This is a study of the effect of harvesting on a system with two competing species. The system is a Ricker-type model that extends the work done by Luis, Elaydi, and Oliveira to include the effect of harvesting on the system. We look at the uniform bound of the system as well as the isoclines and perform a stability analysis of the equilibrium points. We also look at the effects of harvesting on the stability of the system by looking at the bifurcation of the system with respect to harvesting.

One of the most prevalent inherited diseases is cystic fibrosis. This disease is caused by a mutation in a membrane protein, the cystic fibrosis transmembrane conductance regulator (CFTR). CFTR is known to function as a chloride channel that regulates the viscosity of mucus that lines the ducts of a number of organs. Generally, most of the prevalent mutations of CFTR are located in one of two nucleotide binding domains, namely, the nucleotide binding domain 1 (NBD1). However, some mutations in nucleotide binding domain 2 (NBD2) can equally cause cystic fibrosis. In this work, a hierarchical graph is built for NBD2. Using this model for NBD2, we examine the consequence of single point mutations on NBD2. We collate the wildtype structure with eight of the most prevalent mutations and observe how the NBD2 is affected by each of these mutations.

Vancomycin is an antibiotic used for the treatment of systemic infections. It is given intravenously usually every twelve or twenty-four hours. This particular drug has a medium level of boundedness, with approximately fty percent of the drug being free and thus physiologically eective. A physiologically-based pharmacokinetic (PBPK) model was used to better understand the absorption, distribution, and elimination of the drug. Using optimal parameters, the model could be used in the future to test how various factors, such as BMI or excretion levels, might aect the concentration of the antibiotic.

We live in an age when our most private information is becoming exceedingly difficult to keep private. Cryptology allows for the creation of encryptive barriers that protect this information. Though the information is protected, it is not entirely inaccessible. A recipient may be able to access the information by decoding the message. This possible threat has encouraged cryptologists to evolve and complicate their encrypting methods so that future information can remain safe and become more difficult to decode. There are various methods of encryption that demonstrate how cryptology continues to evolve through time. These methods revolve around different areas of mathematics such as arithmetic, number theory, and probability. Another concern that has brought cryptology into everyday use and necessity is user authentication. How does one or a machine know that a user is who they say they are? Living in the age where most of our information is sent and accepted through computers, it is crucial that our information is kept safe, and in the appropriate care.

This thesis studies the behavior of the Eulerian scheme, with "Wave Confinement" (WC), when propagating periodic waves. WC is a recently developed method that was derived from the scheme "vorticity confinement" used in fluid mechanics, and it efficiently solves the linear wave equation. This new method is applicable for numerous simulations such as radio wave propagation, target detection, cell phone and satellite communications. The WC scheme adds a nonlinear term to the discrete wave equation that adds stability with negative and positive diffusion, conserves integral quantities such as total amplitude and wave speed, and it allows wave propagation over long distances with minimal numerical diffusion, which contrasts to other numerical methods where wave propagation is affected by numerical dissipation. Previous studies have shown that WC propagates short pulses/surfaces as thin nonlinear solitary waves. In this thesis, a one-dimensional (1D) periodic wave is propagated by WC using the advection and wave equations. For the advection equation, the parameters and the initial condition (IC) used in WC are analyzed to establish for which conditions the method can be implemented. When the IC is a positive periodic wave, the converged solution consists of a series of hyperbolic secants where the number of cycles of the IC represents the number of hyperbolic secants. Waves with varying signs are analyzed by changing the wave confinement term. For this case, the converged solution is a series of positive and negative hyperbolic secants where each hyperbolic secant is represented by half cycle of the IC. For the wave equation, parameters and different IC's are studied to determine when WC is feasible. For positive periodic waves, the converged solution retains its sinusoidal shape and does not converge to a series of hyperbolic secants. The waves with varying signs, however, converge to a series of hyperbolic secants as seen for the advection equation. WC is stable for various periodic waves for both advection and wave equations, which shows WC is useful for numerically propagating periodic waveforms. Convergence depends on the wave number of the IC and on the parameters (convection speed, positive diffusion, negative diffusion) used in WC.

Malaria is a preventable and treatable blood-borne disease whose complications can be fatal. Although many interventions exist in order to reduce the impacts of malaria, the optimal method of distributing these interventions in a geographical area with limited resources must be determined. This thesis refines a model that uses an integer linear program and a compartmental model of epidemiology called an SIR model of ordinary differential equations. The objective of the model is to find an intervention strategy over multiple time steps and multiple geographic regions that minimizes the number of days people spend infected with malaria. In this paper, we refine the resolution of the model and conduct sensitivity analysis on its parameter values.

The task of finding the beat in music is simple for most people, but surprisingly difficult to replicate in a robot. Progress in this problem has been made using various preprocessing techniques (Hitz 2008; Tomic and Janata 2008). However, a real-time method is not yet available. Methods using a class of oscillators called relay relaxation oscillators are promising. In particular, systems of forced Hopf oscillators (Large 2000; Righetti et al. 2006) have been used with relative success. This work describes current methods of beat tracking and develops a new method that incorporates the best ideas from each existing method and removes the necessity for preprocessing.

Two methods for control of swarms are described. The first of these methods, the Virtual Attractive-Repulsive (VARP) method, is based on potentials defined between swarm elements. The second control method, or the abstraction method, is based on controlling the macroscopic characteristics of a swarm. The derivation of a new control law based on the second method is described. Numerical simulation and analytical interpretation of the result is also presented.

Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the "difference" of any function, from 1st order up to the n-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make n- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied. The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced. First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented.

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to zeta function regularization and explore more fully the relationship between operators in physics and classical zeta functions of mathematics. In so doing, we highlight intriguing connections to number theory that arise.