Simulation of optical field in laser resonators cavity by eigenvector method

In this paper, an Eigenvector method (EM) for the calculation of optical resonator modes and beam propagation is introduced, in which the transit matrix of an optical resonator is obtained by dividing the mirror into finite grids based on the Fresnel–Kirchoff diffractive integral equation. Then, the eigenvectors, representing the multimode characteristics of the resonator, can be calculated by solving the integral matrix eigenequation. The merits of EM include that the considerably simpler procedure of solution of eigenvectors of the matrix eigenequation replaces the complicated iteration in traditional methods, and there is no dependence on the initial field distribution, and a number of modes can be derived once and the discrimination capability of the resonator can be evaluated easily. The examples using EM to simulate con-focal resonators with small or large Fresnel numbers are given, and the calculated results, well matched with Fox–Li method or Lagueree–Gaussian approximation analytical solution, prove that EM is highly feasible and reasonable.     

Characterization of Pygeum africanum bark extracts by HRGC with computer assistance

The components of Pygeum africanum bark extracts, used for the treatment of benign prostatic hypertrophy, were characterized by high resolution gas chromatography (HRGC) and mass spectrometry (MS). Among various compounds n-docosyl trans-ferulate was identified and quantitated by HRGC as a derivative of n-docosanol, which is considered to be one of the active components of the extract. The origin of different Pygeum a. extracts can be studied by eigenvector projection of HRGC profiles with computer assistance.     

Optimal expansion of subspaces for eigenvector approximations

We consider the problem of how to expand a given subspace for approximating an eigenvalue and eigenvector of a matrix A. Specifically, we consider which vector in the subspace, after multiplied by A, provides optimal expansion of the existing subspace for the eigenvalue problem. We determine the optimal vector, when the quality of subspace for approximation is measured by the angle between the subspace and the eigenvector. We have also derived some characterization of the angle that might lead to more practically useful choice of the expansion vector.     

Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations

New methods for computing eigenvectors of symmetric block tridiagonal matrices based on twisted block factorizations are explored. The relation of the block where two twisted factorizations meet to an eigenvector of the block tridiagonal matrix is reviewed. Based on this, several new algorithmic strategies for computing the eigenvector efficiently are motivated and designed. The underlying idea is to determine a good starting vector for an inverse iteration process from the twisted block factorizations such that a good eigenvector approximation can be computed with a single step of inverse iteration.     

A simplex-like method to compute the eigenvalue of an irreducible (max,+)-system

In this paper, we present an alternative method to compute the eigenvalue of an irreducible (max,+)-system. The method resembles the well-known simplex method in linear programming in the sense that the eigenvalue and a corresponding eigenvector are obtained by going along the boundary of a polygon-like set, while increasing the number of equalities in some (max,+)-algebraic eigenvalue–eigenvector expression, until only equalities are left over. The latter is unlike the normal linear programming approach where, going along the boundary of a polygon-like set, a linear functional is optimized.     

Completeness of the System of Root Vectors of 2 × 2 Upper Triangular Infinite-Dimensional Hamiltonian Operators in Symplectic Spaces and Applications

The authors investigate the completeness of the system of eigen or root vectors of the 2 × 2 upper triangular infinite-dimensional Hamiltonian operator H ₀. First, the geometrical multiplicity and the algebraic index of the eigenvalue of H ₀ are considered. Next, some necessary and sufficient conditions for the completeness of the system of eigen or root vectors of H ₀ are obtained. Finally, the obtained results are tested in several examples.     

On solutions of generalized complementarity and eigenvector problems

From a new Fan–Browder type fixed point theorem due to the second author, we deduce an existence theorem for a solution of an equilibrium problem in Section 3. This theorem is applied to generalized complementarity problems in Section 4 and to eigenvector problems in Section 5.     

COMPUTING EIGENVECTORS OF NORMAL MATRICES WITH SIMPLE INVERSE ITERATION

It is well-known that if we have an approximate eigenvalue λ- of a normal matrix A of order n,a good approximation to the corresponding eigenvector u can be computed by one inverse iteration provided the position,say kmax,of the largest component of u is known.In this paper we give a detailed theoretical analysis to show relations between the eigenvecor u and vector xk,k=1,…,n,obtained by simple inverse iteration,i.e.,the solution to the system(A-λI)x=ek with ek the kth column of the identity matrix I.We prove that under some weak conditions,the index kmax is of some optimal properties related to the smallest residual and smallest approximation error to u in spectral norm and Frobenius norm.We also prove that the normalized absolute vector v=|u|/||u||∞ of u can be approximated by the normalized vector of (||x1||2,…||xn||2)^T,We also give some upper bounds of |u(k)| for those “optimal“ indexeds such as Fernando‘s heuristic for kmax without any assumptions,A stable double orthogonal factorization method and a simpler but may less stable approach are proposed for locating the largest component of u.     

Perturbation of the eigenvectors of the graph Laplacian: Application to image denoising

Patch-based denoising algorithms currently provide the optimal techniques to restore an image. These algorithms denoise patches locally in “patch-space”. In contrast, we propose in this paper a simple method that uses the eigenvectors of the Laplacian of the patch-graph to denoise the image. Experiments demonstrate that our denoising algorithm outperforms the denoising gold-standards. We provide an analysis of the algorithm based on recent results on the perturbation of kernel matrices (El Karoui, 2010) [1], [2], and theoretical analyses of patch denoising algorithms (Levin et al., 2012) [3], (Taylor and Meyer, 2012) [4].     

Catenarian Numbers

An integer n is called catenarian if, whenever L/K is an n-dimensional field extension, all maximal chains of fields going from K to L have the same length. Catenarian field extensions and catenarian groups are defined analogously. If n is an even positive integer, 6n is non-catenarian. If n ≥ 3 is odd, there exist infinitely many odd primes p such that p ² n is non-catenarian. A finite-dimensional field extension is catenarian iff its maximal separable subextension is. If q < p are odd primes where q divides p ? 1 (resp., q divides p + 1), every (resp., not every) group of order p ² q is catenarian.