Proyecciones (Antofagasta, On line) <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="" data-original-title="">appropriate credit</a>, provide a link to the license, and <a id="indicate_changes_popup" class="helpLink" tabindex="0" title="" href="" 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="" data-original-title="">technological measures</a> that legally restrict others from doing anything the license permits.</li> </ul> </div> Fractional neutral stochastic integrodifferential equations with Caputo fractional derivative <p>The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic integrodifferential equations driven by Rosenblatt process and Poisson jumps in Hilbert spaces. First we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach.</p> <p>The results are formulated and proved by using the fractional calculus, solution operator and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic differential equations driven by Rosenblatt process and poisson jumps is also been presented. An example is provided to illustrate the theory.</p> K. Ravikumar K. Ramkumar Hamdy Ahmed Copyright (c) 2023 K. Ravikumar, K. Ramkumar, Hamdy Ahmed 2023-05-09 2023-05-09 42 3 549 570 10.22199/issn.0717-6279-4329 Path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold <p>If H is a Hilbert space, the non-compact Stiefel manifold St(n, H) consists of independent n-tuples in H. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the path-connectedness and topological closure of some sets related to the non-compact Stiefel manifold. In the first part, after introducing and proving an essential lemma, we prove that ∩<sub>j∈J</sub> (U(j) + St(n, H)) is path-connected by polygonal paths under a condition on the codimension of the span of the components of the translating J-family. Then, in the second part, we show that the topological closure of St(n, H)∩S contains all polynomial paths contained in S and passing through a point in St(n, H). As a consequence, we prove that St(n, H) is relatively dense in a certain class of subsets which we illustrate with many examples from frame theory coming from the study of the solutions of some linear and quadratic equations which are finite-dimensional continuous frames. Since St(n, L<sup>2</sup>(X, μ; F)) is isometric to, F<sup>F</sup><sub>(X, </sub><sub>μ</sub><sub>), n</sub>, this article is also a contribution to the theory of finite-dimensional continuous Hilbert space frames.</p> Nizar El Idrissi Samir Kabbaj Brahim Moalige Copyright (c) 2023 Nizar El Idrissi, Samir Kabbaj, Brahim Moalige 2023-05-09 2023-05-09 42 3 571 597 10.22199/issn.0717-6279-4818 Quasi-k-normal ring <p>In [4] Wei and Libin defined Quasi normal ring. In this paper we attempt to define Quasi-k-normal ring by using the action of k-potent element. A ring is called Quasi-k-normal ring if ae = 0 ⇒ eaRe = 0 for a ∈ N(R)and e ∈ K(R), where K(R) = {e ∈ R|e<sup>k</sup> = e}. Several analogous results give in [4] is defined here. we find here that a ring is quasi-k-normal if and only if eR(1 − e<sup>k−1</sup>)Re = 0 for each e ∈ K(R). Also we get a ring is quasi-k-normal ring if and only if T<sub>n</sub>(R, R) is quasi-k-normal ring.</p> Kumar Napoleon Deka Helen K. Saikia Copyright (c) 2023 Kumar Napoleon Deka, Helen K. Saikia 2023-05-09 2023-05-09 42 3 599 608 10.22199/issn.0717-6279-4849 Characterization of nonuniform wavelets associated with 𝔄𝔅-MRA on L²(Λ) <p>Ahmad, Bhat and Sheikh characterized composite wavelets based on results of affine and quasi affine frames. We continued their study and provided the characterization of nonuniform composite wavelets based on results of affine and quasi affine frames. Moreover all the nonuniform composite wavelets associated with 𝔄𝔅 -MRA are characterized on L<sup>2</sup>(Λ).</p> Mohd Younus Bhat Shahbaz Rafiq Muddasir A. Lone Altaf A. Bhat Copyright (c) 2023 Mohd Younus Bhat, Shahbaz Rafiq, Muddasir A. Lone, Altaf A. Bhat 2023-05-09 2023-05-09 42 3 609 630 10.22199/issn.0717-6279-4584 On some P-Q modular equations of degree 45 <p>On page 330 of his second notebook, Srinivasa Ramanujan recorded a P-Q modular equation of degree 45, proof of which has been given by Bruce C. Berndt via theory of modular forms. We in this paper, give a simple proof of the same using the identities of Ramanujan and also establish few new P-Q modular equations of degree 45. Further using these, we establish certain new modular equations of signature 3.