European Journal of Pure and Applied Mathematics https://www.ejpam.com/index.php/ejpam <p align="justify"><em>European Journal of Pure and Applied Mathematics</em> is an Open Access international, peer-reviewed electronic journal which is devoted to high-quality and original research in the fields of pure and applied mathematics and their teaching and learning. The aim of the journal is to provide a medium by which a wide range of experience can be exchanged among researchers from diverse fields such as engineering, natural sciences or social sciences. This journal publishes high quality researches in the fields of contemporary pure and applied mathematics with a broad range of topics including algebra, topology, number theory, nonlinear analysis, mathematical methods in operations research, theoretical statistics and econometrics, and theoretical computer science.</p> <p align="justify">Open Access which is a paradigm shift in scholarly publishing means you can publish your research so it is free to access online as soon as it is published, that is, anyone can read (and circulate) freely and cite your work. Many funding agencies mandate publishing your research open access.</p> <p align="justify"><strong>EDITORIAL BOARD</strong></p> <p align="justify">The<strong> </strong>Editorial Board consists of prominent scientists as some are Nobel Laureates.</p> <p align="justify">See <a href="https://ejpam.com/index.php/ejpam/about/editorialTeam">editorial board</a> details.</p> <p><strong>JOURNAL IMPACT </strong></p> <ul> <li>Clarivate Analytics Web of Science - Journal Citation Reports™ (JCR) 2021</li> </ul> <p>Journal Citation Indicator (JCI)*</p> <p>2020: 0.66</p> <p>2019: 0.61</p> <p>* The Journal Citation Indicator is a measure of the average Category Normalized Citation Impact (CNCI) of citable items (articles &amp; reviews) published by a journal over a recent three year period. It is used to help you evaluate journals based on other metrics besides the Journal Impact Factor (JIF).</p> <ul> <li> <p>Scopus</p> </li> </ul> <p>CiteScoreTracker 2021 - April 2022: 0.71</p> <p>CiteScore 2020: 0.5 SNIP 2020: 0.967 </p> <p><strong>ABSTRACTED/INDEXED IN:</strong></p> <p align="justify">Clarivate Analytics Emerging Sources Citation Index (ESCI), Web of Science, <a title="EJPAM Scopus Source Details Page" href="https://www.scopus.com/sourceid/21100981084" target="_blank" rel="noopener">SCOPUS</a>, Mathematical Reviews, EBSCO, Chambridge Scientific Abstracts, IndexCopernicus™, International Abstracts in Operations Research, Ulrich's Periodicals Directory, <a title="Arastirmax Bilimsel Yayın Ä°ndeksi" href="http://www.arastirmax.com/dergi/european-journal-pure-and-applied-mathematics" target="_blank" rel="noopener">Arastirmax</a>.</p> <p align="justify"><strong>PUBLISHER</strong></p> <p align="justify">European Journal of Pure and Applied Mathematics is published by <em>New York Business Global, USA</em>.</p> <p align="justify"><strong>PRESTIGIOUS CONTRIBUTORS</strong></p> <div> <div><span class="gmail_default">In addition to Nobel Laureates and Abel Prize </span>Laureates<span class="gmail_default">, m</span>any researchers, authors and Editorial Board members<span class="gmail_default"> at academic institutions such as University of Oxford, Harvard University, MIT, Stanford University, Princeton University, University of Chicago, Imperial College London and Berkeley, </span>and <span class="gmail_default">other institutions </span>have<span class="gmail_default"> been</span> contribut<span class="gmail_default">ing</span> to EJPAM<span class="gmail_default">. Join them by submitting a paper.</span></div> </div> <div> </div> <div><strong>FUNDING<span class="gmail_default">/GRANTING</span> INSTITUTIONS</strong></div> <div><em> </em></div> <div>Over the course of journal history, a majority of papers published at EJPAM have been financially supported by numerious leading universities and funding agencies all over the world including from University of Oxford, Danish National Research Foundation and Research Council of Norway to Robertson Foundation and King Abdulaziz <span class="gmail_default">City for Science and Technology</span>, to National Natural Science Foundation of China and European Commission.