Mathematics Commons^™

A Weak Galerkin Method For Second Order Elliptic Equations With Polynomial Reductions, Nolisa Malluwawadu

Theses and Dissertations

The second order elliptic equation, which is also known as the diffusion-convection equation, is of great interest in many branches of physics and industry. In this dissertation, we use the weak Galerkin finite element method to study the general elliptic equation with Dirichlet boundary condition and the elliptic equation with mixed boundary conditions. Weak Galerkin finite element schemes are proposed and analyzed. These schemes feature piecewise polynomials of degree k ≥ 1 on each element and piecewise polynomials of degree k − 1 ≥ 0 on each edge or face of the element. Error estimates of optimal order are established …

3-Maps And Their Generalizations, Kevin J. Mccall

Theses and Dissertations

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

Graph Homomorphisms And Vector Colorings, Michael Robert Levet

Theses and Dissertations

A graph vertex coloring is an assignment of labels, which are referred to as colors, such that no two adjacent vertices receive the same color. The vertex coloring problem is NP-Complete [8], and so no polynomial time algorithm is believed to exist. The notion of a graph vector coloring was introduced as an efficiently computable relaxation to the graph vertex coloring problem [7]. In [6], the authors examined the highly symmetric class of 1-walk regular graphs, characterizing when such graphs admit unique vector colorings. We present this characterization, as well as several important consequences discussed in [5, 6]. By appealing …

Geometry Of Derived Categories On Noncommutative Projective Schemes, Blake Alexander Farman

Theses and Dissertations

Noncommutative Projective Schemes were introduced by Michael Artin and J.J. Zhang in their 1994 paper of the same name as a generalization of projective schemes to the setting of not necessarily commutative algebras over a commutative ring. In this work, we study the derived category of quasi-coherent sheaves associated to a noncommutative projective scheme with a primary emphasis on the triangulated equivalences between two such categories.

We adapt Artin and Zhang’s noncommutative projective schemes for the language of differential graded categories and work in Ho (dgcatk), the homotopy category of differential graded categories, making extensive use of Bertrand Toën’s Derived …

Quick Trips: On The Oriented Diameter Of Graphs, Garner Paul Cochran

Theses and Dissertations

In this dissertation, I will discuss two results on the oriented diameter of graphs with certain properties. In the first problem, I studied the oriented diameter of a graph G. Erdos et al. in 1989 showed that for any graph with |V | = n and δ(G) = δ the maximum the diameter could possibly be was 3 n/ δ+1. I considered whether there exists an orientation on a given graph with |G| = n and δ(G) = δ that has a small diameter. Bau and Dankelmann (2015) showed that there is an orientation of diameter 11 n/ δ+1 + …

Local Rings And Golod Homomorphisms, Thomas Schnibben

Theses and Dissertations

The Poincaré series of a local ring is the generating function of the Betti numbers for the residue field. The question of when this series represents a rational function is a classical problem in commutative algebra. Golod rings were introduced by Golod in 1962 and are one example of a class of rings that have rational Poincaré series. The idea was generalized to Golod homomorphisms by Levin in 1975.

In this paper we prove two homomorphisms are Golod. The first is a class of ideals such that the natural projection to the quotient ring is a Golod homomorphism. The second …

Special Fiber Rings Of Certain Height Four Gorenstein Ideals, Jaree Hudson

Theses and Dissertations

Let S be a set of four variables, k a field of characteristic not equal to two such that k contains all square roots, and I a height four Gorenstein ideal of k[S] generated by nine quadratics so that I has a Gorenstein-linear resolution. We define a complex X• so that each module of X• is the tensor product of a certain polynomial ring Q in nine variables and a direct sum of indecomposable k[Sym(S)]-modules and the differential maps are Q- and k[Sym(S)]-module homomorphisms. Work with the Macaulay2 software suggests that H0(X•) is the special fiber ring of I and …

Turán Problems And Spectral Theory On Hypergraphs And Tensors, Shuliang Bai

Theses and Dissertations

Turán problems on uniform hypergraphs have been actively studied for many decades. However, on non-uniform hypergraphs, these problems are rarely considered. We refer a non-uniform hypergraph as an R-hypergraph where R is the set of cardinalities of all edges. An R-graph H is called degenerate if it has the smallest Turán density |R(H)|−1. What do the degenerate R-graphs look like? For the special case R = {r}, the answer to this question is simple: they are r-partite r-uniform hypergraphs. However, it is more intrigue for the other cases. A degenerate hypergraph is called trivial if it is contained in the …

