http://hdl.handle.net/10174/31 2026-08-27T07:20:50Z http://hdl.handle.net/10174/42518 Title: Factorisation of the Classical Nonstandard Bounded Functional Interpretation Authors: Dinis, Bruno; Gaspar, Jaime Editors: Bezzeghoud, Mourad; Carapau, Fernando; Minhós, Feliz; Vaidya, Ashwin Abstract: Functional interpretations are maps of formulas from the language of one theory into the language of another theory, in such away that provability is preserved, proofs are (essentially) translated into proofs, and information about the proofs is gathered. Functional interpretations have many uses, such as: relative consistency results, conservation results and the extraction of computational content from proofs. We prove the factorisation Ust = Kst Bst of Jaime Gaspar and Fernando Ferreira’s classical nonstandard bounded functional interpretation Ust in terms of an extension of Jean-Louis Krivine’s negative translation Kst and Bruno Dinis and Jaime Gaspar’s intuitionistic nonstandard bounded functional interpretation Bst. We also give some applications of the factorisation. 2026-01-01T00:00:00Z http://hdl.handle.net/10174/42198 Title: On the intersection of fixed subgroups of Fn × F m Authors: Carvalho, André Editors: Diekert, Volker Abstract: We prove that, although it is undecidable if a subgroup fixed by an automorphism intersects nontrivially an arbitrary subgroup of F n × F m , there is an algorithm that, taking as input a monomorphism and an endomorphism of F n × F m , decides whether their fixed subgroups intersect nontrivially. The general case of this problem, where two arbitrary endomorphisms are given as input, remains unknown. We show that, when two endomorphisms of a certain type are given as input, this problem is equivalent to the Post Correspondence Problem for free groups. 2026-02-24T00:00:00Z http://hdl.handle.net/10174/41513 Title: Modelos geométricos em bambu Authors: Mateus, Luís; Macias Marques, Pedro Abstract: Construção de modelos geométricos em bambu foi uma das sessões práticas que teve lugar no II Encontro STEAM no passado mês de março na Escola Secundária Camilo Castelo Branco, em Oeiras. Esta sessão, dinamizada pela Associação de Professores de Matemática e pela Associação de Professores de Desenho e Geometria Descritiva, ocorreu em duas edições, cada uma com aproximadamente 12 participantes. A realização da sessão assentou num ponto de partida essencial. A ideia de que a matemática e a geometria podem ser utilizadas para representar, interpretar e agir sobre o mundo que nos rodeia, seja o mundo natural seja o mundo artificializado pelo homem. O processo de construção dos modelos estabeleceu uma relação entre a geometria e a materialidade em que intervêm variadas propriedades físicas como a gravidade, o atrito, a elasticidade, a ductilidade, entre outras, que colocam limites ao que se pode construir, e que, por isso, tornam evidente que qualquer representação que se faça é apenas uma aproximação ao fenómeno que se pretende analisar. 2025-01-01T00:00:00Z http://hdl.handle.net/10174/41411 Title: Bridging meadows and sheaves Authors: Dias, João; Dinis, Bruno; Macias Marques, Pedro Abstract: We bridge sheaves of rings over a topological space with common meadows (algebraic structures where the inverse for multiplication is a total operation). More specifically, we show that, given a topological space X, the subclass of pre-meadows with a, coming from the lattice of open sets of X, and the class of presheaves over X are the same structure. Furthermore, we provide a construction that, given a sheaf of rings F on X, produces a common meadow as a disjoint union of elements of the form F(U) indexed over the open subsets of X. As a consequence, we see that the process of going from a presheaf to a sheaf (called sheafification) allows us to get a way to construct a common meadow from a given pre-meadow as above. 2026-01-01T00:00:00Z