From 35ec151ac8fd6b06c07e300367d9159da5acb983 Mon Sep 17 00:00:00 2001 From: Robert Jeutter Date: Sun, 13 Sep 2020 10:00:13 +0200 Subject: [PATCH] fixed mathematical formulas --- ...n und Diskrete Strukturen - Cheatsheet.pdf | Bin 0 -> 237953 bytes ...n und Diskrete Strukturen - Cheatsheet.tex | 544 ++++++++++-------- ...n und Diskrete Strukturen - cheatsheet.pdf | Bin 5918 -> 0 bytes 3 files changed, 295 insertions(+), 249 deletions(-) create mode 100644 Grundlagen und Diskrete Strukturen - Cheatsheet.pdf rename Grundlagen und Diskrete Strukturen - cheatsheet.tex => Grundlagen und Diskrete Strukturen - Cheatsheet.tex (60%) delete mode 100644 Grundlagen und Diskrete Strukturen - cheatsheet.pdf diff --git a/Grundlagen und Diskrete Strukturen - Cheatsheet.pdf b/Grundlagen und Diskrete Strukturen - Cheatsheet.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e41ef7111c755f2705da760a478ed988c8ec17d6 GIT binary patch literal 237953 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Beweis-Existenz mit Annahme: Der Satz gilt nicht, dann gibt es eine kleinste Zahl n die sich nicht als Produkt von Primzahlen schreiben lässt $\rightarrow$ n weder Primzahl noch 1 $\rightarrow n=m*l$ für $m,l>1 \rightarrow$ m und l sind Produkte von Primzahlen $\rightarrow m*l=$ Produkt von Primzahlen. -Eindeutigkeit mit Annahme: es gibt ein $n>0$ ohen eindeutige Primfaktorzerlegung (PFZ)$\rightarrow$ es gibt ein kleinstes $n>0$ ohne eindeutige PFZ. Kommt eine Primzahl p in beiden Zerlegungen vor, so hat auch $\frac{n}{p}$ zwei versch. PFZen. Man erhält die PFZ von $n'=(1_1-p_1)*b$ aus den PFZen von $q_1-p_1$ und b.. -> Eindeutig bestimmbar. +Eindeutigkeit mit Annahme: es gibt ein $n>0$ ohne eindeutige Primfaktorzerlegung (PFZ)$\rightarrow$ es gibt ein kleinstes $n>0$ ohne eindeutige PFZ. Kommt eine Primzahl p in beiden Zerlegungen vor, so hat auch $\frac{n}{p}$ zwei verschiedene PFZen. Man erhält die PFZ von $n'=(1_1-p_1)*b$ aus den PFZen von $q_1-p_1$ und b.. -> Eindeutig bestimmbar. -## Arithmetik im Restklassenring in $\Z$ -Sei $m>1$ gegeben, $a\equiv b mod m \leftrightarrow m|a-b$ def. Relation auf $\Z$. Die Äquivalenzklasse zu a wird mit $\bar{a}$ bezeichnet, d.h. $\bar{a}=[a]_{/mod m}={x\in \Z: x\equiv a mod m}$, $\Z_m={\bar{a}:a\in\Z}$. Sei dazu $\bar{a}\in\Z_m$ beliebig. +\paragraph{Arithmetik im Restklassenring in $\mathbb{Z}$} +Sei $m > 1$ gegeben, $a\equiv \text{b mod m} \leftrightarrow m|a-b$ def. Relation auf $\mathbb{Z}$. Die Äquivalenzklasse zu a wird mit $\bar{a}$ bezeichnet, d.h. $\bar{a}=[a]_{\text{mod m}}={x\in \mathbb{Z}: x\equiv \text{a mod m}}$, $\mathbb{Z}_m={\bar{a}:a\in \mathbb{Z}}$. Sei dazu $\bar{a}\in \mathbb{Z}_m$ beliebig. -Division mit Rest $\rightarrow$ es gibt eindeutig bestimmt q,r mit $a?q*m+r$ und $0\leq r < m \rightarrow a-r=q*m \rightarrow m| a-r \rightarrow a\equiv r mod m \rightarrow \bar{a}=\bar{r}$. Also tritt $\bar{a}$ in der Liste $\bar{0},\bar{1},...,\bar{m-1}$ auf. Aus $0\leq i < j \leq m-1$ folgt $\bar{i}\not=\bar{j}$. In der Liste $\bar{0},\bar{1},...,\bar{m-1}$ gibt es daher keine Wiederholungen $\rightarrow |\Z_M|=m$. +Division mit Rest $\rightarrow$ es gibt eindeutig bestimmt q,r mit $a=q*m+r$ und $0\leq r < m \rightarrow a-r=q*m \rightarrow m| a-r \rightarrow a\equiv \text{r mod m } \rightarrow \bar{a}=\bar{r}$. Also tritt $\bar{a}$ in der Liste $\bar{0},\bar{1},...,\bar{m-1}$ auf. Aus $0\leq i < j \leq m-1$ folgt $\bar{i}\not=\bar{j}$. In der Liste $\bar{0},\bar{1},...,\bar{m-1}$ gibt es daher keine Wiederholungen $\rightarrow |\mathbb{Z}_M|=m$. -Wir definieren Operationen +,* auf $\Z_m$ durch $\bar{a}+\bar{b}:= \bar{a+b}$ und $\bar{a}*\bar{b}:=\bar{a*b}$ für $a,b\in\Z$. +Wir definieren Operationen +,* auf $\mathbb{Z}_m$ durch $\bar{a}+\bar{b}:= \bar{a+b}$ und $\bar{a}*\bar{b}:=\bar{a*b}$ für $a,b\in \mathbb{Z}$. Wohldefiniert: aus $\bar{a}=\bar{a'}$ und $\bar{b}=\bar{b'}$ folgt $\bar{a+b}=\bar{a'+b'}$. Analog für Multiplikation. -Eigenschaften von $\Z$ mit +,* werden auf $\Z$ mit +,* "vererbt", z.B. Distributivgesetz. +Eigenschaften