rss_2.0Tatra Mountains Mathematical Publications FeedSciendo RSS Feed for Tatra Mountains Mathematical Publications Mountains Mathematical Publications Feed Observations on Ideal Variations of Bornological Covers<abstract> <title style='display:none'>Abstract</title> <p>In this article, we use the notion of ideals to study open covers and related selection principles, and thus, we extend some results in (Caserta et al. 2012; Chandra et al. 2020) where open covers and related selection principles have been investigated using the idea of strong uniform convergence (Beer and Levi, 2009) on a bornology. We introduce the notions of ℐ-<italic>γ</italic><sub>ℬ</sub> <italic>s</italic> -cover, ℐ-strong-ℬ-Hurewicz and ℐ-strong-ℬ-groupable cover. Also, in (<italic>C</italic>(<italic>X</italic>),<italic>τ</italic>s<sub>ℬ</sub>), some properties like ℐ-strictly Frèchet Urysohn, ℐ-Reznichenko property are investigated.</p> </abstract>ARTICLEtrue of Sequences<abstract> <title style='display:none'>Abstract</title> <p>This paper deals with density on the set of natural numbers and its connections to the distribution of sequences. Under the assumption of independence, some formulas are derived.</p> </abstract>ARTICLEtrue Transform and Generalized Lipschitz Classes<abstract> <title style='display:none'>Abstract</title> <p>The aim of this paper is to give necessary and sufficient conditions in terms of the Fourier Laguerre-Bessel transform 𝒲<sub><italic>LB</italic></sub><italic>f</italic> of the function <italic>f</italic> to ensure that <italic>f</italic> belongs to the generalized Lipschitz classes <bold>H</bold><sub><italic>α</italic></sub><sup><italic>k</italic></sup> (<bold>X</bold>) and <bold>h</bold><italic><sup>k</sup><sub>α</sub></italic> (<bold>X</bold>), where <bold>X</bold> =[0, +<italic>∞</italic>) <italic>×</italic> [0, +<italic>∞</italic>).</p> </abstract>ARTICLEtrue Strong Porosity of Some Families of Functions<abstract> <title style='display:none'>Abstract</title> <p>The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology 𝒯 or 𝒜-continuous (i.e., continuous with respect to some special family 𝒜 of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.</p> </abstract>ARTICLEtrue Few Variants of Quasi-Continuity in Bitopological Spaces<abstract> <title style='display:none'>Abstract</title> <p>The purpose of this paper is to introduce a few variants of generalized quasi-continuity of functions defined on a bitopological space and to study their mutual relationship. Moreover, some characterization of sectional quasi-continuous function and its continuity points are investigated.</p> </abstract>ARTICLEtrue Pc-Open Sets and Operation Pc-Separation Axioms in Bitopological Spaces<abstract> <title style='display:none'>Abstract</title> <p>In the present paper, we introduce new types of generalized closed sets called <italic>ij</italic> -pre-generalized closed sets and study some of their properties in bi-topological spaces. Also, we use them to construct new types of separation axioms. Further, we introduce and study the concepts of pairwise operation pc-open sets and pairwise operation pc-separation axioms in bitopological spaces. Several interesting characterizations of different spaces are discussed. The relationships between these spaces are given.</p> </abstract>ARTICLEtrue Fraction Representations of the Generalized Operator Entropy<abstract> <title style='display:none'>Abstract</title> <p>The direct calculation of the generalized operator entropy proves to be difficult due to the appearance of rational exponents of matrices. The main motivation of this work is to overcome these difficulties and to present a practical and efficient method for this calculation using its representation by the matrix continued fraction. At the end of our work, we deduce a continued fraction expansion of the Bregman operator divergence. Some numerical examples illustrating the theoretical result are discussed.</p> </abstract>ARTICLEtrue of Variationally Mcshane Integrable Functions in Locally Convex Space<abstract> <title style='display:none'>Abstract</title> <p>We study measurable real valued multipliers of variationally McShane (resp. McShane) integrable functions defined on a <italic>σ</italic>-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space <italic>X</italic>. We also show: in case <italic>X</italic> is representable by semi-norm then essentially bounded real measurable functions are multipliers of functions which are Pettis integrable as well as integrable by semi-norm. The space of real valued measurable and essentially bounded functions turn out to be precisely the multipliers of variationally McShane (resp. McShane) integrable functions in case <italic>X</italic> is representable by semi-norm.</p> </abstract>ARTICLEtrue Unified Treatment of Generalized Closed Sets in Topological Spaces<abstract> <title style='display:none'>Abstract</title> <p>This paper presents a general unified approach to the notions of generalized closedness in topological spaces. The research concerning the notion of generalized closed sets in topological spaces was initiated by Norman Levine in 1970. In the succeeding years, the concepts of this type of generalizations have been investigated in many versions using the standard generalizations of topologies which has resulted in a large body of literature. However, the methods and results in the past years have become standard and lacking in innovation.