Mathematics Commons^™

Stability Of Planar Detonations In The Reactive Navier-Stokes Equations, Joshua W. Lytle

Theses and Dissertations

This dissertation focuses on the study of spectral stability in traveling waves, with a special interest in planar detonations in the multidimensional reactive Navier-Stokes equations. The chief tool is the Evans function, combined with STABLAB, a numerical library devoted to calculating the Evans function. Properly constructed, the Evans function is an analytic function in the right half-plane whose zeros correspond in multiplicity and location to the spectrum of the traveling wave. Thus the Evans function can be used to verify stability, or to locate precisely any unstable eigenvalues. We introduce a new method that uses numerical continuation to follow unstable …

It Is Better To Be Upside Than Sharpe!, Daniele Dapuzzo

Theses and Dissertations

Based on the assumption that returns in Commercial Real Estate are normally distributed, the Sharpe Ratio has been the standard risk-adjusted performance measure for the past several years. Research has questioned whether this assumption can be reasonably made. The Upside Potential Ratio as a risk-adjusted performance measure is an alternative to measure performance on a risk-adjusted basis but its values differ from the Sharpe Ratio's only in the assumption of skewed returns. We will provide reasonable evidence that CRE returns should not be fitted with a normal distribution and present the Gaussian Mixture Model as our choice of distribution to …

A Continuous Time Model Of Centrally Controlled Motion With Random Switching Terms, J. C. Dallon, L C. Despain, E J. Evans, C P. Grant, Willaim V. Smith

Faculty Publications

This paper considers differential problems with random switching, with specific applications to the motion of cells and centrally coordinated motion. Starting with a differential-equation model of cell motion that was proposed previously, we set the relaxation time to zero and consider the simpler model that results. We prove that this model is well-posed, in the sense that it corresponds to a pure jump-type continuous time Markov process (without explosion). We then describe the model's long-time behavior, first by specifying an attracting steady-state distribution for a projection of the model, then by examining the expected location of the cell center when …

Some Spectral Properties Of A Quantum Field Theoretic Hamiltonian, Devin Burnell Mcghie

Theses and Dissertations

We derive the ground-state eigenvalues and eigenvectors for a simplified version of the 1-D QED single electron-photon model that Glasgow et al recently developed [2]. This model still allows for meaningful interaction between electrons and photons while keeping only the minimum needed to do so. We investigate the interesting spectral properties of this model. We determine that the eigenvectors are orthogonal as one would expect and normalize them.

Steady State Configurations Of Cells Connected By Cadherin Sites, Jared Adam Mcbride

Theses and Dissertations

Many cells employ cadherin complexes (c-sites) on the cell membrane to attach to neighboring cells, as well as integrin complexes (i-sites) to attach to a substrate in order to accomplish cell migration. This paper analyzes a model for the motion of a group of cells connected by c-sites. We begin with two cells connected by a single c-site and analyze the resultant motion of the system. We find that the system is irrotational. We present a result for reducing the number of c-sites in a system with c-sites between pairs of cells. This greatly simplifies the general system, and provides …

American Spread Option Pricing With Stochastic Interest Rate, An Jiang

Theses and Dissertations

In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR …

Periodic Points And Surfaces Given By Trace Maps, Kevin Gregory Johnston

Theses and Dissertations

In this thesis, we consider the properties of diffeomorphisms of R3 called trace maps. We begin by introducing the definition of the trace map. The group B3 acts by trace maps on R3. The first two chapters deal with the action of a specific element of B3,called αn. In particular, we study the fixed points of αn lying on a topological subspace contained in R3, called T . We investigate the duality of the fixed points of the action ofαn, which will be defined later in the thesis.Chapter 3 involves the study of the fixed points of an element called …

Schur Rings Over Projective Special Linear Groups, David R. Wagner

Theses and Dissertations

This thesis presents an introduction to Schur rings (S-rings) and their various properties. Special attention is given to S-rings that are commutative. A number of original results are proved, including a complete classification of the central S-rings over the simple groups PSL(2,q), where q is any prime power. A discussion is made of the counting of symmetric S-rings over cyclic groups of prime power order. An appendix is included that gives all S-rings over the symmetric group over 4 elements with basic structural properties, along with code that can be used, for groups of comparatively small order, to enumerate all …

Existence Of A Periodic Brake Orbit In The Fully Symmetricplanar Four Body Problem, Ammon Si-Yuen Lam

Theses and Dissertations

We investigate the existence of a symmetric singular periodic brake orbit in the equal mass, fully symmetric planar four body problem. Using regularized coordinates, we remove the singularity of binary collision for each symmetric pair. We use topological and symmetry tools in our investigation.

Weak Cayley Table Groups Of Wallpaper Groups, Rebeca Ann Paulsen

Theses and Dissertations

Let G be a group. A Weak Cayley Table mapping ϕ : G → G is a bijection such that ϕ(g1g2) is conjugate to ϕ(g1)ϕ(g2) for all g1, g2 in G. The set of all such mappings forms a group W(G) under composition. We study W(G) for the seventeen wallpaper groups G.

