Mathematics Commons^™

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 …

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.

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.

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 …

Not So Many Non-Disjoint Translations, Andrzej Roslanowski, Vyacheslav V. Rykov

Mathematics Faculty Publications

We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.

On Robust Colorings Of Hamming-Distance Graphs, Isaiah Harney, Heide Gluesing-Luerssen

Mathematics Faculty Publications

H_q(n, d) is defined as the graph with vertex set Zⁿ_q and where two vertices are adjacent if their Hamming distance is at least d. The chromatic number of these graphs is presented for various sets of parameters (q, n, d). For the 4-colorings of the graphs H₂(n, n − 1) a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust 4-colorings of …

Marikina Flood Hazard Models Using Historical Data Of Water Level, Elvira P. De Lara-Tuprio, Evangeline P. Bautista, Reginaldo M. Marcelo, Ramil T. Bataller, Divino Angelo B. Esteban, Yvanne Paolo B. Yutuc

Mathematics Faculty Publications

In this paper, ten-year historical data of water levels recorded at Sto. Niño, Marikina station of MMDA-EFCOS were analysed and processed to determine the number of times per year (annual frequency) that critical levels of the Marikina River near the Sto. Niño station were reached and for how long (duration). Probability distributions for the annual frequency and duration were then fitted to the samples obtained. Monte Carlo simulation was applied in order to generate possible realizations of the random variables. Summary statistics were then obtained from the simulated values. Finally, backtesting using historical data of water levels after the period …

Identification Of Biologically Essential Nodes Via Determinative Power In Logical Models Of Cellular Processes, Trevor Pentzien, Bhanwar L. Puniya, Tomáš Helikar, Mihaela Teodora Matache

Mathematics Faculty Publications

A variety of biological networks can bemodeled as logical or Boolean networks. However, a simplification of the reality to binary states of the nodes does not ease the difficulty of analyzing the dynamics of large, complex networks, such as signal transduction networks, due to the exponential dependence of the state space on the number of nodes. This paper considers a recently introduced method for finding a fairly small subnetwork, representing a collection of nodes that determine the states of most other nodes with a reasonable level of entropy. The subnetwork contains the most determinative nodes that yield the highest information …

When Does The Bombieri–Vinogradov Theorem Hold For A Given Multiplicative Function?, Andrew Granville, Xuancheng Shao

Mathematics Faculty Publications

Let f and g be 1-bounded multiplicative functions for which f ✻ g = 1_.=1. The Bombieri–Vinogradov theorem holds for both f and g if and only if the Siegel–Walfisz criterion holds for both f and g, and the Bombieri–Vinogradov theorem holds for f restricted to the primes.

Gradient Ricci Solitons With A Conformal Vector Field, Ramesh Sharma

Mathematics Faculty Publications

We show that a connected gradient Ricci soliton (M,g,f,λ) with constant scalar curvature and admitting a non-homothetic conformal vector field V leaving the potential vector field invariant, is Einstein and the potential function f is constant. For locally conformally flat case and non-homothetic V we show without constant scalar curvature assumption, that f is constant and g has constant curvature.

Primitive Substitution Tilings With Rotational Symmetries, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Dirk Frettlöh

Mathematics Faculty Publications

This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.

Fractional Generalizations Of Zakai Equation And Some Solution Methods, Sabir Umarov, Fred Daum, Kenric Nelson

Mathematics Faculty Publications

The paper discusses fractional generalizations of Zakai equations arising in filtering problems. The derivation of the fractional Zakai equation, existence and uniqueness of its solution, as well as some methods of solution to the fractional filtering problem, including fractional version of the particle flow method, are presented.

Crystallographic Patterns In Philippine Indigenous Textiles, Ma. Louise Antonette N. De Las Peñas, Agnes Garciano, Debbie Marie Verzosa, Eduard C. Taganap

Mathematics Faculty Publications

The aim of this study was to analyze a representative sample of Philippine indigenous textiles in order to capture the range of symmetries and color symmetries present. This paper examines the existence of symmetries in finite designs, and classifies the plane-group and frieze-group symmetry types of the repeated patterns in woven textiles. The tendency of a particular symmetry to be more or less common than another can indicate relationships between the symmetries and the weaving technique or the culture that produced them. This paper will also examine designs and patterns with color symmetry found in these textiles. The sample consisted …

Rook Placements And Jordan Forms Of Upper-Triangular Nilpotent Matrices, Martha Yip

Mathematics Faculty Publications

The set of n by n upper-triangular nilpotent matrices with entries in a finite field 𝔽_q has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number F_λ(q) of matrices of Jordan type λ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young's lattice and non-attacking rook placements, which leads to a refinement of the formula for F_λ(q).

Initial Value Problems For Caputo Fractional Differential Equations, Paul W. Eloe, Tyler Masthay

Mathematics Faculty Publications

Let n ≥ 1 denote an integer and let n - 1 < α ≤ n: We consider an initial value problem for a nonlinear Caputo fractional differential equation of order α and obtain results analogous to well known results for initial value problems for ordinary differential equations. These results include Picard’s existence and uniqueness theorem, Peano’s existence theorem, extendibility of solutions to the right, maximal intervals of existence, a Kamke type convergence theorem, and the continuous dependence of solutions on parameters. The nonlinear term is assumed to depend on higher order derivatives and solutions are obtained in the space of n - 1 times continuously differentiable functions.

Concavity In Fractional Calculus, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

No abstract provided.

