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Item Temas transversais na formação de professores de matemática : uma ação formativa.(2022) Torisu, Edmilson Minoru; Ferreira, Wenderson MarquesO presente texto tem como objetivo apresentar uma ação formativa realizada com estudantes do curso de licenciatura em Matemática de uma universidade pública brasileira, como parte de uma componente curricular voltada a discussões de temas transversais. Além disso, o estudo investigou impressões dos (as) licenciandos (as) acerca dos impactos de suas participações na ação formativa, para as suas futuras atuações como docentes. Os resultados revelaram que os (as) licenciandos (as) percebem e reconhecem a relevância das discussões acerca dos temas transversais, ressaltando sua importância como oportunidade para se conhecer o contexto do aluno; para se reconhecer e respeitar as diferenças e a elas responder, para evitar processos de exclusão.Item On the direct and inverse zero-sum problems over Cn ⋊s C2.(2022) Avelar, Danilo Vilela; Brochero Martinez, Fabio Enrique; Ribas, SávioLet Cn be the cyclic group of order n. In this paper, we provide the exact values of some zero-sum constants over Cn ⋊s C2 where s 6≡ ±1 (mod n), namely η-constant, Gao constant, and Erdős- Ginzburg-Ziv constant (the latter for all but a “small” family of cases). As a consequence, we prove the Gao’s and Zhuang-Gao’s Conjectures for groups of this form. We also solve the associated inverse problems by characterizing the structure of product-one free sequences over Cn ⋊s C2 of maximum length.Item A note on Bass’ conjecture.(2023) Avelar, Danilo Vilela; Brochero Martinez, Fabio Enrique; Ribas, SávioFor a finite group G, we denote by d(G) and by E(G), respectively, the small Davenport constant and the Gao constant of G. Let Cn be the cyclic group of order n and let Gm,n,s = Cn s Cm be a metacyclic group. In [2, Conjecture 17], Bass conjectured that d(Gm,n,s) = m + n − 2 and E(Gm,n,s) = mn + m + n − 2 provided ordn(s) = m. In this paper, we show that the assumption ordn(s) = m is essential and cannot be removed. Moreover, if we suppose that Bass’ conjecture holds for Gm,n,s and the mn-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2 ≡ s (mod n).Item Extremal product-one free sequences over Cn s C2.(2022) Brochero Martinez, Fabio Enrique; Ribas, SávioLet G be a finite group multiplicatively written. The small Davenport constant of G is the maximum positive integer d(G) such that there exists a product-one free sequence S of length d(G). Let s2 ≡ 1 (mod n), where s ≡ ±1 (mod n). It has been proven that d(Cn s C2) = n (see [13, Lemma 6]). In this paper, we determine all sequences over Cn s C2 of length n which are product-one free. It completes the classification of all product-one free sequences over every group of the form Cn s C2, including the quasidihedral groups and the modular maximal-cyclic groups.Item Existence and nonexistence of solutions to nonlocal elliptic problems.(2022) Bueno, Hamilton Prado; Pereira, Gilberto de Assis; Silva, Edcarlos Domingos da; Ruviaro, RicardoIt is established existence and nonexistence of solutions to nonlocal elliptic problems involving the generalized pseudo-relativistic Hartree equation. Our arguments are based on variational methods together with a fine analysis on the Pohozaev identity.Item Results on a strongly coupled, asymptotically linear pseudo-relativistic Schrödinger system : ground state, radial symmetry and Hölder regularity.