Alles fertig. Alles? Nein, die Related Work leistet weiter erfolgreich Widerstand...
This commit is contained in:
Christoph Daniel Schulze
parent
949d1d7982
commit
17870dac19
8 changed files with 418 additions and 281 deletions
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main.pdf
BIN
main.pdf
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106
main.tex
106
main.tex
|
|
@ -114,7 +114,7 @@ Bla.
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%% Kaffeekränzchen und das Kaffeeproblem
|
||||
\section{Kaffeekränzchen und das Kaffeeproblem}
|
||||
\section{Erster Hauptteil}
|
||||
\label{sec:kaffeeproblem}
|
||||
|
||||
Um das Kaffee-Problem betrachten zu können,
|
||||
|
|
@ -336,7 +336,7 @@ analog zu \autoref{def:kaffeekassentransition}.
|
|||
& = & 0.
|
||||
\end{array}
|
||||
\]
|
||||
Und die bilanzierende Kaffeekasse
|
||||
Die bilanzierende Kaffeekasse
|
||||
ergibt sich also als
|
||||
\( k = (\Delta_1,\ldots,\Delta_n) \).
|
||||
\end{proof}
|
||||
|
|
@ -378,7 +378,7 @@ analog zu \autoref{def:kaffeekassentransition}.
|
|||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%% Visualisierung des Kaffeeproblems
|
||||
\section{Visualisierung des Kaffeeproblems}
|
||||
\section{Zweiter Hauptteil}
|
||||
\label{sec:visualisierung}
|
||||
Im folgenden stellen wir Möglichkeiten vor,
|
||||
das Kaffeeproblem für Gruppen aus 2 oder 3 Personen
|
||||
|
|
@ -390,7 +390,7 @@ Wie bereits in \autoref{beis:2-kaffee-problem} angesprochen,
|
|||
wird nur eine einzelne Komponente der Kaffeekasse benötigt.
|
||||
Somit stellen wir die Kaffeekasse wie in \autoref{fig:2-kaffee-problem} dar.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{figure}[tb]
|
||||
\centering
|
||||
\includegraphics[scale=1]{2KaffeeProblem}
|
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\caption{%
|
||||
|
|
@ -413,71 +413,79 @@ Analog zu \autoref{ssec:2-kaffee-problem} kann
|
|||
für die Darstellung des 3-Kaffee-Problems ein
|
||||
kartesisches Koordinatensystem verwendet werden,
|
||||
in dem die Werte von \(\Delta_1\) und \(\Delta_2\)
|
||||
als Koordinaten eingetragen werden.
|
||||
als Koordinaten eingetragen werden
|
||||
(\autoref{fig:3-kaffee-kartesisch}).
|
||||
Der sich ergebende Wert von \(\Delta_3\)
|
||||
kann auf den dort eingetragenen Diagonalen abgelesen werden,
|
||||
deren Wert nach links steigt und nach rechts fällt.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{figure}[tb]
|
||||
\centering
|
||||
\includegraphics[scale=.6]{3KaffeeKartesisch}
|
||||
\includegraphics[scale=.8]{3KaffeeKartesisch}
|
||||
\caption{%
|
||||
Visualisierung des \(3\)-Kaffee-Problems mit kartesischen Koordianten.
|
||||
Die Darstellung entspricht der Kaffeekasse \(k = (2,-1,-1) \).
|
||||
}
|
||||
\label{fig:3-kaffee-kartesisch}
|
||||
\end{figure}
|
||||
|
||||
In \autoref{fig:3-kaffee-kartesisch} ist eine
|
||||
kartesische Darstellung für die Kaffeekasse
|
||||
\(k = (2,-1,-1) \) gegeben.
|
||||
Die Aktualisierung der Kaffeekasse,
|
||||
und das Ablesen der Lösung des 3-Kaffee-Problems,
|
||||
ist hier allerdings relativ kompliziert.
|
||||
Gibt zum Beispiel \(p_1\) nun \(p_2\) einen Kaffee
|
||||
aus, so muss zunächst im Kopf die
|
||||
neue Kaffeekasse gebildet werden,
|
||||
und anschliessend die neuen Werte
|
||||
für \(\Delta_1\) und \(\Delta_2\)
|
||||
eingetragen werden.
|
||||
Zusätzlich ist der Wert von \(\Delta_3\)
|
||||
nur indirekt zu erkennen.
|
||||
Dieser ergibt sich aus dem negativen Achensabschnittes
|
||||
der fallenden Diagonalen, auf der der Punkt
|
||||
zur Zeit liegt.
