In order to reach the proofs of these properties let us preliminarily determine the variation across a binary collision occurring between particles 1 and 2 of the total directional kinetic energy M([v.sub.1], [v.sub.2], b) (see (25)), namely, the phase-space scalar function [DELTA]M([v.sub.1], [v.sub.2], b) [equivalent to] M([v.sup.(+).sub.1], [v.sup.(+).sub.2]), b)-M([v.sub.1][v.sub.2], b).

Second, (C.2) shows--on the contrary--that binary collisions actually do affect by means of MCBC a velocity-spreading for the 1- and 2-body PDF.

Ghatak, "Simple

binary collision model for Van Hove's Gs(r,t)," Physical Review, vol.

These calculated nuclear reaction cross sections are used in the calculation of average number of projectile participants [[(P).sub.proj]], target participants [[(P).sub.targ]], and binary collision [[(B).sub.C]] through the following simple geometrical consideration [22]:

The average numbers of projectile participants [[(P).sub.proj]], target participants [[(P).sub.targ]], and binary collision [[(B).sub.C]] are used in the shower particle multiplicity calculation.

Schlanges, "Dense plasma temperature equilibration in the

binary collision approximation," Physical Review E--Statistical, Nonlinear, and Soft Matter Physics, vol.

Our results indicate that the degree of such excitations, which predominantly depends on the number of

binary collisions [N.sub.coll], appears to remain almost same for the most central collisions of the Au + Au system in the FAIR energy region.

The map induced on the quotient can be regularized on binary collisions (see [4, 3]), hence the map on the quotient can be extended to a self-map f : [P.sub.1](R)[right arrow] [P.sub.1](R) with three fixed points of index 1.

For d = 2 and n = 3, the three Euler configurations have [mu] = 1, while the two Lagrange points have [mu] = 1, hence the map f induced on the quotient [P.sub.1](C) (again, by regularizing the binary collisions) has Lefschetz number equal to L(f)= 2 - 3 =-1.

Whatever energy is lost is in close

binary collisions. If the velocity is comparable with or greater than thermal speeds, then the particle can lose appreciable amounts of energy in exciting collective oscillations.

The collision integral [[OMEGA].sub.D], which quantitatively summarizes the dynamics of molecular trajectories and

binary collisions for dilute gas mixtures (Hirschfelder et al., 1954), provides a correction to the hard sphere intermolecular potential energy because realistic molecules do not collide like hard spheres when the repulsive part of the potential exhibits some degree of softness.

However, this assumption which is based on a nucleon-nucleon collision in the Glauber model is crude and it looks unrealistic to relate participating nucleons and nucleon-nucleon

binary collisions to soft and hard components at the partonic level.