000938,74 +0.000938,74 +0.000958,74 +0.000963,74 +0.000938,74 +0.001061,74 +0.000938,74 +0.000966,74 +0.000938,74 +0.000938,74 +0.000938,74 +0.000988,74 +0.000938,74 +0.000978,74 +0.000938,74 +0.000961,74 +0.000938,74 +0.000938,74 +0.000938,74 +0.000943,74 +0.000938,74 +0.000938,74 +0.000959,74 +0.000938,74 +0.000963,74 +0.001054,76 +0.001051,76 +0.001090,76 +0.001150,76 +0.001072,76 +0.001172,76 +0.001086,76 +0.001111,76 +0.001071,76 +0.001152,76 +0.001089,76 +0.001051,76 +0.001051,76 +0.001086,76 +0.001061,76 +0.001051,76 +0.001051,76 +0.001075,76 +0.001051,76 +0.001091,76 +0.001112,76 +0.001086,76 +0.001051,76 +0.001071,76 +0.001051,76 +0.001055,76 +0.001051,76 +0.001051,76 +0.001055,76 +0.001051,76 +0.001051,76 +0.001051,76 +0.001053,76 +0.001051,76 +0.001091,76 +0.001051,76 +0.001094,76 +0.001051,76 +0.001051,76 +0.001090,76 +0.001055,76 +0.001052,76 +0.001051,76 +0.001051,76 +0.001118,76 +0.001051,76 +0.001051,76 +0.001183,76 +0.001303,76 +0.001051,76 +0.001112,76 +0.001124,76 +0.001051,76 +0.001051,76 +0.001075,76 +0.001051,76 +0.001553,76 +0.001176,76 +0.001095,76 +0.001416,76 +0.001071,76 +0.001087,76 +0.001051,76 +0.001051,76 +0.001091,76 +0.001083,76 +0.001102,76 +0.001051,76 +0.001092,76 +0.001051,76 +0.001051,76 +0.001071,76 +0.001075,76 +0.001051,76 +0.001216,76 +0.001051,76 +0.001080,76 +0.001051,76 +0.001051,76 +0.001051,76 +0.001114,76 +0.001090,76 +0.001051,76 +0.001053,76 +0.001051,76 +0.001051,76 +0.001051,76 +0.001055,76 +0.001052,76 +0.001094,76 +0.001051,76 +0.001053,76 +0.001051,76 +0.001051,76 +0.001051,76 +0.001115,76 +0.001051,76 +0.001091,76 +0.001053,76 +0.001051,76 +0.001100,78 +0.001097,78 +0.001101,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001099,78 +0.001097,78 +0.001097,78 +0.001122,78 +0.001097,78 +0.001136,78 +0.001156,78 +0.001120,78 +0.001097,78 +0.001097,78 +0.001125,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001099,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001101,78 +0.001097,78 +0.001136,78 +0.001138,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001101,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001119,78 +0.001147,78 +0.001097,78 +0.001167,78 +0.001097,78 +0.001097,78 +0.001136,78 +0.001140,78 +0.001097,78 +0.001097,78 +0.001099,78 +0.001097,78 +0.001097,78 +0.001141,78 +0.001167,78 +0.001148,78 +0.001130,78 +0.001170,78 +0.001122,78 +0.001332,78 +0.001157,78 +0.001139,78 +0.001136,78 +0.001097,78 +0.001121,78 +0.001196,78 +0.001097,78 +0.001096,78 +0.001127,78 +0.001097,78 +0.001097,78 +0.001119,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001160,78 +0.001117,78 +0.001116,78 +0.001097,78 +0.001125,78 +0.001097,78 +0.001097,78 +0.001119,78 +0.001097,78 +0.001097,78 +0.001096,78 +0.001122,78 +0.001097,78 +0.001097,78 +0.001121,78 +0.001136,78 +0.001097,78 +0.001136,78 +0.001128,78 +0.001097,78 +0.001171,78 +0.001097,78 +0.001103,78 +0.001097,78 +0.001117,78 +0.001106,78 +0.001097,78 +0.001097,78 +0.001097,78 +0.001100,78 +0.001264,80 +0.001262,80 +0.001225,80 +0.001223,80 +0.001223,80 +0.001227,80 +0.001223,80 +0.001223,80 +0.001223,80 +0.001225,80 +0.001223,80 +0.001223,80 +0.001227,80 +0.001223,80 +0.001262,80 +0.001283,80 +0.001223,80 +0.001223,80 +0.001227,80 +0.001223,80 +0.001223,80 +0.001223,80 +0.001225,80 +0.001282,80 +0.001223,80 +0.001227,80 +0.001223,80 +0.001712,80 +0.001329,80 +0.001264,80 +0.001255,80 +0.001251,80 +0.001223,80 +0.001599,80 +0.001299,80 +0.001271,80 +0.001335,80 +0.001309,80 +0.001575,80 +0.001408,80 +0.001299,80 +0.001467,80 +0.001480,80 +0.001317,80 +0.001264,80 +0.001394,80 +0.001587,80 +0.001334,80 +0.001425,80 +0.001378,80 +0.001411,80 +0.001429,80 +0.001355,80 +0.001371,80 +0.001385,80 +0.001368,80 +0.001379,80 +0.001418,80 +0.001370,80 +0.001374,80 +0.001350,80 +0.001364,80 +0.001342,80 +0.001407,80 +0.001450,80 +0.001360,80 +0.001466,80 +0.001275,80 +0.001433,80 +0.001438,80 +0.001355,80 +0.001501,80 +0.001334,80 +0.001378,80 +0.001397,80 +0.001461,80 +0.001453,80 +0.001389,80 +0.001323,80 +0.001326,80 +0.001358,80 +0.001449,80 +0.001381,80 +0.001467,80 +0.001347,80 +0.001351,80 +0.001440,80 +0.001414,80 +0.001330,80 +0.001410,80 +0.001383,80 +0.001459,80 +0.001375,80 +0.001542,80 +0.001452,80 +0.001452,80 +0.001384,80 +0.001366,80 +0.001429,80 +0.001353,80 +0.001473,82 +0.001418,82 +0.001347,82 +0.001387,82 +0.001345,82 +0.001437,82 +0.001763,82 +0.001388,82 +0.001389,82 +0.001345,82 +0.001426,82 +0.001455,82 +0.001466,82 +0.001407,82 +0.001468,82 +0.001408,82 +0.001475,82 +0.001458,82 +0.001405,82 +0.001397,82 +0.001423,82 +0.001519,82 +0.001587,82 +0.001440,82 +0.001367,82 +0.001421,82 +0.001491,82 +0.001387,82 +0.002170,82 +0.001483,82 +0.001421,82 +0.001546,82 +0.001426,82 +0.001630,82 +0.001557,82 +0.001396,82 +0.001408,82 +0.001490,82 +0.001365,82 +0.001479,82 +0.001402,82 +0.001574,82 +0.001383,82 +0.001400,82 +0.001366,82 +0.001451,82 +0.001437,82 +0.001522,82 +0.001670,82 +0.001963,82 +0.002133,82 +0.001533,82 +0.001973,82 +0.001660,82 +0.001853,82 +0.001586,82 +0.001375,82 +0.001368,82 +0.001519,82 +0.001350,82 +0.001552,82 +0.001479,82 +0.001522,82 +0.001393,82 +0.001417,82 +0.001377,82 +0.001486,82 +0.001379,82 +0.001348,82 +0.001396,82 +0.001349,82 +0.001388,82 +0.001407,82 +0.001410,82 +0.001679,82 +0.001334,82 +0.001647,82 +0.001530,82 +0.001803,82 +0.001343,82 +0.001370,82 +0.001336,82 +0.001697,82 +0.001429,82 +0.001737,82 +0.001503,82 +0.001345,82 +0.001364,82 +0.001308,82 +0.001612,82 +0.001366,82 +0.001341,82 +0.001347,82 +0.001424,82 +0.001590,82 +0.001366,82 +0.001375,82 +0.001321,82 +0.001353,82 +0.001294,82 +0.001669,84 +0.001526,84 +0.001538,84 +0.001411,84 +0.001462,84 +0.001597,84 +0.001472,84 +0.001640,84 +0.001412,84 +0.001575,84 +0.001458,84 +0.001693,84 +0.001625,84 +0.001592,84 +0.001543,84 +0.001544,84 +0.001556,84 +0.001714,84 +0.001522,84 +0.001623,84 +0.001591,84 +0.001423,84 +0.001478,84 +0.001554,84 +0.001472,84 +0.001471,84 +0.001495,84 +0.001573,84 +0.001593,84 +0.001494,84 +0.001552,84 +0.001578,84 +0.001505,84 +0.001621,84 +0.001543,84 +0.001499,84 +0.001560,84 +0.001557,84 +0.001518,84 +0.001988,84 +0.001971,84 +0.001965,84 +0.001966,84 +0.001569,84 +0.001649,84 +0.001778,84 +0.001657,84 +0.001753,84 +0.001536,84 +0.001505,84 +0.001472,84 +0.001468,84 +0.001448,84 +0.001509,84 +0.001477,84 +0.001544,84 +0.001528,84 +0.001538,84 +0.001554,84 +0.001537,84 +0.001515,84 +0.001594,84 +0.001543,84 +0.001528,84 +0.001484,84 +0.001527,84 +0.001571,84 +0.001508,84 +0.001607,84 +0.001540,84 +0.001538,84 +0.001628,84 +0.001552,84 +0.001569,84 +0.001489,84 +0.001556,84 +0.001618,84 +0.001560,84 +0.001519,84 +0.001590,84 +0.001533,84 +0.001639,84 +0.001880,84 +0.001776,84 +0.001651,84 +0.001795,84 +0.001832,84 +0.001781,84 +0.001668,84 +0.001819,84 +0.001829,84 +0.001789,84 +0.001550,84 +0.001519,84 +0.001526,84 +0.001636,84 +0.001549,84 +0.001489,84 +0.001573,84 +0.001508,84 +0.001689,86 +0.001727,86 +0.001814,86 +0.001746,86 +0.001651,86 +0.001581,86 +0.002784,86 +0.002020,86 +0.001751,86 +0.001838,86 +0.001704,86 +0.001713,86 +0.002233,86 +0.001671,86 +0.001552,86 +0.001590,86 +0.001488,86 +0.001489,86 +0.001621,86 +0.001542,86 +0.001672,86 +0.002154,86 +0.001683,86 +0.001518,86 +0.001719,86 +0.001589,86 +0.001469,86 +0.001469,86 +0.001507,86 +0.001609,86 +0.001761,86 +0.001685,86 +0.001559,86 +0.001546,86 +0.001493,86 +0.001549,86 +0.001502,86 +0.001508,86 +0.001536,86 +0.001624,86 +0.001597,86 +0.001576,86 +0.001566,86 +0.001572,86 +0.001469,86 +0.001510,86 +0.001543,86 +0.001469,86 +0.001513,86 +0.001469,86 +0.001572,86 +0.001678,86 +0.001723,86 +0.001591,86 +0.001565,86 +0.001511,86 +0.001586,86 +0.001469,86 +0.001469,86 +0.001552,86 +0.001868,86 +0.002483,86 +0.002891,86 +0.002797,86 +0.002861,86 +0.002462,86 +0.001964,86 +0.001836,86 +0.001599,86 +0.001622,86 +0.001578,86 +0.001706,86 +0.001705,86 +0.001788,86 +0.001986,86 +0.001600,86 +0.001528,86 +0.001697,86 +0.001970,86 +0.002888,86 +0.002401,86 +0.001572,86 +0.001520,86 +0.001580,86 +0.001508,86 +0.001532,86 +0.001548,86 +0.001624,86 +0.001765,86 +0.001488,86 +0.001627,86 +0.001469,86 +0.001530,86 +0.001525,86 +0.001469,86 +0.001469,86 +0.001512,86 +0.001666,86 +0.001702,86 +0.001659,86 +0.001687,88 +0.001593,88 +0.001635,88 +0.001608,88 +0.001574,88 +0.001612,88 +0.001613,88 +0.001802,88 +0.001868,88 +0.001818,88 +0.001712,88 +0.001593,88 +0.001653,88 +0.001897,88 +0.001593,88 +0.001613,88 +0.001574,88 +0.001771,88 +0.001871,88 +0.001981,88 +0.001962,88 +0.001625,88 +0.001635,88 +0.002564,88 +0.002989,88 +0.002923,88 +0.002775,88 +0.001774,88 +0.001841,88 +0.001674,88 +0.001592,88 +0.001612,88 +0.001573,88 +0.001729,88 +0.001627,88 +0.001949,88 +0.001631,88 +0.001706,88 +0.001599,88 +0.001656,88 +0.001611,88 +0.001573,88 +0.001574,88 +0.001757,88 +0.001927,88 +0.001673,88 +0.001680,88 +0.001703,88 +0.001627,88 +0.001689,88 +0.001610,88 +0.001880,88 +0.001652,88 +0.001959,88 +0.001833,88 +0.001758,88 +0.001791,88 +0.001764,88 +0.001767,88 +0.001668,88 +0.001744,88 +0.001651,88 +0.001574,88 +0.002012,88 +0.001927,88 +0.001750,88 +0.001866,88 +0.001627,88 +0.001799,88 +0.001638,88 +0.001658,88 +0.001789,88 +0.001902,88 +0.001845,88 +0.001852,88 +0.001896,88 +0.001632,88 +0.001619,88 +0.001929,88 +0.002279,88 +0.001912,88 +0.001828,88 +0.001703,88 +0.001683,88 +0.001792,88 +0.001699,88 +0.001671,88 +0.001785,88 +0.001695,88 +0.001674,88 +0.001746,88 +0.001835,88 +0.001816,88 +0.002181,88 +0.002106,88 +0.001822,88 +0.002063,88 +0.001751,88 +0.002060,88 +0.001953,88 +0.002506,90 +0.002729,90 +0.002557,90 +0.002289,90 +0.002198,90 +0.002215,90 +0.002222,90 +0.002952,90 +0.002940,90 +0.002199,90 +0.002351,90 +0.002141,90 +0.001943,90 +0.002165,90 +0.002890,90 +0.002427,90 +0.002257,90 +0.002644,90 +0.002190,90 +0.002735,90 +0.002083,90 +0.002364,90 +0.002731,90 +0.003146,90 +0.002302,90 +0.002487,90 +0.002181,90 +0.002008,90 +0.002087,90 +0.001829,90 +0.001937,90 +0.002182,90 +0.002111,90 +0.002008,90 +0.001927,90 +0.001947,90 +0.002299,90 +0.002398,90 +0.001884,90 +0.001889,90 +0.002031,90 +0.001808,90 +0.001882,90 +0.001899,90 +0.002200,90 +0.002058,90 +0.001909,90 +0.001916,90 +0.001905,90 +0.001876,90 +0.001885,90 +0.001819,90 +0.002004,90 +0.002309,90 +0.002066,90 +0.001904,90 +0.001882,90 +0.001827,90 +0.001815,90 +0.001880,90 +0.001880,90 +0.002356,90 +0.002548,90 +0.001877,90 +0.001901,90 +0.001806,90 +0.001802,90 +0.001874,90 +0.001893,90 +0.002230,90 +0.002034,90 +0.002065,90 +0.001932,90 +0.001997,90 +0.001898,90 +0.002424,90 +0.003206,90 +0.003277,90 +0.003128,90 +0.002408,90 +0.002066,90 +0.002048,90 +0.002269,90 +0.002389,90 +0.002295,90 +0.002206,90 +0.002143,90 +0.002052,90 +0.002010,90 +0.001989,90 +0.002045,90 +0.002232,90 +0.002216,90 +0.002354,90 +0.002202,90 +0.002418,90 +0.002099,90 +0.002110,90 +0.002142,90 +0.002166,90 +0.002204,92 +0.002298,92 +0.002207,92 +0.002173,92 +0.002178,92 +0.002327,92 +0.002337,92 +0.002156,92 +0.002237,92 +0.002421,92 +0.002156,92 +0.002213,92 +0.002182,92 +0.002491,92 +0.002323,92 +0.002225,92 +0.002425,92 +0.002284,92 +0.002338,92 +0.003061,92 +0.003377,92 +0.003534,92 +0.003016,92 +0.002268,92 +0.002223,92 +0.002220,92 +0.002490,92 +0.002097,92 +0.002016,92 +0.002111,92 +0.002131,92 +0.002017,92 +0.001990,92 +0.002122,92 +0.001988,92 +0.001986,92 +0.002012,92 +0.002017,92 +0.001991,92 +0.002005,92 +0.001974,92 +0.002031,92 +0.002031,92 +0.002045,92 +0.002075,92 +0.002085,92 +0.002380,92 +0.002281,92 +0.002188,92 +0.002187,92 +0.002204,92 +0.002148,92 +0.002144,92 +0.002126,92 +0.002146,92 +0.002135,92 +0.002182,92 +0.002274,92 +0.002168,92 +0.002217,92 +0.002235,92 +0.002338,92 +0.002226,92 +0.002195,92 +0.003377,92 +0.003636,92 +0.003556,92 +0.002944,92 +0.002353,92 +0.002290,92 +0.002484,92 +0.002277,92 +0.002147,92 +0.002483,92 +0.002218,92 +0.002122,92 +0.002113,92 +0.002255,92 +0.002275,92 +0.002155,92 +0.002152,92 +0.002163,92 +0.002218,92 +0.002204,92 +0.002177,92 +0.002174,92 +0.002196,92 +0.002190,92 +0.002202,92 +0.002230,92 +0.002163,92 +0.002154,92 +0.002150,92 +0.002293,92 +0.002182,92 +0.002219,92 +0.002143,92 +0.002209,92 +0.002253,92 +0.002173,92 +0.002319,94 +0.002370,94 +0.002364,94 +0.002271,94 +0.002365,94 +0.002355,94 +0.002271,94 +0.003291,94 +0.003712,94 +0.003714,94 +0.003034,94 +0.002388,94 +0.002673,94 +0.002384,94 +0.002423,94 +0.002168,94 +0.002159,94 +0.002336,94 +0.002190,94 +0.002172,94 +0.002268,94 +0.002195,94 +0.002355,94 +0.002301,94 +0.002343,94 +0.002207,94 +0.002375,94 +0.002346,94 +0.002500,94 +0.002342,94 +0.002345,94 +0.002385,94 +0.002242,94 +0.002254,94 +0.002465,94 +0.002337,94 +0.002118,94 +0.002088,94 +0.002141,94 +0.002053,94 +0.002127,94 +0.002154,94 +0.002742,94 +0.002204,94 +0.002145,94 +0.002119,94 +0.002109,94 +0.002133,94 +0.002150,94 +0.002110,94 +0.002025,94 +0.002053,94 +0.002768,94 +0.002178,94 +0.002147,94 +0.002126,94 +0.002086,94 +0.002378,94 +0.002387,94 +0.002090,94 +0.002102,94 +0.002120,94 +0.002158,94 +0.002133,94 +0.002197,94 +0.002690,94 +0.002092,94 +0.002132,94 +0.002095,94 +0.002105,94 +0.002079,94 +0.002148,94 +0.002127,94 +0.002543,94 +0.002073,94 +0.002065,94 +0.002016,94 +0.002084,94 +0.002091,94 +0.001971,94 +0.002219,94 +0.002043,94 +0.002093,94 +0.002011,94 +0.002007,94 +0.001957,94 +0.001953,94 +0.001918,94 +0.002061,94 +0.002174,94 +0.001999,94 +0.001918,94 +0.002002,94 +0.001957,94 +0.001952,94 +0.001918,94 +0.001996,94 +0.002121,94 +0.002045,94 +0.001991,94 +0.002271,96 +0.002186,96 +0.002124,96 +0.002056,96 +0.002083,96 +0.002061,96 +0.002075,96 +0.002041,96 +0.002082,96 +0.002100,96 +0.002065,96 +0.002071,96 +0.002041,96 +0.002045,96 +0.002042,96 +0.002046,96 +0.002042,96 +0.002122,96 +0.002041,96 +0.002087,96 +0.002101,96 +0.002043,96 +0.002041,96 +0.002043,96 +0.002041,96 +0.002128,96 +0.002041,96 +0.002063,96 +0.002041,96 +0.002045,96 +0.002372,96 +0.002088,96 +0.002136,96 +0.002389,96 +0.002162,96 +0.002095,96 +0.002505,96 +0.002663,96 +0.002072,96 +0.002533,96 +0.002530,96 +0.002976,96 +0.003597,96 +0.002478,96 +0.002308,96 +0.002326,96 +0.002289,96 +0.002277,96 +0.002227,96 +0.002305,96 +0.002198,96 +0.002152,96 +0.002522,96 +0.002326,96 +0.002328,96 +0.002126,96 +0.002330,96 +0.002453,96 +0.002437,96 +0.002298,96 +0.002287,96 +0.002595,96 +0.002550,96 +0.002598,96 +0.002488,96 +0.002446,96 +0.002734,96 +0.002717,96 +0.002524,96 +0.002335,96 +0.002321,96 +0.002189,96 +0.002176,96 +0.002150,96 +0.002250,96 +0.002277,96 +0.002221,96 +0.002236,96 +0.002231,96 +0.002248,96 +0.002288,96 +0.002260,96 +0.002340,96 +0.002302,96 +0.002257,96 +0.002349,96 +0.002321,96 +0.002243,96 +0.002259,96 +0.002285,96 +0.002158,96 +0.002152,96 +0.002160,96 +0.002265,96 +0.002317,96 +0.002278,96 +0.002304,96 +0.002213,96 +0.002157,96 +0.002183,96 +0.002445,98 +0.002454,98 +0.002461,98 +0.002463,98 +0.002556,98 +0.002383,98 +0.002484,98 +0.002539,98 +0.002351,98 +0.002377,98 +0.002374,98 +0.002373,98 +0.002304,98 +0.002409,98 +0.002439,98 +0.002416,98 +0.002296,98 +0.002313,98 +0.002352,98 +0.002399,98 +0.002295,98 +0.002355,98 +0.002379,98 +0.002352,98 +0.002363,98 +0.002364,98 +0.002298,98 +0.002357,98 +0.002371,98 +0.002279,98 +0.002246,98 +0.002735,98 +0.002591,98 +0.002416,98 +0.002642,98 +0.002502,98 +0.002411,98 +0.002420,98 +0.002512,98 +0.002377,98 +0.002522,98 +0.002426,98 +0.002718,98 +0.002724,98 +0.002641,98 +0.002545,98 +0.002366,98 +0.002431,98 +0.002570,98 +0.002433,98 +0.002428,98 +0.002452,98 +0.002455,98 +0.002421,98 +0.002828,98 +0.002920,98 +0.003138,98 +0.003385,98 +0.003285,98 +0.002902,98 +0.003023,98 +0.002947,98 +0.002916,98 +0.002957,98 +0.002793,98 +0.003201,98 +0.002843,98 +0.002986,98 +0.002336,98 +0.003146,98 +0.002695,98 +0.002431,98 +0.002494,98 +0.002348,98 +0.002304,98 +0.002355,98 +0.002373,98 +0.002362,98 +0.002358,98 +0.002454,98 +0.002325,98 +0.002288,98 +0.002278,98 +0.002257,98 +0.002223,98 +0.002179,98 +0.002391,98 +0.002197,98 +0.002214,98 +0.002179,98 +0.002314,98 +0.002227,98 +0.002189,98 +0.002274,98 +0.002424,98 +0.002317,98 +0.002325,98 +0.002287,98 +0.002440,98 +0.002235,98 +0.002631,100 +0.003587,100 +0.003203,100 +0.002856,100 +0.002599,100 +0.002795,100 +0.003222,100 +0.002630,100 +0.002563,100 +0.002547,100 +0.002723,100 +0.002813,100 +0.002615,100 +0.002694,100 +0.002470,100 +0.002390,100 +0.002378,100 +0.002357,100 +0.002366,100 +0.002859,100 +0.002374,100 +0.002423,100 +0.002399,100 +0.002359,100 +0.002351,100 +0.002440,100 +0.002509,100 +0.002368,100 +0.002309,100 +0.002358,100 +0.002390,100 +0.002361,100 +0.002445,100 +0.002447,100 +0.002343,100 +0.002413,100 +0.002403,100 +0.002494,100 +0.002308,100 +0.002389,100 +0.002390,100 +0.002425,100 +0.002337,100 +0.002404,100 +0.002424,100 +0.002375,100 +0.002355,100 +0.003358,100 +0.004333,100 +0.004395,100 +0.002452,100 +0.002391,100 +0.002436,100 +0.002365,100 +0.002428,100 +0.002455,100 +0.002320,100 +0.002347,100 +0.002438,100 +0.002372,100 +0.002368,100 +0.002329,100 +0.002793,100 +0.002615,100 +0.002933,100 +0.003123,100 +0.002914,100 +0.002838,100 +0.003154,100 +0.002967,100 +0.003166,100 +0.003947,100 +0.003279,100 +0.003344,100 +0.002560,100 +0.002644,100 +0.003361,100 +0.002568,100 +0.002543,100 +0.002499,100 +0.002502,100 +0.002419,100 +0.002838,100 +0.004438,100 +0.004333,100 +0.003139,100 +0.002395,100 +0.002680,100 +0.002382,100 +0.002357,100 +0.002347,100 +0.002398,100 +0.002343,100 +0.002342,100 +0.002576,100 +0.002307,100 +0.002332,100 +0.002468,100 +0.002346,100 +0.002337,100 +0.002496,102 +0.002542,102 +0.002490,102 +0.002453,102 +0.002558,102 +0.002461,102 +0.002476,102 +0.002542,102 +0.002453,102 +0.002459,102 +0.002561,102 +0.002531,102 +0.002735,102 +0.002487,102 +0.002580,102 +0.002558,102 +0.002536,102 +0.002528,102 +0.002747,102 +0.002652,102 +0.002880,102 +0.002657,102 +0.002634,102 +0.002690,102 +0.002908,102 +0.002496,102 +0.002894,102 +0.002579,102 +0.002784,102 +0.002583,102 +0.002614,102 +0.002550,102 +0.002555,102 +0.002724,102 +0.002518,102 +0.002594,102 +0.002585,102 +0.002763,102 +0.002479,102 +0.002635,102 +0.002453,102 +0.002732,102 +0.002579,102 +0.002501,102 +0.002537,102 +0.002552,102 +0.002534,102 +0.002491,102 +0.002521,102 +0.002630,102 +0.002478,102 +0.002453,102 +0.002599,102 +0.002474,102 +0.002485,102 +0.002490,102 +0.002544,102 +0.002462,102 +0.002457,102 +0.002535,102 +0.002597,102 +0.002591,102 +0.002620,102 +0.002619,102 +0.002475,102 +0.002493,102 +0.002487,102 +0.002495,102 +0.002473,102 +0.002588,102 +0.002487,102 +0.002486,102 +0.002559,102 +0.002453,102 +0.002496,102 +0.002492,102 +0.002496,102 +0.002455,102 +0.002453,102 +0.002457,102 +0.002455,102 +0.002454,102 +0.002496,102 +0.002492,102 +0.002455,102 +0.002457,102 +0.002452,102 +0.002455,102 +0.002617,102 +0.002530,102 +0.002455,102 +0.002492,102 +0.002455,102 +0.002458,102 +0.002453,102 +0.002455,102 +0.002530,102 +0.002486,102 +0.002476,102 +0.002453,102 +0.002644,104 +0.002914,104 +0.002769,104 +0.002818,104 +0.003162,104 +0.002768,104 +0.002623,104 +0.002629,104 +0.002696,104 +0.002602,104 +0.002633,104 +0.002769,104 +0.002594,104 +0.002641,104 +0.002669,104 +0.002631,104 +0.002635,104 +0.002638,104 +0.002594,104 +0.002596,104 +0.002593,104 +0.002675,104 +0.002601,104 +0.002594,104 +0.002599,104 +0.002614,104 +0.002682,104 +0.002697,104 +0.002606,104 +0.002593,104 +0.002598,104 +0.002596,104 +0.002827,104 +0.002739,104 +0.002733,104 +0.002605,104 +0.002660,104 +0.002610,104 +0.002770,104 +0.002754,104 +0.002903,104 +0.002616,104 +0.002653,104 +0.002636,104 +0.002594,104 +0.002617,104 +0.002683,104 +0.002593,104 +0.002621,104 +0.002694,104 +0.002593,104 +0.002931,104 +0.002687,104 +0.002611,104 +0.002615,104 +0.002731,104 +0.002594,104 +0.002599,104 +0.002743,104 +0.002827,104 +0.002593,104 +0.002853,104 +0.002641,104 +0.002677,104 +0.002846,104 +0.002660,104 +0.002593,104 +0.002880,104 +0.002618,104 +0.002593,104 +0.002888,104 +0.002700,104 +0.002593,104 +0.002697,104 +0.002763,104 +0.002742,104 +0.002744,104 +0.002944,104 +0.002612,104 +0.002875,104 +0.002688,104 +0.002594,104 +0.002825,104 +0.002811,104 +0.002790,104 +0.002676,104 +0.002923,104 +0.002650,104 +0.002593,104 +0.002882,104 +0.002773,104 +0.002749,104 +0.002754,104 +0.002698,104 +0.002613,104 +0.002846,104 +0.002832,104 +0.002777,104 +0.002671,104 +0.002618,104 +0.002861,106 +0.002804,106 +0.002767,106 +0.002771,106 +0.002765,106 +0.002768,106 +0.002816,106 +0.003076,106 +0.002747,106 +0.002774,106 +0.002827,106 +0.002822,106 +0.002887,106 +0.003106,106 +0.002937,106 +0.002827,106 +0.002802,106 +0.002752,106 +0.002786,106 +0.002772,106 +0.002751,106 +0.002766,106 +0.002749,106 +0.002751,106 +0.002757,106 +0.002778,106 +0.002750,106 +0.002749,106 +0.002746,106 +0.002751,106 +0.002749,106 +0.002808,106 +0.002755,106 +0.002766,106 +0.002747,106 +0.002748,106 +0.002812,106 +0.003042,106 +0.002754,106 +0.002816,106 +0.002796,106 +0.002796,106 +0.002974,106 +0.002886,106 +0.002746,106 +0.002830,106 +0.002770,106 +0.002814,106 +0.002845,106 +0.002852,106 +0.002921,106 +0.002764,106 +0.002799,106 +0.002937,106 +0.002828,106 +0.002919,106 +0.002782,106 +0.002887,106 +0.002747,106 +0.002781,106 +0.002797,106 +0.002784,106 +0.002768,106 +0.002771,106 +0.002777,106 +0.002746,106 +0.002824,106 +0.002789,106 +0.002930,106 +0.003093,106 +0.002806,106 +0.002859,106 +0.002835,106 +0.002773,106 +0.003069,106 +0.002904,106 +0.002781,106 +0.002778,106 +0.003197,106 +0.002853,106 +0.002909,106 +0.002748,106 +0.003053,106 +0.003029,106 +0.002944,106 +0.003028,106 +0.002922,106 +0.002943,106 +0.002805,106 +0.002842,106 +0.003043,106 +0.002790,106 +0.002891,106 +0.003061,106 +0.002766,106 +0.002845,106 +0.003088,106 +0.002893,106 +0.002975,106 +0.002841,106 +0.003258,108 +0.003219,108 +0.003353,108 +0.003234,108 +0.003371,108 +0.003265,108 +0.003275,108 +0.003025,108 +0.003320,108 +0.003132,108 +0.003040,108 +0.003275,108 +0.003107,108 +0.003192,108 +0.002982,108 +0.002944,108 +0.003102,108 +0.003042,108 +0.003119,108 +0.003389,108 +0.002972,108 +0.002947,108 +0.002959,108 +0.002946,108 +0.003023,108 +0.003080,108 +0.002959,108 +0.003009,108 +0.002963,108 +0.002908,108 +0.003800,108 +0.002959,108 +0.002980,108 +0.002962,108 +0.002907,108 +0.003076,108 +0.003053,108 +0.003043,108 +0.002974,108 +0.002942,108 +0.002948,108 +0.003255,108 +0.003202,108 +0.002996,108 +0.002961,108 +0.002941,108 +0.003798,108 +0.003248,108 +0.003472,108 +0.003765,108 +0.003114,108 +0.003797,108 +0.003137,108 +0.002941,108 +0.003015,108 +0.002948,108 +0.003830,108 +0.003539,108 +0.003319,108 +0.003370,108 +0.003390,108 +0.003863,108 +0.003414,108 +0.003166,108 +0.003177,108 +0.002997,108 +0.004755,108 +0.003511,108 +0.003067,108 +0.003015,108 +0.003014,108 +0.004106,108 +0.003278,108 +0.002963,108 +0.003018,108 +0.003021,108 +0.003756,108 +0.003218,108 +0.004016,108 +0.003551,108 +0.003735,108 +0.003652,108 +0.002973,108 +0.003049,108 +0.003021,108 +0.003233,108 +0.003566,108 +0.003014,108 +0.002982,108 +0.003015,108 +0.002909,108 +0.003506,108 +0.003099,108 +0.003016,108 +0.002987,108 +0.003012,108 +0.003344,108 +0.003108,108 +0.003036,108 +0.002986,108 +0.003121,110 +0.003308,110 +0.003645,110 +0.003130,110 +0.003139,110 +0.003119,110 +0.003193,110 +0.003599,110 +0.003218,110 +0.003943,110 +0.003833,110 +0.004163,110 +0.003342,110 +0.003072,110 +0.003193,110 +0.003109,110 +0.003806,110 +0.003317,110 +0.003172,110 +0.003261,110 +0.003104,110 +0.004003,110 +0.003298,110 +0.003116,110 +0.003148,110 +0.003104,110 +0.003832,110 +0.003439,110 +0.003874,110 +0.003440,110 +0.005118,110 +0.004115,110 +0.003377,110 +0.003371,110 +0.003585,110 +0.004081,110 +0.003263,110 +0.003363,110 +0.005218,110 +0.004327,110 +0.003541,110 +0.003458,110 +0.003406,110 +0.003481,110 +0.004168,110 +0.003612,110 +0.004007,110 +0.003247,110 +0.003406,110 +0.003569,110 +0.003616,110 +0.003771,110 +0.003692,110 +0.003914,110 +0.004257,110 +0.003764,110 +0.003408,110 +0.003849,110 +0.004592,110 +0.003795,110 +0.003294,110 +0.004595,110 +0.005149,110 +0.004321,110 +0.003406,110 +0.004178,110 +0.005475,110 +0.005511,110 +0.003782,110 +0.003560,110 +0.004343,110 +0.003608,110 +0.003263,110 +0.004953,110 +0.004977,110 +0.006301,110 +0.005646,110 +0.004369,110 +0.004527,110 +0.005416,110 +0.004751,110 +0.003676,110 +0.003736,110 +0.003644,110 +0.005144,110 +0.004847,110 +0.005405,110 +0.003992,110 +0.003676,110 +0.004584,110 +0.003187,110 +0.003931,110 +0.004963,110 +0.004742,110 +0.004872,110 +0.003628,110 +0.003541,110 +0.003228,110 +0.003850,110 +0.005009,110 +0.005836,112 +0.006242,112 +0.006106,112 +0.003673,112 +0.003781,112 +0.003425,112 +0.003692,112 +0.004829,112 +0.003334,112 +0.004316,112 +0.004068,112 +0.006432,112 +0.004738,112 +0.003701,112 +0.003584,112 +0.003472,112 +0.003439,112 +0.004773,112 +0.004372,112 +0.004500,112 +0.003843,112 +0.003317,112 +0.004180,112 +0.003587,112 +0.004891,112 +0.003355,112 +0.004367,112 +0.005543,112 +0.004212,112 +0.003753,112 +0.005279,112 +0.006208,112 +0.006200,112 +0.005718,112 +0.004388,112 +0.003853,112 +0.004592,112 +0.005675,112 +0.005520,112 +0.004401,112 +0.004859,112 +0.006026,112 +0.005075,112 +0.004250,112 +0.004091,112 +0.003717,112 +0.003554,112 +0.003785,112 +0.003421,112 +0.003694,112 +0.003385,112 +0.003369,112 +0.003600,112 +0.003460,112 +0.003418,112 +0.005473,112 +0.006749,112 +0.004214,112 +0.003674,112 +0.004082,112 +0.003621,112 +0.003577,112 +0.003567,112 +0.003612,112 +0.004367,112 +0.003855,112 +0.004755,112 +0.004980,112 +0.004530,112 +0.004115,112 +0.004447,112 +0.003744,112 +0.003871,112 +0.003347,112 +0.003526,112 +0.004476,112 +0.005915,112 +0.006213,112 +0.005642,112 +0.003939,112 +0.004915,112 +0.003862,112 +0.003666,112 +0.005661,112 +0.004728,112 +0.004609,112 +0.004075,112 +0.003584,112 +0.003461,112 +0.003718,112 +0.003806,112 +0.004843,112 +0.003594,112 +0.003614,112 +0.003549,112 +0.004039,112 +0.003723,112 +0.003414,112 +0.003796,112 +0.003973,112 +0.004787,114 +0.004136,114 +0.003698,114 +0.003936,114 +0.004186,114 +0.003469,114 +0.003586,114 +0.003563,114 +0.004120,114 +0.003981,114 +0.003613,114 +0.003603,114 +0.004202,114 +0.003610,114 +0.003615,114 +0.003970,114 +0.003713,114 +0.004594,114 +0.003967,114 +0.003705,114 +0.003533,114 +0.005343,114 +0.003616,114 +0.003937,114 +0.003666,114 +0.004701,114 +0.004117,114 +0.003791,114 +0.003927,114 +0.004133,114 +0.003919,114 +0.003650,114 +0.003989,114 +0.005745,114 +0.004907,114 +0.004419,114 +0.006310,114 +0.004878,114 +0.005730,114 +0.004555,114 +0.004293,114 +0.004167,114 +0.003760,114 +0.003707,114 +0.004708,114 +0.004830,114 +0.004394,114 +0.004506,114 +0.004576,114 +0.003709,114 +0.003710,114 +0.003735,114 +0.003963,114 +0.003603,114 +0.003639,114 +0.003622,114 +0.003918,114 +0.003500,114 +0.003536,114 +0.003496,114 +0.003747,114 +0.004614,114 +0.003626,114 +0.003560,114 +0.004132,114 +0.003929,114 +0.003513,114 +0.003639,114 +0.003501,114 +0.004693,114 +0.003638,114 +0.003510,114 +0.003523,114 +0.005249,114 +0.003929,114 +0.003533,114 +0.003540,114 +0.003679,114 +0.004028,114 +0.003837,114 +0.004211,114 +0.004641,114 +0.003909,114 +0.003980,114 +0.003944,114 +0.003978,114 +0.003950,114 +0.003711,114 +0.004290,114 +0.004783,114 +0.003931,114 +0.003651,114 +0.003681,114 +0.003937,114 +0.004409,114 +0.004043,114 +0.004928,114 +0.004647,114 +0.004604,114 +0.004061,114 +0.003837,116 +0.003813,116 +0.003887,116 +0.004217,116 +0.003743,116 +0.003736,116 +0.004412,116 +0.003754,116 +0.003727,116 +0.003773,116 +0.004102,116 +0.004182,116 +0.003631,116 +0.003809,116 +0.003794,116 +0.004108,116 +0.003639,116 +0.003756,116 +0.003689,116 +0.004026,116 +0.003754,116 +0.003671,116 +0.003682,116 +0.003978,116 +0.003798,116 +0.003636,116 +0.003711,116 +0.004586,116 +0.004441,116 +0.004037,116 +0.003830,116 +0.003782,116 +0.004104,116 +0.003632,116 +0.003748,116 +0.003665,116 +0.004122,116 +0.003719,116 +0.003682,116 +0.003697,116 +0.003930,116 +0.003940,116 +0.003639,116 +0.003774,116 +0.003674,116 +0.004208,116 +0.003627,116 +0.003687,116 +0.003675,116 +0.004086,116 +0.003825,116 +0.003594,116 +0.003723,116 +0.004457,116 +0.004397,116 +0.003899,116 +0.003694,116 +0.003790,116 +0.004091,116 +0.003629,116 +0.004212,116 +0.003711,116 +0.004105,116 +0.003695,116 +0.003939,116 +0.003718,116 +0.004045,116 +0.004201,116 +0.003629,116 +0.003789,116 +0.003828,116 +0.004067,116 +0.003635,116 +0.003762,116 +0.003638,116 +0.004158,116 +0.003709,116 +0.003672,116 +0.003669,116 +0.003834,116 +0.003983,116 +0.004511,116 +0.003747,116 +0.003918,116 +0.003984,116 +0.003632,116 +0.003687,116 +0.003700,116 +0.004324,116 +0.003912,116 +0.004117,116 +0.004163,116 +0.004294,116 +0.003923,116 +0.003681,116 +0.003711,116 +0.004289,116 +0.004412,116 +0.003677,116 +0.003650,116 +0.004346,118 +0.003991,118 +0.003824,118 +0.003865,118 +0.003964,118 +0.004073,118 +0.004120,118 +0.003890,118 +0.003943,118 +0.004015,118 +0.003822,118 +0.003899,118 +0.003838,118 +0.004025,118 +0.003836,118 +0.003853,118 +0.003892,118 +0.003903,118 +0.003821,118 +0.003811,118 +0.003981,118 +0.003875,118 +0.003817,118 +0.003806,118 +0.003877,118 +0.003795,118 +0.003856,118 +0.003790,118 +0.003831,118 +0.003843,118 +0.004057,118 +0.004039,118 +0.003847,118 +0.003866,118 +0.003882,118 +0.003810,118 +0.003817,118 +0.003938,118 +0.003862,118 +0.003835,118 +0.003787,118 +0.003868,118 +0.003877,118 +0.003898,118 +0.003814,118 +0.003827,118 +0.003906,118 +0.003855,118 +0.003836,118 +0.003836,118 +0.003881,118 +0.003875,118 +0.003795,118 +0.003789,118 +0.003846,118 +0.003810,118 +0.004143,118 +0.004031,118 +0.003871,118 +0.003842,118 +0.003850,118 +0.003794,118 +0.003790,118 +0.003881,118 +0.003891,118 +0.003850,118 +0.003791,118 +0.003886,118 +0.004017,118 +0.003790,118 +0.003787,118 +0.003825,118 +0.003930,118 +0.003822,118 +0.003789,118 +0.003793,118 +0.003874,118 +0.003793,118 +0.003789,118 +0.003789,118 +0.003855,118 +0.003809,118 +0.004019,118 +0.004125,118 +0.003866,118 +0.003863,118 +0.003847,118 +0.003883,118 +0.003818,118 +0.003886,118 +0.003990,118 +0.003835,118 +0.003814,118 +0.003876,118 +0.003837,118 +0.003814,118 +0.003791,118 +0.003847,118 +0.003900,118 +0.003798,118 +0.004000,120 +0.004003,120 +0.004077,120 +0.003994,120 +0.003974,120 +0.003978,120 +0.004037,120 +0.003980,120 +0.004235,120 +0.004179,120 +0.004089,120 +0.003979,120 +0.003974,120 +0.004010,120 +0.004069,120 +0.003998,120 +0.003995,120 +0.003976,120 +0.004063,120 +0.004007,120 +0.004009,120 +0.003985,120 +0.004014,120 +0.004230,120 +0.003977,120 +0.003984,120 +0.004036,120 +0.004071,120 +0.003974,120 +0.004011,120 +0.004010,120 +0.004069,120 +0.003974,120 +0.004183,120 +0.004231,120 +0.004194,120 +0.003994,120 +0.004074,120 +0.004010,120 +0.004118,120 +0.004167,120 +0.004004,120 +0.003977,120 +0.004067,120 +0.003977,120 +0.003979,120 +0.003978,120 +0.004116,120 +0.004197,120 +0.003985,120 +0.003974,120 +0.004189,120 +0.004266,120 +0.004054,120 +0.004065,120 +0.004118,120 +0.004036,120 +0.004116,120 +0.004106,120 +0.004314,120 +0.003997,120 +0.004074,120 +0.004018,120 +0.004176,120 +0.004099,120 +0.004020,120 +0.004015,120 +0.004070,120 +0.004088,120 +0.004059,120 +0.003995,120 +0.004057,120 +0.004145,120 +0.004015,120 +0.003982,120 +0.004031,120 +0.004070,120 +0.004034,120 +0.003996,120 +0.003981,120 +0.004161,120 +0.004040,120 +0.004191,120 +0.004170,120 +0.004290,120 +0.004079,120 +0.004132,120 +0.004137,120 +0.004106,120 +0.004235,120 +0.004068,120 +0.004007,120 +0.004126,120 +0.004024,120 +0.004011,120 +0.003998,120 +0.004253,120 +0.004004,120 +0.003992,120 +0.003978,120 +0.004298,122 +0.004239,122 +0.004193,122 +0.004273,122 +0.004511,122 +0.004250,122 +0.004320,122 +0.004322,122 +0.004460,122 +0.004300,122 +0.004190,122 +0.004215,122 +0.004357,122 +0.004246,122 +0.004205,122 +0.004235,122 +0.004389,122 +0.004201,122 +0.004194,122 +0.004306,122 +0.004595,122 +0.004326,122 +0.004256,122 +0.004449,122 +0.004713,122 +0.004423,122 +0.004446,122 +0.006128,122 +0.004411,122 +0.004457,122 +0.004512,122 +0.004526,122 +0.004402,122 +0.004343,122 +0.004366,122 +0.004230,122 +0.004355,122 +0.004229,122 +0.004364,122 +0.004227,122 +0.004440,122 +0.004216,122 +0.004329,122 +0.004527,122 +0.004341,122 +0.004345,122 +0.004381,122 +0.004347,122 +0.004344,122 +0.004340,122 +0.004348,122 +0.004345,122 +0.004357,122 +0.004451,122 +0.004466,122 +0.004447,122 +0.004343,122 +0.004389,122 +0.004335,122 +0.004316,122 +0.004315,122 +0.004398,122 +0.004329,122 +0.004331,122 +0.004306,122 +0.004485,122 +0.004364,122 +0.004366,122 +0.004397,122 +0.004434,122 +0.004343,122 +0.004308,122 +0.004346,122 +0.004389,122 +0.004326,122 +0.004366,122 +0.004459,122 +0.004491,122 +0.004347,122 +0.004341,122 +0.004427,122 +0.004351,122 +0.004329,122 +0.004334,122 +0.004406,122 +0.004350,122 +0.004357,122 +0.004361,122 +0.004495,122 +0.004339,122 +0.004342,122 +0.004354,122 +0.004483,122 +0.004325,122 +0.004302,122 +0.004377,122 +0.004309,122 +0.004324,122 +0.004372,122 +0.004457,122 +0.004703,124 +0.004598,124 +0.004443,124 +0.004476,124 +0.004466,124 +0.004408,124 +0.004476,124 +0.004529,124 +0.004481,124 +0.004412,124 +0.004553,124 +0.004512,124 +0.004408,124 +0.004450,124 +0.004573,124 +0.004461,124 +0.007765,124 +0.005173,124 +0.004744,124 +0.004438,124 +0.004704,124 +0.004726,124 +0.004482,124 +0.004423,124 +0.004644,124 +0.004437,124 +0.004417,124 +0.004424,124 +0.004531,124 +0.004459,124 +0.004431,124 +0.004436,124 +0.004615,124 +0.004449,124 +0.004406,124 +0.004666,124 +0.004435,124 +0.004412,124 +0.004413,124 +0.004490,124 +0.004397,124 +0.004397,124 +0.004487,124 +0.004666,124 +0.004561,124 +0.004433,124 +0.004493,124 +0.004520,124 +0.004441,124 +0.004409,124 +0.004517,124 +0.004419,124 +0.004487,124 +0.004420,124 +0.004562,124 +0.004588,124 +0.004433,124 +0.004631,124 +0.004484,124 +0.004458,124 +0.004426,124 +0.004541,124 +0.004425,124 +0.004432,124 +0.004439,124 +0.004638,124 +0.004596,124 +0.004526,124 +0.004487,124 +0.004598,124 +0.004609,124 +0.004424,124 +0.004537,124 +0.004436,124 +0.004412,124 +0.004425,124 +0.004503,124 +0.004560,124 +0.004420,124 +0.004558,124 +0.004533,124 +0.004439,124 +0.004428,124 +0.004719,124 +0.004512,124 +0.004412,124 +0.004443,124 +0.004649,124 +0.004616,124 +0.004498,124 +0.004453,124 +0.004595,124 +0.004456,124 +0.004447,124 +0.004518,124 +0.004489,124 +0.004441,124 +0.004410,124 +0.004513,124 +0.004580,124 +0.004662,126 +0.004719,126 +0.004843,126 +0.004653,126 +0.004644,126 +0.004688,126 +0.004709,126 +0.004654,126 +0.004646,126 +0.004866,126 +0.004883,126 +0.004728,126 +0.004737,126 +0.004719,126 +0.004731,126 +0.004628,126 +0.004735,126 +0.004633,126 +0.004650,126 +0.004696,126 +0.004805,126 +0.004666,126 +0.004765,126 +0.004727,126 +0.004697,126 +0.004747,126 +0.004852,126 +0.004697,126 +0.004709,126 +0.004681,126 +0.004790,126 +0.004838,126 +0.004740,126 +0.004898,126 +0.004784,126 +0.004648,126 +0.004651,126 +0.004750,126 +0.004659,126 +0.004632,126 +0.004671,126 +0.004750,126 +0.004914,126 +0.004887,126 +0.004847,126 +0.004825,126 +0.004675,126 +0.004664,126 +0.004737,126 +0.004657,126 +0.004647,126 +0.004924,126 +0.004950,126 +0.004737,126 +0.004776,126 +0.004728,126 +0.004695,126 +0.004665,126 +0.004781,126 +0.004666,126 +0.004647,126 +0.004690,126 +0.004912,126 +0.004696,126 +0.004686,126 +0.004803,126 +0.004740,126 +0.004676,126 +0.004654,126 +0.004776,126 +0.004657,126 +0.004629,126 +0.004816,126 +0.005045,126 +0.004844,126 +0.004628,126 +0.004776,126 +0.004704,126 +0.004647,126 +0.004736,126 +0.004655,126 +0.004628,126 +0.004640,126 +0.004790,126 +0.004961,126 +0.004783,126 +0.004804,126 +0.004800,126 +0.004720,126 +0.004644,126 +0.004755,126 +0.004661,126 +0.004621,126 +0.004692,126 +0.005043,126 +0.004882,126 +0.004638,126 +0.004839,126 +0.004672,126 +0.004690,126 +0.005225,128 +0.005116,128 +0.005117,128 +0.005175,128 +0.005454,128 +0.005123,128 +0.005107,128 +0.005244,128 +0.005112,128 +0.005123,128 +0.005198,128 +0.005114,128 +0.005077,128 +0.005257,128 +0.005528,128 +0.005315,128 +0.005355,128 +0.005147,128 +0.005106,128 +0.005195,128 +0.005178,128 +0.005105,128 +0.005101,128 +0.005230,128 +0.005409,128 +0.005327,128 +0.005247,128 +0.005288,128 +0.005124,128 +0.005236,128 +0.005127,128 +0.005086,128 +0.005232,128 +0.005578,128 +0.005388,128 +0.005298,128 +0.005190,128 +0.005099,128 +0.005194,128 +0.005261,128 +0.005117,128 +0.005080,128 +0.005250,128 +0.005435,128 +0.005136,128 +0.005334,128 +0.005132,128 +0.005104,128 +0.005230,128 +0.005121,128 +0.005105,128 +0.005130,128 +0.005415,128 +0.005546,128 +0.005168,128 +0.005223,128 +0.005133,128 +0.005122,128 +0.005209,128 +0.005182,128 +0.005085,128 +0.005205,128 +0.005457,128 +0.005141,128 +0.005219,128 +0.005170,128 +0.005112,128 +0.005161,128 +0.005167,128 +0.005112,128 +0.005090,128 +0.005313,128 +0.005500,128 +0.005229,128 +0.005250,128 +0.005178,128 +0.005094,128 +0.005220,128 +0.005137,128 +0.005098,128 +0.005203,128 +0.005323,128 +0.005221,128 +0.005148,128 +0.005209,128 +0.005150,128 +0.005138,128 +0.005215,128 +0.005138,128 +0.005098,128 +0.005285,128 +0.005508,128 +0.005329,128 +0.005309,128 +0.005242,128 +0.005208,128 +0.005242,128 +0.005098,128 +0.005136,128 +0.005154,128 +0.005345,130 +0.005109,130 +0.005101,130 +0.005213,130 +0.005168,130 +0.005101,130 +0.005221,130 +0.005093,130 +0.005125,130 +0.005204,130 +0.005441,130 +0.005309,130 +0.005259,130 +0.005137,130 +0.005116,130 +0.005168,130 +0.005155,130 +0.005108,130 +0.005112,130 +0.005368,130 +0.005092,130 +0.005176,130 +0.005233,130 +0.005154,130 +0.005104,130 +0.005250,130 +0.005092,130 +0.005104,130 +0.005175,130 +0.005286,130 +0.005459,130 +0.005212,130 +0.005176,130 +0.005118,130 +0.005108,130 +0.005281,130 +0.005082,130 +0.005103,130 +0.005340,130 +0.005336,130 +0.005093,130 +0.005294,130 +0.005209,130 +0.005164,130 +0.005152,130 +0.005189,130 +0.005329,130 +0.005140,130 +0.005263,130 +0.005501,130 +0.005242,130 +0.005252,130 +0.005155,130 +0.005076,130 +0.005336,130 +0.005101,130 +0.005096,130 +0.005204,130 +0.005323,130 +0.005099,130 +0.005217,130 +0.005142,130 +0.005118,130 +0.005215,130 +0.005208,130 +0.005089,130 +0.005125,130 +0.005244,130 +0.005430,130 +0.005506,130 +0.005299,130 +0.005211,130 +0.005093,130 +0.005205,130 +0.005145,130 +0.005165,130 +0.005192,130 +0.005181,130 +0.005061,130 +0.005089,130 +0.005094,130 +0.005067,130 +0.005056,130 +0.005117,130 +0.005075,130 +0.005056,130 +0.005119,130 +0.005140,130 +0.005332,130 +0.005271,130 +0.005136,130 +0.005077,130 +0.005105,130 +0.005112,130 +0.005063,130 +0.005056,130 +0.005156,130 +0.005151,130 +0.005054,130 +0.005122,130 +0.005310,132 +0.005291,132 +0.005352,132 +0.005298,132 +0.005291,132 +0.005321,132 +0.005322,132 +0.005535,132 +0.005626,132 +0.005791,132 +0.005476,132 +0.005464,132 +0.005361,132 +0.005298,132 +0.005315,132 +0.005324,132 +0.005383,132 +0.005287,132 +0.005357,132 +0.005293,132 +0.005288,132 +0.005351,132 +0.005338,132 +0.005285,132 +0.005348,132 +0.005316,132 +0.005635,132 +0.005557,132 +0.005343,132 +0.005305,132 +0.005344,132 +0.005322,132 +0.005313,132 +0.005343,132 +0.005423,132 +0.005314,132 +0.005394,132 +0.005388,132 +0.005292,132 +0.005315,132 +0.005354,132 +0.005294,132 +0.005286,132 +0.005360,132 +0.005639,132 +0.005569,132 +0.006009,132 +0.005312,132 +0.005299,132 +0.005374,132 +0.005315,132 +0.005288,132 +0.005413,132 +0.005389,132 +0.005398,132 +0.005396,132 +0.005329,132 +0.005370,132 +0.005364,132 +0.005331,132 +0.005289,132 +0.005350,132 +0.005295,132 +0.005458,132 +0.005831,132 +0.005349,132 +0.005310,132 +0.005362,132 +0.005319,132 +0.005291,132 +0.005315,132 +0.005331,132 +0.005644,132 +0.005542,132 +0.005394,132 +0.005322,132 +0.005315,132 +0.005354,132 +0.005291,132 +0.005286,132 +0.005589,132 +0.005461,132 +0.005522,132 +0.005612,132 +0.005323,132 +0.005350,132 +0.005379,132 +0.005328,132 +0.005358,132 +0.005373,132 +0.005383,132 +0.005312,132 +0.005371,132 +0.005300,132 +0.005297,132 +0.005343,132 +0.005318,132 +0.005354,132 +0.005344,132 +0.005297,132 +0.005727,134 +0.006001,134 +0.005604,134 +0.005544,134 +0.005591,134 +0.005575,134 +0.005543,134 +0.005602,134 +0.005628,134 +0.005538,134 +0.005601,134 +0.005582,134 +0.005604,134 +0.005593,134 +0.005555,134 +0.005542,134 +0.005612,134 +0.005551,134 +0.005696,134 +0.005928,134 +0.005598,134 +0.005536,134 +0.005602,134 +0.005548,134 +0.005540,134 +0.005598,134 +0.005626,134 +0.005551,134 +0.005602,134 +0.005581,134 +0.005547,134 +0.005591,134 +0.005553,134 +0.005532,134 +0.005735,134 +0.006111,134 +0.006117,134 +0.006362,134 +0.006082,134 +0.006000,134 +0.006281,134 +0.006361,134 +0.006285,134 +0.006281,134 +0.005964,134 +0.006317,134 +0.005929,134 +0.006001,134 +0.006070,134 +0.006045,134 +0.006063,134 +0.006067,134 +0.006021,134 +0.006119,134 +0.006383,134 +0.006022,134 +0.006130,134 +0.006054,134 +0.006025,134 +0.006338,134 +0.006325,134 +0.006297,134 +0.006315,134 +0.005940,134 +0.006068,134 +0.006203,134 +0.006115,134 +0.006111,134 +0.006095,134 +0.006063,134 +0.006335,134 +0.006279,134 +0.006346,134 +0.006127,134 +0.006091,134 +0.006185,134 +0.006060,134 +0.005912,134 +0.006329,134 +0.006379,134 +0.006502,134 +0.006095,134 +0.005933,134 +0.006228,134 +0.005815,134 +0.005759,134 +0.005961,134 +0.005614,134 +0.005607,134 +0.005765,134 +0.005981,134 +0.005591,134 +0.005644,134 +0.005614,134 +0.005600,134 +0.005722,134 +0.005538,134 +0.005589,134 +0.005558,134 +0.005544,134 +0.005834,136 +0.005784,136 +0.005778,136 +0.006098,136 +0.006070,136 +0.005775,136 +0.005948,136 +0.006690,136 +0.005806,136 +0.006403,136 +0.006005,136 +0.006340,136 +0.006973,136 +0.006158,136 +0.006490,136 +0.006354,136 +0.005946,136 +0.006281,136 +0.006280,136 +0.006033,136 +0.006958,136 +0.005882,136 +0.006158,136 +0.006321,136 +0.005967,136 +0.006395,136 +0.006145,136 +0.005931,136 +0.006476,136 +0.006284,136 +0.006077,136 +0.006248,136 +0.006011,136 +0.006106,136 +0.006188,136 +0.005986,136 +0.006076,136 +0.006000,136 +0.005937,136 +0.005843,136 +0.005895,136 +0.005877,136 +0.005874,136 +0.005772,136 +0.005883,136 +0.005784,136 +0.005777,136 +0.005908,136 +0.005777,136 +0.005814,136 +0.005822,136 +0.005816,136 +0.005897,136 +0.006165,136 +0.005819,136 +0.005870,136 +0.005808,136 +0.005839,136 +0.005863,136 +0.005844,136 +0.005773,136 +0.005864,136 +0.005783,136 +0.005776,136 +0.005863,136 +0.005780,136 +0.005784,136 +0.005877,136 +0.005782,136 +0.005913,136 +0.006143,136 +0.005873,136 +0.005858,136 +0.005805,136 +0.005825,136 +0.005900,136 +0.005794,136 +0.005851,136 +0.005868,136 +0.005780,136 +0.005806,136 +0.005853,136 +0.005777,136 +0.005772,136 +0.005847,136 +0.005780,136 +0.005777,136 +0.006138,136 +0.005983,136 +0.005805,136 +0.005818,136 +0.005781,136 +0.005883,136 +0.005782,136 +0.005879,136 +0.005866,136 +0.005825,136 +0.005798,136 +0.005845,136 +0.005789,136 +0.006144,138 +0.006206,138 +0.006050,138 +0.006082,138 +0.006384,138 +0.006272,138 +0.006127,138 +0.006058,138 +0.006054,138 +0.006137,138 +0.006182,138 +0.006050,138 +0.006651,138 +0.006438,138 +0.006146,138 +0.006533,138 +0.006085,138 +0.006288,138 +0.006086,138 +0.006083,138 +0.006998,138 +0.006314,138 +0.006406,138 +0.006107,138 +0.006110,138 +0.006393,138 +0.006200,138 +0.006092,138 +0.006353,138 +0.006054,138 +0.006401,138 +0.006207,138 +0.006265,138 +0.006150,138 +0.006073,138 +0.006086,138 +0.006353,138 +0.006384,138 +0.006190,138 +0.006075,138 +0.006109,138 +0.006160,138 +0.006171,138 +0.006062,138 +0.006181,138 +0.006079,138 +0.006195,138 +0.006082,138 +0.006056,138 +0.006119,138 +0.006057,138 +0.006043,138 +0.006311,138 +0.006433,138 +0.006132,138 +0.006261,138 +0.006091,138 +0.006167,138 +0.006054,138 +0.006126,138 +0.006143,138 +0.006052,138 +0.006045,138 +0.006127,138 +0.006047,138 +0.006085,138 +0.006126,138 +0.006050,138 +0.006128,138 +0.006386,138 +0.006225,138 +0.006126,138 +0.006080,138 +0.006061,138 +0.006171,138 +0.006165,138 +0.006168,138 +0.006123,138 +0.006081,138 +0.006128,138 +0.006048,138 +0.006047,138 +0.006126,138 +0.006076,138 +0.006082,138 +0.006268,138 +0.006333,138 +0.006179,138 +0.006075,138 +0.006054,138 +0.006144,138 +0.006149,138 +0.006059,138 +0.006157,138 +0.006082,138 +0.006094,138 +0.006091,138 +0.006047,138 +0.006129,138 +0.006050,138 +0.006337,140 +0.006400,140 +0.006574,140 +0.006676,140 +0.006315,140 +0.006319,140 +0.006404,140 +0.006336,140 +0.006425,140 +0.006379,140 +0.006310,140 +0.006384,140 +0.006309,140 +0.006344,140 +0.006387,140 +0.006314,140 +0.006349,140 +0.006525,140 +0.006626,140 +0.006451,140 +0.006317,140 +0.006307,140 +0.006405,140 +0.006402,140 +0.006386,140 +0.006310,140 +0.006303,140 +0.006385,140 +0.006305,140 +0.006345,140 +0.006347,140 +0.006308,140 +0.006397,140 +0.006469,140 +0.006585,140 +0.006423,140 +0.006313,140 +0.006412,140 +0.006308,140 +0.006395,140 +0.006382,140 +0.006307,140 +0.006343,140 +0.006368,140 +0.006309,140 +0.006386,140 +0.006307,140 +0.006304,140 +0.006382,140 +0.006535,140 +0.006571,140 +0.006345,140 +0.006319,140 +0.006407,140 +0.006317,140 +0.006471,140 +0.006446,140 +0.006339,140 +0.006411,140 +0.006311,140 +0.006303,140 +0.006465,140 +0.006307,140 +0.006342,140 +0.006347,140 +0.006552,140 +0.006571,140 +0.006321,140 +0.006361,140 +0.006429,140 +0.006384,140 +0.006402,140 +0.006311,140 +0.006303,140 +0.006386,140 +0.006306,140 +0.006354,140 +0.006357,140 +0.006309,140 +0.006384,140 +0.006307,140 +0.006558,140 +0.006513,140 +0.006311,140 +0.006354,140 +0.006346,140 +0.006407,140 +0.006385,140 +0.006308,140 +0.006305,140 +0.006381,140 +0.006309,140 +0.006383,140 +0.006312,140 +0.006313,140 +0.006383,140 +0.006480,140 +0.006622,140 +0.006399,140 +0.006320,140 +0.006685,142 +0.006591,142 +0.006702,142 +0.006625,142 +0.006589,142 +0.006665,142 +0.006586,142 +0.006584,142 +0.006666,142 +0.006587,142 +0.006679,142 +0.006590,142 +0.006811,142 +0.006806,142 +0.006596,142 +0.006669,142 +0.006599,142 +0.006663,142 +0.006728,142 +0.006585,142 +0.007004,142 +0.007236,142 +0.007073,142 +0.007474,142 +0.006630,142 +0.006928,142 +0.006732,142 +0.007071,142 +0.006601,142 +0.006587,142 +0.006673,142 +0.006685,142 +0.006709,142 +0.006656,142 +0.006584,142 +0.006663,142 +0.006588,142 +0.006665,142 +0.006650,142 +0.006594,142 +0.006675,142 +0.006671,142 +0.007029,142 +0.006622,142 +0.006624,142 +0.006750,142 +0.007873,142 +0.006733,142 +0.006849,142 +0.006880,142 +0.006839,142 +0.006767,142 +0.006828,142 +0.006757,142 +0.006806,142 +0.006630,142 +0.006778,142 +0.007067,142 +0.006704,142 +0.006746,142 +0.006644,142 +0.006692,142 +0.006747,142 +0.006707,142 +0.006716,142 +0.006660,142 +0.006632,142 +0.006743,142 +0.007173,142 +0.006932,142 +0.006678,142 +0.006988,142 +0.006903,142 +0.006596,142 +0.006914,142 +0.006590,142 +0.007036,142 +0.006830,142 +0.006713,142 +0.006756,142 +0.006656,142 +0.006650,142 +0.006702,142 +0.006673,142 +0.006853,142 +0.006643,142 +0.006941,142 +0.006973,142 +0.006675,142 +0.006820,142 +0.006726,142 +0.006739,142 +0.006694,142 +0.006696,142 +0.006955,142 +0.006669,142 +0.006780,142 +0.006669,142 +0.006774,142 +0.006784,142 +0.006946,144 +0.007180,144 +0.007175,144 +0.006981,144 +0.007011,144 +0.006981,144 +0.007058,144 +0.006979,144 +0.007020,144 +0.006891,144 +0.006974,144 +0.007013,144 +0.007064,144 +0.007010,144 +0.006935,144 +0.007171,144 +0.007264,144 +0.006881,144 +0.007101,144 +0.007048,144 +0.007135,144 +0.006928,144 +0.006969,144 +0.006954,144 +0.006935,144 +0.007018,144 +0.007033,144 +0.007058,144 +0.006911,144 +0.006959,144 +0.007164,144 +0.007034,144 +0.007004,144 +0.006925,144 +0.007048,144 +0.006911,144 +0.006925,144 +0.006982,144 +0.007009,144 +0.006980,144 +0.007044,144 +0.006942,144 +0.007042,144 +0.007044,144 +0.007140,144 +0.007128,144 +0.006992,144 +0.006979,144 +0.006992,144 +0.007026,144 +0.006894,144 +0.006992,144 +0.006908,144 +0.006913,144 +0.007092,144 +0.006903,144 +0.007016,144 +0.006961,144 +0.007095,144 +0.007097,144 +0.007115,144 +0.007031,144 +0.006934,144 +0.007039,144 +0.006954,144 +0.007028,144 +0.006887,144 +0.006999,144 +0.007085,144 +0.006888,144 +0.007021,144 +0.006992,144 +0.007038,144 +0.007296,144 +0.007040,144 +0.007109,144 +0.006944,144 +0.007045,144 +0.006961,144 +0.006940,144 +0.006953,144 +0.006940,144 +0.007063,144 +0.006913,144 +0.007011,144 +0.006932,144 +0.006881,144 +0.007184,144 +0.007128,144 +0.007072,144 +0.006937,144 +0.007065,144 +0.006980,144 +0.006957,144 +0.007054,144 +0.006988,144 +0.007102,144 +0.006926,144 +0.006940,144 +0.007007,144 +0.007310,146 +0.007503,146 +0.007518,146 +0.007639,146 +0.007285,146 +0.007494,146 +0.007291,146 +0.007312,146 +0.007408,146 +0.007292,146 +0.007390,146 +0.007326,146 +0.007384,146 +0.007387,146 +0.007313,146 +0.007532,146 +0.007562,146 +0.007357,146 +0.007306,146 +0.007373,146 +0.007276,146 +0.007394,146 +0.007270,146 +0.007294,146 +0.007332,146 +0.007257,146 +0.007375,146 +0.007292,146 +0.007418,146 +0.007439,146 +0.007513,146 +0.007323,146 +0.007384,146 +0.007416,146 +0.007336,146 +0.007373,146 +0.007230,146 +0.007855,146 +0.007391,146 +0.007525,146 +0.007321,146 +0.007766,146 +0.007705,146 +0.008125,146 +0.007839,146 +0.008005,146 +0.008066,146 +0.007611,146 +0.007603,146 +0.007924,146 +0.007917,146 +0.008697,146 +0.007902,146 +0.008156,146 +0.007777,146 +0.007991,146 +0.007792,146 +0.007690,146 +0.008070,146 +0.007532,146 +0.007425,146 +0.007711,146 +0.007342,146 +0.007745,146 +0.007388,146 +0.007762,146 +0.007380,146 +0.007572,146 +0.008872,146 +0.007442,146 +0.007244,146 +0.007486,146 +0.007478,146 +0.007317,146 +0.007404,146 +0.007255,146 +0.007393,146 +0.007261,146 +0.007373,146 +0.007560,146 +0.007425,146 +0.008690,146 +0.007781,146 +0.007627,146 +0.007529,146 +0.007442,146 +0.007458,146 +0.007404,146 +0.007270,146 +0.007425,146 +0.007276,146 +0.007546,146 +0.007440,146 +0.007283,146 +0.007965,146 +0.008321,146 +0.007419,146 +0.007291,146 +0.007481,146 +0.007402,146 +0.007778,148 +0.007615,148 +0.007758,148 +0.007670,148 +0.007914,148 +0.007726,148 +0.007688,148 +0.008380,148 +0.008756,148 +0.007832,148 +0.007708,148 +0.007808,148 +0.007701,148 +0.007684,148 +0.007601,148 +0.007729,148 +0.007646,148 +0.010019,148 +0.010566,148 +0.008456,148 +0.008756,148 +0.008002,148 +0.007720,148 +0.008182,148 +0.007676,148 +0.007769,148 +0.007617,148 +0.007762,148 +0.007616,148 +0.007896,148 +0.007661,148 +0.008434,148 +0.007774,148 +0.007898,148 +0.007938,148 +0.007794,148 +0.007770,148 +0.007682,148 +0.007740,148 +0.007658,148 +0.007811,148 +0.007639,148 +0.007765,148 +0.008333,148 +0.007766,148 +0.007792,148 +0.008049,148 +0.007656,148 +0.007810,148 +0.007844,148 +0.007735,148 +0.007690,148 +0.007637,148 +0.007724,148 +0.007651,148 +0.007805,148 +0.007719,148 +0.007757,148 +0.007824,148 +0.008019,148 +0.007648,148 +0.007793,148 +0.007652,148 +0.007700,148 +0.007683,148 +0.007605,148 +0.007735,148 +0.007639,148 +0.007750,148 +0.007703,148 +0.007735,148 +0.007821,148 +0.008072,148 +0.007652,148 +0.007801,148 +0.007742,148 +0.007765,148 +0.007639,148 +0.008044,148 +0.008433,148 +0.007989,148 +0.008329,148 +0.008245,148 +0.008221,148 +0.008322,148 +0.008063,148 +0.007857,148 +0.007852,148 +0.007735,148 +0.008152,148 +0.008043,148 +0.008220,148 +0.007921,148 +0.008088,148 +0.008168,148 +0.008056,148 +0.008000,148 +0.008362,148 +0.007909,148 +0.009064,148 +0.008346,150 +0.008128,150 +0.007978,150 +0.007981,150 +0.007997,150 +0.008024,150 +0.008138,150 +0.008056,150 +0.008012,150 +0.008554,150 +0.007982,150 +0.008079,150 +0.008173,150 +0.008247,150 +0.007963,150 +0.007973,150 +0.007973,150 +0.007978,150 +0.007963,150 +0.007982,150 +0.007996,150 +0.008331,150 +0.008317,150 +0.008052,150 +0.008038,150 +0.007984,150 +0.008001,150 +0.008107,150 +0.007991,150 +0.007992,150 +0.008025,150 +0.007979,150 +0.008018,150 +0.008062,150 +0.008459,150 +0.008003,150 +0.008107,150 +0.008063,150 +0.007981,150 +0.007958,150 +0.008063,150 +0.007984,150 +0.008013,150 +0.007974,150 +0.008009,150 +0.008056,150 +0.008926,150 +0.008235,150 +0.008700,150 +0.008811,150 +0.008310,150 +0.008023,150 +0.008036,150 +0.007984,150 +0.008003,150 +0.008007,150 +0.008060,150 +0.007973,150 +0.008231,150 +0.008441,150 +0.008058,150 +0.008011,150 +0.007978,150 +0.008027,150 +0.007937,150 +0.008053,150 +0.007993,150 +0.008003,150 +0.008118,150 +0.008002,150 +0.007976,150 +0.008652,150 +0.008004,150 +0.008149,150 +0.008675,150 +0.008305,150 +0.008023,150 +0.008031,150 +0.007990,150 +0.007980,150 +0.007962,150 +0.008021,150 +0.007980,150 +0.008286,150 +0.008263,150 +0.008098,150 +0.008096,150 +0.008071,150 +0.007968,150 +0.007976,150 +0.007980,150 +0.008035,150 +0.007977,150 +0.008001,150 +0.007974,150 +0.008073,150 +0.008516,150 +0.007981,150 +0.008411,150 +0.008241,150 +0.008650,152 +0.008546,152 +0.008398,152 +0.008301,152 +0.008325,152 +0.008319,152 +0.008371,152 +0.008372,152 +0.008826,152 +0.008335,152 +0.008408,152 +0.008364,152 +0.008326,152 +0.008344,152 +0.008336,152 +0.008302,152 +0.008316,152 +0.008305,152 +0.008350,152 +0.008368,152 +0.008856,152 +0.008355,152 +0.008551,152 +0.008376,152 +0.008291,152 +0.008303,152 +0.008329,152 +0.008321,152 +0.008396,152 +0.008476,152 +0.008333,152 +0.008421,152 +0.009212,152 +0.008451,152 +0.008314,152 +0.008375,152 +0.008240,152 +0.008306,152 +0.008367,152 +0.008485,152 +0.008341,152 +0.008528,152 +0.008268,152 +0.008497,152 +0.008824,152 +0.008429,152 +0.008309,152 +0.008354,152 +0.008250,152 +0.008309,152 +0.008308,152 +0.008352,152 +0.008268,152 +0.008575,152 +0.008266,152 +0.008708,152 +0.008738,152 +0.008758,152 +0.008415,152 +0.008447,152 +0.008256,152 +0.008588,152 +0.008314,152 +0.008504,152 +0.008304,152 +0.008422,152 +0.008275,152 +0.008529,152 +0.008792,152 +0.008401,152 +0.008369,152 +0.008584,152 +0.008314,152 +0.008303,152 +0.008326,152 +0.008382,152 +0.008319,152 +0.008334,152 +0.008260,152 +0.008533,152 +0.008883,152 +0.008571,152 +0.008472,152 +0.008538,152 +0.008479,152 +0.008707,152 +0.008450,152 +0.008625,152 +0.008442,152 +0.008723,152 +0.009167,152 +0.009004,152 +0.009271,152 +0.009005,152 +0.010127,152 +0.010188,152 +0.011063,152 +0.009559,152 +0.009838,152 +0.009961,152 +0.009792,154 +0.009499,154 +0.009653,154 +0.009494,154 +0.009772,154 +0.009646,154 +0.009619,154 +0.009178,154 +0.009863,154 +0.009757,154 +0.009511,154 +0.009677,154 +0.009544,154 +0.009983,154 +0.009500,154 +0.009243,154 +0.009506,154 +0.009279,154 +0.009287,154 +0.009049,154 +0.008765,154 +0.008746,154 +0.008860,154 +0.008931,154 +0.008970,154 +0.008748,154 +0.008955,154 +0.009060,154 +0.009114,154 +0.009070,154 +0.009268,154 +0.009261,154 +0.009288,154 +0.009213,154 +0.009292,154 +0.009037,154 +0.009542,154 +0.009994,154 +0.009146,154 +0.009390,154 +0.009094,154 +0.008783,154 +0.008926,154 +0.008703,154 +0.008679,154 +0.009198,154 +0.008700,154 +0.008618,154 +0.008616,154 +0.008538,154 +0.008603,154 +0.008568,154 +0.008587,154 +0.008565,154 +0.008594,154 +0.008616,154 +0.008779,154 +0.008935,154 +0.008574,154 +0.008673,154 +0.008572,154 +0.008687,154 +0.008556,154 +0.008651,154 +0.008531,154 +0.008601,154 +0.008538,154 +0.008599,154 +0.009100,154 +0.008690,154 +0.008595,154 +0.008651,154 +0.008587,154 +0.008598,154 +0.008538,154 +0.008612,154 +0.008553,154 +0.008637,154 +0.008539,154 +0.008770,154 +0.010257,154 +0.008723,154 +0.008869,154 +0.008579,154 +0.008621,154 +0.008552,154 +0.008607,154 +0.008537,154 +0.008603,154 +0.008534,154 +0.008603,154 +0.008690,154 +0.009026,154 +0.008675,154 +0.009593,154 +0.008742,154 +0.008727,154 +0.008825,154 +0.008665,154 +0.008692,154 +0.008984,156 +0.008931,156 +0.011467,156 +0.011223,156 +0.009128,156 +0.008904,156 +0.008958,156 +0.008882,156 +0.008937,156 +0.009507,156 +0.010447,156 +0.009244,156 +0.009419,156 +0.009403,156 +0.009228,156 +0.009477,156 +0.009392,156 +0.009371,156 +0.009207,156 +0.009178,156 +0.009348,156 +0.009250,156 +0.009315,156 +0.009246,156 +0.009537,156 +0.009041,156 +0.009492,156 +0.009271,156 +0.009272,156 +0.009327,156 +0.009955,156 +0.011658,156 +0.009964,156 +0.011695,156 +0.012714,156 +0.011539,156 +0.010104,156 +0.011475,156 +0.009902,156 +0.009919,156 +0.011891,156 +0.011584,156 +0.010316,156 +0.010424,156 +0.009976,156 +0.011155,156 +0.009782,156 +0.010686,156 +0.013995,156 +0.011084,156 +0.009411,156 +0.010324,156 +0.015872,156 +0.011228,156 +0.009864,156 +0.010520,156 +0.009239,156 +0.010670,156 +0.009977,156 +0.009525,156 +0.009844,156 +0.009151,156 +0.009747,156 +0.009790,156 +0.009815,156 +0.010218,156 +0.010520,156 +0.012190,156 +0.015521,156 +0.009509,156 +0.009140,156 +0.012884,156 +0.009942,156 +0.009681,156 +0.009923,156 +0.009139,156 +0.009531,156 +0.008975,156 +0.009821,156 +0.009291,156 +0.010210,156 +0.009505,156 +0.010075,156 +0.009369,156 +0.008945,156 +0.010025,156 +0.009424,156 +0.009071,156 +0.009673,156 +0.009162,156 +0.009764,156 +0.009041,156 +0.011949,156 +0.009617,156 +0.009074,156 +0.009598,156 +0.009052,156 +0.009570,156 +0.009098,156 +0.010041,156 +0.009708,158 +0.010064,158 +0.009928,158 +0.009806,158 +0.010372,158 +0.009740,158 +0.010422,158 +0.009990,158 +0.012470,158 +0.011446,158 +0.011016,158 +0.013338,158 +0.014195,158 +0.011668,158 +0.011490,158 +0.012130,158 +0.019081,158 +0.015487,158 +0.009681,158 +0.012794,158 +0.010850,158 +0.011093,158 +0.010817,158 +0.011067,158 +0.010525,158 +0.012347,158 +0.013523,158 +0.015113,158 +0.011413,158 +0.011253,158 +0.010759,158 +0.015331,158 +0.016442,158 +0.015369,158 +0.018370,158 +0.018265,158 +0.011872,158 +0.009907,158 +0.011238,158 +0.010150,158 +0.010105,158 +0.010182,158 +0.010156,158 +0.010646,158 +0.010022,158 +0.011163,158 +0.011613,158 +0.012204,158 +0.010184,158 +0.010941,158 +0.009729,158 +0.010241,158 +0.011784,158 +0.012669,158 +0.010926,158 +0.010744,158 +0.009868,158 +0.010450,158 +0.010083,158 +0.009632,158 +0.011187,158 +0.010271,158 +0.010291,158 +0.010722,158 +0.010145,158 +0.010388,158 +0.010175,158 +0.009634,158 +0.010690,158 +0.010194,158 +0.010933,158 +0.010413,158 +0.010111,158 +0.009954,158 +0.009935,158 +0.009513,158 +0.010208,158 +0.009461,158 +0.010033,158 +0.009768,158 +0.009675,158 +0.009992,158 +0.015378,158 +0.009805,158 +0.009962,158 +0.009615,158 +0.009814,158 +0.009782,158 +0.009562,158 +0.009641,158 +0.009396,158 +0.009651,158 +0.009703,158 +0.009545,158 +0.009404,158 +0.009308,158 +0.009399,158 +0.009599,158 +0.009542,158 +0.009624,158 +0.009764,160 +0.009766,160 +0.010419,160 +0.009815,160 +0.009659,160 +0.009766,160 +0.009896,160 +0.009800,160 +0.009621,160 +0.009634,160 +0.009769,160 +0.009709,160 +0.010182,160 +0.010103,160 +0.009729,160 +0.016327,160 +0.018110,160 +0.009952,160 +0.009734,160 +0.009748,160 +0.009848,160 +0.010281,160 +0.009908,160 +0.009824,160 +0.009719,160 +0.009832,160 +0.009633,160 +0.009759,160 +0.009556,160 +0.009787,160 +0.009659,160 +0.010176,160 +0.010140,160 +0.009638,160 +0.010102,160 +0.009776,160 +0.009687,160 +0.009705,160 +0.009597,160 +0.009634,160 +0.009557,160 +0.009941,160 +0.015949,160 +0.018097,160 +0.018372,160 +0.017274,160 +0.017383,160 +0.017845,160 +0.017787,160 +0.017123,160 +0.009852,160 +0.009752,160 +0.009764,160 +0.009808,160 +0.009696,160 +0.009943,160 +0.010198,160 +0.009885,160 +0.009826,160 +0.009636,160 +0.009895,160 +0.009603,160 +0.012460,160 +0.009786,160 +0.009703,160 +0.009974,160 +0.010484,160 +0.009772,160 +0.009750,160 +0.009698,160 +0.009840,160 +0.009727,160 +0.009810,160 +0.009745,160 +0.009768,160 +0.009757,160 +0.012811,160 +0.009725,160 +0.009802,160 +0.009732,160 +0.009781,160 +0.009819,160 +0.009776,160 +0.009768,160 +0.009759,160 +0.009821,160 +0.011373,160 +0.009787,160 +0.009738,160 +0.009744,160 +0.009732,160 +0.009770,160 +0.009758,160 +0.009691,160 +0.009695,160 +0.009737,160 +0.011340,160 +0.009818,160 +0.009742,160 +0.009665,160 +0.010148,162 +0.010251,162 +0.010111,162 +0.010181,162 +0.010023,162 +0.010160,162 +0.011832,162 +0.010147,162 +0.010344,162 +0.009981,162 +0.010106,162 +0.010133,162 +0.010074,162 +0.010133,162 +0.010118,162 +0.010709,162 +0.011354,162 +0.010097,162 +0.010123,162 +0.010064,162 +0.010051,162 +0.010191,162 +0.009993,162 +0.010145,162 +0.010079,162 +0.011254,162 +0.010817,162 +0.010116,162 +0.010152,162 +0.010225,162 +0.010111,162 +0.010227,162 +0.010041,162 +0.010084,162 +0.010101,162 +0.011673,162 +0.010321,162 +0.010229,162 +0.010453,162 +0.010441,162 +0.010382,162 +0.010084,162 +0.010164,162 +0.010015,162 +0.010080,162 +0.011790,162 +0.010053,162 +0.010139,162 +0.010015,162 +0.010101,162 +0.010225,162 +0.009966,162 +0.010188,162 +0.010130,162 +0.010811,162 +0.011149,162 +0.010074,162 +0.010151,162 +0.010192,162 +0.009983,162 +0.010172,162 +0.010086,162 +0.010313,162 +0.010065,162 +0.011288,162 +0.010595,162 +0.010108,162 +0.010117,162 +0.010081,162 +0.010025,162 +0.010051,162 +0.010062,162 +0.009960,162 +0.010127,162 +0.012434,162 +0.010552,162 +0.010108,162 +0.010048,162 +0.010067,162 +0.010161,162 +0.010022,162 +0.010066,162 +0.009953,162 +0.010138,162 +0.011683,162 +0.010006,162 +0.010122,162 +0.010041,162 +0.010038,162 +0.010052,162 +0.009980,162 +0.010051,162 +0.010021,162 +0.010381,162 +0.011370,162 +0.010045,162 +0.010075,162 +0.010103,162 +0.010161,162 +0.010138,162 +0.010413,164 +0.010478,164 +0.010468,164 +0.011700,164 +0.011035,164 +0.010514,164 +0.010441,164 +0.010456,164 +0.010483,164 +0.010397,164 +0.010585,164 +0.010424,164 +0.010495,164 +0.012097,164 +0.010512,164 +0.010464,164 +0.010491,164 +0.010400,164 +0.010487,164 +0.010544,164 +0.010423,164 +0.010437,164 +0.012053,164 +0.010544,164 +0.010522,164 +0.010433,164 +0.010460,164 +0.010495,164 +0.010396,164 +0.010457,164 +0.010480,164 +0.010995,164 +0.011689,164 +0.010565,164 +0.010388,164 +0.010465,164 +0.010401,164 +0.010423,164 +0.010470,164 +0.010378,164 +0.010484,164 +0.012143,164 +0.010831,164 +0.010480,164 +0.010462,164 +0.010407,164 +0.010435,164 +0.010439,164 +0.010430,164 +0.010510,164 +0.011330,164 +0.011362,164 +0.010723,164 +0.010432,164 +0.010464,164 +0.010741,164 +0.010533,164 +0.010585,164 +0.010826,164 +0.010397,164 +0.011904,164 +0.010379,164 +0.010397,164 +0.010446,164 +0.010295,164 +0.010390,164 +0.010363,164 +0.010311,164 +0.010359,164 +0.012203,164 +0.011248,164 +0.010458,164 +0.010732,164 +0.010783,164 +0.010587,164 +0.010392,164 +0.010336,164 +0.010346,164 +0.010453,164 +0.011919,164 +0.010390,164 +0.010326,164 +0.010369,164 +0.010723,164 +0.010331,164 +0.010576,164 +0.010334,164 +0.010446,164 +0.011779,164 +0.010543,164 +0.010437,164 +0.010399,164 +0.010404,164 +0.010340,164 +0.010328,164 +0.010594,164 +0.010360,164 +0.010354,164 +0.012024,164 +0.010405,164 +0.010744,166 +0.010745,166 +0.010723,166 +0.010712,166 +0.010866,166 +0.010727,166 +0.010759,166 +0.012288,166 +0.010952,166 +0.010841,166 +0.010858,166 +0.010729,166 +0.010668,166 +0.010714,166 +0.010732,166 +0.010707,166 +0.012446,166 +0.010702,166 +0.010806,166 +0.010861,166 +0.010752,166 +0.010744,166 +0.010715,166 +0.010670,166 +0.010739,166 +0.011991,166 +0.010997,166 +0.010792,166 +0.010768,166 +0.010704,166 +0.010760,166 +0.010717,166 +0.010701,166 +0.010783,166 +0.011691,166 +0.011293,166 +0.010739,166 +0.010756,166 +0.010708,166 +0.010774,166 +0.010693,166 +0.010799,166 +0.010777,166 +0.010820,166 +0.012218,166 +0.010780,166 +0.010711,166 +0.010703,166 +0.010750,166 +0.010693,166 +0.012657,166 +0.010827,166 +0.011021,166 +0.012073,166 +0.010826,166 +0.010750,166 +0.010721,166 +0.010735,166 +0.010689,166 +0.010798,166 +0.010740,166 +0.010746,166 +0.013313,166 +0.010825,166 +0.010754,166 +0.010796,166 +0.010818,166 +0.010668,166 +0.010738,166 +0.010805,166 +0.010845,166 +0.012153,166 +0.010795,166 +0.010743,166 +0.010758,166 +0.010742,166 +0.010670,166 +0.010774,166 +0.010884,166 +0.010697,166 +0.012186,166 +0.010813,166 +0.010782,166 +0.010716,166 +0.010772,166 +0.010932,166 +0.010732,166 +0.010799,166 +0.010905,166 +0.012334,166 +0.011287,166 +0.010728,166 +0.010781,166 +0.010719,166 +0.010703,166 +0.010891,166 +0.010675,166 +0.010746,166 +0.011915,166 +0.011197,166 +0.011188,168 +0.011176,168 +0.011134,168 +0.011091,168 +0.011122,168 +0.011091,168 +0.011333,168 +0.012676,168 +0.011323,168 +0.011132,168 +0.011158,168 +0.011182,168 +0.011040,168 +0.011103,168 +0.011119,168 +0.011032,168 +0.012494,168 +0.011247,168 +0.011090,168 +0.011127,168 +0.011116,168 +0.011033,168 +0.011132,168 +0.011111,168 +0.011323,168 +0.012560,168 +0.011198,168 +0.011052,168 +0.011112,168 +0.011170,168 +0.011056,168 +0.011182,168 +0.011106,168 +0.011096,168 +0.012608,168 +0.011202,168 +0.011168,168 +0.011096,168 +0.011665,168 +0.011273,168 +0.011239,168 +0.011162,168 +0.011810,168 +0.013018,168 +0.011296,168 +0.013346,168 +0.011301,168 +0.011328,168 +0.011371,168 +0.011118,168 +0.011642,168 +0.013363,168 +0.012294,168 +0.011239,168 +0.011365,168 +0.011286,168 +0.011206,168 +0.011182,168 +0.011193,168 +0.011358,168 +0.012980,168 +0.011323,168 +0.011290,168 +0.011096,168 +0.011324,168 +0.011231,168 +0.011120,168 +0.011345,168 +0.011328,168 +0.013012,168 +0.011295,168 +0.011217,168 +0.011203,168 +0.011287,168 +0.011296,168 +0.011253,168 +0.011104,168 +0.012233,168 +0.012366,168 +0.011131,168 +0.011286,168 +0.011339,168 +0.011209,168 +0.011334,168 +0.011235,168 +0.011186,168 +0.012163,168 +0.011562,168 +0.011245,168 +0.011236,168 +0.011327,168 +0.011180,168 +0.011227,168 +0.011246,168 +0.011389,168 +0.011467,168 +0.011451,168 +0.011252,168 +0.011184,168 +0.011440,168 +0.011803,170 +0.011633,170 +0.011754,170 +0.011733,170 +0.012118,170 +0.011671,170 +0.011704,170 +0.011708,170 +0.011543,170 +0.011757,170 +0.011879,170 +0.012707,170 +0.012234,170 +0.012154,170 +0.012029,170 +0.011819,170 +0.011794,170 +0.011707,170 +0.011739,170 +0.011616,170 +0.011782,170 +0.011894,170 +0.011620,170 +0.011676,170 +0.011737,170 +0.011665,170 +0.013483,170 +0.011864,170 +0.011862,170 +0.011783,170 +0.011887,170 +0.011690,170 +0.011598,170 +0.011614,170 +0.011786,170 +0.011668,170 +0.011574,170 +0.011753,170 +0.012154,170 +0.011741,170 +0.011661,170 +0.011731,170 +0.011587,170 +0.011728,170 +0.011885,170 +0.012014,170 +0.011749,170 +0.012165,170 +0.011731,170 +0.011691,170 +0.011632,170 +0.011787,170 +0.011654,170 +0.011534,170 +0.011671,170 +0.012229,170 +0.011720,170 +0.011710,170 +0.011681,170 +0.011628,170 +0.011850,170 +0.012076,170 +0.012006,170 +0.011998,170 +0.012166,170 +0.011650,170 +0.011619,170 +0.013339,170 +0.011921,170 +0.011944,170 +0.011613,170 +0.012039,170 +0.012333,170 +0.011874,170 +0.011528,170 +0.011718,170 +0.012276,170 +0.011680,170 +0.011720,170 +0.011750,170 +0.011871,170 +0.011891,170 +0.011765,170 +0.011748,170 +0.012251,170 +0.012336,170 +0.011824,170 +0.011639,170 +0.011552,170 +0.012302,170 +0.011869,170 +0.011699,170 +0.011653,170 +0.011691,170 +0.011702,170 +0.011613,170 +0.011693,170 +0.011867,170 +0.011884,170 +0.011810,170 +0.012074,172 +0.011974,172 +0.012033,172 +0.012139,172 +0.012091,172 +0.012195,172 +0.012515,172 +0.012144,172 +0.012095,172 +0.011989,172 +0.012052,172 +0.012197,172 +0.011944,172 +0.012017,172 +0.012634,172 +0.012185,172 +0.012032,172 +0.011984,172 +0.011944,172 +0.011948,172 +0.012013,172 +0.012059,172 +0.012414,172 +0.012556,172 +0.012083,172 +0.012123,172 +0.012035,172 +0.011958,172 +0.012100,172 +0.012146,172 +0.012187,172 +0.012415,172 +0.012021,172 +0.012361,172 +0.012271,172 +0.012159,172 +0.012322,172 +0.012592,172 +0.012602,172 +0.017974,172 +0.012685,172 +0.012590,172 +0.012584,172 +0.013232,172 +0.012474,172 +0.013220,172 +0.012502,172 +0.013119,172 +0.014392,172 +0.013448,172 +0.013295,172 +0.012219,172 +0.013097,172 +0.012303,172 +0.012687,172 +0.012114,172 +0.012033,172 +0.012130,172 +0.012114,172 +0.012187,172 +0.012298,172 +0.012147,172 +0.013105,172 +0.012013,172 +0.012057,172 +0.011979,172 +0.011942,172 +0.011856,172 +0.011861,172 +0.011924,172 +0.012072,172 +0.012259,172 +0.011844,172 +0.011881,172 +0.011843,172 +0.011926,172 +0.011862,172 +0.012586,172 +0.012893,172 +0.012599,172 +0.012343,172 +0.012339,172 +0.012494,172 +0.013816,172 +0.013387,172 +0.012774,172 +0.012310,172 +0.012798,172 +0.012672,172 +0.012516,172 +0.013154,172 +0.013189,172 +0.013596,172 +0.013207,172 +0.012189,172 +0.012599,172 +0.012129,172 +0.011997,172 +0.011920,172 +0.011909,172 +0.012384,174 +0.012414,174 +0.012323,174 +0.012920,174 +0.012416,174 +0.012512,174 +0.012308,174 +0.012340,174 +0.012355,174 +0.012403,174 +0.012300,174 +0.012744,174 +0.012276,174 +0.012355,174 +0.012279,174 +0.012386,174 +0.012333,174 +0.012360,174 +0.012270,174 +0.012730,174 +0.012328,174 +0.012354,174 +0.012453,174 +0.012383,174 +0.012623,174 +0.012689,174 +0.012315,174 +0.012723,174 +0.012302,174 +0.012564,174 +0.012282,174 +0.012302,174 +0.012277,174 +0.012412,174 +0.012272,174 +0.012768,174 +0.012377,174 +0.012371,174 +0.012361,174 +0.013034,174 +0.012987,174 +0.012628,174 +0.013998,174 +0.013698,174 +0.014008,174 +0.013889,174 +0.013980,174 +0.013791,174 +0.013426,174 +0.013223,174 +0.013151,174 +0.013028,174 +0.012541,174 +0.012331,174 +0.012293,174 +0.012333,174 +0.012407,174 +0.012313,174 +0.012564,174 +0.012576,174 +0.012471,174 +0.012335,174 +0.012304,174 +0.012388,174 +0.012524,174 +0.012404,174 +0.012559,174 +0.012668,174 +0.012388,174 +0.012343,174 +0.012292,174 +0.012377,174 +0.012352,174 +0.012424,174 +0.012519,174 +0.012774,174 +0.012341,174 +0.012310,174 +0.012281,174 +0.012314,174 +0.012306,174 +0.012409,174 +0.012421,174 +0.012911,174 +0.012366,174 +0.012398,174 +0.012284,174 +0.012317,174 +0.012333,174 +0.012430,174 +0.012790,174 +0.012808,174 +0.012425,174 +0.012461,174 +0.012323,174 +0.012311,174 +0.012329,174 +0.012341,174 +0.012318,174 +0.012892,174 +0.012960,176 +0.012683,176 +0.012958,176 +0.012756,176 +0.012903,176 +0.015272,176 +0.013608,176 +0.013179,176 +0.012783,176 +0.012762,176 +0.012756,176 +0.013141,176 +0.012996,176 +0.012790,176 +0.013298,176 +0.012712,176 +0.012703,176 +0.012714,176 +0.012706,176 +0.012737,176 +0.012772,176 +0.012755,176 +0.013546,176 +0.013366,176 +0.013516,176 +0.012994,176 +0.012751,176 +0.013049,176 +0.013413,176 +0.012935,176 +0.013297,176 +0.012956,176 +0.012993,176 +0.012823,176 +0.012865,176 +0.012933,176 +0.012857,176 +0.013186,176 +0.013147,176 +0.012913,176 +0.012906,176 +0.012921,176 +0.013084,176 +0.012999,176 +0.012901,176 +0.013274,176 +0.013093,176 +0.013013,176 +0.013214,176 +0.013072,176 +0.013100,176 +0.012980,176 +0.012936,176 +0.013306,176 +0.012931,176 +0.012915,176 +0.012792,176 +0.012953,176 +0.013011,176 +0.012916,176 +0.013064,176 +0.013292,176 +0.013001,176 +0.012893,176 +0.013025,176 +0.015394,176 +0.012886,176 +0.013035,176 +0.013505,176 +0.012926,176 +0.012864,176 +0.012822,176 +0.012955,176 +0.013117,176 +0.013025,176 +0.012799,176 +0.013449,176 +0.012980,176 +0.012970,176 +0.013047,176 +0.013065,176 +0.014701,176 +0.014472,176 +0.013191,176 +0.012983,176 +0.012964,176 +0.012861,176 +0.012866,176 +0.012829,176 +0.012984,176 +0.012747,176 +0.013213,176 +0.012899,176 +0.013713,176 +0.024383,176 +0.023947,176 +0.023787,176 +0.024533,176 +0.020622,176 +0.012850,176 +0.013423,178 +0.013214,178 +0.013369,178 +0.013415,178 +0.013215,178 +0.013210,178 +0.013217,178 +0.013262,178 +0.013397,178 +0.013134,178 +0.013471,178 +0.013426,178 +0.013207,178 +0.013239,178 +0.013152,178 +0.013520,178 +0.013238,178 +0.013340,178 +0.013517,178 +0.013335,178 +0.013132,178 +0.013216,178 +0.013238,178 +0.013494,178 +0.013076,178 +0.013547,178 +0.013249,178 +0.013361,178 +0.013213,178 +0.013097,178 +0.013363,178 +0.013336,178 +0.013284,178 +0.013659,178 +0.013276,178 +0.013364,178 +0.013297,178 +0.013291,178 +0.013249,178 +0.013256,178 +0.013445,178 +0.013561,178 +0.013437,178 +0.013181,178 +0.013125,178 +0.013279,178 +0.013336,178 +0.013215,178 +0.014821,178 +0.013188,178 +0.013328,178 +0.013293,178 +0.013606,178 +0.013379,178 +0.013260,178 +0.014431,178 +0.013638,178 +0.013395,178 +0.013212,178 +0.013191,178 +0.013361,178 +0.013519,178 +0.013701,178 +0.014332,178 +0.013212,178 +0.013294,178 +0.013222,178 +0.013290,178 +0.013266,178 +0.013418,178 +0.014835,178 +0.013408,178 +0.013361,178 +0.013201,178 +0.013219,178 +0.013530,178 +0.013478,178 +0.014482,178 +0.014100,178 +0.013266,178 +0.013233,178 +0.013167,178 +0.013363,178 +0.013241,178 +0.013262,178 +0.014768,178 +0.013224,178 +0.013219,178 +0.013202,178 +0.013827,178 +0.013347,178 +0.013195,178 +0.014322,178 +0.013850,178 +0.013280,178 +0.013235,178 +0.013249,178 +0.013219,178 +0.013212,178 +0.013230,178 +0.015233,180 +0.013626,180 +0.013766,180 +0.013514,180 +0.013690,180 +0.013639,180 +0.013634,180 +0.016223,180 +0.014035,180 +0.013553,180 +0.013603,180 +0.013822,180 +0.013851,180 +0.013732,180 +0.015221,180 +0.013563,180 +0.013429,180 +0.013443,180 +0.013436,180 +0.013512,180 +0.013390,180 +0.014203,180 +0.014348,180 +0.013512,180 +0.013476,180 +0.013443,180 +0.013712,180 +0.013452,180 +0.013452,180 +0.015234,180 +0.013533,180 +0.013496,180 +0.013369,180 +0.013502,180 +0.013443,180 +0.013460,180 +0.014897,180 +0.013679,180 +0.013442,180 +0.013460,180 +0.013442,180 +0.013522,180 +0.013461,180 +0.013933,180 +0.014692,180 +0.013510,180 +0.013534,180 +0.013457,180 +0.013455,180 +0.013410,180 +0.013439,180 +0.015126,180 +0.013476,180 +0.013476,180 +0.013371,180 +0.013891,180 +0.013455,180 +0.013458,180 +0.014949,180 +0.013609,180 +0.013448,180 +0.013444,180 +0.013600,180 +0.013514,180 +0.013525,180 +0.013936,180 +0.014586,180 +0.013585,180 +0.013459,180 +0.013418,180 +0.013584,180 +0.013392,180 +0.013459,180 +0.015049,180 +0.013426,180 +0.013406,180 +0.013403,180 +0.013465,180 +0.013494,180 +0.013467,180 +0.015478,180 +0.014293,180 +0.013425,180 +0.013365,180 +0.013495,180 +0.013476,180 +0.013442,180 +0.013944,180 +0.014480,180 +0.013447,180 +0.013480,180 +0.013425,180 +0.013722,180 +0.013414,180 +0.013474,180 +0.015276,180 +0.013472,180 +0.013435,180 +0.013424,180 +0.013643,180 +0.013902,182 +0.013907,182 +0.015443,182 +0.014029,182 +0.013959,182 +0.013974,182 +0.014015,182 +0.013872,182 +0.013870,182 +0.015263,182 +0.014036,182 +0.013872,182 +0.013946,182 +0.013903,182 +0.014017,182 +0.013900,182 +0.015056,182 +0.017820,182 +0.014901,182 +0.013865,182 +0.013997,182 +0.013866,182 +0.014549,182 +0.019217,182 +0.013981,182 +0.013916,182 +0.013907,182 +0.013986,182 +0.013927,182 +0.013883,182 +0.019150,182 +0.013907,182 +0.013926,182 +0.013915,182 +0.013980,182 +0.013910,182 +0.013977,182 +0.020017,182 +0.014346,182 +0.013972,182 +0.013997,182 +0.013899,182 +0.013899,182 +0.014963,182 +0.018291,182 +0.013958,182 +0.013986,182 +0.013974,182 +0.013918,182 +0.013935,182 +0.016002,182 +0.017890,182 +0.014267,182 +0.014013,182 +0.014023,182 +0.013904,182 +0.013822,182 +0.019177,182 +0.014132,182 +0.013980,182 +0.014011,182 +0.014188,182 +0.013923,182 +0.013921,182 +0.020269,182 +0.014097,182 +0.014227,182 +0.014711,182 +0.014058,182 +0.013927,182 +0.014158,182 +0.019033,182 +0.013974,182 +0.013980,182 +0.013983,182 +0.013825,182 +0.013902,182 +0.016214,182 +0.017030,182 +0.013948,182 +0.013924,182 +0.014053,182 +0.013955,182 +0.013969,182 +0.018330,182 +0.014625,182 +0.013950,182 +0.013899,182 +0.013992,182 +0.013903,182 +0.013896,182 +0.019139,182 +0.014123,182 +0.013892,182 +0.014028,182 +0.013953,182 +0.013911,182 +0.013905,182 +0.019032,182 +0.013962,182 +0.014479,184 +0.014455,184 +0.014369,184 +0.014412,184 +0.014929,184 +0.019210,184 +0.014443,184 +0.014334,184 +0.014386,184 +0.014356,184 +0.014315,184 +0.019437,184 +0.014461,184 +0.014292,184 +0.014367,184 +0.014483,184 +0.014369,184 +0.014329,184 +0.020059,184 +0.014497,184 +0.014341,184 +0.014386,184 +0.014289,184 +0.014306,184 +0.016362,184 +0.017521,184 +0.014389,184 +0.014344,184 +0.014387,184 +0.014368,184 +0.014352,184 +0.019472,184 +0.014474,184 +0.014347,184 +0.015075,184 +0.014360,184 +0.014348,184 +0.014399,184 +0.019423,184 +0.014507,184 +0.014438,184 +0.014416,184 +0.014332,184 +0.015185,184 +0.019410,184 +0.016408,184 +0.015864,184 +0.015536,184 +0.014972,184 +0.015582,184 +0.027270,184 +0.018104,184 +0.015220,184 +0.016124,184 +0.014965,184 +0.014821,184 +0.014793,184 +0.014504,184 +0.014466,184 +0.014458,184 +0.014914,184 +0.014509,184 +0.014484,184 +0.014877,184 +0.014463,184 +0.014389,184 +0.014486,184 +0.014340,184 +0.014570,184 +0.014492,184 +0.014879,184 +0.014335,184 +0.014371,184 +0.014412,184 +0.014258,184 +0.015139,184 +0.014796,184 +0.015086,184 +0.015209,184 +0.014670,184 +0.014514,184 +0.014402,184 +0.014345,184 +0.014389,184 +0.015136,184 +0.014319,184 +0.014330,184 +0.014358,184 +0.014303,184 +0.014298,184 +0.014372,184 +0.014798,184 +0.014356,184 +0.014343,184 +0.014388,184 +0.014325,184 +0.014326,184 +0.014415,184 +0.014743,184 +0.014389,184 +0.015259,186 +0.015109,186 +0.014963,186 +0.015024,186 +0.015551,186 +0.015172,186 +0.015104,186 +0.015000,186 +0.014967,186 +0.014994,186 +0.015068,186 +0.025382,186 +0.029037,186 +0.015744,186 +0.022137,186 +0.015311,186 +0.015295,186 +0.014945,186 +0.014972,186 +0.015185,186 +0.015195,186 +0.014981,186 +0.015459,186 +0.014972,186 +0.014958,186 +0.015054,186 +0.014974,186 +0.014953,186 +0.014940,186 +0.015495,186 +0.015012,186 +0.015054,186 +0.015140,186 +0.015008,186 +0.015085,186 +0.015402,186 +0.015411,186 +0.015059,186 +0.015014,186 +0.015397,186 +0.015012,186 +0.014989,186 +0.015490,186 +0.015117,186 +0.015029,186 +0.015152,186 +0.015804,186 +0.014965,186 +0.015016,186 +0.015574,186 +0.014951,186 +0.014957,186 +0.015062,186 +0.015000,186 +0.015073,186 +0.015221,186 +0.015300,186 +0.014961,186 +0.015027,186 +0.014961,186 +0.014952,186 +0.014975,186 +0.015491,186 +0.015028,186 +0.015032,186 +0.015069,186 +0.014967,186 +0.014969,186 +0.015113,186 +0.015516,186 +0.014974,186 +0.014906,186 +0.015022,186 +0.014984,186 +0.014992,186 +0.015205,186 +0.015358,186 +0.014973,186 +0.015069,186 +0.015046,186 +0.014976,186 +0.015026,186 +0.015461,186 +0.014995,186 +0.014958,186 +0.015029,186 +0.015035,186 +0.014989,186 +0.014946,186 +0.015537,186 +0.018234,186 +0.016711,186 +0.015068,186 +0.015017,186 +0.015000,186 +0.015504,186 +0.014981,186 +0.015016,186 +0.015108,186 +0.015070,186 +0.015702,188 +0.015573,188 +0.016011,188 +0.015475,188 +0.015665,188 +0.015472,188 +0.015559,188 +0.015475,188 +0.015965,188 +0.015482,188 +0.015496,188 +0.015608,188 +0.015600,188 +0.015478,188 +0.015600,188 +0.016119,188 +0.015570,188 +0.015585,188 +0.015472,188 +0.015473,188 +0.015478,188 +0.016057,188 +0.015521,188 +0.015449,188 +0.015591,188 +0.015474,188 +0.015849,188 +0.021293,188 +0.029933,188 +0.029273,188 +0.029209,188 +0.029517,188 +0.029054,188 +0.029225,188 +0.029937,188 +0.029164,188 +0.028743,188 +0.028933,188 +0.029663,188 +0.028790,188 +0.029312,188 +0.029178,188 +0.029934,188 +0.029684,188 +0.029071,188 +0.029666,188 +0.029133,188 +0.029237,188 +0.029800,188 +0.029108,188 +0.029052,188 +0.029270,188 +0.029256,188 +0.029173,188 +0.029406,188 +0.029462,188 +0.029302,188 +0.029582,188 +0.032099,188 +0.029381,188 +0.029352,188 +0.030201,188 +0.025227,188 +0.017845,188 +0.016191,188 +0.016249,188 +0.016641,188 +0.017521,188 +0.016085,188 +0.016326,188 +0.016309,188 +0.016209,188 +0.016283,188 +0.021234,188 +0.015879,188 +0.015915,188 +0.015630,188 +0.015712,188 +0.015935,188 +0.020878,188 +0.016506,188 +0.015879,188 +0.015805,188 +0.015917,188 +0.015749,188 +0.020783,188 +0.015975,188 +0.015752,188 +0.015739,188 +0.015889,188 +0.015739,188 +0.020795,188 +0.015937,188 +0.015899,188 +0.015682,188 +0.017630,188 +0.015786,188 +0.016189,188 +0.015989,188 +0.015832,188 +0.016330,190 +0.016433,190 +0.016326,190 +0.016356,190 +0.016731,190 +0.016684,190 +0.016410,190 +0.016675,190 +0.016609,190 +0.016545,190 +0.016663,190 +0.016281,190 +0.016178,190 +0.016171,190 +0.016348,190 +0.016273,190 +0.016541,190 +0.016366,190 +0.016309,190 +0.016257,190 +0.016565,190 +0.016389,190 +0.016721,190 +0.016352,190 +0.016311,190 +0.016366,190 +0.016356,190 +0.016395,190 +0.016891,190 +0.016458,190 +0.016458,190 +0.016651,190 +0.016730,190 +0.016754,190 +0.017093,190 +0.016874,190 +0.016853,190 +0.017064,190 +0.017028,190 +0.023721,190 +0.018242,190 +0.017103,190 +0.017085,190 +0.016747,190 +0.016366,190 +0.016701,190 +0.016568,190 +0.016367,190 +0.016247,190 +0.016355,190 +0.016363,190 +0.016749,190 +0.016560,190 +0.016463,190 +0.016335,190 +0.016366,190 +0.016223,190 +0.016496,190 +0.016825,190 +0.016327,190 +0.016257,190 +0.016236,190 +0.016298,190 +0.016230,190 +0.016841,190 +0.016357,190 +0.016425,190 +0.016208,190 +0.016314,190 +0.016231,190 +0.016985,190 +0.016357,190 +0.016317,190 +0.016216,190 +0.016243,190 +0.016221,190 +0.016878,190 +0.016325,190 +0.016272,190 +0.016287,190 +0.016271,190 +0.016204,190 +0.017171,190 +0.016317,190 +0.016377,190 +0.016395,190 +0.016442,190 +0.016784,190 +0.017779,190 +0.016077,190 +0.016126,190 +0.016031,190 +0.015989,190 +0.016613,190 +0.016412,190 +0.016562,190 +0.016252,190 +0.016061,190 +0.016037,190 +0.016055,190 +0.017758,192 +0.018130,192 +0.017835,192 +0.017625,192 +0.017569,192 +0.017560,192 +0.018093,192 +0.017590,192 +0.017646,192 +0.017595,192 +0.017589,192 +0.017593,192 +0.018053,192 +0.017857,192 +0.017576,192 +0.018595,192 +0.018000,192 +0.017856,192 +0.017902,192 +0.017668,192 +0.017556,192 +0.017600,192 +0.017589,192 +0.018096,192 +0.017726,192 +0.017848,192 +0.018263,192 +0.018465,192 +0.018469,192 +0.019493,192 +0.019087,192 +0.018881,192 +0.019300,192 +0.018882,192 +0.020475,192 +0.018907,192 +0.018425,192 +0.018675,192 +0.018796,192 +0.019180,192 +0.018455,192 +0.018796,192 +0.019406,192 +0.018863,192 +0.019209,192 +0.019241,192 +0.018335,192 +0.018386,192 +0.018598,192 +0.021133,192 +0.019768,192 +0.019792,192 +0.019381,192 +0.019887,192 +0.019157,192 +0.019405,192 +0.019352,192 +0.018494,192 +0.018213,192 +0.018221,192 +0.018205,192 +0.018568,192 +0.018134,192 +0.017751,192 +0.017784,192 +0.017656,192 +0.018212,192 +0.017853,192 +0.017730,192 +0.017663,192 +0.017586,192 +0.017701,192 +0.017985,192 +0.017648,192 +0.017545,192 +0.017614,192 +0.017532,192 +0.018059,192 +0.017754,192 +0.017547,192 +0.018010,192 +0.017615,192 +0.017596,192 +0.018172,192 +0.017703,192 +0.017613,192 +0.017505,192 +0.017577,192 +0.017640,192 +0.018077,192 +0.017675,192 +0.017876,192 +0.017800,192 +0.017545,192 +0.018172,192 +0.017663,192 +0.017645,192 +0.017551,192 +0.017570,192 +0.017639,192 +0.017723,194 +0.017201,194 +0.017217,194 +0.017284,194 +0.017187,194 +0.017117,194 +0.017673,194 +0.017179,194 +0.017058,194 +0.017121,194 +0.017032,194 +0.017173,194 +0.017470,194 +0.017173,194 +0.017038,194 +0.017035,194 +0.017031,194 +0.017312,194 +0.017448,194 +0.017117,194 +0.017095,194 +0.017161,194 +0.017030,194 +0.017458,194 +0.017410,194 +0.017029,194 +0.017090,194 +0.017150,194 +0.017056,194 +0.017628,194 +0.017249,194 +0.017056,194 +0.017107,194 +0.017040,194 +0.017043,194 +0.017798,194 +0.017358,194 +0.018198,194 +0.018354,194 +0.018124,194 +0.019318,194 +0.019823,194 +0.018403,194 +0.018274,194 +0.017861,194 +0.017662,194 +0.017762,194 +0.017314,194 +0.017294,194 +0.017188,194 +0.017390,194 +0.017211,194 +0.017584,194 +0.017199,194 +0.017207,194 +0.017087,194 +0.017136,194 +0.017488,194 +0.017390,194 +0.017213,194 +0.017064,194 +0.017075,194 +0.017029,194 +0.017756,194 +0.018961,194 +0.017272,194 +0.017127,194 +0.017087,194 +0.017077,194 +0.017945,194 +0.017222,194 +0.017054,194 +0.017034,194 +0.017032,194 +0.017027,194 +0.017971,194 +0.017141,194 +0.017032,194 +0.017031,194 +0.017031,194 +0.017108,194 +0.017596,194 +0.017183,194 +0.017266,194 +0.017126,194 +0.017049,194 +0.017203,194 +0.017470,194 +0.017219,194 +0.017033,194 +0.017059,194 +0.017057,194 +0.017218,194 +0.017503,194 +0.017104,194 +0.017073,194 +0.017095,194 +0.017032,194 +0.017256,194 +0.017648,194 +0.017649,196 +0.017528,196 +0.017527,196 +0.017525,196 +0.017969,196 +0.017686,196 +0.017538,196 +0.017524,196 +0.017531,196 +0.019247,196 +0.020003,196 +0.019093,196 +0.018780,196 +0.018794,196 +0.017793,196 +0.018126,196 +0.017694,196 +0.017561,196 +0.017558,196 +0.017488,196 +0.017509,196 +0.018199,196 +0.017556,196 +0.017535,196 +0.017544,196 +0.017471,196 +0.017794,196 +0.017866,196 +0.017479,196 +0.017464,196 +0.017628,196 +0.017493,196 +0.018101,196 +0.017621,196 +0.017466,196 +0.017535,196 +0.017462,196 +0.017503,196 +0.018121,196 +0.017712,196 +0.017655,196 +0.017517,196 +0.017495,196 +0.017710,196 +0.017903,196 +0.017532,196 +0.017483,196 +0.017467,196 +0.017495,196 +0.018009,196 +0.017698,196 +0.017549,196 +0.017547,196 +0.017500,196 +0.017464,196 +0.018189,196 +0.017632,196 +0.017809,196 +0.017598,196 +0.017607,196 +0.017754,196 +0.017993,196 +0.017522,196 +0.017505,196 +0.017483,196 +0.017663,196 +0.018069,196 +0.017601,196 +0.017471,196 +0.017477,196 +0.017527,196 +0.017463,196 +0.018090,196 +0.017821,196 +0.017606,196 +0.017499,196 +0.017503,196 +0.017560,196 +0.018135,196 +0.017621,196 +0.017612,196 +0.017501,196 +0.017469,196 +0.018569,196 +0.017776,196 +0.017502,196 +0.017464,196 +0.017463,196 +0.017467,196 +0.018028,196 +0.017664,196 +0.018535,196 +0.020325,196 +0.018020,196 +0.018523,196 +0.018185,196 +0.017650,196 +0.017807,196 +0.019012,196 +0.019560,196 +0.021297,198 +0.018879,198 +0.019415,198 +0.018237,198 +0.018151,198 +0.018833,198 +0.018286,198 +0.018125,198 +0.018174,198 +0.018090,198 +0.018107,198 +0.018952,198 +0.018206,198 +0.018143,198 +0.018094,198 +0.018124,198 +0.023542,198 +0.018220,198 +0.018218,198 +0.018148,198 +0.018384,198 +0.022983,198 +0.018703,198 +0.018196,198 +0.018158,198 +0.018113,198 +0.019203,198 +0.022723,198 +0.018422,198 +0.018436,198 +0.018288,198 +0.018514,198 +0.023999,198 +0.018266,198 +0.018654,198 +0.018360,198 +0.018278,198 +0.023968,198 +0.018381,198 +0.018400,198 +0.018328,198 +0.018368,198 +0.024078,198 +0.018356,198 +0.018515,198 +0.018419,198 +0.018498,198 +0.024293,198 +0.020207,198 +0.018954,198 +0.018380,198 +0.018465,198 +0.024021,198 +0.018793,198 +0.018296,198 +0.018294,198 +0.018455,198 +0.021622,198 +0.025022,198 +0.019093,198 +0.018406,198 +0.018466,198 +0.020783,198 +0.022209,198 +0.018543,198 +0.018451,198 +0.018320,198 +0.020868,198 +0.021467,198 +0.018890,198 +0.018451,198 +0.018416,198 +0.018991,198 +0.023501,198 +0.018466,198 +0.018396,198 +0.018622,198 +0.018818,198 +0.019036,198 +0.018999,198 +0.018954,198 +0.018368,198 +0.018850,198 +0.019231,198 +0.018180,198 +0.018262,198 +0.018089,198 +0.018069,198 +0.018296,198 +0.018660,198 +0.018183,198 +0.018072,198 +0.018069,198 +0.018087,198 +0.018824,198 +0.018486,198 +0.018444,198 +0.018565,198 +0.018482,198 +0.018686,198 +0.019614,200 +0.018985,200 +0.018951,200 +0.019013,200 +0.018900,200 +0.019752,200 +0.018977,200 +0.018916,200 +0.019042,200 +0.019185,200 +0.019620,200 +0.019126,200 +0.019041,200 +0.019046,200 +0.019005,200 +0.019648,200 +0.019398,200 +0.018984,200 +0.018921,200 +0.019097,200 +0.019108,200 +0.019361,200 +0.018982,200 +0.018946,200 +0.018984,200 +0.019093,200 +0.019620,200 +0.018948,200 +0.018868,200 +0.018854,200 +0.018931,200 +0.019449,200 +0.019022,200 +0.019014,200 +0.018844,200 +0.018885,200 +0.019446,200 +0.019151,200 +0.019011,200 +0.018906,200 +0.018898,200 +0.018926,200 +0.019673,200 +0.018902,200 +0.018821,200 +0.018833,200 +0.018963,200 +0.019687,200 +0.019112,200 +0.019051,200 +0.018897,200 +0.019042,200 +0.019607,200 +0.019062,200 +0.018903,200 +0.018925,200 +0.018920,200 +0.019317,200 +0.019451,200 +0.018919,200 +0.019014,200 +0.018887,200 +0.019028,200 +0.020660,200 +0.019032,200 +0.018857,200 +0.019030,200 +0.019124,200 +0.022260,200 +0.019522,200 +0.019082,200 +0.019479,200 +0.018689,200 +0.020167,200 +0.018665,200 +0.018714,200 +0.019436,200 +0.018740,200 +0.020238,200 +0.018807,200 +0.018606,200 +0.018601,200 +0.018634,200 +0.019663,200 +0.019381,200 +0.018664,200 +0.018644,200 +0.018625,200 +0.018611,200 +0.020134,200 +0.018772,200 +0.018611,200 +0.018597,200 +0.018606,200 +0.020143,200 +0.018630,200 +0.018587,200 +0.018580,200 +0.018601,200 +0.019625,200 +0.019955,202 +0.019270,202 +0.019276,202 +0.019466,202 +0.020023,202 +0.020513,202 +0.019293,202 +0.019209,202 +0.019191,202 +0.019206,202 +0.020959,202 +0.019258,202 +0.019271,202 +0.019220,202 +0.019201,202 +0.020855,202 +0.019424,202 +0.019376,202 +0.019329,202 +0.019204,202 +0.022095,202 +0.019331,202 +0.019276,202 +0.019260,202 +0.019274,202 +0.020895,202 +0.019298,202 +0.019362,202 +0.019265,202 +0.019339,202 +0.020935,202 +0.019314,202 +0.019344,202 +0.019257,202 +0.019289,202 +0.020819,202 +0.019280,202 +0.019256,202 +0.019269,202 +0.019241,202 +0.020638,202 +0.019632,202 +0.019357,202 +0.019642,202 +0.019539,202 +0.020262,202 +0.020131,202 +0.019248,202 +0.019233,202 +0.019343,202 +0.019285,202 +0.020917,202 +0.019276,202 +0.019215,202 +0.019297,202 +0.019431,202 +0.021968,202 +0.019440,202 +0.019328,202 +0.019243,202 +0.019206,202 +0.019737,202 +0.019363,202 +0.019230,202 +0.019292,202 +0.019199,202 +0.020305,202 +0.019640,202 +0.019347,202 +0.019220,202 +0.019197,202 +0.019628,202 +0.019511,202 +0.019280,202 +0.019240,202 +0.019250,202 +0.019447,202 +0.020550,202 +0.019445,202 +0.019274,202 +0.019406,202 +0.019407,202 +0.019588,202 +0.019275,202 +0.019281,202 +0.019254,202 +0.019371,202 +0.019940,202 +0.019374,202 +0.019312,202 +0.019333,202 +0.019288,202 +0.019764,202 +0.019403,202 +0.019297,202 +0.019224,202 +0.019246,202 +0.019714,202 +0.019373,202 +0.019290,202 +0.020346,204 +0.019918,204 +0.020338,204 +0.019947,204 +0.019849,204 +0.019833,204 +0.020035,204 +0.020295,204 +0.019921,204 +0.019865,204 +0.019769,204 +0.019774,204 +0.020104,204 +0.020178,204 +0.019959,204 +0.019820,204 +0.019834,204 +0.020111,204 +0.020052,204 +0.019878,204 +0.019775,204 +0.019783,204 +0.020072,204 +0.020188,204 +0.019887,204 +0.019811,204 +0.019828,204 +0.020040,204 +0.020067,204 +0.020056,204 +0.019794,204 +0.019855,204 +0.020051,204 +0.020079,204 +0.019893,204 +0.019762,204 +0.019778,204 +0.019882,204 +0.020157,204 +0.019869,204 +0.020606,204 +0.020034,204 +0.020459,204 +0.020252,204 +0.019838,204 +0.019776,204 +0.019760,204 +0.019930,204 +0.020156,204 +0.019840,204 +0.019778,204 +0.019795,204 +0.019914,204 +0.020118,204 +0.019836,204 +0.019809,204 +0.019894,204 +0.019863,204 +0.020166,204 +0.019825,204 +0.019781,204 +0.019731,204 +0.019851,204 +0.020210,204 +0.019788,204 +0.019774,204 +0.019908,204 +0.019780,204 +0.020163,204 +0.019885,204 +0.019818,204 +0.019725,204 +0.019835,204 +0.020265,204 +0.019733,204 +0.019779,204 +0.019784,204 +0.019784,204 +0.020236,204 +0.020084,204 +0.019783,204 +0.019811,204 +0.019831,204 +0.020212,204 +0.019767,204 +0.019805,204 +0.019774,204 +0.019814,204 +0.020358,204 +0.019773,204 +0.019830,204 +0.019771,204 +0.019804,204 +0.020233,204 +0.019854,204 +0.019767,204 +0.019772,204 +0.019890,204 +0.020138,204 +0.019739,204 +0.020474,206 +0.021390,206 +0.020503,206 +0.020879,206 +0.020554,206 +0.020528,206 +0.020522,206 +0.020726,206 +0.020909,206 +0.020593,206 +0.020475,206 +0.020340,206 +0.020749,206 +0.021044,206 +0.020424,206 +0.020461,206 +0.020456,206 +0.020496,206 +0.020875,206 +0.020437,206 +0.020355,206 +0.020425,206 +0.020503,206 +0.020988,206 +0.020476,206 +0.020374,206 +0.020402,206 +0.020521,206 +0.020866,206 +0.020498,206 +0.020491,206 +0.020461,206 +0.020494,206 +0.020731,206 +0.020377,206 +0.020408,206 +0.020478,206 +0.020558,206 +0.020627,206 +0.020435,206 +0.020376,206 +0.020436,206 +0.020589,206 +0.020674,206 +0.020367,206 +0.020386,206 +0.020430,206 +0.020825,206 +0.020477,206 +0.020425,206 +0.020371,206 +0.020455,206 +0.020832,206 +0.020454,206 +0.020433,206 +0.020450,206 +0.020546,206 +0.020845,206 +0.020509,206 +0.020404,206 +0.020361,206 +0.020545,206 +0.020790,206 +0.020547,206 +0.020483,206 +0.020445,206 +0.020497,206 +0.021000,206 +0.020620,206 +0.020432,206 +0.020507,206 +0.020377,206 +0.021019,206 +0.020574,206 +0.020587,206 +0.020427,206 +0.020517,206 +0.021153,206 +0.020512,206 +0.021221,206 +0.020608,206 +0.020599,206 +0.020907,206 +0.020395,206 +0.020442,206 +0.020490,206 +0.020755,206 +0.020848,206 +0.020516,206 +0.020425,206 +0.020515,206 +0.021067,206 +0.020579,206 +0.020578,206 +0.020460,206 +0.020519,206 +0.020952,206 +0.020452,206 +0.020415,206 +0.020555,206 +0.020937,208 +0.021447,208 +0.021110,208 +0.021063,208 +0.021080,208 +0.021059,208 +0.021476,208 +0.021106,208 +0.020988,208 +0.021031,208 +0.021058,208 +0.021473,208 +0.020970,208 +0.020922,208 +0.021063,208 +0.021414,208 +0.021251,208 +0.020921,208 +0.021078,208 +0.020971,208 +0.021575,208 +0.021036,208 +0.021007,208 +0.021015,208 +0.021137,208 +0.021794,208 +0.021007,208 +0.021026,208 +0.021033,208 +0.021063,208 +0.021563,208 +0.021013,208 +0.020908,208 +0.021028,208 +0.021103,208 +0.021375,208 +0.020978,208 +0.021054,208 +0.020989,208 +0.021518,208 +0.020965,208 +0.020957,208 +0.020970,208 +0.020899,208 +0.021561,208 +0.021345,208 +0.020959,208 +0.021026,208 +0.021020,208 +0.021515,208 +0.020916,208 +0.021116,208 +0.021035,208 +0.021011,208 +0.021544,208 +0.020938,208 +0.020971,208 +0.020993,208 +0.021331,208 +0.021146,208 +0.020975,208 +0.021000,208 +0.020880,208 +0.021581,208 +0.020919,208 +0.020933,208 +0.021065,208 +0.021079,208 +0.021627,208 +0.020984,208 +0.021031,208 +0.021143,208 +0.021434,208 +0.021466,208 +0.021069,208 +0.021036,208 +0.020908,208 +0.021261,208 +0.021241,208 +0.020960,208 +0.020993,208 +0.020928,208 +0.021676,208 +0.021072,208 +0.020921,208 +0.021194,208 +0.020953,208 +0.021478,208 +0.020900,208 +0.020952,208 +0.021031,208 +0.020963,208 +0.021564,208 +0.020987,208 +0.021018,208 +0.020997,208 +0.021287,208 +0.021463,208 +0.020961,208 +0.021010,208 +0.021837,210 +0.022359,210 +0.021703,210 +0.021722,210 +0.021748,210 +0.021800,210 +0.022293,210 +0.021726,210 +0.021681,210 +0.021713,210 +0.022189,210 +0.021968,210 +0.021695,210 +0.021828,210 +0.021751,210 +0.022327,210 +0.021710,210 +0.021823,210 +0.021882,210 +0.024530,210 +0.023167,210 +0.021845,210 +0.021955,210 +0.021629,210 +0.022300,210 +0.021750,210 +0.021700,210 +0.021613,210 +0.021707,210 +0.022299,210 +0.021728,210 +0.021733,210 +0.021715,210 +0.022258,210 +0.021667,210 +0.021689,210 +0.021675,210 +0.021617,210 +0.022322,210 +0.021623,210 +0.021723,210 +0.021755,210 +0.021899,210 +0.022125,210 +0.021792,210 +0.021803,210 +0.021650,210 +0.022226,210 +0.021704,210 +0.021738,210 +0.021623,210 +0.021700,210 +0.022291,210 +0.021647,210 +0.021842,210 +0.021684,210 +0.022042,210 +0.021837,210 +0.021654,210 +0.021712,210 +0.021604,210 +0.022472,210 +0.021726,210 +0.021741,210 +0.021754,210 +0.023545,210 +0.022155,210 +0.021975,210 +0.021686,210 +0.021615,210 +0.022360,210 +0.021625,210 +0.022446,210 +0.022446,210 +0.022270,210 +0.022792,210 +0.021866,210 +0.021678,210 +0.021673,210 +0.022255,210 +0.021645,210 +0.021740,210 +0.021689,210 +0.021647,210 +0.022128,210 +0.021689,210 +0.021829,210 +0.021827,210 +0.021905,210 +0.022211,210 +0.021896,210 +0.021644,210 +0.021800,210 +0.022196,210 +0.021754,210 +0.021831,210 +0.021608,210 +0.021701,210 +0.022241,210 +0.021665,210 +0.022333,212 +0.022289,212 +0.022824,212 +0.022219,212 +0.022375,212 +0.022238,212 +0.022223,212 +0.022961,212 +0.022227,212 +0.022258,212 +0.022534,212 +0.022837,212 +0.022326,212 +0.022341,212 +0.022220,212 +0.022228,212 +0.022974,212 +0.022232,212 +0.022573,212 +0.022355,212 +0.022848,212 +0.022325,212 +0.022381,212 +0.022277,212 +0.022785,212 +0.022947,212 +0.022319,212 +0.022236,212 +0.022260,212 +0.022802,212 +0.022282,212 +0.022393,212 +0.022251,212 +0.022384,212 +0.022750,212 +0.022347,212 +0.022354,212 +0.022219,212 +0.022789,212 +0.022340,212 +0.022302,212 +0.022216,212 +0.022453,212 +0.022677,212 +0.022374,212 +0.022228,212 +0.022263,212 +0.022823,212 +0.022218,212 +0.022228,212 +0.022198,212 +0.022395,212 +0.022869,212 +0.022323,212 +0.022361,212 +0.022268,212 +0.022801,212 +0.022334,212 +0.022253,212 +0.022175,212 +0.022361,212 +0.022652,212 +0.022286,212 +0.022231,212 +0.022339,212 +0.023049,212 +0.022210,212 +0.022239,212 +0.022321,212 +0.022345,212 +0.022578,212 +0.022393,212 +0.022224,212 +0.022156,212 +0.022832,212 +0.022247,212 +0.022352,212 +0.022454,212 +0.022399,212 +0.022739,212 +0.022451,212 +0.022183,212 +0.022177,212 +0.022833,212 +0.022297,212 +0.023886,212 +0.022259,212 +0.022521,212 +0.022645,212 +0.022403,212 +0.022249,212 +0.022298,212 +0.022748,212 +0.022594,212 +0.022305,212 +0.022199,212 +0.022525,212 +0.022699,212 +0.022429,212 +0.022883,212 +0.023026,214 +0.023601,214 +0.022894,214 +0.022924,214 +0.022801,214 +0.023424,214 +0.022784,214 +0.022958,214 +0.022937,214 +0.023036,214 +0.023292,214 +0.022977,214 +0.022910,214 +0.022914,214 +0.023308,214 +0.022945,214 +0.022868,214 +0.022820,214 +0.023361,214 +0.022903,214 +0.022967,214 +0.023011,214 +0.022831,214 +0.023594,214 +0.022994,214 +0.022835,214 +0.022873,214 +0.023355,214 +0.023009,214 +0.022910,214 +0.022801,214 +0.023142,214 +0.023126,214 +0.022873,214 +0.022995,214 +0.022831,214 +0.023499,214 +0.022909,214 +0.022804,214 +0.022903,214 +0.023405,214 +0.023121,214 +0.023190,214 +0.023021,214 +0.023059,214 +0.023475,214 +0.022920,214 +0.022871,214 +0.023039,214 +0.023400,214 +0.022957,214 +0.022797,214 +0.022976,214 +0.023346,214 +0.023228,214 +0.022894,214 +0.023023,214 +0.022950,214 +0.023304,214 +0.022880,214 +0.022979,214 +0.022820,214 +0.023251,214 +0.022990,214 +0.022892,214 +0.022999,214 +0.023206,214 +0.023043,214 +0.022884,214 +0.022835,214 +0.022819,214 +0.023492,214 +0.022915,214 +0.022874,214 +0.022853,214 +0.023394,214 +0.022965,214 +0.022915,214 +0.022892,214 +0.023037,214 +0.023072,214 +0.022953,214 +0.022892,214 +0.022866,214 +0.023559,214 +0.022908,214 +0.023742,214 +0.022807,214 +0.023526,214 +0.023016,214 +0.022789,214 +0.023650,214 +0.023282,214 +0.023171,214 +0.022981,214 +0.022893,214 +0.022843,214 +0.023337,214 +0.022931,214 +0.022935,214 +0.023484,216 +0.023960,216 +0.023784,216 +0.023437,216 +0.023560,216 +0.023767,216 +0.023759,216 +0.023548,216 +0.023554,216 +0.023648,216 +0.023907,216 +0.023504,216 +0.023411,216 +0.023361,216 +0.023976,216 +0.023612,216 +0.023527,216 +0.023568,216 +0.024065,216 +0.023642,216 +0.023505,216 +0.023500,216 +0.023761,216 +0.023708,216 +0.023398,216 +0.023390,216 +0.023527,216 +0.023995,216 +0.023649,216 +0.023565,216 +0.023520,216 +0.024105,216 +0.023535,216 +0.023515,216 +0.023435,216 +0.024056,216 +0.023549,216 +0.023436,216 +0.023527,216 +0.023797,216 +0.024015,216 +0.023496,216 +0.023743,216 +0.023574,216 +0.023869,216 +0.023564,216 +0.023412,216 +0.023448,216 +0.024012,216 +0.023638,216 +0.023633,216 +0.023510,216 +0.024206,216 +0.023510,216 +0.023539,216 +0.023430,216 +0.023712,216 +0.023728,216 +0.023420,216 +0.023466,216 +0.023403,216 +0.023832,216 +0.023598,216 +0.023477,216 +0.023488,216 +0.023875,216 +0.023622,216 +0.023453,216 +0.023399,216 +0.024021,216 +0.023578,216 +0.023585,216 +0.023475,216 +0.024381,216 +0.024930,216 +0.023957,216 +0.024402,216 +0.023838,216 +0.024771,216 +0.023503,216 +0.023520,216 +0.023429,216 +0.025026,216 +0.023553,216 +0.023468,216 +0.023556,216 +0.024949,216 +0.023713,216 +0.023406,216 +0.023559,216 +0.024893,216 +0.023608,216 +0.023511,216 +0.023548,216 +0.025018,216 +0.023749,216 +0.023568,216 +0.023493,216 +0.023500,216 +0.025098,216 +0.024321,218 +0.024253,218 +0.024154,218 +0.025959,218 +0.024192,218 +0.024190,218 +0.024139,218 +0.025750,218 +0.024191,218 +0.024242,218 +0.024402,218 +0.030602,218 +0.024774,218 +0.024360,218 +0.024410,218 +0.024888,218 +0.024177,218 +0.024285,218 +0.024374,218 +0.024681,218 +0.024682,218 +0.024262,218 +0.024223,218 +0.024665,218 +0.024532,218 +0.024227,218 +0.024309,218 +0.024826,218 +0.024397,218 +0.024196,218 +0.024256,218 +0.024776,218 +0.024537,218 +0.024244,218 +0.024336,218 +0.024614,218 +0.024646,218 +0.024301,218 +0.024200,218 +0.024366,218 +0.024733,218 +0.024201,218 +0.024230,218 +0.024197,218 +0.024804,218 +0.024310,218 +0.024180,218 +0.024274,218 +0.024824,218 +0.024214,218 +0.024201,218 +0.024154,218 +0.025116,218 +0.024334,218 +0.024319,218 +0.024365,218 +0.024794,218 +0.024186,218 +0.024323,218 +0.024177,218 +0.024657,218 +0.024300,218 +0.024207,218 +0.024112,218 +0.024419,218 +0.024600,218 +0.024227,218 +0.024217,218 +0.024233,218 +0.024734,218 +0.024214,218 +0.025253,218 +0.024190,218 +0.024788,218 +0.024715,218 +0.024626,218 +0.024456,218 +0.024755,218 +0.024196,218 +0.024166,218 +0.024235,218 +0.024681,218 +0.024260,218 +0.024331,218 +0.024221,218 +0.024686,218 +0.024356,218 +0.024330,218 +0.024253,218 +0.024664,218 +0.024319,218 +0.024420,218 +0.025809,218 +0.024817,218 +0.024451,218 +0.024223,218 +0.024401,218 +0.024759,218 +0.024375,218 +0.024219,218 +0.024873,220 +0.025108,220 +0.025061,220 +0.024885,220 +0.024765,220 +0.025074,220 +0.025367,220 +0.024839,220 +0.024765,220 +0.025052,220 +0.025111,220 +0.024817,220 +0.024792,220 +0.025239,220 +0.025257,220 +0.025068,220 +0.024841,220 +0.025115,220 +0.025180,220 +0.024913,220 +0.024890,220 +0.025050,220 +0.025184,220 +0.024911,220 +0.024798,220 +0.025013,220 +0.025192,220 +0.024926,220 +0.024795,220 +0.024976,220 +0.025252,220 +0.024912,220 +0.024793,220 +0.024958,220 +0.025602,220 +0.024894,220 +0.024824,220 +0.024999,220 +0.025398,220 +0.024906,220 +0.024775,220 +0.024963,220 +0.025427,220 +0.024907,220 +0.024832,220 +0.024869,220 +0.025427,220 +0.025238,220 +0.024822,220 +0.024873,220 +0.025321,220 +0.024911,220 +0.024858,220 +0.024902,220 +0.025402,220 +0.024943,220 +0.024956,220 +0.024943,220 +0.025503,220 +0.024872,220 +0.024849,220 +0.024872,220 +0.025515,220 +0.024940,220 +0.024794,220 +0.024854,220 +0.025468,220 +0.025027,220 +0.024804,220 +0.024812,220 +0.025526,220 +0.024885,220 +0.024797,220 +0.024926,220 +0.025620,220 +0.024928,220 +0.024839,220 +0.024926,220 +0.025429,220 +0.025009,220 +0.024797,220 +0.024856,220 +0.025474,220 +0.024913,220 +0.024784,220 +0.024895,220 +0.025441,220 +0.025252,220 +0.024798,220 +0.024784,220 +0.025461,220 +0.024823,220 +0.024757,220 +0.025039,220 +0.026051,220 +0.025156,220 +0.026092,220 +0.024877,220 +0.025423,220 +0.024875,220 +0.025803,222 +0.025612,222 +0.026133,222 +0.025814,222 +0.026533,222 +0.025811,222 +0.026799,222 +0.028307,222 +0.026128,222 +0.026274,222 +0.026490,222 +0.025989,222 +0.025987,222 +0.027747,222 +0.026207,222 +0.025834,222 +0.026027,222 +0.027656,222 +0.025946,222 +0.025867,222 +0.025983,222 +0.027567,222 +0.025950,222 +0.025924,222 +0.025934,222 +0.027547,222 +0.025963,222 +0.025842,222 +0.026992,222 +0.026363,222 +0.025988,222 +0.025937,222 +0.027749,222 +0.025962,222 +0.026520,222 +0.026097,222 +0.027628,222 +0.025700,222 +0.026003,222 +0.025730,222 +0.027521,222 +0.025729,222 +0.026208,222 +0.025815,222 +0.027381,222 +0.025782,222 +0.025681,222 +0.025890,222 +0.027240,222 +0.025703,222 +0.026188,222 +0.028180,222 +0.026576,222 +0.025724,222 +0.025760,222 +0.027488,222 +0.026379,222 +0.026555,222 +0.027810,222 +0.029167,222 +0.026614,222 +0.027743,222 +0.027495,222 +0.029081,222 +0.028671,222 +0.027525,222 +0.026388,222 +0.026126,222 +0.026361,222 +0.026159,222 +0.027152,222 +0.026343,222 +0.026092,222 +0.026325,222 +0.026569,222 +0.026080,222 +0.026654,222 +0.026354,222 +0.026697,222 +0.026229,222 +0.026825,222 +0.027730,222 +0.028674,222 +0.027830,222 +0.028155,222 +0.028284,222 +0.028044,222 +0.026559,222 +0.028011,222 +0.033725,222 +0.027623,222 +0.028418,222 +0.031350,222 +0.028729,222 +0.026083,222 +0.026010,222 +0.032434,222 +0.027779,222 +0.026996,222 +0.028396,222 +0.028949,224 +0.027685,224 +0.026853,224 +0.027342,224 +0.026836,224 +0.026846,224 +0.027353,224 +0.027514,224 +0.035273,224 +0.032047,224 +0.027517,224 +0.026843,224 +0.026461,224 +0.026823,224 +0.026482,224 +0.026429,224 +0.026592,224 +0.026880,224 +0.026321,224 +0.026325,224 +0.026351,224 +0.026876,224 +0.026249,224 +0.026443,224 +0.026446,224 +0.026957,224 +0.026337,224 +0.026385,224 +0.026553,224 +0.026630,224 +0.026286,224 +0.026370,224 +0.026752,224 +0.026410,224 +0.026253,224 +0.026425,224 +0.027212,224 +0.026683,224 +0.026352,224 +0.026306,224 +0.026796,224 +0.026319,224 +0.026409,224 +0.027414,224 +0.027239,224 +0.026580,224 +0.026598,224 +0.026857,224 +0.026380,224 +0.026298,224 +0.026425,224 +0.026794,224 +0.026301,224 +0.026350,224 +0.026489,224 +0.027160,224 +0.026314,224 +0.026349,224 +0.026421,224 +0.026793,224 +0.026374,224 +0.026456,224 +0.026503,224 +0.026953,224 +0.026346,224 +0.026434,224 +0.026791,224 +0.026370,224 +0.026421,224 +0.026372,224 +0.026847,224 +0.026371,224 +0.026353,224 +0.026352,224 +0.026915,224 +0.026435,224 +0.026356,224 +0.026417,224 +0.026870,224 +0.026376,224 +0.026433,224 +0.026624,224 +0.027409,224 +0.026897,224 +0.026468,224 +0.026801,224 +0.026327,224 +0.026391,224 +0.026394,224 +0.026912,224 +0.026554,224 +0.026577,224 +0.026535,224 +0.026772,224 +0.026386,224 +0.026433,224 +0.026597,224 +0.026615,224 +0.026654,224 +0.026921,224 +0.028603,226 +0.027266,226 +0.027135,226 +0.027072,226 +0.027474,226 +0.027131,226 +0.027086,226 +0.027116,226 +0.027478,226 +0.027045,226 +0.027091,226 +0.027592,226 +0.027407,226 +0.027063,226 +0.027063,226 +0.027425,226 +0.027157,226 +0.027344,226 +0.027210,226 +0.027718,226 +0.027114,226 +0.027143,226 +0.027466,226 +0.027005,226 +0.027020,226 +0.026995,226 +0.027437,226 +0.027026,226 +0.026995,226 +0.027142,226 +0.027541,226 +0.026967,226 +0.027073,226 +0.027470,226 +0.026976,226 +0.027179,226 +0.027682,226 +0.027688,226 +0.027113,226 +0.027033,226 +0.027071,226 +0.027531,226 +0.026931,226 +0.027160,226 +0.027355,226 +0.027081,226 +0.026994,226 +0.027038,226 +0.027578,226 +0.027147,226 +0.027222,226 +0.027132,226 +0.027484,226 +0.026990,226 +0.027118,226 +0.027287,226 +0.027333,226 +0.027106,226 +0.027130,226 +0.027430,226 +0.027058,226 +0.026931,226 +0.027072,226 +0.027519,226 +0.027034,226 +0.027094,226 +0.027110,226 +0.027528,226 +0.027124,226 +0.027042,226 +0.027476,226 +0.027152,226 +0.027089,226 +0.027805,226 +0.027947,226 +0.027093,226 +0.027021,226 +0.026933,226 +0.027671,226 +0.027016,226 +0.026995,226 +0.027477,226 +0.027034,226 +0.027993,226 +0.027066,226 +0.027560,226 +0.027294,226 +0.027060,226 +0.027045,226 +0.027400,226 +0.027050,226 +0.026985,226 +0.027514,226 +0.027799,226 +0.027884,226 +0.027555,226 +0.027529,226 +0.026994,226 +0.027625,226 +0.027030,226 +0.028396,228 +0.027789,228 +0.027706,228 +0.028292,228 +0.027713,228 +0.027641,228 +0.027753,228 +0.028166,228 +0.027764,228 +0.027839,228 +0.027917,228 +0.028371,228 +0.027708,228 +0.027819,228 +0.028142,228 +0.027675,228 +0.027806,228 +0.027774,228 +0.028090,228 +0.027691,228 +0.028016,228 +0.028347,228 +0.027715,228 +0.027710,228 +0.027669,228 +0.028135,228 +0.027709,228 +0.027739,228 +0.028289,228 +0.027917,228 +0.027731,228 +0.027709,228 +0.028199,228 +0.027688,228 +0.027750,228 +0.027957,228 +0.028081,228 +0.027754,228 +0.027762,228 +0.028137,228 +0.027730,228 +0.027760,228 +0.027624,228 +0.028232,228 +0.027763,228 +0.027841,228 +0.027998,228 +0.028292,228 +0.027739,228 +0.027804,228 +0.028145,228 +0.027705,228 +0.027735,228 +0.027768,228 +0.028157,228 +0.027802,228 +0.027793,228 +0.028159,228 +0.027800,228 +0.027955,228 +0.027726,228 +0.028202,228 +0.027815,228 +0.027765,228 +0.027927,228 +0.028082,228 +0.027911,228 +0.028507,228 +0.029761,228 +0.029541,228 +0.030357,228 +0.029105,228 +0.030264,228 +0.030174,228 +0.028764,228 +0.028345,228 +0.028749,228 +0.028297,228 +0.031972,228 +0.027970,228 +0.028245,228 +0.027927,228 +0.033471,228 +0.029183,228 +0.030954,228 +0.030335,228 +0.029571,228 +0.029664,228 +0.029525,228 +0.029456,228 +0.029034,228 +0.028302,228 +0.028582,228 +0.027869,228 +0.027958,228 +0.032860,228 +0.054217,228 +0.055905,228 +0.032124,228 +0.030037,228 +0.031212,230 +0.031446,230 +0.029493,230 +0.028981,230 +0.029714,230 +0.029217,230 +0.029330,230 +0.029472,230 +0.028810,230 +0.028966,230 +0.028912,230 +0.029368,230 +0.029060,230 +0.028714,230 +0.029201,230 +0.028925,230 +0.029145,230 +0.029026,230 +0.029022,230 +0.028966,230 +0.028870,230 +0.029391,230 +0.028822,230 +0.029057,230 +0.029450,230 +0.028822,230 +0.029209,230 +0.029746,230 +0.029454,230 +0.028988,230 +0.029127,230 +0.029064,230 +0.029241,230 +0.029435,230 +0.029081,230 +0.031390,230 +0.029498,230 +0.029355,230 +0.029553,230 +0.028982,230 +0.028996,230 +0.028980,230 +0.029188,230 +0.029060,230 +0.028818,230 +0.029422,230 +0.029067,230 +0.028992,230 +0.029280,230 +0.029486,230 +0.029258,230 +0.029168,230 +0.029584,230 +0.029309,230 +0.029174,230 +0.029294,230 +0.028859,230 +0.029155,230 +0.028960,230 +0.029254,230 +0.029074,230 +0.030576,230 +0.030232,230 +0.029177,230 +0.029131,230 +0.030245,230 +0.029385,230 +0.028945,230 +0.028784,230 +0.029233,230 +0.029134,230 +0.029420,230 +0.029514,230 +0.028832,230 +0.028563,230 +0.028667,230 +0.029067,230 +0.028526,230 +0.028438,230 +0.028899,230 +0.029096,230 +0.028589,230 +0.028424,230 +0.029054,230 +0.028619,230 +0.028536,230 +0.028929,230 +0.028593,230 +0.029794,230 +0.038274,230 +0.028631,230 +0.028745,230 +0.028442,230 +0.028873,230 +0.028448,230 +0.028783,230 +0.028461,230 +0.028953,230 +0.028479,230 +0.028411,230 +0.029688,232 +0.029182,232 +0.029007,232 +0.029212,232 +0.030177,232 +0.029792,232 +0.030598,232 +0.033744,232 +0.033655,232 +0.030829,232 +0.032066,232 +0.032223,232 +0.031038,232 +0.030659,232 +0.030033,232 +0.029460,232 +0.029567,232 +0.029924,232 +0.029216,232 +0.029134,232 +0.029829,232 +0.029198,232 +0.029112,232 +0.029550,232 +0.029096,232 +0.029180,232 +0.029056,232 +0.029627,232 +0.029195,232 +0.029088,232 +0.029468,232 +0.029486,232 +0.030023,232 +0.029728,232 +0.029768,232 +0.029106,232 +0.029068,232 +0.029595,232 +0.029140,232 +0.029497,232 +0.029585,232 +0.029460,232 +0.029183,232 +0.029034,232 +0.029641,232 +0.029115,232 +0.029208,232 +0.029817,232 +0.029447,232 +0.029239,232 +0.029114,232 +0.029866,232 +0.029166,232 +0.029146,232 +0.029593,232 +0.029066,232 +0.029145,232 +0.029049,232 +0.029621,232 +0.029042,232 +0.029041,232 +0.029647,232 +0.029164,232 +0.029283,232 +0.029343,232 +0.029674,232 +0.029265,232 +0.029332,232 +0.029687,232 +0.029218,232 +0.029087,232 +0.029494,232 +0.029337,232 +0.029189,232 +0.029091,232 +0.029642,232 +0.029189,232 +0.029207,232 +0.029540,232 +0.029594,232 +0.029035,232 +0.029327,232 +0.032385,232 +0.030660,232 +0.029237,232 +0.029852,232 +0.029299,232 +0.029156,232 +0.029341,232 +0.029562,232 +0.029221,232 +0.029107,232 +0.029742,232 +0.029237,232 +0.029049,232 +0.029655,232 +0.029219,232 +0.029072,232 +0.029165,232 +0.029762,232 +0.030358,234 +0.030127,234 +0.030604,234 +0.030153,234 +0.030130,234 +0.030406,234 +0.030486,234 +0.030233,234 +0.030003,234 +0.030490,234 +0.030202,234 +0.030074,234 +0.030438,234 +0.030215,234 +0.030018,234 +0.031065,234 +0.032074,234 +0.030253,234 +0.030099,234 +0.030677,234 +0.031515,234 +0.031204,234 +0.033240,234 +0.031116,234 +0.030199,234 +0.030712,234 +0.030225,234 +0.030108,234 +0.030597,234 +0.030284,234 +0.030132,234 +0.030359,234 +0.030782,234 +0.030082,234 +0.030109,234 +0.030729,234 +0.030159,234 +0.030105,234 +0.030472,234 +0.030409,234 +0.030108,234 +0.030038,234 +0.030701,234 +0.030097,234 +0.032153,234 +0.032187,234 +0.030610,234 +0.030178,234 +0.030790,234 +0.030180,234 +0.030011,234 +0.030187,234 +0.030570,234 +0.030096,234 +0.030133,234 +0.031568,234 +0.030319,234 +0.030084,234 +0.030822,234 +0.030645,234 +0.030130,234 +0.030539,234 +0.031045,234 +0.030107,234 +0.030286,234 +0.030675,234 +0.030380,234 +0.030718,234 +0.031886,234 +0.032856,234 +0.033020,234 +0.032307,234 +0.031610,234 +0.032002,234 +0.032058,234 +0.031528,234 +0.030505,234 +0.030506,234 +0.030780,234 +0.030169,234 +0.030483,234 +0.030792,234 +0.030206,234 +0.030239,234 +0.031256,234 +0.030443,234 +0.030501,234 +0.030984,234 +0.030521,234 +0.030428,234 +0.030373,234 +0.030966,234 +0.030462,234 +0.030669,234 +0.030869,234 +0.030494,234 +0.030628,234 +0.031014,234 +0.031166,234 +0.033034,234 +0.033777,236 +0.033994,236 +0.033944,236 +0.034148,236 +0.031757,236 +0.030948,236 +0.032556,236 +0.031320,236 +0.031032,236 +0.031013,236 +0.032891,236 +0.032434,236 +0.032233,236 +0.034586,236 +0.033126,236 +0.033744,236 +0.034739,236 +0.031994,236 +0.031318,236 +0.033039,236 +0.031272,236 +0.031136,236 +0.032828,236 +0.031334,236 +0.031468,236 +0.032985,236 +0.031576,236 +0.031285,236 +0.033451,236 +0.035104,236 +0.034135,236 +0.032739,236 +0.032981,236 +0.032480,236 +0.032441,236 +0.032488,236 +0.033266,236 +0.031990,236 +0.031856,236 +0.031373,236 +0.031471,236 +0.031751,236 +0.032149,236 +0.031541,236 +0.031784,236 +0.031390,236 +0.031234,236 +0.031803,236 +0.031537,236 +0.031533,236 +0.032068,236 +0.031771,236 +0.031798,236 +0.031953,236 +0.031569,236 +0.031234,236 +0.031939,236 +0.031545,236 +0.031601,236 +0.031578,236 +0.031932,236 +0.031237,236 +0.031325,236 +0.031721,236 +0.031366,236 +0.031220,236 +0.031750,236 +0.031240,236 +0.032455,236 +0.033766,236 +0.031280,236 +0.030793,236 +0.031328,236 +0.030853,236 +0.030771,236 +0.031968,236 +0.031074,236 +0.030757,236 +0.030916,236 +0.032139,236 +0.032433,236 +0.034592,236 +0.033978,236 +0.032397,236 +0.032334,236 +0.031628,236 +0.031247,236 +0.031339,236 +0.031525,236 +0.032272,236 +0.034320,236 +0.035302,236 +0.033631,236 +0.032443,236 +0.031868,236 +0.036478,236 +0.032029,236 +0.031947,236 +0.030840,236 +0.030834,236 +0.032290,238 +0.031663,238 +0.031577,238 +0.033485,238 +0.033872,238 +0.035361,238 +0.035761,238 +0.032871,238 +0.032131,238 +0.032473,238 +0.031940,238 +0.031745,238 +0.032728,238 +0.032135,238 +0.031720,238 +0.032094,238 +0.033397,238 +0.032657,238 +0.034607,238 +0.034474,238 +0.034204,238 +0.038265,238 +0.033422,238 +0.032119,238 +0.032749,238 +0.032330,238 +0.033634,238 +0.034926,238 +0.036489,238 +0.034506,238 +0.033690,238 +0.032542,238 +0.032052,238 +0.032223,238 +0.033186,238 +0.034131,238 +0.034831,238 +0.034710,238 +0.033838,238 +0.032466,238 +0.032274,238 +0.032050,238 +0.031949,238 +0.032542,238 +0.031693,238 +0.031649,238 +0.032263,238 +0.031716,238 +0.031614,238 +0.032194,238 +0.033197,238 +0.035426,238 +0.035452,238 +0.033799,238 +0.034757,238 +0.033376,238 +0.032357,238 +0.032370,238 +0.033924,238 +0.031932,238 +0.031730,238 +0.033355,238 +0.031602,238 +0.032070,238 +0.033182,238 +0.031728,238 +0.031650,238 +0.033281,238 +0.031678,238 +0.031882,238 +0.034564,238 +0.031760,238 +0.031552,238 +0.033319,238 +0.031600,238 +0.031533,238 +0.032823,238 +0.032028,238 +0.031525,238 +0.031993,238 +0.032756,238 +0.031828,238 +0.031746,238 +0.033238,238 +0.031641,238 +0.031616,238 +0.033137,238 +0.031554,238 +0.031585,238 +0.033085,238 +0.031562,238 +0.031473,238 +0.033401,238 +0.031639,238 +0.031512,238 +0.033092,238 +0.031949,238 +0.031735,238 +0.033181,238 +0.031755,238 +0.032343,240 +0.033862,240 +0.034092,240 +0.032217,240 +0.033336,240 +0.032581,240 +0.032189,240 +0.032208,240 +0.033655,240 +0.032166,240 +0.032460,240 +0.033617,240 +0.032209,240 +0.032333,240 +0.033738,240 +0.032283,240 +0.032250,240 +0.033561,240 +0.033114,240 +0.034166,240 +0.035387,240 +0.034878,240 +0.034778,240 +0.035309,240 +0.032956,240 +0.032670,240 +0.034152,240 +0.032453,240 +0.032523,240 +0.033747,240 +0.032219,240 +0.032256,240 +0.034852,240 +0.032188,240 +0.032216,240 +0.033476,240 +0.032083,240 +0.033007,240 +0.034557,240 +0.032568,240 +0.032249,240 +0.033585,240 +0.032223,240 +0.032403,240 +0.033472,240 +0.032224,240 +0.032070,240 +0.032631,240 +0.032187,240 +0.032076,240 +0.032691,240 +0.032156,240 +0.032070,240 +0.032641,240 +0.032126,240 +0.032065,240 +0.032322,240 +0.032443,240 +0.032428,240 +0.032233,240 +0.032845,240 +0.032217,240 +0.032185,240 +0.032981,240 +0.032174,240 +0.032270,240 +0.032766,240 +0.032613,240 +0.032274,240 +0.032744,240 +0.032155,240 +0.032178,240 +0.032682,240 +0.032216,240 +0.032185,240 +0.032755,240 +0.032373,240 +0.032354,240 +0.032655,240 +0.032702,240 +0.032727,240 +0.033254,240 +0.032606,240 +0.032792,240 +0.033288,240 +0.032707,240 +0.032698,240 +0.032970,240 +0.033131,240 +0.032702,240 +0.032809,240 +0.032904,240 +0.032723,240 +0.032811,240 +0.033252,240 +0.032700,240 +0.032779,240 +0.033331,240 +0.032580,240 +0.032634,240 +0.034677,242 +0.034333,242 +0.035353,242 +0.037132,242 +0.035416,242 +0.035410,242 +0.034834,242 +0.033790,242 +0.034383,242 +0.033788,242 +0.033457,242 +0.034437,242 +0.035468,242 +0.036031,242 +0.038102,242 +0.036129,242 +0.036671,242 +0.038472,242 +0.034601,242 +0.034189,242 +0.036864,242 +0.036533,242 +0.036482,242 +0.037213,242 +0.036504,242 +0.038268,242 +0.036324,242 +0.038244,242 +0.035677,242 +0.035634,242 +0.034792,242 +0.036135,242 +0.036817,242 +0.037817,242 +0.037462,242 +0.036281,242 +0.036753,242 +0.036863,242 +0.035234,242 +0.034721,242 +0.034226,242 +0.034461,242 +0.035540,242 +0.036877,242 +0.036143,242 +0.036095,242 +0.035628,242 +0.034672,242 +0.034271,242 +0.034081,242 +0.034506,242 +0.035000,242 +0.035985,242 +0.035727,242 +0.036704,242 +0.037648,242 +0.035403,242 +0.034006,242 +0.033905,242 +0.034223,242 +0.036224,242 +0.036299,242 +0.038138,242 +0.036845,242 +0.036640,242 +0.036335,242 +0.036843,242 +0.034545,242 +0.033487,242 +0.036115,242 +0.036344,242 +0.036293,242 +0.035307,242 +0.038131,242 +0.045884,242 +0.042915,242 +0.035826,242 +0.036449,242 +0.035700,242 +0.036295,242 +0.039672,242 +0.035614,242 +0.035382,242 +0.034891,242 +0.033864,242 +0.034920,242 +0.035236,242 +0.036033,242 +0.037853,242 +0.036326,242 +0.036606,242 +0.035340,242 +0.038935,242 +0.033753,242 +0.033520,242 +0.038138,242 +0.035472,242 +0.035660,242 +0.035369,242 +0.034935,242 +0.035351,244 +0.035871,244 +0.035230,244 +0.035925,244 +0.038724,244 +0.040129,244 +0.039379,244 +0.038537,244 +0.040497,244 +0.036633,244 +0.035343,244 +0.038747,244 +0.038757,244 +0.045130,244 +0.036858,244 +0.041927,244 +0.039854,244 +0.037641,244 +0.038082,244 +0.043394,244 +0.039307,244 +0.038785,244 +0.037869,244 +0.037780,244 +0.035217,244 +0.034456,244 +0.036203,244 +0.037800,244 +0.040150,244 +0.050008,244 +0.049812,244 +0.041325,244 +0.045832,244 +0.052610,244 +0.052008,244 +0.042045,244 +0.067749,244 +0.072703,244 +0.072166,244 +0.046886,244 +0.038491,244 +0.039296,244 +0.037777,244 +0.049037,244 +0.036943,244 +0.036475,244 +0.036480,244 +0.036279,244 +0.036979,244 +0.035601,244 +0.037431,244 +0.035981,244 +0.035024,244 +0.035396,244 +0.035535,244 +0.035078,244 +0.035980,244 +0.035874,244 +0.035106,244 +0.034888,244 +0.034206,244 +0.034227,244 +0.034895,244 +0.038662,244 +0.034279,244 +0.039300,244 +0.035010,244 +0.034118,244 +0.034979,244 +0.034239,244 +0.034100,244 +0.047447,244 +0.034204,244 +0.034705,244 +0.034445,244 +0.034354,244 +0.034837,244 +0.034082,244 +0.034278,244 +0.034973,244 +0.034284,244 +0.034085,244 +0.034760,244 +0.034259,244 +0.034188,244 +0.034593,244 +0.034133,244 +0.034105,244 +0.034614,244 +0.034142,244 +0.034141,244 +0.034646,244 +0.034292,244 +0.034239,244 +0.034943,244 +0.034119,244 +0.034072,244 +0.034719,244 +0.034357,244 +0.034145,244 +0.035788,246 +0.035212,246 +0.035194,246 +0.035476,246 +0.035298,246 +0.036709,246 +0.035462,246 +0.035223,246 +0.035805,246 +0.035209,246 +0.035165,246 +0.035768,246 +0.035274,246 +0.035219,246 +0.035662,246 +0.035250,246 +0.035914,246 +0.036826,246 +0.038206,246 +0.044112,246 +0.041421,246 +0.043092,246 +0.042983,246 +0.036796,246 +0.039207,246 +0.038178,246 +0.037697,246 +0.039172,246 +0.038369,246 +0.038217,246 +0.037788,246 +0.039724,246 +0.043731,246 +0.039567,246 +0.039280,246 +0.036046,246 +0.035558,246 +0.037500,246 +0.037027,246 +0.037366,246 +0.038814,246 +0.037329,246 +0.036223,246 +0.036504,246 +0.035596,246 +0.036479,246 +0.037131,246 +0.035821,246 +0.036483,246 +0.035860,246 +0.035607,246 +0.036814,246 +0.035801,246 +0.035939,246 +0.036605,246 +0.036062,246 +0.036307,246 +0.035925,246 +0.036517,246 +0.037065,246 +0.036557,246 +0.035529,246 +0.036068,246 +0.036164,246 +0.035511,246 +0.036099,246 +0.035529,246 +0.035457,246 +0.036358,246 +0.035706,246 +0.035986,246 +0.037572,246 +0.035684,246 +0.036861,246 +0.035794,246 +0.035796,246 +0.036792,246 +0.035434,246 +0.035464,246 +0.036024,246 +0.035524,246 +0.035594,246 +0.036032,246 +0.035576,246 +0.036133,246 +0.035656,246 +0.035641,246 +0.036242,246 +0.035668,246 +0.035511,246 +0.036249,246 +0.035679,246 +0.035835,246 +0.036350,246 +0.035769,246 +0.035779,246 +0.036081,246 +0.035742,246 +0.036305,246 +0.035666,246 +0.036538,248 +0.036950,248 +0.036891,248 +0.037282,248 +0.037122,248 +0.035963,248 +0.035966,248 +0.036533,248 +0.035865,248 +0.036491,248 +0.036088,248 +0.035918,248 +0.036324,248 +0.036204,248 +0.036010,248 +0.036408,248 +0.036022,248 +0.035875,248 +0.036484,248 +0.035847,248 +0.035988,248 +0.036638,248 +0.035939,248 +0.036479,248 +0.036101,248 +0.035841,248 +0.037387,248 +0.035967,248 +0.036096,248 +0.036605,248 +0.037319,248 +0.036080,248 +0.036739,248 +0.035905,248 +0.036373,248 +0.036001,248 +0.035857,248 +0.036426,248 +0.036060,248 +0.036116,248 +0.036435,248 +0.036006,248 +0.035934,248 +0.036601,248 +0.036661,248 +0.038456,248 +0.037034,248 +0.036383,248 +0.037245,248 +0.036645,248 +0.036301,248 +0.036790,248 +0.036420,248 +0.036386,248 +0.039304,248 +0.037737,248 +0.036706,248 +0.037622,248 +0.039952,248 +0.039045,248 +0.039536,248 +0.038979,248 +0.037102,248 +0.040524,248 +0.047871,248 +0.039514,248 +0.038784,248 +0.036877,248 +0.036276,248 +0.036576,248 +0.036445,248 +0.036307,248 +0.038011,248 +0.036560,248 +0.036393,248 +0.039587,248 +0.037070,248 +0.038024,248 +0.036550,248 +0.036464,248 +0.039896,248 +0.040696,248 +0.039671,248 +0.037722,248 +0.037045,248 +0.037030,248 +0.036760,248 +0.036654,248 +0.036918,248 +0.036292,248 +0.036506,248 +0.036907,248 +0.036397,248 +0.036666,248 +0.037172,248 +0.036737,248 +0.037231,248 +0.036764,248 +0.037268,248 +0.037170,248 +0.038086,250 +0.038827,250 +0.038990,250 +0.037656,250 +0.039527,250 +0.037633,250 +0.037549,250 +0.038221,250 +0.037844,250 +0.038290,250 +0.037835,250 +0.037638,250 +0.038288,250 +0.037639,250 +0.037595,250 +0.038418,250 +0.037800,250 +0.038053,250 +0.037847,250 +0.037909,250 +0.038100,250 +0.037897,250 +0.037869,250 +0.038147,250 +0.037684,250 +0.038149,250 +0.038995,250 +0.043489,250 +0.045378,250 +0.039512,250 +0.043945,250 +0.041917,250 +0.046253,250 +0.051930,250 +0.061852,250 +0.060873,250 +0.055203,250 +0.056209,250 +0.049196,250 +0.052504,250 +0.054557,250 +0.049361,250 +0.055008,250 +0.048738,250 +0.048519,250 +0.040005,250 +0.052799,250 +0.053327,250 +0.043411,250 +0.044696,250 +0.042284,250 +0.040053,250 +0.043783,250 +0.040671,250 +0.043014,250 +0.041375,250 +0.039464,250 +0.038793,250 +0.037435,250 +0.037281,250 +0.037981,250 +0.037245,250 +0.037327,250 +0.037564,250 +0.037241,250 +0.037451,250 +0.039258,250 +0.037382,250 +0.037922,250 +0.037268,250 +0.037188,250 +0.037485,250 +0.037387,250 +0.038525,250 +0.038325,250 +0.038383,250 +0.038073,250 +0.040982,250 +0.074718,250 +0.038744,250 +0.038323,250 +0.038145,250 +0.037754,250 +0.039391,250 +0.037818,250 +0.038091,250 +0.038323,250 +0.037743,250 +0.038526,250 +0.037601,250 +0.037652,250 +0.038597,250 +0.037943,250 +0.038059,250 +0.037976,250 +0.037702,250 +0.038592,250 +0.037895,250 +0.037840,250 +0.038815,250 +0.038699,252 +0.039045,252 +0.038515,252 +0.038525,252 +0.039196,252 +0.038629,252 +0.038606,252 +0.039003,252 +0.038791,252 +0.039151,252 +0.038370,252 +0.038741,252 +0.039158,252 +0.038488,252 +0.039305,252 +0.038434,252 +0.038216,252 +0.039576,252 +0.038537,252 +0.038717,252 +0.038729,252 +0.038565,252 +0.039139,252 +0.038482,252 +0.038509,252 +0.039110,252 +0.038557,252 +0.039360,252 +0.038386,252 +0.038276,252 +0.039334,252 +0.038254,252 +0.038748,252 +0.038706,252 +0.038297,252 +0.039110,252 +0.038381,252 +0.038306,252 +0.038858,252 +0.038423,252 +0.039294,252 +0.038163,252 +0.037901,252 +0.038654,252 +0.037990,252 +0.038164,252 +0.038542,252 +0.037878,252 +0.038594,252 +0.038343,252 +0.038125,252 +0.038493,252 +0.037896,252 +0.038614,252 +0.038042,252 +0.037831,252 +0.038678,252 +0.038131,252 +0.038024,252 +0.038536,252 +0.042861,252 +0.038422,252 +0.037953,252 +0.037943,252 +0.038348,252 +0.037991,252 +0.038721,252 +0.038117,252 +0.037917,252 +0.038919,252 +0.037938,252 +0.038027,252 +0.039191,252 +0.037861,252 +0.038955,252 +0.040984,252 +0.038092,252 +0.040819,252 +0.038052,252 +0.039542,252 +0.037981,252 +0.037802,252 +0.039468,252 +0.037887,252 +0.037890,252 +0.038429,252 +0.037804,252 +0.038395,252 +0.038648,252 +0.037949,252 +0.038276,252 +0.037851,252 +0.038169,252 +0.039792,252 +0.042231,252 +0.039571,252 +0.038210,252 +0.038467,252 +0.038717,252 +0.038249,252 +0.039936,254 +0.039465,254 +0.039334,254 +0.040065,254 +0.039383,254 +0.040090,254 +0.039435,254 +0.039177,254 +0.040657,254 +0.039187,254 +0.040312,254 +0.039448,254 +0.039488,254 +0.040645,254 +0.040085,254 +0.041609,254 +0.039331,254 +0.039357,254 +0.041614,254 +0.039211,254 +0.041913,254 +0.039449,254 +0.039495,254 +0.041637,254 +0.039478,254 +0.041623,254 +0.039509,254 +0.039517,254 +0.043351,254 +0.039365,254 +0.041827,254 +0.039644,254 +0.039558,254 +0.041600,254 +0.039541,254 +0.041388,254 +0.039354,254 +0.039592,254 +0.041203,254 +0.039634,254 +0.041391,254 +0.039556,254 +0.039549,254 +0.041227,254 +0.039462,254 +0.041692,254 +0.039581,254 +0.039455,254 +0.041425,254 +0.039442,254 +0.041406,254 +0.039491,254 +0.039306,254 +0.042753,254 +0.039219,254 +0.041637,254 +0.039697,254 +0.039207,254 +0.043401,254 +0.039485,254 +0.041004,254 +0.038971,254 +0.039117,254 +0.041379,254 +0.038966,254 +0.040863,254 +0.038944,254 +0.038962,254 +0.042000,254 +0.039254,254 +0.039711,254 +0.039101,254 +0.038850,254 +0.040741,254 +0.039132,254 +0.039386,254 +0.039018,254 +0.038897,254 +0.039604,254 +0.039720,254 +0.039484,254 +0.039226,254 +0.039947,254 +0.039878,254 +0.039104,254 +0.039400,254 +0.040058,254 +0.039104,254 +0.039540,254 +0.039171,254 +0.039434,254 +0.039560,254 +0.039144,254 +0.039675,254 +0.039228,254 +0.039316,254 +0.039891,254 +0.038911,254 +0.039681,254 +0.039048,254 +0.045297,256 +0.045849,256 +0.045395,256 +0.046163,256 +0.045344,256 +0.046896,256 +0.045613,256 +0.048069,256 +0.045930,256 +0.046376,256 +0.045710,256 +0.046088,256 +0.047383,256 +0.045935,256 +0.047501,256 +0.045963,256 +0.047293,256 +0.046090,256 +0.047512,256 +0.045820,256 +0.047847,256 +0.045723,256 +0.047504,256 +0.045714,256 +0.046280,256 +0.048077,256 +0.045847,256 +0.047505,256 +0.045955,256 +0.047295,256 +0.045545,256 +0.047311,256 +0.045926,256 +0.047432,256 +0.045836,256 +0.047683,256 +0.045752,256 +0.047709,256 +0.046274,256 +0.046592,256 +0.047855,256 +0.050953,256 +0.047916,256 +0.045931,256 +0.047817,256 +0.046053,256 +0.049004,256 +0.046315,256 +0.048031,256 +0.046460,256 +0.048593,256 +0.046224,256 +0.047719,256 +0.045832,256 +0.047828,256 +0.046105,256 +0.045767,256 +0.047757,256 +0.045915,256 +0.048175,256 +0.046601,256 +0.047080,256 +0.045218,256 +0.046985,256 +0.045308,256 +0.046867,256 +0.045136,256 +0.047985,256 +0.045647,256 +0.045707,256 +0.046499,256 +0.045497,256 +0.047160,256 +0.045219,256 +0.047095,256 +0.045170,256 +0.046713,256 +0.045684,256 +0.046960,256 +0.045355,256 +0.046547,256 +0.045593,256 +0.045185,256 +0.046832,256 +0.045076,256 +0.047068,256 +0.045091,256 +0.046798,256 +0.045298,256 +0.048013,256 +0.045422,256 +0.046954,256 +0.045552,256 +0.046425,256 +0.046089,256 +0.045377,256 +0.045881,256 +0.045229,256 +0.045887,256 +0.045230,256 +0.042081,258 +0.042614,258 +0.041833,258 +0.042086,258 +0.041492,258 +0.043070,258 +0.042303,258 +0.042069,258 +0.042106,258 +0.041970,258 +0.042518,258 +0.041982,258 +0.042699,258 +0.041835,258 +0.042066,258 +0.042568,258 +0.042002,258 +0.042405,258 +0.042074,258 +0.042611,258 +0.042106,258 +0.041835,258 +0.042608,258 +0.041898,258 +0.042480,258 +0.042001,258 +0.041909,258 +0.042463,258 +0.045149,258 +0.043737,258 +0.042288,258 +0.042411,258 +0.041625,258 +0.042247,258 +0.042165,258 +0.041585,258 +0.042754,258 +0.041842,258 +0.043590,258 +0.041990,258 +0.043031,258 +0.044013,258 +0.041785,258 +0.043605,258 +0.041802,258 +0.043916,258 +0.041727,258 +0.041861,258 +0.043481,258 +0.042282,258 +0.043696,258 +0.042073,258 +0.043473,258 +0.041695,258 +0.041836,258 +0.043509,258 +0.041741,258 +0.043380,258 +0.042463,258 +0.041709,258 +0.042632,258 +0.042106,258 +0.042620,258 +0.042544,258 +0.045255,258 +0.043871,258 +0.041508,258 +0.041967,258 +0.041501,258 +0.041899,258 +0.041971,258 +0.041960,258 +0.041676,258 +0.041295,258 +0.042285,258 +0.041558,258 +0.042110,258 +0.041293,258 +0.041305,258 +0.042215,258 +0.041494,258 +0.041830,258 +0.041343,258 +0.041394,258 +0.042209,258 +0.041295,258 +0.042063,258 +0.043349,258 +0.045494,258 +0.043548,258 +0.041453,258 +0.046725,258 +0.041732,258 +0.046617,258 +0.041177,258 +0.046633,258 +0.041518,258 +0.045840,258 +0.042901,258 +0.041345,258 +0.048255,260 +0.042324,260 +0.047944,260 +0.042678,260 +0.047935,260 +0.042694,260 +0.044666,260 +0.046949,260 +0.042397,260 +0.048135,260 +0.042603,260 +0.047941,260 +0.042574,260 +0.048251,260 +0.042447,260 +0.047803,260 +0.042785,260 +0.042291,260 +0.048307,260 +0.042330,260 +0.048392,260 +0.042397,260 +0.048226,260 +0.042667,260 +0.047873,260 +0.042525,260 +0.042523,260 +0.048155,260 +0.042379,260 +0.049098,260 +0.042286,260 +0.048006,260 +0.042583,260 +0.047957,260 +0.042498,260 +0.043690,260 +0.046701,260 +0.042336,260 +0.048451,260 +0.042400,260 +0.048644,260 +0.042495,260 +0.048564,260 +0.042649,260 +0.045546,260 +0.045037,260 +0.042463,260 +0.048389,260 +0.042364,260 +0.047973,260 +0.042311,260 +0.048661,260 +0.042668,260 +0.047775,260 +0.043000,260 +0.042421,260 +0.049528,260 +0.042978,260 +0.048586,260 +0.042669,260 +0.048119,260 +0.042438,260 +0.048009,260 +0.042556,260 +0.043498,260 +0.047218,260 +0.042378,260 +0.048384,260 +0.042388,260 +0.048305,260 +0.042455,260 +0.048446,260 +0.042351,260 +0.044113,260 +0.047139,260 +0.042322,260 +0.048408,260 +0.042434,260 +0.048234,260 +0.042494,260 +0.048293,260 +0.042480,260 +0.047994,260 +0.042675,260 +0.042313,260 +0.048334,260 +0.042412,260 +0.048358,260 +0.042306,260 +0.048182,260 +0.042543,260 +0.048056,260 +0.042424,260 +0.042349,260 +0.048096,260 +0.042478,260 +0.049024,260 +0.042525,260 +0.048152,260 +0.042303,260 +0.051544,262 +0.045612,262 +0.050535,262 +0.045288,262 +0.048709,262 +0.047252,262 +0.045144,262 +0.051457,262 +0.045062,262 +0.051046,262 +0.045063,262 +0.050513,262 +0.045180,262 +0.051306,262 +0.045181,262 +0.050392,262 +0.045029,262 +0.051435,262 +0.045635,262 +0.050680,262 +0.045062,262 +0.050670,262 +0.045119,262 +0.050427,262 +0.045001,262 +0.046788,262 +0.049032,262 +0.045134,262 +0.050543,262 +0.045412,262 +0.051262,262 +0.045004,262 +0.050438,262 +0.045083,262 +0.051226,262 +0.045002,262 +0.050308,262 +0.045029,262 +0.052155,262 +0.045088,262 +0.050504,262 +0.045078,262 +0.050475,262 +0.045118,262 +0.050361,262 +0.046407,262 +0.045584,262 +0.050035,262 +0.045313,262 +0.051202,262 +0.045060,262 +0.051355,262 +0.045182,262 +0.051282,262 +0.049080,262 +0.046642,262 +0.045015,262 +0.045746,262 +0.045166,262 +0.045918,262 +0.045070,262 +0.045495,262 +0.045338,262 +0.045044,262 +0.045700,262 +0.045056,262 +0.045916,262 +0.045165,262 +0.045669,262 +0.045212,262 +0.045766,262 +0.045216,262 +0.045616,262 +0.045724,262 +0.045022,262 +0.045808,262 +0.045308,262 +0.047476,262 +0.045366,262 +0.045937,262 +0.045114,262 +0.045810,262 +0.045430,262 +0.045258,262 +0.045845,262 +0.045254,262 +0.045789,262 +0.045173,262 +0.045872,262 +0.045020,262 +0.045713,262 +0.045355,262 +0.045629,262 +0.045504,262 +0.045357,262 +0.045746,262 +0.045122,262 +0.045805,262 +0.045116,262 +0.045932,262 +0.045572,264 +0.045377,264 +0.044911,264 +0.045498,264 +0.045480,264 +0.045013,264 +0.045361,264 +0.044835,264 +0.045345,264 +0.044850,264 +0.045363,264 +0.044819,264 +0.045119,264 +0.045038,264 +0.044929,264 +0.045277,264 +0.045018,264 +0.045337,264 +0.044766,264 +0.045071,264 +0.045004,264 +0.045191,264 +0.044777,264 +0.044833,264 +0.045091,264 +0.044788,264 +0.045319,264 +0.044897,264 +0.045363,264 +0.045009,264 +0.045403,264 +0.044783,264 +0.044908,264 +0.045218,264 +0.044848,264 +0.045603,264 +0.044800,264 +0.045850,264 +0.047181,264 +0.045717,264 +0.044871,264 +0.045471,264 +0.044947,264 +0.044940,264 +0.045371,264 +0.044852,264 +0.045318,264 +0.044823,264 +0.045581,264 +0.044927,264 +0.045449,264 +0.045173,264 +0.044880,264 +0.045276,264 +0.045357,264 +0.045468,264 +0.044777,264 +0.045276,264 +0.044682,264 +0.047804,264 +0.046794,264 +0.046725,264 +0.044823,264 +0.044911,264 +0.049118,264 +0.045572,264 +0.045130,264 +0.045017,264 +0.045110,264 +0.044772,264 +0.045407,264 +0.045054,264 +0.045183,264 +0.045040,264 +0.044867,264 +0.045725,264 +0.044958,264 +0.045318,264 +0.044947,264 +0.045226,264 +0.044847,264 +0.045403,264 +0.045587,264 +0.045449,264 +0.045266,264 +0.044929,264 +0.045318,264 +0.044923,264 +0.045372,264 +0.045250,264 +0.045429,264 +0.044795,264 +0.044988,264 +0.045278,264 +0.045132,264 +0.045579,264 +0.044765,264 +0.045500,264 +0.044827,264 +0.045411,264 +0.048093,266 +0.048554,266 +0.047857,266 +0.048832,266 +0.048130,266 +0.048065,266 +0.048130,266 +0.047908,266 +0.048489,266 +0.048060,266 +0.048360,266 +0.047964,266 +0.048440,266 +0.047902,266 +0.048555,266 +0.048409,266 +0.048603,266 +0.047875,266 +0.048451,266 +0.047899,266 +0.048415,266 +0.047878,266 +0.048513,266 +0.048003,266 +0.048460,266 +0.048109,266 +0.048418,266 +0.047827,266 +0.048355,266 +0.048271,266 +0.048101,266 +0.048250,266 +0.047861,266 +0.048418,266 +0.047947,266 +0.048736,266 +0.047898,266 +0.048680,266 +0.047957,266 +0.048521,266 +0.048046,266 +0.048546,266 +0.048040,266 +0.049049,266 +0.047991,266 +0.048726,266 +0.048007,266 +0.048397,266 +0.047939,266 +0.048432,266 +0.047919,266 +0.048511,266 +0.048066,266 +0.048158,266 +0.048293,266 +0.048010,266 +0.048770,266 +0.048032,266 +0.048726,266 +0.047981,266 +0.048417,266 +0.048049,266 +0.048393,266 +0.048031,266 +0.048320,266 +0.048281,266 +0.049239,266 +0.047884,266 +0.048425,266 +0.047931,266 +0.048400,266 +0.047844,266 +0.048448,266 +0.048015,266 +0.048546,266 +0.048001,266 +0.048574,266 +0.048322,266 +0.048336,266 +0.048125,266 +0.048212,266 +0.048421,266 +0.048035,266 +0.048492,266 +0.047906,266 +0.048542,266 +0.048039,266 +0.048647,266 +0.047852,266 +0.048421,266 +0.047796,266 +0.048576,266 +0.047807,266 +0.048873,266 +0.047809,266 +0.048401,266 +0.047913,266 +0.048670,266 +0.048133,266 +0.048864,266 +0.048275,268 +0.048696,268 +0.048144,268 +0.048465,268 +0.048125,268 +0.048255,268 +0.048431,268 +0.048170,268 +0.048705,268 +0.048052,268 +0.048540,268 +0.048015,268 +0.048595,268 +0.048046,268 +0.048586,268 +0.048002,268 +0.049067,268 +0.048110,268 +0.048884,268 +0.048117,268 +0.048584,268 +0.048039,268 +0.048778,268 +0.048048,268 +0.048461,268 +0.048128,268 +0.048589,268 +0.048218,268 +0.048745,268 +0.048078,268 +0.048564,268 +0.048037,268 +0.048153,268 +0.048413,268 +0.047985,268 +0.048706,268 +0.048094,268 +0.048498,268 +0.048325,268 +0.048784,268 +0.048145,268 +0.048614,268 +0.048069,268 +0.048603,268 +0.048081,268 +0.048666,268 +0.048397,268 +0.048522,268 +0.048516,268 +0.048846,268 +0.048086,268 +0.048523,268 +0.048044,268 +0.048577,268 +0.048127,268 +0.048515,268 +0.048060,268 +0.048525,268 +0.048301,268 +0.048229,268 +0.048670,268 +0.048532,268 +0.048557,268 +0.048008,268 +0.048688,268 +0.048004,268 +0.048554,268 +0.048121,268 +0.048588,268 +0.048915,268 +0.048832,268 +0.048046,268 +0.048471,268 +0.048041,268 +0.048648,268 +0.048166,268 +0.048498,268 +0.048032,268 +0.048534,268 +0.048124,268 +0.048592,268 +0.048203,268 +0.048449,268 +0.048046,268 +0.048568,268 +0.048168,268 +0.048328,268 +0.048358,268 +0.048142,268 +0.048514,268 +0.048489,268 +0.048674,268 +0.048018,268 +0.048568,268 +0.048246,268 +0.048617,268 +0.048178,268 +0.048707,268 +0.048113,268 +0.048557,268 +0.051342,270 +0.051541,270 +0.051119,270 +0.051441,270 +0.051187,270 +0.051469,270 +0.050958,270 +0.051311,270 +0.051083,270 +0.051505,270 +0.051255,270 +0.051646,270 +0.050894,270 +0.051497,270 +0.050975,270 +0.051511,270 +0.050891,270 +0.051461,270 +0.051058,270 +0.051288,270 +0.051191,270 +0.051547,270 +0.051195,270 +0.051279,270 +0.051556,270 +0.051011,270 +0.051468,270 +0.050985,270 +0.051403,270 +0.051001,270 +0.051945,270 +0.050997,270 +0.051442,270 +0.051165,270 +0.051447,270 +0.051047,270 +0.051476,270 +0.051056,270 +0.051486,270 +0.051006,270 +0.051561,270 +0.051181,270 +0.051743,270 +0.051012,270 +0.051433,270 +0.050948,270 +0.051489,270 +0.051084,270 +0.051561,270 +0.051276,270 +0.051648,270 +0.050909,270 +0.051372,270 +0.051003,270 +0.051535,270 +0.050986,270 +0.051464,270 +0.050818,270 +0.051522,270 +0.050939,270 +0.051597,270 +0.050956,270 +0.051504,270 +0.050854,270 +0.051526,270 +0.050830,270 +0.051485,270 +0.050898,270 +0.051404,270 +0.056755,270 +0.052415,270 +0.051187,270 +0.051174,270 +0.051398,270 +0.050898,270 +0.051421,270 +0.051019,270 +0.051525,270 +0.050984,270 +0.051634,270 +0.051033,270 +0.051498,270 +0.050999,270 +0.051528,270 +0.050991,270 +0.051460,270 +0.050919,270 +0.051540,270 +0.051221,270 +0.051615,270 +0.050929,270 +0.051637,270 +0.051243,270 +0.051707,270 +0.051443,270 +0.051519,270 +0.050925,270 +0.051655,270 +0.051174,270 +0.051755,270 +0.049435,272 +0.049712,272 +0.049166,272 +0.049631,272 +0.049171,272 +0.049689,272 +0.049117,272 +0.049701,272 +0.049306,272 +0.049876,272 +0.049086,272 +0.049662,272 +0.049139,272 +0.049734,272 +0.049166,272 +0.049868,272 +0.049411,272 +0.049673,272 +0.049128,272 +0.049832,272 +0.049201,272 +0.049724,272 +0.051215,272 +0.050721,272 +0.049178,272 +0.050670,272 +0.049205,272 +0.050658,272 +0.049243,272 +0.050589,272 +0.049139,272 +0.050582,272 +0.049158,272 +0.050957,272 +0.049261,272 +0.050503,272 +0.049055,272 +0.050749,272 +0.049715,272 +0.051716,272 +0.049319,272 +0.050582,272 +0.049194,272 +0.050733,272 +0.049297,272 +0.051079,272 +0.049202,272 +0.050302,272 +0.049431,272 +0.050735,272 +0.049148,272 +0.050486,272 +0.049299,272 +0.050421,272 +0.049318,272 +0.050266,272 +0.049265,272 +0.050209,272 +0.049803,272 +0.051081,272 +0.049932,272 +0.050271,272 +0.049587,272 +0.050158,272 +0.049592,272 +0.050020,272 +0.049635,272 +0.050008,272 +0.050422,272 +0.049948,272 +0.049773,272 +0.049773,272 +0.050027,272 +0.049700,272 +0.050082,272 +0.049474,272 +0.050181,272 +0.049382,272 +0.050677,272 +0.049202,272 +0.052091,272 +0.049543,272 +0.050200,272 +0.049158,272 +0.050576,272 +0.049119,272 +0.050529,272 +0.049232,272 +0.050573,272 +0.049112,272 +0.050611,272 +0.049247,272 +0.050781,272 +0.049196,272 +0.050974,272 +0.049342,272 +0.050484,272 +0.049243,272 +0.050431,272 +0.049244,272 +0.055936,274 +0.053476,274 +0.053482,274 +0.054314,274 +0.052765,274 +0.054135,274 +0.052769,274 +0.054386,274 +0.053011,274 +0.054183,274 +0.052823,274 +0.054154,274 +0.052873,274 +0.054188,274 +0.052746,274 +0.054069,274 +0.052887,274 +0.054350,274 +0.054909,274 +0.053474,274 +0.053989,274 +0.052878,274 +0.054203,274 +0.052980,274 +0.054277,274 +0.052768,274 +0.054341,274 +0.052917,274 +0.054331,274 +0.052932,274 +0.054125,274 +0.052855,274 +0.054084,274 +0.053899,274 +0.053103,274 +0.054212,274 +0.052839,274 +0.055637,274 +0.053230,274 +0.054193,274 +0.052798,274 +0.054121,274 +0.052883,274 +0.054491,274 +0.053176,274 +0.054155,274 +0.052861,274 +0.054216,274 +0.053711,274 +0.053177,274 +0.054143,274 +0.052794,274 +0.054244,274 +0.052822,274 +0.054396,274 +0.052789,274 +0.055424,274 +0.052788,274 +0.054105,274 +0.052730,274 +0.054005,274 +0.052820,274 +0.054166,274 +0.053538,274 +0.053382,274 +0.054193,274 +0.053835,274 +0.055035,274 +0.055093,274 +0.055100,274 +0.054008,274 +0.055025,274 +0.053819,274 +0.054744,274 +0.053556,274 +0.056024,274 +0.054414,274 +0.053970,274 +0.055162,274 +0.053304,274 +0.055780,274 +0.053383,274 +0.055202,274 +0.053354,274 +0.054638,274 +0.053250,274 +0.054771,274 +0.053860,274 +0.054538,274 +0.054270,274 +0.053734,274 +0.054906,274 +0.053493,274 +0.055184,274 +0.053665,274 +0.053948,274 +0.053557,274 +0.054322,274 +0.053787,274 +0.053672,274 +0.054128,276 +0.054896,276 +0.054354,276 +0.054290,276 +0.054894,276 +0.053618,276 +0.054930,276 +0.053730,276 +0.054892,276 +0.054017,276 +0.056243,276 +0.054117,276 +0.056417,276 +0.054062,276 +0.055307,276 +0.055088,276 +0.054532,276 +0.054670,276 +0.053246,276 +0.054359,276 +0.053118,276 +0.054349,276 +0.053165,276 +0.054354,276 +0.053115,276 +0.054216,276 +0.053150,276 +0.054336,276 +0.054530,276 +0.053243,276 +0.055437,276 +0.053164,276 +0.054328,276 +0.053144,276 +0.054764,276 +0.054260,276 +0.054312,276 +0.056166,276 +0.055131,276 +0.053252,276 +0.053969,276 +0.054267,276 +0.053053,276 +0.054094,276 +0.053045,276 +0.053980,276 +0.053225,276 +0.055082,276 +0.053120,276 +0.053623,276 +0.053141,276 +0.054539,276 +0.053074,276 +0.053560,276 +0.053681,276 +0.053762,276 +0.055634,276 +0.053254,276 +0.053405,276 +0.054822,276 +0.054325,276 +0.053122,276 +0.053438,276 +0.053207,276 +0.053634,276 +0.053511,276 +0.053607,276 +0.053119,276 +0.053694,276 +0.053250,276 +0.053541,276 +0.053725,276 +0.053152,276 +0.053525,276 +0.053539,276 +0.053485,276 +0.053115,276 +0.053708,276 +0.053342,276 +0.053600,276 +0.053844,276 +0.053480,276 +0.053116,276 +0.053744,276 +0.053231,276 +0.053467,276 +0.053083,276 +0.053788,276 +0.053221,276 +0.053674,276 +0.053589,276 +0.053146,276 +0.053575,276 +0.053412,276 +0.053608,276 +0.053061,276 +0.053592,276 +0.053129,276 +0.053909,276 +0.052992,276 +0.056686,278 +0.056121,278 +0.056500,278 +0.056478,278 +0.055945,278 +0.056477,278 +0.055941,278 +0.056564,278 +0.055995,278 +0.056528,278 +0.056414,278 +0.056325,278 +0.056512,278 +0.056066,278 +0.056793,278 +0.056123,278 +0.056582,278 +0.055945,278 +0.056602,278 +0.056426,278 +0.056327,278 +0.056475,278 +0.056034,278 +0.056723,278 +0.056029,278 +0.056717,278 +0.056075,278 +0.056660,278 +0.056743,278 +0.056300,278 +0.056511,278 +0.055932,278 +0.056550,278 +0.056184,278 +0.056580,278 +0.056058,278 +0.056266,278 +0.056487,278 +0.056185,278 +0.056611,278 +0.055784,278 +0.056613,278 +0.056066,278 +0.056499,278 +0.056814,278 +0.055931,278 +0.056476,278 +0.056853,278 +0.056435,278 +0.055918,278 +0.056794,278 +0.057376,278 +0.056570,278 +0.056438,278 +0.055998,278 +0.056864,278 +0.056111,278 +0.056630,278 +0.055871,278 +0.056763,278 +0.056253,278 +0.056417,278 +0.056558,278 +0.055982,278 +0.056670,278 +0.055937,278 +0.056469,278 +0.055934,278 +0.056489,278 +0.056263,278 +0.056659,278 +0.056447,278 +0.056029,278 +0.056709,278 +0.056107,278 +0.056422,278 +0.055829,278 +0.056632,278 +0.056431,278 +0.055938,278 +0.056483,278 +0.055816,278 +0.056627,278 +0.055849,278 +0.056359,278 +0.055843,278 +0.057168,278 +0.056457,278 +0.055904,278 +0.056399,278 +0.055899,278 +0.057277,278 +0.055828,278 +0.056306,278 +0.055929,278 +0.056550,278 +0.056387,278 +0.055821,278 +0.056925,278 +0.056030,278 +0.056039,280 +0.054997,280 +0.056157,280 +0.055203,280 +0.055545,280 +0.055597,280 +0.055629,280 +0.055476,280 +0.055143,280 +0.055797,280 +0.055150,280 +0.055650,280 +0.055104,280 +0.055887,280 +0.055714,280 +0.055130,280 +0.055771,280 +0.055378,280 +0.055863,280 +0.055104,280 +0.055756,280 +0.055104,280 +0.055971,280 +0.055739,280 +0.055151,280 +0.055592,280 +0.055293,280 +0.055854,280 +0.055124,280 +0.055650,280 +0.055182,280 +0.055857,280 +0.055306,280 +0.055486,280 +0.055568,280 +0.055182,280 +0.055911,280 +0.055098,280 +0.055473,280 +0.055237,280 +0.055612,280 +0.055152,280 +0.055658,280 +0.055459,280 +0.055312,280 +0.055762,280 +0.055169,280 +0.055615,280 +0.055152,280 +0.055908,280 +0.055204,280 +0.055705,280 +0.055386,280 +0.055221,280 +0.056019,280 +0.055110,280 +0.055731,280 +0.055292,280 +0.056103,280 +0.055521,280 +0.055667,280 +0.055302,280 +0.055408,280 +0.055828,280 +0.055129,280 +0.055558,280 +0.055119,280 +0.055847,280 +0.055210,280 +0.055756,280 +0.055195,280 +0.055800,280 +0.055859,280 +0.055122,280 +0.055436,280 +0.055093,280 +0.055607,280 +0.055062,280 +0.055480,280 +0.055228,280 +0.055440,280 +0.055857,280 +0.055213,280 +0.055463,280 +0.055038,280 +0.055800,280 +0.055128,280 +0.055669,280 +0.055285,280 +0.055366,280 +0.055615,280 +0.055130,280 +0.055400,280 +0.055457,280 +0.055801,280 +0.056098,280 +0.055713,280 +0.055297,280 +0.055464,280 +0.055897,280 +0.061178,282 +0.060940,282 +0.060385,282 +0.061193,282 +0.060907,282 +0.060585,282 +0.060973,282 +0.061083,282 +0.060710,282 +0.060792,282 +0.060495,282 +0.060994,282 +0.060557,282 +0.060853,282 +0.060833,282 +0.060674,282 +0.061016,282 +0.060525,282 +0.060771,282 +0.060985,282 +0.060689,282 +0.060932,282 +0.060415,282 +0.060865,282 +0.065272,282 +0.060795,282 +0.060861,282 +0.060535,282 +0.060739,282 +0.060891,282 +0.060540,282 +0.060519,282 +0.060961,282 +0.060622,282 +0.060677,282 +0.060447,282 +0.060994,282 +0.060663,282 +0.060750,282 +0.060640,282 +0.060824,282 +0.061117,282 +0.060853,282 +0.060488,282 +0.061053,282 +0.060691,282 +0.060881,282 +0.060800,282 +0.060519,282 +0.060955,282 +0.060484,282 +0.060771,282 +0.061039,282 +0.060620,282 +0.061179,282 +0.060412,282 +0.060860,282 +0.061263,282 +0.060412,282 +0.060798,282 +0.060409,282 +0.061082,282 +0.060783,282 +0.060369,282 +0.060791,282 +0.060680,282 +0.060823,282 +0.060749,282 +0.060454,282 +0.060993,282 +0.060702,282 +0.060723,282 +0.060732,282 +0.060771,282 +0.060674,282 +0.060352,282 +0.060997,282 +0.060970,282 +0.060575,282 +0.060690,282 +0.060320,282 +0.061755,282 +0.060964,282 +0.060513,282 +0.060540,282 +0.060371,282 +0.060994,282 +0.061590,282 +0.060600,282 +0.061419,282 +0.060636,282 +0.060621,282 +0.060998,282 +0.060314,282 +0.060860,282 +0.060558,282 +0.060567,282 +0.060689,282 +0.060647,282 +0.060750,282 +0.058921,284 +0.058936,284 +0.059176,284 +0.058603,284 +0.059371,284 +0.058771,284 +0.060202,284 +0.059112,284 +0.058822,284 +0.059130,284 +0.058993,284 +0.059290,284 +0.059066,284 +0.058856,284 +0.059348,284 +0.059093,284 +0.059185,284 +0.058805,284 +0.059290,284 +0.059479,284 +0.059320,284 +0.059182,284 +0.058853,284 +0.059464,284 +0.059027,284 +0.058661,284 +0.058992,284 +0.058628,284 +0.060052,284 +0.058768,284 +0.058889,284 +0.058936,284 +0.058981,284 +0.059222,284 +0.058769,284 +0.059221,284 +0.059170,284 +0.059063,284 +0.059002,284 +0.058687,284 +0.059421,284 +0.058926,284 +0.058686,284 +0.059023,284 +0.058684,284 +0.059015,284 +0.058624,284 +0.059053,284 +0.059106,284 +0.058753,284 +0.059009,284 +0.058660,284 +0.059045,284 +0.058886,284 +0.059208,284 +0.058953,284 +0.058749,284 +0.059335,284 +0.058585,284 +0.059025,284 +0.059127,284 +0.058718,284 +0.059328,284 +0.058785,284 +0.059074,284 +0.059040,284 +0.059038,284 +0.059060,284 +0.058690,284 +0.059024,284 +0.058809,284 +0.062781,284 +0.062612,284 +0.059017,284 +0.062422,284 +0.061937,284 +0.058787,284 +0.062778,284 +0.058929,284 +0.062076,284 +0.062005,284 +0.058849,284 +0.062248,284 +0.058850,284 +0.061965,284 +0.061794,284 +0.060139,284 +0.063142,284 +0.059049,284 +0.061887,284 +0.061915,284 +0.058789,284 +0.061870,284 +0.058607,284 +0.061869,284 +0.062020,284 +0.058899,284 +0.061880,284 +0.058643,284 +0.062002,284 +0.065981,286 +0.063085,286 +0.065705,286 +0.066395,286 +0.063186,286 +0.065908,286 +0.065949,286 +0.062979,286 +0.065761,286 +0.063785,286 +0.064798,286 +0.064375,286 +0.062746,286 +0.063914,286 +0.063989,286 +0.062908,286 +0.064041,286 +0.063173,286 +0.062956,286 +0.063980,286 +0.063076,286 +0.063271,286 +0.063561,286 +0.062711,286 +0.063212,286 +0.063110,286 +0.063045,286 +0.063386,286 +0.062728,286 +0.063221,286 +0.063484,286 +0.063042,286 +0.063231,286 +0.063095,286 +0.062986,286 +0.063989,286 +0.062867,286 +0.063241,286 +0.063624,286 +0.062800,286 +0.063251,286 +0.063172,286 +0.062871,286 +0.063358,286 +0.062740,286 +0.063326,286 +0.063316,286 +0.062790,286 +0.063419,286 +0.063193,286 +0.062935,286 +0.063784,286 +0.063122,286 +0.064094,286 +0.067483,286 +0.062756,286 +0.063161,286 +0.063158,286 +0.062857,286 +0.063157,286 +0.063145,286 +0.062850,286 +0.063292,286 +0.062744,286 +0.063340,286 +0.063324,286 +0.062920,286 +0.063674,286 +0.063221,286 +0.064689,286 +0.063610,286 +0.062748,286 +0.064264,286 +0.063843,286 +0.062842,286 +0.063239,286 +0.063366,286 +0.063586,286 +0.063319,286 +0.063017,286 +0.063736,286 +0.063332,286 +0.062838,286 +0.063847,286 +0.063231,286 +0.063415,286 +0.063301,286 +0.063123,286 +0.062875,286 +0.063330,286 +0.063156,286 +0.063569,286 +0.063378,286 +0.063101,286 +0.063237,286 +0.063325,286 +0.062859,286 +0.063421,286 +0.062976,286 +0.063373,286 +0.065231,288 +0.065049,288 +0.065202,288 +0.065060,288 +0.064777,288 +0.065043,288 +0.064091,288 +0.063515,288 +0.064231,288 +0.064605,288 +0.063848,288 +0.064219,288 +0.063622,288 +0.063975,288 +0.064850,288 +0.064732,288 +0.065307,288 +0.065324,288 +0.064693,288 +0.065179,288 +0.065310,288 +0.064844,288 +0.065290,288 +0.064943,288 +0.065224,288 +0.065342,288 +0.064823,288 +0.065187,288 +0.063950,288 +0.063716,288 +0.064065,288 +0.064224,288 +0.063697,288 +0.064055,288 +0.063800,288 +0.063614,288 +0.064130,288 +0.063993,288 +0.063986,288 +0.064079,288 +0.063585,288 +0.064072,288 +0.065238,288 +0.065154,288 +0.065189,288 +0.065756,288 +0.064803,288 +0.065795,288 +0.065136,288 +0.064741,288 +0.065112,288 +0.064663,288 +0.065022,288 +0.065250,288 +0.064910,288 +0.065307,288 +0.065144,288 +0.064645,288 +0.065135,288 +0.065097,288 +0.064721,288 +0.065196,288 +0.065120,288 +0.064938,288 +0.065088,288 +0.064598,288 +0.065139,288 +0.065346,288 +0.064673,288 +0.065012,288 +0.065111,288 +0.064899,288 +0.064962,288 +0.065290,288 +0.064877,288 +0.065087,288 +0.065402,288 +0.064656,288 +0.065058,288 +0.064950,288 +0.064708,288 +0.064946,288 +0.064646,288 +0.064967,288 +0.065059,288 +0.064819,288 +0.064989,288 +0.065037,288 +0.064656,288 +0.065418,288 +0.065141,288 +0.065037,288 +0.064033,288 +0.063644,288 +0.063892,288 +0.063866,288 +0.064388,288 +0.065006,288 +0.065188,288 +0.064741,288 +0.066894,290 +0.066439,290 +0.065936,290 +0.066189,290 +0.066283,290 +0.065873,290 +0.066252,290 +0.066334,290 +0.066409,290 +0.066338,290 +0.066291,290 +0.065812,290 +0.066247,290 +0.066345,290 +0.065786,290 +0.066483,290 +0.066503,290 +0.066027,290 +0.065965,290 +0.066054,290 +0.066133,290 +0.066230,290 +0.065766,290 +0.066377,290 +0.066257,290 +0.065813,290 +0.066228,290 +0.066223,290 +0.065864,290 +0.066141,290 +0.066147,290 +0.065878,290 +0.066227,290 +0.066032,290 +0.065745,290 +0.066243,290 +0.066123,290 +0.066314,290 +0.066257,290 +0.066300,290 +0.065877,290 +0.066223,290 +0.066203,290 +0.066044,290 +0.066232,290 +0.066328,290 +0.065944,290 +0.066421,290 +0.066077,290 +0.066266,290 +0.066172,290 +0.066210,290 +0.065911,290 +0.066374,290 +0.065973,290 +0.066060,290 +0.066242,290 +0.065823,290 +0.066221,290 +0.066933,290 +0.065836,290 +0.066934,290 +0.066470,290 +0.065773,290 +0.066684,290 +0.066227,290 +0.065889,290 +0.066280,290 +0.066238,290 +0.065969,290 +0.066354,290 +0.066274,290 +0.065947,290 +0.066260,290 +0.066301,290 +0.065937,290 +0.066530,290 +0.066418,290 +0.066328,290 +0.066159,290 +0.066282,290 +0.065979,290 +0.066425,290 +0.066258,290 +0.066737,290 +0.066388,290 +0.066281,290 +0.065948,290 +0.066293,290 +0.066228,290 +0.066327,290 +0.066376,290 +0.066135,290 +0.066085,290 +0.066672,290 +0.066038,290 +0.066262,290 +0.066243,290 +0.065868,290 +0.067703,290 +0.068319,292 +0.065282,292 +0.066127,292 +0.066207,292 +0.065202,292 +0.065459,292 +0.065341,292 +0.065483,292 +0.065563,292 +0.065269,292 +0.065137,292 +0.065553,292 +0.066180,292 +0.065189,292 +0.065532,292 +0.065671,292 +0.065080,292 +0.065441,292 +0.065277,292 +0.066142,292 +0.065382,292 +0.065143,292 +0.065610,292 +0.065445,292 +0.064946,292 +0.065470,292 +0.065479,292 +0.065502,292 +0.065399,292 +0.065523,292 +0.065342,292 +0.065557,292 +0.065380,292 +0.065116,292 +0.065437,292 +0.065276,292 +0.065088,292 +0.065440,292 +0.065563,292 +0.065113,292 +0.065422,292 +0.067850,292 +0.065228,292 +0.065730,292 +0.065117,292 +0.065829,292 +0.065496,292 +0.065036,292 +0.065513,292 +0.065588,292 +0.065061,292 +0.065361,292 +0.065541,292 +0.065403,292 +0.065816,292 +0.065784,292 +0.065190,292 +0.065547,292 +0.065565,292 +0.065511,292 +0.066175,292 +0.065516,292 +0.065099,292 +0.065469,292 +0.065223,292 +0.065155,292 +0.065435,292 +0.065118,292 +0.065490,292 +0.065777,292 +0.064977,292 +0.065431,292 +0.065489,292 +0.065406,292 +0.065351,292 +0.065495,292 +0.067420,292 +0.065514,292 +0.065273,292 +0.065096,292 +0.065467,292 +0.065280,292 +0.065053,292 +0.065718,292 +0.065420,292 +0.065287,292 +0.065960,292 +0.065394,292 +0.065229,292 +0.066330,292 +0.065249,292 +0.065580,292 +0.065533,292 +0.065008,292 +0.065597,292 +0.065550,292 +0.065213,292 +0.065404,292 +0.065527,292 +0.065221,292 +0.070272,294 +0.069894,294 +0.069447,294 +0.069989,294 +0.070190,294 +0.069640,294 +0.069925,294 +0.069958,294 +0.069810,294 +0.069592,294 +0.070002,294 +0.069833,294 +0.069407,294 +0.070111,294 +0.070008,294 +0.069346,294 +0.069899,294 +0.070334,294 +0.074104,294 +0.069582,294 +0.073152,294 +0.072409,294 +0.069674,294 +0.073156,294 +0.072663,294 +0.072446,294 +0.069788,294 +0.072631,294 +0.073153,294 +0.069503,294 +0.072394,294 +0.072529,294 +0.072809,294 +0.069747,294 +0.073212,294 +0.072429,294 +0.069558,294 +0.072261,294 +0.072765,294 +0.072226,294 +0.069624,294 +0.073477,294 +0.072357,294 +0.069487,294 +0.072482,294 +0.072561,294 +0.072795,294 +0.069543,294 +0.073329,294 +0.072326,294 +0.069506,294 +0.072625,294 +0.072249,294 +0.072306,294 +0.069604,294 +0.072639,294 +0.072928,294 +0.069496,294 +0.072464,294 +0.072455,294 +0.104865,294 +0.123952,294 +0.086093,294 +0.070427,294 +0.070317,294 +0.071085,294 +0.070397,294 +0.070592,294 +0.070531,294 +0.070553,294 +0.070144,294 +0.070019,294 +0.070232,294 +0.070128,294 +0.069989,294 +0.070661,294 +0.070622,294 +0.070228,294 +0.070136,294 +0.070735,294 +0.071010,294 +0.070106,294 +0.072252,294 +0.070903,294 +0.070173,294 +0.071147,294 +0.070767,294 +0.070937,294 +0.070518,294 +0.071030,294 +0.070945,294 +0.070058,294 +0.070628,294 +0.070684,294 +0.070469,294 +0.070892,294 +0.070566,294 +0.070380,294 +0.069266,294 +0.070158,294 +0.066924,296 +0.065989,296 +0.066792,296 +0.071059,296 +0.066570,296 +0.066726,296 +0.066726,296 +0.065890,296 +0.066737,296 +0.066712,296 +0.065697,296 +0.067572,296 +0.066782,296 +0.065651,296 +0.066919,296 +0.066653,296 +0.065605,296 +0.066935,296 +0.069359,296 +0.065816,296 +0.066748,296 +0.069562,296 +0.065701,296 +0.066538,296 +0.066545,296 +0.065713,296 +0.066771,296 +0.066588,296 +0.065797,296 +0.066558,296 +0.066710,296 +0.065667,296 +0.066725,296 +0.067926,296 +0.066058,296 +0.067089,296 +0.066650,296 +0.066134,296 +0.067048,296 +0.066233,296 +0.066122,296 +0.066731,296 +0.065608,296 +0.066676,296 +0.066550,296 +0.065618,296 +0.066656,296 +0.066657,296 +0.065731,296 +0.067849,296 +0.066753,296 +0.065648,296 +0.066515,296 +0.066662,296 +0.065703,296 +0.066702,296 +0.066753,296 +0.065804,296 +0.066577,296 +0.066754,296 +0.065707,296 +0.066704,296 +0.066666,296 +0.065873,296 +0.068169,296 +0.066783,296 +0.065783,296 +0.066619,296 +0.066828,296 +0.065808,296 +0.066684,296 +0.066990,296 +0.065875,296 +0.066807,296 +0.065919,296 +0.065617,296 +0.065887,296 +0.065956,296 +0.066141,296 +0.065855,296 +0.065801,296 +0.065621,296 +0.066005,296 +0.067027,296 +0.065780,296 +0.065900,296 +0.065676,296 +0.065972,296 +0.066076,296 +0.065811,296 +0.065789,296 +0.078423,296 +0.070391,296 +0.088537,296 +0.083934,296 +0.074624,296 +0.074875,296 +0.074878,296 +0.072253,296 +0.072570,296 +0.081909,298 +0.079450,298 +0.073185,298 +0.072747,298 +0.072461,298 +0.072164,298 +0.072826,298 +0.072595,298 +0.072652,298 +0.072581,298 +0.072825,298 +0.078690,298 +0.080412,298 +0.079329,298 +0.080308,298 +0.079911,298 +0.077494,298 +0.073448,298 +0.077835,298 +0.073131,298 +0.081008,298 +0.079288,298 +0.080261,298 +0.076548,298 +0.075494,298 +0.079484,298 +0.082832,298 +0.078936,298 +0.079092,298 +0.078865,298 +0.075786,298 +0.079212,298 +0.077011,298 +0.074678,298 +0.074451,298 +0.072372,298 +0.073247,298 +0.072424,298 +0.074625,298 +0.074462,298 +0.076295,298 +0.073360,298 +0.072790,298 +0.073887,298 +0.074409,298 +0.073295,298 +0.072906,298 +0.073051,298 +0.072361,298 +0.072727,298 +0.072425,298 +0.073496,298 +0.072491,298 +0.072717,298 +0.072248,298 +0.073342,298 +0.072399,298 +0.072332,298 +0.072550,298 +0.072592,298 +0.072652,298 +0.073236,298 +0.076327,298 +0.074498,298 +0.073960,298 +0.072665,298 +0.072811,298 +0.073851,298 +0.078967,298 +0.077097,298 +0.076110,298 +0.075197,298 +0.075843,298 +0.075148,298 +0.076042,298 +0.074501,298 +0.075122,298 +0.073719,298 +0.074472,298 +0.074512,298 +0.073973,298 +0.074826,298 +0.073913,298 +0.073266,298 +0.072438,298 +0.073866,298 +0.073519,298 +0.074649,298 +0.072506,298 +0.073378,298 +0.073498,298 +0.072656,298 +0.073167,298 +0.074034,298 +0.073469,298 +0.072580,298 +0.073423,298 +0.073679,298 +0.078404,298 +0.079920,298 diff --git a/buch/papers/multiplikation/code/meas/winograd.txt b/buch/papers/multiplikation/code/meas/winograd.txt new file mode 100644 index 0000000..970a3f4 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/winograd.txt @@ -0,0 +1,110 @@ +0.000001,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000021,16 +0.000011,16 +0.000011,16 +0.000092,32 +0.000092,32 +0.000081,32 +0.000081,32 +0.000081,32 +0.000081,32 +0.000088,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000670,64 +0.000739,64 +0.000609,64 +0.000609,64 +0.000700,64 +0.000648,64 +0.000626,64 +0.000626,64 +0.000626,64 +0.000626,64 +0.005321,128 +0.005286,128 +0.005180,128 +0.005223,128 +0.005249,128 +0.005299,128 +0.005205,128 +0.005268,128 +0.005464,128 +0.005378,128 +0.053123,256 +0.052325,256 +0.052729,256 +0.052930,256 +0.052207,256 +0.053178,256 +0.052122,256 +0.052681,256 +0.052965,256 +0.052486,256 +0.527028,512 +0.525201,512 +0.521822,512 +0.525147,512 +0.525241,512 +0.527725,512 +0.526321,512 +0.526479,512 +0.524020,512 +0.520768,512 +4.732299,1024 +4.617253,1024 +4.647425,1024 +4.519233,1024 +4.917471,1024 +4.564929,1024 +4.870771,1024 +4.555407,1024 +4.727473,1024 +4.559349,1024 +136.409028,2048 +136.390557,2048 +136.541672,2048 +136.598491,2048 +137.720790,2048 +136.825926,2048 +136.367686,2048 +136.650627,2048 +136.642195,2048 +136.622805,2048 diff --git a/buch/papers/multiplikation/code/meas_1024.pdf b/buch/papers/multiplikation/code/meas_1024.pdf Binary files differnew file mode 100644 index 0000000..f489a7d --- /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..ab507a2 --- /dev/null +++ b/buch/papers/multiplikation/code/meas_1024.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 1.280000000000000000e+02 2.560000000000000000e+02 5.120000000000000000e+02 1.024000000000000000e+03 +1.859664916992187500e-05 8.296966552734375000e-05 5.471706390380859375e-04 3.053665161132812500e-03 2.407431602478027344e-02 1.868948936462402344e-01 1.563691616058349609e+00 1.100623321533203125e+01 8.547679090499877930e+01 7.507572824954986572e+02 +8.106231689453125000e-06 9.012222290039062500e-05 7.290840148925781250e-04 4.970788955688476562e-03 2.718997001647949219e-02 2.652802467346191406e-01 1.777865171432495117e+00 1.327002429962158203e+01 1.053971357345581055e+02 8.473208103179931641e+02 +2.098083496093750000e-05 1.742839813232421875e-04 9.438991546630859375e-04 4.754066467285156250e-03 4.852557182312011719e-02 2.204136848449707031e-01 1.447179555892944336e+00 9.938656568527221680e+00 6.396102952957153320e+01 4.614939928054809570e+02 +2.789497375488281250e-05 1.049041748046875000e-04 5.528926849365234375e-04 4.555702209472656250e-03 1.871442794799804688e-02 1.530685424804687500e-01 1.194762229919433594e+00 8.298985958099365234e+00 6.836994743347167969e+01 5.373736469745635986e+02 +1.835823059082031250e-05 7.867813110351562500e-06 1.001358032226562500e-05 5.412101745605468750e-05 4.267692565917968750e-05 1.184940338134765625e-04 2.441406250000000000e-04 6.957054138183593750e-04 2.217054367065429688e-03 1.880884170532226562e-02 diff --git a/buch/papers/multiplikation/code/meas_128.pdf b/buch/papers/multiplikation/code/meas_128.pdf Binary files differnew file mode 100644 index 0000000..c54648f --- /dev/null +++ b/buch/papers/multiplikation/code/meas_128.pdf diff --git a/buch/papers/multiplikation/code/meas_128.txt b/buch/papers/multiplikation/code/meas_128.txt new file mode 100644 index 0000000..f3a5beb --- /dev/null +++ b/buch/papers/multiplikation/code/meas_128.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 1.280000000000000000e+02 +1.239776611328125000e-05 5.507469177246093750e-05 3.888607025146484375e-04 2.762079238891601562e-03 2.097773551940917969e-02 1.672370433807373047e-01 1.410297393798828125e+00 +5.483627319335937500e-06 5.888938903808593750e-05 3.871917724609375000e-04 3.364324569702148438e-03 2.481031417846679688e-02 2.047052383422851562e-01 1.712310314178466797e+00 +1.358985900878906250e-05 1.189708709716796875e-04 6.430149078369140625e-04 5.586385726928710938e-03 3.101944923400878906e-02 1.874091625213623047e-01 1.327976465225219727e+00 +1.978874206542968750e-05 7.224082946777343750e-05 4.618167877197265625e-04 3.294944763183593750e-03 1.755571365356445312e-02 1.360688209533691406e-01 1.028253555297851562e+00 +1.215934753417968750e-05 5.722045898437500000e-06 2.074241638183593750e-05 4.339218139648437500e-05 2.813339233398437500e-05 5.292892456054687500e-05 1.921653747558593750e-04 diff --git a/buch/papers/multiplikation/code/meas_16.pdf b/buch/papers/multiplikation/code/meas_16.pdf Binary files differnew file mode 100644 index 0000000..c2c3834 --- /dev/null +++ b/buch/papers/multiplikation/code/meas_16.pdf diff --git a/buch/papers/multiplikation/code/meas_16.txt b/buch/papers/multiplikation/code/meas_16.txt new file mode 100644 index 0000000..69f85bd --- /dev/null +++ b/buch/papers/multiplikation/code/meas_16.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 +1.549720764160156250e-05 6.914138793945312500e-05 5.259513854980468750e-04 2.841711044311523438e-03 +6.914138793945312500e-06 7.557868957519531250e-05 4.496574401855468750e-04 3.437519073486328125e-03 +1.883506774902343750e-05 1.499652862548828125e-04 8.952617645263671875e-04 4.348516464233398438e-03 +2.694129943847656250e-05 1.082420349121093750e-04 4.131793975830078125e-04 2.580165863037109375e-03 +1.621246337890625000e-05 1.120567321777343750e-05 9.298324584960937500e-06 1.239776611328125000e-05 diff --git a/buch/papers/multiplikation/code/meas_256.pdf b/buch/papers/multiplikation/code/meas_256.pdf Binary files differnew file mode 100644 index 0000000..2eb177b --- /dev/null +++ b/buch/papers/multiplikation/code/meas_256.pdf diff --git a/buch/papers/multiplikation/code/meas_256.txt b/buch/papers/multiplikation/code/meas_256.txt new file mode 100644 index 0000000..62e77cb --- /dev/null +++ b/buch/papers/multiplikation/code/meas_256.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 1.280000000000000000e+02 2.560000000000000000e+02 +1.144409179687500000e-05 5.507469177246093750e-05 3.774166107177734375e-04 3.177404403686523438e-03 2.508044242858886719e-02 2.120554447174072266e-01 1.431464910507202148e+00 1.076412820816040039e+01 +5.722045898437500000e-06 5.745887756347656250e-05 4.494190216064453125e-04 3.611087799072265625e-03 3.317713737487792969e-02 2.292332649230957031e-01 2.090558290481567383e+00 1.306217479705810547e+01 +1.788139343261718750e-05 1.168251037597656250e-04 5.981922149658203125e-04 4.416465759277343750e-03 3.002405166625976562e-02 2.104022502899169922e-01 1.488269329071044922e+00 9.164114713668823242e+00 +1.955032348632812500e-05 7.224082946777343750e-05 3.829002380371093750e-04 2.558946609497070312e-03 2.043128013610839844e-02 1.361320018768310547e-01 1.089214324951171875e+00 8.553364753723144531e+00 +2.384185791015625000e-05 5.245208740234375000e-06 6.437301635742187500e-06 2.455711364746093750e-05 4.148483276367187500e-05 8.702278137207031250e-05 3.793239593505859375e-04 6.709098815917968750e-04 diff --git a/buch/papers/multiplikation/code/meas_32.pdf b/buch/papers/multiplikation/code/meas_32.pdf Binary files differnew file mode 100644 index 0000000..b926095 --- /dev/null +++ b/buch/papers/multiplikation/code/meas_32.pdf diff --git a/buch/papers/multiplikation/code/meas_32.txt b/buch/papers/multiplikation/code/meas_32.txt new file mode 100644 index 0000000..0fdc18d --- /dev/null +++ b/buch/papers/multiplikation/code/meas_32.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 +1.239776611328125000e-05 5.507469177246093750e-05 3.802776336669921875e-04 2.795457839965820312e-03 2.073740959167480469e-02 +5.006790161132812500e-06 5.841255187988281250e-05 3.988742828369140625e-04 3.505229949951171875e-03 2.511668205261230469e-02 +1.335144042968750000e-05 1.149177551269531250e-04 6.387233734130859375e-04 4.088878631591796875e-03 2.969408035278320312e-02 +1.955032348632812500e-05 8.058547973632812500e-05 3.998279571533203125e-04 2.514839172363281250e-03 1.842117309570312500e-02 +1.215934753417968750e-05 8.583068847656250000e-06 6.675720214843750000e-06 2.694129943847656250e-05 2.789497375488281250e-05 diff --git a/buch/papers/multiplikation/code/meas_4096.pdf b/buch/papers/multiplikation/code/meas_4096.pdf Binary files differnew file mode 100644 index 0000000..ecf2cff --- /dev/null +++ b/buch/papers/multiplikation/code/meas_4096.pdf diff --git a/buch/papers/multiplikation/code/meas_4096.txt b/buch/papers/multiplikation/code/meas_4096.txt new file mode 100644 index 0000000..cae1bc6 --- /dev/null +++ b/buch/papers/multiplikation/code/meas_4096.txt @@ -0,0 +1,6 @@ +2.048000000000000000e+03 4.096000000000000000e+03 +6.154183513402938843e+03 4.681333474493026733e+04 +7.375929301261901855e+03 5.846600176072120667e+04 +3.860573610544204712e+03 2.290433094644546509e+04 +4.884613198995590210e+03 4.359707747149467468e+04 +2.157390117645263672e-01 1.491588830947875977e+00 diff --git a/buch/papers/multiplikation/code/meas_512.pdf b/buch/papers/multiplikation/code/meas_512.pdf Binary files differnew file mode 100644 index 0000000..4d8f04b --- /dev/null +++ b/buch/papers/multiplikation/code/meas_512.pdf diff --git a/buch/papers/multiplikation/code/meas_512.txt b/buch/papers/multiplikation/code/meas_512.txt new file mode 100644 index 0000000..1b2089d --- /dev/null +++ b/buch/papers/multiplikation/code/meas_512.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 1.280000000000000000e+02 2.560000000000000000e+02 5.120000000000000000e+02 +1.358985900878906250e-05 5.817413330078125000e-05 4.582405090332031250e-04 3.082036972045898438e-03 2.020335197448730469e-02 1.636352539062500000e-01 1.280331134796142578e+00 1.093638324737548828e+01 8.666778349876403809e+01 +6.198883056640625000e-06 6.270408630371093750e-05 4.820823669433593750e-04 3.279924392700195312e-03 2.462601661682128906e-02 2.034928798675537109e-01 1.630282878875732422e+00 1.372955965995788574e+01 1.104150602817535400e+02 +1.621246337890625000e-05 1.292228698730468750e-04 6.661415100097656250e-04 4.615545272827148438e-03 2.836179733276367188e-02 1.843333244323730469e-01 1.310264825820922852e+00 9.937873125076293945e+00 6.667592120170593262e+01 +2.217292785644531250e-05 7.486343383789062500e-05 4.060268402099609375e-04 2.455949783325195312e-03 1.685857772827148438e-02 1.299629211425781250e-01 1.173750638961791992e+00 8.648802757263183594e+00 6.876212453842163086e+01 +2.431869506835937500e-05 5.006790161132812500e-06 6.914138793945312500e-06 8.106231689453125000e-06 2.717971801757812500e-05 6.461143493652343750e-05 1.480579376220703125e-04 5.280971527099609375e-04 3.390312194824218750e-03 diff --git a/buch/papers/multiplikation/code/meas_64.pdf b/buch/papers/multiplikation/code/meas_64.pdf Binary files differnew file mode 100644 index 0000000..92af29b --- /dev/null +++ b/buch/papers/multiplikation/code/meas_64.pdf diff --git a/buch/papers/multiplikation/code/meas_64.txt b/buch/papers/multiplikation/code/meas_64.txt new file mode 100644 index 0000000..b4fc7a1 --- /dev/null +++ b/buch/papers/multiplikation/code/meas_64.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 1.600000000000000000e+01 3.200000000000000000e+01 6.400000000000000000e+01 +2.145767211914062500e-05 6.175041198730468750e-05 4.422664642333984375e-04 3.235816955566406250e-03 2.289748191833496094e-02 1.855163574218750000e-01 +1.025199890136718750e-05 6.341934204101562500e-05 5.202293395996093750e-04 3.566026687622070312e-03 3.026723861694335938e-02 2.312932014465332031e-01 +2.384185791015625000e-05 1.807212829589843750e-04 6.821155548095703125e-04 4.796504974365234375e-03 2.968001365661621094e-02 2.291278839111328125e-01 +3.504753112792968750e-05 1.106262207031250000e-04 4.322528839111328125e-04 2.696514129638671875e-03 2.188420295715332031e-02 1.477701663970947266e-01 +3.218650817871093750e-05 1.144409179687500000e-05 7.390975952148437500e-06 4.625320434570312500e-05 3.814697265625000000e-05 5.435943603515625000e-05 diff --git a/buch/papers/multiplikation/code/meas_8.pdf b/buch/papers/multiplikation/code/meas_8.pdf Binary files differnew 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 new file mode 100644 index 0000000..6cf6515 --- /dev/null +++ b/buch/papers/multiplikation/code/meas_8.txt @@ -0,0 +1,6 @@ +2.000000000000000000e+00 4.000000000000000000e+00 8.000000000000000000e+00 +1.144409179687500000e-05 5.412101745605468750e-05 3.845691680908203125e-04 +4.768371582031250000e-06 5.698204040527343750e-05 5.209445953369140625e-04 +1.382827758789062500e-05 1.180171966552734375e-04 6.978511810302734375e-04 +1.859664916992187500e-05 7.033348083496093750e-05 3.886222839355468750e-04 +1.525878906250000000e-05 4.529953002929687500e-06 7.390975952148437500e-06 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..9b03a4e --- /dev/null +++ b/buch/papers/multiplikation/einlteung.tex @@ -0,0 +1,51 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Matrizenmultiplikation \label{multiplikation:section:einleitung}} +\rhead{Matrizenmultiplikation} + +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: +Eine $m\times n$-Matrix $\mathbf{A}\in M_{m\times n}(\Bbbk)$ und eine +$n\times p$-Matrix $\mathbf{B}\in M_{n\times p}(\Bbbk)$ haben als Produkt +eine $m\times p$-Matrix $\mathbf{C}=\mathbf{AB}\in M_{m\times p}(\Bbbk)$ 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 $\mathbf{AB}=\mathbf{C}$ wie in Abbildung \ref{multiplikation:fig:mm_viz} visualisiert werden. +Im Fall einer Matrizengr\"osse von $2\times 2$ kann die Matrixgleichung +\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} +explizt als Gleichungen +\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} +der einzelnen Terme geschrieben werden. +\begin{figure} + \center + \includegraphics[]{papers/multiplikation/images/mm_visualisation} + \caption{Grafische Illustration der Matrizenmultiplikation} + \label{multiplikation:fig:mm_viz} +\end{figure} diff --git a/buch/papers/multiplikation/images/algo_tab.pdf b/buch/papers/multiplikation/images/algo_tab.pdf Binary files differnew file mode 100644 index 0000000..7f2bb4f --- /dev/null +++ b/buch/papers/multiplikation/images/algo_tab.pdf diff --git a/buch/papers/multiplikation/images/algo_tab.tex b/buch/papers/multiplikation/images/algo_tab.tex new file mode 100644 index 0000000..50ce392 --- /dev/null +++ b/buch/papers/multiplikation/images/algo_tab.tex @@ -0,0 +1,122 @@ +\documentclass{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{algorithm} +\usepackage{algpseudocode} +\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{multicol} +\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{table}[t] + \begin{tabular}{ll} + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:b1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B1}{$a, b$} + \State \textbf{return} $a+b$ + \EndFunction + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\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} + + \end{minipage} + \end{tabular} +\end{table} + +\begin{table} + \begin{tabular}[t]{ll} + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \label{multiplikation:alg:linear} + \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 + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\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} + \end{minipage} + \end{tabular} +\end{table} + +dhdfh +\end{document} diff --git a/buch/papers/multiplikation/images/bigo.pdf b/buch/papers/multiplikation/images/bigo.pdf Binary files differnew file mode 100644 index 0000000..2519553 --- /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..63fd0fd --- /dev/null +++ b/buch/papers/multiplikation/images/bigo.tex @@ -0,0 +1,112 @@ +\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}[ + xmode=log, ymode=log, + xmin=1e-0, xmax=5000, + ymin=10e-1, ymax=1e7, + grid=both, + major grid style={black!50}, + xlabel = data input size, + ylabel = {time}, + legend pos=north west, + very thick, + yticklabels=\empty, + xticklabels=\empty, + scale only axis=true, + width=12cm, height=8cm, + legend cell align={left} + ] +\addplot [ + domain= 1:5000, + samples=100, + color=red, +] +{1}; +\addlegendentry{$\mathcal{O}(1)$} +\addplot [ + domain= 1:5000, + samples=100, + color=green, +] +{x}; +\addlegendentry{$\mathcal{O}(n)$} +\addplot [ + domain= 1:50000, + samples=100, + color=blue, +] +{x^2}; +\addlegendentry{$\mathcal{O}\left(n^2\right)$} +\addplot [ + domain= 1:500, + samples=100, + color=purple, +] +{x^3}; +\addlegendentry{$\mathcal{O}\left(n^3\right)$} +\addplot [ + domain= 1:500, + samples=100, + color=black, +] +{exp(x) - 1.7}; +\addlegendentry{$\mathcal{O}\left(e^n\right)$} +\addplot [ + domain= 1:5000, + samples=100, + color=orange, +] +{log2(x)+1}; +\addlegendentry{$\mathcal{O}(\log n)$} + +\addplot [ + domain= 1:5000, + samples=100, + color=gray, +] +{x*log2(x)+1}; +\addlegendentry{$\mathcal{O}(n \log n)$} +\end{axis} +\end{tikzpicture} + +\end{document} diff --git a/buch/papers/multiplikation/images/c_meas_4096.pdf b/buch/papers/multiplikation/images/c_meas_4096.pdf Binary files differnew file mode 100644 index 0000000..304015a --- /dev/null +++ b/buch/papers/multiplikation/images/c_meas_4096.pdf diff --git a/buch/papers/multiplikation/images/meas_1024.pdf b/buch/papers/multiplikation/images/meas_1024.pdf Binary files differnew file mode 100644 index 0000000..70c7ec1 --- /dev/null +++ b/buch/papers/multiplikation/images/meas_1024.pdf diff --git a/buch/papers/multiplikation/images/meas_c.pdf b/buch/papers/multiplikation/images/meas_c.pdf Binary files differnew file mode 100644 index 0000000..521151e --- /dev/null +++ b/buch/papers/multiplikation/images/meas_c.pdf diff --git a/buch/papers/multiplikation/images/meas_c.tex b/buch/papers/multiplikation/images/meas_c.tex new file mode 100644 index 0000000..12d3527 --- /dev/null +++ b/buch/papers/multiplikation/images/meas_c.tex @@ -0,0 +1,150 @@ + +\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} +\usepackage{pgfplotstable} +\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}[ +xmode=log, ymode=log, +xmin=30, xmax=10000, +ymin=1e-5, ymax=2e4, +grid=both, +major grid style={black!50}, +xlabel = data input ($n$), +ylabel = {time ($s$)}, +legend pos=north west, +very thick, +scale only axis=true, +width=12cm, height=8cm, + log basis x={10}, + legend cell align={left} +] +\addlegendentry{Winograd} +\addplot[ color=blue, + error bars/.cd, y dir=both, y explicit, +] coordinates { +%(2,1e-07) +%(4,5e-07) +%(8,2.0000000000000003e-06) +%(16,1.1999999999999999e-05) +(32,8.329999999999999e-05) +(64,0.0006479) +(128,0.0052873) +(256,0.052674599999999995) +(512,0.5249752000000001) +(1024,4.671161) +(2048,136.6769777) +(4096,1179.261048) +(8192,10071.512655) +}; +\addlegendentry{Strassen} +\addplot [ color=black, +]coordinates { +%(2,1e-07) +%(4,2.1e-06) +%(8,1.13e-05) +%(16,7.07e-05) +(32,0.0005041) +(64,0.003596) +(128,0.0254481) +(256,0.1781817) +(512,1.2555) +(1024,8.8302371) +(2048,61.9018691) +(4096,414.648901) +(8192,3014.235467) +}; + +\addlegendentry{MM div and conq} +\addplot[ color=green, +] coordinates { +%(2,3e-07) +%(4,1.1e-06) +%(8,8.6e-06) +%(16,7.819999999999999e-05) +(32,0.0005940000000000001) +(64,0.0044339) +(128,0.0348443) +(256,0.29484730000000003) +(512,2.2228507) +(1024,17.659234500000004) +(2048,141.6103936) +(4096,1147.106865) +(8192,9606.402522) +}; + +\addlegendentry{MM} +\addplot [ color=red, +]coordinates { +%(2,0.0) +%(4,3e-07) +%(8,1.8000000000000001e-06) +%(16,1.1999999999999999e-05) +(32,8.93e-05) +(64,0.0006923) +(128,0.0056842) +(256,0.051771500000000005) +(512,0.5062468000000001) +(1024,4.5048086) +(2048,129.2894619) +(4096,1111.312696) +(8192,9376.173434) +}; +\addlegendentry{BLAS} +\addplot[ color=purple, +] coordinates { +%(2,1e-07) +%(4,0.0) +%(8,1e-07) +%(16,3.9e-06) +(32,2.1000000000000002e-05) +(64,0.00018580000000000002) +(128,0.0012649) +(256,0.0096489) +(512,0.0773765) +(1024,0.7643868) +(2048,7.6320993999999995) +(4096,55.845038) +(8192,478.429957) +}; +\end{axis} +\end{tikzpicture} + +\end{document} diff --git a/buch/papers/multiplikation/images/meas_python.pdf b/buch/papers/multiplikation/images/meas_python.pdf Binary files differnew file mode 100644 index 0000000..fe89773 --- /dev/null +++ b/buch/papers/multiplikation/images/meas_python.pdf diff --git a/buch/papers/multiplikation/images/meas_python.tex b/buch/papers/multiplikation/images/meas_python.tex new file mode 100644 index 0000000..ad43cf6 --- /dev/null +++ b/buch/papers/multiplikation/images/meas_python.tex @@ -0,0 +1,145 @@ + +\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} +\usepackage{pgfplotstable} +\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}[ +xmode=log, ymode=log, +xmin=30, xmax=4200, +ymin=0.01, ymax=70000, +grid=both, +major grid style={black!50}, +xlabel = data input ($n$), +ylabel = {time ($s$)}, +legend pos=north west, +very thick, +scale only axis=true, +width=12cm, height=8cm, + log basis x={10}, + legend cell align={left} +] +\addlegendentry{Winograd} +\addplot[ color=blue, +] coordinates { +% (2, 2.7895e-05 ) +% (4, 0.000104904) +% (8, 0.000552893) +% (16, 0.0045557 ) +(32, 0.0187144 ) +(64, 0.153069 ) +(128, 1.19476 ) +(256, 8.29899 ) +(512, 68.3699 ) +(1024,537.374 ) +(2046,4884.61) +(4096,43597.1) +}; +\addlegendentry{Strassen} +\addplot [ color=black, +]coordinates { + % (2,2.09808e-05 ) + % (4,0.000174284 ) + % (8,0.000943899 ) + % (16,0.00475407 ) + (32,0.0485256 ) + (64,0.220414 ) + (128,1.44718 ) + (256,9.93866 ) + (512,63.961 ) +(1024,461.494 ) +(2046,3860.57) +(4096,22904.3) +}; + +\addlegendentry{MM div and conq} +\addplot[ color=green, +] coordinates { + % (2,8.10623e-06 ) + % (4,9.01222e-05 ) + % (8,0.000729084 ) + % (16,0.00497079 ) + (32,0.02719 ) + (64,0.26528 ) + (128,1.77787 ) + (256,13.27 ) + (512,105.397 ) +(1024,847.321 ) +(2046,7375.93) +(4096,58466) +}; + +\addlegendentry{MM} +\addplot [ color=red, +]coordinates { + % (2,1.85966e-05) + % (4,8.29697e-05 ) + % (8,0.000547171) + % (16,0.00305367 ) + (32, 0.0240743 ) + (64, 0.186895 ) + (128, 1.56369 ) + (256, 11.0062 ) + (512, 85.4768) +(1024,750.757 ) +(2046,6154.18) +(4096,46813.3) +}; +% \addlegendentry{NumPy} +% \addplot[ color=blue, +% ] coordinates { +% % (2,1.83582e-05 ) +% % (4,7.86781e-06) +% % (8,1.00136e-05) +% % (16,5.4121e-05 ) +% (32,4.26769e-05) +% (64,0.000118494) +% (128,0.000244141 ) +% (256,0.000695705 ) +% (512,0.00221705 ) +% (1024,0.0188088 ) +% (2046,0.215739) +% (4096,1.49159) +% }; + +\end{axis} +\end{tikzpicture} + +\end{document} diff --git a/buch/papers/multiplikation/images/mm_visualisation.pdf b/buch/papers/multiplikation/images/mm_visualisation.pdf Binary files differnew file mode 100644 index 0000000..9309df1 --- /dev/null +++ b/buch/papers/multiplikation/images/mm_visualisation.pdf diff --git a/buch/papers/multiplikation/images/mm_visualisation.tex b/buch/papers/multiplikation/images/mm_visualisation.tex new file mode 100644 index 0000000..6e8f789 --- /dev/null +++ b/buch/papers/multiplikation/images/mm_visualisation.tex @@ -0,0 +1,45 @@ + + \begin{tikzpicture}[ampersand replacement=\&] + + \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (0,0) + { + A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\ + \vdots \& \& \vdots \& \& \vdots \\ + A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\ + \vdots \& \& \vdots \& \& \vdots \\ + A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\ + }; + + \node [right=0.1 of A] (mul) {$\cdot$}; + + + \matrix (B)[right=0.1 of mul, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] + { + B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\ + }; + + \node [right=0.1 of B] (eq) {$=$}; + + \matrix (C)[right=0.1 of eq, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] + { + C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\ + }; + + + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {}; + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {}; + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {}; + + + \end{tikzpicture} + +\end{document} + diff --git a/buch/papers/multiplikation/images/strassen.pdf b/buch/papers/multiplikation/images/strassen.pdf Binary files differnew file mode 100644 index 0000000..d150125 --- /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..b51a9d5 --- /dev/null +++ b/buch/papers/multiplikation/images/strassen.tex @@ -0,0 +1,231 @@ +\documentclass[border=10pt]{standalone} +\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{pgfplots} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{cite} +\usepackage{url} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage{tikz} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usetikzlibrary{automata,positioning} +\usetikzlibrary{decorations.text} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + +\begin{tikzpicture}[ampersand replacement=\&] + +\foreach \i in {1,...,4} +{ + \small{ + \matrix (X\i)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (0,-\i*5) + { + A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\ + A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\ + A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\ + A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\ + };} + + \foreach \j in {1,...,7} + { + \matrix(M\i\j)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (\j*5,-\i*5) + { + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + }; + } +} + +\huge{ + \node at (-3,-20) {$\mathbf{C}_{22}=$}; + \node at (-3,-15) {$\mathbf{C}_{21}=$} ; + \node at (-3,-10) {$\mathbf{C}_{12}=$} ; + \node at (-3,-5) {$\mathbf{C}_{11}=$} ; + + \node at (5,-2) {$\mathbf{P}$}; + \node at (10,-2) {$\mathbf{Q}$}; + \node at (15,-2) {$\mathbf{R}$}; + \node at (20,-2) {$\mathbf{S}$}; + \node at (25,-2) {$\mathbf{T}$}; + \node at (30,-2) {$\mathbf{U}$}; + \node at (35,-2) {$\mathbf{V}$}; +} + + +\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)] {}; + +% P +\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=gray, fit=(M21-4-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M21-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M21-4-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M21-1-1)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-4-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-4-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-1-1)] {}; + +\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)] {}; + +% Q +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M12-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M12-1-3)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M22-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M22-1-3)] {}; + +\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=(M42-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {}; + +% R + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M13-3-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M13-4-1)] {}; + +\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=gray, fit=(M33-3-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M33-4-1)] {}; + +\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)] {}; + +% S + +\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=gray, fit=(M24-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M24-2-4)] {}; + +\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=gray, fit=(M44-1-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M44-2-4)] {}; + +%T + +\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=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=gray, fit=(M35-4-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M35-4-2)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M45-4-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M45-4-2)] {}; + +% U + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-1-3)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-1-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-3-3)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-3-1)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-1-3)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-1-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-3-3)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-3-1)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-1-3)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-1-1)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-3-3)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-3-1)] {}; + +\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)] {}; + +%V + +\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=gray, fit=(M27-2-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M27-4-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M27-2-2)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M27-4-2)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-2-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-4-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-2-2)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-4-2)] {}; + +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-2-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-4-4)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-2-2)] {}; +\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-4-2)] {}; + + + + + +\end{tikzpicture} + +\end{document} diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex new file mode 100755 index 0000000..8d0c0a8 --- /dev/null +++ b/buch/papers/multiplikation/loesungsmethoden.tex @@ -0,0 +1,506 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Algorithmen} +\rhead{Algorithmen} + +In diesem Abschnitt werden mehrere Algorithmen zur Berechnung der Matrizenmultiplikation vorgestellt, auch werden Bibliotheken zur unkomplizierten Verwendung von vordefinierten Algorithmen gezeigt. + +\subsection{Standardalgorithmus} + +Die Standardmethode ist im Algorithmus \ref{multiplikation:alg:smm} implementiert. +Hierf\"ur wurde die Gleichung \eqref{multiplikation:eq:MM} direkt umgesetzt. +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}\footnotesize\caption{Matrizenmultiplikation} + \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 \cite{multiplikation:DAC} zu markant besseren Laufzeiten. +Die Grundidee ist, dass ein Problem in mehrere, meist simplere und kleinere Teilprobleme aufgeteilt wird. +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 $2^{n-1} \times 2^{n-1}$ aufgeteilt. +Das Matrizenprodukt +\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} +mit \begin{equation} +\mathbf{C}_{ij} = \sum_{k=1}^{2n} \mathbf{A}_{ik} \mathbf{B}_{kj}, +\label{multiplikation:eq:MM_block} +\end{equation} +ist identisch zu der Gleichung \eqref{multiplikation:eq:MM}, f\"ur die Multiplikation der Untermatrizen $\mathbf{A}_{ik}$ und $\mathbf{B}_{kj}$ wird die Matrizenmultiplikation verwendet. + +Der Algorithmus \ref{multiplikation:alg:devide_mm} zeigt den \textit{Divide and Conquer} Ansatz, +Die 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}\footnotesize\caption{Divide and Conquer Matrizenmultiplikation} + \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} \cite{multiplikation:master_theorem} berechnet werden. Das \textit{Master Theorem} bestimmt die Zeitkomplexit\"at von rekursiven Algorithmen. +Ohne auf dieses vertieft einzugehen, bestimmt die Anzahl rekursiver Aufrufe $\mathcal{T} $ der Funktion die Laufzeit. +In diesem Fall wird die Funktion pro Durchlauf acht mal rekursiv aufgerufen, dies f\"uhrt zu +\begin{equation} \label{multiplikation:eq:laufzeitdac} + \mathcal{T}(n) = 8 \cdot \mathcal{T} \left(\frac{n}{2}\right ) + n^2 = \mathcal{O}(n^{\log_2 8}) = \mathcal{O} (n^{3} ), +\end{equation} +also 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{Strassens Algorithmus} + +Strassens Algorithmus \cite{multiplikation:strassen_1969} beschreibt die Matrizenmultiplikation mit einer Vielzahl von Additionen, Subtraktionen und Multiplikationen von Blockmatrizen. +Die sieben grundlegenden Terme +\begin{equation} \label{multiplikation:eq:strassen} +\begin{split} +\text{\textbf{P}} &= \left(\mathbf{A}_{11} + \mathbf{A}_{22}\right ) \cdot \left(\mathbf{B}_{11} + \mathbf{B}_{22}\right ) \\ +\text{\textbf{Q}} &= \left(\mathbf{A}_{21} + \mathbf{A}_{22}\right ) \cdot \mathbf{B}_{11} \\ +\text{\textbf{R}} &= \mathbf{A}_{11} \cdot \left(\mathbf{B}_{12}-\mathbf{B}_{22}\right ) \\ +\text{\textbf{S}} &= \mathbf{A}_{22} \cdot \left(-\mathbf{B}_{11}+\mathbf{B}_{21}\right ) \\ +\text{\textbf{T}} &= \left(\mathbf{A}_{11} + \mathbf{A}_{12}\right ) \cdot \mathbf{B}_{22} \\ +\text{\textbf{U}} &= \left(-\mathbf{A}_{11} + \mathbf{A}_{21}\right ) \cdot \left(\mathbf{B}_{11} + \mathbf{B}_{12}\right ) \\ +\text{\textbf{V}} &= \left(\mathbf{A}_{12} - \mathbf{A}_{22}\right ) \cdot \left(\mathbf{B}_{21} + \mathbf{B}_{22}\right ) +\end{split} +\end{equation} +aus $\mathbf{A}$ und $\mathbf{B}$ werden f\"ur die Berechnung der Bl\"ocke +\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} +der Matrix $\mathbf{C}$ gebraucht. +\begin{algorithm}\footnotesize\caption{Strassen Matrizenmultiplikation} + \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 Methode wird in der Abbildung \ref{multiplikation:fig:strassen} grafisch dargestellt. +Jedes Feld steht f\"ur eine Multiplikation zweier Matrizenelementen von $\mathbf{A}$ oder $\mathbf{B}$ . +Die gr\"unen Felder auf der linken Seite, zeigen die Addition, welche f\"ur den dazugeh\"origen Term ben\"otigt wird. +Die sieben Spalten beschreiben die Matrizen $\mathbf{P,Q,R, \ldots, V}$. +Rote Felder stehen f\"ur eine Subtraktion und die gr\"unen f\"ur eine Addition. +Graue Felder bedeuten, dass die dazugehörige Spalte nicht für die Berechnung benötigt wird. +\begin{figure} + \center + \includegraphics[width=\linewidth]{papers/multiplikation/images/strassen.pdf} + \caption{Der Algorithmus von Strassen verwendet Multiplikationen zur Berechnung der sieben Blockmatrizen $\mathbf{P}$ bis $\mathbf{V}$ aus $\mathbf{A}$ und $\mathbf{B}$, aus denen sich die Blöcke es Produktes $\mathbf{C}=\mathbf{AB}$ ausschliesslich durch Addition und Subtraktion bilden lassen. Die einzelnen Felder in den Quadraten stellen alle möglichen Produkte von Matrizen $\mathbf{A}_{ik}$ und $\mathbf{B}_{jl}$ dar. In den grossen Quadraten am linken Rand sind diejenigen Produkte grün markiert, welche zusammen die entsprechenden Blöcke $\mathbf{C}_{il}$ von $\mathbf{C}$ ergeben. In den Spalten $\mathbf{P}$ bis $\mathbf{V}$ sind die Produkte farblich hervorgehoben, die in der Definition der entsprechenden Matrix vorkommen. Grün und rot symbolisieren die Vorzeichen, mit denen die Produkte kombiniert werden müssen. Graue Felder werden für die Berechnung von $\mathbf{C}_{il}$ nicht benötigt.} + \label{multiplikation:fig:strassen} +\end{figure} + +Die Funktion wird sieben mal rekursiv aufgerufen. +Dies f\"uhrt nach dem \textit{Master Theorem} zu einer Laufzeit von +\begin{equation} \label{multiplikation:eq:laufzeitstrassen} +\mathcal{T}(n) = +7 \cdot \mathcal{T}\left(\frac{n}{2}\right) + n^2 = \mathcal{O}(n^{\log_2 7} ) = \mathcal{O}(n^{2.8074} ) +\end{equation} +und ist somit schneller als die Standardmethode. +Man beachte, dass die Anzahl von Additionen und Subtraktionen gr\"osser und die Anzahl der Multiplikationen kleiner wurde. + +\subsection{Winograds Algorithmus} + +Einen weiteren Ansatz lieferte Shmuel Winograd im Jahre 1968 \cite{multiplikation:winograd_1968}. +Er beschrieb einen neuen Algorithmus f\"ur das Skalarprodukt +\begin{equation} \label{multiplikation:eq:skalar} + \langle x,y \rangle = \sum_{i=1}^{n}x_i y_i. +\end{equation} +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} +die jeweils nur von $x$ und $y$ abhängen. +Dazu werden $2 \cdot \lfloor n/2 \rfloor \leq n$ Multiplikationen benötigt. +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{wenn $n$ gerade}\\ + \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{wenn $n$ ungerade}. + \end{cases} +\end{equation} +Das Skalarprodukt kann also mit $ \lfloor \frac{n+1}{2} \rfloor$ weiteren Multiplikationen berechnet werden. +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. +Die Summen f\"ur $\xi$ und $\eta$ m\"ussen nur einmal berechnet werden. +Für die ursprüngliche Gleichung \eqref{multiplikation:eq:skalar} für das Skalarprodukt benötigt man $Tn$ Multiplikationen. +Damit können wir die Laufzeit der Methode von Winograd mit der Laufzeit der Standardmethode vergleichen. Sie ist kleiner als die Laufzeit für die Standardmethode, wenn gilt +\begin{equation}\label{multiplikation:eq:eff} +\begin{array}{crcl} + & N\lfloor n/2\rfloor + T\lfloor(n+1)/2\rfloor \approx Nn/2 + Tn/2 & \le & Tn \\ +\Leftrightarrow & Nn/2 & \le & Tn/2 \\ +\Leftrightarrow & N & \le & T. +\end{array} +\end{equation} +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 Standardmethode nur die H\"alfte ist. +Mit dem gleichen Ansatz wie in der Gleichung \eqref{multiplikation:eq:eff} aber mit quadratischen Matrizen, muss +\begin{align} + \begin{split} +N=2n, &\quad T = n^2 \\ + 2n &\leq n^2 \\ + 2 &\leq n +\end{split} +\end{align} +sein, damit man etwas einspart. +Die Implementation kann Algorithmus \ref{multiplikation:alg:winograd} entnommen werden. +Falls $m=n=p$, werden $\frac{n^3}{2}$ Multiplikationen benötigt. +Im Abschnitt \ref{muliplikation:sec:bigo} wurde bereits erläutert: falls $n \rightarrow \infty$ können Konstanten vernachlässigt werden und + somit entsteht für diesen Algorithmus wieder die ursprüngliche Laufzeit von $\mathcal{O}(n^3 )$. +\begin{algorithm}\footnotesize\caption{Winograds Matrizenmultiplikation} + \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{Basic Linear Algebra Subprograms (BLAS)} + +Die gebräuchliche Methode f\"ur die Anwendung einer optimierten Matrizenmultiplikation ist die Verwendung einer Subroutine aus den \textit{Basic Linear Algebra Subprograms (BLAS)} \cite{multiplikation:BLAS}. +Die meisten numerischen Bibliotheken von high-level Skriptsprachen wie \texttt{Matlab}, \texttt{NumPy (Python)}, \texttt{GNU Octave} oder \texttt{Mathematica} ben\"utzen eine Form von \textit{BLAS}. + +\textit{BLAS} sind dabei in drei unterschiedliche Levels aufgeteilt. + +\begin{itemize} + \item Level 1 + \begin{itemize} + \item Operationen der Art: $\mathbf{y} \leftarrow \alpha \mathbf{x}+\mathbf{y}$ + \item Dieses Level hat $\mathcal{O}(n)$ Charakteristik + \end{itemize} + \item Level 2 + \begin{itemize} + \item Operationen der Art: $\mathbf{y} \leftarrow \alpha \mathbf{A}\mathbf{x}+\beta \mathbf{y}$ + \item Dieses Level hat $\mathcal{O}(n^2)$ Charakteristik + \end{itemize} + \item Level 3 + \begin{itemize} + \item Operationen der Art: $\mathbf{C} \leftarrow \alpha \mathbf{A}\mathbf{B}+\beta\mathbf{C}$ + \item Dieses Level hat $\mathcal{O}(n^3)$ Charakteristik + \end{itemize} +\end{itemize} + +Die \textit{BLAS} sind auf die modernen Computerprozessoren optimiert und k\"onnen dank einer ausgeklügelter Verwendung der Speicherarchitektur zu erheblichen Leistungsoptimierungen f\"uhren. + + +%\subsubsection{General Matrix Multiplication (GEMM)} +% +%Die \textit{Double-GEMM} \cite{multiplikation:DGEMM} ist definiert als: +% +%\textit{DGEMM performs one of the matrix-matrix operations} +%$$ +% C := \alpha \cdot op( A )\cdot op( B ) + \beta \cdot C, +% $$ +% \textit{where op( X ) is one of} +%$$ +%op( X ) = X \quad \text{ or } \quad op( X ) = X^T, +%$$ +% \textit{alpha and beta are scalars, and A, B and C are matrices, with op( A ) +% an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. +% } + +%Die Implementation von $\alpha\mathbf{A}\mathbf{B} + \beta \mathbf{C} = \mathbf{C}$, wobei $\alpha = 1.0$ und $\beta = 0.0$ in der \texttt{C}-Version von \textit{BLAS}, ist als +%\begin{lstlisting}[style=multiplikationC] +%cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, +% m, n, k, 1, A, m , B, k, 0, C, m); +%\end{lstlisting} +%definiert. + + + +\section{Implementation}\label{multiplikation:section:Implementation} +\rhead{Implementation} + +Folgende Algorithmen wurden jeweils in \texttt{C} und \texttt{Python} implementiert. +\begin{itemize} + \item Standard Matrizenmultiplikation + \item \textit{Divide and Conquer} Matrizenmultiplikation + \item Strassens Matrizenmultiplikation + \item Winograds Matrizenmultiplikation + \item \texttt{BLAS} Matrizenmultiplikation in \texttt{C} + \item \texttt{Numpy} Matrizenmultiplikation in \texttt{Python} +\end{itemize} + +Der Code kann im zum Buch gehörigem \textit{GitHub} \footnote{\url{https://github.com/AndreasFMueller/SeminarMatrizen.git}} Repository gefunden werden. +Anzumerken ist, dass die Matrizenmultiplikation von \texttt{NumPy} als einzige Implementation Multiprocessing und Multithreading verwendet, dies f\"uhrt zu den tiefen Messzeiten. +In Abbildung \ref{multiplikation:fig:python} und Abbildung \ref{multiplikation:fig:c_meas_4096} sind de Messresultate grafisch dargestellt. Die selben Messresultate sind tabellarisch in Tabelle \ref{multiplikation:tab:messung_Python} und Tabelle \ref{multiplikation:tab:messung_C} ersichtlich. + +Die gezeigten Algorithmen haben alle eine Laufzeit der Form $\mathcal{O}(n^k) $. +Bei einer doppelt logarithmischen Darstellung unterscheiden sich diese in Geraden mit unterschiedlichen Steigungen. +Bei den grafisch gezeigten Messresultate, können diese Steigungen gut erkannt werden, wobei die tiefere Laufzeit des Strassen Algorithmus eindrücklich zu sehen ist. +Der benötigte Overhead der Algorithmen zeigt sich in unterschiedlichen $y$-Achsenschnittpunkte. + +In der Messung mit der Programmiersprache \texttt{C} kann ein typischer Cache-Effekt beobachtet wer- +den. +Bei den Algorithmen von Winograd und der Standardmethode hat bei einer Matrizengrösse von $n = 2048$ wohl eine Zeile der Matrix nicht an einer Cache Speicherstelle Platz. +Diese beiden Algorithmen sind die Einzigen, welche \texttt{for}-Schleifen über die ganze Breite der Matrizen verwenden. +Dies führt dazu, dass ganze Zeilen zwischengespeichert werden müssen. +Bei den anderen Algorithmen ist dies nicht der Fall. + +Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{multiplikation:tab:pc_config} aufgelistet. + + +\begin{table} + \begin{center} + \begin{tabular}{r l l l l l} + \hline + \hline + \textbf{n} & \textbf{MM (\textit{s})} & \textbf{MM DC (\textit{s})} & \textbf{Strassen (\textit{s})} & \textbf{Winograd (\textit{s})} & \textbf{BLAS (\textit{s})} \\ + \hline + \multicolumn{6}{c}{} \\ + \textbf{32} & \phantom{000}0.000089 & \phantom{000}0.000594 & \phantom{000}0.0005 & \phantom{0000}0.00008 & \phantom{00}0.000021 \\ + \textbf{64} & \phantom{000}0.00069 & \phantom{000}0.0044 & \phantom{000}0.0036 & \phantom{0000}0.00064 & \phantom{00}0.00018 \\ + \textbf{128} & \phantom{000}0.0057 & \phantom{000}0.035 & \phantom{000}0.025 & \phantom{0000}0.0052 & \phantom{00}0.0012 \\ + \textbf{256} & \phantom{000}0.052 & \phantom{000}0.29 & \phantom{000}0.178 & \phantom{0000}0.053 & \phantom{00}0.0096 \\ + \textbf{512} & \phantom{000}0.51 & \phantom{000}2.22 & \phantom{000}1.25 & \phantom{0000}0.55 & \phantom{00}0.077 \\ + \textbf{1024} & \phantom{000}4.50 & \phantom{00}17.65 & \phantom{000}8.83 & \phantom{0000}4.67 & \phantom{00}0.764 \\ + \textbf{2048} & \phantom{0}129.28 & \phantom{0}141.61 & \phantom{00}61.901 & \phantom{00}136.67 & \phantom{00}7.63 \\ + \textbf{4096} & 1111.31 & 1147.10 & \phantom{0}414.64 & \phantom{0}1179.26 & \phantom{0}55.84 \\ + \textbf{8192} & 9376.17 & 9606.40 & 3014.23 & 10071.51 & 478.42 \\ + \multicolumn{6}{c}{} \\ + \hline + \hline + \end{tabular} + \end{center} + \caption{Laufzeiten der verschieden Algorithmen in der Programmiersprache \texttt{C}} + \label{multiplikation:tab:messung_C} + \end{table} + + + + \begin{table} + \begin{center} + \begin{tabular}{r l l l l l} + \hline + \hline + \textbf{n} & \textbf{MM (\textit{s})} & \textbf{MM DC (\textit{s})} & \textbf{Strassen (\textit{s})} & \textbf{Winograd (\textit{s})} & \textbf{NumPy(\textit{s})} \\ + \hline + \multicolumn{6}{c}{} \\ + \textbf{32} &\phantom{0000}0.0240 & \phantom{0000}0.0271& \phantom{0000}0.04852 & \phantom{0000}0.01871 & 0.0000426 \\ + \textbf{64} &\phantom{0000}0.186 & \phantom{0000}0.265 & \phantom{0000}0.2204 & \phantom{0000}0.1530& 0.000118 \\ + \textbf{128} &\phantom{0000}1.563 & \phantom{0000}1.777 & \phantom{0000}1.447 & \phantom{0000}1.1947 & 0.000244 \\ + \textbf{256} &\phantom{000}11.006 & \phantom{000}13.27 & \phantom{0000}9.938 & \phantom{0000}8.298& 0.000695 \\ + \textbf{512} &\phantom{000}85.476 & \phantom{00}105.397 & \phantom{000}63.961 & \phantom{000}68.360 & 0.00221\\ + \textbf{1024} &\phantom{00}750.757 & \phantom{00}847.321 & \phantom{00}461.494 & \phantom{00}537.374 & 0.0188 \\ + \textbf{2048} &\phantom{0}6154.18 & \phantom{0}7375.93 & \phantom{0}3860.57 & \phantom{0}4884.61 & 0.215 \\ + \textbf{4096} & 46813.30 & 58466.00 & 22904.30 & 43597.10 & 1.49 \\ + \multicolumn{6}{c}{} \\ + \hline + \hline + \end{tabular} + \end{center} + \caption{Laufzeiten der verschieden Algorithmen in der Skriptsprache \texttt{Python}} + \label{multiplikation:tab:messung_Python} + \end{table} + + \begin{table} + \begin{center} + \begin{tabular}{c c c c} + \hline + \hline + \textbf{CPU} & \textbf{OS} & \textbf{GPU } & \textbf{Memory } \\ + \hline + \multicolumn{4}{c}{} \\ + Intel® Core™ i7-4770K CPU & Ubuntu 20.04.2 LTS & Radeon RX 570 & 32 GB 1600 MHz \\ + @ 3.50GHz × 8 & 64-bit & & \\ + \multicolumn{4}{c}{} \\ + \hline + \hline + \end{tabular} + \end{center} + \caption{Messsystem} + \label{multiplikation:tab:pc_config} + \end{table} + +\begin{figure} + \center + \includegraphics[width=\linewidth]{papers/multiplikation/images/meas_c} + \caption{Doppelt logarithmisch dargestellte Laufzeiten, der verschieden Algorithmen, in der Programmiersprache \texttt{C}. + Die Steigung der Messreihe mit Strassens Algorithmus ist deutlich kleiner als deren der anderen Algorithmen. + Die Messung von Winograd ist beinahe gleich wie die Messung mit der Standardmethode, deshalb ist sie nicht gut sichtbar.} + \label{multiplikation:fig:c_meas_4096} +\end{figure} + + +\begin{figure} + \center + \includegraphics[width=\linewidth]{papers/multiplikation/images/meas_python} + \caption{Doppelt logarithmisch dargestellte Laufzeiten, der verschieden Algorithmen, in der Skriptsprache \texttt{Python}. + Die Steigung der Messreihe mit Strassens Algorithmus ist deutlich kleiner als deren der anderen Algorithmen. +} + \label{multiplikation:fig:python} +\end{figure} + +\section{Fazit} +\rhead{Fazit} + +Wie man im Abschnitt \ref{multiplikation:section:Implementation} sehen kann, sind die gezeigten Algorithmen trotz der theoretisch geringeren Zeitkomplexitäten den Implementationen der numerischen Bibliotheken klar unterlegen. +Ein optimierter Speicherzugriff hat einen weitaus grösseren Einfluss auf die Laufzeit als die Zeitkomplexität des Algorithmus. + +Doch haben Entdeckungen wie jene von Strassen und Winograd ihre Daseinsberechtigung. +Nicht auf jeden Computersystemen können die \textit{BLAS} angewandt werden. +Denke man an sehr kleine Mikrocontroller ohne Floatingpoint Recheneinheiten oder auch an \textit{Field Programmable Gate Arrays (FPGA's)}. +Der Overhead der gezeigten Algorithmen ist in allen Fällen grösser als bei der Standardmethode (z.B. sieben rekursive Aufrufe gegenüber drei \texttt{for}-Schleifen). +Um diesem entgegenzuwirken muss der Laufzeitunterschied zwischen Addition und Multiplikation gross genug sein. +Wenn dies gegeben ist und dazu noch grosse Matritzen multipliziert werden, kann die Verwendung der Algorithmen von Strassen oder Winograd zu einer Senkung der Laufzeit führen. diff --git a/buch/papers/multiplikation/main.tex b/buch/papers/multiplikation/main.tex index 42f2768..4a23109 100644..100755 --- a/buch/papers/multiplikation/main.tex +++ b/buch/papers/multiplikation/main.tex @@ -1,36 +1,40 @@ +% !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} +\definecolor{mygreen}{RGB}{28,172,0} % color values Red, Green, Blue +\definecolor{mylilas}{RGB}{170,55,241} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} +\lstdefinestyle{multiplikationC}{ + numbers=left, + belowcaptionskip=1\baselineskip, + breaklines=true, + frame=l, + framerule=0pt, + framesep=-1pt, + xleftmargin=1em, + language=C, + showstringspaces=false, + basicstyle=\ttfamily, + keywordstyle=\bfseries\color{green!40!black}, + commentstyle=\itshape\color{purple!40!black}, + identifierstyle=\color{blue}, + stringstyle=\color{red}, + numberstyle=\ttfamily\tiny, + backgroundcolor=\color{backcolour} +} + +\chapter{Schnelle Matrizenmultiplikation\label{chapter:multiplikation}} +\lhead{Schnelle Matrizenmultiplikation} \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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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..879b210 --- /dev/null +++ b/buch/papers/multiplikation/problemstellung.tex @@ -0,0 +1,137 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Laufzeiten von Algorithmen} +\rhead{Laufzeiten von Algorithmen} +Wegen der breiten Anwendung der Matrizenmultiplikation ist eine effiziente Ausführung dieser Operation von grosser Bedeutung. +Das Ziel dieses Papers ist, verschiedenen Algorithmen der Matrizenmultiplikation vorzustellen. +Gezielt wird auf Algorithmen eingegangen, welche das Problem schneller als der Standardalgorithmus l\"osen. + +\label{muliplikation:sec:bigo} +Die Big $\mathcal{O}$ Notation beschreibt die Laufzeitkomplexit\"at eines Algorithmus in Relation zur Inputgrösse \cite{multiplikation:bigo}. +$f(x) \in \mathcal{O}(g(x))$ besagt, dass die Funktion $f$ nicht wesentlich schneller w\"achst als $g$, wenn $x \rightarrow \infty$. +Dies ist gegeben, falls es für $f \in \mathcal{O}(n^k)$ eine Konstante $C$ gibt, mit $f(n) \leq Cn^k$. +% Es gibt eine Konstante $K$ derart, dass $f(x) \le K g(x)$ für $x\to\infty$. +Vereinfacht werden f\"ur Algorithmen die folgenden Sprechweisen 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} + +Konstanten werden nicht beachtet, eine Laufzeit von $4n^2$ führt, für $n \rightarrow \infty$ zu $\mathcal{O}(n^2)$. +In der Abbildung \ref{multiplikation:fig:bigo} k\"onnen die verschiedenen Laufzeiten miteinander verglichen werden. +Bei einer doppelt logarithmischen Darstellung werden Polynome der Form $f(x) = x^k$ als Gerade und Exponentialfunktionen der Form $f(x) = a^x$ als nach oben gekr\"ummte Kurven abgebildet. + + + +\subsubsection{Beispielalgorithmen} + +Es folgen einige Beispiele von Algorithmen, welche zu einer bestimmten Zeitkomplexit\"atsklasse zugeteilt werden k\"onnen. + + +\begin{table}[t] + \begin{tabular}{ll} + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:b1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B1}{$a, b$} + \State \textbf{return} $a+b$ + \EndFunction + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\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} + \end{minipage} \\ + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \label{multiplikation:alg:linear} + \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 + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\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} + \end{minipage} + \end{tabular} +\end{table} + +%\begin{table} +% \begin{tabular}[t]{ll} + +% \end{tabular} +%\end{table} + +\paragraph{Beschr\"ankter Algorithmus} +Algorithmus \ref{multiplikation:alg:b1} ist ein Beispiel mit beschränkter Laufzeit $\mathcal{O}(1)$ +Da $a$ und $b$ Skalare sind, hat keine Gr\"osse $n$ einen Einfluss auf die Laufzeit. + +Wie erwähnt werden Konstanten nicht beachtet, der Algorithmus \ref{multiplikation:alg:b2} f\"uhrt ebenso zu $\mathcal{O}(1)$ und nicht zu $\mathcal{O}(2)$. + + +\paragraph{Linearer Algorithmus} + +Der Algorithmus \ref{multiplikation:alg:linear} hat ein lineares Verhalten. +Die \texttt{for}-Schleife wird $n$-mal durchlaufen und f\"uhrt deshalb zu $\mathcal{O}(n)$. + +\paragraph{Quadratischer Algorithmus} + +Der Algorithmus \ref{multiplikation:alg:q1} hat ein quadratisches Verhalten. +Die beiden \texttt{for}-Schleifen werden jeweils $n$-mal durchlaufen und f\"uhrt deshalb zu $\mathcal{O} (n^2 )$. + + +\begin{figure} + \center + \includegraphics[]{papers/multiplikation/images/bigo} + \caption{Laufzeiten von verschiedensten Zeitkomplexitäten. Bei einer doppelt logarithmischen Darstellung werden Polynome der Form $f(x) = x^k$ als Gerade und Exponentialfunktionen der Form $f(x) = a^x$ als nach oben gekr\"ummte Kurven dargestellt.} + \label{multiplikation:fig:bigo} +\end{figure} diff --git a/buch/papers/multiplikation/references.bib b/buch/papers/multiplikation/references.bib index 7149fb1..8815386 100644..100755 --- a/buch/papers/multiplikation/references.bib +++ b/buch/papers/multiplikation/references.bib @@ -33,3 +33,70 @@ 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} +} + +@online{multiplikation:master_theorem, + title = {Master theorem (analysis of algorithms)}, + url = {https://en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms)}, + date = {2021-07-28}, + year = {2021}, + month = {7}, + day = {28} +} + + +@online{multiplikation:DAC, + title = {Divide-and-conquer algorithm}, + url = {https://en.wikipedia.org/wiki/Divide-and-conquer_algorithm}, + date = {2021-07-28}, + year = {2021}, + month = {7}, + day = {28} +} + +@online{multiplikation:BLAS, + title = {BLAS (Basic Linear Algebra Subprograms)}, + url = {http://www.netlib.org/blas/}, + date = {2021-08-01}, + year = {2021}, + month = {8}, + day = {01} +} + +@online{multiplikation:DGEMM, + title = {DGEMM}, + url = {http://www.netlib.org/lapack/explore-html/d1/d54/group__double__blas__level3_gaeda3cbd99c8fb834a60a6412878226e1.html#gaeda3cbd99c8fb834a60a6412878226e1}, + date = {2021-08-01}, + year = {2021}, + month = {8}, + day = {01} +} diff --git a/buch/papers/multiplikation/teil0.tex b/buch/papers/multiplikation/teil0.tex deleted file mode 100644 index 082b7f5..0000000 --- a/buch/papers/multiplikation/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{multiplikation:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{multiplikation:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/multiplikation/teil1.tex b/buch/papers/multiplikation/teil1.tex deleted file mode 100644 index 0a6903a..0000000 --- a/buch/papers/multiplikation/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{multiplikation:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{multiplikation:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{multiplikation:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{multiplikation:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{multiplikation:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/multiplikation/teil2.tex b/buch/papers/multiplikation/teil2.tex deleted file mode 100644 index efbf31a..0000000 --- a/buch/papers/multiplikation/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{multiplikation:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{multiplikation:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/multiplikation/teil3.tex b/buch/papers/multiplikation/teil3.tex deleted file mode 100644 index f58508b..0000000 --- a/buch/papers/multiplikation/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{multiplikation:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{multiplikation:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk new file mode 100644 index 0000000..5f14129 --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk @@ -0,0 +1,254 @@ +# Fdb version 3 +["pdflatex"] 1620305767 "algo.tex" "algo.pdf" "algo" 1621586452 + "/dev/null" 1621583990 0 d41d8cd98f00b204e9800998ecf8427e "" + "/etc/texmf/web2c/texmf.cnf" 1619433543 475 c0e671620eb5563b2130f56340a5fde8 "" + "/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc" 1165713224 4850 80dc9bab7f31fb78a000ccfed0e27cab "" + "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm" 1136768653 3584 adb004a0c8e7c46ee66cad73671f37b4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm" 1229303445 688 37338d6ab346c2f1466b29e195316aa4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs5.tfm" 1229303445 684 3a51bd4fd9600428d5264cf25f04bb9a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs7.tfm" 1229303445 692 1b6510779f0f05e9cbf03e0f6c8361e6 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxb.tfm" 1136768653 1020 c53143d3e3747b5c1149bd9a5ecd7b55 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxmi.tfm" 1136768653 1056 e2202af076e43d03fc17f87e104021b0 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmb.tfm" 1136768653 4572 2c370d27bbb031f7592de9d41dc8cfca "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm" 1136768653 4452 0fd0a792eaab7113e4d4f1b941ff0367 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmri.tfm" 1136768653 4640 ce59980bcbe9e6236fab46d0b5212c7e "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm" 1136768653 1004 c0e991f864f31f017ea4ff9e451b76d4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xb.tfm" 1136768653 6892 772bf8e6c154137db8568fa8a47a6ceb "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm" 1136768653 6716 6d25a377562601272906e3bfe6b2817a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm" 1136768653 1080 b674b4ba143004461509a754a0984b67 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm" 1136768653 688 f56006d6e56f46e63d9f63252958b828 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm" 1136768653 2584 cf4a6a7c2a518d47468fe29ef0913ba0 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm" 1232065820 1944 f854e259cb2839e49d4aa2949544a6e1 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm" 1136768653 1180 72784d0ee5a983fba99a0986b31b0493 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm" 1136768653 2408 aec793a3c45e495f7ad15b227c91f508 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm" 1136768653 1268 1d124f224979493f8fd017a7597ea1cd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm" 1136768653 972 2c9ffac4bbd20f91c01aaef9bf3f8710 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm" 1136768653 988 098ca7e8cc5647b9ac21b82dbdce1f01 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm" 1136768653 1084 75e807e9e71f7a312e4e1187dce5e93b "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyatip10.tfm" 1381187214 608 50246cc71b0635b0ba0a5c10a0bf4257 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybsql10.tfm" 1381187214 608 4db60f15ea23b4ec2d796c6d568a63fa "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybtip10.tfm" 1381187214 608 50246cc71b0635b0ba0a5c10a0bf4257 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycirc10.tfm" 1381187214 844 3393210079fb4ed9347e214b3bfd7c1a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmat10.tfm" 1381187214 608 f124f78ed50a1817738d2adb190cf2bd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmbt10.tfm" 1381187214 608 f124f78ed50a1817738d2adb190cf2bd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xydash10.tfm" 1381187214 984 5c01c46b93e3ba8369f3f8edc6e62aef "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyluat10.tfm" 1381187214 608 a3a3bc08980c5126ff2a7a68fb5a64ff "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xylubt10.tfm" 1381187214 608 a3a3bc08980c5126ff2a7a68fb5a64ff "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/rtxmi.pfb" 1232065820 13806 49b888f4605a088e66b9eb4fee320a6e "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/rtxr.pfb" 1136849748 6339 e2b78706efdc360ee6aec9b6e20211a7 "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/txex.pfb" 1136849748 17531 c91f2d6943f51d7c46d6b7b9cedd50ba "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/txsy.pfb" 1136849748 20336 69267d8a81bca8b24c9b42694a4a28f9 "" + "/usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmb8a.pfb" 1136849748 44729 811d6c62865936705a31c797a1d5dada "" + "/usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmr8a.pfb" 1136849748 46026 6dab18b61c907687b520c72847215a68 "" + "/usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmri8a.pfb" 1136849748 45458 a3faba884469519614ca56ba5f6b1de1 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/t1xb.vf" 1136768653 2144 bab2875eda5b2344ea7b1db74ccc03a4 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/t1xr.vf" 1136768653 2140 99e5b3a34695df6221a167ffa8b498d6 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txmi.vf" 1232065820 960 cfcc9d587b40b769f64408b3ca115941 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf" 1136768653 904 e582cae2d8ae3f48a0a520440ebcdb51 "" + "/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b "" + "/usr/share/texlive/texmf-dist/tex/generic/atbegshi/atbegshi.sty" 1575674566 24708 5584a51a7101caf7e6bbf1fc27d8f7b1 "" + "/usr/share/texlive/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty" 1576625341 40635 c40361e206be584d448876bba8a64a3b "" + "/usr/share/texlive/texmf-dist/tex/generic/bitset/bitset.sty" 1576016050 33961 6b5c75130e435b2bfdb9f480a09a39f9 "" + "/usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty" 1576625273 7734 b98cbb34c81f667027c1e3ebdbfce34b "" + "/usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty" 1576625223 8371 9d55b8bd010bc717624922fb3477d92e "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty" 1572645307 492 1994775aa15b0d1289725a0b1bbc2d4c "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifpdf.sty" 1572645307 480 5778104efadad304ced77548ca2184b1 "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty" 1573336935 6902 30fdaf7dc5636b8e3afa306210c45cae "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty" 1572645307 1057 525c2192b5febbd8c1f662c9468335bb "" + "/usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty" 1575499628 8356 7bbb2c2373aa810be568c29e333da8ed "" + "/usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty" 1576625065 31769 002a487f55041f8e805cfbf6385ffd97 "" + "/usr/share/texlive/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty" 1576878844 5412 d5a2436094cd7be85769db90f29250a6 "" + "/usr/share/texlive/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty" 1576624944 13807 952b0226d4efca026f0e19dd266dcc22 "" + "/usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty" 1576624883 18552 1e1cc7b75da0dfaacce7cdcb27d306bf "" + "/usr/share/texlive/texmf-dist/tex/generic/pdfescape/pdfescape.sty" 1576015897 19007 15924f7228aca6c6d184b115f4baa231 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1557692582 992 fb3cda354707a54fda62787a411c7c22 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1546728038 43820 bc6cf5aa959817914ace33f5c6232161 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1557692582 19324 c9a64402f22bd8d81821141a357af653 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1546728038 6038 d639d02574be9a72f3c602c2a3510e02 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1546728038 6948 284bbe3c9a7ca0a826c1c03895e69b9f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1546728038 4883 a6f3eb1f71d8c4affaf43a169828b043 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1546728038 2544 3b1b198fd49f01e328adc9162a07b213 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1576793519 44189 1fd6229dad4c898883516c032f2ca5d2 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1546728038 17311 3092579be20ef0f229c42ad3f09da85c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex" 1546728038 21302 d6c4b340248adbe650ebf6ca76bdccca "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex" 1562964315 9690 7585efa5a591822837f837bc5bc35621 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1576793519 33335 942ccafe284041918d36e54696b98aa7 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1546728038 2965 502761b60f43ab2de5ecb2f4625163ae "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex" 1546728038 5196 f8c5c775d4d6e2cb050392127cabda72 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex" 1576793519 20726 ed6ec1d6f0f35e7a93de4e79af83dbce "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex" 1557692582 35249 144a6b9c4df4644618bb3a0a40472608 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex" 1546728038 21989 266e83c51fe41eb8b8d5e6896dc71cc1 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1546728038 8842 5cc856e132fac404805c6da091779283 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryarrows.code.tex" 1546728038 319 8fc6edce901e074ba09de320a8fc686b "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex" 1546728038 3986 c962be8d57437fcaf853d2babd8ed403 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex" 1546728038 4572 980c82f01c0e3983edadbbc373d304cb "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfit.code.tex" 1546728038 3643 4a4bd51bd85886cc39d4073af8cf77a9 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymatrix.code.tex" 1546728038 4202 e655aa2657da1088ec7745ece2876c4c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex" 1546728038 3937 20cd45386ca23052ce976464f0ada984 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex" 1546728038 919 da625675781832f2b61a7048a51ef656 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex" 1576793519 11544 2a5d66a3270abf4ef673e8a0b7734a90 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1576967981 187592 7922ceab1864698dec4c84978d5b182f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex" 1546728038 31874 d843d507175f2bdfa3abf01f0349dac8 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1546728038 32995 a4d54c043ae5274ceaaddeb36ad43a6f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.code.tex" 1546728038 62281 fd68e6d2c2dc178611c8f4d2d86e79ae "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfint.code.tex" 1557692582 3063 8c415c68a0f3394e45cfeca0b65f6ee6 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex" 1557692582 521 c70cf6ad609de83a27ee7929eb356332 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1557692582 13391 933cab19c6d27039dbfc487330d1005a "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1557692582 104938 15f2d8bdabd6bf9ca70f62cd8e3d4940 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1557692582 10157 218d58ab074e5bd0d027de45ec64cc00 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex" 1576793519 28176 568b081ec39645f2db1a29fbd0c635e2 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1562964315 9054 388d21239a1b6df2cc8beaae31c976b0 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex" 1557692582 3865 cddf7ddc80f018587c55afdcc79fc333 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1557692582 3177 27d85c44fbfe09ff3b2cf2879e3ea434 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1557692582 10925 df50b8a6e5660a585e3a2bf55726dcc8 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1562964315 7787 1750fc3f164703caf31fc8ea9218c67e "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1557692582 3379 cbd0948a550bd7a495a160ca6beee9ed "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1557692582 92405 bba89470858d7b0788a9c09331c39653 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex" 1576793519 36526 453db1f8626a56b5ebb0fad496d6a39f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex" 1576793519 8471 b18959397c76e1e582402ab9f592ed9f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1576793519 21201 46a4dded6619f990ac7347f99fbaac9f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex" 1557692582 16121 9e240115374a8d489f2f786115df83a9 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex" 1576793519 43259 3e05ba63539916af2eaca603c2eda780 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/pgf.revision.tex" 1578520427 465 1f401ab1e7fc6cb7ede39e96c66531fd "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg" 1557692582 926 70ff613fabeb70f5d1673dc0c93987bd "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def" 1557692582 5546 3586827e6032c95512b2a6682d2979a3 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def" 1562964315 12603 c02869ea216d842c29d52fae8738264e "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex" 1557692582 60269 e86bc0081af83a4ad47e4500ee09a2e4 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex" 1557692582 1896 82c274ff520f9e450ccea4e3ef4edc12 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex" 1557692582 7778 a25a32a10ca820357491d4c7b3ac02ea "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex" 1562964315 23777 cb6c8f02f87d86d621f5cb92c44f4998 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex" 1576793519 36815 f7f1772c398f07af2cb741992963045c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex" 1562964315 37439 bd44d50aef702b03193f731207931834 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex" 1557692582 4494 7e5ace0ccf59408f2cf63219a5d36927 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex" 1557692582 7250 03b2b9fb5fa38e7ca5cc3c45860fb210 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex" 1576793519 28309 488ccc6c701bbdd1bf671f708757aa5c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def" 1562964315 6286 1bd76fc45da9929ab2a64f51cba3ab6f "" + "/usr/share/texlive/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty" 1576624663 7008 f92eaa0a3872ed622bbf538217cd2ab7 "" + "/usr/share/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex" 1403829539 2725 fc34ef3ccb37ba15a640e8fca6190bca "" + "/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex" 1417732693 19231 26434a5656c684f5ffb1f26f98006baa "" + "/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex" 1403829539 7677 6f5ce7c1124cad7ec57d05b2562bd8fe "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty" 1312310545 4692 1e1bcf75c622af1eefd9169948208302 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.tex" 1381187214 115380 413d5f789929a45aab7d12ce0d0aee7d "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex" 1312310545 1449 24340b6befc66d28ee1ebb657efb5892 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex" 1312310545 22657 990ce136a3cc15728ba417a2e78b25c8 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex" 1312310545 1374 43fb8dc80dd748631d78096701166d76 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex" 1312310545 4586 edd672434f45626662368282c0322160 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex" 1312310545 109670 d412ee1ff259daefee5e927172e2f9a8 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex" 1337903317 24249 186931a828664624939ab0b347e3952c "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex" 1312310545 9619 b7e4d9a6936ba2ad6119a280abde9641 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyidioms.tex" 1312310545 2907 1ee562fde0b53c9cd16f7a604f33fdf0 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex" 1312310545 10928 c3a572983ccc9fc596b4e9ce454d5652 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex" 1312310545 22583 25b1e7edeee41f181ee9733429da4a9c "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-co.tex" 1312310545 8442 90cb8a3b00c2081384c1ce988d2ba0a3 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-cu.tex" 1312310545 39762 25a964ebb390bcfcd35c040f477eef1d "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-fr.tex" 1312310545 16485 5686b19cc46d046c885428794ed9c114 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-li.tex" 1312310545 2619 1a12b316e2132654e44ba2cd21def637 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-ro.tex" 1312310545 5290 e16fc85c85f64d0a5c04708bf3312d00 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf.tex" 1312310545 18763 e61049d36bdfccb226f22e582d70d368 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrecat.tex" 1312310545 1391 c8763fc8e281cb6ecf697988b6608e4a "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex" 1312310545 7008 cb768d8d63a12d35607cbb3c4e7ba163 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex" 1381187214 3689 0d51788a4141bc66ab896f7ac63495fd "" + "/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty" 1513722769 12604 3dec726c041422879dc3268237f09026 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1523134290 2211 ca7ce284ab93c8eecdc6029dc5ccbd73 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty" 1523134290 5309 0c9ef5db85b924cdbb316f080dfd826e "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty" 1523134290 4161 7f6eb9092061a11f87d08ed13515b48d "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty" 1580683321 85660 baee036978c7a91f4e2bba43f05e5945 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty" 1523134290 4116 32e6abd27229755a83a8b7f18e583890 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty" 1523134290 2432 8ff93b1137020e8f21930562a874ae66 "" + "/usr/share/texlive/texmf-dist/tex/latex/aobs-tikz/tikzlibraryoverlay-beamer-styles.code.tex" 1389658833 4047 82a015585c1ef210fb6750d6322afa7f "" + "/usr/share/texlive/texmf-dist/tex/latex/atveryend/atveryend.sty" 1576191570 19336 ce7ae9438967282886b3b036cfad1e4d "" + "/usr/share/texlive/texmf-dist/tex/latex/auxhook/auxhook.sty" 1576625391 3935 57aa3c3e203a5c2effb4d2bd2efbc323 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/article.cls" 1580683321 20023 e427dd9e17e239bf926ef3aab67fe35e "" + "/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty" 1581632200 4947 0c2888dd88121ae675fc6e82213623ba "" + "/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty" 1580683321 5159 892429808d9e0e2b3548aaefd9a06ed0 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty" 1580683321 5050 8933a39ad74377accd18991c5eb90c58 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo" 1580683321 8446 9874cccac5fee462272c582807dbbf56 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty" 1581112666 2821 2c0928feafd5527387e29a1af774d030 "" + "/usr/share/texlive/texmf-dist/tex/latex/carlisle/slashed.sty" 1137109962 5327 8b3c95b5f71136add36a4a0bb1507594 "" + "/usr/share/texlive/texmf-dist/tex/latex/cite/cite.sty" 1425427964 26218 19edeff8cdc2bcb704e8051dc55eb5a7 "" + "/usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce "" + "/usr/share/texlive/texmf-dist/tex/latex/eso-pic/eso-pic.sty" 1526160256 11991 c1669f88e13f8bb6243df144e456b477 "" + "/usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty" 1548974385 11128 a53805799bebfed6358fc1658a18e41f "" + "/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty" 1578002852 41601 9cf6c5257b1bc7af01a58859749dd37a "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def" 1515537368 17334 520b9b85ad8a2a48eda3f643e27a5179 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty" 1580683321 16932 04729abe63b66ec59ea56edcd722b058 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty" 1580683321 9067 1b996612394a52e1efe89c8bfe8a5892 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/lscape.sty" 1580683321 1753 f80abc75c0e3a4915097779c2649cc98 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty" 1580683321 3976 d7fa7d81d2870d509d25b17d0245e735 "" + "/usr/share/texlive/texmf-dist/tex/latex/hycolor/hycolor.sty" 1580250785 17914 4c28a13fc3d975e6e81c9bea1d697276 "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def" 1579642962 50630 3d9728faf8630190cf601ce2cbe470d9 "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty" 1579642962 238752 60dd338d71b6a4ab2192131f73dc908b "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty" 1579642962 13244 0070bcab7b5a88187847128d22faf4d8 "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def" 1579642962 14134 32b36577d311ddb6522413c7581ee968 "" + "/usr/share/texlive/texmf-dist/tex/latex/jknapltx/mathrsfs.sty" 1137110241 300 12fa6f636b617656f2810ee82cb05015 "" + "/usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd" 1137110241 548 cc4e3557704bfed27c7002773fad6c90 "" + "/usr/share/texlive/texmf-dist/tex/latex/kvoptions/kvoptions.sty" 1575152344 22520 c4c2dab203104295e1e618be7e5c0f5b "" + "/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def" 1580854751 25404 9d60f463a00d154207ec0048dee27cf0 "" + "/usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty" 1581719662 4381 04628f3002bdd1d9c43ef984fd60ae18 "" + "/usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty" 1581719662 81717 e93576ac4b24ce6e121ebd6ec6cf2893 "" + "/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af "" + "/usr/share/texlive/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty" 1575499565 5766 13a9e8766c47f30327caf893ece86ac8 "" + "/usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex" 1546728170 98047 c6fa29828cc60471827afe275c8bd77f "" + "/usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.sty" 1546638616 18060 8cf65af2c4529eed91b5d364b50d3ada "" + "/usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg" 1568236792 1830 bbaba8afaf42cc048ec4d4ff73467521 "" + "/usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty" 1568236792 80511 830f3f1d3ab7448dd84233e9c2f6462c "" + "/usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty" 1568236792 77022 32914f01b528131c47be2a1040d3856d "" + "/usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/pgflibrarymatrix.skeleton.code.tex" 1565039202 19612 007f8469df07e9ef0f680e346cc01945 "" + "/usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/tikzlibrarymatrix.skeleton.code.tex" 1565039202 7267 4d597b08b2429acaa1e526052d9509ed "" + "/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty" 1177890616 3878 6aa7c08ff2621006e0603349e40a30a8 "" + "/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty" 1559339157 5486 a1d954b09782ba0acd8a8abfd98e1028 "" + "/usr/share/texlive/texmf-dist/tex/latex/paralist/paralist.sty" 1485124581 14857 82c76ebe8f06becf69ab309565b2a0cb "" + "/usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty" 1575674318 6575 25396d208d8f2b9395d06ef315d5886c "" + "/usr/share/texlive/texmf-dist/tex/latex/pdfpages/pdfpages.sty" 1580249532 54071 88f1e37dc9e1f95352061a066ed07263 "" + "/usr/share/texlive/texmf-dist/tex/latex/pdfpages/pppdftex.def" 1580249532 6418 197ed301e61ce5b7f446e70345a43a62 "" + "/usr/share/texlive/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty" 1574631863 19963 36fd8e818f9f0f32e2db8413d4970122 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty" 1546728038 1090 d20f587ea9464d1841bd0d13d3ff9856 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty" 1288312291 410 5bf12ea7330e5f12c445332a4fe9a263 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty" 1546728038 21013 e98e1aaaf40d31632787c2bd25d24b57 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty" 1546728038 989 2cf3da8e8ec55131c49389428d565e37 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty" 1203877327 339 592cf35cba3d400082b8a9a5d0199d70 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/math/pgfmath.sty" 1393459310 306 0796eafca5e159e6ec2167a6d22d81b1 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty" 1393459310 443 0b2e781830192df35c0fd357cf13e26e "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgffor.sty" 1393459310 348 8927fde343487e003b01a4c2ca34073b "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty" 1203727794 274 4cad6e665cc93ac2ac979039a94fa1e1 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty" 1203877327 325 2bcd023400636339210573e2b3ee298b "" + "/usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty" 1156702453 857 6c716f26c5eadfb81029fcd6ce2d45e6 "" + "/usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty" 1576624809 9878 9e94e8fa600d95f9c7731bb21dfb67a4 "" + "/usr/share/texlive/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty" 1575674187 9715 b051d5b493d9fe5f4bc251462d039e5f "" + "/usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cfg" 1522098998 1015 662b4d7ad816b857a598284525f5c75e "" + "/usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cls" 1522098998 28890 df75e6d37f47b7e27bff3f37375336b3 "" + "/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty" 1580683321 12560 ce3f59ceae9d9a27bfe037d6bf1d903c "" + "/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty" 1580683321 10216 5efd55f2010055e7b7875afd6a75be82 "" + "/usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty" 1580683321 4120 d1680a5ff60d0aea9c327e07c030f4e9 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/omltxmi.fd" 1137111002 492 e7f8afe4428797548d4301de03a1b15f "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/omstxsy.fd" 1137111002 329 6ac7e19535b9f1d64e4d8e3f77dc30a3 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/omxtxex.fd" 1137111002 312 11fe1916b0a13a81a05234a6fc7f8738 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/ot1txr.fd" 1137111002 1271 4e3afbd8e832f2f9c7f064894e6e68e4 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/t1txr.fd" 1137111002 1242 cbf8a0d4f750f9833a0bfb05fb39f1cb "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/txfonts.sty" 1206746551 50381 d367461010070c7a491b1f6979ab2062 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxexa.fd" 1137111002 310 1b00b0b05685b816e4c6caccce437e0d "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxmia.fd" 1137111002 334 87436a82076ca2e35cd305f852507afc "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsya.fd" 1137111002 310 cee07e4964749ccbc77d84fc49726a79 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyb.fd" 1137111002 310 8c5467c8932c259af51b0f116c9734bd "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyc.fd" 1137111002 310 4b5d6fe830337242ef847b3bff48ba21 "" + "/usr/share/texlive/texmf-dist/tex/latex/url/url.sty" 1388531844 12796 8edb7d69a20b857904dd0ea757c14ec9 "" + "/usr/share/texlive/texmf-dist/tex/latex/varwidth/varwidth.sty" 1238697683 10894 d359a13923460b2a73d4312d613554c8 "" + "/usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty" 1463002160 55589 34128738f682d033422ca125f82e5d62 "" + "/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty" 1417732693 4962 9c1069474ff71dbc47d5006555e352d3 "" + "/usr/share/texlive/texmf-dist/web2c/texmf.cnf" 1581979058 38841 ce3692aa899bb693b90b87eaa5d4d84e "" + "/usr/share/texmf/web2c/texmf.cnf" 1581979058 38841 ce3692aa899bb693b90b87eaa5d4d84e "" + "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1619433582 4770781 1ed1abab22da9c3e2cc82e4db562318b "" + "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1619433611 8255863 afe1ed795207f6401d11bafd6327aa55 "" + "algo.aux" 1620305767 767 9191aef204e325cc808d7c85cedac35f "pdflatex" + "algo.out" 1620305767 43 8eacde2f35419fc00651f55d16e47ae8 "pdflatex" + "algo.tex" 1621585209 3156 4070ef1cd3442b3ab588aedcc8a306bd "" + (generated) + "algo.aux" + "algo.log" + "algo.pdf" + "algo.out" diff --git a/buch/papers/multiplikation/tikz_formulas/algo.fls b/buch/papers/multiplikation/tikz_formulas/algo.fls new file mode 100644 index 0000000..16d387b --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo.fls @@ -0,0 +1,438 @@ +PWD /home/nunigan/Documents/MSE/FS21/SeminarMatrizen/buch/papers/multiplikation/tikz_formulas +INPUT /etc/texmf/web2c/texmf.cnf +INPUT /usr/share/texmf/web2c/texmf.cnf +INPUT /usr/share/texlive/texmf-dist/web2c/texmf.cnf +INPUT /var/lib/texmf/web2c/pdftex/pdflatex.fmt +INPUT algo.tex +OUTPUT algo.log +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex +INPUT /dev/null +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/varwidth/varwidth.