Published by De Gruyter

Volume 67 Issue 3

Issue of Mathematica Slovaca

Contents
Journal Overview

June 7, 2017

Some results in local Hilbert algebras

Ali Soleimani Nasab, Arsham Borumand Saeid

Page range: 541-552

Abstract

In this paper, we study local Hilbert algebras and give some theorems that characterize local Hilbert algebras. Also, we prove that, (i) H is a local Hilbert algebra if and only if the set of all dense elements of H is H ∖ {0}. (ii) H is a local Hilbert algebra if and only if the set of all regular elements of H is a local Boolean algebra. In a local Hilbert algebra H with supremum, we prove that: (i) If F is a Boolean filter of H , then F is a prime filter. (ii) F is a Boolean filter iff F is a maximal filter iff F = H ∖ {0}.

June 7, 2017

A note on residuated po-groupoids and lattices with antitone involutions

Ivan Chajda, Jan Kühr

Page range: 553-560

Abstract

Following [BOTUR, M.—CHAJDA, I.—HALAŠ, R.: Are basic algebras residuated structures? , Soft Comput. 14 (2010), 251—255] we discuss the connections between left-residuated partially ordered groupoids and the so-called basic algebras, which are a non-commutative and non-associative generalization of MV-algebras and orthomodular lattices.

June 5, 2017

The first law of cubology for the Rubik’s Revenge

Stefano Bonzio, Andrea Loi, Luisa Peruzzi

Page range: 561-572

Abstract

We state and prove the first law of cubology of the Rubik’s Revenge and provide necessary and sufficient conditions for a randomly assembled Rubik’s Revenge to be solvable.

June 5, 2017

Conditional associativity in orthomodular lattices

Jeannine J. M. Gabriëls

Page range: 573-584

Abstract

For orthomodular lattices the word problem is still not decided, an important obstacle is the lack of distributivity. But also the absence of the associativity law for operations other than the lattice operations meet and join causes difficulties. In this paper we treat some aspects of associativity in orthomodular lattices. There are six out of 96 orthomodular operations which are associative. We search for orthomodular lattice operations which fulfil the associative identity under some conditions. Specifically we assume that some arguments commute and identify, for each operation, sufficient conditions under which a triple of elements fulfils the respective associativity identity.

June 7, 2017

A monotonicity property for generalized Fibonacci sequences

Toufik Mansour, Mark Shattuck

Page range: 585-592

Abstract

Given k ≥ 2, let a n be the sequence defined by the recurrence a n = α 1 a n –1 + … + α k a n – k for n ≥ k , with initial values a 0 = a 1 = … = a k –2 = 0 and a k –1 = 1. We show under a couple of assumptions concerning the constants α i that the ratio annan− 1n− 1 $\frac{\sqrt[n]{a_n}}{\sqrt[n-1]{a_{n-1}}}$ is strictly decreasing for all n ≥ N , for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the α i are unity or when all of the α i are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.

June 5, 2017

Systems of conditionally independent sets

Pavel Jahoda, Monika Jahodová

Page range: 593-600

Abstract

The concept of conditionally independent sets is introduced in this article. Its link with the concept of the conditional probability and with the concept of a system of independent sets is described here. Furthermore, particular example of a system of conditionally independent sets is given and different ways to create the system of conditionally independent sets are described.

June 5, 2017

On the k-th order lfsr sequence with public key cryptosystems

Bariş Bülent Kirlar, Melek Çİl

Page range: 601-610

Abstract

In this paper, we propose a novel encryption scheme based on the concepts of the commutative law of the k -th order linear recurrences over the finite field 𝔽 q for k > 2. The proposed encryption scheme is an ephemeral-static, which is useful in situations like email where the recipient may not be online. The security of the proposed encryption scheme depends on the difficulty of solving some Linear Feedback Shift Register (LFSR) problems. It has also the property of semantic security. For k = 2, we propose another encryption scheme by using the commutative law of Lucas and Fibonacci sequences having also the same security as proposed for k > 2.