</div> <div> </div> <div>ISSN:1307-5543 <br />Published by New York Business Global, Maryland, USA<br />Inaugural publication: January 2008 <br />The journal is quarterly</div> New York Business Global en-US European Journal of Pure and Applied Mathematics 1307-5543 <p align="justify">Upon acceptance of an article by the journal, the author(s) accept(s) the transfer of copyright of the article to <em>European Journal of Pure and Applied Mathematics.</em></p> <p align="justify"><em>European Journal of Pure and Applied Mathematics will be Copyrihgt Holder.</em></p> <p align="justify">If necessary, authors are responsible for obtaining permissions to reprint previously published figures, tables, and other material.</p> A Note on an Octuple Integral in terms of the Lerch Function https://www.ejpam.com/index.php/ejpam/article/view/4154 The known exact expression for an octuple integral relating to research in the fields of mathematics and physics is summarized. A new closed form expression for this integral is given in terms of the Lerch function. Robert Reynolds Allan Stauffer Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 335 341 10.29020/nybg.ejpam.v15i2.4154 A Variant of Hop Domination in Graphs https://www.ejpam.com/index.php/ejpam/article/view/4352 <p>Let <em>G</em> be a connected graph with vertex and edge sets <em>V (G)</em> and <em>E(G</em>), respectively. A set S ⊆ V (G) is a hop dominating set of <em>G</em> if for each <em>v ∈</em> V (G) \ S, there exists <em>w ∈ S</em> such that<em> d<sub>G</sub>(v,w)</em> = 2. A set <em>S </em>⊆ <em>V (G)</em> is a super hop dominating set if <em>ehpn<sub>G</sub></em>(<em>v, V (G) \ S</em>) <strong>≠ </strong>∅ for<br />each <em>v ∈ </em>V (G) \ <em>S</em>, where <em>ehpn<sub>G</sub></em>(<em>v, </em>V (G) \ S) is the set containing all the external hop private neighbors of <em>v</em> with respect to V (G) \ S<em>.</em> The minimum cardinality of a super hop dominating set of <em>G</em>, denoted by <em>γ<sup>s</sup><sub>h</sub>(G)</em>, is called the super hop domination number of <em>G</em>. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the super hop dominating sets in the join, and lexicographic products of graphs, and determine bounds of the super hop domination number of each of these graphs.</p> Sergio Canoy, Jr. Gemma Puebla Salasalan Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 342 353 10.29020/nybg.ejpam.v15i2.4352 Generalized ωe∗-closed Sets https://www.ejpam.com/index.php/ejpam/article/view/4340 <p>The aim of this paper is to introduce and study a new type of generalized closed sets, called generalized ωe∗ -closed (briefly, gωe∗ -closed) sets, via ωe∗-closure operator. We examine the fundamental properties of the class of these sets. The notion of gωe∗-closed set is weaker than the notions of gωβ-closed set and ωe∗-closed set in the literature. Also, we define and discuss the notions of generalized ωe∗-continuous and generalized ωe∗-irresolute functions.</p> Pınar Şaşmaz Murad Özkoç Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 354 374 10.29020/nybg.ejpam.v15i2.4340 Origami Wrapping with an Equilateral Triangular Prism https://www.ejpam.com/index.php/ejpam/article/view/4319 <p>We study a geometric model to find the relationship of the surface area of rectangular packaging that encapsulates an equilateral triangular prism in a Thai traditional tall wrap pattern and find the minimum area of a rectangular wrapping paper.</p> Prathomjit Khachorncharoenkul Attawut Wongpradit Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 375 389 10.29020/nybg.ejpam.v15i2.4319 Triple Integral involving the Product of the Logarithmic and Bessel Functions expressed in terms of the Lerch Function https://www.ejpam.com/index.php/ejpam/article/view/4203 The aim of the present document is to evaluate a triple integral involving a general class of logarithmic, special function and exponential function. Importance of our resultss lies in the fact that they involve the Bessel function of the First Kind, which<br />is used in a wide range of areas spanning Science and Engineering. Further we establish some special cases Robert Reynolds Allan Stauffer Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 390 396 10.29020/nybg.ejpam.v15i2.4203 Atomic Solution of Poisson Type Equation https://www.ejpam.com/index.php/ejpam/article/view/4318 <p><br />In this paper we find a certain solution of Poisson type fractional differential equation using theory of tensor product of Banach spaces.