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Item Método subgradiente incremental para otimização convexa não diferenciável(Universidade Federal de Goiás, 2014-12-18) Adona, Vando Antônio; Melo, Jefferson Divino Gonçalves de; http://lattes.cnpq.br/8296171010616435; Melo, Jefferson Divino Gonçalves de; Gonçalves, Max Leandro Nobre; Haeser, Gabriel; Ginart, Jorge BarriosWe consider an optimization problem for which the objective function is the sum of convex functions, not necessarily differentiable. We study a subgradient method that executes the iterations incrementally selecting each component function sequentially and processing the subgradient iteration individually. We analyze different alternatives for choosing the step length, highlighting the convergence properties for each case. We also analyze the incremental model in other methods, considering proximal iteration and combinations of subgradient and proximal iterations. This incremental approach has been very successful when the number of component functions is large.Item Inexact variants of the alternating direction method of multipliers and their iteration-complexity analyses(Universidade Federal de Goiás, 2019-03-27) Adona, Vando Antônio; Gonçalves, Max Leandro Nobre; http://lattes.cnpq.br/7841103869154032; Melo, Jefferson Divino Gonçalves de; http://lattes.cnpq.br/8296171010616435; Melo, Jefferson Divino Gonçalves de; Gonçalves, Max Leandro Nobre; Prudente, Leandro da Fonseca; Pérez, Luis Roman Lucambio; Andreani, RobertoThis thesis proposes and analyzes some variants of the alternating direction method of multipliers (ADMM) for solving separable linearly constrained convex optimization problems. This thesis is divided into three parts. First, we establish the iteration-complexity of a proximal generalized ADMM. This ADMM variant, proposed by Bertsekas and Eckstein, introduces a relaxation parameter into the second ADMM subproblem in order to improve its computational performance. We show that, for a given tolerance ρ>0, the proximal generalized ADMM with α in (0, 2) provides, in at most O(1/ρ^2) iterations, an approximate solution of the Lagrangian system associated to the optimization problem under consideration. It is further demonstrated that, in at most O(1/ρ) iterations, an approximate solution of the Lagrangian system can be obtained by means of an ergodic sequence associated to a sequence generated by the proximal generalized ADMM with α in (0, 2]. Second, we propose and analyze an inexact variant of the aforementioned proximal generalized ADMM. In this variant, the rst subproblem is approximately solved using a relative error condition whereas the second one is assumed to be easy to solve. It is important to mention that in many ADMM applications one of the subproblems has a closed-form solution; for instance, l_1-regularized convex composite optimization problems. We show that the proposed method possesses iteration-complexity bounds similar to its exact version. Third, we develop an inexact proximal ADMM whose rst subproblem is inexactly solved using an approximate relative error criterion similar to the aforementioned inexact proximal generalized ADMM. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. Our approach consists of interpreting these ADMM variants as an instance of a hybrid proximal extragradient framework with some special properties. Finally, in order to show the applicability and advantage of the inexact ADMM variants proposed here, we present some numerical experiments performed on a setting of problems derived from real-life applications.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 Ondas de Choques Transicionais Para Modelos Quadráticos de Duas Leis de Conservação(Universidade Federal de Goiás, 2007-11-29) ALMEIDA, Gisele Detomazi; MOTA, Jesus Carlos da; http://lattes.cnpq.br/8457974658695539Transitional shock waves arises in solution of initial values problems for non linear systems of conservation laws that are not strictly hyperbolic. These waves are discontinuous solutions that posses viscous profile but do not conform to the Lax characteristic criterion, where inequalities between the shock propagation speed and the characteristic speeds must to be satisfied. These waves arise as transition between wave groups associated with distinct characteristic families. In this work we studied transitional shock waves for a system of two conservation laws with quadratic fux functions and positive defined viscosity matrix. In particular, we studied the transitional shock waves with viscous profile defined by orbits laying on straightlines. We show from examples, for systems with quadratic fux functions and viscosity matrix chosen in a convenience way, that is necessary to use transitional shock waves to solve the Riemann problem (initial data constant by parts) for these systems.Item Condições de finitude para o produto tensorial não abeliano de grupos(Universidade Federal de Goiás, 2019-08-08) Alves, Thulio de Oliveira; Oliveira, Ricardo Nunes de; http://lattes.cnpq.br/0563210461148997; Oliveira, Ricardo Nunes de; Bastos Júnior, Raimundo de Araújo; Vieira, Ewerton RochaIn this work we are interested in describing the conditions on which the non-abelian tensor product of