Mathematics Commons^™

Characterizing Sparse Graphs By Map Decompositions, Ruth Haas, Audrey Lee, Ileana Streinu, Louis Theran

Mathematics Sciences: Faculty Publications

A map is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into k edge-disjoint maps after: (1) the addition of any ℓ edges; (2) the addition of some ℓ edges. These graphs are identified with classes of sparse graphs; the results are also given in matroidal terms.

On The Simplification Of Certain Q-Multisums, Andrew Sills

Mathematical Sciences: Faculty Publications

Some examples of naturally arising multisum q-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric series.

Transmission Dynamics Of Avian Influenza Among Poultry With And Without Vaccination, Qiao Liang

Mathematics & Statistics ETDs

The continuing avian influenza (AI) out break that began in late 2003 and early 2004 has been disastrous for the poultry industry worldwide. It has resulted in severe socio-economic damage, and it has raised serious concerns for general public health. In this research, we use mathematics to analyze transmission dynamics of AI among poultry. We use a status-based approach to construct systems of differential equations to describe virus transmission dynamics. We develop theoretical means to eradicate the spread of the disease, and we calculate the size of healthy and infected populations during an AI outbreak, and the final population size …

The Minimum Rank Problem Over Finite Fields, Jason Nicholas Grout

Theses and Dissertations

We have two main results. Our first main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs and show that many of these are minimal forbidden subgraphs for any field. Our second main result is a structural characterization of all graphs having minimum rank at most k for any k over any finite field. This characterization leads to a very strong connection to projective geometry and we apply projective …

Infinite Product Group, Keith G. Penrod

Theses and Dissertations

The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda's infinitary product as the fundamental group of a modified wedge product.

Graduate Teaching Assistants' Statistical Knowledge For Teaching, Jennifer Ann Noll

Dissertations and Theses

This dissertation explores graduate teaching assistants’ (TAs’) statistical knowledge for teaching. Data collection methods that enabled the exploration of TA s’ statistical knowledge for teaching include: (a) a task-based web survey administered to 68 TAs from 18 universities across the United States; and, (b) a series of three taskbased interviews with a subset of five TAs from the larger survey population. Through and Strauss, 1967), I investigated the ways in which TAs reason about sampling tasks, and how they think about teaching and student learning in relation to sampling ideas. Building on past research in statistics education on K-12 and …

Pipe Diagrams For Thompson's Group F, Aaron L. Peterson

Theses and Dissertations

We review the definition and standard description of Thompson's Group F. We define the set of pipe diagrams and show that this set forms a group isomorphic to F. We use pipe diagrams to prove two theorems about giving a minimal representation for an arbitrary element of F.

Isomorphisms Of Elliptic Curves Over Extensions Of Finite Fields, Mathew Niemerg

Mathematical Sciences Technical Reports (MSTR)

Our main interest lies in exploring isomorphisms of elliptic curves. In particular, we focus on two curves defined over a base field and look at which extension fields the curves are isomorphic over. Elliptic curves have a fascinating structure behind them. This structure allows for much to be explored and studied.

Double Integral Calculus Of Variations On Time Scales, Gusein Sh. Guseinov, Martin Bohner

Mathematics and Statistics Faculty Research & Creative Works

We consider a version of the double integral calculus of variations on time scales, which includes as special cases the classical two-variable calculus of variations and the discrete two-variable calculus of variations. Necessary and sufficient conditions for a local extremum are established, among them an analogue of the Euler-Lagrange equation.

Fractalized Cyclotomic Polynomials, David P. Roberts

Mathematics Publications

For each prime power p^m, we realize the classical cyclotomic polynomial Φ_p^m(x) as one of a collection of 3^m different polynomials in Z[x]. We show that the new polynomials are similar to Φ_p^m(x) in many ways, including that their discriminants all have the form ±p^c. We show also that the new polynomials are more complicated than Φ_p^m(x) in other ways, including that their complex roots are generally fractal in appearance.

