# International Journal of Pure Mathematics

**E-**ISSN: 2313-0571

Volume 8, 2021

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of NAUN Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Submit a paper | Submission terms | Paper format

**Volume 8, 2021**

**Title of the Paper:** Convex Inequalities and Functional Space with the Convex Semi-Norm

**Authors:** Mykola Yaremenko

**Pages: **66-73

**DOI:**** **10.46300/91019.2021.8.9 **XML**

**Abstract:** In this article, we establish new characterizations of convex functions, prove some connected convex type integral inequality; consider the pair of convex functions as the dual semi-norms in functional space. The properties of the integral operators are considered in the scales of the convex semi-norm under the standard conditions on singular kernels.

**Title of the Paper:** Integer Programming Formulations For The Frobenius Problem

**Authors:** Imdat Kara, Halil Ibrahim Karakas

**Pages: **60-65

**DOI:**** **10.46300/91019.2021.8.8 **XML**

**Abstract:** The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set of positive integers. Our first formulation is used to check if a given positive integer is the Frobenius number of a given set of positive integers. The second formulation aims at finding the Frobenius number directly. The third formulation involves the residue classes with respect to the least member of the given set of positive integers, where a residue table is computed comprising all values modulo that least member, and the Frobenius number is obtained from there. Based on the same approach underlying the third formulation, we propose our fourth formulation which produces the Frobenius number directly. We demonstrate how to use our formulations with several examples. For illustrative purposes, some computa-tional analysis is also presented.

**Title of the Paper:** Remarks on Frobenius Groups

**Authors:** Liguo He, Yubing Cao

**Pages: **58-59

**DOI:**** **10.46300/91019.2021.8.7 **XML**

**Abstract:** Let the finite group G act transitively and non-regularly on a finite set whose cardinality |Ω| is greater than one. Use N to denote the full set of fixed-point-free elements of G acting on along with the identity element. Write H to denote the stabilizer of some α ∈ Ω in G. In the note, it is proved that the subset N is a subgroup of G if and only if G is a Frobenius group. It is also proved G = {N}H, where {N} is the subgroup of G generated by N.

**Title of the Paper:** Infra -α- Compact and Infra -α- Connected Spaces

**Authors:** Raja Mohammad Latif

**Pages: **41-57

**DOI:**** **10.46300/91019.2021.8.6 **XML**

**Abstract:** In 2016 Hakeem A. Othman and Md. Hanif Page introduced a new notion of set in general topology called an infra -α- open set and investigated its fundamental properties and studied the relationship between infra -α- open set and other topological sets. The objective of this paper is to introduce the new concepts called infra -α- compact space, countably infra -α- compact space, infra -α- Lindelof space, almost infra -α- compact space, mildly infra -α- compact space and infra -α- connected space in general topology and investigate several properties and characterizations of these new concepts in topological spaces.

**Title of the Paper:** On Fuzzy L-paracompact Topological Spaces

**Authors:** Francisco Gallego Lupiáñez

**Pages: **38-40

**DOI:**** **10.46300/91019.2021.8.5 **XML**

**Abstract:** The aim of this paper is to study fuzzy extensions of some covering properties defined by L. Kalantan as a modification of some kinds of paracompactness-type properties due to A.V.Arhangels'skii and studied later by other authors. In fact, we obtain that: if (X,T) is a topological space and A is a subset of X, then A is Lindelöf in (X,T) if and only if its characteristic map χ_{A} is a Lindelöf subset in (X,ω(T)). If (X,τ) is a fuzzy topological space, then, (X,τ) is fuzzy Lparacompact if and only if (X,ι(τ)) is L-paracompact, i.e. fuzzy L-paracompactness is a good extension of L-paracompactness. Fuzzy L₂-paracompactness is a good extension of L₂- paracompactness. Every fuzzy Hausdorff topological space (in the Srivastava, Lal and Srivastava' or in the Wagner and McLean' sense) which is fuzzy locally compact (in the Kudri and Wagner' sense) is fuzzy L₂-paracompact

**Title of the Paper:** Table Algebra of Infinite Tables, Multiset Table Algebra, and their Relationship

**Authors:** Iryna Lysenko

**Pages: **34-37

**DOI:**** **10.46300/91019.2021.8.4 **XML**

**Abstract:** The paper is focused on some theoretical questions of the table databases. Two mathematical formalisms such as table algebra of infinite tables and multiset table algebra are considered. Basic definitions referring to these formalisms are given. This paper also addresses the issue of the relationship between table algebra of infinite tables and multiset table algebra. It is proved that table algebra of infinite tables is not a subalgebra of multiset table algebra since it is not closed in relation to some signature operations of multiset table algebra. These signature operations are determined.

**Title of the Paper:** Cartesian Product and Topology on Fuzzy BI-Algebras

**Authors:** Gerima Tefera, Abdi Oli

**Pages: **29-33

**DOI:**** **10.46300/91019.2021.8.3 **XML**

**Abstract:** In this paper, the concepts of homomorphism in fuzzy BI-algebra is intro- duced, and also basic properties of homo- morphisms are investigated. The cartesian product in fuzzy ideals of BI-algebra is investigated with related prop- erties; The concepts of fuzzy topology on BI- algebra elaborated.

**Title of the Paper:** Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds

**Authors:** Richard Selescu

**Pages: **23-28

**DOI:**** **10.46300/91019.2021.8.2 **XML**

**Abstract:** The author proposes two sets of closed analytic functions for the approximate calculus of the complete elliptic integrals of the first and second kinds in the normal form due to Legendre, the respective expressions having a remarkable simplicity and accuracy. The special usefulness of the proposed formulas consists in that they allow performing the analytic study of variation of the functions in which they appear, by using the derivatives. Comparative tables including the approximate values obtained by applying the two sets of formulas and the exact values, reproduced from special functions tables are given (all versus the respective elliptic integrals modulus, k = sin θ). It is to be noticed that both sets of approximate formulas are given neither by spline nor by regression functions, but by asymptotic expansions, the identity with the exact functions being accomplished for the left end k = 0 (θ = 0°) of the domain. As one can see, the second set of functions, although something more intricate, gives more accurate values than the first one and extends itself more closely to the right end k = 1 (θ= 90°) of the domain. For reasons of accuracy, it is recommended to use the first set until θ = 70°.5 only, and if it is necessary a better accuracy or a greater upper limit of the validity domain, to use the second set, but on no account beyond θ = 88°.2.

**Title of the Paper:** Delta – Open Sets and Delta – Continuous Functions

**Authors:** Raja Mohammad Latif

**Pages: **1-22

**DOI:**** **10.46300/91019.2021.8.1 **XML**

**Abstract:** In 1968 Velicko [30] introduced the concepts of δ-closure and δ-interior operations. We introduce and study properties of δ-derived, δ-border, δ-frontier and δ-exterior of a set using the concept of δ-open sets. We also introduce some new classes of topological spaces in terms of the concept of δ-D- sets and investigate some of their fundamental properties. Moreover, we investigate and study some further properties of the well-known notions of δ-closure and δ-interior of a set in a topological space. We also introduce δ-R0 space and study its characteristics. We also introduce δ-R0 space and study its characteristics. We introduce δ-irresolute, δ-closed, pre-δ-open and pre -δ-closed mappings and investigate properties and characterizations of these new types of mappings and also explore further properties of the well-known notions of δ-continuous and δ-open mappings.