Renamed Kaffeekassentransition, adjusted images (again), updated gitignore (generated by gitignore.io)
This commit is contained in:
Nis Börge Wechselberg
parent
738aaab946
commit
8a7019a340
10 changed files with 192 additions and 112 deletions
51
.gitignore
vendored
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.gitignore
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Before Width: | Height: | Size: 20 KiB After Width: | Height: | Size: 20 KiB |
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main.pdf
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59
main.tex
59
main.tex
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@ -51,7 +51,7 @@
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|||
\begin{abstract}
|
||||
Das n-Kaffee-Problem
|
||||
beschreibt die Guthaben-Schulden-Verhältnisse
|
||||
in einer Gruppe von \(n\in\) Personen.
|
||||
in einer Gruppe von \(n\in\naturals\) Personen.
|
||||
Die Verhältnisse werden hierbei in ausgegebenen Kaffees notiert.
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||||
In dieser Arbeit definieren wir das Problem
|
||||
und betrachten Visualisierungen mit dem Ziel,
|
||||
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|
@ -116,7 +116,9 @@ Bla.
|
|||
|
||||
Um das Kaffee-Problem betrachten zu können,
|
||||
müssen wir zunächst eine geeignete Menge von Personen definieren,
|
||||
die das Problem überhaupt interessiert.
|
||||
die das Problem überhaupt interessiert.\footnote{%
|
||||
Hierzu gehört offenbar der Leser,
|
||||
sonst würde er diesen Stuss ja nicht lesen}
|
||||
|
||||
\begin{defi}[Kaffeekränzchen]
|
||||
\label{def:kaffeekraenzchen}
|
||||
|
|
@ -159,14 +161,17 @@ Aus der Definition der \(\Delta_i\) ergibt sich:
|
|||
Gibt eine Person einer anderen nun einen Kaffee aus,
|
||||
müssen wir die Kaffeekasse entsprechend weiterentwickeln.
|
||||
|
||||
\begin{defi}[Kaffeekassentransition]
|
||||
% \begin{defi}[kaffeekassentransitionition]
|
||||
\begin{defi}[Kaffeekassensturz]
|
||||
\label{def:kaffeekassentransition}
|
||||
Sei \(n\in\naturals_{\geq 2}\),
|
||||
\(K\) ein \(n\)-Kaffeekränzchen,
|
||||
\(k\) eine \(K\)-Kaffeekasse
|
||||
sowie \(i \neq j \in {[n]}\) so,
|
||||
dass \(p_i\) \(p_j\) einen Kaffee ausgibt.
|
||||
Die Kaffeekassentransition liefert zu \(k\)
|
||||
% Die Kaffeekassentransition
|
||||
Der Kaffeekassensturz
|
||||
liefert zu \(k\)
|
||||
eine neue Kaffeekasse \(k' = (\Delta_1',\ldots,\Delta_n')\) mit
|
||||
\[ \Delta_x' = \left\{
|
||||
\begin{array}{ll}
|
||||
|
|
@ -267,25 +272,30 @@ in der alle Kaffeeschulden innerhalb des Kaffeekränzchens einzeln ausgewiesen w
|
|||
\item \( \delta_{i,i} = 0 \) für alle \( i \in \naturals_{\leq n} \)
|
||||
\item \( \delta_{i,j} = -\delta_{j,i} \) für alle \( i,j \in \naturals_{\leq n} \)
|
||||
\end{enumerate}
|
||||
Hierbei verhalten sich die einzelnen Kaffeedeltas $\delta_{ij}$ wie folgt:
|
||||
Hierbei verhalten sich die einzelnen Kaffeedeltas $\delta_{i, j}$ wie folgt:
|
||||
\begin{align*}
|
||||
\delta_{ij} < 0 &: \enspace p_j \text{ schuldet } p_i \text{ noch } |\delta_{ij}| \text{ Kaffees.} \\
|
||||
\delta_{ij} = 0 &: \enspace \text{Niemand hat Kaffeeschulden.} \\
|
||||
\delta_{ij} > 0 &: \enspace p_i \text{ schuldet } p_j \text{ noch } \delta_{ij} \text{ Kaffees.}
|
||||
\delta_{i,j} < 0 &: \enspace p_j \text{ schuldet } p_i \text{ noch } |\delta_{ij}| \text{ Kaffees.} \\
|
||||
\delta_{i,j} = 0 &: \enspace \text{Die Kaffeebilanz zwischen }p_i\text{ und }p_j\text{ ist ausgeglichen.} \\
|
||||
\delta_{i,j} > 0 &: \enspace p_i \text{ schuldet } p_j \text{ noch } \delta_{ij} \text{ Kaffees.}
|
||||
\end{align*}
|
||||
\end{defi}
|
||||
|
||||
Für die Verwaltung der expliziten Kaffeekasse definieren wir die Transistion
|
||||
Für die Verwaltung der expliziten Kaffeekasse definieren wir
|
||||
% die Transition
|
||||
den Kassensturz
|
||||
analog zu \autoref{def:kaffeekassentransition}.
