Physical Sciences and Mathematics Commons^™

Efficient Algorithms And Applications In Topological Data Analysis, Junyi Tu

USF Tampa Graduate Theses and Dissertations

Topological Data Analysis (TDA) is a new and fast growing research field developed over last two decades. TDA finds many applications in computer vision, computer graphics, scientific visualization, molecular biology, and material science, to name a few. In this dissertation, we make algorithmic and application contributions to three data structures in TDA: contour trees, Reeb graphs, and Mapper. From the algorithmic perspective, we design a parallel algorithm for contour tree construction and implement it in OpenCL. We also design and implement critical point pairing algorithms to compute persistence diagrams directly from contour trees, Reeb graphs, and Mapper. In terms of …

On Spectral Properties Of Single Layer Potentials, Seyed Zoalroshd

USF Tampa Graduate Theses and Dissertations

We show that the singular numbers of single layer potentials on smooth curves asymptotically behave like O(1/n). For the curves with singularities, as long as they contain a smooth sub-arc, the resulting single layer potentials are never trace-class. We provide upper bounds for the operator and the Hilbert-Schmidt norms of single layer potentials on smooth and chord-arc curves. Regarding the injectivity of single layer potentials on planar curves, we prove that among single layer potentials on dilations of a given curve, only one yields a non-injective single layer potential. A criterion for injectivity of single layer potentials on …

Some Results Concerning Permutation Polynomials Over Finite Fields, Stephen Lappano

USF Tampa Graduate Theses and Dissertations

Let p be a prime, p a power of p and 𝔽_q the finite field with q elements. Any function φ: 𝔽_q → 𝔽_q can be unqiuely represented by a polynomial, 𝔽_φ of degree < q. If the map x ↦ F_φ(x) induces a permutation on the underlying field we say F_φ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F( …

Hamiltonian Formulations And Symmetry Constraints Of Soliton Hierarchies Of (1+1)-Dimensional Nonlinear Evolution Equations, Solomon Manukure

USF Tampa Graduate Theses and Dissertations

We derive two hierarchies of 1+1 dimensional soliton-type integrable systems from two spectral problems associated with the Lie algebra of the special orthogonal Lie group SO(3,R). By using the trace identity, we formulate Hamiltonian structures for the resulting equations. Further, we show that each of these equations can be written in Hamiltonian form in two distinct ways, leading to the integrability of the equations in the sense of Liouville. We also present finite-dimensional Hamiltonian systems by means of symmetry constraints and discuss their integrability based on the existence of sufficiently many integrals of motion.

Putnam's Inequality And Analytic Content In The Bergman Space, Matthew Fleeman

USF Tampa Graduate Theses and Dissertations

In this dissertation we are interested in studying two extremal problems in the Bergman space. The topics are divided into three chapters.

In Chapter 2, we study Putnam’s inequality in the Bergman space setting. In [32], the authors showed that Putnam’s inequality for the norm of self-commutators can be improved by a factor of 1 for Toeplitz operators with analytic symbol φ acting on the Bergman space A2(Ω). This improved upper bound is sharp when φ(Ω) is a disk. We show that disks are the only domains for which the upper bound is attained.

In Chapter 3, we consider the …

A Statistical Analysis Of Hurricanes In The Atlantic Basin And Sinkholes In Florida, Joy Marie D'Andrea

USF Tampa Graduate Theses and Dissertations

Beaches can provide a natural barrier between the ocean and inland communities, ecosystems, and resources. These environments can move and change in response to winds, waves, and currents. When a hurricane occurs, these changes can be rather large and possibly catastrophic. The high waves and storm surge act together to erode beaches and inundate low-lying lands, putting inland communities at risk. There are thousands of buoys in the Atlantic Basin that record and update data to help predict climate conditions in the state of Florida. The data that was compiled and used into a larger data set came from two …

Generalized Phase Retrieval: Isometries In Vector Spaces, Josiah Park

USF Tampa Graduate Theses and Dissertations

In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp.

Resonant Solutions To (3+1)-Dimensional Bilinear Differential Equations, Yue Sun

USF Tampa Graduate Theses and Dissertations

In this thesis, we attempt to obtain a class of generalized bilinear differential equations in (3+1)-dimensions by D_p-operators with p = 5, which have resonant solutions. We construct resonant solutions by using the linear superposition principle and parameterizations of wave numbers and frequencies. We test different values of p in Maple computations, and generate three classes of generalized bilinear differential equations and their resonant solutions when p = 5.

On The Number Of Colors In Quandle Knot Colorings, Jeremy William Kerr

USF Tampa Graduate Theses and Dissertations

A major question in Knot Theory concerns the process of trying to determine when two knots are different. A knot invariant is a quantity (number, polynomial, group, etc.) that does not change by continuous deformation of the knot. One of the simplest invariant of knots is colorability. In this thesis, we study Fox colorings of knots and knots that are colored by linear Alexander quandles. In recent years, there has been an interest in reducing Fox colorings to a minimum number of colors. We prove that any Fox coloring of a 13-colorable knot has a diagram that uses exactly five …

Leonard Systems And Their Friends, Jonathan Spiewak

USF Tampa Graduate Theses and Dissertations

Let $V$ be a finite-dimensional vector space over a field $\mathbb{K}$, and let

\text{End}$(V)$ be the set of all $\mathbb{K}$-linear transformations from $V$ to $V$.

A {\em Leonard system} on $V$ is a sequence

\[(\A ;\B; \lbrace E_i\rbrace_{i=0}^d; \lbrace E^*_i\rbrace_{i=0}^d),\]

where

$\A$ and $\B $ are multiplicity-free elements of \text{End}$(V)$;

$\lbrace E_i\rbrace_{i=0}^d$ and $\lbrace E^*_i\rbrace_{i=0}^d$

are orderings of the primitive idempotents of $\A $ and $\B$, respectively; and

for $0\leq i, j\leq d$, the expressions $E_i\B E_j$ and $E^*_i\A E^*_j$ are zero when $\vert i-j\vert > 1$ and

nonzero when $\vert i-j \vert = 1$.

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Leonard systems arise in connection …