FAPEMIG - Fundação de Amparo a Pesquisa do Estado de Minas Gerais / Neste trabalho, estudamos conceitos de geometria algébrica relacionados a teoria de códigos de Goppa algebricos geometricos (códigos AG). Vimos como o cálculo do semi- grupo de Weierstrass pode ser aplicado na obtencao dos parâmetros de certos cádigos AG. Em particular, calculamos o semigrupo de Weierstrass em dois pontos da curva Xq2r dada pela equacao afim yq + y = xq+1 sobre Fq2r, onde r e um inteiro positivo ímpar e q á uma potencia de um numero primo, e construímos um cádigo AG bipontual sobre Xq2r, cujos parâmetros relativos sao melhores que cádigos AG pontuais comparâveis tambem construídos sobre esta curva. A principal referencia deste trabalho foi [8]. / In this work we study basics concepts of the algebraic geometry related to Algebraic Geometric Goppa codes theory (AG codes). We have seen how the calculation of the Weierstrass semigroup can be applied in obtaining the parameters of certain AG codes. In particular, we calculated the Weierstrass semigroup at two points on the curve Xq2r defined by afim equation yq + y = xq +1 over Fq2r, where r is a positive odd integer and q is a prime power, and construct a two-point AG code over Xq2r whose relative parameters are better than comparable one-point AG code. The main reference of this work was [8]. / Dissertação (Mestrado)

Orientador: Fernando Eduardo Torres Orihuela / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T15:12:25Z (GMT). No. of bitstreams: 1 Castilho_TiagoNunes_M.pdf: 813127 bytes, checksum: 313e9951b003dcd0e0876813659d7050 (MD5) Previous issue date: 2008 / Resumo: Nesta dissertacao estudamos cotas para o numero de pontos racionais de curvas definidas sobre corpos finitos tendo como ponto de partida a teoria de Stohr-Voloch / Abstract: In this work we study upper bounds on the number of rational points of curves over finite fields by using the Stohr-Voloch theory / Mestrado / Algebra Comutativa, Geometria Algebrica / Mestre em Matemática

This thesis takes a closer look at three fundamental Classical Theorems in Real Analysis. First, for the Bolzano Weierstrass Theorem, we will be interested in constructing a convergent subsequence from a non-convergent bounded sequence. Such a subsequence is guaranteed to exist, but it is often not obvious what it is, e.g., if an = sin n. Next, the H¨older Inequality gives an upper bound, in terms of p ∈ [1,∞], for the the integral of the product of two functions. We will find the value of p that gives the best (smallest) upper-bound, focusing on the Beta and Gamma integrals. Finally, for the Weierstrass Polynomial Approximation, we will find the degree of the approximating polynomial for a variety of functions. We choose examples in which the approximating polynomial does far worse than the Taylor polynomial, but also work with continuous non-differentiable functions for which a Taylor expansion is impossible.

Submitted by Luanna Matias (lua_matias@yahoo.com.br) on 2015-02-05T10:44:34Z No. of bitstreams: 2 Tese - Diogo Gonçalves Dias - 2014.pdf.pdf: 490676 bytes, checksum: 3c0940e1fbec55f277f969c4751c5ea6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-02-05T11:02:53Z (GMT) No. of bitstreams: 2 Tese - Diogo Gonçalves Dias - 2014.pdf.pdf: 490676 bytes, checksum: 3c0940e1fbec55f277f969c4751c5ea6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-02-05T11:02:53Z (GMT). No. of bitstreams: 2 Tese - Diogo Gonçalves Dias - 2014.pdf.pdf: 490676 bytes, checksum: 3c0940e1fbec55f277f969c4751c5ea6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-09-29 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / We present hypersurfaces with prescribed normal Gauss map. These surfaces are obtained as the envelope of a sphere congruence where the other envelope is contained in a plane. We introduce classes of surfaces that generalize linear Weingarten surfaces, where the coefficients are functions that depend on the support function and the distance function from a fixed point (in short, DSGW-surfaces). The linear Weingarten surfaces, the Appell’s surfaces and the Tzitzeica’s surfaces are all DSGW-surfaces. From them we obtain new classes of DSGW-surfaces applying inversions and dilatations. For a special class of DSGW-surfaces, which is invariant under dilatations and inversions, we obtain a Weierstrass type representation (in short, EDSGW-surfaces). As application we classify the EDSGW-surfaces of rotation and present a 4-parameter family of complete cyclic EDSGW-surfaces with an isolated singularity and foliated by non-parallel planes. We generalized the EDSGW-surfaces for the case of hypersurfaces in Rn+1, n ≥ 2. We present a representation for these hypersurfaces in the case where the stereographic projection of the normal Gauss map N is given by the identity application. As an application, we will characterize the rotational examples. / Apresentamos parametrizações de hipersuperfícies com aplicação normal de Gauss prescrita. Estas parametrizações são obtidas como o envelope de uma congruência de esferas onde o outro envelope esta contido em um hiperplano. Introduzimos classes de superfícies que generalizam as superfícies de Weingarten linear, onde os coeficientes são funções que dependem da função suporte e da função distância a um ponto fixo (superfícies WGSD). Classes conhecidas destas superfícies são as superfícies de Weingarten linear, as superfícies de Appell e as superfícies de Tzitzéica. A partir delas obtemos novas classes de superfícies WGSD aplicando inversões e dilatações. Para uma classe especial de superfícies WGSD, que é invariante por dilatações e inversoes (superfícies WGSDE), obtemos uma representação tipo Weierstrass, dependendo de duas funções holomorfas. Como aplicação classificamos as superfícies WGSDE de rotação e apresentamos uma família a 4-parâmetros de superfícies WGSDE cíclicas completas com uma singularidade isolada e com planos de folheação não paralelos. Terminamos generalizando as superfícies WGSDE para o hipersuperfícies em Rn+1, n ≥ 2. Apresentaremos uma representação para estas hipersuperfícies no caso em que a projeção estereográfica da normal de Gauss N é dada pela aplicação identidade. Como aplicação, caracterizaremos os exemplos rotacionais.

