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Item Subgradient and gradient methods with feasible inexact projections for constrained convex optimization problems(Universidade Federal de Goiás, 2021-04-30) Aguiar, Ademir Alves; Prudente, Leandro da Fonseca; http://lattes.cnpq.br/4573611419840935; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; Prudente, Leandro da Fonseca; Melo, Jefferson Divino Gonçalves de; Bento, Glaydston de Carvalho; Andreani, RobertoO método subgradiente com uma projeção inexata viável é proposto na presente tese para resolver problemas de otimização convexa com restrições não diferenciáveis. Para realizar a projeção inexata proposta em um conjunto restrito, uma tolerância de erro relativa é introduzida. Além do mais, em cada iteração, o algoritmo requer o cálculo de um subgradiente aproximado da função. Limitantes para a complexidade na iteração e resultados de convergência assintótica para a sequência gerada pelo método empregando os conhecidos tamanhos de passo exógeno, Polyak e dinâmico são estabelecidos. Finalmente, relatamos alguns resultados numéricos a fim de ilustrar o comportamento prático do algoritmo quando o conjunto de restrição é convexo e compacto. Aqui também consideramos um novo método gradiente com projeção inexata usando tolerância de erro relativa. Convergência assintótica e limitantes para a complexidade na iteração do método empregando tamanho de passo constante e de Armijo são apresentados. Resultados numéricos são relatados ilustrando as vantagens potenciais de considerar projeções inexatas em vez de exatas em alguns exemplos de média escala em problemas de mı́ nimos quadrados sobre o espectraedro.Item Conditional gradient methods for multiobjective optimization(Universidade Federal de Goiás, 2021-08-06) Assunção Filho, Pedro Bonfim de; Prudente, Leandro da Fonseca; http://lattes.cnpq.br/4573611419840935; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; Prudente, Leandro da Fonseca; Melo, Jefferson Divino Gonçalves de; Bento, Glaydston de Carvalho; Souza, João Carlos de OliveiraNeste trabalho, analisamos o método do gradiente condicional, também conhecido como método de Frank-Wolfe, para resolver problemas de otimização multiobjetivo restrita. Também propomos e analisamos uma versão generalizada deste método para resolver problemas de otimização composta multiobjetivo que consistem em minimizar simultaneamente várias funções objetivo. Cada função objetiva é a soma de duas funções, uma é considerada continuamente diferenciável e a outra não é necessariamente diferenciável. Ambos os métodos são analisados com três estratégias de obtenção dos tamanhos dos passos, a saber: tipo Armijo, adaptativos e tamanhos decrescentes dos passos. Propriedades de convergência assintótica e limites de complexidade de iteração com e sem suposições de convexidade na função objetivo são estabelecidas. Experimentos numéricos para o método do gradiente condicional são fornecidos para ilustrar a eficácia do método e certificar os resultados teóricos obtidos.Item Concepção, organização e realização de olimpíada interna de matemática, em tempos de ensino remoto(Universidade Federal de Goiás, 2021-12-21) Barbosa, Marco Antônio; Dias, Ivonildes Ribeiro Martins; http://lattes.cnpq.br/8664599889120339; Dias, Ivonildes Ribeiro Martins; Rodrigues, Eunice Cândida Pereira; Souza, Mário José deThe aim of this thesis is to analyze the implementation of a Mathematics Olympics in a private school in the city of Goiânia, Goiás. Considering the COVID-19 pandemic and having Olympic competitions like the Goiás State Mathematics Olympics (OMEG) and Public Schools’ Brazilian Mathematics Olympics (OBEMEP) canceled, which were influential in the weekly training sessions, motivated the coordination to suggest the challenge of creating and implementing a competition associated to the school itself. The project has been developed between the years 2020 and 2021, 2021 being the second edition and having students from the latter years of middle school (6th to 9th grade) as participants. The classes were held remotely, consisting of two classes per week, through zoom.us and the tests, both for the first and the second round, were held on a specific platform belonging to the school. The