This paper is devoted to the integral representation of potentials by exit laws in the framework of sub-Markovian semigroups of kernels acting on L²(m). We mainly investigate subordinated semigroups in Bochner sense by means of subordinators with complete Bernstein functions. As application, we give a representation for the original semigroup.

We consider the functional equation
F(y)−F(x) = (y−x)[f(αx+βy)+f(βx+αy)]
stemming from Gauss quadrature rule. In previous results equations of this type with rational only coefficients α and β were considered. In this paper we allow these numbers to be irrational. We find all solutions of this equation for functions acting on ℝ. However, some results are valid also on integral domains.

We solve the following functional equation
f(x+y+z) + g(x+y) = q(z) + p(y) + h(x),
where f, g, q, p, h are unknown functions transforming a semigroup into a group.

In this paper we show that the rings of regular functions on two real algebraic curves over the same real closed field are Witt equivalent (i.e. their Witt rings are isomorphic) if and only if the curves have the same number of semi-algebraically connected components. Moreover, in the second part of the paper, we prove that every strong isomorphism of Witt rings of rings of regular functions can be extended to an isomorphism of Witt rings of fields of rational functions. This extension is not unique, though.

Let G be a group and S be a subsemigroup in G, generating G as a group. Every element in G is a product of elements from S∪S⁻¹. An equality G = S⁻¹S · · · S⁻¹S allows to define an S-length l(G) of the group G. The note concerns the problem posed by J. Krempa on possible values of l(G). We show that for collapsing groups, supramenable groups and groups of a subexponential growth l(G) ≤ 2. The S-length of a relatively free group can be equal to 1 or 2 or infinity, but it can not be equal to 3. The problem concerning other values is open.

The Euclidean shortest path between two points s and t in the plane with the cellular decomposition in the presence of obstacles is considered. The A* algorithm for a visibility graph (VG) is used to avoid widened obstacles. Computational experiments show that the proposed algorithm is often faster and it analyzes fewer nodes than the classical Dijkstra algorithm.

Report of Meeting. The Eighth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Fûzfa Pihenõpark, Poroszló (Hungary), January 30 – February 2, 2008.