Physical Sciences and Mathematics Commons^™

A Few Problems On The Steiner Distance And Crossing Number Of Graphs, Josiah Reiswig

Theses and Dissertations

We provide a brief overview of the Steiner ratio problem in its original Euclidean context and briefly discuss the problem in other metric spaces. We then review literature in Steiner distance problems in general graphs as well as in trees.

Given a connected graph G we examine the relationship between the Steiner k-diameter, sdiam_k(G), and the Steiner k-radius, srad_k(G). In 1990, Henning, Oellermann and Swart [Ars Combinatoria 12 13-19, (1990)] showed that for any connected graph G, sdiam₃(G) ≤(8/5)srad₃(G) and …

Unconditionally Energy Stable Linear Schemes For A Two-Phase Diffuse Interface Model With Peng-Robinson Equation Of State, Chenfei Zhang

Theses and Dissertations

Many problems in the fields of science and engineering, particularly in materials science and fluid dynamic, involve flows with multiple phases and components. From mathematical modeling point of view, it is a challenge to perform numerical simulations of multiphase flows and study interfaces between phases, due to the topological changes, inherent nonlinearities and complexities of dealing with moving interfaces.

In this work, we investigate numerical solutions of a diffuse interface model with Peng-Robinson equation of state. Based on the invariant energy quadratization approach, we develop first and second order time stepping schemes to solve the liquid-gas diffuse interface problems for …

An Implementation Of The Kapustin-Li Formula, Jessica Otis

Theses and Dissertations

Let R be a regular local ring and take ! to be an isolated singularity on R. Taking Z/2-graded R-modules, X and Y , a matrix factorization of ω is a pair of morphisms (ϕ, ψ) such that ϕ⃝◦ψ = ω and ψ⃝◦ϕ = ω are satisfied in the diagram X ϕ → Y ψ→ X. We will discuss the category of matrix factorizations of ! in R and lead into the homotopy category of matrix factorizations as well as its historical development. Finally, we will conclude with the statement …

Finding Resolutions Of Mononomial Ideals, Hannah Melissa Kimbrell

Theses and Dissertations

In this paper we present two different combinatorial approaches to finding resolutions of polynomial ideals. Their goal is to get resolutions that are as small as possible while still preserving the structure of the zeroth syzygy module. Then we present the idea of a differential graded algebra and discuss when the minimal resolutions of a polynomial ideals admits such a structure.

Congruence Relations Mod 2 For (2 X 4^T + 1)-Colored Partitions, Nicholas Torello

Senior Theses

Let p_r(n) denote the difference between the number of r-colored partitions of n into an even number of distinct parts and into an odd number of distinct parts. Inspired by proofs involving modular forms of the Hirschhorn-Sellers Conjecture, we prove a similar congruence for p_r(n). Using the Jacobi Triple Product identity, we discover a much stricter congruence for p_3(n).

A New Characterization Of V-Posets, Peter Gartland

Theses and Dissertations

In 2016, Hasebe and Tsujie gave a recursive characterization of the set of induced N -free and bowtie-free posets; Misanantenaina and Wagner studied these orders fur- ther, naming them “V-posets”. Here we offer a new characterization of V-posets by introducing a property we refer to as autonomy. A poset P is said to be autonomous if there exists a directed acyclic graph D (with adjacency matrix U ) whose transitive closure is P, with the property that any total ordering of the vertices of D so that Gaussian elimination of UT U proceeds without row swaps is a …

On The Generators Of Quantum Dynamical Semigroups, Alexander Wiedemann

Theses and Dissertations

In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup.

In order to define this class we …

A Development Of Transfer Entropy In Continuous-Time, Christopher David Edgar

Theses and Dissertations

The quantification of causal relationships between time series data is a fundamen- tal problem in fields including neuroscience, social networking, finance, and machine learning. Amongst the various means of measuring such relationships, information- theoretic approaches are a rapidly developing area in concert with other methods. One such approach is to make use of the notion of transfer entropy (TE). Broadly speaking, TE is an information-theoretic measure of information transfer between two stochastic processes. Schreiber’s 2001 definition of TE characterizes information transfer as an informational divergence between conditional probability mass func- tions. The original definition is native to discrete-time stochastic processes …

Successful Pressing Sequences In Simple Pseudo-Graphs, Hays Wimsatt Whitlatch

Theses and Dissertations

Motivated by the study of genomes evolving by reversals, the primary topic of this thesis is “successful pressing sequences” in simple pseudo-graphs. Pressing sequences where first introduced by Hannenhali and Pevzner in 1999 where they showed that sorting signed permutation problem can be solved in polynomial time, therefore demonstrating that the length of a most parsimonious solution to the genome in- version only rearrangement problem can be determined efficiently.

