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This chapter provides an overview of the alpha paleobiodiversity of Angola based on the available fossil record that is limited to the sedimentary rocks, ranging in age from Precambrian to the present. The geological period with the highest paleobiodiversity in the Angolan fossil record is the Cretaceous, with more than 80{%} of the total known fossil taxa, especially marine molluscs, including ammonites as a majority among them. The vertebrates represent about 15{%} of the known fauna and about one tenth of them are species firstly described based on specimens from Angola.

Accumulative Roll Bonding (ARB) was used to fabricate Graphene Oxide-reinforced Al-matrix composites.
Graphene Oxide reinforcement was suspended in a stabilized aqueous solution and applied, prior to each ARB
cycle, through airgun spraying. Different concentrations (graphene oxide/milipore water) were used and for
each concentration, samples produced have undergone up to 5 rolling cycles.
Optical and electron scanning microscopies were used for microstructural characterization which revealed a
non-homogenous deformation of the layers across the composite's thickness.
Although the presence of graphene-oxide promoted an increase in the microhardness, higher values were
obtained with its lowest concentration for similar samples. The number of ARB cycles and the direction of the
tested sections also influenced the microhardness results since the 5-cycle samples and the rolling direction
sections for all the samples achieved higher hardness results. Graphene Oxide revealed to be a major contributor
to the increase of stiffness during bending of the tested samples.

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.

Late Jurassic theropod tracks are very common both in North Africa and Europe. Two recently described ichnotaxa Megalosauripus transjuranicus and Jurabrontes curtedulensis from the Kimmeridgian of Switzerland show the coexistence of two apex predators in the same palaeoenvironment. Similar tracks can be found in tracksites from the Iberian Peninsula and from Morocco. Here, we further explore the similarities among the Swiss ichnotaxa and the other tracks from Germany (Kimmeridgian), Spain (Tithonian-Berriasian), Portugal (Oxfordian-Tithonian) and Morocco (Kimmeridgian) through novel three-dimensional data comparisons. Specimens were grouped in two morphotypes: 1) large and gracile (30 < Foot Length<50 cm) and 2) giant and robust (FL > 50 cm). The analyses show a great morphological overlap among these two morphotypes and the Swiss ichnotaxa (Megalosauripus transjuranicus and Jurabrontes curtedulensis, respectively), even despite the differences in sedimentary environment and age. This suggests a widespread occurrence of similar ichnotaxa along the western margin of Tethys during the Late Jurassic. The new data support the hypothesis of a Gondwana-Laurasia faunal exchange during the Middle or early Late Jurassic, and the presence of migratory routes around the Tethys.

Proof plays a central role in developing, establishing, and communicating mathematical knowledge. Nevertheless it is not such a central element in school mathematics. This article discusses some issues involving mathematical proof in school, intending to characterize the understanding of mathematical proof in school, its function and the meaning and relevance attributed to the notion of simple proof. The main conclusions suggest that the idea of addressing mathematical proof at all levels of school is a recent idea that is not yet fully implemented in schools. It requires an adaptation of the understanding of proof to the age of the students, reducing the level of formality, and allowing the students to experience the different functions of proof and not only the function of verification. Among the different functions of proof, the function of explanation deserves special attention due to the illumination and empowerment that it can bring to the students and their learning. The way this function of proof relates to the notion of simple proof (and the related aesthetic issues) seems relevant enough to make it, in the future, a focus of attention for the teachers who address mathematical proof in the classroom.

The left patience sorting (lPS) monoid, also known in the literature as the Bell monoid, and the right patient sorting (rPS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences.
Presentations for these monoids are given.

Each finite-rank rPS monoid is shown to have polynomial growth and to satisfy a non-trivial identity (dependent on its rank), while the infinite rank rPS monoid does not satisfy a non-trivial identity. The lPS monoids of finite rank have exponential growth and thus do not satisfy non-trivial identities. The
complexity of the insertion algorithms is discussed.

rPS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is $1$ or $2$, they are also biautomatic. lPS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.