Mestrado em Matemática (IME)
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Item Ação de automorfismos livres de pontos fixos(Universidade Federal de Goiás, 2016-05-13) Araujo, Daniel dos Santos; Oliveira, Ricardo Nunes de; http://lattes.cnpq.br/0563210461148997; Silva, Jhone Caldeira; http://lattes.cnpq.br/6848751340618892; Silva, Jhone Caldeira; http://lattes.cnpq.br/6848751340618892; Oliveira, Ricardo Nunes de; http://lattes.cnpq.br/0563210461148997; Acciarri, Cristina; Lima, Aline de SouzaIf a Zn-graded Lie ring L admits a fixed point free automorphism of order n, then L is soluble and the derived length of L is bounded in function only on n. In this work, we study some results about the derived length of the Zn-graded Lie rings and in the particular case that n = 6, we also study properties to the nilpotency class of the lower central series of L. For this, we introduce some basic results of Lie algebras theory and Lie rings, as well preliminary concepts of modules and tensor product. Finally, we study a Lie ring associated to a group once many problems in group theory can be treated by linear methods about Lie algebras and Lie rings.Item Um algoritmo para estimar a dimensão do segundo grupo de homologia de um grupo finitamente apresentado(Universidade Federal de Goiás, 2012-04-12) VIEIRA, Flavio Pinto; SERCONEK, Shirlei; http://lattes.cnpq.br/2718527331501182; ADORNO, Ticianne Proença Bueno; http://lattes.cnpq.br/7769491630579878The main goal of this work is to establish a primary bound for the dimension of the second homology group of a group G, with coefficients in a field k of characteristic p, H2(G;k), using the operating system GAP. It will be presented with several examples, where in some cases, will be calculated the exact dimensions and in other cases only an upper bound.Item Um algoritmo proximal com quase-distância(Universidade Federal de Goiás, 2015-02-25) Assunção Filho, Pedro Bonfim de; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Bento, Glaydston de Carvalho Bento; Cruz Neto, João Xavier da; Ferreira, Orizon PereiraIn this work, based in [1, 18], we study the convergence of method of proximal point (MPP) regularized by a quasi-distance, applied to an optimization problem. The objective function considered not is necessarily convex and satisfies the property of Kurdyka- Lojasiewicz around by their generalized critical points. More specifically, we will show that any limited sequence, generated from MPP, converge the a generalized critical point.Item Algoritmo proximal inexato tipo descida para otimização suave(Universidade Federal de Goiás, 2013-05-20) Godoi, Gean Henrique; Silva, Geci José Pereira da; http://lattes.cnpq.br/9174074436425246; Silva, Geci José Pereira da; Santos, Paulo Sérgio Marques dos; Ferreira, Orizon PereiraThe proximal method is a standard regularization approach in optimization. In this work we focus on a stopping rule of this algorithm, when smoothness is present, so that Newton-like method can be used to solve the subproblems. The basis for our stopping test is a "sufficient" decrease in the objective function where we establish the convergence of the algorithm obtained.Item Alguns problemas de soma-zero com peso(Universidade Federal de Goiás, 2015-03-02) Pereira, Nara Reges Faria de Paiva; Rodrigues, Paulo Henrique de Azevedo; http://lattes.cnpq.br/8910130626123426; Rodrigues, Paulo Henrique de Azevedo; Chagas, Sheila Campos; Chaves, Ana Paula de AraújoIn this work we study some results about Davenport’s constant for finite abelian groups. We also study some weighted zero-sum problems for especific cyclic groups and for this we prove some relations between the invariants sA, gA e hA for such groups.Item Altura e equidistribuição de pontos algébricos(Universidade Federal de Goiás, 2017-06-20) Santos, Jefferson Marques; Chaves, Ana Paula de Araújo; http://lattes.cnpq.br/2332073500640724; Chaves , Ana Paula de Araújo; Lopes , José Othon Dantas; Rodrigues , Paulo Henrique de Azevedo; Oliveira, Ricardo Nunes deThe concept of roots of a polynomial is quite simple but has several applications. This concept extends more generally to the case of "small" algebraic points sequences in a curve. This dissertation aims to estimate the size of algebraic numbers by means of Weil height. In addition to showing that they are distributed evenly around the unit circle, through Bilu Equidistribution Theorem.Item Análise semi-local do método de Gauss-Newton sob uma condição majorante(Universidade Federal de Goiás, 2014-12-18) Aguiar, Ademir