From c412a5070ce026fee2cff0f71e79f5332095bd9b Mon Sep 17 00:00:00 2001 From: salagne Date: Thu, 9 Jul 2020 19:08:20 +0200 Subject: [PATCH] more correcttions --- ana19.tex | 120 ++++++++++++++++++++++++------------------------- analysisII.pdf | Bin 1004407 -> 893484 bytes 2 files changed, 59 insertions(+), 61 deletions(-) diff --git a/ana19.tex b/ana19.tex index 00cb5d5..4369dc5 100644 --- a/ana19.tex +++ b/ana19.tex @@ -3,8 +3,8 @@ \begin{document} \section{Globale Stabilität} -\begin{definition}[Exponentielle Stabilität]s - Sei $y(t)$ die globale Lösung einer AWA. $y(t)$ heißt \underline{exponentiell stabil}, falls Konstanten $\delta, \alpha, A > 0$ sodass $\forall t_{*} \geq t_{0}$ und $\forall w_{*} \in \R^{n}$ mit $\norm{w_{*}} < \delta$ die gestörte AWA +\begin{definition}[Exponentielle Stabilität] + Sei $y(t)$ die globale Lösung einer AWA. $y(t)$ heißt \underline{exponentiell stabil}, falls Konstanten $\delta, \alpha, A > 0$ existieren, sodass $\forall t_{*} \geq t_{0}$ und $\forall w_{*} \in \R^{n}$ mit $\norm{w_{*}} < \delta$ die gestörte AWA \[ v'(t) = f\left(t,v(t)\right), \ \ \ t \geq t_{*}, \ \ \ v(t_{*}) = y(t_{*}) + w_{*} \] @@ -22,17 +22,17 @@ \end{definition} \begin{satz}[Globaler Stabilitätssatz] - Sei $f(t,x)$ einer AWA stetig, lokal Lipschitz-stetig bezüglich $x$ und stark monoton. Dann sind alle Lösungen des AWA global und exponentiell stabil, mit $\delta$ beliebig und $\alpha = A = 1$. \\ - Zusatz: gilt $\sup_{t \geq t_{0}} \norm{f(t,0)} < \infty$, dann sind alle Lösungen gleichmäßig beschränkt. + Sei $f(t,x)$ einer AWA stetig, lokal Lipschitz-stetig bezüglich $x$ und stark monoton. Dann sind alle Lösungen der AWA global und exponentiell stabil, mit $\delta$ beliebig und $\alpha = A = 1$. \\ + Zusatz: Gilt $\sup_{t \geq t_{0}} \norm{f(t,0)} < \infty$, dann sind alle Lösungen gleichmäßig beschränkt. \label{satz:global-stabil} \end{satz} \begin{proof} \begin{enumerate}[1)] - \item Die Lösungen des AWA + \item Die Lösungen der AWA \[ y'(t) = f(t,y(t)), \ \ \ t \geq t_{0}, \ \ \ y(t_{0}) = y_{0}, \] - und des gestörten AWA + und der gestörten AWA \[ v'(t) = f(t,v(t)), \ \ \ t \geq t_{*} \geq t_{0}, \ \ \ v(t_{*}) = y(t_{*}) + w_{*} \] @@ -67,19 +67,19 @@ \] Sei $w(t) \coloneqq v(t) - y(t)$. Dann gilt: \begin{align*} - &\frac{\d{}}{\d{t}} w(t) = 2\left(w'(t),w(t)\right)= 2 \left(f(t,v(t))-f(t,y(t)), v(t) - y(t) \right) + &\frac{\d{}}{\d{t}} \norm{w(t)}^{2} = 2\left(w'(t),w(t)\right)= 2 \left(f(t,v(t))-f(t,y(t)), v(t) - y(t) \right) \intertext{und da $f$ stark monoton ist, folgt:} &\frac{\d{}}{\d{t}} \norm{w(t)}^{2} \leq -2\lambda \norm{w(t)}^{2} \ \ \ \ \ \implies \ \ \ \ \ \frac{\d{}}{\d{t}} \norm{w(t)}^{2} +2\lambda \norm{w(t)}^{2} \leq 0. \end{align*} Ferner gilt: \begin{align*} - \frac{\d{}}{\d{t}} e^{2\lambda (t-t_{*})} \norm{w(t)}^{2} = \underbrace{e^{2\lambda (t-t_{*})}}_{> 0} \underbrace{\left( \frac{\d{}}{\d{t}}\norm{w(t)}^{2} + 2\lambda\norm{w(t)}^{2} \right)}_{\leq 0} \leq 0 + \frac{\d{}}{\d{t}} \left(e^{2\lambda (t-t_{*})} \norm{w(t)}^{2}\right) = \underbrace{e^{2\lambda (t-t_{*})}}_{> 0} \underbrace{\left( \frac{\d{}}{\d{t}}\norm{w(t)}^{2} + 2\lambda\norm{w(t)}^{2} \right)}_{\leq 0} \leq 0 \end{align*} Woraus wir folgern: \begin{salign*} & \int_{t_{*}}^{t} \frac{\d{}}{\d{s}} \left( e^{2\lambda(s-t_{*})}\norm{w(s)}^{2} \right) \d{s} = e^{2\lambda(t-t_{*})}\norm{w(t)}^{2} - \norm{w(t_{*})}^{2} \leq 0 \\ - \implies \ & \norm{w(t)}^{2} \leq e^{-2\lambda(t-t_{*})}\norm{w(t)}^{2} \\ - \implies \ & \norm{w(t)} \ \leq e^{-\lambda(t-t_{*})}\norm{w(t^{*})} \\ + \implies \ & \norm{w(t)}^{2} \leq e^{-2\lambda(t-t_{*})}\norm{w(t_{*})}^{2} \\ + \implies \ & \norm{w(t)} \ \leq e^{-\lambda(t-t_{*})}\norm{w(t_{*})} \\ \implies \ & \norm{v(t) - y(t)} \leq e^{-\lambda(t-t_{*})}\norm{w_{*}}. \end{salign*} @@ -140,9 +140,9 @@ \section{Lineare Systeme von Differentialgleichungen} \begin{definition}[Lineare AWA] - Sei $A(\cdot) \colon I \to \R^{n \times n}$ eine Matrixfunktion, sowie $b(\cdot) \colon I \to \R^{n}$ eine Vektorfunktion. Dann ist eine Lineares AWA der Form: + Sei $A(\cdot) \colon I \to \R^{n \times n}$ eine Matrixfunktion, sowie $b(\cdot) \colon I \to \R^{n}$ eine Vektorfunktion. Dann ist eine lineare AWA der Form: \begin{salign*} - y'(t) &= A(t)y + b(t), \ \ \ t\geq t_{0} \\ + y'(t) &= A(t)y(t) + b(t), \ \ \ t\geq t_{0} \\ y(t_{0}) &= y_{0}. \end{salign*} \end{definition} @@ -183,18 +183,18 @@ \end{proof} \begin{satz}[Homogene lineare Systeme] - Ein homogenes lineares System von DGLs ist der Form + Ein homogenes lineares System von DGLn ist der Form \begin{equation} y'(t) = A(t)y(t). \label{eqq:homog-lin-syst} \end{equation} \begin{enumerate}[1)] \item Die Menge der Lösungen bildet einen Vektorraum $H$. - \item Sei $\{y_{0}^{1},...,y_{0}^{n}\}$ eine Basis des $\R^{n}$. Seien $\{y^{1},...,y^{n}\}$ die Lösungen von AWA: + \item Sei $\{y_{0}^{1},\dots,y_{0}^{n}\}$ eine Basis des $\R^{n}$. Seien $\{y^{1},\dots,y^{n}\}$ die Lösungen von AWA: \[ - (y^{i})' = A(t)y^{i}, \ \ \ y^{i}(t_{0}) = y_{0}^{i}, \ \ \ i\in \{1,...,n\} + (y^{i})' = A(t)y^{i}, \ \ \ y^{i}(t_{0}) = y_{0}^{i}, \ \ \ i\in \{1,\dots,n\} \] - Dann ist $\{y^{1},...,y^{n}\}$ eine Basis von $H$ und es gilt $\dim(H) = n$. - \item Sei $\{y^{1},...,y^{n}\}$ eine Basis des Lösungsraum $H$, dann ist $\{y^{1}(t),...,y^{n}(t)\}$ für $\forall t \geq t_{0}$ eine Basis des $\R^{n}$. + Dann ist $\{y^{1},\dots,y^{n}\}$ eine Basis von $H$ und es gilt $\dim(H) = n$. + \item Sei $\{y^{1},\dots,y^{n}\}$ eine Basis des Lösungsraums $H$, dann ist $\{y^{1}(t),\dots,y^{n}(t)\}$ für $\forall t \geq t_{0}$ eine Basis des $\R^{n}$. \end{enumerate} \end{satz} @@ -210,39 +210,40 @@ Also $\left(\alpha u + \beta v\right) \in H$. \end{itemize} - \item Sei $\{y_{0}^{1},...,y_{0}^{n}\}$ eine Basis des $\R^{n}$. Seien $\{y^{1},...,y^{n}\}$ die eindeutigen Lösungen der AWAn (nach Satz \ref{satz:lineare-awa}). Seien $\alpha_{i} \in \R$, $i \in \{1,...,n\}$ sodass für $t \geq t_{0}$ gilt: + \item Sei $\{y_{0}^{1},\dots,y_{0}^{n}\}$ eine Basis des $\R^{n}$. Seien $\{y^{1},\dots,y^{n}\}$ die eindeutigen Lösungen der AWAn (nach Satz \ref{satz:lineare-awa}). Seien $\alpha_{i} \in \R$, $i \in \{1,\dots,n\}$ sodass für $t \geq t_{0}$ gilt: \[ \sum_{i=1}^{n} \alpha_{i} y^{i}(t) = 0. \] - Für $t = t_{0}$ gilt dann $\sum_{i=1}^{n} \alpha_{i} y_{0}^{i} = 0$, da die $y_{0}^{i}$ linear unabhängig sind, folgt $\alpha_{i} = 0$ für alle $i \in \{1,...,n\}$. Daraus folgt, dass die $y^{i}(t)$ ($i \in \{1,...,n\}$) linear unabhängig sind, also bereits, dass die $y^{i}$ ($i \in \{1,...,n\}$) linear unabhängig sind. \\ + Für $t = t_{0}$ gilt dann $\sum_{i=1}^{n} \alpha_{i} y_{0}^{i} = 0$, da die $y_{0}^{i}$ linear unabhängig sind, folgt $\alpha_{i} = 0$ für alle $i \in \{1,\dots,n\}$. Daraus folgt, dass die $y^{i}(t)$ ($i \in \{1,\dots,n\}$) linear unabhängig sind, also bereits, dass die $y^{i}$ ($i \in \{1,\dots,n\}$) linear unabhängig sind. \\ Da es höchstens $n$ linear unabhängige Anfangswerte gibt, sind nicht mehr als $n$ Funktionen aus $H$ linear abhängig, also $\dim(H) = n$. \item Analog zu 2). \end{enumerate} \end{proof} \begin{definition} - Eine Basis $\{\varphi^{1},...,\varphi^{n}\}$ des Lösungsraums $H$ von $y'(t) = A(t)y(t)$ zu den Anfangswerten $\varphi^{i}(t_{0}) = e^{i}$ heißt \underline{Fundamentalsystem} des linearen Systems von DGLs. \\ - Die Matrix $\phi = \left[\varphi^{1},...,\varphi^{n} \right] $ der Spaltenvektoren $\varphi^{i}$ heißt \underline{Fundamentalmatrix} des linearen Systems von DGLs. \\ + Eine Basis $\{\varphi^{1},\dots,\varphi^{n}\}$ des Lösungsraums $H$ von $y'(t) = A(t)y(t)$ zu den Anfangswerten $\varphi^{i}(t_{0}) = e^{i}$ ($e^i$ Standardbasisvektor) heißt \underline{Fundamentalsystem} des linearen Systems von DGLn. \\ + Die Matrix $\phi = \left[\varphi^{1},\dots,\varphi^{n} \right] $ der Spaltenvektoren $\varphi^{i}$ heißt \underline{Fundamentalmatrix} des linearen Systems von DGLn. \\ Diese Matrix ist regulär und löst die AWA (komponentenweise): \[ - \phi'(t) = A(t)\phi(t), \ \ \ \ t \geq t_{0}, \ \ \ \ \phi(t_{0}) = I. + \phi'(t) = A(t)\phi(t), \ \ \ \ t \geq t_{0}, \ \ \ \ \phi(t_{0}) = \mathbb{I}. \] \end{definition} \begin{satz}[Inhomogene lineare Systeme] - Ein inhomogenes lineare System von DGLs ist der Form + Ein inhomogenes lineare System von DGLn ist der Form \[ - y'(t) = A(t))y(t) + b(t). + y'(t) = A(t)y(t) + b(t). \] Seien $A \colon [t_{0},\infty) \to \R^{n \times n}$, $b \colon [t_{0},\infty) \to \R^{n}$ stetig. Dann gilt: \begin{enumerate}[1)] - \item Die partikuläre Lösung ist für $\text{const} \ c \in R^{n}$: + \item Für konstantes $c \in \R^{n}$ ist \[ - y_{b}(t) = \phi(t) \left( \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + c \right). + y_{b}(t) \coloneqq \phi(t) \left( \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + c \right) \] + eine partikuläre Lösung des inhomogenen linearen Systems. \item Alle Lösungen der inhomogenen Gleichung haben die Form: \[ - y(t) = y_{b}(t) + v(t) + y(t) = y_{b}(t) + v(t), \] wobei $v \in H$ (Lösungsraum des assoziierten homogenen Systems). \item Gilt $c = y_{0}$, dann gilt $y_{b}(t_{0}) = y_{0}$. @@ -258,7 +259,7 @@ Also ist $y_{b}$ eine Lösung des inhomogenen Systems und gilt $c=y_{0}$, dann löst $y_{b}$ die AWA $y' = Ay + b$, $y(t_{0}) = y_{0}$. Daraus folgt 1) und 3). \item Sei $y$ eine zweite Lösung des inhomogenen Systems. Für $w=y-y_{b}$ gilt dann: \[ - w' = y' - y_{b}' = Ay + b - (Ay_{b} + b) = A(y-y_{0}) = Aw. + w' = y' - y_{b}' = Ay + b - (Ay_{b} + b) = A(y-y_{b}) = Aw. \] Also bereits $w \in H$. \end{enumerate} @@ -268,14 +269,14 @@ \begin{enumerate}[(1)] \item Die Lösung $y_{b}(t) = \phi(t) \left( \int_{t_{0}}^{t} \phi^{-1}(s)b(s) \d{s} + y_{0} \right)$ entspricht genau der Lösung \[ - y(t) = \exp\left( \int_{t_{0}}^{t} a(s) \d{s} \right) \left(y_{0} + \int_{t_{0}}^{t} \exp\left( -\int_{t_{0}}^{t} a(s) \d{s} \right)b(t) \d{s} \right) + y(t) = \exp\left( \int_{t_{0}}^{t} a(s) \d{s} \right) \left(y_{0} + \int_{t_{0}}^{t} \exp\left( -\int_{t_{0}}^{t} a(s) \d{s} \right)b(\tau) \d{\tau} \right) \] der skalaren linearen AWA $y'(t) = a(t)y(t) + b(t)$ ($t \geq t_{0}$) (Variation der Konstanten). \item Für lineare DGL mit konstanten Koeffizienten \[ y'(t) = Ay(t), \ \ \ \ \ A \in \R^{n\times n} \] - gibt es eine Lösungstheorie, die auf algebraischen Argumente zurückgreift. + gibt es eine Lösungstheorie, die auf algebraische Argumente zurückgreift. \end{enumerate} \end{bem} @@ -288,7 +289,7 @@ \end{salign*} Gesucht ist eine stetig differenzierbare Lösung $y \colon I \to \R^{n}$ die beide Bedingungen erfüllt. Die zweite Bedingung lässt sich verallgemeinern zu einer Mehrpunkt RWA: \[ - r\left( y(t_{1}),...,y(t_{k}) \right) = 0. + r\left( y(t_{1}),\dots,y(t_{k}) \right) = 0. \] \end{bem} @@ -299,7 +300,7 @@ \] für $t \in [0,\pi]$. Dies ist äquivalent zu folgenden System: \begin{salign*} - y_{1} = y, y_{2} = y', && \begin{cases} + y_{1} = y,\ y_{2} = y', && \begin{cases} y_{1}' = y_{2} \\ y_{2}' = -y_{1} \end{cases}. @@ -311,13 +312,13 @@ \begin{enumerate}[1)] \item Für $y(0) = y(\pi)$, $y'(0)=y'(\pi)$. $r = \begin{pmatrix} y_{1}(0)-y_{1}(\pi) \\ y_{2}(0)-y_{2}(\pi) - \end{pmatrix} = 0$ ist die Lösung des RWP $y(t) \equiv 0$, $t \in [0,\pi]$ eindeutig. + \end{pmatrix} = 0$ ist die Lösung des RWA $y(t) \equiv 0$, $t \in [0,\pi]$ eindeutig. \item Für $y(0) = 0$, $y(\pi) = 0$, $r = \begin{pmatrix} y_{1}(0) \\ y_{1}(\pi) - \end{pmatrix} = 0$ hat das RWP unendlich viele Lösungen $y(t) = c_{1}\sin(t)$ ($t \in [0,t]$). + \end{pmatrix} = 0$ hat das RWA unendlich viele Lösungen $y(t) = c_{1}\sin(t)$ ($t \in [0,t]$). \item Für $y(0) = 0$, $y(\pi) = 1$, $r = \begin{pmatrix} y_{1}(0) \\ y_{1}(\pi) -1 - \end{pmatrix} = 0$ hat das RWP keine Lösung. + \end{pmatrix} = 0$ hat das RWA keine Lösung. \end{enumerate} \end{bsp} @@ -330,7 +331,7 @@ \end{definition} \begin{bem} - Eine Lösung des inhomogenen DLG-System ist von der Form: + Eine Lösung des inhomogenen DGL-System ist von der Form: \[ y(t,s) = \varphi^{0}(t) + \sum_{i=1}^{n} s_{i}\varphi^{i}(t) = \varphi^{0}(t) + \phi(t)s. \] @@ -338,13 +339,13 @@ \[ (\varphi^{0})'(t) = A(t)\varphi^{0}(t) + f(t), \ \ \ \ t \geq a, \ \varphi^{0}(a) = 0. \] - $\varphi^{i}$ ($i \in \{1,...,n\}$) lösen die AWA: + $\varphi^{i}$ ($i \in \{1,\dots,n\}$) lösen die AWA: \[ (\varphi^{i})'(t) = A(t)\varphi^{i}(t), \ \ \ \ t \geq a, \ \varphi^{i}(a) = e^{i}. \] - $\varphi^{0}, \varphi^{i}$ ($i \in \{1,...,n\}$) sind eindeutige Lösungen, außerdem: + $\varphi^{0}, \varphi^{i}$ ($i \in \{1,\dots,n\}$) sind eindeutige Lösungen, außerdem: \[ - \phi(t) = \left[\varphi^{1}(t),...,\varphi^{n}(t) \right]. + \phi(t) = \left[\varphi^{1}(t),\dots,\varphi^{n}(t) \right]. \] Offenbar löst $y(t,s)$ die DGL: \begin{salign*} @@ -356,7 +357,7 @@ \] Dies lässt sich umformen zu: \[ - B_{a}(\varphi^{0}(a) + \phi(a)s) + B_{b}(\varphi^{0}(b) + \phi(b)s) = g. + B_{a}(\underbrace{\varphi^{0}(a)}_{=0} + \underbrace{\phi(a)}_{=\mathbb{I}}s) + B_{b}(\varphi^{0}(b) + \phi(b)s) = g. \] Also: \[ @@ -365,7 +366,7 @@ \end{bem} \begin{satz}[Existenzsatz für lineare RWA] - Die lineare RWA besitzt eine eindeutige Lösung $y(t)$ für beliebige $f(t)$ und $g$ genau dann wenn: $B_{a} + B_{b}\phi(b) \in \R^{n\times n}$ regulär ist, bzw. die assoziierte RWA nur die triviale Lösung $y \equiv 0$ hat. + Die lineare RWA besitzt eine eindeutige Lösung $y(t)$ für beliebige $f(t)$ und $g$ genau dann, wenn $B_{a} + B_{b}\phi(b) \in \R^{n\times n}$ regulär ist, bzw. die assoziierte homogene RWA nur die triviale Lösung $y \equiv 0$ hat. \end{satz} \begin{proof} @@ -376,9 +377,9 @@ eindeutig lösbar für $s \in \R^{n}$ und somit löst $y(t,s)$ die RWA. \\ \glqq $\Rightarrow$ \grqq: Die Lösung der RWA kann man darstellen als \[ - y(t) = \varphi^{0}(t) + \phi(t)s, \ \ \ s \in \R^{n}. + y(t) = \varphi^{0}(t) + \phi(t)s, \ \ \ s \in \R^{n}, \] - weil die Funktionen $\{ \varphi^{1},...,\varphi^{n}\}$ eine Basis des Lösungsraum des assoziierten homogenen DGL bilden. \\ + weil die Funktionen $\{ \varphi^{1},\dots,\varphi^{n}\}$ eine Basis