(2022) Bueno, Hamilton Prado; Mamani, Guido Gutierrez; Medeiros, Aldo Henrique de Souza; Pereira, Gilberto de AssisIn this paper we consider the asymptotically linear, strongly coupled nonlinear system ⎧ ⎪⎨ ⎪⎩ √ −∆ + m2 u = u 2 + v 2 1 + s(u2 + v 2) u + λv, √ −∆ + m2 v = u 2 + v 2 1 + s(u2 + v 2) v + λu, where m > 0, 0 < λ < m and 0 < s < 1/(λ + m) are constants. By applying the Nehari–Pohozaev manifold, we prove that our system has a ground state solution. We also prove that solutions of this system are radially symmetric and belong to C0,μ(RN ) for some 0 < μ < 1 and each N > 1.Item Pohozaev-type identities for a pseudo-relativistic schrodinger operator and applications.(2019) Bueno, Hamilton Prado; Pereira, Gilberto de Assis; Medeiros, Aldo Henrique de SouzaIn this paper we prove a Pohozaev-type identity for both the prob- lem (−∆ + m2 ) su = f(u) in RN and its harmonic extension to R N+1 + when 0 < s < 1. So, our setting includes the pseudo-relativistic operator √ −∆ + m2 and the results showed here are original, to the best of our knowledge. The identity is first obtained in the extension setting and then “translated” into the original problem. In order to do that, we develop a specific Fourier trans- form theory for the fractionary operator (−∆ + m2 ) s , which lead us to define a weak solution u of the original problem if the identity (S) Z RN (−∆ + m2 ) s/2u(−∆ + m2 ) s/2 vdx = Z RN f(u)vdx is satisfied by all v ∈ Hs (RN ). The obtained Pohozaev-type identity is then applied to prove both a result of nonexistence of solution to the case f(u) = |u| p−2u if p ≥ 2 ∗ s and a result of existence of a ground state, if f is modeled by κu3/(1+u 2 ), for a constant κ. In this last case, we apply the Nehari-Pohozaev manifold introduced by D. Ruiz. Finally, we prove that positive solutions of (−∆ + m2 ) su = f(u) are radially symmetric and decreasing with respect to the origin, if f is modeled by functions like t α, α ∈ (1, 2 ∗ s − 1) or tln t.Item Demonstrações dos teoremas de Ceva, Menelaus e Desargues no Plano de Argand-Gauss.(2022) Cruz, Breno Arcanjo Fernandes da; Souza, Gil Fidelix de; Ferreira, Geraldo César GonçalvesNeste trabalho fazemos uso da boa estrutura do conjunto dos números complexos, essencialmente de sua geometria, para promover o estudo de objetos e a obtenção de resultados da Geometria Plana. A boa estrutura do conjunto dos números complexos permite o estudo de resultados clássicos, por exemplo, os Teoremas de Ceva, Menelaus e Desargues. O primeiro Teorema estabelece condições necessárias e suficientes para que três cevianas sejam concorrentes, o segundo estabelece condições de colinearidade para um conjunto de pontos, ou para a concorrência de um conjunto de segmentos e o terceiro refere-se a triângulos projetivos e pode ser visto como uma consequência do primeiro resultado.Item De Pick a Euler : exemplos e demonstrações.(2022) Gomides, Amanda Figueiredo; Ferreira, Geraldo César GonçalvesPara além das formas tradicionais de cálculo de áreas de figuras planas, existem métodos pouco explorados e difundidos. O teorema de Pick é um desses métodos, que relaciona número de pontos internos e de borda de um polígono simples (sem buracos e cujos lados não se cruzam) inscrito em uma malha quadriculada para calcular sua área. E a partir da associação deste teorema com a Fórmula de Euler para figuras planas poligonais, que relaciona seu número de faces, arestas, vértices e buracos, podemos criar uma generalização do Teorema de Pick que abrange polígonos não simples, desde que inscritos em uma malha quadriculada.Item Soluções para a equação geral de Poisson unidimensional.