preparation followed the specifications of OBEMEP regarding both educational content and the specifics of the test for the first and second round. The classes have been developed in accordance with the National Common Curriculum Base (BNCC), which defines the regulatory compliance meant to achieve the essential educational objectives supposed to be developed during the basic education period, and inspired in the problem-solving methodology. The process described includes all stages from its original idea to prizes. Worth mentioning are classes and references, the challenges encountered in dealing with a remote class system - until then unknown to Basic Education, the promptness of the work ensured by the Mathematics department and the success of the students, who were dedicated to studying mathematics, in spite of the difficulties stemming from the pandemic. Furthermore, this paper hopes to inspire Mathematics teachers from Basic Education to implement in their own schools internal competitions which promote commitment to studying and the dissemination of mathematics and Olympic competitions.Item Números binomiais: aplicações ao ensino e extensões(Universidade Federal de Goiás, 2021-08-24) Barros, Gabriella da Silva; Vargas, Tiago Moreira; http://lattes.cnpq.br/1884655261136497; Vargas, Tiago Moreira; Vargas Júnior, Valdivino; Lima, Thaynara Arielly de; Belisário, Hugo Leonardo da SilvaIn this work, we approach the binomial coefficients, which are seen in basic education in the 2nd grade of high school, when students are presented with the contents of combinatorial analysis. When binomial coefficients are presented, students study the definition and emer-gence of these coefficients in the Triangle of Pascal and Newton’s Binomial. In order to expose the binomial coefficients, we present and demonstrate some of their relevant properties and some of their extensions. Initially, we present the counting principles, after exposing the defi-nition of the binomial coefficients and their properties, we also present the relationship of these coefficients with the Fibonacci sequence and the Lucas Theorem. In addition, we pro-pose some activities that can be used by the high school mathematics teacher that use the bi-nomial coefficients as well as their properties. We finish with a construction of the Pascal Tri-angle and the development of the Newton Binomial in the Geogebra that can be used by the teacher during the presentation of these contents.Item Sobre estruturas gradiente Einstein-type produto torcido(Universidade Federal de Goiás, 2022-04-13) Batista, Elismar Dias; Adriano, Levi Rosa; http://lattes.cnpq.br/3206466156270217; Adriano, Levi Rosa; Corro, Armando Mauro Vasquez; Santos, João Paulo dos; Gomes, José Nazareno Vieira; Ribeiro Júnior, Ernani de SousaIn this work, we study gradient Einstein-type structures immersed in a warped product space of an interval I and a Riemannian manifold M, with a potential function h given by the height function. First, we obtained the necessary conditions for an Einstein-type gradient structure immersed in a warped product to be minimal, totally umbilic or totally geodesic. In addition, we provide triviality results for the potential function h. Next, we characterize the rotational hypersurfaces having a gradient Einstein-type structure in warped product, where the fiber is a space form. We also study particular cases of gradient Einstein-type structures in warped product spaces, namely, gradient Ricci-harmonic solitons (GRHS). In this case, we prove triviality results for the potential and warped functions when they reach a maximum or minimum. Finally, we provide a family non-enumerable of non-trivial geodesically complete examples of GRHS considering the base and fiber of a warped product conformal to a semi-Euclidean space invariant under the action of a translation group of codimension one.