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> On the mixed multifractal formalism for vector-valued measures <p>The multifractal formalism for vector-valued measures holds when-ever the existence of corresponding Gibbs-like measures, supported on the singularities sets holds. We tried through this article to improve a result developed by Menceur et al. in [29] and to suggest a new sufficient condition for a valid mixed multifractal formalism for vector-valued measures. We describe a necessary condition of validity for the formalism which is very close to the sufficient one.</p> Bilel Selmi Anouar Ben Mabrouk Copyright (c) 2022 Bilel Selmi, Anouar Ben Mabrouk 2022-08-26 2022-08-26 41 5 1015 1032 10.22199/issn.0717-6279-4187 Lie (Jordan) centralizers on alternative algebras <p>In this article, we study Lie (Jordan) centralizers on alternative algebras and prove that every multiplicative Lie centralizer has proper form on alternative algebras under certain assumptions.</p> Aisha Jabeen Bruno Ferreira Copyright (c) 2022 Aisha Jabeen, Bruno Ferreira 2022-08-26 2022-08-26 41 5 1035 1050 10.22199/issn.0717-6279-4789 Equitable chromatic number of weak modular product of Some graphs <p>An equitable coloring of a graph G is a proper coloring of the vertices of G such that the number of vertices in any two color clases differ by at most one. The equitable chromatic number χ=(G) of a graph G is the minimum number of colors needed for an equitable coloring of G. In this paper, we obtain the equitable chromatic number of weak modular product of two graphs G and H, denoted by G o H.</p> <p>First, we consider the graph G o H, where G is the path graph, and H be any simple graph like the path, the cycle graph, the complete graph. Secondly, we consider G and H as the complete graph and cycle graph respectively. Finally, we consider G as the star graph and H be the complete graph and star graph.</p> K. Kaliraj R. Narmadha Devi J. Vernold Vivin Copyright (c) 2022 K. Kaliraj, R. Narmadha Devi, J. Vernold Vivin 2022-08-26 2022-08-26 41 5 1051 1062 10.22199/issn.0717-6279-5140 Spectral analysis for finite rank perturbation of diagonal operator in non-archimedean Banach space of countable type <p>In this paper we are concerned with the spectral analysis for the classes of finte rank perturbations of diagonal operators in the form A = D +F where D is a diagonal operator and F &nbsp;is an operator of finite rank in the non archimedean Banach space of countable type. Using the theory of Fredholm operators in non archimedean setting and the concept of essential spectrum for linear operators, we compute the spectrum of A.</p> Abdelkhalek El amrani Aziz Blali Mohamed Amine Taybi Copyright (c) 2022 Abdelkhalek El amrani, Aziz Blali, Mohamed Amine Taybi 2022-08-26 2022-08-26 41 5 1063 1074 10.22199/issn.0717-6279-4984 On the cohomological equation of a linear contraction <p>In this paper, we study the discrete cohomological equation of a contracting linear automorphism A of the Euclidean space <strong>R</strong><sup>d</sup>. More precisely, if δ is the cobord operator defined on the Fréchet space E = C<sup>l</sup> (<strong>R</strong><sup>d</sup>) (0 ≤ l ≤ ∞) by: δ(h) = h − h ◦ A, we show that:</p> <ul> <li>If E = C<sup>0</sup>(<strong>R</strong><sup>d</sup>), the range δ (E) of δ has infinite codimension and its closure is the hyperplane E<sub>0</sub> consisting of the elements of E vanishing at 0. Consequently, H<sup>1</sup> (A, E) is infinite dimensional non Hausdorff topological vector space and then the automorphism A is not cohomologically C<sup>0</sup>-stable.</li> <li>If E = C<sup>l </sup>(<strong>R</strong><sup>d</sup>), with 1 ≤ l ≤ ∞, the space δ (E) coincides with the closed hyperplane E<sub>0</sub>. Consequently, the cohomology space H<sup>1</sup> (A, E) is of dimension 1 and the automorphism A is cohomologically C<sup>l</sup>-stable.