groups is finite, thus we show that if G and H are groups that act compatibly on each other, then the tensor product of G and H is finite, if and only if the set of all tensor is finite. As an immediate consequence of this result, if G and H are finite, then the tensor product of G and H is finite. Furthermore we describe finite conditions for specific subgroups of the tensor product when G and H are FC-groups and also when H=G and G is a BFC-group.Item Convergência do Método do Ponto Proximal para Funções que Satisfazem a Desigualdade de Łojasiewicz(Universidade Federal de Goiás, 2012-06-27) AMARAL, José Henrique Salazar do; BENTO, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394This paper presents an analysis of convergence of the proximal point method for functions that satisfy the inequality of Lojasiewicz.Item Sobre uma Construção Relacionada ao Quadrado Tensional não-Abeliano de um Grupo(Universidade Federal de Goiás, 2011-07-01) ANDRADE, Agenor Freitas de; RODRIGUES, Paulo Henrique de Azevedo; lattes.cnpq.br/8910130626123426; OLIVEIRA, Ricardo Nunes de; http://lattes.cnpq.br/0563210461148997Let G and Gj be isomorphic groups. We study the group V (G) which is an extension of the non-abelian tensor square of a group G, G G. Looking for V (G) as an operator in the class of groups, we observe that this operator preserves some properties of the group G such as finiteness, nilpotency and solubility. For a p-group finite G we find an upper bound for the order of G G. Finally, we verified computationally, for some groups, and that the results and also the bounds for the orders of the groups shown here are actually respected.Item A coexistência de quatro ciclos limite em campos vetoriais seccionalmente lineares em R3(Universidade Federal de Goiás, 2012-07-30) ANDRADE, Kamila da Silva; MEDRADO, João Carlos da Rocha; http://lattes.cnpq.br/5021927574622286In this work we study continuous, symmetric and piecewise linear vector fields on R3, we investigate the existence of limit cycles using the closing equations method. More specifically, we study a two parameters family of this vector fields and we show the coexistence of four limit cycles and too, its realization on Chua s circuit.Item Superfícies translacionais no espaço isotrópico(Universidade Federal de Goiás, 2019-03-01) Andrade, Thamara Policarpo Mendes de; Corro, Armando Mauro Vasquez; http://lattes.cnpq.br/4498595305431615; Corro, Armando Mauro Vasquez; Pereira, Rosane Gomes; Carretero, José Luis TeruelTranslation Surfaces are obtained by translating curves contained in non-parallel planes. In this paper, the results of the Aydin, M. E; Ergut, M. Affine Translation Surfaces in the Isotropic Space [3]. Are considered the Affine Translation Surfaces type 1 in the Isotropic 3- Space I3, obtained for translating of two curves in the planes not necessarily orthogonal. The objective was to characterize the Weingarten Affine Translation Surfaces, which satisfy certain conditions with respect to Gaussian and mean curvatures. In addition, results were obtained for Translation Surfaces satisfying \Delta _{I,II }ri = \lambda_{i}ri, finding explicit solutions for the parameterization of such surfaces. Some examples are presented, as well as their respective graphs that were plotted using the Mathematical software.Item Criptografia de curvas elípticas(Universidade Federal de Goiás, 2017-03-15) Angulo, Rigo Julian Osorio; Chaves, Ana Paula de Araújo; http://lattes.cnpq.br/2332073500640724; Chaves, Ana Paula de Araújo; http://lattes.cnpq.br/2332073500640724; Rodrigues, Paulo Henrique de Azevedo; Godinho, Hemar TeixeiraAccording to history, the main objective of cryptography was always to provide security in communications, to keep them out of the reach of unauthorized entities. However, with the advent of the era of computing and telecommunications, applications of encryption expanded to offer security, to the ability to: verify if a message was not altered by a third party, to be able to verify if a user is who claims to be, among others. In this sense, the cryptography of elliptic curves, offers certain advantages over their analog systems, referring to the size of the keys used, which results in the storage capacity of the devices with certain memory limitations. Thus, the objective of this work is to offer the necessary mathematical tools for the understanding of how elliptic curves are used in public key cryptography.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 Programação linear e suas aplicações: definição e métodos de soluções(Universidade Federal de Goiás, 2013-03-18) Araújo, Pedro Felippe da Silva; Cruz, José Yunier Bello; http://lattes.cnpq.br/8377200040018415; Cruz, José Yunier Bello; Sandoval, Wilfredo Sosa; Melo, Jefferson Divino Gonçalves deProblems involving the idea of optimization are found in various elds of study, such as, in Economy is in search of cost minimization and pro t maximization in a rm or country, from the available budget; in Nutrition is seeking to redress the essential nutrients daily with the lowest possible cost, considering the nancial capacity of the individual; in Chemistry studies the pressure and temperature minimum necessary to accomplish a speci c chemical reaction in the shortest