Mathematics Commons^™

Exploring Diagonals In The Calkin-Wilf Tree, Matthew Gagne

Undergraduate Review

For centuries, people have been interested in patterns. Even in that which appears random, humans have been trying to understand the underlying order of things. Mathematicians throughout time have studied many phenomena, including infinite sequences of numbers and have been able, at times, to see structure. Many have found the satisfaction, even joy, of discovering patterns in sequences. A typical way to describe this is by a recursive formula. A recursive definition defines a term in the sequence using the previous terms in the sequence. Even more satisfying than a recursive formula is a closed formula. With this, one can …

The Fibonacci Sequence And Hosoya's Triangle, Jeffrey Lee Smith

Theses Digitization Project

The purpose of this thesis is to study the Fibonacci sequence in a context many are unfamiliar with. A triangular array of numbers, similar looking to Pascal's triangle, was constructed a few decades ago and is called Hosoya's triangle. Each element within the triangle is created using Fibonacci numbers.

Whitney's 2-Isomorphism Theorem For Hypergraphs, Eric Anthony Taylor

Theses Digitization Project

This study will examine a fundamental theorem from graph theory: Whitney's 2-Isomorphism Theorem. Whitney's 2-Isomorphism theorem characterizes when two graphs have isomorphic cycle matroids.

Comparing The Algebraic And Analytical Properties Of P-Adic Numbers With Real Numbers, Joseph Colton Wilson

Theses Digitization Project

This study will provide a glimpse into the world of p-adic numbers, which encompasses a different way to measure the distance between rational numbers. Simple calculations and surprising results are examined to help familiarize the reader to the new space.

A Study Of Finite Symmetrical Groups, Patrick Kevin Martinez

Theses Digitization Project

This study discovered several important groups that involve the classical and sporadic groups. These groups appeared as finite homomorphic images of the progenitors 3*8 : PGL₂(7), 2*¹⁴ : L₃ (2), 5*³ : S₃ and 7*2 : m S₃.

The Banach-Tarski Paradox, Matthew Jacob Norman

Theses Digitization Project

The purpose of this thesis is to establish the history and motivation leading up to the Banach-Tarski Paradox, as well as its proof. This study discusses the early history of set theory as it is documented as well as the necessary basics of set theory in order to further understand the contents within. Set theory not only proved to be for the mathematical at heart but also struck interest into the mind of philosophers, theologians, and logicians.

Behavior Of Solutions For Bernoulli Initial-Value Problems, Carlos Marcelo Sardan

Theses Digitization Project

The purpose of this project is to investigate blow-up properties of solutions for specific initial-value problems that involve Bernoulli Ordinary Differential Equations (ODE's). The objective is to find conditions on the coefficients and on the initial-values that lead to unbounded growth of solutions in finite time.

Complex Dynamics In Predator-Prey Models With Nonmonotonic Functional Response And Seasonal Harvesting, Jicai Huang, Jing Chen, Yijun Gong, Weipeng Zhang

Mathematics Faculty Articles

In this paper we study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant-yield for prey, it is shown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established by numerical simulations. When the harvesting is seasonal for both species, sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. …

Review: Cyclicity Results For Some Antianalytic Toeplitz Operators Acting On H^P, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

The Graphic Nature Of Gaussian Periods, William Duke, Stephan Ramon Garcia, Bob Lutz '13

Pomona Faculty Publications and Research

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework with which to study the properties of certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, this approach reveals that these classical objects display a dazzling array of visual patterns of great complexity and remarkable subtlety.

Curvelets And The Radon Transform, Jill Dickerson

Electronic Theses and Dissertations

Computed Tomography (CT) is the standard in medical imaging field. In this study, we look at the curvelet transform in an attempt to use it as a basis for representing a function. In doing so, we seek a way to reconstruct a function from the Radon data that may produce clearer results. Using curvelet decomposition, any known function can be represented as a sum of curvelets with corresponding coefficients. It can be shown that these corresponding coefficients can be found using the Radon data, even if the function is unknown. The use of curvelets has the potential to solve partial …

Smooth And Non-Smooth Traveling Wave Solutions Of Some Generalized Camassa-Holm Equations, Taslima Rehman

Electronic Theses and Dissertations

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, …

A Reduction Theorem For The Kripke-Joyal Semantics: Forcing Over An Arbitrary Category Can Always Be Replaced By Forcing Over A Complete Heyting Algebra, Imants Barušs, Robert Woodrow

Psychology

No abstract provided.

Problem Solving And Its Elements In Forming Proof, Joanna Mamona-Downs, Martin Downs

The Mathematics Enthusiast

The character of the mathematics education traditions on problem solving and proof are compared, and aspects of problem solving that occur in the processes of forming a proof, which are not well represented in the literature, are portrayed.

From Urysohn's Universal Space To A Universal Space-Time, Asuman Güven Aksoy, Zachary Glassman '14, Olga M. Kosheleva, Vladik Kreinovich

CMC Faculty Publications and Research

A known Urysohn’s result shows that there exists a universal metric space, i.e., a metric space into every other (separable) metric space can be isomorphically embedded. Moreover, this universal metric space can be selected to be ultra-homogeneous — every isomorphism of its two finite subsets can be extended to the isomorphism of the whole space. Starting with Einstein’s theories of Special and General relativity, space-times are described by a different type of structure — a set (of events) equipped with the proper time τ (a, b) between points a and b; such spaces are known as space-times with kinematic metric, …

Review: On Some Bergman Shift Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Transmission Rate In Partial Differential Equation In Epidemic Models, Alaa Elkadry

Theses, Dissertations and Capstones

The rate at which susceptible individuals become infected is called the transmission rate. It is important to know this rate in order to study the spread and the effect of an infectious disease in a population. This study aims at providing an understanding of estimating the transmission rate from mathematical models representing the population dynamics of an infectious diseases using two different methods. Throughout, it is assumed that the number of infected individuals is known. In the first chapter, it includes historical background for infectious diseases and epidemic models and some terminology needed to understand the problems. Specifically, the partial …

Dg And Hdg Methods For Curved Structures, Li Fan

Wayne State University Dissertations

We introduce and analyze discontinuous Galerkin methods

for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking.

We also prove that, when polynomials of degree $k$ are used,

{\em all} the numerical traces superconverge with a rate of order

h ^2k+1.

Based on the superconvergent phenomenon and we show how to

post-process them in an element-by-element fashion

to obtain a far better approximation. Indeed, we prove that,

if polynomials …

Qualitative Properties Of Solutions Of Fully Nonlinear Equations And Overdetermined Problems, Jiuyi Zhu

Wayne State University Dissertations

In section 2 of part I, We study the maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We

obtain the radial symmetry and monotonicity properties for

nonnegative viscosity solutions of fully nonlinear equations under some asymptotic decay rate at infinity. Our symmetry and monotonicity results also

apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum

principle for viscosity solutions to fully nonlinear elliptic equations is established. In section 3, We establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the …

Two-Time-Scale Systems In Continuous Time With Regime Switching And Their Applications, Yousef Talafha

Wayne State University Dissertations

This dissertation is focuses on near-optimal controls for stochastic differential equation with regime switching. The random switching is presented by a continuous-time Markov chain. We use the idea of relaxed control and mean of martingale formulation to show a weak convergence result.

The first chapter is devoted to the study of stochastic Li´enard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Li´enard equation and the discrete component is a Markov chain with a finite …