|
||||
Die Aktualisierung der Darstellung muss erfolgen,
|
||||
indem der Punkt den Änderungen in \(\Delta_1\) und \(\Delta_2\) entsprechend
|
||||
verschoben wird.
|
||||
Man kommt hier zuweilen nicht drum herum,
|
||||
nachzudenken.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
Um dieses schwerwiegende Manko zu beheben
|
||||
bietet sich eine Visualisierung
|
||||
in einem gradlinigen, nichtorthogonalen Koordinatensystem an,
|
||||
wie in \autoref{fig:3-kaffee-schief} dargestellt.
|
||||
Die Achsen schneiden sich hier in einem Winkel von \(60\degree\).
|
||||
Die Darstellung unterscheidet sich prinzipiell
|
||||
nicht sonderlich von der Darstellung im orthogonalen Koordinatensystem.
|
||||
Allerdings ist das Verschieben der Markierung simpler:
|
||||
um einen ausgegebenen Kaffee einzutragen
|
||||
muss die Markierung "`mit dem Kaffee"'
|
||||
von der ausgebenden Person weg
|
||||
hin zur empfangenen Person verschoben werden.
|
||||
|
||||
\begin{figure}[tb]
|
||||
\centering
|
||||
\includegraphics[scale=.6]{3KaffeeProblem}
|
||||
\includegraphics[scale=.8]{3KaffeeProblem}
|
||||
\caption{%
|
||||
Visualisierung des \(3\)-Kaffee-Problems mit schiefen Koordianten
|
||||
Visualisierung des \(3\)-Kaffee-Problems mit schiefen Koordinaten.
|
||||
Die Darstellung entspricht der Kaffeekasse \(k = (2,-1,-1) \).
|
||||
}
|
||||
\label{fig:3-kaffee-schief}
|
||||
\end{figure}
|
||||
|
||||
Als alternative Darstellung kann ein
|
||||
geradliniges, nichtorthgonales Koordiantensystem
|
||||
gewählt werden. In \autoref{fig:3-kaffee-schief}
|
||||
schneiden sich die Achsen
|
||||
mit einem Winkel von 60\degree.
|
||||
Bei dieser Art des Koordinatensystems
|
||||
können die selben Koordinaten verwendet werden,
|
||||
die bereits im kartesischen Koordinatensystem berechnet wurden.
|
||||
Allerdings können hier die Namen der Personen an den Seiten
|
||||
plaziert werden, um eine einfacher Transition
|
||||
zu ermöglichen.
|
||||
Nun kann hier auch die Markierung wieder
|
||||
``mit dem Kaffee'', von der ausgebenden Person weg
|
||||
und auf die empfangende Person zu, verschoben werden.
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%% Zusammenfassung
|
||||
\section{Zusammenfassung}
|
||||
\section{Schluss}
|
||||
\label{sec:zusammenfassung}
|
||||
|
||||
Zusammenfassung des Papers.
|
||||
Mögliche Future Work:
|
||||
\begin{itemize}
|
||||
\item Kaffeeproblem für \(n>3\) visualisieren
|
||||
\item Wir haben noch keinen Beweis dafür, dass die Visualisierung in einem Dreieck unmöglich oder zumindest blöd ist.
|
||||
\end{itemize}
|
||||
In dieser durch die Einleitung eingeleiteten,
|
||||
durch beiden Hauptteile angereicherten
|
||||
und im Schluss abgeschlossenen Arbeit
|
||||
haben wir das n-Kaffee-Problem eingeführt,
|
||||
Probleme an den Haaren herbeigezogen
|
||||
und schließlich Visualisierungen entwickelt.\footnote{%
|
||||
Visualisierungen, die sich übrigens prima an Whiteboards machen.}
|
||||
Mit Hilfe des Kaffeesatzes konnten wir Kaffeekassen
|
||||
auf kleinere Kaffeekassen reduzieren
|
||||
und so auch komplexere Konstellationen visualisieren.
|
||||
Wir haben bilanzierende und explizite Kaffeekassen
|
||||
voneinander abgegrenzt
|
||||
und das Kaffee-Paradoxon nicht nur entdeckt,
|
||||
sondern auch aufgeklärt.
|
||||
|
||||
Offen bleibt,
|
||||
ob man einfache Visualisierungen auch für \(n\)-Kaffeekränzchen
|
||||
mit \(n > 3\) entwickeln kann.
|
||||
Des Weiteren hatten wir keine Lust mehr,
|
||||
weiter nach noch einfacheren Visualisierungen
|
||||
für \(n < 4\) zu finden.
|
||||
Wir überlassen es dem geneigten wissenschaftlichen Nachwuchs,
|
||||
darüber Köpfe zu zerbrechen.
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
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Reference in a new issue