Theses/Dissertations - Mathematics

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    Chaos on non-compact metric spaces.
    (2023-08) Mohn, Jasmin, 1995-; Raines, Brian Edward, 1975-
    Distributional chaos, ω-chaos, and the specification property have been the subject of much inquiry in dynamical systems in the last 50 years. Several results link the specification property to both distributional chaos and ω-chaos in compact dynamical systems. We focus our study of distributional chaos on a shift space over a countable alphabet with the product topology and the usual shift map, which is known as the Baire Space. We prove that on the Baire Space subshifts of finite type exhibit dense distributional chaos and subshifts of bounded type that are perfect and have a dense set of periodic points also have distributional chaos. We also show that in this context a subshift with a generalized form of the specification property must have distributional chaos. For ω-chaos, we focus on Lindelöf dynamical systems. In particular, if a system exhibits a generalized version of the specification property and has at least three points with mutually separated orbit closures, then the system exhibits dense ω-chaos.
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    A polynomial approximation to the inverse of a matrix.
    (2023-08) Henningsen, Joel A., 1992-; Morgan, Ronald Benjamin,1958-
    We develop a surprisingly accurate polynomial approximation to the inverse of a large matrix, i.e. p(A) ⇡ A1. This polynomial is implemented using the roots of the GMRES polynomial, or harmonic Ritz values, and can be used to solve multiple right-hand sides as well as trace estimates to the inverse of a matrix. Issues with stability that arise with applying a high-degree polynomial are discussed. In addition, we give bounds on the error of our approximate inverse and illustrate how accurate the approximation can be in practice.
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    From efficient high-order methods to scientific machine learning.
    (May 2023) Marchena-Menendez, Jorge, 1996-; Kirby, Robert C.
    We aim to improve and solve some limitations of the current Finite Element Methods (FEM) packages. We focus on the efficient implementation of implicit time discretization schemes, achieving computational savings by exploiting the structure of serendipity elements and developing a wrapper that integrates Machine Learning with FEM packages. First, we focus on implicit Runge-Kutta methods which possess high-order accuracy and important stability properties. Implementation difficulties and the high expense of solving the coupled algebraic system at each time step are frequently cited as impediments. We present Irksome, a high-level library for manipulating UFL (Unified Form Language) expressions of semidiscrete variational forms to obtain UFL expressions for the coupled Runge-Kutta stage equations at each time step. Irksome works with the Firedrake package to enable the efficient solution of the resulting coupled algebraic systems. Secondly, we develop some additive Schwarz methods (ASM) based on solving local patch problems with serendipity elements, allowing us to obtain the same order of accuracy as rectangular tensor-product elements with many fewer degrees of freedom (DoFs). Adapting arguments from Pavarino for the tensor-product case, we prove that patch smoothers give conditioning estimates independent of the polynomial degree for a model problem. We combine this with a low-order global operator to provide an optimal two-grid method, with conditioning estimates independent of the mesh size and polynomial degree. The theory holds for two- and three-dimensional serendipity elements and can be extended to multigrid algorithms. Lastly, we introduce Torchfire, a differentiable programming interface that combines PyTorch and Firedrake to perform model-constrained deep learning of solutions of parameterized PDEs and PDE-constrained inverse problems. It leverages PyTorch's high-level interface for training neural networks, its automatic differentiation module for computing derivatives, and Firedrake's capabilities for computing finite element residuals. Torchfire provides differentiable wrappers of Firedrake-based computations for use in PyTorch, enabling users to write Firedrake code linked to the network without causing a disconnection of the graph for backpropagation. Numerical experiments using Firedrake, PETSc, and PyTorch, respectively confirm our theory and demonstrate the efficacy of the software and our solver techniques for each case.
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    Hubbard trees and their properties.
    (May 2023) Hammon, Cordell, 1996-; Meddaugh, Jonathan.
