https://repositorio.ufu.br/handle/123456789/5473 Fri, 17 Apr 2026 04:23:32 GMT 2026-04-17T04:23:32Z https://repositorio.ufu.br/handle/123456789/48576 Title: A fatoração de xⁿ-1 em binômios e trinômios irredutíveis sobre os corpos com 2 e 4 elementos Abstract: Let q be a prime power and Fq the finite field with q elements. The factorization of polynomials over finite fields plays a fundamental role in several areas, such as error-correcting coding theory and cryptography. A particularly important polynomial is xⁿ-1, since, for example, each of its irreducible factors corresponds to a cyclic code. Under certain conditions, the polynomial xⁿ-1 factors exclusively into irreducible binomials and trinomials over Fq. In this case, we say that the pair (n, q) is 3-sparse. The aim of this work is to give a complete classification of the pairs (n, 2) and (n, 4) that are 3-sparse. Thu, 19 Feb 2026 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/48576 2026-02-19T00:00:00Z https://repositorio.ufu.br/handle/123456789/46700 Title: Alguns critérios de caoticidade de operadores em espaços de Fréchet Abstract: In this work, we study some dynamical properties of continuous linear operators on Fréchet spaces, focusing on the concepts of hypercyclicity, chaoticity, and frequent hypercyclicity. We study important conditions and results that characterize hypercyclic operators, chaotic operators, and frequently hypercyclic operators, and we explore the Hypercyclicity Criterion, Kitai Criterion, Gethner–Shapiro Criterion, Chaoticity Criterion, and the Frequent Hypercyclicity Criterion. We study the relationship that these criteria have with other notions of chaos, especially topological transitivity, as well as the notions of mixing and weakly mixing. Furthermore, we present examples, in each one of these cases, to show that such criteria provide sufficient, but not necessary, conditions for determining each of these notions. Wed, 30 Jul 2025 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/46700 2025-07-30T00:00:00Z https://repositorio.ufu.br/handle/123456789/46683 Title: Uma introdução aos polinômios sobre corpos finitos Abstract: In this work, we study topics in finite field theory. Initially, we explore the structure of finite fields, covering from basic properties to representations of their elements, such as polynomial, cyclotomic, and matrix representations. Next, we focus on the study of polynomials over finite fields, highlighting: the order-defined polynomials, linked to the structure of multiplicative groups; the primitive polynomials; and the irreducible polynomials, fundamental for factorization and construction of field extensions. Finally, we examine q-polynomials, which generalize classical polynomial structures. Thu, 24 Jul 2025 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/46683 2025-07-24T00:00:00Z https://repositorio.ufu.br/handle/123456789/45330 Title: Uma nova abordagem da espaçabilidade em conjuntos de sequências e funções Abstract: In this work we explore a new approach to spaceability in sequences and functions sets. We investigate this more restrictive notion in certain sets of absolutely p-summable sequences and measurable functions. Furthermore, we study the spaceability of sets of surjective and injective operators between sequence spaces. Wed, 26 Mar 2025 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/45330 2025-03-26T00:00:00Z