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 56

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”
A saturação Lipschitz de uma álgebra
(Universidade Federal do Espírito Santo, 2022-12-19) Schultz Netto, Guilherme; Silva, Thiago Filipe da; https://orcid.org/0000-0002-3152-0987; http://lattes.cnpq.br/5049713215002090; Dalbelo, Thais Maria; Hernandes, Marcelo Escudeiro
The first description of the Lipschitz saturation of an algebra was given by (PHAM; TEISSIER, 1969) and later expanded upon by various mathematicians. Here we will recap the work done by Lipman in (LIPMAN, 1975), where he approached in a purely algebraic way (in the sense of not resorting to geometric tools) the set (which we will show to be, in fact, a ring), displaying seven initial properties and then expanding the concept to, among other things, compare such a structure with the saturation defined by Zariski, for example. Therefore, let’s review the prerequisites and demonstrate the compatibility of Lipschitz saturation with inclusions, direct limits and products, functorality, descents by plane algebras and contractions.
A Transformada de Fourier para o Laplaciano Generalizado
(Universidade Federal do Espírito Santo, 2024-03-05) Ramos Junior, Jomar Ferreira; Valentim, Fábio Júlio da Silva; https://orcid.org/0000-0003-2405-7696; http://lattes.cnpq.br/8745134398831488; https://orcid.org/0009-0006-1032-5169; http://lattes.cnpq.br/1800635453022041; Aranda, José Miguel Mendoza; https://orcid.org/; http://lattes.cnpq.br/8615067875072268; Silva, Jean Carlos da; https://orcid.org/; http://lattes.cnpq.br/9490078990099931
This academic dissertation aims primarily to contribute to the enhancement of understanding of the Fourier Theory applied to the generalized Laplacian. The proposed methodology involves the construction of an orthonormal basis of eigenfunctions for the operator, based on the appropriate choice of Green’s functions. The central problem consists of finding the solution u(x) that satisfies certain boundary conditions for the equation Lu = f, using a series representation of the eigenfunctions of the operator L. The dissertation addresses fundamental aspects such as the definition of the domain of the generalized Laplacian, the analysis of Green’s functions and their applications in solving partial differential equations, as well as transformations for the generalized Laplacian. The interest in consolidating the Fourier Theory for the generalized Laplacian aims to provide a deeper understanding of the properties of this operator and its relation to Fourier Theory, establishing a foundation for future research, including more complex cases such as the differential operator in reverse order. This work represents a significant contribution to the understanding of the theory of the generalized Laplacian and its connections with Fourier Theory.
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.
Atratores Globais Pré-Compactos Para Sistemas Dinâmicos Impulsivos
(Universidade Federal do Espírito Santo, 2021-10-04) Batista, Ester Félix; Demuner, Daniela Paula; https://orcid.org/0009-0009-8807-3784; http://lattes.cnpq.br/6010222845541176; https://orcid.org/; http://lattes.cnpq.br/6869829314147181; Ferreira, Jaqueline da Costa; https://orcid.org/; http://lattes.cnpq.br/6307931438193441; Jimenez, Manuel Francisco Zuloeta; https://orcid.org/0000-0002-5771-4448; http://lattes.cnpq.br/6115398975075240
The present research studies the existence of global attractors for impulsive dynamical systems. It introduces initially the basic theory of semigroups, which are continuous evolutionary processes, and their properties searching for conditions to ensure the global attractor existence. It presents the impulsive dynamical systems that studies the behavior of the evolution process that undergo state variations of short duration, which can be considered instantaneous. It exhibits the definition of precompact global attractor for impulsive dynamical systems and presents results that ensure the upper and lower semicontinuity of global attractors for a family of impulsive dynamical systems.
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 de hiperfícies com curvatura normal nula em Sn xR e Hn x R
(Universidade Federal do Espírito Santo, 2023-01-24) Ramos, Gustavo Panin; Passamani, Apoena Passos; https://orcid.org/0000-0003-1639-430X; http://lattes.cnpq.br/4845419392362758; https://orcid.org/0009-0001-7736-0819; http://lattes.cnpq.br/4128628460165034; Onnis, Irene Ignazia; https://orcid.org/0000-0003-0045-2173; http://lattes.cnpq.br/9456144794670433; Batoreo, Marta Jakubowicz; https://orcid.org/0009-0007-8085-4776; http://lattes.cnpq.br/1461528961848898
In this work we expose some results in Riemannian geometry and an foliation theory, proving the Frobenius Theorem. However our focus is the description of all hypersurfaces of the spaces S n × R and Hn × R when regarded as submanifolds with codimension two of the spaces R n+2 and Hn+2 that have flat normal bundle curvature. This result was presented in the paper On a class of hypersurfaces in S n × R and Hn × R whose author is Ruy Tojeiro. Our goal is to detail the proofs presented therein.
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 Álgebras Antiassociativas
(Universidade Federal do Espírito Santo, 2021-02-18) Kuster, Crislaine; Fehlberg Júnior, Renato; https://orcid.org/0000-0002-1718-5027; http://lattes.cnpq.br/4297634330123270; https://orcid.org/0009000699526142; http://lattes.cnpq.br/7975527282917016; Macedo, Tiago Rodrigues; https://orcid.org/0000-0001-8340-8711; http://lattes.cnpq.br/6725002674545546; Locateli, Ana Claudia; https://orcid.org/0000-0003-0840-626X; http://lattes.cnpq.br/9432637928506871; Kaygorodov, Ivan; https://orcid.org/0000000320849215; http://lattes.cnpq.br/0810152953225890
The purpose of this work is to classify pure antiassociative algebras, that is, that do not satisfy the identity xyz=0, of dimensions 4 and 5 over C.
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.
Descrição de sistemas impulsivos e um estudo de propriedades recursivas
(Universidade Federal do Espírito Santo, 2022-09-01) Santos, Lucas Venancio da Silva; Souto, Ginnara Mexia; https://orcid.org/; http://lattes.cnpq.br/2095148182435653; https://orcid.org/; http://lattes.cnpq.br/; Demuner, Daniela Paula; https://orcid.org/; http://lattes.cnpq.br/6010222845541176; Ferreira, Jaqueline da Costa; https://orcid.org/; http://lattes.cnpq.br/6307931438193441; Jimenez, Manuel Francisco Zuloeta; https://orcid.org/; http://lattes.cnpq.br/
abstract
Dinâmica Hamiltoniana Simplética: Capacidade de Hofer-Zehnder
(Universidade Federal do Espírito Santo, 2019-09-20) Ferreira, Brayan Cuzzuol; Batoreo, Marta Jakubowicz; https://orcid.org/; http://lattes.cnpq.br/1461528961848898; https://orcid.org/; http://lattes.cnpq.br/; Passamani, Apoena Passos; https://orcid.org/; http://lattes.cnpq.br/4845419392362758; Ramos, Vinicius Gripp Barros; https://orcid.org/; http://lattes.cnpq.br/
This dissertation presents some fundamental theorems on Symplectic Geometry including Darboux's theorem and Gromov's nonsqueezing theorem. Furthermore, we study the existence of periodic orbits of hamiltonians systems. The Hofer-Zehnder capacity was the m

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