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\title{General Mathematics and Computational Science I}
\author{Exercise 19}
\date{November 22, 2005}
\maketitle

\begin{enumerate}

\item Solve the following linear programming problem using the simplex
method.

Maximize
$
  z = 8 \, x_3 - 3 \, x_4 + 10 \, x_5
$
subject to 
\begin{gather*}
  x_1 -2 \, x_3 + x_4 - x_5 = -1 \,, \\
  x_2 -3 \, x_3 + x_4 - x_5 = -2 \,, \\
  x_i \geq 0 \text{ for } i=1,\dots,5 \,.
\end{gather*}

\emph{Note:} Use artificial variables to find an initial set of
feasible basic variables.


\item In the Kac ring model, $N$ sites are placed around a circle.
The sites are populated with $B$ black balls and $W=N-B$ white balls
at random.  Moreover, $n$ markers are placed around the circle at
random; the number of blace balls at a marked site is denoted $b$, the
number of white balls at a marked site is $w$.
\begin{center}
\includegraphics[width=6cm]{ring.pdf}
\end{center}
Let $\mu$ denote the probability that a site has a marker on it.
Explain why
\begin{equation*}
  \mu = \frac{n}{N} = \frac{b}{B} = \frac{w}{W} \,.
\end{equation*}

\end{enumerate}
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