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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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