by Ho Wing Kin. / Thesis submitted in: July 1997. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-92). / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- The Euler-Lagrange Equation for Capillary Surfaces --- p.6 / Chapter 1.2 --- The Capillary Tube --- p.12 / Chapter 2 --- Comparison Principles --- p.16 / Chapter 2.1 --- The General Comparison Principle --- p.16 / Chapter 2.2 --- Applications --- p.24 / Chapter 3 --- Existence Criteria --- p.30 / Chapter 3.1 --- Necessary Conditions --- p.31 / Chapter 3.2 --- Sufficient Conditions --- p.35 / Chapter 4 --- Uniqueness --- p.53 / Chapter 4.1 --- General Bounded Domains Case --- p.53 / Chapter 4.2 --- Infinite Strip Case --- p.56 / Chapter 5 --- Gradient Estimate of Surfaces of Constant Mean Curvature --- p.66 / Chapter 5.1 --- The Gradient Estimate --- p.67 / Chapter 5.2 --- Behavior as R→ 1 --- p.70 / Chapter 5.3 --- Existence of the Comparison Surfaces --- p.76 / Chapter 5.4 --- Ro is Best Possible --- p.82 / Appendix Mean Curvature --- p.85

O trabalho realizado consiste no desenvolvimento das equações pertinentes à teoria de cascas, implementação de programas de elementos finitos em Matlab e resolução de problemas de casca numéricos e analíticos. No desenvolvimento da teoria de cascas é evidenciada a aplicação da curvatura da estrutura no cálculo das forças e momentos resultantes para uma superfície de casca simplificada a uma estrutura localmente bidimensional. O problema de casca é resolvido analiticamente para um caso de curvatura simples e comparado com a resolução por elementos finitos em um programa desenvolvido pelo autor, em que a utilização da curvatura para o cálculo das forças e momentos resultantes é inserida na programação. A análise de elementos finitos é realizada para casos de casca com curvatura simples e dupla curvatura e de duas formas, sendo que a primeira utiliza elementos cujas normais médias são empregadas na montagem da matriz de rigidez, constituindo um elemento de casca e a segunda aplica normais a cada nó de cada elemento, tendo-se assim uma superfície facetada, com comportamento de placa em cada elemento. Os resultados obtidos mostram que o impacto da aplicação da curvatura em geral é pequeno nas regiões mais críticas para as forças e momentos resultantes, como na região de engaste. Porém, algumas regiões da casca apresentam grandes variações, e caso sejam de importância para o usuário, cabe uma análise mais detalhada em que o emprego da curvatura possa ser considerado. / This work presents the development of the shell theory equations, implementation of finite element programs in Matlab and the resolution of numerical and analytical shell problems. Along with the development of the shell theory, the application of the curvature of the structure in the calculation of stress and couple resultants for a shell structure simplified to a bidimensional problem become clear. The shell problem is solved analytically, by means of the application of the shell equations in a shell with simple curvature, and this solution is compared with the numerical solution using the finite element program implemented, considering the curvature of the structure for the stress and couple resultants. Finite element analysis is performed for the simple and double curvature cases of shells, and in two distinct ways, the first one considering averaged normals for neighbor elements, which produces shell elements, the second one using normals to each node of each element, which results in locally flat elements, behaving as plates. Results obtained show that the impact of the application of the curvature in the resultants is usually small in the most critical points, such as the crimp. However, some regions of the shell present huge variation, and further analysis is recommended, since the application of the curvature can be important.