Alves; Gonçalves, Max Leandro Nobre; http://lattes.cnpq.br/7841103869154032; Gonçalves, Max Leandro Nobre; Gonçalves, Douglas Soares; Melo, Jefferson Divino Gonçalves deIn this dissertation we present a semi-local convergence analysis for the Gauss-Newton method to solve a special class of systems of non-linear equations, under the hypothesis that the derivative of the non-linear operator satisfies a majorant condition. The proofs and conditions of convergence presented in this work are simplified by using a simple majorant condition. Another tool of demonstration that simplifies our study is to identify regions where the iteration of Gauss-Newton is “well-defined”. Moreover, special cases of the general theory are presented as applications.Item Aplicações da álgebra linear nas cadeias de Markov(Universidade Federal de Goiás, 2013-04-11) Silva, Carlos Eduardo Vitória da; Mota, Jesus Carlos da; http://lattes.cnpq.br/8457974658695539; Mota, Jesus Carlos da; Seimetz, Rui; Smith, Ole PeterThe theory of linear algebra and matrices and systems particularly are linear math topics that can be applied not only within mathematics itself, but also in various other areas of human knowledge, such as physics, chemistry, biology, all engineering, psychology, economy, transportation, administration, statistics and probability, etc... The Markov chains are used to solve certain problems in the theory of probability. Applications of Markov chains in these problems, depend directly on the theory of matrices and linear systems. In this work we use the techniques of Markov Chains to solve three problems of probability, in three distinct areas. One in genetics, other in psychology and the other in the area of mass transit in a transit system. All work is developed with the intention that a high school student can read and understand the solutions of three problems presented.Item Aplicações para o princípio de indução matemática(Universidade Federal de Goiás, 2014-09-26) Silva Júnior, Normando; Silva, Maxwell Lizete da; http://lattes.cnpq.br/9078818338290451; Silva, Maxwell Lizete da; http://lattes.cnpq.br/9078818338290451; Krindges, André; Lima, Lidiane dos Santos MonteiroThis paper sought to systematically present knowledge on the number theory in a clear and acessible way to a broader community than the mathematical academics, the principle of mathematical induction PMI, was the background and the main tool to all demonstrations so to permeate almost all the results and, in each section, at least one numerical example was given, thus making it easier to readers beginning their studies in math and always seeking to at least encourage the investigative feeling on all readers. The continuous fractions, subject commonly overlooked, yet with vast applications in Physics and Calculus, proved to be familiar with the Fibonacci numbers. Sequentially, two classic game problems were presented, The Hanoi Tower and the problem of the false coin, which could, from simple examples be generalized demonstrating solutions for the problems at any given natural number. Finally, the linear recurrences of second order and the higher order arithmetic progression were shown to have deep connections with the Fibonacci sequence, so then, these numbers became the main motivators for all the paper, that prizes for demonstrable results through PMI or related to this sequence of numbers, and always sought to strengthen the admiration of the dialogues with branches apparently so fixed that are the familiar through the appearance of the Fibonacci numbers in these topics.Item Uma apresentação policíclica para o multiplicador de Schur e o quadrado tensorial não abeliano de um grupo policíclico(Universidade Federal de Goiás, 2015-03-19) Silva, Jefferson dos Santos e; Dias, Ivonildes Ribeiro Martins; Silva, Jhone Caldeira; Silva, Jhone Caldeira; Dias, Ivonildes Ribeiro Martins; Oliveira, Ricardo Nunes de; Rocco, Noraí RomeuIn this work, based on [9], describes an effective method for computing a consistent polycyclic presentation for the nonanbeian tensor square G G of a group G given by a consistent polycyclic presentation.Item Bifurcações em sistemas dinâmicos suaves por partes(Universidade Federal de Goiás, 2015-03-06) Tsujii, Marcos; Tonon , Durval José; http://lattes.cnpq.br/3688981956532711; Tonon, Durval José; http://lattes.cnpq.br/3688981956532711; Mizukoshi, Marina Tuyako; Carvalho, Tiago deIn this work, we will study the dynamics in smooth vector elds, in vector elds near the boundary and in