update code'

git pus"g

diff --git a/a2/1.tex b/a2/1.tex
index b294ecc..94a01cb 100644 (file)
--- 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}