一些连续统的孤立链回归点

对树或图上的连续自映射,本文证明了孤立链回归点是终周期点,如孤立边回归不在临界点的轨道中且它的轨道中也不含有临界点,则它还是周期点,我们还证明了上述性质可以扩张到一类特殊的λ－dendroid上。     

On farthest points of sets

For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and su.cient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.     

On the stability of the vertical rotation of a solid suspended on a rod

The problem of the motion of a dynamically symmetric solid suspended from a fixed point by a weightless rod and two ball and socket joints one of which is fixed at the fixed point O', and the other is on the body axis of symmetry at the point O is considered. The question of the stability of the uniform body rotation when points O' and O, and the body centre of inertia C lie on the same vertical, and at the same time point O may be either above or below point O', and point C either above or below point O, is discussed. An analysis of the necessary and sufficient conditions for stability is given. The set of all the system's parameters is reduced to three independent dimensionless parameters L, Ω and β, and in the plane (L, Ω), for fixed values of β, the regions for which the unperturbed rotation is steady, or steady to a first approximation, or non-steady are indicated. The regions for which the body rotation is steady to a first approximation when the point O is situated higher than the point O', and the point C lies higher or lower than the point O are determined.

The sufficient conditions for the vertical rotation of a dynamically symmetric body suspended on a filament were obtained in /1/ and investigated for the cases where in non-perturbed motion the point C is below point O, when points C and O coincide, and when the length of the filament is zero (Lagrange gyroscope). In /2/ an analysis is given of the sufficient conditions for stability obtained in /1/, and also the necessary conditions for the cases where in a non-perturbed motion point C is located above point O.     

Preservation of the existence of fixed and coincidence points under homotopy of mappings of ordered sets

In problems of topology and analysis, well-known theorem on the preservation by any continuous homotopy of the property of a mapping to have a fixed point and the property of a pair of mappings to have a coincidence point are extensively applied. Thus, for contraction mappings and some of their generalizations, Frigon’s results on the preservation of the property to have a fixed point by a homotopy of a special type are known. This paper presents theorems on the preservation by order homotopy of the property of a pair of mappings to have a coincidence point. As a corollary, conditions under which such a homotopy preserves the property of a mapping to have a fixed point are obtained.     

Families of paths in the plane that connect all point pairs

A mesh is a family of paths in the plane connecting every pair of points. A crossing point of a mesh is a point which is an interior point to more than one path. A simple crossing point is an interior point of exactly two paths. We give an example of a mesh with only simple crossing points. We characterize subsets of the plane that can be the set of crossing points of a mesh. Our emphasis is on constructive methods.     

一类面积函数的最小值点的特性

本文运用Ｒｉｅｍａｎｎ－Ｓｔｉｅｌｊｅｏ积分的基本性质讨论了一类由连续单调函数所确定的面积函数的最小值问题，得到了该问题的最小值点恰为区问的中点。其变形问题的最小值点的函数值与函数在两端点的函数值的均值相等。     

向量值映射锥鞍点存在定理

本文首次通过借助Kakutani-Fan-Glicksberg固定点定理和非线性标量化,研究了向量值映射的Benson锥鞍点定理.然后,通过使用该Kakutani-Fan-Glicksberg固定点定理,同样得到了集值映射的锥松鞍点定理.     

Convergence Properties of a Smoothing Approach for Mathematical Programs with Second-Order Cone Complementarity Constraints

This paper discusses the convergence properties of a smoothing approach for solving the mathematical programs with second-order cone complementarity constraints (SOCMPCCs). We first introduce B-stationary, C-stationary, M(orduckhovich)-stationary, S-stationary point, SOCMPCC-linear independence constraint qualification (denoted by SOCMPCC-LICQ), second-order cone upper level strict complementarity (denoted by SOC-ULSC) condition at a feasible point of a SOCMPCC problem. With the help of the projection operator over a second-order cone, we construct a smooth optimization problem to approximate the SOCMPCC. We demonstrate that any accumulation point of the sequence of stationary points to the sequence of smoothing problems, when smoothing parameters decrease to zero, is a C-stationary point to the SOCMPCC under SOCMPCC-LICQ at the accumulation point. We also prove that the accumulation point is an M-stationary point if, in addition, the sequence of stationary points satisfy weak second order necessary conditions for the sequence of smoothing problems, and moreover it is a B-stationary point if, in addition, the SOC-ULSC condition holds at the accumulation point.     

