Mathematics Commons^™

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.

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.

Probabilistic Modeling Of Student Interactions During A Passing Period At The University Of Dayton, Allyson Pacifico

Honors Theses

The University of Dayton is composed of five colleges and schools: College of Arts and Sciences, School of Law, School of Business Administration, School of Education and Health Sciences, and School of Engineering. The University of Dayton is composed of about 11,000 students on campus who all have distinct class schedules and paths they take between their classes. In this study, I wanted to know the probability of meeting my friends with a different class schedule as I walk between classes. The data consisted of one to two students from each college, except for the School of Law, who documented …

Alcoholism: A Mathematical Model With Media Awareness Campaigns, Erik H. Ander, Zeynep Teymuroglu

Undergraduate Mathematics Day: Past Content

In this paper, we study how media awareness campaigns influence the spread and persistence of drinking behavior in a community. Here, we present a compartmental population model with an additional differential equation to describe the dynamics of media awareness campaigns in combating problem drinking ([10], [12], [21]). Our model indicates a basic reproductive number, R0, where there exists an asymptotically stable drinking-free equilibrium if R0 < 1, and a unique endemic state, which appears to be stable when R0 > 1. We found that the following two components affect the basic reproductive number: the strength of peer influence of problem drinkers on susceptibles and the average overall time spent in the problem drinking environment. Furthermore, …

How One’S Risk Preferences Affect Their Investment Decisions, Kari Hayes, Anna Petrick

Undergraduate Mathematics Day: Past Content

The purpose of our project was to display how our personal risk preferences affect our investment decisions, if we invested on two assets: one risky asset (stock) and one risk-free asset (bank account). We considered the problem in both discrete and continuous case. In particular, the stock price follows a multinomial tree in the discrete case; and follows a Geometric Brownian motion in the continuous case. We then found the expected value of the stocks at varying times. By setting what we expect our bank account to be at those times equal to these expected values, we solved for the …

Finite Sum Representations Of Elements In R And R2, Lewis T. Dominguez, Rachelle R. Bouchat

Undergraduate Mathematics Day: Past Content

In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a master’s student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2.

Mathematics With Only Rods, Jianqiao Mao, Zheng Yang

Undergraduate Mathematics Day: Past Content

We discuss in this expository paper the rod system used in ancient China based on the mathematical classic work of Sun Zi, with a focus on application to solving systems of linear equations. The mathematics involved is authentic and beautiful, and we believe it is also of interest from historical, cultural, and pedagogical perspectives.

Generalized Catalan Numbers And Objects: X; Y Equivalence Classes And Polyominoes, Emily S. Dautenhahn, Hannah E. Pieper

Undergraduate Mathematics Day: Past Content

No abstract provided.

Baseball: Defense Or No?, Jacob D. Stemmerich

Undergraduate Mathematics Day: Past Content

Defense wins championships, or so they say. How do baseball organizations find the right defenders to win games? FanGraphs has published a series of metrics that teams throughout Major Leage Baseball use to quantify players’ fielding prowess. Baseball analysts use Wins Above Replacement, WAR, to predict who should be the league most valuable player, MVP. This uses defensive metrics to quantify how many runs the player produces when the team wins. The paper will discuss the metrics that already exist, and the technology that has been developed to analyze these metrics and other measurements of a player’s defensive skills.

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, …

Magic Polygons And Their Properties, Victoria Jakicic, Rachelle Bouchat

Undergraduate Mathematics Day: Past Content

Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define magic polygons. Furthermore, this paper will examine the existence of magic polygons, along with several other properties inherent to magic polygons.

Math Department Newsletter, 2017, University Of Dayton. Department Of Mathematics

Department of Mathematics Newsletters

No abstract provided.

Conference Program, University Of Dayton

Summer Conference on Topology and Its Applications

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications.

On Product Stability Of Asymptotic Property C, Gregory C. Bell, Andrzej Nagórko

Summer Conference on Topology and Its Applications

Asymptotic property C is a dimension-like large-scale invariant of metric spaces that is of interest when applied to spaces with infinite asymptotic dimension. It was first described by Dranishnikov, who based it on Haver's topological property C. Topological property C fails to be preserved by products in very striking ways and so a natural question that remained open for some 10+ years is whether asymptotic property C is preserved by products. Using a technique inspired by Rohm we show that asymptotic property C is preserved by direct products of metric spaces.

Domains And Probability Measures: A Topological Retrospective, Michael Mislove

Summer Conference on Topology and Its Applications

Domain theory has seen success as a semantic model for high-level programming languages, having devised a range of constructs to support various effects that arise in programming. One of the most interesting - and problematic - is probabilistic choice, which traditionally has been modeled using a domain-theoretic rendering of sub-probability measures as valuations. In this talk, I will place the domain-theoretic approach in context, by showing how it relates to the more traditional approaches such as functional analysis and set theory. In particular, we show how the topologies that arise in the classic approaches relate to the domain-theoretic rendering. We …

Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, Thomas Weighill, Jerzy Dydak

Summer Conference on Topology and Its Applications

Coarse geometry is the study of the large scale behaviour of spaces. The motivation for studying such behaviour comes mainly from index theory and geometric group theory. In this talk we introduce the notion of (hybrid) large scale normality for large scale spaces and prove analogues of Urysohn’s Lemma and the Tietze Extension Theorem for spaces with this property, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which …

Braid Group Actions On Rational Maps, Eriko Hironaka, Sarah Koch

Summer Conference on Topology and Its Applications

Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if …

Entropy In Topological Groups, Part 1, Dikran Dikranjan

Summer Conference on Topology and Its Applications

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another.

