A Comparison-estimate of the second Rauch type for Ricci curvature

We establish a comparison-estimate of the second Rauch type for Ricci curvature. As an application, we get a result of local splitting for Riemannian manifolds. This work is partly supported by the National Natural Science Foundation (10371047) of China.     

A remark on the mean curvature of a graph-like hypersurface in hyperbolic space

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space Hn+1. We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the (n−1)-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.     

On real hypersurfaces in a quaternionic hyperbolic space in terms of the derivative of the second fundamental tensor

The purpose of this paper is to give a characterization of real hypersurfaces of type A₀, A in a quaternionic hyperbolic space QH ^m by the covariant derivative of the second fundamental tensor. This revised version was published online in June 2006 with corrections to the Cover Date.     

Bayes empirical Bayes estimation for discrete exponential families

Bayes-empiric Bayes estimation of the parameter of certain one parameter discrete exponential families based on orthogonal polynomials on an interval (a, b) is introduced. The resulting estimator is shown to be asymptotically optimal. The application of this method to three special distributions, the binomial, Poisson and negative binomial, is discussed.The first author was supported by NSF grant DCR-8504620.     

On a conjecture of W.E. Clark and M.E.H. Ismail

W.E. Clark and M.E.H. Ismail conjectured the inequality

On some hypersurfaces of the homogeneous nearly Kähler

In this paper, we study hypersurfaces of the homogeneous nearly Kähler manifold with typical properties. We first show that in the NK there exist neither totally umbilical hypersurfaces nor hypersurfaces of parallel second fundamental form. Then we investigate hypersurfaces of such that its shape operator A and induced almost contact structure ϕ satisfy the condition , and as the main result, a complete classification of this remarkable family of hypersurfaces in is presented.     

Analysis of Velázquez’s solution to the mean curvature flow with a type II singularity

Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons’ cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C⁰ sense to a minimal hypersurface which is tangent to Simons’ cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.     

Well-Conditioned Spectral Collocation Methods for Problems in Unbounded Domains

Based on the generalized Laguerre and Hermite functions, we construct two types of Birkhoff-type interpolation basis functions. The explicit expressions are derived, and fast and stable algorithms are provided for computing these basis functions. As applications, some well-conditioned collocation methods are proposed for solving various second-order differential equations in unbounded domains. Numerical experiments illustrate that our collocation methods are more efficient than the standard Laguerre/Hermite collocation approaches.     

Efficient Laguerre and Hermite Spectral Methods for Odd-Order Differential Equations in Unbounded Domains

Laguerre dual-Petrov-Galerkin spectral methods and Hermite Galerkin spectral methods for solving odd-order differential equations in unbounded domains are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Numerical results demonstrate the effectiveness of the suggested approaches.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.