SCHWEITZER 2004

The Lindelöf number $L(X)$ of a topological space $X$ is the least infinite cardinal $\lambda$ with the property that every open covering of $X$ has a subcovering of cardinality at most $\lambda$ . Prove that if evert non-countably infinite subset of a first countable space $X$ has a point of condensation, then $L(X)=\sup L(A)$ , where $A$ runs over the separable closed subspaces of $X$ . (A point of condensation of a subset $H\subseteq X$ is a point $x\in X$ such that any neighbourhood of $x$ intersects $H$ in a non-countably infinite set.)

Write $t(G)$ for the number of complete quadrilaterals in the graph $G$ and $e_G(S)$ for the number of edges spanned by a subset $S$ of vertices of $G$ . Let $G_1, G_2$ be two (simple) graphs on a common underlying set $V$ of vertices, $|V|-n$ , and assume that $|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}$ holds for any subset $S\subseteq V$ . Prove that $|t(G_1)-t(G_2)|\le \frac{n^4}{1000}$ .

Prove that there is a constant $c>0$ such that for any $n>3$ there exists a planar graph $G$ with $n$ vertices such that every straight-edged plane embedding of $G$ has a pair of edges with ratio of lengths at least $cn$ .

Determine all totally multiplicative and non-negative functions $f\colon\mathbb{Z}\rightarrow \mathbb{Z}$ with the property that if $a, b\in \mathbb{Z}$ and $b\neq 0$ , then there exist integers $q$ and $r$ such that $a-qb+r$ and $f(r)<f(b)$ .

Let $G$ be a non-solvable finite group and let $\varepsilon > 0$ . Show that there exist a positive integer $k$ and a word $w\in F_k$ such that $w$ assumes the value $1$ with probability less than $\varepsilon$ when its $k$ arguments are considered to be independent and uniformly distributed random variables with values in $G$ . (We write $F_k$ for the free group generated by $k$ elements.)

Is is true that if the perfect set $F\subseteq [0,1]$ is of zero Lebesgue measure then those functions in $C^1[0,1]$ which are one-to-one on $F$ form a dense subset of $C^1[0,1]$ ? (We use the metric $$d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|$$ to define the topology in the space $C^1[0,1]$ of continuously differentiable real functions on $[0,1]$ .)

Suppose that the closed subset $K$ of the sphere $$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$ is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$ . Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between $$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$ and $K$ is less than $\varepsilon$ .

Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$ , any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$ .

Let $F$ be a smooth (i.e. $C^{\infty}$ ) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an almost-immersion if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$ , respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$ . Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.

Let $\mathcal{N}_p$ stand for a $p$ dimensional random variable of standard normal distribution. For $a\in\mathbb{R}^p$ , let $H_p(a)$ stand for the expectation $E|\mathcal{N}_p+a|$ . For $p>1$ , prove that $$H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r$$