[pt] Este trabalho aborda o problema da estabilidade de sistemas lineares, invariantes no tempo, a tempo discreto, com o espaço de estado sendo um espaço de Hilbert complexo e separável de dimensão infinita. São investigadas condições necessárias e/ou suficientes para quatro conceitos diferentes de estabilidade: estabilidade assintótica uniforme e estabilidade assintótica forte, estabilidade assintótica fraca e estabilidade limitada. Identifica-se e analisa-se as conexões entre os problemas de estabilidade e dois problemas em aberto da teoria de operadores em espaços de Hilbert: o problema do subespaço invariante e o problemas da similaridade e contração. Diversos resultados, oriundos de tentativas de solução para os dois problemas acima, ou motivados por aquelas tentativas, são utilizadas para fornecer caracterizações adicionais (principalmente caracterizações espectrais) para os quatro conceitos de estabilidade em questão. / [en] This work deals with the stability problem for time- invariant discrete linear systems evolving in a separable infinite-dimensional Hilbert space. Necessary and/or sufficient conditions for uniform, strong and weak asymptotic stability, as well as to bounded stability problems to two open problems in operator theory, namely, the invariant subspace and the similarity to contractions, are identified and analysed in detail. Several results from the many attempts, of solving the above mentioned open problems, or motivated by those attempts, are used to supply additional characterizations (mainly spectral characterization) for the four stabilty concepts under consideration.

We describe the space of polynomial fields tangent to a given an algebraic curve. / Se da una descripción del espacio de campos polinomiales tangentes a una curva algebraica dada.

Orientador: Adriana Cesar de Mattos / Banca: Marcus Vinicius Maltempi / Banca: Oscar João Abdounur / Resumo: As origens da Teoria dos Invariantes na Inglaterra e o Mécanique Analytique de Lagrange (1788), é um trabalho voltado principalmente a entender uma possível influência que levou George Boole em 1841, a escrever o artigo Exposition of a General Theory of Linear Transformations e verificar se a motivação que o fez produzir este trabalho é igual ou diferente da motivação que ele exerceu sobre Arthur Cayley e consequentemente sobre James Joseph Sylvester. O presente trabalho apresenta um estudo das origens da Teoria dos Invariantes, no século XIX na Inglaterra. De acordo com os historiadores da Matemática o marco do início desta Teoria foi a publicação de George Boole em 1841. Assumimos este artigo como referência principal para realizar nossa pesquisa. Analisamos "antes" e "após" esta publicação de 1841. Concluímos que o Mécanique Analytique de Lagrange, foi a principal motivação para George Boole escrever seu trabalho e, certamente, George Boole foi uma grande influência para Arthur Cayley no que condiz com a escolha do assunto "invariantes" bem como o desenvolvimento desta Teoria por Cayley / Abstract: The origins of the theory of invariants in England and Mécanique Analytique of Lagrange (1788), is a work geared primarily to understand a possible influence that led George Boole in 1841, writing the article Exposition of the General Theory of Linear Transformations and verify that the motivation that did produce this work is equal or different of the motivation that he exerted on Arthur Cayley and James Joseph Sylvester consequently. This paper presents a study of the Invariant Theory origins, in the nineteenth century in England. According to historians of Mathematics the beginning of this Theory was the publication in 1841 of George Boole. We have taken this article as a reference to our research. We have proposed to analyzed "before" and "after" this publication, 1841. We conclude that the Mécanique Analytique Lagrange, was the essential motivation for George Boole write his work, and certainly George Boole was a great influence to Arthur Cayley in which matches the choice of subject "invariants" as well as the development of this Theory by Cayley / Mestre

[pt] A geometria diferencial provê um conjunto de medidas invariantes sob a ação de um grupo de transformações, em particular rígidas, afins e projetivas. Os invariantes por transformações rígidas são usados em quase todas as aplicações de computação gráfica e modelagem geométrica. O caso afim, por ser mais geral, permite estender essas ferramentas. Neste trabalho, propriedades geométricas são apresentadas no caso de superfícies paramétricas ou implícitas, em particular, a métrica afim, os vetores co-normal e normal afins e as curvaturas Gaussiana e média afins. Alguns resultados usuais de geometria Euclidiana, como a fórmula de Minkowski, são estendidos para o caso afim. Esse estudo permite definir estimadores das estruturas afins no caso de isossuperfícies. Porém, um cálculo direto dessas estruturas resulta em um grande número de operações e instabilidade numérica. Uma redução geométrica é proposta, obtendo fórmulas mais simples e mais estáveis numericamente. As propriedades geométricas incorporadas no Marching Cubes são analisadas e discutidas. / [en] Differential Geometry provides a set of measures invariant under a set of transformations, in particular rigid, affine, and projective. The invariants by rigid motions are using almost all applications of computer graphics and geometric modeling. The affine case, since it is more general, allows to extend these tools. In this work, geometric properties are presented in the case of parametric or implicit surfaces, in particular the affine metric, the conormal and normal vectors, and the affine Gaussian and mean curvatures. Some usual results of Euclidean geometry, as the Minkowski formula, are extended for the affine case. This study allows to define estimators of affines structure in the case of isosurfaces. Although, the direct calculation of these structures greatly increases the number of operations and numerical instabilities. A geometrical reduction is proposed obtaining a much simpler and numerical stabler formulae. The geometrical properties are incorporated in the Marching Cubes algorithms, then they are analyzed and discussed.

