Seja L um campo vetorial complexo não singular definido em um aberto do plano. Treves provou que se L é localmente resolúvel então L é localmente integrável. Para campos planares hipoelíticos, vale uma propriedade adicional, a saber, toda integral primeira (restrita a um aberto suficientemente pequeno) é uma aplicação injetiva (e aberta); isto, por sua vez, implica que toda solução da equação homogênea Lu = 0 é localmente da forma u = h 0 Z, com h holomorfa, sendo Z uma integral primeira do campo. O problema central de interesse desta dissertação é a questão global correspondente, ou seja, a exisatência de integrais primeiras globais injetoras e a representação dde soluções globais por composições da integral primeira com uma função holomorfa / Let L be a nonsingular complex vector field defined on an open subset of the plane. Treves proved that if L is locally solvable then L is locally integrable. For hypoelliptic planar vector fields an additional property holds, namely, every first integral (restricted to a sufficiently small open set) is an injective (and open) mapping; this, on its turn, implies that each solution of the homogeneous equation Lu = 0 is locally of the form u = h Z, where h is holomorphic and Z is a first integral of the vector eld. The central problem of interest in this work is the corresponding global question, that is, the existence of global, injective first integrals and the representation of global solutions as compositions of the first integral with a holomorphic function

Em Computação Gráfica, campos vetoriais possuem diversas aplicações desde a síntese e mapeamento de texturas à animações de fluidos, produzindo efeitos amplamente utilizados na indústria do entretenimento. Para produzir tais campos, é preferível o uso de ferramentas de design em vez de simulações numéricas não só devido ao menor custo computacional mas, principalmente, por prover liberdade ao artista ao sintetizar o campo de acordo com a sua necessidade. Atualmente, na literatura, existem bons métodos de design de campos vetoriais em superfícies de objetos tridimensionais porém, o design no interior desses objetos ainda é pouco estudado, principalmente quando o campo de interesse possui propriedades específicas. O objetivo deste trabalho é desenvolver uma técnica para sintetizar campos vetoriais, com características do movimento de fluidos incompressíveis, no interior de domínios. Em uma primeira etapa, o método consiste na interpolação dos vetores de controle, com uma certa propriedade desejada, em todo o domínio. Posteriormente, o campo obtido é modificado para respeitar a geometria do contorno. / Vector fields are important to an wide range of applications on the field of Computer Graphics, from the synthesis and mapping of textures to fluid animation, producing effects widely used on the entertainment industry. To produce such fields, design tools are prefered over numerical simulations not only for its lower computational cost, but mainly by providing freedom to the artist in the creation process. Nowadays, good methods of vector field design over surfaces exist in literature, however there is only a few studies on the synthesis of vector fields of the interior of objects and even fewer when specific properties of the field are required. This work presents a technique to synthesize vector fields with properties of imcompressible fluids motion in the interior of objects. On a first step, the method consists in interpolating control vectors with a certain desired property throughout the whole domain and later the resulting field is modified to properly fit the boundary geometry of the object.

Neste trabalho estudamos índices de campos de vetores em variedades regulares e em variedades com singularidades isoladas. O principal resultado e o Teorema de Poincaré-Hopf que relaciona a característica de Euler de uma variedade com o índice de Poincaré-Hopf do campo. Para intersecções completas com singularidades isoladas, vemos também algumas variações deste teorema que relacionam a característica de Euler com o índice de Schwartz, o índice GSV e o número de Milnor da fibra genérica / In this work we study some indices of vector fields on regular manifolds, and on manifolds with isolated singularity. The main result is the Poincare-Hopf Theorem, which connects the Euler characteristic with the Poincare-Hopf index of the field. For complete intersections with isolated singularities, we also study some variations of this theorem, which connects the Euler characteristic with the Schwartz index, the GVS index and the Milnor number of the generic fiber

Neste trabalho estudamos índices de campos de vetores em variedades regulares e em variedades com singularidades isoladas. O principal resultado e o Teorema de Poincaré-Hopf que relaciona a característica de Euler de uma variedade com o índice de Poincaré-Hopf do campo. Para intersecções completas com singularidades isoladas, vemos também algumas variações deste teorema que relacionam a característica de Euler com o índice de Schwartz, o índice GSV e o número de Milnor da fibra genérica / In this work we study some indices of vector fields on regular manifolds, and on manifolds with isolated singularity. The main result is the Poincare-Hopf Theorem, which connects the Euler characteristic with the Poincare-Hopf index of the field. For complete intersections with isolated singularities, we also study some variations of this theorem, which connects the Euler characteristic with the Schwartz index, the GVS index and the Milnor number of the generic fiber

