Mathematics Commons^™

Lipschitz Restrictions Of Continuous Functions And A Simple Construction Of Ulam-Zahorski C1 Interpolation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We present a simple argument that for every continuous function f : R → R its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the C 1 free interpolation theorem, that for every continuous function f : R → R there exists a continuously differentiable function g : R → R which agrees with f on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with the Lebesgue measure theory.

Minimal Degrees Of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates, Krzysztof Ciesielski

Faculty & Staff Scholarship

A rational function of the form x α1 1 x α2 2 ⋯x αn n x β1 1 +x β2 2 +⋯+x βn n is a Genocchi-Peano example, GPE, provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree D(n) of a GPE of n-variables equals 2 ⌊ e 2 e2−1 n⌋ + 2i for some i ∈ {0, 1, 2}. We also investigate the minimal degree Db(n) of a bounded GPE of n-variables and note that D(n) ≤ Db(n) ≤ n(n + 1). Finding better bounds for numbers Db(n) remains an …

Bounded Point Derivations On Certain Function Spaces, Stephen Deterding

Theses and Dissertations--Mathematics

Let 𝑋 be a compact subset of the complex plane and denote by 𝑅^𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿^𝑝(𝑋) norm. We show that if a point 𝑥₀ admits a bounded point derivation on 𝑅^𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥₀. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if …

Polytopes Associated To Graph Laplacians, Marie Meyer

Theses and Dissertations--Mathematics

Graphs provide interesting ways to generate families of lattice polytopes. In particular, one can use matrices encoding the information of a finite graph to define vertices of a polytope. This dissertation initiates the study of the Laplacian simplex, P_G, obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. The Laplacian simplex is extended through the use of a parallel construction with a finite digraph D to obtain the Laplacian polytope, P_D.

Basic properties of both families of simplices, P_G and P …

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Theses and Dissertations--Mathematics

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H¹-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, …

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.

Analytical And Numerical Investigations Of The Kudryashov Generalized Kdv Equation, William Hilton

Electronic Theses and Dissertations

This thesis concerns an analytical and numerical study of the Kudryashov Generalized Korteweg-de Vries (KG KdV) equation. Using a refined perturbation expansion of the Fermi-Pasta-Ulam (FPU) equations of motion, the KG KdV equation, which arises at sixth order, and general higher order KdV equations are derived. Special solutions of the KG KdV equation are derived using the tanh method. A pseudospectral integrator, which can handle stiff equations, is developed for the higher order KdV equations. The numerical experiments indicate that although the higher order equations exhibit complex dynamics, they fail to reach energy equipartition on the time scale considered.

Analysis Of High Performance Scientific Programming Workflows, Withana Kankanamalage Umayanganie Klaassen

Open Access Theses & Dissertations

Substantial time is spent on building, optimizing and maintaining large-scale software that is run on supercomputers. However, little has been done to utilize overall resources efficiently when it comes to including expensive human resources. The community is beginning to acknowledge that optimizing the hardware performance such as speed and memory bottlenecks contributes less to the overall productivity than does the development lifecycle of high-performance scientific applications. Researchers are beginning to look at overall scientific workflows for high performance computing. Scientific programming productivity is measured by time and effort required to develop, configure, and maintain a simulation experiment and its constituent …

Engaging Students In The Practice Of Statistics Through Undergraduate Research, Debra L. Hydorn

Mathematics Articles

As statisticians, we engage in a variety of activities, some of which are regularly integrated into our undergraduate courses. However, the individual courses that comprise a mathematics or statistics degree program might not provide students with experiences in the broader range of activities that define the practice of statistics. To remedy this situation, faculty can consider developing and mentoring undergraduate research projects. This article briefly discusses the skills that comprise statistical practice along with some course and program options for helping students to develop these skills. Then, types of undergraduate research projects in statistics are described to help faculty generate …

Analysis Of A Vector-Borne Diseases Model With A Two-Lag Delay Differential Equation, Y. Qaddura, Nsoki Mavinga

Mathematics & Statistics Faculty Works

We are concerned with the stability analysis of equilibrium solutions for a two-lag delay differential equation which models the spread of vector-borne diseases, where the lags are incubation periods in humans and vectors. We show that there are some values of transmission and recovery rates for which the disease dies out and others for which the disease spreads into an endemic. The proofs of the main stability results are based on the linearization method and the analysis of roots of transcendental equations. We then simulate numerical solutions using MATLAB. We observe that the solution could possess chaotic and sometimes unbounded …

The Impact Of Spike-Frequency Adaptation On Balanced Network Dynamics, Victor J. Barranca, Han Huang , '19, Sida Li , '18

Mathematics & Statistics Faculty Works

A dynamic balance between strong excitatory and inhibitory neuronal inputs is hypothesized to play a pivotal role in information processing in the brain. While there is evidence of the existence of a balanced operating regime in several cortical areas and idealized neuronal network models, it is important for the theory of balanced networks to be reconciled with more physiological neuronal modeling assumptions. In this work, we examine the impact of spike-frequency adaptation, observed widely across neurons in the brain, on balanced dynamics. We incorporate adaptation into binary and integrate-and-fire neuronal network models, analyzing the theoretical effect of adaptation in the …

Dynamical Analysis Of An N−H−T Cosmological Quintessence Real Gas Model With A General Equation Of State, Emil Prodanov, Rossen Ivanov

Articles

The cosmological dynamics of a quintessence model based on real gas with general equation of state is presented within the framework of a three-dimensional dynamical system describing the time evolution of the number density, the Hubble parameter, and the temperature. Two global first integrals are found and examples for gas with virial expansion and van der Waals gas are presented. The van der Waals system is completely integrable. In addition to the unbounded trajectories, stemming from the presence of the conserved quantities, stable periodic solutions (closed orbits) also exist under certain conditions and these represent models of a cyclic Universe. …

Distribution Of Zeros Of Nondegenerate Functions On Short Cuttings Ii, Vasili Ivanovich Bernik, Natalia Viktorovna Budarina, Artyom Vadimovich Lunevich, Hugh O’Donnell

Articles

No abstract provided.

