Connected covering numbers

A connected covering is a design system in which the corresponding {\em block graph} is connected. The minimum size of such coverings are called {\em connected coverings numbers}. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with {\em Tur\'an systems}. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas.Comment: 15 pages, 4 figures, 1 tabl

Matroid toric ideals: complete intersection, minors and minimal systems of generators

In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be sufficient for $d = 2$ (leading to a criterion for determining whether $\mathcal M$ is binary) and for $d = 3$. Finally, we characterize all matroids $\mathcal M$ such that $I_{\mathcal M}$ has a unique minimal set of binomial generators.Comment: 9 page

Exponential type of hypercyclic entire functions

In this paper the exponential type of hypercyclic entire functions with respect to a sequence (Φn(D)) of differential operators is considered, where every Φn is an entire function of exponential type. We prove that under suitable conditions certain rates of growth are possible for hypercyclicity while others are not. In particular, our statements extend the negative part of a sharp result on growth of D-hypercyclic entire functions due to Grosse-Erdmann, and are related to a result by Chan and Shapiro about the existence of Φ(D)-hypercyclic functions in certain Hilbert spaces of entire functions.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí

Families of strongly annular functions: linear structure

A function f holomorphic in the unit disk D is called strongly annular if there exists a sequence of concentric circles in D expanding out to the unit circle such that f goes to infinity as |z| goes to 1 through these circles. The residuality of the family of strongly annular functions in the space of holomorphic functions on D is well known, and it is extended here to certain classes of functions. This important topological property is enriched in this paper by studying algebraic-topological properties of the mentioned family, in the modern setting of lineability. Namely, we prove that although this family is clearly nonlinear, it contains, except for the zero function, large vector subspaces as well as infinitely generated algebras. Similar results are obtained for strongly annular functions on the whole complex plane and for weighted Bergman spaces.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia e Innovación (MICIN). EspañaMinisterio de Ciencia y Tecnología (MCYT). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER

Universality of holomorphic functions bounded on closed sets

In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator.Plan Andaluz de Investigación (Junta de Andalucía)Dirección General de Enseñanza Superior (DGES). EspañaMinisterio de Ciencia y Tecnología (MCYT). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER