非退化扩散过程的极性的必要性

设Ｘ（ｔ）是一Ｎ维非退化扩散过程．设 Ｅ（０，∞）和 Ｆ ＲＮ都为紧集．本文给出了：Ｐ（Ｘ－１（Ｆ）∩Ｅ≠φ）＞０，Ｐ（Ｘ－１（Ｆ）≠φ）＞０和Ｐ（Ｘ（Ｅ）≠φ）＞０的充分条件．证明了：ｉ）设 Ｎ≥ ３，ａ）若 ｄｉｍ（Ｆ）＜Ｎ－２，则 Ｐ（Ｘ－１（Ｆ）＝φ）＝１； ｂ）若ｄｉｍ（Ｆ）＞Ｎ－２，则 Ｐ（Ｘ－１（Ｆ）≠φ）＞０； ｃ）存在 Ｆ１ ＲＮ，Ｆ２ ＲＮ，ｄｉｍ（Ｆ１）＝ｄｉｍ（Ｆ２）＝Ｎ－２，但有Ｐ（Ｘ－１（Ｆ１）＝φ）＝１，Ｐ（Ｘ－１（Ｆ２）≠φ）＞０．ｉｉ）设Ｎ＝１，ａ）若ｄｉｍ（Ｅ）＞１／２，则ｘ∈Ｒ１，Ｐ（Ｘ－１（ｘ）∩Ｅ≠φ）＞０；ｂ）存在Ｅ（０，∞），ｄｉｍ（Ｅ）＝１／２，使得ｘ∈Ｒ１，Ｐ（Ｘ－１（ｘ）∩Ｅ≠φ）＞０．以上这些结果，不仅仅是Ｂｒｏｗｎ运动的推广，即使就Ｂｒｏｗｎ运动的情形而言，其中有些结果也是新的．     

The Inner Radius of Univalency of the Rhom busDom ain

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＮａｒｒａｔａｔｉｏｎｏｆｔｈｅＴｈｅｏｒｅｍＩｆｆｕｎｃｔｉｏｎｆｉｓａｎａｌｙｔｉｃｉｎｄｏｍａｉｎＤａｎｄｆ′（ｚ）≠０ａｔｐｏｉｎｔｚ，ｗｅｄｅｆｉｎｅｔｈｅｓｃｈｗａｒｚｉａｎｄｅｒｉｖａｔｉｖｅｏｆｆａｓＳｆ（ｚ）＝（ｆ″（ｚ）ｆ′（ｚ））′－１２（ｆ″（ｚ）ｆ′（ｚ））２．（１．１）　　Ｆｏｒａｌｏｃａｌｌｙｕｎｉｖａｌｅｎｔｈｏｌｏｍｏｒｐｈｉｃｆｕｎｃｔｉｏｎ，ｉｔｓＳｃｈｗａｒｚｉａｎｄｅｒｉｖａｔｉｖｅｉｓｃｌｅａｒ．Ａｔｐｏｌｅｓ，ｔ…     

关于Klippert的一个猜测

本文证明文［２］中的猜测不成立设ｆ为区间［０，１］上实值函数，满足条件（ａ）ｆ在［０，１］上有界，（ｂ）ｆ在［０，１］上连续，（ｃ）ｆ（ｘ）不存在，试问是否存在？１９８２年Ｓ．Ｒｉｃｃｉ［１］给出Ｆ′＋（０）存在的例子：最近Ｊ．ｋｌｉｐｐｅｒｔ［２］给出Ｆ′＋（０）不存在的例子：Ｊ．ｋｌｉｐｐｅｒｔ提出如下猜测：如果ｆ满足（ａ）－（ｃ）并且ｆ的相邻零点之间距离依，α＞１，趋近于零，则Ｆ′＋（０）＝０，我们证明上述猜测不真，此外，我们给出Ｆ′＋（０），存在的一个充分条件。     

关于随机动力系统的Fatou－Julia猜测

设Ｆ＝｛ｆ１，…，ｆＭ｝是一个次数大于１的多项式集合。我们证明了在一定条件下Ｆａｔｏｕ集Ｆ（Ｆ）没有游荡区域。更确切地说，对于Ｆ（Ｆ）的每一个分支Ω，存在整数ｍ≥０，ｎ≥０，ｍ≠ｎ和λ∈∑Ｍ使得Ｗｍα（Ω）和Ｗｎα（Ω）落入Ｆ（Ｆ）的同一分支     

