Nearest q-Flat to m Points

Recently, Bradley and Mangasarian studied the problem of finding the nearest plane to m given points in ⁿ in the least square sense. They showed that the problem reduces to finding the least eigenvalue and associated eigenvector of a certain n×n symmetric positive-semidefinite matrix. We extend this result to the general problem of finding the nearest q-flat to m points, with 0qn–1.     

Polynomial Solution of Non-Central Potentials

We show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in certain class of non-central potentials can be easily calculated analytically in a simple and elegant manner by using Nikiforov and Uvarov (NU) method. We discuss the generalized Coulomb and harmonic oscillator systems. We study the Hartmann Coulomb and the ring-shaped and compound Coulomb plus Aharanov–Bohm potentials as special cases. The results are in exact agreement with other methods.     

Titchmarsh's asymptotic formula for periodic eigenvalues and an extension to the p-Laplacian

An asymptotic formula for periodic eigenvalues, due to Titchmarsh, is taken to a higher degree of accuracy using a Prüfer transformation method, not depending on Floquet theory. With the idea of the rotation number, this method is then extended to the p-Laplacian.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. I. Asymptotic Expansion Formulas

We consider the spectral problem for the random Schrödinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We obtain the explicit almost sure asymptotic expansion formulas for the extreme eigenvalues and eigenfunctions in the intermediate rank case, provided the upper distributional tails of potential decay at infinity slower than the double exponential function. For the fractional-exponential tails (including Weibull’s and Gaussian distributions), extremal type limit theorems for eigenvalues are proved, and the strong influence of parameters of the model on a specification of normalizing constants is described. In the proof we use the finite-rank perturbation arguments based on the cluster expansion for resolvents. The results of our paper illustrate a close connection between extreme value theory for spectrum and extremal properties of i.i.d. potential. On the other hand, localization properties of the corresponding eigenfunctions give an essential information on long-time intermittency for the parabolic Anderson model.     

On the Accuracy of Gaussian Approximation in Hilbert Space

The bound on remainder term in CLT for a sum of independent random variables taking values in Hilbert space is obtained. This bound sharpens the result due to Bentkus and Götze.     

Classification of hypersurfaces with constant Laguerre eigenvalues in R~n

Let x:M → Rn be an umbilical free hypersurface with non-zero principal curvatures.Then x is associated with a Laguerre metric g,a Laguerre tensor L,a Laguerre form C,and a Laguerre second fundamental form B,which are invariants of x under Laguerre transformation group.An eigenvalue of Laguerre tensor L of x is called a Laguerre eigenvalue of x.In this paper,we classify all oriented hypersurfaces with constant Laguerre eigenvalues and vanishing Laguerre form.     

Rayleigh–Ritz variation method and connected‐moments polynomial approach

We show that the connected‐moments polynomial approach proposed recently is equivalent to the well known Rayleigh–Ritz variation method in the Krylov space. We compare the latter with one of the original connected‐moments methods by means of a numerical test on an anharmonic oscillator. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

     

A new method for estimating bounds of eigenvalues

A new method for estimating the bounds of eigenvalues is presented. In order to show that the method proposed is as effective as Qiu's, an undamping spring-mass system with 5 nodes and 5 degrees of freedom is given. To illustrate that the present method can be applied to structures which cannot be treated by non-negative decomposition, a plane frame with 202 nodes and 357 beam elements is given. The results show that the present method is effective for estimating the bounds of eigenvalues and is more common than Qiu's. Project supported by the National Natural Science Foundation (No. 19872028) and the Mechanical Technology Development Foundation of China.