    A Hubbard tree is a set of points at the core of dendritic Julia sets. These trees encapsulate all the information about the larger Julia set, but in a much smaller, easier to understand structure. We discuss the structure of Hubbard trees, in particular, we provide a useful definition of branch point and endpoint. Afterwards, we demonstrate the existence of some trees that cannot be Hubbard trees in any meaningful context. Later, we turn our attention to the structure of inverse limits of Hubbard trees, making use of the definition of branch point and endpoint tenured earlier in the work and demonstrate that one Hubbard tree, with minor alterations, can generate infinitely many mutually non-homeomorphic inverse limits.
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    On certain results of Fourier analysis on non-abelian discrete groups.
    (August 2022) Chuah, Chian Yeong, 1991-; Mei, Tao, 1974-
    The topics of this thesis lie in the intersection of harmonic analysis, functional analysis and operator algebras. Recently, advancement made in the area of quantum information has ignited a wave of interest in the non-commutative aspects of analysis. This, in turn, helped to connect the tools in operator algebras with the techniques used in harmonic analysis to produce various deep results. The first part of the thesis will discuss about the characterization of positive definite, radial functions on some non-commutative groups. The subject of positive definite functions is widely studied due to its link with the area of Fourier Multipliers. So, a characterization of positive, definite functions will help in identifying the class of Fourier Multipliers which is completely positive. There is also some surprising consequences which slightly improves the classical Schoenberg-Bochner theorem. The second part of the thesis focus on the non-commutative Khintchine inequality and its relation with the Z2 property. The classical Khintchine inequality has been studied extensively and greatly improved by many mathematicians. In our setting, we show that the existence of the Z2 property in an orthonormal system will result in its matrix-valued coefficient function satisfying a Khintchine-type inequality. For the case of an abelian group, we also obtain some partial converse result on that, where the existence of a Khtinchine-type inequality will imply that the group has finite Z2 constants. In some examples, the optimal constants can be computed.
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    Shifts of hereditarily finite type, a subclass of shifts of finite type.
    (August 2022) Reynolds, Mads, 1973-; Meddaugh, Jonathan.
    We introduce the class of hereditarily finite type shift spaces. Hereditarily finite type shift spaces are the subclass of shift spaces such that all of their projections are of finite type. We provide a method to construct a graph for projections of these spaces. We establish that the question of whether a shift space is of hereditarily finite type is answerable for mixing shifts of finite type. We then present a family of shifts of finite type, derived from a family of labeled graphs, and classify these spaces as hereditarily finite type or not, under the assumption that only a single edge label in the graph of said space is used more than once.
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    Mean field games of controls and moderate interactions.
    (August 2022) Mullenix, Alan B., 1988-; Graber, P. Jameson.
    Mean Field Game theory, a quickly growing field emerging around 2007, seeks to answer qualitative and quantitative questions about high population competitive dynamics. A typical Mean Field Game is comprised of a large pool of identical, weak, rational agents, each seeking to maximize a payout (or, equivalently, minimize a cost). When an agent selects a strategy, it alters the state of the playing field for all other agents, that is, the actions of all agents must be factored into the strategy choice of any agent. The main object of interest is the Nash Equilibria of the game: a strategy choice that, if chosen by all players, no single player may improve their outcome by adjusting their own strategy. The question of these Nash Equilibria can be cast as solutions of a coupled system of partial differential equations: a backward in time Hamilton Jacobi Equation that encapsulates the strategy costs, the unknown being the value function u(x,t), and a forward in time Fokker-Planck type equation governing the evolution m(x,t): the probability distribution of player states. We look at two adaptations of the traditional Mean Field Game. First, a Potential Mean Field Game of Controls, in which agents have knowledge of not only the current states of other agents, but also their current adjustment to strategy. While the potential structure allows for the use of convex analysis techniques, we introduce a new, spatially non-local coupling component that depends on both the distribution of player states and the feedback. We also consider a Mean Field Game of Moderate Interactions, in which agents are more intensely affected by those in their immediate vicinity. This augments the standard Mean Field Game by introducing a local coupling term that has interaction with the gradient of the value function. We establish, in each case, well posedness results under generic assumptions on the data. In the conclusion we remark on the possible generalizations and further directions to take each work.