Neste trabalho, nós abordamos alguns resultados de T. Fukuda e de N. Dutertre e T. Fukui sobre a topologia das singularidades de Morin. Em particular, apresentamos uma nova prova para o Teorema de Dutertre-Fukui [2, Theorem 6.2], para o caso em que N = Rn, usando a Teoria de Morse para variedades com bordo. Baseados nas propriedades de um n-campo de vetores gradiente (∇ f1; : : : ∇fn) de uma aplicação de Morin f : M → Rn, com dim M ≥ n, na segunda parte deste trabalho, nós introduzimos o conceito de n-campos de Morin para n-campos de vetores que não são necessariamente gradientes. Nós também generalizamos o resultado de T. Fukuda [3, Theorem 1], que estabelece uma equivalência módulo 2 entre a característica de Euler de uma variedade diferenciável M e a característica de Euler dos conjuntos singulares de uma aplicação de Morin definida sobre M, para o contexto dos n-campos de Morin. / In this work, we revisit results of T. Fukuda and N. Dutertre and T. Fukui on the topology of Morin maps. In particular, we give a new proof for Dutertre-Fukui\'s Theorem [2, Theorem 6.2] when N = Rn, using Morse Theory for manifolds with boundary. Based on the properties of a gradient n-vector field (∇ f1; : : : ∇ fn) of a Morin map f : M → Rn, where dim M ≥ n, in the second part of this work, we introduce the concept of Morin n-vector field for n-vector fields V = (V1; : : : ; Vn) that are not necessarily gradients. We also generalize the result of T. Fukuda [3, Theorem 1], which establishes a module 2 equivalence between Euler\'s characteristic of a manifold M and Euler\'s characteristic of the singular sets of a Morin map defined on M, to the context of Morin n-vector fields.

Neste trabalho, nós abordamos alguns resultados de T. Fukuda e de N. Dutertre e T. Fukui sobre a topologia das singularidades de Morin. Em particular, apresentamos uma nova prova para o Teorema de Dutertre-Fukui [2, Theorem 6.2], para o caso em que N = Rn, usando a Teoria de Morse para variedades com bordo. Baseados nas propriedades de um n-campo de vetores gradiente (∇ f1; : : : ∇fn) de uma aplicação de Morin f : M → Rn, com dim M ≥ n, na segunda parte deste trabalho, nós introduzimos o conceito de n-campos de Morin para n-campos de vetores que não são necessariamente gradientes. Nós também generalizamos o resultado de T. Fukuda [3, Theorem 1], que estabelece uma equivalência módulo 2 entre a característica de Euler de uma variedade diferenciável M e a característica de Euler dos conjuntos singulares de uma aplicação de Morin definida sobre M, para o contexto dos n-campos de Morin. / In this work, we revisit results of T. Fukuda and N. Dutertre and T. Fukui on the topology of Morin maps. In particular, we give a new proof for Dutertre-Fukui\'s Theorem [2, Theorem 6.2] when N = Rn, using Morse Theory for manifolds with boundary. Based on the properties of a gradient n-vector field (∇ f1; : : : ∇ fn) of a Morin map f : M → Rn, where dim M ≥ n, in the second part of this work, we introduce the concept of Morin n-vector field for n-vector fields V = (V1; : : : ; Vn) that are not necessarily gradients. We also generalize the result of T. Fukuda [3, Theorem 1], which establishes a module 2 equivalence between Euler\'s characteristic of a manifold M and Euler\'s characteristic of the singular sets of a Morin map defined on M, to the context of Morin n-vector fields.

Neste trabalho calculamos a classe de Euler de uma folheação umbílica em um ambiente com forma de curvatura apropriada. Combinamos o teorema de Hopf-Milnor e o número de Euler de uma folheação, definido por Connes, para mostrar como a geometria da folheação influencia na topologia da variedade folheada, bem como na topologia da folheação. Além disso, exibimos uma lista de invariantes topológicos para campos vetoriais unitários em hipersuperfícies fechadas do espaço Euclidiano, e mostramos como estes invariantes podem ser empregados como obstruções a certas folheações com geometria prescrita. / In this work we compute the Euler class of an umbilic foliation on a manifold with suitable curvature form. We combine the Hopf-Milnor theorem and the Euler number of a foliation, defined by Connes, in order to show how the geometry of the foliation influences the topology of the foliated space as well as the topology of the foliation. Besides, we exhibit a list of topological invariants for unit vector fields on closed Euclidean hypersurfaces, and show how these invariants may be employed as obstructions to certain foliations with prescribed geometry.