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 Geometria de Moulton x geometria euclidiana: trilhas de aprofundamento para o ensino médio(Universidade Federal de Goiás, 2023-03-24) Borges, Charlles de Morais; Smith, Ole Peter; http://lattes.cnpq.br/9938244088922275; Smith, Ole Peter; Silva, Geci José Pereira da; Souza, Flávio Raimundo deWith the advent of the approval of the National Common Curriculum Base for Secondary Education - BNCC, which brings learning based on Skills and Competences, and made the content no longer the protagonist of the teaching-learning process, this is what the present work addresses . For this, we present a little of the trajectory of this BNCC homologation process, its main paths and resolutions. We present how this Base was structured, enunciating the expected Skills and Competences, as well as proposing an educational product in the form of a Deepening Track in Mathematics, for the student's formation. The purpose of the deepening trail proposed here is to present the model of geometry proposed by Moulton, as a way of provoking a comparison with the geometry proposed by Euclid, evidencing the existence of other geometries, the so-called non-Euclidean geometries that, in most cases , are not introduced to students until the end of high school. Thus, through skills already acquired in Elementary School in relation to the study of Plane Geometry, lead the student to review already known concepts and propose challenges, so that through scientific research, he can further develop his skills and competences that lead them to propose solutions and conjecture new ideas about the proposed subject. We begin with a presentation of primitive concepts and postulates of the Euclidean model, and then we present primitive concepts of the model proposed by Moulton. We show how its structure was defined for the plane, the behavior of straight lines and their equations, how to calculate the distances between two points and also how to determine the measure of angles between straight lines. For this, we make use of Cartesian Geometry, structured by René Descartes, as a link in the comparison between these two models. The concepts to be worked on are already well known in Euclidean Geometry and, from this other model, we are going to show some of the transformations and modifications undergone by these classic figures of Plane Geometry, namely: parallel and perpendicular lines, the triangle and its notable elements , circle, ellipse, parabola and hyperbola. With the intention of making a broad initial exploration of the transformations and modifications undergone by these figures, we will also present the step-by-step of a constructive process of some conics, using dynamic geometry software, in this case, Geogebra, in order to also develop computational and programming skills, which are expected in the new National Base.Item Uma proposta para o ensino de vetores no novo ensino médio usando álgebra linear: matrizes-linha e suas propriedades(Universidade Federal de Goiás, 2023-03-31) Brito, Marcos Filipe de Oliveira; Souza, Marcelo Almeida de; http://lattes.cnpq.br/1343419041226215; Souza, Marcelo Almeida de; Faria, Elisabeth Cristina de; Vieira, Vanda DomingosIn this dissertation, the proposal is presented that the teaching of vectors in High School is carried out through Mathematics, and in a consistent way, namely, we propose that a contextualized study of vectors be carried out after working with a notion of Linear Algebra using row matrices and its properties. We carried out a survey at BNCC and DC-GOEM to verify the feasibility of inserting this proposal. With vectors, many geometry problems can be solved quickly and elegantly (demonstrations of properties and theorems), and in the classroom using ICT, such as the free software GeoGebra. We present several applications of vectors using GeoGebra.Item Trajectory control for two-dimensional piecewise linear dynamical systems(Universidade Federal de Goiás, 2022-08-25) Cabrera, Marly Tatiana Anacona; Cristiano, Rony; http://lattes.cnpq.br/5692970734994664; Cristiano, Rony; Tonon, Durval José; Pagano, Daniel JuanOs Sistemas Descontínuos Suaves por Partes (DPWS) são usados para descrever diversos fenômenos em diferentes áreas de estudo tais como física, biologia, química, engenharia, medicina, etc. Neste sistema, o movimento dinâmico do sistema é caracterizado por períodos de evolução suave interrompidos por eventos causados por algum tipo de descontinuidade, que pode ser dada pela natureza do sistema ou introduzida por alguma lei de controle descontínua. Uma característica importante desta classe de sistemas é a