Mathematics Commons^™

An Exposition Of The Deterministic Polynomial-Time Primality Testing Algorithm Of Agrawal-Kayal-Saxena, Robert Lawrence Anderson

Theses and Dissertations

I present a thorough examination of the unconditional deterministic polynomial-time algorithm for determining whether an input number is prime or composite proposed by Agrawal, Kayal and Saxena in their paper [1]. All proofs cited have been reworked with full details for the sake of completeness and readability.

Matrix Representations Of Automorphism Groups Of Free Groups, Ivan B. Andrus

Theses and Dissertations

In this thesis, we study the representation theory of the automorphism group Aut (Fn) of a free group by studying the representation theory of three finite subgroups: two symmetric groups, Sn and Sn+1, and a Coxeter group of type Bn, also known as a hyperoctahedral group. The representation theory of these subgroups is well understood in the language of Young Diagrams, and we apply this knowledge to better understand the representation theory of Aut (Fn). We also calculate irreducible representations of Aut (Fn) in low dimensions and for small n.

Statistical Properties Of Thompson's Group And Random Pseudo Manifolds, Benjamin M. Woodruff

Theses and Dissertations

The first part of our work is a statistical and geometric study of properties of Thompson's Group F. We enumerate the number of elements of F which are represented by a reduced pair of n-caret trees, and give asymptotic estimates. We also discuss the effects on word length and number of carets of right multiplication by a standard generator x0 or x1. We enumerate the average number of carets along the left edge of an n-caret tree, and use an Euler transformation to make some conjectures relating to right multiplication by a generator. We describe a computer algorithm which produces …

Classifying Homotopy Types Of One-Dimensional Peano Continua, Mark H. Meilstrup

Theses and Dissertations

Determining the homotopy type of one-dimensional Peano continua has been an open question of some interest. We give a complete invariant of the homotopy type of such continua, which consists of a pair of subspaces together with a relative homology group. Along the way, we describe reduced forms for one-dimensional Peano continua.

Three Pension Cost Methods Under Varying Assumptions, Linda S. Grizzle

Theses and Dissertations

A pension plan administrator promises certain benefits in the future in exchange for labor today. In order to budget for this expense and create more security for the participant, the administrator uses a pension cost method. Each cost method assigns a portion of the future liability to the current year. This is called the normal cost. We calculate the normal cost under three cost methods using different annuity, interest and inflation assumptions. Then we make comparisons between cost methods as well as between assumption changes. The cost methods considered in this paper are the unit credit cost method, projected unit …

Explicit Computations Supporting A Generalization Of Serre's Conjecture, Brian Francis Hansen

Theses and Dissertations

Serre's conjecture on the modularity of Galois representations makes a connection between two-dimensional Galois representations and modular forms. A conjecture by Ash, Doud, and Pollack generalizes Serre's to higher-dimensional Galois representations. In this paper we discuss an explicit computational example supporting the generalized claim. An ambiguity in a calculation within the example is resolved using a method of complex approximation.

The Lie Symmetries Of A Few Classes Of Harmonic Functions, Willis L. Petersen

Theses and Dissertations

In a differential geometry setting, we can analyze the solutions to systems of differential equations in such a way as to allow us to derive entire classes of solutions from any given solution. This process involves calculating the Lie symmetries of a system of equations and looking at the resulting transformations. In this paper we will give a general background of the theory necessary to develop the ideas of working in the jet space of a given system of equations, applying this theory to harmonic functions in the complex plane. We will consider harmonic functions in general, harmonic functions with …

Hopf Bifurcations And Horseshoes Especially Applied To The Brusselator, Steven R. Jones

Theses and Dissertations

In this paper we explore bifurcations, in particular the Hopf bifurcation. We study this especially in connection with the Brusselator, which is a model of certain chemical reaction-diffusion systems. After a thorough exploration of what a bifurcation is and what classifications there are, we give graphic representations of an occurring Hopf bifurcation in the Brusselator. When an additional forcing term is added, behavior changes dramatically. This includes the introduction of a horseshoe in the time map as well as a strange attractor in the system.

