Department of Mathematicshttps://hdl.handle.net/2104/47732023-12-01T14:07:01Z2023-12-01T14:07:01Z551On certain results of Fourier analysis on non-abelian discrete groups.https://hdl.handle.net/2104/123162023-09-22T08:01:38Zdc.title: On certain results of Fourier analysis on non-abelian discrete groups.
dc.description.abstract: 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.
Shifts of hereditarily finite type, a subclass of shifts of finite type.https://hdl.handle.net/2104/123132023-09-22T08:04:15Zdc.title: Shifts of hereditarily finite type, a subclass of shifts of finite type.
dc.description.abstract: 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.
Mean field games of controls and moderate interactions.https://hdl.handle.net/2104/123122023-09-22T08:05:26Zdc.title: Mean field games of controls and moderate interactions.
dc.description.abstract: 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.
Characterizing smoothness of solutions with respect to boundary data of Dirichlet-nonlocal-Dirichlet boundary value problems for third order ordinary differential equations.https://hdl.handle.net/2104/123022023-09-22T08:01:30Zdc.title: Characterizing smoothness of solutions with respect to boundary data of Dirichlet-nonlocal-Dirichlet boundary value problems for third order ordinary differential equations.
dc.description.abstract: 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.
Generalized inverse limits and the intermediate value property.https://hdl.handle.net/2104/120602022-08-29T16:15:20Z2021-07-26T00:00:00Zdc.title: Generalized inverse limits and the intermediate value property.
dc.description.abstract: 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.
2021-07-26T00:00:00ZOn functions related to the spectral theory of Sturm--Liouville operators.https://hdl.handle.net/2104/119392022-08-30T04:08:18Z2022-04-01T00:00:00Zdc.title: On functions related to the spectral theory of Sturm--Liouville operators.
dc.description.abstract: 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.
2022-04-01T00:00:00ZApproximation and interpolation with Bernstein polynomials.https://hdl.handle.net/2104/117162022-08-29T17:30:03Z2021-11-03T00:00:00Zdc.title: Approximation and interpolation with Bernstein polynomials.
dc.description.abstract: 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.
2021-11-03T00:00:00ZOn various notions of the shadowing property in non-compact spaces.https://hdl.handle.net/2104/116062022-08-30T04:40:57Z2021-07-25T00:00:00Zdc.title: On various notions of the shadowing property in non-compact spaces.
dc.description.abstract: 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.
2021-07-25T00:00:00ZGroup automorphisms of incidence algebras.https://hdl.handle.net/2104/115832022-08-29T14:30:48Z2021-07-01T00:00:00Zdc.title: Group automorphisms of incidence algebras.
dc.description.abstract: 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,≤).
2021-07-01T00:00:00ZPoncelet polygons through the lenses of orthogonal polynomials on the unit circle, finite Blaschke products, and numerical ranges.https://hdl.handle.net/2104/115672022-08-29T16:16:20Z2021-08-06T00:00:00Zdc.title: Poncelet polygons through the lenses of orthogonal polynomials on the unit circle, finite Blaschke products, and numerical ranges.
dc.description.abstract: 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.
2021-08-06T00:00:00Z