June 7, 2017

On parametric spaces of bicentric quadrilaterals

Farzali Izadi, Foad Khoshnam, Allan J. MacLeod, Arman Shamsi Zargar

Page range: 611-622

Abstract

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by E a : y 2 = x 3 + ( a 4 − 4 a 3 − 2 a 2 − 4 a + 1) x 2 + 16 a 4 x which have, in general, (ℤ/8ℤ), and in rare cases (ℤ/2ℤ × ℤ/8ℤ) as torsion subgroups. We show the existence of infinitely many elliptic curves E a of rank at least two with torsion subgroup ℤ/8ℤ, parametrised by the points of an elliptic curve of rank at least one, and give five particular examples of rank 5. We, also, show the existence of a subfamily of E a whose torsion subgroup is ℤ/2ℤ × ℤ/8ℤ.

June 5, 2017

Character sums with an explicit evaluation

Badria Alsulmi, Vincent Pigno, Christopher Pinner

Page range: 623-630

Abstract

We show that for Dirichlet characters χ , χ 1 , …, χ s mod p m the sum ∑ x1=1pm… ∑ xs=1pmχ 1(x1)… χ s(xs)χ (A1x1+⋯ +Asxs+Bx1w1⋯ xsws) $$\begin{array}{} \displaystyle\sum\limits_{x_1=1}^{p^m} \dots \sum\limits_{x_s=1}^{p^m}\chi_1(x_1)\dots \chi_s(x_s) \chi(A_1x_1 + \dots + A_sx_s + Bx_1^{w_1}\cdots x_s^{w_s}) \end{array} $$ has a simple evaluation when m is sufficently large, for m ≥ 2 if p∤ 2A1⋯ AsB(1− w1− ⋯ − ws). $$p\nmid 2A_1\cdots A_sB(1-w_1-\cdots -w_s). $$

June 5, 2017

Some remarks on formal power series and formal Laurent series

Dariusz Bugajewski, Xiao-Xiong Gan

Page range: 631-644

Abstract

In this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.—BUGAJEWSKI, D.: On formal Laurent series , Bull. Braz. Math. Soc. 42 (2011), 415–437], is not continuous. We also show some applications of fixed point theorems to some nonlinear mappings defined on the space of formal power series or on the space of formal Laurent series. Finally, in the second part of the paper, we propose another approach to study of dot product and multiplication of formal Laurent series, in particular establishing integral representation of dot product and convolution representation of multiplication.

June 5, 2017

The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations

Irina Gelbukh

Page range: 645-656

Abstract

We study b1′ $b_{1}'$ ( M ), the co-rank of the fundamental group of a smooth closed connected manifold M . We calculate this value for the direct product of manifolds. We characterize the set of all possible combinations of b1′ $b_{1}'$ ( M ) and the first Betti number b 1 ( M ) by explicitly constructing manifolds with any possible combination of b1′ $b_{1}'$ ( M ) and b 1 ( M ) in any given dimension. Finally, we apply our results to the topology of Morse form foliations. In particular, we construct a manifold M and a Morse form ω on it for any possible combination of b1′ $b_{1}'$ ( M ), b 1 ( M ), m ( ω ), and c ( ω ), where m ( ω ) is the number of minimal components and c ( ω ) is the maximum number of homologically independent compact leaves of ω .

June 7, 2017

On prime and semiprime generalized hyperaction of hypermonoid

Muhammad Shabir, Salma Shaheen

Page range: 657-670

Abstract

In this paper, we introduce the notion of generalized hyperaction of hypermonoid (GHS-act) on a set, establish its basic properties and prove some fundamental results. We also define the notion of prime, semiprime, irreducible and maximal GHS-subacts and characterize hypermonoid in terms of these GHS-subacts.

June 7, 2017

On O’Malley ϱ-upper continuous functions

Stanisław Kowalczyk, Katarzyna Nowakowska

Page range: 671-684

Abstract

In the paper we present some properties of functions which are ϱ -upper continuous in O’Malley’s sense. Connections with the standard ϱ -upper continuous functions are studied and the maximal additive class and the maximal multiplicative class for this family of functions are described.

