Г—顺从群和Banach模

设G是局部紧群,Г是G×G中的一个闭子群.在本文中,定义L∞(G)为左BanachL1(Г)一模,给出G为Г-顺从群的一个充分必要条件和关于U(G,Г)空间的一个因子分解定理.     

初等Abel群的加法定理

本文中用Kneser's定理得到下列结论一个新的简单证法.设G为初等Abel p-群(运算用加法),S={a1,a2,…,an)为G的一个n项不含有零然的元素列(元素可允许重复),∣s∣=n=pm-1 p-2,,其中p为素数,若对G的任意子群H,S最多含有∣H∣-1项,则:(1)当m=2时,∑0(S)=G;(2)当m 3时,∑(S)=G.特别有(1)Olson'猜想r(ZpZp)=2p-2;(2)r(mZp)=c(mZp)=pm-1 p-2,m 3.     

关于哈密顿线图的一个注记

一、 引言令 G 是顶点集合为 V(G)且边集合为 E(G)的简单图.图 G 的线图 L(G)是顶点集合为 E(G)的图,L(G)的两个顶点,e_1和 e_2是相邻接的当且仅当 e_1和 e_2在 G中有一个公共顶点.图 G 的一条通道是点与边的一个交替序列 v_0,e_1,v_1,…,v_(n-1),e_n,v_n 其中 e_i(i=     

一道浙江高考题的深度解析及纰漏

题目(2010年高考浙江理科第21题)已知m>1,直线ι:X-my-m2/2=0,椭圆c:x2/m2+y2=1,F1,F2分别为椭圆C的左右焦点. (Ⅰ)当直线ι过右焦点F2时,求直线ι的方程; (Ⅱ)设直线ι与椭圆C交于A,B两点,△AF1F2,△BF1F2的重心分别为G,H.若原点O在以线段GH为直径的圆内,求实数m的取值范围.     

一类二分图的k-匹配数

设G是一个具有二分类(X_1,X_2)的简单偶图,|X_1|=|X_2|=n,如果对于给定的c>0,|M(S)|≥(1+c)|S|对任意满足|S|≤n/2的S(?)X_i(i=1,2)都成立,其中N(S)是S的邻集,则称G是(n,c)-扩张图.给出了(n,c)-扩张图的k-匹配数与完美匹配数之比的顺从界.     

具有性质Γ的有限von Neumann代数的交叉积北大核心CSCD

令M为可分的Ⅱ_(1)型von Neumann代数.我们证明了,如果M具有性质Γ,G为可数顺从群且α是G在M上保迹的真外作用,则交叉积M■αG为具有性质Γ的Ⅱ_(1)型von Neumann代数.     

自同构群的阶为2tp2q(t=1,2,3)的有限Abel群G

本文研究了自同构群A(G)阶为2tp2g(t=1,2,3)(p,g为不同的奇素数)的有限Abel群G的构造.利用有限Abel群G的自同构群的阶和有限Abel群的性质,获得以下结果:当t=1时,G最多有6型;当t=2时,G最多有32型;当t=3时,G最多有82型.     

完全多部图K_(1*r,3*(k-2))的m数

如果对一个图G的每个顶点v,任给一个k-列表L(v),使得G要么没有正常列表染色,要么至少有两种正常列表染色,则称图G具有M(k)性质.定义图G的m数为使得图G具有M(k)性质的最小整数k,记为m(G).已有研究表明,当k=3,4时,图K_(1*r,3*(k-2))具有M(k)性质,且当r≥2时,m(K_(1*r,3*(k-2)))=k.本文将上述结论推广到每一个k,证明了对任意r∈N~+,k≥3,图K_(1*r,3*(k-2))具有M(k)性质,且当k≥4,r≥(k-2)时,m(K_(1*r,3*(k-2)))=k.此外,得到图K_(1,3,3,3)的m数为4,该图是图K_(1*r,3*(k-2))中r=1,k=5时的特殊情况,同时也是现有研究中尚未解决的一个问题.     

