Physical Sciences and Mathematics Commons^™

Applied Analysis For Learning Architectures, Himanshu Singh

USF Tampa Graduate Theses and Dissertations

Modern data science problems revolves around the Koopman operator Cφ (or Composition operator) approach, which provides the best-fit linear approximator to the dynamical system by which the dynamics can be advanced under the discretization. The solution provided by Koopman in the data driven methods is in the sense of strong operator topology, which is nothing better then the point-wise convergence of data (snapshots) in the underlying Hilbert space. Chapter 2 provides the details about the aforementioned issues with essential counter-examples. Thereafter, provable convergence guarantee phenomena is demonstrated by the Liouville weighted composition operators Af,φ over the Fock space by providing …

Classification Of Finite Topological Quandles And Shelves Via Posets, Hitakshi Lahrani

USF Tampa Graduate Theses and Dissertations

The objective of this dissertation is to investigate finite topological quandles and topological shelves. Precisely, we give a classification of both finite topological quandles and topological shelves using the theory of posets. For quandles with more than one orbit, we prove the following Theorem.

Proposition 0.0.1. Let X be a finite quandle with n orbits X₁, ... , X_n. Then any right continuous poset on X is n-partite with vertex sets X₁, ... , X_n.

For connected quandles, we prove the following Theorem.

Theorem 0.0.2. There is no T …

Rational Functions Of Degree Five That Permute The Projective Line Over A Finite Field, Christopher Sze

USF Tampa Graduate Theses and Dissertations

Rational functions over a finite field Fq induce mappings from the projective line P1(Fq) to itself. Rational functions that permute the projective line are called permutation rational functions (PRs). The notion of permutation rational functions is a natural extension of the permutation polynomials which have been studied for over a century. Recently, PRs of degrees up to four have been determined. This dissertation is a project aimed at determining PRs of degree five.

Rational functions of degree five (excluding those that are equivalent to polynomials) are divided into five cases according to the factorization of their denominators. Our main results …

Accelerating Multiparametric Mri For Adaptive Radiotherapy, Shraddha Pandey

USF Tampa Graduate Theses and Dissertations

MR guided Radiotherapy (MRgRT) marks an important paradigm shift in the field of radiotherapy. Superior tissue contrast of MRI offers better visualization of the abnormal lesions, as a result precise radiation dose delivery is possible. In case of online treatment planning, MRgRT offers better control of intratumoral motion and quick adaptation to changes in the gross tumor volume. Nonetheless, the MRgRT process flow does suffer from some challenges that limit its clinical usability. The primary aspects of MRgRT workflow are MRI acquisition, tumor delineation, dose map prediction and administering treatment. It is estimated that the acquisition of MRI takes around …

Boundary Behavior Of Analytic Functions And Approximation Theory, Spyros Pasias

USF Tampa Graduate Theses and Dissertations

In this Thesis we deal with problems regarding boundary behavior of analytic functions and approximation theory. We will begin by characterizing the set in which Blaschke products fail to have radial limits but have unrestricted limits on its complement. We will then proceed and solve several cases of an open problem posed in \cite{Da}. The goal of the problem is to unify two known theorems to create a stronger theorem; in particular we want to find necessary and sufficient conditions on sets $E_1\subset E_2$ of the unit circle such that there exists a bounded analytic function that fails to have …

Data-Driven Analytical Predictive Modeling For Pancreatic Cancer, Financial & Social Systems, Aditya Chakraborty

USF Tampa Graduate Theses and Dissertations

Pancreatic cancer is one of the most deathly disease and becoming an increasingly commoncause of cancer mortality. It continues giving rise to massive challenges to clinicians and cancer researchers. The combined five-year survival rate for pancreatic cancer is extremely low, about 5 to 10 percent, owing to the fact that a large number of the patients are diagnosed at stage IV when the disease has metastasized. Our study investigates if there exists any statistical significant difference between the median survival times and also the survival probabilities of male and female pancreatic cancer patients at different cancer stages, and irrespective of …

On Simultaneous Similarity Of D-Tuples Of Commuting Square Matrices, Corey Connelly

USF Tampa Graduate Theses and Dissertations

It has been shown by B. Shekhtman that when any d-tuple A of pairwise commuting N × N matrices with complex entries is cyclic, then A is simultaneously similar to the d-tuple of commuting N × N matrices B if and only if B is cyclic, and the sets of polynomials in d variables which annihilate A and B are equivalent.

