# Scientific Journals and Yearbooks Published at SAS

## Tatra Mountains Mathematical Publications

Volume 49, 2011, No. 2

Content:

Another proof of Hurewicz theorem.

Miroslav Repický 1)

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 1) Matematický ústav SAV, Jesenná 5; 041 54 Košice; Slovensko. repicky@kosice.upjs.sk

Hurewicz scheme, $D$-proper mapping, analytic set

A Hurewicz theorem says that every coanalytic non-$Gδ$ set $C$ in a Polish space contains a countable set $Q$ without isolated points such that $\overline Q\cap C=Q$. We present another elementary proof of this theorem and generalize it for $κ$-Suslin sets. As a consequence, under Martin's Axiom, we obtain a characterization of $\boldsymbolΣ12$ sets that are the unions of less than the continuum closed sets.

How to cite (APA format):
Repický, M. (2011). Another proof of Hurewicz theorem. Tatra Mountains Mathematical Publications, 49(2), 1-7.

One more difference between measure and category.

Jacek Hejduk 1)

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 1) Chair of Real Functions Faculty of Mathematics, University of Łódź , Banacha 22; 90-238ô Łódź; POLAND. heiduk@math.uni.lodz.pl

density operator, abstract density topology, measurable space

The first part of the paper contains some ideas of the density topologies in the measurable spaces. The second part is devoted to the difference between measure and category for the abstract density space related to the separation axioms.

How to cite (APA format):
Hejduk, J. (2011). One more difference between measure and category. Tatra Mountains Mathematical Publications, 49(2), 9-15.

Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in $(l)$-groups.

Antonio Boccuto 1), Xenofon Dimitriou 2), Nikolas Papanastassiou 3)

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 1) Dipartimento di Matematica, Università di Perugia; Via Vanvitelli 1; 06123 Perugia; ITALY. boccuto@yahoo.it 2) Department of Mathematics University of Athens , Department of Mathematics; University of Athens, Panepistimiopolis; Athens 15784; GREECE. xenofon11@gmail.com; 3) Department of Mathematics University of Athens , Panepistimiopolis; Athens 15784; GREECE. npapanas@math.uoa.gr

{$(l)$-group, ideal, ideal $(D)$-convergence, limit theorem

Some Brooks-Jewett, Vitali-Hahn-Saks and Nikod\'{y}m convergence type theorems in the context of $(l)$-groups with respect to ideal convergence are proved. Moreover, an example is given.

How to cite (APA format):
Boccuto, A, Dimitriou, X, Papanastassiou, N. (2011). Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in $(l)$-groups. Tatra Mountains Mathematical Publications, 49(2), 17-26.

On some new subfamilies of classical spaces of absolutely $p$-summable sequences.

Roman Wituła 1), Damian Słota 1)

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 1) Institute of Mathematics, Silesian University of Technology; Kaszubska 23; PL–44-100 Gliwice; POLAND. roman.witula@polsl.pl

almost $l^p$, exactly $l^p$, exmost $l^p$

In this paper the properties of some new subfamilies of the spaces $lp(\mathbb{C})$ are discussed.

How to cite (APA format):
Wituła, R, Słota, D. (2011). On some new subfamilies of classical spaces of absolutely $p$-summable sequences. Tatra Mountains Mathematical Publications, 49(2), 27-48.

Measures and idempotents in the non-commutative situation.

Reinhard Börger 1)

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 1) Fakultät für Mathematik und Informatik, , Fern Universität, 58084 Hagen, Germany. reinhard.boerger@fernuni-hagen.de

sequential space, weakly Hausdorff sequential orthomodular poset (WHSOP), sequentially convex space, polymeasure, tensor product, sequentially convex algebra, idempotent, multiplicative measure, involution

We investigate measures on sequential orthomodular posets with values in a vector space or a (not necessarily commutative) algebra with reasonable sequential topologies, using a universal property. Unfortunately, the universal measure and the universal multiplicative measure need not coincide any more as in the commutative situation. This may have applications in quantum physics.

How to cite (APA format):
Börger, R. (2011). Measures and idempotents in the non-commutative situation. Tatra Mountains Mathematical Publications, 49(2), 49-58.

A note on the Kluávnek Integral.