existência de um ponto de pseudo-equilíbrio que está sobre a superfície onde ocorre a descontinuidade, de tal modo que ele pode ser alcançado em tempo finito. O presente trabalho considera uma família de sistemas lineares por partes bidimensionais, com a descontinuidade dada por uma linha reta. Inicialmente, o comportamento dinâmico das trajetórias do sistema é totalmente caracterizado para os casos em que existe um pseudo-equilíbrio estável. Em seguida, são fornecidas condições sobre os parâmetros do sistema para garantir estabilidade em tempo finito e com no máximo uma comutação. Os resultados obtidos são aplicados a um sistema DPWS bidimensional que descreve a dinâmica de um tipo de conversor de potência denominado Buck Converter, onde se deseja que a partir de uma tensão inicial o sistema atinja uma tensão de saída desejada em tempo finito e com no máximo uma comutação.Item Hipersuperfícies de tipo Ribaucour(Universidade Federal de Goiás, 2022-09-27) Cárdenas Mendez, Milton Javier; Vasquez Corro, Armando Mauro; http://lattes.cnpq.br/4498595305431615; Vasquez Corro, Armando Mauro; Leandro Neto, Benedito; Carrion Riveros, Carlos Maber; Santos, João Paulo dos; Adriano, Levi RosaIn this work we define the Ribaucour-type surfaces (in short, RT-surfaces),These surfaces satisfy a relationship similar to the Ribaucour surfaces that are related to the Élie Cartan problem. This class furnishes what seems to be the first examples of pairs of noncongruent surfaces in Euclidean space such that, under a diffeomorphism, lines of curvatures are preserved and principal curvatures are switched. We show that every compact and connected RT-surface is the sphere with center at the origin. We obtain present a Weierstrass type representation for RT-surfaces with prescribed Gauss map which depends on two holomorphic functions. We give explicit examples of RT-surfaces. Also, we use this representation to classify the RT-surfaces of rotation. We define the GRT-surfaces which are a generalization of the RT-surfaces, show a local parameterization of this class of surfaces and classify them in the in which case they are of rotation and generalize as RT-surfaces for the case of hypersurfaces in Rn+1, display a parameterizationção for the rotational cases and analyze the general of generatrix curves when hypersurfaces and rotation behavior are 3-dimensional.Item Complexidade por iteração do método HPE e sua versão acelerada para otimização convexa(Universidade Federal de Goiás, 2023-04-10) Chagas, Marcus Vinícius de Morais; Melo, Jefferson Divino Gonçalves de; http://lattes.cnpq.br/8296171010616435; Melo, Jefferson Divino Gonçalves de; Gonçalves, Max Leandro Nobre; Bento, Glaydston de Carvalho; Alves, Maicon MarquesIn this work, we analyze the Hybrid Proximal Extragradiente (HPE) method to find zeroes of maximal monotone operators and its accelerated version Accelerated Hybrid Proximal Extragradient (A-HPE) to solve convex optimization problems whose objective function is given by the sum of two other convex functions, one differentiable with Lipschitz gradient and another one not necessarily differentiable. The HPE method was introduced by Solodov and Svaiter, it consists of an inexact version of the proximal point method having its proximal subproblems inexactly solved using a relative error criterion followed by an extragradient step. The HPE can also be seen as a framework, in the sense that many other methods for minimizing convex functions and more generally to find zeroes of maximal monotone operators can be seen as instances of the HPE method, such as the extragradient method, regularized Newton type method, ADMM, etc. In this work, we will analyze both the asymptotic convergence of the HPE method and its iteration-complexity.We will also analyze the iteration-complexity of the A-HPE method proposed by Monteiro and Svaiter. The A-HPE is a first-order accelerated method, i.e., it uses only information of the functional values and the first derivative or subgradients of the objective function and has optimal iteration-complexity.