by Guan-xin Li. / "January 1999." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 223-225). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Neste trabalho, estuda-se propriedades e implementações de duas importantes representações de formas denominadas curvatura e eixos de simetria (ou esqueletos), relacionando-as ao clássico modelo de processamento da informação visual proposto por Marr. Para tanto, fez-se necessária uma releitura dos critérios para a representação de formas já existentes, além da proposição e comparação de algumas abordagens numéricas através das quais curvaturas e esqueletos são calculados, com especial atenção dada a características como robustez e invariância. No caso da curvatura, que é uma medida clássica da geometria diferencial sem divergências quanto a sua definição e interpretação, foi dado maior enfoque à robustez dos métodos quanto à presença de ruídos de diversas naturezas e quanto a sensibilidade na escolha dos parâmetros que levaram aos resultados mais precisos. Além disso, propôs-se uma versão não-derivativa da curvatura (denominada circularidade local) a fim de contornar o problema advindo das instabilidades do cálculo de derivadas de dados reais amostrados. Ainda relativamente à curvatura, um método baseado na transformada de Fourier bidimensional foi proposto e comparado à abordagem unidimensional já estabelecida e a outro método padrão (&#946-Splines/ Medioni) de cálculo de curvatura de contornos digitais. Já no caso dos esqueletos, a existência de diversas interpretações geométricas e matemáticas para o conceito e as inúmeras abordagens numéricas para o cálculo dos mesmos demandaram a revisão da literatura sob a luz do modelo inicialmente introduzido para que se pudesse impor uniformidade à terminologia e situar as diferentes abordagens e implementações dentro de um panorama comum, sendo que dois novos métodos para o cálculo de esqueletos foram criados quando do desenvolvimento deste trabalho (esqueletos por dilatações exatas e esqueletos afins baseados em áreas). / In this work, properties and implementations of two important shape representa¬tions had been studied: curvature and symmetry axis (or skeletons) and they had been related to the Marr\'s classical visual information processing model. Thus, it was necessary a re-interpretation of previously existent criteria for shape represen¬tation in addition to the proposition and comparison of some numerical approaches for curvature and skeletons calculations with special attention given to robustness and invariance characteristics. In the curvature case, a classic measure from dif¬ferential geometry with no divergence about its definition and interpretation, we focused mainly on the numerical methods robustness in presence of several noise categories and on the best parameter set choice sensibility. Additionally, a no derivative version of curvature (also called local circularity) was proposed in order to circumvent problems related to the sampled real data derivatives numerical ins¬tabilities. Still talking about curvature, a 2D Fourier transform based method had been proposed and assessed in comparison to the 1D approach and other standard digital contour curvature calculation method (&#946-Splines/ Medioni). In the skeleton\'s case, the existence of several geometrical and mathematical interpretations of the concept and the innumerable numeric approaches to their calculation demanded a literature survey regarding the initially introduced model to impose uniformity to the terminology and to situate the different approaches and implementations into a common framework where two new methods to skeleton\'s calculations had been created (exact dilations based skeletons and area based affine skeletons).

Les phénomènes d’interactions choc/couche limite sont dimensionnants pour de nombreuses applications des domaines de l’aéronautique et du spatial. Ils peuvent être associés à la formation de décollements instationnaires à basse fréquence qui n’ont été étudiés jusqu’à présent qu’en géométrie plane. La présente étude vise à caractériser ce type d’interaction en configuration cylindrique. Un outil de simulation numérique directe,basé sur l’extension de schémas hybrides aux différences finies de haute précision (centrés optimisés6/WENO 5) en géométrie curviligne, a été développé et validé à travers divers cas test standards. Une première partie de l’étude se focalise sur l’influence d’un effet de courbure transverse sur le développement des propriétés d’une couche limite supersonique à Mach 3. Il est montré que l’augmentation de la courbure relative de la couche limite tend à réduire l’énergie de fluctuation à basse fréquence près de la paroi, tout en renforçant les perturbations à hautes fréquences dans la zone externe de la couche limite. En comparaison avec le cas plan, la courbure transverse induit une ré-organisation notable des structures de la couche limite et un comportement différent des invariants d’anisotropie des contraintes, mais ne conduit qu’à une légère modification des distributions de contraintes et de l’équilibre global d’énergie cinétique turbulente. Une seconde partie de l’étude se concentre sur la zone d’interaction avec une rampe de compression et le mouvement instationnaire du choc en géométrie de révolution complète. La déformation azimutale du choc est caractérisée dans son mouvement. Elle apparaît essentiellement associée à la fluctuation de la ligne de décollement et l’organisation des structures tourbillonnaires amont. Il est montré que l’énergie des modes azimutaux de pression pariétale fluctuante est plus amplifiée pour les modes d’ordre plus élevé. La contribution à l’effort latéral associé au mode 1 apparaît plus particulièrement marquée à basses fréquences dans la zone amont au point de décollement et à moyennes fréquences en aval de la zone de recollement sur la rampe où les niveaux les plus élevés de fluctuations sont observés. Il est montré que les fluctuations à basses fréquences sont en revanche portées par des modes azimutaux d’ordre de plus en plus élevé à travers la zone d’interaction. / Shock wave/boundary layer interactions (SWBLI) are present in various aerospace engineering applications.They can be associated with separated regions yielding low-frequency unsteadiness, which have mainly been studied in planar geometries. The present study aims at characterizing this type of interaction in a cylindrical configuration. A direct numerical simulation solver has been developed and validated with various test cases. It is based on a high-order finite difference based hybrid schemes (6th order centered scheme/5thorder WENO), extended to curvilinear geometries. Transverse curvature effects on properties of spatially developing supersonic boundary layer at Mach 3 are first examined. It is shown that the increase of the relative curvature of the boundary layer tends to reduce the fluctuation energy at lower frequencies near the wall, while reinforcing the perturbations at higher frequencies in the upper zone of the boundary layer.In comparison with the planar case, the transverse curvature leads to a significant re-organization of the boundary layer structures and a subsequent modified behavior of the invariants of anisotropy turbulent stress tensor. It however only leads to slightly modified distributions of Reynolds stress and a rather similar overall balance of turbulent kinetic energy through the boundary layer. The second part of this study is dedicated to the unsteady motions of the shock/separation zone in a cylinder/compression flare configuration for which the full cylindrical geometry is taken into account. The shock distortions in the azimutal direction appears to be mainly associated to the organization of the upstream vortex structures and the subsequent azimutal fluctuations of the separation line. It is shown that the energy of the fluctuating wall pressure is more amplified for higher order azimutal modes. The contributions to lateral forces, associated to the first mode, are dominated by low-frequencies only upstream of the separation line in the intermittent region. They become more dominant in the middle frequency range downstream of the reattachment zone on the ramp. It is also shown that the low-frequency activity at the wall is progressively due to higher order azimuthal modes through the interaction zone.