Made available in DSpace on 2015-05-15T11:46:02Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 666060 bytes, checksum: 1ad661f6cc42df5f3ee67db9a939af86 (MD5) Previous issue date: 2011-07-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we obtain Weierstrass-type representations for immersed surfaces in Heisenberg space, endowed with a left-invariant metric. We will consider the Riemannian and Lorentzian case. We will define two complex functions (spinors) satisfying a linear Dirac-type equation, obtaining thus a representation for immersed surfaces with prescribed mean curvature. The same will enable us write a representation of minimal immersion in terms of a harmonic Gauss map. / Neste trabalho obtemos uma representações tipo Weierstrass para superfícies imersas no espaço de Heisenberg, dotado com uma métrica invariante à esquerda. Consideraremos os casos Riemanniano e Lorentziano. Definimos duas funções complexas (spinors), satisfazendo uma equação linear tipo Dirac que usamos para obter uma representação para superfícies imersas com curvatura média prescrita. A mesma possibilita escrever uma representação de imersões mínimas em termos de uma aplicação de Gauss harmônica.

Orientador: Fernando Eduardo Torres Orihuela / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T08:34:02Z (GMT). No. of bitstreams: 1 VillanuevaZevallos_JuanElmer_D.pdf: 1127069 bytes, checksum: 8ac303abd191b4c264038dcd1ce40be1 (MD5) Previous issue date: 2008 / Resumo: Os principais tópicos aqui considerados são do tipo aritmético. Introduzimos e estudamos semigrupos que generalizam os chamados semigrupos de Arf. Além de seu interesse particular, eles podem ser usados para esclarecer a estrutura de anéis de semigrupos no sentido de Lipman. Também calculamos os valores exatos dos pesos de semigrupos usando o número de lacunas pares. Isto está relacionado ao recobrimento duplo de curvas e tem interesse no estudo de moduli e constelação de curvas. / Abstract: The main topics considered here are of arithmetical type. We introduce and study semigroups that generalize the so-called Arf semigroups. Apart from being interesting by their own, they may be used to clarify the structure of semigroup rings in the sense of Lipman. We also compute the true value of the weights of semigroups by using the number of even gaps. This is related to double covering of curves and is useful to the study of moduli and constellation of curves. / Doutorado / Geometria Algebrica / Doutor em Matemática

In this paper we consider sequences of best approximation. We first examine the rho best approximation function and its applications, through an example in approximation theory and two new examples in calculating n-widths. We then further discuss approximation theory by examining a modern proof of Weierstrass's Theorem using Dirac sequences, and providing a new proof of Chebyshev's Equioscillation Theorem, inspired by the de La Vallee Poussin Theorem. Finally, we examine the limits of approximation theorem by looking at Bernstein Lethargy theorem, and a modern generalization to infinite-dimensional subspaces. We all note that smooth functions are bounded by Jackson's Inequalities, but see a newer proof that a single non-differentiable point can make functions again susceptible to lethargic rates of convergence.

We study two dimension estimates regarding linear series on algebraic curves. First, we generalize the classical Brill-Noether theorem to many cases where the Brill-Noether number is negative. Second, we extend results of Eisenbud, Harris, and Komeda on the existence of Weierstrass points with certain semigroups, by refining their dimension estimate in light of combinatorial considerations. Both results are proved by constructing chains of elliptic curves, joined at pairs of points differed by carefully chosen orders of torsion, and smoothing these chains. These arguments lead to several combinatorial problems of separate interest. / Mathematics

Thesis (Ph. D.)--Universiteit van Amsterdam, 1983. / Includes bibliographical references.

In this thesis we summarise several results in the literature which show the approximation capabilities of multilayer feedforward artificial neural networks. We show that multilayer feedforward artificial neural networks are capable of approximating continuous and measurable functions from Rn to R to any degree of accuracy under certain conditions. In particular making use of the Stone-Weierstrass and Hahn-Banach theorems, we show that a multilayer feedforward artificial neural network can approximate any continuous function to any degree of accuracy, by using either an arbitrary squashing function or any continuous sigmoidal function for activation. Making use of the Stone-Weirstrass Theorem again, we extend these approximation capabilities of multilayer feedforward artificial neural networks to the space of measurable functions under any probability measure.