A signed permutation is an integer permutation where each entry is given a sign: plus or minus. A reversal in a signed permutation is the operation of reversing a subword and …

Dynamical Entropy Of Quantum Random Walks, Duncan Wright

Theses and Dissertations

In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy.

Entropy is an essential notion in physics and information theory. Motivated by the study of …

On The Characteristic Polynomial Of A Hypergraph, Gregory J. Clark

Theses and Dissertations

We consider the computation of the adjacency characteristic polynomial of a uniform hypergraph. Computing the aforementioned polynomial is intractable, in general; however, we present two mechanisms for computing partial information about the spectrum of a hypergraph as well as a methodology (and in particular, an algo- rithm) for combining this information to determine complete information about said spectrum. The first mechanism is a generalization of the Harary-Sachs Theorem for hypergraphs which yields an explicit formula for each coefficient of the characteristic polynomial of a hypergraph as a weighted sum over a special family of its subgraphs. The second is a …

Classification Of Non-Singular Cubic Surfaces Up To E-Invariants, Mohammed Alabbood

Theses and Dissertations

In this thesis, we use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3, q) from 6 points in general position in PG(2, q) for q = 17, 19, 23, 29, 31. We classify the cubic surfaces with twenty-seven lines in three dimensions (up to e- invariants) by introducing computational and geometrical procedures for the classi- fication. All elliptic and hyperbolic lines on a non-singular cubic surface in PG(3, q) for q = 17, 19, 23, 29 …

An Examination Of Kinetic Monte Carlo Methods With Application To A Model Of Epitaxial Growth, Dylana Ashton Wilhelm

Theses and Dissertations

Through the assembly of procedural information about physical processes, the kinetic Monte Carlo method offers a simple and efficient stochastic approach to model the temporal evolution of a system. While suitable for a variety of systems, the approach has found widespread use in the simulation of epitaxial growth. Motivated by chem- ically reacting systems, we discuss the developments and elaborations of the kinetic Monte Carlo method, highlighting the computational cost associated with realizing a given algorithm. We then formulate a solid-on-solid bond counting model of epitax- ial growth which permits surface atoms to advance the state of the system through …

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 …

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 + …

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 …

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 …

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 …

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 …

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.

Subdivision Of Measures Of Squares, Dylan Bates

Theses and Dissertations

The primary goal of our work is to establish a method to relate simple measures to a given set of moments. We calculate the moments of squares via linear polynomial weight measures and straight line cuts and use this to calculate the centre of mass of the square. The one-to-one correspondence that is found is needed to represent surfaces with gaps, which can estimate arbitrary measures on squares. From this, a subdivision scheme is developed, which successively quadrisects squares and uses the relation to estimate the new measures in order to provide a good representation of the original surface. One …

Convergence And Rate Of Convergence Of Approximate Greedy-Type Algorithms, Anton Dereventsov

Theses and Dissertations

In this dissertation we study the questions of convergence and rate of convergence of greedy-type algorithms under imprecise step evaluations. Such algorithms are in demand as the issue of calculation errors appears naturally in applications.

We address the question of strong convergence of the Chebyshev Greedy Algorithm (CGA), which is a generalization of the Orthogonal Greedy Algorithm (also known as the Orthogonal Matching Pursuit), and show that the class of Banach spaces for which the CGA converges for all dictionaries and objective elements is strictly between smooth and uniformly smooth Banach spaces.

We analyze an application-oriented modification of the CGA, …

Deep Learning: An Exposition, Ryan Kingery

Theses and Dissertations

In this paper we describe and survey the field of deep learning, a type of machine learning that has seen tremendous growth and popularity over the past decade for its ability to substantially outperform other learning methods at important tasks. We focus on the problem of supervised learning with feedforward neural networks. After describing what these are we give an overview of the essential algorithms of deep learning, backpropagation and stochastic gradient descent. We then survey some of the issues that occur when applying deep learning in practice. Last, we conclude with an important application of deep learning to the …

Polynomials Of Small Mahler Measure With No Newman Multiples, Spencer Victoria Saunders

Theses and Dissertations

A Newman polynomial is a polynomial with coefficients in f0;1g and with constant term 1. It is known that the roots of a Newman polynomial must lie in the slit annulus fz 2C: f��1 1 such that if a polynomial f (z) 2 Z[z] has Mahler measure less than s and has no nonnegative real roots, then it must divide a Newman polynomial. In this thesis, we present a new upper bound on such a s if it exists. We also show that there are infinitely many monic polynomials that have distinct Mahler measures which all lie below f, have …