In this thesis, the critical phenomenon in Berry curvature of nonlinear systems that occurs at phase boundaries is described by using the Bogoliubov excitation of the semiquantal dynamics. Its is shown that when the critical boundary in the parameter space is crossed, the nonlinear geometric phase of the Bogloubov excitations surrounding the elliptic fixed points experiences non-analytic behavior. / 在本論文，我們利用半古典動力學的博戈留波夫激發研究非線性系統的貝里曲率在相邊界上出現的臨界現象。結果顯示，當參數空間中的臨界曲面被越過，環繞橢圓不動點的博戈留波夫激發的非線性幾何相位發生非解析行為。 / Kam, Chon Fai = 非線性系統的貝里曲率 / 甘駿暉. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 49-56). / Abstracts also in Chinese. / Title from PDF title page (viewed on 18, October, 2016). / Kam, Chon Fai = Fei xian xing xi tong de Beili qu lu / Gan Junhui. / Detailed summary in vernacular field only.

This thesis is about some extensions of the ideas and techniques used in integrable field theories to deal with non-integrable theories. It is presented in two parts. The first part deals with gauge theories in 3 and 4 dimensional space-time; we propose what we call the integral formulation of them, which at the end give us a natural way of defining the conserved charges that are gauge invariant and do not depend on the parametrisation of space-time. The definition of gauge invariant conserved charges in non-Abelian gauge theories is an open issue in physics and we think our solution might be a first step into its full understanding. The integral formulation shows a deeper connection between different gauge theories: they share the same basic structure when written in the loop space. Moreover, in our construction the arguments leading to the conservation of the charges are dynamical and independent of the particular solution. In the second part we discuss the recently introduced concept called quasi-integrability: one observes soliton-like configurations evolving through non-integrable equations having properties similar to those expected for integrable theories. We study the case of a model which is a deformation of the non-linear Schr¨odinger equation consisting of a more general potential, connected in a way with the integrable one. The idea is to develop a mathematical approach to treat more realistic theories, which is in particular very important from the point of view of applications; the NLS model appears in many branches of physics, specially in optical fibres and Bose-Einstein condensation. The problem was treated analytically and numerically, and the results are interesting. Indeed, due to the fact that the model is not integrable one does not find an infinite number of conserved charges but, instead, a set of infinitely many charges that are asymptotically conserved, i.e., when two solitons undergo a scattering process the charges they carry before the collision change, but after the collision their values are recovered. / Essa tese discute algumas extensões de ideias e técnicas usadas em teorias de campos integráveis para tratar teorias que não são integráveis. Sua apresentação é feita em duas partes. A primeira tem como tema teorias de calibre em 3 e 4 dimensões; propomos o que chamamos de equação integral para uma tal teoria, o que nos permite de maneira natural a construção de suas cargas invariantes de calibre, e independentes da parametrização do espaço-tempo. A definição de cargas conservadas in variantes de calibre em teorias não-Abelianas ainda é um assunto em aberto e acreditamos que a nossa solução pode ser um primeiro passo em seu entendimento. A formulação integral mostra uma conexão profunda entre diferentes teorias de calibre: elas compartilham da mesma estrutura básica quando formuladas no espaço dos laços. Mais ainda, em nossa construção os argumentos que levam `a conservação das cargas são dinâmicos e independentes de qualquer solução particular. Na segunda parte discutimos o recentemente introduzido conceito de quasi-integrabilidade: em (1 + 1) dimensões existem modelos não integráveis que admitem soluções solitonicas com propriedades similares `aquelas de teorias integráveis. Estudamos o caso de um modelo que consiste de uma deformação (não-integrável) da equação de Schrödinger não-linear (NLS), proveniente de um potencial mais geral, obtido a partir do caso integrável. O que se busca é desenvolver uma abordagem matemática sistemática para tratar teorias mais realistas (e portanto não integráveis), algo bastante relevante do ponto de vista de aplicações; o modelo NLS aparece em diversas áreas da física, especialmente no contexto de fibra ótica e condensação de Bose-Einstein. O problema foi tratado de maneira analítica e numérica, e os resultados se mostram interessantes. De fato, sendo a teoria não integrável não é encontrado um conjunto com infinitas cargas conservadas, mas, pode-se encontrar um conjunto com infinitas cargas assintoticamente conservadas, i.e., quando dois solitons colidem as cargas que eles tinham antes tem os seus valores alterados, mas após a colisão, os valores inicias, de antes do espalhamento, são recobrados.

