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\centerline{\LARGE M.Sc.~Math~Workshop --- Assignment \#2}
\centerline{\large HUJI Spring 1998}
\centerline{Dror Bar-Natan}

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\begin{enumerate}
\setcounter{enumi}{6}

\item Prove: If $\lambda>0$ is irrational then there exists a continuous
  function $\phi:[0,1]\to[0,1]$ so that for every $\epsilon>0$ and
  every 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 $|f(x,y)-g(\phi(x)+\lambda\phi(y))|<\epsilon$ on a set of area
  at least $1-\epsilon$ in $[0,1]\times[0,1]$. (Notice the different
  order of the quantifiers relative to Q2).

\item Can you completely cover a disk of diameter 100 with 99 rectangles of
  sides $100\times 1$?

\item Let $A$ and $B$ be two $n\times n$ matrices over ${\mathbf C}$.
  \begin{enumerate}
  \item Prove that $[A,B]\eqdef AB-BA\neq I$.
  \item Prove that if $[A,[A,B]]=0$, then $[A,B]$ is nilpotent.
  \end{enumerate}

\item Find a 2-variable polynomial that is always positive on ${\mathbf
  R}^2$, and has exactly two critical points, both of which are minima.
% (x*x*y - x - 1)^2 + (x*x - 1)^2

\item Let $C_n=\{(z_1,\ldots,z_n)\in{\mathbf C}^n:\,\forall 1\leq<j\leq
  n,\, z_i\neq z_j$ be the configuration space of $n$ distinct points
  in the complex plane ${\mathbf C}$, and let the $n$th pure braid
  group $PB_n=\pi_1(C_n)$ be the fundamental group of $C_n$.  Prove
  that $PB_n\simeq PB_{n-1}\ltimes F_{n-1}$ where $F_{n-1}$ denotes
  the free group on $n-1$ generators. Deduce that $PB_n\simeq F_1\ltimes
  F_2\ltimes\cdots\ltimes F_{n-1}$.

\item Let $A_n=\{1,2,3\}^n$ and let $A=A_1$. A function $f:A_n\to A$ is
  called {\em injective}, if whenever $x$ and $y$ are different,
  $f(x)\neq f(y)$. Obviously, injective functions exist only if $n=1$,
  and in this case, they are simply permutations $\pi:A\to A$. A function
  $f:A_n\to A$ is called {\em weakly injective}, if whenever $x$ and $y$
  are {\em totally different}, meaning that $x_i\neq y_i$ for {\em all}
  $1\leq i\leq n$, one has $f(x)\neq f(y)$. Prove that if $f:A_n\to A$
  is weakly injective then for some permutation $\pi:A\to A$ and $1\leq
  i\leq n$, one has $f(x)=\pi(x_i)$ for all $x\in A_n$.

\item $f$ is a real valued function on the reals, and it is known that at
  any point at least one derivative of $f$ vanishes (possibly different
  derivatives at different points). Prove that $f$ is a polynomial.

\item What is the configuration space of the machine $M_3$ in the picture
  below?
  \begin{figure}[htpb]
  \[ \eepic{M3}{0.75} \]
  \caption{The machine $M_3$.}
  \end{figure}

\end{enumerate}

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