When Numerical Analysis Crosses Paths With Catalan And Generalized Motzkin Numbers, Paul W. Eloe, Catherine Kublik

Mathematics Faculty Publications

We study a linear doubly indexed sequence that contains the Catalan numbers and relates to a class of generalized Motzkin numbers. We obtain a closed form formula, a generating function and a nonlinear recursion relation for this sequence. We show that a finite difference scheme with compact stencil applied to a nonlinear differential operator acting on the Euclidean distance function is exact, and exploit this exactness to produce the nonlinear recursion relation. In particular, the nonlinear recurrence relation is obtained by using standard error analysis techniques from numerical analysis. This work shows a connection between numerical analysis and number theory, …

Global Dynamic Scenarios For Competitive Maps In The Plane, Elliott J. Bertrand, M.R.S. Kulenovic

Mathematics Faculty Publications

In this paper we present some global dynamic scenarios for general competitive maps in the plane. We apply these results to the class of second-order autonomous difference equations whose transition functions are decreasing in the variable \(x_{n}\) and increasing in the variable \(x_{n-1}\).We illustrate our results with the application to the difference equation ...

Global Dynamics Of Higher-Order Transcendental-Type Generalized Beverton–Holt Equations, Elliott J. Bertrand, M.R.S. Kulenovic

Mathematics Faculty Publications

We investigate generalized Beverton–Holt difference equations of order k.

A Supply Chain Profile Of A School-Based Feeding Program Using The Centralized Kitchen Model, Eden Delight Miro, J. Lemuel Martin, Leslie Lopez, Joselito Secson, Carmela Oracion, Jhoel Loanzon

Mathematics Faculty Publications

There has recently been renewed interest and a growing demand for school feeding programs. In the Philippines, the government, through the Department of Education (DepEd), Department of Social Welfare and Development (DSWD), and non-government organizations such as the Ateneo Center for Educational Development (ACED), has initiated such programs to address the prevalence of malnutrition among Filipino school-age children. In 2011, ACED introduced the ACED Blueplate Centralized Kitchen (ABCK) model for large-scale school feeding. This study aims to provide a supply chain profile of the first and largest city-wide implementation of the ABCK model in the Philippines to date, which is …

String C-Groups Of Order 1024, Yasushi Gomi, Mark L. Loyola, Ma. Louise Antonette N. De Las Peñas

Mathematics Faculty Publications

This paper determines the nondegenerate string C-groups of order 1024. For groups of rank 3, we use the technique of central extension of string C-groups of order 512. For groups of rank at least 4, we compute for quotients of universal string C-groups.

Some Remarks On Ricci Solitons, Ramesh Sharma, S Balasubramanian, N. Uday Kiran

Mathematics Faculty Publications

We obtain an intrinsic formula of a Ricci soliton vector field and a differential condition for the non-steady case to be gradient. Next we provide a condition for a Ricci soliton on a Kaehler manifold to be a Kaehler–Ricci soliton. Finally we give an example supporting the first result.

The Classification Of Infinite Abelian Groups With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language L∞ω, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in L∞ω.

Matching And Independence Complexes Related To Small Grids, Benjamin Braun, Wesley K. Hough

Mathematics Faculty Publications

The topology of the matching complex for the 2 x n grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes Ind(Δ^m_n) that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain Ind(Δ^m_n). Furthermore, we determine the Euler characteristic of Ind(Δ^m_n) and prove that several homology groups of Ind(Δ^m_n) are non-zero.

Equivariant Iterated Loop Space Theory And Permutative G–Categories, Bertrand J. Guillou, J. Peter May

Mathematics Faculty Publications

We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V–fold loop G–spaces to several avatars of a recognition principle for infinite loop G–spaces. We then explain what genuine permutative G–categories are and, more generally, what E_∞–G–categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt–Priddy–Quillen theorem as a statement about genuine G–spectra and use it to give a new, categorical proof of the tom Dieck splitting theorem …

Sasakian Manifolds With Purely Transversal Bach Tensor, Amalendu Ghosh, Ramesh Sharma

Mathematics Faculty Publications

We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature ≥2n(2n+1), equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S³. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π₁(M).

Matroid Configurations And Symbolic Powers Of Their Ideals, A. V. Geramita, B. Harbourne, J. Migliore, Uwe Nagel

Mathematics Faculty Publications

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid …

The Double Pivot Simplex Method, Fabio Vitor, Todd Easton

Mathematics Faculty Publications

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time …

Categorical Models For Equivariant Classifying Spaces, Bertrand J. Guillou, J. Peter May, Mona Merling

Mathematics Faculty Publications

Starting categorically, we give simple and precise models for classifying spaces of equivariant principal bundles. We need these models for work in progress in equi- variant infinite loop space theory and equivariant algebraic K–theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.

Challenges In Modeling Complexity Of Neglected Tropical Diseases: A Review Of Dynamics Of Visceral Leishmaniasis In Resource Limited Settings, Swati Debroy, Olivia F. Prosper, Austin Mishoe, Anuj Mubayi

Mathematics Faculty Publications

Objectives: Neglected tropical diseases (NTD), account for a large proportion of the global disease burden, and their control faces several challenges including diminishing human and financial resources for those distressed from such diseases. Visceral leishmaniasis (VL), the second-largest parasitic killer (after malaria) and an NTD affects poor populations and causes considerable cost to the affected individuals. Mathematical models can serve as a critical and cost-effective tool for understanding VL dynamics, however, complex array of socio-economic factors affecting its dynamics need to be identified and appropriately incorporated within a dynamical modeling framework. This study reviews literature on vector-borne diseases and collects …