des Lösungsraum des assoziierten homogenen DGL bilden. \\ Für homogene RWA gilt $g - B_{b}\varphi^{0}(b) = 0$ und die Gleichung für $s$ lautet \[ \left( B_{a} +B_{b}\phi(b)\right)s = 0 @@ -390,28 +391,25 @@ Die Lösung einer linearen RWA ist im Kern die Lösung eines LGS. \end{bem} -\begin{bem} - Nun betrachten wir die nichtlineare RWA - \[ - y' = f(t,y), \ \ \ t \in [a,b], \ \ \ \ \ r(y(a),y(b)) = 0. - \] - Frage: falls die nichtlineare RWA eine Lösung $y(t)$ besitzt, ist diese lokal eindeutig? \\ - Also existiert eine Umgebung $U_{R}(y) = \{ v \in C[a,b] \ | \ \norm{y-v}_{\infty} < R\}$, sodass es keine andere Lösung $\tilde{y} \neq y$ existiert? \\ - Wir führen folgende Notationen ein: - \begin{salign*} - f_{x}'(t,x) &= \left( \frac{\partial f_{i}(t,x)}{\partial x_{j}} \right)_{i,j=1}^{n} \\ - r_{x}'(t,x) &= \left( \frac{\partial r_{i}(x,y)}{\partial x_{j}} \right)_{i,j=1}^{n} \\ - r_{y}'(t,x) &= \left( \frac{\partial r_{i}(x,y)}{\partial y_{j}} \right)_{i,j=1}^{n} - \end{salign*} - für die Jacobi-Matrizen von $f(t,\cdot)$, $r(\cdot,\cdot)$ ein. -\end{bem} +Nun betrachten wir die nichtlineare RWA + \[ + y' = f(t,y), \ \ \ t \in [a,b], \ \ \ \ \ r(y(a),y(b)) = 0. + \] +Frage: falls die nichtlineare RWA eine Lösung $y(t)$ besitzt, ist diese lokal eindeutig? \\ +Also existiert eine Umgebung $U_{R}(y) = \{ v \in C[a,b] \ | \ \norm{y-v}_{\infty} < R\}$, sodass keine andere Lösung $\tilde{y} \neq y$ existiert? \\ +Wir führen die Notationen + \begin{salign*} + f_{x}'(t,x) &= \left( \frac{\partial f_{i}(t,x)}{\partial x_{j}} \right)_{i,j=1}^{n} \\ + r_{x}'(t,x) &= \left( \frac{\partial r_{i}(x,y)}{\partial x_{j}} \right)_{i,j=1}^{n} \\ + r_{y}'(t,x) &= \left( \frac{\partial r_{i}(x,y)}{\partial y_{j}} \right)_{i,j=1}^{n} + \end{salign*} +für die Jacobi-Matrizen von $f(t,\cdot)$, $r(\cdot,\cdot)$ ein. \begin{satz}[Lokale Eindeutigkeit] - Eine Lösung $y$ von nichtlinearen RWA ist lokal eindeutig genau dann wenn: \\ - die lineare RWA + Eine Lösung $y$ von nichtlinearen RWA ist lokal eindeutig genau dann, wenn die lineare RWA \begin{salign*} & v'(t) = f_{x}'(t,y(t))v(t), \ \ \ \ \ \ t \in I, \\ - & r_{x}'(y(a),y(b))v(a) + r_{y}'(y(a),y(b))v(b) = 0. + & r_{x}'(y(a),y(b))\cdot v(a) + r_{y}'(y(a),y(b))\cdot v(b) = 0. \end{salign*} nur die triviale Lösung $v \equiv 0$ besitzt. \end{satz} diff --git a/analysisII.pdf b/analysisII.pdf index 0a506e3e33b6126207a813c0d5e73160b468a06a..9b69dd7dbdffea0d30a8d41f6386a177f813d362 100644 GIT binary patch delta 625839 zcmV)TK(W8~rai30HjpF(F*YERLEb2Tjgv`F!!Qtq@BI}%CI=6(9mmOyt!x#jD!Cxw z5LzcS6i3Q})PK*|Mk*ESC9x;--Zy?p4$8%xk*1#{q#+48|MPxaP7cm^MwkN5WJ=1K za9IFL3nHa}f)%7(k^AV#R-0{a`j`qPqRW`_46l#n6{;0s3<@c^Q&Yhq!zGn}9E1=- zJ@48YP>Q0{&M#CneTk)vszGU}DN58?&f#1R_$%GeIcNfpHcbug%ae 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