Diffusion processes provide a natural way of modelling a variety of physical and economic phenomena. It is often the case that one is unable to observe a diffusion process directly, and must instead rely on noisy observations that are discretely spaced in time. Given these discrete, noisy observations, one is faced with the task of inferring properties of the underlying diffusion process. For example, one might be interested in inferring the current state of the process given observations up to the present time (this is known as the filtering problem). Alternatively, one might wish to infer parameters governing the time evolution the diffusion process. In general, one cannot apply Bayes’ theorem directly, since the transition density of a general nonlinear diffusion is not computationally tractable. In this thesis, we investigate a novel method of simplifying the problem. The stochastic differential equation that describes the diffusion process is replaced with a simpler ordinary differential equation, which has a random driving noise that approximates Brownian motion. We show how one can exploit this approximation to improve on standard methods for inferring properties of nonlinear diffusion processes.

Master of Science / Department of Mathematics / Nathan Albin / This work utilizes finite differences to approximate the first derivative of non-periodic smooth functions. Math literature indicates that stabilizing Partial Differential Equation solvers based on high order finite difference approximations of spatial derivatives of a non-periodic function becomes problematic near a boundary. Hagstrom and Hagstrom have discovered a method of introducing additional grid points near a boundary, which has proven to be effective in stabilizing Partial Differential Equation solvers. Hagstrom and Hagstrom demonstrated their method for the case of the one-dimensional advection equation using spatial derivative approximations of even orders up to twenty-second order. In this dissertation, we explore the efficacy of the Hagstrom and Hagstrom method for the same Partial Differential Equation with spatial derivative approximations of odd orders and orders higher than twenty-two and report the number and locations of additional grid points required for stability in each case.

Doctor of Philosophy / Department of Family Studies and Human Services / Jared R. Anderson / The current study tests a theoretical model exploring the relationship between hooking up and marital quality and whether this relationship is mediated by sexual satisfaction and communication using public-use data from currently married participants in Wave IV of the National Longitudinal Study of Adolescent Health (Add Health, n = 1,729). Gender proved to significantly moderate the association between the variables in the model, but college education did not. The results indicate that hooking up has a direct negative relationship with marital quality for men that is not mediated by either sexual satisfaction or communication. The results for women revealed no direct relationship between hooking up and marital quality, but an indirect influence via communication.

Through perturbation analysis, a study of the role of Brinkman viscosity in the propagation of finite amplitude harmonic waves is carried out. Interplay between various parameters, namely, frequency, Reynolds number and beta are investigated. For systems with physically realizable Reynolds numbers, departure from the Darcy Jordan model (DJM) is noted for high frequency signals. Low and high frequency limiting cases are discussed, and the physical parameters defining the acoustic propagation are obtained. Through numerical analyses, the roles of Brinkman viscosity, the Darcy coefficient, and the coefficient of nonlinearity on the evolution of finite amplitude harmonic waves is stud- ied. An investigation of acoustic blow-ups is conducted, showing that an increase in the magnitude of the nonlinear term gives rise to blow-ups, while an increase in the strength of the Darcy and/or Brinkman terms mitigate them. Finally, an analytical study via a regular perturbation expansion is given to support the numerical results. In order to gain insight into the formation and evolution of nonlinear standing waves un- der the Brinkman model, a numerical analysis is conducted on the weakly nonlinear model based on Brinkman’s equation. We develop a finite difference scheme and conduct a param- eter study. An examination of the Brinkman, Darcy, and nonlinear terms is carried out in the context of their roles on shock formation. Finally, we compare our findings to those of previous results found in similar nonlinear equations in other fields. So as to better understand the behavior of finite-amplitude harmonic waves under a Brinkman-based poroacoustic model, approximations and transformations are used to recast the Brinkman equation into the damped Burger’s equation. An examination is carried out for the two special solutions of the damped Burger’s equation: the approximate solution to the damped Burger’s equation and the boundary value problem given an initial sinusoidal pulse. The effects of the Darcy coefficient, Reynolds number, and nonlinear coefficient on these solutions are investigated.

PT-symmetric Hamiltonians proposed by Bender and Boettcher can have real energy spectra. As an extension of the Hermitian Hamiltonian, PT-symmetric systems have attracted a great interest in recent years. Understanding the underlying mathematical structure of these theories sheds insight on outstanding problems of physics. These problems include the nature of Higgs particles, the properties of dark matter, the matter-antimatter asymmetry in the universe, and neutrino oscillations. Furthermore, PT-phase transition has been observed in lasers, optical waveguides, microwave cavities, superconducting wires and circuits. The objective of this thesis is to extend the iterative method of solving Schrodinger equation used for an harmonic oscillator systems to Hamiltonians with PT-symmetric potentials. An important aspect of this approach is the high accuracy of eigenvalues and the fast convergence. Our method is a combination of Hill determinant method [8] and the power series expansion. eigenvalues and the fast convergence. One can transform the Schrodinger equation into a secular equation by using a trial wave function. A recursion structure can be obtained using the secular equation, which leads to accurate eigenvalues. Energy values approach to exact ones when the number of iterations is increased. We obtained eigenvalues for a set of PT-symmetric Hamiltonians.

