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In 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris, for which he prepared 23 problems that mathematicians should solve during the twentieth century. It was found that there was a note on a 24th Problem focusing on the problem of simplicity of proofs. One of the lines of research that was generated from this problem was the identification of proofs. In this article, we present a possible method for exploring the identification of proofs based on the membership problem original from the theory of polynomial rings. To show this, we start by giving a complete worked-out example of a membership problem, that is, the problem of checking if a given polynomial belongs to an ideal generated by finitely many polynomials. This problem can be solved by considering Gröbner bases and the corresponding reductions. Each reduction is a simplification of the polynomial and it corresponds to a rewriting step. In proving that a polynomial is a member of an ideal, a rewriting process is used, and many different such processes can be considered. To better illustrate this, we consider a graph where each rewriting step corresponds to an edge, and thus a path corresponds to a rewriting process. In this paper, we consider the identification of paths, within the context of the membership problem, to propose a criterion of identification of proofs.
This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert’s 24th problem’.

The potential of using different representations is widely recognized, but not much is known about how teachers use them nor about the impact of the technology on such use. The goal of this study is to characterize the teachers’ representational fluency when teaching functions at high school level, discussing, at the same time, the impact in the use of representations resulting from the use of technology. Adopting a qualitative approach, I analyze one teacher’s practice. The results suggest that algebraic and graphical representations are seen as more important, that tabular representation is assumed as irrelevant and that the access to technology impacts the learning, the representations used and how they are used.

This study focusses on the criteria used by pre-service teachers of Mathematics to choose interdisciplinary tasks. The pre-service teachers’ knowledge is assumed as the basis of the actions taken and used as the origin of the choices and approaches observed. The study adopts a qualitative and interpretative methodology and the data were collected using class observation and interviews. The analysis is guided by the Application and Pedagogical Content Knowledge, a model inspired on TPACK (from Mishra and Koehler) and MKT (from Ball and colleagues). The conclusions point to an appreciation of the mathematical part of the tasks and to a devaluation of the remaining components. This suggests difficulty in articulating and integrating different domains of knowledge and points to a fragmented view of the potential of using mathematical applications.

Nowadays, adhesively bonded structures have received exhaustive attention mainly because, contrary to mechanical joints, they are able to avoid stress concentration. When a material is externally bonded to another structural member to improve the strength or stiffness of the latter, the adhesive joint is supposed to perform well for a long time, independently of the type of loading the bonded joint will be subjected to. However, studies dedicated to this topic are scarce when it comes to the influence of thermal action. The influence of temperature variations on bonded joints is not yet well understood, so more studies are needed to improve the current level of knowledge. The present study aims to develop an analytical solution capable of simulating the interfacial bond behaviour between two structural materials subjected to thermal loading. The complete debonding processes of such adhesively bonded joints are estimated based on a bi-linear bond–slip relationship. The proposed analytical model is validated by the numerical simulation of several examples, where some parameters previously identified as potentially affecting the bond behaviour are investigated. A commercial software based on the Finite Element Method (FEM) is used to support those examples in which either the analytical or the numerical simulations agreed very well. © 2019 Elsevier Ltd

A produção de betão com substituição de agregados naturais por agregados reciclados minimiza os impactes da indústria da construção. Contudo, o recurso a este tipo de agregado é limitado devido a dúvidas dos diferentes agentes da indústria da construção e à ausência de regulamentação específica para o projecto de betão com agregados reciclados. A percepção de betões com agregados reciclados como um material heterogéneo com comportamento imprevisível é o principal entrave, quer para a aceitação da indústria, quer para o desenvolvimento de regulamentação. Diferentes aspectos relacionados com o efeitos da variabilidade das propriedades de betão com agregados reciclados na regulamentação de estruturas são abordados conceptualmente: a heterogeneidade dos agregados reciclados, o efeito desta heterogeneidade nas propriedades mecânicas de betão e os efeitos da variabilidade das propriedades de betão na sua fiabilidade estrutural. Os primeiros resultados de uma campanha experimental desenvolvida especificamente para abordar o efeito de agregados grossos reciclados na fiabilidade são apresentados e as suas implicações no projecto de estruturas discutidas. São apresentadas sugestões de desenvolvimentos futuros que visam a calibração de coeficientes parciais de segurança que possibilitem o projecto de estruturas de betão armado dimensionadas segundo o formato do Eurocódigo 2 e mantendo os mesmos níveis de segurança de betão convencional.