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\def\Var{\operatorname{Var}}

\begin{document}

\title{General Mathematics and Computational Science II}
\author{Midterm Exam}
\date{March 6, 2007}
\maketitle

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\begin{enumerate}

\item A point $P$ lies on the arc $AB$ of the circle circumscribing an
equilateral triangle $ABC$; see figure.  Prove that $PC=PA + PB$.

\emph{Hint:} By the \emph{arc central angle theorem}, the angles
$\angle APC$ and $\angle ABC$ are equal.

\hfill (10)

\begin{center}
\includegraphics[width=10cm]{ivanovProblem7.eps}
\end{center}

%\newpage
\item A point $P$ lies on the line segment $AB$ such that $AP = 2 \, PB$; see
figure.  The coordinates of the points are given by $\boldsymbol{a}$,
$\boldsymbol{v}$, and $\boldsymbol{b}$, respectively.

Find an expression for $\boldsymbol{v}$ in terms of $\boldsymbol{a}$
and $\boldsymbol{b}$.  \hfill(5)

\begin{center}
\includegraphics[width=10cm]{midtermproblem.eps}
\end{center}

%\newpage
\item Describe the symmetry group of the square.  How many elements
does it have?  Give an example of a minimal generating set.
\hfill(10)

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\item 
\begin{enumerate}
\item Are the even integers $\{2k \colon k \in \mathbb{Z}\}$ a
subgroup of $(\mathbb{Z},+)$?  The odd integers $\{2k +1 \colon k \in
\mathbb{Z}\}$?
\item Show that the identity element $e$ of a group $(G,{\circ})$ is
the unique element which satisfies $e \circ a = a \circ e = a$ for all
$a \in G$.
\end{enumerate}
\hfill(5+5)

%\newpage
\item Consider an ensemble of Kac rings with $N$ sites.  Initially,
each member of the ensemble has the same configuration of black and
white balls, while the distribution of markers among the edges of the
rings is random; each edge carries a marker with probability $\mu$.
Recall from class that the difference between the number of black
balls and the number of white balls, averaged over the ensemble, is
given by
\[
  \langle \Delta(t) \rangle = (1-2\,\mu)^t \, \Delta(0) 
\]
for $0\leq t \leq N$. 
\begin{enumerate}
\item Show that $\Var[\Delta(N)] = (1-(1-2 \mu)^{2N}) \,
\Delta^2(0)$. 
\item Is $\langle \Delta(N) \rangle$ a good description of $\Delta(N)$
for a ``typical'' member of the ensemble?  Briefly explain your
answer.
\end{enumerate}
\hfill (7+3)

%\newpage
\item Set up a Kac ring in its initial state $S_0$ with a random
distribution of markers and balls and assume that $\Delta$, the
difference between the number of black and the number of white balls,
is positive.

First, turn the ring by one clockwise step into some state $S_1$.  We
demonstrated in class that $\Delta$ is more likely to decrease than to
increase during this first step.

Now turn the ring one step anti-clockwise.  Since the dynamics is
time-reversible, we are reverting back to $S_0$, hence $\Delta$ is
more likely to increase than to decrease in this backward step.

Now turn the ring another step anti-clockwise into a state $S_{-1}$.
Is $\Delta$ now more likely to increase or to decrease?  Explain!
\hfill (5)

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\end{enumerate}


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