Physical Sciences and Mathematics Commons^™

Thermodynamically Consistent Hydrodynamic Phase Field Models And Numerical Approximation For Multi-Component Compressible Viscous Fluid Mixtures, Xueping Zhao

Theses and Dissertations

Material systems comprising of multi-component, some of which are compressible, are ubiquitous in nature and industrial applications. In the compressible fluid flow, the material compressibility comes from two sources. One is the material compressibility itself and another is the mass-generating source. For example, the compressibility in the binary fluid flows of non-hydrocarbon (e.g. Carbon dioxide) and hydrocarbons encountered in the enhanced oil recovery (EOR) process, comes from the compressibility of the gas-liquid mixture itself. Another example of the mixture of compressible fluids is growing tissue, in which cell proliferation and cell migration make the material volume changes so that it …

Identifying Prevalent Mathematical Pathways To Engineering In South Carolina, Eliza Gallagher, Christy Brown, D. Andrew Brown, Kristin Kelly Frady, Patrick Bass, Michael A. Matthews, Thomas T. Peters, Robert J. Rabb, Ikhalfani Solan, Ronald W. Welch, Anand K. Gramopadhye

Faculty Publications

National data indicate that initial mathematics course placement in college is a strong predictor of persistence to degree in engineering, with students placed in calculus persisting at nearly twice the rate of those placed below calculus. Within the state of South Carolina, approximately 95% of engineering-intending students who initially place below calculus are from in-state. In order to make systemic change, we are first analyzing system-wide data to identify prevalent educational pathways within the state, and the mathematical milestones along those pathways taken by students in engineering and engineering-related fields. This paper reports preliminary analysis of that data to understand …

Numerical Methods For A Two-Species Competition-Diffusion Model With Free Boundaries, Shuang Liu, Xinfeng Liu

Faculty Publications

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce …

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 + …

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 …

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 …

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 …

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 …

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 …