Matemática

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http://repositorio.ufes.br/handle/10/17495

Programa de Pós-Graduação em Matemática

Centro: CCE
Telefone: (27) 4009 2474
URL do programa: http://www.matematica.ufes.br/pos-graduacao/PPGMAT

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Agora exibindo 1 - 20 de 37

A conjectura de Euler sobre somas de potências quárticas de números inteiros
(Universidade Federal do Espírito Santo, 2017-07-11) Lopes, Gislayni Telles Vieira Santana; Oliveira, José Gilvan de; Passamani, Apoenã Passos; Kaygorodov, Ivan; Conte, Luciane Quoos
In 1772, Leonard Euler conjectured that the sum of n powers of positive integers of a given exponent n would also be such a power. However, if the number of powers in this sum is less than the exponent, then such sum could not result in an exponent power n. In the present work we will focus on the n = 4 case of the Euler’s conjecture. In a first approach, we will present a counterexample to the conjecture, that is, we will display positive whole solution for the diophantine equation a 4 +b 4 +c 4 = e 4 , which is equivalent to verify that the set of rational points of the surface S1 : r 4 + s 4 + t 4 = 1 is not empty. We will use the theory of elliptic curves and concepts of Number Theory, such as quadratic reciprocity and Legendre’s theorem, in the construction of a method to obtain the counterexample. In a second approach, we will use the group structure of an elliptic curve to show that there is an infinity of positive integer solutions for the above Diophantine equation if we add a quartic power of an integer in that sum.
A estrutura de semigrupos numéricos esparsos
(Universidade Federal do Espírito Santo, 2017-01-01) Burock, Katherine Pereira; Oliveira, José Gilvan de; Tizziotti, Guilherme Chaud; Contiero, André Luís; Bayer, Valmecir Antonio dos Santos
Sparse semigroups will be studied in this dissertation by analyzing their classifications and properties, such as their upper limits for the genus, the interaction between the single and double leaps, the influence of the genus on the leaps, the influence of the parity of the Frobenius number and also the classification of limit sparse semigroups. At the end we will give an introduction to ?-sparse semigroups by analyzing their structure and trying to extend some notions and properties as a natural generalization of the sparse semigroups. For this we will review other published works on the subject, where the main reference used was [1] “On the structure of numerical sparse semigroups and applications”
Acoplamentos: uma primeira visão e algumas aplicações
(Universidade Federal do Espírito Santo, 2016-06-24) Oliveira, Weverthon Lobo de; Valentim, Fábio Júlio da Silva; Gonçalves Junior, Etereldes; Romero, Freddy Rolando Hernandez
We will cover the Couplings Theory showing applications in Probability and analysis of problems, more specifically, exhibit applications in two contexts, which are the Markov Chain and Optmal Transport Problem. Chapter 1 presents some preliminary ideas which serve as a base for understanding this work, we will cover Probability notions, Markov Chains, Topology, Continuous Functions and semicontinuous and Convex Analysis. In the next two chapters will be presented the main contexts with their respective applications, more precisely, reserve the Chapter 2 to display Couplings and some of its applications in Markov chains, highlighting the calculation of mixing time of a Markov Chain and the proof of Theorem of Convergence via couplings. Chapter 3 will discuss the Transportation Problem Great, which can be divide in two problems, the Monge problem and Kantorovich problem. the difference will be presented between the two problems, conditions for solution of existence and the relationship between the two problems and finally present an suggestion algorithm that solves Kantorovich problem and demonstrate the isoperimetric inequality via Monge problem.
Álgebra diferencial e equações diferenciais polinomiais
(Universidade Federal do Espírito Santo, 2015-01-01) Silva, Flávio da; Bayer, Valmecir Antônio dos Santos; Merlo, Leandro Colau; Alves, Magno Branco
In this dissertation we study systems of parametric differential polynomial equations. The main result is the determination of an explicit expression for an implicit such equation.
Alguns teoremas limites para sequências de variáveis aleatórias
(Universidade Federal do Espírito Santo, 2014-10-16) Waiandt, Euclésio Rangel; Valentim, Fábio Júlio da Silva; Demuner, Daniela Paula; Gonçalves, Ana Patricia Carvalho
The Central Limit Theorem and the Law of Large Numbers are among the most important results of probability theory. The first one seeks conditions under which v????-E???? ?? ???????? converges in distribution to the normal distribution with parameters 0 and 1, when ?? tends to infinity, where ???? is the sum of ?? independent random variables. At the same time, the second gives conditions such that ????-E???? ?? converges to zero, or equivalently, that ???? ?? converges to the expectation of the random variables, if they are identically distributed. In both cases, the sequences discussed are of the type ????+???? ???? , where ???? > 0 and ???? are real constants. Characterizing the possible limits of such sequences is one of the goals of this dissertation, as they not only converge to a degenerated random variable or a random variable with normal distribution, as the Law of Large Numbers and the Central Limit Theorem, respectively. Thus, we are naturally led to the study of infinitely divisible and stable distributions and their limits theorems. This becomes the main objective of this dissertation. In order to prove the theorems, the method of Lyapunov is applied as the main strategy, which analyzes the convergence of the sequence of characteristic functions related to the random variables. So we carry out a detailed approach of such functions in this research.
