commit 7d95052b284deb2c354266570bd89c23ad3328c3
Author: Nis Boerge Wechselberg <enbewe@enbewe.de>
Date:   Thu Sep 11 22:07:17 2014 +0200

    Initial commit

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diff --git a/main.pdf b/main.pdf
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+\documentclass[10pt,a4paper,oneside]{scrartcl}
+\usepackage[utf8]{inputenc}
+\usepackage[german]{babel}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{graphicx}
+
+\newtheorem{defi}{Definition}
+\newtheorem*{beis}{Beispiel}
+
+\author{Nis BÃ¶rge Wechselberg, Christoph Daniel Schulze}
+\title{Beschreibung und Veranschaulichung des n-Kaffee-Problems}
+
+\begin{document}
+  \maketitle
+  \begin{abstract}
+    Das n-Kaffee-Problem beschreibt die Guthaben-Schulden-VerhÃ¤ltnisse in einer Gruppe von n Personen. Die VerhÃ¤ltnisse werden hierbei in ausgegebenen Kaffees notiert.
+    
+    FÃ¼r zwei beteiligte Personen ist dieses VerhÃ¤ltnis noch trivial zu betrachten. Es wird in diesem Artikel versucht die Visualisierung auch fÃ¼r mehr Personen mÃ¶glichst einfach zu halten und eine manuelle grafische Dokumentation zu ermÃ¶glichen.
+  \end{abstract}
+  
+  \begin{defi}[Kaffeerunde]
+    Als \textbf{Kaffeerunde K} bezeichnen wir die betrachtete Menge von \( n \in \mathbb{N} \) Personen. Hierbei werden die Personen als \( p_i, i < n\) bezeichnet und es gilt \( K = \{ p_0, \ldots, p_{n-1} \} \).
+  \end{defi}
+
+  \begin{beis}[2-Kaffee-Problem]
+    Sind nur 2 Personen \( p_0 \) und \( p_1 \) an der Kaffeerunde beteiligt, lÃ¤sst sich das 2-Kaffee-Problem als \( x \in \mathbb{Z} \) beschreiben. Hierbei gilt:
+    \begin{description}
+      \item[x = 0] Das VerhÃ¤ltnis ist ausgeglichen, niemand hat Kaffeeschulden.
+      \item[x $>$ 0] \( p_1 \) schuldet \( p_0 \) noch x Kaffees.
+      \item[x $<$ 0] \( p_0 \) schuldet \( p_1 \) noch x Kaffees.
+    \end{description}
+
+  	\begin{figure}[hbtp]
+      \centering
+      \includegraphics[scale=1]{2KaffeeProblem}
+      \caption{Beispiel fÃ¼r das 2-KÃ¶rper-Problem, $p_1$ schuldet $p_0$ noch 2 Kaffee}
+    \end{figure}
+  \end{beis}
+
+  \begin{defi}[n-Kaffee-Problem]
+    Das allgemeine \textbf{n-Kaffee-Problem} zu einer Kafferunde K mit n Personen lÃ¤sst sich definieren als \( x \in \mathbb{Z}^{n-1} \), also \( x = (x_0, \ldots, x_{n-1}) \). Zu diesem Tupel werden die Kaffeefunktion \( k_{i,j} : \mathbb{Z}^{n-1} \rightarrow \mathbb{Z}^{n-1} \) definiert durch
+    \[ k_{i,j}(x_0,\ldots,x_{n-1}) = 
+      \left\{ \begin{array}{cc}
+        (x_0,\ldots,x_{i-1},x_i +1, x_{i+1},\ldots,x_{n-1}) & $, falls $ i < n \\ 
+        (x_0-1, \ldots, x_{n-1}-1) & $, falls $ i = n
+      \end{array} \right.
+    \]
+    Diese Kaffeefunktion \( k_i \) gibt die VerÃ¤nderung der Guthaben-Schulden-VerhÃ¤ltnisse innerhalb der Kaffeerunde an, wenn \( p_i \) ein Kaffee \textbf{ausgegeben wird}.
+  \end{defi}
+
+  \begin{beis}[3-Kaffee-Problem]
+  Foo
+  	\begin{figure}[hbtp]
+      \centering
+      \includegraphics[scale=1]{3KaffeeProblem}
+      \caption{}
+    \end{figure}
+  \end{beis}
+
+\end{document}
