Digital Commons Network^™

The Effect Of Fixed Time Delays On The Synchronization Phase Transition, Shaizat Bakhytzhan

USF Tampa Graduate Theses and Dissertations

Nature is full of synchronization phenomena, which are essential to many scientific fields like biology, chemistry, physics, and neuroscience. The Kuramoto model is a well-known theoretical model that helps explain the fundamental ideas behind synchronization dynamics [6]. Nevertheless, in practical situations, systems frequently display intrinsic latency, which can greatly impact their behavior during synchronization. This insight inspired our work, which looks at the results of adding temporal delays to the Kuramoto model. In particular, we investigate how the system’s synchronization dynamics are affected by delays. We shed light on the mechanisms underpinning synchronization in the face of temporal delays and …

Quandle Rings, Idempotents And Cocycle Invariants Of Knots, Dipali Swain

USF Tampa Graduate Theses and Dissertations

Quandles are sets with self-distributive binary operations that axiomatize the three Reidemeister movesin classical knot theory. In an attempt to bring ring theoretic techniques to the study of quandles, a theory of quandle rings analogous to the classical theory of group rings where several interconnections between quandles and their associated quandle rings have been explored. Functoriality of the construction implies that morphisms of quandle rings give a natural enhancement of the well-known quandle coloring and quandle 2 cocycle invariant of knots and links.

The dissertation is structured into two main parts. In the first part, we delve into quandle rings …

On The Subelliptic And Subparabolic Infinity Laplacian In Grushin-Type Spaces, Zachary Forrest

USF Tampa Graduate Theses and Dissertations

This thesis poses the ∞-Laplace equation in Grushin-type spaces. Grushin-type spaces G are defined by the vector fields which serve as a basis for their tangent spaces; by weighting the canonical (Euclidean) directional vectors {∂/∂x_i}ⁿ_i=1 by functions ρ_i that obey certain technical assumptions, we produce a class of metric spaces in which certain directions may not be accessible at all points in the space. We prove the existence and uniqueness of viscosity solutions to both Dirichlet problems and Cauchy-Dirichlet problems involving the∞-Laplacian over bounded Grushin-type domains. The main tool in proving uniqueness of …

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 …

Matrix Models Of 2d Critical Phenomena, Nathan Hayford

USF Tampa Graduate Theses and Dissertations

The 2D Ising model has played an important role in the theory of phase transitions, as one of only ahandful of exactly solvable models in statistical mechanics. The original model, introduced in the 1920s, has a rich mathematical structure. It thus came as a pleasant surprise when physicists studying matrix models of 2D gravity found that, coupled to quantum gravity, the planar Ising model still had an elegant solution. The methods used by V. Kazakov and his collaborators involved the method of orthogonal polynomials. However, these methods were formal, and no direct analytic derivation of the phase transition has been …

Recovering Generators Of Principal Ideals Using Subfield Structure And Applications To Cryptography, William Youmans

USF Tampa Graduate Theses and Dissertations

The principal ideal problem (PIP) is the problem of determining if a given ideal of a number field is principal, and if so, of finding a generator.Algorithms for resolving the PIP can be efficiently adapted to solve many hard problems in algebraic number theory, such as the computation of the class group, unit group, or $S$-unit group of a number field. The PIP is also connected to the search for approximate short vectors, known as the $\gamma$-Shortest Vector Problem ($\gamma$-SVP), in certain structured lattices called ideal lattices, which are prevalent in cryptography. We present an algorithm for resolving the PIP …

Data-Driven Learning Algorithm Via Densely-Defined Multiplication Operators And Occupation Kernels., John Kyei

USF Tampa Graduate Theses and Dissertations

Consider a nonautonomous nonlinear evolution $\dot{x}=f(x,t,\mu)$, where the vector $x(t) \in \mathbb{R}^n$ represents the state of the dynamical system at time $t$, $\mu$ contains system parameters, and $f(\cdot)$ represents a dynamic constraint. In most practical applications, the nonlinear dynamic constraint $f$ is unknown analytically. The problem of approximating $f$ directly from data measurements generated by the system is a main goal of this manuscript. In the postulates of the Nonlinear Autoregressive (NAR) framework, we show that the problem of approximating $f$ can be studied through symbols of densely defined multiplication operators over a Reproducing Kernel Hilbert Spaces (RKHS). In this …

Exploring The Vulnerability Of A Neural Tangent Generalization Attack (Ntga) - Generated Unlearnable Cifar-10 Dataset, Gitte Ost

USF Tampa Graduate Theses and Dissertations

Nowadays, a massive amount of data is generated and stored on servers and cloudsfrom various applications daily. Preventing these data from unauthorized use often becomes necessary and critical in various real-world applications. Many researchers have studied this crucial problem and developed different methods for this purpose. Among them, Neural Tangent Generalization Attack (NTGA) is one of the most efficient methods to make a dataset unlearnable, which means that the dataset is not learnable by machine learning/deep learning methods. That is, the NTGA-generated dataset is protected against unauthorized use. In this thesis, we explore the vulnerability of an NTGA-generated unlearnable CIFAR-10 …

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 …

Methods In Discrete Mathematics To Study Dna Rearrangement Processes, Lina Fajardo Gómez

USF Tampa Graduate Theses and Dissertations

In this work, we introduce novel tools to study DNA recombination pathways and measure their complexity. Genome rearrangement in some ciliate species can be modeled by subword pattern deletions in double-occurrence words (DOWs), words where each symbol appears exactly twice. The iterated deletions can be represented by a graph whose vertices are DOWs connected by an edge if one word can be obtained from the other through a pattern deletion. On this graph, called the “word graph”, we build a complex comprised of cells defined by Cartesian products of simplicial digraphs where we define a boundary operator and compute homology …

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 …

Application Of The Riemann-Hilbert Method To Soliton Solutions Of A Nonlocal Reverse-Spacetime Sasa-Satsuma Equation And A Higher-Order Reverse-Time Nls-Type Equation, Ahmed Ahmed

USF Tampa Graduate Theses and Dissertations

For many years, the study of integrable systems has been one of the most fascinating branches of mathematics and has been thought to be an interesting area for both mathematicians and physicists alike.Many natural phenomena can be predicted by using integrable systems, particularly by studying their different solutions, as well as analyzing and exploring their properties and structures. They are commonly found in nonlinear optics, plasmas, ocean and water waves, gravitational fields, and fluid dynamics. Typical examples of integrable systems include the Korteweg-de Vries (KdV) equation, the nonlinear Schrödinger (NLS) equation, and the Kadomtsev-Petviashvili (KP) equation. Solitons are intrinsic solutions …

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 …