von $\mathbb{Z}$ mit +,* werden auf $\mathbb{Z}$ mit +,* "vererbt", z.B. Distributivgesetz. -Satz: Sei $m\geq 2$ dann ist $\Z_m$ mit +,* ein kommutativer Ring mit $\bar{1}\not=\bar{0}$. Genau dann ist $\Z_m$ sogar ein Körper, wenn m eine Primzahl ist. +Satz: Sei $m\geq 2$ dann ist $\mathbb{Z}_m$ mit +,* ein kommutativer Ring mit $\bar{1}\not=\bar{0}$. Genau dann ist $\mathbb{Z}_m$ sogar ein Körper, wenn m eine Primzahl ist. Satz: Genau dann gibt es einen Körper mit n ELementen, wenn n eine Primzahl ist. D.h.. wenn $n=p^a$ ist für eine Primzahl p und $a\geq 1$. -## Konstruktion von $\mathbb{Q}$ aus $\Z$ -Sei $M=\Z x(\Z /{0}$ die Menge von Brüchen. Durch $(a,b)\sim (c,d)\leftrightarrow ad=bc$ wird Äquivalenzrelation auf M durchgefühert. Schreibweise für die Äquivalenzklassen $\frac{a}{b}$ Die Elemente von $\mathbb{Q} :{\frac{a}{b}:a,b\in\Z, b\not=0}$ heißten rationale Zahlen. +\paragraph{Konstruktion von $\mathbb{Q}$ aus $\mathbb{Z}$} +Sei $M=\mathbb{Z} x(\mathbb{Z} /{0}$ die Menge von Brüchen. Durch $(a,b)\sim (c,d)\leftrightarrow ad=bc$ wird Äquivalenzrelation auf M durchgeführt. Schreibweise für die Äquivalenzklassen $\frac{a}{b}$ Die Elemente von $\mathbb{Q} :{\frac{a}{b}:a,b\in\mathbb{Z}, b\not=0}$ heißten rationale Zahlen. Definiere Operationen +,* auf $\mathbb{Q}$ wie folgt: -- $\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{b*d}$ (wohldefiniert) -- $\frac{a}{b}*\frac{c}{d} = \frac{a*c}{b*d}$ +\begin{itemize} + \item $\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{b*d}$ (wohldefiniert) + \item $\frac{a}{b}*\frac{c}{d} = \frac{a*c}{b*d}$ +\end{itemize} Satz: $\mathbb{Q}$ mit +,* ist ein Körper. -Durch $\frac{a}{b}\leq\frac{c}{d}$ wird eine totale Ordnung auf $\mathbb{Q}$ definiert. Konstruktion von $\R$ aus $\mathbb{Q}$ mit Dedchin-Schnitten. -### Ring der formalen Potenzreihe -Sei k ein Körper (oder nur ein Ring mit 1+0). Eine Folge $(a_0, a_1,...,a:n)\in K^{\N}$ mit Einträgen aus K heißt formale Potenzreihe. Die Folge (0,1,0,0,...) wird mit x bezeichnet. Statt $K^{\N}$ schreibt man $K[[x]]$. $(0_0,a_1,a_2,...)$ heißt Polynom in x, falls es ein $d\in\N$ gibt mit $a_j=0$ f.a. $j0$, dann ist $p_B:B\rightarrow [0,1]; p_B(\omega)=\frac{p(\omega)}{p(B)}$ eine Verteilung auf B, denn $\sum p_b(\omega)=\sum \frac{p(\omega)}{p(B)}=\frac{1}{p(B)} \sum p(\omega)= \frac{1}{p(B)} p(B)= 1$. $p_B$ ist die durch B bedingte Verteilung. Für $A\subseteq \Omega$ gilt $p_B(A\cap B)=\sum p_B(\omega)=\sum\frac{p(\omega)}{p(B)}= \frac{p(A\cap B)}{p(B)}:= p(A|B)$ ("p von A unter B") bedingte Wahrscheinlichkeit von A unter B. @@ -555,95 +600,98 @@ Satz (Bayer, erweitert): $A_1,...,A_k,B$ wie eben, $p(B)>0$. Für $i\in {1,...,k Bespiel: In einem Hut liegen drei beidseitig gefärbte Karten. Jemand zieht ("zufällig") eine Karte und leg sie mit einer ("zufälligen") Seite auf den Tisch. Karten rot/rot, rot/blau und blau/blau. Gegeben er sieht rot, wie groß ist die Wahrscheinlichkeit, dass die andere Seite auch rot ist? p(unten rot | oben rot) = p(unten rot und oben rot)/p(oben rot) = $\frac{p\binom{r}{r}}{p(\binom{r}{r}\binom{r}{b})}=\frac{\frac{2}{6}}{\frac{3}{6}}=\frac{2}{3}$ -Eine Funktion $X:\Omega \rightarrow \R$ heißt (reellwertige) Zufallsvariable. Weil $\Omega$ endlich ist, ist auch $X(\Omega)={X(\omega): \omega \in \Omega}\subseteq \R$ endlich. Durch $p_x(x):=p(X=x):=p({\omega \in \Omega: X(\omega)=x})$ wird ein Wahrscheinlichkeitsraum $(X(\Omega),p_x)$ definiert; denn $\sum p_x(x)=p(\Omega)=1$. $p_x$ heißt die von X induzierte Verteilung. $X(\Omega)$ ist meist erheblich kleiner als $\Omega$. -Beispiel: Augensumme beim Doppelwurf: $X:\Omega\rightarrow \R, X((i,j))=i+j \rightarrow X(\Omega)={2,3,4,...,12}$ +Eine Funktion $X:\Omega \rightarrow \mathbb{R}$ heißt (reellwertige) Zufallsvariable. Weil $\Omega$ endlich ist, ist auch $X(\Omega)={X(\omega): \omega \in \Omega}\subseteq \mathbb{R}$ endlich. Durch $p_x(x):=p(X=x):=p({\omega \in \Omega: X(\omega)=x})$ wird ein Wahrscheinlichkeitsraum $(X(\Omega),p_x)$ definiert; denn $\sum p_x(x)=p(\Omega)=1$. $p_x$ heißt