Cadeias de Markov: tempo de mistura, cuttoff e redes
(Universidade Federal do Espírito Santo, 2016-02-19) Carneiro, Filipe Ribeiro; Valentim, Fábio Júlio da Silva; Romero, Freddy Rolando Hernandez; Costalonga, João Paulo
This work deals with Markov chains in discrete time and finite state space. We treat the convergence of these objects, and define the total variation distance and studied some of their properties , as well as ways of estimating the mixing time, for example, using the eigenvalues of the transition matrix. Present is the one by one relationship between reversible Markov chains and graphs, and how the network theory can help in Markov Chains context. We also define and show some results concerning the so called Cutoff phenomenon, concluding by exhibiting a counter-example due to Aldous.
Caracterização geométrica do espaço moduli de conexões ASD do fibrado de Hopf quatérnio
(Universidade Federal do Espírito Santo, 2017-03-06) Maroja, Aaron Aragon; Câmara, Leonardo Meireles; Bursztyn, Henrique; Batoréo, Marta Jakubowicz
In the early 80’s, C.C. Taubes and K. Uhlenbeck provided the analytical foundations so that the solutions of the Yang-Mills equations, called instantons, would have a geometrical use, yet to be found in the same period. Simon Donaldson has built then a theory based on certain aspects of these solutions over 4-dimensional, oriented, closed, differentiable manifolds. In this context, one can isolate a class of connections, called anti-self-dual, that necessarily sastify the Yang-Mills equation. The collection of all such, modulo a natural equivalence relation, namely gauge equivalence, is called the moduli space M of the bundle and its study has led to astonishing insights into the structure of smooth 4- manifolds. This work is set to study in detail a particular example of Donaldson’s Theory on the Hopf bundle over the 4-dimensional manifold ?? 4 . We arrive at the BPST instantons of such bundle via the Cartan canonical 1-form on ????(2). Once these are in hand, we use the conformal invariance of anti-self-dual equations to write down a 5-parameter family of such connections. By making use of a theorem of Atiyah, Hitchin and Singer, we assert that every element of the moduli space M is uniquely represented by a connection in this family. From this we obtain a concrete realization of M as the open unit ball in R 5 . In particular, M is a 5-dimensional manifold with a natural compactification whose boundary is a copy of the base space ?? 4 .
Classificação de estruturas de Nambu lineares e p-formas singulares
(Universidade Federal do Espírito Santo, 2012-08-13) Almeida, Carla Rodrigues; Alves, Magno Branco; Câmara, Leonardo Meireles; Bursztyn, Henrique; Corrêa Júnior, Maurício Barros
The aim of this work is to study the foliations that arise from Nambu structures and present the relationship between differential forms and some of this structures. More specifically, to make a study of the Poisson geometry and of singular foliations, emphasiz-ing the case of the simplectic foliation that arises from the Poisson structure and then, to present the Nambu geometry, studying the case of the foliations that arise from the this structures of order grater than or equal to three. In this particular case, we shall show how this Nambu structures are related with differential formas and, by this relationship, classify linear Nambu structure through a result of classification of integrable differential p-forms
Conexões afins e a teoria de Cartan-Einstein
(Universidade Federal do Espírito Santo, 2016-07-12) Xavier, Roberta Meschese; Câmara, Leonardo Meireles; Macarini, Leonardo Magalhães; Vieira, Matheus Brioschi Herkenhoff
The Cartan-Einstein theory of gravitation is a modified version of the General Theory of Relativity. While Einstein’s theory was developed according to the hypothesis that the relativity of space-time has zero torsion, Cartan allows torsion and relate it to the angular momentum of the matter several years before the discovery of the spin of the electron. Cartan’s articles, in particular Sur les variétés the affine connexion et la théorie de la Généralisée relativité, which is the basis of this work, contains important new mathematical ideas that have influenced the development of differential geometry and, in particular, led to the general theory of affine connections. Essentially these are geometrical objects on a differentiable manifolds that connect nearby tangent spaces. In this dissertation we study the invariance of the laws of classical and relativistic mechanics in continuous media and the geometry of space-time from the standpoint of affine connections.
