Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem

This paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1 and v_2 solve two equations with different coefficients). We shall obtain an estimate of ||v_1-v_2||_\infty with an explicit dependence on the L^\infty-distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of ||v_1-v_2||_{C^2(\R^n)}. Afterwards, the former result will be crucial in the estimate of the rate of convergence for the homogenization of Bellman equations. In some regular cases, we shall obtain the same rate of convergence established in the monographs [11,26] for regular linear problems

On the puzzle of space weathering alteration of basaltic asteroids

The majority of basaltic asteroids are found in the inner main belt, although a few have also been observed in the outer main belt and near-Earth space. These asteroids -referred to as V-types- have surface compositions that resemble that of the 530km sized asteroid Vesta. Besides the compositional similarity, dynamical evidence also links many V-type asteroids to Vesta. Moreover, Vesta is one of the few asteroids to have been identified as source of specific classes of meteorites, the howardite, eucrite, diogenite achondrites (HEDs). Despite the general consensus on the outlined scenario, several questions remain unresolved. In particular, it is not clear if the observed spectral diversity among Vesta, V-types and HEDs is due to space weathering, as is thought to be the case for S-type asteroids. In this paper, SDSS photometry is used to address the question of whether the spectral diversity among candidate V-types and HEDs can be explained by space weathering. We show that visible spectral slopes of V-types are systematically redder with respect to HEDs, in a similar way to what is found for ordinary chondrite meteorites and S-types. On the assumption that space weathering is responsible for the slope mismatch, we estimated an upper limit for the reddening timescale of about 0.5Ga. Nevertheless, the observed slope mismatch between HEDs and V-types poses several puzzles to understanding its origin. The implication of our findings is also discussed in the light of Dawn mission to Vesta.Comment: Accepted by ApJ

Stellar populations in the Magellanic Clouds: looking through the dust

We present the first results of our study of stellar populations in the Large and Small Magellanic Clouds based on multi-band WFPC2 observations of "random" fields taken as part of the "pure parallel" programme carried out with the HST as a service to the community.Comment: Two pages, one figure, to appear in the proceedings of IAU Symposium 241 "Stellar Populations as Building Blocks of Galaxies", eds. A. Vazdekis and R. Peletie

A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks

Three definitions of viscosity solutions for Hamilton-Jacobi equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to establish their equivalence

Stationary Mean Field Games systems defined on networks

We consider a stationary Mean Field Games system defined on a network. In this framework, the transition conditions at the vertices play a crucial role: the ones here considered are based on the optimal control interpretation of the problem. We prove separately the well-posedness for each of the two equations composing the system. Finally, we prove existence and uniqueness of the solution of the Mean Field Games system

Continuous dependence estimates and homogenization of quasi-monotone systems of fully nonlinear second order parabolic equations

Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic equations. As by-product of these estimates, we get an H\"older estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation. In the second part of the paper we employ similar techniques to study the periodic homogenization of quasi-monotone systems of fully nonlinear second-order uniformly parabolic equations. Finally, some examples are discussed

The Luminosity Function of Omega Centauri

Deep HST-WFPC2 observations of the stellar population in the globular cluster Omega Cen (NGC 5139) have been used to measure the luminosity function of main sequence stars at the low-mass end. Two fields have been investigated, located respectively ~4.6' and ~7' away from the cluster center, or near the half-light radius of this cluster (~4.8'). The color-magnitude diagrams derived from these data show the cluster main sequence extending to the detection limit at I ~ 24. Information on both color and magnitude is used to build the luminosity functions of main sequence stars in these fields and the two independent determinations are found to agree very well with each other within the observational uncertainty. Both functions show a peak in the stellar distribution around M_I ~ 9 followed by a drop at fainter magnitudes well before photometric incompleteness becomes significant, as is typical of other globular clusters observed with the HST. This result is at variance with previous claims that the luminosity function of Omega Cen stays flat at low masses, but is in excellent agreement with recent near-IR NICMOS observations of the same cluster.Comment: To appear in The Astronomical Journal. Contains 7 pages, 4 figures, prepared with the AAS LaTeX macr

NGC2298: a globular cluster on its way to disruption

We have studied the stellar main sequence (MS) of the globular cluster NGC2298 using deep HST/ACS observations in the F606W and F814W bands covering an area of 3.4' x 3.4' around the cluster centre or about twice the cluster's half-mass radius. The colour-magnitude diagram that we derive in this way reveals a narrow and well defined MS extending down to the 10 sigma detection limit at m_606~26.5, m_814~25, corresponding to stars of ~0.2 Msolar. The luminosity function (LF) obtained with these data, once corrected for the limited effects of photometric incompleteness, reveals a remarkable deficiency of low-mass stars as well as a radial gradient, in that the LF becomes progressively steeper with radius. Using the mass-luminosity relation appropriate for the metallicity of NGC2298, we derive the cluster's global mass function (GMF) by using a multi-mass Michie-King model. Over the range 0.8 - 0.2 Msolar, the number of stars per unit mass decreases following a power-law distribution of the type dN/dm \propto m^(0.5), where, for comparison, typical halo clusters have dN/dm \propto m^(-1.5). If the IMF of NGC2298 was similar to that of other metal poor halo clusters, like e.g. NGC6397, the present GMF that we obtain implies that this object must have lost of order 85% of its original mass, at a rate much higher than that suggested by current models based on the available cluster orbit. The latter may, therefore, need revision.Comment: 9 pages, 6 figures, accepted for publication in Astronomy and Astrophysic