</p> G. Sharath Copyright (c) 2023 G. Sharath 2023-05-09 2023-05-09 42 3 631 650 10.22199/issn.0717-6279-4956 Generation of anti-magic graphs from binary graph products <p>An anti-magic labeling of a graph G is a one-to-one correspondence between E(G) and {1, 2, ··· , |E|} such that the vertex-sum for distinct vertices are different. Vertex-sum of a vertex u ∈ V (G) is the sum of labels assigned to edges incident to the vertex u. It was conjectured by Hartsfield and Ringel that every tree other than K<sub>2</sub> has an anti-magic labeling. In this paper, we consider various binary graph products such as corona, edge corona and rooted products to generate anti-magic graphs. We prove that corona products of an anti-magic regular graph G with K<sub>1</sub> and K<sub>2</sub> are anti-magic. Further, we prove that rooted product of two anti-magic trees are anti-magic. Also, we prove that rooted product of an anti-magic graph with an anti-magic tree admits anti-magic labeling.</p> P. Ragukumar Copyright (c) 2023 P. Ragukumar 2023-05-10 2023-05-10 42 3 651 661 10.22199/issn.0717-6279-5220 Periodic orbits of Linear flows on connected Lie groups <p>Our main goal is to study the periodic orbits of linear flows on a real, connected Lie group. Since each linear flow φ<sub>t</sub> has a derivation associated 𝒟, we show that the existence of periodic orbits of φ<sub>t</sub> is based on the eigenvalues of the derivation 𝒟. From this, we study periodic orbits of a linear flow on noncompact, semisimple Lie groups, and we work with periodic orbits of a linear flow on a connected, simply connected, solvable Lie groups of dimension 2 or 3.</p> Simão Nicolau Stelmastchuk Copyright (c) 2023 Simão Nicolau Stelmastchuk 2023-05-10 2023-05-10 42 3 663 680 10.22199/issn.0717-6279-5261 Exponential stability and instability in nonlinear differential equation with multiple delays <p>Inequalities regarding the solutions of the nonlinear differential equation with multiple delays</p> <p>x<sup>l</sup>(t) = a(t)f(x(t)) +Σ<sup>n</sup><sub>i=1</sub>b<sub>i</sub>(t)f(x(t − h<sub>i</sub>)),</p> <p>are obtained by means of Lyapunov functionals. These inequalities are then used to obtain sufficient conditions that guarantee exponential decay of solutions to zero of the multi delay nonlinear differential equation. In addition, we obtain a criterion for the instability of the zero solution. The results generalizes some results in the literature.</p> Adusei-Poku Afful Ernest Yankson Copyright (c) 2023 Adusei-Poku Afful, Ernest Yankson 2023-05-10 2023-05-10 42 3 681 693 10.22199/issn.0717-6279-4197 On βκ-normal spaces <p>A topological space X is called βκ-normal if for every pair of disjoint regularly closed sets A and B, there exist disjoint open sets U and V of X such that <img src="" alt="" width="50" height="22" /> = A, <img src="" alt="" width="49" height="22" /> = B and <img src="" alt="" width="12" height="22" /> ∩ <img src="" alt="" width="12" height="22" /> = ∅. In this paper, we investigated a weaker form of normality called βκ-normality which is simultaneous generalization of normality, κ-normality and almost β-normality. Some new decomposition of normality is obtained in terms weakly β-normal spaces.</p> Sumit Singh Manoj Kumar Rana Copyright (c) 2023 Sumit Singh, Manoj Kumar Rana 2023-05-10 2023-05-10 42 3 695 712 10.22199/issn.0717-6279-4277 Existence and uniqueness of the generalized solution of a non-homogeneous hyperbolic differential equation modeling the vibrations of a dissipating elastic rod <p>The purpose of this mathematical paper is to establish a qualitative research of the existence and uniqueness of the generalized solution to a non-homogeneous hyperbolic partial differential equation problema</p> <p><img src="" alt="" width="149" height="41" /></p> <p>subject to the contour condition u = 0 over Σ, and with initial conditions u(x, 0) = u<sub>0</sub>(x) in Ω, ∂u<sub>t</sub>(x, 0) = u<sub>1</sub>(x) in Ω. In the development of the research, the deductive method of Faedo-Garleskin and Medeiro is used to demonstrate the existence of the generalized solution that consists in the construction of approximate solutions in a finite dimensional space, obtaining a succession of approximate solutions to the non-homogeneous hyperbolic problem, that is, by means of a priori estimations, these successions of approximate solutions are passed to limit in a suitable topology. Then the initial conditions are verified and the uniqueness of the generalized solution is proved.