Today's society, which is strongly based on knowledge and interaction with information, has a key component in technological innovation, a fundamental tool for the development of the current teaching methodologies. Nowadays, there are a lot of online resources, such as MOOCs (Massive Open Online Courses) and distance learning courses. One aspect that is common to all of these is a high dropout rate: about 90% in MOOCs and 50% in the courses of the Spanish National Distance Education University, among other examples. In this paper, we analyze a number of actions undertaken in the Master's Degree in Computational Mathematics at Universitat Jaume I in Castellón, Spain. These actions seem to help decrease the dropout rate in distance learning; the available data confirm their effectiveness.

We consider the valueμ(ν) = limm→∞ m −1 a(mL), where a(mL) is the last value of the vanishing sequence of H 0 (mL) along a divisorial or irrational valuation ν centered at O P 2 ,p , L (respectively, p) being a line (respectively, a point) of the projective plane P 2 over an algebraically closed field. This value contains, for valuations, similar information as that given by Seshadri constants for points. It is always true thatμ(ν) ≥ 1/vol(ν) and minimal valuations are those satisfying the equality. In this paper, we prove that the Greuel-Lossen-Shustin Conjecture implies a variation of the Nagata Conjecture involving minimal valuations (that extends the one stated in [15] to the whole set of divisorial and irrational valuations of the projective plane) which also implies the original Nagata's conjecture. We also provide infinitely many families of minimal very general valuations with an arbitrary number of Puiseux exponents, and an asymptotic result that can be considered as evidence in the direction of the above mentioned conjecture [15].

In the first part of this paper we introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module M over the polynomial ring K [X 1 , . . . , X n ] by reducing the problem to the computation of the finite set of the new defined Hilbert partitions. In the second part we show how Hilbert partitions may be used for computing the Stanley depth of the module M. In particular, we answer two open questions posed by Herzog in [8].

ABSTRACT. In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients I/J of monomial ideals J ⊂ I, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in [IKMF15]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

We prove that the Newton-Okounkov body of the flag E • := {X = X r ⊃ E r ⊃ {q}}, defined by the surface X and the exceptional divisor E r given by any divisorial valuation of the complex projective plane P 2 , with respect to the pull-back of the line-bundle O P 2 (1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular.

Abstract. Let Γ = α, β be a numerical semigroup. In this article we consider several relations between the so-called Γ-semimodules and lattice paths from (0, α) to (β, 0): we investigate isomorphism classes of Γ-semimodules as well as certain subsets of the set of gaps of Γ, and finally syzygies of Γ-semimodules. In particular we compute the number of Γ-semimodules which are isomorphic with their k-th syzygy for some k.

In this article we mainly consider the positively Zgraded polynomial ring R = F[X, Y ] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be the Hilbert series of some R-module of positive depth. In the generic case, that is deg(X) and deg(Y ) being coprime, this criterion can be formulated in terms of the numerical semigroup generated by those degrees.

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