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.     

平面C-B样条的奇拐点分析

平面C-B样条曲线是三次均匀B样条的推广.通过移动C-B样条曲线段的一个控制点而固定其余三个控制点的方法,讨论了在曲线上形成零曲率点的移动控制点的轨迹,得到了C-B样条曲线段的尖点判别曲线、拐点判别区域,同时也给出了在曲线段上生成重结点的移动控制点的轨迹区域.     

A numerical method for branch points of a system of nonlinear algebraic equations

A numerical method is proposed for determining the branch points of a system of nonlinear algebraic equations which depend on a parameter. The basic idea of the present method is to replace a part of the original equations in the neighborhood of the branch point by a set of new equations until the point is reduced to a regular point, and hence its accurate point can easily be obtained with second-order convergence. An illustrative example is solved to show the effectiveness of the present algorithm.     

Fixed point sets of homeomorphisms of compact surfaces

Every closed and non-empty subset of a compact surfaceS can be the fixed point set of a homeomorphism, andS also admits fixed point free homeomorphisms if it does not have the fixed point property. A partial extension to higher dimensions states that every closed and non-empty subset of a compactn-manifold can be the fixed point set of a surjective self-map. This research was partially supported by the National Research Council of Canada (Grant A 7579).     

The generalized center problem of degenerate resonant singular point

In this paper, generalized center condition and integrability of degenerate resonant singular point for a class of complex polynomial differential system were studied. The method was based on a homeomorphic transformation of the degenerate singular point into elementary singular point, which allows us to compute the generalized singular point quantities and determine the generalized center condition for the origin. In the end, we obtained the necessary and sufficient conditions of generalized complex center of degenerate resonant singular point.     

对数螺线、黄金分割与斐波那契数列的完美统一的

本文对对数螺线、黄金分割与斐波那契数列之间的关系进行研究,把线段上黄金分割点的定义推广到射线上的黄金分割点列,发现过极轴上任意一点有且仅有一条特殊的对数螺线与极轴的交点所成的点列为黄金分割点列,并把这个点列所对应的坐标定义为黄金分割数列,我们发现首项为1/√15黄金分割数列无限逼近于斐波那契数列。     

On the equivalence of singular point quantities and the integrability of a fine critical singular point

In this paper, the equivalence of singular point quantities and the integrability of a fine critical singular point are discussed. For an extensive class of complex autonomous differential systems, the theoretical basis for the methods of using formal series and integrating factors to calculate the singular point quantities for the origin (a fine critical singular point) and judge its integrability is obtained. Two recursion formulas, which are much simpler than focal values for computation because the operation is rational, for computing singular point quantities are introduced.     

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

This paper deals with the problems of bifurcation of limit cycles and pseudo-isochronous center conditions at degenerate singular point in a class of septic polynomial differential system. We solve the problems by an indirect method, i.e., we transform the degenerate singular point into an elementary singular point. Then we construct a septic system which allows the appearance of eight limit cycles in the neighborhood of degenerate singular point. Finally, we investigate the pseudo-isochronous center conditions at degenerate singular point for the system. As far as we know, this is the first time that an example of septic system with eight limit cycles bifurcating from degenerate singular point is given, and it is also the first time the pseudo-isochronous center conditions at degenerate singular point in a septic system are discussed.     

Self-similar point vortices and confinement of vorticity

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a su?cient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed.     

The weakly strongly exposed points of Orlicz spaces

In this paper, criteria in order that a point in an Orlicz space be a weakly strongly exposed point, or a *-weakly strongly exposed point are given, respectively.     

The weakly strongly exposed points of Orlicz spaces

In this paper, criteria in order that a point in an Orlicz space be a weakly strongly exposed point, or a *-weakly strongly exposed point are given, respectively. The subject supported by NSF of China.     

广义次似凸集值优化的鞍点定理

讨论广义次似凸集值优化的鞍点定理.给出广义次似凸集值映射的两个性质.定义广义次似凸集值优化的Fritz-John鞍点和Kuhn-Tucker鞍点.获得一系列广义次似凸集值优化的鞍点定理.