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.

Magister Scientiae Dentium - MSc(Dent) / Background: A close marginal adaptation between the tooth and the bracket base is important since it provides the space for the adhesive. In order to withstand and resist the orthodontic forces exerted as well as everyday forces like mastication and oral hygiene practices, the adhesive material to the bracket must have sufficient sheer bond strength. This means that no deformations, cracks, or fractures should occur within the bracket material and adhesive (Keizer et al., 1976). Aim: The aim of this study was to determine the accuracy of the marginal adaptation of the bracket bases of seven different brands of orthodontic brackets to the tooth surface of a right upper first premolar (ie. Abzil, Forestadent, GAC, Gemini, IMD, Ormco and Victory LP). Method: This research was an in vitro, descriptive comparison study. Fifteen caries and crack free intact human first premolars were used. The teeth were obtained from patients requiring extractions for orthodontic purposes and collected from Tygerberg Oral Health Centre. A convenience sample method was used, where whenever an upper caries free premolar was extracted, the parent was asked if the tooth could be used for this study and consent was obtained. The crowns of the 15 teeth were cleaned and polished with pumice and rubber cups for 10 seconds (as the clinician would do prior to bracket cementation). The same 15 teeth were used with the seven different brands of brackets in order to establish a comparison of the adaptability of the brackets. The brackets were placed at a set orthodontic prescription of 4mm (measured from the slot area of the bracket to the tip of the buccal cusp of the tooth) on the upper first premolars. After bracket placement, the dontrix gauge was applied to the bracket to engage the slot area. The brackets were held in place with a constant force of 0.70 Newton (N) by the dontrix gauge. This allowed for reproducibility for the seven brackets with all fifteen teeth. In order to assess the space between the brackets and the teeth no adhesive was used. The space between the margins of the bracket and tooth interface was viewed under the Stereomicroscope (Carl Zeiss microscope, Zeiss Stemi508) at 50 times magnification. A two way mixed measures ANOVA was run to determine whether there were differences between the seven brackets placed at six points on the tooth surface. Results: GAC had the smallest overall mean measurement between bracket base and tooth surface followed by Ormco and Gemini respectively. Abzil had the largest overall mean measurement for the six points around the bracket

Orthokeratology (OK) is the reduction, modification or elimination of myopia through application of contact lenses. With the development of high Dk/t lens materials, overnight therapy has become the modality of choice for OK. Overnight OK lens wear has been previously investigated in terms of its efficacy to reduce myopia. However, the underlying effects of overnight OK lens wear on the human cornea have received less attention. As well as the clinical efficacy of overnight OK, this study investigated the effects of overnight OK on topographical corneal thickness and the overnight corneal edema response, and corneal tissue changes with overnight OK. Eighteen subjects participated as the OK lens-wearing group, wearing BE lenses (UltraVision, Brisbane, Queensland) in both eyes. A further ten subjects participated as control subjects, wearing conventional rigid lenses (J-Contour, UltraVision) in the right eye (RE) only. The left eye (LE) acted as a non-lens-wearing control. Both groups wore lenses overnight only, with no lens wear during the day. Measurements were conducted at baseline then on day 1, 4, 10, 30, 60, and 90 for the OK lens-wearing eyes; and up to day 30 for the control group, in the morning (after overnight lens wear) and in the evening (after 8-10 hours of lens removal). Variables measured included best vision sphere (BVS), unaided logMAR visual acuity (VA), refractive astigmatism, apical corneal power (ACP), simulated K readings (Medmont E300 corneal topographer), topographical corneal thickness (Holden-Payor optical pachometer), and keratocyte and endothelial cell densities (ConfoScan2 confocal microscope). Approximately 75% of myopia was corrected after the first night of OK lens wear and the changes in refractive error stabilised by day 10. By day 90, myopia reduction averaged 2.54 ?? 0.63 D. This was associated with significant improvement in unaided VA of about 82% after the first night of lens wear. There was no change in refractive astigmatism over the 3-month period. There was significant reduction in ACP in the OK lens-wearing eyes after the first night of lens wear, which accounted for more than 70% of the total ACP change over the 3-month period (RE: -2.16 ?? 0.53 D; LE: -2.11 ?? 0.86 D). There was significant central epithelial thinning (about 30%) and significant thickening (about 3%) in the mid-peripheral stroma in the OK lens-wearing eyes. Significant central epithelial thinning was found after the first night of lens wear while thickening in the mid-peripheral stroma reached statistical significance by day 4. Further analysis suggests that topographical corneal thickness changes account for the refractive error changes with overnight OK lens wear, rather than corneal bending. The central overnight stromal edema response was significantly reduced in the OK lens-wearing eyes (1.2 ?? 0.5%) to a level lower than in the conventional RGP (6.2 ?? 1.2%) and non-lens-wearing eyes (2.5 ?? 0.9%) in the control group. Mid-peripheral and peripheral stromal edema responses showed similar levels to those predicted based on lens Dk/t. A single overnight wear of BE and Paragon Corneal Refractive Therapy (CRT) lenses showed that the edema response to BE lens wear is significantly less than in the CRT lens-wearing eyes (BE: 2.5 ?? 0.7%; CRT 3.5 ?? 1.3%) immediately on eye opening. No significant changes were found in either central stromal keratocyte or endothelial cell densities in either OK or control groups over the study period. In conclusion, overnight OK lens wear induces significant reductions in myopia after the first night of lens wear associated with improvement in unaided VA. Overnight OK lens wear causes significant thinning in the central epithelium and significant mid-peripheral stromal thickening which results in flattening of the central cornea and steepening in the mid-periphery. Although there were no significant changes in central stromal keratocyte and endothelial cell densities, thinning of the central epithelial layer raises concerns regarding the safety of the procedure, especially with the alarming number of corneal infections reported recently in the literature.

This work is an investigation into the curvature related potential for flux to flow in the radial direction in the back-iron of laminated axial flux machine cores. Analytical and numerical models are presented. Analysis based on these models has shown that, in practical axial flux machines, the radial component of the flux density can be neglected with respect to the flux density distribution in the core back-iron. It has also been found that if the core permeability, core conductivity and number of poles are sufficiently high then power loss due to curvature related cross-lamination flux is negligible compared to normal eddy current losses. A closed form expression to predict losses due to curvature related radial flux is also presented. This expression allows axial flux machine designers to make quick assessment of the need to consider these losses when designing axial flux machines.