[pt] Nesta teses, nós estudamos H-superfícies, isto é, superfícies tendo curvatura media constante, imersas em variedades homogêneas simplesmente conexas de dimensão 3. Nós focamos nossa atenção no estudo de existência de H multigráficos. Também estudamos a H-superfícies invariantes por um grupo a um parâmetro de isometrias que estão imersas no espaço PSL(2) (R, T). / [en] In this thesis we study H-surfaces, that is, surfaces having constant mean curvature, immersed in homogeneous simply connected 3-manifold. We focus our attention in the study of existence of H multigraphs. We also study the H-surfaces invariant by one-parameter group of isometries which are immersed in the space]PSL2(R, T).

Comenzamos con un breve recordatorio sobre algunas nociones de conjuntos algebraicos, morfismos racionales y regulares. Por otro lado, veremos que la forma de Weierstrass de una cúbica tiene asociado dos elementos importantes. El primero es el discriminante τ que nos permite decidir si una cúbica es singular o no. El segundo elemento, muy importante en este trabajo, es el invariante j, cuyo nombre se debe a que éste no varía a pesar de los cambios de coordenadas que se realicen en la curva. Este elemento cobra gran importancia pues nos ayuda a reconocer cuando dos curvas elípticas son isomorfas. Y además, también nos permite contar el número de automorfismos sobre una curva elíptica dada. / We start with a brief reminder on some notions of algebraic sets, rational and regular maps. On the other hand, we will see that the Weierstrass form of a cubic has two important elements associated to it. The first is the discriminant τ that allows us to decide whether a cubic is singular or not. The second element, very important in this work, is the j invariant, whose name is due to the fact that it does not vary despite the changes in coordinates that are made in the curve. This element is crutial because it helps us to recognize when two elliptic curves are isomorphic. And in addition, it also allows us to count the number of automorphisms on a given elliptic curve.

Les travaux de ma thèse s'articulent en trois parties distinctes.Dans la première partie j'étudie les mesures de Malliavin-Shavguldize sur les difféomorphismes du cercle et de l'intervalle. Il s'agit de mesures de type " Haar " pour ces groupes de dimension infinie : elles furent introduites il a une vingtaine d'années pour permettre une étude de leur théorie des représentations. Un premier chapitre est dédié à recueillir les résultats présents dans la littérature et et les représenter dans une forme plus étendue, avec un regard particulier sur les propriétés de quasi-invariance de ces mesures. Ensuite j'étudie de problèmes de nature plus dynamique : quelle est la dynamique qu'on doit s'attendre d'un difféomorphisme choisi uniformément par rapport à une mesure de Malliavin-Shavguldize ? Je démontre en particulier qu'il y a une forte présence des difféomorphismes de type Morse-Smale.La partie suivante vient de mon premier travail publié, obtenu en collaboration avec Andrés Navas. Inspirés d'un théorème récent de Avila et Kocsard sur l'unicité des distributions invariantes par un difféomorphisme lisse minimal du cercle, nous analysons le même problème en régularité faible, avec des argument plus géométriques.La dernière partie est constituée des résultats récemment obtenus avec Mikhail Khristoforov et Victor Kleptsyn. Nous abordons les problèmes reliés à la gravité quantique de Liouville en étudiant des espaces auto-similaires qui sont la limite de graphes finis. Nous démontrons qu'il est possible de trouver des distances aléatoires non-triviales sur ces espaces qui sont compatibles avec la structure auto-similaire.

