BDTD - Biblioteca Digital de Teses e Dissertações
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Doctoral Thesis Fundamentos de uma análise matemática fuzzy baseada em números fuzzy e ordens admissíveis(Universidade Federal do Rio Grande do Norte, 2024-02-26) Cárcamo, Nicolás Eduardo Zumelzu; Bedregal, Benjamin René Callejas; Villarroel, José Edmundo Mansilla; http://lattes.cnpq.br/4601263005352005; Santiago, Regivan Hugo Nunes; Matamala, Roberto Antonio Díaz; Dimuro, Graçaliz Pereira; Paiva, Rui Eduardo BrasileiroThe notion of admissible orders in interval fuzzy logic emerged in 2010 with the aim of providing a minimum criterion that a total order in the set of closed subintervals of the unitary interval [0,1] should meet to be used in applications of this fuzzy theory. Later, this same idea was adapted to other extensions of fuzzy logic. In this thesis, we take the idea of admissible orders outside the context of extensions of fuzzy logic. In fact, here we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, that is, a total order that refines this partial order. We pay special attention to the partial order proposed by Ram´ık and R´ımanek in 1985. Furthermore, we present a method to construct admissible orders over fuzzy ´ numbers from admissible orders defined for intervals, considering a upper dense sequence, and we prove that these orders are admissible with respect to Ram´ık and R´ımanek ordering. ´ From such admissible orders we study fundamental concepts of Mathematical Analysis in the context of fuzzy numbers. The objective is to take the first steps towards the development of a mathematical analysis of fuzzy numbers in certain admissible orders in a robust and wellfounded way, preserving as much as possible properties of traditional mathematical analysis. In this way, we introduce the notion of Riemann integral over fuzzy numbers, called fuzzy Riemann integral, considering admissible orders, and we study properties and characterizations of this integral. We formalize the concepts of vector space without inverses and ordered vector space without inverses, a type of hyperstructures, which generalizes the conventional notion of ordered vector spaces. It is worth noting that the space of triangular fuzzy numbers (TFN) and TFNs with some orders are examples of both hyperstructures. Furthermore, we introduce the notion of increasing functions of average type over fuzzy numbers equipped with admissible orders in general, characterizing them as idempotent, and in particular, in the ordered vector space without inverses. Finally, we introduce the concept of weighted vector-fuzzy graphs and use tools built from average-like functions in the ordered vector space without inverses, to solve types of shortest path problems in weighted graphs.Master Thesis Construção de um modelo molecular magnético: uma proposta para o ensino da ligação química(Universidade Federal do Rio Grande do Norte, 2023-12-11) Barreto, Wallyson Rodrigo da Silva; Silva Júnior, Carlos Neco da; https://orcid.org/0000-0001-9015-1934; http://lattes.cnpq.br/0917006004895117; http://lattes.cnpq.br/6306294560730177; Pontes, Ana Cristina Facundo de Brito; Hussein, Fabiana Roberta Gonçalves e Silva; Oliveira, Otom Anselmo de; https://orcid.org/0000-0003-0409-4000; http://lattes.cnpq.br/7903084274135177In this work, a prototype called “Magnetic Molecular Model” was developed, produced from plastic spheres in 3D printers with magnets inserted internally inside each sphere to simulate the attractive forces that exist between atoms in the formation of a chemical bond. The material was designed with the aim of minimizing learning difficulties related to understanding the formation of chemical bonds between atoms, which students normally do not understand that there are attractive and repulsive forces in the formation of the bond. To think about production, an investigation was carried out in different Chemistry textbooks, whether at higher or secondary level (Plano Nacional do Livro Didático, 2007), to identify the highest frequency of examples of molecules that form chemical bonds that appear in the books quite frequently. After this stage, the educational product was applied to 15 Chemistry teachers in initial training and 13 in continuing training in the form of a mini-course. As a first result, it was possible to verify the acceptance of the model by the participants who stated that through its use it is possible to work on concepts related to chemical bonds, such as molecular geometry, electronegativity and the different bonds that occur between atoms. Still as the main result, the participants brought that the sensorial perception of the attractive and repulsive forces existing between structures analogous to chemical molecules, not presented in other molecular models present in the commercial market and that this factor is highlighted for the use of this resource in the Basic Education classroom.