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/varwidth/varwidth.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/mathrsfs.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/mathrsfs.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/txfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/txfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xy.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyrecat.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyidioms.tex +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xydash10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyatip10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybtip10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybsql10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycirc10.tfm +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifpdf.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifpdf.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmat10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmbt10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyluat10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xylubt10.tfm +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-co.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-cu.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-fr.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-li.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-ro.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/paralist/paralist.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/paralist/paralist.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pdfescape/pdfescape.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pdfescape/pdfescape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hycolor/hycolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hycolor/hycolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/auxhook/auxhook.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/auxhook/auxhook.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/kvoptions/kvoptions.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/kvoptions/kvoptions.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/url/url.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/url/url.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bitset/bitset.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bitset/bitset.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/atbegshi/atbegshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/atbegshi/atbegshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/atveryend/atveryend.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/atveryend/atveryend.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/pgf.revision.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/pgf.revision.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfint.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/carlisle/slashed.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/carlisle/slashed.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pdfpages.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pdfpages.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eso-pic/eso-pic.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eso-pic/eso-pic.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pppdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pppdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/cite/cite.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/cite/cite.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymatrix.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymatrix.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/aobs-tikz/tikzlibraryoverlay-beamer-styles.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/aobs-tikz/tikzlibraryoverlay-beamer-styles.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/tikzlibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/tikzlibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/pgflibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/pgflibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfit.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfit.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +INPUT algo.aux +INPUT algo.aux +OUTPUT algo.aux +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omltxmi.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omltxmi.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omstxsy.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omstxsy.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omxtxex.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omxtxex.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxexa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxexa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/t1txr.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/t1txr.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/ot1txr.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/ot1txr.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsya.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsya.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyb.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyb.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs5.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxmia.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxmia.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyc.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyc.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty +INPUT algo.out +INPUT algo.out +INPUT algo.out +INPUT algo.out +OUTPUT algo.pdf +INPUT ./algo.out +INPUT ./algo.out +OUTPUT algo.out +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/lscape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/lscape.sty +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txmi.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmri.tfm +INPUT /var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/t1xr.vf +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/t1xb.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txmi.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmri.tfm +INPUT algo.aux +INPUT ./algo.out +INPUT ./algo.out +INPUT /usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/rtxmi.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/rtxr.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/txex.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/txsy.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmb8a.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmr8a.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmri8a.pfb diff --git a/buch/papers/multiplikation/tikz_formulas/algo.pdf b/buch/papers/multiplikation/tikz_formulas/algo.pdf Binary files differnew file mode 100644 index 0000000..f711224 --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo.pdf diff --git a/buch/papers/multiplikation/tikz_formulas/algo.tex b/buch/papers/multiplikation/tikz_formulas/algo.tex new file mode 100755 index 0000000..1e437c2 --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo.tex @@ -0,0 +1,131 @@ +\documentclass[border=10pt,varwidth]{standalone} +\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{cite} +\usepackage{url} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage{tikz} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usetikzlibrary{automata,positioning} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + +$ +A= +\begin{bmatrix} +A_{11} & A_{12}\\ +A_{21} & A_{22} +\end{bmatrix}, +B= +\begin{bmatrix} +B_{11} & B_{12}\\ +B_{21} & B_{22} +\end{bmatrix}, +C= +\begin{bmatrix} +C_{11} & C_{12}\\ +C_{21} & C_{22} +\end{bmatrix} +$ + +\medskip +$ +A \cdot B = C +$ + +\medskip +$ +C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\ +C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\ +C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\ +C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22} +$ + +\medskip +\begin{math} +\begin{aligned} +\text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\ +\text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\ +\text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\ +\text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\ +\text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\ +\text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12})) \\ +\text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\ +\end{aligned} +\end{math} + + +\medskip +\begin{math} +\begin{aligned} +C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\ +C_{21} &= \text{II} + \text{IV} \\ +C_{12} &= \text{III} + \text{V}\\ +C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\ +\end{aligned} +\end{math} + + +\medskip +\begin{math} +\begin{aligned} +C_{11} &= \text{II} + \text{IV} \\ +C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22})C_{21} \\ +C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\ +C_{11} &= A_{11}B_{11} + A_{12}B_{21} +\end{aligned} +\end{math} + +\section{Winograd} + +$ +x_1 y_1 + x_2 y_2 = (x_1 +y_2)(y_1 + x_2)-x_1 x_2 - y_1 y_2 +$ + +$ +x = (x_1, \cdots, x_n), y=(y_1, \cdots, y_n) +$ + +\[ +\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j} +\] + +\[ +\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j} +\] + +\[ +\langle x,y \rangle = +\begin{cases} + \displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if $n$ is even}\\ +\displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if $n$ is odd} +\end{cases} +\] + +\end{document} diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk new file mode 100644 index 0000000..ddfa880 --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk @@ -0,0 +1,245 @@ +# Fdb version 3 +["pdflatex"] 1621585121 "algo_graph.tex" "algo_graph.pdf" "algo_graph" 1621585184 + "/dev/null" 1621583990 0 d41d8cd98f00b204e9800998ecf8427e "" + "/etc/texmf/web2c/texmf.cnf" 1619433543 475 c0e671620eb5563b2130f56340a5fde8 "" + "/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc" 1165713224 4850 80dc9bab7f31fb78a000ccfed0e27cab "" + "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm" 1136768653 3584 adb004a0c8e7c46ee66cad73671f37b4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm" 1229303445 688 37338d6ab346c2f1466b29e195316aa4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs5.tfm" 1229303445 684 3a51bd4fd9600428d5264cf25f04bb9a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs7.tfm" 1229303445 692 1b6510779f0f05e9cbf03e0f6c8361e6 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxmi.tfm" 1136768653 1056 e2202af076e43d03fc17f87e104021b0 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm" 1136768653 4452 0fd0a792eaab7113e4d4f1b941ff0367 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmri.tfm" 1136768653 4640 ce59980bcbe9e6236fab46d0b5212c7e "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm" 1136768653 1004 c0e991f864f31f017ea4ff9e451b76d4 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm" 1136768653 6716 6d25a377562601272906e3bfe6b2817a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm" 1136768653 1080 b674b4ba143004461509a754a0984b67 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm" 1136768653 688 f56006d6e56f46e63d9f63252958b828 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm" 1136768653 2584 cf4a6a7c2a518d47468fe29ef0913ba0 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm" 1232065820 1944 f854e259cb2839e49d4aa2949544a6e1 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm" 1136768653 1180 72784d0ee5a983fba99a0986b31b0493 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm" 1136768653 2408 aec793a3c45e495f7ad15b227c91f508 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm" 1136768653 1268 1d124f224979493f8fd017a7597ea1cd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm" 1136768653 972 2c9ffac4bbd20f91c01aaef9bf3f8710 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm" 1136768653 988 098ca7e8cc5647b9ac21b82dbdce1f01 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm" 1136768653 1084 75e807e9e71f7a312e4e1187dce5e93b "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyatip10.tfm" 1381187214 608 50246cc71b0635b0ba0a5c10a0bf4257 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybsql10.tfm" 1381187214 608 4db60f15ea23b4ec2d796c6d568a63fa "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybtip10.tfm" 1381187214 608 50246cc71b0635b0ba0a5c10a0bf4257 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycirc10.tfm" 1381187214 844 3393210079fb4ed9347e214b3bfd7c1a "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmat10.tfm" 1381187214 608 f124f78ed50a1817738d2adb190cf2bd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmbt10.tfm" 1381187214 608 f124f78ed50a1817738d2adb190cf2bd "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xydash10.tfm" 1381187214 984 5c01c46b93e3ba8369f3f8edc6e62aef "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyluat10.tfm" 1381187214 608 a3a3bc08980c5126ff2a7a68fb5a64ff "" + "/usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xylubt10.tfm" 1381187214 608 a3a3bc08980c5126ff2a7a68fb5a64ff "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/rtxr.pfb" 1136849748 6339 e2b78706efdc360ee6aec9b6e20211a7 "" + "/usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmr8a.pfb" 1136849748 46026 6dab18b61c907687b520c72847215a68 "" + "/usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmri8a.pfb" 1136849748 45458 a3faba884469519614ca56ba5f6b1de1 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/t1xr.vf" 1136768653 2140 99e5b3a34695df6221a167ffa8b498d6 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txmi.vf" 1232065820 960 cfcc9d587b40b769f64408b3ca115941 "" + "/usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf" 1136768653 904 e582cae2d8ae3f48a0a520440ebcdb51 "" + "/usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b "" + "/usr/share/texlive/texmf-dist/tex/generic/atbegshi/atbegshi.sty" 1575674566 24708 5584a51a7101caf7e6bbf1fc27d8f7b1 "" + "/usr/share/texlive/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty" 1576625341 40635 c40361e206be584d448876bba8a64a3b "" + "/usr/share/texlive/texmf-dist/tex/generic/bitset/bitset.sty" 1576016050 33961 6b5c75130e435b2bfdb9f480a09a39f9 "" + "/usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty" 1576625273 7734 b98cbb34c81f667027c1e3ebdbfce34b "" + "/usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty" 1576625223 8371 9d55b8bd010bc717624922fb3477d92e "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty" 1572645307 492 1994775aa15b0d1289725a0b1bbc2d4c "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifpdf.sty" 1572645307 480 5778104efadad304ced77548ca2184b1 "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty" 1573336935 6902 30fdaf7dc5636b8e3afa306210c45cae "" + "/usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty" 1572645307 1057 525c2192b5febbd8c1f662c9468335bb "" + "/usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty" 1575499628 8356 7bbb2c2373aa810be568c29e333da8ed "" + "/usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty" 1576625065 31769 002a487f55041f8e805cfbf6385ffd97 "" + "/usr/share/texlive/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty" 1576878844 5412 d5a2436094cd7be85769db90f29250a6 "" + "/usr/share/texlive/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty" 1576624944 13807 952b0226d4efca026f0e19dd266dcc22 "" + "/usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty" 1576624883 18552 1e1cc7b75da0dfaacce7cdcb27d306bf "" + "/usr/share/texlive/texmf-dist/tex/generic/pdfescape/pdfescape.sty" 1576015897 19007 15924f7228aca6c6d184b115f4baa231 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1557692582 992 fb3cda354707a54fda62787a411c7c22 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1546728038 43820 bc6cf5aa959817914ace33f5c6232161 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex" 1557692582 19324 c9a64402f22bd8d81821141a357af653 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex" 1546728038 6038 d639d02574be9a72f3c602c2a3510e02 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex" 1546728038 6948 284bbe3c9a7ca0a826c1c03895e69b9f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex" 1546728038 4883 a6f3eb1f71d8c4affaf43a169828b043 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex" 1546728038 2544 3b1b198fd49f01e328adc9162a07b213 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex" 1576793519 44189 1fd6229dad4c898883516c032f2ca5d2 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex" 1546728038 17311 3092579be20ef0f229c42ad3f09da85c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex" 1546728038 21302 d6c4b340248adbe650ebf6ca76bdccca "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex" 1562964315 9690 7585efa5a591822837f837bc5bc35621 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex" 1576793519 33335 942ccafe284041918d36e54696b98aa7 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex" 1546728038 2965 502761b60f43ab2de5ecb2f4625163ae "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex" 1546728038 5196 f8c5c775d4d6e2cb050392127cabda72 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex" 1576793519 20726 ed6ec1d6f0f35e7a93de4e79af83dbce "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex" 1557692582 35249 144a6b9c4df4644618bb3a0a40472608 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex" 1546728038 21989 266e83c51fe41eb8b8d5e6896dc71cc1 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex" 1546728038 8842 5cc856e132fac404805c6da091779283 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryarrows.code.tex" 1546728038 319 8fc6edce901e074ba09de320a8fc686b "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex" 1546728038 3986 c962be8d57437fcaf853d2babd8ed403 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex" 1546728038 4572 980c82f01c0e3983edadbbc373d304cb "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfit.code.tex" 1546728038 3643 4a4bd51bd85886cc39d4073af8cf77a9 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymatrix.code.tex" 1546728038 4202 e655aa2657da1088ec7745ece2876c4c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex" 1546728038 3937 20cd45386ca23052ce976464f0ada984 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex" 1546728038 919 da625675781832f2b61a7048a51ef656 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex" 1576793519 11544 2a5d66a3270abf4ef673e8a0b7734a90 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex" 1576967981 187592 7922ceab1864698dec4c84978d5b182f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex" 1546728038 31874 d843d507175f2bdfa3abf01f0349dac8 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex" 1546728038 32995 a4d54c043ae5274ceaaddeb36ad43a6f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.code.tex" 1546728038 62281 fd68e6d2c2dc178611c8f4d2d86e79ae "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfint.code.tex" 1557692582 3063 8c415c68a0f3394e45cfeca0b65f6ee6 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex" 1557692582 521 c70cf6ad609de83a27ee7929eb356332 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex" 1557692582 13391 933cab19c6d27039dbfc487330d1005a "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex" 1557692582 104938 15f2d8bdabd6bf9ca70f62cd8e3d4940 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex" 1557692582 10157 218d58ab074e5bd0d027de45ec64cc00 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex" 1576793519 28176 568b081ec39645f2db1a29fbd0c635e2 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex" 1562964315 9054 388d21239a1b6df2cc8beaae31c976b0 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex" 1557692582 3865 cddf7ddc80f018587c55afdcc79fc333 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex" 1557692582 3177 27d85c44fbfe09ff3b2cf2879e3ea434 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex" 1557692582 10925 df50b8a6e5660a585e3a2bf55726dcc8 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex" 1562964315 7787 1750fc3f164703caf31fc8ea9218c67e "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex" 1557692582 3379 cbd0948a550bd7a495a160ca6beee9ed "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex" 1557692582 92405 bba89470858d7b0788a9c09331c39653 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex" 1576793519 36526 453db1f8626a56b5ebb0fad496d6a39f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex" 1576793519 8471 b18959397c76e1e582402ab9f592ed9f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex" 1576793519 21201 46a4dded6619f990ac7347f99fbaac9f "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex" 1557692582 16121 9e240115374a8d489f2f786115df83a9 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex" 1576793519 43259 3e05ba63539916af2eaca603c2eda780 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/pgf.revision.tex" 1578520427 465 1f401ab1e7fc6cb7ede39e96c66531fd "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg" 1557692582 926 70ff613fabeb70f5d1673dc0c93987bd "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def" 1557692582 5546 3586827e6032c95512b2a6682d2979a3 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def" 1562964315 12603 c02869ea216d842c29d52fae8738264e "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex" 1557692582 60269 e86bc0081af83a4ad47e4500ee09a2e4 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex" 1557692582 1896 82c274ff520f9e450ccea4e3ef4edc12 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex" 1557692582 7778 a25a32a10ca820357491d4c7b3ac02ea "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex" 1562964315 23777 cb6c8f02f87d86d621f5cb92c44f4998 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex" 1576793519 36815 f7f1772c398f07af2cb741992963045c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex" 1562964315 37439 bd44d50aef702b03193f731207931834 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex" 1557692582 4494 7e5ace0ccf59408f2cf63219a5d36927 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex" 1557692582 7250 03b2b9fb5fa38e7ca5cc3c45860fb210 "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex" 1576793519 28309 488ccc6c701bbdd1bf671f708757aa5c "" + "/usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def" 1562964315 6286 1bd76fc45da9929ab2a64f51cba3ab6f "" + "/usr/share/texlive/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty" 1576624663 7008 f92eaa0a3872ed622bbf538217cd2ab7 "" + "/usr/share/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex" 1403829539 2725 fc34ef3ccb37ba15a640e8fca6190bca "" + "/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex" 1417732693 19231 26434a5656c684f5ffb1f26f98006baa "" + "/usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex" 1403829539 7677 6f5ce7c1124cad7ec57d05b2562bd8fe "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty" 1312310545 4692 1e1bcf75c622af1eefd9169948208302 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xy.tex" 1381187214 115380 413d5f789929a45aab7d12ce0d0aee7d "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex" 1312310545 1449 24340b6befc66d28ee1ebb657efb5892 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex" 1312310545 22657 990ce136a3cc15728ba417a2e78b25c8 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex" 1312310545 1374 43fb8dc80dd748631d78096701166d76 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex" 1312310545 4586 edd672434f45626662368282c0322160 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex" 1312310545 109670 d412ee1ff259daefee5e927172e2f9a8 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex" 1337903317 24249 186931a828664624939ab0b347e3952c "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex" 1312310545 9619 b7e4d9a6936ba2ad6119a280abde9641 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyidioms.tex" 1312310545 2907 1ee562fde0b53c9cd16f7a604f33fdf0 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex" 1312310545 10928 c3a572983ccc9fc596b4e9ce454d5652 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex" 1312310545 22583 25b1e7edeee41f181ee9733429da4a9c "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-co.tex" 1312310545 8442 90cb8a3b00c2081384c1ce988d2ba0a3 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-cu.tex" 1312310545 39762 25a964ebb390bcfcd35c040f477eef1d "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-fr.tex" 1312310545 16485 5686b19cc46d046c885428794ed9c114 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-li.tex" 1312310545 2619 1a12b316e2132654e44ba2cd21def637 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-ro.tex" 1312310545 5290 e16fc85c85f64d0a5c04708bf3312d00 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf.tex" 1312310545 18763 e61049d36bdfccb226f22e582d70d368 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrecat.tex" 1312310545 1391 c8763fc8e281cb6ecf697988b6608e4a "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex" 1312310545 7008 cb768d8d63a12d35607cbb3c4e7ba163 "" + "/usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex" 1381187214 3689 0d51788a4141bc66ab896f7ac63495fd "" + "/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty" 1513722769 12604 3dec726c041422879dc3268237f09026 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c "" + "/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1523134290 2211 ca7ce284ab93c8eecdc6029dc5ccbd73 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty" 1523134290 5309 0c9ef5db85b924cdbb316f080dfd826e "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty" 1523134290 4161 7f6eb9092061a11f87d08ed13515b48d "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty" 1580683321 85660 baee036978c7a91f4e2bba43f05e5945 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty" 1523134290 4116 32e6abd27229755a83a8b7f18e583890 "" + "/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty" 1523134290 2432 8ff93b1137020e8f21930562a874ae66 "" + "/usr/share/texlive/texmf-dist/tex/latex/aobs-tikz/tikzlibraryoverlay-beamer-styles.code.tex" 1389658833 4047 82a015585c1ef210fb6750d6322afa7f "" + "/usr/share/texlive/texmf-dist/tex/latex/atveryend/atveryend.sty" 1576191570 19336 ce7ae9438967282886b3b036cfad1e4d "" + "/usr/share/texlive/texmf-dist/tex/latex/auxhook/auxhook.sty" 1576625391 3935 57aa3c3e203a5c2effb4d2bd2efbc323 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/article.cls" 1580683321 20023 e427dd9e17e239bf926ef3aab67fe35e "" + "/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty" 1581632200 4947 0c2888dd88121ae675fc6e82213623ba "" + "/usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty" 1580683321 5159 892429808d9e0e2b3548aaefd9a06ed0 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty" 1580683321 5050 8933a39ad74377accd18991c5eb90c58 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo" 1580683321 8446 9874cccac5fee462272c582807dbbf56 "" + "/usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty" 1581112666 2821 2c0928feafd5527387e29a1af774d030 "" + "/usr/share/texlive/texmf-dist/tex/latex/carlisle/slashed.sty" 1137109962 5327 8b3c95b5f71136add36a4a0bb1507594 "" + "/usr/share/texlive/texmf-dist/tex/latex/cite/cite.sty" 1425427964 26218 19edeff8cdc2bcb704e8051dc55eb5a7 "" + "/usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce "" + "/usr/share/texlive/texmf-dist/tex/latex/eso-pic/eso-pic.sty" 1526160256 11991 c1669f88e13f8bb6243df144e456b477 "" + "/usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty" 1548974385 11128 a53805799bebfed6358fc1658a18e41f "" + "/usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty" 1578002852 41601 9cf6c5257b1bc7af01a58859749dd37a "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg" 1459978653 1213 620bba36b25224fa9b7e1ccb4ecb76fd "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def" 1515537368 17334 520b9b85ad8a2a48eda3f643e27a5179 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty" 1580683321 16932 04729abe63b66ec59ea56edcd722b058 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty" 1580683321 9067 1b996612394a52e1efe89c8bfe8a5892 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/lscape.sty" 1580683321 1753 f80abc75c0e3a4915097779c2649cc98 "" + "/usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty" 1580683321 3976 d7fa7d81d2870d509d25b17d0245e735 "" + "/usr/share/texlive/texmf-dist/tex/latex/hycolor/hycolor.sty" 1580250785 17914 4c28a13fc3d975e6e81c9bea1d697276 "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def" 1579642962 50630 3d9728faf8630190cf601ce2cbe470d9 "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty" 1579642962 238752 60dd338d71b6a4ab2192131f73dc908b "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty" 1579642962 13244 0070bcab7b5a88187847128d22faf4d8 "" + "/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def" 1579642962 14134 32b36577d311ddb6522413c7581ee968 "" + "/usr/share/texlive/texmf-dist/tex/latex/jknapltx/mathrsfs.sty" 1137110241 300 12fa6f636b617656f2810ee82cb05015 "" + "/usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd" 1137110241 548 cc4e3557704bfed27c7002773fad6c90 "" + "/usr/share/texlive/texmf-dist/tex/latex/kvoptions/kvoptions.sty" 1575152344 22520 c4c2dab203104295e1e618be7e5c0f5b "" + "/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def" 1580854751 25404 9d60f463a00d154207ec0048dee27cf0 "" + "/usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty" 1581719662 4381 04628f3002bdd1d9c43ef984fd60ae18 "" + "/usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty" 1581719662 81717 e93576ac4b24ce6e121ebd6ec6cf2893 "" + "/usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af "" + "/usr/share/texlive/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty" 1575499565 5766 13a9e8766c47f30327caf893ece86ac8 "" + "/usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex" 1546728170 98047 c6fa29828cc60471827afe275c8bd77f "" + "/usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.sty" 1546638616 18060 8cf65af2c4529eed91b5d364b50d3ada "" + "/usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg" 1568236792 1830 bbaba8afaf42cc048ec4d4ff73467521 "" + "/usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty" 1568236792 80511 830f3f1d3ab7448dd84233e9c2f6462c "" + "/usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty" 1568236792 77022 32914f01b528131c47be2a1040d3856d "" + "/usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/pgflibrarymatrix.skeleton.code.tex" 1565039202 19612 007f8469df07e9ef0f680e346cc01945 "" + "/usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/tikzlibrarymatrix.skeleton.code.tex" 1565039202 7267 4d597b08b2429acaa1e526052d9509ed "" + "/usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty" 1177890616 3878 6aa7c08ff2621006e0603349e40a30a8 "" + "/usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty" 1559339157 5486 a1d954b09782ba0acd8a8abfd98e1028 "" + "/usr/share/texlive/texmf-dist/tex/latex/paralist/paralist.sty" 1485124581 14857 82c76ebe8f06becf69ab309565b2a0cb "" + "/usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty" 1575674318 6575 25396d208d8f2b9395d06ef315d5886c "" + "/usr/share/texlive/texmf-dist/tex/latex/pdfpages/pdfpages.sty" 1580249532 54071 88f1e37dc9e1f95352061a066ed07263 "" + "/usr/share/texlive/texmf-dist/tex/latex/pdfpages/pppdftex.def" 1580249532 6418 197ed301e61ce5b7f446e70345a43a62 "" + "/usr/share/texlive/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty" 1574631863 19963 36fd8e818f9f0f32e2db8413d4970122 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty" 1546728038 1090 d20f587ea9464d1841bd0d13d3ff9856 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty" 1288312291 410 5bf12ea7330e5f12c445332a4fe9a263 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty" 1546728038 21013 e98e1aaaf40d31632787c2bd25d24b57 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty" 1546728038 989 2cf3da8e8ec55131c49389428d565e37 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty" 1203877327 339 592cf35cba3d400082b8a9a5d0199d70 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/math/pgfmath.sty" 1393459310 306 0796eafca5e159e6ec2167a6d22d81b1 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty" 1393459310 443 0b2e781830192df35c0fd357cf13e26e "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgffor.sty" 1393459310 348 8927fde343487e003b01a4c2ca34073b "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty" 1203727794 274 4cad6e665cc93ac2ac979039a94fa1e1 "" + "/usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty" 1203877327 325 2bcd023400636339210573e2b3ee298b "" + "/usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty" 1156702453 857 6c716f26c5eadfb81029fcd6ce2d45e6 "" + "/usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty" 1576624809 9878 9e94e8fa600d95f9c7731bb21dfb67a4 "" + "/usr/share/texlive/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty" 1575674187 9715 b051d5b493d9fe5f4bc251462d039e5f "" + "/usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cfg" 1522098998 1015 662b4d7ad816b857a598284525f5c75e "" + "/usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cls" 1522098998 28890 df75e6d37f47b7e27bff3f37375336b3 "" + "/usr/share/texlive/texmf-dist/tex/latex/tools/array.sty" 1580683321 12560 ce3f59ceae9d9a27bfe037d6bf1d903c "" + "/usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty" 1580683321 10216 5efd55f2010055e7b7875afd6a75be82 "" + "/usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty" 1580683321 4120 d1680a5ff60d0aea9c327e07c030f4e9 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/omltxmi.fd" 1137111002 492 e7f8afe4428797548d4301de03a1b15f "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/omstxsy.fd" 1137111002 329 6ac7e19535b9f1d64e4d8e3f77dc30a3 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/omxtxex.fd" 1137111002 312 11fe1916b0a13a81a05234a6fc7f8738 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/ot1txr.fd" 1137111002 1271 4e3afbd8e832f2f9c7f064894e6e68e4 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/t1txr.fd" 1137111002 1242 cbf8a0d4f750f9833a0bfb05fb39f1cb "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/txfonts.sty" 1206746551 50381 d367461010070c7a491b1f6979ab2062 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxexa.fd" 1137111002 310 1b00b0b05685b816e4c6caccce437e0d "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxmia.fd" 1137111002 334 87436a82076ca2e35cd305f852507afc "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsya.fd" 1137111002 310 cee07e4964749ccbc77d84fc49726a79 "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyb.fd" 1137111002 310 8c5467c8932c259af51b0f116c9734bd "" + "/usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyc.fd" 1137111002 310 4b5d6fe830337242ef847b3bff48ba21 "" + "/usr/share/texlive/texmf-dist/tex/latex/url/url.sty" 1388531844 12796 8edb7d69a20b857904dd0ea757c14ec9 "" + "/usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty" 1463002160 55589 34128738f682d033422ca125f82e5d62 "" + "/usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty" 1417732693 4962 9c1069474ff71dbc47d5006555e352d3 "" + "/usr/share/texlive/texmf-dist/web2c/texmf.cnf" 1581979058 38841 ce3692aa899bb693b90b87eaa5d4d84e "" + "/usr/share/texmf/web2c/texmf.cnf" 1581979058 38841 ce3692aa899bb693b90b87eaa5d4d84e "" + "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1619433582 4770781 1ed1abab22da9c3e2cc82e4db562318b "" + "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1619433611 8255863 afe1ed795207f6401d11bafd6327aa55 "" + "algo_graph.aux" 1621585123 662 b2b94621371df8d9296b8bf5bec1b851 "pdflatex" + "algo_graph.out" 1621585122 0 d41d8cd98f00b204e9800998ecf8427e "pdflatex" + "algo_graph.tex" 1621585144 5895 0e03594e6e25b7f3671b72694de0d3f4 "" + (generated) + "algo_graph.out" + "algo_graph.pdf" + "algo_graph.aux" + "algo_graph.log" diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.fls b/buch/papers/multiplikation/tikz_formulas/algo_graph.fls new file mode 100644 index 0000000..bd1c14e --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.fls @@ -0,0 +1,485 @@ +PWD /home/nunigan/Documents/MSE/FS21/SeminarMatrizen/buch/papers/multiplikation/tikz_formulas +INPUT /etc/texmf/web2c/texmf.cnf +INPUT /usr/share/texmf/web2c/texmf.cnf +INPUT /usr/share/texlive/texmf-dist/web2c/texmf.cnf +INPUT /var/lib/texmf/web2c/pdftex/pdflatex.fmt +INPUT algo_graph.tex +OUTPUT algo_graph.log +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/shellesc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifluatex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/xkeyval/xkeyval.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex +INPUT /dev/null +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/standalone/standalone.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/article.cls +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/size10.clo +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/geometry/geometry.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifvtex.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty +INPUT /usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/psnfss/times.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/mathrsfs.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/mathrsfs.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3kernel/expl3.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/l3packages/xparse/xparse.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/lipsum/lipsum.ltd.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/amsmath/amscd.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphicx.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/graphics.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/trig.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-def/pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/fancyhdr/fancyhdr.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/textcomp.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/txfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/txfonts.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xy.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xy.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyrecat.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyidioms.tex +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xydash10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyatip10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybtip10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xybsql10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycirc10.tfm +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifpdf.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/iftex/ifpdf.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyall.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycurve.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyframe.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycmtip.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xytips.tex +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmat10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xycmbt10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xyluat10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/xypic/xylubt10.tfm +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyline.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyrotate.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xycolor.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xymatrix.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xyarrow.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xygraph.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-co.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-cu.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-fr.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-li.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/xypic/xypdf-ro.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/paralist/paralist.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/paralist/paralist.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvsetkeys/kvsetkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/kvdefinekeys/kvdefinekeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pdfescape/pdfescape.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pdfescape/pdfescape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hycolor/hycolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hycolor/hycolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/letltxmacro/letltxmacro.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/auxhook/auxhook.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/auxhook/auxhook.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/kvoptions/kvoptions.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/kvoptions/kvoptions.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/intcalc/intcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/etexcmds/etexcmds.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/url/url.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/url/url.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bitset/bitset.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bitset/bitset.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/bigintcalc/bigintcalc.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/atbegshi/atbegshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/atbegshi/atbegshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/atveryend/atveryend.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/atveryend/atveryend.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/array.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-latex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/ms/everyshi.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfrcs.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/pgf.revision.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/pgf.revision.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/systemlayer/pgfsys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/xcolor/xcolor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics-cfg/color.cfg +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfint.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathconstruct.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathusage.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorescopes.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransformations.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorequick.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreobjects.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepathprocessing.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreexternal.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorelayers.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcoretransparency.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepatterns.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/basiclayer/pgfcorerdf.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleshapes.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmoduleplot.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-0-65.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/compatibility/pgfcomp-version-1-18.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgffor.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/utilities/pgfkeys.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pgf/math/pgfmath.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarytopaths.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/carlisle/slashed.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/carlisle/slashed.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pdfpages.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pdfpages.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/base/ifthen.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/tools/calc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eso-pic/eso-pic.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/eso-pic/eso-pic.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pppdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdfpages/pppdftex.def +INPUT /usr/share/texlive/texmf-dist/tex/latex/cite/cite.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/cite/cite.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymatrix.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarymatrix.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarypositioning.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/aobs-tikz/tikzlibraryoverlay-beamer-styles.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/aobs-tikz/tikzlibraryoverlay-beamer-styles.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/tikzlibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/tikzlibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/pgflibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/matrix-skeleton/pgflibrarymatrix.skeleton.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfit.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryfit.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibrarybackgrounds.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryautomata.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikzlibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/generic/pgf/libraries/shapes/pgflibraryshapes.multipart.code.tex +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/lstmisc.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/listings/listings.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/multirow/multirow.sty +INPUT algo_graph.aux +INPUT algo_graph.aux +OUTPUT algo_graph.aux +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omltxmi.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omltxmi.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omstxsy.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omstxsy.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omxtxex.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/omxtxex.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxexa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxexa.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/t1txr.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/t1txr.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/share/texlive/texmf-dist/tex/context/base/mkii/supp-pdf.mkii +INPUT /usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/ot1txr.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/ot1txr.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsya.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsya.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyb.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyb.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/jknapltx/ursfs.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs7.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs5.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxmia.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxmia.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyc.fd +INPUT /usr/share/texlive/texmf-dist/tex/latex/txfonts/utxsyc.fd +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/refcount/refcount.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty +INPUT /usr/share/texlive/texmf-dist/tex/generic/gettitlestring/gettitlestring.sty +INPUT algo_graph.out +INPUT algo_graph.out +INPUT algo_graph.out +INPUT algo_graph.out +INPUT ./algo_graph.out +INPUT ./algo_graph.out +OUTPUT algo_graph.out +OUTPUT algo_graph.pdf +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/pdflscape/pdflscape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/lscape.sty +INPUT /usr/share/texlive/texmf-dist/tex/latex/graphics/lscape.sty +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs5.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/t1xr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsy.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txex.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsya.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyb.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/rsfs/rsfs10.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txmia.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txsyc.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/txexa.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txmi.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmri.tfm +INPUT /var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txmi.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxmi.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmri.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/txr.vf +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxptmr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/txfonts/rtxr.tfm +INPUT /usr/share/texlive/texmf-dist/fonts/vf/public/txfonts/t1xr.vf +INPUT algo_graph.aux +INPUT ./algo_graph.out +INPUT ./algo_graph.out +INPUT /usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/txfonts/rtxr.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmr8a.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmri8a.pfb diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf Binary files differnew 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=\&] + + \foreach \i in {1,...,4} + { + \small{ + \matrix (X\i)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (0,-\i*5) + { + A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\ + A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\ + A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\ + A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\ + };} + + \foreach \j in {1,...,7} + { + \matrix(M\i\j)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (\j*5,-\i*5) + { + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + }; + } + } + +\huge{ + \node at (-3,-20) {$C_{22}=$}; + \node at (-3,-15) {$C_{21}=$} ; + \node at (-3,-10) {$C_{12}=$} ; + \node at (-3,-5) {$C_{11}=$} ; + + \node at (5,-2) {I}; + \node at (10,-2) {II}; + \node at (15,-2) {III}; + \node at (20,-2) {IV}; + \node at (25,-2) {V}; + \node at (30,-2) {VI}; + \node at (35,-2) {VII}; + } + + + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-1-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-2-2)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-3-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-4-2)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-1-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-2-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-3-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-4-4)] {}; + + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M14-1-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M14-2-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-2)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-2-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-4-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-2-2)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-4-2)] {}; + + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M23-3-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M23-4-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-2)] {}; + + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M34-1-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M34-2-4)] {}; + + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {}; +\end{tikzpicture} + + + +\end{document} diff --git a/buch/papers/munkres/figures/MatrixA.png b/buch/papers/munkres/figures/MatrixA.png Binary files differnew file mode 100644 index 0000000..45a71a4 --- /dev/null +++ b/buch/papers/munkres/figures/MatrixA.png diff --git a/buch/papers/munkres/figures/Matrixdarstellung.png b/buch/papers/munkres/figures/Matrixdarstellung.png Binary files differnew file mode 100644 index 0000000..91a376d --- /dev/null +++ b/buch/papers/munkres/figures/Matrixdarstellung.png diff --git a/buch/papers/munkres/figures/Netzwerkdarstellung.png b/buch/papers/munkres/figures/Netzwerkdarstellung.png Binary files differnew file mode 100644 index 0000000..6c20bf4 --- /dev/null +++ b/buch/papers/munkres/figures/Netzwerkdarstellung.png diff --git a/buch/papers/munkres/figures/Ungarische_Methode_Beispiel.png b/buch/papers/munkres/figures/Ungarische_Methode_Beispiel.png Binary files differnew file mode 100644 index 0000000..242db77 --- /dev/null +++ b/buch/papers/munkres/figures/Ungarische_Methode_Beispiel.png diff --git a/buch/papers/munkres/figures/Ungarische_Methode_Beispiel_Zuw.png b/buch/papers/munkres/figures/Ungarische_Methode_Beispiel_Zuw.png Binary files differnew file mode 100644 index 0000000..73217d3 --- /dev/null +++ b/buch/papers/munkres/figures/Ungarische_Methode_Beispiel_Zuw.png diff --git a/buch/papers/munkres/figures/beispiel_munkres.png b/buch/papers/munkres/figures/beispiel_munkres.png Binary files differnew file mode 100644 index 0000000..2303708 --- /dev/null +++ b/buch/papers/munkres/figures/beispiel_munkres.png diff --git a/buch/papers/munkres/figures/bipartiter_graph.png b/buch/papers/munkres/figures/bipartiter_graph.png Binary files differnew file mode 100644 index 0000000..87c164c --- /dev/null +++ b/buch/papers/munkres/figures/bipartiter_graph.png diff --git a/buch/papers/munkres/figures/ganzzahlige_punkte.png b/buch/papers/munkres/figures/ganzzahlige_punkte.png Binary files differnew file mode 100644 index 0000000..5689825 --- /dev/null +++ b/buch/papers/munkres/figures/ganzzahlige_punkte.png diff --git a/buch/papers/munkres/main.tex b/buch/papers/munkres/main.tex index 4dd20fa..201e70b 100644 --- a/buch/papers/munkres/main.tex +++ b/buch/papers/munkres/main.tex @@ -3,29 +3,11 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:munkres}} -\lhead{Thema} +\chapter{Das Zuordnungsproblem und der Munkres-Algorithmus\label{chapter:munkres}} +\lhead{Das Zuordnungsproblem und der Munkres-Algorithmus} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Marc Kühne} -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/munkres/teil0.tex} \input{papers/munkres/teil1.tex} diff --git a/buch/papers/munkres/teil0.tex b/buch/papers/munkres/teil0.tex index de522c7..0578429 100644 --- a/buch/papers/munkres/teil0.tex +++ b/buch/papers/munkres/teil0.tex @@ -3,20 +3,8 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{munkres: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{munkres:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - +\section{Einleitung\label{munkres:section:teil0}} +\rhead{Einleitung} +Im Bereich der Unternehmensplanung (Operations Research) gibt es verschiedene Fragestellungen. Eine davon ist das sogenannte Transportproblem. Zum Transport einheitlicher Objekte von mehreren Angebots- zu mehreren Nachfrageorten ist ein optimaler, d. h. kostenminimaler Plan zu finden, wobei die vorhandenen und zu liefernden Mengen an den einzelnen Standorten gegeben sowie die jeweiligen Transportkosten pro Einheit zwischen allen Standorten bekannt sind. +Nun gibt es im Bereich des klassischen Transportproblems Sonderfälle. Ein Sonderfall ist z.B. das Zuordnungsproblem. diff --git a/buch/papers/munkres/teil1.tex b/buch/papers/munkres/teil1.tex index f4f5e39..aad45cc 100644 --- a/buch/papers/munkres/teil1.tex +++ b/buch/papers/munkres/teil1.tex @@ -3,53 +3,85 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 +\section{Beschrieb des Zuordnungsproblems \label{munkres: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 + +Das Spezielle an einem Zuordnungsproblem ist, dass es an jedem Ort nur eine Einheit angeboten bzw. nachgefragt wird. Es werden hier nicht Mengen möglichst kostenminimal von einem zum anderen +Ort transportiert, sondern es geht um die kostenminimale Zuordnung von z.B. Personen oder Bau-Maschinen auf bestimmte Orte, Stellen oder Aufgaben. +Um dieses Problem in einer einfachen, händischen Art und Weise zu lösen wurde der Munkres-Algorithmus, auch die Ungarische Methode genannt, entwickelt. Diese Methode ist ein weiteres Hauptthema dieses Kapitels. + +\subsection{Zuordnungsproblem an einem konkreten Beispiel +\label{munkres:subsection:bonorum}} +Als Beispiel betrachten wir den Fall, wo ein Bauunternehmer einen Bauingenieur beauftragt, eine optimale Transportroute für die Umplatzierung seiner Kräne zu eruieren. Das heisst, die Transportstrecke für die Umplatzierung seine Kräne +soll möglichst klein werden. +Die Frage lautet: Wie sind die Kräne umzusetzen, damit deren Transportstrecke minimal wird? Bei der normalen Optimierung dürfen normalerweise beliebige reelle Werte $\mathbb{R}$ angenommen werden. +Beim Beispiel mit den Kräne gibt es aber ein Problem. Bei der Suche nach der optimalen Lösung darf nur die Methode der ganzzahligen Optimierung gewählt werden. Materialien kann man aufteilen, jedoch Maschinen nicht. Die Bauarbeiter auf der neuen Baustelle benötigen einen ganzen Kran und nicht nur einen halben Kran. Es muss immer ein ganzer Kran (Anzahl 1) von A nach B oder gar kein Kran (Anzahl 0) verschoben werden. +Für solche Optimierungsprobleme für reelle Variablen sind verschiedene Verfahren entwickelt worden, die im Allgemeinen auch sehr effizient sind. Das reelle Problem ist also in einer einfachen Art und Weise lösbar. Doch das Problem bleibt, wie in der Illustration oben ersichtlich. Es kann mit ganzzahligen Punkten kein Optimum erzielt werden. Das Ziel ist es an das Optimum so nah wie möglich heranzukommen und dies ist eine vergleichsweise träge und langsame Angelegenheit. + +\begin{figure} +\centering +\includegraphics[width=8cm]{papers/munkres/figures/ganzzahlige_punkte} +\caption{Problem der Ganzzahligkeit.