Irreducible numerical semigroups with multiplicity three and four

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in (Blanco-Puerto, 2011). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups as those that are generated by a generalized arithmetic progression.Comment: 18 page

On the net reproduction rate of continuous structured populations with distributed states at birth

We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least!) in three different ways. One approach is to study spectral properties of a parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.Comment: To appear in Computers and Mathematics with Application

On pseudo-Frobenius elements of submonoids of Nd

Producción CientíficaIn this paper we study those submonoids of Nd with a nontrivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of Nd and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.Ministerio de Economía, Industria y Competitividad (project MTM2017-84890-P)Ministerio de Economía, Industria y Competitividad - Fondo Europeo de Desarrollo Regional (project MTM2015-65764-C3-1-P

Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities

We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a xed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new xed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any nite dimension n. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial dierential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile- adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process

Asymptotic behaviour of a structured population model on a space of measures

In this paper we consider a physiologically structured population model with distributed states at birth, formulated on the space of non-negative Radon measures. Using a characterisation of the pre-dual space of bounded Lipschitz functions, we show how to apply the theory of strongly continuous positive semigroups to such a model. In particular, we establish the exponential convergence of solutions to a one-dimensional global attractor