Made available in DSpace on 2015-03-26T13:45:36Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1441623 bytes, checksum: 546ade192993fd13d15e3039ab577882 (MD5) Previous issue date: 2013-07-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study the classical invariants of Legendrian Knots Theory and we show that these are not complet. To do this we introduce a notion of a Basic Knot Theory like their classical invariants, Thurston-Bennequin number and Maslov number. Then we discuss a new tool developed by Chekanon and denoted by DGA (Differential Graduated Algebra), wich will help us in the proof of the incompletness of classical invariants of legendrian knots. / Neste trabalho, estudaremos os invariantes clássicos da Teoria de Nós Legendreanos e mostraremos que estes não são completos. Para tal introduzimos uma noção básica da Teoria de Nós Legendreanos, assim como seus invariantes clássicos, o número de Thurston-Bennequin e o número de Maslov. Em seguida discutiremos uma nova ferramenta desenvolvida por Chekanov, a Álgebra Diferencial Graduada, denotada por DGA (Differential Graduated Algebra), que nos auxiliar na prova da incompletude dos invariantes clássicos de nós legendreanos.

Résumé : L’ombre sur l’imagerie satellitaire à très haute résolution spatiale a suscité relativement peu de travaux de recherche en télédétection. La présente étude a pour objectif la détection des zones d’ombre sur l’imagerie IKONOS via l’exploitation des caractéristiques des couleurs de l’ombre. Ces caractéristiques découlent des couleurs photométriques invariantes issues des transformées couleurs. L’approche proposée a pour but d’évaluer dans un premier temps, le potentiel des transformées RGB normalisé, IHS (Intensity, Hue, Saturation), HSV (Hue, Saturation, Value), C[indice inférieur 1] C[indice inférieur 2] C[indice inférieur 3], L[indice inférieur 1] L[indice inférieur 2] L[indice inférieur 3] et M[indice inférieur 1] M[indice inférieur 2] M[indice inférieur 3] (Gevers and Smeulders, 1999) à discriminer les zones d’ombre. Cette évaluation s’est effectuée sur deux images simples dont la première est une image d’extérieur affectée par les effets atmosphériques, et la deuxième est une image d’intérieur, illuminée par une source de lumière artificielle. Les effets atmosphériques, notamment la dispersion, conduisent à la saturation maximale des zones d’ombre sur l’image d’extérieur. Cette conclusion a été vérifiée sur l’image IKONOS et exploitée pour développer une approche pour la détection de l’ombre sur ce genre de données. La transformée IHS a été retenue pour générer les couleurs photométriques H et S respectivement la teinte et la saturation. Deux dérivées I-S et H+l/I+1 ont été calculées. Ces dernières permettent le rehaussement des zones d’ombre sur l’image IKONOS. Trois méthodes de seuillage ont été appliquées sur les deux dérivées afin de particulariser les régions d’ombre. Les deux images finales issues du seuillage ont été fusionnées. La dernière étape de la recherche a consisté en une validation des résultats. Cette opération a démontré la robustesse de l’approche avec une précision globale moyenne de 80%. La confusion entre les zones d’ombre et les régions sombres est la principale faiblesse de l’approche proposée. Cependant, son amélioration est possible grâce à l’intégration d’autres types d’informations comme la texture et la prise en compte du voisinage. // Abstract : The shadow phenomena on the satellite imagery with very high spatial resolution has been the subject of a few research tasks in remote sensing. The aim of the present study is the detection of the shadow’s areas on IKONOS imagery with the use of the shadow’s colors characteristics. These characteristics were derived from the invariant photometric colors. The purpose of the suggested approach is to evaluate initially, the potential of normalized color RGB, IHS (Intensity, Hue, Saturation), HSV (Hue, Saturation, Value), C[subscript 1] C[subscript 2] C[subscript 3], L[subscript 1] L[subscript 2] L[subscript 3] et M[subscript 1] M[subscript 2] M[subscript 3] (Gevers and Smeulders, 1999) transform colors to discriminate shadow’s areas. This evaluation was carried out on two simple images whose first is an outside scene affected by the atmospheric effects, and the second is an interior one illuminated by artificial light source. The atmospheric effects, in particular dispersion, lead to the maximum saturation of shadow’s zones on the outside image. This conclusion was checked on IKONOS image and exploited to develop a method to detect the shadow on this kind of data. IHS transform was retained to generate the photometric colors hue (H) and saturation (S). Two derived images I-S and H+l/I+1 were calculated. They allow the raising of the shadow’s areas on IKONOS image. Three methods of thresholding were applied to the two derived images in order to differentiate the areas of interest. The two final images resulting from the thresholding were joined. The last stage of research consisted of a validation of the results. This operation showed the robustness of the approach with 80% of precision. Confusion between the shadow’s areas and the dark ones is the principal weakness of the suggested approach. However, this new technique can be improved by introducing other information like texture and nearest neighbor analysis.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.