Relação de alcance em dígrafos transitivos e a propriedade Z

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2016.

Optimal reparametrizations in the square root velocity framework

The square root velocity framework is a method in shape analysis to define a distance between curves and functional data. Identifying two curves, if the differ by a reparametrization leads to the quotient space of unparametrized curves. In this paper we study analytical and topological aspects of this construction for the class of absolutely continuous curves. We show that the square root velocity transform is a homeomorphism and that the action of the reparametrization semigroup is continuous. We also show that given two $C^1$-curves, there exist optimal reparametrizations realising the minimal distance between the unparametrized curves represented by them.

Rota's universal operators and invariant subspaces in Hilbert spaces

A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Semigrupos numéricos

Um semigrupo numérico é um submonoide de (N, +) tal que o seu complementar em N é finito. Neste trabalho estudamos alguns invariantes de um semigrupo numérico S tais como: multiplicidade, dimensão de imersão, número de Frobenius, falhas e conjunto Apéry de S. Caracterizamos uma apresentação minimal para um semigrupo numérico S e descrevemos um método algorítmico para determinar esta apresentação. Definimos um semigrupo numérico irredutível como um semigrupo numérico que não pode ser expresso como intersecção de dois semigrupos numéricos que o contenham propriamente. A finalizar este trabalho, estudamos os semigrupos numéricos irredutíveis e obtemos a decomposição de um semigrupo numérico em irredutíveis. ABSTRACT: A numerical semigroup is a submonoid of (N, +) such that its complement of N is finite. ln this work we study some invariants of a numerical semigroup S such as: multiplicity, embedding dimension, Frobenius number, gaps and Apéry set of S. We characterize a minimal presentation of a numerical semigroup S and describe an algorithmic procedure which allows us to compute a minimal presentation of S. We define an irreducible numerical semigroup as a numerical semigroup that cannot be expressed as the intersection of two numerical semigroups properly containing it. Concluding this work, we study and characterize irreducible numerical semigroups, and describe methods for computing decompositions of a numerical semigroup into irreducible numerical semigroups.