Cugliandolo-Kurchan equations for dynamics of Spin-Glasses

We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations intensively studied by Cugliandolo and Kurchan. Research partially supported by NSF grants #DMS-0406042, #DMS-FRG-0244323     

Aging of spherical spin glasses

Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for (hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging” phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures. Received: 8 July 1999 / Revised version: 2 June 2000 / Published online: 6 April 2001     

Reporting computational experiments in mathematical programming

This paper presents a set of recommended standards for the presentation of computational experiments in mathematical programming.The authors are members of the Committee on Algorithms of the Mathematical Programming Society.     

Dual to primal conversion in geometric programming

The aim of this paper is not to derive new results, but rather to provide insight that will hopefully aid researchers involved in the design and coding of algorithms for geometric programs. The main contributions made here are: (i) a computationally useful interpretation of the Lagrange multipliers associated with the dual orthogonality constraints, (ii) a computationally useful interpretation of the Lagrange multiplier associated with the dual normality constraint, and (iii) an analysis of the much-avoided issue of subsidiary problems.This work was supported in part by the National Research Council of Canada, Grant No. A3552.The author would like to acknowledge the contribution of an anonymous referee, whose constructive criticism led to this improved version of the original paper.     

Study of the gel films of Acetobacter Xylinum cellulose and its modified samples by 1H NMR cryoporometry and small-angle X-ray scattering

Gel films of Acetobacter Xylinum cellulose and its modified samples have been investigated by 1H nuclear magnetic resonance (NMR) cryoporometry and small-angle X-ray scattering. The joint use of these two methods made it possible to characterize the sizes of aqueous pores in gel films and estimate the sizes of structural inhomogeneities before and after the sorption of polyvinylpyrrolidone and Se₀ nanoparticles (stabilized by polyvinylpyrrolidone) into the films. According to small-angle X-ray scattering data, the sizes of inhomogeneities in a gel film change only slightly upon the sorption of polyvinylpyrrolidone and nanoparticles. The impregnated material is sorbed into water-filled cavities that are present in the gel film. ¹H NMR cryoporometry allowed us to reveal the details of changes in the sizes of small aqueous pores during modifications.     

Determination of the size and phase composition of silver nanoparticles in a gel film of bacterial cellulose by small-angle X-ray scattering,electron diffraction,and electron microscopy

The nanoscale structural features in a composite (gel film of Acetobacter Xylinum cellulose with adsorbed silver nanoparticles, stabilized by N-polyvinylpyrrolidone) have been investigated by small-angle X-ray scattering. The size distributions of inhomogeneities in the porous structure of the cellulose matrix and the size distributions of silver nanoparticles in the composite have been determined. It is shown that the sizes of synthesized nanoparticles correlate with the sizes of inhomogeneities in the gel film. Particles of larger size (with radii up to 100 nm) have also been found. Electron microscopy of thin cross sections of a dried composite layer showed that large particles are located on the cellulose layer surface. Electron diffraction revealed a crystal structure of silver nanoparticles in the composite.     

Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix whose (i, j) entry is , where (x _ij , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, , and σ is a deterministic function. For random diagonal D _N independent of and with appropriate rescaling a _N , we prove that converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. Supported in part by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada and a University of Saskatchewan start-up grant. Research partially supported by NSF grant #DMS-0806211.     

Random polynomials having few or no real zeros

Consider a polynomial of large degree whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly real zeros with probability as through integers of the same parity as the fixed integer . In particular, the probability that a random polynomial of large even degree has no real zeros is . The finite, positive constant is characterized via the centered, stationary Gaussian process of correlation function . The value of depends neither on nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability one may specify also the approximate locations of the zeros on the real line. The constant is replaced by in case the i.i.d. coefficients have a nonzero mean.

     

Thick points for intersections of planar sample paths

Let denote the number of visits to of the simple planar random walk , up to step . Let be another simple planar random walk independent of . We show that for any , there are points for which . This is the discrete counterpart of our main result, that for any , the Hausdorff dimension of the set of thick intersection points for which , is almost surely . Here is the projected intersection local time measure of the disc of radius centered at for two independent planar Brownian motions run until time . The proofs rely on a ``multi-scale refinement' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius centered at by for general sets .

     

Quenching of giant hysteresis effects in La(1-z)Y(z)Hx switchable mirrors

Hydrogen atoms adsorbed on TiO2(110)-(1x1) surfaces have been characterized by scanning tunneling microscopy (STM) combined with electron stimulated desorption (ESD) technique. Certain amounts of H atoms are unexpectedly found on the TiO2 surfaces annealed at 900 K. Two forms of adsorption were discriminated in STM images from the different sensitivity to ESD and tentatively assigned to hydroxyl-type (O-H) and hydride-type (Ti-H) species.