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    Characterizing smoothness of solutions with respect to boundary data of Dirichlet-nonlocal-Dirichlet boundary value problems for third order ordinary differential equations.
    (August 2022) Ickes, Henry, 1992-; Henderson, Johnny.
    The goal of our paper is to establish analogues of Peano’s Theorem for Dirichlet-nonlocal-Dirichlet conditions in third order ordinary differential equations. We begin by establishing the difference quotient, followed by a telescoping sum, allowing us to set up a linear system of equations in two unknowns to establish the existence of the limit as h → 0. We finish by employing Cramer’s Rule, and list some quick corollaries for each variation of Peano’s Theorem.
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    Generalized inverse limits and the intermediate value property.
    (2021-07-26) Dunn, Tavish J., 1993-; Ryden, David James, 1971-
    We introduce and discuss various notions of the intermediate value property applicable to upper-semicontinuous set-valued functions f : [0, 1] → 2^[0,1]. In the first part, we present sufficient conditions such that an inverse limit of a sequence of bonding functions of this type is a continuum. In the second part, we examine the relationship between the dynamics of an upper-semicontinuous function with the intermediate value property and the topological structure of the corresponding inverse limit. In particular, we present conditions under which the existence of a cycle of period not a power of 2 implies indecomposability in the inverse limit and vice-versa. Lastly, we show that these conditions are sharp by constructing a family of upper-semicontinuous functions with the intermediate value property and cycles of all periods, yet admits a hereditarily decomposable inverse limit.
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    On functions related to the spectral theory of Sturm--Liouville operators.
    (2022-04-01) Stanfill, Jonathan, 1989-; Gesztesy, Fritz, 1953-
    Functions related to the spectral theory of differential operators have been extensively studied due to their many applications in mathematics and physics. In this dissertation, we will consider spectral ζ-functions, ζ-regularized functional determinants, and Donoghue m-functions associated with Sturm--Liouville operators. We apply our results to an array of examples, including regular Schrödinger operators as well as Jacobi and generalized Bessel operators in the singular context. We begin by employing a recently developed unified approach to the computation of traces of resolvents and ζ-functions to efficiently compute values of spectral ζ-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions τ. Furthermore, we give the full analytic continuation of the ζ-function through a Liouville transformation and provide an explicit expression for the ζ-regularized functional determinant in terms of a particular set of a fundamental system of solutions of τy = zy. Next we turn to Donoghue m-functions. Assuming the standard local integrability hypotheses on the coefficients of the singular Sturm--Liouville differential equation τ, we study all corresponding self-adjoint realizations in L^2((a,b); rdx) and systematically construct the associated Donoghue m-functions in all cases where τ is in the limit circle case at least at one interval endpoint a or b. Finally, we construct Donoghue m-functions for the Jacobi differential operator in L^2 ((-1,1); (1-x)^alpha(1+x)^beta dx) associated with the differential expression τ alpha,beta = - (1-x)^-alpha(1+x)^-beta(d/dx)((1-x)^alpha+1((1+x)^beta+1)(d/dx), x E (-1,1), alpha,beta E R, whenever at least one endpoint, x = pm1, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's m-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case.
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    Approximation and interpolation with Bernstein polynomials.
    (2021-11-03) Allen, Larry J., 1993-; Kirby, Robert C.
    Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. In this dissertation, we investigate fundamental problems in approximation theory and numerical analysis involving Bernstein polynomials. We begin by developing a structured decomposition of the inverse of the matrices related to approximation and interpolation. These matrices are highly ill-conditioned, and so we introduce a nonstandard matrix norm to study the conditioning of the matrices, showing that the conditioning in this case is better than in the standard 2-norm. We conclude by giving an algorithm for enforcing bounds constraints on the approximating polynomial. Extensions of the interpolation problem and constrained approximation problem to higher dimensions are also considered.
