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\centerline{\LARGE M.Sc.~Math~Workshop --- Assignment \#3}
\centerline{\large HUJI Spring 1998}
\centerline{Dror Bar-Natan}
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\begin{enumerate}
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\item Prove: If $\lambda>0$ is irrational and $\epsilon>0$ then there
exists 5 continuous functions $\phi_i:[0,1]\to[0,1]$ $(1\leq i\leq 5)$
so that for every continuous function $f:[0,1]\times[0,1]\to{\bold R}$
there exists a continuous function $g:[0,1+\lambda]\to{\bold R}$ so that
\[
|f(x,y)-\sum_{i=1}^5 g(\phi_i(x)+\lambda\phi_i(y))|
< \left(\frac23+\epsilon\right)|f(x,y)|
\]
for every $x,y\in[0,1]$.
\item A {\em quantum probability space} is a pair $({\mathcal H}, v)$
where $\mathcal H$ is a Hilbert space and $v\in\mathcal H$ is a unit
vector. A {\em random variable} on $\mathcal H$ is a self-adjoint
operator $\mathcal H\to\mathcal H$. We say that $\langle v,A^nv\rangle$
is the expectation value of the $n$th power of the random variable $A$
(if this quantity exists, namely if $v$ is in the domain of definition
of $A^n$). In particular, we set $E(A)=\langle v, Av\rangle$ to be
the {\em expectation} of $A$, and $V(A)=\langle v, A^2v\rangle-\langle
v, Av\rangle^2$ to be the {\em variance} of $A$. Prove that if $P$
and $Q$ are random variables on some quantum probability space
$({\mathcal H}, v)$, and $P$ and $Q$ satisfy $[P,Q]=PQ-QP=iI$, then
$V(P)V(Q)\geq\frac{1}{4}$. It is a good idea to start with the
simplifying assumption $E(P)=E(Q)=0$.
\item Prove that a finite group of affine transformations always has a
fixed point.
\item Prove that the area of any planar section of a perfect tetrahedron is
at most the area of a face of that tetrahedron.
\item Prove that any knot in ${\mathbf R}^3$ is the boundary of some
double-sided (non-M\"obius) surface embeded in ${\mathbf R}^3$.
\item A rectangle $R$ is tiled (presented as a disjoint union, not
minding about 1-dimensional boundaries) with (possibly different)
semi-integral rectangles --- rectangles at least one of whose sides is of
integral length. Prove that $R$ itself is semi-integral.
\end{enumerate}
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