In categories of commutative universal algebras of given types we discover full subcategories in which direct products coincide with tensor products.

It is shown that the values of two highest multiplicities of Peano maps cannot be the values of openess.
This adds a new detail to the results by Hurewicz (1933), concerning dimension raising maps, when they are restricted to maps from the closed interval into the plane.

In this note we establish some characterizations of (single valued) unctions, that assume values in a Banach space, generating K-Schur concave sums. These results improve some theorems obtained in [13] and [11]. Moreover we prove that a set-valued function is K-concave if and only of it is K-t-concave and K-quasi concave (where t is a fixed number in (0,1)). This result improves the theorems obtained in [11], [2], [5] and extends the theorem of [3].

Some characterizations of Wright-convex stochastic processes are presented.

We consider the classical property (N) of Luzin for various mappings in connection with a measure extension problem. We give some examples of Borel measurable mappings and of Lebesgue measurable mappings which transform all compact sets with measure zero into sets with measure zero but do not have the property (N) of Luzin.

We give conditions under which every measurable function is the limit almost everywhere of a sequence of continuous functions.

The general solution of the functional equation
f(x) = f(x+1) + f(x(x+1)),
considered both on (0,+∞) and ℝ, are studied. Constructions of odd and even solutions are given.

We consider equation (6) in the class of continuous functions f: I→ℝ satisfying (7), where f is a non-trivial real interval and n,k are fixed positive integers. The obtained results are applied to get the solutions of the system of functional equations (3)-(5) in the class of pairs of functions f,g: I→ℝ such that f is continuous. Some connections between solutions of the equations and a class of subsemigroups of some Lie groups are established as well.

Addition formulas for generalized trigonometric functions corresponding to a given symmetric bounded and convex planar set containing the origin as an inner point are derived. Connections with the theory of characters on (semi) groups are considered. Hyers-Ulam stability of a suitable system of functional equations is investigated. It is also shown that superstability phenomenon fails to hold for that system.

In his work on the Gołąb-Schinzel equation, K. Baron shows a theorem concerning continuous complex-valued solutions, defined on the
complex plane. In this note, we will give a direct proof of this theorem, which does not use the form of the general solution of the Gołąb-Schinzel equation.

We find all of continuous, homeomorphic and Ck solutions of the system of the Abel equations
ϕ(f(x)) = ϕ(x)+a
ϕ(g(x)) = ϕ(x)+b   for x∈ℝ²,
where a,b are linearly independent vectors and f,g are commutable orientation preserving homeomorphisms of the plane onto itself satisfying some condition which is equivalent to the fact that there exists a homeomorphic solution of the system above.

A system of two simultaneous functional equations in a single variable, related to a generalized Gołąb-Schinzel functional equation, is considered.

In inner product spaces the Ptolemaic inequality (1) and the quadrilateral inequality (2) are well known. By using the identity (3), we derive here (2) from (1). The rest of the paper is devoted to some comments on (1), (2), (3).