拟圆的三个充要条件

设D是(-R2)中的Jordan域,D*=(-R2)\(-D)是D的外部,本文证明了拟圆的下面三个充要条件(1)D是拟圆当且仅当D和D*都是弱拟凸域;(2)D是拟圆当且仅当D和D*都是弱Cigar域;(3)D是拟圆当且仅当D是弱一致域.     

拟共形映射和弱John域

作为John域的推广,本文定义了弱John域,并讨论了弱John域与拟圆、弱John域与拟共形映射之间的关系,得到(1)若R2中的Jordan域D和它的外部D*=R2\D均是弱John域,则D是拟圆;(2)R2中的弱John域是拟共不变的;(3)R2中的有界拟圆必是弱John域.最后构造例子说明R2中的无界拟圆不一定是弱John域.     

拟共形映射和弱John域

作为John域的推广,本文定义了弱John域,并讨论了弱John域与拟圆、弱John域与拟共形 映射之间的关系,得到(1)若(?)。中的Jordan域D和它的外部 均是弱John域,则D 是拟圆;(2)R2中的弱John域是拟共不变的;(3)R2中的有界拟圆必是弱John域．最后构造例子 说明R2中的无界拟圆不一定是弱John域．     

关于拟圆性质的评注

设D是扩充复平面(R)2中的单连通Jordan真子域,D*=(R)2\(D)是D的外部.本文肯定并证明了K.Hag在1999年提出的如下两个悬而未决的问题(1)D是Cigar域当且仅当D是OLC域;(2)D是Turning域当且仅当D*是Cigar域.     

关于拟圆性质的评注

设D是扩充复平面R-2中的单连通Jordan真子域,D*=R-2\D-是D的外部.本文肯定并证明 了K.Hag在1999年提出的如下两个悬而未决的问题:(1)D是Cigar域当且仅当D是OLC域; (2)D是Turning域当且仅当D*是Cigar域.     

Lip_J-扩张域和一致域(英文)

本文证明了如下两个结果:(1)域D(?)R~n是一致域当且仅当D是Lip_J-扩张域;(2)Jordan域D(?)R~2是拟圆当且仅当对在D上满足|f′(z)|≤d(z,(?)D)~(-1)的任意的解析函数f恒有f∈Lip_J(D),其中J(x_1,x_2)=1/2 log(1+|x_1-x_2|/d(x_1,(?)D))(1+|x_1-x_2|/d(x_2,(?)D)),x_1,x_2∈D.     

Apollonian度量与Apollonian边界条件

证明了(1)(-R)n中真子域D上的Apollonian度量aD是拟共形映射的拟不变量;(2)(-R)n中严格一致域是拟共形不变的;(3)(-R)2中的Jordan域D是拟圆当且仅当D是严格一致域.作为应用,进一步得到了Apollonian边界条件,拟共形映射和局部Lipschitz映射之间的关系.     

双曲测地线与拟圆

设D是R2中的有界Jordan域,证明了D是拟圆当且仅当存在常数M≥1,对D中任意 两条不相交的闭双曲测地线段α1,α2,恒有mod(△(α1,α2;R2))≤Mmod(△(α1,α2;D)).     

Apollonian度量与Apollonian边界条件

证明了(1)■中真子域D上的Apollonian度量αD是拟共形映射的拟不变量;(2)■中严格一致域是拟共形不变的;(3)■中的Jordan域D是拟圆当且仅当D是严格一致域,作为应用,进一步得到了Apollonian边界条件,拟共形映射和局部Lipschitz映射之间的关系。     

N-弱拟Armendariz环

本文研究了N-弱拟Armendariz环的基本性质以及与一些特殊环的关系.利用某些矩阵环的特殊性质,得到了环R是N-弱拟Armendariz环当且仅当环T_n(R)是N-弱拟Armendariz环,推广了弱拟-Armendariz环的相应结果.     

Two Suficient and Necesary Conditions for Quasidisks

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REMARKS ON JOHN DISKS

Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.     

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ? be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ?[Γ] is not a weakly factorial domain.     

Hyperbolically convex constricted domains

A subdomain G in the unit disk D is called hyperbolically convex if the non-euclidean segment between any two points in G also lies in G. We introduce the concept of constricted domain relative to the hyperbolic geometry of D and prove that a hyperbolic convex domain is constricted if and only if it is not a quasidisk. Also examples are given to illustrate these ideas.     

Monoid Domain Constructions of Antimatter Domains

An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient field K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X; ?⁺] is an antimatter GCD domain. We also show that a GCD domain D is antimatter if and only if P^?1 = D for each maximal t-ideal P of D.     

Milnor方图中的w-凝聚性

整环R称为ω-凝聚整环,是指R的每个有限型理想是有限表现型的.本文证明了ω-凝聚整环是v-凝聚整环,且若(RDTF,M)是Milnor方图,则在Ⅰ型情形,R是ω-凝聚整环当且仅当D和T都是ω-整环,且T_M是赋值环;对于Ⅱ-型情形,R是ω-凝聚整环当且仅当D是域,[F:D]<∞,M是R的有限型理想,T是ω-凝聚整环,且R_M是凝聚整环.     

Remarks on Centers of Rings

It is proved that for matrices A,B in the n by n upper triangular matrix ring Tn(R) over a domain R,if AB is nonzero and central in Tn(R) then AB =BA.The n by n full matrix rings over right Noetherian domains are also shown to have this property.In this article we treat a ring property that is a generalization of this result,and a ring with such a property is said to be weakly reversible-over-center.The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains.The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally.We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.     

On v-Domains and Star Operations

Let * be a star operation on an integral domain D. Let f (D) be the set of all nonzero finitely generated fractional ideals of D. Call D a *-Prüfer (respectively, (*, v)-Prüfer) domain if (FF ^?1)* = D (respectively, (F ^v F ^?1)* = D) for all F ∈  f (D). We establish that *-Prüfer domains (and (*, v)-Prüfer domains) for various star operations * span a major portion of the known generalizations of Prüfer domains inside the class of v-domains. We also use Theorem 6.6 of the Larsen and McCarthy book [30 Larsen , M. D. , McCarthy , P. J. ( 1971 ). Multiplicative Theory of Ideals . New York : Academic Press . [Google Scholar]], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for *-Prüfer domains, and which cannot be. We also show that in a *-Prüfer domain, each pair of *-invertible *-ideals admits a GCD in the set of *-invertible *-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prüfer v-multiplication domains. We also link D being *-Prüfer (or (*, v)-Prüfer) with the group Inv*(D) of *-invertible *-ideals (under *-multiplication) being lattice-ordered.