Although relativistic physics tend to omit nondynamical "absolute objects" such as a flat metric tensor or a preferred time foliation, there exist interesting questions related to such entities, such as worries about the "flow" of time in special relativity, and the apparent disappearance of time altogether in canonical general relativity. This latter problem is related to the lack of a fixed causal structure with repect to which one might posit "equal-time" commutation relations, for example. In view of these issues, we consider whether including a flat background metric, and perhaps a preferred foliation, is physically worthwhile. We show how a derivation of Einstein's equations from flat spacetime can be generalized to include a preferred foliation, the possible significance of which we discuss, though ultimately we suggest why such a foliation might be present in metaphysics and yet absent from physics. We also derive a new "slightly bimetric" class of theories using the flat spacetime approach. However, such derivations are only formally special relativistic, because they give no heed to the flat metric's causal structure, which the curved effective metric might well violate. After reviewing the history of this problem, we introduce new variables to give a kinematic description of the relation between the two null cones. Then we propose a method to enforce special relativistic causality by using the guage freedom to restrict the configuration space suitably. Consequences for exact solutions, such as the Schwarzschild solution and its 'singularity,' are discussed. Advantages and difficulties regarding adding a mass term to the theory are discussed briefly. / text

This paper gives an overview of the mathematical and practical tools that can be used in turbo-machinery RANS simulation to realize unstructured mesh adaptation. It first presents the concept of metric and recalls that the hessian of the physical flow properties can become, thanks to small modifications, both a metric and a upper bound of the P1 projection error. The resulting metric is then studied on a simple 2D case. In a second part, the industrial application of this concept is addressed and the tools used to overcome the turbo-machinery specificities are explained. Finally, some 2D and 3D results are presented.

Visual sensitivity to small color difference is an important factor for precision color matching. Small color differences can be measured by the line element theory in terms of color distances between a color point and neighborhoods of points in a color space. This theory gives a smooth positive definite symmetric metric tensor which describes threshold of color differences by ellipsoids in three dimensions and ellipses in two dimensions. The metric tensor is also known as the Riemannian metric tensor. In regard to the color differences, there are many color difference formulas and color spaces to predict visual difference between two colors but, it is still challenging due to the nonexistence of a perfect uniform color space. In such case, the Riemannian metric tensor can be used as a tool to study the performance of various color spaces and color difference metrics for measuring the perceptual color differences. It also computes the shortest length or the distance between any two points in a color space. The shortest length is called a geodesic. According to Schrödinger's hypothesis geodesics starting from the neutral point of a surface of constant brightness correspond to the curves of constant hue. The chroma contours are closed curves at constant intervals from the origin measured as the distance along the constant hue geodesics. This hypothesis can be utilized to test the performance of color difference formulas to predict perceptual attributes (hue and chroma) and distribution of color stimulus in any color space. In this research work, a method to formulate line element models of color difference formulas the ΔE*ab, the ΔE*uv, the OSA-UCS ΔEE and infinitesimal approximation of CIEDE2000 (ΔE00) is presented. The Jacobian method is employed to transfer their Riemannian metric tensors in other color spaces. The coefficients of such metric tensors are used to compute ellipses in two dimensions. The performance of these four color difference formulas is evaluated by comparing computed ellipses with experimentally observed ellipses in different chromaticity diagrams. A method is also developed for comparing the similarity between a pair of ellipses. The technique works by calculating the ratio of the area of intersection and the area of union of a pair of ellipses. Similarly, at a fixed value of lightness L*, hue geodesics originating from the achromatic point and their corresponding chroma contours of the above four formulas in the CIELAB color space are computed by solving the Euler-Lagrange equations in association with their Riemannian metrics. They are compared with with the Munsell chromas and hue circles at the Munsell values 3, 5 and 7. The result shows that neither formulas are fully perfect for matching visual color difference data sets. However, Riemannized ΔE00 and the ΔEE formulas measure the visual color differences better than the ΔE*ab and the ΔE*uv formulas at local level. It is interesting to note that the latest color difference formulas like the OSA-UCS ΔEE and the Riemannized ΔE00 do not show better performance to predict hue geodesics and chroma contours than the conventional CIELAB and CIELUV color difference formulas and none of these formulas fit the Munsell data accurately

On the tangent bundle of a Riemannian manifold the most natural choice of metric tensor field is the Sasaki metric. This immediately brings up the question of infinitesimal symmetries associated with the inherent geometry of the tangent bundle arising from the Sasaki metric. The elucidation of the form and the classification of the Killing vector fields have already been effected by the Japanese school of Riemannian geometry in the sixties. In this thesis we shall take up the conformal vector fields of the Sasaki metric with the help of relatively advanced techniques.

