This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words and pairs of tableaux.

We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number $n$, the $n$-th Bell number is given by the number of standard tableaux containing $n$ symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a `hook length formula' that gives the number of standard tableaux of a given shape and deduce some consequences.

This paper constructs presentations via finite complete rewriting systems for plactic monoids of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$, using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic.

https://docentes.fct.unl.pt/lblm/files/jib-2019-196-110694.pdf

Isolated theropod teeth are some of the most common fossils in the dinosaur fossil record and are continually reported in the literature. Recently developed quantitative methods have improved our ability to test the affinities of isolated teeth in a repeatable framework. But in most studies, teeth are diagnosed on qualitative characters. This can be problematic because the distribution of theropod dental characters is still poorly documented, and often restricted to one lineage. To help in the identification of isolated theropod teeth, and to more rigorously evaluate their taxonomic and phylogenetic potential, we evaluated dental features in two ways. We first analyzed the distribution of 34 qualitative dental characters in a broad sample of taxa. Functional properties for each dental feature were included to assess how functional similarity generates homoplasy. We then compiled a quantitative data matrix of 145 dental characters for 97 saurischian taxa. The latter was used to assess the degree of homoplasy of qualitative dental characters, address longstanding questions on the taxonomic and biostratigraphic value of theropod teeth, and explore the major evolutionary trends in the theropod dentition.

In smaller phylogenetic datasets for Theropoda, dental characters exhibit higher levels of homoplasy than non-dental characters, yet they still provide useful grouping information and optimize as local synapomorphies of smaller clades. In broader phylogenetic datasets, the degree of homoplasy displayed by dental and non-dental characters is not significantly different. Dental features on crown ornamentations, enamel texture and tooth microstructure have significantly less homoplasy than other dental features and can be used to identify many theropod taxa to ‘family’ or ‘sub-family’ level, and some taxa to genus or species. These features should, therefore, be a priority for investigations seeking to classify isolated teeth.

Our observations improve the taxonomic utility of theropod teeth and in some cases can help make isolated teeth useful as biostratigraphic markers. This proposed list of dental features in theropods should, therefore, facilitate future studies on the systematic paleontology of isolated teeth.

https://docentes.fct.unl.pt/lblm/files/bioelectrochem-2019-125-8.pdf

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.

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’.

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.

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.

The list of approximations to the Normal cumulative distribution function is long and, eventually, not fully known due to the large number of published articles in the last decades. In this paper we will present new improvements in some well known approximations, without increasing the complexity of the formulas.

https://docentes.fct.unl.pt/lblm/files/chemeurj-2019-25-4309.pdf

The current curricular guidelines for mathematics education in Portugal emphasize the relevance of working with different representations of functions to promote understanding. Given this relevance, we seek understanding about the notion of function held by 37 basic education pre-service teachers in their first year of a master’s course. Data were collected through a task focusing on identifying functions in situations based on different representations. The content analysis technique was then adopted in the search for an understanding of the justifications given by the participants. The results achieved suggest it is easier for the pre-service teachers to identify examples that are not functions than examples that are functions. There is also a tendency for greater accuracy in the identification of examples expressed by tables than by algebraic expressions. The justifications presented show a notion of function as a relation between values of two non-empty sets, but without guaranteeing that this relation is single-valued.