Mathematics electronic theses and dissertations (MU)
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The items in this collection are the theses and dissertations written by students of the Department of Mathematics. Some items may be viewed only by members of the University of Missouri System and/or University of Missouri-Columbia. Click on one of the browse buttons above for a complete listing of the works.
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Item Green's function estimates and elliptic measures for some linear elliptic equations with singular drifts.(University of Missouri--Columbia, 2025) Pathak, Aritro; Montgomery-Smith, StephenThe purpose of this thesis is to study pointwise estimates on Green's functions and elliptic measures for linear elliptic partial differential equations that have a drift term that diverges at the boundary in several different ways. These questions are of fundamental importance in its own right in partial differential equations, with future applications in other settings such as the time-independent Schroedinger equation, as well as being important for questions of solvability of rough Dirichlet and Neumann problems for a large class of elliptic operators with singular lower order terms.Item Orbit separation and invariant theory : degree bounds and algorithms(University of Missouri--Columbia, 2025) Katz, Joshua; Edidin, DanThe goal of this thesis is to contribute to the growing number of applications of computational invariant theory to problems in signal processing and statistics. Our focus is on problems of both sample complexity and the construction of algorithms for explicit signal recovery. The primary statistical model we study is the multi-reference alignment (MRA) problema fundamental challenge in statistical signal processing--in which one seeks to recover an unknown signal from many noisy and transformed copies. We use tools from invariant theory, particularly the method of moments, to analyze signal identifiability and to develop practical recovery algorithms.Item Solitary vortex carrying capillary-gravity waves(University of Missouri--Columbia, 2025) Slease, Gregory; Walsh, SamuelWe prove the existence of solitary wave solutions to the incompressible, irrotational Euler equations carrying a point or hollow vortex in finite depth. Our analysis treats waves influenced by both gravity and surface tension, i.e. capillary-gravity waves. The second section of the paper is dedicated the formulation of the problem for a wave-borne point vortex, followed by the proof of existence of solutions. Moreover, we provide the leading order terms of the asymptotic form of these solutions. Finally, the hollow vortex problem is treated in the third section; this is achieved by using vortex disingularization. Our techniques specifically address fast moving waves with O(1) vortex strength.Item Some conditional results involving arithmetic functions(University of Missouri--Columbia, 2025) Sinha, Saloni; Banks, WilliamIn analytic number theory, conditional results often serve as a precursor to unconditional results. In this thesis, we present two types of conditional results. First, we establish conditional estimates on twisted sums of certain arithmetic functions such as generalized von Mangoldt and Mobius functions under the Riemann hypothesis, and we also present a strong converse. Second, we investigate a discrete negative moment of the zeta function, obtaining a lower bound that supports a previously conjectured estimate. Both results rely on the use of arithmetic functions and connect to broader problems. While the results we obtain are conditional, they provide a framework that could lead to unconditional estimates through alternative methods.Item Optimal frames, the Hadamard conjecture, and Williamson matrices of order an odd multiple of 4(University of Missouri--Columbia, 2025) Nguyen, Raymond; Montgomery-Smith, StephenThis dissertation concerns two distinct areas of research: frame theory and Hadamard matrices. In the area of frame theory, we consider four optimization problems posed by Cahill & Casazza [CC2022] related to total coherence, total volume, and nuclear energy. These problems were solved for cases in which equiangular Parseval frames exist and/or the dimension is two [CC2022]. In this paper, we explore cases in which equiangular Parseval frames do not exist or the dimension is greater than two. We also introduce a quantity called the nuclear potential, pose a related optimization problem, and show that the solutions to it are precisely the equal norm Parseval frames. In the area of Hadamard matrices, we provide a new framework for finding Williamson matrices of order a multiple of 4. By combining this framework with new techniques, we generate examples of Williamson matrices of orders 72, 76, 84, 92, 100, 108, 116, and 124. In particular, we present the first examples, to our knowledge, of Williamson matrices of orders 92 and 116 constructed without relying on Williamson matrices of lower order. Based on the data collected, we conjecture that Williamson matrices exist in all orders that are an odd multiple of 4 (thus, Hadamard matrices exist in all orders that are an odd multiple of 16.) A proof of this conjecture would serve as a partial solution to the Hadamard conjecture by confirming the existence of Hadamard matrices in all orders in an arithmetic sequence. In addition, it would immediately have an impact on fields outside of mathematics including coding theory, cryptography, data compression and storage, design and analysis of experiments, and signal processing [HW1978] [SWW2005] [ASEA2011].