Fixed Points Of Abelian Actions, John Franks, Michael Handel, Kamlesh Parwani

Faculty Research and Creative Activity

We prove that if $\F$ is an abelian group of $C^1$ diffeomorphisms isotopic to the identity of a closed surface $S$ of genus at least two then there is a common fixed point for all elements of $\F.$

Stochastic Modeling Of Retail Mortgage Loans Based On Past Due, Prepaid, And Default States, Chang Liu

Doctoral Dissertations

Stochastic models were developed that provide important measures related to retail mortgages and credit cards for the management of a bank. Based on Markov theory, two models were developed that predict mortgage portfolio size and expected duration of stay in each of the states, which are defined according to the criteria of Basel Accord II and the Federal Reserve Bank. Also, to facilitate comparisons among different types of credit products and different time periods, a model was developed to generate a health index for a retail mortgage. This model could be easily extended, using multivariate regression or multivariate time series …

As Flat As Possible, Jon T. Jacobsen

All HMC Faculty Publications and Research

How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students …

On A Theorem Of Shchepin And Repovš Concerning The Smoothness Of Compacta, Ethan Akin, William Ott

Mathematics and Statistics Faculty Research & Creative Works

We show that a theorem of Shchepin and Repovš concerning the smoothness of compacta follows from the theory of semicontinuous relations. © 2007 Elsevier B.V. All rights reserved.

Suboptimality Conditions For Mathematical Programs With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich

Mathematics Research Reports

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal …

On The Use Of Quasi-Newton Methods For The Minimization Of Convex Quadratic Splines, William Howard Thomas Ii

Mathematics & Statistics Theses & Dissertations

In reformulating a strictly convex quadratic program with simple bound constraints as the unconstrained minimization of a strictly convex quadratic spline, established algorithms can be implemented with relaxed differentiability conditions. In this work, the positive definite secant update method of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) is investigated as a tool to solve the unconstrained minimization problem. It is shown that there is a linear convergence rate and, for nondegenerate problems, the process terminates in a finite number of iterations. Numerical examples are provided.

Proof Of The Barker Array Conjecture, James A. Davis, Jonathan Jedwab, Ken W. Smith

Department of Math & Statistics Faculty Publications

Using only elementary methods, we prove Alquaddoomi and Scholtz’s conjecture of 1989, that no s × t Barker array having s, t > 1 exists except when s = t = 2.

Scattered Homoclinics To A Class Of Time-Recurrent Hamiltonian Systems, Gregory S. Spradlin

Mathematics - Daytona Beach

A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zero whose support is spread out over the real line. © EDP Sciences, SMAI 2007.

Discrete Kato-Type Theorem On Inviscid Limit Of Navier-Stokes Flows, Wenfang Cheng, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

The inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics. Here we present a somewhat surprising result related to numerical approximation of the problem. More precisely, we show that numerical solutions of the incompressible Navier-Stokes equations converge to the exact solution of the Euler equations at vanishing viscosity and vanishing mesh size provided that small scales of the order of U in the directions tangential to the boundary are not resolved in the scheme. Here is the kinematic viscosity of the fluid and U is the typical velocity taken to be …

Note On The Emergence Of Large Scale Coherent Structure Under Small Scale Random Bombardments: The Discrete Case, Andrew Majda, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We continue our study on mathematical justification of the emergence of large-scale coherent structure in a two-dimensional fluid system under small scale random bombardments. We treat the case of small-scale random bombardments at discrete times which is different from our earlier work [Commun. Pure Appl. Math. 59, 467 (2006)], where we approximated the small-scale random kicks by a continuous in time random process. in the absence of geophysical effects, the large-scale structure emerging out of the small-scale random forcing is the same as the case of continuous in time forcing that we studied before. © 2007 American Institute of Physics.

Balanced Proper Orthogonal Decomposition For Model Reduction Of Infinite Dimensional Linear Systems, John R. Singler, Belinda A. Batten

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we extend a method for reduced order model derivation for finite dimensional systems developed by Rowley to infinite dimensional systems. The method is related to standard balanced truncation, but includes aspects of the proper orthogonal decomposition in its computational approach. The method is also applicable to nonlinear systems. The method is applied to a convection diffusion equation.

