Vedecké časopisy a ročenky vydávané na pôde SAV

The relations between inclusion quasi-continuity (upper inclusion quasi-continuity, lower inclusion quasi-continuity) and somewhat continuity (upper somewhat continuity, lower somewhat continuity) of the multifunction $F:X \rightarrow Y$, where $X$ is a bitopological space are considered.

A special kind of stable points for an established family of functions which are continuous in suitable topologies introduced in $\mathbb{R}$ is considered.

Some common properties (continuity,quasicontinuity, symmetrical quasicontinuity, …) of functions whose graphs have the same closures are investigated.

In this article we introduce and investigate some typical property in the space of all Baire 1 functions. In the class of derivatives this property implies that primitive functions are monotone in some intervals. Moreover, we observe that the quasicontinuity implies this property.

We consider the notion of decomposable hull of a subset of the space of vector measures and give some elementary properties. The relation between decomposability of the set of measure selections of a multimeasure and additivity of multimeasure is studied. We give an example of multimeasure with nonconvex and nondecomposable set of measure selections.

Under the assumption $L=V$ we construct a Hamel bases $H₁$ and $H₂$ of $\real$ and a continuous bijection $f\colon H₁ \to \real\setminus H₂$.

A function from the plane to the plane is axial if it does not change one coordinate. We show that every function is a superposition of three axial functions both measurable and with the Baire property. Every Borel function is a composition of three Borel axial functions.

The sums and the maximums of functions from the class $\M \cap \Bα$, for each countable ordinal $α$ are characterized. Moreover, the maximal additive class and the maximal class with respect to maximums for these classes of functions are characterized as well.

The sup-measurability of multivalued function $F$ of two variables in metric spaces with diferentiation basis is studied. It is shown that if all $x$-sections of $F$ have (Z) property and all $y$-sections of $F$ are measurable, then $F$ is sup-measurable.

This paper introduces the notion of a cluster system and a cluster operation acting on multifunctions, as a base tool for investigation of continuity properties of multifunctions. The main results of the paper concern the lower quasi continuity and the lower semi continuity of a cluster multifunction and the properties of a multifunction having the closed graph with respect to a given cluster system.

In this paper the concepts of $I$-limit points, $I$-cluster points and $I$-limit superior and limit inferior of double sequences are introduced. We prove some basic properties.

In the paper we investigate $f$-density operators and $f$-density topologies generated by functions $f$ satisfying the condition $\liminf\limits_{x\rightarrow 0⁺}((f(x)) / (x))>0$. We compare them with topologies generated by sequences.

The structure of the set $\mathcal B$ of all ranges of additive generators of associative functions is studied here. Sufficient conditions for $\cap_n=1^∞ A_n\in \mathcal B$, where $A_n\in \mathcal B$ for all $n\in N$, are introduced. Examples of elements of $\mathcal B$ which are nowhere dense in $[0,∞]$ and contain $0$ as an accumulation point are presented.

The paper contains investigation of the density topologies generated by functions $f$ such that $\limsup_{x\to 0⁺} ((x) / (f(x))) >0$.

Let $A=(a_nk)$ be a regular matrix and $S(A)$ be the set of all sequences $x=(x_k)$ such that the series $∑_k=1^∞a_nkx_k$ converges for each $n=1,2,…$ It is shown that the convergence field $F(A)$ of the matrix $A$ is a $σ$-porous set in the linear metric space $S(A)$ endowed with the Fr

I. Dobrakov introduced in: [\textit{On submeasures I}, Dissertat. Math. \textbf{112} (1974), 5–35], the notion of a submeasure defined on the ring of sets. This type of a submeasure is now known as the Dobrakov submeasure. In this paper we develop some limit techniques to create new Dobrakov submeasures from the old ones in the case when elements of the ring $\mathcal{R}$ are subsets of the real line.

A continuous and nowhere monotone function for which every set having the Baire property, on which it is one-to-one, is of first category, is constructed here. Further, it is shown that every continuous and nowhere monotone function has the above property. We also show that an analogous result fails to hold if one does not assume that a set on which the function is one-to-one is a Baire set. \par In the second part of the paper we investigate the Lebesgue measure of a set on which a continuous nowhere monotone function is one-to-one. Here the situation turns out to be more varied. For each $η\in[0,1)$ we construct a continuous function non-monotone on any interval with the following property: there exists a set of measure $η$ on which this function is one-to-one, and every set on which the function is one-to-one has measure smaller or equal to $η$.

A generalization of the Herglotz theorem is established for a monotone $σ$-complete partially ordered vector spaces. Some version of this theorem may be proved without the assumption of the monotone $σ$-completeness.

We present a moment problem in the context of fuzzy sets. A generalization of the Hausdorff moment theorem is formulated and proved for fuzzy double sequences