A Quest For Positive Definite Matrices Over Finite Fields, Erin Patricia Hanna

Theses and Dissertations

Positive definite matrices make up an interesting and extremely useful subset of Hermitian matrices. They are particularly useful in exploring convex functions and finding minima for functions in multiple variables. These matrices admit a plethora of equivalent statements and properties, one of which is an existence of a unique Cholesky decomposition. Positive definite matrices are not usually considered over finite fields as some of the definitions and equivalences are quickly seen to no longer hold. Motivated by a result from the theory of pressing sequences, which almost mirrors an equivalent statement for positive definite Hermitian matrices, we consider whether any …

States And The Numerical Range In The Regular Algebra, James Patrick Sweeney

Theses and Dissertations

In this dissertation we investigate the algebra numerical range defined by the Banach algebra of regular operators on a Dedekind complete complex Banach lattice, i.e., V (Lr(E), T) = {Φ(T) : Φ ∈ Lr(E)∗, ||Φ|| = 1 = Φ(I)}. For T in the center Z(E) of E we prove that V (Lr(E), T) = co(σ(T)). For T ⊥ I we prove that V (Lr(E), T) is a disk centered at the origin. We then consider the part of V (Lr(E), T) obtained by restricting ourselves to positive states Φ ∈ Lr(E)∗. In this case we show that we get a …

Classical And Quantum Kac’S Chaos, Rade Musulin

Theses and Dissertations

In 1956 Kac studied the Boltzmann equation, an integro-differential equation which describes the density function of the distribution of the velocities of the molecules of dilute monoatomic gases under the assumption that the energy is only transferred via collisions between the molecules. In an attempt at a solution to the Boltzmann equation, Kac introduced a property of the density function that he called the “Boltzmann property" which describes the behavior of the density function at a given fixed time as the number of particles tends to infinity. The Boltzmann property has been studied extensively since then, and has been abstracted …

Theory, Computation, And Modeling Of Cancerous Systems, Sameed Ahmed

Theses and Dissertations

This dissertation focuses on three projects. In Chapter 1, we derive and implement the compact implicit integration factor method for numerically solving partial differential equations. In Chapters 2 and 3, we generalize and analyze a mathematical model for the nonlinear growth kinetics of breast cancer stem cells. And in Chapter 4, we develop a novel mathematical model for the HER2 signaling pathway to understand and predict breast cancer treatment. Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential …

Parametric And Non-Parametric Regression Models With Applications To Climate Change, Osita Eluemuno Onyejekwe

Theses and Dissertations

In this dissertation we have studied the climate factors that contribute to climate change using univariate and multivariate parametric methods as well as nonparametric models. In this study, we have three major contributions. First, the extent of mountain glaciers around the globe and their responses to climate factors are investigated using multivariate methods and we have proposed a predictive model to estimate the mountain glacier response to climate factors. Second, we have addressed the important problem of bandwidth selection in presence of correlated noise in nonparametric regression analysis. We have proposed a denoising method based on an ensemble bandwidth optimization …

On Logconcavity Of Multivariate Discrete Distributions, Majed Ghazi Alharbi

Theses and Dissertations

The contribution of this dissertation to the literature is twofold. First, we use a geometric perspective to present all possible subdivisions of R³ into tetrahedra with disjoint interiors and adopt a combinatorial approach to obtain a special subdivision of Rⁿ into simplices with disjoint interiors, where two simplices are called neighbors if they share a common facet. We then use the neighborhood relationship of the simplices in each subdivision to fully describe the sufficient conditions for the strong unimodality/logconcavity of the trivariate discrete distributions and further extend these results to present a new sufcient condition for the strong unimodality/logconcavity of …

Hankel Partial Contraction, Contractive Completion, Moore-Penrose Inverse, Extremal Case, Manuel A. Villarreal Jr.

Theses and Dissertations

In this article we find concrete necessary and sufficient conditions for the existence of contractive completions of Hankel partial contractions of size 3x3 non-extremal case.

