Optimal frames, the Hadamard conjecture, and Williamson matrices of order an odd multiple of 4
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This 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].
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Ph. D.