Thesis (Ph. D.)--Florida State University, 2005. / Advisor: Dr. John R. Quine, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed Sept. 19, 2005). Document formatted into pages; contains x, 69 pages. Includes bibliographical references.

Thesis (MSc (Physical and Mathematical Analysis))--University of Stellenbosch, 2009. / In this thesis we aim to show that the Schr odinger equation, which is a boundary eigenvalue problem, can have a discrete and real energy spectrum (eigenvalues) even when the Hamiltonian is non-Hermitian. After a brief introduction into non-Hermiticity, we will focus on solving the Schr odinger equation with a special class of non-Hermitian Hamiltonians, namely PT - symmetric Hamiltonians. PT -symmetric Hamiltonians have been discussed by various authors [1, 2, 3, 4, 5] with some of them focusing speci cally on obtaining the real and discrete energy spectrum. Various methods for solving this problematic Schr odinger equation will be considered. After starting with perturbation theory, we will move on to numerical methods. Three di erent categories of methods will be discussed. First there is the shooting method based on a Runge-Kutta solver. Next, we investigate various implementations of the spectral method. Finally, we will look at the Riccati-Pad e method, which is a numerical implemented analytical method. PT -symmetric potentials need to be solved along a contour in the complex plane. We will propose modi cations to the numerical methods to handle this. After solving the widely documented PT -symmetric Hamiltonian H = p2 􀀀(ix)N with these methods, we give a discussion and comparison of the obtained results. Finally, we solve another PT -symmetric potential, illustrating the use of paths in the complex plane to obtain a real and discrete spectrum and their in uence on the results.

To understand the physical mechanism underlying the cross-field transport of mass and energy in magnetoplasmas is a long-standing problem in fusion research. Woods (1987) has recently developed a second-order transport theory which has been used to explain a number of transport-related phenomena observed in tokamaks. Here, we apply second-order transport theory to the reverse field pinch (RFP) and a phenomenon observed in tokamaks known as 'snakes'. Expressions for the mass and energy confinement times in the RFP, τ_p and τ_e, are deduced and agreement with experimental results from HBTX is found. For typical operating conditions the times τ_p ~ 0.1ms and τ_e ~ 0.2ms are observed in HBTX. Second-order transport theory predicts τ_p ~ 0.4ms and τ_e ~ 0.4ms for this machine. Scaling laws for β_p versus η_e,β_p versus I_φ and τ_e versus I_φ are compared with measurements from HBTX and agree well with observation. Snakes are large density perturbations observed in JET after fuel pellet injection. Typical snakes in JET are remarkably stable and are found to have density decay times longer than predictions based on neoclassical theory (Stringer 1987). After their formation, snakes have even been observed to grow (Weller <i>et al.</i> 1987), which suggests the presence of an inward diffusion mechanism. There is also some evidence for a temperature depression in the snakes region. An explanation of the stability and energy balance in snakes based on second-order transport theory is proposed.

Orientadores: Helena Judith Nussenzveig Lopes, Milton da Costa Lopes Filho / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T02:48:33Z (GMT). No. of bitstreams: 1 Silva_LuizAlbertoViana_D.pdf: 1535000 bytes, checksum: dc971611cbda75b2a6dd72a0bd80d05f (MD5) Previous issue date: 2012 / Resumo: Um problema clássico em aberto é determinar se, em domínios com fronteira, soluções das equações de Navier-Stokes convergem, em um sentido apropriado, a uma solução das equações de Euler quando a viscosidade do fluido tende a zero. Baseados nesta importante questão, Kelliher, Lopes Filho e Nussenzveig Lopes examinaram, em [21], o comportamento de escoamentos com viscosidade pequena em domínios limitados com fronteira afastada, e descreveram condições precisas para que o escoamento limite fosse regido pelas equações de Euler no espaço todo. O presente trabalho é uma continuação natural do artigo mencionado, pois analisamos a dinâmica de escoamentos tridimensionais incompressíveis com viscosidade pequena em torno de obstáculos distantes. Mais precisamente, apresentamos uma estimativa fina que indica um comportamento assintótico para famílias de soluções das equações de Navier-Stokes em termos da viscosidade do escoamento e da localização do obstáculo, e contrastamos a referida estimativa com aquela demonstrada no contexto dos escoamentos em domínios limitados / Abstract: It is a classical open problem to determine if the vanishing viscosity limit can be established in the presence of boundaries. Based on this important issue, Kelliher, Lopes Filho and Nussenzveig Lopes studied in [21] the behavior of viscous incompresible flow in an expanding bounded domain when the viscosity is very small. To be more precise, these three authors described conditions under which the limiting flow satisfies the full space Euler equations. The present work is natural continuation of the aforementioned research since we consider 3D incompressible viscous flows around a distant obstacle along with the vanishing viscosity limit. Specificly, we obtain such a polynomial decay which shows an asymptotic behavior of families of 3D incompressible viscous flows, in the exterior of a single smooth obstacle, in terms of both the obstacle position and the small viscosity. Our approach allows us to compare our rate of convergence to that ones proved in [21] / Doutorado / Matematica / Doutor em Matemática

The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.

Renewal theory began development in the early 1940s, as the need for it in the industrial engineering sub-discipline operations research had risen. In time, the theory found applications in many stochastic processes. In this thesis I investigated the effect of seasonal effects on Poisson and non-Poisson renewal processes in the form of perturbations. It was determined that the statistical analysis methods developed at UNT Center for Nonlinear Science can be used to detect the effects of seasonality on the data obtained from Poisson/non-Poisson renewal systems. It is proved that a perturbed Poisson process can serve as a paradigmatic model for a case where seasonality is correlated to the noise and that diffusion entropy method can be utilized in revealing this relation. A renewal model making a connection with the stochastic resonance phenomena is used to analyze a previous neurological experiment, and it was shown that under the effect of a nonlinear perturbation, a non-Poisson system statistics may make a transition and end up in the of Poisson basin of statistics. I determine that nonlinear perturbation of the power index for a complex system will lead to a change in the complexity characteristics of the system, i.e., the system will reach a new form of complexity.

In this dissertation, we study the dynamics, fractal geometry and the topology of the Julia set of functions in the family H which is a set in the class S, the Speiser class of entire transcendental functions which have only finitely many singular values. One can think of a function from H as a generalized expanding function from the cosh family. We shall build a version of thermodynamic formalism for functions in H and we shall show among others, the existence and uniqueness of a conformal measure. Then we prove a Bowen's type formula, i.e. we show that the Hausdorff dimension of the set of returning points, is the unique zero of the pressure function. We shall also study conjugacies in the family H, perturbation of functions in the family and related dynamical properties. We define Perron-Frobenius operators for some functions naturally associated with functions in the family H and then, using fundamental properties of these operators, we shall prove the important result that the Hausdorff dimension of the subset of returning points depends analytically on the parameter taken from a small open subset of the n-dimensional parameter space.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERO. / Includes bibliographical references. / by Edward Gregory Taylor. / Ph.D.

A nonlinear second order differential equation related to vortex motion is derived. This equation is analyzed using various numerical and analytical techniques including finding approximate solutions using a perturbative approach.

The present study was undertaken to determine the effects of the aerodynamics on a close earth orbit and reentry trajectory. The aerodynamic influence is compared to the effects of the earth’s oblateness, the sun, and the moon. In order to obtain maximum accuracy and computational speed Encke’s perturbative procedures were used during orbital periods and Cowell’s integration procedures during thrust and reentry periods. / M.S.