Mestrado em Matemática (IME)

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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, Benedito
In 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. Keywords
Policiclos em sistemas de Filippov planares
(Universidade Federal de Goiás, 2023-07-27) Souza, Alessandra Carlos de; Gomide, Otávio Marçal Leandro; http://lattes.cnpq.br/6665788071640310; Gomide, Otávio Marçal Leandro; Cristiano, Rony; Lima, Dahisy Valadão de Souza
In this work, we study the local structure of planar Filippov systems around low codi mension Σ−singularities and we analyze systems presenting polycycles passing through Σ−singularities. In this way, we analyze Poincaré maps (associated with such polycycles) and determine bifurcation diagrams of Filippov systems around these minimal sets. More specifically, we study the generic bifurcation of a Filippov system around a global con nection passing through a visible fold-regular singularity, the so-called critical crossing cycle and we show that, under smale pertubations, such connection breaks originating étther a sliding cycle or a crossing limit cycle. We also study a planar Filippov system model around a certain Σ−singularity called Fold-Cusp, where a fold and a cusp meet and we show the existence of a critical crossing cycle bifurcations from such singularity in an unfolding of this system. In addition, we exhibit the bifurcation diagram of this unfolding.
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 Marques
In 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.
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 de
In 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.
Sistemas descontínuos lento-rápidos e aplicações
(Universidade Federal de Goiás, 2023-01-26) Fernandes, Vitória Chaves; Euzébio, Rodrigo Donizete; http://lattes.cnpq.br/9213320273714493; Euzébio, Rodrigo Donizete; Tonon, Durval José; Buzzi, Claudio Aguinaldo
In this work we study dynamical systems focused on two areas: discontinuous systems and singular perturbation problems. We analyze the intersection of these two areas through some theoretical results. In the first moment, we will present a theory similar to Fenichel's Theory for singularly perturbed discontinuous systems, later we will show that a system obtained via regularization can be associated with a singular perturbation problem. In addition, we will study a mathematical modeling in the area of climatology, with the objective of analyzing the bifurcations of singularities and the existence of a periodic orbit for certain specific parameters. In this model, we cannot apply Fenichel's theory, for this reason we use an ad-hoc application of Fenichel's Theory to demonstrate the desired results. Finally, we will present some unpublished results for the climatological model.
Unicidade dos solitons de Ricci gradiente estáveis
(Universidade Federal de Goiás, 2022-11-18) Souza, Vítor Emanoel Resplandes de; Leandro Neto, Benedito; http://lattes.cnpq.br/3393448440968708; Leandro Neto, Benedito; Barboza, Marcelo Bezerra; Santos, João Paulo dos
We present the Ricci solitons, objects that appear as self-similar solutions of the Ricci flow. We show some properties and results related to these solitons. In addition, we study [2] and prove that a three-dimensional gradient steady Ricci soliton that is asymptotic to the Bryant soliton must be isometric to the Bryant soliton. This theorem was proved by Brendle [1], and more generally stated by Cao et al. However Cao states the result without any proof, thus, the ultimate goal of our research is to demonstrate the theorem stated by Cao by generalizing Brendle’s result to dimensions greater than or equal to 3.
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 de
A 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.
Métodos de primeira ordem acelerados
(Universidade Federal de Goiás, 2022-08-18) Ribeiro, Douglas Nascimento; Melo, Jefferson Divino Gonçalves de; http://lattes.cnpq.br/8296171010616435; Melo, Jefferson Divino Gonçalves de; Alves, Maicon Marques; Gonçalves, Max Leandro Nobre; Ferreira, Orizon Pereira
The analysis of the efficiency of algorithms to solve optimization problems is fundamental for the improvement and design of algorithms with better computational performances. Such efficiency can be measured, for example, by the “speed” in which the sequence generated by the algorithm converges to a solution of the problem at hand. From the work of Nesterov and Nemirovski in the 80's, the efficiency of an algorithm was also considered through its iteration complexity, that is, the number of iterations necessary to obtain an "approximate solution" for the problem. In this work, we will analyze the iteration complexity of the algorithms: Iterative Shrinkage-Thresholding (ISTA), Fast Iterative Shrinkage-Thresholding (FISTA) and an accelerated Forward-Backward Nesterov type method. This study will be be carried out from a theoretical and computational point of view.