This thesis offers a further generalization of this result, demonstrating the necessary and sufficient conditions for the simultaneous similarity of n-cyclic d-tuples of commuting square complex-valued matrices.

Stability Analysis Of Delay-Driven Coupled Cantilevers Using The Lambert W-Function, Daniel Siebel-Cortopassi

USF Tampa Graduate Theses and Dissertations

A coupled delay-feedback system of two cantilevers can yield greater sensitivity than that of asingle cantilever system, with potential applications in atomic force microscopy. The Lambert W-function analysis concept for delay differential equations is used to more accurately model the behavior of specific configurations of these cantilever systems. We also use this analysis concept to find parameters which yield stability for greater parameter ranges, of the delay differential equations. The Q factor, or quality factor, is the ratio of energy stored in the system, to the energy lost per fixed oscillation/movement cycle. Having stability of the cantilevers corresponds to the …

A Functional Optimization Approach To Stochastic Process Sampling, Ryan Matthew Thurman

USF Tampa Graduate Theses and Dissertations

The goal of the current research project is the formulation of a method for the estimation and modeling of additive stochastic processes with both linear- and cycle-type trend components as well as a relatively robust noise component in the form of Levy processes. Most of the research in stochastic processes tends to focus on cases where the process is stationary, a condition that cannot be assumed for the model above due to the presence of the cyclical sub-component in the overall additive process. As such, we outline a number of relevant theoretical and applied topics, such as stochastic processes and …

Advances And Applications Of Optimal Polynomial Approximants, Raymond Centner

USF Tampa Graduate Theses and Dissertations

The history of optimal polynomial approximants (OPAs) dates back to the engineering literature of the 1970s. Here, these polynomials were studied in the context of the Hardy space H^2(X), where X denotes the open unit disk D or the bidisk D^2. Under certain conditions, it was thought that these polynomials had all of their zeros outside the closure of X. Hence, it was suggested that these polynomials could be used to design a stable digital filter. In recent mathematics literature, OPAs have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the …

Symbolic Computation Of Lump Solutions To A Combined (2+1)-Dimensional Nonlinear Evolution Equation, Jingwei He

USF Tampa Graduate Theses and Dissertations

This thesis aims to consider a (2+1)-dimensional nonlinear evolution equation and its lump solutions. Byusing symbolic computation, two classes of lump solutions are presented. And for two specific chosen examples, we will show three-dimensional plots and density plots to exhibit dynamical features of the lump solution, which are made by Maple plot tools.

Riemann-Hilbert Problems For Nonlocal Reverse-Time Nonlinear Second-Order And Fourth-Order Akns Systems Of Multiple Components And Exact Soliton Solutions, Alle Adjiri

USF Tampa Graduate Theses and Dissertations

We first investigate the solvability of an integrable nonlinear nonlocal reverse-time six-component fourth-order AKNS system generated from a reduced coupled AKNS hierarchy under a reverse-time reduction. Riemann-Hilbert problems will be formulated by using the associated matrix spectral problems, and exact soliton solutions will be derived from the reflectionless case corresponding to an identity jump matrix. Secondly, we present the inverse scattering transform for solving a class of eight-component AKNS integrable equations obtained by a specific reduction associated with a block matrix spectral problem. The inverse scattering transform based on Riemann-Hilbert problems is presented along with a jump matrix taken to …

Long-Time Asymptotics For Mkdv Type Reduced Equations Of The Akns Hierarchy In Weighted L2 Sobolev Spaces, Fudong Wang

USF Tampa Graduate Theses and Dissertations

The long-time asymptotics of nonlinear integrable partial differential equations is one of the important research areas in the field of integrable systems. The main tool to analyze the long-time behaviors is the so-called nonlinear steepest descent method, or Deift-Zhou's method, which was born in 1993. To apply Deift-Zhou's method, one first uses the inverse scattering transform to formulate the nonlinear PDEs in terms of an oscillatory 2 by 2 matrix Riemann-Hilbert problem (RHP). After about 15 years of development, a generalized version of Deift-Zhou's method, the ∂—steepest method, came out. The ∂—steepest descent method is a useful method for analyzing …