} +\label{munkres:Vr2} +\end{figure} + + +\subsection{Zuordnungsproblem abstrakt +\label{munkres:subsection:bonorum}} + +In einem Zuordnungsproblem sind alle Angebots- und Bedarfsmengen gleich 1 \begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{munkres:equation1} +a_{i}=b_{j}=1 \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{munkres: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{munkres: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{munkres: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. +Das Ziel ist es die Gesamtkosten zu minimieren. Mit Hilfe einer $n\times n$ Matrix +\[ +A += +\begin{pmatrix} +a_{11}&a_{12}&\dots &a_{1n}\\ +a_{21}&a_{22}&\dots &a_{2n}\\ +\vdots&\vdots&\ddots&\vdots\\ +a_{n1}&a_{n2}&\dots &a_{nn} +\end{pmatrix} +\in \mathbb{R}^{n,n} +\] +kann der Faktor Kosten mit in die Rechnung eingebracht werden. +In den Zellen dieser Matrix sind die Zahlen $a_{i,j}$ dargestellt, welche den Weg in z.B. Kilometer beschreiben. +Sie entstehen, wenn man z.B. einem Kran $i$ dem Einsatzort $j$ zuordnet. + +\subsection{Alternative Darstellungen des Zuordnungsproblems +\label{munkres:subsection:bonorum}} +\subsubsection{Netzwerk} +Ein (Fluss- oder Transport-) Netzwerk (engl. network) ist ein zusammenhängender Graph, bei dem jede Kante einen Fluss aufnehmen kann und jede Kante eine Kapazität für den Fluss hat. Die Menge des Flusses auf einer Kante kann die Kapazität der Kante nicht überschreiten. Ein Fluss muss die Einschränkung erfüllen, dass die Menge des Flusses in einen Knoten gleich der Menge des Flusses aus ihm heraus ist. Ein Fluss-Netzwerk (engl. flow network) ist ein Netzwerk, dessen Kanten zusätzlich Kosten pro Mengeneinheit des Flusses zugeordnet sind. Typischerweise will man einen Fluss durch die Kanten bestimmen, der den Einschränkungen des Netzwerks genügt und dessen Gesamtkosten minimal sind. Im Bild 21.2 dargestellt sind in den eckigen Klammern links die externen Flüsse $[1]$ für jeden Kran und in den eckigen Klammern rechts eine $[-1]$ für jeden Baustellenort. Die Kosten sind entlang der Kanten als Zahlen in Klammern dargestellt. +\subsubsection{Matrix} +Im Bild 21.3 ist eine typische $4\times 4$ Matrix dargestellt. Die Zeilen A1 bis A4 betreffen z.B. vier bestehende Maschinenlager eines Unternehmers. In den Spalten B1 bis B4 sind vier neue Baustellenorte zugewiesen. Die Zahlen in der Matrix bedeuten z.B. die Distanz in Kilometer von dem jeweiligen Lager zur jeweiligen Baustelle. +\subsubsection{Bitpartiter Graph} +Ein bipartiter Graph ist ein mathematisches Modell für Beziehungen +zwischen den Elementen zweier Mengen. Es eignet sich sehr gut zur Untersuchung von Zuordnungsproblemen. Zwischen zwei Gruppen von Objekten wird hierbei eine eindeutige Zuordnung hergestellt. Der Graph ist in Abbildung 21.4 ersichtlich. +\begin{itemize} +\item 3 = Anzahl der Knoten aus Menge A. +\item 3 = Anzahl der Knoten aus Menge B. +\end{itemize} + + +\begin{figure} +\centering +\includegraphics[width=5cm]{papers/munkres/figures/Netzwerkdarstellung} +\caption{Typische Netzwerkdarstellung eines Zuordnungsproblems.} +\label{munkres:Vr2} +\end{figure} +\begin{figure} +\centering +\includegraphics[width=5cm]{papers/munkres/figures/Matrixdarstellung} +\caption{Typische 4x4 Matrixdarstellung eines Zuordnungsproblems.} +\label{munkres:Vr2} +\end{figure} +\begin{figure} +\centering +\includegraphics[width=5cm]{papers/munkres/figures/bipartiter_graph} +\caption{$K_{3,3}$ vollständig bipartiter Graph mit 3 Knoten pro Teilmenge.} +\label{munkres:Vr2} +\end{figure} diff --git a/buch/papers/munkres/teil2.tex b/buch/papers/munkres/teil2.tex index 23536b9..2fe24f8 100644 --- a/buch/papers/munkres/teil2.tex +++ b/buch/papers/munkres/teil2.tex @@ -3,38 +3,10 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 +\section{Schwierigkeit der Lösung (Permutationen) \label{munkres: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{munkres: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. +\rhead{Schwierigkeit der Lösung (Permutationen)} +Eine Permutation ist eine Anordnung von Objekten in einer bestimmten Reihenfolge oder eine Umordnung von Objekten aus einer vorgegebenen Reihung. Ist eine optimale Zuordnung gefunden, so steht in jeder Zeile und jeder Spalte der Matrix genau ein Element, das zur optimalen Lösung gehört, eine solche Gruppe von Positionen wird auch als Transversale der Matrix bezeichnet. +Die Problemstellung kann auch so formuliert werden, dass man die Zeilen- oder die Spaltenvektoren so umordnet soll, dass die Summe der Elemente in der Hauptdiagonale maximal wird. Hieraus wird sofort ersichtlich, dass es in einer $n$×$n$-Matrix genau so viele Möglichkeiten gibt, die Zeilen- bzw. Spaltenvektoren zu ordnen, wie es Permutationen von $n$ Elementen gibt, also $n!$. Außer bei kleinen Matrizen ist es nahezu aussichtslos, die optimale Lösung durch Berechnung aller Möglichkeiten zu finden. Schon bei einer 10×10-Matrix gibt es nahezu 3,63 Millionen (3.628.800) zu berücksichtigende Permutationen. diff --git a/buch/papers/munkres/teil3.tex b/buch/papers/munkres/teil3.tex index b67ad74..fd25a74 100644 --- a/buch/papers/munkres/teil3.tex +++ b/buch/papers/munkres/teil3.tex @@ -3,38 +3,86 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 +\section{Der Munkres-Algorithmus (Ungarische Methode) \label{munkres:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum +\rhead{Der Munkres-Algorithmus (Ungarische Methode)} + +Mit der ungarischen Methode können also Optimierungsprobleme gelöst +werden, die bei gewichteten Zuordnungen in bipartiten Graphen entstehen. +Mit ihr kann die eindeutige Zuordnung von Objekten aus zwei Gruppen so +optimiert werden, dass die Gesamtkosten minimiert werden bzw.~der +Gesamtgewinn maximiert werden kann. + +\subsection{Geschichte +\label{munkres:subsection:malorum}} +Die Ungarische Methode wurde 1955 von Harold Kuhn entwickelt und veröffentlicht. +Der Name ``Ungarische Methode'' ergab sich, weil der Algorithmus +weitestgehend auf den früheren Arbeiten zweier ungarischer Mathematiker +basierte: Dénes Kőnig und Jenő Egerváry. +James Munkres überprüfte den Algorithmus im Jahr 1957 und stellte fest, +dass der Algorithmus (stark) polynomiell ist. +Seitdem ist der Algorithmus auch als Kuhn-Munkres oder +Munkres-Zuordnungsalgorithmus bekannt. +Die Zeitkomplexität des ursprünglichen Algorithmus war $O(n^4)$, +später wurde zudem festgestellt, dass er modifiziert werden kann, +um eine $O(n^3)$-Laufzeit zu erreichen. + +\subsection{Besondere Leistung der Ungarischen Methode +\label{munkres:subsection:malorum}} +Die Ungarische Methode ist ein kombinatorischer Optimierungsalgorithmus, der das Zuordnungsproblem +in polynomieller Zeit löst. +Der Begriff polynomielle Laufzeit bedeutet, dass die Laufzeit des Programms +wie $n^2$, $n^3$, $n^4$, etc.~wächst und vernünftig skaliert. $n$ ist hierbei die ''Grösse'' des Problems. + +\subsection{Unterschiedliche Anzahl von Quellen und Zielen +\label{munkres:subsection:malorum}} +Es gibt Fälle, in welchen das Ausgangsproblem keine quadratische Form besitzt. Das ist z. B. dann der Fall, wenn drei Mitarbeiter vier verschiedene Eignungstests absolvieren müssen. In diesem Fall wird in der Ungarischen Methode die Matrix künstlich mittels einer Dummy Position zu einem Quadrat ergänzt. Dummy-Positionen werden dann mit der größten vorhandenen Zahl aus der Matrix besetzt. Beispielsweise wird eine $3\times 4$ zu einer $4\times 4$-Matrix. + +\subsection{Beispiel eines händischen Verfahrens +\label{munkres:subsection:malorum}} + +Die ungarische Methode kann in einem einfachen händischen Beispiel erläutert werden. Wir gehen von der Kostenmatrix $A$ aus. Diese Matrix wird in mehreren Schritten immer weiter reduziert. Anschliessend erfolgen mehrere Zuordnungen. Hierbei ist zu beachten, dass jede Zeile und jede Spalte immer genau eine eindeutige Zuordnung ergibt. Es gibt Situationen, in denen man nichts mehr tun muss, um eine optimale Zuordnung zu finden. Eine optimale Zuordnung ohne zusätzliche Kosten ist eine Auswahl genau eines Feldes in jeder Zeile und Spalte, welches 0 enthält. Das Ziel des Algorithmus ist also, die Matrix so zu ändern, dass genügend Nullen in der Matrix vorkommen. Es ist zudem wichtig, dass man nach jeder Modifikation der Matrix testet, ob man bereits eine Zuordnung machen kann, also genügend Nullen hat. +Das Vorgehen wird in den nachfolgenden Schritten 1-6 beschrieben und auch in der Abbildung 21.5 dargestellt. + +\begin{enumerate} +\item Man beginnt mit der Zeilen-Reduktion. Pro Zeile eruiert man die kleinste Zahl. Diese kleinste Zahl, jeweils in rot markiert, wird bei allen anderen Ziffern in der jeweiligen Zeile subtrahiert. Mit dieser Subtraktion zieht man die unvermeidbaren Kosten ab, die man hat, um eine Baustelle zu erreichen. Man erkennt, dass die Nullen mit zwei Linien abdeckbar sind. Das heisst es gibt zwei Spalten bei denen noch keine Zuordnungen möglich sind. + +\item Auch im zweiten Schritt werden mittels der Spalten-Reduktion die unvermeidbaren Weg-Kosten abgezogen. Man zieht die kleinste Zahl, wiederum in rot markiert, in jeder Spalte von allen Zahlen in der Spalte ab. +Die Nullen können somit mit drei Linien abgedeckt werden. Im Idealfall hat die Matrix in jeder Zeile und Spalte bereits genügend viele Nullen, so dass man bereits eine Zuordnung ohne Mehrkosten machen kann. Dies ist jedoch noch nicht der Fall. Es sollen weitere Nullen in die Matrix hineingebracht werden. + +\item Es bleiben jetzt einige Felder übrig, für die noch keine Zuordnung möglich ist. Die kleinste Ziffer wird dabei aus den noch nicht mit blau markierten Zahlen ausgewählt werden. Im Beispiel ist es die Zahl 1. Das Feld mit dem kleinsten Eintrag beinhaltet die Kosten, die unvermeidlich sind, wenn man für diese Felder auch noch eine Zuordnung machen will. Um neue Nullen zu bekommen, lagert man jetzt die Kosten auf die anderen Zeilen und Spalten um. Dies tut man, indem man in allen nicht abgedeckten Feldern die minimalen Kosten subtrahiert und in den blau markierten Kreuzungspunkten dazu addiert. +Dieser Schritt 3 muss so oft wiederholt werden, bis genügend viele Nullen in der Matrix vorhanden sind. + +\item In Schritt 4 sollen jetzt möglichst viele Nullen markiert werden, welche freistehend sind. +Freistehend bedeutet, dass sowohl in der jeweiligen Zeile und Spalte keine andere markierte Null vorhanden ist. + +\item Alle markierten Nullen werden jetzt in eine 1 umgewandelt. Die restlichen Ziffern in der Matrix, exklusiv die einsen, sollen jetzt ignoriert und durch eine Null ersetzt werden. + +\item Zu guter Letzt werden überall wo eine 1 steht, die Zahlen aus der Ausgangsmatrix eingefügt. Nach Einsetzen der Zahlen können die in rot markierten Zahlen aufsummiert werden. Man erhält den minimalsten Transportweg von total 13 Kilometer. +\end{enumerate} + +\begin{figure} +\centering +\includegraphics[width=8cm]{papers/munkres/figures/Ungarische_Methode_Beispiel.png} +\caption{Händisches Beispiel des Munkres Algorithmus, minimalster Transportweg.} +\label{munkres:Vr2} +\end{figure} + +\subsection{Zuordnung der Kräne \label{munkres: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. +Als Resultat des Munkres-Algorithmus werden in Abbildung 21.6 nebst dem minimalsten Transportweg auch die optimalste Zuweisung der Kräne auf die neuen Standorte ersichtlich. +Es können die folgenden Zuordnungen aus der Matrix abgelesen werden: +\begin{itemize} +\item Der Kran von Baustelle A1 soll zur Baustelle B2. +\item Der Kran von Baustelle A2 soll zur Baustelle B3. +\item Der Kran von Baustelle A3 soll zur Baustelle B4. +\item Der Kran von Baustelle A4 soll zur Baustelle B1. +\end{itemize} +\begin{figure} +\centering +\includegraphics[width=3cm]{papers/munkres/figures/Ungarische_Methode_Beispiel_Zuw.png} +\caption{Händisches Beispiel des Munkres Algorithmus, Zuweisung der Kräne } +\label{munkres:Vr2} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/munkres/teil4.tex b/buch/papers/munkres/teil4.tex new file mode 100644 index 0000000..9a27227 --- /dev/null +++ b/buch/papers/munkres/teil4.tex @@ -0,0 +1,9 @@ +% +% teil4.tex -- Beispiel-File für Teil 4 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{- +\label{munkres:section:teil4}} +\rhead{-} + diff --git a/buch/papers/munkres/teil5.tex b/buch/papers/munkres/teil5.tex new file mode 100644 index 0000000..b938c50 --- /dev/null +++ b/buch/papers/munkres/teil5.tex @@ -0,0 +1,8 @@ +% +% teil5.tex -- Beispiel-File für Teil 5 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{- +\label{munkres:section:teil5}} +\rhead{-} diff --git a/buch/papers/punktgruppen/Makefile b/buch/papers/punktgruppen/Makefile index f92dc95..03ad15a 100644 --- a/buch/papers/punktgruppen/Makefile +++ b/buch/papers/punktgruppen/Makefile @@ -11,11 +11,16 @@ SOURCES := \ symmetry.tex TIKZFIGURES := \ + tikz/atoms-grid-still.tex \ + tikz/atoms-grid-force.tex \ + tikz/atoms-piezo-still.tex \ + tikz/atoms-piezo-force-vertical.tex \ + tikz/atoms-piezo-force-horizontal.tex \ tikz/combine-symmetries.tex \ tikz/lattice.tex \ - tikz/piezo-atoms.tex \ tikz/piezo.tex \ tikz/projections.tex \ + tikz/stereographic-projections.tex \ tikz/symmetric-shapes.tex FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES)) @@ -28,7 +33,7 @@ figures/%.pdf: tikz/%.tex pdflatex --output-directory=figures $< .PHONY: standalone -standalone: standalone.tex $(SOURCES) +standalone: standalone.tex $(SOURCES) $(FIGURES) mkdir -p standalone cd ../..; \ pdflatex \ diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index 8cde9d7..fbb073e 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -11,8 +11,15 @@ dependencies-punktgruppen = \ papers/punktgruppen/crystals.tex \ papers/punktgruppen/piezo.tex \ papers/punktgruppen/references.bib \ - papers/punktgruppen/tikz/combine-symmetries.tex \ + papers/punktgruppen/tikz/atoms-grid-force.tex \ + papers/punktgruppen/tikz/atoms-grid-still.tex \ + papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex \ + papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex \ + papers/punktgruppen/tikz/atoms-piezo-still.tex \ + papers/punktgruppen/tikz/combine-symmetries.tex \ papers/punktgruppen/tikz/lattice.tex \ papers/punktgruppen/tikz/piezo-atoms.tex \ papers/punktgruppen/tikz/piezo.tex \ - papers/punktgruppen/tikz/projections.tex + papers/punktgruppen/tikz/projections.tex \ + papers/punktgruppen/tikz/stereographic-projections.tex \ + papers/punktgruppen/tikz/symmetric-shapes.tex diff --git a/buch/papers/punktgruppen/crystals.tex b/buch/papers/punktgruppen/crystals.tex index 1aec16f..0a9d3b6 100644 --- a/buch/papers/punktgruppen/crystals.tex +++ b/buch/papers/punktgruppen/crystals.tex @@ -1,8 +1,7 @@ \section{Kristalle} -%einleitung sollte noch an das ende von der Symmetrie angepasst werden -Unter dem Begriff Kristall sollte sich jeder ein Bild machen können. -Wir werden uns aber nicht auf sein Äusseres fokussieren, sondern was ihn im Inneren ausmacht. -Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. +Eine nicht allzu häufig gestellte Frage ist, wie ein Kristall definiert ist. +Um zu klären, was ein Kristall mit Symmetrien zu tun hat, ist jedoch genau diese Frage äusserst relevant. +Glücklicherweise ist das Innere eines Kristalles relativ einfach definiert. \begin{definition}[Kristall] Ein Kristall besteht aus Atomen, welche sich in einem Muster arrangieren, welches sich in drei Dimensionen periodisch wiederholt. \end{definition} @@ -12,117 +11,160 @@ Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. \includegraphics[]{papers/punktgruppen/figures/lattice} \caption{ Zweidimensionales Kristallgitter. - \texttt{TODO: make wider and shorter} \label{fig:punktgruppen:lattice} } \end{figure} \subsection{Kristallgitter} Ein zweidimensionales Beispiel eines solchen Muster ist Abbildung \ref{fig:punktgruppen:lattice}. -Für die Überschaubarkeit haben wir ein simples Motiv eines einzelnen grauen Punktes gewählt und betrachten dies nur in Zwei Dimensionen. -Die eingezeichneten Vektoren $\vec{a}$ und $\vec{b}$ sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. -Wird ein beliebiger grauer Gitterpunkt in \ref{fig:punktgruppen:lattice} gewählt -und um eine ganzzahlige Linearkombination von $\vec{a}$ und $\vec{b}$ verschoben, -endet er zwangsweise auf einem Gitterpunkt, wenn nicht wieder am selben Ort. -Im Dreidimensionalen-Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor $\vec{c}$ also +Für die Überschaubarkeit haben wir ein simples Motiv eines einzelnen grauen Punktes dargestellt und betrachten dies nur in zwei Dimensionen. +Die eingezeichneten Vektoren \(\vec{a}_1\) und \(\vec{a}_2\) sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. +Wird ein beliebiger grauer Gitterpunkt in Abbildung \ref{fig:punktgruppen:lattice} gewählt und um eine ganzzahlige Linearkombination von \(\vec{a}_1\) und \(\vec{a}_2\) verschoben, endet er zwangsweise auf einem Gitterpunkt, wenn nicht wieder am selben Ort. +Im dreidimensionalen Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor \(\vec{a}_3\) also \[ - \vec{r} = n_1 \vec{a} + n_2 \vec{b} + n_3 \vec{c} + \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3 = \sum_i n_i \vec{a}_i \] -erreicht werden sofern $\{n_1,n_2,n_3\} \in \mathbb{Z}$ sind. -Sind die Vektoren $\vec{a}$ , $\vec{b}$ , $\vec{c}$ gegeben , -ist ein Kristallgitter eindeutig beschrieben, weswegen sie auch als Grundvektoren bekannt sind. +erreicht werden sofern \(n_1,n_2,n_3 \in \mathbb{Z}\) sind. +Sind die Vektoren \(\vec{a}_1\), \(\vec{a}_2\), \(\vec{a}_3\) gegeben, ist ein Kristallgitter eindeutig beschrieben, weswegen sie auch als Grundvektoren bekannt sind. -\subsection{Translationssymmetrie} +\subsection{Translationssymmetrie} Da sich das ganze Kristallgitter wiederholt, wiederholen sich auch dessen Eigenschaften periodisch mit den Grundvektoren. -Sollte man sich auf einem Gitterpunkt in einem Kristall aufhalten, ist es unmöglich zu wissen, auf welchem Gitterpunkt man sich befindet, -da die Umgebungen aller Punkte Identisch sind. -Mit anderen worten: Jedes Kristallgitter $ G $ ist \emph{Translationssymmetrisch} in der Translation +Sollte man sich auf einem Gitterpunkt in einem Kristall aufhalten, ist es unmöglich zu wissen, auf welchem Gitterpunkt man sich befindet, da die Umgebungen aller Punkte identisch sind. +Mit anderen Worten: Jedes Kristallgitter $ G $ ist \emph{translationssymmetrisch} in der Translation \[ - Q_i(G) = G + \vec{a_i} -\] wobei der Vektor $a_i$ ein Grundvektor sein muss. + \vec{Q}_i(G) = G + \vec{a}_i, +\] +wobei der Vektor $\vec{a}_i$ ein Grundvektor sein muss. Da die Translationssymmetrie beliebig oft mit allen Grundvektoren angewendet werden kann, können wir auch sagen, dass alle Verschiebungen um eine Linearkombination -der Vektoren $\vec{a}$ , $\vec{b}$ und $\vec{c}$ erlaubt sind oder kurz, um $\vec{r}$. -Verschiebungen um $\vec{r}$ bewirken demnach keine Veränderungen, -solange wir ein unendlich grosses Kristallgitter verschieben. +der Vektoren $\vec{a}_1$ , $\vec{a}_2$ und $\vec{a}_3$ erlaubt sind. +Dabei sollte erwähnt werden, dass eine Translationssymmetrie nur in unendlich grossen Kristallgittern besteht. -\subsection{Limitierte Kristallsymmetrien} +\subsection{Einschränkungen durch Kristallsymmetrien} \label{sec:punktgruppen:Translationssymmetrie} Die Translationssymmetrie ist wohl keine grosse Überraschung, wenn man die Abbildung \ref{fig:punktgruppen:lattice} betrachtet. - Was nicht direkt ersichtlich ist, ist das auch wenn die Grundvektoren frei gewählt werden können, - können nur Rotationssymmetrische Kristalle bestimmter Rotationswinkel erzeugt werden. - + Was nicht direkt ersichtlich ist, ist dass bei beliebigen Grundvektoren nicht beliebige Symmetrien erstellt werden können. + Dies weil die Translationssymmetrie eines Kristalles weitere Symmetrien deutlich einschränkt. + \begin{figure} \centering \includegraphics[]{papers/punktgruppen/figures/combine-symmetries} \caption{ Translations und Rotationssymmetrisches Kristallgitter - \texttt{TODO: make wider and change color (yellow)} } \label{fig:punktgruppen:rot-geometry} \end{figure} - \subsubsection{Translationssymmetrie $Q$ in Kombination mit Rotationssymmetrie $C_\alpha$} % Müssen uns auf eine schreibweise für Symmetrie Operationen einigen oder sicher am Ende überprüfen - In Abbildung \ref{fig:punktgruppen:rot-geometry} Sehen wir Gitterpunkte und deren Zusammenhänge. +\begin{satz} \label{thm:punktgruppen:crystal-restriction} + Die Rotationssymmetrien eines Kristalls sind auf 2-fach, 3-fach, 4-fach und 6-fach beschränkt. + Mit anderen Worten: Es sind nur Drehwinkel von + 0\(^{\circ}\), + 60\(^{\circ}\), + 90\(^{\circ}\), + 120\(^{\circ}\) und + 180\(^{\circ}\) + m\"oglich. +\end{satz} + +\begin{proof} + In Abbildung \ref{fig:punktgruppen:rot-geometry} sehen wir Gitterpunkte und deren Zusammenhänge. \begin{itemize} - \item $A$ ist unser erster Gitterpunkt. - - \item $A'$ ist gegeben, weil wir $A$ mit der Translation $Q$ um einen Grundvektor verschieben und wir wissen, - dass nach einer Translation wieder ein Gitterpunkt an der Verschobenen Stelle sein muss. - \item $B$ entsteht, weil wir die Rotationssymmetrie $C_\alpha$ auf den Punkt $A$ anwenden. - Dadurch dreht sich das ganze Gitter um den Winkel $\alpha$. - Für uns bedeutet dies lediglich, dass unser zweiter Punkt $A'$ abgedreht wird. - An der neuen Position von $A'$ muss also auch ein Punkt sein, um die Rotationssymmetrie zu erfüllen. - \item $B$ ist unser Name für diesen neuen Punkt. - Da auch die Eigenschaften des Kristallgittes periodisch mit dem Gitter sein müssen, dürfen wir $C_\alpha$ auch auf $A'$ anwenden. - Also wenden wir $C_\alpha$ invertiert - \footnote{Eine Rotationssymmetrie muss auch in die inverse Richtung funktionieren. - Genauere Überlegungen hierzu werden dem Leser überlassen, da sich die Autoren nicht explizit mit dieser Frage Auseinander gesetzt haben.} - auch auf $A'$ an. - Dies dreht $A$ auf einen neuen Punkt. - \item $B'$ ist kein zufälliger Name für diesen neuen Punkt, denn wir wissen, dass zwischen allen Punkten eine Translationssymmetrie bestehen muss. - Die Translationssymmetrie zwischen $B$ und $B'$ ist hier als $Q'$ bezeichnet. + \item \(A\) ist unser erster Gitterpunkt. + + \item \(A'\) ist gegeben, weil wir \(A\) mit der Translation \(\vec{Q}\) um einen Grundvektor verschieben und wir wissen, + dass nach einer Translation wieder ein Gitterpunkt an der verschobenen Stelle sein muss. + \item \(B\) entsteht, weil wir die Rotationssymmetrie \(C_n\) auf den Punkt \(A\) anwenden. + Dadurch dreht sich das ganze Gitter um den Winkel \(360^\circ/n\). + Für uns bedeutet dies lediglich, dass unser zweiter Punkt \(A'\) abgedreht wird. + An der neuen Position \(B\) von \(A'\) muss also auch ein Punkt des Gitters sein, um die Rotationssymmetrie zu erfüllen. + \item \(B\) ist unser Name für diesen neuen Punkt. + Da auch die Eigenschaften des Kristallgitters periodisch mit dem Gitter sein müssen, dürfen wir \(C_n\) auch auf \(A'\) anwenden. + Also wenden wir \(C_n^{-1}\) auch auf \(A'\) an. + Dies dreht \(A\) auf einen neuen Punkt. + \item \(B'\) ist kein zufälliger Name für diesen neuen Punkt, denn wir wissen, dass zwischen allen Punkten eine Translationssymmetrie bestehen muss. + Die Translationssymmetrie zwischen \(B\) und \(B'\) ist hier als \(\vec{Q}'\) bezeichnet. \end{itemize} Mit den gegebenen Punkten lassen sich geometrische Folgerungen ziehen. - Wir beginnen, indem wir die Länge der Translation $Q$ mit jener von $Q'$ vergleichen. - Aus Abbildung \ref{fig:punktgruppen:rot-geometry} ist ersichtlich, dass $|Q| = |Q'|+ 2x$. - Ist $Q$ ein Grundvektor so muss $|Q'|$ ein ganzes vielfaches von $|Q|$ sein. Also + Wir beginnen, indem wir die Länge der Verschiebung \(|\vec{Q}| = Q\) setzen und \(|\vec{Q}'| = Q'\). + Aus Abbildung \ref{fig:punktgruppen:rot-geometry} ist ersichtlich, dass \(Q' = Q + 2x\). + Da \(\vec{Q}\) eine Translation um ein Grundvektor ist , muss \(\vec{Q}'\) ein ganzes Vielfaches von \(\vec{Q}\) sein. + Demnach ist auch die Länge \[ - |Q'| = n|Q| = |Q| + 2x + Q' = nQ = Q + 2x . \] - Die Strecke $x$ lässt sich auch mit hilfe der Trigonometrie und dem angenommenen Rotationswinkel $\alpha$ ausdrücken: + Die Strecke \(x\) lässt sich auch mit Hilfe der Trigonometrie und dem angenommenen Rotationswinkel \(\alpha\) ausdrücken: \[ - n|Q| = |Q| + 2|Q|\sin(\alpha - \pi/2) + nQ = Q + 2Q\sin(\alpha - \pi/2) . \] - Wir können mit $|Q|$ dividieren um unabhängig von der Läge des Grundvektors zu werden, - was auch Sinn macht, da eine Skalierung eines Kristalles seine Symmetrieeigenschaften nicht tangieren soll. - Zusätzlich können wir den Sinusterm vereinfachen. + Wir können durch \(Q\), dividieren um unabhängig von der Läge des Grundvektors zu werden, was auch Sinn macht, + da eine Skalierung eines Kristalles seine Symmetrieeigenschaften nicht tangiert. + Zusätzlich können wir den Sinusterm vereinfachen. Somit wird \[ - n = 1 - 2\cos\alpha - \alpha = \cos^{-1}\left(\frac{1-n}{2}\right) + n = 1 - 2\cos\alpha \quad\text{oder}\quad + \alpha = \cos^{-1}\left(\frac{1-n}{2}\right). \] Dies schränkt die möglichen Rotationssymmetrien auf - \[ + \( \alpha \in \left\{ 0^\circ, 60^\circ, 90^\circ, 120^\circ, 180^\circ\right\} - \] + \) ein. +\end{proof} \begin{figure} \centering - \includegraphics[]{papers/punktgruppen/figures/projections} - \caption{Kristallklassen mit zugehöriger Schönfliesnotation} - \label{fig:punktgruppen:Kristallkassen} + \includegraphics[height=6cm]{papers/punktgruppen/figures/stereographic-projections} + \caption{ + Stereografische Projektion einer \(C_{i}\) Symmetrie. Es wird eine Linie vom magentafarbenen Punkt auf der oberen Hälfte der Kugel zum Südpol gezogen. + Wo die Linie die Ebene schneidet (\(z = 0\)), ist die Projektion des Punktes. + Die Koordinaten der Projektionen sind einfach zu berechnen: ein Punkt auf eine Kugel mit Radius \(r\) mit den Koordinaten \(x, y, z,\) wird auf \(xr/(r + z), yr/(r + z)\) projiziert. + Für den orangefarbenen Punkt unterhalb des Äquators wird die Linie zum Nordpol gezogen und die Projektionsformel hat stattdessen einen Nenner von \(r - z\). + } + \label{fig:punktgruppen:stereographic-projections} \end{figure} \subsection{Kristallklassen} -Vorgehend wurde gezeigt, dass in einem zweidimensionalen Kristallgitter nicht alle Symmetrien möglich sind. -Mit weiteren ähnlichen überlegungen gezeigt werden kann, dass Kristalle im dreidimensionalen Raum -\footnote{Alle $17$ möglichen zweidimensionalen Symmetrien sind als Wandmustergruppen bekannt} -nur auf genau $32$ Arten punktsymmetrisch sein können. -Diese $32$ möglichen Punktsymmetrien scheinen durchaus relevant zu sein, denn sie werden unter anderem als Kristallklassen bezeichnet. -Eine mögliche Art, die Klassen zu benennen ist nacht dem Mathematiker Arthur Moritz Schönflies, -welcher sich mit der Klasifizierung dieser Symmetrien auseinandergesetzt hat. -Auf der Abbildung \ref{fig:punktgruppen:Kristallkassen} sind die möglichen Punktsymmetrien mit deren Schönfliesnotation aufgelistet. -Als Darstellungsmethode wurde die stereographische Projektion gewählt, wobei $5$ Klassen aus Gründen der Überschaubarkeit nicht gezeichnet wurden. + +Im vorausgegangenen Abschnitt wurde gezeigt, dass in einem zweidimensionalen Kristallgitter nicht alle Symmetrien möglich sind. + Mit weiteren ähnlichen Überlegungen kann gezeigt werden, dass Kristalle im dreidimensionalen Raum nur auf genau 32 Arten rein punktsymmetrische Symmetriegruppen bilden können. + Diese 32 möglichen Symmetriegruppen scheinen durchaus relevant zu sein, denn sie werden unter anderem als Kristallklassen bezeichnet. + Die 32 möglichen Kristallklassen sind auf Abbildung \ref{fig:punktgruppen:kristallklassen} zu sehen. + Die Darstellung von dreidimensionalen Punktsymmetrien wurde mit der stereographischen Projektion ermöglicht (siehe Abbildung \ref{fig:punktgruppen:stereographic-projections}), wobei die gestrichelten Klassen aus Gründen der Überschaubarkeit nicht im Detail gezeichnet wurden. + + +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/projections} + \caption{Kristallklassen mit zugehörigem Schönflies-Symbol} + \label{fig:punktgruppen:kristallklassen} +\end{figure} + +\subsubsection{Schönflies-Symbolik} + +Jede der 32 Kristallklassen auf der Abbildung \ref{fig:punktgruppen:kristallklassen} ist mit ihrem zugehörigen Schönflies-Symbol bezeichnet. + Die Schönflies-Symbolik stammt von dem Mathematiker Arthur Moritz Schönflies, welcher sich unter anderem mit der Klasifizierung der Punktgruppen auseinandergesetzt hat. + Er hat Untergruppen gebildet, welche als Grossbuchstaben in Abbildung \ref{fig:punktgruppen:kristallklassen} zu sehen sind. + \begin{itemize} + \item In Kristallen ist nur die Drehgruppe \(C\), Diedergruppe \(D\), Drehspiegelgruppe \(S\), Tetraedergruppe \(T\) und die Oktaedergruppe \(O\) zu finden. + Es gäbe auch die Ikosaedergruppe \(I\) und die Kugelgruppe \(K\), diese sind aber nach Satz \ref{thm:punktgruppen:crystal-restriction} nicht kompatibel mit der Translationssymmetrie eines Kristalles und daher in der Kristallographie nicht relevant. + \item Dank Abschnitt \ref{sec:punktgruppen:Translationssymmetrie} wissen wir, wieso in Abbildung \ref{fig:punktgruppen:kristallklassen} auf \(C\) nur ganz bestimmte Subskripte folgen. + Ist im Subskript eine Zahl \(n\) zu finden, steht dies für eine \(n\)-fache Symmetrie. + Daher darf \(C_5\) auf der Abbildung \ref{fig:punktgruppen:kristallklassen} nicht vorkommen, da \(360^\circ/5 = 72^\circ\) was nach Satz \ref{thm:punktgruppen:crystal-restriction} keine mögliche Rotationssymmetrie eines Kristalles ist. + \item Sind im Subskript Buchstaben, definieren diese weitere Symmetrieeigenschaften der Klasse. + Für die folgenden Betrachtungen müssen wir uns Abbildung \ref{fig:punktgruppen:kristallklassen} genauer ansehen. + Dabei ist mit horizontal flach auf dem Papier gemeint. + \begin{itemize} + \item[\(h\)] bezeichnet eine horizontale Spiegelebene und + \item[\(v\)] eine Symmetrieebene, was eine Spiegelebene ist, die sich mit der Symmetrie mitdreht. + Zum Beispiel hat \(C_{3v}\) eine vertikale Spiegelebene, die durch die 3-fache Drehsymmetrie als 3 Spiegelebenen erscheinen. + \item[\(s\)] ist ein spezielles Subskript um die beiden Symmetriegruppen \(C_{1v}\) und \(C_{1h}\) zu beschreiben, weil \(C_{1v} = C_{1h}\). + \item[\(d\)] symbolisiert eine diagonale Symmetrieebene. + Es wird ersichtlich wie diagonal gemeint ist, wenn man \(D_2\) zu \(D_{2d}\) vergleicht. + \item[\(i\)] steht für ein Inversionszentrum. Hat eine Symmetriegruppe ein Inversionszentrum, bedeutet dies dass sie im Ursprung punktsymmetrisch ist. + \end{itemize} + \end{itemize} +Zu beachten ist jedoch, dass manche Symmetriegruppen mit mehreren Schönflies-Symbolen beschieben werden können. + \(C_{3i}\) beschreibt genau das selbe wie \(S_6\), da eine dreifache Rotationssymmetrie mit einem Inversionszentrum einer sechsfachen Drehspiegelsymmetrie entspricht. + +%% vim:spell spelllang=de showbreak=.. breakindent linebreak: diff --git a/buch/papers/punktgruppen/figures/atoms-grid-force.pdf b/buch/papers/punktgruppen/figures/atoms-grid-force.pdf Binary files differnew file mode 100644 index 0000000..b3e6215 --- /dev/null +++ b/buch/papers/punktgruppen/figures/atoms-grid-force.pdf diff --git a/buch/papers/punktgruppen/figures/atoms-grid-still.pdf b/buch/papers/punktgruppen/figures/atoms-grid-still.pdf Binary files differnew file mode 100644 index 0000000..752014d --- /dev/null +++ b/buch/papers/punktgruppen/figures/atoms-grid-still.pdf diff --git a/buch/papers/punktgruppen/figures/atoms-piezo-force-horizontal.pdf b/buch/papers/punktgruppen/figures/atoms-piezo-force-horizontal.pdf Binary files differnew file mode 100644 index 0000000..313dc69 --- /dev/null +++ b/buch/papers/punktgruppen/figures/atoms-piezo-force-horizontal.pdf diff --git a/buch/papers/punktgruppen/figures/atoms-piezo-force-vertical.pdf b/buch/papers/punktgruppen/figures/atoms-piezo-force-vertical.pdf Binary files differnew file mode 100644 index 0000000..9a86b7c --- /dev/null +++ b/buch/papers/punktgruppen/figures/atoms-piezo-force-vertical.pdf diff --git a/buch/papers/punktgruppen/figures/atoms-piezo-still.pdf b/buch/papers/punktgruppen/figures/atoms-piezo-still.pdf Binary files differnew file mode 100644 index 0000000..83b6590 --- /dev/null +++ b/buch/papers/punktgruppen/figures/atoms-piezo-still.pdf diff --git a/buch/papers/punktgruppen/figures/combine-symmetries.pdf b/buch/papers/punktgruppen/figures/combine-symmetries.pdf Binary files differindex 13f7330..6cd4e64 100644 --- a/buch/papers/punktgruppen/figures/combine-symmetries.pdf +++ b/buch/papers/punktgruppen/figures/combine-symmetries.pdf diff --git a/buch/papers/punktgruppen/figures/lattice.pdf b/buch/papers/punktgruppen/figures/lattice.pdf Binary files differindex 6565be5..712d6f4 100644 --- a/buch/papers/punktgruppen/figures/lattice.pdf +++ b/buch/papers/punktgruppen/figures/lattice.pdf diff --git a/buch/papers/punktgruppen/figures/piezo-atoms.pdf b/buch/papers/punktgruppen/figures/piezo-atoms.pdf Binary files differdeleted file mode 100644 index 63da7a9..0000000 --- a/buch/papers/punktgruppen/figures/piezo-atoms.pdf +++ /dev/null diff --git a/buch/papers/punktgruppen/figures/piezo.pdf b/buch/papers/punktgruppen/figures/piezo.pdf Binary files differindex ca6192b..904250a 100644 --- a/buch/papers/punktgruppen/figures/piezo.pdf +++ b/buch/papers/punktgruppen/figures/piezo.pdf diff --git a/buch/papers/punktgruppen/figures/projections.pdf b/buch/papers/punktgruppen/figures/projections.pdf Binary files differindex c9369b2..202fc8d 100644 --- a/buch/papers/punktgruppen/figures/projections.pdf +++ b/buch/papers/punktgruppen/figures/projections.pdf diff --git a/buch/papers/punktgruppen/figures/stereographic-projections.pdf b/buch/papers/punktgruppen/figures/stereographic-projections.pdf Binary files differnew file mode 100644 index 0000000..7598265 --- /dev/null +++ b/buch/papers/punktgruppen/figures/stereographic-projections.pdf diff --git a/buch/papers/punktgruppen/figures/symmetric-shapes.pdf b/buch/papers/punktgruppen/figures/symmetric-shapes.pdf Binary files differindex 0b3ba54..3a8d9dd 100644 --- a/buch/papers/punktgruppen/figures/symmetric-shapes.pdf +++ b/buch/papers/punktgruppen/figures/symmetric-shapes.pdf diff --git a/buch/papers/punktgruppen/intro.tex b/buch/papers/punktgruppen/intro.tex index 24212e7..e3f0226 100644 --- a/buch/papers/punktgruppen/intro.tex +++ b/buch/papers/punktgruppen/intro.tex @@ -1,14 +1,16 @@ \section{Einleitung} -Es gibt viele Möglichkeiten sich in Kristallen zu verlieren. -Auch wen man nur die mathematischen Betrachtunngsweisen berüksichtigt, hat man noch viel zu viele Optionen sich mit Kristallen zu beschäftigen. -In diesem Kapitel ist daher der Fokus ``nur'' auf die Symmetrie gelegt. -Zu beginn werden wir zeigen was eine Symmetrie ausmacht und dass sie noch weit mehr in sich verbirgt als nur schön auszusehen. -Die vorgestellten Symmetrien sind äusserst gut geeignet um die Grundeigenschaften eines Kristalles zu Beschreiben. -Mit etwas kiffligen geometrischen Überlegungen kann man zeigen wass in der Welt der Kristallographie alles möglich ist oder nicht. -Die Einschränkungen sind durchaus wilkommen, dank ihnen halten sich die möglichen Kristallgitter in Grenzen und Lassen sich Kategorisieren. -Kategorien sind nicht nur für einen besseren Überblich nützlich, sondern kann man aus ihnen auch auf Physikalische Eigenschaften schliessen, als spannendes Beispiel: Die Piezoelektrizität. -Die Piezoelektrizität ist vielleicht noch nicht jedem bekannt, sie versteckt sich aber in diversen Altagsgegenständen zum Beispiel sorgen sie in den meisten Feuerzeugen für die Zündung. -Ein Funken Interesse ist hoffentlich geweckt um sich mit dem scheinbar trivialen thema der Symmetrie auseinander zu setzten. - +Es gibt viele Möglichkeiten sich in Kristallen zu verlieren. +Auch wenn man nur die mathematischen Betrachtungsweisen berücksichtigt, hat man noch viel zu viele Optionen, sich mit Kristallen zu beschäftigen. +In diesem Kapitel wird daher der Fokus ``nur'' auf die Symmetrie gelegt. +Zu Beginn werden wir zeigen, was eine Symmetrie ausmacht und dass sie noch weit mehr in sich verbirgt als nur schön auszusehen. +Die vorgestellten Symmetrien sind äusserst gut geeignet, um die Grundeigenschaften eines Kristalles zu beschreiben. +Mit etwas kniffligen geometrischen Überlegungen kann man zeigen, was in der Welt der Kristallographie alles möglich ist oder nicht. +Diese erlauben alle möglichen Kristalle nach ihren Symmetrien in erstaunlich wenige Klassen zu kategorisieren. +Kategorien sind nicht nur für einen besseren Überblick nützlich, sondern kann man aus ihnen auch auf physikalische Eigenschaften schliessen. +Als spannendes Beispiel: Die Piezoelektrizität. +Piezoelektrizität beschreibt einen Effekt, ohne welchen diverse Altagsgegenständen nicht besonders nützlich wären. +Zum Beispiel sorgt er in den allermeisten Feuerzeugen für die Zündung. +Hiermit ist hoffentlich ein Funken Interesse geweckt um sich mit dem scheinbar trivialen Thema der Symmetrie auseinander zu setzten. +%% vim:linebreak breakindent showbreak=.. spell spelllang=de: diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index a6e246c..556fc2b 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -18,6 +18,8 @@ \nocite{punktgruppen:pinter-algebra} \nocite{punktgruppen:sands-crystal} \nocite{punktgruppen:lang-elt2} +\nocite{punktgruppen:ouchem} +\nocite{punktgruppen:restriction} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/piezo.tex b/buch/papers/punktgruppen/piezo.tex index e6b595a..1cf9b98 100644 --- a/buch/papers/punktgruppen/piezo.tex +++ b/buch/papers/punktgruppen/piezo.tex @@ -1,6 +1,5 @@ \section{Piezoelektrizität} -Die Piezoelektrizität ist per Definition spannend. -Sie beschreibt die Eigenschaft, dass gewisse Kristalle eine elektrische Spannung erzeugen, wenn machanischer Druck auf sie ausgeübt wird. +Die Piezoelektrizität ist die spannende Eigenschaft, dass gewisse Kristalle eine elektrische Spannung erzeugen, wenn mechanischer Druck auf sie ausgeübt wird. \begin{figure} \centering @@ -10,65 +9,69 @@ Sie beschreibt die Eigenschaft, dass gewisse Kristalle eine elektrische Spannung \end{figure} \subsection{Polarisierung} -Piezoelektrizität basiert darauf, dass zwischen den Oberflächen des Kristalles ein Ladungsungleichgewicht entsteht siehe Abbildung\ref{fig:punktgruppen:basicPiezo}. -Dieses Ungleichgewicht resultiert, -weil durch den mechanischen Druck auf der einen Oberfläche des Kristalles positiv Ione näher an die Oberfläche gelangen, -wärend auf der gegenüberliegenden Oberfläche sich mehr negative Ionen Sammeln. -Das sich die atomare Struktur eines Kristalles unter Druck genau so verformt ist nicht bei jedem Kristall gegeben. + +Piezoelektrizität basiert darauf, dass zwischen den Oberflächen des Kristalles ein Ladungsungleichgewicht entsteht (siehe Abbildung\ref{fig:punktgruppen:basicPiezo}). +Dieses Ungleichgewicht resultiert, weil durch den mechanischen Druck auf der einen Oberfläche des Kristalles positive Ionen näher an die Oberfläche gelangen, wärend auf der gegenüberliegenden Seite dasselbe mit negativen Ionen passiert. +Es besitzt jedoch nicht jeder Kristall eine atomare Struktur, welche sich unter Druck genau so verformt. Der Aufbau und somit auch die Symmetrie des Kristalles sind daher relevant für die Entstehung dieses Effektes. + \begin{figure} \centering - \includegraphics[]{papers/punktgruppen/figures/piezo-atoms} + \begin{tabular}{c |c} + \subfigure[][\label{fig:punktgruppen:atoms-piezo}]{\includegraphics{papers/punktgruppen/figures/atoms-piezo-still}} & + \subfigure[][\label{fig:punktgruppen:atoms-grid}]{\includegraphics{papers/punktgruppen/figures/atoms-grid-still}} \\ + \subfigure[][\label{fig:punktgruppen:atoms-piezo-fv}]{\includegraphics{papers/punktgruppen/figures/atoms-piezo-force-vertical}} + \hspace{2mm} + \subfigure[][\label{fig:punktgruppen:atoms-piezo-fh}]{\includegraphics{papers/punktgruppen/figures/atoms-piezo-force-horizontal}} + \hspace{3mm} & \hspace{3mm} + \subfigure[][\label{fig:punktgruppen:atoms-grid-f}]{\includegraphics{papers/punktgruppen/figures/atoms-grid-force}} \\ + \end{tabular} \caption{ Kristallstrukturen mit und ohne piezoelektrischer Eigenschaft. - \texttt{TODO: adapt figure for paper with subfigure markers.} } \label{fig:punktgruppen:atomPiezo} \end{figure} \subsection{Atomarer Aufbau} -Die Polarisation resultiert über eine gesamte Oberfläche eines Kristalles, entscheidend ist aber der atomare Aufbau. + +Die Polarisation entsteht an der Oberfläche eines Kristalles, die Erklärung dazu finden wir jedoch im atomaren Aufbau. Wir wollen dazu die verschiedenen Kristallstrukturen auf Abbildung \ref{fig:punktgruppen:atomPiezo} diskutieren. -In Abbildung \ref{fig:punktgruppen:atomPiezo} gilt für alle Strukturen, dass rote Kreise Positive Ionen und blaue negative Ionen repräsentieren. -%liste oder anderes format?.. -Struktur$(a)$ zeigt ein piezoelektrisches Material in Ruhe. Struktur $(b)$ ist dasselbe Kristallgitter, jedoch wird es senkrecht belastet. -Eingezeichnet ist auch das elektrische Feld, welches entsteht, weil mitlleren Ladungsträger weiter auseinander gerdrückt werden. -Als hilfe zur Vorstellung kann man $(b)$ zwischen zwei leitende Platten setzen, -so wird ersichtlich, dass mit wachsendem Druck eine negative Ladung an die rechte Platte gedrückt wird, -während sich die positiven Ionen weiter entfernen. -$(d)$ ist nicht piezoelektrisch. -Dies wird ersichtlich, wenn man $(d)$ unterdruck setzt und sich die Struktur zu $(e)$ verformt. -Setzt man $(e)$ gedanklich auch zwischen zwei leitende Platten scheint es als würden rechts mehr Positive Ionen in die Platte gedrückt werden -und links umgekehrt. -Dies ist aber nicht mehr der Fall, wenn der Kristall nach oben und periodisch wiederholt. -Struktur $(c)$ zeigt $(a)$ in unter horizontaler Belastung. -Was in zwischen $(b)$ und $(c)$ zu beobachten ist, ist dass das entstandene Ladungsdifferenz orthogonal zu der angelegten Kraft entsteht, -im Gegensatz zu $(b)$. -Daraus kann man schlissen, dass $(a)$ keine Rotationssymmetrie von $90^\circ$ besitzen kann, weil die Eigenschaften ändern bei einer $90^\circ$ Drehung. -Das Fehlen dieser Rotationssymmetrie kann mit betrachten von $(a)$ bestätigt werden. - -\subsection{Punktsymmetrie}\footnote{In der Literatur wird ein Punktsymmetrisches Kristallgitter oft als Kristallgitter mit Inversionszentrum bezeichnet.} -Piezoelektrische Kristalle können nicht Punktsymmetrisch sein. +In Abbildung \ref{fig:punktgruppen:atomPiezo} gilt für alle Strukturen, dass rote Kreise positive Ionen und blaue negative Ionen repräsentieren. +Struktur \subref{fig:punktgruppen:atoms-piezo} zeigt ein piezoelektrisches Material in Ruhe. +Struktur \subref{fig:punktgruppen:atoms-piezo-fv} ist dasselbe Kristallgitter, jedoch wird es senkrecht belastet. +Eingezeichnet ist auch das elektrische Feld, welches entsteht, weil die Ladungsträger ganz links und rechts weiter auseinander gedrückt werden. +Als Hilfe zur Vorstellung kann man \subref{fig:punktgruppen:atoms-piezo-fv} zwischen zwei leitende Platten setzen, so wird ersichtlich, dass mit wachsendem Druck eine negative Ladung an die rechte Platte gedrückt wird, während sich die positiven Ionen weiter entfernen. + + +Die Struktur \subref{fig:punktgruppen:atoms-grid} ist nicht piezoelektrisch. +Dies wird ersichtlich, wenn man \subref{fig:punktgruppen:atoms-grid} unter Druck setzt und sich die Struktur zu \subref{fig:punktgruppen:atoms-grid-f} verformt. +Setzt man \subref{fig:punktgruppen:atoms-grid-f} gedanklich auch zwischen zwei leitende Platten, scheint es, als würden rechts mehr positive Ionen in die Platte gedrückt werden und links umgekehrt. +Dies ist aber nicht mehr der Fall, wenn sich die Struktur nach oben und unten periodisch wiederholt. + + +Struktur \subref{fig:punktgruppen:atoms-piezo-fh} zeigt \subref{fig:punktgruppen:atoms-piezo} in unter horizontaler Belastung. +Was zwischen \subref{fig:punktgruppen:atoms-piezo-fv} und \subref{fig:punktgruppen:atoms-piezo-fh} zu beobachten ist, dass die entstandene Ladungsdifferenz orthogonal zu der angelegten Kraft entsteht, im Gegensatz zu \subref{fig:punktgruppen:atoms-piezo-fh}. +Daraus kann man schliessen, dass \subref{fig:punktgruppen:atoms-piezo} keine Rotationssymmetrie von \(90^\circ\) besitzen kann, weil die Eigenschaften der Struktur sich bei einer \(90^\circ\) Drehung ändern. +Das Fehlen dieser Rotationssymmetrie bestätigt sich auch wenn \subref{fig:punktgruppen:atoms-piezo} als Hexagon betrachtet wird. + + +\subsection{Punktsymmetrie} + +Piezoelektrische Kristalle können nicht punktsymmetrisch sein. Kristallgitter, bei welchen eine Punktspiegelung eine symmetrische Operation ist, können keine piezoelektrische Kristalle bilden. -Auf Abbildung \ref{fig:punktgruppen:atomPiezo} ist bewusst $(a)$ ein nicht Punktsymmetrischer Kristall mit einem Punktsymmetrischen $(d)$ verglichen worden. -Als vereinfachte Erklärung kann mann sich wieder das Bild vor augen führen, eines Kristalles, -welcher unter Druck auf der einen Seite negative und der anderen Seite positive Ionen an seine Oberfläche verdrängt. -Spiegelt man nun den Kristall um den Gitterpunkt in der mitte des Kristalles, so würden die negativen Ionen auf den Positiven auf der anderen seite landen, -was der Definition einer Symmetrie deutlich widerspricht. +Auf Abbildung \ref{fig:punktgruppen:atomPiezo} ist bewusst \subref{fig:punktgruppen:atoms-piezo} ein nicht punktsymmetrischer Kristall mit einem punktsymmetrischen \subref{fig:punktgruppen:atoms-grid} verglichen worden. +Als vereinfachte Erklärung kann man sich wieder das Bild eines Kristalles wie \subref{fig:punktgruppen:atoms-piezo} vor Augen führen, welcher unter Druck auf der einen Seite negative und der anderen Seite positive Ionen an seine Oberfläche verdrängt. +Spiegelt man nun den Kristall um den Gitterpunkt in der Mitte des Kristalles, so würden die negativen Ionen auf den positiven auf der anderen Seite landen, was der Definition einer Symmetrie deutlich widerspricht. + \subsection{Vom Kristall zum Feuer} -Piezoelektrizität hat durchaus nutzen im Alltag. -Feuerzeuge welche nicht auf dem Prinzip beruhen einen Zündstein abzuschleifen, -sonder ohne Verschleiss auf Knopfdruck einen Zündfunken erzeugen, basieren auf dem Prinzip der Piezoelektrizität. -Drückt der Nutzende auf den Zündknopf spannt sich eine Feder bis zu einer Konfigurierten Spannung. -Wird vom Nutzenden weiter gedrückt entspannt sich die Feder schlagartig und beschleunigt mit der gespeicherten Energie ein Hammer, -welcher auf das Piezoelement aufschlägt. -Der augenblicklich hohe Druck sorgt an den Piezokontakten für eine eben so Kurze aber hohe elekrische Spannung. + +Piezoelektrizität hat durchaus Nutzen im Alltag. +Feuerzeuge welche nicht auf dem Prinzip beruhen einen Zündstein abzuschleifen, sondern ohne Verschleiss auf Knopfdruck einen Zündfunken erzeugen, basieren auf dem Prinzip der Piezoelektrizität. +Drückt der Nutzende auf den Zündknopf, spannt sich eine Feder bis zu einer konfigurierten Spannung. +Drückt der Nutzende stärker zu, entspannt sich die Feder schlagartig und beschleunigt mit der gespeicherten Energie ein Hammer, welcher auf das Piezoelement aufschlägt. +Der augenblicklich hohe Druck sorgt an den Piezokontakten für eine eben so kurze aber hohe elektrische Spannung. Die Spannung reicht aus, um eine Funkenstrecke zu überwinden und so eine entflammbares Gas zu entzünden. -Sollten Sie also eines Tages in die Situation geraten, in welcher Sie zwei verschiedene Kristalle vor sich haben -und ein piezoelektrisches Feuerzeug bauen müssen, -wobei Sie aber wissen, dass einer eine Punktsymmetrie aufweist, -versuche sie es mit dem anderen. -Ich muss aber anmerken, dass aus den $21$ möglichen Kristallsymmetrien ohne Punktsymmetrie einer nicht piezoelektrisch ist. -ein wenig glück brauchen Sie also immer noch. +Sollte der Leser eines Tages in die Situation geraten, in welcher er zwei verschiedene Kristalle vor sich hat und ein piezoelektrisches Feuerzeug bauen musst, wobei bekannt ist, dass der eine eine Punktsymmetrie aufweist, empfiehlt es sich, sich mit dem anderen zu versuchen. + diff --git a/buch/papers/punktgruppen/references.bib b/buch/papers/punktgruppen/references.bib index 9edb8bd..7928b22 100644 --- a/buch/papers/punktgruppen/references.bib +++ b/buch/papers/punktgruppen/references.bib @@ -26,10 +26,29 @@ @book{punktgruppen:lang-elt2, title = {Elektrotechnik 2}, - author = {Hans-Dieter Lang}, + author = {Hans-Dieter Lang Ph.D}, publisher = {Fachhochschule Ostschweiz Rapperswil}, year = {2020}, month = {2}, inseries = {Vorlesungsskript zum Modul ELT}, } +@online{punktgruppen:ouchem, + title = {Symmetry in Crystallography}, + author = {Dept. of Chemistry \& Biochemistry{,} Chemical Crystallography Laboratory{,} University of Oklahoma}, + year = {2019}, + month = {11}, + day = {17}, + url = {http://archive.today/2021.07.22-083802/http://xrayweb.chem.ou.edu/notes/symmetry.html}, + urldate = {2021-07-22}, +} + +@online{punktgruppen:restriction, + title = {Structure of Materials: Allowed Rotational Symmetry in Crystals}, + author = {Silvija Gradecak-Garaj{,} Massachusetts Institute of Technology (MIT)}, + year = {2020}, + month = {4}, + day = {9}, + url = {https://www.youtube.com/watch?v=Ia2eHF1ZKoI}, + urldate = {2021-07-30}, +} diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index 1dc6f98..4a8d911 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -1,175 +1,137 @@ \section{Symmetrie} Das Wort Symmetrie ist sehr alt und hat sich seltsamerweise von seinem -ursprünglichen griechischen Wort -\(\mathrm{\Sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\) -\footnote{\emph{Symmetr\'ia}: ein gemeinsames Mass habend, gleichmässig, -verhältnismässig} fast nicht verändert. In der Alltagssprache mag es ein -locker definierter Begriff sein, aber in der Mathematik hat Symmetrie eine sehr -präzise Bedeutung. +ursprünglichen griechischen Wort \(\mathrm{\Sigma\upsilon\mu\mu\varepsilon\tau\rho\iota\alpha}\)\footnote{\emph{Symmetr\'ia}: ein gemeinsames Mass habend, gleichmässig,verhältnismässig} fast nicht verändert. +In der Alltagssprache mag es ein locker definierter Begriff sein, in der Mathematik hat Symmetrie jedoch eine sehr präzise Bedeutung. \begin{definition}[Symmetrie] - Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer - bestimmten Operation invariant ist. + Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer bestimmten Operation invariant ist. \end{definition} -Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit -einigen geometrischen Beispielen beginnen. Wie wir jedoch später sehen werden, -ist das Konzept der Symmetrie eigentlich viel allgemeiner. +Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit einigen geometrischen Beispielen beginnen. +Wie wir jedoch später sehen werden, ist das Konzept der Symmetrie eigentlich viel allgemeiner. \begin{figure} - \centering - \includegraphics{papers/punktgruppen/figures/symmetric-shapes} - \caption{ - Beispiele für geometrisch symmetrische Formen. - \label{fig:punktgruppen:geometry-example} - } + \centering + \includegraphics{papers/punktgruppen/figures/symmetric-shapes} + \caption{ + Beispiele für geometrisch symmetrische Formen. + \label{fig:punktgruppen:geometry-example} + } \end{figure} \subsection{Geometrische Symmetrien} -In Abbildung \ref{fig:punktgruppen:geometry-example} haben wir einige Formen, -die offensichtlich symmetrisch sind. Zum Beispiel hat das Quadrat eine Gerade, an -deren es gespiegelt werden kann, ohne sein Aussehen zu verändern. Regelmässige -Polygone mit \(n\) Seiten sind auch gute Beispiele, um eine diskrete -Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um -einen Punkt um einen bestimmten Winkel \(360^\circ/n\) die Figur unverändert -lässt. Das letzte Beispiel auf der rechten Seite ist eine unendliche -Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für -\(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen. Dies ist -hoffentlich ausreichend, um die Bedeutung hinter der Notation zu verstehen, die -nun eingeführt wird. - -% Vieleicht eine kurze Einführung in für die Definition, ich habe das gefühl, dass in der Definition die Symmetrie-Operation und die Gruppe auf einmal erklährt wird -\subsubsection{Symetriegruppe} -\texttt{TODO: review this paragraph, explain what is \(\mathds{1}\).} +In Abbildung \ref{fig:punktgruppen:geometry-example} haben wir einige Formen, die offensichtlich symmetrisch sind. +Zum Beispiel hat das Quadrat eine Gerade, an der es gespiegelt werden kann, ohne sein Aussehen zu verändern. +Regelmässige Polygone mit \(n\) Seiten sind auch gute Beispiele, um eine diskrete Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um einen Punkt um einen bestimmten Winkel \(360^\circ/n\) die Figur unverändert lässt. +Das letzte Beispiel auf der rechts ist eine unendliche Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für den Drehwinkel \(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen. Ein Objekt kann mehr als nur eine Symmetrie aufweisen. -Als Beispiel, kann das Quadrat in Abbildung \ref{fig:punktgruppen:geometry-example} -nicht nur um $\sigma$ sondern auch Diagonal gespiegelt werden oder um $90^\circ$ gedreht werden. -Fässt man die möglichen Symmetrien zusammen, entsteht eine Symmetriegruppe. +Zum Beispiel kann das Quadrat in Abbildung \ref{fig:punktgruppen:geometry-example} nicht nur um \(\sigma\) sondern auch diagonal gespiegelt werden oder um \(90^\circ\) gedreht werden. +Fasst man die möglichen Symmetrien zusammen, entsteht eine Symmetriegruppe. \begin{definition}[Symmetriegruppe] - Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt. - Bei einer anderen Operation \(h\) definieren wir die Komposition \(h\circ g\) - als die Anwendung der Operationen nacheinander. Alle Operationen bilden unter - Komposition eine Gruppe, die Symmetriegruppe genannt wird. -\end{definition} % ich lese diese Definition ein wenig holprig, vieleicht können wir sie zusammen anschauen - -% Nach meinem Geschmack könne es hier auch eine einleitung wie mein Beispiel geben dammit man den Text flüssiger lesen kann -\begin{definition}[Zyklische Untergruppe, Erzeuger] - Sei \(g\) ein Element einer Symmetriegruppe \(G\). Alle möglichen - Kompositionen von \(g\) und \(g^{-1}\) bilden eine sogenannte zyklische - Untergruppe von \(G\), und \(g\) wird ihr Erzeuger genannt. Die erzeugte - Untergruppe \(\langle g \rangle\) wird mit spitzen Klammern um den Erzeuger - bezeichnet. + Seien \(g\) und \(h\) umkehrbare Operationen, sogenannte Symmetrieoperationen, die ein mathematisches Objekt unverändert lassen. + Die Komposition \(h\circ g\) definieren wir als die Anwendung der Operationen nacheinander. + Alle möglichen Symmetrieoperationen bilden unter Komposition eine Gruppe, die Symmetriegruppe genannt wird. \end{definition} -Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. -Bezeichnen wir mit \(r\) eine Drehung im Gegenuhrzeigersinn von \(360^\circ/n\) -um einen Punkt. Diese Definition reicht aus, um die gesamte Symmetriegruppe -\[ - C_n = \langle r \rangle - = \left\{\mathds{1}, r, r^2, \ldots, r^{n-1}\right\} -\] -der Drehungen eines \(n\)-Gons zu definieren. Das liegt daran, -dass wir durch die mehrfache Verwendung von \(r\) jeden Winkel erzeugen, der -die Rotationssymmetrie bewahrt. Hier die Potenzen von \(r\) sind als -wiederholte Komposition gemeint, dass heisst \(r^n = r\circ r \circ \cdots -r\circ r\). Wenn wir diese Idee nun erweitern, können wir mit einem -Erzeugendensystemen komplexere Strukturen aufbauen. +Eine Gruppe benötigt ausserdem auch zwingend ein neutrales Element, welches wir mit \(\mathds{1}\) bezeichnen. +Die Anwendung der neutralen Operation ist gleichbedeutend damit, alles unverändert zu lassen. +Weiterhin muss in einer Gruppe für jede Operation \(g\) auch eine inverse Operation \(g^{-1}\) vorkommen, die rückgängig macht, was \(g\) getan hat. +Somit ist \(\mathds{1}\) auch äquivalent dazu, eine Operation und dann ihre Inverse anzuwenden. + Die Definition der Symmetriegruppe ist mit der Kompositionsoperation gegeben, sie wird aber auch oft als Multiplikation geschrieben. +Das liegt daran, dass in manchen Fällen die Zusammensetzung algebraisch durch eine Multiplikation berechnet wird. +Die Verwendung einer multiplikativen Schreibweise ermöglicht es, einige Ausdrücke kompakter zu schreiben, z.B. +durch Verwendung von Potenzen \(r^n = r\circ r \circ \cdots r\circ r\) für eine wiederholte Komposition. -\begin{definition}[Erzeugendensysteme] - % please fix this unreadable mess - Jede Gruppe kann durch eines oder mehrere ihrer Elemente generiert werden. - Wir lassen \(g_1, g_2, \ldots, g_n\) erzeugenden Elemente einer - Symmetriegruppe sein. Da es mehrere Erzeuger gibt, müssen auch die - sogenannte Definitionsgleichungen gegeben werden, die die - Multiplikationstabelle vollständig definieren. Die Gleichungen sind ebenfalls - in den Klammern angegeben. Die erzeugende Elementen zusammen mit der - Definitionsgleichungen bauen ein Erzeugendensysteme. +\begin{definition}[Zyklische Untergruppe, Erzeuger] + Sei \(g\) ein Element einer Symmetriegruppe \(G\). + Alle möglichen Kompositionen von \(g\) und \(g^{-1}\) bilden eine sogenannte zyklische Untergruppe von \(G\), wobei \(g\) Erzeuger der Untergruppe genannt wird. + Die von \(g\) erzeugte Untergruppe \(\langle g \rangle = \{ g^k : k \in \mathbb{Z} \}\) wird mit spitzen Klammern bezeichnet. \end{definition} +\begin{beispiel} + Um die Syntax zu verstehen, betrachten wir eine durch \(a\) erzeugte Gruppe \(G = \langle a \rangle\). + Das bedeutet, dass \(G\) die Elemente \(a, aa, aaa, \ldots\) sowie \(a^{-1}, a^{-1}a^{-1}, \ldots\) und ein neutrales Element \(\mathds{1} = aa^{-1}\) enthält. +\end{beispiel} +\begin{beispiel} + Als anschaulicheres Beispiel können wir eine zyklische Untergruppe des \(n\)-Gon formalisieren. + Wir bezeichnen mit \(r\) eine Drehung im Gegenuhrzeigersinn von \(360^\circ/n\) um einen Punkt. + Diese Definition reicht aus, um die gesamte Symmetriegruppe + \[ + C_n = \langle r \rangle + = \{\mathds{1}, r, r^2, \ldots, r^{n-1}\} + \] + der Drehungen eines \(n\)-Gons zu erzeugen. + Das liegt daran, dass wir durch die mehrfache Verwendung von \(r\) jeden Winkel erzeugen k\"onnen, der die Rotationssymmetrie bewahrt. + In ähnlicher Weise, aber weniger interessant, enthält die Reflexionssymmetriegruppe \(\langle\sigma\rangle\) nur \(\left\{\mathds{1}, \sigma\right\}\), weil \(\sigma^2 = \mathds{1}\). +\end{beispiel} -\texttt{TODO: should put examples for generators?