June 7, 2017

Uniqueness theorems for finitely additive probabilities on quantum structures

Marina Svistula

Page range: 685-690

Abstract

The proofs of uniqueness theorems, presented here, allow to extend the earlier results. For example, the following hold: let μ and ν be two finitely additive probabilities on a structure L̃ (for example, L̃ is a pseudo-effect algebra), and let μ be convex-ranged; if there exists an element a ∈ L̃ with 0 < μ ( a ) < 1 and such that μ ( a ) = μ ( b ) ⇒ ν ( a ) = ν ( b ) whenever b ∈ L̃ , then ν = μ .

June 6, 2017

Some results on differential-difference analogues of Brück conjecture

Xianjing Dong, Kai Liu

Page range: 691-700

Abstract

In this paper, we will consider value sharing problems on a transcendental entire function f with its differential-difference polynomials. We establish some results that can be viewed as the differential-difference analogues of Brück conjecture.

June 7, 2017

Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments

Simona Fišnarová, Robert Mařík

Page range: 701-718

Abstract

In this paper we derive oscillation criteria for the second order half-linear neutral differential equation [r(t)Φ (z′(t))]′+c(t)Φ (x(σ (t)))=0,z(t)=x(t)+b(t)x(τ (t)), $$ \displaystyle \Bigl[r(t)\Phi(z'(t))\Bigr]'+c(t)\Phi(x(\sigma(t)))=0, \quad z(t)=x(t)+b(t)x(\tau(t)), $$ where Φ( t ) = | t | p −2 t , p ≥ 2, is a power type nonlinearity. We improve recent results published in the literature by obtaining better oscillation constants and removing the usual condition σ ( τ ( t )) = τ ( σ ( t )). Two methods (comparison method and Riccati equation method) are used.

June 7, 2017

On support, separation and decomposition theorems for t-Wright-concave functions

Andrzej Olbryś

Page range: 719-730

Abstract

In the present paper we prove a separation theorem for t -Wright convex and t -Wright concave functionals. We also consider the problem of supporting the t -Wright concave functions by t -Wright affine ones. As a consequence of these results we prove some decomposition theorems for t -Wright convex functions.

June 5, 2017

A characterization of holomorphic bivariate functions of bounded index

Richard F. Patterson, Fatih Nuray

Page range: 731-736

Abstract

The following notion of bounded index for complex entire functions was presented by Lepson. function f ( z ) is of bounded index if there exists an integer N independent of z , such that max{l:0≤ l≤ N}|f(l)(z)|l!≥ |f(n)(z)|n!for alln. $$ \max\limits_{\{l: 0\leq l\leq N\}} \left \{ \frac{|{f^{(l)}(z)}|}{l!}\right \} \geq \frac{|{f^{(n)}(z)}|}{n!}\quad\text{for all}\,\, n. $$ The main goal of this paper is extend this notion to holomorphic bivariate function. To that end, we obtain the following definition. A holomorphic bivariate function is of bounded index, if there exist two integers M and N such that M and N are the least integers such that max{(k,l):0,0≤ k,l≤ M,N}|f(k,l)(z,w)|k!l!≥ |f(m,n)(z,w)|m!n!for allmandn. $$ \max\limits_{\{(k,l): 0,0\leq k, l\leq M, N\}} \left \{ \frac{|{f^{(k,l)}(z,w)}|}{k!\,l!}\right \} \geq \frac{|{f^{(m,n)}(z,w)}|}{m!\,n!}\quad\text{for all}\,\, m \, \text{and}\,\, n. $$ Using this notion we present necessary and sufficient conditions that ensure that a holomorphic bivariate function is of bounded index.

