A characterization of discrete unimodality with applications to variance upper bounds

Bertin and Theodorescu (1984,Statist. Probab. Lett.,2, 23–30) developed a characterization of discrete unimodality based on convexity properties of a discretization of distribution functions. We offer a new characterization of discrete unimodality based on convexity properties of a piecewise linear extension of distribution functions. This reliance on functional convexity, as in Khintchine's classic definition, leads to variance dilations and upper bounds on variance for a large class of discrete unimodal distributions. These bounds are compared to existing inequalities due to Muilwijk (1966,Sankhy, Ser. B,28, p. 183), Moors and Muilwijk (1971,Sankhy, Ser. B,33, 385–388), and Rayner (1975,Sankhy, Ser. B,37, 135–138), and are found to be generally tighter, thus illustrating the power of unimodality assumptions.     

Action on flag varieties: 2-Dimensional case

LetA be a finite dimensional commutative semisimple algebra over a fieldk and letV be a finitely generatedA-module. We examine the action of the general linear group GL _A (V) on the set of flags ofk-subspaces ofV. Also, let (V, B) be a finitely generated symplectic module overA. We also investigate the action of the symplectic group Sp _A (V, B) on the set of flags ofB-isotropick-subspaces ofV, whereB=°B is thek-symplectic form induced by a nonzerok-linear map :A k. In both cases, the orbits are completely classified in terms of certain integer invariants provided that dim _{k A}=2.This work is partially supported by a KOSEF research grant.     

Kinetic theory of granular shear flow: Constitutive relations for the hard-disk model

A kinetic theory for the constitutive Theological relations of rapid granular shear flow of hard circular disks, characterized by a coefficient of restitutione and a surface roughness coefficient, is formulated. From a set of general constitutive equations for single-particle dynamical variables, the approximate expressions for the limit of small and large dimensionless dissipative parameterR _t are obtained. HereR _t is defined as the ratio /, where is the fluctuation of translational velocity from the mean flow velocity, is the diameter of a disk, and is the shear rate. At smallR _t the theoretical predictions can be compared with exact computer simulation results of granular dynamics that are also reported. The agreement between theory and simulation is better than expected; the present theory is accurate up to high packing density in this region ofR _t .     

Condensed matter effects on nuclear fusion rates in laboratory and astrophysical environments

Previously overlooked condensed matter effects (CME) can significantly influence nuclear fusion rates in both laboratory and astrophysical environments. In dense plasmas, the ensemble of fusing particles has a significant exchange of kinetic and potential energies. Thus, there are diminished effective flux velocities resulting in a significant selective reduction of fusion rates. Our CME predictions are testable in laboratory experiments and have broad-ranging implications on the fusion rates for stellar media in general. By calculating reaction rates forp(p, e ⁺ v _e ) D and⁷Be(p, )⁸B in the sun, we show that CME help to solve the solar neutrino problem.