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Structure Preserving Reduced-Order Models Of Hamiltonian Systems, Megan Alice Mckay

Theses and Dissertations

Large-scale dynamical systems are expensive to simulate due to the computational cost accrued y the substantial number of degrees of freedom. To accelerate repeated numerical simulations of the systems, proper orthogonal decomposition reduced order models (POD-ROMs) have been developed. When applied to Hamiltonian systems, however, special care must be taken when performing the reduced order modeling to keep their energy-preserving nature. This work presents a survey of several structure-preserving reduced order models (SP-ROMs). In addition, this work employs the discrete empirical interpolation method (DEIM) and develops an SP-DEIM model for nonlinear Hamiltonian systems. The wave equation is considered as a …

Poset Ramsey Numbers For Boolean Lattices, Joshua Cain Thompson

Theses and Dissertations

For each positive integer n, let Qn denote the Boolean lattice of dimension n. For posets P, P', define the poset Ramsey number R(P,P') to be the least N such that for any red/blue coloring of the elements of Q_N, there exists either a subposet isomorphic to P with all elements red, or a subposet isomorphic to P' with all elements blue.

Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q₂, Q_n) ≤ 2n + 2 and R(Q …

The Existence And Quantum Approximation Of Optimal Pure State Ensembles, Ryan Thomas Mcgaha

Theses and Dissertations

In this manuscript we study entanglement measures defined via the convex roof construction. In the first chapter we build the notion of an entanglement measure from the ground up and discuss various issues that arise when trying to measure the amount of entanglement present in an arbitrary density operator. Through this introduction we will motivate the use of the convex roof construction. In the second chapter we will show that the infimum in the convex roof construction is achieved for a specific set of entanglement measures and provide canonical examples of such measures. We also describe LOCC operations via a …

Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock

Theses and Dissertations

This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0 \pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if $\bfrac{n}{5} = 1$ and $\bfrac{n}{7} = …

Covering Systems And The Minimum Modulus Problem, Maria Claire Cummings

Theses and Dissertations

A covering system or a covering is a set of linear congruences such that every integer satisfies at least one of these congruences. In 1950, Erdős posed a problem regarding the existence of a finite covering with distinct moduli and an arbitrarily large minimum modulus. This remained unanswered until 2015 when Robert Hough proved an explicit bound of 1016 for the minimum modulus of any such covering. In this thesis, we examine the use of covering systems in number theory results, expand upon the proof of the existence of an upper bound on the minimum modulus in the case of …

Tangled Up In Tanglegrams, Drew Joseph Scalzo

Theses and Dissertations

Tanglegrams are graphs consisting of two rooted binary plane trees with the same number of leaves and a perfect matching between the two leaf sets. A Tanglegram drawing is a special way of drawing a Tanglegram; and a Tanglegram is called planar if it has a drawing such that the matching edges do not cross. In this thesis, we will discuss various results related to the construction and planarity of Tanglegrams, as well as demonstrate how to construct all the Tanglegrams of size 4 by looking at two types of rooted binary trees - Caterpillar and Complete Binary Trees. After …

Results On Select Combinatorial Problems With An Extremal Nature, Stephen Smith

Theses and Dissertations

This dissertation is split into three sections, each containing new results on a particular combinatorial problem. In the first section, we consider the set of 3-connected quadrangulations on n vertices and the set of 5-connected triangulations on n vertices. In each case, we find the minimum Wiener index of any graph in the given class, and identify graphs that obtain this minimum value. Moreover, we prove that these graphs are unique up to isomorphism.

In the second section, we work with structures emerging from the biological sciences called tanglegrams. In particular, our work pertains to an invariant of tanglegrams called …