Abstract: The paper can be considered as a complement to previous papers of the authors. An insight into applied asymptotic analysis of boundary value problems in singularly perturbed domains is presented. As a result, the asymptotic expansions of eigenvalues are obtained and discussed in terms of integral attributes of the geometrical perturbations including the virtual mass tensor, polarizati...

The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. This may be seen for example as a simplified scalar cross-sectional model of the problem for localised modes in photonic crystal fibers. For a high contrast small size periodicity and a finite size defect we consider the critical (the so called double poros-ity type) scaling. We employ (high contra...

We study the instability of algebraic solitons for integrable nonlinear equations in one spatial dimension that include modified KdV, focusing NLS, derivative NLS, and massive Thirring equations. We develop the analysis of the Evans function that defines eigenvalues in the corresponding Lax operators with algebraically decaying potentials. The standard Evans function generically has singulariti...

Usually, principal components analysis is carried out by calculating the eigenvalues and eigenvectors of the correlation matrix. With N cases and P variables, if we write X for the N × P matrix which has been standardised so that columns have zero mean and unit standard deviation, we find the eigenvalues and eigenvectors of the P × P matrix X.X (which is N or (N − 1) times the correlation matri...

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor, and the adjacency tensor has at most one H-eigenva...

Computing the roots of a scalar polynomial, or the eigenvalues of a matrix polynomial, expressed in the Chebyshev basis {Tk(x)} is a fundamental problem that arises in many applications. In this work, we analyze the backward stability of the polynomial rootfinding problem solved with colleague matrices. In other words, given a scalar polynomial p(x) or a matrix polynomial P (x) expressed in the...

The scattering of a time-harmonic plane wave in an inhomogeneous medium is modeled by the scattering problem for the Helmholtz equation. A transmission eigenvalue is a wavenumber at which the scattering operator has a non-trivial kernel or cokernel. Because many sampling methods for locating scatterers succeed only at wavenumbers that are not transmission eigenvalues, they have been studied for...

Transfer matrices are used widely for the dynamic analysis of engineering structures, increasingly so for static analysis, and are particularly useful in the treatment of repetitive structures for which, in general, the behaviour of a complete structure can be determined through the analysis of a single repeating cell, together with boundary conditions if the structure is not of infinite extent...

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The spiked covariance model has gained increasing popularity in high-dimensional data analysis. A fundamental problem is determination of the number eigenvalues, K. For estimation K, most attention focused on use top eigenvalues sample matrix, and there little investigation into proper ways using bulk to estimate We propose a principled approach incorporating Our method imposes working residual...