The problem of constructing data-based, predictive, reduced models for the Kuramoto–Sivashinsky equation is considered, under circumstances where one has observation data only for a small subset of the dynamical variables. Accurate prediction is achieved by developing a discrete-time stochastic reduced system, based on a NARMAX (Nonlinear Autoregressive Moving Average with eXogenous input) representation. The practical issue, with the NARMAX representation as with any other, is to identify an efficient structure, i.e., one with a small number of terms and coefficients. This is accomplished here by estimating coefficients for an approximate inertial form. The broader significance of the results is discussed.

A dissertation submitted for the fulfilment of the requirements of the degree of Master of Science to the Faculty of Science, Witwatersrand University, Johannesburg. June 2016. / The combination of effective numerical techniques and scientific intuition to find new and novel types of materials is the process used in the discovery of materials for future technologies. Adding to that, being able to calculate the radiative lifetimes of excitons, exciton properties, and the optical properties by using efficient numerical techniques gives an estimation and identification of the best candidate materials for a solar cell. This approach is inexpensive and stable. Present ab initio methods based on Many-body perturbation theory and density functional theory are capable of predicting these properties with a high enough level of accuracy for most cases. The electronic properties calculated using GaAs as a reference system and the 3D hybird perovskite CH3NH3PbI3 are based on density functional theory. The optical properties are investigated by calculating the dielectric function. The theoretical framework of the radiative lifetime of excitons and calculating the exciton properties are based on Wannier model of the exciton and the Bethe-Salpeter equation. / MT2017

Thesis advisor: Thomas O'Hare / Psychopathy has received a marked increase in attention in the research literature over the past 2 decades since the validation and standardization of assessment tools designed to measure this construct, particularly the Psychopathy Checklist-measures (Hare, 1991/2003; Hart, Cox, & Hare, 1995; and Forth, Kosson, & Hare, 2003). Psychopathy has been identified as the best single predictor of violence among adult offenders (Hart, 1998). Such findings have led some to conclude that "psychopathy is the most important psychological construct for policy and practice in the criminal justice system" (Harris, Skilling, & Rice, 2001). Despite the overwhelming evidence of substantial societal and individual costs attributable to this disorder, little is known about psychosocial precursors of psychopathy. This study examines risk factors related to the development of psychopathy, as measured by the PCL:SV, in a sample of 446 psychiatric patients using structural equation modeling (SEM). The final SEM includes five predictor variables measuring early-life physical abuse, paternal antisocial behavior, and cognitive ability. Severe physical abuse (&beta; = 0.17, <italic>p</italic> = .043), biological father's alcohol abuse history (&beta; = .16, <italic>p</italic> =.004), biological father's arrest history (&beta; = 0.13, <italic>p</italic> = .02), and the subject's cognitive ability (&beta; = -0.18, <italic>p</italic> < .001) were found predictive of psychopathy in this sample. Post hoc analyses comparing male and female subjects, and black and white subjects, indicate different causal pathways in the development of psychopathy among these groups. Future research designed to assess these potentially different causal pathways are recommended. Implications to clinical theory, practice, and policy are also discussed. / Thesis (PhD) — Boston College, 2009. / Submitted to: Boston College. Graduate School of Social Work. / Discipline: Social Work.

Financial mathematics is a branch of mathematics that assesses the risk and value of various financial instruments. Banks, companies, and other institutions mitigate their risk through financial instruments known as derivatives,that derive their value from some underlying asset. The equations that arise from pricing and modeling can be very complex, leading to the necessity of numerical methods. This project studied the use of certain numerical methods for the pricing of a particular type of option called an Asian option. Asian options can provide favorable risk profiles because the payout is determined based on the average value over a time period, rather than the final value. The price of an Asian option is governed by a partial differential equation in three variables: stock price, average price over the current time interval, and time. The solution method was first to discretize the partial differential equation into a system of ordinary differential equations. Next, the ODE system was integrated using a stiff-ODE solver available in MATLAB. Enhancements to this solution method include specifying the sparsity pattern, implementing an iterative linear solver (GMRES) in place of MATLAB's built-in direct linear solver, and using preconditioning to improve the solution characteristics of that solver.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.