possible time; in Engineering seeks the lowest cost for the construction of an aluminium alloy mixing various raw materials and restrictions obeying minimum and maximum of the respective elements in the alloy. All examples cited, plus a multitude of other situations, seek their Remedy by Linear Programming. They are problems of minimizing or maximizing a linear function subject to linear inequalities or Equalities, in order to nd the best solution to this problem. For this show in this paper methods of problem solving Linear Programming. There is an emphasis on geometric solutions and Simplex Method, to form algebraic solution. Wanted to show various situations which may t some of these problems, some general cases more speci c cases. Before arriving eventually in solving linear programming problems, builds up the eld work of this type of optimization, Convex Sets. There are presentations of de nitions and theorems essential to the understanding and development of these problems, besides discussions on the e ciency of the methods applied. During the work, it is shown that there are cases which do not apply the solutions presented, but mostly t e ciently, even as a good approximation.Item A curvatura Gaussiana via ângulo de contato de superfícies imersas em S3(Universidade Federal de Goiás, 2015-02-27) Argote, Fernando Arnulfo Zuñiga; Corro, Armando Mauro Vasquez; http://lattes.cnpq.br/4498595305431615; Corro, Armando Mauro Vasquez; Santos, João Paulo dos; Pieterzack, Maurício DonizettiIn this work we refer to the study of a geometric invariant surfaces immersed in Euclidean 3-dimensional sphere S3. Such invariant, known as angle contact, is the complementary angle between the distribution of contact d and the tangent space of the surface. Montes and Verderesi [22] characterized the minimal surfaces in S3 with constant contact angle and Almeida, Brazil and Montes [4] studied some properties of immersed constant mean curvature into a round sphere S3 with constant contact angle. The our aim of this work is to deduce a general formula involving the Gaussian curvature, the mean curvature and the contact angle of surfaces immersed in Euclidean sphere 3-dimensional, which shows that the surface is flat if the contact angle is constant. Moreover, we deduce that the Clifford tori are the unique compact surfaces with constant mean curvature having such propriety. KeywordsItem 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 Fluidos perfeitos estáticos com simetrias(Universidade Federal de Goiás, 2019-04-25) Barboza, Marcelo Bezerra; Pina, Romildo da Silva; http://lattes.cnpq.br/2675728978857991; Corro, Armando Mauro Vasquez; Leandro Neto, Benedito; Manfio, Fernando; Marrocos, Marcus Antônio Mendonça; Pina, Romildo da SilvaThis work presents a two step procedure that is virtually capable of producing an infinite number of exact solutions to Einstein's equation of a perfect fluid on a static manifold. These steps could roughly be described as: 1) classifying the symmetries of the referred equation that convert it into a second order non linear ordinary differential equation of very specific nature -- whose solutions are a whole lot easier to come up with than those of the original problem, and 2) solving this ordinary equation -- which quite explains the need for the word `virtually' above, since not all solutions of the ordinary equation are known to its exact form. Finally, in the last chapter, we utilize a Theorem due to Liouville to determine the rigid motions of Riemannian metrics on euclidean space that do admit symmetries in a translational group and also belong to the conformal class of the flat metric.Item Sobre uma classe de álgebras associadas a duas famílias de grafos orientados(Universidade Federal de Goiás, 2015-03-02) Barboza, Marcelo Bezerra; Lima, Aline de Souza; Silva, Jhone Caldeira; http://lattes.cnpq.br/6848751340618892; Silva, Jhone Caldeira; Lima, Aline de Souza; Chagas, Sheila Campos; Oliveira, Ricardo Nunes deGiven a directed layered graph , we present the algebra A() as a quotient of the free associative or tensor algebra (with unit, over an arbitrarily fixed field of scalars), freely generated by the set of edges in . We calculate the Hilbert series associated with the grading on A() coming from degree in the tensor algebra. We also calculate the group of automorphisms of A() that preserve the (ascending) filtration associated with the grading mentioned above. Despite the fact the main results within this notes remain true for a relatively large class of directed graphs, we stay close to the ones Dn and Ln, n 3, that is, those consisting, respectively, on the Hasse diagram of the partially ordered sets of faces in a regular polygon containing n edges and the power set of {1, . . . , n}. The work teaching us all of the above is [1], by Colleen Duffy.Item Sequências monótonas aplicadas a um problema de cauchy para um sistema de reação-difusão-convecção(Universidade Federal de Goiás, 2015-08-07) Barros, Carlos Eduardo Rosado de; Mota, Jesus Carlos da; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4787068H6; Mota, Jesus Carlos da; Bastos, Waldemar Donizete; Silva, Maxwell LizeteIn this work, mainly based on the articles, [1], [2] and [7], one studies a