PEREIRA, Rodrigo Mendes.Geometria métrica e topologia de superfícies algebricamente parametrizadas. 2016. 68 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-06-09T17:57:35Z No. of bitstreams: 1 2016_tese_rmpereira.pdf: 663687 bytes, checksum: 396b9fef564fece1bd791fcf3677b5e5 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Oi Andrea, Favor informar ao aluno para fazer as seguintes correções: No Resumo e Abstract não tem recuo. Os termos das palavras-chave e abstract são separadas e finalizadas por ponto. nAS Referências, o sobrenome dos autores são em caixa alta. Ex: WHITNEY, Hassler. Dois ou três autores: SOBRENOME, Nome; SOBRENOME, Nome Rocilda on 2017-06-13T12:22:02Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-06-19T17:03:38Z No. of bitstreams: 1 2016_tese_rmpereira.pdf: 663687 bytes, checksum: 396b9fef564fece1bd791fcf3677b5e5 (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Oi Andrea, Estou devolvendo novamente o arquivo do Rodrigo, pois ele não fez as correções. Elton disse que ele veio ontem e foi dito o quais eram. Estou enviando as mesmas. Favor repassar pra ele. Caso ele não tenha entendido são essas as correções: Sumário a Abstracts não tem o recuo. As palavras-chave e keywords são separados e finalizados por ponto. E Referências : Os sobrenomes dos autores são em caixa alta: BALLETSEROS, J .J .Nuno. BIRBRAIR, L. ; FERNANDES, A.; GRANDJEAN, V. SE FOR MAIS DE 3 AUTORES INDICA-SE APENAS O PRIMEIRO ACRESCENTANDO A EXPRESSÃO ET AL. Birbrair, L ET AL. on 2017-06-21T14:06:19Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-06-22T12:40:49Z No. of bitstreams: 1 2016_tese_rmpereira.pdf: 661684 bytes, checksum: c8673d7c6d1002a51d5c45584e60f6b7 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-06-22T14:20:44Z (GMT) No. of bitstreams: 1 2016_tese_rmpereira.pdf: 661684 bytes, checksum: c8673d7c6d1002a51d5c45584e60f6b7 (MD5) / Made available in DSpace on 2017-06-22T14:20:44Z (GMT). No. of bitstreams: 1 2016_tese_rmpereira.pdf: 661684 bytes, checksum: c8673d7c6d1002a51d5c45584e60f6b7 (MD5) Previous issue date: 2016-07-20 / In this work, we study the singularities of the surfaces obtained as an image of a plane by an algebraic or analytic application in the spaces R3 and R4. We study the topological and metric properties of the knots that are obtained as a link of this surfaces. Normal embedding criterion are obtained for the surfaces and, in addition, the connections between the topology of the knot and normal embedding are investigated. We will also give a description of the tangent cones of these surfaces at the singular points. / Nesse trabalho, estudamos as singularidades das superfícies obtidas como imagem de um plano por uma aplicação algébrica ou analítica no espaços R3 e R4. Estudamos as propriedades topológicas e métricas dos nós que são obtidos como link de tais superfícies. É obtido critérios de mergulho normal para a superfície-imagem e, além disso, é investigado as ligações entre a topologia do nó e o mergulho normal. Faremos ainda uma descrição dos cones tangentes destas superfícies nos pontos singulares.