Topology-related issues are becoming increasingly important in Computer Graphics. This research examines the use of topological analysis for solving two important problems in 3D Graphics: surface parameterization, and vector field design on surfaces. Many applications, such as high-quality and interactive image synthesis, benefit from the solutions to these problems. Surface parameterization refers to segmenting a 3D surface into a number of patches and unfolding them onto a plane. A surface parameterization allows surface properties to be sampled and stored in a texture map for high-quality and interactive display. One of the most important quality measurements for surface parameterization is stretch, which causes an uneven sampling rate across the surface and needs to be avoided whenever possible. In this thesis, I present an automatic parameterization technique that segments the surface according to the handles and large protrusions in the surface. This results in a small number of large patches that can be unfolded with relatively little stretch. To locate the handles and large protrusions, I make use of topological analysis of a distance-based function on the surface. Vector field design refers to creating continuous vector fields on 3D surfaces with control over vector field topology, such as the number and location of the singularities. Many graphics applications make use of an input vector field. The singularities in the input vector field often cause visual artifacts for these applications, such as texture synthesis and non-photorealistic rendering. In this thesis, I describe a vector field design system for both planar domains and 3D mesh surfaces. The system provides topological editing operations that allow the user to control the number and location of the singularities in the vector field. For the system to work for 3D meshes surface, I present a novel piecewise interpolating scheme that produces a continuous vector field based on the vector values defined at the vertices of the mesh. I demonstrate the effectiveness of the system through several graphics applications: painterly rendering of still images, pencil-sketches of surfaces, and texture synthesis.

Βασικό αντικείμενο της εργασίας είναι ο έλεγχος και η ευστάθεια ομάδων αποτελούμενων από κινούμενα ρομπότ. Για το σκοπό αυτό καταγράφονται και παρουσιάζονται, αναλυτικά, μέθοδοι και τρόποι που εξυπηρετούν προς την κατεύθυνση αυτή. Η εργασία χωρίζεται σε τέσσερα κεφάλαια, από τα οποία, τα τρία πρώτα έχουν θεωρητικό χαρακτήρα, σε αντίθεση με το τέταρτο κεφάλαιο που είναι πρακτικού περιεχομένου. Το πρώτο κεφάλαιο αποτελεί, κατά μία έννοια, εισαγωγή στο θέμα του ελέγχου ρομπότ, καθώς παρουσίαζεται σε αυτό μία μέθοδος με την οποία επιτυγχάνεται ο έλεγχος και ο σχεδιασμός κίνησης για ένα και μόνο ρομπότ, σε περιβάλλον εμποδίων. Με τον τρόπο αυτό δίνεται μία βάση και ένα θεωρητικό πλαίσιο, για την περαιτέρω μελέτη, που παρουσιάζεται στα επόμενα κεφάλαια και αφορά ομάδες από κινούμενα ρομπότ. Στο δεύτερο κεφάλαιο γίνεται η παρουσίαση μίας μεθόδου με την οποία μπορεί να καθοριστεί ένας σχηματισμός αποτελούμενος από ρομπότ, ικανός να εκτελέσει διάφορες επιθυμητές κινήσεις και κατόπιν, αφού εξασφαλιστεί αυτή η ικανότητα, να κατασκευάστεί ένα κατάλληλο σύστημα ελέγχου για την πραγματοποιήση των κινήσεων αυτών. Στο τρίτο κεφάλαιο, που ολοκληρώνει και το θεωρητικό μέρος της εργασίας αυτής, γίνεται η καταγραφή μιας μεθόδου για την εξέταση της ευστάθειας σχηματισμών ρομπότ κατά την εκτέλεση κινήσεων στο χώρο, σε περιβάλλον εμποδιών. Η μέθοδος αυτή συναντάται με τον αγγλικό όρο, Leader-to-Formation Stability (LFS) και σχετίζεται με τον βαθμό διατήρησης της μορφής του σχηματισμού και των σφαλμάτων σχηματισμού εντός επιτρεπτών ορίων. Στο τέταρτο κεφάλαιο, γίνεται η παρουσίαση ενός προγράμματος σε γλώσσα Matlab, με το οποίο επιτυγχάνεται η προσομοίωση κινήσεων ενός ή πολλών ρομπότ στο επίπεδο, σε περιβάλλον εμποδίων. Το πρόγραμμα συναντάται εξ ολοκλήρου και στο συνοδευτικό CD της εργασίας. / Basic object of this work is the control and the stability of teams constituted of moving robots. For this aim they are recorded and are presented, analytically, methods and ways that they serve to this direction. The work is separated in four chapters, from which, the three first have theoretical character, contrary to the fourth chapter that is of practical content. The first chapter constitutes, at a significance, import in the subject of robot control , as is presented in this, a method with which are achieved the control and the planning of movement for one and alone robot, in environment of obstacles. With this way is given a base and a theoretical frame, for the further study, that is presented in the next capitals and concerns teams of moving robots. In the second chapter comes the presentation of a method with which it can be determined a formation of robots, that is capable to execute various, desirable movements and then, after is ensured this faculty, is been constructed a suitable system of control for the realisation of this movements. In the third chapter, that it completes also the theoretical part of this work, comes the recording of a method for the examination of stability of formations of robots, at the implementation of movements in the space, in environment of obstacles. This method is met with the English term, Leader-to-Formation Stability (LFS) and is related with the degree of maintenance of form of the formation and faults of formation inside permissible limits. In the fourth chapter, comes the presentation of a program in Matlab language , with which is achieved the simulation of movements of one or many robots on the surface, in environment of obstacles. The program is met entirely also in the accompanying CD of this work.

