Mestrado em Matemática (IME)

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http://200.137.215.59/tede/handle/tde/289

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Superfícies harmônicas no 3-Espaço Euclidiano
(Universidade Federal de Goiás, 2013-08-30) Barbosa, Sarah Oliveira; Corro, Armando Mauro Vasquez; Corro, Armando Mauro Vasquez; Riveros, Carlos Maber Carrion; Lemes, Max Valério
Embargado
Métodos de regularização cúbica com aproximações preguiçosas da hessiana
(Universidade Federal de Goiás, 2025-02-25) Gehlen Filho, Vilmar; Gonçalves, Max Leandro Nobre; http://lattes.cnpq.br/7841103869154032; Gonçalves, Max Leandro Nobre; Melo, Jefferson Divino Gonçalves de; Grapiglia, Geovani Nunes; Santos, Luiz Rafael dos
In this work, we present variants of the Cubic Regularization Newton's (CRN) method incorporating lazy Hessian approximations for solving general non-convex optimization problems (0-3). We propose two approaches: the first (Algorithm 1) employs the exact gradient while reusing the same Hessian approximation for a block of $ m $ iterations, whereas the second (Algorithm 2) extends this idea by additionally allowing the use of inexact gradients. Implementations of methods, where information about derivatives are computed through finite difference strategies, are presented. One interesting feature of our algorithms is that the regularization parameter and the accuracy of the derivative approximations (when they are updated) are jointly adjusted using a nonmonotone line search criterion. We establish first-order complexity results for both methods. Specifically, for a given precision $\epsilon$, it is shown that the Algorithm~1 and Algorithm~2 require at most {$\mathcal{O}\left( m^{1/2} \epsilon^{-3/2}\right)$} outer iterations to generate an $\epsilon-$approximate critical point for aforementioned problem. When the derivatives are computed by finite difference approaches, we show that Algorithm~1 (resp. Algorithm~2) needs at most {$\mathcal{O}\left((n+m)m^{-1/2}\epsilon^{-3/2}+(n+m)\right)$} (resp. {$\mathcal{O}\left((n^2+mn)m^{-1/2}\epsilon^{-3/2}+(n^2+mn)\right)$}) gradient and function (resp. function) evaluations to generate an $\epsilon$-approximate critical point, where $n$ is the dimension of the domain of the objective function.
Hipersuperfícies de Einstein em espaços de produto torcido
(Universidade Federal de Goiás, 2025-03-07) Moyses, Junior Rodrigues; Adriano, Levi Rosa; http://lattes.cnpq.br/3206466156270217; Adriano , Levi Rosa; Sampaio Junior, Valter Borges; Silva, Tarcísio Castro
with n ≥ 2, where I ⊂ R is an open interval, ω is the warping function and Qn ε denotes the simply connected space form of constant sectional curvature ε = −1,0,1. Knowing that constant sectional curvature manifolds (CSC, for short) are the simplest examples of Einstein manifolds, through references such as [5], [7], [11], [13], [14] and [19] we seeked conditions for an Einstein hypersurface of I ×ω Qn ε is necessarily CSC. For certain warping functions ω, we studied the existence of these hypersurfaces through the class of rotational hypersurfaces. Furthermore, using the theory of isoparametric hypersurfaces, we characterized a special type of Einstein hypersurfaces - called ideal - as local graphs of parallel hypersurfaces of Qn ε in which, under certain conditions in main curvatures, are CSC.
Curvas esféricas com curvatura proporcional a função altura
(Universidade Federal de Goiás, 2024-09-27) Furtado, Miriam Cristina Ferreira; Reis, Hiuri Fellipe Santos dos; http://lattes.cnpq.br/9444154235787960; Freitas, Bruno Rodrigues de; http://lattes.cnpq.br/4201351441514126; Freitas, Bruno Rodrigues de; Reis, Hiuri Fellipe Santos dos; Leandro Neto, Benedito; Sampaio Junior, Valter Borges
In the work, we analyze the study by [17] on the characterization of soliton solutions of the Curve Shortening Flow (CSF) on revolution surfaces in the space R³, demonstrating that such solutions asymptote the parallels that are geodesics. Furthermore, we will address the work carried out by [8] that characterizes soliton solutions on the unitary sphere S². We will apply a similar methodology to describe curves with curvature depends on the height function
O quadrado tensorial não-abeliano, construções relacionadas e p-grupos
(Universidade Federal de Goiás, 2024-03-18) Souza, Sofia de Siqueira e; Oliveira, Ricardo Nunes de; http://lattes.cnpq.br/0563210461148997; Oliveira, Ricardo Nunes de; Melo, Emerson Ferreira de; Bastos Junior, Raimundo de Araujo
Let G and Gφ be two isomorphic groups. We study the group ν(G) that is an extension of the non-abelian tensor square of G, G⊗G. We show how ν(G) preserves properties from the group G in question. We also study potent and powerful finite p-groups. We prove that the non-abelian tensor square of G and the k-th term of the lower central series of ν(G) are potently embedded in ν(G), when G is a potent finite p-group. Moreover, for an odd prime p, if G is a potent p-group, then exp(ν(G)) divides p · exp(G)
Sistemas lento-rápido por partes e aplicações
(Universidade Federal de Goiás, 2024-03-18) Silva, Luan Lima da; Euzébio, Rodrigo Donizete; http://lattes.cnpq.br/9213320273714493; Tonon, Durval José; http://lattes.cnpq.br/3688981956532711; Tonon, Durval José; Euzébio, Rodrigo Donizete; Goncalves, Luiz Fernando; Carvalho, Tiago de
This dissertation presents a study of piecewise-smooth (or discontinuous) vector fields, with an emphasis on the Filippov’s convention and the Sotomayor-Teixeira’s regulariza tion process. We also pay special attention to slow-fast systems, which are a class of 1- parameter smooth vector fields defined on more than one time scale and which establish relations with discontinuous vector fields. Initially, results and definitions are presented from the classical theory of smooth vector fields. The aim is to establish a natural con nection between the set of smooth vector fields and the set of discontinuous vector fields. Then we present the main concepts and results associated with slow-fast systems and piecewise-smooth systems. Finally, a piecewise-smooth vector field model applied to the study of climate dynamics is presented and discussed.
Desigualdade de Trudinger-moser em variedades riemannianas completas e não compactas
(Universidade Federal de Goiás, 2024-03-15) França, Júlio Cesar Pereira; Macedo, Abiel Costa; http://lattes.cnpq.br/6413790814030608; Macedo, Abiel Costa; Carvalho, Marcos Leandro Mendes; Oliveira, José Francisco Alves de
In this work, we will address the validity of the Trudinger-Moser inequality on complete and non-compact Riemannian manifolds. More precisely, we will discuss the proofs provided by Yang and by Li and Lu for the subcritical and critical cases.
Cohomology and partial differential equations
(Universidade Federal de Goiás, 2024-06-20) Marinho, Artur Jorge; Silva, Kaye Oliveira da; http://lattes.cnpq.br/3634338534144726; Silva, Kaye Oliveira da; Siciliano, Gaetano; Macedo, Abiel Costa
This text deals with elliptic partial differential equation problems by the Morse theoric point of view. Since Morse theory has a connection with some concepts from algebraic topology, cohomology theory is employed to show existence result for a class of equations. The concept of cohomological index will be useful to show multiple solution results for the Brezis-Nirenberg problem for the p-Laplacian.
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.

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