DSpace Coleção: PAPGM

A search for linearity in the universe of topological vector spaces

2026-05-17T06:10:36Z

Título: A search for linearity in the universe of topological vector spaces Autor(es): Ribeiro, Geivison dos Santos Orientador: Pellegrino, Daniel Marinho Abstract: This thesis provides criteria (both negative and positive) that contribute to the existing literature and address open problems within the notions of (α, β)-lineability/spaceability, (α, β)-dense lineability, pointwise lineability, and [S]-lineability. In our exploration, we began by investigating the behavior of algebraic and topological structures present in the set of unbounded, continuous, and integrable functions on the interval [0, ∞). This investigation was initiated by Calderón-Moreno, Gerlach-Mena, and Prado-Bassas, where they demonstrated, among other results, that the set A := f ∈ C [0, ∞) ∩ L1 [0, ∞) : lim sup x→∞ |f (x)| = ∞ is lineable. To better understand the dimensional relationships in this environment, we employed new techniques and gained additional insights into both the topological and algebraic structure of this set. Specifically, we proved its pointwise spaceability (and thus, spaceability). Additionally, we demonstrated that the set Lp[0, 1] \ S q∈(p,∞)Lq[0, 1], para p ∈ (0, ∞), although (1,c)-spaceable (see [21]), is not (א0,c)-spaceable. We established a general criterion for negative results concerning (α, β)-spaceability and verified that the set N D[0, 1] of nowhere differentiable functions cannot be (α, β)-spaceable for any infinite cardinal α. We also provided criteria for positive results, showing in particular that the sets l∞ \ F, where F ∈ {c, c0}, and Lp[0, 1] \ S q∈(p,∞)Lq[0, 1], for p ∈ (0, ∞), are (α,c)-espaceable if and only if α is finite. We introduced the notion of (α, β)-dense lineability and provided a criterion to demonstrate in particular that the set Lp[0, 1] \ S q∈(p,∞)Lq[0, 1], for p ∈ (0, ∞) is (α, β)- dense lineable for every 0 ≤ α ≤ β and max {α, א0} ≤ β ≤ c. Our findings highlight that the geometry of the studied sets alone is insufficient and that the type of topology considered in each environment also plays a crucial role. Editor: Universidade Federal da Paraíba Tipo: Tese

Sombreamento e estabilidade estrutural forte em Dinâmica Linear

2026-02-27T06:08:44Z

Título: Sombreamento e estabilidade estrutural forte em Dinâmica Linear Autor(es): Azevedo, Mateus Vinicius Santos de Orientador: Costa Júnior, Fernando Vieira Abstract: In this work, we present the class of hyperbolic operators and their connection with the well-established concepts of shadowing and strong structural stability in discrete dynamical systems and ergodic theory. We then examine a characterization of weighted shift operators on c0 or ℓp spaces (where 1 ≤ p < ∞) satisfying the shadowing property, as demonstrated by N. C. Bernardes Jr. and A. Messaoudi (2020). Furthermore, we analyze how F. Bayart (2021) established that, in this framework, the shadowing property is equivalent to strong structural stability. Editor: Universidade Federal da Paraíba Tipo: Dissertação

Applications of geometric identities to rigidity problems

2026-04-21T01:29:43Z

Título: Applications of geometric identities to rigidity problems Autor(es): Araújo, Murilo Chavedar de Souza Orientador: Freitas, Allan George de Carvalho Abstract: In this thesis, we explore the applications of integral identities, such as the Reilly-type and Pohozaev-type identities, in various geometric contexts, highlighting their roles in obtaining inequalities and rigidity results for specific classes of Riemannian manifolds. First, we consider the context of V -static manifolds, which are Riemannian manifolds with boundary, constant scalar curvature, and a metric that is a critical point of the volume functional with a fixed boundary metric. In this context, we employ our Reilly-type identity to establish Heintze-Karcher and Minkowski inequalities for bounded domains. Furthermore, we examine the rigidity phenomena associated with these inequalities, especially in cases where equality is achieved, shedding light on the geometric structure of these manifolds. Additionally, we obtain an inequality for domains in m-quasi Einstein manifolds along with a rigidity characterization. This inequality is motivated by the stability of the Wang-Yau energy. Finally, we turn our attention to weighted overdetermined problems on Riemannian manifolds with density. By studying a Poisson problem associated with the weighted Laplacian, we derive a Heintze-Karcher inequality and a Soap Bubble-type theorem that characterize geodesic balls in these weighted spaces. By imposing Dirichlet and Neumann boundary conditions, we also establish a Serrin-type result in generalized cones and convex cones of Euclidean space, identifying metric balls as the unique solutions to the underlying overdetermined problem. Editor: Universidade Federal da Paraíba Tipo: Tese

Nested Hilbert schemes on Hirzebruch surfaces and quiver varieties

2024-12-05T06:05:08Z

Título: Nested Hilbert schemes on Hirzebruch surfaces and quiver varieties Autor(es): Santos, Pedro Henrique dos Orientador: Bruzzo, Ugo Abstract: Hilbert schemes were introduced by Grothendieck. They are a fundamental example of the notion of moduli spaces of geometric structures. The work of Nakajima on the properties of the Hilbert schemes of points of the complex plane has been the basis of many works that try to understand the properties of Hilbert schemes of other 2- dimensional varieties and also for higher dimensions. Furthermore, the nested Hilbert scheme of points on the complex plane was studied by von Flach, Jardim and Lanza. Moreover, Bartocci, Bruzzo, Lanza and Rava obtained a quiver description to the Hilbert scheme of points of the total space Ξn of appropriate line bundles over the projective line. In this work we show that the nested Hilbert scheme of points on the last varieties, parameterizing pairs of nested 0-cycles, is the quiver variety associated with a suitable quiver with relations, generalizing previous work about nested Hilbert schemes on the complex plane, in one direction, and about the Hilbert schemes of points of Ξn in another direction. Editor: Universidade Federal da Paraíba Tipo: Tese