Bounds on Mahalanobis norms and their applications

AbstractLocal and global bounds for ratios of norms, and minimal and maximal norms, are constructed for pairs and ensembles of quadratic norms of Rk, with corresponding results for Mahalanobis distance functions. These support envelopes for distributions of certain quadratic forms in Gaussian variates. Applications are noted in the use of quadratic classification rules and in assessing hit probabilities in ballistic systems

Majorants and Minorants for Elliptic Measures on Rk

AbstractLet μ(· ; Σ, G1) and μ(· ; Ω, G2) be elliptically contoured measures on Rk centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G1, G2). Elliptical measures vm(·) and vM(·), depending on (Σ, Ω, G1, G2), are constructed such that Vm(C) ≤ {μ(C; Σ, G1), μ(C; Ω, G2)} for every symmetric convex set C ⊂ Rk with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems

Two-body correlations in Bose condensates

We formulate a method to study two-body correlations in a condensate of N identical bosons. We use the adiabatic hyperspheric approach and assume a Faddeev like decomposition of the wave function. We derive for a fixed hyperradius an integro-differential equation for the angular eigenvalue and wave function. We discuss properties of the solutions and illustrate with numerical results. The interaction energy is for N~20 five times smaller than that of the Gross-Pitaevskii equation

Spectral Duality for Planar Billiards

For a bounded open domain $\Omega$ with connected complement in ${\bf R}^2$ and piecewise smooth boundary, we consider the Dirichlet Laplacian $-\Delta_\Omega$ on $\Omega$ and the S-matrix on the complement $\Omega^c$. We show that the on-shell S-matrices ${\bf S}_k$ have eigenvalues converging to 1 as $k\uparrow k_0$ exactly when $-\Delta_\Omega$ has an eigenvalue at energy $k_0^2$. This includes multiplicities, and proves a weak form of ``transparency'' at $k=k_0$. We also show that stronger forms of transparency, such as ${\bf S}_{k_0}$ having an eigenvalue 1 are not expected to hold in general.Comment: 33 pages, Postscript, A

Topological Defects, Orientational Order, and Depinning of the Electron Solid in a Random Potential

We report on the results of molecular dynamics simulation (MD) studies of the classical two-dimensional electron crystal in the presence disorder. Our study is motivated by recent experiments on this system in modulation doped semiconductor systems in very strong magnetic fields, where the magnetic length is much smaller than the average interelectron spacing $a_0$, as well as by recent studies of electrons on the surface of helium. We investigate the low temperature state of this system using a simulated annealing method. We find that the low temperature state of the system always has isolated dislocations, even at the weakest disorder levels investigated. We also find evidence for a transition from a hexatic glass to an isotropic glass as the disorder is increased. The former is characterized by quasi-long range orientational order, and the absence of disclination defects in the low temperature state, and the latter by short range orientational order and the presence of these defects. The threshold electric field is also studied as a function of the disorder strength, and is shown to have a characteristic signature of the transition. Finally, the qualitative behavior of the electron flow in the depinned state is shown to change continuously from an elastic flow to a channel-like, plastic flow as the disorder strength is increased.Comment: 31 pages, RevTex 3.0, 15 figures upon request, accepted for publication in Phys. Rev. B., HAF94MD

The Effect of Splayed Pins on Vortex Creep and Critical Currents

We study the effects of splayed columnar pins on the vortex motion using realistic London Langevin simulations. At low currents vortex creep is strongly suppressed, whereas the critical current j_c is enhanced only moderately. Splaying the pins generates an increasing energy barrier against vortex hopping, and leads to the forced entanglement of vortices, both of which suppress creep efficiently. On the other hand splaying enhances kink nucleation and introduces intersecting pins, which cut off the energy barriers. Thus the j_c enhancement is strongly parameter sensitive. We also characterize the angle dependence of j_c, and the effect of different splaying geometries.Comment: 4 figure

Growth, competition and cooperation in spatial population genetics

We study an individual based model describing competition in space between two different alleles. Although the model is similar in spirit to classic models of spatial population genetics such as the stepping stone model, here however space is continuous and the total density of competing individuals fluctuates due to demographic stochasticity. By means of analytics and numerical simulations, we study the behavior of fixation probabilities, fixation times, and heterozygosity, in a neutral setting and in cases where the two species can compete or cooperate. By concluding with examples in which individuals are transported by fluid flows, we argue that this model is a natural choice to describe competition in marine environments.Comment: 29 pages, 14 figures; revised version including a section with results in the presence of fluid flow

Screening current effects in Josephson junction arrays

The purpose of this work is to compare the dynamics of arrays of Josephson junctions in presence of magnetic field in two different frameworks: the so called XY frustrated model with no self inductance and an approach that takes into account the screening currents (considering self inductances only). We show that while for a range of parameters the simpler model is sufficiently accurate, in a region of the parameter space solutions arise that are not contained in the XY model equations.Comment: Figures available from the author