reactiondiffusion- convection system, related the propagation of a combustion front through a porous medium, giving origin a Cauchy problem. Such a problem has been approached by the methode of the monotone iterations, which leads to an unique time-global solution.Item Generalized vector equilibrium problems and algorithms for variational inequality in hadamard manifolds(Universidade Federal de Goiás, 2016-10-20) Batista, Edvaldo Elias de Almeida; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Ferreira, Orizon Pereira; http://lattes.cnpq.br/0201145506453251; Bento, Glaydston de Carvalho; http://lattes.cnpq.br/1089906772427394; Cruz Neto, João Xavier da; Peréz, Luís Román Lucambio; Alves, Maicon MarquesIn this thesis, we study variational inequalities and generalized vector equilibrium problems. In Chapter 1, several results and basic definitions of Riemannian geometry are listed; we present the concept of the monotone vector field in Hadamard manifolds and many of their properties, besides, we introduce the concept of enlargement of a monotone vector field, and we display its properties in a Riemannian context. In Chapter 2, an inexact proximal point method for variational inequalities in Hadamard manifolds is introduced, and its convergence properties are studied; see [7]. To present our method, we generalize the concept of enlargement of monotone operators, from a linear setting to the Riemannian context. As an application, an inexact proximal point method for constrained optimization problems is obtained. In Chapter 3, we present an extragradient algorithm for variational inequality associated with the point-to-set vector field in Hadamard manifolds and study its convergence properties; see [8]. In order to present our method, the concept of enlargement of maximal monotone vector fields is used and its lower-semicontinuity is established to obtain the convergence of the method in this new context. In Chapter 4, we present a sufficient condition for the existence of a solution to the generalized vector equilibrium problem on Hadamard manifolds using a version of the KnasterKuratowski-Mazurkiewicz Lemma; see [6]. In particular, the existence of solutions to optimization, vector optimization, Nash equilibria, complementarity, and variational inequality is a special case of the existence result for the generalized vector equilibrium problem.Item Gradiente ricci solitons e variedades de Einstein com métrica produto torcido(Universidade Federal de Goiás, 2016-03-31) Batista, Elismar Dias; Souza, Marcelo Almeida de; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4797616Y0; Pina, Romildo da Silva; http://buscatextual.cnpq.br/buscatextual/visualizacv.do?id=K4782462D0; Pina, Romildo da Silva; Souza, Marcelo Almeida dea; Santos, João Paulo dosThis work is based on the articles [26] and [27], where we studied Einstein manifolds and gradient Ricci soliton with twisted product structure. As a result, we prove the following: if M is an Einstein warped product space with nonpositive scalar curvature and compact base, then M is a Riemannian product space. Besides, we show that the Riemannian product Rp×F is a gradient Ricci soliton if and only if F is Ricci soliton gradient. Then, we show that the warped product R×f B is gradient Ricci solitons with f ′′ 6= 0, therefore F is Einstein. By using these results, we build nontrivial examples of gradient Ricci soliton where the fiber is either an Einstein manifold or a nontrivial gradient Ricci soliton.Item Soluções clássicas para um problema de combustão em meios porosos com n camadas(Universidade Federal de Goiás, 2019-08-16) Batista, Marcos Roberto; Mota, Jesus Carlos da; http://lattes.cnpq.br/8457974658695539; Mota, Jesus Carlos da; Silva, Edcarlos Domingos da; Carvalho, Marcos Leandro Mendes; Ercole, Grey; Santos, Marcelo Martins dosIn this work, we study the classical solutions for a parabolic system of reaction-diffusion-convection equations, coupled to a system of ordinary differential equations, with boundary and initial conditions in a bounded domain. The coupled system models the propagation of a combustion front through a porous medium with n layers, where the dependent variables are the temperatures and the fuel concentrations in each layer. Problems for parabolic equations systems coupled with Ordinary Differential Equations (ODEs) system, where the coupling occurs in both the reaction functions and the associated differential operator coefficients, are little known in the literature. In classical theory in general, the coupling appears only in the reaction functions. Initially, using the Monotone Iterative Method, we prove the existence and uniqueness of a global solution in time for the particular case where the fuel concentrations in each layer are known functions. Next, we show the existence of the local solution in time for the complete problem when the concentrations are unknown functions. The proof is obtained by defining an operator in the Continuous Hölder function set and showing that it has a fixed point that is a local solution to the problem. This solution can be extended to a global solution in time for the problem, provided that the spatial derivatives of the temperature in each layer are bounded functions.