Published by De Gruyter

Volume 64 Issue 5

October 2014

Issue of Mathematica Slovaca

Contents
Journal Overview

Open Access November 15, 2014

The pasting constructions for effect algebras

Yongjan Xie, Yongming Li, Aili Yang

Page range: 1051-1074

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Abstract

The aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.

Open Access November 15, 2014

BM-Algebras and related topics

Andrzej Walendziak

Page range: 1075-1082

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Abstract

Some connections between BM-algebras and its related topics are studied. It is proved that the class of medial BH-algebras coincides with the class of BM-algebras. Moreover, the congruence lattice of a BM-algebra is investigated.

Open Access November 15, 2014

The generalized q-Pilbert matrix

Emrah Kiliç, Helmut Prodinger

Page range: 1083-1092

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Abstract

A generalized q-Pilbert matrix from[KILIÇ, E.-PRODINGER, H.: The q-Pilbert matrix, Int. J. Comput. Math. 89 (2012), 1370–1377] is further generalized, introducing one additional parameter. Explicit formulæ are derived for the LU-decomposition and their inverses, as well as the Cholesky decomposition. The approach is to use q-analysis and to leave the justification of the necessary identities to the q-version of Zeilberger’s celebrated algorithm. However, the necessary identities have appeared already in disguised form in the paper referred above, so that no new computations are necessary.

Open Access November 15, 2014

States on bounded commutative residuated lattices

Michiro Kondo

Page range: 1093-1104

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Abstract

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X, (1)If s is a state, then X/ker(s) is an MV-algebra.(2)If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra. Moreover we show that for a state s on X, the following statements are equivalent: (i)s is a state-morphism on X.(ii)ker(s) is a maximal filter of X.(iii)s is extremal on X.

Open Access November 15, 2014

A categorical equivalence for bounded distributive quasi lattices satisfying: x ∨ 0 = 0 ⇒ x = 0

Hector Freytes, Antonio Ledda

Page range: 1105-1122

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Abstract

In this work, we investigate a categorical equivalence between the class of bounded distributive quasi lattices that satisfy the quasiequation x∨0 = 0 =⇒ x = 0, and a category whose objects are sheaves over Priestley spaces.

Open Access November 15, 2014

On left regular and intra-regular ordered semigroups

Niovi Kehayopulu, Michael Tsingelis

Page range: 1123-1134

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Abstract

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.

Open Access November 15, 2014

Quasi-Monte Carlo integration in unanchored Sobolev spaces

Jana Fialová

Page range: 1135-1144

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Abstract

We find a concrete sequence of N points, for which the squared worst-case quasi-Monte Carlo error in the Hilbert space of continuous functions defined on [0, 1] with square integrable second derivative is smaller than for the centered regular lattice point set.

Open Access November 15, 2014

The exponential Diophantine equation x2 + (3n 2 + 1)y = (4n 2 + 1)z

Jianping Wang, Tingting Wang, Wenpeng Zhang

Page range: 1145-1152

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Abstract

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x 2 + (3n 2 + 1)y = (4n 2 + 1)z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n 3 + 3n, 1, 3).

Open Access November 15, 2014

The maximal class with respect to maximums for the family of upper semicontinuous strong Świca¸tkowski functions

Paulina Szczuka

Page range: 1153-1164

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Abstract

The main goal of this paper is to characterize the maximal class with respect to maximums for the family of upper semicontinuous strong Świşatkowski functions.

Open Access November 15, 2014

On a strengthened multidimensional Hilbert-type inequality

Mario Krnić

Page range: 1165-1182

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Abstract

The main objective of this paper is a study of the general refinement and converse of the multidimensional Hilbert-type inequality in the so-called quotient form. Such extensions are deduced with the help of the sophisticated use of the well-known Hölder’s inequality. The obtained results are then applied to homogeneous kernels with the negative degree of homogeneity. Also, we establish the conditions under which the constant factors involved in the established inequalities are the best possible. Finally, we consider some particular settings with homogeneous kernels and weighted functions. In such a way we obtain both refinements and converses of some actual results, known from the literature.

