We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.

Let ℓ > 2 be a prime number. Let K be a number field containing a unique ℓ-adic prime and assume that its class is an ℓth power in the class group C_K. The main theorem of the paper gives a sufficient condition for a finite set of primes of K to be the wild set of some Hilbert self-equivalence of K of degree ℓ.

Some reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest and reverses of Hölder and Schwarz trace inequalities are also given.

In paper [4] there are considered random dynamical systems with randomly chosen jumps acting on Polish spaces. The intensity of this process is a constant λ. In this paper we formulate criteria for the existence of an invariant measure and asymptotic stability for these systems in the case when λ is not constant but a Lipschitz function.

The aim of this manuscript is to establish fixed point results satisfying contractive conditions of rational type in the setting of complex valued metric spaces. The derived results generalize and extend some well known results in the existing literature.

In the paper two types of stability and of b-stability of functional equations are distinguished.

In this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].

We describe some algorithms, without using resolution of singularities, that establish the rationality of the motivic Igusa zeta series of certain hypersurfaces that are non-degenerated with respect to their Newton polyhedron. This includes, in any characteristic, the motivic rationality for polydiagonal hypersurfaces, vertex singularities, binomial hypersurfaces, and Du Val singularities.

Let (S,+) be a commutative semigroup, σ: S→S be an endomorphism with σ² = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f,g: S→K of the functional equation
f(x+y) + f(x+σ(y)) + g(x+y) = 2f(x) + g(x)g(y)   for x,y∈S.
We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

We investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear elliptic divergence second order equations in domains extending to infinity along a cone. We find an exponent of the solution decreasing rate: we derive the estimate of the weak solution modulus for our problems near the infinity under assumption that leading coefficients of the equations do not satisfy the Dini-continuity condition.

Almost complete description of abelian groups (A,+, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

Report of Meeting. The Sixteenth Debrecen-Katowice Winter Seminar Hernádvécse (Hungary), January 27–30, 2016.