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Various Dynamical Regimes, And Transitions From Homogeneous To Inhomogeneous Steady States In Nonlinear Systems With Delays And Diverse Couplings, Ryan Roopnarain

Electronic Theses and Dissertations, 2020-2023

This dissertation focuses on the effects of distributed delays modeled by 'weak generic kernels' on the collective behavior of coupled nonlinear systems. These distributed delays are introduced into several well-known periodic oscillators such as coupled Landau-Stuart and Van der Pol systems, as well as coupled chaotic Van der Pol-Rayleigh and Sprott systems, for a variety of couplings including diffusive, cyclic, or dynamic ones. The resulting system is then closed via the 'linear chain trick' and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. A variety of dynamical regimes and transitions between …

Population Persistence And Disease Invasion In Heterogenous Networks, Poroshat Yazdanbakhshghahyazi

Electronic Theses and Dissertations, 2020-2023

The problem of understanding how biological species and infectious diseases can persist and spread in heterogeneous networks has brought a wide attention, recently highlighted due to the COVID-19 pandemic. This dissertation investigates the connection between the structures of heterogeneous networks and population persistence/disease invasion. To do so, we propose a new index for network heterogeneity by employing the Laplacian matrix of population dispersal and its corresponding group inverse. The network growth rate and reproduction number can be evaluated using the network average and the network heterogeneity index as the first and second order approximation, respectively. We also illustrate the impact …

Extensions Of The General Solution To The Inverse Problem Of The Calculus Of Variations, And Variational, Perturbative And Reversible Systems Approaches To Regular And Embedded Solitary Waves, Ranses Alfonso Rodriguez

Electronic Theses and Dissertations, 2020-2023

In the first part of this Dissertation, hierarchies of Lagrangians of degree two, three or four, each only partly determined by the choice of leading terms and with some coefficients remaining free, are derived. These have significantly greater freedom than the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation since our existence conditions are for individual coefficients in the Lagrangian. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations. Families of regular and embedded solitary waves are derived for some of …

Mathematical Model For Giraffe Population Dynamics, Huntir Bass

Electronic Theses and Dissertations, 2020-2023

Since the 1980s the overall giraffe population has dropped at least 40% causing some researchers to label this rapid decline as the "Silent Extinction." Due to this plummet, understanding the behaviors of the giraffe population is absolutely necessary before they are on the brink of extinction. Through the usage of mathematical modeling methodologies, a general model is created to illustrate the relationship between juvenile and adult female giraffes through numerous interaction parameters. Variations on specific variables generate different simulations, which allows more biological accuracy. With each variation having an established coexistence equilibrium between the juvenile and adult female populations, the …

The Parker Problem In Hall Magnetohydrodynamics Analytical And Numerical Solutions, Chad Malott

Electronic Theses and Dissertations, 2020-2023

In this thesis we follow on the mathematical aspects of the previous work of Shivamoggi (2009) on the Parker problem in Hall magnetohydrodynamics (MHD). We will present an analysis involving detailed analytical and numerical solutions to the Parker problem in Hall MHD. We give an analytical formulation for the Parker problem in Hall MHD, involving an initial value problem (IVP) associated with a first order Riccati equation (RE). We present Mathematica software exact solutions directly with special functions and more straightforward solutions that use the change of variables and power series methods without special functions. We give an asymptotic formulation …

Cholera Transmission Dynamic Model With Environmental Impacts Of Plankton Reservoirs, Sweety Sarker

Electronic Theses and Dissertations, 2020-2023

Cholera is an acute disease that is a global threat to the world and can kill people within a few hours if left untreated. In the last 200 years, seven pandemics occurred, and, in some countries, it remains endemic. The World Health Organization (WHO) declared a global initiative to prevent cholera by 2030. Cholera dynamics are contributed by several environmental factors such as salinity level of water, water temperature, presence of plankton especially zooplankton such as cladocerans, rotifers, copepods, etc. Vibrio cholerae (V. cholerae) bacterium is the main reason behind the cholera disease and the growth of V. cholerae depends …

Function Approximation Guarantees For A Shallow Neural Network Trained By Gradient Flow, Russell Gentile

Electronic Theses and Dissertations, 2020-2023

This work features an original result linking approximation and optimization theory for deep learning. Several examples from recent literature show that, given the same number of learnable parameters, deep neural networks can approximate richer classes of functions, with better accuracy than classical methods. The bulk of approximation theory results though, are only concerned with the infimum error for all possible parameterizations of a given network size. Their proofs often rely on hand-crafted networks, where the weights and biases are carefully selected. Optimization theory indicates that such models would be difficult or impossible to realize with standard gradient-based training methods. The …

Optimal Impulse Controls With Changing Running Cost And Applications In Mortgage Refinance, Yuchen Cao

Electronic Theses and Dissertations, 2020-2023

Almost all home buyers have mortgages and it is quite common to have mortgage refinanced. There are two main reasons that make people decide to refinance the mortgage: (i) need some cash for urgent purposes, and (ii) lower the monthly payment. In this dissertation, we are not going to discuss (i), and we are investigating problems related to (ii). To begin with, let us intuitively make the following observations: If the interest rate remains the same as the current mortgage interest rate, then the monthly payment will automatically lower if you start a new mortgage with the same term, say, …

The Effects Of Viscous Damping On Rogue Wave Formation And Permanent Downshift In The Nonlinear Schrödinger Equation, Evelyn Smith

Honors Undergraduate Theses

This thesis investigates the effect of viscous damping on rogue wave formation and permanent downshift using the higher-order nonlinear Schrödinger equation (HONLS). The strength of viscous damping is varied and compared to experiments with only linear damped HONLS.

Stability analysis of the linear damped HONLS equation shows that instability stabilizes over time. This analysis also provides an instability criterion in the case of HONLS with viscous damping.

Numerical experiments are conducted in the two unstable mode regime using perturbations of the Stokes wave as initial data. With only linear damping permanent downshift is not observed and rogue wave formation is …

Multicolor Ramsey And List Ramsey Numbers For Double Stars, Jake Ruotolo

Honors Undergraduate Theses

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. For a graph H, the k-color Ramsey number r(H; k) of H is the smallest integer n such that every k-edge-coloring of K_n contains a monochromatic copy of H. Despite active research for decades, very little is known about Ramsey numbers of graphs. This is especially true for r(H; k) when k is at least 3, also known as the multicolor Ramsey number of …