We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin’s Invariant Subspace theorem...

Let A denote an n n real symmetric matrix-valued function depending on a vector of real parameters, x 2 < m. Assume that A is a twice continuously diierentiable function of x, with the second derivative satisfying a Lipschitz condition. Consider the following optimization problem: minimize the largest eigenvalue of A(x). Let x denote a minimum. Typically, the maximum eigen-value of A(x) is mult...

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is ...

The main theme of this paper is to develop a novel eigenvalue optimization framework for learning a Mahalanobis metric. Within this context, we introduce a novel metric learning approach called DML-eig which is shown to be equivalent to a well-known eigenvalue optimization problem called minimizing the maximal eigenvalue of a symmetric matrix (Overton, 1988; Lewis and Overton, 1996). Moreover, ...

In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above rela...

Positive systems are systems in which the state variables take only nonnegative values for all times. In this paper, we derive eigenvalue regions for discrete and continuous-time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix. c © 2003 Elsevier B.V. All rights reserved.

We consider the operator of taking the 2pth derivative of a function with zero boundary conditions for the function and its first p − 1 derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for p = 1 and p = 3.

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.

We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue asymptotic depends on the geometry of the cusp and on the boundary conditions.

In this paper we show that the conjecture of Lemmens and Seidel 1973 for systems equiangular lines with common angle arccos⁡(1/5) is true. Our main tool forbidden subgraphs smallest eigenvalue −5.