</p> Jhony Alfonso Chávez Delgado Augusto Becerra Castañeda Luis César Méndez Avalos Eduardo Rodríguez Delgado Luis Asunción López Puycan Copyright (c) 2023 Jhony Alfonso Chávez Delgado, Augusto Becerra Castañeda, Luis César Méndez Avalos, Eduardo Rodríguez Delgado, Luis Asunción López Puycan 2023-05-15 2023-05-15 42 3 713 726 10.22199/issn.0717-6279-4478 On the Wiener index and the hyper-Wiener index of the Kragujevac trees <p>In this paper, the Wiener index and the hyper-Wiener index of the Kragujevac trees is computed in term of its vertex degrees. As application, we obtain an upper bond and a lower bound for the Wiener index and the hyper-Wiener index of these trees.</p> Abbas Heydari Copyright (c) 2023 Abbas Heydari 2023-05-15 2023-05-15 42 3 727 739 10.22199/issn.0717-6279-4804 On graded G2-absorbing and graded strongly G2-absorbing second submodules <p>In this paper, we introduce the concepts of graded G2-absorbing and graded strongly G2-absorbing second submodules of graded modules over graded commutative rings. We give a number of results concerning these classes of graded submodules and their homogeneous components.</p> Shatha Alghueiri Khaldoun Al-Zoubi Copyright (c) 2023 Shatha Alghueiri, Khaldoun Al-Zoubi 2023-05-15 2023-05-15 42 3 741 756 10.22199/issn.0717-6279-5145 On uniform-ultimate boundedness and periodicity results of solutions to certain second order non-linear vector differential equations <p>In this paper, we employ the second method of Lyapunov to examine sufficient conditions for the uniform-ultimate boundedness of solutions and existence of at least one periodic solution to the following second order vector differential equation:</p> <p>Ẍ+ F(X, Ẋ ) Ẋ + H(X) = P(t, X, Ẋ ),</p> <p>when the non-linear term H(X) is: (i) differentiable, (ii) non-necessarily differentiable. The results contain in this paper are new and complement related ones in the literature.</p> Adetunji Adeyanju Copyright (c) 2023 Adetunji Adeyanju 2023-05-16 2023-05-16 42 3 757 773 10.22199/issn.0717-6279-5421 k-Zumkeller Graphs through Splitting of Graphs <p>Let G = (V,E) be a simple graph with vertex set V and edges set E. A 1−1 function f : V → N is said to induce a k-Zumkeller graph G if the induced edge function <em>f </em><sup>∗</sup> : E → N defined by <em>f</em> <sup>∗</sup>(xy) = f(x)f(y) satisfies the following conditions:</p> <ol> <li><em>f</em> <sup>∗</sup>(xy) is a Zumkeller number for every xy ∈ E.</li> <li>The total distinct Zumkeller numbers on the edges of G is k.</li> </ol> <p>In this article, we compute k-Zumkeller graphs through the graph splitting operation on path, cycle and star graphs.</p> M. Kalaimathi B. J. Balamurugan Copyright (c) 2023 M. Kalaimathi , B. J. Balamurugan 2023-05-16 2023-05-16 42 3 775 794 10.22199/issn.0717-6279-5723 On domination in the total torsion element graph of a module <p>Let R be a commutative ring with non-zero unity and M be a unitary R-module. Let T(M) be the set of torsion elements of M. Atani and Habibi [6] introduced the total torsion element graph of M over R as an undirected graph T(Γ(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ T(M). The main objective of this paper is to study the domination properties of the graph T(Γ(M)). The domination number of T(Γ(M)) and its induced subgraphs T or(Γ(M)) and T of(Γ(M)) has been determined. Some domination parameters of T(Γ(M)) are also studied. In particular, the bondage number of T(Γ(M)) has been determined. Finally, it has been proved that T(Γ(M)) is excellent, domatically full and well covered under certain conditions.</p> Jituparna Goswami Moytri Sarmah Copyright (c) 2023 Jituparna Goswami, Moytri Sarmah 2023-05-16 2023-05-16 42 3 795 814 10.22199/issn.0717-6279-4904