Dynamical exchange corrections to the dielectric function in a two-dimensional electron gas

The effect of dynamical exchange interactions on the dielectric function of a two-dimensional electron gas is studied using a variational approach. Exchange effects are introduced via the local-field correction. The variationally obtained local-field factor is compared to the earlier perturbative result to first order in the electron-electron interaction.     

Laplacians and the Cheeger Inequality for Directed Graphs

We consider Laplacians for directed graphs and examine their eigenvalues. We introduce a notion of a circulation in a directed graph and its connection with the Rayleigh quotient. We then define a Cheeger constant and establish the Cheeger inequality for directed graphs. These relations can be used to deal with various problems that often arise in the study of non-reversible Markov chains including bounding the rate of convergence and deriving comparison theorems.Received September 8, 2004     

The spline collocation method for parabolic boundary integral equations on smooth curves

Summary. We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces. Received May 17, 2001 / Revised version received February 19, 2002 / Published online April 17, 2002     

Crystal Structure of Adduct 2-Phenyl-imidazo [ 4,5-f ] 1,10-phenanthroline Methanol

The structure of the title adduct comprises a phenanthroline derivative 2-phenyl-imidazo[4,5-f]1,10-phenanthroline and a methanol.The composition of the crystalline adduct was characterized as C19H12N4.CH3OH.It belongs to orthorhombic system,space group Pna21 with a=1.3693(4)nm,b=2.2988(7)nm,c=0.51338(15)nm,V=1.6160(8)nm^3.Z=4,and final R1=0.0423.wR2=0.1012 .Crystal structure shows that all the 19 carbon atoms and 4 nitrogen atoms are coplanar.The bond length data indicated that a very extensive conjugation system was formed.This conjugation makes the compound being a potentially excellent energy transformer used for luminescent materials.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

A preconditioned domain decomposition algorithm for the solution of the elliptic Neumann problem

A new preconditioned conjugate gradient (PCG)-based domain decomposition method is given for the solution of linear equations arising in the finite element method applied to the elliptic Neumann problem. The novelty of the proposed method is in the recommended preconditioner which is constructed by using cyclic matrix. The resulting preconditioned algorithms are well suited to parallel computation.     

An optical bridge reflectometer for sensitive measurement of ion implantation dose

An instrument is described that can be used to monitor, with unprecedented sensitivity, changes in the optical reflectivity due to crystailine damage incurred during ion implantation. It is shown that at the shot-noise limit, changes in the optical reflectivity of silicon as small as 5·10^–7 can be measured in a 10 Hz bandwidth with a signal-to-noise ratio of 100, corresponding to an extrapolated uniform implantation dose of 5·10⁸ cm^–2 for ¹¹B⁺ at 50 keV in silicon.     

An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem

Summary. This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: , for , subject to , where . We assume that on , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter . We present numerical results in support of this result. Received February 25, 1994     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)