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Nis Boerge Wechselberg 2014-09-11 22:07:17 +02:00
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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 Börge 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-Verhältnisse in einer Gruppe von n Personen. Die Verhältnisse werden hierbei in ausgegebenen Kaffees notiert.
Für zwei beteiligte Personen ist dieses Verhältnis noch trivial zu betrachten. Es wird in diesem Artikel versucht die Visualisierung auch für mehr Personen möglichst einfach zu halten und eine manuelle grafische Dokumentation zu ermöglichen.
\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, lässt sich das 2-Kaffee-Problem als \( x \in \mathbb{Z} \) beschreiben. Hierbei gilt:
\begin{description}
\item[x = 0] Das Verhältnis 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 für das 2-Körper-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 lässt 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 Veränderung der Guthaben-Schulden-Verhältnisse 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}