Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2007. / Includes bibliographical references (p. 48).

Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references.

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Chi-Wang Shu. Includes bibliographical references (leaves 83-87). Also available online.

Introduction: The discovery of new solutions to Einstein's field equations has long been a problem in General Relativity. However due to new techniques of Newman and Penrose [1], Carter [2] and others there has been a considerable proliferation of new solutions in recent times. Consequently a new problem has arisen. How are we to interpret the new solutions physically? The tools available, despite a spate of papers in the past fifteen years, remain inadequate although often sophisticated. Any attempts at physical interpretations of metrics are beset with difficulties. There is always the possibility that two entirely different physical pictures will emerge. For example a direct approach would be to attempt an "infilling" of the metric, that is, an extension of the metric into the region occupied by the gravitating matter. However even for the Kerr [1] metric the infilling is by no means unique, in fact a most natural "infilling" turns out to be unphysical (Israel [1]). Yet few people would doubt the physical significance of the Kerr metric. Viewed in this light our attempt to discuss, among other things, the physical interpretation of type D metrics is slightly ambitious. However the problems with regard to this type of metric are not as formidable as for most of the other metrics, since we have been able to integrate the geodesic equations. Nevertheless it is still not possible to produce complete answers to all the questions posed. After a chapter on Mathematical preliminaries the study divides naturally into four sections. We start with an outline of the Hamilton-Jacobi theory of Rund [1] and then go on to show how this theory can be applied to the Carter [2] metrics. In the process we lay a foundation in the calculus of variations for Carter's work. This leads us to the construction of Killing tensors for all but one of the Kinnersley [1] type D vacuum metrics and the Cartei [2] metrics which are not necessarily vacuum metrics. The geodesic equations, for these metrics, are integrated using the Hamilton-Jacobi procedure. The remaining chapters are devoted to the Kinnersley [1] type D vacuum metrics. We omit his class I metrics since these are the Schwarzschild metrics, and have been studied in detail before. Chapter three is devoted to a general study of his class II a metric, a generalisation of the Kerr [1] and NUT (Newman, Tamburino and Unti [1]) metrics. We integrate the geodesic equations and discuss certain general properties: the question of geodesic completeness, the asymptotic properties, and the existence of Killing horizons. Chapter four is concerned with the interpretation of the new parameter 'l', that arises in the class II a and NUT metrics. This parameter was interpreted by Demianski and Newman [1] as a magnetic monopole of mass. Our work centers on the possibility of obtaining observable effects from the presence of 'l'. We have been able to show that its presence is observable, at least in principle, from a study of the motion of particles in the field. In the first place, if l is comparable to the mass of the gravitating system, a comparatively large perihelion shift is to be expected. The possibility of anomalous behaviour in the orbits of test particles, quite unlike anything that occurs in a Newtonian or Schwarzschild field, also arises. In the fifth chapter the Kinnersley class IV metrics are considered. These metrics, which in their simplest form have been known for some time, present serious problems and no interpretations have been suggested. Our discussion is essentially exploratory and the information that does emerge takes the form of suggestions rather than conclusions. Intrinsically the metrics give the impression that interesting results should be obtainable since they are asymptotically flat in certain directions. However the case that we have dealt with does not appear to represent a radiation metric.

In this thesis, we consider second-order fully nonlinear partial differential equations of elliptic type. Our aim is to develop computational methods using convergent difference schemes for stationary Hamilton-Jacobi equations with Dirichlet and Neumann type boundary conditions in arbitrary two-dimensional domains. First, we introduce the notion of viscosity solutions in both continuous and discontinuous frameworks. Next, we review Barles-Souganidis approach using monotone, consistent, and stable schemes. In particular, we show that these schemes converge locally uniformly to the unique viscosity solution of the first-order Hamilton-Jacobi equations under mild assumptions. To solve the scheme numerically, we use Euler map with some initial guess. This iterative method gives the viscosity solution as a limit. Moreover, we illustrate our numerical approach in several two-dimensional examples.

I present two differential-algebraic (DA) methods for approximately solving the Hamilton- Jacobi (HJ) equation. I use the “automatic differentiation” property of DA to convert the nonlinear partial-differential HJ equation into a initial-value problem for a DA-valued first-order ordinary differential equation (ODE), the “HJ/DA equation”. The solution of either form of the HJ/DA equation is equivalent to a perturbative expansion of Hamilton’s principle function about some reference trajectory (RT) through the system. The HJ/DA method also extracts the equations of motion for the RT itself. Hamilton’s principle function generates the canonical transformation, or mapping, between the initial and final state of every trajectory through the system. Since the map is represented by a generating function, it must automatically be symplectic, even in the presence of round-off error. The DA-valued ODE produced by either form of HJ/DA is equivalent tc a hierarchically-ordered system of real-valued ODEs without “feedback” terms; therefore the hierarchy may be truncated at any (arbitrarily high) order without loss of self consistency. The HJ/DA equation may be numerically integrated using standard algorithms, if all mathematical operations are done in DA. I show that the norm of the DA-valued part of the solution is bounded by linear growth. The generating function may be used to track either particles or the moments of a particle distribution through the system. In the first method, all information about the perturbative dynamics is contained in the DA-valued generating function. I numerically integrate the HJ/DA equation, with the identity as the initial generating function. A difficulty with this approach is that not all canonical transformations can be represented by the class of generating functions connected to the identity; one finds that with the required initial conditions, the generating function becomes singular near caustics or foci. One may continue integrating through a caustic by using a Legendre transformation to obtain a new (but equivalent) generating function which is singular near the identity, but nonsingular near the caustic. However the Legendre transformation is a numerically costly procedure, so one would not want to do this often. This approach is therefore not practical for systems producing periodic motions, because one must perform a Legendre transformation four times per period. The second method avoids the caustic problem by representing only the nonlinear part of the dynamics by a generating function. The linearized dynamics is treated separately via matrix techniques. Since the nonlinear part of the dynamics may always be represented by a near-identity transformation, no problem occurs when passing through caustics. I successfully verify the HJ/DA method by applying it to three problems which can be solved in closed form. Finally, I demonstrate the method’s utility by using it to optimize the length of a lithium lens for minimum beam divergence via the moment-tracking technique. / Ph. D.

Hamilton-Jacobi equations are a class of partial differential equations that arise in many areas of science and engineering. Originating from classical mechanics, they are widely used in various fields such as optimal control theory, quantitative finance, and game theory. Stochastic homogenization is a phenomenon used to study the behavior of solutions to partial differential equations in stationary ergodic media, aiming to understand how these solutions average out or 'homogenize' over large scales. This process results in effective deterministic descriptions, called effective Hamiltonians, which capture the essential behavior of the system. We consider nonconvex Hamilton-Jacobi equations in one space dimension. We provide a fully constructive proof of homogenization, which yields a formula for the effective Hamiltonian. Our proof employs sublinear correctors, functions extensively discussed in the literature. The proof involves strong induction: we first show homogenization for our base cases, then use gluing processes to generalize the solution for the strong induction. Finally, we extend the result to a wide class of functions. We study the properties of the resulting effective Hamiltonian and investigate the occurrence of flat pieces. Additionally, we develop a Python-based computational tool that performs the same homogenization steps in a computing environment, returning the effective Hamiltonian along with its graph and properties. / Mathematics

No description available.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 145-151).