Neste trabalho determinamos relações entre a obstrução de Euler de uma função analítica com singularidade isolada f e o número de Milnor de f definido por Bruce e Roberts para funções definidas em espaços singulares. Apresentamos também uma generalização da obstrução de Euler de uma função analítica com singularidade isolada para o caso de uma aplicação \'f : (V, 0) seta (\'C POT. k\', 0) onde (V, 0) é o germe de uma variedade analítica complexa, equidimensional de dimensão \' n > OU = k\' , e uma fórmula para calcular a obstrução de Euler de k-referenciais, em termos da obstrução de Euler de f / In this work we determine relations between the local Euler obstruction of an analytic function singular at the origin to the case of a analytic map \'f : (V, 0) seta (\'C POT. k, 0\'), where (V, 0) is the germ of a complex analytic variety, equidimensional of dimension \' n > OU = k\', and a formula which computes the local Euler obstruction, defined for k-frames, in the local Euler obstruction of f

In the present dissertation, we characterize the range of the attenuated Radon transform of zero, one, and two tensor fields, supported in strictly convex set. The approach is based on a Hilbert transform associated with A-analytic functions of A. Bukhgeim. We first present new necessary and sufficient conditions for a function to be in the range of the attenuated Radon transform of a sufficiently smooth function supported in the convex set. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of A. Bukhgeim. We then uses the range characterization of the Radon transform of functions to characterize the range of the attenuated Radon transform of vector fields as they appear in the medical diagnostic techniques of Doppler tomography. As an application we determine necessary and sufficient conditions for the Doppler and X-ray data to be mistaken for each other. We also characterize the range of real symmetric second order tensor field using the range characterization of the Radon transform of zero tensor field.

Nous nous intéressons à l'étude des singularités réelles à l'aide d'arguments provenant de l'intégration motivique. Une telle démarche a été initiée par S. Koike et A. Parusiński puis poursuivie par G. Fichou. Afin de donner une classification des singularités réelles, T.-C. Kuo a défini la notion d'équivalence blow-analytique. Il s'agit d'une relation d'équivalence pour les germes analytiques réels n'admettant pas de module continu pour les singularités isolées. Cette notion est étroitement liée à la notion d'applications analytiques par arcs définie par K. Kurdyka. Il est donc naturel d'adapter des arguments provenant de l'intégration motivique pour l'étude de l'équivalence blow-analytique. La difficulté réside désormais dans le fait de trouver des méthodes permettant de montrer que deux germes sont équivalents et de construire des invariants permettant de distinguer deux germes qui ne sont pas dans la même classe. Nous travaillons avec une variante plus algébrique de cette notion, l'équivalence blow-Nash introduite par G. Fichou. La première partie de la thèse consiste en un théorème d'inversion donnant des conditions pour que l'inverse d'un homéomorphisme blow-Nash soit encore blow-Nash. L'intérêt d'un tel énoncé est que de telles applications apparaissent dans la définition de l'équivalence blow-Nash. La seconde partie est consacrée à l'étude d'une nouvelle fonction zêta motivique. Il s'agit d'associer à un germe analytique une série formelle. Cette fonction zêta motivique généralise les fonctions zêta de Koike-Parusiński et de Fichou et admet une formule de convolution. Il s'agit d'un invariant pour l'équivalence blow-Nash. / The main purpose of this thesis is to study real singularities using arguments from motivic integration as initiated by S. Koike and A. Parusiński and then continued by G. Fichou. In order to classify real singularities, T.-C. Kuo introduced the blow-analytic equivalence which is an equivalence relation on real analytic germs without moduli for isolated singularities. This notion is closely related to the notion of arc-analytic maps introduced by K. Kurdyka, thus it is natural to adapt arguments from motivic integration to the study of the relation. The difficulty lies in finding efficient ways to prove that two germs are equivalent and in constructing invariants that distinguish germs which are not in the same class. We focus on the blow-Nash equivalence, a more algebraic notion which was introduced by G. Fichou. The first part of this thesis consists in an inverse theorem for blow-Nash maps. Under certain assumptions, this ensures that the inverse of a homeomorphism which is blow-Nash is also blow-Nash. Such maps are involved in the definition of the blow-Nash equivalence. In the second part, we associate a power series to an analytic germ, called the zeta function of the germ. This construction generalizes the zeta functions of Koike-Parusiński and Fichou. Furthermore, it admits a convolution formula while being an invariant for the blow-Nash equivalence.