We consider the problem of finding a bounded-curvature path in the plane from one configuration αs to another configuration αt that avoids the interior of a set of polygonal obstacles Ε. We call any such path from αs to αt a feasible path. In this thesis, we develop algorithms to find feasible paths that have explicit guarantees on when they will return a feasible path. We phrase our guarantees and run time analysis in terms of the complexity of the desired solution (see k and λ below). In a sense, our algorithms are output sensitive, which is particularly desirable because there are no known bounds on the solution complexity amid arbitrary polygonal environments. Our first major result is an algorithm that given Ε, αs, αt, and a positive integer k either (i) verifies that every feasible path has a descriptive complexity greater than k or (ii) outputs a feasible path. The run time of this algorithm is bounded by a polynomial in n (the total number of obstacle vertices in Ε), m (the bit precision of the input), and k. This result complements earlier work by Fortune and Wilfong: their algorithm considers paths of arbitrary descriptive complexity (it has no dependence on k), but it never outputs a path, just whether or not a feasible path exists. Our second major result is an algorithm that given E, αs, αt, a length λ, and an approximation factor Ε, either (i) verifies that every feasible path has length greater than λ or (ii) constructs a feasible path that is at most (1+ Ε) times longer than the shortest feasible path. The run time of this algorithm is bounded by a polynomial in n, m, Ε-1, and λ. This algorithm is the result of applying the techniques developed earlier in our thesis to the previous approximation approaches. A shortcoming of these prior approximation algorithms is that they only search a special class of feasible paths. This restriction implies that the path that they return may be arbitrarily longer than the shortest path. Our algorithm returns a true approximation because we search for arbitrary shortest paths. / Science, Faculty of / Computer Science, Department of / Graduate

No description available.

The purpose of this thesis is to study in depth the relationship between the curvature of space-time and the other geometrical objects which naturally arise in general relativity. Most of the results obtained apply to the generic case. Chapter 1 contains a discussion of certain aspects of fibre bundle theory required in later chapters which may be unfamiliar to many relativists, while chapter 2 contains preliminary material on curvature in relativity and proves a continuity property of the algebraic classification of the Weyl and energy-momentum tensors. Chapter 3 describes the generic behaviour of the Riemann, Weyl and energy-momentum tensors, and chapter 5 goes on to use this description to investigate the relationship of the Riemann tensor to the metric, conformal class and connection of space-time in the generic case. In particular it is proved that the Riemann tensor uniquely and continuously determines the connections. The information obtained in chapter 3 on the algebraic type of curvature in the general case is related in chapter 4 to the topology of the underlying manifold. In chapter 6 a topology is defined on the set of sectional curvatures of all Lorentz metrics on a given manifold. The remainder of the chapter attempts to do for the sectional curvature what was done for the Riemann tensor in chapter 5 but, because sectional curvature is more difficult to handle, the results obtained are necessarily more modest.

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering, 2017. / The cardinal aspects of supersonic and hypersonic propulsion intake design involve understanding the internal shock wave structures forming therein. A study was conducted to explore the effects of internal surface curvature and entry deflection angle on steady axisymmetric shock waves. Very little is known about these influences with only Curved Shock Theory, produced by M¨ older, providing analytical insight directly after a curved shock wave. The shock waves and accompanying flow fields which were generated were studied via experimental and numerical means. Radius normalised internal radii of curvature of 1, 1.5 and 2 with entry deflection angles of 0◦, 4◦ and 8◦ were investigated between a Mach number range of Mach 2.4 and 3.6. Experimental results were produced using a blow down supersonic wind tunnel facility and were captured via shadowgraph and schlieren flow visualisation techniques. The numerical simulations were validated using the experimental results. A self similar curved shock wave shape equation was presented with an empirical model which uses flow Mach number and internal radius of curvature in order to produce the resulting curved shock shape. Curved Shock Theory streamlines were used to try predict the internal surfaces that produced the curved shocks but results did not correlate. This was due to extreme streamline curvature curving the streamlines when the shock angle approached the Mach angle. Very good agreement was however found between the theoretical and numerical streamlines at lower curvatures. The higher the internal surface curvature and entry deflection angle, the greater the flow fields were impacted. Steeper characteristics formed as a result, curving the shock wave more noticeably. Both the internal surface curvature and entry deflection angle were found to have an effect on the trailing edge expansion fans which then altered the shape of downstream shock wave structures. The highest curvature models produced steady double reflection patterns due the flow being turned in onto itself by the imposed internal surface curvature. The effects of conical and curved internal surfaces were explored for additional insight into the presence of flow-normal curvature and the curving of the attached shock waves. / XL2018