Catalogue Of Extreme Wave Events In Ireland: Revised And Updated For 14 680 Bp To 2017, Laura Cooke, Emiliano Renzi, John M. Dudley, Colm Clancy, FréDéRic Dias

Articles

This paper aims to extend and update the survey of extreme wave events in Ireland that was previously carried out by O’Brien et al. (2013). The original catalogue high- lighted the frequency of such events dating back as far as the turn of the last ice age and as recent as 2012. Ireland’s marine territory extends far beyond its coastline and is one of the largest seabed territories in Europe. It is therefore not surprising that extreme waves have continued to occur reg- ularly since 2012, particularly considering the severity of weather during the winters of 2013–2014 and 2015–2016. In …

Resale Value Of Individual Sets Of Magic The Gathering, Nichole Bolas Mlodzik

Honors Program Theses

Magic: the Gathering is a strategy-based fantasy card game developed by Richard Garfield in 1993. The appeal of playing the game is found in using cardboard printed cards to cast spells to achieve a victory over one or more opponents. This thesis will provide an analysis of six MTG sets, calculating the average value of one pack of fourteen cards. The relationships between the average resale value for each set will be evaluated using statistical graphical analysis. Conclusions will be drawn regarding the average resale value of a pack of fourteen cards in each of six sets

Symmetry Of Numerical Range And Semigroup Generation Of Infinite Dimensional Hamiltonian Operators, Junjie Huang, Jie Liu, Alatancang Chen

Turkish Journal of Mathematics

This paper deals with the infinite dimensional Hamiltonian operator with unbounded entries. Using the core of its entries, we obtain the conditions under which the numerical range of such an operator is symmetric with respect to the imaginary axis. Based on the symmetry above, a necessary and sufficient condition for generating $C_0$ semigroups is further given.

An Operational Matrix Method For Solving Linear Fredholm‒Volterra Integro-Differential Equations, Şuayi̇p Yüzbaşi, Nurbol Ismailov

Turkish Journal of Mathematics

The aim of this paper is to propose an efficient method to compute approximate solutions of linear Fredholm‒Volterra integro-differential equations (FVIDEs) using Taylor polynomials. More precisely, we present a method based on operational matrices of Taylor polynomials in order to solve linear FVIDEs. By using the operational matrices of integration and product for the Taylor polynomials, the problem for linear FVIDEs is transformed into a system of linear algebraic equations. The solution of the problem is obtained from this linear system after the incorporation of initial conditions. Numerical examples are presented to show the applicability and the efficiency of the …

An Effective Application Of Differential Quadrature Method Based Onmodified Cubic B-Splines To Numerical Solutions Of The Kdv Equation, Ali̇ Başhan

Turkish Journal of Mathematics

In this study, numerical solutions of the third-order nonlinear Korteweg--de Vries (KdV) equation are obtained via differential quadrature method based on modified cubic B-splines. Five different problems are solved. To show the accuracy of the proposed method, $L_{2}$ and $L_{\infty }$ error norms of the problem, which has an analytical solution, and three lowest invariants are calculated and reported. The obtained solutions are compared with some earlier works. Stability analysis of the present method is also given.

Applications Of Extended Watson's Summation Theorem, Young Sup Kim, Sebastien Gaboury, Arjun Kumar Rathie

Turkish Journal of Mathematics

In this research paper, several interesting applications of the extended classical summation theorem are given. As special cases, we recover several known results available in the literature.

Joint Densities Of Hitting Times For Finite State Markov Processes, Tomasz R. Bielecki, Monique Jeanblanc, Ali̇ Devi̇n Sezer

Turkish Journal of Mathematics

For a finite state Markov process $X$ and a finite collection $\{ \Gamma_k, k \in K \}$ of subsets of its state space, let $\tau_k$ be the first time the process visits the set $\Gamma_k$. In general, $X$ may enter some of the $\Gamma_k$ at the same time and therefore the vector $\bm\tau :=(\tau_k, k \in K)$ may put nonzero mass over lower dimensional regions of ${\mathbb R}_+^{ K }$; these regions are of the form $R_s=\{{\bm t} \in {\mathbb R}_+^{ K }: t_i = t_j, ~~i,j \in s(1) \} \cap \bigcap_{l=2}^{ s } \{{\bm t}:t_m < t_i = t_j,~~ i,j \in s(l), m \in s(l-1) \}$ where $s$ is any ordered partition of the set $K$ and $s(j)$ denotes the $j^{th}$ subset of $K$ in the partition $s$. When $ s < K $, the density of the law of $\bm\tau$ over these regions is said to be ``singular'' because it is with respect to the $ s $-dimensional Lebesgue measure over the region $R_s.$ We derive explicit/recursive and simple to compute formulas for these singular densities and their corresponding tail probabilities over all $R_s$ as $s$ ranges over ordered partitions of $K$. We give a numerical example and indicate the relevance of our results to credit risk modeling.

The Natural Brackets On Couples Of Vector Fields And $1$-Forms, Miroslav Doupovec, Jan Kurek, Wlodzimierz Mikulski

Turkish Journal of Mathematics

All natural bilinear operators transforming pairs of couples of vector fields and $1$-forms into couples of vector fields and $1$-forms are found. All natural bilinear operators as above satisfying the Leibniz rule are extracted. All natural Lie algebra brackets on couples of vector fields and $1$-forms are collected.