(2022) Figueiredo, Joyce Kelly; Martins, Eder Marinho; Ferreira, Wenderson MarquesA Equação Geral de Poisson Unidimensional é uma equação diferencial com diversas aplicações físicas. À vista disso, há interesse em saber se o problema de valores de contorno definido a partir dela possui solução. Nesse trabalho, provaremos a existência de solução positiva e obteremos uma fórmula geral para esta. A abordagem que apresentamos aqui é diferente da usual vista na literatura.Item Algumas contribuições do GeoGebra para uma abordagem corporificada da convergência de sequências numéricas no cálculo.(2013) Fonseca, Daila Silva Seabra de Moura; Franchi, Regina Helena de Oliveira LinoEste artigo apresenta algumas contribuições do software GeoGebra para a criação de ambientes de exploração, de modo a corporificar a convergência de sequências numéricas. Discute atividades desenvolvidas em uma turma de Cálculo como parte de uma pesquisa de Mestrado que teve como principais referenciais teóricos o Pensamento Matemático Avançado e os Três Mundos da Matemática. Os resultados indicam que o uso do GeoGebra contribuiu para a corporificação da convergência, na medida em que tornou possível a construção e a manipulação das sequências de forma dinâmica, e também para o estabelecimento de relações entre suas diferentes representações, por meio da conexão entre as janelas de visualização algébrica, gráfica e numérica.Item On a singular minimizing problem.(2018) Ercole, Grey; Pereira, Gilberto de AssisFor each q ∈ (0, 1) let λq(Ω) := inf k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and Z Ω |v| q dx = 1, where p > 1 and Ω is a bounded and smooth domain of R N , N ≥ 2. We first show that 0 < μ(Ω) := lim q→0+λq(Ω)|Ω| p q < ∞, where |Ω| = R Ω dx. Then, we prove that μ(Ω) = min (k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and lim q→0+ 1 |Ω| Z Ω |v| q dx 1 q = 1) and that μ(Ω) is reached by a function u ∈ W1,p 0 (Ω), which is positive in Ω, belongs to C 0,α(Ω), for some α ∈ (0, 1), and satisfies − div(|∇u| p−2 ∇u) = μ(Ω)|Ω| −1 u −1 in Ω, and Z Ω log udx = 0. We also show that μ(Ω)−1 is the best constant C in the following log-Sobolev type inequality exp 1 |Ω| Z Ω log |v| p dx ≤ C k∇vk p Lp(Ω) , v ∈ W1,p 0 (Ω) and that this inequality becomes an equality if, and only if, v is a scalar multiple of u and C = μ(Ω)−1.Item Extremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness.(2022) Costa, Felipe; Souza, Gil Fidelix de; Montenegro, MarcosThe paper concerns positive solutions for the Dirichlet problem −Lu = ΛF(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n ≥ 2, u = (u1, ..., um) : Ω → Rm, m ≥ 1, Lu = (L1u1, ..., Lmum), where each Li denotes a uniformly elliptic linear operator of second order in nondivergence form in Ω, Λ = (λ1, ..., λm) ∈ Rm, F = (f1, ..., fm) : Ω × Rm → Rm and ΛF(x, u) = (λ1f1(x, u), ..., λmfm(x, u)). For a general class of maps F we prove that there exists a hypersurface Λ∗ in Rm + := (0, ∞)m such that tuples Λ ∈ Rm + below Λ∗ correspond to minimal positive strong solutions of the above system. Stability of these solutions is also discussed. Already for tuples above Λ∗, there is no nonnegative strong solution. The shape of the hypersurface Λ∗ depends on growth on u of the nonlinearity F in a sense to be specified. When Λ ∈ Λ∗ and the coefficients of each operator Li are slightly smooth, the problem admits a unique minimal nonnegative weak solution, called extremal solution. Furthermore, when F depends only on u and all Li are Laplace operators, we investigate the L∞ regularity of this solution for any m ≥ 1 in dimensions 2 ≤ n ≤ 9 for balls and n = 2 and n = 3 for convex domains.Item Existence of a positive solution for a class of non-local elliptic problem with critical growth in Rn.