一类三次系统的奇点量公式与可积性条件

本研究一类二阶复域微分自治系统，得到原点的前四个奇点量公式以及原点领城存在的正则积分的充分必要条件。     

一类五次系统的奇点量和可积性条件

本文采用代数运算方法研究了一类五次系统的原点奇点量和可积性条件，并给出了该系统的15个基本Lie-不变量。     

一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量，然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

一类Z_2对称五次微分系统的中心条件和极限环分支

本文研究了一类Z2对称五次微分系统的中心条件和小振幅极限环分支.通过前6阶焦点量的计算,获得了原点为中心的充要条件,并证明系统从原点分支出的小振幅极限环的个数至多为6.最后通过构造后继函数,给出系统具有6个围绕原点的小振幅极限环的实例.     

Lienard方程全局稳定性的条件

在本文中，利用Ｆｉｌｉｐｐｏｖ变换，我们给出了判定Ｌｉｅｎａｒｄ系统的原点是全局稳定的两个准则．     

一类2n+1次多项式微分系统的局部极限环分支

研究了一类2n 1次多项式微分系统在原点的局部极限环分支问题,通过计算与理论推导得出了该系统原点的奇点量表达式,确定了系统原点的中心条件以及最高阶细焦点的条件,并在此基础上构造出系统在原点分支出4个极限环的实例.     

一类五次多项式系统无穷远点等时中心条件与极限环分支

研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性.     

具有两焦点的二次系统的极限环

本文给出二次系统在两个焦点周围的极限环不能同时多于一个的判别法，同时也顺便得到在原点周围的极限环的唯一性判别法。     

关于系统(x)=h(y)-F(x),(y)=-g(x) 的中心

本文给出了广义Ｌｉｅｎａｒｄ系统的原点为局部中心和全局中心的几个充分条件及原点为全局中心的充要条件,所得结果推广和改进了文献［１－６］的结果．     

Birkhoff系统的一类新型守恒量

给出了Birkhoff系统的一类新型守恒量。首先，建立了Birkhoff系统的运动方程及其Mei对称性的定义和判据；其次，给出了系统的一类新型守恒量的存在定理，并导出了用于确定无限小生成元的广义Killing方程；最后，建立了守恒定理的逆定理     

FOCAL QUANTITIES AND BIFURCATION OFLIMIT CYCLES FOR E~1_3 SYSTEM

1IntroductiollInthispapersweconsiderthenumberofthelimitcycleforthefollowingcubicpolynomialdifferentialsystemwiththeEdstyleFOrthepurposeofanalysingthenumberoflimitcycles,itisnecessarytocalculatethefocalquantities.FOrsystem(1))weperformfirstthefollowinghomeomorphictransformationThen,system(l)iscarriedintothefollowingsystemwhereIn[1],usingPoincaremethodandWuelimination,wedesignaspecialal-gorithmtocomputingthefocalvaluesforplanardynamicsystems.Byouralgorithm,weobtainthefollowingformularofcalcul…     

焦点量与鞍点量的关系

对于一般情形,给出焦点量和鞍点量计算与约化的Maple算法,从而统一了焦点量和鞍点量的计算,并给出细焦点与细鞍点的变换,利用变换推导了焦点量和鞍点量的关系．     

与平面凸集几何量有关的不等式(英文)

本文研究了平面凸集几何量之间的关系.通过定义新几何量:平面凸集的最大内接正三角形T和最大内接正方形S,分别获得了T(S)的边长、凸集的直径和面积之间的关系式.     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

UBIQUITOUS PROPERTIES OF CERTAIN OPERATIONAL QUANTITIES OF LINEAR RELATIONS IN NORMED SPACES

Abstract

Let X and Y be normed spaces and T a linear relation in X x Y. Certain known operational quantities of linear operators are extended to linear relations. Properties of T such as precompactness, strict singularity, upper semi-Fredholmness and partial continuity are defined and characterised in terms of the given quantities, and certain uniform boundedness properties of these quantities derived.     

焦点量算法与中心条件推导

对于一般情形, 基于后继函数法给出焦点量计算的递推公式;基于形式级数法给出焦点量计算和化简的Maple算法;给出了时间可逆条件的推导算法,给出了一类五次系统时间可逆的充要条件.     

Conserved Quantities and Symmetries Related to Stochastic Dynamical Systems

The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.     

Sixteen large-amplitude limit cycles in a septic system

In this paper, bifurcation of limit cycles from the infinity of a two-dimensional septic polynomial differential system is investigated. Sufficient and necessary conditions for the infinity to be a center are derived and the fact that there exist 16 large amplitude limit cycles bifurcated from the infinity is proved as well. The study relays on making use of a recursive formula for computing the singular point quantities of the infinity. As far as we know, this is the first example of a septic system with 16 limit cycles bifurcated from the infinity.     

The loop quantities and bifurcations of homoclinic loops

The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.     

一类三次Kolmogorov系统的极限环分支

本文研究了一类三次Kolmogorov系统，得出了该系统可分支出三个极限环，且其中有两个是稳定的，同时给出了其中心条件.