树上p-Laplacian算子的第一混合特征值

本文研究了树上以唯一根为Dirichlet边界的离散加权p-Laplacian算子的第一特征值问题,该特征值亦为一类加权Hardy不等式的最优常数.通过研究此特征值对应的特征函数的性质,给出具有混合边界的特征值的三种变分公式,该结果可以看做经典变分公式的重要补充.作为应用,给出了此特征值的上、下界的基本估计,并给用两个例子例证相关结果.     

半直线上L~p-Poincaré不等式最优常数的估计

运用分析方法给出半直线上测度有限时在Dirichlet边界条件下L~p-Poincaré不等式最优常数的变分公式,并运用迭代方法结合变分公式得到了最优常数的显式估计.     

半直线上第一Dirichlet特征值的对偶变分公式

目标是建立半直线上二阶椭圆算子的第一Dirichlet特征值的对偶变分公式, 并给出该特征值的仅依赖于算子系数的显式估计. 还可处理相应的离散情形以及连续情形的高阶特征值问题.     

四阶测度微分方程第一特征值的极小值问题

给定有限的弹性力,引起梁弯曲的最小压缩负荷是多少?从数学上讲,这对应于四阶测度微分方程在Dirichlet边界条件下的第一特征值的极小化问题.本文将建立四阶测度微分方程第一特征值的变分刻画,然后给出第一特征值极小化问题的完整解答.从而在数学上回答了上述物理问题.     

一维情形第一特征值的变分公式及逼近定理

给出了一维椭圆算子和一类 Markov 链的第一特征值的完全变分公式和逼近程序.     

跳过程的主特征值

本文给出了一般跳过程主特征值下界的一个变分公式．对于马氏链,所设条件常满足;所得估计在轻微条件下可达到精确．从这个意义上讲,所得到的公式是彻底的．     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

第一特征值对偶变分公式的分析证明(一维情形) *

第一非零特征值是自共轭算子谱的主项 ,在各种应用中起着重要作用 .关于此特征值 ,有熟知的变分公式 (称为极大极小原理 ) ,它对于上界估计特别有效 .对于下界估计的新的对偶变分公式 ,就一维情形给出分析证明 (原来的证明使用的是概率方法 ) ,并作了若干扩充 .     

一类Dirichlet边值逆问题

给出解析函数的一类Dirichlet边值逆问题的数学提法.依据解析函数Dirichlet边值问题和广义Dirichlet边值问题的理论,讨论了此边值逆问题的可解性.利用解析函数Dirichlet边值问题的Schwarz公式,给出了该边值逆问题的可解条件和解的表示式.     

一类二阶微分方程的特征值估计及其反问题

借助Rouché定理及渐近分析的方法,给出了边界条件含有特征参数的一类二阶微分方程的特征值渐近公式.运用特征值渐近公式给出了特征值反问题的一个惟一性结果及重构公式.     

Completely J —positive linear systems of finite order

Completely J — positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J — positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J — symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J — symmetric realizations are discussed.     

The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank

In this paper we show that the geodesic flow on a compact locally symmetric space of nonpositive curvature has a unique invariant measure of maximal entropy. As an application to dynamics we show that closed geodesics are uniformly distributed with respect to this measure. Furthermore, we prove that the volume entropy is minimized at a compact locally symmetric space of nonpositive curvature among all conformally equivalent metrics with the same total volume.     

A Decomposition Theorem for Lévy Processes on Local Fields

The study of Lévy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Lévy processes on local fields is given in terms of a structure result for measures on local fields and a Lévy–Khinchine representation. It is shown that a measure on a local field can be decomposed into three parts: a spherically symmetric measure, a totally non-spherically symmetric measure and a singular measure. We show that if the Radon–Nikodym derivative of the absolutely continuous part of a Lévy measure on a local field is locally constant, the Lévy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Lévy processes with single balls as support of their Lévy measure, end a singular Lévy process. These processes are independent. Explicit formulae for the transition function are obtained.     

集合的对称差及其测度

集合的对称差是集合的基本运算之一,它在测度论及其应用中扮演着一个重要的角度,本文深入地对集合的对称差进行讨论,研究了它的性质,通过集合的不交分解揭示了若干个集合的对称差的本质,给出了关于集合的对称差的测度计算公式。     

An Isoperimetric Result for the Gaussian Measure and Unconditional Sets

The paper studies an isoperimetric problem for the Gaussianmeasure and coordinatewise symmetric sets. The notion of boundarymeasure corresponding to the uniform enlargement is considered,and it is proved that symmetric strips or their complementshave minimal boundary measure.     

Symmetric Gauss–Lobatto and Modified Anti-Gauss Rules

The present paper is concerned with symmetric Gauss–Lobatto quadrature rules, i.e., with Gauss–Lobatto rules associated with a nonnegative symmetric measure on the real axis. We propose a modification of the anti-Gauss quadrature rules recently introduced by Laurie, and show that the symmetric Gauss–Lobatto rules are modified anti-Gauss rules. It follows that for many integrands, symmetric Gauss quadrature rules and symmetric Gauss–Lobatto rules give quadrature errors of opposite sign.     

Fourier transform of symmetric gauss measures on the Heisenberg group

An explicit form is derived for the Fourier transform of symmetric Gauss measures on the Heisenberg group at the Schrödinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two symmetric Gauss measures to be a symmetric Gauss measure and for commutability of two symmetric Gauss measures. Moreover, necessary and sufficient conditions are presented for the convolution of two symmetric Gauss convolution semigroups to be a convolution semigroup.     

Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane

It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length.     

Fourier Transform of Symmetric Gauss Measures on the Heisenberg Group

An explicit form is derived for the Fourier transform of symmetric Gauss measures on the Heisenberg group at the Schrödinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two symmetric Gauss measures to be a symmetric Gauss measure and for commutability of two symmetric Gauss measures. Moreover, necessary and sufficient conditions are presented for the convolution of two symmetric Gauss convolution semigroups to be a convolution semigroup. May 8, 2001     

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??This paper establishes limsup type law of the iterated logarithm of the occupation measure, using the asymptotic equivalence relation between the occupation measure and the number of excursion process of a symmetric Cauchy process. Furthermore, by using the density theorem and the economic coverage method, it derives the exact Hausdorff measure for the range of a symmetric Cauchy process in \mathbb{R}.