C2(4,k) 中的强支撑可迹图

设2≤h≤3,l0,k≥0是整数,C_h(l,k)是由h-边连通简单图组成的集合,图G∈C_h(l,k)当且仅当对图G的任意一个二边割或三边割X,图G-X的每个分支都至少有︱V(G)-k︱/l个点.设e=u_1v_1和e'=u_2v_2是图G的两条边.若e≠e',G(e,e')是将图G中的边e=u_1v_1和e'=u_2v_2分别用路u_1v_ev_1和u_2v_e'v_2替换得到的图(其中,v_e,v_e'是不在V(G)中的两个新的点).若e=e',G(e,e')是将图G中的边e=u_1v_1用路u_1v_ev_1替换得到的图,也记作G(e).若对任意的e,e'∈E(G),G(e,e')都有支撑(v_e,v_e')迹,则称图G是强支撑可迹的.作者证明了,若图G∈C_2(4,k)且|V(G)|5k,则要么图G是强支撑可迹图,要么存在e,e'∈E(G),使得G(e,e')可以收缩成一个有限图类F中的图.当k=4时,F被完全确定了.     

K2Q的某些有限阶元素

本文证明了若n≥2,则G2n3m(Q)是K2Q的子群当且仅当n=2,m=0,并且通过改进[1]的方法,还证明了G25(Q),G49(Q)和G27(Q)都不是K2Q的子群,从而部分地证实了Browkin的一个著名猜想.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

非自伴算子代数间的等距代数同构

设Ｇｉ是满足第二可数性公理的、Ｈａｕｓｄｏｒｆｆ的、顺从的、ｒ－离散的、主的局部紧群胚，并且有一个紧开Ｇ－集覆盖；设Ｐｉ是Ｇｉ中含Ｇ＿ｉ￣０的开闭集，且满足及相应的模是具有性质ＤＣ的Ｃ（Ｇｉ）的子代数（ｉ＝１，２）．本文证明从Ａ（Ｐ１）到Ａ（Ｐ２）上的每一个等距代数同构可以扩张成从Ｃ（Ｇ１）到Ｃ（Ｇ２）上的Ｃ－同构，进一步，可以对Ｃ（Ｇ２）重新坐标化，使得这个Ｃ－同构可由一个群胚同构生成．     

On amenability of two-generator groups

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not su?cient for amenabilty.     

Connes-Amenability And Normal, Virtual Diagonals For Measure Algebras I

It is proved that the measure algebra M(G) of a locally compactgroup G is Connes-amenable if and only if G is amenable.     

Doubly stochastic graph matrices, II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))-¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

Doubly stochastic graph matrices,II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))?¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

Weak amenability of Segal algebras

Let be a locally compact abelian group, and let . We show that the Segal algebra is always weakly amenable, but that it is amenable only if is discrete.

     

THE NORMALITY OF CAYLEY GRAPHS OF SIMPLE GROUP A5 WITH SMALL VALENCIES

Let G be a finite group and S a subset of G not containing the identity element 1. We define the Cayley (di)graph X = Cay(G, S) of G with respect to S by V(X) = G,E(X) = {(g, sg) [ g ∈ G, s ∈ S}. A Cayley (di)graph X = Cay(G, S) is called normal if GR A = Aut(X). In this paper we prove that if S = {a, b, c} is a 3-generating subset of G = A5 not containing the identity 1, then X = Cay(G, S) is a normal Cayley digraph.     

THE MINIMAL CLOSED NON-TRIVIAL IDEALS OF TOEPLITZ ALGEBRAS ON DISCRETE GROUPS

61. IntroductionLet G be a discrete (not necessarily abelian) group. For any subset G of G, we saythat (G, G ) is a quasi-partial ordered group if e 6 G , G ' G G G and G = G ' G ',where e is the unit of G and G ' = {g--' I g e G }; further, (G, G ) is referred to as aquasi-ordered group if G = G u G '. Note that when G7 = G n G ' = {e}, a quasi-partial ordered group (resp. quasi-ordered group) (G, G ) is known as a pajrtially ordered(resp. ordered) group.Let { 6, I g e G } b…     

有限Abel群的一个特征

本文证明了有限群Ｇ是Ａｂｅｌ群当且仅当Ｇ_ｒ满足下列条件：（Ⅰ） Ｇ有一个幂自同构 ａ使得 ＣＧ（ａ）是一个初等 ＡｂｅｌＺ一群．（Ⅱ）Ｇ没有子群与２－群＜ａ,ｂ｜ａ~２~ｎ＝ｂ~２~ｍ＝１,ａ~ｂ＝ａ~（１＋２）~（ｎ－１）＞同构,其中ｎ≥３,ｎ≥ｍ．利用该结果,作者还证明若有限群Ｇ有一个幂自同构ａ使得Ｃ_Ｇ（ａ）是一个初等Ａｂｅｌ２－群,则Ｇ是幂零群