Afin de pouvoir apparier de manière précise les couleurs il est essentiel de prendre en compte la sensibilité visuelle à percevoir de petites différences de couleur. Les petites différences de couleur peuvent être mesurées par des ellipses qui décrivent les différences justes observables (just noticeable difference - JND). Ces ellipses décrivent la faculté du Système Visuel Humain à discriminer des couleurs très peu différentes. D'un point de vue mathématique, ces ellipses peuvent être modélisées par une fonction différentielle positive de forme quadratique, caractéristique de ce que l'on appelle communément une métrique Riemannienne. La métrique Riemannienne peut être considérée comme un outil utile pour évaluer l'adéquation, la robustesse et la précision, d'un espace couleur ou d'une métrique couleur, à décrire, à mesurer, correctement les différences de couleur telles qu'elles sont perçues par le Système Visuel Humain. L'un des particularités de cette métrique est qu'elle modélise la plus petite distance qui sépare deux couleurs dans un espace couleur par une ligne géodésique. Selon l'hypothèse de Schrödinger les lignes géodésiques qui partent d'un point neutre d'une surface de luminosité constante décrivent des courbes de teinte constante. Les contours de chrominance (chroma) forment alors des courbes fermées à intervalles constants à partir de ce point neutre situées à une distance constante des lignes géodésiques associées à ces teintes constances. Cette hypothèse peut être utilisée pour tester la robustesse, la précision, des formules mathématiques utilisées pour mesurer des différences couleur (color difference formulas) et pour prédire quelle valeurs peuvent prendre tel ou tel attribut perceptuel, ex. la teinte et la saturation (hue and chroma), ou telle distribution de stimulus couleur, dans n'importe quel espace couleur. Dans cette thèse, nous présentons une méthode qui permet de modéliser les éléments de ligne (lignes géodésiques), correspondants aux formules mathématiques Delta E * ab, Delta E * uv, OSA-UCS Delta EE utilisées pour mesurer des différences couleur, ainsi que les éléments de ligne correspondants à l'approximation infinitésimales du CIEDE2000. La pertinence de ces quatre formules mathématiques a été évaluée par comparaison, dans différents plans de représentation chromatique, des ellipses prédites et des ellipses expérimentalement obtenues par observation visuelle. Pour chacune de ces formules mathématiques, nous avons également testé l'hypothèse de Schrödinger, en calculant à partir de la métrique Riemannienne, les lignes géodésiques de teinte et les contours de chroma associés, puis en comparant les courbes calculées dans l'espace couleur CIELAB avec celles obtenues dans le système Munsell. Les résultats que nous avons obtenus démontrent qu'aucune de ces formules mathématiques ne prédit précisément les différences de couleur telles qu'elles sont perçues par le Système Visuel Humain. Ils démontrent également que les deux dernières formules en date, OSA-UCS Delta EE et l'approximation infinitésimale du CIEDE2000, ne sont pas plus précises que les formules conventionnelles calculées à partir des espaces couleur CIELAB et CIELUV, quand on se réfère au système Munsell (Munsell color order system) / Visual sensitivity to small color difference is an important factor for precision color matching. Small color differences can be measured by the line element theory in terms of color distances between a color point and neighborhoods of points in a color space. This theory gives a smooth positive definite symmetric metric tensor which describes threshold of color differences by ellipsoids in three dimensions and ellipses in two dimensions. The metric tensor is also known as the Riemannian metric tensor. In regard to the color differences, there are many color difference formulas and color spaces to predict visual difference between two colors but, it is still challenging due to the nonexistence of a perfect uniform color space. In such case, the Riemannian metric tensor can be used as a tool to study the performance of various color spaces and color difference metrics for measuring the perceptual color differences. It also computes the shortest length or the distance between any two points in a color space. The shortest length is called a geodesic. According to Schrödinger's hypothesis geodesics starting from the neutral point of a surface of constant brightness correspond to the curves of constant hue. The chroma contours are closed curves at constant intervals from the origin measured as the distance along the constant hue geodesics. This hypothesis can be utilized to test the performance of color difference formulas to predict perceptual attributes (hue and chroma) and distribution of color stimulus in any color space. In this research work, a method to formulate line element models of color difference formulas the ΔE*ab, the ΔE*uv, the OSA-UCS ΔEE and infinitesimal approximation of CIEDE2000 (ΔE00) is presented. The Jacobian method is employed to transfer their Riemannian metric tensors in other color spaces. The coefficients of such metric tensors are used to compute ellipses in two dimensions. The performance of these four color difference formulas is evaluated by comparing computed ellipses with experimentally observed ellipses in different chromaticity diagrams. A method is also developed for comparing the similarity between a pair of ellipses. The technique works by calculating the ratio of the area of intersection and the area of union of a pair of ellipses. Similarly, at a fixed value of lightness L*, hue geodesics originating from the achromatic point and their corresponding chroma contours of the above four formulas in the CIELAB color space are computed by solving the Euler-Lagrange equations in association with their Riemannian metrics. They are compared with with the Munsell chromas and hue circles at the Munsell values 3, 5 and 7. The result shows that neither formulas are fully perfect for matching visual color difference data sets. However, Riemannized ΔE00 and the ΔEE formulas measure the visual color differences better than the ΔE*ab and the ΔE*uv formulas at local level. It is interesting to note that the latest color difference formulas like the OSA-UCS ΔEE and the Riemannized ΔE00 do not show better performance to predict hue geodesics and chroma contours than the conventional CIELAB and CIELUV color difference formulas and none of these formulas fit the Munsell data accurately