} \\ +Wenn wir diese Idee nun erweitern, können wir mit einem Erzeugendensystem +komplexere Strukturen aufbauen. -Die Reflexionssymmetriegruppe ist nicht so interessant, da sie nur -\(\left\{\mathds{1}, \sigma\right\}\) enthält. Kombiniert man sie jedoch mit -der Rotation, erhält man die so genannte Diedergruppe -\[ - D_n = \langle r, \sigma : r^{n-1} = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle - = \left\{ - \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} - \right\}. -\] +%@Naoki Are you ok with my grammar fixes I'm not 101% shore how to use the word Erzeugendensystem? +\begin{definition}[Erzeugendensystem] + Jede diskrete Gruppe kann durch eines oder mehrere ihrer Elemente generiert werden. + Wir lassen \(g_1, g_2, g_3, \ldots\) erzeugenden Elemente einer Symmetriegruppe sein. + Da es mehrere Erzeuger gibt, müssen auch die sogenannten Definitionsgleichungen gegeben werden, die die Multiplikationstabelle vollständig definieren. + Die Gleichungen sind ebenfalls in den Klammern angegeben. + Die erzeugenden Elementen bauen zusammen mit den Definitionsgleichungen ein Erzeugendensystem. +\end{definition} +\begin{beispiel} + Wir werden nun alle Symmetrien eines \(n\)-Gons beschreiben, was bedeutet, dass wir die Operationen \(r\) und \(\sigma\) kombinieren. + Die Definitionsgleichungen sind \(r^n = \mathds{1}\), \(\sigma^2 = \mathds{1}\) und \((\sigma r)^2 = \mathds{1}\). + Die ersten beiden sind ziemlich offensichtlich. + Die letzte wird oft auch als Inversion bezeichnet, weil die Anwendung von \(\sigma r\) dasselbe ist wie das Ziehen einer Linie von einem Punkt, die durch den Ursprung geht, und das Verschieben des Punktes auf die andere Seite des Nullpunkts. + Wenn man dies zweimal macht, geht man zurück zum Anfangspunkt. + Daraus ergibt sich die so genannte Diedergruppe + \begin{align*} + D_n &= \langle r, \sigma : r^n = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle \\ + &= \{ + \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} + \}. \qedhere + \end{align*} +\end{beispiel} -Die Symmetrieoperationen, die wir bis jetzt besprochen haben, haben immer -mindestens einen Punkt gehabt, der wieder auf sich selbst abgebildet wird. Im -Fall der Rotation war es der Drehpunkt, bei der Spiegelung die Punkte der -Spiegelachse. Dies ist jedoch keine Voraussetzung für eine Symmetrie, da es -Symmetrien gibt, die jeden Punkt zu einem anderen Punkt verschieben können. -Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, nennt man -Punktsymmetrie. +Die Symmetrieoperationen, die wir bis jetzt besprochen haben, haben immer mindestens einen Punkt gehabt, der wieder auf sich selbst abgebildet wird. +Im Fall der Rotation war es der Drehpunkt, bei der Spiegelung die Punkte der Spiegelachse. +Dies ist jedoch keine Voraussetzung für eine Symmetrie, da es Symmetrien gibt, die jeden Punkt zu einem anderen Punkt verschieben können. + Diesen Spezialfall, bei dem immer mindestens ein Punkt unverändert bleibt, nennt man Punktsymmetrie. \begin{definition}[Punktgruppe] - Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens - einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine - Punktgruppe ist. + Wenn es einen Punkt gibt, der von jeder Gruppenoperation unverändert gelassen wird, ist die Symmetriegruppe eine Punktgruppe. \end{definition} \subsection{Algebraische Symmetrien} -Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich -möglich ist, Gleichungen zu schreiben. Die naheliegende Frage ist dann, könnte -es sein, dass wir bereits etwas haben, das dasselbe tut? Natürlich, ja. +Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich möglich ist, Gleichungen zu schreiben. +Die anschliessende Frage ist dann, ob wir bereits mathematische Objekte haben, mit denen wir Gleichungen schreiben, die sich auf die gleiche Weise verhalten. +Die Antwort lautet natürlich ja. Um es formaler zu beschreiben, werden wir einige Begriffe einführen. \begin{definition}[Gruppenhomomorphismus] - Seien \(G\) und \(H\) Gruppe mit unterschiedlicher Operation \(\diamond\) - bzw. \(\star\). Ein Homomorphismus\footnote{ Für eine ausführlichere - Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist - eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt - \(f(a\diamond b) = f(a) \star f(b)\). Man sagt, dass der Homomorphismus - \(f\) \(G\) in \(H\) transformiert. + \(G\) und \(H\) seien Gruppen mit unterschiedlichen Operationen \(\diamond\) bzw. + \(\star\). + Ein Homomorphismus\footnote{ Für eine ausführlichere Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt \(f(a\diamond b) = f(a) \star f(b)\). + Man sagt, dass der Homomorphismus \(f\) \(G\) in \(H\) transformiert. \end{definition} \begin{beispiel} - Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen - Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem - komplexen Einheitskreis. Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) - ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. + Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht genau dem komplexen Einheitskreis. + Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. \end{beispiel} \begin{definition}[Darstellung einer Gruppe] - Die Darstellung einer Gruppe ist ein Homomorphismus, der eine Symmetriegruppe - auf eine Menge von Matrizen abbildet. - \[ - \Phi: G \to \operatorname{GL}_n(\mathbb{R}). - \] - Äquivalent kann man sagen, dass ein Element aus der Symmetriegruppe auf einen - Vektorraum \(V\) wirkt, indem man definiert \(\Phi : G \times V \to V\). + Die Darstellung einer Gruppe ist ein Homomorphismus + \[ + \Phi: G \to \operatorname{GL}_n(\mathbb{R}), + \] + der eine Symmetriegruppe auf eine Menge von Matrizen abbildet. + Äquivalent kann man sagen, dass ein Element aus der Symmetriegruppe auf einen Vektorraum \(V\) wirkt, indem man \(\Phi : G \times V \to V\) definiert. \end{definition} \begin{beispiel} - Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine - Drehung von \(2\pi k/n\) um den Ursprung dar. Die mit der Matrix - \[ - \Phi(r^k) = \begin{pmatrix} - \cos(2\pi k/n) & -\sin(2\pi k/n) \\ - \sin(2\pi k/n) & \cos(2\pi k/n) - \end{pmatrix} - \] - definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von - \(C_n\). In diesem Fall ist die erste Gruppenoperation die Komposition und - die zweite die Matrixmultiplikation. Man kann überprüfen, dass \(\Phi(r^2 - \circ r) = \Phi(r^2)\Phi(r)\). + Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine Drehung von \(2\pi k/n\) um den Ursprung dar. + Die mit der Matrix + \[ + \Phi(r^k) = \begin{pmatrix} + \cos(2\pi k/n) & -\sin(2\pi k/n) \\ + \sin(2\pi k/n) & \cos(2\pi k/n) + \end{pmatrix} + \] + definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von \(C_n\). + In diesem Fall ist die erste Gruppenoperation die Komposition und die zweite die Matrixmultiplikation. + Man kann überprüfen, dass \(\Phi(r^2 \circ r) = \Phi(r^2)\Phi(r)\). \end{beispiel} - -\texttt{TODO: rewrite section on translational symmetry.} -%% TODO: title / fix continuity -% Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren: -% eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr -% nützlich ist, nämlich die Translationssymmetrie. Von einem mathematischen -% Objekt \(U\) wird gesagt, dass es eine Translationssymmetrie \(Q(x) = x + a\) -% hat, wenn es die Gleichung -% \[ -% U(x) = U(Q(x)) = U(x + a), -% \] -% für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche -% Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine -% zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\) -% dann \(\vec{v}(t) = \vec{v}(t + \tau)\). - -% \subsection{Sch\"onflies notation} - -% vim:ts=2 sw=2 spell spelllang=de: diff --git a/buch/papers/punktgruppen/tikz/atoms-grid-force.tex b/buch/papers/punktgruppen/tikz/atoms-grid-force.tex new file mode 100644 index 0000000..05742cf --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-grid-force.tex @@ -0,0 +1,42 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { + \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ + }; + + \foreach \d in {NW, N, NE} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {SW, S, SE} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-grid-still.tex b/buch/papers/punktgruppen/tikz/atoms-grid-still.tex new file mode 100644 index 0000000..4e43856 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-grid-still.tex @@ -0,0 +1,33 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] { + \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\ + }; + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex new file mode 100644 index 0000000..e4c3f93 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex @@ -0,0 +1,47 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {}; + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \draw[orange, very thick, <-] (C6) to ++(.7,0); + \draw[orange, very thick, <-] (C3) to ++(-.7,0); + + \node[black] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex new file mode 100644 index 0000000..892ab42 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex @@ -0,0 +1,52 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {}; + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \foreach \d in {C1, C2} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {C4, C5} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \node[black] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[red!50, right = of E] {\(+\)}; + \node[blue!50, left = of E] {\(-\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex new file mode 100644 index 0000000..2eb78ba --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex @@ -0,0 +1,34 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} { + \node[charge, \t] (C\i) at (\x:1.5cm) {}; + } + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {}; + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/combine-symmetries.tex b/buch/papers/punktgruppen/tikz/combine-symmetries.tex index 84e0a76..fa669ae 100644 --- a/buch/papers/punktgruppen/tikz/combine-symmetries.tex +++ b/buch/papers/punktgruppen/tikz/combine-symmetries.tex @@ -13,6 +13,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, dot/.style = { draw, circle, thick, black, fill = gray!40!white, minimum size = 2mm, @@ -45,7 +46,7 @@ (A2) ++(-.5,0) arc (180:60:.5); \draw[red!80!black, dashed, thick, ->] (A2) to (B2); - \draw[yellow!50!orange, thick, ->] + \draw[cyan!40!blue, thick, ->] (B1) to node[above, midway] {\(\vec{Q}'\)} (B2); \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {}; diff --git a/buch/papers/punktgruppen/tikz/lattice.tex b/buch/papers/punktgruppen/tikz/lattice.tex index 9c05af3..a6b1876 100644 --- a/buch/papers/punktgruppen/tikz/lattice.tex +++ b/buch/papers/punktgruppen/tikz/lattice.tex @@ -13,23 +13,24 @@ \begin{document} \begin{tikzpicture}[ - dot/.style = { - draw, circle, thick, black, fill = gray!40!white, - minimum size = 2mm, - inner sep = 0pt, - outer sep = 1mm, - }, + >=latex, + dot/.style = { + draw, circle, thick, black, fill = gray!40!white, + minimum size = 2mm, + inner sep = 0pt, + outer sep = 1mm, + }, ] \begin{scope} - \clip (-2,-2) rectangle (3,4); + \clip (-2,-2) rectangle (7,2); \foreach \y in {-7,-6,...,7} { \foreach \x in {-7,-6,...,7} { \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {}; } } \end{scope} - \draw[black, thick] (-2, -2) rectangle (3,4); + \draw[black, thick] (-2, -2) rectangle (7,2); \draw[red!80!black, thick, ->] (N00) to node[midway, below] {\(\vec{a}_1\)} (N10); diff --git a/buch/papers/punktgruppen/tikz/piezo-atoms.tex b/buch/papers/punktgruppen/tikz/piezo-atoms.tex index 82a2710..1811392 100644 --- a/buch/papers/punktgruppen/tikz/piezo-atoms.tex +++ b/buch/papers/punktgruppen/tikz/piezo-atoms.tex @@ -13,6 +13,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, node distance = 2mm, charge/.style = { circle, draw = black, thick, diff --git a/buch/papers/punktgruppen/tikz/piezo.tex b/buch/papers/punktgruppen/tikz/piezo.tex index 1d16ab7..6542f26 100644 --- a/buch/papers/punktgruppen/tikz/piezo.tex +++ b/buch/papers/punktgruppen/tikz/piezo.tex @@ -12,12 +12,14 @@ \usetikzlibrary{calc} \begin{document} -\begin{tikzpicture} +\begin{tikzpicture}[ + >=latex, + ] \begin{scope}[ node distance = 0cm ] \node[ - rectangle, fill = gray!60!white, + rectangle, fill = gray!20!white, minimum width = 3cm, minimum height = 2cm, ] (body) {\(\vec{E}_p = \vec{0}\)}; @@ -43,9 +45,9 @@ xshift = 7cm ] \node[ - rectangle, fill = gray!40!white, + rectangle, fill = gray!20!white, minimum width = 3cm, minimum height = 1.5cm, - ] (body) {\(\vec{E}_p = \vec{0}\)}; + ] (body) {\(\vec{E}_p \neq \vec{0}\)}; \node[ draw, rectangle, thick, black, fill = red!50, diff --git a/buch/papers/punktgruppen/tikz/projections.tex b/buch/papers/punktgruppen/tikz/projections.tex index a763e77..e8a4a2e 100644 --- a/buch/papers/punktgruppen/tikz/projections.tex +++ b/buch/papers/punktgruppen/tikz/projections.tex @@ -13,6 +13,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, classcirc/.style = { draw = gray, thick, circle, minimum size = 12mm, @@ -43,7 +44,7 @@ \node[classcirc] (C2h) {} node[classlabel] {\(C_{2h}\)}; & \node[classcirc] (D2) {} node[classlabel] {\(D_{2}\)}; \\ - \node[classcirc] (D3d) {} node[classlabel] {\(D_{3d}\)}; & + \node[classcirc] (D3d) {} node[classlabel] {\(C_{3v}\)}; & \node[classcirc] (C2v) {} node[classlabel] {\(C_{2v}\)}; & \node[classcirc] (D2h) {} node[classlabel] {\(D_{2h}\)}; & \node[classcirc] (D3) {} node[classlabel] {\(D_{3}\)}; & diff --git a/buch/papers/punktgruppen/tikz/stereographic-projections.tex b/buch/papers/punktgruppen/tikz/stereographic-projections.tex new file mode 100644 index 0000000..7d612fb --- /dev/null +++ b/buch/papers/punktgruppen/tikz/stereographic-projections.tex @@ -0,0 +1,108 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{tikz-3dplot} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + +\tdplotsetmaincoords{60}{130} +\pgfmathsetmacro{\l}{2} + +\begin{tikzpicture}[ + >=latex, + tdplot_main_coords, + dot/.style = { + black, fill = black, circle, + outer sep = 0, inner sep = 0, + minimum size = 1mm + }, + round/.style = { + draw = orange, thick, circle, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt, + }, + cross/.style = { + cross out, draw = magenta, thick, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt + }, + ] + + % origin + \coordinate (O) at (0,0,0); + + % poles + \coordinate (NP) at (0,0,\l); + \coordinate (SP) at (0,0,-\l); + + % axis + % \draw[->] (O) -- ++(1.5*\l,0,0); + % \draw[->] (O) -- ++(0,1.5*\l,0); + % \draw[->] (O) -- ++(0,0,1.5*\l); + + % gray unit circle + \tdplotdrawarc[gray, thick]{(O)}{\l}{0}{360}{}{}; + \draw[gray, dotted] (-\l, 0, 0) to (\l, 0, 0); + \draw[gray, dotted] (0, -\l, 0) to (0, \l, 0); + + % meridians + \foreach \phi in {0, 30, 60, ..., 150}{ + \tdplotsetrotatedcoords{\phi}{90}{0}; + \tdplotdrawarc[lightgray, dashed, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{}; + } + + % dot above and its projection + \pgfmathsetmacro{\phi}{120} + \pgfmathsetmacro{\theta}{60} + + \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\pz}{cos(\theta)*\l}) + + \coordinate (A) at (\px,\py,\pz); + \coordinate (Aproj) at ({\px * \l / (\l + \pz)}, {\py * \l / (\l + \pz)}, 0); + + % dot below and its projection + \pgfmathsetmacro{\phi}{-60} + \pgfmathsetmacro{\theta}{120} + + \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\pz}{cos(\theta)*\l}) + + \coordinate (B) at (\px,\py,\pz); + \coordinate (Bproj) at ({\px * \l / (\l - \pz)}, {\py * \l / (\l - \pz)}, 0); + + % projection lines + \draw[gray] (A) to (SP); + \draw[gray] (SP) to (O) to (Aproj); + + \draw[gray] (B) to (NP); + \draw[gray] (NP) to (O) to (Bproj); + + % dots + \draw (O) node[dot] {}; + \draw (SP) node[dot] {}; + \draw (NP) node[dot] {}; + \draw (A) node[dot, fill = magenta, minimum size = 1.5mm] {}; + \draw (B) node[dot, fill = orange, minimum size = 1.5mm] {}; + + % projection markers + \draw[very thick, magenta] + (Aproj) ++(.15,0) to ($(Aproj)+(-.15, 0)$) + (Aproj) ++(0,.15) to ($(Aproj) +(0, -.15)$); + + \tdplotdrawarc[orange, very thick]{(Bproj)}{.1}{0}{360}{}{}; + +\end{tikzpicture} +\end{document} +% vim:ts=2 sw=2 et: diff --git a/buch/papers/punktgruppen/tikz/symmetric-shapes.tex b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex index b2c051f..688fb61 100644 --- a/buch/papers/punktgruppen/tikz/symmetric-shapes.tex +++ b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex @@ -14,6 +14,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, node distance = 2cm, shapetheme/.style = { very thick, draw = black, fill = magenta!20!white, diff --git a/buch/papers/reedsolomon/Makefile b/buch/papers/reedsolomon/Makefile index 9c96e88..4be963e 100644 --- a/buch/papers/reedsolomon/Makefile +++ b/buch/papers/reedsolomon/Makefile @@ -4,6 +4,52 @@ # (c) 2020 Prof Dr Andreas Mueller # -images: - @echo "no images to be created in reedsolomon" +SOURCES := \ + anwendungen.tex \ + codebsp.tex \ + decmitfehler.tex \ + decohnefehler.tex \ + dtf.tex \ + einleitung.tex \ + endlichekoerper.tex \ + hilfstabellen.tex \ + idee.tex \ + main.tex \ + packages.tex \ + rekonstruktion.tex \ + restetabelle1.tex \ + restetabelle2.tex \ + standalone.tex \ + zusammenfassung.tex + +TIKZFIGURES := \ + tikz/polynom2.tex \ + tikz/fourier.tex + +FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES)) + + +all: images standalone + + +.PHONY: images +images: $(FIGURES) + +figures/%.pdf: tikz/%.tex + mkdir -p figures + pdflatex --output-directory=figures $< + +.PHONY: standalone +standalone: standalone.tex $(SOURCES) $(FIGURES) + mkdir -p standalone + cd ../..; \ + pdflatex \ + --halt-on-error \ + --shell-escape \ + --output-directory=papers/reedsolomon/standalone \ + papers/reedsolomon/standalone.tex; + cd standalone; \ + bibtex standalone; \ + makeindex standalone; + diff --git a/buch/papers/reedsolomon/Makefile.inc b/buch/papers/reedsolomon/Makefile.inc index 6a676f8..ea51f7a 100644 --- a/buch/papers/reedsolomon/Makefile.inc +++ b/buch/papers/reedsolomon/Makefile.inc @@ -6,9 +6,17 @@ dependencies-reedsolomon = \ papers/reedsolomon/packages.tex \ papers/reedsolomon/main.tex \ - papers/reedsolomon/references.bib \ - papers/reedsolomon/teil0.tex \ - papers/reedsolomon/teil1.tex \ - papers/reedsolomon/teil2.tex \ - papers/reedsolomon/teil3.tex + papers/reedsolomon/einleitung.tex \ + papers/reedsolomon/idee.tex \ + papers/reedsolomon/dtf.tex \ + papers/reedsolomon/endlichekoerper.tex \ + papers/reedsolomon/codebsp.tex \ + papers/reedsolomon/decohnefehler.tex \ + papers/reedsolomon/decmitfehler.tex \ + papers/reedsolomon/rekonstruktion.tex \ + papers/reedsolomon/zusammenfassung.tex \ + papers/reedsolomon/anwendungen.tex \ + papers/reedsolomon/hilfstabellen.tex \ + papers/reedsolomon/references.bib + diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex b/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex new file mode 100644 index 0000000..038e93e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex @@ -0,0 +1,33 @@ +% polynome1 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\teiler}{40} +\begin{document} + + +\begin{tikzpicture}[>=latex,thick] + + \begin{axis}[ + axis lines = left, + xlabel = \(x\), + ylabel = {\(f(x)\)}, + ] + %Below the red parabola is defined + \addplot[ + color=blue, + ] + coordinates { + (0,23.1)(10,27.5)(20,32)(30,37.8)(40,44.6)(60,61.8)(80,83.8)(100,114) + }; + %Here the blue parabola is defined + + \end{axis} +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/reedsolomon/anwendungen.tex b/buch/papers/reedsolomon/anwendungen.tex index 83e0f94..b9b1d69 100644 --- a/buch/papers/reedsolomon/anwendungen.tex +++ b/buch/papers/reedsolomon/anwendungen.tex @@ -6,14 +6,38 @@ \section{Anwendungen des Reed-Solomon-Codes \label{reedsolomon:section:anwendung}} \rhead{Anwendungen} -\textcolor{red}{Platzierung der Bilder? Quellenangabe der Bilder?} -In den vorherigen Abschnitten haben wir betrachtet, wie Reed-Solomon-Codes in der Theorie Funktionieren. +In den vorherigen Abschnitten haben wir betrachtet, wie Reed-Solomon-Codes in der Theorie funktionieren. In diesem Abschnitt werden wir einige Anwendungen vorstellen, bei denen ein Reed-Solomon-Code zum Einsatz kommt. + +Dabei teilen all diese Anwendungen das gleiche Problem: Die Daten können nur durch einen (höchst Wahrscheinlichen) fehlerbehafteten Kanal empfangen werden. Es gibt keine andere Methode, an diese Daten zu kommen, als über diesen Kanal. + +In der Netzwerktechnik zum Beispiel ist es üblich, dass bei Paketverluste oder beschädigt empfangene Datenpaketen diese einfach noch einmal innert wenigen Millisekunden angefordert werden können. +In der Raumfahrt ist dies nicht möglich, da aufgrund der beschränkten Speichermöglichkeit die gesammelten Daten so rasch wie möglich zur Erde gesendet werden. +Diese Daten wiederum brauchen aufgrund der grossen Distanz Stunden bis die Daten beim Empfänger ankommen. +Fehlerhafte Daten kann also auf Grund der Zeitverzögerung nicht mehr angefordert werden. + +Bei CDs oder DVDs gibt es zwar kein zeitliches Problem, jedoch erschweren Kratzer, Verschmutzungen oder Produktionsfehler das Lesen einer solchen Disk. +Da vor allem Produktionsfehler und Kratzer irreversibel sind und die Disk nicht nach jedem Kratzer ersetzt werden muss, so wird die korrekte Ausgabe der gespeicherten Information durch die Fehlerkorrektur sichergestellt. + +Einen ähnlichen Ansatz verfolgen QR-Codes, wobei die Information auch dann noch gelesen werden kann wenn der Code nicht mehr vollständig vorhanden ist. + +%Wie man sieht, eignen sich Reed-Solomon-Codes vor allem für Anwendungen, bei der die Informationen nicht auf einen Anderen Weg beschafft werden kann. +% +% +%, bei denen die Wahrscheinlichkeit hoch ist, dass während der Übertragung +% +%Es ist deshalb umso wichtiger die Daten Codiert zu lesen um so gleich die Lesefehler zu korrigieren. +% +% da aufgrund der grossen Distanz Stunden vergehen können bis gesendete Daten auf der Erde empfangen werden kann. +% + Obwohl alle diese Codes nach dem gleichen Prinzip arbeiten gibt es starke Unterschiede in deren Funktionsweise. Dies kommt vor allem daher, da die Codes nur Ressourcen zur Verfügung haben, die von der Hardware bereitstellt wird, auf denen die Codes implementiert wurden. Diese Codes bedienen sich daher verschiedener Tricks und Optimierungen um möglichst effizient zu arbeiten. -% + +Um die Fähigkeit eines verwendeten Reed-Solomon-Codes zu beschreiben verwendet man die Notation ($n$,$k$), wobei $n$ die Grösse des Nachrichtenblocks angibt und $k$ die Anzahl der Stellen, die für Nutzdaten gebraucht werden können. + %Dies kommt vor allem daher, da diese Codes an ihre Hardware gebunden sind, auf denen sie implementiert worden sind. %Deshalb wurden diese Codes stark optimiert damit sie möglichst Effizient arbeiten können. % @@ -45,8 +69,23 @@ Diese Codes bedienen sich daher verschiedener Tricks und Optimierungen um mögli %In den letzten abschnitten haben wir uns ausführlich die Funktionsweise des Reed-Solomon-Codes angeschaut. In diesem Abschnitt möchten wir dem Leser ein paar bekannte beispiele vorstellen, in denen Reed-Solomon-Codes zum einsatz kommen. Es sei jedoch angemerkt, dass diese Anwendungen in der Umsetzung oft ein wenig anderst funktionieren als hier vorgestellt. Dies wurde vor allem wegen technischen optimierungen realisiert. (technische tricks und finessen), von der logik jedoch sehr stark an unserem Beispiel orientieren \subsection{Raumfahrt} -Obwohl Reed-Solomon-Codes bereits in den 1960er entwickelt wurden fanden sie erstmals Anwendung in der Voyager Raumsonde der NASA. Die Daten der zwei im Jahre 1977 gestarteten Sonden werden mit einem RS(255,233)-Code \textcolor{red}{benötigt das weitere erklärungen, wie z.b. 255: grösse nachrichtenblock, 233: anzahl der nutzbaren daten ?} zusammen mit einem konventionellen Faltungscode übertragen. +Obwohl Reed-Solomon-Codes bereits in den 1960er entwickelt wurden fanden sie erstmals Anwendung in der Voyager Raumsonde der NASA. Die Daten der zwei im Jahre 1977 gestarteten Sonden (siehe Abbildung \ref{fig:voyager}) werden mit einem ($255$,$233$)-Code +Codiert. +Der Nachrichtenblock hat somit eine Länge von $255$ Zahlen, wovon $233$ als Nutzlast zur Verfügung stehen. +Damit ist es möglich bis zu $11$ Fehler im Nachrichtenblock zu korrigieren. +Der Codierte Nachrichtenblock wird in kleinere Blöcke aufgeteilt, mit einem Faltungscode erneut Codiert und anschliessend gesendet. +Ein Faltungscode ist wie ein Reed-Solomon-Code in der Lage Fehler zu korrigieren, +Codiert seine Information aber auf eine andere weise. Aus jedem unterteilten Block wird vor dem Versenden ein Paritätsbit erzeugt und dem Block angehängt. Anhand diesem Paritätsbit überprüft der Empfänger, ob bei der Übertragung der Block beschädigt wurde. Ist dies der Fall, wird der Block bei der Decodierung nicht beachtet. Diese so entstandenen ``Lücken'' im Datenstrom werden wiederum vom Reed-Solomon-Code korrigiert. Dieses Zusammenspiel beider Codes garantiert so eine hohe Robustheit gegenüber Übertragungsfeher. +% +% Funktioniert aber nach einem ganz anderen Prinzip. +% +%Durch diese doppelte Codierung wird eine äusserst hohe Übertragungssicherheit garantiert. +% +%Dabei steht die Zahl 255 für grösse des Nachrichtenblocks, der die Anzahl 233 +% +% +% \textcolor{red}{benötigt das weitere Erklärungen, wie z.b. 255: grösse Nachrichtenblock, 233: anzahl der nutzbaren daten ?} zusammen mit einem konventionellen Faltungscode übertragen. Eine von der Sonde gesendete Nachricht hat eine Blockgrösse von 255 Zeichen, wovon 233 für die Nutzdaten gebraucht werden können. Dieser Code ist somit in der Lage 11 Fehler in einem Nachrichtenblock zu korrigieren. % % Die zwei im Jahre 1977 gestarteten Sonden senden Daten mit der Hilfe eines RS(255,233)-Code für die digitalen Bilder sowie einem konventionellen Faltungscode. % @@ -56,14 +95,14 @@ Obwohl Reed-Solomon-Codes bereits in den 1960er entwickelt wurden fanden sie ers \begin{figure} \centering \includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/Voyager_Sonde} - \caption{Voyager Raumsonde} + \caption{Mit einer Entfernung von über 22.8 Milliarden Kilometer ist die Voyager 1 Raumsonde das am weitesten entfernte, von Menschen erschaffene Objekt. Obwohl ihre Schwestersonde Voyager 2 zuerst ins All gestartet wurde befindet Sie sich ``nur'' 19 Milliarden Kilometer weit weg von der Erde. Aufgrund abnehmender Batterieleistung werden die beiden Sonden ihre wissenschaftlichen Aktivitäten etwa 2025 einstellen, bleiben aber bis in die 2030er mit uns in Kontakt.} \label{fig:voyager} \end{figure} \subsection{CD/DVD} Compact discs verwenden sogar zwei ineinander verschachtelte Reed-Solomon-Codes, einen (32,28)-Code und einen (28,24)-Code. -Beide Codes sind in der Lage, Fehler aus dem jeweils anderen gelesenen Block zu korrigieren. Dieses spezielle zusammenspielen dieser beiden Codes werden auch Cross-interleaved Reed-Solomon-Codes (CIRC) genannt. -Diese Vorgehensweise erzielt eine hohe Robustheit gegenüber Produktionsfehler oder Verschmutzung auf der Disc. Bei CD's sind diese in der Lage bis zu 4000 fehlerhafte Bits am Stück (ca. $2.5mm$) zu erkennen und zu korrigieren. +Beide Codes sind in der Lage, Fehler aus dem jeweils anderen gelesenen Block zu korrigieren. Dieses spezielle Zusammenspielen dieser beiden Codes werden auch Cross-interleaved Reed-Solomon-Codes (CIRC) genannt. +Diese Vorgehensweise erzielt eine hohe Robustheit gegenüber Produktionsfehlern oder Verschmutzung auf der Disc. Bei CDs sind diese in der Lage, bis zu 4000 fehlerhafte Bits am Stück (ca. $2.5mm$) zu erkennen und zu korrigieren. Die Digital Video Disc funktioniert nach dem selben Konzept mit grösseren Codeblöcken. Die DVD verwendet einen (208,192)-Code und einen (182,172)-Code. @@ -72,13 +111,25 @@ Die Digital Video Disc funktioniert nach dem selben Konzept mit grösseren Codeb \begin{figure} \centering - \includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/Compact_Disc} - \caption{Compact Disc} + \subfigure[]{ + \includegraphics[width=0.45\textwidth]{papers/reedsolomon/images/Compact_Disc} + } + \subfigure[]{ + \includegraphics[width=0.45\textwidth]{papers/reedsolomon/images/Compact_Disc_zoomed_in} + } + \caption{CDs kamen 1982 auf den Markt. Sie funktioniert durch das Einpressen oder Einbrennen von Punkten und Strichen, die die Daten repräsentieren. Gelesen werden diese wiederum durch die Reflektion eines Lasers an diesen Punkten und Strichen.} \label{fig:cd} \end{figure} \subsection{QR-Codes} -Quick Response Codes funktionieren nach einem sehr ähnlichen Prinzip wie in unserem Beispiel, nur dass QR-Codes in einem $\mathbb{F}_{256}$ Körper arbeiten. Je nach grösse der Codierung ist der QR-Code im Endeffekt robuster gegen Beschädigungen. Bei Low Level Codes können 7\% der Daten Wiederhergestellt werden, beim High Level Code sind das sogar 30\%. +Quick Response Codes oder auch QR-Codes funktionieren nach einem sehr ähnlichen Prinzip wie in unserem Beispiel der Abschnitte \ref{reedsolomon:section:codebsp} - \ref{reedsolomon:section:rekonstruktion} nur das QR-Codes in einem $\mathbb{F}_{256}$ Körper arbeiten. Die physische Grösse eines Codes ist stark abhängig von der Menge an codierten Daten sowie dem verwendeten Fehlerkorrektur-Level. Es ist so auf dem ersten Blick nicht ersichtlich, wie viel Nutzinformationen ein Qr-Code enthält. Die QR-Codes in Abbildung \ref{fig:qr} zeigen jeweils die Gleiche Information mit unterschiedlichem Fehlerkorrektur-Level. Codes mit einem höheren Korrektur-Level können auch für Designer-Codes Zweckentfremdet werden. Dabei wird z.B. das Firmenlogo oder einen Schriftzug über den Qr-Code gelegt, ohne das die Funktion des Codes beeinträchtigt wird. Ein Beispiel dazu ist unter Abbildung \ref{fig:designqr} zu finden. + +% + +%So kann auf den ersten Blick nicht +% +% +% funktionieren nach einem sehr ähnlichen Prinzip wie in unserem Beispiel, nur dass QR-Codes in einem $\mathbb{F}_{256}$ Körper arbeiten. Je nach grösse der Codierung ist der QR-Code im Endeffekt robuster gegen Beschädigungen. Bei Low Level Codes können 7\% der Daten Wiederhergestellt werden, beim High Level Code sind das sogar 30\%. \begin{figure} \centering @@ -88,6 +139,30 @@ Quick Response Codes funktionieren nach einem sehr ähnlichen Prinzip wie in uns \subfigure[]{ \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/qrcode_l} } - \caption{(a) High Level Code, (b) Low Level Code} +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode_ohnelogo} +% } +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode} +% } + \caption{Anhand der grösse würde man darauf schliessen, dass bei (a) mehr Informationen Codiert sind als bei (b). Tatsächlich aber beinhalten beide Codes die gleiche Information. Das liegt daran, da die Fehlerkorrekturfähigkeit von QR-Codes sich in insgesamt vier Levels aufteilen lassen. Der höchste Fehlerkorrektur-Level, der bei (a) angewendet wurde, ist in der Lage, bis zu 30\% der Daten wiederherzustellen. Der kleinste Level schafft etwa 7\%, der in (b) veranschaulicht wird. Da die Grösse also nichts über die Menge an Daten aussagt, könnte es sich bei (a) auch um einen Code mit viel Nutzdaten und kleinem Fehlerkorrektur-Level handeln. Der Unterschied ist von Auge nicht sichtbar.} \label{fig:qr} \end{figure} + +\begin{figure} + \centering +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/qrcode_h} +% } +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/qrcode_l} +% } + \subfigure[]{ + \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode_ohnelogo} + } + \subfigure[]{ + \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode} + } + \caption{Während (a) noch einen unveränderten QR-Code repräsentiert, handelt es sich bei (b) nun um einen Designer-QR-Code. Beide Codes verfügen über einen mittleren Fehlerkorrektur-Level von theoretisch 15\%. Da bei (b) jetzt einen Teil des Codes durch ein Logo verdeckt wird, schränkt sich die Fehlerkorrekturfähigkeit je nach Grösse des verdeckten Teils mehr oder weniger stark ein. Unser Designer-Code in (b) ist nur noch in der Lage etwa 9\% des Codes zu rekonstruieren.} + \label{fig:designqr} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex index 8430ebd..eb4e82f 100644 --- a/buch/papers/reedsolomon/codebsp.tex +++ b/buch/papers/reedsolomon/codebsp.tex @@ -76,7 +76,7 @@ dar. \subsection{Der Ansatz der diskreten Fouriertransformation \label{reedsolomon:subsection:diskFT}} -In einem vorherigen Abschnitt \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. +Im vorherigen Abschnitt \ref{reedsolomon:section:dtf} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. Wir wählen deshalb eine Zahl $a$, die die gleichen Aufgaben haben soll wie $e^{\frac{j}{2 \pi}}$ in der diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ ist. Dazu soll die Potenz von $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken. Dazu ändern wir die Darstellung von \[ diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex index 025be3a..9647775 100644 --- a/buch/papers/reedsolomon/dtf.tex +++ b/buch/papers/reedsolomon/dtf.tex @@ -1,30 +1,124 @@ % -% teil3.tex -- Beispiel-File für Teil 3 +% dtf.tex -- Idee mit DFT % -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Diskrete Fourien Transformation +\section{Übertragung mit Hilfe der diskrten Fourier-Transformation \label{reedsolomon:section:dtf}} \rhead{Umwandlung mit DTF} -Um die Polynominterpolation zu umgehen, gehen wir nun über in die Fourientransformation. -Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauch -für den Reed-Solomon-Code. Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert. -wobei sie dann bei späteren Berchnungen ganz nütlich ist. +Die Grundidee eines fehlerkorrigierenden Code ist, dass Informationen eines Datenpunktes +durch die Codierung auf viele übertragene Werte verteilt werden. +Die Decodierung ist in der Lage, den ursprünglichen Datenwert zu rekonstruieren, +sogar wenn einzelne wenige übertragene Werte beschädigt worden sind. +\par +Die Fourier-Transformation transformiert einen einzelnen Wert, +eine Dirac-Funktion, auf ein Spektrum, welches sich über die ganze Frequenzachse erstreckt. +Aus der Filtertheorie ist bekannt, dass der ursprüngliche Impuls mehr oder weniger rekonstruierbar ist, + vorausgesetzt, es gehen nicht zu viele Frequenzen bei der Übertragung verloren. +\par +Es liegt daher nahe zu versuchen, die Fourier-Transformation +für Codierung und Decodierung zu verwenden. -\subsection{Übertragungsabfolge -\label{reedsolomon:subsection:Übertragungsabfolge}} -Das Signal.... sind die Daten, Zahlen welche übertragen werden sollen. -Das speziell ist das wir 100 Punkte übertragen und von 64 bis 100, -werden nur Null Punkte übertragen, dies weiss auch unser Empfänger. -Nun wird das Signal in Abbildung... codiert... -Somit wird die Information jedes Punktes auf das ganze spektrum von 0 bis 100 übertragen. -Kommen nuun drei Fehler... hinzu zu diesem codierten Signal sind diese nicht zu erkennen. -Nach dem Empfangen... und decodieren ... erkennt man die fehlerhafte information in den Punkten 64 bis 100. -Filtert man nur diese Punkte heraus und Transformiert sie mit Fourier erhält man die stellen an denen die Fehler sich eingeschlichen haben. +\subsection{Beispiel mit Fehlerkorrektur mit Fourier-Transformation +\label{reedsolomon:subsection:sendbsp}} +Das folgende Beispiel soll zeigen, wie die Idee der Fehlerkorrektur umgesetzt wurde. +Die Fehlererkennung des Reed-Solomon-Codes funktioniert nach einem sehr Ähnlichen Prinzip. -\subsection{Diskrete Fourientransformation Zusamenhang -\label{reedsolomon:subsection:dtfzusamenhang}} -Die Diskrete Fourientransformation ist definiert als -.... +%Das folgende Beispiel soll zeigen, wie Fehlerkorrektur möglich ist. +%Dieses auf eine Art, die der Funktionsweise des Reed-Solomon-Codes, +%der später erklärt wird, analog ist. +\par +Der Auftrag besteht darin, 64 Datenwerte zu übertragen, 32 Fehler erkennen können und bis zu 16 Fehler zu rekonstruieren. +Mit Hilfe der Fourier-Transformation werden die \textcolor{blue}{blauen Datenpunkte} transformiert, +zu den \textcolor{darkgreen}{grünen Übertragungspunkten}. +Durch eine Rücktransformation können die \textcolor{blue}{blauen Datenpunkte} wieder rekonstruiert werden. +\begin{figure}%[!ht] + \centering + \resizebox{\textwidth}{!}{ + \includegraphics[width=\textwidth]{papers/reedsolomon/figures/fourier} + %\input{papers/reedsolomon/tikz/plotfftraw.tex} + } + \caption{Übertragungsabfolge \ref{reedsolomon:subsection:sendbsp}} + \label{fig:sendorder} +\end{figure} +In der Abbildung \ref{fig:sendorder} wird eine Übertragung Schritt für Schritt illustriert. +In der folgenden Aufzählung werden diese einzelne Schritte erklärt und erläutert: +\begin{enumerate}[(1)] + \item Das Signal besteht aus 64 zufälligen, ganzzahligen Datenwerten zwischen 0 und 10. + Für die Rekonstruktion werden zusätzliche Datenwerte benötigt, wir fügen deshalb 32 Werte hinzu. + Diese setzen wir willkürlich alle auf Null und nennen sie Fehlerkorrekturstellen. + Wir erhalten so einen erweiterten Signalvektor der Länge $N =96$. + \item Mit der Fourier-Transformation wird der ganze Signalvektor codiert. + Dadurch wird jede Informationseinheit auf alle Punkte des Spektrums verteilt. + \item Wir dürfen annehmen, dass bei der Übertragung, nur einzelne übertragene + Werte durch Fehler verändert werden. + \par + Im Beispiel sind dies die Werte an den Stellen 6, 20 und 74 (\textcolor{red}{rote Kurve}), + die um einen Betrag verändert werden. + Dieser ist bis zu 150-mal kleiner als die ursprünglich codierten Werte. + Der Empfänger erkennt daher im allgemeinen nicht, ob und wo Übertragungsfehler aufgetreten sind. + \item Ohne Übertragungsfehler kann der Signalvektor durch die inverse Fourier-Transformation vollständig + wiederhergestellt werden. + Dazu gehören auch die Nullen an den Fehlerkorrekturstellen 64 - 96. + \par + Sind Übertragungsfehler aufgetreten, werden an diesen Stellen die Werte von Null abweichen. + Somit haben wir bereits Fehler erkannt. + \item Die Werte an den Fehlerkorrekturstellen 64 - 96, die nicht mehr Null sind, nennen wir das Syndrom. + Im Syndrom steckt nur Information über die Fehler, sie werden durch die inverse Fourier-Transformation erzeugt. + \item Um die Fehler zu rekonstruieren, kann man versuchen, die Information im Syndrom mit Fourier-Transformation zu transformieren. + Da das Syndrom nur ein Teil der Fehlerinformation ist, liefert die Fourier-Transformation eine Approximation der Fehler. + Diese Approximation der Fehler ist genau genug, um die Fehlerstellen zu lokalisieren. +\end{enumerate} +Im Beispiel haben wir mit dem Syndrom nur etwa ein Drittel der Fehlerinformation, es ist daher zu erwarten, +dass die Fehlerwerte auch nur ein Drittel so gross sind. +\par +Damit können die Fehler korrigiert und die Originaldaten wiederhergestellt werden. +Der Rekonstruktionsauftrag ist damit erfolgreich ausgeführt. +\subsection{Fourier-Transformation und Polynome\label{reedsolomon:subsection:ftandpolynom}} +Im Abschnitt \externaldocument{papers/reedsolomon/idee}\ref{reedsolomon:section:polynomansatz} +wurden Werte eines Polynoms zur Codierung verwendet. +Die 7 Übertragungspunkte könnten ein Polynom +\begin{equation} + \textcolor{darkgreen}{p(x)} + = + \textcolor{blue}{a_0} + \textcolor{blue}{a_1}x + \textcolor{blue}{a_2}x^2 + + \textcolor{gray}{a_3}x^3 + \textcolor{gray}{a_4}x^4 + \textcolor{gray}{a_5}x^5 + + \textcolor{gray}{a_6}x^6 +\label{reedsolomon:equationpoly} +\end{equation} +sechsten Grades bestimmen. +Durch die Wahl von $\textcolor{gray}{a_3=0}$, $\textcolor{gray}{a_4=0}$, $\textcolor{gray}{a_5=0}$, $\textcolor{gray}{a_6=0}$ +erzeugen wir die für die Fehlerkorrektur nötige Redundanz, ganz analog zum Schritt (1) im Beispiel. +\par +Die Analogie geht aber noch weiter. + Schreibt man + \( w = + e^{-\frac{2\pi j}{N} k}\) + \label{reedsolomon:DFT_summand}, damit wird aus der Formel + \begin{equation} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + ,\label{reedsolomon:DFT} + \end{equation} + für die diskrete-Fourier-Transformation das Polynom + \begin{equation} + q(w)= + \frac{{f}_0}{N} + \frac{{f}_1}{N} w^1 + \frac{{f}_2}{N} w^2 + \dots + \frac{{f}_{N-1}}{N} w^{N-1}. + \label{reedsolomon:DFT_polynom} + \end{equation} + Im Beispiel werden aber Werte des Polynoms + \begin{equation} + \textcolor{darkgreen}{q(w)}= + \frac{\textcolor{blue}{{f}_0}}{N} + \frac{\textcolor{blue}{{f}_1}}{N} w^1 + \frac{\textcolor{blue}{{f}_2}}{N} w^2 + \dots + + \frac{\textcolor{blue}{{f}_{63}}}{N} w^{63} + \frac{\textcolor{gray}{{f}_{64}}}{N} w^{64} + \textcolor{gray}{\dots} + \frac{\textcolor{gray}{{f}_{N-1}}}{N} w^{N-1} + \label{reedsolomon:DFT_polynom2} + \end{equation} + für verschiedene \( w = e^{-\frac{2\pi j}{N} k}, k=1, \dots ,N-1\) übermittelt. +Das Syndrom entstand durch die Wahl ${f_{64}}=0$ bis ${f}_{N-1}=0$ (graue Koeffizenten). +\par +Die Polynominterpolation und die Fourier-Transformation rechnen beide mit reellen Zahlen. +Wenn die Approximation nicht mehr genügend gut ist um die Fehler zu erkennen und rekonstruieren, +dann brauchen wir andere Varianten. +Um dieser Approximation zu entkommen, verlassen wir die reellen Zahlen und gehen zum endlichen Körpern, oder auch Galois-Körper genannt. +Dieser bietet uns einige Vorteile.
\ No newline at end of file diff --git a/buch/papers/reedsolomon/einleitung.tex b/buch/papers/reedsolomon/einleitung.tex index 3d40db1..f99ad82 100644 --- a/buch/papers/reedsolomon/einleitung.tex +++ b/buch/papers/reedsolomon/einleitung.tex @@ -6,14 +6,12 @@ \section{Einleitung \label{reedsolomon:section:einleitung}} \rhead{Einleitung} -Der Reed-Solomon-Code ist entstaden im ... vom .. um, -das Problem der Daten Übertragung zu lösen. -In deiesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt. -Es wird jedoch nicht auf die technische Umsetzung oder Implementierung eingegangen. -Um beim Daten Übertragen fehler zu erkennen könnte man die Daten jeweils doppelt senden, -und so jeweilige Fehler zu erkennen. -Doch dies braucht schnell unmengen an Daten, wenn man nach vielen Fehler absichern möchte. -Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise. +Der Reed-Solomon-Code wurde von den beiden Mathematiker Irving S. Reed und Gustave Solomon im Jahre 1960 entwickelt. +Dabei haben sie das Problem der Fehlerhaften Datenübertragung gelöst. +In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge und +Funktionsweise des Reed-Solomon-Code erklärt. +Es wird jedoch nicht auf die technische Umsetzung oder Implementierung eingegangen, jedoch wird im Abschnitt \ref{reedsolomon:section:anwendung} einige Anwendungen des Reed-Solomon-Codes vorgestellt. + diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex index 1d196fd..3019dd7 100644 --- a/buch/papers/reedsolomon/endlichekoerper.tex +++ b/buch/papers/reedsolomon/endlichekoerper.tex @@ -3,21 +3,63 @@ % % (c) 2021 Michael Steiner, Hochschule Rapperswil % -\section{Reed-Solomon in Endlichen Körpern +\section{Reed-Solomon in endlichen Körpern \label{reedsolomon:section:endlichekoerper}} \rhead{Reed-Solomon in endlichen Körpern} -\[ -\textcolor{red}{\text{TODO: (warten auf den 1. Teil)}} -\] -Das Rechnen in endlichen Körpern bietet einige Vorteile: +Im vorherigen Abschnitt haben wir gesehen, dass wir die Fehler mittels Approximation suchen und somit nur ungefähre Angaben haben, wo sich Fehler aufhalten. +Um dies zu ändern wechseln wir vom komplexen Zahlenraum in endliche Körper. +In endlichen Körpern gibt es keine Approximationen wie bei den rationalen und reellen Zahlen. +Alle Zahlen sind richtig oder falsch, ``fast richtig'' gibt es nicht. +Zudem beschränken sich die arithmetischen Rechenoperationen auf das Addieren und Multiplizieren. +Wir können also nur ganze Zahlen als Resultat erhalten. +Dies erleichtert auch die Umsetzung auf ein digitales System, da Computer in der Regel lieber mit ganzen als mit gebrochenen oder komplexen Zahlen arbeiten. -\begin{itemize} - \item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten. - - \item Digitale Fehlerkorrektur: lässt sich nur in endlichen Körpern umsetzen. - -\end{itemize} +Um jetzt eine Nachricht in einem endlichen Körpern zu konstruieren gehen, wir im Grunde gleich vor wie im Beispiel aus dem Abschnitt \ref{reedsolomon:subsection:sendbsp}. +Eine Nachricht besteht aus einem Nutzdatenteil und einem Fehlerkorrekturteil. +Diese Nachricht wird codiert, übertragen und beim Empfänger wieder decodiert. +In endlichen Körpern können wir jedoch nicht mehr die Fouriertransformation zur Hilfe nehmen. +Wir müssen also eine Alternative finden, welche die gleichen Eigenschaften wie die Fouriertransformation aufweist, aber im endlichen Körper verwendet werden kann. +Auch beim Decodieren müssen wir uns etwas einfallen lassen, wenn die Vorgehensweise mit dem Lokator auch in endlichen Körpern funktionieren soll. Die folgenden Abschnitte widmen sich deshalb der genaueren Betrachtung eines Reed-Solomon-Codes und wie er in endlichen Körpern funktioniert. -Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen. - -Nun stellt sich die Frage, wie wir eine fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes ``Lokatorpolynom'' generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. +% +%Damit all diese Probleme möglichst verständlich +% +% +%Um all diese Probleme und möglichst +% +% +%um Fehler zu erkennen und mittels Lokatorpolynom +% +% +% ein Lokatorpolynom zu finden. +% +% +% +% Eine Nachricht besteht aus einem Nutzdatenanteil und einem Fehlerkorrekturteil, +% +% +% +%In diesem Zahlenraum gibt es nur Natürliche Zahlen und es darf nur Addiert oder Multipliziert werden. +%Der grosse Vorteil an endlichen Körper ist, dass dich der einfacher Digital umsetzen lässt. +% +% +%Dieser Zahlenraum bringt eine Menge von neuen Regeln mit sich. +%So gibt es dort nur Natürliche Zahlen und die Arithmetischen Rechenoperationen sind beschränkt auf die Addition und Multiplikation. +% +% +% +%\[ +%\textcolor{red}{\text{TODO: (warten auf den 1. Teil)}} +%\] +%Das Rechnen in endlichen Körpern bietet einige Vorteile: +% +%\begin{itemize} +% \item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten. +% +% \item Digitale Fehlerkorrektur: lässt sich nur in endlichen Körpern umsetzen. +% +%\end{itemize} +% +%Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen. +% +%Nun stellt sich die Frage, wie wir eine fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes ``Lokatorpolynom'' generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. diff --git a/buch/papers/reedsolomon/experiments/f.m b/buch/papers/reedsolomon/experiments/f.m index 6bdc741..bf2587c 100644 --- a/buch/papers/reedsolomon/experiments/f.m +++ b/buch/papers/reedsolomon/experiments/f.m @@ -1,8 +1,8 @@ -# -# f.m -- Reed-Solomon-Visualisierung mit FFT -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# +% +% f.m -- Reed-Solomon-Visualisierung mit FFT +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + N = 64; b = 32; l = N + b; @@ -51,6 +51,7 @@ syndrom(1:N,1) = zeros(N,1) plot(abs(syndrom)); xlim([1, l]); title("Syndrom"); + pause() locator = abs(fft(syndrom)) @@ -59,3 +60,13 @@ plot(locator); xlim([1, l]); title("Locator"); pause() + + +writematrix([transpose(counter), abs(signal)], 'signal.txt') +writematrix([transpose(counter), abs(codiert)], 'codiert.txt') +writematrix([transpose(counter), fehler], 'fehler.txt') +writematrix([transpose(counter), abs(empfangen)], 'empfangen.txt') +writematrix([transpose(counter), abs(decodiert)], 'decodiert.txt') +writematrix([transpose(counter), abs(syndrom)], 'syndrom.txt') +writematrix([transpose(counter), locator], 'locator.txt') + diff --git a/buch/papers/reedsolomon/experiments/plot.tex b/buch/papers/reedsolomon/experiments/plot.tex new file mode 100644 index 0000000..4b156bb --- /dev/null +++ b/buch/papers/reedsolomon/experiments/plot.tex @@ -0,0 +1,103 @@ +% polynome1 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usepackage{pgfplotstable} +\usepackage{filecontents} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\x}{10} +\newcommand{\y}{-8} +\begin{document} + +\begin{tikzpicture}[] + + %--------------------------------------------------------------- + %Knote + \matrix[draw = none, column sep=20mm, row sep=4mm]{ + \node(signal) [] { + \begin{tikzpicture} + \begin{axis}[ + title = {\Large {Signal}}, + xlabel={Anzahl Übertragene Zahlen}, + xtick={0,20,40,64,80,98},] + \addplot[blue] table[col sep=comma] {signal.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(codiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Codiert}}] + \addplot[] table[col sep=comma] {codiert.txt}; + \end{axis} + \end{tikzpicture}}; \\ + + &\node(fehler) [] { + \begin{tikzpicture} + \begin{axis}[scale=0.6, title = {\Large {Fehler}}] + \addplot[red] table[col sep=comma] {fehler.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(decodiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Decodiert}}] + \addplot[blue] table[col sep=comma] {decodiert.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(empfangen) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Empfangen}}] + \addplot[] table[col sep=comma] {empfangen.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(syndrom) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Syndrom}}] + \addplot[blue] table[col sep=comma] {syndrom.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(locator) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Locator}}] + \addplot[] table[col sep=comma] {locator.txt}; + \end{axis} + \end{tikzpicture}};\\ + }; + %------------------------------------------------------------- + %FFT & IFFT deskription + + \draw[thin,gray,dashed] (0,12) to (0,-12); + \node(IFFT) [scale=0.7] at (0,12.3) {IFFT}; + \draw[<-](IFFT.south west)--(IFFT.south east); + \node(FFT) [scale=0.7, above of=IFFT] {FFT}; + \draw[->](FFT.north west)--(FFT.north east); + + \draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); + %Arrows + \draw[ultra thick, ->] (signal.east) to (codiert.west); + \draw[ultra thick, ->] (codiert.south) to (fehler.north); + \draw[ultra thick, ->] (fehler.south) to (empfangen.north); + \draw[ultra thick, ->] (empfangen.west) to (decodiert.east); + \draw[ultra thick, ->] (syndrom.east) to (locator.west); + \draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; + \draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + + %item + \node[circle, draw, fill =lightgray] at (signal.north west) {1}; + \node[circle, draw, fill =lightgray] at (codiert.north west) {2}; + \node[circle, draw, fill =lightgray] at (fehler.north west) {3}; + \node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; + \node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; + \node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; + \node[circle, draw, fill =lightgray] at (locator.north west) {7}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/reedsolomon/figures/fourier.pdf b/buch/papers/reedsolomon/figures/fourier.pdf Binary files differnew file mode 100644 index 0000000..4995141 --- /dev/null +++ b/buch/papers/reedsolomon/figures/fourier.pdf diff --git a/buch/papers/reedsolomon/figures/plotfft.pdf b/buch/papers/reedsolomon/figures/plotfft.pdf Binary files differnew file mode 100644 index 0000000..80adafb --- /dev/null +++ b/buch/papers/reedsolomon/figures/plotfft.pdf diff --git a/buch/papers/reedsolomon/figures/polynom2.pdf b/buch/papers/reedsolomon/figures/polynom2.pdf Binary files differnew file mode 100644 index 0000000..55a50ac --- /dev/null +++ b/buch/papers/reedsolomon/figures/polynom2.pdf diff --git a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex index 4a7716a..daa2913 100644 --- a/buch/papers/reedsolomon/idee.tex +++ b/buch/papers/reedsolomon/idee.tex @@ -1,58 +1,157 @@ % -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% idee.tex -- Polynom Idee % \section{Idee \label{reedsolomon:section:idee}} \rhead{Problemstellung} -Das Problem liegt darin Informationen, Zahlen, -zu Übertragen und Fehler zu erkennen. -Beim Reed-Solomon-Code kann man nicht nur Fehler erkenen, -man kann sogar einige Fehler korrigieren. +Um Fehler in einer Datenübertragung zu erkennen, könnte man die Daten jeweils doppelt senden, +also den gleiche Wert immer zweimal versenden. +Tritt ein Fehler ein wird sich dies in der Differenz der beiden Werten bemerkbar machen. +Aber wie erkennen wir, welcher nun der richtige ist? Die Lösung ist simpel: Wir übertragen den Wert einfach dreimal. +Wenn jetzt ein Fehler auftritt, kann durch die beiden unveränderten Werten den richtigen bestimmt werden. +Doch was machen wir, wenn bei dieser Übertragung zwei Fehler auftreten? +Oder noch schlimmer: Was wenn zweimal derselbe Fehler auftritt? Die beiden Fehlerhaften Werte überstimmen bei der Evaluierung den gesendeten Datenwert, der dann unwiderruflich verloren geht. +Wir könnten dies noch steigern mit vier, fünf oder mehr gleichen Übertragenen Werte. Dies erhöht zwar die Robustheit der gesendeten Daten, führt aber auch dazu, dass wir durch die Mehrfachübertragung nur sehr wenige Nutzdaten versenden können. +Gerade in unserer heutigen Zeit wäre dies ein enorm grosses Problem und aus diesem Grund wurden alternative Ansätze ausgearbeitet um dieses grundlegende Problem zu lösen. +% +% +%Gerade in der heutigen modernen Zeit bei dem hohen bedarf an Daten würden unsere Kommunikationssysteme bei weitem nicht ausreichen um den einen einzigen Datenwert mehrfach zu übertragen +% +% Gerade in der Heutigen modernen Zeit bei diesem enormen mass an daten die wir alle tagtäglich anfordern Währe dies wohl unmöglich, wenn wir die daten auf diese Weise +% +% +% +% +% +%Wenn es uns gelingt, Fehler nach Ihrer Übertragung zu erkennen, dann könnten wir in einem neuen Ansatz den fehlerhaft empfangenen Wert noch einmal anfordern. +%Wir stellen fest, dass für viele alltägliche Anwendungen völlig ausreichend ist. +% +%Was ist, wenn wir aber eine Datenquelle haben, von der wir nur einmalig lesen können? +% +% +% +%Beim Übertragen von drei Werten können wir maximal 2 Fehler erkennen aber nicht mehr korrigieren. +%Wenn wir noch mehr Werte +% +%Wir Übertragen Ziemlich viele Werte für so wenige Nutzdaten. Hinzu kommt, dass wir bei dieser Vorgehensweise gerade mal bestimmen können, dass überhaupt Fehler aufgetreten sind +% +% +%Wir haben also drei Werte die bestimmt einen Fehler korrigieren können, was ziemlich viele Werte um einen Fehler zu korrigieren. +% +% um so jeweils einzelne Fehler zu erkennen. +%Wenn jedoch mehr als nur ein Fehler erkannt werden und sogar noch das Original rekonstruiert werden soll, dann sollen die Daten drei oder vierfach versendet werden. +%Doch nur schon um einen Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach versendet. +%Das Hauptproblem ist, dass Informationen Fehlerfrei Übertragen werden sollen. Um dies zu erreichen muss gleich nach dem Empfangen Fehler erkannt und korrigiert werden. +% +%Das Problem liegt darin, Informationen oder Zahlen beim Übertragen gleichzeitig noch +% +%Das Problem liegt darin, das Informationen oder Zahlen zu Übertragen und gleichzeitig Fehler zu erkennen +% +% +%Das Problem liegt darin Informationen, Zahlen, zu Übertragen und Fehler zu erkennen und zu korrigieren. +%Der Unterschied des Fehler Erkennens und Korrigirens, ist das beim Erkennen nur die Frage beantwortet wird: Ist die Übertragung fehlerhaft oder nicht? +%Beim Korrigieren werden Fehler erkannt und dann zusätzlich noch die Originalwerte rekonstruiert. +%Eine weitere Möglichkeit wäre, dass der Empfänger nach einer fehlerhaften Übertragung die selben Daten nochmals anfordert. +%Dies führt wieder zu unerwünschten mehrfachen Übertragung. +%In Anwendungen des Reed-Solomon-Codes Abschnitt \externaldocument{papers/reedsolomon/anwendungen} \ref{reedsolomon:section:anwendung} +% ist diese vom Empfänger gesteuerte erneute Übertragen meistens nicht sinnvoll oder sogar unmöglich. +%Der Reed-Solomon-Code macht dies Übertragung auf eine andere, clevere Weise. +\subsection{Polynom-Ansatz +\label{reedsolomon:section:polynomansatz}} \rhead{Polynom-Ansatz} -Eine Idee ist die Daten, -ein Polynom zu bilden und dieses dann mit bestimmten Punkten überträgt. -Nehmen wir als beisbiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5}, -welche uns dann das Polynom +Eine zentrale Idee des Reed-Solomon-Code ist, aus den Daten ein Polynom zu bilden. +Mit dieser Polynomfunktion wird dann eine Anzahl von Werten übertragen. +\begin{beispiel} Nehmen wir die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1} und \textcolor{blue}{5}, welche übertragen werden sollen. Daraus bilden wir das Polynom \begin{equation} p(x) = -2x^2 + 1x + 5 +\textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5}. \label{reedsolomon:equation1} \end{equation} -ergeben. -Übertragen werden nun die stellen 1, 2, 3\dots 7 dieses Polynomes. -Grafisch sieht man dies dann im Abbild //TODO -Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger erkennen, welches das Richtige Polynom ist. -Der Leser/Empfänger weiss, mit welchem Grad das Polynom entwickelt wurde. -\subsection{Beispiel} -Für das Beispeil aus der Gleichung \ref{reedsolomon:equation1}, -ist ein Polynome zweiten Grades durch drei Punkte eindeutig bestimmbar. -Hat es Fehler in der Übertragunge gegeben, kann man diese erkennen, -da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen. -Ab wie vielen Fehler ist das Polynom nicht mehr erkennbar beim Übertragen von 7 Punkten? -Bei 2 Fehlern kann man noch eindeutig bestimmen, dass das Polynom mit 4 Punkten, -gegenüber dem mit 5 Punkten falsch liegt. -Werden es mehr Fehler kann nur erkennt werden das das Polynom nicht stimmt. -Das Orginale Polynom kann aber nicht mehr gefunden werden. -Dabei sollten mehr Übertragungspunkte gegeben werden. - -\section{Fehlerbestimmung -\label{reedsolomon:section:Fehlerbestimmmung}} -So wird ein Muster indentifiziert, welches genau vorherbestimmen kann, -wie gross das Polynom sein muss und wie viele Übertragungspunkte gegeben werden müssen. -Durch ein klein wenig Überlegung ist klar das die anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast), -die dan Entschlüsselt werden sollen den Grad des Polynoms minus 1 ergeben. -Für die Anzahl an Übertragungspunkte, muss bestimmt werden wieviel Fehler erkennt und korrigiert werden sollen. -Mit Hilfe der Tabelle.... sieht man das es bei $$t$$ Fehlern und $$k$$ Nutzlast, -für das Übertragen $$k+2t$$ Punkte gegben werden müssen. - -Ein toller Nebeneffekt ist das dadurch auch $$2t$$ Fehler erkannt werden. -um zurück auf unser Beispiel zu kommen, -können von den 7 Übertragungspunkten bis zu $$2t = 2*2 = 4 $$ Punkten falsch liegen -und es wird kein eindeutiges Polynom 2ten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert. - -Ein Polynom durch Punkt mit Polynom Interpolation zu rekonstruieren ist schwierig und Fehleranfällig. +\par +Ein Polynom zweiten Grades ist durch drei Punkte eindeutig bestimmbar. +Bei einer fehlerlosen Übertragung können wir mit 3 übertragenen Werten + das Polynom durch Polynominterpolation volständig rekonstruieren. +Wir brauchen Polynominterpolation als Methode, um aus den Punkten wieder ein Polynom zu bilden. +Die Koeffizente des rekonstruierten Polynoms sind dann unsere gesendeten Zahlen \textcolor{blue}{2}, \textcolor{blue}{1} und \textcolor{blue}{5}. +\par +Wie können wir nun Fehler erkennen oder sogar korrigieren? +Versuchen wir doch, mehr Werte zu übertragen, wie zum Beispiel 7 Werte. +Übertragen werden nun die \textcolor{darkgreen}{grünen Werte} + des \textcolor{blue}{blauen Polynomes} an den Stellen 1, 2, 3, \dots , 7. +In Abbildung \ref{fig:polynom} ist das zu den \textcolor{blue}{Datenpunkten} gehörige Polynom blau dargestellt, +die \textcolor{darkgreen}{übertragenen Werte} des Polynoms sind grün, wobei diese Punkte aufgrund von Übertragungsfehler jetzt eine Parabel darstellen. +Die Fehlerhaften Punkte lassen sich sehr einfach bestimmen, weil diese nicht auf der ursprünglichen Funktion liegen. +Somit können die roten Punkte auf der Parabel durch die grauen ersetzt werden und sind damit korrigiert. + +Bisher konnten wir von 7 Zahlen zwei Fehler erkennen und korrigieren. Können wir in diesem Beispiel noch mehr Fehler korrigieren? +Wir erhöhen dazu die Fehleranzahl Schritt für Schritt: +\begin{itemize} + \item[\textit{1 Fehler}:] Bei einem Fehler können konkurrenzierende, aber falsche Polynome zusammen mit zwei originalen Punkten entstehen. + Dabei können aber maximal 3 Punkte auf diesem Konkurrenzpolynom sein. + Da 6 > 3 ist haben wir unser originales Polynom gefunden. + \item[\textit{2 Fehler}:] Bei Zwei Fehlern kann ein Fehler mit zwei originalen Punkten ein konkurrenzierendes, aber falsches Polynom bilden. + Da der zweite \textcolor{red}{Fehler} frei wählbar ist, kann dieser auch auf dem \textcolor{gray}{Konkurrenzpolynom} liegen, wie in der Abbilbung \ref{fig:polynom} zu sehen ist. + Nun haben wir, ein \textcolor{blue}{originales Polynom} mit \textcolor{darkgreen}{5} übereinstimmenden und ein konkurrenzierendes mit 4 Punkten. + Da 5 noch grösser als 4 ist, können wir sagen, welches das Originalpolynom ist. + \item[\textit{3 Fehler}:] Bei Drei kann genau wie bei 1 oder 2 Fehler, ein konkurenzierendes Polynom mit einem Fehler und zwei originalen Punkten bestimmt werden. + Auch hier sind die anderen Fehler frei wählbar und liegen auf dem Konkurrenzpolynom. + Nun ist es so das 5 Punkte auf diesem konkurenzierenden Polynom und 4 Punkte auf dem originalen. + Das Originalpolynom kann nicht mehr gefunden werden. + \item[\textit{4 Fehler}:] Bei Vier kann noch erkannt werden, dass Fehler aufgetreten sind, da 3 originale Punkte das ursprüngliche Polynom ergeben. + Somit haben wir mindestens 2 verschieden Polynome, was bedeutet, dass Fehler entstanden sind. + \item[\textit{5 Fehler:}] Bei Fünf kann mit den 2 originalen Punkte das Originale Polynom nicht mehr erkannt werden und + somit kann auch keine Aussage mehr gemacht werden, ob Fehler aufgetreten sind oder nicht. +\end{itemize} + +\begin{figure}%[!ht] + \centering + %\includegraphics[width=\textwidth]{papers/reedsolomon/figures/polynom2} + \input{papers/reedsolomon/tikz/polynomraw.tex} + \caption{Polynom $p(x)$ von der Gleichung\eqref{reedsolomon:equation1}} + \label{fig:polynom} +\end{figure} +\qedhere +\end{beispiel} + +\section{Anzahl Übertragungswerte bestimmen +\label{reedsolomon:section:Fehlerkorrekturstellen}} +Um zu bestimmen, wie viele zusätzliche \textcolor{darkgreen}{Übertragungspunkte} notwendig sind um die Fehler zu korrigieren, + muss man zuerst wissen, wie viele \textcolor{blue}{Datenwerte} gesendet und wie viele \textcolor{red}{Fehler} erkannt werden sollen. +Die Anzahl Datenwerte ergeben die Anzahl Polynomkoeffizenten \textcolor{blue}{$k$} und somit den Grad $k-1$ des Polynoms. +Die Bestimmung der Anzahl der Fehler \textcolor{red}{$t$}, welche korrigiert werden können, braucht Redundanz. +Bilden wir verschieden grosse Polynome und untersuchen diese mit unterschiedlich vielen Fehlern erkennt man allmählich ein Muster. + +\begin{table}%[!ht] + \centering + \begin{tabular}{ c c | c} + \hline + Nutzlas & Fehler & Übertragen \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3 & 3 & 9 Werte eines Polynoms vom Grad 2 \\ + \hline + $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ + \hline + \end{tabular} + \caption{ Fehlerkorrekturstellen Bestimmung.