Physical Sciences and Mathematics Commons^™

Widely Digitally Delicate Brier Primes And Irreducibility Results For Some Classes Of Polynomials, Thomas David Luckner

Theses and Dissertations

This dissertation considers three different sections of results. In the first part of the dissertation, a result on consecutive primes which are widely digitally delicate and Brier numbers is discussed. Making use of covering systems and a theorem of D. Shiu, M. Filaseta and J. Juillerat showed that for every positive integer k, there exist k consecutive widely digitally delicate primes. They also noted that for every positive integer k, there exist k consecutive primes which are Brier numbers. We show that for every positive integer k, there exist k consecutive primes that are both widely digitally …

Deep Learning Methods For Some Problems In Scientific Computing, Yuankai Teng

Theses and Dissertations

Deep learning has emerged as a powerful approach for solving complex problems in scientific computing due to the increasing availability of large-scale data and computational resources. This thesis explores the potential of deep learning methods for three specific problems in scientific computing: (i) reducing the dimensions of variables in function approximation, (ii) solving linear reaction-diffusion equations, and (iii) finding the parametric representations of parameters in the numerical schemes for solving time-dependent partial differential equations.

For the first problem, a novel deep learning architecture is developed for reducing the dimensions of variables in function approximation. The proposed method achieves state-of-the-art performance …

Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon

Senior Theses

This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.

Extreme Covering Systems, Primes Plus Squarefrees, And Lattice Points Close To A Helix, Jack Robert Dalton

Theses and Dissertations

This dissertation considers three different topics.

In the first part, we prove that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also, if the least modulus is 3, the least common multiple of the moduli is at least 120; finally, if the least modulus is 4, the least common multiple of the moduli is at least 360. The constants 60, 120, and 360 are best possible, they cannot be replaced by larger constants. We also show that there do not exist distinct covering systems with all of the moduli in …

Poset Ramsey Numbers For Boolean Lattices, Joshua Cain Thompson

Theses and Dissertations

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets P, P', define the poset Ramsey number R(P,P') to be the least N such that for any red/blue coloring of the elements of Q_N, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to P' with all elements blue.

Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q₂, Q_n) ≤ 2n + 2 and R(Q …

Structure Preserving Reduced-Order Models Of Hamiltonian Systems, Megan Alice Mckay

Theses and Dissertations

Large-scale dynamical systems are expensive to simulate due to the computational cost accrued y the substantial number of degrees of freedom. To accelerate repeated numerical simulations of the systems, proper orthogonal decomposition reduced order models (POD-ROMs) have been developed. When applied to Hamiltonian systems, however, special care must be taken when performing the reduced order modeling to keep their energy-preserving nature. This work presents a survey of several structure-preserving reduced order models (SP-ROMs). In addition, this work employs the discrete empirical interpolation method (DEIM) and develops an SP-DEIM model for nonlinear Hamiltonian systems. The wave equation is considered as a …

Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock

Theses and Dissertations

This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0 \pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if $\bfrac{n}{5} = 1$ and $\bfrac{n}{7} = …

Covering Systems And The Minimum Modulus Problem, Maria Claire Cummings

Theses and Dissertations

A covering system or a covering is a set of linear congruences such that every integer satisfies at least one of these congruences. In 1950, Erdős posed a problem regarding the existence of a finite covering with distinct moduli and an arbitrarily large minimum modulus. This remained unanswered until 2015 when Robert Hough proved an explicit bound of 1016 for the minimum modulus of any such covering. In this thesis, we examine the use of covering systems in number theory results, expand upon the proof of the existence of an upper bound on the minimum modulus in the case of …

Tangled Up In Tanglegrams, Drew Joseph Scalzo

Theses and Dissertations

Tanglegrams are graphs consisting of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. A Tanglegram drawing is a special way of drawing a Tanglegram; and a Tanglegram is called planar if it has a drawing such that the matching edges do not cross. In this thesis, we will discuss various results related to the construction and planarity of Tanglegrams, as well as demonstrate how to construct all the Tanglegrams of size 4 by looking at two types of rooted binary trees - Caterpillar and Complete Binary Trees. After …