</li> </ul> Régis Leclercq Abdellatif Zeggar Copyright (c) 2022 Régis Leclercq, Abdellatif Zeggar 2022-09-13 2022-09-13 41 5 1075 1091 10.22199/issn.0717-6279-4559 I-convergent triple fuzzy normed spaces <p>In this paper we introduce the lacunary ideal convergence of triple sequences in fuzzy normed spaces and the relation between lacunary convergence and lacunary ideal convergence is investigated for triple sequences in fuzzy normed spaces. Concept of limit point and cluster point for triple sequences in fuzzy normed spaces and theorems related to these concepts are also given.</p> Tanweer Jalal Ishfaq Ahmad Malik Copyright (c) 2022 Tanweer Jalal, Ishfaq Ahmad Malik 2022-09-13 2022-09-13 41 5 1093 1109 10.22199/issn.0717-6279-4867 Stratonovich-Henstock integral for the operator-valued stochastic process <p>In this paper, we introduce the Stratonovich-Henstock integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also formulate a version of Ito's formula for this integral.</p> Recson Canton Mhelmar Labendia Tin Lam Toh Copyright (c) 2022 Recson Canton, Mhelmar Labendia, Tin Lam Toh 2022-09-13 2022-09-13 41 5 1111 1130 10.22199/issn.0717-6279-5018 Implications of Some Types of Pairwise Closed Graphs <p>The main goal of this paper is to introduce and look into some of the fundamental properties of pairwise strongly closed, pairwise strongly -closed and pairwise quasi -closed graphs. Some characterizations and several properties concerning these graphs are obtained. We also investigate relationships between&nbsp; (i,j)-strongly alpha -closed graph&nbsp; G(f) and&nbsp; (i,j)-weakly alpha -continuous. We study relationships between&nbsp; (i,j) strongly alpha -closed&nbsp; (i,j)-quasi alpha-closed graphs with covering properties. The concepts of pairwise -closed and pairwise quasi H-closed relatively are stated.</p> Hend Bouseliana Adem Kılıçman Copyright (c) 2022 Hend Bouseliana, Adem Kılıçman 2022-09-13 2022-09-13 41 5 1131 1139 10.22199/issn.0717-6279-5062 Some characterizations of frames in ℓ²(I; H) and topological applications <p>We propose in this article some characterizations of the notion of frame in ℓ<sup>²</sup>(<em>I</em>; <em>H</em>). The first one is general, and depends on a procedure of inserting a family of vectors instead of x in the definition of a frame. This allows us to define the analysis, synthesis and frame operator on the space ℓ<sup>²</sup>(<em>I</em>; <em>H</em>) instead of <em>H</em>. The second one is specific to ℓ<sup>²</sup> (<em>I</em>; <em>C</em><sup>k</sup>) and relate it to the freeness of the finite set of components of the frame. The third one concerns normalised tight frames in ℓ<sup>²</sup>(<em>I</em>; <em>C</em><sup>k</sup>). Afterwards, we give an example of a frame in ℓ<sup>²</sup>(<em>I</em>; <em>C</em><sup>²</sup>) using another sufficient condition in dimension 2. We conclude with some topological applications of these characterizations.</p> Nizar El Idrissi Samir Kabbaj Brahim Moalige Copyright (c) 2022 Nizar El Idrissi, Samir Kabbaj, Brahim Moalige 2022-09-13 2022-09-13 41 5 1141 1152 10.22199/issn.0717-6279-4043 Combination labelings of graphs related to several cycles and paths <p>Suppose that G = (V (G), E(G)) is a graph and |V (G)| = p. If there exists a bijective function <em>f</em> : V (G) → {1, 2, 3, ..., p} such that an <em>f</em> <sup>c</sup> : E(G) → N defined by <em>f</em> <sup>c</sup>(uv) = (<sup>f(u)</sup><sub>f(v)</sub>)when <em>f</em> (u) &gt; f(v) and <em>f </em><sup>c</sup>(uv) = (<sup>f(u)</sup><sub>f(v)</sub>)when <em>f </em>(v) &gt; <em>f</em> (u) is an injection function, then <em>f</em> is called a combination labelings and G is called a combination graph.</p> <p>This article considers a suitable bijective function f and prove that G(C<sub>n</sub>, C<sub>m</sub>, P<sub>k</sub>) which are graphs related to two cycles and one path containing three parameters, are combination graphs.