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 On the classification of Ricci solitons and Yamabe solitons(Universidade Federal de Goiás, 2023-03-20) Contreras, Jeferson Arley Poveda; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Santos, João Paulo do; Ribeiro Júnior, Ernani de Sousa; Tenenblat, Keti; Barboza, Marcelo BezerraIn this work, we will study the self-similar solutions of both Ricci flow and Yamabe flow. These solutions are also known as Ricci and Yamabe soliton, respectively. Inspired by the divergence equation used by Robinson in his demonstration of the uniqueness of static black holes and by Brendle’s classification of steady Ricci solitons, we will make some important characterizations of these solitons. We prove that four-dimensional gradient Yamabe solitons must have a Yamabe metric, provided that an asymptotic condition holds. Inspired by the geometry of the cigar soliton, we demonstrate that a gradient steady Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton N n−1×R, where N n−1 is Ricci flat, or isometric to the Bryant soliton. In the final Chapter, we prove some rigidity results for shrinking and expanding Ricci solitons.Item Variedades quasi-Einstein com simetrias e generalizações(Universidade Federal de Goiás, 2021-12-05) Correia, Paula Gonçalves; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Leandro Neto, Benedito; Adriano, Levi Rosa; Corro, Armando Mauro Vasquez; Santos, João Paulo dos; Ferraioli, Diego CatalanoIn this work we study quasi-Einstein warped products and generalized quasi-Einstein manifolds locally conformally flat. We prove that in some quasi-Einstein semi-Riemannian warped product the fiber is necessarily an Einstein manifold. We provide all quasi-Einstein manifolds when r-Bakry-Émery tensor is null, the base is conformal to a n-dimensional pseudo-Euclidean space, invariant under the action of the (n-1)-dimensional translation group, the potential function depends only on the base and the fiber is Ricci-flat. As an application, we provide a family of Ricci-flat warped product whose base is not locally conformally flat. Considering generalized m-quasi-Einstein manifolds conformal to the pseudo-Euclidean space, we used an ansatz method to obtain the most general symmetry group of maximal dimension in which there are solutions to the system of equations that describe such manifolds. In addition, using the same method, we show that there is no different invariant of smaller dimension on a generalized m-quasi-Einstein manifold.Item Uma desigualdade de Minkowski e soluções exatas para o espaço-tempo estático de Einstein-Maxwell(Universidade Federal de Goiás, 2023-02-24) Costa, Ana Paula de Melo da; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Cruz, Cícero Tiarlos Nogueira; Adriano, Levi Rosa; Silva, Maria de Andrade Costa e; Batista, Rondinelle MarcolinoWe study the static Einstein-Maxwell space when it is conformal to an n-dimensional pseudo-Euclidean space, which is invariant under the action of an (n−1)-dimensional translation group. We also provide a complete classification of such space. Moreover, we prove a Minkowski-like inequality for an asymptotically flat static Einstein- Maxwell (electrovacuum) space-time using as an approach the inverse mean curvature flow (IMCF).Item Análise do discriminante em equações polinomiais de terceiro grau a partir de funções simétricas(Universidade Federal de Goiás, 2020-11-09) Costa, Debora Naiure Araujo; Gomes, Alacyr José; http://lattes.cnpq.br/5994366998164007; Silva, Geci José Pereira da; Belisário, Hugo Leonardo da Silva; Gomes, Alacyr JoséThe objective of this work is to make a study on the characteristics of the roots of polynomial equations of degrees 2 and 3, studying the discriminant written in function of invariants that depend only on the coefficients of an initial equation. We will start by remembering some important definitions and propositions about polynomials, symmetric functions of the roots and develop the resolution of the cubic by radicals using two different methods. Finally, it will be possible to analyze the discriminant through algebraic manipulations that we will do, using the content presented in the previous chapters, and then we will write the discriminant from the roots of the equations, which enable to develop a proof for the theorems that relate the roots of the equations 2º and 3º degrees with their discriminants.Item On the effect of existence of static