Quadratic Neural Network Architecture As Evaluated Relative To Conventional Neural Network Architecture, Reid Taylor

Senior Theses

Current work in the field of deep learning and neural networks revolves around several variations of the same mathematical model for associative learning. These variations, while significant and exceptionally applicable in the real world, fail to push the limits of modern computational prowess. This research does just that: by leveraging high order tensors in place of 2nd order tensors, quadratic neural networks can be developed and can allow for substantially more complex machine learning models which allow for self-interactions of collected and analyzed data. This research shows the theorization and development of mathematical model necessary for such an idea to …

The Existence And Quantum Approximation Of Optimal Pure State Ensembles, Ryan Thomas Mcgaha

Theses and Dissertations

In this manuscript we study entanglement measures defined via the convex roof construction. In the first chapter we build the notion of an entanglement measure from the ground up and discuss various issues that arise when trying to measure the amount of entanglement present in an arbitrary density operator. Through this introduction we will motivate the use of the convex roof construction. In the second chapter we will show that the infimum in the convex roof construction is achieved for a specific set of entanglement measures and provide canonical examples of such measures. We also describe LOCC operations via a …

Results On Select Combinatorial Problems With An Extremal Nature, Stephen Smith

Theses and Dissertations

This dissertation is split into three sections, each containing new results on a particular combinatorial problem. In the first section, we consider the set of 3-connected quadrangulations on n vertices and the set of 5-connected triangulations on n vertices. In each case, we find the minimum Wiener index of any graph in the given class, and identify graphs that obtain this minimum value. Moreover, we prove that these graphs are unique up to isomorphism.

In the second section, we work with structures emerging from the biological sciences called tanglegrams. In particular, our work pertains to an invariant of tanglegrams called …

Simulation Of Pituitary Organogenesis In Two Dimensions, Chace E. Covington

Theses and Dissertations

The pituitary gland is a vital part of the endocrine system found in all vertebrates and is responsible for the production of hormones that influence many physiological processes in the organism’s body. Although much has been learned of pituitary organogenesis, studying the dynamics of the cells in the developing pituitary gland is difficult. Pituitary organogenesis has been studied through “snapshots” of a developing pituitary gland by removing and viewing the pituitary glands of different specimens. Thus, how the individual cells in the developing pituitary gland behave and interact with one another is not fully understood. To aid in understanding pituitary …

Trimming Complexes, Keller Vandebogert

Theses and Dissertations

We produce a family of complexes called trimming complexes and explore applications. We first study ideals defining type 2 compressed rings with socle minimally generated in degrees s and 2s − 1 for s > 2. We prove that all such ideals arise as trimmings of grade 3 Gorenstein ideals and show that trimming complexes yield an explicit free resolution. In particular, we give bounds on parameters arising in the Tor-algebra classification and construct explicit ideals attaining all intermediate values for every s. This partially answers a question of realizability of Tor-algebra structures posed by Avramov. Next, we study how …

Polynomials, Primes And The Pte Problem, Joseph C. Foster

Theses and Dissertations

This dissertation considers three different topics. In the first part of the dissertation, we use Newton Polygons to show that for the arithmetic functions g(n) = n t , where t ≥ 1 is an integer, the polynomials defined with initial condition P g 0 (X) = 1 and recursion P g n (X) = X n Xn k=1 g(k)P g n−k (X) are X/ (n!) times an irreducible polynomial. In the second part of the dissertation, we show that, for 3 ≤ n ≤ 8, there are infinitely many 2-adic integer solutions to the Prouhet-Tarry-Escott (PTE) problem, that are …

A Numerical Investigation Of Fractional Models For Viscoelastic Materials With Applications On Concrete Subjected To Extreme Temperatures, Murray Macnamara

Theses and Dissertations

Materials exhibiting both elastic and viscous properties have been termed the name viscoelastic materials and have been modeled using a combination of integer order derivatives affixed in varying ways called viscoelastic models. This results in highly complicated numerical procedures necessitating highly expensive computational time which we will show. To that end the use of fractional derivatives were researched and determined to be the ideal solution for modeling these materials, of which this paper is focused on exploring. Such research began as a theoretical study, however over time the applied benefits were discovered and utilized and have since been expanded on, …