當絶熱地改變哈密頓量的參數時，波函數會獲得一個幾何相位，既 Berry相。它可以表示為參數空間內一個局域的規範場，叫作 Berry曲率。Berry曲率在凝聚態物理的許多領域的研究中起著至關重要的作用，例如量子霍爾效應以及拓撲絶緣體。因此它已成為固體的最基本的性質之一。在量子霍爾效應中，霍爾電導可以表示為 Berry曲率在布里淵區上的積分。這個積分是一個量子化的 Chern數，並且反映了系統的拓撲結構。然而由於時間反演對稱性，拓撲絶緣體的霍爾電導等於零。因此對時間反演不變絶緣體的 Berry曲率的直接以及非破壞性的測量已經成為凝聚態物理中的重要問題。 / 在這篇論文中，我們提出標準的非線性光譜學可以用來探測時間反演不變絶緣體的性質，而且非線性光譜學的實驗比直流實驗更可控。通過計算，我們發現當遠紅外光和 THz光入射到樣品上時，系統的三階光學響應不為零，這與輸運實驗的結果相比形成了鮮明的對比。更重要的是響應函數正比於能帶的非阿貝爾 Berry曲率。這個結果提供了直接測量時間反演不變系統的 Berry曲率的可能性。 / 對具有(近似的 )空間旋轉對稱性的時間反演不變絶緣體，我們發現三階光學響應與等能球面的 Berry曲率通量直接相關。由於 Berry曲率通量給出了能帶簡併點處的奇異單子的拓撲電荷，因此人們可以利用這種方法直接測量能帶的拓撲結構。作為一個例子，這個方法被應用於 III-V族化合物半導體的八帶模型，並給出了一個拓撲電荷為 3的非線性響應。 / Berry phase, a geometric phase acquired by a wave function by adiabatically varying the parameters of the Hamiltonian, can be expressed in terms of a local gauge field in parameter space, called Berry curvature. The Berry curvature plays an essential role in many fields of condensed matter physics, such as the quantum Hall eect and in the study of Topological insulators (TI) and hence it has become one of the most fundamental properties of solids. In Quantum Hall eect, the Hall conductance can be expressed as an integral of the Berry curvature over the Brillouin zone, which is a quantized Chern number and reflects the topology of the system. However in TI, the Hall conductance is equal to zero as a result of the time-reversal (TR) symmetry. Thus, the direct and nondestructive measurement of the Berry curvature of a TR invariant insulator is an important issue in condensed matter physics. / In this thesis, we show that the standard nonlinear optical spectroscopy, being more experimentally controllable than DC experiments, can be used to detect the bulk properties of TR invariant insulators. Through a general calculation, we nd that, when optical and terahertz light fields are employed, the third order optical eect is nonzero compared with the transport method. And the susceptibility is exactly proportional to the non-Abelian Berry curva-ture of the energy band, which provides the possibility of determining Berry curvature directly. / For the TR invariant insulator with (approximate) rotational symmetry, the third order optical susceptibility is related to the the Berry curvature flux through the iso-energy sphere, which gives the topological charge of the monopole at the degeneracy point. Hence it enables one to measure the topo¬logical property of the energy band explicitly. As an example, the method is applied to the eight-band model of III-V compound semiconductors and gives a quantized susceptibility with topological charge equal to 3. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yang, Fan = 時間反演不變絶緣體的Berry曲率的非線性光學測量 / 楊帆. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 77-[80]). / Abstracts also in Chinese. / Yang, Fan = Shi jian fan yan bu bian jue yuan ti de Berry qu lu de fei xian xing guang xue ce liang / Yang Fan. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction of Berry phase --- p.1 / Chapter 1.1.1 --- Basic concepts of the Berry phase and Berry curvature --- p.2 / Chapter 1.1.2 --- Degeneracy and monopole --- p.5 / Chapter 1.1.3 --- Berry phase in Bloch bands --- p.7 / Chapter 1.1.4 --- Non-Abelian Berry curvature --- p.8 / Chapter 1.2 --- Quantum Hall effect and topological insulator --- p.10 / Chapter 1.2.1 --- Anomalous velocity and Quantum Hall effect --- p.11 / Chapter 1.2.2 --- Topological insulator --- p.14 / Chapter 1.3 --- Introduction of nonlinear optics --- p.16 / Chapter 1.3.1 --- Nonlinear optical susceptibilities --- p.16 / Chapter 1.3.2 --- Density matrix formalism --- p.19 / Chapter 1.3.3 --- Diagrammatic analysis of nonlinear optical processes --- p.21 / Chapter 1.4 --- Polarization operator of band electrons --- p.24 / Chapter 1.5 --- Outline of this thesis --- p.26 / Chapter 2 --- Third-order Optical Response of a General Insulator --- p.28 / Chapter 2.1 --- Introduction --- p.28 / Chapter 2.2 --- Microscopic mechanism --- p.30 / Chapter 2.3 --- Third-order nonlinear susceptibility --- p.31 / Chapter 2.3.1 --- A general model --- p.31 / Chapter 2.3.2 --- Perturbative calculation I --- p.35 / Chapter 2.3.3 --- Perturbative calculation II --- p.40 / Chapter 2.3.4 --- Total response --- p.43 / Chapter 2.4 --- Diagrammatic calculation of the third-order response --- p.45 / Chapter 2.5 --- Application to topological insulators --- p.56 / Chapter 2.6 --- Summary --- p.59 / Chapter 3 --- Nonlinear Optical Measurement of Topological Charge --- p.61 / Chapter 3.1 --- Introduction --- p.61 / Chapter 3.2 --- Third-order response with resonant interband transitions --- p.62 / Chapter 3.3 --- Third-order response and topological charge in a rotationally symmetric insulator --- p.66 / Chapter 3.4 --- Quantized susceptibility of III-V compound semiconductors --- p.70 / Chapter 3.5 --- Summary --- p.74 / Chapter 4 --- Summary and Conclusions --- p.75 / Bibliography --- p.77 / Chapter A --- Calculation of equation (2.32) --- p.81 / Chapter B --- Proof of formula (3.20) --- p.89 / Chapter C --- Third-order response with multiple conduction and valence bands --- p.92

Poon Chi Cheung. / Bibliography: leaves 66-69 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1986

by Lam Chi Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 58-63). / Abstract also in Chinese. / Chapter Chapter 0 --- Introduction --- p.6 / Chapter Chapter 1 --- Preliminaries --- p.10 / Chapter Chapter 2 --- Uniform L∞ Bounds and Blow-up Behavior --- p.20 / Chapter Chapter 3 --- Branch Bubbling and Pre-branch Bubbling Sequences --- p.32 / Chapter Chapter 4 --- Related Problems --- p.46 / Bibliography --- p.58

Du, Yangge. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 88-91). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.

This thesis contains the author's results on the evolution of convex hypersurfaces by positive powers of the Gauss curvature. We first establish interior estimates for strictly convex solutions by deriving lower bounds for the principal curvatures and upper bounds for the Gauss curvature. We also investigate the optimal regularity of weakly convex translating solutions. The interesting case is when the translator has flat sides. We prove the existence of such translators and show that they are of optimal class C^1,1. Finally, we classify all closed self-similar solutions of the Gauss curvature flow which is closely related to the asymptotic behavior.