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 Pereira
The 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.
Um estudo sobre automorfismos potências com centralizadores 2-grupos abelianos elementares
(Universidade Federal de Goiás, 2022-09-09) Souza, Gabriella Cristina de; Silva, Jhone Caldeira; http://lattes.cnpq.br/6848751340618892; Silva, Jhone Caldeira; Lima, Igor dos Santos; Oliveira, Ricardo Nunes de
Let $\varphi$ be an automorphism of a group $G$. We denote by $C_G(\varphi)$ the centralizer of $\varphi$ in $G$, that is, the subgroup of the fixed points of $\varphi$ to $G$. It is known that various properties of $G$ are in a certain sense close to the corresponding properties of the subgroup $C_{G}(\varphi)$. In the case where $\varphi$ is a power automorphism, we have that all elements having order 2 are fixed by $\varphi$. For this reason, we consider the case where $C_{G}(\varphi)$ is an elementary abelian $2$-group. A power automorphism $\varphi$ is said to be a pre-fixed-point-free power automorphism if $C_{G}(\varphi)$ is an elementary abelian $2$-group. When a group $G$ admits a pre-fixed-point-free power automorphism, we say that $G$ is an $E$-group. In this work, we determine all $E$-groups and their pre-fixed-point-free by power automorphisms. In particular, we use some results on power automorphisms to show a characterization of finite abelian groups.
A conjectura de Wilf do ponto de vista da profundidade de um semigrupo numérico e outros invariantes
(Universidade Federal de Goiás, 2022-08-26) Tôledo, Anna Carolina Gomes; Tenório, Wanderson; http://lattes.cnpq.br/6406888404650319; Tenório, Wanderson; Sepulveda Castellanos, Alonso; Souza, Matheus Bernardini de
In this work, it will be presented the so-called Wilf’s conjecture, which asks about a relation between invariants of numerical semigroups. Starting with examples of known families of numerical semigroups satisfying the conjecture, it will be shown the Eliahou’s approach [5] that corroborates to the validity of the conjecture through tools from the depth of a numerical semigroup. This result greatly contributes with the methods for the conjecture and determines a new invariant to be studied in numerical semigroups: the Eliahou number. It will be presented some basic concepts and properties about numerical semigroups, as well as as the prerequisites for the method of Eliahou to prove that numerical semigroups with depth q ≤ 3 satisfy the Wilf’s conjecture.
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 Juan
Os 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.
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 Duran
In 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.
Superfícies focais e pontos umbílicos Darbouxianos segundo a análise de Allvar Gullstrand
(Universidade Federal de Goiás, 2019-02-13) Ferreira, Samuel Carlos de Souza; Garcia, Ronaldo Alves; http://lattes.cnpq.br/5680428710939826; Garcia, Ronaldo Alves; Freitas, Bruno Rodrigues de; Cruz, Douglas Hilário da; Silva, Débora Lopes da
Allvar Gullstrand (1862-1930) was a Swedish ophthalmologist who received the Nobel Prize in Physiology orMedicine for his studies in images and refraction of light in the eye (EHINGER; GRZYBOWSKI, 2011). For the success of his research, Gullstrand would need advanced knowledge in Physics and Geometry. In order to understand the human eye, Gullstrand (1900) studied the focal surfaces and their respective singularities using the fourth order derivatives of a surface (NORDENSON, 1962, p. 285). The main goal of this dissertation was to study Gullstrand (1904) to understand the relation between the umbilic points and focal surfaces. In this sense, there are two possibilities for the focal surfaces, say M1 and M2, which depend on the type of the Darbouxian umbilic point, being that M1 is related to the types D1, D2 or D3 and M2 to the type D3. The relation between type of Darbouxian umbilic points and the focal surface will depend on the analysis of the ridges (critical points of the principal curvatures) that pass through the respective umbilic point where, for points of type D1 or D2 passes only one ridge and through the point of type D3 passes one or three ridges.