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    On various notions of the shadowing property in non-compact spaces.
    (2021-07-25) Grigsby, Ian, 1993-; Meddaugh, Jonathan.
    We discuss various notions of the shadowing property on non-compact spaces. In the first part, we discuss the shadowing property acting on general sequence spaces. We develop criteria upon the weights of the space in which shadowing of particular types of orbits occurs. Then we move on to operators acting on Fréchet spaces, and show that an system exhibits the shadowing property if its Waelbroeck spectrum misses the unit circle. Additionally, we discuss the non-uniform pseudo-orbit tracing property, a variant of the shadowing property which allows for different error tolerances depending on where a point lies in the space.
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    Group automorphisms of incidence algebras.
    (2021-07-01) Rebrovich, Jack, 1994-; Herden, Daniel.
    Let (P,≤) be an arbitrary partially ordered set and I(P) its incidence space. Then FI(P) denotes the associated finitary incidence algebra, where FI(P) = I(P) for locally finite posets (P,≤). We investigate the group of units U(FI(P)) of incidence algebras and its normal subgroups. This includes the structure and properties of maximal abelian, normal subgroups, and criteria for solvability and nilpotency. Lastly, we will classify the automorphism group of the group of units when (P,≤) = (Z,≤).
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    Poncelet polygons through the lenses of orthogonal polynomials on the unit circle, finite Blaschke products, and numerical ranges.
    (2021-08-06) Poe, Taylor A., 1994-; Simanek, Brian Z., 1985-; Martinez-Finkelshtein, Andrei.
    The topics in this thesis fall in the intersection of projective geometry, complex analysis, and linear algebra. Each of these three fields gives a canonical construc-tion of families of polygons inscribed in the unit circle with interlacing vertices. In projective geometry, we consider Poncelet curves which admit a family of circum-scribed polygons. From the perspective of linear algebra, we consider the eigenvalues of a one parameter family of matrices called CMV matrices. Complex analysis gives two equivalent constructions of such polygons. One involves the preimages of points on T under a Blaschke product. The other involves the zero sets of paraorthogonal polynomials on the unit circle. In all three cases, the intersection of the polygons in this family forms part of an algebraic curve, which will be the primary focus of our study. A large part of this dissertation is dedicated to making each of these constructions precise in order to prove their equivalence. Particular attention is given to the notion of foci of an algebraic curve. In the latter chapters, we will focus on the specific case in which this inscribed curve (the intersection of the family of polygons) is an ellipse. We will give necessary and sufficient conditions in terms of the constructions of these families of n-gons for small n for which the inscribed curve is an ellipse. We will also provide algorithms for constructing such families of polygons with prescribed foci for the corresponding ellipse. We will conclude with possible directions for similar progress for larger n.
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    Forcing ℵ1-free groups to be free.
    (2021-04-12) Pasi, Alexandra V., 1993-; Herden, Daniel.
    ℵ1-free groups, abelian groups whose countable subgroups are free, are objects of both algebraic and set-theoretic interest. Illustrating this, we note that ℵ1-free groups, and in particular the question of when ℵ1-free groups are free, were central to the resolution of the Whitehead problem as undecidable. In elucidating the relationship between ℵ1-freeness and freeness, we prove the following result: an abelian group G is ℵ1-free in a countable transitive model of ZFC (and thus by absoluteness, in every transitive model of ZFC) if and only if it is free in some generic model extension. We would like to answer the more specific question of when an ℵ1-free group can be forced to be free while preserving the cardinality of the group. For groups of size ℵ1, we establish a necessary and sufficient condition for when such forcings are possible. We also identify both existing and novel forcings which force such ℵ1-free groups of size ℵ1 to become free with cardinal preservation. These forcings lay the groundwork for a larger project which uses forcing to explore various algebraic properties of ℵ1-free groups and develops new set-theoretical tools for working with them.
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    Orbits, pseudo orbits, and the characteristic polynomial of q-nary quantum graphs.
