Viscosity solutions, almost everywhere solutions and explicit formulas

Consider the differential inclusion in . We exhibit an explicit solution that we call fundamental. It also turns out to be a viscosity solution when properly defining this notion. Finally, we consider a Dirichlet problem associated to the differential inclusion and we give an iterative procedure for finding a solution.

     

Smooth solutions to a class of free boundary parabolic problems

We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive , if the data are smooth.

     

Cauchy Problem of Some Doubly Degenerate Parabolic Equations with Initial Datum a Measure

This paper discusses the Cauchy problem of the equation

Existence of Entire Solutions of a Singular Semilinear Elliptic Problem

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang‘s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems.     

The Attribute Based Hill Climber

In this paper we introduce the Attribute Based Hill Climber, a parameter-free algorithm that provides a concrete, stand-alone implementation of a little used technique from the Tabu Search literature known as regional aspiration. Results of applying the algorithm to two classical optimisation problems, the Travelling Salesman Problem and the Quadratic Assignment Problem, show it to be competitive with existing general purpose heuristics in these areas.     

Selecting distances in arrangements of hyperplanes spanned by points

In this paper we consider a problem of distance selection in the arrangement of hyperplanes induced by n given points. Given a set of n points in d-dimensional space and a number k, , determine the hyperplane that is spanned by d points and at distance ranked by k from the origin. For the planar case we present an O(nlog²n) runtime algorithm using parametric search partly different from the usual approach [N. Megiddo, J. ACM 30 (1983) 852]. We establish a connection between this problem in 3-d and the well-known 3SUM problem using an auxiliary problem of counting the number of vertices in the arrangement of n planes that lie between two sheets of a hyperboloid. We show that the 3-d problem is almost 3SUM-hard and solve it by an O(n²log²n) runtime algorithm. We generalize these results to the d-dimensional (d4) space and consider also a problem of enumerating distances.     

Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm

The nonlinear optical characteristics of silicate glasses doped with silver nanoparticles are investigated by Z-scan technique using second harmonic radiation of picosecond Nd:YAG laser (=532 nm, =55 ps). The real and imaginary parts of third-order nonlinear susceptibility of silver-contained glasses were measured. It was found that the sign of Im ⁽³⁾was negative due to saturated absorption and the sign of Re ⁽³⁾ was changed from negative (self-defocusing) to positive (self-focusing) with growth of laser radiation intensity. The mechanisms responsible for saturated absorption and nonlinear refraction are discussed.     

Nonlinear refraction and nonlinear absorption of As2S3 aqueous solution

We report for the first time to our knowledge on the preparation of colloidal solution of chalcogenide semiconductor As₂S₃ by laser ablation and the measurements of its nonlinear-optical characteristics using Z-scan method at the wavelength of Nd:YAG laser radiation ( = 1064 nm, = 25 ns). The nonlinear refractive index was measured to be –7.5 × 10^–18 m² W^–1. Nonlinear absorption coefficient of chalcogenide solution was measured to be 1 cm GW^–1.     

An Inverse Problem for an Harmonic Oscillator Perturbed by Potential: Uniqueness

Consider the perturbed harmonic oscillator Ty = -y" + x²y + q(x)y on L²(R) where the real potential q satisfy some assumption on infinity (the case q L²(R), (t+1)^-rdt), r < 1 is covered).