Wong, Sen. / Thesis submitted in: November 2006. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaf 85). / Abstracts in English and Chinese. / Chapter 1 --- From algebraic sets to Preschemes --- p.5 / Chapter 1.1 --- Affine varieties and prevarieties --- p.5 / Chapter 1.2 --- Affine schemes and preschemes --- p.13 / Chapter 1.3 --- Morphisms --- p.16 / Chapter 1.4 --- Prevarieties vs Preschemes --- p.29 / Chapter 1.5 --- Summary and Products --- p.37 / Chapter 2 --- More on prevarieties --- p.43 / Chapter 2.1 --- Introduction --- p.43 / Chapter 2.2 --- Subprevarieties and product of prevarieties --- p.48 / Chapter 2.3 --- Dominating and birational morphisms --- p.60 / Chapter 2.4 --- Complete varieties --- p.64 / Chapter 2.5 --- Chow's Theorem --- p.68 / Chapter 3 --- From preschemes to schemes --- p.71 / Chapter 3.1 --- Introduction --- p.71 / Chapter 3.2 --- Open subpreschemes and Open immersions --- p.74 / Chapter 3.3 --- Closed subpreschemes and Closed immersions --- p.76 / Chapter 3.4 --- Schemes --- p.79 / Chapter A --- Hilbert's Nullstellensatz --- p.81 / Bibliography --- p.85

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaf 37). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaf 37). Also available on the Internet.

Inaugural dissertation--Universität Regensburg, 1998.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Includes bibliographical references. / Ph.D.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaf 71). Issued in print and online. Available via ProQuest Digital Dissertations.

M.Sc. (Applied Mathematics) / Explicit schemes for integrating ODEs and time–dependent partial differential equations (in the method of lines–MoL–approach) are very well–known to be stable as long as the maximum sizes of their timesteps remain below a certain minimum value of the spatial grid spacing. This is the Courant– Friedrich’s–Lewy (CFL) condition. These schemes are the ones traditionally being used for performing simulations in Numerical Relativity (NR). However, due to the above restriction on the timestep, these schemes tend to be so much inadequate for simulating some of the highly probable and astrophysically interesting phenomenae. So, it is of interest this currernt moment to seek or find integrating schemes that may help numerical relativists to somehow circumvent the CFL restriction inherent in the use of explicit schemes. In this quest, a more natural starting point appears to be implicit schemes. These schemes possess a highly desireable stability property – they are unconditionally stable. There also exists a combination of implicit and explicit (IMEX) schemes. Some researchers have already started exploring (since 2009, 2011) these for NR purposes. We report on the implementation of two implicit schemes (implicit Euler, and implicit midpoint rule) for Einstein’s evolution equations. For low computational costs, we concentrated on spherical symmetry. The integration schemes were successfully implemented and showed satisfactory second order convergence patterns on the systems considered. In particular, the Implicit Midpoint Rule proved to be a little superior to the implicit Euler scheme.