die von X induzierte Verteilung. $X(\Omega)$ ist meist erheblich kleiner als $\Omega$. +Beispiel: Augensumme beim Doppelwurf: $X:\Omega\rightarrow \mathbb{R}, X((i,j))=i+j \rightarrow X(\Omega)={2,3,4,...,12}$ -Satz: Seien $(\Omega_1, p_1),(\Omega_2, p_2)$ Wahrscheinlichkeitsräume und $(\Omega, p)$ ihr Produktraum. Sei $X:\Omega_1\rightarrow\R,Y:\Omega_2\rightarrow \R$, fasse X,Y als ZVA in $\Omega$ zusammen $X((\omega_1,\omega_2))=X(\omega_1)$ und $Y((\omega_1,\omega_2))=Y(\omega_2)$; d.h. X,Y werden auf $\Omega$ "fortgesetzt". Dann sind X,Y stochastisch unabhängig in $(\Omega, p)$ (und $p(X=x)=p_1(X=x), p(Y=y)=p_2(Y=y)$). +Satz: Seien $(\Omega_1, p_1),(\Omega_2, p_2)$ Wahrscheinlichkeitsräume und $(\Omega, p)$ ihr Produktraum. Sei $X:\Omega_1\rightarrow\mathbb{R},Y:\Omega_2\rightarrow \mathbb{R}$, fasse X,Y als ZVA in $\Omega$ zusammen $X((\omega_1,\omega_2))=X(\omega_1)$ und $Y((\omega_1,\omega_2))=Y(\omega_2)$; d.h. X,Y werden auf $\Omega$ "fortgesetzt". Dann sind X,Y stochastisch unabhängig in $(\Omega, p)$ (und $p(X=x)=p_1(X=x), p(Y=y)=p_2(Y=y)$). -## Erwartungswert, Varianz, Covarianz -Sei $X:\Omega\rightarrow \R$ ZVA im Wahrscheinlichkeitsraum $(\Omega, p)$. $E(X)=\sum_{x\in X(\Omega)}x p(X=x)=\sum_{\omega in Omega} X(\omega)p(\omega)$ "E verhält sich wie Integral"; E(x) heißt Erwartungswert von x. +\paragraph{Erwartungswert, Varianz, Covarianz} +Sei $X:\Omega\rightarrow \mathbb{R}$ ZVA im Wahrscheinlichkeitsraum $(\Omega, p)$. $E(X)=\sum_{x\in X(\Omega)}x p(X=x)=\sum_{\omega in Omega} X(\omega)p(\omega)$ "E verhält sich wie Integral"; E(x) heißt Erwartungswert von x. -Linearität des Erwartungswertes: $E(x+y)=E(x)+E(y)$ und $E(\alpha x)=\alpha E(x)$. -Ist $X:\Omega\rightarrow \R$ konstant gleich c, so ist $E(x)=\sum x*p(X=x)=c*p(X=x)=c*1=c$. - -Die Varianz von X: $Var(X)=E((X-E(X))^2)$ heißt Varianz von X (um E(X)). -Die Covarianz: $Cov(X,Y)=E((X-E(X))*(Y-E(Y)))$ heißt Covarianz von X und Y. -Der Verschiebungssatz: $Cov(X,Y)=E(X*Y)-E(X)*E(Y)$ +Linearität des Erwartungswertes: $E(x+y)=E(x)+E(y)$ und $E(\alpha x)=\alpha E(x)$.\ +Ist $X:\Omega\rightarrow \mathbb{R}$ konstant gleich c, so ist $E(x)=\sum x*p(X=x)=c*p(X=x)=c*1=c$.\ +Die Varianz von X: $Var(X)=E((X-E(X))^2)$ heißt Varianz von X (um E(X)).\ +Die Covarianz: $Cov(X,Y)=E((X-E(X))*(Y-E(Y)))$ heißt Covarianz von X und Y.\ +Der Verschiebungssatz: $Cov(X,Y)=E(X*Y)-E(X)*E(Y)$\ $Var(X)=Cov(X,X)=E(X*X)-E(X)E(X)=E(X^2)-(E(X))^2$ Seien X,Y stochastisch unabhängig ($\leftrightarrow p(X=x \wedge Y=y)=p(X=x)*p(Y=y)$) -$E(X)*E(Y)=\sum_{x\in X(\Omega)} x*p(X=x)* \sum_{y\in Y(\Omega)} y*p(Y=y)=\sum_{x\in X(\Omega)} \sum_{y\in Y(\Omega)} xy*p(X=x)p(Y=y)=\sum_{Z\in\R} z*p(X*Y=Z) = E(X*Y)$ -Sind X,Y stochastisch unanhängig ZVA, so ist $E(X)*E(Y)=E(X*Y)$; folglich $Cov(X,Y)=0$ +$E(X)*E(Y)=\sum_{x\in X(\Omega)} x*p(X=x)* \sum_{y\in Y(\Omega)} y*p(Y=y)=\sum_{x\in X(\Omega)} \sum_{y\in Y(\Omega)} xy*p(X=x)p(Y=y)=\sum_{Z\in\mathbb{R}} z*p(X*Y=Z) = E(X*Y)$. +Sind X,Y stochastisch unabhängig ZVA, so ist $E(X)*E(Y)=E(X*Y)$; folglich $Cov(X,Y)=0$ Satz: Seien X,Y ZVA, dann gilt $Var(X+Y)=Var(x)+Var(Y)+2*Cov(X,Y)$. Sind insbesondere X,Y unabhängig gilt: $Var(X+Y)=Var(X)+Var(Y)$. -Sei $(\Omega, p)$ Wahrscheinlichkeitsraum, $X:\Omega\rightarrow \R$ Zufallsvariable heißt Bernoulliverteilt im Parameter p falls $p(X=1)=p$ und $p(X=0)=1-p$, $p\in [0,1]$. $E(X)=\sum x*p(X=x)= 1*p(X=1)=p$ +Sei $(\Omega, p)$ Wahrscheinlichkeitsraum, $X:\Omega\rightarrow \mathbb{R}$ Zufallsvariable heißt Bernoulliverteilt im Parameter p falls $p(X=1)=p$ und $p(X=0)=1-p$, $p\in [0,1]$. $E(X)=\sum x*p(X=x)= 1*p(X=1)=p$ Für $X:\Omega\rightarrow {0,1}$ ist $X^2=X$: $Var(X)=E(X^2)-E(X)^2 = p-p^2 = p(1-p)=p*q$ -## Binominalkoeffizienten -Sei N eine Menge, dann ist $\binom{N}{k} := (x \subseteq N: \text{x hat genau k Elemente } (|x|=k) )$ für $k\in \N$. Für $n\in \N$ sei $\binom{n}{k}:=|(\binom{1,...,k}{k})$. +\paragraph{Binominalkoeffizienten} +Sei N eine Menge, dann ist $\binom{N}{k} := (x \subseteq N: \text{x hat genau k Elemente } (|x|=k) )$ für $k\in \mathbb{N}$. Für $n\in \mathbb{N}$ sei $\binom{n}{k}:=|(\binom{1,...,k}{k})$. Satz: $\binom{n}{0}={n}{n}=1$ f.a. $n\geq 0$ $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ f.a. $n\geq 0,k\geq 1, k\geq n-1$ Jede n-elementige Menge N ist $\binom{N}{0}=(\emptyset), \binom{N}{n}={N}\rightarrow \binom{n}{0}=\binom{n}{n}=1$. Den zweiten Teil der Behauptung zeigt man induktiv über n. +\paragraph{Binominalsatz} +$(a+b)^n = \sum_{k=0}^n a^k b^{n-k}$ für $a,b\in \mathbb{R}$. +Für $n\in \mathbb{N}$ sei $n!