Texte extr. de: Th.--Madrid, 1968. / Bibliogr. p. 195-227.

Made available in DSpace on 2016-06-02T20:15:24Z (GMT). No. of bitstreams: 1 4096.pdf: 1596453 bytes, checksum: 49914b2cdea816c0ab5585fb684460e5 (MD5) Previous issue date: 2011-06-17 / Universidade Federal de Sao Carlos / In this thesis, we explore two sort of quantum correlations: entanglement and quantum discord. We present a geometric method to identify and measure the degree of entanglement based on symmetries of vectors and matrices associated with the two-qubits density operator of quantum states. We introduce a new basis of parameters describing the density operator, and this procedure allows us to establish the Peres-Horodecki separability criterion in terms of squared distances that obey the Minkowski metric, giving a more general interpretation of this criterion as well as building a quantifier of entanglement. In this method, if the squared distance is of the kind timelike, i.e. non-negative, the two-qubit system is separable. Otherwise, if it is spacelike, namely, the squared distance is negative, the two qubits are entangled. Such squared distances are invariant by unitary transformations and can be represented graphically in a hyperbolic parameterized phase space, allowing a suitable graphic representation, i.e., in a phase space where the system trajectories can be drawn. The method is generalized to a larger class of states having at most seven independent parameters, the D-7 manifold class. Using group theory methods we classify these states according to the symmetries of seven generators, where one of them commutes with the others. We illustrate the method and the theory by presenting several two-qubit systems found in the literature. This same notation is used to calculate the quantum discord for states whose 4 × 4 matrices belong to the D-7 manifold class, providing a more explicit condition of minimization of entropy. We calculate the dissipative dynamics of two-qubits quantum discord under local noisy environments. Choosing initial conditions that manifest the so-called sudden death of entanglement, we compare the dynamics of entanglement with that of quantum discord and we show that in cases where the entanglement suddenly disappears, quantum discord vanishes only in the asymptotic limit. / Nesta tese exploramos dois tipos de correlações quânticas: o emaranhamento e a discórdia. Apresentamos um método geométrico de caracterização e quantificação do emaranhamento baseado em simetrias de vetores e matrizes associados ao operador densidade dos estados quânticos de dois qubits. Introduzimos uma nova base de parâmetros que descrevem o operador densidade, e este procedimento nos permite estabelecer o critério de separabilidade de Peres-Horodecki em termos de distâncias quadráticas que obedecem a métrica de Minkowski, proporcionando uma interpretação mais geral deste critério bem como a construção de um quantificador de emaranhamento. Neste método, quando as distâncias quadráticas forem não negativas, o sistema é dito separável, por outro lado, quando forem negativas o sistema é dito emaranhado. Tais distâncias quadráticas são invariantes por transformações unitárias e podem ser representadas graficamente em um espaço de fase hiperbólico parametrizado, onde uma análise quantitativa pode ser realizada e até mesmo trajetórias podem ser traçadas. O método é generalizado para uma classe maior de estados com até sete parâmetros independentes, que nomeamos de estados de variedade D-7, através do uso de teoria de grupos, onde classificamos os estados de acordo com as simetrias de seus sete geradores, sendo que um deles comuta com todos os outros. Para ilustrar o método proposto, uma série de exemplos presentes na literatura são estudados. Esta mesma notação é empregada no cálculo da discórdia quântica para estados de variedade D-7, proporcionando uma abordagem mais explícita da condição de minimização da entropia. A dinâmica dissipativa da discórdia para um sistema de dois qubits imersos em reservatórios individuais é calculada e, escolhendo condições iniciais que manifestem o fenômeno de morte súbita do emaranhamento, comparamos as duas dinâmicas (emaranhamento e discórdia) e mostramos que nos casos onde o emaranhamento desaparece subtamente, a discórdia quântica desaparece somente no limite assintótico.