</p> Rouba Shatnawi Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 397 402 10.29020/nybg.ejpam.v15i2.4318 On Nowhere Dense Sets https://www.ejpam.com/index.php/ejpam/article/view/4283 <p>We introduce two types of strongly nowhere dense sets, namely (s, v)-strongly nowhere dense set, (s, v)*-strongly nowhere dense set and analyze their characteristics in a bigeneralized topological space (BGTS). Further, it is also given some relations between these two types of strongly nowhere dense sets along with its various properties for (s, v)*-strongly nowhere dense set. Finally, the necessary and sufficient condition is found between \mu-strongly nowhere dense set and (s, v)*-strongly nowhere dense set in a BGTS.</p> Preecha Yupapin Vadakasi Subramanian Yasser Farhat Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 403 414 10.29020/nybg.ejpam.v15i2.4283 On (Λ, p)-closed Sets and the Related Notions in Topological spaces https://www.ejpam.com/index.php/ejpam/article/view/4274 <p>This article deals with the concepts of Λp-sets and (Λ, p)-closed sets which are defined by utilizing the notions of preopen sets and preclosed sets. We also introduce and characterize some new low separation axioms. Characterizations of Λp-R0 spaces are given. Moreover, we introduce the concept of weakly (Λ, p)-continuous functions. In particular, several characterizations of weakly (Λ, p)-continuous functions are established.</p> Chawalit Boonpok Chokchai Viriyapong Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 415 436 10.29020/nybg.ejpam.v15i2.4274 A Triple Integral Involving the Struve Function Hv(t) Expressed in terms of the Hurwitz-Lerch Zeta Function https://www.ejpam.com/index.php/ejpam/article/view/4227 <p>The main focus of the present paper is to establish a triple integral involving the Struve function in terms of the Hurwitz-Lerch zeta function by using our contour integral method. We also consider some useful examples as special cases of this integral involving the Struve function to give the applications of our main results.</p> Robert Reynolds Allan Stauffer Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 437 442 10.29020/nybg.ejpam.v15i2.4227 A Note on Strongly δθ-I-continuous Functions https://www.ejpam.com/index.php/ejpam/article/view/4317 <p>In this article, we investigate some properties of strongly δθ-I-continuous functions and other types of related functions. Specially, we characterize strongly δθ-I-continuous, we investigate their relationship with other types of functions, and we study the behavior of certain topological notions under the action of these functions.</p> José Sanabria Rafael Lozada-Yavina José Tormet Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 443 453 10.29020/nybg.ejpam.v15i2.4317 Derivations in Differentially δ-prime Rings https://www.ejpam.com/index.php/ejpam/article/view/4344 <p>Let R be an associative ring with identity. In this paper we extend the J.H. Maynes results, which he treatised in [28]. In particular, we prove that if R is a δ-prime ring with charR non equal 2 and I is a nonzero δ-ideal of R, where δ ∈ D, c ∈ R and [c, δ(c)] in the center of R, then R is commutative.</p> Iman Taha Rohaidah Masri Ahmad Alkhalaf Rawdah Tarmizi Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 454 466 10.29020/nybg.ejpam.v15i2.4344 Hop Independent Sets in Graphs https://www.ejpam.com/index.php/ejpam/article/view/4350 <p>Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop independent set of G if any two distinct vertices in S are not at a distance two from each other, that is, dG(v, w) 6= 2 for any distinct vertices v, w ∈ S. The maximum cardinality of a hop independent set of G, denoted by αh(G), is called the hop independence number of G. In this paper, we show that the absolute difference of the independence number and the hop independence number of a graph can be made arbitrarily large. Furthermore, we determine the hop independence numbers of some graphs including those resulting from some binary operations of graphs.</p> Javier Hassan Sergio R. Canoy, Jr. Alkajim A. Aradais Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 467 477 10.29020/nybg.ejpam.v15i2.4350 e*-Essential Small Submodules and e*-Hollow Modules https://www.ejpam.com/index.php/ejpam/article/view/4301 <p>The purpose of this paper is to introduce new concepts in a module  over a ring , the first one is called e*- small essential submodule, which is a generalization of the small submodule, the second concept is called e*-radical submodule which is a generalization of the radical submodule and the last concept is called e* - hollow module which is a generalization of the hollow module. We will prove some properties of all these concepts.