# Scientific Journals and Yearbooks Published at SAS

## Article List

## Tatra Mountains Mathematical Publications

Volume 49, 2011, No. 2

Content:

- Repický, M.
**Another proof of Hurewicz theorem.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 1-7. - Hejduk, J.
**One more difference between measure and category.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 9-15. - Boccuto, A. - Dimitriou, X. - Papanastassiou, N.
**Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in $(l)$-groups.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 17-26. - Wituła, R. - Słota, D.
**On some new subfamilies of classical spaces of absolutely $p$-summable sequences.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 27-48. - Börger, R.
**Measures and idempotents in the non-commutative situation.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 49-58. - Tkačik, Š. - Riečan, B.
**A note on the Kluávnek Integral.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 59-65. - Khurana, S.
**A decomposition of bounded, weakly measurable functions.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 67-70. - Szczuka, P.
**Sums and products of extra strong Świątkowski functions.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 71-79. - Tulone, F.
**Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 81-88. - Debiève, C. - Duchoň, M.
**Functions with bounded variation in locally convex space.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 89-98. - Duchoň, M.
**A generalized Bernstein approximation theorem.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 99-109. - Ftorek, B. - Marčoková, M.
**Markov type polynomial inequality for some generalized Hermite weight.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 111-118. - Borsík, J.
**Generalized oscillations for generalized continuities.**

In*Tatra Mountains Mathematical Publications*. Vol. 49, no. 2 (2011), p. 119-125.

Another proof of Hurewicz theoremFulltext Miroslav Repický ^{1)}
Hurewicz scheme, $D$-proper mapping, analytic setA Hurewicz theorem says that every coanalytic non-$G
_{δ}$ 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 $\boldsymbolΣ^{1}_{2}$ sets that are the unions of less than the continuum closed sets.Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 1-7. | ||||||||

One more difference between measure and categoryFulltext Jacek Hejduk ^{1)}
density operator, abstract density topology, measurable spaceThe 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 9-15. | ||||||||

Brooks-Jewett-type theorems for the pointwise ideal convergence of measures with values in $(l)$-groupsFulltext Antonio Boccuto ^{1)}, Xenofon Dimitriou ^{2)}, Nikolas Papanastassiou ^{3)}
{$(l)$-group, ideal, ideal $(D)$-convergence, limit theoremSome 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 17-26. | ||||||||

On some new subfamilies of classical spaces of absolutely $p$-summable sequencesFulltext Roman Wituła ^{1)}, Damian Słota ^{1)}
almost $l^p$, exactly $l^p$, exmost $l^p$In this paper the properties of some new subfamilies of the spaces $l
^{p}(\mathbb{C})$ are discussed.Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 27-48. | ||||||||

Measures and idempotents in the non-commutative situationFulltext Reinhard Börger ^{1)}
sequential space, weakly Hausdorff sequential orthomodular poset (WHSOP), sequentially
convex space, polymeasure, tensor product, sequentially convex algebra,
idempotent, multiplicative measure, involutionWe 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 49-58. | ||||||||

A note on the Kluávnek IntegralFulltext Štefan Tkačik ^{1)}, Beloslav Riečan ^{2)}
Lebesgue integral, Archimedean integral, summation of infinite seriesFor 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 59-65. | ||||||||

A decomposition of bounded, weakly measurable functionsFulltext Surjit Singh Khurana ^{1)}
liftings, weakly measurable functions, weakly equivalent functions, vector measures with finite variationsLet $(X, \mathcal{A}, μ)$ 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 $F χ_{A}$ has range in a closed separable subspace of $E$ (so $F χ_{A}\colon X \to E$ is strongly measurable) and for any $ B \in \mathcal{A}$ with $ μ(B) >0$ and $B \cap A = \emptyset$, $F χ_{B}$ cannot be weakly equivalent to a $E$-valued strongly measurable function. Some known results are obtained as corollaries.Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 67-70. | ||||||||

Sums and products of extra strong Świątkowski functionsFulltext Paulina Szczuka ^{1)}
Darboux function, quasi-continuous function, strong Świątkowski
function, extra strong Świątkowski function, sum of functions, product of functionsIn 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 71-79. | ||||||||

Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric spaceFulltext Francesco Tulone ^{1)}
Henstock-Kurzweil integral, Perron integral, Lebesgue integral, derivation basis, compact zero-dimensional metric space, major and minor functionA 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 81-88. | ||||||||

Functions with bounded variation in locally convex spaceFulltext Camille Debiève ^{1)}, Miloslav Duchoň ^{2)}
locally convex space, bounded variation, vector-valued measure on Borel subsetsThe 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 89-98. | ||||||||

A generalized Bernstein approximation theoremFulltext Miloslav Duchoň ^{1)}
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 $f$,
$$ B ^{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 $\{x_{i} ≥ 0$, $i=1,…,m$, $∑_{i=1}^{m} x_{i}=1 \}$ 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.Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 99-109. | ||||||||

Markov type polynomial inequality for some generalized Hermite weightFulltext Branislav Ftorek ^{1)}, Mariana Marčoková ^{2)}
Markov type inequality, weight function, generalized Hermite polynomialsIn 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 $\alpha =0$.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 111-118. | ||||||||

Generalized oscillations for generalized continuitiesFulltext Ján Borsík ^{1)}
generalized topology, generalized continuity, generalized oscillationLet $(X,{\mathfrak{g}})$ 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.
Tatra Mountains Mathematical Publications. Volume 49, 2011, No. 2: 119-125. |