piecewise-smooth vector elds and each of their most popular types of bifurcations up to now.Item Campos vetoriais suaves por partes: modelos predador-presa(Universidade Federal de Goiás, 2015-03-06) Silva, Lucyjane de Almeida; Medrado, João Carlos da Rocha; Medrado, João Carlos da Rocha; Lima, Maurício Firmino Silva; Tonon, Durval JoséIn this work we study the global qualitative behavior of three predator-prey models. We analyze the existence of limit cycle and canard cycle and we investigate the kinds of bifurcation that can occur. In the first model, Gause predator-prey with a refuge, we analyze the effects of a prey refuge on the ecosystem qualitative behavior. Employing the carrying capacity of the prey population in the Gause Model with a refuge we obtain the second model, for which we analyze the effects of the carrying capacity and we compare the results. In the third model we consider the continuous threshold harvesting strategies ocurring when the predator density is above a certain threshold. We note that the model has a complex dynamics with multiple internal equilibria and different types of bifurcation.Item Uma caracterização da planaridade de uma família de funções binomiais sobre corpos finitos via curvas algébricas(Universidade Federal de Goiás, 2022-08-26) Chu, Daniel; Tenório, Wanderson; http://lattes.cnpq.br/6406888404650319; Tenório, Wanderson; Tizziotti, Guilherme Chaud; Cunha, Gregory DuranIn this work, we study a family of binomial functions given by $$ f_{a,b}(x)=ax^{2^{2^m}+1}+bx^{2^m+1}, \quad \mbox{with } \ a,b\in\mathbb{F}_{q^3}^\times, $$ over a finite field of characteristic 2. The aim consists to show that is possible to relate the planarity of the family above to a cubic plane projective curve $\mathcal{C}_{a,b}$. From this method, it is possible establish a characterization of the pairs $(a,b)\in(\mathbb{F}_{q^3}^\times)^2$ such that the function $f_{a,b}(x)$ is planar.Item Centros Persistentes(Universidade Federal de Goiás, 2010-03-05) ROCHA, Valdomiro; MEDRADO, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286The problem of destingnishing whether a monodromic critical point with imaginary eigenvalues of a family of a planar analitical vector field is a center or a focus was already solved by Lyapunov. This is the famous center-focus problem which was solved by calculating the so-called Lyapunov constants and see whether or not they are zero. We present a few ways to calculate them acording the approaches that they use: camputation of a Lyapunov function; use of normal forms; computation of the power of expansion of a solution in polar coordinates; use of the algebraic structure of Lyapunov constants; method of Lyapunov-Schmit and Melnikov functions. Despite all of the above the centerfocus problem for a simple family as the cube is resisting all attempts at solution. For this reason the centers, we propose to grade the in three levels in order to make the problem more feasible.Item Ciclos limite de grande amplitude em sistemas lineares definidos por partes(Universidade Federal de Goiás, 2023-01-10) Becatti, Fernanda dos Anjos Félix; Freitas, Bruno Rodrigues de; http://lattes.cnpq.br/4201351441514126; Freitas, Bruno Rodrigues de; Cristiano, Rony; Carvalho, Tiago deIn this paper we deal with a family of piecewise smooth planar linear systems with two zones, we study about the maximum number of limit cycles that can be obtained when studying the orbit at infinity. We start from a canonical form with 12 parameters and reduce the number of parameters to 5 for the analysis. We investigate questions related to stability, characterization of the orbits and other properties that can be obtained by studying the family of systems under consideration.Item Ciclos limite em sistemas lineares suaves por parte(Universidade Federal de Goiás, 2018-03-07) Silva, Ana Maria Alves da; Euzébio, Rodrigo Donizete; http://lattes.cnpq.br/9213320273714493; Buzzi, Claudio Aguinaldo; Tonon, Durval José; Euzébio, Rodrigo DonizeteIn this work we will study limit cycles in piecewise smooth linear systems. We begin studying the case where the separation curve is a polygonal and we give an example of a system having seven limit cycles. The existence of an arbitrary number of limit cycles is also proved for these systems, as well as an example of a system with 10 limit cycles. The existence of n limit cycles, n 2 N, is also studied through a perturbation in the separation curve. Finally, we study limit cycles in planar piecewise linear systems presenting a twofold