Mathematics Commons^™

2016 (Spring), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2016 Spring Colloquium.

Domain Representability And Topological Completeness, Matthew D. Devilbiss

Honors Theses

Topological completeness properties seek to generalize the definition of complete metric space to the context of topologies. Chapter 1 gives an overview of some of these properties. Chapter 2 introduces domain theory, a field originally intended for use in theoretical computer science. Finally, Chapter 3 examines how this computer-scientific notion can be employed in the study of topological completeness in the form of domain representability. The connections between domain representability and other topological completeness properties are subsequently examined.

Asymptotically Periodic Solutions Of Volterra Integral Equations, Muhammad Islam

Mathematics Faculty Publications

We study the existence of asymptotically periodic solutions of a nonlinear Volterra integral equation. In the process, we obtain the existence of periodic solutions of an associated nonlinear integral equation with infinite delay. Schauder's fixed point theorem is used in the analysis.

Smallest Eigenvalues For A Right Focal Boundary Value Problem, Paul W. Eloe, Jeffrey T. Neugebauer

Mathematics Faculty Publications

We establish the existence of smallest eigenvalues for the fractional linear boundary value problems D^α₀₊u+λ₁p(t)u = 0 and D^α₀₊u+λ₂q(t)u = 0, 0

Sphere Representations, Stacked Polytopes, And The Colin De Verdière Number Of A Graph, Lon Mitchell, Lynne Yengulalp

Mathematics Faculty Publications

We prove that a k-tree can be viewed as a subgraph of a special type of (k + 1)- tree that corresponds to a stacked polytope and that these “stacked” (k + 1)-trees admit representations by orthogonal spheres in R k+1. As a result, we derive lower bounds for Colin de Verdi`ere’s µ of complements of partial k-trees and prove that µ(G) + µ(G) > |G| − 2 for all chordal G.

Existence Of Periodic Solutions For A Quantum Volterra Equation, Muhammad Islam, Jeffrey T. Neugebauer

Mathematics Faculty Publications

The objective of this paper is to study the periodicity properties of functions that arise in quantum calculus, which has been emerging as an important branch of mathematics due to its various applications in physics and other related fields. The paper has two components. First, a relation between two existing periodicity notions is established. Second, the existence of periodic solutions of a q-Volterra integral equation, which is a general integral form of a first order q-difference equation, is obtained. At the end, some examples are provided. These examples show the effectiveness of the relation between the two periodicity notions that …

Positive Solutions For A Singular Fourth Order Nonlocal Boundary Value Problem, John M. Davis, Paul W. Eloe, John R. Graef, Johnny Henderson

Mathematics Faculty Publications

Positive solutions are obtained for the fourth order nonlocal boundary value problem, u⁽⁴⁾=f(x,u), 0 < x < 1, u(0) = u''(0) = u'(1) = u''(1) - u''(2/3)=0, where f(x,u) is singular at x = 0, x=1, y=0, and may be singular at y=∞. The solutions are shown to exist at fixed points for an operator that is decreasing with respect to a cone.

Almost Automorphic Solutions Of Delayed Neutral Dynamic Systems On Hybrid Domains, Murat Adıvar, Halis Can Koyuncuoğlu, Youssef Raffoul

Mathematics Faculty Publications

We study the existence of almost automorphic solutions of the delayed neutral dynamic system on hybrid domains that are additively periodic. We use exponential dichotomy and prove uniqueness of projector of exponential dichotomy to obtain some limit results leading to sufficient conditions for existence of almost automorphic solutions to neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of the coefficient matrices in the system. Hence, we significantly improve the results in the existing literature. Finally, we also provide an existence result for an almost periodic solutions of the system.

Stochastic Models Of Evidence Accumulation In Changing Environments, Alan Veliz-Cuba, Zachary P. Kilpatrick, Krešimir Josić

Mathematics Faculty Publications

Organisms and ecological groups accumulate evidence to make decisions. Classic experiments and theoretical studies have explored this process when the correct choice is fixed during each trial. However, we live in a constantly changing world. What effect does such impermanence have on classical results about decision making? To address this question we use sequential analysis to derive a tractable model of evidence accumulation when the correct option changes in time. Our analysis shows that ideal observers discount prior evidence at a rate determined by the volatility of the environment, and the dynamics of evidence accumulation is governed by the information …

Necessary And Sufficient Conditions For Stability Of Volterra Integro-Dynamic Equation Systems On Time Scales, Youssef Raffoul

Mathematics Faculty Publications

In this research we establish necessary and sufficient conditions for the stability of the zero solution of scalar Volterra integro-dynamic equation on general time scales. Our approach is based on the construction of suitable Lyapunov functionals. We will compare our findings with known results and provides application to quantum calculus.

