Research Papers of
BS Mathematics 4-1

This paper used the Adomian Decomposition Method (ADM) to generate approximate solutions for modified Cauchy-Euler differential equations. This method provides the necessary conditions as it is applied to the modified Cauchy-Euler differential equations. An example is presented to test the ADM’s accuracy and efficiency. ADM is one of the valuable tools in mathematical analysis as it can provide versatile solutions for differential equations.

Keywords: Adomian decomposition method, Cauchy-Euler differential equation, modified Cauchy-Euler differential equation.

In this paper, we give some characterization of V-regular semigroup by restricting the C-set of 𝓠-regular semigroup to the set of idempotents. Furthermore, by restricting to the set of idempotents, we construct the surjective V-subhomomorphism which will be used in showing the V-subdirect product. Consequently, we construct E-unitary cover for V-regular semigroup. As a result, the E-unitary cover for V-regular cover will require lesser conditions making every E-unitary cover a subdirect product for V-regular semigroup.

Keywords: V-regular semigroup, C-set, 𝓠-regular semigroup, V-subhomomorphism, V-subdirect product, E-unitary cover

In this paper, we explore the concept of largest order maximal sum-free sets in ℤn, with the main purpose of applying and validating the findings from Renato de Amorim's study entitled "On Sum-Free Subsets of Abelian Groups". Here, categorizing  ℤn into three types based on divisibility properties is necessary to identify patterns. Since we have partitioned ℤn into three types, we found out that these types have certain set generators. We present that the largest order maximal sum-free sets can be characterized if and only if S(ℤn)=mS where m≤n, gcd(m,n)=1,|S|=μ(G)|G| and S follows the lemmas for each type. Additionally, using Euler's totient function, a sharp upper bound for the number of largest order maximal sum-free sets in ℤn is established.

Keywords: largest order maximal sum-free sets, Euler's Totient function, ℤn

A 3-maniplex, or a maniplex of rank 3, is defined by a connected properly 3-colored simple graph where each vertex, also called a flag, is incident to exactly three edges of distinct colors 0, 1, and 2. The concern of polytopality, which involves having the properties of an abstract polytope, arises when performing operations such as mixing. The mix of polytopes either constructs another polytope or does not. This mixing operation is analogous to the parallel product of maps, where the induced poset of maps must exhibit the properties of an abstract polyhedron to become polytopal. The mix of maniplexes generalizes both the mix of polytopes and the parallel product of maps. This study will determine the necessary and sufficient condition for a mix of 3-maniplexes to be polytopal by utilizing the Component Intersection Property (CIP) of Garza-Vargas and Hubard. By establishing this condition, we classify maniplexes according to polytopality.

Keywords: 3-maniplex, flags, maps, abstract n-polytopes, polytopality, CIP

A k-orbit polyhedron is a special type of abstract polytope defined by the group action acting on its flags. In this paper, we prove the existence of k-orbit polyhedron for every even integer k ≥ 4 given that every face is at least a triangle. This investigation is limited to three-dimensional spaces and cases where polyhedra exhibit non-degeneracy. Through concepts of combinatorial techniques and theorems of group theory, it is revealed that the number of flags of a polyhedron can be partitioned into disjoint k equivalence classes with equal sizes, leading to the derivation of the k flag orbit equation. Subsequently, we have shown the existence of even k-orbit polyhedra through explicit constructions subdivided into cases where k = 4; k = 6; k ≡ 0 (mod 4), and k ≡ 2 (mod 4), which makes up the even integer k ≥ 4. Having this study as supplementary to precedent studies, we found that k-orbit n-polytopes exist for every integer k for n = 3. Our study further showed that k-orbit polyhedra for even k ≥ 4 are not unique, indicating that there exists an infinite number of such polyhedra.