Open Access November 15, 2014

Some class of analytic functions related to conic domains

Stanisława Kanas, Dorina Răducanu

Page range: 1183-1196

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Abstract

For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where $$\mathbb{U}$$ denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R qλ f is defined. Applying R qλ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.

Open Access November 15, 2014

Subordination and superordination results associated with the generalized hypergeometric function

Nanjundan Magesh, Jagadeesan Jothibasu, Sami Murthy

Page range: 1197-1216

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Abstract

In the present investigation, we obtain some subordination and superordination results involving Hadamard products for certain normalized analytic functions in the open unit disk. Relevant connections of the results, which are presented in this paper, with various other known results are also pointed out.

Open Access November 15, 2014

Some results on the entire function sharing problem

BaoQin Chen, Sheng Li

Page range: 1217-1226

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Abstract

Our main result is as follows: let f and a be two entire functions such that $$\max \{ \rho _2 (f),\rho _2 (a)\} < \tfrac{1} {2}$$. If f and f (k) a CM, and if ρ(a (k) − a) < ρ(f − a), then f (k) − a = c(f − a) for some nonzero constant c. This result is applied to improve a result of Gundersen and Yang.

Open Access November 15, 2014

Oscillation of second-order nonlinear differential equations with damping

Tongxing Li, Yuriy Rogovchenko, Shuhong Tang

Page range: 1227-1236

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Abstract

We study oscillatory properties of solutions to a class of nonlinear second-order differential equations with a nonlinear damping. New oscillation criteria extend those reported in [ROGOVCHENKO, Yu. V.—TUNCAY, F.: Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Anal. 69 (2008), 208–221] and improve a number of related results.

Open Access November 15, 2014

Estimates on the dimension of an exponential attractor for a delay differential equation

Sakineh Habibi

Page range: 1237-1248

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Abstract

We study the long time behavior of delay differential equation, considered in a bounded domain in ℝd. Using the short trajectory method to prove the existence of the exponential attractor. Also we have estimates on the fractal dimension of an exponential attractor.

Open Access November 15, 2014

Existence of non-trivial solutions for systems of n fourth order partial differential equations

Shapour Heidarkhani

Page range: 1249-1266

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Abstract

In this paper, employing a very recent local minimum theorem for differentiable functionals due to Bonanno, the existence of at least one nontrivial solution for a class of systems of n fourth order partial differential equations coupled with Navier boundary conditions is established.

Open Access November 15, 2014

On the generalization of density topologies on the real line

Jacek Hejduk, Renata Wiertelak

Page range: 1267-1276

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Abstract

The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

Open Access November 15, 2014

Principal bundle structure on jet prolongations of frame bundles

J. Brajerčík, M. Demko, D. Krupka

Page range: 1277-1290

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Abstract

In this paper, we introduce the structure of a principal bundle on the r-jet prolongation J r FX of the frame bundle FX over a manifold X. Our construction reduces the well-known principal prolongation W r FX of FX with structure group G nr. For a structure group of J r FX we find a suitable subgroup of G nr. We also discuss the structure of the associated bundles. We show that the associated action of the structure group of J r FX corresponds with the standard actions of differential groups on tensor spaces.

Open Access November 15, 2014

A representation of skew effect algebras

Ivan Chajda

Page range: 1291-1297

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Abstract

It was shown by the author and R. Halaš that every effect algebra can be organized into a conditionally residuated structure. Skew effect algebras were introduced as a non-associative modification of effect algebras. Hence, there is natural question if a similar characterization by a certain residuated structure is possible. For this we use the so-called skew residuated structure introduced recently by the author and J. Krňávek. It is shown that this is really a suitable tool for the representation.

About this journal

Objective
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca.

It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc.　 The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.

Topics

Algebra and number theory
Analysis
Graph theory
Probability theory and mathematical statistics
Topology, measure theory and integration
Many-valued logic and quantum structures
Differential equations and dynamic systems
Physics
Soft computing
Cryptography
Biology
Economy
Measuring

Article formats
Original research articles, short notes

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