Groups Acting on Tensor Products

Groups preserving a distributive product are encountered often in algebra. Examples include automorphism groups of associative and nonassociative rings, classical groups, and automorphism groups of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving tensor products over semisimple and semiprimary rings, and present effective algorithms to construct generators for these groups. We also discuss applications of our methods to algorithmic problems for which all currently known methods require an exponential amount of work. (C) 2013 Elsevier B.V. All rights reserved.

Strong Dynamical Heterogeneity and Universal Scaling in Driven Granular Fluids

Large-scale simulations of two-dimensional bidisperse granular fluids allow us to determine spatial correlations of slow particles via the four-point structure factor S-4 (q, t). Both cases, elastic (epsilon = 1) and inelastic (epsilon < 1) collisions, are studied. As the fluid approaches structural arrest, i.e., for packing fractions in the range 0.6 <= phi <= 0.805, scaling is shown to hold: S-4 (q, t)/chi(4)(t) = s(q xi(t)). Both the dynamic susceptibility chi(4)(tau(alpha)) and the dynamic correlation length xi(tau(alpha)) evaluated at the alpha relaxation time tau(alpha) can be fitted to a power law divergence at a critical packing fraction. The measured xi(tau(alpha)) widely exceeds the largest one previously observed for three-dimensional (3d) hard sphere fluids. The number of particles in a slow cluster and the correlation length are related by a robust power law, chi(4)(tau(alpha)) approximate to xi(d-p) (tau(alpha)), with an exponent d - p approximate to 1.6. This scaling is remarkably independent of epsilon, even though the strength of the dynamical heterogeneity at constant volume fraction depends strongly on epsilon.