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<p> Hawking's Singularity Theorem establishes the existence of a cosmological singularity in a spacetime for which no global assumptions about causality are made. This theory has been useful for predicting the occurrence of singularities in a spacetime without solving Einstein's field equation. This paper is an exposition of the tools and some of the theory required to prove and apply Hawking's theorem. Emphasis is placed on practical methods for applying this result to the flat, dust-filled Robertson-Walker spacetime, and the black hole interior of the Kruskal extension of the Schwarzschild spacetime.</p> / Thesis / Master of Science (MSc)

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A lot of attention has been drawn recently to global log canonical thresholds of Fano varieties, which are algebraic counterparts of the α-invariant of Tian for smooth Fano varieties. In particular, global log canonical thresholds are related to the existence of Kahler-Einstein metrics on Fano varieties. The purpose of this thesis is to apply techniques from singularity theory in order to compute the global log canonical thresholds of all Del Pezzo surfaces of degree 1 with Du Val singularities, as well as the global log canonical thresholds of all Del Pezzo surfaces of Picard rank 1 with Du Val singularities. As a consequence, it is proven that Del Pezzo surfaces of degree 1 with Du Val singularities admit a Kahler-Einstein metric if the singular locus consists of only A1, or A3, or A4 type Du Val singular points.

by Lo, Ka-wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 76-81 (2nd gp.)). / Acknowledgments --- p.i / Abstract --- p.ii / Notations --- p.iii / List of Figures --- p.v / List of Tables --- p.vii / Chapter 1. --- Introduction / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Literature Review --- p.4 / Chapter 1.3 --- Objective --- p.10 / Chapter 2. --- Comparison of Different Approaches / Chapter 2.1 --- Sample Manipulator --- p.11 / Chapter 2.1.1 --- Force Decomposition Method --- p.12 / Chapter 2.1.2 --- Forward Rate Kinematics Base Method --- p.15 / Chapter 2.1.3 --- Grassmann Geometry Method --- p.18 / Chapter 2.2 --- Comparison Criteria --- p.20 / Chapter 2.2.1 --- Computational Complexity --- p.20 / Chapter 2.2.2 --- Scope of Application --- p.22 / Chapter 2.3 --- Summary --- p.23 / Chapter 3. --- Enumeration of Configuration Singularity / Chapter 3.1 --- Novel 6 DOF --- p.25 / Chapter 3.1.1 --- Result Analysis --- p.31 / Chapter 3.2 --- A 3 DOF with Symmetric Base --- p.33 / Chapter 3.2.1 --- Result Analysis --- p.35 / Chapter 3.3 --- A 3 DOF with Non-Symmetric Base --- p.36 / Chapter 3.3.1 --- Result Analysis --- p.37 / Chapter 3.4 --- A New Model of 6-SPS Defined by Kong et al --- p.40 / Chapter 3.5 --- A New Class of 6-SPS Platform-Type Parallel Manipulator --- p.45 / Chapter 3.5.1 --- The Hexagonal Base --- p.46 / Chapter 3.5.2 --- The Pentagonal Base --- p.50 / Chapter 3.5.3 --- The Tetragonal Base --- p.52 / Chapter 3.5.4 --- The Triangular Base --- p.55 / Chapter 3.6 --- Summary --- p.59 / Chapter 4. --- Numerical Analysis / Chapter 4.1 --- Parameter Analysis --- p.60 / Chapter 4.1.1 --- One Unknown Variable --- p.61 / Chapter 4.1.2 --- Two Unknown Variables --- p.63 / Chapter 4.2 --- Critical Value of Ratio R/q --- p.69 / Chapter 4.3 --- Summary --- p.72 / Chapter 5. --- Conclusions and Future Work / Chapter 5.1 --- Conclusions --- p.73 / Chapter 5.2 --- Future Work --- p.75 / References --- p.76 / Appendix --- p.82

Chan, Kai Leung. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leave 78). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

"Certain physical problems in electrostatics, magnetostatics, and heat transfer give rise to elliptic boundary value problems with transmission conditions on a layer. We focus on a particular problem with a second order transmission condition, representing an infinitely conductive layer. To approximate irregular layers that may naturally arise, a sequence of layers that converge to the fractal von Koch curve is considered. The solution to this transmission problem with a prefractal layer exhibits singularities due to the transmission condition across the layer as well as the reentrant corners introduced in the domain by the prefractal curve. To solve this problem numerically using a finite element method, the mesh must be adjusted to account for these singularities. We exhibit a general algorithm for creating a finite element discretization of the domain that results in linear convergence of the numerical solution to the true solution in a suitable norm."

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An m charge in the n dimensional Euclidean space is a linear functional acting on m dimensional polyhedral chains and satisfying the following continuity condition. The value of the linear functional approaches zero on chains whose normal masses are bounded and whose flat norms asymptotically vanish. Our main theorem relates m charges to pairs of continuous differential forms. Luzin's theorem states that every measurable function on the line is the derivative of a continuous, almost everywhere differentiable function. We show this can be improved in several dimensions. Finally we prove that a compact subset C of the n dimensional Euclidean space does not support the distributional divergence of a bounded measurable vector field if and only if C has vanishing (n-1) dimensional Hausdorff measure.

The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points.