horizons(Universidade Federal de Goiás, 2021-12-17) Coutinho, Fernando Soares; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Corro, Armando Mauro Vasquez; Ribeiro Júnior, Ernani de Sousa; Santos, João Paulo dos; Reis, Hiuri Fellipe Santos dosNeste trabalho estudamos os efeitos da existência do horizonte estático. Em geral, o horizonte estático é definido como o conjunto onde a função lapso, para uma variedade estática, é identicamente nula. Este conjunto está fisicamente relacionado com o horizonte de eventos, a fronteira de um buraco negro. No primeiro capítulo, estudamos a conjectura da bola fluida. Construímos uma fórmula de divergência de Robinson para o espaço tempo fluido perfeito estático. Inspirado por esta conjectura, um resultado de rigidez para o fator espacial de um espaço-tempo fluido perfeito estático satisfazendo algumas condições de contorno é provado, desde que uma equação de estado seja válida. No segundo capítulo, o objetivo é investigar a geometria do espaço-tempo fluido perfeito estático em variedades compactas com bordo. Fornecemos uma estimativa de bordo para espaço-tempo fluido perfeito estático e estabelecemos uma fórmula do tipo Böchner para uma grande classe de espaços que incluem o espaço-tempo fluido perfeito estático, métricas críticas do funcional de volume, espaços estáticos e métricas CPE. Mais ainda, como consequência desta fórmula obtemos um resultado do tipo “gap” para um espaço tempo fluído perfeito estático compacto.Item Problemas variacionais com norma L2 prescrita(Universidade Federal de Goiás, 2021-03-01) Cunha, Gabriel Neves; Silva, Kaye Oliveira da; http://lattes.cnpq.br/3634338534144726; Silva, Kaye Oliveira da; Siciliano, Gaetano; Miyagaki, Olimpio HiroshiIn this study, we searched for standing waves weak solutions for the Maxwell- Schrodinger- poisson system of partial differential equations by variational methods, minimizing the energy functional associated to the problem with constrained L2 norm.Item Nilpotência de grupos e álgebras de Lie admitindo grupos de Frobenius de automorfismos(Universidade Federal de Goiás, 2021-04-07) Dal Berto, Lucas Matheus de Lima; Silva, Jhone Caldeira; http://lattes.cnpq.br/6848751340618892; Silva, Jhone Caldeira; Rodrigues, Paulo Henrique De Azevedo; Lima, Igor dos SantosLet A be a group acting on a group G. It is known that some properties of G are influenced by CG(A), for example, suppose that a Frobenius group FH acts on a finite group G, we known that if CG(F) = 1 and CG(H) is nilpotent, then G is nilpotent and, adding the hypothesis that F is cyclic, we have that the nilpotency class of G is bounded in terms of the order of H and the nilpotency class of CG(H). Until now, it was not evident, considering the hypotheses mentioned above, if the nilpotency class of G could be made independent of the order of H. In this dissertation, we show that exists a family G of finite nilpotent groups, of unbounded nilpotency class, such that each group in G admits a metacyclic Frobenius group of automorphisms so that CG(F) = 1 and CG(H) is abelian, thus evidencing the essential dependency of the order of H in the nilpotency class of G.Item Melnikov theory for Filippov systems with an algebraic switching curve(Universidade Federal de Goiás, 2022-05-25) Erazo, Gerardo Homero Anacona; Gomide, Otávio Marçal Leandro; http://lattes.cnpq.br/6665788071640310; Gomide, Otávio Marçal Leandro; Andrade, Kamila da Silva; Ramírez Cespedes, Oscar AlexanderA existência de ciclos limite em sistemas dinâmicos é um tópico de pesquisa muito atrativo devido a suas amplas aplicações. A importância do desenvolvimento deste tema pode ser justificada por um dos itens da famosa lista de problemas proposta por David Hilbert que permanece em aberto e consiste em encontrar o número máximo de ciclos limite de um sistema planar polinomial. Em virtude de sua relevância, atualmente há um grande interesse em estender a busca de ciclos limite a outros tipos de sistemas, como os famigerados sistemas de Filippov. Neste trabalho, estudamos uma técnica para encontrar ciclos limite em sistemas dinâmicos suaves por partes, chamada Teoria de Melnikov, e aplicamos tal técnica para encontrar cotas inferiores para o número máximo de ciclos limite de sistemas polinomiais.