积分方程特征值的一个下界估计式及其应用

导出了具有正定对称核的积分方程最小特征值的一个下界估计式，为此为基础，获得了相应特征值的一种算法。     

The first eigenvalue of the Dirac operator on Kähler manifolds

If M^2m is a closed Kähler spin manifold of positive scalar curvature R, then each eigenvalue λ of type r (r {1, …, [(m + 1)/2]}) of the Dirac operator D satisfies the inequality λ² ≥ rR₀/4r − 2, where R₀ is the minimum of R on M^2m. Hence, if the complex dimension m is odd (even) we have the estimation

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for the first eigenvalue of D. In the paper is also considered the limiting case of the given inequalities. In the limiting case with m = 2r − 1 the manifold M^2m must be Einstein. The manifolds S², S² × S², S² × T², P³( ), F( ), P³( ) × T² and F( ³) × T², where F( ³) denotes the flag manifold and T² the 2-dimensional flat torus, are examples for which the first eigenvalue of the Dirac operator realizes the limiting case of the corresponding inequality. In general, if M^2m is an example of odd complex dimension m, then M^2m × T² is an example of even complex dimension m + 1. The limiting case is characterized by the fact that here appear eigenspinors of D² which are Kählerian twistor-spinors.     

一类混合图的结构及其特征空间

主要讨论具有如下性质的一类连通混合图G:其所有非奇异圈恰有一条公共边,且除了该公共边的端点外,任意两个非奇异圈没有其它交点.本文给出了图G的结构性质,建立了其最小特征值λ1(G)(以及相对应的特征向量)与某个简单图的代数连通度(以及Fiedler向量)之间联系,并应用上述联系证明了λ1(■)≤α(G),其中G是由G通过对其所有无向边定向而获得,α(■)为■的代数连通度.     

The Principal Eigenvalue for Jump Processes

A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented. The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions. Received November 15, 1999, Accepted March 9, 2000     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Bound states of a system of two fermions on a one-dimensional lattice

We consider the Hamiltonian of a system of two fermions on a one-dimensional integer lattice. We prove that the number of bound states N(k) is a nondecreasing function of the total quasimomentum of the system k ∈ [0, π]. We describe the set of discontinuity points of N(k) and evaluate the jump N(k +0) − N(k) at the discontinuity points. We establish that the bound-state energy z _n (k) increases as the total quasimomentum k ∈ [0, π] increases. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 1, pp. 47–57, April, 2006.     

一种基于特征值反求的复合材料层合板波动特性的近似修改方法

采用一种特征值反求形式的方法用来近似层状复合板的波动特性修改。通过敏感度分析,确定层合板中能最显著地改变结构波动特性的结构参数为设计变量。采用二阶泰勒展式近似特征值和特征频率相对于设计变量的改变率。计算了不同结构变化时的色散特性,并与精确计算结果进行了比较。结果证明本方法对于层状复合板波动特性修改的近似是可行并且有效的。     

对称循环M-矩阵的若干性质

研究实对称循环M-矩阵A的特征值范围、逆矩阵A<'-1>的存在性以及讨论了矩阵(A<'-1>)<'k>的极限性质和(A<'-1>)<'k>的||·||<,2>性质,最后给出了A<'-1>摄动定理.     

The first eigenvalue of the p‐Laplace operator under powers of mean curvature flow

In this paper, we show the following main results. Let (Mⁿ,g(t)), t ∈ [0,T), be a solution of the unnormalized H^k ? flow on a closed manifold, and λ_1,p(t) be the first eigenvalue of the p‐Laplace operator. If there exists a nonnegative constant ε such that in M × [0,T) and in M × [0,T),then λ_1,p(t) is increasing and the differentiable almost everywhere along the unnormalized H^k ? flow on [0,T). At last, we discuss some useful monotonic quantities. Copyright © 2013 John Wiley & Sons, Ltd.