} + \label{tab:fehlerkorrekturstellen} +\end{table} +\par +Es müssen mehr Punkte auf dem \textcolor{blue}{originalen Polynom} liegen, als auf dem konkurenzierenden. +Somit braucht man für die Übertragung pro \textcolor{red}{Fehler} zwei Übertragungspunkte mehr. +Wie in der Tabelle \ref{tab:fehlerkorrekturstellen} ersichtlich ist ergeben sich diese Anzahl an \textcolor{darkgreen}{Punkte} für die Übertragung. +\begin{equation} + \textcolor{darkgreen}{u}= + \textcolor{blue}{k}+2\textcolor{red}{t}. + \label{reedsolomon:equation2} +\end{equation} + +Ein Nebeneffekt ist, dass auch $2t$ Fehler erkannt werden können, die aber nicht korrigiert werden können. +Um die Polynomkoeffizenten nach der Übertragung zu rekonstruieren, haben wir jedes mal die Polynominterpolationsmethode angewendet. +Diese Polynominterpolation ist leider schwierig zu berechnen und sehr fehleranfällig. +Es wäre daher einfacher, wenn wir eine alternative Vorgehensweise finden könnten. + diff --git a/buch/papers/reedsolomon/images/Compact_Disc_zoomed_in.png b/buch/papers/reedsolomon/images/Compact_Disc_zoomed_in.png Binary files differnew file mode 100644 index 0000000..69556d0 --- /dev/null +++ b/buch/papers/reedsolomon/images/Compact_Disc_zoomed_in.png diff --git a/buch/papers/reedsolomon/images/designer_qrcode.png b/buch/papers/reedsolomon/images/designer_qrcode.png Binary files differnew file mode 100644 index 0000000..a9e0505 --- /dev/null +++ b/buch/papers/reedsolomon/images/designer_qrcode.png diff --git a/buch/papers/reedsolomon/images/designer_qrcode_ohnelogo.png b/buch/papers/reedsolomon/images/designer_qrcode_ohnelogo.png Binary files differnew file mode 100644 index 0000000..fe4251d --- /dev/null +++ b/buch/papers/reedsolomon/images/designer_qrcode_ohnelogo.png diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index 6bd04f2..017fe94 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -8,29 +8,9 @@ \begin{refsection} \chapterauthor{Joshua Bär und Michael Steiner} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - % Joshua \input{papers/reedsolomon/einleitung.tex} \input{papers/reedsolomon/idee.tex} -\input{papers/reedsolomon/teil2.tex} \input{papers/reedsolomon/dtf.tex} % Michael @@ -45,6 +25,13 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren \nocite{reedsolomon:weitz} \nocite{reedsolomon:informationkommunikation} +\nocite{reedsolomon:voyager_programm} +\nocite{reedsolomon:voyager} +\nocite{reedsolomon:cd_wiki} +\nocite{reedsolomon:cd} +\nocite{reedsolomon:strichepunkte} +\nocite{reedsolomon:qr_wiki} +\nocite{reedsolomon:qr} %\nocite{reedsolomon:mendezmueller} \printbibliography[heading=subbibliography] diff --git a/buch/papers/reedsolomon/packages.tex b/buch/papers/reedsolomon/packages.tex index 3643731..40c6ea3 100644 --- a/buch/papers/reedsolomon/packages.tex +++ b/buch/papers/reedsolomon/packages.tex @@ -8,3 +8,7 @@ % following example %\usepackage{packagename} +\usepackage{pgfplots} +\usepackage{filecontents} +\usepackage{xr} + diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib index 731bd35..b84b5a4 100644 --- a/buch/papers/reedsolomon/references.bib +++ b/buch/papers/reedsolomon/references.bib @@ -23,3 +23,65 @@ volume = {1} } +@online{reedsolomon:voyager_programm, + title = {Information über das Voyager Programm}, + url = {https://de.wikipedia.org/wiki/Voyager-Programm}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:voyager, + title = {Bild der Voyager Raumsonde}, + url = {https://en.wikipedia.org/wiki/Voyager_1}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:cd_wiki, + title = {Alles über die CD}, + url = {https://de.wikipedia.org/wiki/Compact_Disc}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:cd, + title = {Abbildung einer CD}, + url = {https://www.stickpng.com/img/electronics/compact-discs/stack-compact-disc}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:strichepunkte, + title = {Abbildung der Striche und Punkte einer CD}, + url = {https://www.researchgate.net/figure/The-readable-area-of-a-CD-is-magnified-in-order- to-see-the-pit-and-land-sizing-The_fig7_303401629}, + date = {2021-07-26}, + year = {2021}, + month = {7}, + day = {26} +} + +@online{reedsolomon:qr_wiki, + title = {Funktionsweise des QR-Codes}, + url = {https://de.wikipedia.org/wiki/QR-Code}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:qr, + title = {Tool zum erstellen von QR-Codes}, + url = {https://www.qrcode-generator.ch}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/standalone.tex b/buch/papers/reedsolomon/standalone.tex new file mode 100644 index 0000000..c850d1f --- /dev/null +++ b/buch/papers/reedsolomon/standalone.tex @@ -0,0 +1,30 @@ +\documentclass{book} + +\input{common/packages.tex} + +% additional packages used by the individual papers, add a line for +% each paper +\input{papers/common/addpackages.tex} + +% workaround for biblatex bug +\makeatletter +\def\blx@maxline{77} +\makeatother +\addbibresource{chapters/references.bib} + +% Bibresources for each article +\input{papers/common/addbibresources.tex} + +% make sure the last index starts on an odd page +\AtEndDocument{\clearpage\ifodd\value{page}\else\null\clearpage\fi} +\makeindex + +%\pgfplotsset{compat=1.12} +\setlength{\headheight}{15pt} % fix headheight warning +\DeclareGraphicsRule{*}{mps}{*}{} + +\begin{document} + \input{common/macros.tex} + \def\chapterauthor#1{{\large #1}\bigskip\bigskip} + \input{papers/reedsolomon/main.tex} +\end{document} diff --git a/buch/papers/reedsolomon/standalone/standalone.pdf b/buch/papers/reedsolomon/standalone/standalone.pdf Binary files differnew file mode 100644 index 0000000..dfa9eea --- /dev/null +++ b/buch/papers/reedsolomon/standalone/standalone.pdf diff --git a/buch/papers/reedsolomon/teil2.tex b/buch/papers/reedsolomon/teil2.tex deleted file mode 100644 index b2adc9f..0000000 --- a/buch/papers/reedsolomon/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{reedsolomon: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{reedsolomon: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/reedsolomon/tikz/Makefile b/buch/papers/reedsolomon/tikz/Makefile new file mode 100644 index 0000000..1753f37 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/Makefile @@ -0,0 +1,7 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +fourier.pdf: fourier.tex + pdflatex fourier.tex diff --git a/buch/papers/reedsolomon/tikz/codiert.txt b/buch/papers/reedsolomon/tikz/codiert.txt new file mode 100644 index 0000000..4a481d8 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/codiert.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,24.0186013379153 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,4.84567134895776 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 +61,20.0667752354841 +62,12.6951848921471 +63,17.6292439561917 +64,21.377558326432 +65,9.54321919833388 +66,28.3035029562577 +67,38.467599433197 +68,11.3577742151117 +69,17.9815599423852 +70,23.2496572716993 +71,21.7594021268945 +72,15.0332963783729 +73,35.9073375743642 +74,43.6179140616243 +75,10.4219666223433 +76,4.84567134895776 +77,6.89948680823017 +78,16.5769107917917 +79,23.217268502288 +80,4.58257569495584 +81,10.1736266929292 +82,13.5324640263625 +83,44.9845039238714 +84,32.1114345314062 +85,24.5237955316204 +86,15.6627133315015 +87,12.3214904922455 +88,18.2010889773887 +89,37.3066577242559 +90,24.0186013379153 +91,69.1793197789512 +92,23.8408857476069 +93,12.1189172092243 +94,41.9840308053375 +95,131.570790435043 diff --git a/buch/papers/reedsolomon/tikz/decodiert.txt b/buch/papers/reedsolomon/tikz/decodiert.txt new file mode 100644 index 0000000..f6221e6 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/decodiert.txt @@ -0,0 +1,96 @@ +0,6.05208333333333 +1,6.02602539785853 +2,0.0261327016093151 +3,5.98927158561317 +4,4.019445724874 +5,0.0247005083663722 +6,4.97798278395618 +7,1.95246440445439 +8,0.974000110512201 +9,2.00528527696027 +10,1.00071804528155 +11,1.97630907888264 +12,0.0232923747656228 +13,6.01302820392331 +14,3.03567381915226 +15,5.02435590137329 +16,7.00526061008995 +17,5.00739608089369 +18,5.02211514480064 +19,4.02175864806658 +20,1.00236543833726 +21,4.98147315261261 +22,8.97728828610336 +23,8.98481304394618 +24,2.98958333333333 +25,1.98491220960989 +26,5.97728835934715 +27,5.98144124907561 +28,4.00163839998525 +29,2.02176249296313 +30,9.02210713874162 +31,1.00742763919872 +32,1.00557258081044 +33,1.02435888848794 +34,2.03577412756745 +35,6.01302820392331 +36,5.97917574041123 +37,0.976310374034338 +38,9.00062625447998 +39,7.00515849238528 +40,6.97396416790894 +41,0.95256880864368 +42,8.97794719866783 +43,9.01850701506487 +44,10.0194409579917 +45,8.98926601525997 +46,7.9866590265379 +47,5.02603060999077 +48,2.05208333333333 +49,4.02603841132848 +50,0.986882897867895 +51,0.0177592928994285 +52,9.01944131204563 +53,3.0185365665612 +54,2.97803642439316 +55,2.95243072164649 +56,4.97396651395488 +57,6.00516695947321 +58,0.0143895905726619 +59,7.97630812771393 +60,5.97917574041123 +61,9.01298821331865 +62,3.03567381915226 +63,4.02435609145793 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/tikz/empfangen.txt b/buch/papers/reedsolomon/tikz/empfangen.txt new file mode 100644 index 0000000..38c13b0 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/empfangen.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,23.6290258699579 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,5.55320238736303 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 +61,20.0667752354841 +62,12.6951848921471 +63,17.6292439561917 +64,21.377558326432 +65,9.54321919833388 +66,28.3035029562577 +67,38.467599433197 +68,11.3577742151117 +69,17.9815599423852 +70,23.2496572716993 +71,21.7594021268945 +72,15.0332963783729 +73,35.9073375743642 +74,44.6135417384784 +75,10.4219666223433 +76,4.84567134895776 +77,6.89948680823017 +78,16.5769107917917 +79,23.217268502288 +80,4.58257569495584 +81,10.1736266929292 +82,13.5324640263625 +83,44.9845039238714 +84,32.1114345314062 +85,24.5237955316204 +86,15.6627133315015 +87,12.3214904922455 +88,18.2010889773887 +89,37.3066577242559 +90,24.0186013379153 +91,69.1793197789512 +92,23.8408857476069 +93,12.1189172092243 +94,41.9840308053375 +95,131.570790435043 diff --git a/buch/papers/reedsolomon/tikz/fehler.txt b/buch/papers/reedsolomon/tikz/fehler.txt new file mode 100644 index 0000000..23f1a83 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/fehler.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,2 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,2 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,1 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/tikz/fourier.pdf b/buch/papers/reedsolomon/tikz/fourier.pdf Binary files differnew file mode 100644 index 0000000..7e0198b --- /dev/null +++ b/buch/papers/reedsolomon/tikz/fourier.pdf diff --git a/buch/papers/reedsolomon/tikz/fourier.tex b/buch/papers/reedsolomon/tikz/fourier.tex new file mode 100644 index 0000000..7b4ccea --- /dev/null +++ b/buch/papers/reedsolomon/tikz/fourier.tex @@ -0,0 +1,139 @@ +% +% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{pgfplots} +\usepackage{pgfplotstable} +\usepackage{csvsimple} +\usepackage{filecontents} + +\def\plotwidth{7.5cm} +\def\plotheight{5.5cm} +\def\xverschiebung{4.5cm} +\def\yverschiebung{-7cm} +\def\yyverschiebung{-14cm} + +\def\marke#1{ + \coordinate (M) at (-0.8,4.6); + \fill[color=lightgray] (M) circle[radius=0.3]; + \draw (M) circle[radius=0.3]; + \node at (M) {#1}; +} + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\begin{document} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=blue!10] (-5.7,-14.5) rectangle (2.6,5.0); +\fill[color=darkgreen!10] (2.6,-14.5) rectangle (11.1,5.0); + +\draw[dashed,line width=2pt,color=lightgray] (2.6,4.9) -- (2.6,-14.4); +\coordinate (B) at (2.6,-1.3); +\node[color=gray] at (B) [rotate=90,above] {Zeitbereich\strut}; +\node[color=gray] at (B) [rotate=90,below] {Frequenzbereich\strut}; + +\begin{scope}[xshift=-\xverschiebung,yshift=0cm] + \begin{axis} + [title = {\large Signal\strut}, + xtick={0,32,64,96}, + axis background/.style={fill=white}, + width=\plotwidth,height=\plotheight] + \addplot[blue,line width=1pt] table[col sep=comma] + {tikz/signal.txt}; + \end{axis} + \marke{1} +\end{scope} + +\begin{scope}[xshift=\xverschiebung,yshift=0cm] + \begin{axis}[title = {\large Codiert\strut}, + xtick={0,32,64,96}, + axis background/.style={fill=white}, + width=\plotwidth,height=\plotheight] + \addplot[color=black!60!green,line width=1pt] + table[col sep=comma] + {tikz/codiert.txt}; + \end{axis} + \marke{2} + \draw[->,line width=1pt] (3,-0.4) -- node[right] {Übertragung} (3,-2.2); +\end{scope} + +\definecolor{pink}{rgb}{0.6,0.2,1} + +\begin{scope}[xshift=-\xverschiebung,yshift=\yverschiebung] + %\fill[color=pink!20] (4.65,0.35) ellipse (1.1cm and 0.5cm); + \begin{axis}[title = {\large Decodiert\strut}, + xtick={0,32,64,96}, + axis background/.style={fill=white}, + width=\plotwidth,height=\plotheight] + \addplot[blue,line width=1pt] + table[col sep=comma] {tikz/decodiert.txt}; + \end{axis} + \marke{4} + \draw[color=pink] (4.65,0.35) ellipse (1.1cm and 0.5cm); + \draw[->,color=pink,line width=1pt] + (4.65,-0.15) to[out=-90,in=90] (3,-2.2); +\end{scope} + +\begin{scope}[xshift=\xverschiebung,yshift=\yverschiebung] + \begin{axis}[title = {\large Empfangen {\color{red} mit Fehlern}\strut}, + xtick={0,96}, + axis background/.style={fill=white}, + axis y line*=left, + width=\plotwidth,height=\plotheight] + \addplot[color=black!60!green,line width=1pt] + table[col sep=comma] + {tikz/empfangen.txt}; + \end{axis} + \begin{axis}[xtick={6,20,74}, axis y line*=right, + width=\plotwidth,height=\plotheight] + \addplot[red,line width=1pt] + table[col sep=comma] {tikz/fehler.txt}; + \end{axis} + \marke{3} +\end{scope} + +\begin{scope}[xshift=-\xverschiebung,yshift=\yyverschiebung] + \begin{axis}[title = {\large \color{pink}Syndrom\strut}, + xtick={0,32,64,96}, + axis background/.style={fill=white}, + width=\plotwidth,height=\plotheight] + \addplot[pink,line width=1pt] + table[col sep=comma] {tikz/syndrom.txt}; + \end{axis} + \marke{5} +\end{scope} + +\begin{scope}[xshift=\xverschiebung,yshift=\yyverschiebung] + % Beschriftung Rechts + \begin{axis}[axis x line= none, axis y line*=right, ytick={0.3}, + xtick={0,32,64,96}, + axis background/.style={fill=white}, + width=\plotwidth,height=\plotheight] + \addplot[color=black!60,line width=1pt] {0.3}; + \end{axis} + \begin{axis}[title = {\large Lokator\strut},axis y line*=left, + xtick={0,6,20,74,96}, + width=\plotwidth,height=\plotheight] + \addplot[gray,line width=1pt] + table[col sep=comma] {tikz/locator.txt}; + \end{axis} + \marke{6} +\end{scope} + +% Fourier-Transformations-Pfeile + +\draw[->,line width=1pt] (1.8,2) -- node[above] {DFT\strut} (3.8,2); + +\begin{scope}[yshift=\yverschiebung] +\draw[<-,line width=1pt] (1.8,2) -- node[above] {DFT$\mathstrut^{-1}$} (3.8,2); +\end{scope} + +\begin{scope}[yshift=\yyverschiebung] +\draw[->,line width=1pt] (1.8,2) -- node[above] {DFT\strut} (3.8,2); +\end{scope} + +\end{tikzpicture} +\end{document} diff --git a/buch/papers/reedsolomon/tikz/locator.txt b/buch/papers/reedsolomon/tikz/locator.txt new file mode 100644 index 0000000..b28988c --- /dev/null +++ b/buch/papers/reedsolomon/tikz/locator.txt @@ -0,0 +1,96 @@ +0,0.0301224340567056 +1,0.141653026854885 +2,0.138226631799377 +3,0.0339903276086929 +4,0.310585462557496 +5,0.551427312631385 +6,0.628514858396814 +7,0.51102386251559 +8,0.275861355940449 +9,0.0502396354182268 +10,0.090185502547573 +11,0.110759344849756 +12,0.0684618905063001 +13,0.0362855426992259 +14,0.0697096919781468 +15,0.109288539370248 +16,0.0923187999496653 +17,0.0512198536768088 +18,0.274192386987782 +19,0.51349614953654 +20,0.633154426602466 +21,0.553283743533942 +22,0.307840573214514 +23,0.0341664350328392 +24,0.140270857957 +25,0.138527177682831 +26,0.029637547736156 +27,0.0816962563186052 +28,0.0944383203811073 +29,0.0263932110686261 +30,0.0585881348402056 +31,0.0737117341599984 +32,0.0239973937701886 +33,0.0464215468420038 +34,0.0616218854220964 +35,0.0221963086695009 +36,0.0390764778127646 +37,0.0537637218396934 +38,0.0208333333333332 +39,0.0343107696069045 +40,0.0483441215964552 +41,0.0198077862118806 +42,0.0311207395968725 +43,0.0444955089373458 +44,0.0190533549944159 +45,0.0290049795038723 +46,0.0417536642697558 +47,0.0185261550443084 +48,0.0277059929762261 +49,0.0398606084144816 +50,0.0181978813094817 +51,0.0271098219177584 +52,0.0386836665079729 +53,0.0180518611046889 +54,0.0272138992557141 +55,0.0381891287148314 +56,0.0180809085252469 +57,0.0281418959420061 +58,0.0384596362516637 +59,0.0182864418432272 +60,0.0302250788423173 +61,0.0397874837986351 +62,0.0186786556701694 +63,0.0342489348284216 +64,0.0429932815348666 +65,0.0192777878591759 +66,0.0422808966931999 +67,0.0506815964680563 +68,0.0201167847752226 +69,0.0615048274405271 +70,0.0744953894508454 +71,0.021246054596492 +72,0.142602265816215 +73,0.273502052865436 +74,0.325309673287599 +75,0.272705389655349 +76,0.149074257381345 +77,0.0247199397628712 +78,0.0680137859566976 +79,0.075388270873485 +80,0.0273637831604903 +81,0.0407867704453274 +82,0.0632964886441949 +83,0.0309749128751093 +84,0.0315202035072035 +85,0.0627625211892184 +86,0.0360843918243497 +87,0.02794920551495 +88,0.0677921493367236 +89,0.0437167157553067 +90,0.0270640150996317 +91,0.0783380025231622 +92,0.0561293738314281 +93,0.0278742033265809 +94,0.0981443889498639 +95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/tikz/plotfft.tex b/buch/papers/reedsolomon/tikz/plotfft.tex new file mode 100644 index 0000000..77c4dc3 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/plotfft.tex @@ -0,0 +1,104 @@ +% +% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT +% +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{pgfplots} +\usepackage{pgfplotstable} +\usepackage{csvsimple} +\usepackage{filecontents} + + + +\begin{document} +\begin{tikzpicture}[] + + %--------------------------------------------------------------- + %Knote + \matrix(m) [draw = none, column sep=25mm, row sep=2mm]{ + + \node(signal) [] { + \begin{tikzpicture} + \begin{axis} + [title = {\Large {Signal}}, + xtick={0,20,40,64,80,98}] + \addplot[blue] table[col sep=comma] {tikz/signal.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(codiert) [] { + \begin{tikzpicture}[] + % Beschriftung Rechts + \begin{axis}[axis x line= none, axis y line*=right,ytick={0}] + \addplot[color=white] {0}; + \end{axis} + + \begin{axis}[ title = {\Large {Codiert}}, axis y line*=left] + \addplot[color=black!60!green] table[col sep=comma] {tikz/codiert.txt}; + \end{axis} + \end{tikzpicture}}; \\ + + \node(decodiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Decodiert}}] + \addplot[blue] table[col sep=comma] {tikz/decodiert.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(empfangen) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Empfangen \space + \space Fehler}}, + xtick={0,40,60,100}, axis y line*=left] + \addplot[color=black!60!green] table[col sep=comma] {tikz/empfangen.txt}; + \end{axis} + \begin{axis}[xtick={7,21,75}, axis y line*=right] + \addplot[red] table[col sep=comma] {tikz/fehler.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(syndrom) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Syndrom}}] + \addplot[black] table[col sep=comma] {tikz/syndrom.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(locator) [] { + \begin{tikzpicture} + % Beschriftung Rechts + \begin{axis}[axis x line= none, axis y line*=right, ytick={0.3}]; + \addplot[color=black!60] {0.3}; + \end{axis} + + \begin{axis}[title = {\Large {Locator}},axis y line*=left] + \addplot[gray] table[col sep=comma] {tikz/locator.txt}; + \end{axis} + \end{tikzpicture}};\\ + }; + %------------------------------------------------------------- + %FFT & IFFT deskription + + \draw[thin,gray,dashed] (0,9) to (0,-9); + \node(IFFT) [scale=0.9] at (0,9.3) {IFFT}; + \draw[stealth-](IFFT.south west)--(IFFT.south east); + \node(FFT) [scale=0.9, above of=IFFT] {FFT}; + \draw[-stealth](FFT.north west)--(FFT.north east); + + %Arrows + \draw[thick, ->] (signal.east) to (codiert.west); + \draw[thick, ->] (codiert.south) to (empfangen.north); + \draw[thick, ->] (empfangen.west) to (decodiert.east); + \draw[thick, ->] (syndrom.east) to (locator.west); + \draw[thick](decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; + \draw[thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + + %item + \node[circle, draw, fill =lightgray] at (signal.north west) {1}; + \node[circle, draw, fill =lightgray] at (codiert.north west) {2}; + \node[circle, draw, fill =lightgray] at (empfangen.north west) {3}; + \node[circle, draw, fill =lightgray] at (decodiert.north west) {4}; + \node[circle, draw, fill =lightgray] at (syndrom.north west) {5}; + \node[circle, draw, fill =lightgray] at (locator.north west) {6}; +\end{tikzpicture} +\end{document}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/tikz/plotfftraw.tex b/buch/papers/reedsolomon/tikz/plotfftraw.tex new file mode 100644 index 0000000..db35734 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/plotfftraw.tex @@ -0,0 +1,81 @@ + +\begin{tikzpicture}[] + + %--------------------------------------------------------------- + %Knote + \matrix(m) [draw = none, column sep=25mm, row sep=2mm]{ + + \node(signal) [] { + \begin{tikzpicture} + \begin{axis} + [title = {\Large {Signal}}, + xtick={0,20,40,64,80,98}] + \addplot[blue] table[col sep=comma] {tikz/signal.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(codiert) [] { + \begin{tikzpicture}[] + \begin{axis}[ title = {\Large {Codiert \space + \space Fehler}}, + xtick={0,40,60,100}, axis y line*=left] + \addplot[green] table[col sep=comma] {tikz/codiert.txt}; + \end{axis} + \begin{axis}[xtick={7,21,75}, axis y line*=right] + \addplot[red] table[col sep=comma] {tikz/fehler.txt}; + \end{axis} + \end{tikzpicture}}; \\ + + \node(decodiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Decodiert}}] + \addplot[blue] table[col sep=comma] {tikz/decodiert.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(empfangen) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Empfangen}}] + \addplot[green] table[col sep=comma] {tikz/empfangen.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(syndrom) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Syndrom}}] + \addplot[black] table[col sep=comma] {tikz/syndrom.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(locator) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Locator}}] + \addplot[gray] table[col sep=comma] {tikz/locator.txt}; + \end{axis} + \end{tikzpicture}};\\ + }; + %------------------------------------------------------------- + %FFT & IFFT deskription + + \draw[thin,gray,dashed] (0,9) to (0,-9); + \node(IFFT) [scale=0.9] at (0,9.3) {IFFT}; + \draw[stealth-](IFFT.south west)--(IFFT.south east); + \node(FFT) [scale=0.9, above of=IFFT] {FFT}; + \draw[-stealth](FFT.north west)--(FFT.north east); + + \draw[thick, ->,] (codiert)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); + %Arrows + \draw[thick, ->] (signal.east) to (codiert.west); + \draw[thick, ->] (codiert.south) to (empfangen.north); + \draw[thick, ->] (empfangen.west) to (decodiert.east); + \draw[thick, ->] (syndrom.east) to (locator.west); + \draw[thick](decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; + \draw[thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + + %item + \node[circle, draw, fill =lightgray] at (signal.north west) {1}; + \node[circle, draw, fill =lightgray] at (codiert.north west) {2+3}; + \node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; + \node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; + \node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; + \node[circle, draw, fill =lightgray] at (locator.north west) {7}; +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/tikz/polynom2.tex b/buch/papers/reedsolomon/tikz/polynom2.tex new file mode 100644 index 0000000..80557fb --- /dev/null +++ b/buch/papers/reedsolomon/tikz/polynom2.tex @@ -0,0 +1,60 @@ +% polynome +%------------------- + +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{pgfplots} + + +\begin{document} +% Teiler für das Skalieren der Grafik /40 +\newcommand{\teiler}{40} + + +%////////////////////////////////////// + +\begin{tikzpicture}[>=latex,thick,] + \draw[color=blue, line width=1.4pt] + plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); + + \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; + \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; + + \def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + + \def\hellpunkt#1{ + \fill[color=lightgray] #1 circle[radius=0.08]; + \draw[gray] #1 circle[ radius=0.07]; + } + + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); + + + \punkt{(1,8/\teiler)} + \hellpunkt{(2,15/\teiler)} + \hellpunkt{(3,26/\teiler)} + \punkt{(4,41/\teiler)} + \punkt{(5,60/\teiler)} + \punkt{(6,83/\teiler)} + \punkt{(7,110/\teiler)} + + + + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,37.66/\teiler)} + + \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; + \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; +\end{tikzpicture} +\end{document} diff --git a/buch/papers/reedsolomon/tikz/polynomraw.tex b/buch/papers/reedsolomon/tikz/polynomraw.tex new file mode 100644 index 0000000..02968fd --- /dev/null +++ b/buch/papers/reedsolomon/tikz/polynomraw.tex @@ -0,0 +1,50 @@ +% polynomraw + +\newcommand{\teiler}{40} + + +%////////////////////////////////////// + +\begin{tikzpicture}[>=latex,thick,] + \draw[color=blue, line width=1.4pt] + plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); + + \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; + \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; + + \def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + + \def\hellpunkt#1{ + \fill[color=lightgray] #1 circle[radius=0.08]; + \draw[gray] #1 circle[ radius=0.07]; + } + + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); + + + \punkt{(1,8/\teiler)} + \hellpunkt{(2,15/\teiler)} + \hellpunkt{(3,26/\teiler)} + \punkt{(4,41/\teiler)} + \punkt{(5,60/\teiler)} + \punkt{(6,83/\teiler)} + \punkt{(7,110/\teiler)} + + + + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,37.66/\teiler)} + + \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; + \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/tikz/signal.txt b/buch/papers/reedsolomon/tikz/signal.txt new file mode 100644 index 0000000..c4fa5f8 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/signal.txt @@ -0,0 +1,96 @@ +0,6 +1,6 +2,0 +3,6 +4,4 +5,0 +6,5 +7,2 +8,1 +9,2 +10,1 +11,2 +12,0 +13,6 +14,3 +15,5 +16,7 +17,5 +18,5 +19,4 +20,1 +21,5 +22,9 +23,9 +24,3 +25,2 +26,6 +27,6 +28,4 +29,2 +30,9 +31,1 +32,1 +33,1 +34,2 +35,6 +36,6 +37,1 +38,9 +39,7 +40,7 +41,1 +42,9 +43,9 +44,10 +45,9 +46,8 +47,5 +48,2 +49,4 +50,1 +51,0 +52,9 +53,3 +54,3 +55,3 +56,5 +57,6 +58,0 +59,8 +60,6 +61,9 +62,3 +63,4 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,0 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/tikz/syndrom.txt b/buch/papers/reedsolomon/tikz/syndrom.txt new file mode 100644 index 0000000..8ca9eed --- /dev/null +++ b/buch/papers/reedsolomon/tikz/syndrom.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,0 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,0 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/tikz/tikz/codiert.txt b/buch/papers/reedsolomon/tikz/tikz/codiert.txt new file mode 100644 index 0000000..4a481d8 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/codiert.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,24.0186013379153 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,4.84567134895776 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 +61,20.0667752354841 +62,12.6951848921471 +63,17.6292439561917 +64,21.377558326432 +65,9.54321919833388 +66,28.3035029562577 +67,38.467599433197 +68,11.3577742151117 +69,17.9815599423852 +70,23.2496572716993 +71,21.7594021268945 +72,15.0332963783729 +73,35.9073375743642 +74,43.6179140616243 +75,10.4219666223433 +76,4.84567134895776 +77,6.89948680823017 +78,16.5769107917917 +79,23.217268502288 +80,4.58257569495584 +81,10.1736266929292 +82,13.5324640263625 +83,44.9845039238714 +84,32.1114345314062 +85,24.5237955316204 +86,15.6627133315015 +87,12.3214904922455 +88,18.2010889773887 +89,37.3066577242559 +90,24.0186013379153 +91,69.1793197789512 +92,23.8408857476069 +93,12.1189172092243 +94,41.9840308053375 +95,131.570790435043 diff --git a/buch/papers/reedsolomon/tikz/tikz/decodiert.txt b/buch/papers/reedsolomon/tikz/tikz/decodiert.txt new file mode 100644 index 0000000..f6221e6 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/decodiert.txt @@ -0,0 +1,96 @@ +0,6.05208333333333 +1,6.02602539785853 +2,0.0261327016093151 +3,5.98927158561317 +4,4.019445724874 +5,0.0247005083663722 +6,4.97798278395618 +7,1.95246440445439 +8,0.974000110512201 +9,2.00528527696027 +10,1.00071804528155 +11,1.97630907888264 +12,0.0232923747656228 +13,6.01302820392331 +14,3.03567381915226 +15,5.02435590137329 +16,7.00526061008995 +17,5.00739608089369 +18,5.02211514480064 +19,4.02175864806658 +20,1.00236543833726 +21,4.98147315261261 +22,8.97728828610336 +23,8.98481304394618 +24,2.98958333333333 +25,1.98491220960989 +26,5.97728835934715 +27,5.98144124907561 +28,4.00163839998525 +29,2.02176249296313 +30,9.02210713874162 +31,1.00742763919872 +32,1.00557258081044 +33,1.02435888848794 +34,2.03577412756745 +35,6.01302820392331 +36,5.97917574041123 +37,0.976310374034338 +38,9.00062625447998 +39,7.00515849238528 +40,6.97396416790894 +41,0.95256880864368 +42,8.97794719866783 +43,9.01850701506487 +44,10.0194409579917 +45,8.98926601525997 +46,7.9866590265379 +47,5.02603060999077 +48,2.05208333333333 +49,4.02603841132848 +50,0.986882897867895 +51,0.0177592928994285 +52,9.01944131204563 +53,3.0185365665612 +54,2.97803642439316 +55,2.95243072164649 +56,4.97396651395488 +57,6.00516695947321 +58,0.0143895905726619 +59,7.97630812771393 +60,5.97917574041123 +61,9.01298821331865 +62,3.03567381915226 +63,4.02435609145793 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/tikz/tikz/empfangen.txt b/buch/papers/reedsolomon/tikz/tikz/empfangen.txt new file mode 100644 index 0000000..38c13b0 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/empfangen.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,23.6290258699579 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,5.55320238736303 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 +61,20.0667752354841 +62,12.6951848921471 +63,17.6292439561917 +64,21.377558326432 +65,9.54321919833388 +66,28.3035029562577 +67,38.467599433197 +68,11.3577742151117 +69,17.9815599423852 +70,23.2496572716993 +71,21.7594021268945 +72,15.0332963783729 +73,35.9073375743642 +74,44.6135417384784 +75,10.4219666223433 +76,4.84567134895776 +77,6.89948680823017 +78,16.5769107917917 +79,23.217268502288 +80,4.58257569495584 +81,10.1736266929292 +82,13.5324640263625 +83,44.9845039238714 +84,32.1114345314062 +85,24.5237955316204 +86,15.6627133315015 +87,12.3214904922455 +88,18.2010889773887 +89,37.3066577242559 +90,24.0186013379153 +91,69.1793197789512 +92,23.8408857476069 +93,12.1189172092243 +94,41.9840308053375 +95,131.570790435043 diff --git a/buch/papers/reedsolomon/tikz/tikz/fehler.txt b/buch/papers/reedsolomon/tikz/tikz/fehler.txt new file mode 100644 index 0000000..23f1a83 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/fehler.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,2 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,2 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,1 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/tikz/tikz/locator.txt b/buch/papers/reedsolomon/tikz/tikz/locator.txt new file mode 100644 index 0000000..b28988c --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/locator.txt @@ -0,0 +1,96 @@ +0,0.0301224340567056 +1,0.141653026854885 +2,0.138226631799377 +3,0.0339903276086929 +4,0.310585462557496 +5,0.551427312631385 +6,0.628514858396814 +7,0.51102386251559 +8,0.275861355940449 +9,0.0502396354182268 +10,0.090185502547573 +11,0.110759344849756 +12,0.0684618905063001 +13,0.0362855426992259 +14,0.0697096919781468 +15,0.109288539370248 +16,0.0923187999496653 +17,0.0512198536768088 +18,0.274192386987782 +19,0.51349614953654 +20,0.633154426602466 +21,0.553283743533942 +22,0.307840573214514 +23,0.0341664350328392 +24,0.140270857957 +25,0.138527177682831 +26,0.029637547736156 +27,0.0816962563186052 +28,0.0944383203811073 +29,0.0263932110686261 +30,0.0585881348402056 +31,0.0737117341599984 +32,0.0239973937701886 +33,0.0464215468420038 +34,0.0616218854220964 +35,0.0221963086695009 +36,0.0390764778127646 +37,0.0537637218396934 +38,0.0208333333333332 +39,0.0343107696069045 +40,0.0483441215964552 +41,0.0198077862118806 +42,0.0311207395968725 +43,0.0444955089373458 +44,0.0190533549944159 +45,0.0290049795038723 +46,0.0417536642697558 +47,0.0185261550443084 +48,0.0277059929762261 +49,0.0398606084144816 +50,0.0181978813094817 +51,0.0271098219177584 +52,0.0386836665079729 +53,0.0180518611046889 +54,0.0272138992557141 +55,0.0381891287148314 +56,0.0180809085252469 +57,0.0281418959420061 +58,0.0384596362516637 +59,0.0182864418432272 +60,0.0302250788423173 +61,0.0397874837986351 +62,0.0186786556701694 +63,0.0342489348284216 +64,0.0429932815348666 +65,0.0192777878591759 +66,0.0422808966931999 +67,0.0506815964680563 +68,0.0201167847752226 +69,0.0615048274405271 +70,0.0744953894508454 +71,0.021246054596492 +72,0.142602265816215 +73,0.273502052865436 +74,0.325309673287599 +75,0.272705389655349 +76,0.149074257381345 +77,0.0247199397628712 +78,0.0680137859566976 +79,0.075388270873485 +80,0.0273637831604903 +81,0.0407867704453274 +82,0.0632964886441949 +83,0.0309749128751093 +84,0.0315202035072035 +85,0.0627625211892184 +86,0.0360843918243497 +87,0.02794920551495 +88,0.0677921493367236 +89,0.0437167157553067 +90,0.0270640150996317 +91,0.0783380025231622 +92,0.0561293738314281 +93,0.0278742033265809 +94,0.0981443889498639 +95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/tikz/tikz/signal.txt b/buch/papers/reedsolomon/tikz/tikz/signal.txt new file mode 100644 index 0000000..c4fa5f8 --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/signal.txt @@ -0,0 +1,96 @@ +0,6 +1,6 +2,0 +3,6 +4,4 +5,0 +6,5 +7,2 +8,1 +9,2 +10,1 +11,2 +12,0 +13,6 +14,3 +15,5 +16,7 +17,5 +18,5 +19,4 +20,1 +21,5 +22,9 +23,9 +24,3 +25,2 +26,6 +27,6 +28,4 +29,2 +30,9 +31,1 +32,1 +33,1 +34,2 +35,6 +36,6 +37,1 +38,9 +39,7 +40,7 +41,1 +42,9 +43,9 +44,10 +45,9 +46,8 +47,5 +48,2 +49,4 +50,1 +51,0 +52,9 +53,3 +54,3 +55,3 +56,5 +57,6 +58,0 +59,8 +60,6 +61,9 +62,3 +63,4 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,0 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/tikz/tikz/syndrom.txt b/buch/papers/reedsolomon/tikz/tikz/syndrom.txt new file mode 100644 index 0000000..8ca9eed --- /dev/null +++ b/buch/papers/reedsolomon/tikz/tikz/syndrom.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,0 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,0 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index b1588ff..8e0d36d 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -1,17 +1,18 @@ \section{Einleitung\label{spannung:section:Einleitung}} \rhead{Einleitung} Das Hook'sche Gesetz beschreibt die Beziehung von Spannung und Dehnung von linear-elastischen Materialien im Eindimensionalen. -In diesem Kapitel geht es darum das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. +In diesem Kapitel geht es darum, das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen. In jedem erdenklichen Punkt im Dreidimensionalen herrscht daher ein entsprechender individueller Spannungszustand. Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus. -Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen. +Darum werden Vektoren, Matrizen und Tensoren zu Hilfe gezogen. Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden. Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. -Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. -Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. -In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein, +Um die mathematischen und physikalischen Berechnungen anwenden zu können, +müssen vorerst ein paar spezifische Bedingungen vorausgesetzt und Annahmen getroffen werden. +Ebenfalls gilt es, ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. +In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannungen, Dehnungen und Verformungen an elastischen Materialien ein, wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden, z.~B.~\cite{spannung:Grundlagen-der-Geotechnik}. \section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} @@ -29,7 +30,7 @@ Belastet man den Boden mit einer Spannung so wird diese in den Boden geleitet und von diesem kompensiert. Im Boden entstehen unterschiedlich hohe Zusatzspannungen. Diese Zusatzspannung breitet sich räumlich im Boden aus. -Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bild4} breitet sich die Zusatzspannung zwiebelartig aus. +Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{fig:Bild4} breitet sich die Zusatzspannung zwiebelartig aus. \begin{figure} \centering @@ -38,11 +39,11 @@ Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bi \label{fig:Bild4} \end{figure} -Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{spannung:Bild5}). -Wie diese Geometrie der Ausbreitung ist, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{fig:Bild5}). +Wie diese Geometrie der Ausbreitung aussieht, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. Diese Zusatzspannung $\sigma$ ist im Wesentlichen abhängig von $(x,y,t)$. Je nach Modell werden noch andere Parameter berücksichtigt. -Das können beispielsweise jenste Bodenkennwerte oder auch der Wassergehalt sein. +Das können beispielsweise verschiedene Bodenkennwerte oder auch der Wassergehalt sein. \begin{figure} \centering @@ -72,18 +73,18 @@ berechnet werden mit: t &= \text{Tiefe [\si{\meter}]} \\ s &= \text{Setzung, Absenkung [m].} \end{align*} -Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen-der-Geotechnik}] beschrieben. +Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch \cite{spannung:Grundlagen-der-Geotechnik} beschrieben. In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. -Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{spannung:Bild3}). +Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{fig:Bild3}). Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$. Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnungen mit Hilfe einer Spannungsgleichung berechnet werden. Diese beruht auf Annahmen nach Hooke auf einem linear-elastischen Boden. -Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. +Generell wird im Bauingenieurwesen oder auch im Maschinenbau versucht, manche Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. \begin{figure} \centering \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} - \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat} + \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welche kompliziertere Spannungsausbreitung zur Folge hat} \label{fig:Bild3} \end{figure} diff --git a/buch/papers/spannung/main.tex b/buch/papers/spannung/main.tex index bbdf730..d2aeda9 100644 --- a/buch/papers/spannung/main.tex +++ b/buch/papers/spannung/main.tex @@ -3,7 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:spannung}} +\chapter{Dreidimensionaler Spannungszustand\label{chapter:spannung}} \lhead{Dreiachsiger Spannungszustand} \begin{refsection} \chapterauthor{Adrian Schuler und Thomas Reichlin} diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index 7647252..089c28e 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,9 +1,10 @@ \section{Der Spannungszustand\label{spannung:section:Der Spannungsustand}} \rhead{Der Spannungszustand} -Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{spannung:Bild2}). +Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{fig:Bild2}). Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material. -Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt. -Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. +Um alle Spannungen eines Punktes darstellen zu können, +stellt man sich modellhaft ein infinitesimales Bodenelement in Form eines Würfels vor. +Man spricht auch von einem Elementarwürfel. \begin{figure} \centering @@ -15,19 +16,19 @@ Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten im Betrag die selben Spannungen aufweisen, sodass der Elementarwürfel im Gleichgewicht ist. Wäre dieses Gleichgewicht nicht vorhanden, käme es zu Verschiebungen und Drehungen. -Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$. +Das infinitesimale Bodenteilchen hat die Koordinatenachsen $1$, $2$, $3$. Veränderungen der Normalspannungen können durch Schubspannungen kompensiert werden und umgekehrt. -So sind insgesamt neun verschiedene Spannungen möglich, wobei drei Normal- und sechs Schubspannungen sind. +So sind insgesamt neun verschiedene Spannungen möglich, konkret sind dies drei Normal- und sechs Schubspannungen. Normalspannungen wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. Alle Beträge dieser neun Spannungen am Elementarwürfel bilden den Spannungszustand. -Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetz berechnet werden. +Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetzes berechnet werden. Daher gibt es auch den entsprechenden Dehnungszustand. \section{Spannungszustand\label{spannung:section:Spannungsustand}} \rhead{Spannungszustand} -Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{spannung:Bild1}). +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{fig:Bild1}). Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. Nach Hooke gilt: \[ @@ -59,7 +60,7 @@ mit A &= \text{Fläche [\si{\meter\squared}].} \end{align*} Diese Beziehung gilt bei linear-elastischen Materialien, welche reversible Verformungen zulassen. -Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. +Es ist praktisch, die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. \begin{figure} \centering \includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} @@ -73,10 +74,10 @@ Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich E\cdot\varepsilon \] beschreiben. -Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch +Im Falle, dass $E$ nicht konstant ist, wird dieser durch \[ E = -\frac{\Delta\sigma}{\Delta\varepsilon} +\frac{\text{d}\sigma}{\text{d}\varepsilon} \] -ausgedrückt werden.
\ No newline at end of file +ausgedrückt.
\ No newline at end of file diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 74516c1..647b452 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,8 +1,8 @@ \section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}} \rhead{Skalare, Vektoren, Matrizen und Tensoren} -Der Begriff Tensor kann als Überbegriff, der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden. +Der Begriff Tensor kann als Überbegriff der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden. Allerdings sind noch höhere Stufen dieser Objekte beinhaltet. -Ein Skalar, ein Vektor oder eine Matrix ist daher auch ein Tensor. +Skalare, Vektoren oder Matrizen sind daher auch Tensoren. Ein Skalar ist ein Tensor 0. Stufe. Mit einem Vektor können mehrere Skalare auf einmal beschrieben werden. Ein Vektor hat daher die Stufe 1 und ist höherstufig als ein Skalar. @@ -14,11 +14,10 @@ Jede Stufe von Tensoren verlangt andere Rechenregeln. So zeigt sich auch der Nachteil von Tensoren mit Stufen höher als 2. Man ist also bestrebt höherstufige Tensoren mit Skalaren, Vektoren oder Matrizen zu beschreiben. -Der Begriff Tensor wurde 1840 von Rowan Hamilton in die Mathematik eingeführt. +In den 40er Jahren vom 19. Jahrhundert wurde der Begriff Tensor von Rowan Hamilton in die Mathematik eingeführt. James Clerk Maxwell hat bereits mit Tensoren operiert, ohne den Begriff Tensor gekannt zu haben. Erst Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. Er hat in der Elastizitätstheorie als erstes Tensoren eingesetzt und beschrieben. Auch Albert Einstein hat solche Tensoren eingesetzt, um in der Relativitätstheorie die Änderung der 4D Raumzeit beschreiben zu können. \cite{spannung:Tensor} -\cite{spannung:Voigtsche-Notation} diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 6326eab..8620afe 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -3,7 +3,7 @@ Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen. \begin{figure} \centering - \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} + \includegraphics[width=0.30\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} \caption{Beispiel eines Spannungszustandes; Vergrösserung eines infinitesimalen Bodenteilchen} \label{fig:infinitesimalerWuerfel} \end{figure} @@ -49,7 +49,7 @@ Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\ dargestellt werden und beschreibt den gesamten Dehnungszustand. Der Spannungs- und Dehnungstensor 2. Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. -Gemäss der Hadamard-Algebra dürfen Zeile um Zeile in eine Spalte notiert werden, sodass es einen Spaltenvektor ergibt. +Man darf Zeile um Zeile in eine Spalte notieren, sodass es einen Spaltenvektor ergibt. So ergibt sich der Spannungsvektor \[ @@ -79,7 +79,7 @@ So ergibt sich der Spannungsvektor \sigma_{33} \end{pmatrix} \] -und Dehnungsvektor +und der Dehnungsvektor \[ \overline{\varepsilon} = @@ -140,14 +140,6 @@ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{ \end{pmatrix} \] geschrieben werden kann. -Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. -Folglich gilt: -\[ -\overline{\overline{C}} -= -\overline{\overline{C}}~^{T} -. -\] Die allgemeine Spannungsgleichung lautet nun: \[ \vec\sigma @@ -155,8 +147,7 @@ Die allgemeine Spannungsgleichung lautet nun: \overline{\overline{C}}\cdot\vec{\varepsilon} . \] - -Als Indexnotation +Sie kann ebenfalls als Indexnotation \[ \sigma_{ij} = @@ -164,7 +155,15 @@ Als Indexnotation \sum_{l=1}^3 C_{ijkl}\cdot\varepsilon_{kl} \] -kann dies ebenfalls geschrieben werden. +geschrieben werden. +Der Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. +Folglich gilt: +\[ +\overline{\overline{C}} += +\overline{\overline{C}}~^{T} +. +\] Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert. Da dieser Modul durch die eindimensionale Betrachtung definiert ist, @@ -221,7 +220,7 @@ definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich \end{pmatrix} . \] -Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als +Die Normalspannung $\sigma_{22}$ lässt sich zum Beispiel als \[ \sigma_{22} = @@ -229,11 +228,13 @@ Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als \] berechnen. +Reduzierte Spannungs- und Dehnungsgleichungen + Man betrachte nun die Eigenschaften des Elastizitätstensors. Dieser ist quadratisch und symmetrisch, die verschiedenen Einträge wechseln sich aber miteinander ab. Es ergeben sich keine Blöcke mit einheitlichen Einträgen. -Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch Elastizitätstensor symmetrisch sind. +Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch der Elastizitätstensor symmetrisch sind. Wäre dem nicht so, würde sich das Material je nach Richtung unterschiedlich elastisch verhalten. Diese Symmetrie setzt daher voraus, dass \[ @@ -399,7 +400,7 @@ Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit \] beschreiben. Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert. -Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist: +Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \begin{equation} \begin{pmatrix} \sigma_{11}\\ @@ -433,7 +434,7 @@ Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist: Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben. -Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben. +Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung die Einträge verschoben haben. Da nach Voigt zuerst die Normalspannungen und anschliessend die Schubspannungen notiert worden sind, ergeben sich die $3\times3$ Blöcke. Man betrachte als Beispiel die Berechnung von $\sigma_{33}$. @@ -441,8 +442,8 @@ Es ist ersichtlich, dass die Schubdehnungen keinen Einfluss auf $\sigma_{33}$ ha Der Einfluss der zu $\sigma_{33}$ äquivalenten Dehnung $\varepsilon_{33}$ hat den grössten Einfluss. Die anderen Normalspannungen $\sigma_{11}$ und $\sigma_{22}$ haben einen unter anderem mit $\nu$ korrigierten Einfluss. -Von $\overline{\overline{C}}$ bildet man noch die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$ um die Gleichung umstellen zu können. -Dadurch erhält man die Dehnungsgleichung: +Von $\overline{\overline{C}}$ bildet man die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$, mithilfe des Gauss - Jordan Algorithmus, um die Gleichung umstellen zu können. +Durch einige Berechnungsschritte erhält man die Dehnungsgleichung: \[ \vec{\varepsilon} diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 3e456c3..a9080ea 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -30,7 +30,7 @@ q \label{spannung:Invariante_q} . \end{equation} -Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt. +Diese Zusammenhänge werden im Skript \cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik} aufgezeigt. Die hydrostatische Spannung $p$ kann gemäss Gleichung \eqref{spannung:Invariante_p} als \[ p @@ -38,28 +38,28 @@ p \frac{\sigma_{11}+2\sigma_{33}}{3} \] vereinfacht werden. -Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q}als +Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q} als \[ q = \sigma_{11}-\sigma_{33} \] -vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten. +vereinfacht. Man kann $p$ als Druck und $q$ als Schub betrachten. -Die Invarianten können mit der Spannungsformel \eqref{spannung:Spannungsgleichung} berechnet werden. +Die Invarianten $p$ und $q$ können mit der Spannungsgleichung \eqref{spannung:Spannungsgleichung} berechnet werden. Durch geschickte Umformung dieser Gleichung, lassen sich die Module als Faktor separieren. Dabei entstehen spezielle Faktoren mit den Dehnungskomponenten. So ergibt sich \[ -\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} +\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{\displaystyle{p}} = -\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{v}} +\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\displaystyle{{\varepsilon_{v}}}} \] und \[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} +\overbrace{\sigma_{11}-\sigma_{33}}^{\displaystyle{q}} = -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{s}} +\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\displaystyle{\varepsilon_{s}}} . \] Die Faktoren mit den Dehnungskomponenten können so mit @@ -79,8 +79,8 @@ eingeführt werden, mit \varepsilon_{v} &= \text{Hydrostatische Dehnung [-]} \\ \varepsilon_{s} &= \text{Deviatorische Dehnung [-].} \end{align*} -Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglichen werden. -Die deviatorische Dehnung $\varepsilon_{s}$ kann mit einer Verzerrung verglichen werden. +Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression und +die deviatorische Dehnung $\varepsilon_{s}$ mit einer Verzerrung verglichen werden. Diese zwei Gleichungen kann man durch die Matrixschreibweise \begin{equation} @@ -90,8 +90,8 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise \end{pmatrix} = \begin{pmatrix} - \frac{3E}{2(1+\nu)} & 0 \\ - 0 & \frac{E}{3(1-2\nu)} + \displaystyle{\frac{3E}{2(1+\nu)}} & 0 \\ + 0 & \displaystyle{\frac{E}{3(1-2\nu)}} \end{pmatrix} \begin{pmatrix} \varepsilon_{s}\\ @@ -100,9 +100,11 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise \label{spannung:Matrixschreibweise} \end{equation} vereinfachen. -Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. -Änderungen des Spannungszustandes können mit dieser Gleichung vollumfänglich erfasst werden. +Änderungen des Spannungszustandes können mit diesen Gleichungen vollumfänglich erfasst werden. +Diese Spannungsgleichung mit den zwei Einträgen ($p$ und $q$) ist gleichwertig +wie die ursprüngliche Spannungsgleichung mit den neun Einträgen +($\sigma_{11}$, $\sigma_{12}$, $\sigma_{13}$, $\sigma_{21}$, $\sigma_{22}$, $\sigma_{23}$, $\sigma_{31}$, $\sigma_{32}$, $\sigma_{33}$). Mit dieser Formel \eqref{spannung:Matrixschreibweise} lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. -Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. +Ein solcher Versuch, der oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index 2f2e4ce..00b2d4f 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -1,6 +1,6 @@ -\section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}} -\rhead{Oedometer-Versuch} -Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. +\section{Oedometrischer Elastizitätsmodul\label{spannung:section:Oedometrischer Elastizitätsmodul}} +\rhead{Oedometrischer Elastizitätsmodul} +Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{\text{OED}}$ bestimmt werden. Dieser beschreibt ebenfalls das Verhältnis zwischen Spannung und Dehnung, allerdings unter anderen Bedingungen. Diese Bedingung ist das Verhindern der seitlichen Verformung, sprich der Dehnung in Richtung $1$ und $2$. Es wird ein Probeelement mit immer grösseren Gewichten belastet, welche gleichmässig auf das Material drücken. @@ -43,8 +43,8 @@ Diese lautet nun: \end{pmatrix} = \begin{pmatrix} - \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{3(1-2\nu)} + \displaystyle{\frac{E_{\text{OED}}}{(1+\nu)}} & 0 \\ + 0 & \displaystyle{\frac{E_{\text{OED}}}{3(1-2\nu)}} \end{pmatrix} \begin{pmatrix} \varepsilon_{11}\\ @@ -52,28 +52,28 @@ Diese lautet nun: \end{pmatrix} . \] -Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen +Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{\text{OED}}$ und die seitlichen Spannungen $\sigma_{33}$ mit den zwei Gleichungen \[ \sigma_{11}-\sigma_{33} = -\frac{E_{OED}}{(1+\nu)}\cdot\varepsilon_{11} +\frac{E_{\text{OED}}}{(1+\nu)}\cdot\varepsilon_{11} \] und \[ \sigma_{11}+2\sigma_{33} = -\frac{E_{OED}}{3(1-2\nu)}\cdot\varepsilon_{11} +\frac{E_{\text{OED}}}{3(1-2\nu)}\cdot\varepsilon_{11} \] berechnen. -Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen. +Mit diesen Gleichungen hat man das Gleichungssystem um $E_{\text{OED}}$ und $\sigma_{33}$ zu berechnen. Die Poisson-Zahl muss als Kennwert gemäss der Bodenklasse gewählt werden. -Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{spannung:DiagrammOedometer-Versuch}). +Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{fig:DiagrammOedometer-Versuch}). Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. -Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. +Mit diesem ermittelten $E_{\text{OED}}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} + \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} \caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material} \label{fig:DiagrammOedometer-Versuch} \end{figure}
\ No newline at end of file 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..8994066 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -1,118 +1,98 @@ -\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} -1 & \text{Kante von $j$ nach $i$} \\ 0 & \text{keine Kante von $j$ nach $i$} -\end{matrix} - \right. +A_{i,j} = \begin{cases} +1&\quad\text{Kante von $j$ nach $i$}\\ +0&\quad\text{keine Kante von $j$ nach $i$} +\end{cases} \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\dots 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} +Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt, so entsteht die Link-Matrix +\[ P_{i,j}=\frac{A_{i,j}}{\sum_{k=1}^{n}A_{k,j}} \] 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} +\( \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1\dots n\right\} \) 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} -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. +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: +\( \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\) +und somit allgemein: +\( \Vec{r_i} = P^i\cdot\Vec{r_0}\) +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. |
