Taking advantage from the so-called Lemma on small eigenvalues by Colin de Verdière, we study ramification for multiple of Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior turns out to depend expansion suitable associated eigenfunctions. We treat case planar domains details, thanks a generalization u-capacity which compute dimension 2. In this are proved split essentia...

This paper is concerned with the problem of designing discrete-time control systems with closed-loop eigenvalues in a prescribed region of stability. First, we obtain a state feedback matrix which assigns all the eigenvalues to zero, and then by elementary similarity operations we find a state feedback which assigns the eigenvalues inside a circle with center   and radius. This new algorithm ca...

A generalization of Gershgorin's theorem is developed for the eigenvalue problem Ax = XBx and is applied to obtain perturbation bounds for multiple eigenvalues. The results are interpreted in terms of the chordal metric on the Riemann sphere, which is especially convenient for treating infinite eigenvalues.

Refined estimates for finite element or, more generally, Galerkin approximations of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems are presented. More specifically, refined results on the asymptotic behavior of the eigenvalue and eigenvector errors are proved. Both simple and multiple eigenvalues are treated.