Mathematica Slovaca

Expanding lattice ordered abelian groups to Riesz spaces

Antonio Di Nola, Giacomo Lenzi, Gaetano Vitale — Wed, 16 Feb 2022 00:00:00 +0000

First we give a necessary and sufficient condition for an Abelian lattice ordered group to allow expansion into a Riesz space (or vector lattice). So we build a totally ordered Abelian group with two non-isomorphic Riesz spatial structures, thus improving a previous document in which the example was an Abelian trellis group not totally ordered. This answers a question raised by Conrad in 1975. We also give a partial solution to another problem considered in the same document. Finally, we apply our results to MV-algebras and Riesz MV-algebras.

Ordinary generating functions of binary products of third-order recurrence relations and 2-orthogonal polynomials

Hind Merzouk, Ali Boussayoud, Abdelhamid Abderrezzak — Wed, 16 Feb 2022 00:00:00 +0000

In this paper, we introduce the new generating functions for the product of some numbers and 2-orthogonal polynomials by making use of the symmetrizing endomrphism operators π_{e₂e₃}π_{e₁e₂} to the series ∑_{n=0}^\infty S_{n}(A₃)e₁ⁿzⁿ.

Quartic polynomials with a given discriminant

Jiří Klaška — Wed, 16 Feb 2022 00:00:00 +0000

Let $0\ne D\in Z$ and let $Q_D$ be the set of all monic quartic polynomials $x^4 + ax^3 + bx^2 + cx + d \in Z[x]$ with the discriminant equal to D. In this paper we will devise a method for determining the set $Q_D$. Our method is strongly related to the theory of integral points on elliptic curves. The well-known Mordell’s equation plays an important role as well in our considerations. Finally, some new conjectures will be included inspired by extensive calculations on a computer.

Joint approximation by Dirichlet $L$-functions

Antanas Laurinčikas; Darius Šiaučiūnas — Wed, 16 Feb 2022 00:00:00 +0000

In the paper, collections of analytic functions are simultaneously approximated by collections of shifts of Dirichlet $L$-functions $\left(L(s+i\gamma_1(\tau), \chi_1),\right.$ $\left. \dots, L(s+i\gamma_r(\tau), \chi_r)\right)$, with arbitrary Dirichlet characters $\chi_1,\dots, \chi_r$. The differentiable functions $\gamma_1(\tau), \dots, \gamma_r(\tau)$ and their derivatives satisfy certain growth conditions. The obtained results extend those of [17].

Generalizations of Hardy type inequalities by Taylor's formula

Kristina Krulić Himmelreich — Wed, 16 Feb 2022 00:00:00 +0000

In this paper we use Taylor's formula to prove new Hardy-type inequalities involving convex functions. We give new results that involve the Hardy–Hilbert inequality, Pólya-Knopp inequality and bounds for the identity related to the Hardy-type functional. At the end, mean value theorems of Cauchy type are given.

Certain estimates of normalized analytic functions

Swati Anand, Naveen Kumar Jain, Sushil Kumar — Wed, 16 Feb 2022 00:00:00 +0000

Let φ be a normalized convex function defined on open unit disk ⅅ. For a unified class of normalized analytic functions which satisfy the second order differential subordination f'(z)+ α z f''(z) ≺ φ(z) for all z ∈ ⅅ, we investigate the distortion theorem and growth theorem. Further, the bounds on initial logarithmic coefficients, inverse coefficient and the second Hankel determinant involving the inverse coefficients are examined.

Oscillation of second order delay differential equations with nonlinear nonpositive neutral term

Blanka Baculíková, B. Sudha, K. Thangavelu, E. Thandapani — Wed, 16 Feb 2022 00:00:00 +0000

This paper deals with oscillation of a second order delay di erential equations with a nonlinear nonpositive neutral term. Some new oscillation criteria and three examples are presented which improve and generalize the results reported in the literature.

Existence and multiplicity of radially symmetric k-admissible solutions for Dirichlet problem of k-Hessian equations

Zhiqian He, Liangying Miao — Wed, 16 Feb 2022 00:00:00 +0000

In this paper, we study the existence and multiplicity of radially symmetric k-admissible solutions for the k-Hessian equation with 0-Dirichlet boundary condition

S_k(D² u)= f(-u) in B
u=0 on ∂B

and the corresponding one-parameter problem, where B is a unit ball in Rⁿ with n ≥ 1, k∈ {1, ... , n}, f: [0, +∞) → [0, +∞) is continuous. We show that the k-admissible solutions are not convex, so we construct a new cone and obtain the existence of triple and arbitrarily many k-admissible solutions via the Leggett-Williams' fixed point theorem.

Existence and asymptotic periodicity of solutions for neutral integro-differential evolution equations with infinite delay

Jianbo Zhu, Xianlong Fu — Wed, 16 Feb 2022 00:00:00 +0000

In this work, making use of the theory of resolvent operators and Banach fixed point theorem, we first discuss the existence and regularity of mild solutions for neutral partial functional integro-differential equations with infinite delay. We assume that the linear part of the considered equation generates a resolvent operator and the nonlinear function satisfies Lipschitz conditions. Then we investigate the asymptotic periodicity of mild solutions under asymptotic periodic assumption on the nonlinear function. The obtained results extend somewhat the related conclusions in literature. In the end, an example is presented to illustrate the obtained results.