</p> Hiba Baanoon Wasan Khalid Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 478 485 10.29020/nybg.ejpam.v15i2.4301 Closed Ideals and Annihilators of Distributive Dual Weakly Complemented Lattice https://www.ejpam.com/index.php/ejpam/article/view/4329 <p>The goal of this paper is study closed ideals and annihilators over the class of distributive dual weakly complemented lattices<br>(DDWCL's). The algebraic structure of ideals, closed ideals and dense ideals are shown. The connection between closed ideals and<br>annihilators in this class is obtained.</p> Eman Rezk Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 486 495 10.29020/nybg.ejpam.v15i2.4329 Exact Solution for Nonlinear Oscillators with Coordinate-Dependent Mass https://www.ejpam.com/index.php/ejpam/article/view/4306 In this work, we aim to obtain an exact solution for a nonlinear oscillator with coordinate position- dependent mass. The equation of motion of the nonlinear oscillator under investigation becomes exact after making reduction of order. The obtained solution was expressed in terms of position and time. Initial conditions were applied, in addition to modiOed initial condition. Finally, Oxed points where studied with their stability, and some plots desribing the system where presented. Ata Abu- As'ad Jihad Asad Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 496 510 10.29020/nybg.ejpam.v15i2.4306 Extensions of Two Classical Poisson Limit Laws to Non-stationary Independent Sequences https://www.ejpam.com/index.php/ejpam/article/view/4323 <p>In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences (Xn)n≥1 of binomial random variables (rv’s) to a Poisson law is classical and easy to prove. A version of such a result concerning sequences (Yn)n≥1 of negative binomial rv’s also exists. In both cases, Xn and Yn −n are by-row sums Sn[X] and Sn[Y ] of arrays of Bernoulli rv’s and corrected geometric rv’s respectively. When considered in the general frame of asymptotic theorems of by-row sums of rv’s of arrays, these two simple results in the independent and identically distributed scheme can be generalized to non-stationary data and beyond to nonstationary and dependent data. Further generalizations give interesting results that would not be found by direct methods. In this paper, we focus on generalizations to the non-stationary independent data. Extensions to dependent data will be addressed later.</p> Aladji Babacar Niang Harouna Sangar´e Tchilabalo Abozou Kpanzou Gane Samb Lo Nafy Ngom Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 511 527 10.29020/nybg.ejpam.v15i2.4323 Weakly (Λ, sp)-continuous Multifunctions https://www.ejpam.com/index.php/ejpam/article/view/4303 <p>This paper is deals with the concept of weakly (Λ, sp)-continuous multifunctions. In particular, some characterizations of weakly (Λ, sp)-continuous multifunctions are investigated.</p> Chawalit Boonpok Chokchai Viriyapong Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 528 536 10.29020/nybg.ejpam.v15i2.4303 Bifurcation Analysis of a Mathematical Model for the Covid-19 Infection among Pregnant and Non-Pregnant Women https://www.ejpam.com/index.php/ejpam/article/view/4312 <p>A mathematical study and analysis of the covid-19 disease is essential in the control and eradication of the dreadful covid-19 infection. This research work is focused on the covid-19 infection among the women population. The women population was divided into seven compartments. These compartments were built into a deterministic model presented as a system of ordinary differential equations. Basic mathematical analyses such as positivity of solution, the disease-free equilibrium and the basic reproduction number, were performed on the model. The model assumes only positive solutions and the total population size is bounded. The next generation matrix approach was employed in generating the basic reproduction number R0. The sensitivity analysis revealed that among the most sensitive parameters of R0 are the effective contact rate for Covid-19 transmission, the modification parameter accounting for increased susceptibility to covid-19 infection by pregnant women, and the transmission coefficient of the infectious pregnant women. The bifurcation analysis revealed a forward bifurcation which guarantees the stability of the disease-free<br />equilibrium when R0 is lowered below unity. Numerical simulations, using Mathematica Version 12.0 package, are given to show the effects of the most sensitive parameters of the basic reproduction number on the number of infectious cases of covid-19.