504,311 research outputs found
Quasilinear eigenvalues
In this work, we review and extend some well known results for the
eigenvalues of the Dirichlet Laplace operator to a more general class of
monotone quasilinear elliptic operators. As an application we obtain some
homogenization results for nonlinear eigenvalues.Comment: 23 pages, Rev. UMA, to appea
Limit theorems for sample eigenvalues in a generalized spiked population model
In the spiked population model introduced by Johnstone (2001),the population
covariance matrix has all its eigenvalues equal to unit except for a few fixed
eigenvalues (spikes). The question is to quantify the effect of the
perturbation caused by the spike eigenvalues. Baik and Silverstein (2006)
establishes the almost sure limits of the extreme sample eigenvalues associated
to the spike eigenvalues when the population and the sample sizes become large.
In a recent work (Bai and Yao, 2008), we have provided the limiting
distributions for these extreme sample eigenvalues. In this paper, we extend
this theory to a {\em generalized} spiked population model where the base
population covariance matrix is arbitrary, instead of the identity matrix as in
Johnstone's case. New mathematical tools are introduced for establishing the
almost sure convergence of the sample eigenvalues generated by the spikes.Comment: 24 pages; 4 figure
Bounds for eigenvalue ratios of the Laplacian
For a bounded domain with a piecewise smooth boundary in an
-dimensional Euclidean space , we study eigenvalues of the
Dirichlet eigenvalue problem of the Laplacian. First we give a general
inequality for eigenvalues of the Laplacian. As an application, we study lower
order eigenvalues of the Laplacian and derive the ratios of lower order
eigenvalues of the Laplacian.Comment: 14 page
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