</p> <p>The basic notion used in this conception is the closure operator designated by a family ℬ ⊆ 𝒫(<italic>X</italic>), which need not be a Kuratowski operator. Here, we introduce a general conception of natural extensions of families ℬ ⊆ 𝒫 (<italic>X</italic>), denoted by ℬ ᐊ 𝒦, which are determined by other families 𝒦 ⊆ 𝒫(<italic>X</italic>). Precisely, <disp-formula> <alternatives> <graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2023-0028_eq_001.png"/> <mml:math xmlns:mml="" display="block"><mml:mrow><mml:mi>ℬ</mml:mi><mml:mo>⊲</mml:mo><mml:mi>𝒦</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo> <mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>ℬ</mml:mi></mml:msup><mml:mo>⊆</mml:mo><mml:msup><mml:mrow><mml:mover accent="true"><mml:mi>A</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow><mml:mi>𝒦</mml:mi></mml:msup></mml:mrow> <mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math> <tex-math>\mathcal{B} \triangleleft \mathcal{K} = \left\{ {A \subseteq X:{{\bar A}^\mathcal{B}} \subseteq {{\bar A}^\mathcal{K}}} \right\},</tex-math> </alternatives> </disp-formula> where <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2023-0028_eq_002.png"/> <mml:math xmlns:mml="" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mo>…</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mi>𝒜</mml:mi></mml:msup></mml:mrow></mml:math> <tex-math>{\overline {\left( \ldots \right)} ^\mathcal{A}}</tex-math> </alternatives> </inline-formula> denotes the closure operator designated by 𝒜 ⊆ 𝒫(<italic>X</italic>).</p> <p>We prove that the collection of all generalizations ℬ ᐊ 𝒦, where ℬ, 𝒦 ⊆ 𝒫 (<italic>X</italic>), forms a Boolean algebra. In this theory, the family of all generalized closed sets in a topological space <italic>X</italic>(𝒯 )is equal to 𝒞 ᐊ 𝒯, where 𝒞 is the family of all closed subsets of X. This concept gives tools that enable the systemizing and developing of the current research area of this topic. The results obtained in this general conception easily extend and imply well-known theorems as obvious corollaries. Moreover, they also give many new results concerning relationships between various types of generalized closedness studied so far in a topological space. In particular, we prove and demonstrate in a graph that in a topological space <italic>X</italic>(𝒯) there exist only nine different generalizations determined by the standard generalizations of topologies. The tools introduced in this paper enabled us to show that many generalizations studied in the literature are improper.</p> </abstract>ARTICLEtrue Strongly Star -Compactness of Topological Spaces<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we introduce strongly star g-compactness as a topological covering property and compare its structure to other topological properties that have analogous structures. The characteristics of a strongly star g-compact subset and strongly star g-compact subspace are looked at. Finally, some finite intersection-like characteristics that will result in some situations akin to strongly star g-compactness are presented.</p> </abstract>ARTICLEtrue Properties of Meager Ideals and Cardinal Invariants<abstract> <title style='display:none'>Abstract</title> <p>We study two particular modifications of the P-property of ideals and related cardinal invariants cof<sup>𝒥</sup> (ℐ)and cov<sup>+</sup>(ℐ). We give some results on the existence of P (𝒥)-ideals or non-P (𝒥)-ideals regarding specific classes of ideals, particularly meager ideals on <italic>ω</italic>. We also provide values of the cardinal invariant cof<sup>𝒥</sup> (ℐ) describing the smallest families ensuring P(𝒥) for particular critical ideals. Moreover, we obtain a simple way of proving strict inequalities Fin &lt;<sub><italic>K</italic></sub> 〈𝒜 〉 &lt;<sub><italic>K</italic></sub> Fin <italic>×</italic> Fin for any MAD family 𝒜 using the weak P-ideal notion.</p> </abstract>ARTICLEtrue Trajectories Smoothing by Evolving Curves<abstract> <title style='display:none'>Abstract</title> <p>When analyzing cell trajectories, we often have to deal with noisy data due to the random motion of the cells and possible imperfections in cell center detection. To smooth these trajectories, we present a mathematical model and numerical method based on evolving open-plane curve approach in the Lagrangian formulation. The model contains two terms: the first is the smoothing term given by the influence of local curvature, while the other attracts the curve to the original trajectory. We use the flowing finite volume method to discretize the advection-diffusion partial differential equation. The PDE includes the asymptotically uniform tangential redistribution of curve grid points. We present results for macrophage trajectory smoothing and define a method to compute the cell velocity for the discrete points on the smoothed curve.