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Heißer, schwarzer Kaffee\footnote{% Wenn man da Milch reintut ist er nicht mehr schwarz, Junge!} kann also völlig zu Recht als das Fundament -des Fortschritts der Menschheit +menschlichen Fortschitts angesehen werden. Um die immer wieder notwendigen und erholsamen Unterbrechungen @@ -214,7 +214,7 @@ Betrachten wir das Beispiel \(n=2\). einen Kaffee für \(p_2\) bezahlt hat, dann ergibt sich der Zustand \(k = (-2,2)\). - Es lässt sich leicht erkennen, + Es lässt sich leicht erkennen, dass stets $\Delta_1 = - \Delta_2$ gelten muss. Somit können wir ohne Informationsverlust die zweite Komponente der Kaffekasse vernachlässigen @@ -261,11 +261,11 @@ Das ist die Aussage des folgenden Satzes. Die in \autoref{def:kaffeekasse} definierte Kaffeekasse modelliert einen gemeinsamen Kaffeepool innerhalb des Kaffeekränzchens. Diese Kaffeekasse kann auch als \emph{bilanzierende Kaffeekasse} bezeichnet werden. -Als alternative Notation könnte auch eine explizite Kaffeekasse vorgehalten werden, +Als alternative Notation könnte auch eine explizite Kaffeekasse vorgehalten werden, in der alle Kaffeeschulden innerhalb des Kaffeekränzchens einzeln ausgewiesen werden. \begin{defi}[Explizite Kaffeekasse] - \label{def:explizitekaffeekasse} + \label{def:explizitekaffeekasse} Sei \(n \in \naturals_{\geq2}\) und \(K\) ein \(n\)-Kaffeekränzchen. Eine \emph{explizite K-Kaffeekasse} ist die Matrix \( \kappa \in \naturals^{n\times n} \) wobei gilt: @@ -281,7 +281,7 @@ in der alle Kaffeeschulden innerhalb des Kaffeekränzchens einzeln ausgewiesen w \end{align*} \end{defi} -Für die Verwaltung der expliziten Kaffeekasse definieren wir +Für die Verwaltung der expliziten Kaffeekasse definieren wir % die Transition den Kassensturz analog zu \autoref{def:kaffeekassentransition}. @@ -309,28 +309,30 @@ analog zu \autoref{def:kaffeekassentransition}. \end{defi} \begin{satz} - Sei \(n \in \naturals\), \(K\) ein \(n\)-Kaffeekränzchen + Sei \(n \in \naturals\), \(K\) ein \(n\)-Kaffeekränzchen und \( \kappa \) eine explizite \(K\)-Kaffeekasse. Dann existiert eine entsprechende, bilanzierende \(K\)-Kaffeekasse. \end{satz} \begin{proof} - Sei \(n \in \naturals\), \(K\) ein \(n\)-Kaffeekränzchen + Sei \(n \in \naturals\), \(K\) ein \(n\)-Kaffeekränzchen und \(\kappa\) eine explizite \(K\)-Kaffeekasse. Dann existieren laut \autoref{def:explizitekaffeekasse} \(\delta_{i,j} \in \integers\) - für alle \(i,j \in \naturals_{\leq n}\) als Einträge in \(\kappa\). Setze nun + für alle \(i,j \in \naturals_{\leq n}\) als Einträge in \(\kappa\). Setze nun \[ \Delta_i = \sum_{l=1}^{n} \delta_{i,l} . \] - Dann ergibt sich die bilanzierende Kaffeekasse als - \( k = (\Delta_1,\ldots,\Delta_n) \). + Dann gilt + \[ \sum_ \] + ergibt sich die bilanzierende Kaffeekasse als + \( k = (\Delta_1,\ldots,\Delta_n) \) \end{proof} \begin{beob}[Kaffeeparadoxon] \label{beob:kaffeparadoxon} - Betrachten wir das 3-Kaffeekränzchen \( K = {p_1, p_2, p_3} \) + Betrachten wir das 3-Kaffeekränzchen \( K = {p_1, p_2, p_3} \) und ihre explizite Kaffeekasse \(\kappa\). - Nehmen wir an \(p_2\) hat bisher + 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 + Somit ergibt sich \[\kappa = \left( \begin{array}{rrr} 0 & 1 & 1 \\ @@ -434,14 +436,14 @@ 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. +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 +plaziert werden, um eine einfacher Transition zu ermöglichen. -Nun kann hier auch 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.