$C(X)$上的开点与紧开拓扑

In this note we define a new topology on C(X),the set of all real-valued continuous functions on a Tychonoff space X.The new topology on C(X) is the topology having subbase open sets of both kinds:[f,C,ε[={g E C(X):|f(x)-g(x)| ε for every x∈C} and[U,r]~-={g∈C(X):g~(-1)(r)∩U≠φ},where f∈C(X),C∈KC(X)={nonempty compact subsets of X},ε 0,while U is an open subset of X and r∈R.The space C(X) equipped with the new topology T_(kh) which is stated above is denoted by C_(kh)(X).Denote X_0={x∈X:x is an isolated point of X} and X_c={x∈X:x has a compact neighborhood in X}.We show that if X is a Tychonoff space such that X_0=X_c,then the following statements are equivalent:(1) X_0 is G_δ-dense in X;(2) C_(kh)(X) is regular;(3) C_(kh)(X) is Tychonoff;(4) C_(kh)(X) is a topological group.We also show that if X is a Tychonoff space such that X_0=X_c and C_(kh)(X) is regular space with countable pseudocharacter,then X is σ-compact.If X is a metrizable hemicompact countable space,then C_(kh)(X) is first countable.     

On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.     

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

Families of Semiopen Sets and Their Accompanied Topologies

Introduction Let (X, U) be a topological space. A set sX is called a semiopen set of U, if there exists u∈U such that usu~-, where u~- expresses the clo sure of u. The family of all semiopen sets of U is called the family of semiopen sets of U, and the topology U an accompanied topology of. The     

THE （u＋К）-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS

Let Н be a complex,separable,infinite dimensional Hilbert space,T∈L(Н),(U+κ)(T) denotes the (U+κ)-orbit of T,i.e.,(U+κ)(T)={R^-1 TR:R is invertible and of the form unitary plus compact}.Let Ω be an analytic and simply connected Cauchy domain in C and n∈N,A（Ω,n）denotes the class of operators,each of which satisfies (i) T is essentially normal;(ii)σ(T)=Ω^-,ρF（T）∩σ（T）=Ω;（iii）ind(λ-T)=-n,nul(λ-T)=0,(λ∈Ω）。it is proved that given T1,T2∈A(Ω,n）and c>0,there exists a compact operator K with ||K||<ε such that T1+K∈(u+κ)(T2),this result generalizes a result of P.S.Guinand and L.Marcoux[6,15],Furthermore,the authors give a character of the norm closure of (u+κ)(T),and prove that for each T∈А(Ω,n）,there exists a compact (SI) perturbation of T whose norm can be arbitrarily small.     

Normal systems over ANR’s, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di-mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U ) = w(X ) (where "w"is the topological weight) for each open nonempty subset U of X , then X itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W (X, ) = {(x n ) ∞ n=1 ∈ X ω : x n = for almost all n} is homeomorphic to a pre-Hilbert space E with E ～ = ΣE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.     

ON THE EQUIVALENCE PRINCIPLE FOR CONTRACTIVE MAPPINGS

The main result is:Theorem 1.Let T be a continuous selfmapping of a complete metric space(X,d)andhave the unique fixed point property.If there exists n:X→N(the set of all positive integers)which is locally bounded such that for each x∈X and for all r∈N,r≥n(x),D(O_T(T~(n(x))x,0,r))≤(?)(D(O_T(x,0,r))),orD(O_T(T~(n(x))x,0,r))≤(?)(D(O_T(x,0,r))),where(?),and(?)are contractive gauge functions,then(a)T has a unique fixed point x~*;(b)For each x∈X,T~nx→x~* as n→∞;(e)There exists a neighborhood U(x~*)of x~* such that(?)T~n(U(x~*))={x~*};(d)x~*is stable;(e)For any given C∈(0,1)there exists a metric d~* topologically equivalent to d suchthat T is a Banach contraction under d~* with Lipschitz constant C.By Theorem 1 it is shown that many contractive type mappings defined in[1—26]aretopologically equivalent to each other.     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

关于Banach空间中渐近非扩张型半群的不动点与渐近行为

Let X be a Banach space with a weakly continuous duality map Jφ,C a non-empty weakly compact convex subset of X, and T:(T(t):t∈S} an asymptotically nonexpansive type semigroup on C. In this paper, the inequality K∩F(T)≠0 is characterized, where K is a subset of C and F(T) is the set of all common fixed points of T. Furthermore, it is shown that an almost-orbit {u(t):t∈S} of T converges weakly to a point in F(T) if and only if {u(t):t∈S}is weakly asymptotically regular.     

Non-wandering sets of the powers of dendrite maps

Let(X, d) be a metric space and f be a continuous map from X to X. Denote by EP(f)and Ω(f) the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold:(1) If x ∈Ω(f)-Ω(f~n) for some n ≥ 2,then x ∈ EP(f).(2) Ω(f) is contained in the closure of EP(f). The aim of this note is to show that the above results do not hold for maps of dendrites D with Card(End(D)) = ?0(the cardinal number of the set of positive integers).     

Besov空间中双参数Volterra型重分数过程的弱逼近

In this paper, we prove that two-parameter Volterra multifractional process can be approximated in law in the topology of the anisotropic Besov spaces by the family of processes{B_n(s,t)},n∈N defined by B_n(s,t)=∫_0~s ∫_0~tk_(a(s))(s,u)K_(β(t))(t,u)θ_(n(u,v))dudv,here {θ_n(u, v)}n∈N is a family of processes, converging in law to a Brownian sheet as n→∞,based on the well known Donsker's theorem.     