Graphs Whose Minimal Rank Is Two: The Finite Fields Case, Wayne Barrett, Hein Van Der Holst, Raphael Loewy

Faculty Publications

Let F be a finite field, G = (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with p^t elements, p does not = 2, it is shown that mr(F,G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most …

Hyperbolic Sets That Are Not Locally Maximal, Todd L. Fisher

Faculty Publications

This paper addresses the following topics relating to the structure of hyperbolic sets: First, hyperbolic sets that are not contained in locally maximal hyperbolic sets. Second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a 4-dimensional manifold. The third example is volume preserving and constructed on R4. We show that every hyperbolic …

Wave Scattering From Infinite Cylindrical Obstacles Of Arbitrary Cross-Section, Matthew B. Weber

Theses and Dissertations

In this work the scattering of an incident plane wave propagating along a plane perpendicular to the xy-plane is studied. The wave is scattered from an infinitely long cylindrical object of arbitrary cross-section. Due to the arbitrary geometry of the obstacle, a finite differences numerical method is employed to approximate the solution of the scattering problems. The wave equation is expressed in terms of generalized curvilinear coordinates. Boundary conforming grids are generated using elliptic grid generators. Then, a explicit marching in time scheme is implemented over these grids. It is found that as time grows the numerical solution converges to …

How Cellular Movement Determines The Collective Force Generated By The Dictyostelium Discoideum Slug, J. C. Dallon, H. G. Othmer

Faculty Publications

How the collective motion of cells in a biological tissue originates in the behavior of a collection of individuals, each of which responds to the chemical and mechanical signals it receives from neighbors, is still poorly understood. Here we study this question for a particular system, the slug stage of the cellular slime mold Dictyostelium discoideum. We investigate how cells in the interior of a migrating slug can effectively transmit stress to the substrate and thereby contribute to the overall motive force. Theoretical analysis suggests necessary conditions on the behavior of individual cells, and computational results shed light on experimental …

A Numerical Scheme For Mullins-Sekerka Flow In Three Space Dimensions, Sarah Marie Brown

Theses and Dissertations

The Mullins-Sekerka problem, also called two-sided Hele-Shaw flow, arises in modeling a binary material with two stable concentration phases. A coarsening process occurs, and large particles grow while smaller particles eventually dissolve. Single particles become spherical. This process is described by evolving harmonic functions within the two phases with the moving interface driven by the jump in the normal derivatives of the harmonic functions at the interface. The harmonic functions are continuous across the interface, taking on values equal to the mean curvature of the interface. This dissertation reformulates the three-dimensional problem as one on the two-dimensional interface by using …

Psl(2,7)-Extensions With Certain Ramification At Two Primes, Glen E. Simpson

Theses and Dissertations

We conduct a parallel Hunter search in order to find a degree 7 number field K ramified at primes q and p with discriminant d(K)=q^6 p^2 where q=11 and 2

A New Approach To Lie Symmetry Groups Of Minimal Surfaces, Robert D. Berry

Theses and Dissertations

The Lie symmetry groups of minimal surfaces by way of planar harmonic functions are determined. It is shown that a symmetry group acting on the minimal surfaces is isomorphic with H × H^2 — the analytic functions and the harmonic functions. A subgroup of this gives a generalization of the associated family which is examined.