Made available in DSpace on 2015-05-15T11:46:16Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 713593 bytes, checksum: 9a4eca1dfe2fca22fc63a3364773af08 (MD5) Previous issue date: 2013-09-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Cook's local infuence approach based on normal curvature is an important diagnostic tool for assessing local infuence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of infuence measures for objective functions at a point with a nonzero rst derivative. The aim of this paper is to develop a diferential-geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and ane curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. / A Abordagem de influência local de Cook [2] com base em curvatura normal é uma importante ferramenta de diagnóstico para avaliar a influência local de pequenas perturbações de um modelo estatístico. No entanto, tem sido desenvolvida nenhuma abordagem rigorosa para abordar duas questões fundamentais: a escolha de uma perturbação apropriada e o desenvolvimento de medidas de influência para funções objetos em um ponto com a primeira derivada diferente de zero. O objetivo deste trabalho é desenvolver uma estrutura diferencial-geométrica de um modelo de perturbação (chamado de variedade de perturbação) e utilizar o tensor métrico associado e as curvaturas afins para resolver esses problemas. Vamos mostrar que o tensor métrico da variedade de perturbação fornece informações importantes sobre a seleção de uma perturbação apropriada de um modelo.

Made available in DSpace on 2015-05-15T11:46:23Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 490053 bytes, checksum: c9cf689fde66aed86d7eb372eeb05045 (MD5) Previous issue date: 2009-05-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Cook's local infuence approach based on normal curvature is an important diagnostic tool for assessing local infuence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of infuence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a diferential-geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. / Esta dissertação é dedicada ao estudo do sistema de Navier-Stokes sob ponto de vista da teoria do controle. Primeiramente estudamos a controlabilidade das aproximações de Galerkin do sistema de Navier-Stokes. Utilizando argumentos de dualidade e de ponto fixo, mostramos que, com hipóteses adequadas sobre a base de Galerkin, estas aproximações, finito dimensionais, são exatamente controláveis. Passando ao modelo em dimensão infinita, analisamos a controlabilidade sobre trajetórias. Isto é feito usando uma desigualdade do tipo Calerman para o sistema de Navier-Stokes linearizado e uma versão do teorema da função inversa. Dessa forma, temos um resultado de controlabilidade local exata para o sistema de Navier-Stokes.