Keywords: polyhedra, non-degenerate polyhedra, group action, flag orbit, k-orbit polyhedron

Power Fibonacci Sequence is defined as Fibonacci sequence G with G0 = 1 and G1 = a, where a must be a root of f(x) = x² − x − 1 (mod m), such that for some modulus m, G ≡ 1, a, a², a³,... (mod m). In this paper, we investigated the existence of k-triad sequence on power Fibonacci sequences modulo m, where k-triad is defined as given any integer k, the three numbers a1, a2, a3 are said to form a k-triad, if the numbers a1a2 + k, a1a3 + k, a2a3 + k are all perfect squares. By determining its necessary and sufficient conditions, we proved that letting the sequence Gn ≡ an (mod m) be a power Fibonacci sequence with a = u + 2, then the subsequence {G2n} is a k-triad sequence if and only if k = u² + 3u + 1; and letting sequence Gn ≡ an (mod m) be a power Fibonacci sequence with a = u + 2, then the subsequence {G2n+1} is a k-triad sequence if and only if k = −u² − 3u − 1. Using the concept of Fibonacci sequence, k-triads, modulo m, discovered findings by Hoggatt and Bergum, Cassini’s identity, and the two (2) theorems we proved, the main goal was attained.

Keywords: Fibonacci sequence, Power Fibonacci sequence, k-triads, modulo m

This study focuses on the study of the dominating sets existing on Generalized Ottomar graphs. Generalized Ottomar graph is made up of a heart, feet, and a  path. A heart is the center of the Okn,m  which consist of n vertices. The number of feet is the same as the number of vertices of heart connected by a simple path. In order to study this graph, we cover the dominating sets in specific types: Perfect, Total, and Equitable domination, and these holds due to the results presented by Haynes T. W., Hedetniemi, S. and Slater, P.. In this paper, we apply some of the methods introduced by Cockayne, E. J. and Hedetniemi, S. T.  and Mark Caay and Esperanza Arugay. We compute for the dominating numbers with respect to the types of domination whereas varies depends the definition. Note that we use as well the paper of Mr. Mark L. Caay to compute the dominating numbers for perfect equitable. Domination on Graphs and tables for dominating sets are presented. At the end of the paper, we show that the the domination existing on a simple ottomar graph can be extended to a generalized ottomar graph. In particular we show that the domination number for total and equitable is equal to the minimum number of adjacent vertices on the graphs which is also on the generalized ottomar graph.

Keywords: Ottomar graphs, dominations, graphs

A Diophantine equation is a polynomial equation that contains two or more unknown variables. The extension ring ℤm[√d] can be defined as the set {a + b√d: a,b ∈ ℤm}  where d is a square-free number. The paper by Arevalo et al. finds the number of solutions for the equation x3 = b in ℤp[i]. In this paper, we aim to generalize the work of Arevalo et al. by determining the number of solution for the equation xn = c in ℤm[√d], where c ∈ ℤm\{0}, x ∈ ℤm[√d]={a + b√d: a,b ∈ ℤm}, d is a square-free number, and n ∈ ℤ+. This study utilizes the concepts of the Legendre symbol, quadratic residue, quadratic non-residue, and propositions on finite fields. The study first proves lemmas and propositions about the nth roots of unity before obtaining the main theorems. We proved that for gcd(c,m)=1 and [c/m]n=1, the number of roots of the equation  xn = c is fn(ℤm[√d]), and if [c/m]n=0, then there is no solution. On the other hand, if gcd(c,m)≠1, the number of solutions of xn = c is exactly the same as yn=μ. Additionally, for m = p1k1p2k2 ... puku where ku≥1, we have fn(ℤm[√d]) = ∏ku=1(fn(ℤm[√d])). 

Keywords: diophantine, Legendre symbol, nth roots of unity, quadratic residue, quadratic nonresidue.

Superstochastic matrices are square matrices with non-negative entries where all its column, row, diagonal, and anti-diagonal sums are equal to 1. The numerical range of a matrix refers to the set of all complex numbers that can be formed from the matrix and vectors. In this paper, we analyzed the numerical range of superstochastic matrices under two cases based on the properties of the matrix entries: (I) identical entries and (II) distinct entries.

Keywords: superstochatic matrix, square matrix, numerical range