June 5, 2017

Asymptotic expansion of double Laplace-type integrals with a curve of minimal points and application to an exit time problem

Abdallah Benaissa, Moussa Benlahcene

Page range: 737-750

Abstract

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ . Complete asymptotic expansions will be constructed if that order is constant along γ , and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

June 7, 2017

The geometry of two-valued subsets of L_p-spaces

Anthony Weston

Page range: 751-758

Abstract

Let 𝓜(Ω, μ ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ ). Let B ⊂ 𝓜(Ω, μ ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p -negative type when viewed as a metric subspace of L p (Ω, μ ) if and only if B is an affinely independent subset of 𝓜(Ω, μ ) (when 𝓜(Ω, μ ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ ) has strict p -negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p -negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ ) that have p -negative type but not q -negative type for any q > p . Such sets preclude the existence of certain types of isometry into L p -spaces.

June 7, 2017

Hemi-slant submanifolds as warped products in a nearly Kaehler manifold

Viqar Azam Khan, Kamran Khan

Page range: 759-772

Abstract

The present article is devoted to the study of conditions on a hemi-slant submanifold of a nearly Kaehler manifold under which the submanifold is a warped product submanifold.

June 5, 2017

A comparison of two tests in multivariate models

Jitka Machalová, Lubomír Kubáček

Page range: 773-784

Abstract

In multivariate models the test usually used for a linear hypothesis is the Wilks lambda test. There does not seem to be a need for any alternative. However, if the covariance matrix is parameterized by a few parameters only, and the parameters can be estimated, the problem arises whether a “plug-in” test, which approximates the test in a situation where the covariance matrix is totally known, is more suitable. A partial answer to this question is given in the paper. The sensitivity approach is used in order to check how this principle works in a situation where a plug-in procedure can be compared with another, already established, procedure.

June 7, 2017

Modelling prescription behaviour of general practitioners

Šárka Hudecová, Michal Pešta, Daniel Hlubinka

Page range: 785-802

Abstract

The health care costs have been rapidly rising in recent years and the public health insurance companies are highly interested in prescription policies of general practitioners. In our contribution, a complex model for prescription behaviour of Belgian general practitioners is built using the structural equation modelling (SEM) framework. The model involves a large number of prescribed medicament groups as well as doctors’ and patients’ characteristics. As one of the results, a relatively small number of medicament groups, which effectively describe the prescription behaviour of a given doctor, is obtained. These indicators are consequently used in a generalized linear model for predicting the drug expenses per patient. Such a model can be used as a useful guideline for the expenses’ assessment of a particular practitioner.

June 5, 2017

Characterization of congruence lattices of principal p-algebras

Abd El-Mohsen Badawy

Page range: 803-810

Abstract

A characterization of the congruence lattice of a principal p -algebra satisfying a certain condition in terms of congruence pairs is given.

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

Information on submission process

Volume 67 Issue 3

Issue of Mathematica Slovaca

Some results in local Hilbert algebras

Abstract

A note on residuated po-groupoids and lattices with antitone involutions

Abstract

The first law of cubology for the Rubik’s Revenge

Abstract

Conditional associativity in orthomodular lattices

Abstract

A monotonicity property for generalized Fibonacci sequences

Abstract

Systems of conditionally independent sets

Abstract

On the k-th order lfsr sequence with public key cryptosystems

Abstract

On parametric spaces of bicentric quadrilaterals

Abstract

Character sums with an explicit evaluation

Abstract

Some remarks on formal power series and formal Laurent series

Abstract

The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations

Abstract

On prime and semiprime generalized hyperaction of hypermonoid

Abstract

On O’Malley ϱ-upper continuous functions

Abstract

Uniqueness theorems for finitely additive probabilities on quantum structures

Abstract

Some results on differential-difference analogues of Brück conjecture

Abstract

Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments

Abstract

On support, separation and decomposition theorems for t-Wright-concave functions

Abstract

A characterization of holomorphic bivariate functions of bounded index

Abstract

Asymptotic expansion of double Laplace-type integrals with a curve of minimal points and application to an exit time problem

Abstract

The geometry of two-valued subsets of Lp-spaces

Abstract

Hemi-slant submanifolds as warped products in a nearly Kaehler manifold

Abstract

A comparison of two tests in multivariate models

Abstract

Modelling prescription behaviour of general practitioners

Abstract

Characterization of congruence lattices of principal p-algebras

Abstract

The geometry of two-valued subsets of L_p-spaces