Some of the most challenging eigenvalue problems arise in the stability analysis of solutions to parameter-dependent nonlinear partial diierential equations. Surface tension gradients along the free boundary of the oat-zone in crystal growth give rise to a stationary thermocapillary convection. Loss of stability leads to undesirable material imperfections. Stability analyses employing both ener...

Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according to the Bures measure. The eigenvalues of the random density matrices are analysed: we derive the eigenvalue distribution for the Bures ensemble which is shown to be broader then the quarter-circle di...

To state our results precisely, we introduce some notation. Throughout this paper, let F be a given Boolean function over N variables, and we assume that it is given as a CNF formula with M clauses and that it has P2N satisfying assignments, where P will be referred as the sat. assignment ratio of F . Furthermore, we introduce two parameters1 δ > 0 and ε > 0, and consider the following situatio...

The eigenvalue problem of the Laplace-Beltrami operators on curved surfaces plays an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting (LTL) method with the configuration equation to develop a new effective and convergent algori...

We consider the sloshing problem for an incompressible, inviscid, irrotational fluid in an open container, including effects due to surface tension on the free surface. We restrict ourselves to a constant contact angle and seek time-harmonic solutions of the linearized problem, which describes the time-evolution of the fluid due to a small initial disturbance of the surface at rest. As opposed ...

SUMMARY The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial diierential eigenvalue problems in two-dimensional or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the...

Given a linear self-adjoint partial diierential operator L, the smallest few eigenvalues and eigenfunctions of L are computed by the homotopy (continuation) method. The idea of the method is very simple. From some initial operator L0 with known eigenvalues and eigenfunctions, deene the homotopy H (t) = (1 ? t)L0 + tL; 0 t 1. If the eigenfunctions of H (t0) are known, then they are used to deter...

The parameter σ(G) of a graph G stands for the number of Laplacian eigenvalues greater than or equal to the average degree of G. In this work, we address the problem of characterizing those graphs G having σ(G) = 1. Our conjecture is that these graphs are stars plus a (possible empty) set of isolated vertices. We establish a link between σ(G) and the number of anticomponents of G. As a by-produ...

This publication is a contribution to basic research in image comparison using eigenvalue spectra as features. The differential-geometric approach of eigenvalue spectrum based descriptors is naturally applicable to shape data, but so far little work has been done to transfer it to the setting of image data painted on a rectangle or general curved surface. We present a new semi-global feature de...

The concept of "{pseudospectra for matrices, introduced by Trefethen and his co-workers, has been studied extensively since 1990. In this paper, "{ pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of "{pseudospectra for matrix pencils will be discussed. As an application,...