Infinite Horizon Noncooperative Differential Games with Non-Smooth Costs

In the present paper, we consider a class of two players infinite horizon differential games, with piecewise smooth costs exponentially discounted in time. Through the analysis of the value functions, we study in which cases it is possible to establish the existence Nash equilibrium solutions in feedback form. We also provide examples of piecewise linear costs whose corresponding games have either infinitely many Nash equilibria or no solutions at all.Comment: 17 pages, 5 figure

Infinite Horizon Noncooperative Differential Games

For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.Comment: 25 pages, 7 figure

Sum-factorization techniques in Isogeometric Analysis

The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one for matrix-free methods, and study the behavior of their computational complexity in terms of the spline order $p$. Opposed to the standard approach, these algorithms do not apply the idea element-wise, but globally or on macro-elements. If this approach is applied to Gauss quadrature, the computational complexity grows as $p^{d+2}$ instead of $p^{2d+1}$ as previously achieved.Comment: 34 pages, 8 figure

Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time

The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.Comment: 13 pages, 3 figures. in v2: Fixed few misprint

The boundary Riemann solver coming from the real vanishing viscosity approximation

We study a family of initial boundary value problems associated to mixed hyperbolic-parabolic systems: v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x = \epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx} The conservative case is, in particular, included in the previous formulation. We suppose that the solutions $v^{\epsilon}$ to these problems converge to a unique limit. Also, it is assumed smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $A$ can be $0$. Second, we take into account the possibility that $B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added Section 3.1.2. Minor changes in other section

Mid-infrared colour gradients and the colour-magnitude relation in Virgo early-type galaxies

We make use of Spitzer imaging between 4 and 16 micron and near-infrared data at 2.2 micron to investigate the nature and distribution of the mid-infrared emission in a sample of early-type galaxies in the Virgo cluster. These data allow us to conclude, with some confidence, that the emission at 16 micron in passive ETGs is stellar in origin, consistent with previous work concluding that the excess mid-infrared emission comes from the dusty envelopes around evolved AGB stars. There is little evidence for the mid-infrared emission of an unresolved central component, as might arise in the presence of a dusty torus associated with a low-luminosity AGN. We nonetheless find that the 16 micron emission is more centrally peaked than the near-infrared emission, implying a radial stellar population gradient. By comparing with independent evidence from studies at optical wavelengths, we conclude that a metallicity that falls with increasing radius is the principal driver of the observed gradient. We also plot the mid-infrared colour-magnitude diagram and combine with similar work on the Coma cluster to define the colour-magnitude relation for absolute K-band magnitudes from -26 to -19. Because a correlation between mass and age would produce a relation with a gradient in the opposite sense to that observed, we conclude that the relation reflects the fact that passive ETGs of lower mass also have a lower average metallicity. The colour-magnitude relation is thus driven by metallicity effects. In contrast to what is found in Coma, we do not find any objects with anomalously bright 16 micron emission relative to the colour-magnitude relation. Although there is little overlap in the mass ranges probed in the two clusters, this may suggest that observable ``rejuvenation'' episodes are limited to intermediate mass objects.Comment: 8 pages, 4 figure

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields $$u_t + f(u)_x = 0,$$ then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families $$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N,$$ and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure \mu_{i,\jump} is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\mathrm{jump}}$ is positive

Pre-MS depletion, accretion and primordial 7Li

We reconsider the role of pre-main sequence (pre-MS) Li depletion on the basis of new observational and theoretical evidence: i) new observations of Halpha emissions in young clusters show that mass accretion could be continuing till the first stages of the MS, ii) theoretical implications from helioseismology suggest large overshooting values below the bottom of the convective envelopes. We argue here that a significant pre-MS 7Li destruction, caused by efficient overshoot mixing, could be followed by a matter accretion after 7Li depletion has ceased on MS thus restoring Li almost to the pristine value. As a test case we show that a halo dwarf of 0.85 Msun with an extended overshooting envelope starting with an initial abundance of A(Li) = 2.74 would burn Li completely, but an accretion rate of the type 1e-8xe^{-t/3e6} Msun yr$^{-1}$ would restore Li to end with an A(Li) = 2.31. A self-regulating process is required to produce similar final values in a range of different stellar masses to explain the PopII Spite plateau. However, this framework could explain why open cluster stars have lower Li abundances than the pre-solar nebula, the absence of Li in the most metal poor dwarfs and a number of other features which lack of a satisfactory explanation.Comment: To be published in Memorie della Societ\`a Astronomica Italiana Supplementi Vol. 22, Proceedings of Lithium in the cosmos, Iocco F., Bonifacio P., Vangioni E., ed

Hyperbolic Balance Laws with a Non Local Source

This paper is devoted to hyperbolic systems of balance laws with non local source terms. The existence, uniqueness and Lipschitz dependence proved here comprise previous results in the literature and can be applied to physical models, such as Euler system for a radiating gas and Rosenau regularization of the Chapman-Enskog expansion.Comment: 26 page

Some new well-posedness results for continuity and transport equations, and applications to the chromatography system

We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up of the BV norm at the initial time. We apply these results (valid in any space dimension) to the k x k chromatography system of conservation laws and to the k x k Keyfitz and Kranzer system, both in one space dimension.Comment: 33 pages, minor change