On donne la derivee DF de la fonction a valeur tensorielle d'un tenseur dans le cas general

/pdf/the-derivative-of-a-tensor-valued-function-of-a-tensor-1gz6bonvxa.pdf

Perturbation theory of eigenvalue problems

The derivative of a tensor-valued function of a tensor

The spectrum of a selfadjoint, C' linear elliptic partial differential operator on a compact manifold contains only isolated eigenvalues, each having finite multiplicity. It is sometimes the case that these multiplicities are unbounded; this is common in problems arising in applications because of the high degree of symmetry usually present. The main theorem shows that the property of having only simple eigenvalues is generic for operators obtained by varying the zeroth order part of a given operator. The purpose of this article is to provide a proof of the following theorem. Main theorem. Let M be a compact, connected C' manifold without boundary. Let L be a sel/adjoint, C' linear elliptic differential operator on M. The set Ep e Cw(M): all eigenvalues of L + p are simple} is residual in C'(M). The term elliptic operator used in this article will always mean the operator is selfadjoint, C' and linear. The main theorem can be summarized by saying that almost all elliptic operators obtained by varying the zeroth order part of a given operator have only simple eigenvalues. For terminology, see ? 1. The main theorem was first announced in [1] for second-order operators, and the proof is the one given in [2]; it uses perturbation theory. A proof using transversality theory, applicable to operators whose eigenfunctions satisfy the strong unique continuation property, has been obtained by K. Uhlenbeck [6]. The theorem is particularly useful in obtaining genericity results about the eigenfunctions (see [1] , [2], [6]). 1. Notation; outline of proof. The manifold M and the operator L will be fixed throughout as in the theorem. For standard terminology on elliptic operators, see [3], [4]. Received by the editors June 4, 1973 and, in revised form, January 23, 1974. AMS (MOS) subject classifications (1970). Primary 35P05, 35J30; Secondary 47A55, 47B25, 58G99.

/pdf/genericity-of-simple-eigenvalues-for-elliptic-pde-s-1ipjh3mf3f.pdf

Genericity of simple eigenvalues for elliptic PDE’s

The theories of the dilatation, r ..-->.. re/sup i theta/, and translation, x ..-->.. x + iq, transformations as related to the Stark problem are reviewed, and new results obtained. Results for the hydrogen atom n = 1 and n = 2 levels and the /sup 1/P/sup 0/, 2s2p H/sup -/ shape resonance in dc fields are presented, and the extension to the ac Stark effect made. Spectral estimates are made using the technique of the numerical range and via discussion of several model problems, using both coordinate rotation and coordinate translation. 45 references.

Complex coordinates and the Stark effect

The idea of subspace fitting provides a popular framework for different applications of parameter estimation and system identification. Previously, some algorithms have been suggested based on similar ideas, for a sensor array processing problem where the underlying data model is not low rank. We show that two of these algorithms (DSPE and DISPARE) fail to give consistent estimates and introduce a general class of subspace fitting-like algorithms for consistent estimation of parameters from a possibly full-rank data model. The asymptotic performance is analyzed, and an optimally weighted algorithm is derived. The result gives a lower bound on the estimation performance for any estimator based on a low-rank approximation of the linear space spanned by the sample data. We show that in general, for full-rank data models, no subspace-based method can reach the Cramer-Rao lower bound (CRB).

background

A generalization of weighted subspace fitting to full-rank models

/pdf/hardy-poincare-rellich-and-uncertainty-principle-4rpnmxygqy.pdf

Perturbation theory of eigenvalue problems

Citations

The derivative of a tensor-valued function of a tensor

Genericity of simple eigenvalues for elliptic PDE’s

Complex coordinates and the Stark effect

A generalization of weighted subspace fitting to full-rank models

Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds

Related Papers (5)

Perturbation theory for linear operators

Methods of Modern Mathematical Physics

Matrix Analysis

The algebraic eigenvalue problem

Matrix computations