Submitted by DANIELE ALESSANDRA REGHINI GAZETTA null (daniellygaze@hotmail.com) on 2016-01-15T17:36:23Z No. of bitstreams: 1 diss-daniele.pdf: 783553 bytes, checksum: e593f1ebb872fff02a080d05283744d5 (MD5) / Rejected by Ana Paula Grisoto (grisotoana@reitoria.unesp.br), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: No campo “Versão a ser disponibilizada online imediatamente” foi informado que seria disponibilizado o texto completo porém no campo “Data para a disponibilização do texto completo” foi informado que o texto completo deverá ser disponibilizado apenas 6 meses após a defesa. Caso opte pela disponibilização do texto completo apenas 6 meses após a defesa selecione no campo “Versão a ser disponibilizada online imediatamente” a opção “Texto parcial”. Esta opção é utilizada caso você tenha planos de publicar seu trabalho em periódicos científicos ou em formato de livro, por exemplo e fará com que apenas as páginas pré-textuais, introdução, considerações e referências sejam disponibilizadas. Se optar por disponibilizar o texto completo de seu trabalho imediatamente selecione no campo “Data para a disponibilização do texto completo” a opção “Não se aplica (texto completo)”. Isso fará com que seu trabalho seja disponibilizado na íntegra no Repositório Institucional UNESP. Por favor, corrija esta informação realizando uma nova submissão. Agradecemos a compreensão. on 2016-01-15T19:12:27Z (GMT) / Submitted by DANIELE ALESSANDRA REGHINI GAZETTA null (daniellygaze@hotmail.com) on 2016-01-16T16:43:56Z No. of bitstreams: 2 diss-daniele.pdf: 783553 bytes, checksum: e593f1ebb872fff02a080d05283744d5 (MD5) daniele-dissert.pdf: 585710 bytes, checksum: 222237614b39411bc9b9a3e82ad6ab17 (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2016-01-18T16:33:44Z (GMT) No. of bitstreams: 1 gazetta_dar_me_sjrp.pdf: 783553 bytes, checksum: e593f1ebb872fff02a080d05283744d5 (MD5) / Made available in DSpace on 2016-01-18T16:33:44Z (GMT). No. of bitstreams: 1 gazetta_dar_me_sjrp.pdf: 783553 bytes, checksum: e593f1ebb872fff02a080d05283744d5 (MD5) Previous issue date: 2016-01-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O principal resultado dessa dissertação é o Teorema de Poincaré-Bendixson para campos de vetores planares suaves por partes, que nos diz quais são os tipos de conjuntos limite. Estudaremos também detalhes a respeito dos conceitos de conjuntos minimais e caóticos em campos de vetores planares suaves por partes. / The main result of this work is the Poincaré - Bendixson Theorem for planar piecewise smooth vector fields, which tell us what kind of limit sets arise in this context. We will also study details about the concepts of minimal and chaotic sets in planar piecewise smooth vector fields.