=n(n-1)(n-2)...*3*2*1=\prod i$; für $n\in\mathbb{N}$ und $k\geq 0$ sei $[\binom{n}{k}]=\frac{n!}{k!(n-k)!}$ -## Binominalsatz -$(a+b)^n = \sum_{k=0}^n a^k b^{n-k}$ für $a,b\in \R$ -Für $n\in \N$ sei $n!=n(n-1)(n-2)...*3*2*1=\prod i$; für $n\in\N$ und $k\geq 0$ sei $[\binom{n}{k}]=\frac{n!}{k!(n-k)!}$ - -Satz: $\binom{n}{0}=\binom{n}{n}=1$ für jedes $n\in\N$, $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$ für $k\geq 1$ und $k\leq n-1$. -Zweiter Teil: $[\binom{n-1}{k}]+[\binom{n-1}{k-1}]=\frac{n!}{k!(n-k)!} = [\binom{n}{k}]$. Also stimmen die Rekursionsgleichungen von $\binom{n}{k}$ und $[\binom{n}{k}]$ überein sowie $\binom{n}{k}=[\binom{n}{k}]$. Folglich ist die Anzahl k-elementiger Teilmengen eine n-elementige Menge gleich $\frac{n!}{k!(n-k)!}. +Satz: $\binom{n}{0}=\binom{n}{n}=1$ für jedes $n\in\mathbb{N}$, $\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}$ für $k\geq 1$ und $k\leq n-1$. +Zweiter Teil: $[\binom{n-1}{k}]+[\binom{n-1}{k-1}]=\frac{n!}{k!(n-k)!} = [\binom{n}{k}]$. Also stimmen die Rekursionsgleichungen von $\binom{n}{k}$ und $[\binom{n}{k}]$ überein sowie $\binom{n}{k}=[\binom{n}{k}]$. Folglich ist die Anzahl k-elementiger Teilmengen eine n-elementige Menge gleich $\frac{n!}{k!(n-k)!}$. Seien $X_1,...,X_n$ unabhängige ZVAen, alle $X_i$ seien Bernoulli-Verteilt im Parameter $p[0,1]$, d.h. $p(X_1=1)=p$, $p(X_i=0)=(1-p)$. Dann ist $X_i=X_1+X_2+...+X_n$ ebenfalls reelwertige ZVA. Im Fall $X_i:\Omega\rightarrow {0,1}$ ist $X:\Omega\rightarrow {0,1,...,n}$. Die Verteilung von X ergibt sich wie folgt, für $k\in {0,1,...,n}$: $p(X=k)=\binom{n}{k}*p^k(1-p)^{n-k}$ Eine ZVA heißt binominalverteilt in den Parametern n und p falls gilt: $p(X=k)=\binom{n}{k}p^k (1-p)^{n-k}$ für $k\in{0,1,...,n}$; schreibe $X\sim L(n,p)$. Sonst ist X Bernoulliverteilt (genau dann wenn $X\sim L(1,p)$). -## Erwartungswert und Varianz -Sei $X\sim L(n,p)$ OBdA $X=X_1,+...+X_n$ wobei $X_i$ unabhängig und Bernoulliverteilt. +\paragraph{Erwartungswert und Varianz} +Sei $X\sim L(n,p)$ OBdA $X=X_1,+...+X_n$ wobei $X_i$ unabhängig und Bernoulliverteilt.\ +$E(X)=n*p$, $E(X_i)=p$\ +$Var(X)=n*p*(1-p)$, $Var(X_i)=p*(1-p)$ -$E(X)=n*p$, $E(X_i)=p$ +\paragraph{Multinominalverteilung} +$\binom{N}{k_1,...,k_n}$ sei Menge der Abbildungen $f:N\rightarrow {1,...,r}$ mit $k1,...,k_r\geq 0$, $k_1+...+k_r=|\mathbb{N}|$ und $f^{-1}[{j}]=k_j \binom{n}{k_1,...,k_r} = |\binom{N}{k_1,...,k_r}$. -$Var(X)=nÜp*(1-p)$, $Var(X_i)=p*(1-p)$ - -## Multinominalverteilung -$\binom{N}{k_1,...,k_n}$ sei Menge der Abbildungen $f:N\rightarrow {1,...,r}$ mit $k1,...,k_r\geq 0$, $k_1+...+k_r=|\N|$ und $f^{-1}[{j}]=k_j \binom{n}{k_1,...,k_r} = |\binom{N}{k_1,...,k_r}$. - -## Hypergeometrische Verteilung +\paragraph{Hypergeometrische Verteilung} Beispiel: Urne mit zwei Sorten Kugeln; N Gesamtzahl der Kugeln, M Gesamtzahl Kugeln Sorte 1, N-M Gesamtzahl Kugeln Sorte 2, $n\leq N$ Anzahl Elemente einer Stichprobe. X Anzahl der Kugeln Sorte 1 in einer zufälligen n-elementigen Stichprobe. -$p(X=k)=\frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}} -Eine ZVA $X:\Omega\rightarrow \R$ heißt hypergeometrisch Verteilt in den Parametern M,N,n falls $p(X=k)$ für alle $k\geq 0, k\geq M$ gilt. +$p(X=k)=\frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}$ +Eine ZVA $X:\Omega\rightarrow \mathbb{R}$ heißt hypergeometrisch Verteilt in den Parametern M,N,n falls $p(X=k)$ für alle $k\geq 0, k\geq M$ gilt. $E(X)=\sum_{x=0}^M \frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}=...=n*\frac{M}{N}$ $Var(X)=E(X^2)-E(X)^2 =...