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\centerline{\LARGE M.Sc.~Math~Workshop --- Assignment \#8}
\centerline{\large HUJI Spring 1998}
\centerline{Dror Bar-Natan}
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\begin{enumerate}
\setcounter{enumi}{36}
\item Prove that if $n^c$ is an integer for every $n$, then $c$ is an
integer.
\item Prove that in any finite partition of the integers to at least two
arithmetic progressions, at least two of the progressions will have the
same skip.
\item Compute the fundamental group and describe the universal covering
space of the topological space obtained by identifying the three edges of a
triangle as in the figure below:
\[ \eepic{triangle}{0.5} \]
\item Let $\gamma_{1,2}:S^1\to{\bold R}^3$ be two non-intersecting smooth
paths in ${\bold R}^3$. Set
\[ l(\gamma_1,\gamma_2) = \int_{S^1\times S^1}dt_1dt_2\,
\frac{
\left(\dot{\gamma}_1(t_1)\times\dot{\gamma}_2(t_2)\right)
\cdot\left(\gamma_1(t_1)-\gamma_2(t_2)\right)
}{
\left|\gamma_1(t_1)-\gamma_2(t_2)\right|^3
},
\]
where $t_{1,2}$ are parameters in $S^1$, $\dot\gamma$ denotes the derivative
of $\gamma$ with respect to its parameter (i.e., its tangent vector),
$\times$ and $\cdot$ denote the vector and scalar products in ${\bold R}^3$
(respectively), and $|\ |$ denotes the Euclidean norm.
\begin{enumerate}
\item Prove that $l$ is a link invariant (that is, prove that it
is invariant under deformations of $\gamma_{1,2}$ in the spirit of
assignment \#6). (One of many possibilities is to consider the map
$S^1\times S^1\to S^2$ given by $(t_1,t_2)\mapsto\frac{\gamma_1(t_1)
- \gamma_2(t_2)}{\left|\gamma_1(t_1) - \gamma_2(t_2)\right|}$, and to
study the pullback of the volume form on $S^2$ via this map).
\item Find a simple combinatorial formula for computing $l$ given a generic
planar projection of $\gamma_{1,2}$.
\end{enumerate}
\item Let $f:{\bold R}\to{\bold C}$ be a unit-norm $L^2$ function, and let
$\tilde{f}$ be the Fourier transform of $f$ (the Fourier transform is
normalized so that $L^2$ norms are preserved). Let $\mu$ be the
distribution on ${\bold R}$ whose density function is $|f|^2$, let $V=\int
x^2d\mu-\left(\int xd\mu\right)^2$ be
the variance of $\mu$, and make similar definitions for $\tilde{\mu}$ and
$\tilde{V}$ starting from $\tilde{f}$.
\begin{enumerate}
\item Prove that $V\cdot\tilde{V}\geq\frac14$.
\item Determine all the functions $f$ for which the above inequality is an
equality.
\end{enumerate}
Hint: you may want to recall Q16.
\end{enumerate}
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