A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance $\rho(d)$ ($d$ is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar's problem. Using classical potential theory for the harmonic case, we show that $1+\rho(d)$ is equal to the largest positive zero of a certain sequence of monic polynomials of degree $2d-1$ with integer coefficients which we call Gonchar polynomials. Rather surprisingly, $\rho(2)$ is the Golden ratio and $\rho(4)$ the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.Comment: 12 pages, 6 figures, 1 tabl

All functions are (locally) ss-harmonic (up to a small error) - and applications

The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Stress corrosion cracking in Al-Zn-Mg-Cu aluminum alloys in saline environments

Copyright 2013 ASM International. This paper was published in Metallurgical and Materials Transactions A, 44A(3), 1230 - 1253, and is made available as an electronic reprint with the permission of ASM International. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplications of any material in this paper for a fee or for commercial purposes, or modification of the content of this paper are prohibited.Stress corrosion cracking of Al-Zn-Mg-Cu (AA7xxx) aluminum alloys exposed to saline environments at temperatures ranging from 293 K to 353 K (20 °C to 80 °C) has been reviewed with particular attention to the influences of alloy composition and temper, and bulk and local environmental conditions. Stress corrosion crack (SCC) growth rates at room temperature for peak- and over-aged tempers in saline environments are minimized for Al-Zn-Mg-Cu alloys containing less than ~8 wt pct Zn when Zn/Mg ratios are ranging from 2 to 3, excess magnesium levels are less than 1 wt pct, and copper content is either less than ~0.2 wt pct or ranging from 1.3 to 2 wt pct. A minimum chloride ion concentration of ~0.01 M is required for crack growth rates to exceed those in distilled water, which insures that the local solution pH in crack-tip regions can be maintained at less than 4. Crack growth rates in saline solution without other additions gradually increase with bulk chloride ion concentrations up to around 0.6 M NaCl, whereas in solutions with sufficiently low dichromate (or chromate), inhibitor additions are insensitive to the bulk chloride concentration and are typically at least double those observed without the additions. DCB specimens, fatigue pre-cracked in air before immersion in a saline environment, show an initial period with no detectible crack growth, followed by crack growth at the distilled water rate, and then transition to a higher crack growth rate typical of region 2 crack growth in the saline environment. Time spent in each stage depends on the type of pre-crack (“pop-in” vs fatigue), applied stress intensity factor, alloy chemistry, bulk environment, and, if applied, the external polarization. Apparent activation energies (E a) for SCC growth in Al-Zn-Mg-Cu alloys exposed to 0.6 M NaCl over the temperatures ranging from 293 K to 353 K (20 °C to 80 °C) for under-, peak-, and over-aged low-copper-containing alloys (~0.8 wt pct), they are typically ranging from 20 to 40 kJ/mol for under- and peak-aged alloys, and based on limited data, around 85 kJ/mol for over-aged tempers. This means that crack propagation in saline environments is most likely to occur by a hydrogen-related process for low-copper-containing Al-Zn-Mg-Cu alloys in under-, peak- and over-aged tempers, and for high-copper alloys in under- and peak-aged tempers. For over-aged high-copper-containing alloys, cracking is most probably under anodic dissolution control. Future stress corrosion studies should focus on understanding the factors that control crack initiation, and insuring that the next generation of higher performance Al-Zn-Mg-Cu alloys has similar longer crack initiation times and crack propagation rates to those of the incumbent alloys in an over-aged condition where crack rates are less than 1 mm/month at a high stress intensity factor

Regularity of Infinity for Elliptic Equations with Measurable Coefficients and Its Consequences

This paper introduces a notion of regularity (or irregularity) of the point at infinity for the unbounded open subset of \rr^{N} concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the A-harmonic measure of the point at infinity is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of \rr^{N}, N\ge 3 is established in terms of the Wiener test for the regularity of the point at infinity. It coincides with the Wiener test for the regularity of the point at infinity in the case of Laplace equation. From the topological point of view, the Wiener test at infinity presents thinness criteria of sets near infinity in fine topology. Precisely, the open set is a deleted neigborhood of the point at infinity in fine topology if and only if infinity is irregular.Comment: 20 page