Mathematics
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Item Locally Recoverable Codes From Algebraic Curves(2018) Ballentine, Sean Frederick; Barg, Alexander; Haines, Thomas; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Locally recoverable (LRC) codes have the property that erased coordinates can be recovered by retrieving a small amount of the information contained in the entire codeword. An LRC code achieves this by making each coordinate a function of a small number of other coordinates. Since some algebraic constructions of LRC codes require that $n \leq q$, where $n$ is the length and $q$ is the size of the field, it is natural to ask whether we can generate codes over a small field from a code over an extension. Trace codes achieve this by taking the field trace of every coordinate of a code. In this thesis, we give necessary and sufficient conditions for when the local recoverability property is retained when taking the trace of certain LRC codes. This thesis also explores a subfamily of LRC codes with hierarchical locality (H-LRC) which have tiers of recoverability. We provide a general construction of codes with 2 levels of hierarchy from maps between algebraic curves and present several families from quotients of curves by a subgroup of automorphisms. We consider specific examples from rational, elliptic, Kummer, and Artin-Schrier curves and examples of asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.Item Nonlinear Analysis of Phase Retrieval and Deep Learning(2017) Zou, Dongmian; Balan, Radu V; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Nonlinearity causes information loss. The phase retrieval problem, or the phaseless reconstruction problem, seeks to reconstruct a signal from the magnitudes of linear measurements. With a more complicated design, convolutional neural networks use nonlinearity to extract useful features. We can model both problems in a frame-theoretic setting. With the existence of a noise, it is important to study the stability of the phaseless reconstruction and the feature extraction part of the convolutional neural networks. We prove the Lipschitz properties in both cases. In the phaseless reconstruction problem, we show that phase retrievability implies a bi-Lipschitz reconstruction map, which can be extended to the Euclidean space to accommodate noises while remaining to be stable. In the deep learning problem, we set up a general framework for the convolutional neural networks and provide an approach for computing the Lipschitz constants.Item Spherical two-distance sets and related topics in harmonic analysis(2014) Yu, Wei-Hsuan; Barg, Alexander; Benedetto, John Joseph; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation is devoted to the study of applications of harmonic analysis. The maximum size of spherical few-distance sets had been studied by Delsarte at al. in the 1970s. In particular, the maximum size of spherical two-distance sets in $\mathbb{R}^n$ had been known for $n \leq 39$ except $n=23$ by linear programming methods in 2008. Our contribution is to extend the known results of the maximum size of spherical two-distance sets in $\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46, 78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and $20$ since 1973. We use the semidefinite programming method to find the maximum size for equiangular line sets in $\mathbb{R}^n$ when $24 \leq n \leq 41$ and $n=43$. We suggest a method of constructing spherical two-distance sets that also form tight frames. We derive new structural properties of the Gram matrix of a two-distance set that also forms a tight frame for $\mathbb{R}^n$. One of the main results in this part is a new correspondence between two-distance tight frames and certain strongly regular graphs. This allows us to use spectral properties of strongly regular graphs to construct two-distance tight frames. Several new examples are obtained using this characterization. Bannai, Okuda, and Tagami proved that a tight spherical designs of harmonic index 4 exists if and only if there exists an equiangular line set with the angle $\arccos (1/(2k-1))$ in the Euclidean space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We show nonexistence of tight spherical designs of harmonic index $4$ on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite programming method. We also derive new relative bounds for equiangular line sets. These new relative bounds are usually tighter than previous relative bounds by Lemmens and Seidel.Item Three Dimensional Edge Detection Using Wavelet and Shearlet Analysis(2012) Schug, David Albert; O'Leary, Dianne P; Easley, Glenn R; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Edge detection determines the boundary of objects in an image. A sequence of images records a 2D representation of a scene changing over time, giving 3D data. New 3D edge detectors, particularly ones we developed using shearlets and hybrid shearlet-Canny algorithms, identify edges of complicated objects much more reliably than standard approaches, especially under high noise conditions. We also use edge information to track the position and velocity of objects using an optimization algorithm.