The first eigenvalue of a spherical cap

We find upper and lower bounds for the first eigenvalue of the Laplacian on the two-sphere from which a disk has been removed, with Dirichlet conditions imposed on the resulting boundary. When the radius of the disk tends to zero our lower bound is sharper than that obtained by Del Grosso, Gerardi, and Marchetti in the preceding paper.     

A partial inverse Sturm-Liouville problem: Matching a step function potential to a finite eigenvalue list

We investigate a certain inverse problem involving the Sturm-Liouville equation. In particular, given a finite list of target values, when can a potential function of a given form be found that produces these numbers as eigenvalues?     

Bounds for the First eigenvalue of a spherical cap

We study upper and lower bounds for the lowest eigenvalueλ of the Laplace operator on a spherical capC _θ inm-dimensional space (m ≥ 3). We prove that these bounds are sharp by finding asymptotic expressions forλ asθ → π and asθ → 0.     

A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

Let M be a closed hypersurface in a simply connected rank-1 symmetric space . In this paper, we give an upper bound for the first eigenvalue of the Laplacian of M in terms of the Ricci curvature of and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of M.     

The behavior of the principal eigenvalue of a mixed elliptic problem with respect to a parameter

In this paper we determine the exact asymptotic behavior of the principal eigenvalue of a mixed elliptic eigenvalue problem which depends on a positive parameter λ when λ→∞. We analyze the case in which the problem is considered in a smooth bounded domain Ω of RN, and also the case of planar domains which are smooth except for a finite number of corner points.     

A sharp lower bound for the first eigenvalue on Finsler manifolds

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.     

An inverse spectral problem for a nonsymmetric differential operator: Reconstruction of eigenvalue problem

In this paper, an algorithm is established to reconstruct an eigenvalue problem from the given data satisfying certain conditions. These conditions are proved to be not only necessary but also sufficient for the given data to coincide with the spectral characteristics corresponding to the reconstructed eigenvalue problem.     

A characteristic property of spherical caps

It is proved that given H ≥ 0 and an embedded compact orientable constant mean curvature H surface M included in the half space z ≥ 0, not everywhere tangent to z = 0 along its boundary , the inequality

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.     

An inexact inverse iteration for large sparse eigenvalue problems

In this paper, we propose an inverse inexact iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the linear convergence of inverse iteration. We also propose two practical formulas for the accuracy bound which are used in actual implementation. © 1997 John Wiley & Sons, Ltd.     

一类正则Sturm—Liouville问题特征的渐近分析

考虑[0,π]上一类带一般分离型边界条件的正则Sturm—Liouville问题特征的渐近表示,利用Frechet导数,对特征值进行精细的分析,清楚地给出了方程系数q（x）及边界条件中常数cot α,col β对特征值的影响．使结论更具一般化．     

On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion:

−du^″(x)=g(x)u(x)−u²(x)     

确定Sturm-Liouville问题中系数的反问题

讨论确定Sturm－Liouville问题中算子的系数α（x）或者q（x）的问题．在一定条件下,系数α（x）或者q（x）可由数据或者数据唯一确定,这里的u_j（x）满足而构成L²（0,1）的一个基,α和β为给定的实数.     

Several sharp upper bounds for the largest laplacian eigenvalue of a graph

Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G,respectively.In this paper,we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D~(-1/2) KD~(1/2) and U~(-1/2)BU~(1/2),respectively,where D is the degree diagonal matrix of G and U=diag(d_u,d_v,:uv∈E(G)). And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G.Moreover,we determine all extremal graphs which achieve these upper bounds.Finally, some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.     

A note on backward perturbations for the hermitian eigenvalue problem

Through different orthogonal decompositions of computed eigenvectors we can define different Hermitian backward perturbations for a Hermitian eigenvalue problem. Certain optimal Hermitian backward perturbations are studied. The results show that not all the optimal Hermitian backward perturbations are small when the computed eigenvectors have a small residual and are close to orthonormal.Dedicated to Åke Björck on the occasion of his 60th birthdayThis work was supported by the Swedish Natural Science Research Council under Contract F-FU 6952-302 and the Department of Computing Science, Umeå University.     

An eigenvalue bound for the Laplacian of a graph

We present a lower bound for the smallest non-zero eigenvalue of the Laplacian of an undirected graph. The bound is primarily useful for graphs with small diameter.