Nesta Tese, nossos objetos de estudo s?o as hipersuperf?cies tipo-espa?o imersas num espa?o-tempo com alguma curvatura de ordem superior constante. Em rela??o a estas hipersuperf?cies, abordamos quest?es como exist?ncia, unicidade e estimativa de quantidades geom?tricas associadas as mesmas. Mais precisamente, desenvolvemos f?rmulas tipo-Minkowski para hipersuperf?cies tipo-espa?o compactas com bordo imersas no espa?o de De Sitter e tendo alguma curvatura de ordem superior constante. Em seguida, aplicamos estas f?rmulas para estabelecer uma rela??o entre a curvatura m?dia e a geometria do bordo. Obtemos, tamb?m, uma estimativa sharp para a fun??o altura de hipersuperf?cies tipo-espa?o compactas imersas no espa?o de Lorentz-Minkowski Ln+1 com alguma curvatura de ordem superior constante n?o-nula. Como aplica??o desta estimativa,tratamos sobre a natureza de um fim de uma hipersuperf?cie tipo-espa?o completa de Ln+1. Finalmente, estudamos a exist?ncia e unicidade de gr?ficos verticais completos com curvatura m?dia constante tanto no Steady State space Hn+1 como no espa?o hiperb?lico Hn+1. Como consequ?ncia deste estudo, obtemos resultados tipo-Bernstein em H3 e H3.

One way of modeling human knowledge is by using multidimensional spaces, in which an object is represented as a point in the space, and the distances among the points reflect the similarities among the represented objects. The distances are measured with some metric, commonly some instance of the Minkowski metric. The instances differ with the magnitude of the so-called r-parameter. The instances most commonly mentioned in the literature are the ones where r equals 1, 2 and infinity. Cognitive scientists have found out that different metrics are suited to describe different dimensional combinations. From these findings an important distinction between integral and separable dimensions has been stated (Garner, 1974). Separable dimensions, e.g. size and form, are best described by the city-block metric, where r equals 1, and integral dimensions, such as the color dimensions, are best described by the Euclidean metric, where r equals 2. Developmental psychologists have formulated a hypothesis saying that small children perceive many dimensional combinations as integral whereas adults perceive the same combinations as separable. Thus, there seems to be a shift towards increasing separability with age or maturity. Earlier experiments show the same phenomenon in adult short-term learning with novel stimuli. In these experiments, the stimuli were first perceived as rather integral and were then turning more separable, indicated by the Minkowski-r. This indicates a shift towards increasing separability with familiarity or skill. This dissertation aims at investigating the generality of this phenomenon. Five similarity-rating experiments are conducted, for which the best fitting metric for the first half of the session is compared to the last half of the session. If the Minkowski-r is lower for the last half compared to the first half, it is considered to indicate increasing separability. The conclusion is that the phenomenon of increasing separability during short-term learning cannot be found in these experiments, at least not given the operational definition of increasing separability as a function of a decreasing Minkowski-r. An alternative definition of increasing separability is suggested, where an r-value ‘retreating’ 2.0 indicates increasing separability, i.e. when the r-value of the best fitting metric for the last half of a similarity-rating session is further away from 2.0 compared to the first half of the session.

The purpose of this study was to see how the fundamental properties of hyperbolic geometry applies in physics.