</p> Jonathan Tsetimi Marcus Ifeanyi Ossaiugbo Augustine Atonuje Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 537 556 10.29020/nybg.ejpam.v15i2.4312 The Fractional Differential Equations with Uncertainty by Conformable Derivative https://www.ejpam.com/index.php/ejpam/article/view/4299 <p>We provide a fractional order fuzzy fractional differential equation q ∈ (0, 1]. A fuzzy fractional integral and a fuzzy conformable derivative are shown and proved. To prove fuzzy solutions for fractional differential equations with fuzzy beginning values and deterministic or fuzzy functions, two alternative techniques are used. The application has been submitted.</p> Atimad Harir Said Melliani Lalla Saadia Chadli Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 557 571 10.29020/nybg.ejpam.v15i2.4299 (Λ, sp)-open Sets in Topological Spaces https://www.ejpam.com/index.php/ejpam/article/view/4276 <p>This paper is concerned with the concepts of s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets. Some properties of s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets and b(Λ, sp)-open sets are discussed. In particular, the relationships between s(Λ, sp)-open sets, p(Λ, sp)-open sets, α(Λ, sp)-open sets, β(Λ, sp)-open sets, b(Λ, sp)-open sets and other related sets are established. Moreover, several characterizations of Λsp-extremally disconnected spaces are investigated.</p> Chawalit Boonpok Jeeranunt Khampakdee Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 572 588 10.29020/nybg.ejpam.v15i2.4276 Some Properties of Weak Separation Axioms in Coc-Compact Sets https://www.ejpam.com/index.php/ejpam/article/view/4298 <p>In this paper, we introduce some separation axioms in coc-compact set, namely coc-T0-space, coc-T 1/4- space, coc-T 3/8- space, cocT1/2- space, coc-T5/8- space, coc-Di- space, coc-Ri- space for i = 0, 1, weak coc-D1- space and weak coc-R0- space, and we study some relations between them, also we prove that some of these separation axioms have ”hereditary property”.</p> Fuad A AbuShaheen Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 589 601 10.29020/nybg.ejpam.v15i2.4298 On the Wiener and Harary Index of Splitting Graphs https://www.ejpam.com/index.php/ejpam/article/view/4316 <p>In this study we define a graph operation on a finite simple graph G = (V, E) called the S-splitting graph of G where S is a non-empty subset of vertices of G. If S = V , then it is the splitting graph of G defined by E. Sampathkumar, and H.B. Walikar in the 1980’s. This paper investigates the Wiener and Harary indices of the S-splitting graph of G for some families of graph.</p> Francis Joseph H. Campena ̃ Ma. Christine G. Egan John Rafael M. Antalan Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 602 619 10.29020/nybg.ejpam.v15i2.4316 A Quadruple Integral Involving the Hermite polynomial Hn(x): Derivation and Evaluation https://www.ejpam.com/index.php/ejpam/article/view/4204 <p>A closed form expression of a quadruple integral involving the Hermite polynomial Hn(x) is derived. Special cases are expressed in terms of special functions and fundamental constants. All the results in this work are new.</p> Robert Reynolds Allan Stauffer Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 620 625 10.29020/nybg.ejpam.v15i2.4204 On Almost α(Λ, sp)-continuous Multifunctions https://www.ejpam.com/index.php/ejpam/article/view/4277 <p>Our main purpose is to introduce the notion of almost α(Λ, sp)-continuous multifunctions. Moreover, some characterizations of almost α(Λ, sp)-continuous multifunctions are established.</p> Chawalit Boonpok Jeeranunt Khampakdee Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 626 634 10.29020/nybg.ejpam.v15i2.4277 Monophonic Eccentric Domination Numbers of Graphs https://www.ejpam.com/index.php/ejpam/article/view/4354 <p>Let G be a (simple) undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a monophonic eccentric dominating set if every vertex in V (G) \ S has a monophonic eccentric vertex in S. The minimum size of a monophonic eccentric dominating set in G is called the monophonic eccentric domination number of G. It is shown that the absolute difference of the domination number and monophonic eccentric domination number of a graph can be made arbitrarily large. We characterize the monophonic eccentric dominating sets in graphs resulting from the join, corona, and lexicographic product of two graphs and determine bounds on their monophonic eccentric domination numbers.