Variable-Order Fractional Partial Differential Equations: Analysis, Approximation And Inverse Problem, Xiangcheng Zheng

Theses and Dissertations

Variable-order fractional partial differential equations provide a competitive means in modeling challenging phenomena such as the anomalous diffusion and the memory effects and thus attract widely attentions. However, variable-order fractional models exhibit salient features compared with their constant-order counterparts and introduce mathematical and numerical difficulties that are not common in the context of integer-order and constant-order fractional partial differential equations.

This dissertation intends to carry out a comprehensive investigation on the mathematical analysis and numerical approximations to variable-order fractional derivative problems, including variable-order time-fractional, space-fractional, and space-time fractional partial differential equations, as well as the corresponding inverse problems. Novel techniques …

Diameter Of 3-Colorable Graphs And Some Remarks On The Midrange Crossing Constant, Inne Singgih

Theses and Dissertations

The first part of this dissertation discussing the problem of bounding the diameter of a graph in terms of its order and minimum degree. The initial problem was solved independently by several authors between 1965 − 1989. They proved that for fixed δ ≥ 2 and large n, diam(G) ≤ 3n+ O(1). In 1989, Erdős, Pach, Pollack, and Tuza conjectured that the upper bound on the diameter can be improved if G does not contain a large complete subgraph K_k.

Let r, δ ≥ 2 be fixed integers and let G be a connected graph with n vertices …

Finite Axiomatisability In Nilpotent Varieties, Joshua Thomas Grice

Theses and Dissertations

Study of general algebraic systems has long been concerned with finite basis results that prove finite axiomatisability of certain classes of general algebras. In the 1970’s, Bjarni Jónsson speculated that a variety generated by a finite algebra might be finitely based provided the variety has a finite residual bound (that is, a finite bound on the cardinality of subdirectly irreducible algebras in the variety). As such, most finite basis results since then have had the hypothesis of a finite residual bound. However, Jónsson also speculated that it might be sufficient to replace the finite residual bound with the weaker hypothesis …

An Ensemble-Based Projection Method And Its Numerical Investigation, Shuai Yuan

Theses and Dissertations

In many cases, partial differential equation (PDE) models involve a set of parameters whose values may vary over a wide range in application problems, such as optimization, control and uncertainty quantification. Performing multiple numerical simulations in large-scale settings often leads to tremendous demands on computational resources. Thus, the ensemble method has been developed for accelerating a sequence of numerical simulations. In this work we first consider numerical solutions of Navier-Stokes equations under different conditions and introduce the ensemblebased projection method to reduce the computational cost. In particular, we incorporate a sparse grad-div stabilization into the method as a nonzero penalty …

Connections Between Extremal Combinatorics, Probabilistic Methods, Ricci Curvature Of Graphs, And Linear Algebra, Zhiyu Wang

Theses and Dissertations

This thesis studies some problems in extremal and probabilistic combinatorics, Ricci curvature of graphs, spectral hypergraph theory and the interplay between these areas. The first main focus of this thesis is to investigate several Ramsey-type problems on graphs, hypergraphs and sequences using probabilistic, combinatorial, algorithmic and spectral techniques:

• The size-Ramsey number Rˆ(G, r) is defined as the minimum number of edges in a hypergraph H such that every r-edge-coloring of H contains a monochromatic copy of G in H. We improved a result of Dudek, La Fleur, Mubayi and Rödl [ J. Graph Theory 2017 ] on the size-Ramsey number …

Preparing For The Future: The Effects Of Financial Literacy On Financial Planning For Young Professionals, Tanay Singh

Senior Theses

Purpose – Many people between the age of 20 and 34 have not considered planning financially for the future in any significant capacity and in doing so, they limit their potential savings. The purpose of this study is to examine what financial expectations are for people in the early stages of their career and determine if improving financial literacy and revealing financial realities helps to produce more accurate or realistic expectations. Ultimately, the goal is to better prepare participants in the study for the working world and increased responsibilities outside of the college/university environment by getting them to start thinking …

Counting Number Fields By Discriminant, Harsh Mehta

Theses and Dissertations

The central topic of this dissertation is counting number fields ordered by discriminant. We fix a base field k and let Nd(k,G;X) be the number of extensions N/k up to isomorphism with Nk/Q(dN/k) ≤ X, [N : k] = d and the Galois closure of N/k is equal to G.