Modeling The Influence Of El Niño On Parasite Transmission In Sand Crab Populations And Seabird Abundance Along The Californian Coast, Aboubacar Dio Seck

Theses and Dissertations

Pacific mole crabs (Emerita analoga) are one of the most important and abundant

invertebrates in sandy beach environments. Consequently, they are a common food source

for seabirds and sea otters. Since the mole crab serves as the primary intermediate host for

acanthocephalan parasites, they have been linked to a number of mortality events. It is

currently estimated that 13-16% of deaths in the threatened California sea otter population

have been caused by infection. In addition, unusually high loads of acanthocephalan

parasites have been linked to episodic deaths of thousands of surf scoters. Studies suggest

that acanthocephalan development and transmission may …

Spaces Of Weakly Holomorphic Modular Forms In Level 52, Daniel Meade Adams

Theses and Dissertations

Let M^#_k(52) be the space of weight k level 52 weakly holomorphic modular forms with poles only at infinity, and S^#_k(52) the subspace of forms which vanish at all cusps other than infinity. For these spaces we construct canonical bases, indexed by the order of vanishing at infinity. We prove that the coefficients of the canonical basis elements satisfy a duality property. Further, we give closed forms for the generating functions of these basis elements.

Weakly Holomorphic Modular Forms In Level 64, Christopher William Vander Wilt

Theses and Dissertations

Let M^#_k(64) be the space of weakly holomorphic modular forms in level 64 and weight k which can have poles only at infinity, and let S^#_k(64) be the subspace of M^#_k(64) consisting of forms which vanish at all cusps other than infinity. We explicitly construct canonical bases for these spaces and show that the coefficients of these basis elements satisfy Zagier duality. We also compute the generating function for the canonical basis.

Busemann G-Spaces, Cat(K) Curvature, And The Disjoint (0, N)-Cells Property, Clarke Alexander Safsten

Theses and Dissertations

A review of geodesics and Busemann G-spaces is given. Aleksandrov curvature and the disjoint (0, n)-cells property are defined. We show how these properties are applied to and strengthened in Busemann G-spaces. We examine the relationship between manifolds and Busemann G-spaces and prove that all Riemannian manifolds are Busemann G-spaces, though not all metric manifolds are Busemann G-spaces. We show how Busemann G-spaces that also have bounded Aleksandrov curvature admit local closest-point projections to geodesic segments. Finally, we expound local properties of Busemann G-spaces and define a new property which we call the symmetric property. We show that Busemann …

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 …

Time Dependent Solution For An Incompressible Viscous Fluid Flow In A Cavity, Bishnu Parajuli

Theses and Dissertations

The Finite Element Method is used to find the time dependent solution for an incompressible viscous fluid flow in a cavity in two dimension. The governing equations for this system are the continuity equation and the momentum equation. We use finite element method to obtain the dependent variables. Here we derive weak formulation and develop finite element model by using Galerkin Method. Then we compute velocity components of the fluid for a square cavity. The numerical results are presented.

Multi-Type Branching Processes Model Of Nosocomial Epidemic, Zeinab Nageh Mohamed

Theses and Dissertations

The potency of an infectious disease to spread between different types of susceptible individuals in a hospital determines the fate of controlling nosocomial epidemics. I use a multi-type branching process with a joint negative binomial offspring distribution to study nosocomial epidemics. In particular, I estimate the basic reproduction number R0 and study its relationship with the offspring distribution’s parameters at different and fixed number of generations. Also, I study the effect of contact tracing on estimates of R0.

A New Approach To Ramanujan's Partition Congruences, Mayra C. Huerta

Theses and Dissertations

MacMahon provided Ramanujan and Hardy a table of values for p(n) with the partitions of the first 200 integers. In order to make the table readable, MacMahon grouped the entries in blocks of five. Ramanujan noticed that the last entry in each block was a multiple of 5. This motivated Ramanujan to make the following conjectures, p(5n+4) ≡ 0 (mod 5); p(7 n+5) ≡ 0 (mod 7); p(11n+6) ≡ 0 (mod 11) which he eventually proved.

