常曲率平面上Fujiwara-Bol定理的注记

本文研究常曲率平面上的凸集,研究常曲平面上的凸集方法.根据Grinberg-Ren-Zhou的思想方法,我们给出著名的常曲率平面上Fujiwara-Bol定理的简单证明.     

关于凝聚映像的几点注记

1.引言文中,假定 X 是 Banach 空间,Y 是赋范线性空间。对于 E(?)Y,用 B(E),C(E),cf(E)分别表示 E 的不空的有界,闭,闭凸的子集组。E(?),(?)E 则分别表示 E 的内核(内点全体)及 E 的边界(即(?)E=(?)\E(?))。     

REMARKS ON THE WILANSKY PROPERTY

Abstract

A separable FK-space E has the Wilansky Property if whenever F is an FK-space contained and dense in E with F^β = E^β then F = E. In 1987 G. Bennett and W. Stadler independently showed that if E and E^B are both Bk—AK spaces then E has the Wilansky Property. In 1990 D. Noll relaxed the AK condition by arguing if E, E^f are Bk—Ad spaces and if E^β is separable then E has the Wilansky Property. In this note we show that Noll's result is in fact equivalent to the original Bennett/Stadler result.     

ON THE VITALI-HAHN-SAKS-NIKODYM THEOREM

Abstract

The classical Vitali-Hahn-Saks-Nikodym Theorem [5, Thm. I.4.8] gives a limit criterion for when a sequence of strongly additive vector measures on a σ-field of sets having their range in a Banach space can be expected to be uniformly strongly additive. In [16, Cor. 8], Saeki proved that the limit condition on the sequence of vector measures could be substantially weakened as long as the Banach space in play is “good enough”. Saeki's result was based upon his work on a class of set functions too large to have Rosenthal's Lemma at his disposal. In Section 2, we prove Saeki's result with Rosenthal's Lemma at the basis of our work and then augment our characterization of Banach spaces enjoying Saeki's result in [1] with another natural equivalent condition. In Section 3 we extend Saeki's result to Boolean algebras having the Subsequential Interpolation property.     

REMARKS ON BRUALDI＇S THEOREM FOR SPECTRUM INCLUSION REGION OF AN IRREDUCIBLE MATRIX

This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition for a boundary point of Brualdi‘s inclusion region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.     

关于Wielandt-Hoffman定理

关于正规矩阵的任意扰动,有下述定理成立. 定理1.设A为n阶正规矩阵,C为n阶任一矩阵.A的特征值为λ_1,…,λ_n,C的特征值为μ_1…,μ_n.C~H表示C的转置共轭,||·||_2与||·||_F分别表示矩阵的谱范数与Frobenius范数.记     

关于Lax-Milgram定理和Cea定理的一个注记

本文研究抽象变分问题（不必要求具有强制性）的Galerhin方法，利用泛函分析理论证明了：若变分问题的Galerkin逼近问题存在唯一解，那么它本身的解存在唯一且可由Galerhin逼近解无限逼近的充要条件是其Galerkin逼近格式具有某种稳定性．此结果是对Lax-Milgram定理和Cea定理的补充，可以应用于不必具有强制性的变分问题．     

THOUGHTS ON THE CANTOR-BERNSTEIN THEOREM

The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation f U g^?1. Considering Tarski's Fixpoint Theorem as the implicit basic ingredient of such proofs. We examine several classical proofs/starting with Dedekind (1887), and illuminate their common feature by means of the categorical notion of a natural fixpoint. We consider a categorical form (CBT) of the theorem (with h ? f Ug^?1) in a variety of contexts, obtaining some examples of categories where CBT holds and others where it fails. Among other results we prove for a topos E, (1) CBT holds if E is Boolean, and conversely if E has a natural number object; (2) The Axiom of Choice in E implies a dual version of CBTI and conversely if E has splitting supports and a natural number object.     

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.     