|
||||
|
||||
\begin{defi}[Explizite Kaffeekassentransition]
|
||||
% \begin{defi}[Explizite Kaffeekassentransition]
|
||||
\begin{defi}[Expliziter Kaffeekassensturz]
|
||||
\label{def:explizitekaffeekassentransition}
|
||||
Sei \(n \in \naturals_{\geq 2}\),
|
||||
\(K\) ein Kaffeekränzchen über \(n\),
|
||||
\(\kappa\) eine explizite \(K\)-Kaffeekasse
|
||||
sowie \(i \neq j \in {[n]}\) so,
|
||||
dass \(p_i\) \(p_j\) einen Kaffee ausgibt.
|
||||
Die explizite Kaffeekassentransition liefert zu \(\kappa\)
|
||||
% Die explizite Kaffeekassentransition
|
||||
Der explizite Kaffeekassensturz
|
||||
liefert zu \(\kappa\)
|
||||
eine neue explizite Kaffeekasse \( \kappa' \in \integers^{n\times n} \) mit
|
||||
\[ \delta_{k,l}' = \left\{
|
||||
\begin{array}{ll}
|
||||
|
|
@ -315,7 +325,8 @@ analog zu \autoref{def:kaffeekassentransition}.
|
|||
\begin{beob}[Kaffeeparadoxon]
|
||||
\label{beob:kaffeparadoxon}
|
||||
Betrachten wir das 3-Kaffeekränzchen \( K = {p_1, p_2, p_3} \)
|
||||
und ihre explizite Kaffeekasse \(\kappa\). Bisher hat \(p_2\)
|
||||
und ihre explizite Kaffeekasse \(\kappa\).
|
||||
Nehmen wir an \(p_2\) hat bisher
|
||||
jeweils einen Kaffee für \(p_1\) und \(p_3\) bezahlt.
|
||||
Weiter hat \(p_3\) \(p_1\) einen Kaffee ausgegeben.
|
||||
Somit ergibt sich
|
||||
|
|
@ -333,8 +344,9 @@ analog zu \autoref{def:kaffeekassentransition}.
|
|||
|
||||
Bilden wir zu \(\kappa\) nun die bilanzierende Kaffeekasse \(k\),
|
||||
so ergibt sich \(k = (2,-2,0) \).
|
||||
Hierbei ist zu erkennen, das nur 2 Kaffees benötigt werden, um die Kaffeeschulden
|
||||
auszugleichen.
|
||||
Hierbei ist unmittelbar zu erkennen,
|
||||
das nur 2 Kaffees benötigt werden,
|
||||
um die Kaffeeschulden auszugleichen.
|
||||
|
||||
Dieses Phänomen, welches wir als \emph{Kaffeeparadoxon} bezeichnen, lässt sich
|
||||
in dem Auftreten von \emph{transitiven Kaffeeschulden} begründen.
|
||||
|
|
@ -383,7 +395,7 @@ als Koordinaten eingetragen werden.
|
|||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=.7\textwidth]{3KaffeeKartesisch}
|
||||
\includegraphics[scale=.6]{3KaffeeKartesisch}
|
||||
\caption{%
|
||||
Visualisierung des \(3\)-Kaffee-Problems mit kartesischen Koordianten.
|
||||
}
|
||||
|
|
@ -410,7 +422,7 @@ zur Zeit liegt.
|
|||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=.7\textwidth]{3KaffeeProblem}
|
||||
\includegraphics[scale=.6]{3KaffeeProblem}
|
||||
\caption{%
|
||||
Visualisierung des \(3\)-Kaffee-Problems mit schiefen Koordianten
|
||||
}
|
||||
|
|
@ -420,16 +432,15 @@ zur Zeit liegt.
|
|||
Als alternative Darstellung kann ein
|
||||
geradliniges, nichtorthgonales Koordiantensystem
|
||||
gewählt werden. In \autoref{fig:3-kaffee-schief}
|
||||
In diesem System schneiden sich die
|
||||
Achsen mit einem Winkel von 60\degree.
|
||||
Bei korrekter Beschriftung der Seiten, können hier
|
||||
die selben Koordinaten verwendet werden,
|
||||
die bereits im kartesischen Koordinatensystem
|
||||
bereechnet wurden.
|
||||
Allerdings können hier die Personen an den Seiten
|
||||
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.
|
||||
Auch hier kann nun die Markierung wieder
|
||||
Nun kann hier auch die Markierung wieder
|
||||
``mit dem Kaffee'', von der ausgebenden Person weg
|
||||
und auf die empfangende Person zu, verschoben werden.
|
||||
|
||||
|
|
|
|||
2
n-kaffee-problem.sublime-project
Normal file
2
n-kaffee-problem.sublime-project
Normal file
|
|
@ -0,0 +1,2 @@
|
|||
{
|
||||
}
|
||||
Loading…
Reference in a new issue