Die Dissertation behandelt zwei Gebiete der Geometrie endlichdimensionaler Banach-Räume (Minkowski-Geometrie). Der erste Schwerpunkt liegt dabei auf Winkelmassen und Winkelhalbierenden. Dafür gibt es verschiedene Verallgemeinerungen dieser Euklidischen Konzepte, die im allgemeinen in Minkowski-Räumen verschieden sind. Es werden alle Minkowski-Räume charakterisiert, in welchen zwei dieser Konzepte für alle möglichen Winkel das selbe Maß oder die selben Winkelhalbierenden liefern. Der zweite Teil der Dissertation behandelt die Einbettung von metrischen Räumen in Minkowski-Räume. Dabei steht die Einbettung in beliebige geeignete Minkowski-Räume fester Dimension im Mittelpunkt. Hauptergebnis ist hier die vollständige Klassifikation aller 2-Abstands-Mengen in Minkowski-Ebenen, d.h., aller möglichen Mengen von Punkten einer Minkowski-Ebene, so dass zwischen diesen Punkten nur zwei verschiedene positive Abstandswerte auftreten. / This dissertation deals with two geometric subjects in finite dimensional Banach spaces (Minkowski geometry). The first topics are angle measures and angular bisectors. There are several possibilities to generalize these Euclidean concepts, which yield in general distinct geometrical objects in Minkowski spaces. A characterization is given for Minkowski spaces, for which two such concepts yield for all possible angles the same angular measure or the same angular bisector. The second part of the dissertation deals with embeddings of metric spaces into Minkowski spaces. It focuses on embeddings into some arbitrary suitable Minkowski space of prescribed dimension. The major result is the complete classification of all 2-distance sets in Minkowski planes, i.e., of all subsets of points of a Minkowski plane such that there are only two different positive distance values between these points.

The central objective of this work is to present an analysis of the asymptotic conformal Killing vectors in asymptotically-flat space-times of general relativity. This problem has been examined by two different methods; in Chapter 5 the asymptotic expansion technique originated by Newman and Unti [31] leads to a solution for asymptotically-flat spacetimes which admit an asymptotically shear-free congruence of null geodesics, and in Chapter 6 the conformal rescaling technique of Penrose [54] is used both to support the findings of the previous chapter and to set out a procedure for solution in the general case. It is pointed out that Penrose's conformal technique is preferable to the use of asymptotic expansion methods, since it can be established in a rigorous manner without leading to the possible convergence difficulties associated with asymptotic expansions. Since the asymptotic conformal symmetry groups of asymptotically flat space-times Are generalisations of the conformal group of Minkowski space-time we devote Chapters 3 and 4 to a study of the flat space case so that the results of later chapters may receive an interpretation in terms of familiar concepts. These chapters fulfil a second, equally important, role in establishing local isomorphisms between the Minkowski-space conformal group, 90(2,4) and SU(2,2). The SO(2,4) representation has been used by Kastrup [61] to give a physical interpretation using space-time gauge transformations. This appears as part of the survey of interpretative work in Chapter 7. The SU(2,2) representation of the conformal group has assumed a theoretical prominence in recent years. through the work of Penrose [9-11] on twistors. In Chapter 4 we establish contact with twistor ideas by showing that points in Minkowski space-time correspond to certain complex skew-symmetric rank two tensors on the SU(2,2) carrier space. These objects are, in Penrose's terminology [91, simple skew-symmetric twistors of valence [J. A particularly interesting aspect of conformal objects in space-time is explored in Chapter 8, where we extend the work of Geroch [16] on multipole moments of the Laplace equation in 3-space to the consideration. of Q tý =0 in Minkowski space-time. This development hinges upon the fact that multipole moment fields are also conformal Killing tensors. In the final chapter some elementary applications of the results of Chapters 3 and 5 are made to cosmological models which have conformal flatness or asymptotic conformal flatness. In the first class here we have 'models of the Robertson-Walker type and in the second class we have the asymptotically-Friedmann universes considered by Hawking [73]. / University of Bradford Research Studenship

In this paper, we study Bézier surfaces in View the MathML source three-dimensional Minkowski space. In particular, we focus on timelike and spacelike cases for Bézier surfaces. We also deal with the Plateau¿Bézier problem in View the MathML source, obtaining conditions over the control net to be extremal of the Dirichlet function for both timelike and spacelike Bézier surfaces. Moreover, we provide interesting examples showing the behavior of the Plateau¿Bézier problem in View the MathML source and illustrating the relationship between it and the corresponding Plateau¿Bézier problem in the Euclidean space R3.

