From: Mart Lubbers <mart@martlubbers.net>
Date: Wed, 16 Dec 2015 19:32:39 +0000 (+0100)
Subject: update code'
X-Git-Url: https://git.martlubbers.net/?a=commitdiff_plain;h=cd29bc230f39614f057810c37520ce20f75d76d0;p=ar1516.git

update code'
git pus"g
---

diff --git a/a2/1.tex b/a2/1.tex
index b294ecc..94a01cb 100644
--- a/a2/1.tex
+++ b/a2/1.tex
@@ -1,5 +1,5 @@
 \section{Problem 1}
-{\em Three non-self-supporting villages $A$, $B$ and $C$ in the middle of
+\emph{Three non-self-supporting villages $A$, $B$ and $C$ in the middle of
 nowhere consume one food package each per time unit. The required food packages
 are delivered by a truck, having a capacity of $300$ food packages. The
 locations of the villages are given in the following picture, in which the
@@ -19,28 +19,30 @@ city $B$, city $C$, the truck, the trucks location and a help variable to
 record how much food is dumped in the city. The distance between two cities is
 described with the function $dist(x, y)$. For all cities except $S$ a variable
 is defined that states the maximum capacity. 
-$$a_{max}=120\wedge b_{max}=120\wedge c_{max}=200$$
+$$a_{\max}=120\wedge b_{\max}=120\wedge c_{\max}=200$$
 Iteration $0$ depicts the initial state and can be described with a simple
 conjuction:
-$$a_0=40\wedge b_0=30\wedge c_0=145\wedge t_0=300\wedge l_0=S\wedge d_0=0$$.
+$$a_0=40\wedge b_0=30\wedge c_0=145\wedge t_0=300\wedge l_0=S\wedge d_0=0$$
 
 After that all iteration can be formalized as
-\begin{align}
+$$\begin{array}{lrl}
 	\bigwedge_{i\in I} \Bigg(
 	% Bent in S
-		& \Bigg[l_i=S\wedge d_i=0\wedge t_i=300\wedge
-		\bigvee_{k\in\{A,B\}}\bigg(l_{i-1}=k\wedge
-		\bigwedge_{s\in\{A,B,C\}}s_i=s_{i-1}-dist(S, k)\bigg)\Bigg]\vee\\
+	& \Bigg[ & l_i=S\wedge d_i=0\wedge t_i=300\wedge\\
+	& & \bigvee_{k\in\{A,B\}}
+		\bigg(l_{i-1}=k\wedge
+		\bigwedge_{s\in\{A,B,C\}}
+			s_i=s_{i-1}-dist(S, k)\bigg)\Bigg]\vee\\
 	% Bent in A,B,C
-		& \bigvee_{k\in\{a,b,c\}}
-			\Bigg[l_i=k\wedge d_i>0\wedge d_i<t_{i-1}\wedge d_i<k_{max}-k_{i-1}
-			t_i=t_i-d_i\wedge k_i=k_i+d_i\wedge\\
-		& \qquad\qquad\bigwedge_{k'\in\{k\}\setminus\{S,A,B,C\}}\bigg(
-		l_{i-1}=k'\wedge
+	& \bigvee_{k\in\{a,b,c\}}\Bigg[ & l_i=k\wedge\\
+	& & d_i>0\wedge d_i<t_{i-1}\wedge d_i<k_{\max}-k_{i-1}\wedge\\
+	& & t_i=t_{i-1}-d_i\wedge k_i=k_{i-1}+d_i\wedge\\
+	& & \bigwedge_{k'\in\{k\}\setminus\{S,A,B,C\}}
+		\bigg(l_{i-1}=k'\wedge
 		\bigwedge_{s\in\{k\}\setminus\{A,B,C\}}
-			s_i=s_{i-1}-dist(k, s)\bigg)\Bigg]\\
-	\nonumber\Bigg)
-\end{align}
+			s_i=s_{i-1}-dist(k, k')\bigg)\Bigg]\\
+	& \Bigg)
+\end{array}$$
 
 \begin{enumerate}
 	\item This part of the formula describes what happens if the truck is in
@@ -52,14 +54,14 @@ After that all iteration can be formalized as
 \end{enumerate}
 
 \begin{enumerate}[a.]
-	\item {\em Show that it is impossible to deliver food packages in such a
+	\item \emph{Show that it is impossible to deliver food packages in such a
 		way that each of the villages consumes one food package per time unit
 		forever.}
-	\item {\em Show that this is possible if the capacity of the truck is
+	\item \emph{Show that this is possible if the capacity of the truck is
 		increased to 320 food packages. (Note that a finite graph contains an
 		infinite path starting in a node $v$ if and only if there is a path
 		from $v$ to a node $w$ for which there is a non-empty path from $w$ to
 		itself.)}
-	\item {\em Figure out whether it is possible if the capacity of the truck
+	\item \emph{Figure out whether it is possible if the capacity of the truck
 		is set to $318$.}
 \end{enumerate}