关于迭代函数方程f~2(x)=af(x) bx的通解

设λ的二次三项式λ２－ａλ－ｂ的两个零点为λ１＝ｒ，λ２＝ｓ（ａ及ｂ为实数）．对０＜ｒ＜ｓ，ｒ＜０＜ｓ≠－ｒ及ｒ＝ｓ≠０这三种情形，Ｊ．Ｍａｔｋｏｗｓｋｉ与ＷｅｉｎｉａｎＺｈａｎｇ在“Ｍｅｔｈｏｄｏｆｃｈａｒａｃｔｅｒｉｓｔｉｃｓｆｏｒｆｕｎｃｔｉｏｎａｌｅｑｕａｔｉｏｎｓｉｎｐｏｌｙｎｏｍｉａｌｆｏｒｍ”一文中给出了迭代函数方程ｆ２（ｘ）＝ａｆ（ｘ）＋ｂｘ，对任ｘ∈Ｒ；ｆ∈Ｃ０（Ｒ，Ｒ）（１）的通解，并证明了当ｒ及ｓ非实数时方程（１）无解．对ｒ＝－ｓ≠０的情形，Ｍ．Ｋｕｃｚｍａ已给出了方程（１）的通解．本文则对ｒ＜ｓ＜０及ｒｓ＝０这两种情形给出了方程（１）的通解．此外，本文还给出了ｒ＜０＜ｓ≠－ｒ时关于方程（１）的通解的一个简洁的证明     

群环上的幂自由模

本文证明了：设Ｒ为ｃｈａｒＲ≠０，Ｇ为有限生成的Ａｂｅｌ群，则：Ｐ∈Ｆ＝（ＲＧ）当且仅当ｓ＞０，使得Ｐ^ｓ∈Ｆ＝（ＲＧ）．     

关于随机动力系统的Fatou—Julia猜测

设ξ＝｛ｆ１，…，ｆＭ｝是一个镒数大于１的多项式集合。我们证明了在一定条件下Ｆａｔｏｕ集Ｆ没有游荡区域。更确切地说，对于Ｆ的每一个分支Ω，存在整数ｍ≥０，ｎ≥０，ｍ≠ｎ和λ∈ΣＭ使得Ｗ＾ｍα（Ω）和Ｗ＾ｎα（Ω）落入Ｆ同一分支。     

基于校正函数的一类具三角结构的非线性系统的自适应控制

１引言 考虑如下的具有二角结构的作线性系统ＬＰＴＳ（ｌｉｎｅａｒｌｙ　ｐａｒａｍｅｔｅｒｉｚｅｄ ｔｒｉａｎｇｕｌａｒｓｔｒｕｃｔｕｒｅ）．其中ｘ＝（ｘ１,．……ｘｎ）Ｔ０∈Ｒｎ为系统的状态,ｕ∈Ｒ为控制输入,θ∈Ｒｐ为未知常值参数向量,γｉ０及γｉ′,β′（ｘ）β０为已知的光滑非线性函数,γｉ０（０）＝０,γｉ′（０）＝０,β０（０）≠０;（ｉ＝１,…,ｎ）,对这样的系统,Ｄ．Ｓｅｔｏ,Ａ．Ｍ．Ａｎｎａｓｗａｍｙ和Ｊ．Ｂａｉｌｌｉｅｕｌ在［１］中进行了讨论,但是他们对同一参数用了多个估计,使得系…     

利用对称变换求二次曲线的切线

利用对称变换求二次曲线的切线程涛（乌鲁木齐铁路局教育学院８３００１１）设Ｃ是一条非退化的二次曲线，其方程为Ｆ（ｘ，ｙ）＝０；Ｍ（ｈ，ｋ）是Ｃ上一点．对Ｃ上的点Ｐ（ｘ，ｙ），作以Ｍ为中心的对称变换Ｔ：ＭＰ′＝ＰＭ，Ｔ下的象就是Ｃ′：Ｆ（２ｈ—ｘ，２ｋ－...     