La cristallisation en solution est un procédé largement utilisé dans l'industrie comme opération de séparation et de purification qui a pour but de produire des solides avec des propriétés spécifiques. Les propriétés concernant la taille et la forme ont un impact considérable sur la qualité finale des produits. Il est donc primordial de pouvoir déterminer la distribution granulométrique (DG) des cristaux en formation. En utilisant une caméra in situ, il est possible de visualiser en temps réel les projections 2D des particules 3D présentes dans la suspension. La projection d'un objet 3D sur un plan 2D entraîne nécessairement une perte d'informations : déterminer sa taille et sa forme à partir de ses projections 2D n’est donc pas aisé. C'est tout l'enjeu de ce travail: caractériser géométriquement et morphométriquement des objets 3D à partir de leurs projections 2D. Tout d'abord, une méthode basée sur le maximum de vraisemblance des fonctions de densité de probabilité de mesures géométriques projetées a été développée pour déterminer la taille d'objets 3D convexes. Ensuite, un descripteur de forme stéréologique basé sur les diagrammes de forme a été proposé. Il permet de caractériser la forme d'un objet 3D convexe indépendamment de sa taille et a notamment été utilisé pour déterminer les facteurs d'anisotropie des objets 3D convexes considérés. Enfin, une combinaison des deux études précédentes a permis d'estimer à la fois la taille et la forme des objets 3D convexes. Cette méthode a été validée grâce à des simulations, comparée à une méthode de la littérature et utilisée pour estimer des DGs d'oxalate d'ammonium qui ont été comparées à d’autres méthodes granulométriques. / Solution crystallization processes are widely used in the process industry as separation and purification operations and are expected to produce solids with desirable properties. The properties concerning the size and the shape are known to have a considerable impact on the final quality of products. Hence, it is of main importance to be able to determine the granulometry of the crystals (CSD) in formation. By using an in situ camera, it is possible to visualize in real time the 2D projections of the 3D particles in the suspension.The projection of a 3D object on a 2D plane necessarily involves a loss of information. Determining the size and the shape of a 3D object from its 2D projections is therefore not easy. This is the main goal of this work: to characterize geometrically and morphometrically 3D objects from their 2D projections. First of all, a method based on the maximum likelihood estimation of the probability density functions of projected geometrical measurements has been developed to estimate the size of 3D convex objects. Then, a stereological shape descriptor based on shape diagrams has been proposed. It enables to characterize the shape of a 3D convex object independently of its size and has notably been used to estimate the value of the anisotropy factors of the 3D convex objects. At last, a combination of the two previous studies has allowed to estimate both the size and the shape of the 3D convex objects. This method has been validated with simulated data, has been compared to a method from the literature and has been used to estimate size distributions of ammonium oxalate particles crystallizing in water that have been compared to other CSD methods.

El objetivo fundamental de este trabajo ha sido el estudio del sistema fundamental de paralelos de un cuerpo convexo (conjunto compacto y convexo) en el espacio euclídeo n-dimensional. Se ha llevado a cabo siguiendo tres líneas diferentes: el estudio del polinomio de Steiner y el polinomio alternado Steiner desde el punto de vista algebraico de sus raíces y la conjetura de Matheron; el estudio de la diferenciabilidad de las quermassintegrales asociadas al cuerpo con respecto al parámetro de definición del sistema completo de paralelos y, por último, el estudio de las quermassintegrales del cuerpo desde el punto de vista analítico proporcionado por la identificación del cuerpo convexo con su función soporte, las propiedades de ésta cuando el cuerpo es suficientemente suave y la desigualdad de Brunn-Minkowski. / The aim of this work consists on studying the full system of parallel bodies of a convex body (compact and convex set) in the n-dimensional Euclidean space. It has been carried out following three different lines of work: the study of the Steiner polynomial and the alternating Steiner polynomial from the algebraic point of view of its roots; the study of the differentiability of the quermassintegrals associated to the convex body with respect to the parameter that defines the full system of parallel bodies and finally, the study of the quermassintegrals from the analytical point of view provided by the identification of a convex body with its support function, the properties of this function when the body is smooth enough and the Brunn-Minkowski inequality