Proyecciones (Antofagasta, On line) https://www.revistaproyecciones.cl/index.php/proyecciones <p align="justify">La revista&nbsp;Proyecciones. Journal of Mathematics es una publicación científica, sin fines de lucro, oficial de la Universidad Católica del Norte, Antofagasta, Chile. Fue fundada en 1982 y depende del Departamento de Matemáticas de la Universidad Católica del Norte.<br>Proyecciones. Journal of Mathematics edita un volumen con 5 números al año.</p> Universidad Católica del Norte. en-US Proyecciones (Antofagasta, On line) 0717-6279 <div id="deed-conditions" class="row"> <ul class="license-properties col-md-offset-2 col-md-8" dir="ltr"> <li class="license by"> <p><strong>Attribution</strong> — You must give <a id="appropriate_credit_popup" class="helpLink" tabindex="0" title="" href="https://creativecommons.org/licenses/by/4.0/deed.en" data-original-title="">appropriate credit</a>, provide a link to the license, and <a id="indicate_changes_popup" class="helpLink" tabindex="0" title="" href="https://creativecommons.org/licenses/by/4.0/deed.en" data-original-title="">indicate if changes were made</a>. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.<span id="by-more-container"></span></p> </li> </ul> </div> <div class="row"> <ul id="deed-conditions-no-icons" class="col-md-offset-2 col-md-8"> <li class="license"><strong>No additional restrictions</strong> — You may not apply legal terms or <a id="technological_measures_popup" class="helpLink" tabindex="0" title="" href="https://creativecommons.org/licenses/by/4.0/deed.en" data-original-title="">technological measures</a> that legally restrict others from doing anything the license permits.</li> </ul> </div> A linear time algorithm for minimum equitable dominating set in trees https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4552 <p><em>Let G = (V, E) be a graph. A subset D<sup>e</sup> of V is said to be an equitable dominating set if for every v </em><em>∈</em><em> V \ D<sup>e</sup> there exists u </em><em>∈</em><em> D<sup>e</sup> such that uv </em><em>∈</em><em> E and |deg(u) </em><em>−</em><em> deg(v)| </em><em>≤</em><em> 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γ<sup>e</sup>. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.</em></p> Sohel Rana Sk. Md. Abu Nayeem Copyright (c) 2021 Sohel Rana, Sk. Md. Abu Nayeem http://creativecommons.org/licenses/by/4.0 2021-07-25 2021-07-25 40 4 805 814 10.22199/issn.0717-6279-4552 Mappings and decompositions of pairwise continuity on (i, j)-almost Lindelöf and (i, j)-weakly Lindelöf spaces https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3253 <p><em>The purpose of this paper is to study the eect of mappings and some decompositions of pairwise continuity on (i, j)-almost Lindelöf spaces and (i, j)-weakly Lindelöf spaces. The main results are that an (i, j)-continuous image of an (i, j)-almost Lindelöf space is (i, j)-almost Lindelof and a pairwise almost continuous image of an (i, j)-weakly Lindelöf space is (i, j)-weakly Lindelöf. We also show that (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf properties are bitopological properties.</em></p> Zabidin Salleh Adem Kiliçman Copyright (c) 2021 Zabidin Salleh, Adem Kiliçman http://creativecommons.org/licenses/by/4.0 2021-07-25 2021-07-25 40 4 815 836 10.22199/issn.0717-6279-3253 Fixed point theorems in fuzzy metric spaces for mappings with Bγ,µ condition https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3333 <p><em>In this paper we prove some fixed point theorems in fuzzy metric spaces for a class of generalized nonexpansive mappings satisfying B<sub>γ,µ</sub> condition. We introduce a type of convexity in fuzzy metric spaces with respect to an altering distance function and prove convergence results for some iteration schemes to the fixed point. The results are supported by suitable examples.</em></p> Nilakshi Goswami Bijoy Patir Copyright (c) 2021 Nilakshi Goswami, Bijoy Patir http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 837 857 10.22199/issn.0717-6279-3333 On graded primary-like submodules of graded modules over graded commutative rings https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3467 <p><em>Let G be a group with identity e. Let R be a G-graded commutative ring andM a graded R-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to and extra properties of these graded submodules.