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

Origin Gaps and the Eternal Sunshine of the Second-Order Pendulum

The rich experiences of an intentional, goal-oriented life emerge, in an unpredictable fashion, from the basic laws of physics. Here I argue that this unpredictability is no mirage: there are true gaps between life and non-life, mind and mindlessness, and even between functional societies and groups of Hobbesian individuals. These gaps, I suggest, emerge from the mathematics of self-reference, and the logical barriers to prediction that self-referring systems present. Still, a mathematical truth does not imply a physical one: the universe need not have made self-reference possible. It did, and the question then is how. In the second half of this essay, I show how a basic move in physics, known as renormalization, transforms the "forgetful" second-order equations of fundamental physics into a rich, self-referential world that makes possible the major transitions we care so much about. While the universe runs in assembly code, the coarse-grained version runs in LISP, and it is from that the world of aim and intention grows.Comment: FQXI Prize Essay 2017. 18 pages, including afterword on Ostrogradsky's Theorem and an exchange with John Bova, Dresden Craig, and Paul Livingsto