    (2020-07-17) Hudgins, Victoria Kathryn, 1989-; Harrison, Jonathan M.
    Quantum graphs provide a simple model of quantum mechanics in systems with complex geometry and can be used to study quantum chaos. We evaluate the variance of the coefficients of a quantum binary graph’s associated characteristic polynomial, which is related to the quantum graph’s spectrum. This variance can be written as a finite sum over pairs of short pseudo orbits on the graph with the same topological and metric lengths.To account for all pairs of this type, we first count the numbers of primitive periodic orbits and primitive pseudo orbits on general q-nary graphs by exploiting properties of Lyndon words. We then classify the primitive pseudo orbits on binary graphs by their numbers of self-intersections, the number of repetitions of each self-intersection, and the lengths of self-intersections, in order to determine the contributions of primitive pseudo orbit pairs to the variance. By arranging the sum in a way that considers the contribution of each primitive pseudo orbit paired with all possible partners, we can evaluate the sum over all pairs of primitive pseudo orbits and then use the graph’s ergodicity to asymptotically determine the variance in the limit of large binary graphs. The Bohigas-Giannoni-Schmit conjecture suggests spectral statistics of generic quantum graphs are typically modeled by those of random matrices, in the limit of large graphs. However, we show that, for families of binary graphs, there is a uniform family-specific deviation from random matrix behavior in the variance of coefficients of the characteristic polynomial. Related results for the variance of the coefficients of the characteristic polynomial for general q-nary quantum graphs are also investigated.
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    An explicit description of Pieri inclusions.
    (2020-06-09) Miller, John Anderson, II, 1991-; Hunziker, Markus.
    By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fløstad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups. We give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.
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    Oscillating satellites about the straight line equilibrium points.
    (1952) Clayton, Robert Lee
    The problem as studied in this thesis is the behavior of a satellite at the three equilibrium points. When a satellite is displaced from rest at one of these points it will either oscillate or rapidly leave the system. The equations of motion were first set up between the satellite and two other rotating bodies. These equations were then expanded in series and certain restrictions placed upon them. With these restrictions, constants of integration were found, and the integrated equations of motion were also found. An application was then made to the Solar System in which the Sun and Jupiter were taken as the major bodies. The masses of these two bodies were then substituted into the equations of motion and equations were found for the three points of equilibrium. The orbits were then plotted at the three points.
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    Polynomial preconditioning with the minimum residual polynomial.
    (2019-12-17) Loe, Jennifer A., 1992-; Morgan, Ronald Benjamin, 1958-
    Krylov subspace methods are often used to solve large, sparse systems of linear equations Ax=b. Preconditioning can help accelerate the Krylov iteration and reduce costs for solving the problem. We study a polynomial preconditioner p(A) based upon the minimum residual polynomial from GMRES. The polynomial can improve the eigenvalue distribution of A for better convergence. We demonstrate a power basis method of generating the polynomial and how it can be unstable. Then we develop a new method to generate the polynomial which uses harmonic Ritzvalues as roots. We discuss sources of roundoff error and give a procedure to make the polynomial stable through adding extra copies of roots. Furthermore, we implement the polynomial into the software Trilinos. We use this implementation to test the polynomial composed with other preconditioners and give considerations for parallel computing.
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    Preconditioning mixed finite elements for tide models.
    (2019-07-24) Kernell, Oliver Tate, 1990-; Kirby, Robert C.
    We describe finite element methods for the linearized rotating shallow water equations which govern tides. Symplectic Euler and Crank-Nicolson time-stepping strategies have good energy preservation properties, which is desirable for tide modeling, but require careful treatment of linear algebra. For symplectic Euler, we have to invert the Raviart-Thomas element mass matrix at every time step. Thus we give estimates for the eigenvalues of these mass matrices. Crank-Nicolson, being fully implicit, has a more complicated system of equations which requires inverting the entire system. For this, we present an effective block preconditioner using parameter-weighted norms in H(div). We give results that are nearly dependent of the given constants. Finally, we provide numerical results that confirm this theory.