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Let be an -dimensional () simply connected Hadamard manifold. If the radial Ricci curvature of is bounded from below by with respect to some point , where is the Riemannian distance on to , is a nonpositive continuous function on , then the first nonzero Neumann eigenvalues of the Laplacian on the geodesic ball , with center and radius , satisfy
where is the solution to
For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues λ of the fixed membrane for any n the following inequality holds [...] where λ(o) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.
We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.
We study the propagation of electromagnetic waves in a
guide the section of which is a thin annulus. Owing to the presence of a
small parameter, explicit approximations of the TM and TE eigenmodes are
obtained. The cases of smooth and non smooth boundaries are presented.
In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, and . Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations.
Let be an odd function of a class such that and increases on . We approximate the positive solution of on with homogeneous Dirichlet boundary conditions by the solution of on with adequate non-homogeneous Dirichlet conditions. We show that the error tends to zero exponentially fast, in the uniform norm.
Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and increases on
[0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on with homogeneous Dirichlet boundary conditions by the
solution of on ]0,L[2 with adequate
non-homogeneous Dirichlet conditions.
We show that the error uL - u
tends to zero exponentially fast, in the uniform norm.
By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally,...
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