含有不可数个无界变差点的一维连续函数

在单位区间[0,1]上构造了图像长度为无穷的一维连续函数.该函数含有不可数个但Lebesgue测度为0的无界变差点.所有无界变差点组成的集合中每一点皆为该集合的聚点.     

n维空间的Peano曲线

那汤松在文献[1]里,叙述了 Hilbert 关于平面上的 Peano 曲线的构造,并向读者提出了几个问题,其中第四个问题是:“构造三维空间的 Peano 曲线,即在给定的区间[0,1]上,构造这样的三个连续函数(?)(t)、(?)(t)、(?)(t),使所有的点((?)(t),(?)(t),(?)(t))的集合与立方体[0,1]×[0,1]×[0,1]重合”.本文将更一般地、解析地给出在 Jordan 意义下的 n 维欧氏空间的 Peano 曲线 (n≥2),即在区间 [0,1]上,给出 n 个连续函数x_1(t),x_2(t),…,x_n(t),使所有的点 (x_1(t),x_2(t),…,x_n(t))的集合与 n 维立方体[0,1]×[0,1]×…×[0,1] 重合.首先,在区间[0,2]上定义函数     

一种非线性函数的迭代动力分析

对一种非线性函数迭代的动力性质进行了分析,结果显示在函数的定义区间[－2,2]内,点χ的移动轨道具有如下性质：（i）点χ的移动轨道具有各种复杂的周期轨道,所有周期轨道形成的集合在区间[－2,2]内稠密,（ii）点χ的移动轨道或者在区间（－2,2）内稠密,或者经有限次映射后最终进入由所有的周期轨道形成的集合中。     

中点凸性与广义凸函数相关集合的对称性问题

研究十二类广义凸函数相关集合的对称性问题及其对应函数的中点凸性问题.证明了其中的十个集合具有对称性,应用反例说明了余下的两个集合不必具有对称性.证明了六个集合对应的函数具有中点凸性,应用反例说明了余下六个集合对应的函数不必具有中点凸性.     

关于齐次Cantor集的一个注记

设c是[0,1]上Hausdorff测度为正有限的齐次Cantor集类,本文证明了     

可列集合套

[1]中给出模糊集的分解定理与表现定理，指出可利用集合套H＝{H（λ）|λ∈[0,1]}刻化模糊集。本文定义可列集合套H＝{H（a）|a∈Q}，Q是（0，1）的可列稠密子集。相应地给出新的分解定理与表现定理。指出可利用可列集合套刻化的模糊集。     

稠密的s维分形集

对任意给定的0≤s≤1,本文构造Cantor型集E_s,使dim_H E_s=s,且E_s在[0,1]内稠密。     

β-变换下攀援集和distal集的测度性质

设β1为实数,T_β为[0,1]的β变换.攀援集的任何两个点随着时间的转移会越来越接近但同时又总能在任意长时间后保持一定的距离.证明了在Lebesgue测度意义下关于T_β的攀援集是一个零测集.Distal点对的两个点表示随着时间的转移始终保持着一定的距离.如果固定其中一个点x_0,所有满足x∈[0,1)且lim inf n→∞|T_β~n(x)-T_β~n(x_0)|0的点称为关于x_0的distal集,如果把这个集合记为R_β(x_0),根据Borel-Cantelli引理得到R_β(x_0)的Lebesgue测度为零.     

双格半群在模糊逻辑中的典例

分析了n元模糊逻辑函数集合中的偏序结构,论证了该集合M-={f|f:[0,1]n→[0,1],(A)x∈[0,1]n,f(x)∈[0,1]}是一个双格半群.并且M-关于其上定义的等值关系构成的商集W={Cf|(A)g∈Cf(∈)M-,f(x)=g(x),f,g∈M,x∈[0,1]n}也构成一个双格半群.     

向量值测度产生的集值测度及性质

由有界变差向量值测度的值域,通过取凸包和闭包,构造了L[0,1],L2[0,1]和C[0,1]空间上的有界变差紧凸集值测度,结果由欧氏空间推广到函数空间.     

Banach空间上凸集的凸性

以Banach空间的一般凸集为研究对象,将Banach空间的凸性研究推广到了内部非空的凸集上．打破了从单位球出发研究Banach空间几何的具有局限性的研究方法,给出了严格凸集的若干特征刻画及性质,并得到了严格凸集和光滑集之间的对偶定理．     

Some Characterization of Locally Nonconical Convex Sets

A closed convex set Q in a local convex topological Hausdorff spaces X is called locally nonconical (LNC) if for every x, y Q there exists an open neighbourhood U of x such that . A set Q is local cylindric (LC) if for x, y Q, x y, z (x, y) there exists an open neighbourhood U of z such that U Q (equivalently: bd(Q) U) is a union of open segments parallel to [x, y]. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication LNC LC was proved in general, while the inverse implication was proved in case of Hilbert spaces.     

SOME RESULTS ON BEST APPROXIMATION IN SPACES WITH CONVEX STRUCTURE

Some generalizations of the result proved by S.P. Singh [J. Approx. Theory 25（1979）, 89-90] are presented in convex metric spaces. The results proved contain several known results on the subject.     

Some results on best approximation in spaces with convex structure

Some generalizations of the result proved by S.P Singh[J.Approx.Theory 25(1979),89-90]are presented in convex metric spaces.The results proved contain several known results on the subject.     

On the Geometry of Locally Nonconical Convex Sets

A convex subset Q of a Hausdorff topological vector space is called locally nonconical (LNC) if for every two points x,yQ there is a relative neighborhood U of x in Q such that U+ (y-x) Q. A geometric characterization (Theorem 2.2) of closed LNC sets with nonempty interior in a Hilbert space is supplied. It states that any proper line segment ]x,y[ contained in bd(Q), the topological boundary of Q, lies inside a relative neighborhood in bd(Q) composed of parallel line segments. It is shown that one half of this characterization, at least, generalizes to the setting of a locally convex Hausdorff topological vector space (LCHTVS). This leads to the observation that the set ext(Q) of extreme points of any LNC set Q in an LCHTVS is closed. Finally, it is proven that, in the same setting, all LNC sets are uniformly stable and, hence, stable.     

一种新的集合序列收敛性

本我们引进了无穷维Ｂａｎａｃｈ空间中的一种新的集合序列收敛性概念,讨论了它与其它收敛性概念的关系。另外,我们还研究了集合序列根限的Ｍｉｎｋｏｗｓｋｉ和。     

Covering a Convex Polygon by Triangles

According to a theorem of A. V. Bogomolnaya, F. L. Nazarov and S. E. Rukshin, if n points are given inside a convex n-gon, then the points and the sides of the polygon can be numbered from 1 to n so that the triangles spanned by the ith point and the ith side(i=1....,n ) cover the polygon. In this paper, we prove that the same can be done without assuming that the given points are inside the convex n-gon. We also show that in the general case at least [(n/3)] mutually nonoverlapping triangles can be constructed in the same manner.     

E-凸函数的若干特征

讨论了一类广义的凸集和凸函数:E-凸集和E-凸函数的若干性质,并给出E-凸函数的一个判别准则．     

关于Ky Fan的极大极小不等式

本文继续研究Ｋｙ Ｆａｎ型的极大极小不等式，给出了一睦非凸性的新结果。     

实分析教学杂记

以区间长度、曲边梯形及凸多边形概念为基础分别引入点集测度、下方图形与凸集概念.