Štefan Tkačik 1), Beloslav Riečan 2)

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 1) Department of Mathematics, Faculty of Education; Catholic University in Ružomberok; Hrabovská cesta 1; 034 01 Ružomberok; Slovakia. stefan.tkacik@fedu.ku.sk 2) Department of Medical Informatics; Faculty of Natural Science; Matej Bel University , Tajovského 40; 974 01 Bansk8 Bystrica; SLOVAKIA. riecan@fpv.umb.sk

Lebesgue integral, Archimedean integral, summation of infinite series

For real valued functions, Igor Kluv\'anek has introduced an integral, called Archimedean, which is equivalent to the Lebesgue integral. It is based on the summation of infinite series and it avoids measure. Modifying the definition by Kluv\'anek, we introduce an integral which has some good and some bad properties and we compare it with the Lebesgue integral.

How to cite (APA format):
Tkačik, Š, Riečan, B. (2011). A note on the Kluávnek Integral. Tatra Mountains Mathematical Publications, 49(2), 59-65.

A decomposition of bounded, weakly measurable functions.

Surjit Singh Khurana 1)

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 1) Department of Mathematics, The University of Iowa; Iowa City; Iowa 52242; U. S. A.. khurana@math.uiowa.edu

liftings, weakly measurable functions, weakly equivalent functions, vector measures with finite variations

Let $(X, \mathcal{A}, μ)$ be a complete probability space, $ρ$ a lifting, $\mathcal{T}ρ$ the associated Hausdorff lifting topology on $X$ and $E$ a Banach space. Suppose $F\colon (X, \mathcal{T}ρ) \to E''σ$ be a bounded continuous mapping. It is proved that there is an $A \in \mathcal{A}$ such that $F χA$ has range in a closed separable subspace of $E$ (so $F χA\colon X \to E$ is strongly measurable) and for any $B \in \mathcal{A}$ with $μ(B) >0$ and $B \cap A = \emptyset$, $F χB$ cannot be weakly equivalent to a $E$-valued strongly measurable function. Some known results are obtained as corollaries.

How to cite (APA format):
Khurana, S. (2011). A decomposition of bounded, weakly measurable functions. Tatra Mountains Mathematical Publications, 49(2), 67-70.

Sums and products of extra strong Świątkowski functions.

Paulina Szczuka 1)

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 1) Bydgoszcz Academy, pl. Weyssenhoffa 11; PL–85-072 Bydgoszcz; POLAND. paulinaszczuka@wp.pl

Darboux function, quasi-continuous function, strong Świątkowski function, extra strong Świątkowski function, sum of functions, product of functions

In this paper we present a characterization of sums of extra strong Świątkowski functions, and we examine some functions which can be written as the product of extra strong \'Swi\c{a}tkowski functions.

How to cite (APA format):
Szczuka, P. (2011). Sums and products of extra strong Świątkowski functions. Tatra Mountains Mathematical Publications, 49(2), 71-79.

Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space.

Francesco Tulone 1)

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 1) Department of Mathematics, University of Palermo; via Archirafi 34; I–90123 Palermo; ITALY. tulone @math.unipa.it

Henstock-Kurzweil integral, Perron integral, Lebesgue integral, derivation basis, compact zero-dimensional metric space, major and minor function

A Henstock-Kurzweil type integral on a compact zero-dimensional metric space is investigated. It is compared with two Perron type integrals. It is also proved that it covers the Lebesgue integral.

How to cite (APA format):
Tulone, F. (2011). Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space. Tatra Mountains Mathematical Publications, 49(2), 81-88.

Functions with bounded variation in locally convex space.

Camille Debiève 1), Miloslav Duchoň 2)

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 1) Département de Mathématiques;, Université Catholique de Louvain; 2, Chemin du Cyclotron; B--1348 Louvain-la-Neuve; BELGIUM Pure et Appl.; Chemin du Cyclotron 2; B-1348 Louvain-la-Neuve; BELGIQUE. camille.debieve@uclouvain.be 2) Mathematical Institute SAS, Štefánikova 49; 814 73 Bratislava; SLOVAKIA. duchon@mat.savba.sk

locally convex space, bounded variation, vector-valued measure on Borel subsets

The present paper is concerned with some properties of functions with values in locally convex vector space, namely functions having bounded variation and generalizations of some theorems for functions with values in locally convex vector spaces replacing Banach spaces, namely {Theorem}: If $X$ is a sequentially complete locally convex vector space, then the function $x(·):[a,b] \to X$ having a bounded variation on the interval $[a,b]$ defines a vector-valued measure $m$ on borelian subsets of $[a,b]$ with values in $X$ and with the bounded variation on the borelian subsets of $[a,b]$; the range of this measure is also a weakly relatively compact set in $X$. This theorem is an extension of the results from Banach spaces to locally convex spaces.

