On the Convergence Rate of Vanishing Viscosity Approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound \big\|u(t,\cdot)-u^\ve(t,\cdot)\big\|_{\L^1}= \O(1)(1+t)\cdot \sqrt\ve|\ln\ve| on the distance between an exact BV solution $u$ and a viscous approximation u^\ve, letting the viscosity coefficient \ve\to 0. In the proof, starting from $u$ we construct an approximation of the viscous solution u^\ve by taking a mollification u*\phi_{\strut \sqrt\ve} and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed \ve. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.Comment: 34 pages, 5 figure

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

An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space Dimensions

The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed.Comment: 12 pages, 5 figure

COMPASS Results on Collins and Sivers Asymmetries

In the list of the main items studied by the CERN COMPASS experiment there are the transverse spin and momentum effects visible in the azimuthal distributions of hadrons produced in the deep inelastic scattering. In the years 2002-2004 COMPASS has collected data with a 6LiD target with the polarization oriented transversely with respect to the muon beam direction for about 20% of the running time; in 2007, COMPASS has used for the first time a proton NH_3 target with the data taking time equally shared between longitudinal and transverse polarization of the target. After reviewing the results obtained with the deuteron, the new results for the Collins and Sivers asymmetries of the proton will be presented.Comment: 5 pages, 3 figure, presented at DIS 2009 - QCD and Deep Inelastic Scattering, Madrid, Spain, April 26-30, 200

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

Hyperbolic systems of conservation laws in one space dimension

Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations

Physics Results from COMPASS

The COMPASS Experiment at the CERN SPS has a broad physics program focused on the nucleon spin structure and on hadron spectroscopy, using muon and hadron beams. Main objectives for the spin program with the muon beam are the direct measurement of the gluon contribution to the spin of the nucleon, semi-inclusive measurements, and the measurement of the transverse spin distribution $\Delta_T q$. The COMPASS apparatus consists of a two-stage large acceptance spectrometer designed for high data rates and equipped with high-resolution tracking, particle identification and electromagnetic and hadronic calorimetry. The data taking is ongoing since 2002 and till now was mainly devoted to the spin programme using a 160 GeV$/c$ naturally polarized, $\mu^+$ beam and a polarized \mrf{^6LiD} target. First physics results from the 2002 and 2003 runs are presented.Comment: 9 pages, 4 figures, Plenary talk at the "16th International Spin Physics Symposium", October 10-16, 2004, Trieste, Italy; to be published in the Conference Proceedings, World Scientifi

Approximation in FEM, DG and IGA: A Theoretical Comparison

In this paper we compare approximation properties of degree $p$ spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, \smooth {p-1} splines provide better a priori error bounds for the approximation of functions in $H^{p+1}(0,1)$. Our result holds for all practically interesting cases when comparing \smooth {p-1} splines with \smooth {-1} (discontinuous) splines. When comparing \smooth {p-1} splines with \smooth 0 splines our proof covers almost all cases for $p\ge 3$, but we can not conclude anything for $p=2$. The results are generalized to the approximation of functions in $H^{q+1}(0,1)$ for $q<p$, to broken Sobolev spaces and to tensor product spaces.Comment: 21 pages, 4 figures. Fixed typos and improved the presentatio