This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy Almost Axisymmetric Flows with a forcing term model.

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AC/DC Buck converters have been used widely in many applications from cell phones to vehicle battery chargers. Due to their importance many researchers have been studying their behavior to improve their efficiency and reduce their size and/or cost. One of the most common defects of these converters, whether they are used for high power or low power applications, is the unwanted ripples in the input voltage across the input stage. It is believed that these ripples are caused by the interaction between the converter itself or its controller with the rectifier required to change the AC input into DC followed by an input filter. Many strategies have been suggested to tackle this problem. A new strategy to improve the controller of that converter was suggested by M. Lenells [1] and it was based on a Hamiltonian model for the 3-phase AC/DC converter together with its rectifier. As a first step, we simulated this model for a single-phase DC/DC buck converter only using the so-called S-Functions in MATLAB/SIMULINK. Then we could find a control law that would reduce the ripples in the input voltage and keep the output voltage constant simultaneously. In this report, we present this model and its simulation to pave the way for the control and simulation of the 3-phase AC/DC converter.

Let M be smooth symplectic closed manifold and L a closed Lagrangian submanifold of M. It was shown by Ozan that Ham(M,L): the relative Hamiltonian diffeomorphisms on M fixing the Lagrangian submanifold L setwise is a subgroup which is equal to the kernel of the restriction of the flux homomorphism to the universal cover of the identity component of the relative symplectomorphisms. In this thesis we show that Ham(M,L) is a non-simple perfect group, by adopting a technique due to Thurston, Herman, and Banyaga. This technique requires the diffeomorphism group be transitive where this property fails to exist in our case.

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2008. / Kaynakça var.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 93-103).

The influence of matter described by the Vlasov equation, on the evolution of anisotropy in the spatially-homogeneous universes, called the Bianchi cosmologies, is studied. Due to the spatial-homogeneity, the Einstein equations for each Bianchi Type are reduced to a set of coupled ordinary differential equations, which has Hamiltonian form with the metric components being the canonical coordinates. In the vacuum Bianchi cosmologies, it is known that a curvature potential, which comes from the symmetries of the three-dimensional Lie groups, determines the basic properties of the evolution of anisotropy. In this work, matter potentials are constructed for Vlasov matter. They are obtained by first introducing a new matter action principle for the Vlasov equation, in terms of a conjugate pair of functions, and then enforcing the symmetry to obtain a reduction. This yields an expression for the matter potential in terms of the phase space density, which is further reduced by assuming cold streaming matter. Some vacuum Bianchi cosmologies and Type I with Vlasov matter are compared. It is shown that the Vlasov-matter potential for cold streaming matter results in qualitatively distinct dynamics from the well-known vacuum Bianchi cosmologies. / text

This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situations. Different techniques are used depending on the particular problem under study; dynamical systems methods are applied to cosmological models with spatial homogeneity; Hamiltonian methods are used in connection with dynamical systems to find global monotone quantities determining the asymptotic states; Fuchsian methods are used to quantify the structure of singularities in spacetimes without symmetries. All these separate methods of analysis provide insights about different facets of the structure of the equations, while at the same time they show the relationships between those facets when the different methods are used to analyze overlapping areas. The thesis consists of two parts. Part I reviews the areas of mathematics and cosmology necessary to understand the material in part II, which consists of five papers. The first two of those papers uses dynamical systems methods to analyze the simplest possible homogeneous model with two tilted perfect fluids with a linear equation of state. The third paper investigates the past asymptotic dynamics of barotropic multi-fluid models that approach a `silent and local' space-like singularity to the past. The fourth paper uses Hamiltonian methods to derive new monotone functions for the tilted Bianchi type II model that can be used to completely characterize the future asymptotic states globally. The last paper proves that there exists a full set of solutions to Einstein's field equations coupled to an ultra-stiff perfect fluid that has an initial singularity that is very much like the singularity in Friedman models in a precisely defined way. / <p>Status of the paper "Perfect Fluids and Generic Spacelike Singularities" has changed from manuscript to published since the thesis defense.</p>

This report provides an introduction to geometric mechanics, which seeks to model the behavior of physical mechanical systems using differential geometric objects. In addition to its elegance as a method of representation, this formulation also admits the application of powerful analytical techniques from geometry as an aid to understanding these systems. In particular, it reveals the fundamental role that symplectic geometry plays in mechanics (something which is not at all obvious from the traditional Newtonian formulation), and in the case of systems exhibiting symmetry, leads to an elucidation of conservation and reduction laws which can be used to simplify the analysis of these systems. The contribution here is primarily one of exposition. Geometric mechanics was developed as an aid to understanding physics, and we have endeavored throughout to highlight the physical principles at work behind the mathematical formalism. In particular, we show quite explicitly the entire development of mechanics from first principles, beginning with Newton's laws of motion and culminating in the geometric reformulation of Lagrangian and Hamiltonian mechanics. Self-contained presentations of this entire range of material do not appear to be common in either the physics or the mathematics literature, but we feel very strongly that this is essential in order to understand how the more abstract mathematical developments that follow actually relate to the real world. We have also attempted to make many of the proofs contained herein more explicit than they appear in the standard references, both as an aid in understanding and simply to make them easier to follow, and several of them are original where we feel that their presentation in the literature was unacceptably opaque (this occurs primarily in the presentation of the geometric formulation of Lagrangian mechanics and the appendix on symplectic geometry). Finally, we point out that the fields of geometric mechanics and symplectic geometry are vast, and one could not hope to get more than a fragmentary glimpse of them in a single work, which necessiates some parsimony in the presentation of material. The subject matter covered herein was chosen because it is of particular interest from an applied or engineering perspective in addition to its mathematical appeal. / text

Isolated and Dynamical horizons are used to generate boundary conditions upon the lapse and shift vectors. Numerous results involving the Hamiltonian of General relativity are derived, including a self-contained derivation of the Hamiltonian equations of general relativity using both a direct 'brute force' method of directly computing Lie derivatives, as well as the standard Hamil- tonian approach. Conclusions are compared to numerous examples, including the Kerr, Schwarzschild-De Sitter, McVittie, and Vaiyda spacetimes. / text