SCHWEITZER 1983

Let $I$ be an ideal of the ring $R$ and $f$ a nonidentity permutation of the set $\{ 1,2,\ldots, k \}$ for some $k$ . Suppose that for every $0 \not= a \in R, aI \not= 0$ and $Ia \not=0$ hold; furthermore, for any elements $x_1,x_2,\ldots ,x_k \in I$ , $$x_1x_2\ldots x_k=x_{1f}x_{2f}\ldots x_{kf}$$ holds. Prove that $R$ is commutative.

R. Wiegandt

For which cardinalities $\kappa$ do antimetric spaces of cardinality $\kappa$ exist? $(X,\varrho)$ is called an $\textit{antimetric space}$ if $X$ is a nonempty set, $\varrho : X^2 \rightarrow [0,\infty)$ is a symmetric map, $\varrho(x,y)=0$ holds iff $x=y$ , and for any three-element subset $\{a_1,a_2,a_3 \}$ of $X$ $$\varrho(a_{1f},a_{2f})+\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f})$$ holds for some permutation $f$ of $\{1,2,3 \}$ .

V. Totik

Let $T$ be a bounded linear operator on a Hilbert space $H$ , and assume that $\|T^n \| \leq 1$ for some natural number $n$ . Prove the existence of an invertible linear operator $A$ on $H$ such that $\| ATA^{-1} \| \leq 1$ .

E. Druszt

Prove that any identity that holds for every finite $n$ -distributive lattice also holds for the lattice of all convex subsets of the $(n-1)$ -dimensional Euclidean space. (For convex subsets, the lattice operations are the set-theoretic intersection and the convex hull of the set-theoretic union. We call a lattice $n$ - $\textit{distributive}$ if $$x \wedge (\bigvee_{i=0}^n y_i)=\bigvee_{j=0}^n(x \wedge (\bigvee_{0\leq i \leq n, i \not= j\ }y_i))$$ holds for all elements of the lattice.)

A. Huhn

Prove that if $E \subset \mathbb{R}$ is a bounded set of positive Lebesgue measure, then for every $u < 1/2$ , a point $x=x(u)$ can be found so that $$|(x-h,x+h) \cap E| \geq uh$$ and $$|(x-h,x+h) \cap (\mathbb{R} \setminus E)| \geq uh$$ for all sufficiently small positive values of $h$ .

K. I. Koljada

Let $M^n \subset \mathbb{R}^{n+1}$ be a complete, connected hypersurface embedded into the Euclidean space. Show that $M^n$ as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is, $M^n$ can be decomposed to a direct product of the form $M^n=M^k \times \mathbb{R}^{n-k} \;(k<n)$ as well, where $M^k$ is a hypersurface in some $(k+1)$ -dimensional subspace $E^{k+1} \subset \mathbb{R}^{n+1} , \mathbb{R}^{n-k}$ is the orthogonal complement of $E^{k+1}$ .

Z. Szabo