Maria Vaz Pinto

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals are defined by assigning linear weights to monomials. We examine Simis and normally torsion-free ideals, relate some of the properties of monomial ideals and weighted monomial ideals, and present a structure theorem for edge ideals of [ ]-uniform clutters whose ideal of covers is Simis in degree [ ]. One of our main results is a combinatorial classification of when the dual of the edge ideal of a weighted oriented graph is Simis in degree [ ].

Let G be a graph obtained by taking r >= 2 paths and identifying all first vertices and identifying all last vertices. We compute the Castelnuovo-Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. This invariant is known for several special families of graphs such as trees, cycles, complete graphs and complete bipartite graphs. For bipartite graphs, it is also known that the computation of the regularity can be reduced to the 2-connected case. Thus, we focused on the first case of a bipartite graph where the regularity was unknown. We also prove new inequalities relating the Castelnuovo-Mumford regularity of S/I(X) with the combinatorial structure of G, for a general graph.

The aim of this work is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we give degree formulas for the generalized Hamming weights of Reed--Muller-type codes, and we determine the minimum distance of toric codes over hypersimplices, and the 1st and 2nd generalized Hamming weights of squarefree evaluation codes.

Operacionalizamos o constructo consciência metalinguística e apresentamos três testes (THAM's, testes de habilidades metalinguísticas), provas ou recursos, que permitem analisar esta competência. Consideramos esta capacidade importante e passível de ser promovida desde a infância. Assim, a proposta aventa ferramentas para três faixas etárias distintas: para crianças, o THAM-1, para adolescentes, o THAM-2 e para adultos, o THAM-3. Estes recursos são adaptações para o português europeu dos MAT's (acrónimo de Metalinguistic Awareness Tests), originais em versão italiana, de Pinto (cf. Matel project), com versões em várias línguas, mundialmente disseminados, mesmo por via de um projeto europeu, MATEL. Ao nível nacional, já se encontram em circulação, pela publicação de um livro, estando em curso diversas investigações.

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. The case we consider is the first case where the regularity was unknown, following earlier computations (by several authors) of the regularity when G is a tree, cycle, complete graph or complete bipartite graph, but specially in light of the reduction of the computation of the regularity in the bipartite case to the computation of the regularity of the blocks of G. We also prove new inequalities relating the Castelnuovo--Mumford regularity of S/I(X) with the combinatorial structure of G, for a general graph.