Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples

Continuum description of profile scaling in nanostructure decay

The relaxation of axisymmetric crystal surfaces with a single facet below the roughening transition is studied via a continuum approach that accounts for step energy g_1 and step-step interaction energy g_3>0. For diffusion-limited kinetics, free-boundary and boundary-layer theories are used for self-similar shapes close to the growing facet. For long times and g_3/g_1 < 1, (a) a universal equation is derived for the shape profile, (b) the layer thickness varies as (g_3/g_1)^{1/3}, (c) distinct solutions are found for different g_3/_1, and (d) for conical shapes, the profile peak scales as (g_3/g_1)^{-1/6}. These results compare favorably with kinetic simulations.Comment: 4 pages including 3 figure

Novel continuum modeling of crystal surface evolution

We propose a novel approach to continuum modeling of the dynamics of crystal surfaces. Our model follows the evolution of an ensemble of step configurations, which are consistent with the macroscopic surface profile. Contrary to the usual approach where the continuum limit is achieved when typical surface features consist of many steps, our continuum limit is approached when the number of step configurations of the ensemble is very large. The model can handle singular surface structures such as corners and facets. It has a clear computational advantage over discrete models.Comment: 4 pages, 3 postscript figure

Decay of one dimensional surface modulations

The relaxation process of one dimensional surface modulations is re-examined. Surface evolution is described in terms of a standard step flow model. Numerical evidence that the surface slope, D(x,t), obeys the scaling ansatz D(x,t)=alpha(t)F(x) is provided. We use the scaling ansatz to transform the discrete step model into a continuum model for surface dynamics. The model consists of differential equations for the functions alpha(t) and F(x). The solutions of these equations agree with simulation results of the discrete step model. We identify two types of possible scaling solutions. Solutions of the first type have facets at the extremum points, while in solutions of the second type the facets are replaced by cusps. Interactions between steps of opposite signs determine whether a system is of the first or second type. Finally, we relate our model to an actual experiment and find good agreement between a measured AFM snapshot and a solution of our continuum model.Comment: 18 pages, 6 figures in 9 eps file

A Local Computation Approximation Scheme to Maximum Matching

We present a polylogarithmic local computation matching algorithm which guarantees a (1-\eps)-approximation to the maximum matching in graphs of bounded degree.Comment: Appears in Approx 201

On computational irreducibility and the predictability of complex physical systems

Using elementary cellular automata (CA) as an example, we show how to coarse-grain CA in all classes of Wolfram's classification. We find that computationally irreducible (CIR) physical processes can be predictable and even computationally reducible at a coarse-grained level of description. The resulting coarse-grained CA which we construct emulate the large-scale behavior of the original systems without accounting for small-scale details. At least one of the CA that can be coarse-grained is irreducible and known to be a universal Turing machine.Comment: 4 pages, 2 figures, to be published in PR

Modeling and simulation of Li-Ion conduction in POLY(Ethylene Oxide

Prof. Moshe Israeli passed away on Feburary 18, 2007. This paper is dedicated to his memory Polyethylene oxide (PEO) containing a lithium salt (e.g. LiI) serves as a solid polymer electrolyte (SPE) in thin-film batteries and its ionic conductivity is a key parameter of their performance. We model and simulate Li + ion conduction in a single PEO molecule. Our simplified stochastic model of ionic motion is based on an analogy between protein channels of biological membranes that conduct Na +, K +, and other ions, and the PEO helical chain that conducts Li + ions. In contrast with protein channels and salt solutions, the PEO is both the channel and the solvent for the lithium salt (e.g., LiI). The mobile ions are treated as charged spherical Brownian particles. We simulate Smoluchowski dynamics in channels with a radius of ca 0.1nm and study the effect of stretching and temperature on ion conductivity. We assume that each helix (molecule) forms a random angle with the axis between these electrodes and the polymeric film is composed of many uniformly distributed oriented boxes that include molecules with the same direction. We further assume that mechanical stretching aligns the molecular structures in each box along the axis of stretching (intra-box alignment). Our model thus predicts the PEO conductivity as a function of the stretching, the salt concentration and the temperature. The computed enhancement of the ionic conductivity in the stretch direction is in good agreement with experimental results. The simulation results are also in qualitative agreement with recent theoretical and experimental results.