Physical Sciences and Mathematics Commons^™

Digital Simulations For Grade 7 To 10 Mathematics, Ma. Louise Antonette N. De Las Peñas, Debbie Marie Verzosa, Maria Alva Q. Aberin, Len Patrick Dominic M. Garces, Flordeliza F. Francisco, Evangeline P. Bautista, Mark Anthony C. Tolentino, Winfer C. Tabares

Mathematics Faculty Publications

This article describes a Department of Science and Technology – Philippine Council for Industry, Energy and Emerging Technology (DOST-PCIEERD) project aimed to facilitate the implementation of the mathematical objectives raised by the Department of Education’s (DepEd) K to 12 program in the Philippines through the use of innovative digital technologies. In particular, a selection of application software (“apps”) were created for Grade 7 to 10 mathematics that covered topics indicated in the five strands outlined in the K to 12 program – namely (1) number, (2) geometry, (3) measurement, (4) patterns and algebra, and (5) statistics and probability. The design …

A Generalization Of Schroter's Formula To George Andrews, On His 80th Birthday, James Mclaughlin

Mathematics Faculty Publications

We prove a generalization of Schroter's formula to a product of an arbitrary number of Jacobi triple products. It is then shown that many of the well-known identities involving Jacobi triple products (for example the Quintuple Product Identity, the Septuple Product Identity, and Winquist's Identity) all then follow as special cases of this general identity. Various other general identities, for example certain expansions of (q; q)(infinity) and (q; q)(infinity)(k), k >= 3, as combinations of Jacobi triple products, are also proved.

Quasilinearization And Boundary Value Problems At Resonance, Kareem Alanazi, Meshal Alshammari, Paul W. Eloe

Mathematics Faculty Publications

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Local And Global Color Symmetries Of A Symmetrical Pattern, Ma. Louise Antonette N. De Las Peñas, Agatha Kristel Abila, Eduard C. Taganap

Mathematics Faculty Publications

This study addresses the problem of arriving at transitive perfect colorings of a symmetrical pattern P consisting of disjoint congruent symmetric motifs. The pattern P has local symmetries that are not necessarily contained in its global symmetry group G. The usual approach in color symmetry theory is to arrive at perfect colorings of P ignoring local symmetries and considering only elements of G. A framework is presented to systematically arrive at what Roth [Geom. Dedicata (1984), 17, 99–108] defined as a coordinated coloring of P, a coloring that is perfect and transitive under G, satisfying the condition that the coloring …

Two Dimensional Search Algorithms For Linear Programming, Fabio Torres Vitor

Mathematics Faculty Publications

Linear programming is one of the most important classes of optimization problems. These mathematical models have been used by academics and practitioners to solve numerous real world applications. Quickly solving linear programs impacts decision makers from both the public and private sectors. Substantial research has been performed to solve this class of problems faster, and the vast majority of the solution techniques can be categorized as one dimensional search algorithms. That is, these methods successively move from one solution to another solution by solving a one dimensional subspace linear program at each iteration. This dissertation proposes novel algorithms that move …

Mittag–Leffler Stability Of Systems Of Fractional Nabla Difference Equations, Paul W. Eloe, Jaganmohan Jonnalagadda

Mathematics Faculty Publications

Mittag-Leffler stability of nonlinear fractional nabla difference systems is defined and the Lyapunov direct method is employed to provide sufficient conditions for Mittag-Leffler stability of, and in some cases the stability of, the zero solution of a system nonlinear fractional nabla difference equations. For this purpose, we obtain several properties of the exponential and one parameter Mittag-Leffler functions of fractional nabla calculus. Two examples are provided to illustrate the applicability of established results.

