Unfoldings of saddle-nodes and their Dulac time

In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem A) we prove uniform regularity by which orbits and their derivatives arrive at a node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a building block in the study of bifurcations of critical periods in a neighbourhood of a polycycle. Finally, we apply Theorems A and B to the study of critical periods of the Loud family of quadratic centers and we prove that no bifurcation occurs for certain values of the parameters (Theorem C)

Foliations and webs inducing Galois coverings

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations

La atención a la dependencia en España

ResumenHasta el momento, tanto en España como en la mayoría de los países de la Unión Europea, el fenómeno de la dependencia se ha considerado implícitamente un problema de índole fundamentalmente privada que debe ser resuelto por las familias. Así, ya sea mediante cuidadores informales o contratando la ayuda de profesionales, bien en el ámbito doméstico o en residencias asistidas, son las propias personas dependientes y sus familias las que hoy por hoy soportan la mayor parte de los costes implicados. Así las cosas, la preocupación actual reside en que los cambios sociales que se avecinan, principalmente el incremento de la esperanza de vida y la mayor participación laboral de las mujeres, acentúan la necesidad de organizar colectivamente lo que hasta el momento se ha resuelto en el ámbito familiar.En este contexto, coincidiendo con el anuncio del gobierno de una «Ley de Dependencia» para finales de 2005, el presente trabajo se propone analizar brevemente las cuatro cuestiones que nos parecen más importantes para centrar el debate: el alcance actual de los problemas de dependencia y su posible evolución futura, las características del presente modelo de atención y sus posibles problemas, el papel que desempeñan y deberían desempeñar los servicios sanitarios en materia de dependencia y, finalmente, las ventajas y los inconvenientes de las distintas alternativas de que dispone la Administración si pretende aumentar su grado de implicación en este ámbito.AbstractTo date, both in Spain and virtually all the other European Union (EU) countries, dependency has been seen to be a fundamentally private problem to be dealt with by the family concerned. In this way, whether through informal carers or contracted professionals, in the domestic environment or in care homes, it is the dependent person themselves and their families who currently bear the majority of the costs. In light of this, current concern lies in the social change that is coming on, mainly the accelerated aging process and the increased participation of middle-aged women in the labour market, which heighten the need for collective organisation of that which until now has been resolved within family circles.In this context, at the same time that the Government announces to issue a «Long Term Care Law» by the end of 2005, our paper briefly analyzes what we consider the four crucial issues in this area: the current scope of dependency problems and its possible future evolution, the characteristics of the current spanish long-term care system and its main problems; the role that health services should have in the dependency issue; and finally, the benefits and drawbacks of the main alternatives that the Administration could manage in case it intends to increase its involvement in this field