We establish two main results in this work. In the first result we establish upper bounds for N_|G| (k,G;X) in the case that G is a finite group with an abelian normal subgroup. Further, we establish upper bounds for the case N _|F| (k,G;X) where G is a Frobenius …

Rationality Questions And The Derived Category, Alicia Lamarche

Theses and Dissertations

This document is roughly divided into four chapters. The first outlines basic preliminary material, definitions, and foundational theorems required throughout the text. The second chapter, which is joint work with Dr. Matthew Ballard, gives an example of a family of Fano arithmetic toric varieties in which the derived category is able to detect the existence of k-rational points. More succinctly, we show that if X is a generalized del Pezzo variety defined over a field k, then X contains a k-rational point (and is in fact k-rational, that is, birational to Pnk ) if and only if Db(X) admits a …

Distance Related Graph Invariants In Triangulations And Quadrangulations Of The Sphere, Trevor Vincent Olsen

Theses and Dissertations

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. I provide asymptotic upper bounds and sharp lower bounds for the Wiener index of simple triangulations and quadrangulations with given connectivity. Additionally, I make conjectures for the extremal triangulations and quadrangulations which maximize the Wiener index based on computational evidence. If σ(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness and proximity of G are defined as the largest and smallest value of σ(v) over all vertices v of G, respectively. …

Windows And Generalized Drinfeld Kernels, Robert R. Vandermolen

Theses and Dissertations

We develop a generalization of a construction of Drinfeld, first inspired by the Qconstruction of Ballard, Diemer, and Favero. We use this construction to provide kernels for Grassmann flops over an arbitrary field of characteristic zero. In the case of Grassmann flops this generalization recovers the kernel for a Fourier-Mukai functor on the derived category of the associated global quotient stack studied by Buchweitz, Leuschke, and Van den Bergh. We show an idempotent property for this kernel, which after restriction, induces a derived equivalence over any twisted form of a Grassmann flop.

Two Inquiries Related To The Digits Of Prime Numbers, Jeremiah T. Southwick

Theses and Dissertations

This dissertation considers two different topics. In the first part of the dissertation, we show that a positive proportion of the primes have the property that if any one of their digits in base 10, including their infinitely many leading 0 digits, is replaced by a different digit, then the resulting number is composite. We show that the same result holds for bases b 2 {2, 3, · · · , 8, 9, 11, 31}.

In the second part of the dissertation, we show for an integer b ≥ 5 that if a polynomial ƒ( x) with non-negative coefficients …

Moving Off Collections And Their Applications, In Particular To Function Spaces, Aaron Fowlkes

Theses and Dissertations

The main focus of this paper is the concept of a moving off collection of sets. We will be looking at how this relatively lesser known idea connects and interacts with other more widely used topological properties. In particular we will examine how moving off collections act with the function spaces Cp(X), C0(X), and CK (X). We conclude with a chapter on the Cantor tree and its moving off connections.

Many of the discussions of important theorems in the literature are expressed in terms that do not suggest the concept …

Numerical Methods For A Class Of Reaction-Diffusion Equations With Free Boundaries, Shuang Liu

Theses and Dissertations

The spreading behavior of new or invasive species is a central topic in ecology. The modelings of free boundary problems are widely studied to better understand the nature of spreading behavior of new species. From mathematical modeling point of view, it is a challenge to perform numerical simulations of free boundary problems, due to the moving boundary, the stiffness of the system and topological changes.

In this work, we design numerical methods to investigate the spreading behavior of new species for a diffusive logistic model with a free boundary and a diffusive competition system with free boundaries. We develop a …

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 …