Equivalência entre o método de Melnikov e o método de Averaging para campos de vetores suaves por partes quase-integráveis
(Universidade Federal de Goiás, 2022-05-23) Tunubalá Sánchez, Angela Carolina; Tonon, Durval José; http://lattes.cnpq.br/3688981956532711; Tonon, Durval José; Cristiano, Rony; Pessoa, Claudio Gomes
In this work the equivalence between the Melnikov method and the Averaging method will be studied for a piecewise smooth near-integrable systems in n-dimensional spaces (n ≥ 2). We present an introduction of both methods, then we direct the study of these methods to piecewise smooth near-integrable systems and we show the equivalence between them, where a key step of solving this problem is to construct an appropriate change of coordinates, which transforms the perturbed piecewise smooth system into a periodic system. We also present the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems and we will finish the work with some applications of the results presented in a class of three-dimensional autonomous systems.
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 Alexander
A 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.
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 Santos
Let 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.
Concentração e multiplicidade de soluções positivas para o p-Laplaciano fracionário envolvendo não linearidade côncavo-convexa
(Universidade Federal de Goiás, 2021-03-04) Oliveira, Jefferson Luís Arruda; Silva, Edcarlos Domingos da; http://lattes.cnpq.br/7817014732764711; Silva, Edcarlos Domingos da; Furtado, Marcelo Fernandes; Ruviaro, Ricardo
In this work, we study the existence and multiplicity of positive solutions to the elliptical problem with concave-convex nonlinearities, given by, {𝜀𝑠𝑝(−Δ)𝑝𝑠𝑢+𝑉(𝑥)|𝑢|𝑝−2𝑢=𝜆𝑓(𝑥)|𝑢|𝑞−2𝑢+𝑔(𝑥)|𝑢|𝑟−2𝑢,in 𝑅𝑁,𝑢 ∈𝑊𝑠,𝑝(𝑅𝑁). where 𝜀,𝜆>0 are positive parameters, 𝑁>𝑠𝑝 with 𝑠 ∈(0,1) fixed, 1<𝑞<𝑝<𝑟<𝑝𝑠∗ e 𝑝𝑠∗= 𝑁𝑝𝑁−𝑠𝑝. We consider adequate hypotheses about the functions 𝑓, 𝑔 and the potential 𝑉, to conclude a result of existence and multiplicity of solutions to the above problem. More specifically we demonstrate the existence of 𝑘 +1 positive solutions for the problem studied, where 𝑘 is a integer positive number.
Equação de Choquard: existência de soluções de energia mínima para uma classe de problemas não locais envolvendo potenciais limitados ou ilimitados
(Universidade Federal de Goiás, 2021-02-05) Lima, Eduardo Dias; Silva, Edcarlos Domingos da; http://lattes.cnpq.br/7817014732764711; Silva, Edcarlos Domingos da; Silva, Maxwell Lizete da; Melo Júnior, José Carlos de Albuquerque
In this work, we present a study on the existence of a minimum energy solution for the following non-linear Choquard equation { −Δ𝑢+𝑉(𝑥)𝑢=(∫𝑄(𝑦)𝐹(𝑢(𝑦))𝑑𝑦|𝑥−𝑦|𝜇𝑅𝑁)𝑢∈𝐷1,2(𝑅𝑁) 𝑄(𝑥)𝑓(𝑢(𝑥)) where 𝑁≥3,0<𝜇<𝑁,𝑉∈ 𝐶(𝑅𝑁,[0,+∞)),𝑄∈𝐶(𝑅𝑁,(0,+∞)),𝑓∈𝐶(𝑅,𝑅) and 𝐹(𝑡)=∫𝑓(𝑠)𝑑𝑠𝑡0. The non-linearity 𝑓:𝑅→𝑅 is continuous and has asymptotically linear behavior at infinity. Using some assumptions for the Nehari manifold N and some other inequalities, established at work, the equation above has a minimal energy solution. Moreover, through the Pohoˇzaev’s manifold, we guarantee existence of a minimal energy solution in the zero mass case. For the elaboration of this work, we follow the article [11] by S. Chen and S. Yuan.
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 Hiroshi
In 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.

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