Roughly speaking, entropy is a non-negative real number or infinity assigned to a "selfmap" T of a "space" X, where the "space" X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The "selfmap" T can be, …

Pseudo-Contractibility, Felix Capulín, Leonardo Juarez-Villa, Fernando Orozco

Summer Conference on Topology and Its Applications

Let X, Y be topological spaces and let f, g:X→ Y be mappings, we say that f is pseudo-homotopic to g if there exist a continuum C, points a, b ∈ C and a mapping H:X ×C → Y such that H(x, a)=f(x) and H(x, b)=g(x) for each x ∈ X. The mapping H is called a pseudo-homotopy between f and g. A topological space X is said to be pseudo-contractible if the identity mapping is pseudo-homotopic to a constant mapping in X. i.e., if there exist a continuum C, points a, b ∈ C, x₀ ∈ X and …

Normal Images Of A Product And Countably Paracompact Condensation, Jila Niknejad

Summer Conference on Topology and Its Applications

In 1997, Buzjakova proved that for a pseudocompact Tychonoff space X and λ = | βX|⁺, X condenses onto a compact space if and only if X×(λ+1) condenses onto a normal space. This is a condensation form of Tamano's theorem. An interesting problem is to determine how much of Buzjakova's result will hold if "pseudocompact" is removed from the hypothesis.

In this talk, I am going to show for a Tychonoff space X, there is a cardinal λ such that if X×(λ+1) condenses onto a normal space, then X condenses onto a countably paracompact space.

On The Tightness And Long Directed Limits Of Free Topological Algebras, Gábor Lukács, Rafael Dahmen

Summer Conference on Topology and Its Applications

For a limit ordinal λ, let (A_α)_{α < λ} be a system of topological algebras (e.g., groups or vector spaces) with bonding maps that are embeddings of topological algebras, and put A = ∪_{α < λ} A_α. Let (A, T) and (A, A) denote the direct limit (colimit) of the system in the category of topological spaces and topological algebras, respectively. One always has T ⊇ A, but the inclusion may be strict; however, if the tightness of A is smaller than the cofinality of λ, then A=T.

In 1988, Tkachenko proved …

A New Class Of Dendrites Having Unique Second Symmetric Product, David Maya, José G. Anaya, Fernando Orozco Zitli

Summer Conference on Topology and Its Applications

The second symmetric product of a continuum X, F₂(X), is the hyperspace consisting of all nonempty subsets of X having at most two points. A continuum X has unique hyperspace F₂(X) provided that each continuum Y satisfying that F₂(X) and F₂(Y) are homeomorphic must be homeomorphic to X. In this talk, a new class of dendrites having unique F₂(X) will be presented.

On The Axiomatic Systems Of Steenrod Homology Theory Of Compact Spaces, Leonard Mdzinarishvili, Anzor Beridze

Summer Conference on Topology and Its Applications

The Steenrod homology theory on the category of compact metric pairs was axiomatically described by J.Milnor. In Milnor, the uniqueness theorem is proved using the Eilenberg-Steenrod axioms and as well as relative homeomorphism and clusres axioms. J. Milnor constructed the homology theory on the category Top²_C of compact Hausdorff pairs and proved that on the given category it satisfies nine axioms - the Eilenberg-Steenrod, relative homeomorphis and cluster axioms (see theorem 5 in Milnor). Besides, he proved that constructed homology theory satisfies partial continuity property on the subcategory Top²_CM (see theorem 4 in Milnor) and the …

On Roitman's Principle For Box Products, Hector Alonso Barriga-Acosta

Summer Conference on Topology and Its Applications

One of the oldest problems in box products is if the countable box product of the convergent sequence is normal. It is known that consistenly (e.g., b=d, d=c) the answer is affirmative. A recent progress is due to Judy Roitman that states a combinatorial principle which also implies the normality and holds in many models.

Although the countable box product of the convergent sequence is normal in some models of b < d < c, Roitman asked what happen with her principle in this models. We answer that Roitman's principle is true in some models of b < d < c.

A Compact Minimal Space Whose Cartesian Square Is Not Minimal, Jan P. Boronski, Alex Clark, Piotr Oprocha

Summer Conference on Topology and Its Applications

A compact metric space X is called minimal if it admits a minimal homeomorphism; i.e. a homeomorphism h:X→ X such that the forward orbit {hⁿ(x):n=1, 2, ...} is dense in X, for every x ∈ X. In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from "A compact minimal space Y such that its square YxY is not minimal" whose existence answer the following long standing problem in the negative.

Problem. Is minimality preserved under Cartesian product in the class of compact spaces?

Note that for the fixed point property this question …

Liouville Numbers And One-Sided Ergodic Hilbert Transformations, David Constantine, Joanna Furno

Summer Conference on Topology and Its Applications

We examine one-sided ergodic Hilbert transforms for irrational circle rotations and some mean-zero functions. Our approach uses continued fraction expansions to specify rotations by Liouville numbers for which the transformation has everywhere convergence or divergence.