Geometria integral i convexitat en espais de curvatura holomorfa constant

En aquest treball es tracten qüestions de la geometria integral clàssica a l'espai hiperbòlic i projectiu complex i a l'espai hermític estàndard, els anomenats espais de curvatura holomorfa constant. La geometria integral clàssica estudia, entre d'altres, l'expressió en termes geomètrics de la mesura de plans que tallen un domini convex fixat de l'espai euclidià. Aquesta expressió es dóna en termes de les integrals de curvatura mitja. Un dels resultats principals d'aquest treball expressa la mesura de plans complexos que tallen un domini fixat a l'espai hiperbòlic complex, en termes del que definim com volums intrínsecs hermítics, que generalitzen les integrals de curvatura mitja. Una altra de les preguntes que tracta la geometria integral clàssica és: donat un domini convex i l'espai de plans, com s'expressa la integral de la s-èssima integral de curvatura mitja del convex intersecció entre un pla i el convex fixat? A l'espai euclidià, a l'espai projectiu i hiperbòlic reals, aquesta integral correspon amb la s-èssima integral de curvatura mitja del convex inicial: se satisfà una propietat de reproductibitat, que no es té en els espais de curvatura holomorfa constant. En el treball donem l'expressió explícita de la integral de la curvatura mitja quan integrem sobre l'espai de plans complexos. L'expressem en termes de la integral de curvatura mitja del domini inicial i de la integral de la curvatura normal en una direcció especial: l'obtinguda en aplicar l'estructura complexa al vector normal. La motivació per estudiar els espais de curvatura holomorfa constant i, en particular, l'espai hiperbòlic complex, es troba en l'estudi del següent problema clàssic en geometria. Quin valor pren el quocient entre l'àrea i el perímetre per a successions de figures convexes del pla que creixen tendint a omplir-lo? Fins ara es coneixia el comportament d'aquest quocient en els espais de curvatura seccional negativa i que a l'espai hiperbòlic real les fites obtingudes són òptimes. Aquí provem que a l'espai hiperbòlic complex, les cotes generals no són òptimes i optimitzem la superior.

Sharp weighted bounds for fractional integral operators

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp bounds are obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.

Mujeres en la casa, la escuela y la fábrica. El singular discurso de la domesticidad de la escuela economista española (1861-1909)

Los textos de la Escuela Economista Española (segunda mitad del XIX) contienen una valoración del papel de las mujeres en la economía y la sociedad transgresor frente al discurso dominante, que defendía un único y exclusivo rol para todas las mujeres: el hogar y la maternidad. La mayoría de los miembros de esta corriente económica defienden el trabajo femenino en las fábricas, basándose en argumentos salariales; e incluso demandan una formación profesional para aquellas que en muchos casos ni tan siquiera eran alfabetizadas por ser mujeres. Los textos de estos economistas transmiten nuevas ideas sobre el papel económico y social de las mujeres en una España dominada por un discurso que negaba la necesidad del trabajo femenino para las familias trabajadoras.

Singular integrals on Ahlfors-David regular subsets of the Heisenberg group

"Vegeu el resum a l'inici del document del fitxer adjunt."

Selection bias and unobservable heterogeneity applied at the wage equation of European married women

This paper utilizes a panel data sample selection model to correct the selection in the analysis of longitudinal labor market data for married women in European countries. We estimate the female wage equation in a framework of unbalanced panel data models with sample selection. The wage equations of females have several potential sources of.

A characterization of two weight norm inequalities for maximal singular integrals

"Vegeu el resum a l'inici del document del fitxer adjunt."

Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball

"Vegeu el resum a l'inici del document del fitxer adjunt."

Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.

Lp theory for the multidimensional aggregation equation

"Vegeu el resum a l'inici del document del fitxer adjunt."

Optimal exponent heat balance and refined integral methods applied to Stefan problems

When using a polynomial approximating function the most contentious aspect of the Heat Balance Integral Method is the choice of power of the highest order term. In this paper we employ a method recently developed for thermal problems, where the exponent is determined during the solution process, to analyse Stefan problems. This is achieved by minimising an error function. The solution requires no knowledge of an exact solution and generally produces significantly better results than all previous HBI models. The method is illustrated by first applying it to standard thermal problems. A Stefan problem with an analytical solution is then discussed and results compared to the approximate solution. An ablation problem is also analysed and results compared against a numerical solution. In both examples the agreement is excellent. A Stefan problem where the boundary temperature increases exponentially is analysed. This highlights the difficulties that can be encountered with a time dependent boundary condition. Finally, melting with a time-dependent flux is briefly analysed without applying analytical or numerical results to assess the accuracy.

Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions

We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soulé. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soulé as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that generalizes a result of Bismut, Gillet and Soulé and was already proved by Zha. This result can be combined with the arithmetic Grothendieck-Riemann-Roch theorem for submersions to extend this theorem to arbitrary projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincaré lemma for the complex of currents with fixed wave front set, and second we prove that certain direct images of Bott-Chern classes are closed.

Application of standard and refined heat balance integral methods to one-dimensional Stefan problems

The work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems. These include standard test problems, both with one and two phase changes, which have exact solutions to enable us to test the accuracy of the approximate solutions. We also consider situations where no analytical solution is available and compare these to numerical solutions. It is popular to use a quadratic profile as an approximation of the temperature, but we show that a cubic profile, seldom considered in the literature, is far more accurate in most circumstances. In addition, the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases. We assess which integral method is better for various problems, showing that it is largely dependent on the specified boundary conditions.

Política pública en Francia: prohibición general o parcial del velo integral en el territorio nacional: Proyecto de ley del martes 11 de mayo de 2010

La República francesa se ofende por los burkas y de los niqab deambulando por las calles. Ve a sus principios rectores de Liberté, Égalité, Fraternité mofados, a su democracia desafiada, a su convivencia arriesgada. Por ello, el Gobierno francés ha impulsado desde hace un año una política pública que persigue la prohibición del velo integral -llámese burka o niqab- en el conjunto del espacio público. Años atrás, Francia, país que tanto aprecia los debates, se vio inmersa en la cuestión más general de la identidad nacional desde las reacciones de las banlieues en el 2005. Y por ello, Nicolas Sarkosy, en su programa electoral camino de las presidenciales en el 2007, ya incluía como una prioridad el tema de los valores y principios genuinamente franceses y europeos. Y en esta perspectiva, el Ministerio de Inmigración añadió sus calificativos de Integración e Identidad Nacional. El burka en el espacio público ha asaltado los medios de comunicación y la sociedad en su conjunto, especialmente desde hace dos o tres años. Es un problema sin precedentes en las democracias asentadas -aunque podríamos recordar el motín de Esquilache en 1766 bajo el reinado de Carlos III- sin una Jurisprudencia asentada al respecto, ni un marco normativo que lo prevea directamente. Y por ello, la élite política se divide estos días entre los prudentes y los atrevidos, los que se conforman con una prohibición parcial y los que persiguen una prohibición absoluta. Se trata de una política pública atípica, pues los bienes afectados son en realidad valores, principios y conceptos abstractos. Es la República francesa y la cohesión social los que se ven desafiados. Veremos en un primer momento, la etapa de identificación como problema público y su inclusión en la agenda política, por ende desde un punto de vista técnico, analizaremos brevemente esta política pública Top down limitada, y en el último apartado compararemos los argumentos de las dos alternativas que dispone actualmente el Parlamento para aprobar o no una ley al respecto. Por su parte, la Comisión Parlamentaria alegará principios de índole más filosófica que pragmática, mientras que el Consejo de Estado se centrará exclusivamente en los términos jurídicos y en la Jurisprudencia reduciendo en gran medida el impulso inicial del Gobierno. Quedará por ver estos días de debate en l’Assemblée Nationale1 si la prohibición será total o parcial.

A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics

To describe the collective behavior of large ensembles of neurons in neuronal network, a kinetic theory description was developed in [13, 12], where a macroscopic representation of the network dynamics was directly derived from the microscopic dynamics of individual neurons, which are modeled by conductance-based, linear, integrate-and-fire point neurons. A diffusion approximation then led to a nonlinear Fokker-Planck equation for the probability density function of neuronal membrane potentials and synaptic conductances. In this work, we propose a deterministic numerical scheme for a Fokker-Planck model of an excitatory-only network. Our numerical solver allows us to obtain the time evolution of probability distribution functions, and thus, the evolution of all possible macroscopic quantities that are given by suitable moments of the probability density function. We show that this deterministic scheme is capable of capturing the bistability of stationary states observed in Monte Carlo simulations. Moreover, the transient behavior of the firing rates computed from the Fokker-Planck equation is analyzed in this bistable situation, where a bifurcation scenario, of asynchronous convergence towards stationary states, periodic synchronous solutions or damped oscillatory convergence towards stationary states, can be uncovered by increasing the strength of the excitatory coupling. Finally, the computation of moments of the probability distribution allows us to validate the applicability of a moment closure assumption used in [13] to further simplify the kinetic theory.

A one-dimensional Keller-Segel equation with a drift issued from the boundary

We investigate in this note the dynamics of a one-dimensional Keller-Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This note is completed with a brief introduction to a more realistic model (still one-dimensional).