A review of geodesics and Busemann G-spaces is given. Aleksandrov curvature and the disjoint (0, n)-cells property are defined. We show how these properties are applied to and strengthened in Busemann G-spaces. We examine the relationship between manifolds and Busemann G-spaces and prove that all Riemannian manifolds are Busemann G-spaces, though not all metric manifolds are Busemann G-spaces. We show how Busemann G-spaces that also have bounded Aleksandrov curvature admit local closest-point projections to geodesic segments. Finally, we expound local properties of Busemann G-spaces and define a new property which we call the symmetric property. We show that Busemann G-spaces which have the disjoint (0,n)-cells property for every value of n cannot have the symmetric property.

Space curves have a variety of uses within mathematics, and much attention has been paid to calculating quantities related to such objects. The quantities of curvature and energy are of particular interest to us. While the notion of curvature is well-known, the Mobius energy is a much newer concept, having been first defined by Jun O'Hara in the early 1990s. Foundational work on this energy was completed by Freedman, He, and Wang in 1994, with their most important result being the proof of the energy's conformal invariance. While a variety of results have built those of Freedman, He, and Wang, two topics remain largely unexplored: the interaction of curvature and Mobius energy and the generalization of the Mobius energy to curves with a varying thickness. In this thesis, we investigate both of these subjects. We show two fundamental results related to curvature and energy. First, we show that any simple, closed, twice-differentiable curve can be transformed in an energy-preserving and length-preserving way that allows us to make the pointwise curvature arbitrarily large at a point. Next, we prove that the total absolute curvature of a twice-differentiable curve is uniformly bounded with respect to conformal transformations. This is accomplished mainly via an analytic investigation of the effect of inversions on total absolute curvature. In the second half of the thesis, we define a generalization of the Mobius energy for simple curves of varying thickness that we call the "nonuniform energy." We call such curves "weighted knots," and they are defined as the pairing of a curve parametrization and positive, continuous weight function on the same domain. We then calculate the first variation formulas for several different variations of the nonuniform energy. Variations preserving the curve shape and total weight are shown to have no minimizers. Variations that "slide" the weight along the curve are shown to preserve energy is special cases.

In this study we provide a new proof of C¹,α boundary regularity for finite perimeter sets with flat boundary which are local minimizers of a variational mean curvature formula. Our proof is provided for curvature term H∈LΩ. The proof is a generalization of Cafarelli and C#243;rdoba's method, and combines techniques from geometric measure theory and the theory of viscosity solutions which have been developed in the last 50 years. We rely on the delicate interplay between the global nature of sets which are variational minimizers of a given functional, and the pointwise local nature of comparison surfaces which satisfy certain PDE. As a heuristic, in our proof we can consider the curvature as an error term which is estimated and controlled at each point of the calculation.