The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of `plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same
evaluation (that is, contain the same number of each generating symbol). This paper focusses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-$n$ case. For the hypoplactic monoid, this is $n-1$; for the sylvester and taiga monoids, at least $n-1$ and at most $n$; for the stalactic monoid, $3$ (except for ranks $1$ and $2$, when it is respectively $0$ and $1$); for the plactic monoid, at least $n-1$ and at most $2n-3$. The current state of knowledge, including new and previously-known results, is summarized in a table.

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.

The main goal of this work is to characterize how the knowledge of pre-service teachers about the characteristics of the tasks and the role of technology evolves. Based on a case study carried out around a pair of pre-service teachers, the main conclusions point to the contribution of the reflection around a set of six tasks on Functions selected by the pre-service teachers. Central to this reflection was an analyze of the role technology can play in tasks, the comments made by the colleagues to their tasks and some experiences on modeling and open-ended tasks. These elements provided the development of a greater awareness regarding aspects such as the level of structuring of the task and its degree of challenge. And this was determinant for an appropriation of the different characteristics of the tasks and to the development of the pre-service teachers’ knowledge.

This paper reports the application of additive manufacturing technology to fabricate bidimensional
lightweight composite meshes capable of demonstrating auxetic properties
(negative Poisson’s ratio (NPR)) in combination with negative thermal expansion (NTE)
behaviour, using as constituent materials polymers that do not exhibit NTE behaviour. To
describe the combination of NPR and NTE characteristics, the designation of ‘anepectic’ is being
proposed. Each mesh, obtained from varying either the material combination or the design
parameters, was tested on a heated silicone bath to study the effects of the different combinations
on the coefficient of thermal expansion (CTE). It was found that all meshes studied demonstrated
a successful combination of NPR and NTE behaviours, and it was revealed that there is a
possibility to tailor the meshes to activate the NTE behaviour within a chosen range of
temperatures. For an extreme case, a Poisson’s ratio of −0.056, along with a CTE of
−1568×10^−6 K−1 has been achieved.

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

In this paper we characterize the monoids of endomorphisms of the semigroups PO_n and POI_n of all order-preserving partial transformations and of all order-preserving partial permutations, respectively, of a finite n-chain.