= n*\frac{M}{N}(1-\frac{M}{N})(\binom{N-n}{N-1})$ -\section{Elementare Graphentheorie + +\section{Elementare Graphentheorie} $G=(V,E)$ heißt Graph mit Eckenmenge $V(G)=V$ und Kantenmenge $E(G)=E\subseteq {{x,y}:x\not=y \in V}$. Veranschaulichung als Punkte in der Ebene (V) mit "Verknüpfungslinien" von x nach y. Bsp $G=({1,2,3,4},{12,13,14,15,16})$. -$P=x_0,...,x_e$ Folge pw verschiedener Ecken mit $x_{i-1},...,x_i \in E(k)$ für $i\in{1,...,l}$ heißt ein Weg von $x_0$ nach $x_e$ der Länge l. Für $(a,b)\in V(G)$ heißt $d_G(a,b)=min{l: es_gibt_einen_a,b-Weg_der_Länge_l}$ Abstand von a nach b. Falls es keinen a,b-Weg gibt, definiere $d_G(a,b)=+\infty$. +$P=x_0,...,x_e$ Folge pw verschiedener Ecken mit $x_{i-1},...,x_i \in E(k)$ für $i\in{1,...,l}$ heißt ein Weg von $x_0$ nach $x_e$ der Länge l. Für $(a,b)\in V(G)$ heißt $d_G(a,b)=min(l: \text{ es gibt einen a,b-Weg der Länge l} )$ Abstand von a nach b. Falls es keinen a,b-Weg gibt, definiere $d_G(a,b)=+\infty$. $a\sim b \leftrightarrow$ es gibt einen a,b-Weg in G wird eine Äquivalenzrelation auf V(G) definiert. DIe Äquivalenzklassen heißen (Zusammenhangs-) Komponenten von G. -G heißt zusammenhängend, wenn G höchstens eine Komponente besitzt. $d_G: V(G) x V(G) \leftrightarrow \R_{\geq 0}$ ist eine Matrix -1. $d_G(x,y)=0 \leftrightarrow x=y$ f.a. $x,y \in V(G)$ -2. $d_G(x,y)=d_G(y,x)$ f.a. $x,y\in V(F)$ -3. $d_G(x,z)\leq d_G(x,y) + d_G(y,z))$ f.a. $x,y,z \in V(G)$ +G heißt zusammenhängend, wenn G höchstens eine Komponente besitzt. $d_G: V(G) x V(G) \leftrightarrow \mathbb{R}_{\geq 0}$ ist eine Matrix +\begin{itemize} + \item $d_G(x,y)=0 \leftrightarrow x=y$ f.a. $x,y \in V(G)$ + \item $d_G(x,y)=d_G(y,x)$ f.a. $x,y\in V(F)$ + \item $d_G(x,z)\leq d_G(x,y) + d_G(y,z))$ f.a. $x,y,z \in V(G)$ +\end{itemize} Für $A\subseteq V(G)$ sei $G[A]:= (A, {x,y\in E(G):x,y\in A})$. Für $F\subseteq E(G)$ sei $G[F]:=(V(G), F)$. $G[A]$ bzw $G[F]$ heißt von A bzw F induzierte Teilgraph. Ein Graph H mit $V(H)\subseteq V(G)$ und $E(H)\subseteq E(G)$ heißt Teilgraph von G, schreibweise $H\leq G$. $\leq$ ist Ordnung, denn: -1. $G\leq G$ -2. $H\leq G \wedge G\leq H \rightarrow H=G$ -3. $H\leq G \wedge G=L \rightarrow H\leq L$ +\begin{itemize} + \item $G\leq G$ + \item $H\leq G \wedge G\leq H \rightarrow H=G$ + \item $H\leq G \wedge G=L \rightarrow H\leq L$ +\end{itemize} Ist $P=x_0,...,x_p$ Weg, so heißt auch der Teilgraph ein Weg von $x_0$ nach $x_e$. Graphen G, H heißen isomorph, falls es einen Isomorphismus von V(G) nach V(H) gibt. Das heißt eine Bijektion. $V(G)\rightarrow V(H)$ mit $f(x)f(y)\in E(H)\leftrightarrow x,y \in E(G)$. Es gilt: -1. $G\cong G$ -2. $G\cong H \rightarrow H \cong G$ -3. $G\cong H \wedge H\cong L \rightarrow G\cong L$ +\begin{itemize} + \item $G\cong G$ + \item $G\cong H \rightarrow H \cong G$ + \item $G\cong H \wedge H\cong L \rightarrow G\cong L$ +\end{itemize} -Eine Folge $C=x_0,x_1,...,x_{l-1}$ von Ecken mit $x_i,x_{i+1}\in E(G)$ für $i\in {0,...,l-2}$ und $x_{l-1}x_0 \in E(G)$ heißt Kreis in G der Länge l, falls $x_0,...,x_{l-1}$ pw versceiden sind. Bsp: Kreis der Länge 5. +Eine Folge $C=x_0,x_1,...,x_{l-1}$ von Ecken mit $x_i,x_{i+1}\in E(G)$ für $i\in {0,...,l-2}$ und $x_{l-1}x_0 \in E(G)$ heißt Kreis in G der Länge l, falls $x_0,...,x_{l-1}$ pw verschieden sind. Bsp: Kreis der Länge 5. -EIn Teilgraph H des Graphen G (also $H\leq G$) heißt aufspannend, falls $V(H)=V(G)$. Für eine Ecke $x\in V(G)$ sei $d_G(x)=|{x,y\in E(G), y\in V(G)}|$ die Anzahl der mit x indizierten Kanten, der sogenannte Grad von x in G. +Ein Teilgraph H des Graphen G (also $H\leq G$) heißt aufspannend, falls $V(H)=V(G)$. Für eine Ecke $x\in V(G)$ sei $d_G(x)=|{x,y\in E(G), y\in V(G)}|$ die Anzahl der mit x indizierten Kanten, der sogenannte Grad von x in G. Weiter $N_G(x):={x\in V(G): xy \in E(G)}$ die Menge der nachbarn von x in G. Hier gilt: $|N_G(x)=d_G(x)|$. @@ -652,9 +700,11 @@ In jedem Graph G gilt $\sum_{x\in V(G)} d_G(x)=2|E(G)|$. Der Durchschnittsgrad v Ein Graph ist ein Baum wenn G zusammenhängend und G-e nicht