Discrete Models And Algorithms For Analyzing Dna Rearrangements, Jasper Braun

USF Tampa Graduate Theses and Dissertations

In this work, language and tools are introduced, which model many-to-many mappings that comprise DNA rearrangements in nature. Existing theoretical models and data processing methods depend on the premise that DNA segments in the rearrangement precursor are in a clear one-to-one correspondence with their destinations in the recombined product. However, ambiguities in the rearrangement maps obtained from the ciliate species Oxytricha trifallax violate this assumption demonstrating a necessity for the adaptation of theory and practice.

In order to take into account the ambiguities in the rearrangement maps, generalizations of existing recombination models are proposed. Edges in an ordered graph model …

On Some Problems On Polynomial Interpolation In Several Variables, Brian Jon Tuesink

USF Tampa Graduate Theses and Dissertations

Polynomial approximation is a long studied process, with a history dating back to the 1700s, At which time Lagrange, Newton and Taylor developed their famed approximation methods. At that time, it was discovered that every Taylor projection (projector) is the pointwise limit of Lagrange projections. This leaves open a rather large and intriguing question, What happens in several variables?

To this end we define a linear idempotent operator to be an ideal projector whenever its kernel is an ideal. No matter the number of variables, Taylor projections and Lagrange projections are always ideal projectors, and it is well known that …

On The P(X)-Laplace Equation In Carnot Groups, Robert D. Freeman

USF Tampa Graduate Theses and Dissertations

In this thesis, we examine the p(x)-Laplace equation in the context of Carnot groups. The p(x)-Laplace equation is the prototype equation for a class of nonlinear elliptic partial differential equations having so-called nonstandard growth conditions. An important and useful tool in studying these types of equations is viscosity theory. We prove a p()-Poincar´e-type inequality and use it to prove the equivalence of potential theoretic weak solutions and viscosity solutions to the p(x)-Laplace equation. We exploit this equivalence to prove a Rad´o-type removability result for solutions to the p-Laplace equation in the Heisenberg group. Then we extend this result to the …

Clustering Methods For Gene Expression Data Of Oxytricha Trifallax, Kyle Houfek

USF Tampa Graduate Theses and Dissertations

Clustering is a data analysis method which is used in a large variety of research fields. Many different algorithms exist for clustering, and none of them can be considered universally better than the others. Different methods of clustering are expounded upon, including hierarchical clustering and k-means clustering. Topological data analysis is also described, showing how topology can be used to infer structural information about the data set. We discuss how one finds the validity of clusters, as well as an optimal clustering method, and conclude with how we used various clustering methods to analyze transcriptome data from the ciliate Oxytricha …

Global And Stochastic Dynamics Of Diffusive Hindmarsh-Rose Equations In Neurodynamics, Chi Phan

USF Tampa Graduate Theses and Dissertations

This dissertation consisting of three parts is the study of the open problems of global dynamics of diffusive Hindmarsh-Rose equations, random dynamics of the stochastic Hindmarsh-Rose equations with multiplicative noise and additive noise respectively, and synchronization of boundary coupled Hindmarsh-Rose neuron networks.

In Part I (Chapters 2, 3 and 4) of this dissertation, we study the global dynamics for the single neuron model of diffusive and partly diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain. The existence of global attractors as well as its regularity and structure are established by showing the absorbing properties and the asymptotically compact characteristics, especially …

Restricted Isometric Projections For Differentiable Manifolds And Applications, Vasile Pop

USF Tampa Graduate Theses and Dissertations

The restricted isometry property (RIP) is at the center of important developments in compressive sensing. In R^N, RIP establishes the success of sparse recovery via basis pursuit for measurement matrices with small restricted isometry constants δ_2s < 1=3. A weaker condition, δ_2s < 0:6246, is actually sufficient to guarantee stable and robust recovery of all s-sparse vectors via l₁-minimization. In infinite Hilbert spaces, a random linear map satisfies a general RIP with high probability and allow recovering and extending many known compressive sampling results. This thesis extends the known restricted isometric projection of sparse datasets of vectors embedded in the Euclidean spaces RN down into low-dimensional subspaces R^m ,m << N …