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\title{General Mathematics and CPS II}
\author{Exercise 10}
\date{March 7, 2012}
\maketitle

\emph{Note:} This exercise is due March 21, 2012.  This material will
\emph{not} be appear on the midterm exam.

\begin{enumerate}

\item Newton's second law of mechanics for a particle of mass $m$
situated at position $x(t)$ moving with velocity $v(t)$ and subject to
a force $F(x(t))$ can be written
\begin{gather*}
  \frac{\d x}{\d t} = v \,, \\
  m \, \frac{\d v}{\d t} = F(x(t)) \,.
\end{gather*}

Use the chain rule of calculus to show that the particle satisfies the
same equation with $t$ replaced by the reversed time $r=-t$ and $v$
replaced by $-v$.

\item Show that in a time-discrete, reversible, system with a finite
number of states any orbit must return to its initial state after a
finite number of steps.

\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 on the edges between the
sites at random; the number of black balls just before a marked edge
is denoted $b$, the number of white balls just before a marked edge is
$w$.  

Let $\mu$ denote the probability that an edge has a marker on it.
Explain why
\begin{equation*}
  \mu = \frac{n}{N} = \frac{b}{B} = \frac{w}{W} \,.
\end{equation*}
\item Show that 
\[
  \sum_{k=0}^n \binom{n}{k} \, k \, x^k \, y^{n-k} 
  = n \, x \, (x+y)^{n-1} 
\]
and
\[
  \sum_{k=0}^n \binom{n}{k} \, k^2 \, x^k \, y^{n-k} 
  = n \, x \, (x+y)^{n-1} + n \, (n-1) \, x^2 \, (x+y)^{n-2} \,.
\]

\emph{Hint:} Differentiate the binomial theorem.
\end{enumerate}
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