</p> Anabel Gamorez Sergio Canoy Jr. Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 635 645 10.29020/nybg.ejpam.v15i2.4354 Bipolar Soft Generalized Topological Structures and Their Application in Decision Making https://www.ejpam.com/index.php/ejpam/article/view/4353 <p>The basic of bipolar soft set theory stands for a mathematical instrument that brings<br />together the soft set theory and bipolarity. Its definition is based on two soft sets, a set that<br />provides positive information and other that gives negative. This paper mainly aims at defining<br />a new bipolar soft generalized topological space; setting out of the point that the collection of<br />bipolar soft sets forms the basis for the definition of the new concept is defined. Added to that,<br />an investigation has been made of the four concepts of bipolar soft generalized, namely g-interior,<br />g-closure, g-exterior and g-boundary. Furthermore, the main properties of bipolar soft generalized<br />topological space (BSGT S) are established. This paper also attends to the discussion of the<br />relations between these new definitions and the application of the given bipolar soft generalized<br />topological spaces in a decision-making problem where an algorithm for this application has been<br />suggested. Finally, to clarify and substantiate what the current work subsumes, some examples<br />have been provided.</p> Hind Saleh Baravan Asaad Ramadhan Mohammed Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 646 671 10.29020/nybg.ejpam.v15i2.4353 Closed Extension Topological Spaces https://www.ejpam.com/index.php/ejpam/article/view/4379 <p>Let ( X , τ ) be a topological space and p ̸∈ X. Put Xp = X ∪ { p }. Define a topology τ⋆ on Xp by τ⋆ = { ∅ } ∪ {U ∪ { p } : U ∈ τ }. The space ( Xp, τ⋆) is called the closed extension space of ( X , τ ). We present new results about the closed extension topological spaces. Mainly weaker versions of normality.</p> Dina Abuzaid Suad Al-Qarhi Lutfi Nooruddin Kalantan Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 672 680 10.29020/nybg.ejpam.v15i2.4379 Spectral Dichotomy Methods of a Matrix with respect to the General Equation of the Parabola https://www.ejpam.com/index.php/ejpam/article/view/4348 <p>This paper presents methods of spectral dichotomy of a matrix which compute spectral projectors on the subspace associated with the eigenvalues external to the parabolas described by a general equation. These methods are modifications of the one proposed in [A. N. Malyshev and M. Sadkane, SIAM J. MATRIX ANAL. APPL. 18 (2), 265-278, 1997] which uses the spectral dichotomy method of a matrix with respect to the imaginary axis. Theoretical and algorithmic aspects of the methods are developed. Numerical results obtained by applying methods presented on matrices are reported.</p> Seydou Traoré Dosso Mouhamadou Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 681 725 10.29020/nybg.ejpam.v15i2.4348 Existence and Uniqueness of Solutions for Nonlinear Fractional Integro-Differential Equations with Nonlocal Boundary Conditions https://www.ejpam.com/index.php/ejpam/article/view/4366 <p>In this paper, the existence and uniqueness of solutions of fractional integro-differential equations with nonlocal boundary conditions is investigated. We establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle.</p> M.J. Mardanov Y.A. Sharifov Humbet Aliyev Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 726 735 10.29020/nybg.ejpam.v15i2.4366 On Weakly Connected Closed Geodetic Domination in Graphs Under Some Binary Operations https://www.ejpam.com/index.php/ejpam/article/view/4356 <p>Let G be a simple connected graph. For S ⊆ V (G), the weakly connected closed geodetic dominating set S of G is a geodetic closure IG[S] which is between S and is the set of all vertices on geodesics (shortest path) between two vertices of S. We select vertices of G<br />sequentially as follows: Select a vertex v1 and let S1 = {v1}. Select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Then successively select vertex vi ∈/ IG[Si−1] and let Si = {v1, v2, ..., vi} for i = 1, 2, ..., k until we select a vertex vk in the given manner that yields IG[Sk] = V (G). Also, the subgraph weakly induced ⟨S⟩w by S is connected where ⟨S⟩w = ⟨N[S], Ew⟩ with Ew = {u, v ∈E(G) : u ∈ S or v ∈ S} and S is a dominating set of G. The minimum cardinality of weakly connected closed geodetic dominating set of G is denoted by γwcg(G). In this paper, the authors show and investigate the concept weakly connected closed geodetic dominating sets of some graphs<br />and the join, corona, and Cartesian product of two graphs are characterized. The weakly connected closed geodetic domination numbers of these graphs are determined. Also, some relationships between weakly connected closed geodetic dominating set, weakly connected closed geodetic set, geodetic dominating set, and geodetic connected dominating set are established.