Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes

The folk questions in Lorentzian Geometry, which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime $(M,g)$ admits a smooth time function $\tau$ whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting $M= \R \times {\cal S}$, $g= - \beta(\tau,x) d\tau^2 + \bar g_\tau$, (b) if a spacetime $M$ admits a (continuous) time function $t$ (i.e., it is stably causal) then it admits a smooth (time) function $\tau$ with timelike gradient $\nabla \tau$ on all $M$.Comment: 9 pages, Latex, to appear in Commun. Math. Phys. Some comments on time functions and stably causal spacetimes are incorporated, and referred to gr-qc/0411143 for further detail

U-operators

Inspired by a statement of W. Luh asserting the existence of entire functions having together with all their derivatives and antiderivatives some kind of additive universality or multiplicative universality on certain compact subsets of the complex plane or of, respectively, the punctured complex plane, we introduce in this paper the new concept of U-operators, which are defined on the space of entire functions. Concrete examples, including differential and antidifferential operators, composition, multiplication and shift operators, are studied. A result due to Luh, Martirosian and Müller about the existence of universal entire functions with gap power series is also strengthened.Plan Andaluz de Investigación (Junta de Andalucía

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

Sequences of differential operators: exponentials, hypercyclicity and equicontinuity

In this paper, an eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of C N are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as before is also studied, and it is even characterized if the domain is C N. The results obtained extend or improve earlier work of several authors.Dirección General de Enseñanza Superior (DGES). EspañaJunta de Andalucí