限制性多源点偏心距增广问题

给定一个赋权图$G=（V,E;w,c）$以及图$G$的一个支撑子图$G_{1}=（V,E_{1}）$,这里源点集合$S=\{s_{1},s_{2},\cdots,s_{k}\}\subseteq V$,权重函数$w：E\rightarrow\mathbb{R}^{+}$,费用函数$c：E\setminus E_{1}\rightarrow\mathbb{Z}^{+}$和一个正整数$B$,本文考虑两类限制性多源点偏心距增广问题,具体叙述如下：（1）限制性多源点最小偏心距增广问题是要寻找一个边子集$E_{2}\subseteq E\setminus E_{1}$,满足约束条件$c（E_{2}）$$\leq$$B$,目标是使得子图$G_{1}\cup E_{2}$上源点集$S$中顶点偏心距的最小值达到最小;（2）限制性多源点最大偏心距增广问题是要寻找一个边子集$E_{2}\subseteq E\setminus E_{1}$,满足约束条件$c（E_{2}）$$\leq$$B$,目标是使得子图$G_{1}\cup E_{2}$上源点集$S$中顶点偏心距的最大值达到最小。本文设计了两个固定参数可解的常数近似算法来分别对上述两类问题进行求解。     

限制性多源点偏心距增广问题

给定一个赋权图$G=（V,E;w,c）$以及图$G$的一个支撑子图$G_{1}=（V,E_{1}）$,这里源点集合$S=\{s_{1},s_{2},\cdots,s_{k}\}\subseteq V$,权重函数$w：E\rightarrow\mathbb{R}^{+}$,费用函数$c：E\setminus E_{1}\rightarrow\mathbb{Z}^{+}$和一个正整数$B$,本文考虑两类限制性多源点偏心距增广问题,具体叙述如下：（1）限制性多源点最小偏心距增广问题是要寻找一个边子集$E_{2}\subseteq E\setminus E_{1}$,满足约束条件$c（E_{2}）$$\leq$$B$,目标是使得子图$G_{1}\cup E_{2}$上源点集$S$中顶点偏心距的最小值达到最小;（2）限制性多源点最大偏心距增广问题是要寻找一个边子集$E_{2}\subseteq E\setminus E_{1}$,满足约束条件$c（E_{2}）$$\leq$$B$,目标是使得子图$G_{1}\cup E_{2}$上源点集$S$中顶点偏心距的最大值达到最小。本文设计了两个固定参数可解的常数近似算法来分别对上述两类问题进行求解。     

从Hardy空间到Zygmund-型空间的加权微分复合算子

设φ为单位圆盘D上的解析自映射,H(D)表示D上的所有解析函数的集合,u∈H(D).研究了从Hardy空间到Zygmund-型空间及小Zygmund-型空间的加权微分复合算子D_(φ,u)~n,的有界性和紧性,其中n∈N_0.     

在${\mathbb{R}}^m$中重对数广义律的精确速率

令\{$X$, $X_n$, $n\ge 1$\}是期望为${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$和协方差阵为${\rm Cov}(X,X)=\sigma^2I_m$的独立同分布的随机向量列, 记$S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. 对任意$d>0$和$a_n=o((\log\log n)^{-d})$, 本文研究了${{\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$的一类加权无穷级数的重对数广义律的精确速率.     

$L^{p}_{[0,1]}$空间M\"{u}ntz有理逼近的Jackson型估计 (英)

设$\Lambda=\{\lambda_{n}\}_{n=1}^{\infty}$为正的实数数列, 且当$n\rightarrow\infty$时, 有$\lambda_{n}\searrow 0$.本文给出了当 $\lambda_{n}\leq Mn^{-\frac{1}{2}},\;n=1,2, \cdots ,$(其中$M>0$为一正常数)时M\"{u}ntz系统$\{x^{\lambda_n}\}$的有理函数在$ L_{[0,1]} ^{p}$空间的逼近速度,主要结论为$R_{n} (f, \Lambda )_{L^{p}}\leq C_M \omega (f, n^{-\frac{1}{2}})_{L^{p}},\;1 \leq p \leq \infty.$     

关于杨全会和陈永高提出的一个问题

对任意的正整数与集合,令为解的个数.杨全会和陈永高证明了:若整数且,则不存在集合使得对所有充分大的整数成立,其中.对整数和,定义为满足对所有整数成立的集合的个数.杨全会和陈永高证明了是有限的,且.同时,他们问对任意整数,是否存在使得对所有整数成立.在本文中,我们给出了在时的准确公式.从而推出在时成立.     

High dimensional reducible hyperplane sections with multigenera $\leq$ 1

Let X be an algebraic submanifold of the complex projective space $\mathbb{P}^N$ of dimension $n \geq 5$. We describe those $X \subset \mathbb{P}^N$ whose intersection with some hyperplane is a smooth simply normal crossing divisor $A_{1} + \cdots + A_{r}$ with $r \geq 2$ such that $g(A_{k}, L_{A_k}) \leq 1$ for $k=1,\ldots, r$.Received: 14 December 2001     

某些Ruzsa数$R_m$的新上界

For A ■ Z m and n ∈ Z m ,let σ A (n) be the number of solutions of equation n = x + y,x,y ∈ A.Given a positive integer m,let R m be the least positive integer r such that there exists a set A ■ Z m with A + A = Z m and σ A (n) ≤ r.Recently,Chen Yonggao proved that all R m ≤ 288.In this paper,we obtain new upper bounds of some special type R kp 2 .