A Contractor Based on Convex Interval Taylor

International audienceInterval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function andby Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called X-Newton, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen's variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint. The X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic

Collision Spectroscopy. I. Analysis of the scattering of He+ by Ne and Ar

Experimental data on the differential scattering of He+ by Ne and Ar in the energy range from 10 eV to 100 keV are plotted in a reduced coordinate system suggested by a scaling law for the forward scattering. The resulting curves are used to determine the interaction potential. The repulsive interaction dominating at higher energies shows pronounced shell-structure effects; leading to the deduction of the screening constants for the L and M shells of Ar and for the K and L shells of Ne. At lower energies a polarization attrac­tion appears, allowing deduction of the polarizabilities of Ne and Ar. A simple analytic potential is constructed, including a polarizability term appropriately damped inside the outer shell, which fits the data over the entire range. In addition to the pure elastic scattering, effects of inelastic interactions are diagnosed. A prominent curve crossing is located and the scattering pattern arising from it is interpreted by a semiclassical theory. In collisions with closer encounters, a different type of inelastic. process appears which apparently involves a more intense coupling than the curve crossing and which appears to open up a number of competing inelastic channels.Supported in part by the National Aeronautics and Space Administration and by the U.S. Army Research Offic