An invariant distribution in static granular media

We have discovered an invariant distribution for local packing configurations in static granular media. This distribution holds in experiments for packing fractions covering most of the range from random loose packed to random close packed, for beads packed both in air and in water. Assuming only that there exist elementary cells in which the system volume is subdivided, we derive from statistical mechanics a distribution that is in accord with the observations. This universal distribution function for granular media is analogous to the Maxwell-Boltzmann distribution for molecular gasses.Comment: 4 pages 3 figure

New Variants of Lattice Problems and Their NP-Hardness

We introduce some new variants of lattice problems: Quadrant-SVP, Quadrant-CVP and Quadrant-GapCVP'. All of them are NP-hard under deterministic reductions from subset sum problem. These new type of lattice problems have potential in construction of cryptosystems. Moreover, these new variant problems have reductions with standard SVP (shortest vector problem) and CVP (closest vector problem), this feature gives new way to study the complexity of SVP and CVP, especially for the proof of NP-hardness of SVP under deterministic reductions, which is an open problem up to now. ? 2014 Springer International Publishing.EI

A fourth power discrepancy mean

Let S be a bounded closed convex plane set with sufficiently smooth boundary curve. The area of S is the number of integer points in S minus a correction, the local discrepancy. Kendall’s classic paper introduced the Fourier transform of the local discrepancy and found the best possible mean square estimate. We obtain a corresponding fourth power estimate, valid merely under a C 2 smoothness condition