zusammenhängend für jedes $e\in E(G)$ "G ist minimal zusammenhängend" Graph G ist ein Baum wenn G kreisfrei und Graph G+xy nicht kreisfrei für jedes $xy \not\in E(G)$ G ist Baum, wenn -- G ist kreisfrei und zusammenhängend -- G kreisfrei und $|E(G)|=|V(G)|-1$ -- G zusammenhängend und $|E(G)|=|V(G)|-1$ +\begin{itemize} + \item G ist kreisfrei und zusammenhängend + \item G kreisfrei und $|E(G)|=|V(G)|-1$ + \item G zusammenhängend und $|E(G)|=|V(G)|-1$ +\end{itemize} Jeder Baum mit wenigstens einer Ecke besitzt eine Ecke vom Grad $\leq 1$, ein sog. Blatt ("jeder Baum besitzt ein Blatt"). $\rightarrow E(G)=|V(G)|-1$ für jeden Baum also $d(G)=\frac{2|V(G)| -2}{|V(G)|}<2$. @@ -670,27 +720,29 @@ Ein Spannbaum T von G heißt Breitensuchbaum von G bei $x\in V(G)$ falls $d_F(z, Ein Spannbaum T von G heißt Tiefensuchbaum von G bei $x\in V(G)$ falls für jede Kante zy gilt: z liegt auf dem y,x-Weg in T oder y liegt auf dem z,t-Weg in T. Satz: Sei G zusammenhängender Graph $x\in V(G)$. -(X) sind $x_0,...,x_{e-1}$ schon gewählt und gibt es ein $+ \in {0,..., e-1}$ so, dass $x_+$ einen Nachbarn y in V(G)\{$x_0,...,x_{e-1}$}, so setze $x_e=y$ und $f(e):=t$; iteriere mit $e+1$ statt e. +(X) sind $x_0,...,x_{e-1}$ schon gewählt und gibt es ein $+ \in (0,..., e-1)$ so, dass $x_{+}$ einen Nachbarn y in $V(G)\ (x_0,...,x_{e-1} )$, so setze $x_e=y$ und $f(e):=t$; iteriere mit $e+1$ statt e. Dann ist $T:=({x_0,...,x_e},{x_j*x_{f(j)}: j\in {1,...,e}})$ ein Spannbaum -- (X) wird in + stets kleinstmöglich gewählt, so ist T ein Breitensuchbaum -- wird in (X) + stets größtmöglich gewählt, so ist T ein Tiefensuchbaum +\begin{itemize} + \item (X) wird in + stets kleinstmöglich gewählt, so ist T ein Breitensuchbaum + \item wird in (X) + stets größtmöglich gewählt, so ist T ein Tiefensuchbaum +\end{itemize} -## Spannbäume minimaler Gewichte +\paragraph{Spannbäume minimaler Gewichte} G Graph, $F \subseteq E(G)$ heißt kreisfrei, falls G(F) kreisfrei ist. Lemma (Austauschlemma für Graphen): Seien F, F' zwei kreisfreie Kantenmengen in Graph G und $|F|<|F'|$, dann gibt es ein $e \in F'/F$ so, dass $F\vee {e}$ kreisfrei ist. -G, $\omega:E(G)\rightarrow \R$ (Gewichtsfunktion an den Kanten). Für $F\subseteq E(G)$ sei $\omega (F)=\sum \omega (e)$, speziell $\omega (\emptyset)=0$. +G, $\omega:E(G)\rightarrow \mathbb{R}$ (Gewichtsfunktion an den Kanten). Für $F\subseteq E(G)$ sei $\omega (F)=\sum \omega (e)$, speziell $\omega (\emptyset)=0$. Für einen Teilgraphen H von G sei $\omega (G)=\omega (E(G))$. Ein Spannbaum minimalen Gewichts ist ein Spannbaum T von G mit $\omega (T)\leq \omega (S)$ für jeden Spannbaum S von G. -Satz (Kruskal): Sei G zuständiger Graph, $\omega:E(G)\rightarrow \R$; Setze $F=\emptyset$. Solange es eine Kante $e\in E(G)\\F$ gibt so, dass $F \vee {e}$ kreisfrei ist, wähle e mit minimalem Gewicht $\omega(e)$, setzte $F=F\vee {e}$, iterieren. Das Verfahren endet mit einem Spannbaum $T=G(F)$ minimalen Gewichts. +Satz (Kruskal): Sei G zuständiger Graph, $\omega:E(G)\rightarrow \mathbb{R}$; Setze $F=\emptyset$. Solange es eine Kante $e\in E(G)/F$ gibt so, dass $F \vee (e)$ kreisfrei ist, wähle e mit minimalem Gewicht $\omega(e)$, setzte $F=F\vee {e}$, iterieren. Das Verfahren endet mit einem Spannbaum $T=G(F)$ minimalen Gewichts. -Beweis: Weil G endlich ist endet das Verfahren mit einem maximal kreisfreien Graphen T. Seien $e_1,...,e_n$ die Kanten von T in der Reihenfolge ihres Erscheinens, sei S Spannbaum minimalen Gewichts und $f_1,...,f_m$ die Kanten in Reihenfolge aufsteigenden Gewichts. Angenommen (redactio ad absurdum) $\omega(T)>\omega(S)$. Dann gibt es ein $i\in{1,...,m}$ mit $\omega(e_i)>\omega(f_i)$. Wähle i kleinstmöglich, dann ist $F={e_1,...,e_{i-1}}$ und $F'={f_1,...,f_i}$ kreisfrei. Nach Austaschlemma gibt es ein $f\in F'/F$ so, dass $F\vee {f}$ kreisfrei ist. Also ist f ein Kandidat bei der Auswahl von $e_i$ gewesen, also $\omega(e_i)\leq \omega(f)$ (Fehler!). Folglich ist $\omega(T)\leq \omega(S) \Rightarrow \omega(T)=\omega(S)$ also T und S Spannbaum mit minimalen Gewichten. +Beweis: