The object of this paper is to obtain necessary and sufficient conditions to characterize the matrices in classes (l_∞^y(p,s), l_∞(q)), (c₀^y(p,s), l_∞(q)), (l_∞^y(p,s), c₀(q)), and (c₀^y(p,s), c₀(q)) which will fill up a gap in the existing literature.

Answering a question posed by Wu Wei Chao [2], we determine all solutions f : ℝ→ℝ of the equation
f(x²+y+f(y) ) = f(x)² + 2y,   x,y∈ℝ.

This paper presents open problems on the relation between additive and multiplicative structures of cyclotomic fields.

The extensions of the well-known Sperner's result on antichains of subsets of a given finite set for divisors of a positive integers are shown to hold also for sets of regular systems of divisors of elements of arithmetical semigroups.

The sufficient conditions of string stability of singularly perturbed, nonlinear stochastic systems are established. The excitations are assumed to be parametric white noise. In this case the objective is to analyze composite systems in their lower order subsystems and in terms of their interconnecting structure and the perturbation parameter e. An example is given to illustrate the results.

In this note we give necessary and sufficient conditions for a hyperharmonic function to be a Bloch function.

Let X, Y be real linear spaces. We are looking for a function G : X²→ℝ such that the equation
f(x+y) = G(x,y)[f(x)+f(y)]
is equivalent to the orthogonal Cauchy equation
x⟂y ⇒ f(x+y)=f(x)+f(y).
Several kinds of orthogonalities are considered. The quotient ‖(x-y)/(x+y)‖ closely connected with the James orthogonality plays here a distinguished role. Similar problems are considered for the Ptolemaic equation
x⟂y ⇒ f(x+y)f(x-y) = f(x)²+f(y)².
As a result a characterization of inner product spaces is obtained.

Problem session. The 4th Czech and Polish Conference on Number Theory Cieszyn, June 11-14, 2002.

Report of Meeting. The Second Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities January 30 - February 2, 2002, Hajdúszoboszló, Hungary.