Constant-Sign Solutions Of A System Of Integral Equations With Integrable Singularities, Ravi P. Agarwal, Donal O'Regan, Patricia J.Y. Wong

Mathematics and System Engineering Faculty Publications

We consider the following systems of Fredholm integral equations and the system of Volterra integral equations where the nonlinearities fi, 1 ≤ i ≤ n may be singular in the independent variable and may also be singular at uj = 0, j ∈ {1, 2,..., n}. Our aim is to establish criteria such that the above systems have at least one constant-sign solution (u1, u2,..., un), i.e., for each 1 ≤ i ≤ n, Θiui ≥ 0 where Θi ∈ {1,-1} is fixed. © 2007 Rocky Mountain Mathematics Consortium.

Determination Of Interphase Line Tension In Langmuir Films, Jacob R. Wintersmith '06, Lu Zou, Andrew J. Bernoff, James C. Alexander, J. Adin Mann Jr.

All HMC Faculty Publications and Research

A Langmuir film is a molecularly thin film on the surface of a fluid; we study the evolution of a Langmuir film with two coexisting fluid phases driven by an interphase line tension and damped by the viscous drag of the underlying subfluid. Experimentally, we study a 4^′-8-alkyl[1,1^′-biphenyl]-4-carbonitrile (8CB) Langmuir film via digitally imaged Brewster angle microscopy in a four-roll mill setup which applies a transient strain and images the response. When a compact domain is stretched by the imposed strain, it first assumes a bola shape with two tear-drop shaped reservoirs connected by a thin …

Nonlinear Dynamics In Combinatorial Games: Renormalizing Chomp, Eric J. Friedman, Adam S. Landsberg

WM Keck Science Faculty Papers

We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to …

Positive Solutions For Classes Of Multiparameter Elliptic Semipositone Problems, Scott Caldwell, Alfonso Castro, Ratnasingham Shivaji, Sumalee Unsurangsie

All HMC Faculty Publications and Research

We study positive solutions to multiparameter boundary-value problems of the form

-Δu = λg(u)+μf(u) in Ω

u = 0 on ∂Ω

where λ>0, μ>0, Ω⊆Rⁿ; n≥2 is a smooth bounded domain with ∂Ω in class C² and Δ is the Laplacian operator. In particular, we assume g(0)>0 and superlinear while f(0)

Teach A Course In The Math Of Voting And Choice, Karl-Dieter Crisman

ACMS Conference Proceedings 2007

Many mathematics instructors at the college level are looking for a curricular option that has the potential to serve a number of different constituencies. It could be to encourage more students to take math courses, or to give worthwhile options to students who need to take math but who are not ready for calculus (or its sequence). On the other hand, one may wish to add a new course for majors outside of the typical offerings, or even to prepare students for undergraduate research. The mathematics of voting and choice is ideally suited to meet all these needs in the …

Bach (To The Calculus Of) Variations, Charles R. Hampton

ACMS Conference Proceedings 2007

While it is quite common for professionals (doctors, lawyers, academics, etc) to be talented in many ways, including musical talent, there is a special connection between music and mathematics. Musicians collectively are not more talented in mathematics than other professionals and other academics. This paper examines the connections between math and music, particularly calculus and the music of Johann Sebastian Bach.

Chaos Theory And Metaphysical (In) Determinism, Tim Rogalsky

ACMS Conference Proceedings 2007

This paper will begin by introducing the issues that arise from chaos theory for the Christian mathematician and scientist: What is at stake in this debate? It will then briefly review chaos theory, by means of two examples. It will then introduce the metaphysical interpretations given to chaos theory by three different scientist-theologians. The paper will conclude with a brief introduction to open theists, and analyze their use of chaos theory to supper their theological claims.

Closed-Neighborhood Anti-Sperner Graphs, John P. Mcsorley, Alison Marr, Thomas D. Porter, Walter D. Wallis

Articles and Preprints

For a simple graph G let N_G[u] denote the closed-neighborhood of vertex u ∈ V (G). Then G is closed-neighborhood anti-Sperner (CNAS) if for every u there is a v ∈ V (G)\{u} with N_G [u] ⊆ N_G [v] and a graph H is closed-neighborhood distinct (CND) if every closed-neighborhood is distinct, i.e., if N_H[u] ≠ N_H[v] when u ≠ v, for all u and v ∈ V (H).

In this paper we …

The Probability Of Relatively Prime Polynomials, Arthur T. Benjamin, Curtis D. Bennett

All HMC Faculty Publications and Research

Euclid does integers