Mathematics Commons^™

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 …

Benson's Theorem For Partial Geometries, Ellen J. Kamischke

Dissertations, Master's Theses and Master's Reports - Open

In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values …

Computer Programming To Advance Gravitational Lensing, Alex Roche

Undergraduate Review

The purpose of this research was to create a computer code that would numerically test a Poisson equation relating the mass distribution of a lens galaxy cluster to weak gravitational shear. Einstein’s theory of general relativity predicts that space-time is bent by massive objects, and in weak gravitational lensing, galaxy clusters act as lenses. The observable result is that galaxies far behind the gravitational lens will appear slightly more elliptical than they actually are. The ellipticity of the background galaxies is quantifiable and is directly related to the weak gravitational shear, and the shear is used to determine the mass …

A Note On Chaos In Product Maps, Risong Li, Xiaoliang Zhou

Turkish Journal of Mathematics

In this paper, we mainly discuss how chaos conditions on semi-flows carry over to their products. We show that if two semi-flows (or even one of them) are sensitive, so does their product. On the other side, the product of two topologically transitive semi-flows need not be topologically transitive. We then provide several sufficient conditions under which the product of two chaotic semi-flows is chaotic in the sense of Devaney. Also, stronger forms of sensitivity and transitivity for product systems are studied. In particular, we introduce the notion of ergodic sensitivity and prove that for any given two (not-necessarily continuous) …

Total Acquisition In Graphs, Timothy D. Lesaulnier, Noah Prince, Paul S. Wenger, Douglas B. West, Pratik Worah

Faculty Publications & Research

Let G be a weighted graph in which each vertex initially has weight 1. A total acquisition move transfers all the weight from a vertex u to a neighboring vertex v, under the condition that before the move the weight on v is at least as large as the weight on u. The (total) acquisition number of G, written a_t(G), is the minimum size of the set of vertices with positive weight after a sequence of total acquisition moves. Among connected n-vertex graphs, a_t(G) is maximized by trees. …

An Approach To Discrete Quantum Gravity, S. Gudder

Mathematics Preprint Series

This article presents a simplified version of the author’s previous work. We first construct a causal growth process (CGP). We then form path Hilbert spaces using paths of varying lengths in the CGP. A sequence of positive operators on these Hilbert spaces that satisfy certain normalization and consistency conditions is called a quantum sequential growth process (QSGP). The operators of a QSGP are employed to define natural decoherence functionals and quantum measures. These quantum measures are extended to a single quantum measure defined on a suitable collection of subsets of a space of all paths. Continuing our general formalism, we …

A Dynamics For Discrete Quantum Gravity, Stan Gudder

Mathematics Preprint Series

This paper is based on the causal set approach to discrete quantum gravity. We first describe a classical sequential growth process (CSGP) in which the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the “completed” universe is given by a path through a discretely growing chain of causets. We then quantize the CSGP by forming a Hilbert space H on the set of paths. The quantum dynamics is governed by a sequence of positive operators ρn on H that satisfy normalization and consistency conditions. …

Visualizing Chaos, Margaret Peterson

Honors Theses, 1963-2015

Chaos is typically visualized on an infinite 2D plane. By using stereographic projection, my colleague Preston Hardy and I utilized a third dimension to plot basin maps of iterative root finding methods on a subset of the complex plane onto a sphere. These spheres are then shaded in accordance to the speed in which the particular initial point converges, creating images that can be used to visualize all basins of attraction on the complex plane on a finite 3D surface. The resulting images are used to explore efficiency of root finding methods as well as evaluating the choice of addition …

The Growth In Normal Subgroups Under Direct Products, Whitney Radil

Honors Theses, 1963-2015

This paper will consider the growth of the number of normal subgroups in a nonabelian group under the direct product operation. The groups considered in this paper are dihedral groups and semidirect products with a similar structure. Many proofs will rely on the use of a key property of normal subgroups, namely closure under conjugation, to predict this growth.

Nim On Groups, Marie Meyer

Honors Theses, 1963-2015

The traditional game of Nim comprises of two players who take turns removing objects from distinct piles. The player who takes the last object is the winner. We consider the game Nim on Cayley graphs of finite groups, where the piles are located on the vertices and the number of objects in each pile is denoted as the weight of the vertex. In this version of the game, a player wins by trapping the opponent on a vertex with weight zero so he or she is unable to further reduce the weight of that vertex. We examine winning strategies for …

With One Breath, Jessica Solfest

Honors Theses, 1963-2015

Biologists know that the cardiovascular and respiratory systems interact and coordinate via the baroreflex and other pathways, but there is still much to discover regarding the dynamical aspects of their communication. Synchronization patterns between the respiratory and cardiovascular systems evince their interaction. This thesis focuses on the synchronization of cardiovascular and respiratory activity. I analyze the frequency of this synchrony to determine patterns and characterize the systems’ adaptability to changes in the environment. I applied time series analysis and Markov processes to dynamical systems in an attempt to unearth meaning behind cardiorespiratory communicative mysteries. Through this research, I found that …

Ioana's Superrigidity Theorem And Orbit Equivalence Relations, Samuel Coskey

Mathematics Faculty Publications and Presentations

We give a survey of Adrian Ioana’s cocycle superrigidity theoremfor profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana’s theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas’s theorem that the classification of torsion-free abelian groups of …

Navigating The Seas Of Mathematics Education: New Waves In Research To Improve Student Learning, Jinfa Cai, Sasha Wang

Mathematics Faculty Publications and Presentations

This issue focuses on research in the domain of mathematics education. Although mathematics has been a subject of study for many centuries, mathematics education is a relatively new field of scholarly inquiry, having been established as an independent field of research only in the early twentieth century. The most significant milestone was the establishment of the International Commission on Mathematical Instruction (ICMI) in 1908. Since 1969, ICMI has organized the International Congress on Mathematical Education, a quadrennial international meeting whose aim is to present the current states and trends in mathematics education research and in the practice of mathematics teaching …