Weil G endlich ist endet das Verfahren mit einem maximal kreisfreien Graphen T. Seien $e_1,...,e_n$ die Kanten von T in der Reihenfolge ihres Erscheinens, sei S Spannbaum minimalen Gewichts und $f_1,...,f_m$ die Kanten in Reihenfolge aufsteigenden Gewichts. Angenommen (redactio ad absurdum) $\omega(T)>\omega(S)$. Dann gibt es ein $i\in{1,...,m}$ mit $\omega(e_i)>\omega(f_i)$. Wähle i kleinstmöglich, dann ist $F={e_1,...,e_{i-1}}$ und $F'={f_1,...,f_i}$ kreisfrei. Nach Austauschlemma gibt es ein $f\in F'/F$ so, dass $F\vee {f}$ kreisfrei ist. Also ist f ein Kandidat bei der Auswahl von $e_i$ gewesen, also $\omega(e_i)\leq \omega(f)$ (Fehler!). Folglich ist $\omega(T)\leq \omega(S) \Rightarrow \omega(T)=\omega(S)$ also T und S Spannbaum mit minimalen Gewichten. -## Das Traveling Salesman Problem -G sei Graph (vollständig) auf n Ecken, d.h. $xy\in E(G) \forall x\not =y$ aus V(G) und $\omega*E(G)\rightarrow \R$. Finde aufspannenden Kreis C von G minimalen Gewichts. Zusatzannahme (metrische TSP) $\omega(xz)\leq \omega(xy)+\omega(yz)$. +\subsection{Das Traveling Salesman Problem} +G sei Graph (vollständig) auf n Ecken, d.h. $xy\in E(G) \forall x\not =y$ aus V(G) und $\omega*E(G)\rightarrow \mathbb{R}$. Finde aufspannenden Kreis C von G minimalen Gewichts. Zusatzannahme (metrische TSP) $\omega(xz)\leq \omega(xy)+\omega(yz)$. Finde einen aufspannenden Kreis C, der um einen Faktor von höchstens zwei von einem aufspannenden Kreis D minimalen Gewichts abweicht ($\omega(C)\leq 2 \omega(D)$) sog. Approximationsalgorithmus mit Gütefaktor $\leq$. Konstruiere eine Folge$x_0,...,x_m$ mit der Eigenschaft, dass jede Kante von T genau zweimal zum Übergang benutzt wird, d.h. zu $e\in E(T)$ existieren $i\not = j$ mit $e=x_i x_{i+1}$ und $e=x_j x_{j+1}$ und zu jedem k existieren $e\in E(T)$ mit $e=x_k x_{k+1}$. Das Gewicht dieser Folge sei $\sum \omega(x_i x_{i+1})= 2\omega(T)$. @@ -699,20 +751,14 @@ Eliminiere Mehrfachnennungen in der Folge. Gibt es $i\not= j$ mit $x_j=x_i$ so s G Graph, $k\geq 0$. Eine Funktion $f:V(G)\rightarrow C$ mit $|C|\leq k$ heißt k-Färbung, falls $f(x)\not = f(y)$ für $xy\in E(G)$. G heißt k-färbbar, falls G eine k-Färbung besitzt. Das kleinste $k\geq 0$ für das G k-färbbar ist heißt dramatische Zahl von G, Bezeichnung $X(G)$. - - Satz (Tuga): Sei $k\geq 2$ und G ein Graph ohne Kreise eine Lösung $l\equiv 1 mod k$, dann ist G k-faltbar. G 2-färbbar $\leftrightarrow$ G hat keine Kreise ungerader Länge. Ein Graph heißt bipartit mit den Klassen A,B falls $(x\in A \wedge y\in B)\vee (x\in B \wedge y\in A)$ für alle $xy \in E(G)$ gilt. Genau dann ist G bipoartit mit gewissen Klassen A,B wenn G 2-färbbar ist. -Satz (Hall) "Heiratssatz": Sei G bipartit mit Klassen A,B. Dann gilt G hat ein Matching von A $\leftrightarrow |N_G(X)|\leq |X|$ für alle $X\subseteq A$. +Satz (Hall): Sei G bipartit mit Klassen A,B. Dann gilt G hat ein Matching von A $\leftrightarrow |N_G(X)|\leq |X|$ für alle $X\subseteq A$. Satz: "$\rightarrow$" sei M Matching von A in G $\rightarrow |N_G(X)| \leq N_{G[M]}(X)=|X|$. "$\leftarrow$" Induktiv über $|V(G)|$. Ein schneller Zeuge für die Existenz eines Matchings von A im bipartiten Graphen G mit Klassen A,B ist das Matching selbst. Ein schneller Zeuge für die nicht-existenz eines Matchings ist ein $X\subseteq A$ mit $|N_G(X)| < |X|$. Das Entscheidungsproblem "hat ein bipartiter Graph ein Matching?" ist im NP und zugleich in co-NP. Also ist auch Problem "ist ein Graph 2-färbbar?" in NP und co-NP. Das Problem "ist ein Graph 3-färbbar" ist in NP. Es ist sogar NP-vollständig, d.h. jedes Problem in NP (jedes Entscheidungsproblem mit schnellen Zeugen für Ja) lässt sich in Polynomalzeit in dieses Färbungsproblem überführen. -Ein weiteres Problem in NP ist: aussagenlogische Formel gegeben $F=C_1 \wedge C_2 \wedge ... \wedge C_m$, jedes $C_b$ ist von der Form $P\vee Q \vee R$ mit $P=x_i; Q=x_j; R=x_2$ oder $P=\neg x_i; Q=\neg x_j; R=\neg x_2$. Auch dieses Problem ist NP-vollständig. -- SAT ist die "Mutter aller Probleme in NP" -- Färbbarkeit lässt sich darauf zurückführen und umgekehrt - - +\end{multicols} \end{document} \ No newline at end of file diff --git a/Grundlagen und Diskrete Strukturen - cheatsheet.pdf b/Grundlagen und Diskrete Strukturen - cheatsheet.pdf deleted file mode 100644 index b7b77eb9ddc31d45404cd396b75534fd95317859..