Conjecturas em anéis de grupo
(Universidade Federal do Espírito Santo, 2018-03-07) Costa, Vagner Pereira; Fehlberg Junior, Renato; Serdà, Javier Sánchez; Silva, Thiago Filipe da
Group rings have a very rich algebraic structure, since to explore it we must resort to techniques other than group theory and ring theory; we must also resort to the theory of algebraic numbers, the representation of groups and algebras and other algebraic theories. Among the subjects of interest in group rings, we highlight some conjectures that will be the objects of study of the present dissertation: the isomorphism problem, the normalizer problem and the Zassenhaus conjectures. On the isomorphism problem and the normalizer problem, we will prove its validity in some particular cases and it will be presented the known counterexamples. On the Zassenhaus conjectures, we will enunciate and present for which group classes they were proved. We will show how these conjectures relate to the isomorphism problem.
Cotas superiores para o número de pontos racionais e aplicações às torres de corpos de funções
(Universidade Federal do Espírito Santo, 2010-08-19) Silva, Thiago Filipe da; Oliveira, José Gilvan de; Noseda, Francesco; Conte, Luciane Quoos
Curvas nodais maximais via curvas de Fermat
(Universidade Federal do Espírito Santo, 2009-01-01) Profilo, Stanley; Bayer, Valmecir Antonio dos Santos; Oliveira, José Gilvan de; Fantin, Silas
We study the rational projective nodal plane curves in the projective plane P2(C) by using the Fermat curve Fn : Xn+Y n+Zn = 0. We deal with the theory of dual curves in the projective plane and a special type of group action of Zn x Zn on the Fermat curve and its dual to construct, for any positive integer n maior ou igual a 3, a rational nodal plane curve of degree equal to n -1. A rational nodal plane curve is a projective rational plane curve (that is, a genus zero curve) that presents as singularities only nodal points, that is, singularities of multiplicity two with distinct tangents. The basic reference is the paper "On Fermat Curves and Maximal Nodal Curves"by Matsuo OKA published in Michigan Math. Journal, v.53. in 2005.
Decomposição de Helmholtz-Hodge via funções de Green
(Universidade Federal do Espírito Santo, 2018-10-11) Cordeiro, José Eduardo; Carmo, Fabiano Petronetto do; Paiva Neto, Afonso; Gonçalves Junior, Etereldes
The Helmholtz-Hodge Decomposition (HHD) describes a vector field as the sum of an incompressible, an irrotational, and a harmonic vector field. Unfortunately, for bounded domains, the HHD is not uniquely defined, traditionally, boundary conditions are imposed to obtain a unique solution, but this imposition may not give a compatible decomposition. This work exposes the natural HHD, which is defined without assuming boundary conditions a priori. Using Green’s functions on an infinite extension of the vector field combined with an influence analysis of the components, it’s possible to generate a unique decomposition without assuming boundary conditions. As a result, it enables a reliable analysis without problems generated by boundary conditions.
Existência de solução de energia mínima para uma equação de Schrödinger não linear
(Universidade Federal do Espírito Santo, 2012-03-06) Rocha, Karlo Fernandes; Xavier, Magda Soares; Silva, João Pablo Pinheiro da; Furtado, Marcelo Fernandes
In this work we study the existence of solution of a quasilinear Schr¨odinger equation in R N , demonstrated by Ruiz and Siciliano. By working in an appropriated functions space, by using a variational identity demonstrated by Pucci and Serrin, a set M containing all nontrivial solutions of the equation is obtained. By using a concentrationcompactness result due to Lions, it is possible to prove that the infimum of the functional associated with the equation, restricted to the set M, is achieved at some u which is a positive ground state solution
Existência de solução para uma equação de Schrodinger quasilinear
(Universidade Federal do Espírito Santo, 2010-11-26) Ribeiro, Maico Felipe Silva; Xavier, Magda Soares; Silva, Elves Alves de Barros e; Furtado, Marcelo Fernandes
In this paper we study the existence of solution of a quasilinear stationary Schrodinger equation in the autonomous and nonautonomous cases. These results were demonstrated by Colin and Jeanjean. Applying a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is well defined in the usual Sobolev space H1(RN).The existence of solution for the autonomous case is obtained as a consequence of a result due to Berestycki and Lions. In the nonautonomous case, we show that the associated functional satisfies the mountain pass geometric hypotheses. Using a version of Mountain Pass Theorem without the compactness condition, we obtain a Cerami sequence in the minimax level weakly convergent to a solution v0. In the proof that v0 is nontrivial, the main tool is a concentration-compactness result due to Lions
Formas modulares e o problema dos números congruentes
(Universidade Federal do Espírito Santo, 2015-10-29) Reis, Alexandre Silva dos; Oliveira, José Gilvan de; Conte, Luciane Quoos; Bayer, Valmecir Antonio dos Santos
Complex lattices, complex tori and elliptic curves are objects that although having different structures and nature, are equivalent. It is possible by means of a complex lattice to obtain a complex torus and hence, to obtain an elliptic curve; and that “path”’ can also be done in reverse. This connection will be the main object of study in this work, which will also address a careful manner some issues related to it, such as the special linear group, modular forms and modular curves. Finally, as an application of the concepts and tools studied, the congruent numbers problem is considered. This problem besides being closely related to elliptic curves, has a relationship with the famous Birch and Swinnerton-Dyer conjecture, one of the Millennium Problems.