A Randomized Controlled Trial: Attachment-Based Family And Nondirective Supportive Treatments For Youth Who Are Suicidal, Guy S. Diamond, Roger R. Kobak, E. Stephanie Krauthamer Ewing, Suzanne A. Levy, Joanna L. Herres, Jody M. Russon, Robert J. Gallop

Mathematics Faculty Publications

Objective: To evaluate the efficacy of attachment-based family therapy (ABFT) compared with a family-enhanced nondirective supportive therapy (FE-NST) for decreasing adolescents’ suicide ideation and depressive symptoms. Method: A randomized controlled trial of 129 adolescents who are suicidal ages 12- to 18-years-old (49% were African American) were randomized to ABFT (n ¼ 66) or FE-NST (n ¼ 63) for 16 weeks of treatment. Assessments occurred at baseline and 4, 8, 12, and 16 weeks. Trajectory of change and clinical recovery were calculated for suicidal ideation and depressive symptoms. Results: There was no significant between-group difference in the rate of change in …

A Zero-Suppressed Binary Decision Diagram Approach For Constrained Path Enumeration, Renzo Roel P. Tan, Jun Kawahara, Agnes Garciano, Immanuel Sin

Mathematics Faculty Publications

Combinatorial optimization over graphs has been the subject of research. Recently, the solution of such problems by enumeration using a compact data structure called the zero-suppressed binary decision diagram was proposed and studied. The paper augments the existing frontier-based search method of construction and puts forth a technique for accommodating additional constraints during computation. The shortest and longest path problems for the Osaka Metro transit network are simultaneously solved as demonstration. Furthermore, a comparison of the approach with a conventional integer programming method is presented towards justifying the effectiveness of the algorithm.

Weighted Composition Operators On The Hilbert Hardy Space Of A Half-Plane, Valentin Matache

Mathematics Faculty Publications

Operators of type f→ψf∘ϕ acting on function spaces are called weighted composition operators. If the weight function ψ is the constant function 1, then they are called composition operators. We consider weighted composition operators acting on the Hilbert Hardy space of a half-plane and study compactness, boundedness, invertibility, normality and spectral properties of such operators.

Impartial Achievement Games For Generating Nilpotent Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben

Mathematics Faculty Publications

We study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form T×H, where T is a 2-group and H is a group of odd order. This includes all nilpotent and hence abelian groups.

Isolating And Quantifying The Role Of Developmental Noise In Generating Phenotypic Variation, Maria Kiskowski, Tilmann Glimm, Nickolas Moreno, Tony Gamble, Ylenia Chiari

Mathematics Faculty Publications

Genotypic variation, environmental variation, and their interaction may produce variation in the developmental process and cause phenotypic differences among individuals. Developmental noise, which arises during development from stochasticity in cellular and molecular processes when genotype and environment are fixed, also contributes to phenotypic variation. While evolutionary biology has long focused on teasing apart the relative contribution of genes and environment to phenotypic variation, our understanding of the role of developmental noise has lagged due to technical difficulties in directly measuring the contribution of developmental noise. The influence of developmental noise is likely underestimated in studies of phenotypic variation due to …

Quasilinearization And Boundary Value Problems At Resonance For Caputo Fractional Differential Equations, Saleh S. Almuthaybiri, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

The quasilinearization method is applied to a boundary value problem at resonance for a Caputo fractional differential equation. The method of upper and lower solutions is first employed to obtain the uniqueness of solutions of the boundary value problem at resonance. The shift argument is applied to show the existence of solutions. The quasilinearization algorithm is then developed and sequences of approximate solutions are constructed that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two applications are provided to illustrate the main results.

Avery Fixed Point Theorem Applied To A Hammerstein Integral Equation, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

Abstract. We apply a recent Avery et al. fixed point theorem to the Hammerstein integral equation (see paper for equation). Under certain conditions on G, we show the existence of positive and positive symmetric solutions. Examples are given where G is a convolution kernel and where G is a Green’s function associated with different boundary-value problem.