(2022) Leme, Leandro Correia Paes; Rodrigues, Bruno MendesIn this article, we consider the following non-local elliptic equation with critical growth ⎧⎪⎨⎪⎩− a + b RN |∇u| 2 dx p−1 2 Δu = λk(x)uq + u2∗−1, x ∈ RN , u ∈ D1,2(RN ), where N ≥ 3, λ > 0, 2∗:= 2N N−2 , 1 < p ≤ q < 2∗ − 1, a ≥ 0, b ≥ 0 and k(x) ∈ L 2∗ 2∗−q−1 (RN ) is a nonnegative function. Using variational methods and concentration-compactness principle, we obtain a positive solution.Item Existence and multiplicity results for an elliptic problem involving cylindrical weights and a homogeneous term μ.(2019) Assunção, Ronaldo Brasileiro; Miyagaki, Olimpio Hiroshi; Leme, Leandro Correia Paes; Rodrigues, Bruno MendesWe consider the following elliptic problem ⎧⎨ ⎩ − div |∇u| p−2 ∇u |y| ap = μ |u| p−2 u |y| p(a+1) + h(x) |u| q−2 u |y| bq + f(x, u) in Ω, u = 0 on ∂Ω, in an unbounded cylindrical domain Ω := {(y, z) ∈ Rm+1 × RN−m−1 ; 0 1, 1 ≤ mItem Asymptotic behaviour as p → ∞ of least energy solutions of a (p, q(p))-Laplacian problem.(2019) Alves, Claudianor Oliveira; Ercole, Grey; Pereira, Gilberto de AssisWe study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem −(Δp + Δq(p))u = λp|u(xu)| p−2u(xu)δxu in Ω u = 0 on ∂Ω, where xu is the (unique) maximum point of |u|, δxu is the Dirac delta distribution supported at xu, limp→∞ q(p) p = Q ∈ (0, 1) if N 0 is such that min ∇u∞ u∞ : 0 ≡ u ∈ W1,∞(Ω) ∩ C0(Ω) limp→∞(λp) 1/p < ∞.Item Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation.(2019) Belchior, Pedro; Bueno, Hamilton Prado; Miyagaki, Olimpio Hiroshi; Pereira, Gilberto de AssisAbstract. With appropriate hypotheses on the nonlinearity f , we prove the existence of a ground state solution u for the problem − + m2u + V u = W ∗ F (u) f (u) in RN, where V is a bounded potential, not necessarily continuous, and F the primitive of f . We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.Item On spacetime foliations and electromagnetic knots.(2022) Silva, Wanderson Costa e; Costa, Érico Goulart de Oliveira; Ottoni, José EloyThe present paper has a number of distinct purposes. First is to give a description of a class of electromagnetic knots from the perspective of foliation theory. Knotted solutions are then interpreted in terms of two codimension-2 foliations whose knotted leaves intersect orthogonally everywhere in spacetime. Secondly, we show how the foliations give rise to field lines and how the topological invariants emerge. The machinery used here emphasizes intrinsic properties of the leaves instead of observer dependent quantities—such as a time function, a local rest frame or a Cauchy hypersurface. Finally, we discuss the celebrated Hopf–Rañada solution in details and stress how the foliation approach may help in future developments of the theory of electromagnetic knots. We conclude with several possible applications, extensions and generalizations.Item Generators of finite fields with prescribed traces.(2021) Reis, Lucas; Ribas, SávioThis paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.Item The {1, s}-weighted Davenport constant IN Ckn.(2021) Brochero Martinez, Fabio Enrique; Ribas, SávioLet G be a finite abelian group and let ∅ 6= A ⊂ Z. The A-weighted Davenport constant of G is the smallest positive integer DA(G) such that every sequence x1 · . . . · xDA(G) over G has a non-empty subsequence (xji )i such that ε1xj1 + ε2xj2 + . . . + εtxjt = 0 for some ε1, ε2, . . ., εt ∈ A. In this paper, we obtain both upper and lower bounds for D{1,s}(Ck n), where Cn denotes the cyclic group of order n, s2 ≡ 1 (mod n) and s 6≡ ±1 (mod n). These bounds become sharp in some “small” cases.