The purpose of this thesis is to give new proofs for Ramanujan's partition …

Investigating The Mathematical Dispositions And Self-Efficacy For Teaching Mathematics Of Preservice Teachers, Jasmine M. Cruz

Theses and Dissertations

The study of the individual's beliefs and the role and influence they have on the individual's actions and behaviors, have long been examined and investigated by educators and psychologists. Moreover, researchers have overwhelmingly claimed and demonstrated that the beliefs held by teachers significantly influences their behavior and educational practices in the classroom. This thesis study investigates the mathematical disposition and self-efficacy for teaching mathematics of preservice teachers. The study's primary goals are to discover if there is a relationship or association between a teacher's mathematical disposition(MD) and his/her self-efficacy for teaching mathematics (SEFTM), and if there are significant differences between …

Problem Book On Higher Algebra And Number Theory, Ryanto Putra

Theses and Dissertations

This book is an attempt to provide relevant end-of-section exercises, together with their step-by-step solutions, to Dr. Zieschang's classic class notes Higher Algebra and Number Theory. It's written under the notion that active hands-on working on exercises is an important part of learning, whereby students would see the nuance and intricacies of a math concepts which they may miss from passive reading. The problems are selected here to provide background on the text, examples that illuminate the underlying theorems, as well as to fill in the gaps in the notes.

Mathematical Modeling Of Mers-Cov Nosocomial Epidemic, Adriana Quiroz

Theses and Dissertations

This thesis concerns about the analysis and modeling of spread of an infectious disease inside a hospital. We begin from the basic knowledge of the simple models: SIR and SEIR, to show an appropriate understanding of the epidemic dynamic process. We consider the Middle East Respiratory Syndrome Corona Virus (MERS-CoV), in Saudi Arabia, to introduce MERS-CoV SEIR ward model by developing different systems of equations in each ward (unit). We use the Next Generation Matrix method to calculate the basic reproduction number R0. Simulations of different scenarios are done using different combination of parameters.

To model MERS-CoV we established …

Disease Modeling Using Fractional Differential Equations And Estimation, Daniel P. Medina

Theses and Dissertations

Ordinary differential equations has been the most conventional approach when modeling spread of infectious diseases. Effective research has shown that using fractional-order differentiation can be a very useful and efficient extension for some mathematical models. In this thesis, fractional calculus is used to depict an SEIR model with a system of fractional-order differential equations. I also simulate the fractional-order SEIR using integer-order numerical methods. I also establish the estimation framework and show that it is accurately working.

Coupled Telegraph And Sir Model Of Information And Diseases, Jose De Jesus Galarza

Theses and Dissertations

In this work, the effect of information propagation on disease spread and vaccination uptake through networks is studied. In this model the information reaches different people at different distances from the center of information containing the health data. We use a pair of Telegraph equations to depict the vaccine and disease information propagation on a network embedded into a straight line. The Telegraph equation is coupled with an SIR (Susceptible-Infected-Recovered) model to examine the anticipated mutual influence. Numerical simulations and stability analysis were made to study the model. We show how the propagation of information about the disease impacts the …

On The Existence Of Non-Free Totally Reflexive Modules, J. Cameron Atkins

Theses and Dissertations

For a standard graded Cohen-Macaulay ring R, if the quotient R/(x) admits nonfree totally reflexive modules, where x is a system of parameters consisting of elements of degree one, then so does the ring R. A non-constructive proof of this statement was given in [16]. We give an explicit construction of the totally reflexive modules over R obtained from those over R/(x).

We consider the question of which Stanley-Reisner rings of graphs admit nonfree totally reflexive modules and discuss some examples. For an Artinian local ring (R,m) with m3 = 0 and containing the complex numbers, we describe an explicit …

A Family Of Simple Codimension Two Singularities With Infinite Cohen-Macaulay Representation Type, Tyler Lewis

Theses and Dissertations

A celebrated theorem of Buchweitz, Greuel, Knörrer, and Schreyer is that the hypersurface singularities of finite representation type, i.e. the hypersurface singularities admitting only finitely many indecomposable maximal Cohen-Macaulay modules, are exactly the ADE singularities. The codimension 2 singularities that are the analogs of the ADE singularities have been classified by Frühbis-Krühger and Neumer, and it is natural to expect an analogous result holds for these singularities. In this paper, I will present a proof that, in contrast to hypersurfaces, Frühbis-Krühger and Neumer’s singularities include a subset of singularities of infinite representation type.