singularity in R2, as well as limit cycles sorrounding a T-singularity.Item Ciclos limite para a equação de Abel generalizada(Universidade Federal de Goiás, 2009-10-30) Belisário, Hugo Leonardo da Silva; Garcia, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826In this work we conducted a study on the equations of the type dx dt = nå i=0 ai(t)xi; (A) where ai 2 C1, i = 0; ;n and 0 t 1. An equation of the form (A) is called a generalized Abel equation. Our study refers to the problem proposed by C. Pugh: There is a natural number N depending only on n, such that the equation (A) has at most N limit cycles? Initially we study the problem of C. Pugh for n = 1 and n = 2, for which the equation (A) has at most one and two limit cycles, respectively. For n = 3, A. Lins Neto shows that if a3(t) does not change sign on [0;1], then the equation (A) has at most three limit cycles. Also A. Lins Neto shows that, given a natural number l, it is possible to construct an equation of the form (A) with n = 3 that has at least l limit cycles. Still for n = 3, A. Gasull and J. Llibre study the problem of C. Pugh considering that a2(t) does not change sign on [0;1], and M. J. Alvarez, A. Gasull and H. Giacomini also study the problem of C. Pugh considering that there are real numbers a and b such that aa3(t)+ba2(t) does not change sign on [0;1] and a1(t) = a0(t) = 0. Besides this, we study some more general results studied by A. Gasull and A. Guillamon.Item Classificação das superfícies de revolução tipo Delaunay completas(Universidade Federal de Goiás, 2008-05-30) SOUZA, Luiz Gustavo Alves de; PINA, Romildo da Silva; http://lattes.cnpq.br/2675728978857991In this dissertation we have studied the articles The Surfaces of Delaunay, by James Eells, and Classi¯cation des Surfaces de Type Delaunay, by Ricardo Sa Earp and Eric Toubiana. Based on the ¯rst work we have classi¯ed the surfaces of revolution of constant mean curvature known as surfaces of Delaunay. By using the second one we have looked at special surfaces of Weingarten, whose mean and gaussian curvatures satis¯es the relation H = f(H2 ¡ K); where f is an elliptic function of class C1. We have classi¯ed the complete surfaces of revolution, that satis¯es the Weingarten relation. They are known as surfaces of Delaunay TypeItem Classificação de superfícies de translação, homotéticas e separáveis com curvaturas constantes no espaço euclidiano(Universidade Federal de Goiás, 2024-01-26) Muñoz González, Alejandra; Adriano, Levi Rosa; http://lattes.cnpq.br/3206466156270217; Adriano, Levi Rosa; Ribeiro Júnior, Ernani de Sousa; Leandro Neto, BeneditoIn this work, we study some classes of surfaces with constant Gaussian (K) or mean curvature (H) in Euclidean space R3. In the first part, we investigate surfaces obtained as the sum of two curves or as graphs of the product of two functions. We consider the problem of finding all surfaces of these types with constant Gaussian curvature (CGC).We extend the results to non-degenerate surfaces in Lorentz-Minkowski space. In the second part, we consider surfaces with constant Gaussian curvature given by an implicit equation of the form f (x) + g(y) + h(z) = 0, where f , g, and h are real functions of one variable. If K = 0, we show that the surface is a surface of revolution, a cylindrical surface, or a conical surface, obtaining explicit parametrizations of these surfaces. If K ̸= 0, the surface is a surface of revolution. KeywordsItem Classificação e construção de superfícies mínimas de translação em formas espaciais(Universidade Federal de Goiás, 2022-10-27) Silva, Marcos Gomes da; Adriano, Levi Rosa; http://lattes.cnpq.br/3206466156270217; Adriano, Levi Rosa; Tokura, Willian Isao; Corro, Armando Mauro Vasquez; Lima, Ronaldo Freire deA translation surface of Euclidean space is the sum of two regular curves and , called the generating curves. In this paper we classify the minimal translation surfaces of and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations of the generating curves, any minimal translation surface is described as , where is a curve parameterized by arc length s, its curvature is a positive solution of the autonomous ODE and its torsion is Here and are constants such that the cubic equation has three real roots , and . Furthermore in the half-space model of hyperbolic space, that is, with the hyperbolic metric, a translation surface that writes as or , where f and g are smooth functions, we prove that the only minimal translation surfaces are totally geodesic planes.