Aproximation properties of λ-Bernstein-Kantorovich-Stancu operators

Murat Bodur, Nesibe Manav, Fatma Taşdelen — Wed, 16 Feb 2022 00:00:00 +0000

The goal of this paper is to construct a new type of Bernstein operators depending on the shape parameter λϵ[-1,1]. For these new type operators a uniform convergence result is presented. Furthermore, order of approximation in the sense of local approximation is investigated and Voronovskaja type theorem is proved. Lastly, some graphical results are given to show the rate of convergence of constructed operators to a given function f.

A Korovkin type approximation theorem for Balázs type Bleimann, Butzer and Hahn operators via power series statistical convergence

Dilek Söylemez — Wed, 02 Mar 2022 00:00:00 +0000

In this paper, we obtain a Korovkin type approximation theorem for power series statistical convergence of functions belonging to the class produced by multivariable modulus of continuity function. As an application of this theorem, we construct a non-tensor product Balazs type BBH operator which does not converge in ordinary sense. Moreover, we study promised approximation properties of this operator and compute the rate of convergence. Finally, we prove that our new approximation result works but its classical case fails.

Hyperbolic geometry for non-differential topologists

Piotr Niemiec, Piotr Pikul — Wed, 16 Feb 2022 00:00:00 +0000

A soft presentation of hyperbolic spaces (as metric spaces), free of di erential apparatus, is off ered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and hyperbolic and Euclidean spaces are the only locally compact geodesic (i.e., convex) metric spaces that are three-point homogeneous.

Set star-Menger and set strongly star-Menger spaces

Ljubiˇsa D. R. Koˇcinac, Şukran Konca, Sumit Singh — Wed, 16 Feb 2022 00:00:00 +0000

Motivated by the Arhangeľskii "s-Lindelöf cardinal function" definition, Kočinac and Konca defined and studied set covering properties and set star covering properties. In this paper, we present results on the star covering properties called set star-Menger and set strongly star-Menger. We investigate the relationship among set star-Menger, set strongly star-Menger and other related properties and study the topological properties of set star-Menger and set strongly star-Menger properties.

Some characterizations of mixed renewal processes

Demetrios P. Lyberopoulos, Nikolaos D. Macheras — Wed, 16 Feb 2022 00:00:00 +0000

Some characterizations of mixed renewal processes in terms of exchangeability and of different types of regular conditional probabilities are given. As a consequence, an existence result for mixed renewal processes, providing also a new construction for them, is obtained. As an application, some concrete examples of constructing such processes are presented and the corresponding regular conditional probabilities are explicitly computed.

The U family of distributions: Properties and applications

Wed, 16 Feb 2022 00:00:00 +0000

In this article, we develop a new general family of distributions aimed at unifying some well-established lifetime distributions and offering new work perspectives. A special family member based on the so-called modified Weibull distribution is highlighted and studied. It differs from the competition with a very flexible hazard rate function exhibiting increasing, decreasing, constant, upside- down bathtub and bathtub shapes. This panel of shapes remains rare and particularly desirable for modeling purposes. We provide the main mathematical properties of the special distribution, such as a tractable infinite series expansion of the probability density function, moments of several kinds (raw, incomplete, probability weighted ...) with discussions on the skewness and kurtosis. The stochastic ordering structure and stress-strength parameter are also considered, as well as the basics of the order statistics. Then, an emphasis is put on the inferential features of the related model. In particular, the estimation of the model parameters is employed by the maximum likelihood method, with a simulation study to confirm the suitability of the approach. Three practical data sets are then analyzed. It is observed that the proposed model gives better fits than other well-known lifetime models derived from the Weibull model.

A new method for generalizing Burr and related distributions

Tanujit Chakraborty, Suchismita Das, Swarup Chattopadhyay — Wed, 16 Feb 2022 00:00:00 +0000

A new method has been proposed to generalize Burr-XII distribution, also called Burr distribution, by adding an extra parameter to an existing Burr distribution for more
exibility. In this method, the exponent of the Burr distribution is modeled using a nonlinear function of the data and one additional parameter. The models of this newly introduced generalized Burr family can signifficantly increase the exibility of the former Burr distribution with respect to the density and hazard rate shapes. Families expanded using the method proposed here is heavy-tailed and belongs to the maximum domain of attractions of the Frechet distribution. The method is further applied to yield three-parameter classical Pareto and generalized exponentiated distributions which shows the broader application of the proposed idea of generalization. A relevant model of the new generalized Burr family has been considered in detail, with particular emphasis on the hazard functions, stochastic orders, estimation procedures, and testing methods are derived. Finally, as empirical evidence, the new distribution is applied to the analysis of large-scale heavy-tailed network data and compared with other commonly used distributions available for fitting degree distributions of networks. Experimental results suggest that the proposed Burr distribution with nonlinear exponent better fits the large-scale heavytailed networks better than the popularly used Marhsall-Olkin generalization of Burr and exponentiated Burr distributions.

On Mahler's classification of formal power series over a finite field

Gülcan Kekeç — Wed, 02 Mar 2022 00:00:00 +0000

Let K be a finite field, K(x) be the field of rational functions in x over K and $\mathbb{K}$ be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U₁-number in $\mathbb{K}$ are U_m-numbers in $\mathbb{K}$, where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.