Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.Comment: Minor changes and journal-re

Mean free paths and in-medium scattering cross sections of energetic nucleons in neutron-rich nucleonic matter within the relativistic impulse approximation

The mean free paths and in-medium scattering cross sections of energetic nucleons in neutron-rich nucleonic matter are investigated using the nucleon optical potential obtained within the relativistic impulse approximation with the empirical nucleon-nucleon scattering amplitudes and the nuclear densities obtained in the relativistic mean field model. It is found that the isospin-splitting of nucleon mean free paths, sensitive to the imaginary part of the symmetry potential, changes its sign at certain high kinetic energy. The in-medium nucleon-nucleon cross sections are analytically and numerically demonstrated to be essentially independent of the isospin asymmetry of the medium and increase linearly with density in the high energy region where the relativistic impulse approximation is applicable.Comment: 13 pages, 6 figure

Ordinary differential equations which linearize on differentiation

In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.Comment: 9 page

Circumstantial evidence for a soft nuclear symmetry energy at supra-saturation densities

Within an isospin- and momentum-dependent hadronic transport model it is shown that the recent FOPI data on the $\pi^-/\pi^+$ ratio in central heavy-ion collisions at SIS/GSI energies (Willy Reisdorf {\it et al.}, NPA {\bf 781}, 459 (2007)) provide circumstantial evidence suggesting a rather soft nuclear symmetry energy \esym at $\rho\geq 2\rho_0$ compared to the Akmal-Pandharipande-Ravenhall prediction. Some astrophysical implications and the need for further experimental confirmations are discussed.Comment: Version to appear in Phys. Rev. Let

Deformation of Quantum Dots in the Coulomb Blockade Regime

We extend the theory of Coulomb blockade oscillations to quantum dots which are deformed by the confining potential. We show that shape deformations can generate sequences of conductance resonances which carry the same internal wavefunction. This fact may cause strong correlations of neighboring conductance peaks. We demonstrate the relevance of our results for the interpretation of recent experiments on semiconductor quantum dots.Comment: 4 pages, Revtex, 4 postscript figure

Effect of symmetry energy on two-nucleon correlation functions in heavy-ion collisions induced by neutron-rich nuclei

Using an isospin-dependent transport model, we study the effects of nuclear symmetry energy on two-nucleon correlation functions in heavy ion collisions induced by neutron-rich nuclei. We find that the density dependence of the nuclear symmetry energy affects significantly the nucleon emission times in these collisions, leading to larger values of two-nucleon correlation functions for a symmetry energy that has a stronger density dependence. Two-nucleon correlation functions are thus useful tools for extracting information about the nuclear symmetry energy from heavy ion collisions.Comment: Revised version, to appear in Phys. Rev. Let

Lie point symmetries of difference equations and lattices

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations

The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.Comment: 29 pages. Plain TeX. Phyzzx needed. An example and some references added. To appear in J. Phys.