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 5918 zcmV+(7vbn7P((&8F)lRBNG9VxzARr(L zFGgu>bY*fNFGg%(bY((gI0ZL#}C=jVvXkl*qn$=wE+lHgaY(T(l)=Z$DX<>6x?dGxU3t zp5OcKo-co%KEL{5Js_Q!zYpg&&RrQ(arOn&$fj;G3lRLi}r1 zEL!&_a&Pzs_7Ql;y~ImG6iU2N;Q2u!cKd0Ngs{OIhB)f~F$n#rzkGYOoC#XQxc`0t z_|N4p4FnBnfRIJ7M zgEan(mjZY3o5|Lp02}!_*a`Lgm@2T}%14m)NazKmeBf8d8j(j)q#k(Zpv@5Zu@})u z;3`G#2ncdi4;Th6(Vz`3q8>P}fh%r;EA-+xQ4buIa&#nUJ9O-cu@^)oF-{ztQZFQ- z1-7wiq|^IomPvkrvMid&K|R#QrZ6ZU_{zm<--}VuogUfE+i7-}f%-MTMUjV4(iyJ% zTfsF%DDRb(tsKPs_A?2VLD!3xwIS>h%i#X zel#@vqtb8r+d8Tw_Ci48pvneJOe**si1k!NKj_;gt;#9|v^ zewt3x2?MMFIYJ;pDv^7+oadJ_21bd`dq(xSa@XIK3=uznPcJ2>g&b5uK>BGmQxpgi z4@JnJz}R97V6M`UCyY2QwltuORgl2NxQY^piV}zxCDd}-4?OCZ;X*@hQ>HIUx~@ui zd&4LMfT&@gGD)w|#kEY)W|M64hM^6MJ1AXlP%=^fni>MMF_oSRDlHdtM*Xx=J{S_E zvTK|XdYUgYD;?9C#Z zH=47MdN@e-f*{9%_#O$6fdgl=#F?8g-D&KGAPZ%|SQJ?+nhEH68IHX6sqe~KAnL&f z4!wxyhoJ^X;Mm59BJaE6dSe?68&ro)*I?B`msFxri*8f)NIVo`16#8+!fd3FwC7yB zYE;q`i2!6Qj)*A}Pv=||E|-OHZa**ynJ>~=(I^&>Jm6$^vCy)hb7vB*WkC(Qkllr^ zP4;v!f*SdYy#yzF%*}UXH^yM1D6as9V2k;Ze@MsbA&FqFF$ZgHW&u|OaNQiLRA?%ekXvh%3HFY#rLEM-%~SjjMEo)GT4)jo>xzV zwT~dVH&YPl8;#|dm#8#H6$~6q0ZSf-bkn5G3ib57&*BkB9azchp{V~<#Xt%v2!Yd# z(eGm6fR1Nek?Gu8VSMeQz_FGgpRGn#!uv>}VEVYS93tvRoc~prqvCkmFTOig>|jFU zWCsiSLNn$&j8^N(yizd2m=~MOD}Ak1XBt9`gD?`+CM4E5OD9b1P3>}O)$hA-DpWj&i`nR#% z+BF|ozR$lH`L46yy;s8!QB|@t6LbPO{y_?VTd>Y4iJK+rdhW`JIe+_>HwNA)&>rk+yd#e~}(S11VPm18o7H;vsPFri@6fHzs-O3Co=?Uz zS=&k#Iv`U%_53I{3Li!jOjgBI$4hhpYg&Y>vS+H=TQ@@JU(7PuSt*t`;%_e;ZU5^u zd+p;u?fFA43y7E58(u$Lq^h;XJ1}x>&(XTGpq{Nm?5c5@jr^6Qbi#aL6orD)QOHtl zFYx7*r^`+dFQ7qBd+5Pn7(`M3Y|silg*J|`r(z6RRO9L8}OlZ))zh{aho+lo3vb3 z&PAgAR2Mcf^R3>d6Q~XS9pF2vrULiVV65XeZQ*xSP&{c{;h<~|JFq#d*gQlybmDB% zc6X$^xUc;ny%tv4KP`$xk%wa3qu8%T`0Jvq#6>9#hIs=*?Q9P4-gT^F#=hA!>fX59 zwbdK%*AgV3&k}qN4s%P8d`3%f7}wF9$5IGqsHj6?ntX~pHvTI~RhPMo`@-NkC5wU$ z*&K0HhxlI7z5VH1z1aByclTWE!cQS}$$zNS#s7Iy$EtIomr%^}BI2vuUOVu#oJ~%| zFo~RkmV>)ldYR6~`68RheEY|x9Ce(RS;P_I=val$6kpij3F-s zZlrXT|LtU17?heThi@L2!!CU1azM$PH6#6(%VKz_)Ugc%Cy+{$;L>rpsD}2AUjTTcp?8`0*8x z;aJVxiziZ9BDq_-PQ8}&W`ZLpy60+W$~s(YNO;{DSKp&?sGW0LGHPMOPXolL?>i)d zLi!PvG4Nv!FX~US3l0V@`Huv&atiDph5YkK{4LA*@l^RN7SA4wzZG*K5OOD-A>|~M zeQYZBW6ScGIg?i4Ky7)<$XUt_6S}F1HR)#7?YFa_Ivx*e3jgtDCfHKir$W}X8r%F@ z{q6BIy^)iVf>+|ijW~P&*=TGY)GOxf?Bz`UgLrUV+uB{4ML^)2UiR`W68INQtz6kL2-u#EuhaeodJ`X~cvZ%f#aPQY(jX zTdLvSTYFet&i2AuVhp!Dq_Qfdw-it&y-G+O=UZ;O*h(?C>Ro#dm$0ooQ~_G6{pl(Y zy2}-Kpds^73r>XT6vB~0uVL$D@&>Va;nXOEy%^zkjY6#&^$!L1TV@4Uh7qc(>bolL z3`qs@u~HFRRTKrpg!C+}Jd1Ts+;FXuAms%MS!y+m^h4$8?Dur09hzes8KSP@9tJdz z0GN~=SEq4&g;?&8w{mZrTITH*)qMDAdY8)E0?HG0E|(eccG{n#NNjklqqbUq6(k@t zNQm@qtnWMhr*`V>1F$UOBzB~bitT$(2t37iI z7cEz)RD|`=lb-6A&07$+3M)$0VCB%(N(7Pd!Qh)UeXD z%}Up#RWtW!z(}lFZ@$-}pB9S^DtevUhk6`F`=nJhPaT|1ya1z36<~*0r7l+r+f;~r zww^-mBHiu1Xl3iYSD)SKLny$ywMQS8M zT?ce({Y zAhf1q%(h62@N9mgbiHz3SNfG5d6^nB>CEb9jO5fCgz3aAt5j}9I-z&EpQ#^AGTnG} zgNEiwM<=_-gZnG5eK~FDty~7XucpedKBF(DUtp_yN5ax-W>KMXKU=z5((Y=P6+HP`181bn*2TguojBS( zr&?)b&A$?B-m|G%QabzE7`u$9__=uD=chMXkf6&?L9pcGeT|-=EfRx$^=Peu8ZFjow=}y4 zT>Piegim7)?%`dudD6#6%{cmtl@Sh17L4Vmpgno7Cy2-|)^-CvN(gPNrGaK(TN3K|bpM#O7V- zZAs6=j-ahE6?MO6cb|%BFH&{TT2F|#V$ubSkvrt5M&qA}BHJumEN*14v&IaJtSjwL zU9kqcs3RoieOzvOqRJ|~4lrqJ18yBV#oZm=cA?Ye%3e>2R-rhIn=^D*9u<~7_H43b zeaK1C;v~9B2cJF0kxS?6iSp`i+30i(<}k9sZc7QVIk@*sCEQJuX3U2hA|k2-T1`e4 z{3&mFt4r_NsufmiaJ@cY)+g4x5arCOoZdXPZkx_#^@SBz9vFWEEHtB^#$-wMUfd#_*`42>w`AveY=vdQC~E?izDI=dnjEm#{A$ z$--bZDfk_w_E~X_Z*EzUR%@E2ld1`%HhH|t*s^z3TgkERhb)&CZ@kLpH{*r6huYDd zo1|Cs{IYtrjYQ(*6NbgNE*ldPFRo4uzZkF)2N5XH#k1<|5Z9Y0aHH;$jpK|<+YHiK}tf(s%1qiE=?9s#1ng(_mubw`{3I2DfGEA`E^-z4!~i9+Z>q z-s9)@-rZw+_`RSP2G~P{^eFQAI}(1ox%cW9zjq0L1dGCt3Vz^^-tguLPY9;HXZL;- zuLm;?h+2ys_99_NBE0IDo3sq8SZI-mH=jD7r4ry;tkhCnuv$wqEU-cZHa6)77Ow`? 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