# Vedecké časopisy a ročenky vydávané na pôde SAV

## Tatra Mountains Mathematical Publications

Volume 40, 2008, No. 2

Content:

 On generalizations of the notion of continuity of multifunctions in bitopological spaces Andrzej Rychlewicz 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 1-11. The stable points and the attractors of Darboux functions Helena Pawlak, Ryszard J. Pawlak A special kind of stable points for an established family of functions which are continuous in suitable topologies introduced in $\mathbb{R}$ is considered. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 13-24. On a functional relation defined by the equality of the closures of graphs Zbigniew Grande, Halina Wiśniewska Some common properties (continuity,quasicontinuity, symmetrical quasicontinuity, …) of functions whose graphs have the same closures are investigated. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 25-33. Of a subfamily of derivatives Ewa Strońska 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 35-42. Note on decomposable sets Jolanta Olko 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 43-51. Injective continuous images of Hamel bases Andrzej Nowik Under the assumption $L=V$ we construct a Hamel bases $H1$ and $H2$ of $\real$ and a continuous bijection $f\colon H1 \to \real\setminus H2$. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 53-58. Note on axial functions on the plane Marcin Szyszkowski 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 59-62. Algebraic properties of the classes $\mathcal {M} \cap \mathcal {B}α$ Katarzyna Chmielewska 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 63-69. On sup-measurability of multivalued functions with the (Z) property in second variable Grażyna Kwiecińska 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 71-80. Continuity properties of cluster multifunctions and closed graph theorems Milan Matejdes 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 81-89. On extremal $I$-limit points of double sequences Pratulananda Das, Prasanta Malik 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 91-102. Density of topologies generated by functions and by sequences Małgorzata Filipczak, Tomasz Filipczak In the paper we investigate $f$-density operators and $f$-density topologies generated by functions $f$ satisfying the condition $\liminf\limitsx\rightarrow 0+((f(x)) / (x))>0$. We compare them with topologies generated by sequences. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 103-115. Intersections of ranges of additive generators of associative functions Peter Viceník The structure of the set $\mathcal B$ of all ranges of additive generators of associative functions is studied here. Sufficient conditions for $\capn=1∞ An\in \mathcal B$, where $An\in \mathcal B$ 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 117-131. On the density topologies generated by functions Jacek Hejduk The paper contains investigation of the density topologies generated by functions $f$ such that $\limsupx\to 0+ ((x) / (f(x))) >0$. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 133-141. Convergence fields of regular matrix transformations $2$ Pavel Kostyrko Let $A=(ank)$ be a regular matrix and $S(A)$ be the set of all sequences $x=(xk)$ such that the series $∑k=1∞ankxk$ 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 Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 143-147. On Dobrakov net submeasures Ján Haluška, Ondrej Hutník 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 149-160. The one-to-one restrictions of functions Aleksandra Karasińska 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 $η$. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 161-169. A Herglotz theorem in ordered vector spaces which are not a lattice Miloslav Duchoň, Peter Maličký 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. Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 171-183. Moment problem for double fuzzy sequences Miloslav Duchoň, Camille Debiève 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 Fulltext Tatra Mountains Mathematical Publications. Volume 40, 2008, No. 2: 185-192.