Bochner-Kähler And Bach Flat Manifolds, Amalendu Ghosh, Ramesh Sharma

Mathematics Faculty Publications

We have classified Bochner-Kähler manifolds of real dimension > 4, which are also Bach flat. In the 4-dimensional case, we have shown that, if the scalar curvature is harmonic, then it is constant. Finally, we show that the gradient of scalar curvature of any Bochner-Kähler manifold is an infinitesimal harmonic transformation, and if it is conformal then the scalar curvature is constant.

K -Isocoronal Tilings, Eduard C. Taganap, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

In this article, a framework is presented that allows the systematic derivation of planar edge-to-edge k-isocoronal tilings from tile-s-transitive tilings, s k. A tiling T is k-isocoronal if its vertex coronae form k orbits or k transitivity classes under the action of its symmetry group. The vertex corona of a vertex x of T is used to refer to the tiles that are incident to x. The k-isocoronal tilings include the vertex-k-transitive tilings (k-isogonal) and k-uniform tilings. In a vertex-k- transitive tiling, the vertices form k transitivity classes under its symmetry group. If this tiling consists of regular polygons then …

Graceful Labeling Of Triangular Extension Of Complete Bipartite Graph, Sarbari Mitra, Soumya Bhoumik

Mathematics Faculty Publications

For positive integers m, n, K m,n represents the complete bipartite graph. We name the graph G = K m,n ⊙ K2 as triangular extension of complete bipartite graph K m,n , since there is a triangle hanging from every vertex of K m,n . In this paper we show that G is graceful when m = n = 2ℓ, for any integer ℓ.

L(2, 1)-Labeling Of Circulant Graphs, Sarbari Mitra, Soumya Bhoumik

Mathematics Faculty Publications

An L(2, 1)-labeling of a graph Γ is an assignment of non-negative integers to the vertices such that adjacent vertices receive labels that differ by at least 2, and those at a distance of two receive labels that differ by at least one. Let λ12(Γ) denote the least λ such that Γ admits an L(2, 1)-labeling using labels from {0, 1, . . ., λ}. A Cayley graph of group G is called a circulant graph of order n, if G = Zn. In this paper initially we investigate the upper bound for the span of the L(2, 1)-labeling for …

Quasilinearization And Boundary Value Problems For Riemann-Liouville Fractional Differential Equations, Paul W. Eloe, Jaganmohan Jonnalagadda

Mathematics Faculty Publications

We apply the quasilinearization method to a Dirichlet boundary value problem and to a right focal boundary value problem for a RiemannLiouville fractional differential equation. First, we sue the method of upper and lower solutions to obtain the uniqueness of solutions of the Dirichlet boundary value problem. Next, we apply a suitable fixed point theorem to establish the existence of solutions. We develop a quasilinearization algorithm and construct sequences of approximate solutions that converge monotonically and quadratically to the unique solution of the boundary value problem. Two examples are exhibited to illustrate the main result for the Dirichlet boundary value …

Comparison Of Green's Functions For A Family Of Boundary Value Problems For Fractional Difference Equations, Paul W. Eloe, Catherine Kublik, Jeffrey T. Neugebauer

Mathematics Faculty Publications

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.

The Large Contraction Principle And Existence Of Periodic Solutions For Infinite Delay Volterra Difference Equations, Paul W. Eloe, Jaganmohan Jonnalagadda, Youssef Raffoul

Mathematics Faculty Publications

In this article, we establish sufficient conditions for the existence of periodic solutions of a nonlinear infinite delay Volterra difference equation. (See paper for equation.)

We employ a Krasnosel’skii type fixed point theorem, originally proved by Burton. The primary sufficient condition is not verifiable in terms of the parameters of the difference equation, and so we provide three applications in which the primary sufficient condition is verified.

Analysis Of Optimal Superconvergence Of A Local Discontinuous Galerkin Method For Nonlinear Second-Order Two-Point Boundary-Value Problems, Mahboub Baccouch

Mathematics Faculty Publications

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1toward the derivatives of Gauss-Radau projections of …