This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic var...

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structure...

We study the boundary value problem −div((|∇u|1 + |∇u|2)∇u) = λ|u|u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in R with smooth boundary, λ is a positive real number, and the continuous functions p1, p2, and q satisfy 1 < p2(x) < q(x) < p1(x) < N and maxy∈Ω q(y) < Np2(x) N−p2(x) for any x ∈ Ω. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ...

We establish universality of local eigenvalue correlations in unitary random matrix ensembles 1 Zn | det M | 2α e −ntr V (M) dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x| 2α e −nV (x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local ...

In the numerical solution of large-scale eigenvalue problems, Davidson-type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson-type methods, the factorization is replaced by iterative linear sol...

Optical eigenvalue communication is a promising technique for overcoming the Kerr nonlinear limit in optical systems. The associated with Schrödinger equation remains invariant during fiber-based dispersive transmission. However, practical applications involving use of such systems are limited by occurrence fiber loss and amplified noise that induce distortion. Thus, several time-domain neural-...

Abstract In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate applicability our main theoretical findings.

In this work we calculate the eigenvalues obtained by preconditioning the discrete Helmholtz operator with Sommerfeld-like boundary conditions on a rectilinear domain, by a related operator with boundary conditions that permit the use of fast solvers. The main innovation is that the eigenvalues for two and three-dimensional domains can be calculated exactly by solving a set of one-dimensional e...

The main contribution of this paper is an error representation formula for eigenvalue approximations for positive definite operators defined as quadratic forms. The formula gives an operator theoretic framework for treating discrete eigenvalue approximation/estimation problems for unbounded positive definite operators independent of the multiplicity. Furthermore, by the use of the error represe...