Browsing by Author "Hardt, Robert M"

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    Local Criteria in Polyhedral Minimizing Problems
    (2023-04-21) Valfells, Asgeir; Hardt, Robert M
    This thesis will discuss two polyhedral minimizing problems and the necessary local criteria we find any such minimizers must have. We will also briefly discuss an extension of a third minimizing problem to higher dimension. The first result we present classifies the three-dimensional piecewise linear cones in $\mathbb{R}^4$ that are mass minimizing w.r.t. Lipschitz maps in the sense of Almgren's $M(0,\delta)$ sets as in Taylor's classification of two-dimensional soap film singularities. There are three that arise naturally by taking products of $\mathbb{R}$ with lower dimensional cases and earlier literature has demonstrated the existence of two with 0-dimensional singularities. We classify all possible candidates and demonstrate that there are no p.l. minimizers outside these five. The second result we present is an assortment of criteria for edge-length minimizing polyhedrons. The aim is to get closer to answering a 1957 conjecture by Zdzislaw Melzak, that the unit volume polyhedron with least edge length was a triangular right prism, with edge length $2^{2/3}3^{11/6}\approx 11.896$. We present a variety of variational arguments to restrict the class of minimizing candidates.
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    Spectral Analysis of One-Dimensional Operators
    (2015-02-25) Fillman, Jacob D; Damanik, David T; Hardt, Robert M; Cox, Steven J
    We study the spectral analysis of one-dimensional operators, motivated by a desire to understand three phenomena: dynamical characteristics of quantum walks, the interplay between inverse and direct spectral problems for limit-periodic operators, and the fractal structure of the spectrum of the Thue-Morse Hamiltonian. Our first group of results comprises several general lower bounds on the spreading rates of wave packets defined by the iteration of a unitary operator on a separable Hilbert space. By using tools within the class of CMV matrices, we apply these general lower bounds to deduce quantitative lower bounds for the spreading of the time-homogeneous Fibonacci quantum walk. Second, we construct several classes of limit-periodic operators with homogeneous Cantor spectrum, which connects problems from inverse and direct spectral analysis for such operators. Lastly, we precisely characterize the gap structure of the canonical periodic approximants to the Thue-Morse Hamiltonian, which constitutes a first step towards understanding the fractal structure of its spectrum. This thesis contains joint work with David Damanik, Milivoje Lukic, Paul Munger, and Robert Vance.
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    Two Variants on the Plateau Problem
    (2016-03-11) Funk, Quentin A; Hardt, Robert M
    The Plateau problem in $\mbb{R}^3$ begins with a given simple, closed curve $\gamma$, and asks to find a surface $M$ with $\partial M= \gamma$ that minimizes area among all surfaces with $\gamma$ as their boundary. In 1960 Federer and Fleming generalized this idea and analyzed the currents developed by De Rham. They proved certain subclasses of currents (in particular, the integral currents) can be used as a powerful tool in area and volume minimization problems. In this thesis, we approach two generalizations--first, we prove a Homological Plateau problem in the singular setting of semi-algebraic geometry using the tools of geometric measure theory. We obtain similar results to those of Federer and Fleming even in this more singular case. Second, an earlier solution to the Plateau problem was achieved independently by Douglas and Rado in 1931 and 1933, respectively, using mappings from the two-dimensional disk. In the second chapter we generalize this mapping to a so-called ``multiple-valued'' mapping of the disk. Multiple-valued maps are a cornerstone of the regularity theorems of F. Almgren and are interesting in their own right for many problems in the geometric calculus of variations. We prove existence and regularity for these Plateau solutions under fairly general conditions. We also produce a class of examples and analyze a degenerate case.