</em></p> Khaldoun Al-Zoubi Mohammed Al-Dolat Copyright (c) 2021 Khaldoun Falah Al-Zoubi, Mohammed Al-Dolat http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 859 871 10.22199/issn.0717-6279-3467 Lyapunov-type inequality for a Riemann-Liouville type fractional boundary value problem with anti-periodic boundary conditions https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3488 <p><em>In this article, we establish a Lyapunov-type inequality for a two-point Riemann-Liouville type fractional boundary value problem associated with well-posed anti-periodic boundary conditions. As an application, we estimate a lower bound for the eigenvalue of the corresponding fractional eigenvalue problem. </em></p> Jagan Mohan Jonnalagadda Debananda Basua Copyright (c) 2021 Jagan Mohan Jonnalagadda, Debananda Basua http://creativecommons.org/licenses/by/4.0 2021-07-25 2021-07-25 40 4 873 884 10.22199/issn.0717-6279-3488 A new approach for Volterra functional integral equations with non-vanishing delays and fractional Bagley-Torvik equation https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4010 <p><em>A numerical technique for Volterra functional integral equations (VFIEs) with non-vanishing delays and fractional Bagley-Torvik equation is displayed in this work. The technique depends on Bernstein polynomial approximation. Numerical examples are utilized to evaluate the accurate results. The findings for examples figs and tables show that the technique is accurate and simple to use.</em></p> Fateme Ghomanjani Copyright (c) 2021 Fateme Ghomanjani http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 885 903 10.22199/issn.0717-6279-4010 Total irregularity strength of some cubic graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3715 <p><em>Let G = (V;E) be a graph. A total labeling ψ : V ⋃ E → {1, 2, ....k} is called totally irregular total k-labeling of G if every two distinct vertices u and v in V (G) satisfy wt(u) ≠wt(v); and every two distinct edges u<sub>1</sub>u<sub>2</sub> and v<sub>1</sub>v<sub>2</sub> in E(G) satisfy wt(u<sub>1</sub>u<sub>2</sub>) ≠ wt(v<sub>1</sub>v<sub>2</sub>); where wt(u) = ψ (u) + ∑<sub>uv∊E(G)</sub> ψ(uv) and wt(u<sub>1</sub>u<sub>2</sub>) = ψ(u<sub>1</sub>) + ψ(u<sub>1</sub>u<sub>2</sub>) + ψ(u<sub>2</sub>): The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G): In this paper, we determine the exact value of the total irregularity strength of cubic graphs.</em></p> Muhammad Ibrahim S. Khan Muhammad Ahsan Asim Muhammad Waseem Copyright (c) 2021 Muhammad Ibrahim, Muhammad Ahsan Asim, S. Khan, Muhammad Waseem http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 905 918 10.22199/issn.0717-6279-3715 Line graph of unit graphs associated with finite commutative rings https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4112 <p><em>For a given graph G, its line graph denoted by L(G) is a graph whose vertex set V (L(G)) = E(G) and {e<sub>1</sub>, e<sub>2</sub>} ∈ E(L(G)) if e1 and e2 are incident to a common vertex in G. Let R be a finite commutative ring with nonzero identity and G(R) denotes the unit graph associated with R. In this manuscript, we have studied the line graph L(G(R)) of unit graph G(R) associated with R. In the course of the investigation, several basic properties, viz., diameter, girth, clique, and chromatic number of L(G(R)) have been determined. Further, we have derived sufficient conditions for L(G(R)) to be Planar and Hamiltonian</em></p> Pranjali Amit Kumar Pooja Sharma Copyright (c) 2021 Pranjali, Amit Kumar, Pooja Sharma http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 919 926 10.22199/issn.0717-6279-4112 Square root stress-sum index for graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4362 <p><em>The stress of a vertex is a node centrality index, which has been introduced by Shimbel (1953). The stress of a vertex in a graph is the number of geodesics (shortest paths) passing through it. In this paper, we introduce a new topological index for graphs called square root stress sum index using stresses of vertices. Further, we establish some inequalities, prove some results and compute stress-sum index for some standard graphs.