</p> </abstract>ARTICLEtrue Polylogarithm Functions<abstract> <title style='display:none'>Abstract</title> <p>We investigate a discrete analogue of the polylogarithm function. Difference and summation relations are obtained, as well as its connection to the discrete hypergeometric series.</p> </abstract>ARTICLEtrue Stability of Integro-Differential Volterra Equation on Time Scales<abstract> <title style='display:none'>Abstract</title> <p>We study the Volterra integro-differential equation on time scales and provide sufficient conditions for boundness of all solutions of considered equation. Using that result, we present the conditions for exponential stability of considered equation. All the results proved on the general time scale include results for both integral and discrete Volterra equations.</p> </abstract>ARTICLEtrue Properties of Solutions to Fourth-Order Difference Equations on Time Scales<abstract> <title style='display:none'>Abstract</title> <p>We provide sufficient criteria for the existence of solutions for fourth-order nonlinear dynamic equations on time scales <disp-formula> <alternatives> <graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2023-0016_eq_001.png"/> <mml:math xmlns:mml="" display="block"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>a</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>b</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math> <tex-math>{\left( {a\left( t \right){x^{{\Delta ^2}}}\left( t \right)} \right)^{{\Delta ^2}}} = b\left( t \right)f\left( {x\left( t \right)} \right) + c\left( t \right),</tex-math> </alternatives> </disp-formula> such that for a given function <italic>y</italic> : 𝕋 → ℝ there exists a solution <italic>x</italic> : 𝕋 → ℝ to considered equation with asymptotic behaviour <inline-formula> <alternatives> <inline-graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2023-0016_eq_002.png"/> <mml:math xmlns:mml="" display="inline"><mml:mrow><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mi>β</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> <tex-math>x\left( t \right) = y\left( t \right) + o\left( {{1 \over {{t^\beta }}}} \right)</tex-math> </alternatives> </inline-formula>. The presented result is applied to the study of solutions to the classical Euler–Bernoulli beam equation, which means that it covers the case 𝕋 = ℝ.</p> </abstract>ARTICLEtrue of the Samuelson Economic Cycle Model<abstract> <title style='display:none'>Abstract</title> <p>The paper presents an analysis of modifications of the standard discrete Samuelson model with determination of the stability of the equilibrium point. The stability region of the equilibrium point depending on the parameters contained in each model is determined using the Schur-Cohn stability criterion.</p> </abstract>ARTICLEtrue Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations<abstract> <title style='display:none'>Abstract</title> <p>We investigate the discrete equations of the form <disp-formula> <alternatives> <graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2023-0014_eq_001.png"/> <mml:math xmlns:mml="" display="block"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math> <tex-math>\Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n \right)}}} \right) + {b_n}.</tex-math> </alternatives> </disp-formula> Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.</p> </abstract>ARTICLEtrue, Uniqueness and Ulam-Hyers-Rassias Stability of Differential Coupled Systems with Riesz-Caputo Fractional Derivative<abstract> <title style='display:none'>Abstract</title> <p>This article deals with the existence, uniqueness and Ulam-Hyers--Rassias stability results for a class of coupled systems for implicit fractional differential equations with Riesz-Caputo fractional derivative and boundary conditions. We will employ the Banach’s contraction principle as well as Schauder’s fixed point theorem to demonstrate our existence results. We provide an example to illustrate the obtained results.</p> </abstract>ARTICLEtrue Fold Bifurcations from Spectral Intervals of Higher Strict Multiplicity<abstract> <title style='display:none'>Abstract</title> <p>We provide two sufficient criteria for the bifurcation of bounded entire or homoclinic solutions to nonautonomous difference equations depending on a single real parameter. Our analysis is based on a nonhyperbolic solution, whose variational equation possesses exponential dichotomies on semiaxes ensuring that the corresponding critical spectral interval of the dichotomy spectrum has strict multiplicity &gt; 1. This extends earlier results on the fold bifurcation.</p> </abstract>ARTICLEtrue for an Inverse Quantum-Dirac Problem with Given Weyl Function<abstract> <title style='display:none'>Abstract</title> <p>In this work, we consider a boundary value problem for a <italic>q</italic>-Dirac equation. We prove orthogonality of the eigenfunctions, realness of the eigenvalues, and we study asymptotic formulas of the eigenfunctions. We show that the eigenfunctions form a complete system, we obtain the expansion formula with respect to the eigenfunctions, and we derive Parseval’s equality. We construct the Weyl solution and the Weyl function. We prove a uniqueness theorem for the solution of the inverse problem with respect to the Weyl function.</p> </abstract>ARTICLEtrue