关于Kahan定理的一个注记

设A∈C~(n×n),B∈C~(k×k)均为Hermite矩阵,它们的特征值分别为{λ_j}_(j=1)~n和{μ_j}_(j=1)~k(k≤n);Q∈~(n×k)为列满秩矩阵.令 (1) 则存在A的k个特征值λ_(j_2),λ_(j_2),…,λ_(j_k),使得 (2) 其中σ_k为Q的最小奇异值,||·||_2表示矩阵的谱范数.这是著名的Kahan定理·1996年曹志浩等在[2]中将(2)加强为 (3) 这是Kahan的猜想.在本文中,我们讨论将Kahan定理中“B为k阶Hermite矩阵”改为B为k阶(任意)方阵后,特征值的扰动估计,有以下结果. 定理 设A∈C~(n×n)为Hermite矩阵,其特征值为{λ_j}_(j=1)~n,B∈C~(k×k)的特征值为{μ_j}_(j=1)~k,而Q∈C~(n×k)为列满秩矩阵.则存在A的k个特征值λ_(j_1),λ_(j_2),…,λ_(j_k),使得     

关于一个定理的改进

本文利用归纳法和调和函数的平均值公式，并通过寻求函数的最大值，把调和函数k阶偏导数的局部估计式的系数由Ck=（2^n＋1 nk）^k/a（n）缩小到Ck=（n＋k）^n＋k＇/a（b）b^n-k。     

半质环的交换性定理

任意半质PI－环的交换性问题是环的交换性理论的一个主要内容,本文利用亚直不可约环和分式环的性质,研究了任意半质PI－环的交换性问题,证明了半质PI－环的交换性定理。     

ON THE COMPARISON THEOREM OF THE GTOR METHOD

1IntroductionInordertogeneralizetheclasificiterativemethodssuchasJacobimethod,Gauss--SeidelInethod,relaxedJacobimethod,SORmethodandAORmethod,Kuang[2]presentsatwasparameteriterativemethodcalledTORmethod,whichiseffectivetogivethenumericalsolutionofpartialdifferentialequations.Wang['1extendstheTORmethodtothegelleralizedTOR(GTOR)methodandimprovestheresultsof[1,6,71.LetAx=b,(1.1)whereA=D--E--F,E=EI E2.ThentheiterationmatrixoftheGTORiterativemethodisgivenby['1:LGTOR(cr,g,A)=(D~crE…     

关于Lax-Milgram定理和Céa定理的一个注记

本文研究抽象变分问题(不必要求具有强制性)的Galerhin方法,利用泛函分析理论证明了:若变分问题的Galerkin逼近问题存在唯一解,那么它本身的解存在唯一且可由Galerhin逼近解无限逼近的充要条件是其Galerkin逼近格式具有某种稳定性.此结果是对Lax-Milgram定理和C啨a定理的补充,可以应用于不必具有强制性的变分问题.     

ON THE MYSOVSKICH THEOREM OF NEWTONS METHOD

In a real or oompbx Banach space X, let P be an operator with Lipsohitz continnous Frechet derivative P', and \[{X_*} \in X\] such that \[P({X_*}) = 0\] and \[{P^'}{({X_*})^{ - 1}}\] exists. It is shown that a ball with center \[{X_*}\] and best possible radius such that the theorem of Mysoyskich guarantees convergenee of Newton's method to \[{X_*}\] starting from any point \[{x_0}\] in ihe ball (theorem 3). In comparison with the corresponding results of Rall's work on Kantorovich theorem, the radius obtained is smaller than that from Kantorovich theorem. Therefore we suggest here an improved form of Mysoyskich theorem (theorem 1) . Thus, the corresponding value of the radius is augmented beyond that from Kantorovich theorem (theorem 2).     

ON THE ERROR BOUND IN SLEPIANWOLF THEOREM

In this paper,we discuss the problem of estimation of the probability of error inSlepian-Wolf Theorem.We give both an upper bound and a lower bound of the error exponentof the best code C(R_x,R_y).For a main part of the achievable rates,we have determined theerror exponent completely,for the others,our estimation is accurate.     

ON THE GENERALIZATIONS OF DEN JOY-WOLFF THEOREM

We consider generalizations of Denjoy-Wolff theorem on strongly pseudo-convex domains. Beardon [3] gave a general Denjoy-Wolff theorem in hyperbolic domain in the complex plane. Our main results are generalizations of Beardon’s result in a higher dimensional setting.     

ON THE PUTNAM-FUGLEDE THEOREM OF NON-NORMAL OPERATORS

In this paper we have extended the Putnam-Fuglede Theorem of nomal operators anddiscussed the condition for the Putnam-Fuglede Theorem holding.We have proved that ifA and B~* are hyponomal operators and AX=XB,then A~*X=XB~*;that if A and B~* aresemi-hyponomal operators and X is