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 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].Item Multiparameter persistence modules(University of Missouri--Columbia, 2025) Nategh, Mehdi; Qin, Zhenbo; Wang, ShuguangIn this dessertation, Hausdorff and Gromov-Hausdorff distances are discussed. Building on higher dimensional partitions and their Young diagrams, a framework for N d -indexed multiparameter persistence modules is established and decomposabil-ity and key properties of barcode admissible modules are explored. Relationships between the rank invariant, and the N d -indexed persistence modules that admit bar-codes are investigated and, a necessary and sufficient condition for two persistence modules admitting barcodes to share the same rank invariant is shown. A connection between the barcode admissibility of an N d -indexed persistence module and those of the persistence sub-modules associated to the connected components of its Hasse graph is discussed. Lastly, an algorithm which verifies whether an N d -indexed persis-tence module admits a barcode is developed. An extension of Forman’s discrete Morse theory is presented. Some of the exist-ing theorems are corrected and the collapse theory under group actions is explored. Additionally, in connection to persistence homology, a Morse-theoretic proof of the structure theorem is offered.Item Existence and formulas regarding the epsilon multiplicity(University of Missouri--Columbia, 2025) Landsittel, Stephen Douglas; Cutkosky, Steven DaleThe volume of an mR-primary graded family of ideals is the natural generalization of the Hilbert-Samuel multiplicity to arbitrary mR-primary graded families. Lazarsfeld and Mustata, and more generally, Cutkosky, have developed a formula (called volume equals multiplicity) for showing that the volume of a graded family is a limit of Hilbert-Samuel multiplicities. We show that the analogue of this fact for non mRprimary graded families holds in the case that the graded family arises from an ideal. We accomplish this by first proving a more general multiplicity formula.