From Subcompact To Domain Representable, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

No abstract provided.

2015 (Fall), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2015 Fall Colloquium.

Infographics And Mathematics: A Mechanism For Effective Learning In The Classroom, Ivan Sudakov, Thomas Bellsky, Svetlana Usenyuk, Victoria V. Polyakova

Physics Faculty Publications

This work discusses the creation and use of infographies in an undergraduate mathematics course. Infographies are a visualization of information combining data, formulas, and images. This article discusses how to form an infographic and uses infographics on topics within mathematics and climate as examples. It concludes with survey data from undergraduate students on both the general use of infographics and on the specific infographics designed by the authors.

2015 (Summer), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2015 Summer Colloquium.

Root Cover Pebbling On Graphs, Claire A. Sonneborn

Honors Theses

Consider a graph, G, with pebbles on its vertices. A pebbling move is defined to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The cover pebbling number of a graph, γ(G), is the minimum number of pebbles such that, given any configuration of γ(G) pebbles on the vertices of G, pebbling moves can be used to place one pebble on each vertex of G. We define the root vertex of a graph and fix an initial configuration of pebbles on G where we place all pebbles on the root …

2015 (Spring), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2015 Spring Colloquium.

Qualitative Theory Of Functional Differential And Integral Equations, Muhammad Islam, Cemil Tunc, Mouffak Benchohra, Bingwen Lui, Samir H. Saker

Mathematics Faculty Publications

Functional differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some varying quantities and their rates of change in space and/or time (expressed as derivatives or differences) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. In some cases, this differential equation (called an equation of motion) may be solved explicitly. In fact, differential equations play an important role in modelling virtually every physical, technical, biological, ecological, and epidemiological process, from celestial motion, to bridge design, …

Bounded, Asymptotically Stable, And L^1 Solutions Of Caputo Fractional Differential Equations, Muhammad Islam

Mathematics Faculty Publications

The existence of bounded solutions, asymptotically stable solutions, and L¹ solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. …

Sobriety In Delta Not Sober, Joe Mashburn

Mathematics Faculty Publications

We will show that the space delta not sober defined by Coecke and Martin is sober in the Scott topology, but not in the weakly way below topology.

Multi-Term Linear Fractional Nabla Difference Equations With Constant Coefficients, Paul W. Eloe, Zi Ouyang

Mathematics Faculty Publications

We shall consider a linear fractional nabla (backward) fractional difference equation of Riemann–Liouville type with constant coefficients. We apply a transform method to construct solutions. Sufficient conditions in terms of the coefficients are given so that the solutions are absolutely convergent. The method is known for two-term fractional difference equations; the method is new for fractional equations with three or more terms. As a corollary, we exhibit new summation representations of a discrete exponential function, at, t = 0; 1; : : : .

Sixteenth Kenneth C. Schraut Memorial Lecture (Poster), University Of Dayton. Department Of Mathematics

Kenneth C. Schraut Memorial Lectures

No abstract provided.

Riemannian Geometry (Abstract), Chikako Mese

Kenneth C. Schraut Memorial Lectures

Riemannian Geometry studies the geometry of curved spaces. It originated with the ideas of the Bernhard Riemann in the 19th century extending Gaussian geometry, or the study of geometry of curves and surfaces contained in 3 dimensional Euclidean space.

2015 Program And Abstracts, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

No abstract provided.

The Graph Theory Origin Story (Abstract), Daniel Roberts

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

Many research questions in pure mathematics arise from considerations of real world problems. Part of the job of a mathematician is to ask this type of question.

2015 Undergraduate Mathematics Day Poster, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

No abstract provided.

2014 (Fall), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2014 Fall Colloquium.

2014 (Summer), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2014 Colloquium.

2014 (Spring), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2014 Spring Colloquium.

A Generalization Of Poincaré-Cartan Integral Invariants Of A Nonlinear Nonholonomic Dynamical System, Muhammad Usman, M. Imran

Mathematics Faculty Publications

Based on the d'Alembert-Lagrange-Poincar\'{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincar\'{e}-Hamilton equations, and study a version of corresponding Poincar\'{e}-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincar\'{e} variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the …

Lyapunov Functionals That Lead To Exponential Stability And Instability In Finite Delay Volterra Difference Equations, Catherine Kublik, Youssef Raffoul

Mathematics Faculty Publications

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation.

Also, by displaying a slightly different Lyapunov functional, we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is the relaxing of the condition |a(t)| < 1. Moreover, we provide examples in which we show that our theorems provide an improvement of some recent results.