SCHWEITZER 2019

Prove that if every subspace of a Hausdorff space $X$ is $\sigma$ -compact, then $X$ is countable.

Let $R$ be a noncommutative finite ring with multiplicative identity element $1$ . Show that if the subring generated by $I \cup \{1\}$ is $R$ for each nonzero ideal $I$ then $R$ is simple.

Prove that there are infinitely many integers $m$ , $n$ , such that $1 < m < n$ , and the greatest common divisors $(m, n)$ , $(m, n+1)$ , $(m+1, n)$ and $(m+1, n+1)$ are all greater than $\sqrt{n}/999$ .

An $n \times m$ matrix is nice if it contains every integer from $1$ to $mn$ exactly once and $1$ is the only entry which is the smallest both in its row and in its column. Prove that the number of $n \times m$ nice matrices is $(nm)!n!m!/(n+m-1)!$ .

Let $S \subset \mathbb{R}^d$ be a convex compact body with nonempty interior. Show that there is an $\alpha > 0$ such that if $S = \cap_{i \in I} H_i$ , where $I$ is an index set and $(H_i)_{i \in I}$ are halfspaces, then for any $P \in \mathbb{R}^d$ , there is an $i \in I$ for which $\mathrm{dist}(P, H_i) \ge \alpha \mathrm{dist}(P, S)$ .

Let $d$ be a positive integer and $1 < a \le (d+2)/(d+1)$ . For given $x_0, x_1,\dots, x_d \in (0, a-1)$ , let $x_{k+1} = x_k (a - x_{k-d})$ , $k \ge d$ . Prove that $\lim_{k \to \infty} x_k = a-1$ .

Given a polynomial $P$ , assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$ .

Let $f: \mathbb{R} \to \mathbb{R}$ be a measurable function such that $f(x+t) - f(x)$ is locally integrable for every $t$ as a function of $x$ . Prove that $f$ is locally integrable.

Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$ . Prove that $$\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n}$$ holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$ .