Ultraconnected And Critical Graphs, Jason Nicholas Grout

Theses and Dissertations

We investigate the ultraconnectivity condition on graphs, and provide further connections between critical and ultraconnected graphs in the positive definite partial matrix completion problem. We completely characterize when the join of graphs is ultraconnected, and prove that ultraconnectivity is preserved by Cartesian products. We completely characterize when adding a vertex to an ultraconnected graph preserves ultraconnectivity. We also derive bounds on the number of vertices which guarantee ultraconnectivity of certain classes of regular graphs. We give results from our exhaustive enumeration of ultraconnected graphs up to 11 vertices. Using techniques involving the Lovász theta parameter for graphs, we prove certain …

A Forbidden Subgraph Characterization Problem And A Minimal-Element Subset Of Universal Graph Classes, Michael D. Barrus

Theses and Dissertations

The direct sum of a finite number of graph classes H_1, ..., H_k is defined as the set of all graphs formed by taking the union of graphs from each of the H_i. The join of these graph classes is similarly defined as the set of all graphs formed by taking the join of graphs from each of the H_i. In this paper we show that if each H_i has a forbidden subgraph characterization then the direct sum and join of these H_i also have forbidden subgraph characterizations. We provide various results which in many cases allow us to exactly …

Problems Related To The Zermelo And Extended Zermelo Model, Benjamin Zachary Webb

Theses and Dissertations

In this thesis we consider a few results related to the Zermelo and Extended Zermelo Model as well as outline some partial results and open problems related thereto. First we will analyze a discrete dynamical system considering under what conditions the convergence of this dynamical system predicts the outcome of the Extended Zermelo Model. In the following chapter we will focus on the Zermelo Model by giving a method for simplifying the derivation of Zermelo ratings for tournaments in terms of specific types of strongly connected components. Following this, the idea of stability of a tournament will be discussed and …

Lattices And Their Applications To Rational Elliptic Surfaces, Gretchen Rimmasch

Theses and Dissertations

This thesis discusses some of the invariants of rational elliptic surfaces, namely the Mordell-Weil Group, Mordell-Weil Lattice, and another lattice which will be called the Shioda Lattice. It will begin with a brief overview of rational elliptic surfaces, followed by a discussion of lattices, root systems and Dynkin diagrams. Known results of several authors will then be applied to determine the groups and lattices associated with a given rational elliptic surface, along with a discussion of the uses of these groups and lattices in classifying surfaces.

A Thermoviscoelastic Beam Model For Brakes, Kenneth Kuttler, J. Bajkowski, J. R. Fernandez, M. Shillor

Faculty Publications

A model for the dynamic thermomechanical behavior of a viscoelastic beam which is in frictional contact with a rigid rotating wheel is presented. It describes a simple braking system in which the wheel comes to a stop as a result of the frictional traction generated by the beam. Friction is modelled with a temperature and slip rate dependent coefficient of friction. Frictional heat generation is taken into account as well as the wheel temperature evolution, and the wear of the beam's contacting end. The model is formulated as a variational inequality. A FEM numerical scheme for the model is described, …

The Topology Of Hyperbolic Attractors On Compact Surfaces, Todd L. Fisher

Faculty Publications

Suppose M is a compact surface and M is a nontrivial mixing hyperbolic attractor for some f 2 Diff(M). We show that if is a hyperbolic set for some g 2 Diff(M), then is a nontrivial mixing hyperbolic attractor or repeller for g.

Minimal Graphs In R3 Over Convex Domains, Michael Dorff

Faculty Publications

Krust established that all conjugate and associate surfaces of a minimal graph over a convex domain are also graphs. Using a convolution theorem from the theory of harmonic univalent mappings, we generalize Krust's theorem to include the family of convolution surfaces which are generated by taking the Hadamard product or convolution of mappings. Since this convolution involves convex univalent analytic mappings, this family of convolution surfaces is much larger than just the family of associated surfaces. Also, this generalization guarantees that all the resulting surfaces are over close-toconvex domains. In particular, all the associate surfaces and certain Goursat transformation surfaces …

Sandwich Theorem And Calculation Of The Theta Function For Several Graphs, Marcia Ling Riddle

Theses and Dissertations

This paper includes some basic ideas about the computation of a function theta(G), the theta number of a graph G, which is known as the Lovasz number of G. theta(G^c) lies between two hard-to-compute graph numbers omega(G), the size of the largest lique in a graph G, and chi(G), the minimum number of colors need to properly color the vertices of G. Lovasz and Grotschel called this the "Sandwich Theorem". Donald E. Knuth gives four additional definitions of theta, theta_1, theta_2, theta_3, theta_4 and proves that they are all equal.

