We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain Ω(h) perturbed by a small (with diameter O(h)) Lipschitz cavern ωh in a smooth boundary ∂Ω = ∂Ω(0). The case of the hole ωh inside the domain but very close to the boundary ∂Ω is under consideration as well. It is proven that the main correc...

In this study the interaction between self-excited and paramet rically excited oscillations in two-degree-of-freedom systems with quadratic nonlinearities is investigated. The fundamental parametric resonance of the first mode and 3:1 internal resonance is considered, followed by 1:2 internal and parametric resonances of the second mode. The method of multiple time scales is applied to derive f...

For a nonsingular integer matrix A, we study the growth of the order of A modulo N . We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values o...

The research in this paper is motivated by a recent work of I. Barany and J. Solymosi [I. Barany and J. Solymosi. Gershgorin disks for multiple eigenvalues of non-negative matrices. Preprint arXiv no. 1609.07439, 2016.] about the location of eigenvalues of nonnegative matrices with geometric multiplicity higher than one. In particular, an answer to a question posed by Barany and Solymosi, about...

Proofs of convergence of adaptive finite element methods for approximating eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai et. al. in [8], who proved convergence and optimality of AFEM for eigenvalues of multiplicity greater than one. There it...

Modal dispersion data i.e. mode travel time as a function of frequency have been used in inversion schemes that estimate sediment properties in a range-independent environment. This method has been extended to include range-dependent environment. Simulation studies show that range-dependent sediment properties can be extracted if mode dispersion data are obtained for multiple source/receiver lo...

This paper looks at the iterative solution of complex, non-Hermitian systems of linear equations associated with a lattice QCD problem in particle physics. Let the n by n system of equations be Ax = b. We use a version of the GMRES method that deflates eigenvalues and can solve systems with multiple right-hand sides. We give examples that show deflating eigenvalues can make a significant improv...

Computing the spectral decomposition of a normal matrix is among the most frequent tasks to numerical mathematics. A vast range of methods are employed to do so, but all of them suffer from instabilities when applied to degenerate matrices, i.e., those having multiple eigenvalues. We investigate the spectral representation’s effectivity properties on the sound formal basis of computable analysi...

In this paper, we develop a perturbation analysis for stability spectra (Lyapunov exponents and Sacker-Sell spectrum) for products of operators on a Hilbert space (both real and complex) based upon the discrete QR technique. Error bounds are obtained in both the integrally separated and non-integrally separated cases that correspond to distinct and multiple eigenvalues, respectively, for a sing...

We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in M nonoverlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. We present a general finite element method to obtain a numerical solution of the eigenvalue problem, starting from a nonstandard formally equivalen...