Generalized Jacobi Identities and Ball-Box Theorem for Horizontally Regular Vector Fields

Consider a family $\mathcal{H}:= \{X_{j} =: f_{j}\cdot\nabla: j=1,\ldots , m\}$ of C ¹ vector fields in ? ⁿ and let s∈?. We assume that for all p∈{1,…,s} and j ₁,…,j _p ∈{1,…,m} the horizontal derivatives $X_{j_{1}}X_{j_{2}}\cdots X_{j_{p-1}}f_{j_{p}}$ exist and are Lipschitz continuous with respect to the control distance defined by  $\mathcal{H}$ . Then we show that different notions of commutator agree. This involves an accurate analysis of some algebraic identities involving nested commutators which seem to have an independent interest. Our principal applications are a ball-box theorem, the doubling property, and the Poincaré inequality for Hörmander vector fields under an intrinsic “horizontal regularity” assumption on their coefficients.     

一类灰色脉冲随机时滞系统的几乎必然指数鲁棒稳定性

为了研究一类灰色脉冲随机时滞系统几乎必然指数稳定性的问题,首先利用Razumikhin方法和Lyapunov函数,给出了脉冲随机泛函微分系统几乎必然指数稳定的条件,然后基于此条件和时变灰矩阵的连续矩阵覆盖的分解技术,得到了该类灰色脉冲随机时滞系统几乎必然指数稳鲁棒定性的判据,最后通过一个数值例子说明了得判据是有效的和实用的.     

On a Class of Generalized Vector Quasiequilibrium Problems with Set-Valued Maps

This paper considers the following generalized vector quasiequilibrium problem: find a point (z ₀,x ₀) of a set E×K such that x ₀∈A(z ₀,x ₀) and

Duality Results for Generalized Vector Variational Inequalities with Set-Valued Maps

In this paper, we introduce new dual problems of generalized vector variational inequality problems with set-valued maps and we discuss a link between the solution sets of the primal and dual problems. The notion of solutions in each of these problems is introduced via the concepts of efficiency, weak efficiency or Benson proper efficiency in vector optimization. We provide also examples showing that some earlier duality results for vector variational inequality may not be true. This work was supported by the Brain Korea 21 Project in 2006.     

一类BAM神经网络的概周期解的存在性和指数收敛性

利用不动点理论和推广的Halanay时滞微分不等式,研究一类双向联想记忆神经网络的概周期解的存在性和全局指数收敛性,给出其判定的充分性条件.     

一类平面三次向量场的全局拓扑结构

本文利用中心投影变换的思想证明了一类具有星形结点的平面三次向量场的几何性质依赖于无穷远处的几何性质.研究了该向量场的全局拓扑结构,得到了该向量场不考虑极限环的存在性时有27类不同的全局拓扑等价类,以及存在赤道闭轨线的充要条件和存在至少一个极限环的条件.     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

Bi-center Problem in a Class of Z2-equivariant Quintic Vector Fields

In this paper, we study the center problem for Z2-equivariant quintic vector ?elds. First of all, for convenience in analysis, the system is simpli?ed by using some transformations. When the system has two nilpotent points at (0,±1) with multiplicity three, the ?rst seven Lyapunov constants at the singular points are calculated by applying the inverse integrating factor method. Then, ?fteen center conditions are obtained for the two nilpotent singular points of the system to be centers, and the su?ciency of the ?rst seven center conditions are proved. Finally, the ?rst ?ve Lyapunov constants are calculated at the two nilpotent points (0,±1) with multiplicity ?ve by using the method of normal forms, and the center problem of this system is partially solved.     

一类积分方程的正解的存在性与可积性

讨论了与加权Hardy-Littlewood-Sobolev不等式有密切联系的一类积分方程：（？）证明了此类积分方程在L~（n（p-1）/（n-λ-β））（R~n）∩L~（q0）（R~n）中存在唯一的正解,并利用迭代技巧得到了正解的可积区间L~5（R~n）,s∈[min{qo,n（p-1）/（n-λ-β）},∞].     

对称扩散过程的对数Sobolev不等式、指数可积性和熵的指数衰减性(英)

本文讨论了对称扩散过程的指数可积性，熵的指数衰减性与Sobolev不等式之间的等价关系．并给出了对称扩散过程的对数Sobolev不等式成立的一个充分条件．     

Necessary and Sufficient Conditions for the Uniform Integrability of the Stochastic Exponential

Journal of Theoretical Probability - We establish necessary and sufficient conditions for uniform integrability of the stochastic exponential $${{{mathcal {E}}}}(M)$$ , where M is a continuous...     

On the Integrability of Generalized Almost para-Norden and para-Hermitian Structures

In this paper, we investigate the \(\nabla \) integrability of generalized almost para-Norden/para-Hermitian structures on the big-tangent bundle of smooth manifolds endowed with some additional structures. We also study their invariance under a B-field transformation of the first and second types, respectively. Finally, we obtain a relation between \(\nabla \) integrability and the standard Courant integrability of generalized almost para-Norden or para-Hermitian structures, and we give some conditions for the equivalence of these kinds of integrability, in some particular cases.     

一类有界函数的可积问题探讨

讨论闭区间上有无穷多个不连续点的有界函数的可积问题,所得结果是"闭区间上不连续的点仅有有限个的函数可积"的推广.     

一类三次Kolmogorov系统可积性的判断方法

研究了一类系统可积性的判断方法问题,特别地,给出了易于计算的一类任意的1:-q型三次Kolmogorov系统原点的鞍点量的代数递推公式,同时还给出了一个相应的例子.     

Exponential Attractor for a Class of Nonclassical Diusion Equation

In this paper, we consider the asymptotic behavior of solutions for a class of nonclassical diffusion equation. We show the squeezing property and the existence of exponential attractor for this equation. We also make the estimates on its fractal dimension and exponential attraction.     

一类高阶非线性系统的几乎干扰解耦问题

对于一类高阶下三角形结构的非线性系统内稳几乎干扰解耦问题 ( ADD)的可解性 ,本文按 L2 —L2 p增益 ,使用“加幂积分器”方法 ,没有对系统中的函数作任何增长性条件限制 ,证明了文中非线性系统的ADD问题有一个连续状态反馈解 .并且可以显式解出此连续状态反馈控制器使得干扰对输出的影响可限制到任一精度且保持系统内稳 .