We study the topology of the space Ωp of admissible paths between two points e (the origin) and p on a step-two Carnot group G:As it turns out, Ωp is homotopy equivalent to an infinite dimensional sphere and in particular it is contractible. The energy function:is defined by J(γ) = 1 2 I γ 2 ; critical points of this function are sub-Riemannian geodesics between e and p. We study the asymptotic of the number of geodesics and the topology of the sublevel sets:the number of geodesics joining e and p is bounded and the homology of Ω s p stabilizes to zero for s large enough. A completely different behavior is experienced for the generic vertical p. In this case we show that J is a Morse-Bott function: geodesics appear in isolated families (critical manifolds), indexed by their energy. Denoting by l the corank of the horizontal distribution on G, we prove that: Card{Critical manifolds with energy less than s} ≤ O(s) l . Despite this evidence, Morse-Bott inequalities b(Ω s p ) ≤ O(s) l are far from being sharp and we show that the following stronger estimate holds: b(Ω s p ) ≤ O(s) l−1 . Thus each single Betti number bi(Ω s p ) (i > 0) becomes eventually zero as s → ∞, but the sum of all of them can possibly increase as fast as O(s) l−1 . In the case l = 2 we show that indeed b(Ω s p ) = τ (p)s + o(s) (l = 2). The leading order coefficient τ (p) can be analytically computed using the structure constants of the Lie algebra of G.Using a dilation procedure, reminiscent to the rescaling for Gromov-Hausdorff limits, we interpret these results as giving some local information on the geometry of G (e.g. we derive for l = 2 the rate of growth of the number of geodesics with bounded energy as p approaches e along a vertical direction).
Abstract:In order to assess water efficiency options on the European scale, a multi-criteria integrative hydro-economic modeling framework has been developed. With this framework, it is possible to assess combinations of measures which could help reducing the gap between water demand and water availability, while taking into account ecological, water quality, flood risk and economic aspects. The assessed measures include water retention, water savings and nutrient reduction measures. The presented work was carried out within the framework of the "Blueprint to safeguard Europe's waters" policy initiative of the European Commission. Contrary to earlier studies concentrating on single measures in single river basins, this study shows that this modeling environment can evaluate combinations of measures in multiple river basins that meet the considered objectives, and in general can improve various water quantity and quality indicators as compared to the baseline situation. However, additional work is needed on for example quantifying the economics of damage and benefits before the modelling environment may be used for policy advice.
Resumen: La inestabilidad laboral hace cada vez más complicada la transición a la vida adulta de los jóvenes, especialmente a la hora de dejar el hogar familiar. En este artículo se profundiza cómo los mileuristas residentes en Barcelona de entre 25 y 34 años perciben y representan la inestabilidad laboral a partir de sus experiencias personales. El objetivo del estudio es explorar la influencia de la precariedad sobre la inserción laboral, el bienestar y los proyectos biográficos de estos jóvenes y qué papel desempeña la familia como "amortiguador social" de las externalidades negativas referidas a sus trabajos inciertos e inseguros. A través de las entrevistas realizadas a 40 jóvenes-adultos sobre estos temas, conseguimos evidenciar un amplio abanico de estrategias de emancipación que se corresponde a sus distintas formas de vivir la
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