</p> Jamil Hamja Imelda S. Aniversario Helen M. Rara Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 736 752 10.29020/nybg.ejpam.v15i2.4356 E-Bayesian Estimation under Loss Functions in Competing Risks https://www.ejpam.com/index.php/ejpam/article/view/4351 <p>Using gamma prior distribution of which shape hyperparameter has beta distributio and rate parameter has three different distributions over a finite interval, we studied the E-Bayesian estimation of one scale parameter of Gompertz distribution based on progressively type I censored sample from the competing risks model subject to K independent causes. The estimators obtained<br />generalize those issued from the quadratic loss, entropy loss and DeGroot loss functions.</p> Didier Alain Njamen Njomen Thiery Donfack Joseph Ngatchou-Wandji Georges Nguefack-Tsague Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 753 773 10.29020/nybg.ejpam.v15i2.4351 Results about P-Normality https://www.ejpam.com/index.php/ejpam/article/view/4387 <p>A topological space X is called P-normal if there exist a normal space Y and a bijective function f : X −→ Y such that the restriction f|A: A −→ f(A) is a homeomorphism for each paracompact subspace A ⊆ X. In this paper we present some new results on P-normality. We<br />study the invariance and inverse invariance of P-normality as a topological property. We also investigate the Alexandroff Duplicate of a P-normal space, the closed extension of a P-normal space, the discrete extension of a P-normal space and the Dowker topological space. Furthermore, we introduce a new property related to P-normality which we call strong P-normality.</p> Lutfi Kalantan Mai Mansouri Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 774 783 10.29020/nybg.ejpam.v15i2.4387 New Spectral Idea for Conjugate Gradient Methods and its Global Convergence Theorems https://www.ejpam.com/index.php/ejpam/article/view/4364 <p>Recently, the unconstrained optimization conjugate gradient methods have been widely utilized, especially for problems that are known as large-scale problems. This work proposes a new spectral gradient coefficient obtained from a convex linear combination of two different gradient coefficients to solve unconstrained optimization problems. One of the most essential features of<br />our suggested strategy is to guarantee the suitable subsidence direction of the line search precision. Furthermore, the proposed strategy is more effective than previous conjugate gradient approaches and stationery, which have been observed in the test problem. However, when it is compared to other conjugate gradient methods, such as FR methods, the proposed method confirmed the globally convergent, indicating that it can be used in scientific data computation.</p> Aseel Qasim Zinah Salih Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 784 795 10.29020/nybg.ejpam.v15i2.4364 Toeplitz Matrix and Nyström Method for Solving Linear Fractional Integro-differential Equation https://www.ejpam.com/index.php/ejpam/article/view/4384 <p>In this paper, the Volterra-Fredholm integral equation is derived from a linear integro-differential equation with a fractional order 0 &lt; α &lt; 1 using Riemann–Liouville fractional integral. The existence and uniqueness of the solution are proved using the Picard method. Popular numerical methods; the Toeplitz matrix, and the product Nystr ̈om are used in the solution. These methods will prove their effective in solving this type of equation. Two examples are solved using the mentioned methods and the estimation error is calculated. Finally, a comparison between the numerical results is made.</p> Sameeha Raad Khawlah AlQurashi Copyright (c) 2022 European Journal of Pure and Applied Mathematics https://creativecommons.org/licenses/by-nc-sa/4.0 2022-04-30 2022-04-30 15 2 796 809 10.29020/nybg.ejpam.v15i2.4384