</p> Aiewcharoen Busakorn Ratinan Boonklurb Sakulwat Promvichitkul Copyright (c) 2022 Aiewcharoen Busakorn, Ratinan Boonklurb, Sakulwat Promvichitkul 2022-09-27 2022-09-27 41 5 1153 1172 10.22199/issn.0717-6279-5250 Orbit equivalence of linear systems on manifolds and semigroup actions on homogeneous spaces <p>In this paper we introduce the notion of orbit equivalence for semi-<br>group actions and the concept of generalized linear control system on<br>smooth manifold. The main goal is to prove that, under certain condi-<br>tions, the semigroup system of a generalized linear control system on a<br>smooth manifold is orbit equivalent to the semigroup system of a linear<br>control system on a homogeneous space.</p> João Augusto Navarro Cossich R. M. Hungaro O. G. Rocio A. J. Santana Copyright (c) 2022 João Augusto Navarro Cossich, R. M. Hungaro, O. G. Rocio, A. J. Santana 2022-09-27 2022-09-27 41 5 1173 1198 Fractional metric dimension of generalized prism graph <p>Fractional metric dimension of connected graph $G$ was introduced by Arumugam et al. in [Discrete Math. 312, (2012), 1584-1590] as a natural extension of metric dimension which have many applications in different areas of computer sciences for example optimization, intelligent systems, networking and robot navigation. In this paper fractional metric dimension of generalized prism graph $P_{m}\times C_{n}$ is computed using combinatorial criterion devised by Liu et al. in [ Mathematics, 7(1), (2019), 100].</p> Nosheen Goshi Sohail Zafar Tabasam Rashid Copyright (c) 2022 Nosheen Goshi, Sohail Zafar, Tabasam Rashid 2022-09-27 2022-09-27 41 5 1199 1212 10.22199/issn.0717-6279-4722 On even-odd meanness of super subdivision of some graphs <p>Graph Labeling is a significant area of graph theory that is used in a variety of applications like coding hypothesis, x-beam crystallography, radar, cosmology, circuit design, correspondence network tending to, and database administration. This study provides a general overview of graph naming in heterogeneous fields, however it primarily focuses on graph subdivision. The even vertex odd meanness of super subdivide of various graphs is discussed in this study. The graphs generated by super subdivided of path, cycle, comb, crown, and planar grid are even-odd mean graphs, according to our proof.</p> Mohamed Basher Muhammad Kamran Siddiqui Copyright (c) 2022 Mohamed Basher, Muhammad Kamran Siddiqui 2022-09-27 2022-09-27 41 5 1213 1228 10.22199/issn.0717-6279-5302 Bounds for absolute values and imaginary parts of matrix eigenvalues via traces <p>Let λ<sub>1</sub>(A), λ<sub>2</sub>(A), ..., λ<sub>n</sub>(A) be the eigenvalues of an n × n-matrix A taken with their algebraic multiplicities. We suggest new bounds for |λ<sub>j</sub> (A) − <sup>trace(A)/ n</sup> | and |Im λ<sub>j</sub> (A) − <sup>Im trace(A)/n</sup> | (j = 1, ..., n), which refine the previously published results.</p> <p> </p> Michael Gil' Copyright (c) 2022 Michael Gil' 2022-09-27 2022-09-27 41 5 1229 1237 10.22199/issn.0717-6279-5349 Decomposition dimension of corona product of some classes of graphs <p>For an ordered <em>k</em>-decomposition <em>D</em> = {G<sub>1</sub>, G<sub>2</sub>,...,G<sub>k</sub>} of a connected graph G = (V,E), the <em>D</em>-representation of an edge e is the k-tuple γ(e/D)=(d(e, G<sub>1</sub>), d(e, G<sub>2</sub>), ...,d(e, G<sub>k</sub>)), where d(e, G<sub>i</sub>) represents the distance from e to G<sub>i</sub>. A decomposition D is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). In this paper, the decomposition dimension of corona product of the path P<sub>n</sub> and cycle C<sub>n</sub> with the complete graphs K<sub>1</sub> and K<sub>2</sub> are determined.</p> T. Reji R. Ruby Copyright (c) 2022 T. Reji , R. Ruby 2022-09-27 2022-09-27 41 5 1239 1250 10.22199/issn.0717-6279-5466