Mathematical physics

In this thesis we will discuss connections between Hamiltonian systems with a periodic kick and planar diffeomorphisms. After a brief overview of Hamiltonian theory we will focus, as an example, on derivations of the Hâenon map that can be obtained by considering kicked Hamiltonian systems. We will conclude with examples of Hâenon maps of interest.

Core-collapse supernovae (CCSN) are among the most energetic explosions in the universe, liberating ~1053 erg of gravitational binding energy of the stellar core. Most of this energy ( ~99%) is emitted in neutrinos and only 1% is released as electromagnetic radiation in the visible spectrum. Energy radiated in the form of gravitational waves (GWs) is about five orders smaller. Nevertheless, this energy corresponds to a very strong GW signal and, because of this CCSN are considered as one of the prime sources of gravitational waves for interferometric detectors. Gravitational waves can give us access to the electromagnetically hidden compact inner core of supernovae. They will provide valuable information about the angular momentum distribution and the baryonic equation of state, both of which are uncertain. Furthermore, they might even help to constrain theoretically predicted SN mechanisms. Detection of GW signals and analysis of the observations will require realistic signal predi ctions from the non-parameterized relativistic numerical simulations of CCSN. This dissertation presents the gravitational wave signature of core-collapse v supernovae. Previous studies have considered either parametric models or nonexploding models of CCSN. This work presents complete waveforms, through the explosion phase, based on first-principles models for the first time. We performed 2D simulations of CCSN using the CHIMERA code for 12, 15, and 25M non-rotating progenitors. CHIMERA incorporates most of the criteria for realistic core-collapse modeling, such as multi-frequency neutrino transport coupled with relativistic hydrodynamics, eective GR potential, nuclear reaction network, and an industry-standard equation of state.

A discrete formalism for General Relativity was introduced in 1961 by Tulio Regge in the form of a piecewise-linear manifold as an approximation to (pseudo-)Riemannian manifolds. This formalism, known as Regge Calculus, has primarily been used to study vacuum spacetimes as both an approximation for classical General Relativity and as a framework for quantum gravity. However, there has been no consistent effort to include arbitrary non-gravitational sources into Regge Calculus or examine the structural details of how this is done. This manuscript explores the underlying framework of Regge Calculus in an effort elucidate the structural properties of the lattice geometry most useful for incorporating particles and fields. Correspondingly, we first derive the contracted Bianchi identity as a guide towards understanding how particles and fields can be coupled to the lattice so as to automatically ensure conservation of source. In doing so, we derive a Kirchhoff-like conservation principle that identifies the flow of energy and momentum as a flux through the circumcentric dual boundaries. This circuit construction arises naturally from the topological structure suggested by the contracted Bianchi identity. Using the results of the contracted Bianchi identity we explore the generic properties of the local topology in Regge Calculus for arbitrary triangulations and suggest a first-principles definition that is consistent with the inclusion of source. This prescription for extending vacuum Regge Calculus is sufficiently general to be applicable to other approaches to discrete quantum gravity. We discuss how these findings bear on a quantized theory of gravity in which the coupling to source provides a physical interpretation for the approximate invariance principles of the discrete theory.

Stable and metastable phases of Fe and Al and structural anomalies of Zn and Cd have been studied by epitaxial Bain path (EBP) and minimum path (MNP) first-principles procedures, based on finding equilibrium structures from minimizing the Gibbs free energy G with respect to structure at a given hydrostatic pressure p and temperature T . The main accomplishments are as follows. (1) This dissertation illustrates the effectiveness of the MNP procedure for finding stable and metastable phases of an element by studying four Bravais phases of Fe including body-centered tetragonal (bct), body-centered cubic (bcc), face-centered cubic (fcc) and rhombohedral (rh) phases. The determination of absolute stability using MNP is new; MNP finds all the elastic constants cjj of a given state and the eigenvalues of the elastic constants matrix, which determine the absolute stability of the state.

A Shock wave as represented by the Riemann problem and a Point-blast explosion are two key phenomena involved in a supernova explosion. Any hydrocode used to simulate supernovae should be subjected to tests consisting of the Riemann problem and the Point-blast explosion. L. I. Sedov's solution of Point-blast explosion and Gary A. Sod's solution of a Riemann problem have been re-derived here from one dimensional fluid dynamics equations . Both these problems have been solved by using the idea of Self-similarity and Dimensional analysis. The main focus of my research was to subject the CHIMERA supernova code to these two hydrodynamic tests. Results of CHIMERA code for both the blast wave and Riemann problem have then been tested by comparing with the results of the analytic solution.

Spectral decomposition is a method of expressing functions as a harmonic series, and can be used for the simplification of complicated physical problems. This type of analysis requires knowledge of the function at all points on a circle or sphere. In problems where the function is known only at discreet points, regular intervals in a rectangular grid, for example, numerical methods must be employed to compute approximate coefficients for the harmonic expansion. In this paper, we investigate numerical methods for computing Fourier coefficients of a two dimensional function at a fixed radius, and spherical harmonic coefficients in three dimensions on a sphere of fixed radius.