</em></p> K. B. Mahesh R. Rajendra P. Siva Kota Reddy Copyright (c) 2021 Dr. Mahesh, Dr. Rajendra, Dr. Siva Kota Reddy Polaepalli http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 927 936 10.22199/issn.0717-6279-4362 Basarab loop and the generators of its total multiplication group https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4430 <p><em>A loop (Q; ·) is called a Basarab loop if the identities: (x · yx<sup>ρ</sup>)(xz) = x · yz and (yx) · (x<sup>λ</sup>z · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping T<sub>x</sub> are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ T<sub>x</sub> is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop</em></p> Temitope Jaiyéolá Gideon Effiong Copyright (c) 2021 T.G. Jaiyeola, PhD, Gideon Effiong http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 939 962 10.22199/issn.0717-6279-2021-4430 Graphs of edge-to-vertex detour number 2 https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4454 <p><em>For two vertices u and v in a graph G = (V,E), the detour distance D(u, v) is the length of a longest u − v path in G. A u − v path of length D(u, v) is called a u − v detour. For subsets A and B of V , the detour distance D(A,B) is defined as D(A,B) = min{D(x, y) : x ∈ A, y ∈ B}. A u − v path of length D(A,B) is called an A-B detour joining the sets A,B ⊆ V where u ∈ A and v ∈ B. A vertex x is said to lie on an A − B detour if x is a vertex of some A−B detour. A set S ⊆ E is called an edge-to-vertex detour set if every vertex of G is incident with an edge of S or lies on a detour joining a pair of edges of S. The edge-to-vertex detour number dn<sub>2</sub>(G) of G is the minimum order of its edge-to-vertex detour sets and any edge-to-vertex detour set of order dn<sub>2</sub>(G) is an edge-to-vertex detour basis of G. Graphs G of size q for which dn<sub>2</sub>(G) = 2 are characterized.</em></p> A. P. Santhakumaran Copyright (c) 2021 A. P. Santhakumaran http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 963 979 10.22199/issn.0717-6279-4454 λ-quasi Cauchy sequence of fuzzy numbers https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4466 <p><em>In this paper we introduce the λ -quasi Cauchy sequence of fuzzy numbers. We obtain the relation between strongly λ-quasi Cauchy convergence and statistically λ -quasi Cauchy convergence for fuzzy numbers.</em></p> Achyutananda Baruah Bipan Hazarika Copyright (c) 2021 Achyutananda Baruah, Bipan Hazarika http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 981 988 10.22199/issn.0717-6279-4466 Equitably strong nonsplit equitable domination in graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4497 <p><em>In a simple, finite and undirected graph G with vertex set V and edge set E, Prof. Sampathkumar defined degree equitability among vertices of G. Two vertices u and v are said to be degree equitable if |deg(u) − deg(v)| ≤ 1. Equitable domination has been defined and studied in [7]. V.R.Kulli and B.Janakiram defined strong non - split domination in a graph [12]. In this paper, the equitable version of this new type of domination is studied</em></p> P. Nataraj R. Sundareswaran V. Swaminathan Copyright (c) 2021 P. Nataraj, Sundareswaran Raman, V. Swaminathan http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 989 999 10.22199/issn.0717-6279-4497 The probability of an automorphism of an abelian group fixing a group element https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4513 <p>In this paper, we consider the probability of an automorphism of a finite abelian group fixing a group element. Explicit computations are made to find the fusion classes of a finite abelian groups. The probability of an automorphism fixing a group element is obtained in terms of fusion classes. We also compute the bounds of the probability for some particular cases.</p> Harsha Arora Copyright (c) 2021 Harsha Arora http://creativecommons.org/licenses/by/4.0 2021-07-26 2021-07-26 40 4 1001 1010 10.22199/issn.0717-6279-2021-4513 Lyapunov-type inequality for higher order left and right fractional p-Laplacian problems https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4366 <p><em>In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.</em></p> Alberto Cabada Rabah Khaldi Copyright (c) 2021 Alberto Cabada, Rabah Khaldi http://creativecommons.org/licenses/by/4.0 2021-08-03 2021-08-03 40 4 1031 1040 10.22199/issn.0717-6279-4366 Fiedler vector analysis for particular cases of connected graphs https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/4790 <p><em>In this paper, some subclasses of block graphs are considered in order to analyze Fiedler vector of its members. Two families of block graphs with cliques of fixed size, the block-path and block-starlike graphs, are analyzed. Cases A and B of classification for both families were considered, as well as the behavior of the algebraic connectivity when some vertices and edges are added for particular cases of block-path graphs.</em></p> Daniel Traciná Claudia Justel Copyright (c) 2021 Public Domain https://creativecommons.org/publicdomain/zero/1.0/ 2021-08-03 2021-08-03 40 4 1041 1051 10.22199/issn.0717-6279-4790