Machine Learning For Disease Prediction, Abraham Jacob Frandsen

Theses and Dissertations

Millions of people in the United States alone suffer from undiagnosed or late-diagnosed chronic diseases such as Chronic Kidney Disease and Type II Diabetes. Catching these diseases earlier facilitates preventive healthcare interventions, which in turn can lead to tremendous cost savings and improved health outcomes. We develop algorithms for predicting disease occurrence by drawing from ideas and techniques in the field of machine learning. We explore standard classification methods such as logistic regression and random forest, as well as more sophisticated sequence models, including recurrent neural networks. We focus especially on the use of medical code data for disease prediction, …

Weakly Holomorphic Modular Forms In Prime Power Levels Of Genus Zero, David Joshua Thornton

Theses and Dissertations

Let N ∈ {8,9,16,25} and let M^#₀(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M^#₀(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results …

Isomorphisms Of Landau-Ginzburg B-Models, Nathan James Cordner

Theses and Dissertations

Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted A and B) that are constructed from a nondegenerate quasihomogeneous polynomial W and a related group of symmetries G. In 2013, Tay proved that given two polynomials W1, W2 with the same quasihomogeneous weights and same group G, the corresponding A-models built with (W1, G) and (W2, G) are isomorphic. An analogous theorem for isomorphisms between orbifolded B-models remains to be found. This thesis investigates isomorphisms between B-models using polynomials in two variables in search of such a theorem. In particular, several examples are given showing the relationship between continuous …

Geometric Optimization, Analysis, And Design, Denise Halverson

Journal of Undergraduate Research

1. Evaluation of how well the academic objectives of the proposal were met

The goals for the mentoring environment were reached:

• Academic Development – Each student had the opportunity to work on a cutting edge research problem, as indicated below. Papers are in various stages of completion, a few submitted or accepted for publication, but all have drafted papers.
• Community Development – Several students were able to attend the MAA Math Fest in both 2014 and 2015. Most students gave presentations at the CPMS Student Research Conference. My students have been able to interact with engineering student working on similar …

Modeling Individual Health Care Utilization, Matthew Aaron Webb

Theses and Dissertations

Health care represents an increasing proportion of global consumption. We discuss ways to model health care utilization on an individual basis. We present a probabilistic, generative model of utilization. Leveraging previously observed utilization levels, we learn a latent structure that can be used to accurately understand risk and make predictions. We evaluate the effectiveness of the model using data from a large population.

The Principles Of Effective Teaching Student Teachershave The Opportunity To Learn In An Alternativestudent Teaching Structure, Danielle Rose Divis

Theses and Dissertations

Research has shown that the focus of mathematics student teaching programs is typically classroom management and non-mathematics specific teaching strategies. However, the redesigned BYU student teaching structure has proven to help facilitate a greater focus on mathematics-specific pedagogy and student mathematics during post-lesson reflection meeting conversations. This study analyzed what specific principles of NCTM’s standards of effective teaching were discussed in the reflection meetings of this redesigned structure. This study found that the student teachers extensively discussed seven of the eight principles NCTM considers to be necessary for effective mathematics teaching. Other pedagogical principles pertaining to student mathematical learning not …

Construction And Isomorphism Of Landau-Ginzburg B-Model Frobenius Algebras, Matthew Robert Brown

Theses and Dissertations

Landau-Ginzburg Mirror Symmetry provides for the construction of two algebraic objects, called the A- and B-models. Special cases of these models–constructed using invertible polynomials and abelian symmetry groups–are well understood. In this thesis, we consider generalizations of the B-model, and specifically address the associativity of the multiplication in these models. We also prove an explicit B-model isomorphism for a class of polynomials in three variables.

Spectral Graph Theory For Weighted Digraphs, Alexander Zaitzeff, Jeffrey Humpherys

Journal of Undergraduate Research

For digraphs weighted and unweighted, one important application is ranking: Given a directed graph, whether it be the Internet or a social network, which node (representing a web page or a person) is the most important? There are many different methods to find answer this question. A few are highest indegree, closeness centrality¹, betweeness centrality², eigenvector centrality, Katz Centrality³, and PageRank⁴. Our idea is to use sparsity, or the idea that in a network only has a few important nodes, to determine the ranking on a graph.

The Solenoid And Warsawanoid Are Sharkovskii Spaces, Tyler Willes Hills

Theses and Dissertations

We extend Sharkovskii's theorem concerning orbit lengths of endomorphisms of the real line to endomorphisms of a path component of the solenoid and certain subspaces of the Warsawanoid. In particular, Sharkovskii showed that if there exists an orbit of length 3 then there exist orbits of all lengths. The solenoid is the inverse limit of double covers over the circle, and the Warsawanoid is the inverse limit of double covers over the Warsaw circle. We show Sharkovskii's result is true for path components of the solenoid and certain subspaces of the Warsawanoid.

