A note on the eigenvalue ratio of vibrating strings

For vibrating strings with symmetric single-well densities, it is known that the ratio λ₂/λ₁1 is maximized when the density is constant. In this note, we extend this result to a class of symmetric densities.      

Eigenvalue ratios of vibrating strings

Summary New results for the eigenvalue ratios of vibrating strings are presented partially in connection with previous results concerning Schr?dinger operators.     

子空间上的对称正定及对称半正定阵的左右特征值反问题

文[1][2][3]中讨论AX＝B的对称阵逆特征值问题,文[4][5][6]中讨论了半正定阵的逆特征值问题。本文讨论了空间了子空间上的对称正定及对称半正定阵的左右特征值反问题,给出了解存在的充分条件及解的表达式。     

实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

Bisection acceleration for the symmetric tridiagonal eigenvalue problem

We present new algorithms that accelerate the bisection method for the symmetric tridiagonal eigenvalue problem. The algorithms rely on some new techniques, including a new variant of Newton's iteration that reaches cubic convergence (right from the start) to the well separated eigenvalues and can be further applied to acceleration of some other iterative processes, in particular, of the divide-and-conquer methods for the symmetric tridiagonal eigenvalue problem. This revised version was published online in June 2006 with corrections to the Cover Date.     

关于周期对称三对角矩阵的广义特征值反问题

在给定部分特征值、部分特征向量及附加条件下提出了一类反问题,并给出了此问题解存在性的证明及求解的算法.     

Sharp bounds for the first eigenvalue of symmetric Markov processes and their applications

By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.     

The inverse eigenvalue problem for symmetric quasi anti-bidiagonal matrices

In this paper we construct the symmetric quasi anti-bidiagonal matrix that its eigenvalues are given, and show that the problem is also equivalent to the inverse eigenvalue problem for a certain symmetric tridiagonal matrix which has the same eigenvalues. Not only elements of the tridiagonal matrix come from quasi anti-bidiagonal matrix, but also the places of elements exchange based on some conditions.     

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 upper densities of symmetric perfect sets

In this paper, we give the exact upper densities of Hausdorff measures of a class of symmetric perfect sets.     

On a recursive inverse eigenvalue problem

Let s ₁, ..., s _n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s _i is an eigenvalue of the leading principal submatrix A _i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s ₁, ..., s _n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A _i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A _i and A _{i + 1}.     

Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions

Based on a linear finite element space,two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed.Some relationships between the finite element method and the finite difference method are addressed,too.     

An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

Based on the theory of inverse eigenvalue problem, a correction of an approximate model is discussed, which can be formulated as NX=XΛ, where X and Λ are given identified modal and eigenvalues matrices, respectively. The solvability conditions for a symmetric skew-Hamiltonian matrix N are established and an explicit expression of the solutions is derived. For any estimated matrix C of the analytical model, the best approximation matrix to minimize the Frobenius norm of C − N is provided and some numerical results are presented. A perturbation analysis of the solution is also performed, which has scarcely appeared in existing literatures. Supported by the National Natural Science Foundation of China(10571012, 10771022), the Beijing Natural Science Foundation (1062005) and the Beijing Educational Committee Foundation (KM200411232006, KM200611232010).     

The equation of the p-adic closed strings for the scalar tachyon field

We investigate the structure of solutions of boundary value problems for a non-linear pseudodifferential equation describing the dynamics(rolling)of p-adic closed strings for a scalar tachyon field.     

Spectral gap and logarithmic Sobolev constant for continuous spin systems

The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.