Research Papers of
BS Mathematics 4-2

This paper undertakes an investigation into the Approximation of the Core Inverse of Square Real Matrices. The Core inverse, denoted as A(#), serves as a unique matrix that satisfies specific conditions, which holds significant importance in mathematical applications, particularly when classical matrix inversion is impossible. The study presents a thorough exploration of an extensively known generalized inverse, namely the Moore-Penrose inverse, as a foundational concept. Methodologically, we initiate our inquiry by formulating an optimization problem akin to the methodology employed in approximating the Moore-Penrose inverse. Additionally, we integrate the Tikhonov Regularization technique to enhance the stability of solutions, particularly in cases where matrices exhibit ill-posed properties. Furthermore, our approach encompasses the application of gradient, symmetric, and proximity operators to reach the desired outcomes. Ultimately, we provide a formula for approximating the Coreinverse, thus offering valuable contributions to the fields of linear algebra, analysis, and optimization.

Keywords: core inverse, Moore-Penrose inverse, Tikhonov regularization, gradient, symmetric and proximity operators

Fibonacci partial words are sequences over a finite alphabet that may contain symbols, called holes, wherein its length follows from the Fibonacci numbers. Its combinatorial properties play an important role in some areas of mathematics. Using the existing results of Fibonacci partial words, the researchers came up with a new variety of partial words. In this paper, we introduced the concept of Fibonacci-Lucas words and Fibonacci-Lucas partial words governed by the recursive relations, ln = fn-1fn+1 and l◇n = f◇n-1f◇n+1, respectively. We used the idea and one of the identities of the Lucas sequence to form the new variety of word. We have established some properties of Fibonacci-Lucas partial words such as compatibility and periodicity. Furthermore, we have proven the compatibility of two Fibonacci-Lucas partial words with certain conditions on their initial Fibonacci partial words. Moreover, we have shown the common strong period of the compatible Fibonacci-Lucas partial words. At the end of the study, we have concluded that the strong periodicity, compatibility and containment properties of Fibonacci-Lucas partial words are somehow connected with each other. Aside from that, we also observed the importance of holes while considering the relationships of the mentioned properties.

Keywords: Fibonacci-Lucas words, Fibonacci-Lucas partial words, strong periodicity, compatibility, containment

The power Fibonacci sequence in ℤm or ℤm[√δ]  where δ is a square-free integer is defined as a Fibonacci sequence Fn where F0 = 1 and F1 = a for some a ∈ ℤm or a ∈ {x + y√δ| x,y ∈ ℤm} such that Fn ≡ an (mod m), for all n ∈ ℕ ∪ {0}. The Horadam sequence, on the other hand, is a generalization of the Fibonacci sequence, defined Hn = uHn-1 + vHn-2 for n ≥ 2, where H0 = a and H1 = b, such that a, b are real numbers, and u, v are nonzero numbers. In this study, we investigated the existence of power Horadam sequences in ℤm and ℤm[√d] where d is a square-free integer and u2 + 4v is odd, as well as the number of such sequences for a given m. We proved that with specific conditions, and if m is odd such that m = p1q1p2q2...pkqk, where p is an odd prime, then (N,u,v,ℤm[√d]) can either be 2|s|, (2pnμn∕2) · (2piμi) · 2|s|, or 0. Similarly, if m is even such that m = 2rp1q1p2q2...pkqk, where r ≥ 1, p is an odd prime, and u is odd, then (N,u,v,ℤm[√d]) can either be 2|s|+1, (2pnμn∕2) · (2piμi) · 2|s|+1, or 0 for n = 1,...,t, i = t+1,...,l, j = l+1,...,s, such that k = |t|+|l|+|s|; t,l,s ≥ 0. Applying the concepts of the Legendre Symbol, Quadratic residue, Hensel's Lemma, and the three (3) propositions and seventeen (17) lemmas, the main goal was accomplished.

Keywords: Horadam sequence, Fibonacci sequence, Legendre symbol, quadratic residue, modulo 

 The concept of prime and composite numbers is important in the field of mathematics and cryptography. In this paper, we introduce a new method to determine the sufficient and necessary conditions that determine the compositeness of Mersenne number Mp of the form 2p-1 when p is prime. Using quadratic equation, Vieta's formulas, and Fermat's Little Theorem, the results reveal that for p≥11, Mp is composite if and only if there exists s ∈ ℕ, more specifically, s≤(√(2p-1)+1)∕2p + 1∕p satisfying the modular equation (sp+1)2 - 2p-2 ≡ 0 (mod 2sp+1)  given that (sp+1)2 - 2p-2<0. Moreover, Mp is composite if there exists an integer λ  within ( ((p+1)2  - 2p-2 ) ∕ (2p+1),0 ) such that (2λ-1)2+Mp is a perfect square. Corollaries were also developed to improve the results by eliminating values of λ that do not satisfy the given condition. These findings offer significant advancement in identifying composite Mersenne numbers and contribute to the broader understanding of number theory.

Keywords: Mersenne number, composite numbers, prime numbers

Research on sigma coloring within the complements of graphs in a complete graph offers profound insights for future exploration. This study investigates the effects of edge deletions and vertex coloring on the complement of the bi-star graph within a complete graph, significantly advancing graph coloring theory. By analyzing the sigma chromatic number of these complements, this research establishes foundational insights and practical applications in network design and traffic management. Building upon prior studies, which sparked initial explorations into sigma chromatic numbers, our investigation focuses on vertex coloring where each vertex’s “color sum” reflects the sum of colors assigned to its adjacent vertices. We explore the sigma chromatic number σ(G) as the minimum number of colors required to ensure adjacent vertices have distinct color sums. Our findings contribute new insights into sigma coloring within the complement of bi-star graphs B̅l,m,n, enhancing our understanding of complex graph structures and their practical implications.

Keywords: sigma coloring, bi-star graph, complement of a graph, complete graph

A prime divisor graph denoted as Gs originated from set S = nk+1 is a graph containing prime vertices, and will only form an edge when the common multiple of two vertices is in the set S. A set D(Gs) is said to be a dominating set of Gs if D⊆ V(Gs) and every vertex not in D or V(Gs)\D is adjacent to at least one vertex in D. A relatively prime dominating set of Gs, denoted as Drp(Gs), is a dominating set of the connected graph of Gs such that all pairs of vertices u,v∈D(Gs), satisfy GCD(deg(u), deg(v)) = 1, and its minimum cardinality is called the relatively prime domination number denoted as γrp(Gs) .This study presents the asymptotic behavior of relatively prime dominating sets by studying the degrees of the prime divisor graph and illustrates its behavior as the number of vertices increases.

Keywords: prime divisor graphs, relatively prime domination number, relatively prime dominating set, asymptotic behavior