How to cite (APA format):
Debiève, C, Duchoň, M. (2011). Functions with bounded variation in locally convex space. Tatra Mountains Mathematical Publications, 49(2), 89-98.

A generalized Bernstein approximation theorem.

Miloslav Duchoň 1)

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 1) Mathematical Institute SAS, Štefánikova 49; 814 73 Bratislava; SLOVAKIA. duchon@mat.savba.sk

Bernstein polynomial, Bernstein approximation theorem, generalized

The present paper is concerned with some generalizations of Bernstein's approximation theorem. One of the most elegant and elementary proofs of the classic result, for a function $f(x)$ defined on the closed interval $[0,1]$, uses the Bernstein's polynomials of $f$,

$$Bn(x)=Bnf(x)=∑k=0n f(((k) / (n)))\binom{n}{k}xk(1-x)n-k$$

We shall concern the $m$-dimensional generalization of the Bernstein's polynomials and the Bernstein's approximation theorem by taking an $(m-1)$-dimensional simplex in cube $[0,1]m$. This is motivated by the fact that in the field of mathematical biology naturally arouse dynamic systems determined by quadratic mappings of standard" $(m-1)$-dimensional simplex $\{xi ≥ 0$, $i=1,…,m$, $∑i=1m xi=1 \}$ to self. The last condition guarantees saving of the fundamental simplex. Then there are surveyed some other the $m$-dimensional generalizations of the Bernstein's polynomials and the Bernstein's approximation theorem.

How to cite (APA format):
Duchoň, M. (2011). A generalized Bernstein approximation theorem. Tatra Mountains Mathematical Publications, 49(2), 99-109.

Markov type polynomial inequality for some generalized Hermite weight.

Branislav Ftorek 1), Mariana Marčoková 2)

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 1) Department of Applied Mathematics, Faculty of Mechanical Engineering; University of Žilina; Universitná 1; SK--010-26 Žilina; SLOVAKIA. branislav.ftorek@fstroj.uniza.sk 2) Department of Mathematics, Faculty of Science University of Žilina, Univerzitná 1; SK--010-26 Žilina, SLOVAKIA. mariana.marcokova@fpv.uniza.sk

Markov type inequality, weight function, generalized Hermite polynomials

In this paper we study some weighted polynomial inequalities of Markov type in $L^2$-norm. We use the properties of the system of generalized Hermite polynomials $\{H^{(\alpha)} _n (x)\}_{n=0}^{\infty}$. The polynomials $H^{(\alpha)} _n (x)$ are orthogonal in $\mathbb{R}=(-\infty,\infty )$ with respect to the weight function $$W(x)=|x|^{2\alpha } { e}^{- x^2},\qquad \alpha > -{1\over 2}.$$ The classical Hermite polynomials $H_n (x)$ present the special case for $\alpha =0$.

How to cite (APA format):
Ftorek, B, Marčoková, M. (2011). Markov type polynomial inequality for some generalized Hermite weight. Tatra Mountains Mathematical Publications, 49(2), 111-118.

Generalized oscillations for generalized continuities.

Ján Borsík 1)

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 1) Matematický ústav SAV, Grešákova 6; 040 01 Košice; Slovensko. borsik@saske.sk

generalized topology, generalized continuity, generalized oscillation

Let $(X,{\mathfrak{g}})$ be a generalized topological space, $(Y,d)$ a metric one and $f\colon X\to Y$ a function. We can define a generalized oscillation of $f$ at $x\in X$ as $k_f^{\mathfrak{g}}(x)=\inf\{ \DeclareMathOperator{diam} f(A): A\in{\mathfrak{g}}, x\in A\}$. We discuss some properties of the generalized oscillation.

How to cite (APA format):
Borsík, J. (2011). Generalized oscillations for generalized continuities. Tatra Mountains Mathematical Publications, 49(2), 119-125.