椭圆一个定理的又一初等证明

定理　椭圆Ｃ：ｘ２ａ２＋ｙ２ｂ２＝１（ａ＞ｂ＞０）有且仅有两条对称轴：直线ｘ＝０和ｙ＝０．文［１］指出,这个定理的证明一般要用到仿射几何知识,同时文［１］给出了一个初等证明．笔者再给出这个定理的又一种初等证明如下．定理的证明　易验证直线ｘ＝０和ｙ＝０均是椭圆Ｃ的对称轴．因点Ｂ（０,ｂ）关于直线ｘ＝ｋ（ｋ≠０）的对称点Ｂ′（２ｋ,ｂ）不在椭圆Ｃ图１上,故直线ｘ＝ｋ（ｋ≠０）不是椭圆Ｃ的对称轴．设Ｆ１,Ｆ２是椭圆Ｃ的两个焦点,椭圆Ｃ的长轴Ａ１Ａ２关于直线ｌ：ｙ＝ｋｘ＋ｎ（ｋ,ｎ至少有一个不等于零）的…     

Day范数的c0（Γ,Χ）的暴露点和光滑点

设Γ为非空集,Ｘ为实Ｂａｎａｃｈ空间,本研究赋Ｄａｙ范数的ｃ０（Γ,Ｘ）的光滑点,强光滑点,暴露点和强暴露点。     

Renormings concerning exposed points and non-smoothness

Intuitively, non-smooth points might look like exposed points. However, in this paper we show that real Banach spaces having dimension greater than or equal to three can be equivalently renormed to obtain non-smooth points which are also non-exposed.     

Properties of symmetric matrices

We find the extreme points and the smooth points of the unit ball of the Banach space n_℘ of all real symmetric (n×n)-matrices.     

THE SUPPORT POINTS AND EXTREME POINTS OF THE UNIT BALL OF H''''p

Byσthe author denotes the areal measure on the unit disk D=z: |z|<1. Let H'p={f(z): f(z) is analytic in D and 1/π∫D|f(z)|p dσ<+∞}. Let B(H＇p)={f:f∈H＇p and 1/π∫D|f(z)|p dσ1. This article researches the support points and extreme points of B(H'p).     

Recurrent points and non-wandering points of graph maps

Let G be a graph and f:G→G be a continuous map. Denote by P(f), R(f) and Ω(f) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L=(x,y) is an open arc contained in an edge of G such that {fm(x),fk(y)}⊂(x,y) for some m,k∈N, then R(f)∩(x,y)≠∅; (2) Any isolated point of P(f) is also an isolated point of Ω(f); (3) If x∈Ω(f)−Ω(fn) for some n∈N, then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17] and [19] on interval maps or tree maps.     

A short note about exposed points in real banach spaces

We express the set of exposed points in terms of rotund points and non-smooth points.As long as we have Banach spaces each time＂bigger＂,we consider sets of non-smooth points each time＂smaller＂.     

关于迭代函数不动点的研究

In this paper the following results are got 1. We prove that the order definitionwhich is given by J.F. Trawb on the iterative function is wrong. 2. A existent theorem on the repulsive point is given. 3. We give some theorems to judge a fixed point of iterative function. They all imporve the Ostrowski's theorems.     

星形函数族的一个子族的极值点与支撑点

设T={f(z):f(z)在单位圆盘|z|<1上解析,f(z)=z ,an是实数, |an|≥|an 1|且|an|≤1}．该文找出了解析函数族T的极值点与支撑点.     

On the number of triple points of an immersed surface with boundary

Given a generic immersionf:S ¹→S ² of a circle into the sphere, we find the best possible lower estimation for the number of triple points of a generic immersionF: (M, S ¹)→(B ³,S ²) extendingf, whereM is an oriented surface with boundary ∂M=S ¹,B ³ is the 3-dimensional ball with boundaryS ². Supported by the Hungarian National Science and Research Foundation OTKA 2505 Supported by the Hungarian National Science and Research Foundation OTKA T4232 This article was processed by the author using the Springer-Verlag T_EX P Jour1g macro package 1991.     

Extreme points, exposed points, differentiability points in CL-spaces

This paper presents a property of geometric and topological nature of Gateaux differentiability points and Fréchet differentiability points of almost CL-spaces. More precisely, if we denote by a maximal convex set of the unit sphere of a CL-space , and by the cone generated by , then all Gateaux differentiability points of are just n-s, and all Fréchet differentiability points of are (where n-s denotes the non-support points set of ).