Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so provide powerful combinatorial tools to work with them. This paper constructs an analogous `quasi-crystal' structure for the hypoplactic monoid, whose elements can be identified with quasi-ribbon tableaux and whose connection with the theory of quasi-symmetric functions echoes the connection of the plactic monoid with the theory of symmetric functions. This quasi-crystal structure and the associated quasi-Kashiwara operators are shown to interact just as neatly with the combinatorics of quasi-ribbon tableaux and with the hypoplactic version of the Robinson--Schensted--Knuth correspondence. A study is then made of the interaction of the crystal graph of the plactic monoid and the quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal structure is applied to prove some new results about the hypoplactic monoid.Comment: 55 pages. Minor revision to fix typos, add references, and discuss an open questio

The nuclear pseudospin symmetry along an isotopic chain

We investigate the isospin dependence of pseudospin symmetry in the chain of tin isotopes (from $^{120}$Sn until $^{170}$Sn). Using a Woods-Saxon parametrization of the nuclear potential for these isotopes we study in detail the effect of the vector-isovector $\rho$ and Coulomb potentials in the energy splittings of neutron and proton pseudospin partners in the isotopic chain. We conclude that the realization of nuclear pseudospin symmetry does not change considerably with the mass number, and is always favored for neutrons. We also find that the $\rho$ potential accounts for essentially all the pseudospin isospin asymmetry observed and that the Coulomb potential plays a negligible role in this asymmetry. This can be explained by the dynamical nature of pseudospin symmetry in nuclei, namely the dependence of the pseudospin splittings on the shape of the nuclear mean-field potential.Comment: 4 pages, 4 figures, to be published in Brazilian Journal of Physic

Analysis of engaged online collaborative discourse: a methodological approach

The purpose of this chapter is to present a reflection on collaborative learning mediated by the computer, discussing some difficulties and methodological constraints that we encounter when we try to analyze the interactions that occurred in this collaboration in an online course and the level of involvement in ollaborative discourse produced by participants. For we apply the Speech Involvement Scale Collaborative Computer-mediated Conference.Projeto MEDEIAinfo:eu-repo/semantics/publishedVersio

Nuclear Matter Properties in Derivative Coupling Models Beyond Mean - Field Approximation

The structure of infinite nuclear matter is studied with two of the Zimanyi - Moszkowski (ZM) models in the framework of a relativistic approximation which takes into account Hartree terms and beyond and is compared with the results which come out of the relativistic Hartree - Fock approach in the linear Walecka model. The simple treatment applied to these models can be used in substitution to the more complicated Dirac - Brueckner - Hartree - Fock method to perform future calculations in finite nuclei.Comment: 11 pages including 1 table, 1 figure (available upon request