First I am going to describe the proof of the …

Bounding The Number Of Graphs Containing Very Long Induced Paths, Steven Kay Butler

Theses and Dissertations

Induced graphs are used to describe the structure of a graph, one such type of induced graph that has been studied are long paths.

In this thesis we show a way to represent such graphs in terms of an array with two colors and a labeled graph. Using this representation and the techniques of Polya counting we will then be able to get upper and lower bounds for graphs containing a long path as an induced subgraph.

In particular, if we let P(n,k) be the number of graphs on n+k vertices which contains P_n, a path on n vertices, as …

Every Three-Point Set Is Zero Dimensional, David L. Fearnley, J. W. Lamoreaux, David L. Fearnley

Faculty Publications

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all n > 3, there are n-point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

Odd Perfect Numbers Have A Prime Factor Exceeding 10^7, Paul M. Jenkins

Faculty Publications

It is proved that every odd perfect number is divisible by a prime greater than 10^7.

Modeling The Effects Of Transforming Growth Factor-Beta On Extracellular Matrix Alignment In Dermal Wound Repair, J. C. Dallon, J. A. Sherratt, P. K. Maini

Faculty Publications

We present a novel mathematical model for collagen deposition and alignment during dermal wound healing, focusing on the regulatory effects of TGF. Our work extends a previously developed model which considers the interactions between fibroblasts and extracellular matrix, composed of collagen and a fibrin based blood clot, by allowing fibroblasts to orient the collagen matrix, and produce and degrade the extracellular matrix, while the matrix can direct the fibroblasts and control their speed. Here we extend the model by allowing a time varying concentration of TGF to alter the properties of the fibroblasts. Thus we are able to simulate experiments …

Blowup In A Mass-Conserving Convection-Diffusion Equation With Superquadratic Nonlinearity, Todd L. Fisher, Christopher P. Grant

Faculty Publications

A nonlinear convection-diffusion equation with boundary conditions that conserve the spatial integral of the solution is considered. Previous results on nite-time blowup of solutions and on decay of solutions to the corresponding Cauchy problem were based on the assumption that the nonlinearity obeyed a power law. In this paper, it is shown that assumptions on the growth rate of the nonlinearity, which take the form of weak superquadraticity and strong superlinearity criteria, are suffcient to imply that a large class of nonnegative solutions blow up in nite time.

Gravitational Descendants And The Moduli Space Of Higher Spin Curves, Tyler J. Jarvis, Takashi Kimura, Arkady Vaintrob

Faculty Publications

The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of r-spin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a priori analog in the theory of Gromov-Witten invariants), in terms of the multiplicativity of the virtual class. We prove that the Descent Axiom holds in the convex case, and consequently in genus zero.

Biological Implications Of A Discrete Mathematical Model For Collagen Deposition And Alignment In Dermal Wound Repair, J. C. Dallon, J. A. Sherratt, P. K. Maini, M. W. Ferguson

Faculty Publications

We develop a novel mathematical model for collagen deposition and alignment during dermal wound healing. We focus on the interactions between fibroblasts, modelled as discrete entities, and a continuous extracellular matrix composed of collagen and a fibrin based blood clot. There are four basic interactions assumed in the model: fibroblasts orient the collagen matrix, fibroblasts produce and degrade collagen and fibrin and the matrix directs the fibroblasts and determines the speed of the cells. Several factors which influence the alignment of collagen are examined and related to current anti-scarring therapies using transforming growth factor " The most influential of these …