Compression Bodies And Their Boundary Hyperbolic Structures, Vinh Xuan Dang

Theses and Dissertations

We study hyperbolic structures on the compression body C with genus 2 positive boundary and genus 1 negative boundary. We consider individual hyperbolic structures as well as special regions in the space of all such hyperbolic structures. We use some properties of the boundary hyperbolic structures on C to establish an interesting property of cusp shapes of tunnel number one manifolds. This extends a result of Nimershiem in [26] to the class of tunnel number one manifolds. We also establish convergence results on the geometry of compression bodies. This extends the work of Ito in [13] from the punctured-torus case …

Alternating Direction Implicit Method With Adaptive Grids For Modeling Chemotaxis In Dictyostelium Discoideum, Christopher F. Loomis

Theses and Dissertations

Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons …

A Nonabelian Landau-Ginzburg B-Model Construction, Ryan Thor Sandberg

Theses and Dissertations

The Landau-Ginzburg (LG) B-Model is a significant feature of singularity theory and mirror symmetry. Krawitz in 2010, guided by work of Kaufmann, provided an explicit construction for the LG B-model when using diagonal symmetries of a quasihomogeneous, nondegenerate polynomial. In this thesis we discuss aspects of how to generalize the LG B-model construction to allow for nondiagonal symmetries of a polynomial, and hence nonabelian symmetry groups. The construction is generalized to the level of graded vector space and the multiplication developed up to an unknown factor. We present complete examples of nonabelian LG B-models for the polynomials x^2y + y^3, …

Nonlocally Maximal Hyperbolic Sets For Flows, Taylor Michael Petty

Theses and Dissertations

In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.

Topics Pertaining To The Group Matrix: K-Characters And Random Walks, Randall Dean Reese

Theses and Dissertations

Linear characters of finite groups can be extended to take k operands. The basics of such a k-fold extension are detailed. We then examine a proposition by Johnson and Sehgal pertaining to these k-characters and disprove its converse. Probabilistic models can be applied to random walks on the Cayley groups of finite order. We examine random walks on dihedral groups which converge after a finite number of steps to the random walk induced by the uniform distribution. We present both sufficient and necessary conditions for such convergence and analyze aspects of algebraic geometry related to this subject.

Analysis Of Multiple Collision-Based Periodic Orbits In Dimension Higher Than One, Skyler C. Simmons

Theses and Dissertations

We exhibit multiple periodic, collision-based orbits of the Newtonian n-body problem. Many of these orbits feature regularizable collisions between the masses. We demonstrate existence of the periodic orbits after performing the appropriate regularization. Stability, including linear stability, for the orbits is then computed using a technique due to Roberts. We point out other interesting features of the orbits as appropriate. When applicable, the results are extended to a broader family of orbits with similar behavior.

Evaluation And Refinement Of Generalized B-Splines, Ian Daniel Henriksen

Theses and Dissertations

In this thesis a method for direct evaluation of Generalized B-splines (GB-splines) via the representation of these curves as piecewise functions is presented. A local structure is introduced that makes the GB-spline curves more amenable to the integration used in constructing bases of higher degree. This basis is used to perform direct computation of piecewise representation of GB-spline bases and curves. Algorithms for refinement using these local structures are also developed.

Octahedral Extensions And Proofs Of Two Conjectures Of Wong, Kevin Ronald Childers

Theses and Dissertations

Consider a non-Galois cubic extension K/Q ramified at a single prime p > 3. We show that if K is a subfield of an S_4-extension L/Q ramified only at p, we can determine the Artin conductor of the projective representation associated to L/Q, which is based on whether or not K/Q is totally real. We also show that the number of S_4-extensions of this type with K as a subfield is of the form 2^n - 1 for some n >= 0. If K/Q is totally real, n > 1. This proves two conjectures of Siman Wong.

Investigations Into Non-Degenerate Quasihomogeneous Polynomials As Related To Fjrw Theory, Scott C. Mancuso

Theses and Dissertations

The motivation for this paper is a better understanding of the basic building blocks of FJRW theory. The basics of FJRW theory will be briefly outlined, but the majority of the paper will deal with certain multivariate polynomials which are the most fundamental building blocks in FJRW theory. We will first describe what is already known about these polynomials and then discuss several properties we proved as well as conjectures we disproved. We also introduce a new conjecture suggested by computer calculations performed as part of our investigation.

The Fourier Coefficients Of Modular Forms, Kyle Pratt, Dr. Paul Jenkins

Journal of Undergraduate Research

Modular forms are complex analytic functions with remarkable properties. Modular forms possess interesting and surprising connections to many different branches of mathematics. For example, it is well-known that Andrew Wiles’ proof of Fermat’s Last Theorem, a conjecture that had been unresolved for more than three centuries, utilized modular forms in a crucial way.

Zeros Of Poincare Series Of Level 2, Andrew Haddock, Paul Jenkins

Journal of Undergraduate Research

Introduction Poincaré series are a certain type of modular form. Modular forms are complex-valued functions that satisfy certain symmetry properties. There are many different types of modular forms, and one way to classify modular forms is by their level, such as 1, 2, 3, etc. They are of much interest as a research subject because they are connected in surprising ways to many different fields in number theory—e.g., elliptic curves, quadratic forms, and partition functions, to name just a few. As we gain more insight into the various properties of modular forms, we gain more insight into how these various …