Método dos elementos finitos através da análise isogeométrica: uma introdução
(Universidade Federal do Espírito Santo, 2016-06-24) Gomes Filho, Hélio; Gonçalves Junior, Etereldes; Carmo, Fabiano Petronetto do; Sousa, Fabrício Simeoni de
The Isogeometric Analysis is a method that combine the Finite Elements Method with NonUniform Rational Basis Spline (NURBS). The NURBS is used to describe the geometry with great flexibility, and it can also work as basis functions. The main concepts of NURBS are presented in this study, and how to apply the method to solve ordinary and partial differential equations. A comparison between the isogeometric analysis and the classical finite elements method is showed to contrast the error behavior in both methods, and the advantages of describe exactly a domain. The elasticity problem in two-dimensional and three-dimensional model are performed as application of the isogeometric analysis, and as exemple it was developed a model of a shell based on a real structure.
Métodos de projeção multidimensional
(Universidade Federal do Espírito Santo, 2013-05-10) Dal Col Júnior, Alcebíades; Carmo, Fabiano Petronetto do; Gonçalves Junior, Etereldes; Nonato, Luís Gustavo
The problem we are interested in solving comes from a area of knowledge called data visualization. In our studies, groups of objects are analyzed to produce the input data of our problem, each object is represented by attributes, have so a list of attributes for each object. The idea is to represent, through these lists of attributes, objects through points in R 2 so that we can conduct a group of objects. As we said each object is represented by a list of attributes, this may be interpreted as a point of a multidimensional space. For example, if they are considered m valued attributes for all objects can interpret them as points in a space of dimension m, or m-dimensional. But we want to produce a visualization of the data on the computer screen through points in R 2 , it was then performs a process known as multidimensional projection, that is obtaining points in a low dimensional space representing points in a high dimensional space preserving neighborhood relations as much as possible. Various methods of multidimensional projection are found in the literature. In this work, study and implement methods NNP, Force, LSP, PLP and LAMP. These methods deal with the problem in different ways: geometrically; linear systems, in particular, laplacian systems; and mappings related orthogonal. The lists of attributes associated with the groups of objects are called dataset. Two sets of data in this paper present trends grouping known a priori, therefore were used to give credibility to our implementations of the methods. Two other data set are studied and these were not provided with such feature, the methods of multidimensional projection are then used to define trends grouping for these two data sets.
Movimentos recorrentes e quase periódicos em sistemas semidinâmicos impulsivos
(Universidade Federal do Espírito Santo, 2017-05-29) Coswosck, Vinicius Bassi; Demuner, Daniela Paula; Bonotto, Everaldo de Mello; Valentim, Fábio Júlio da Silva
In this work, we study the theory of impulsive semidynamic systems theory. These systems describe the evolution processes subject to quickly variations of state, and can be considered instantaneous. In the first part of this work is introduced the theory of semidynamic systems. These are not subject to variations of state because they are continuous. In the second part are presented the impulsive semi-dynamic systems, a generalization of the theory of impulsive semi-dynamic systems. To study the almost periodic recurrent movements of the impulsive semi-dynamic systems, in the third and fourth part, concepts of minimal sets, asymptotic points and Zhukoviskij stability are studied.
Multiplicidade e concentração de soluções positivas para uma equação elíptica quasilinear
(Universidade Federal do Espírito Santo, 2012-03-07) Oliveira Junior, José Carlos de; Xavier, Magda Soares; Silva, João Pablo Pinheiro da; Furtado, Marcelo Fernandes
In this work, we study results on existence and concentration of positive solutions for a Schrödinger equation in R N involving the p-laplacian operator with 2 = p < N, a subcritical nonlinearity, a positive parameter ? and a potencial a(x) satisfying some hypotheses. Such problem was rst studied by Bartsch and Wang [5] in the case of laplacian operator (p = 2). We present versions of the results of [5] in the case of the p-laplacian, which were demonstrated by Furtado [17, 18].

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