一类缺项算子矩阵的四类点谱的扰动

有界线性算子的点谱可进一步细分为4类,分别为$\sigma_{p1}$, $\sigma_{p2}$, $\sigma_{p3}$ 和$\sigma_{p4}$.设 $H, K$为无穷维可分的Hilbert空间,用$M_C$表示$2\times 2$上三角算子矩阵$\left(\begin{array}{cc} A & C \\ 0 & B \\ \end{array} \right)$,对于给定的 $A\in B(H),~B\in B(K)$,描述了集合$\bigcap\limits_{C\in B(K,H)}\sigma_{p1}(M_C)$, $\bigcap\limits_{C\in B(K,H)}\sigma_{p2}(M_C)$, $\bigcap\limits_{C\in B(K,H)}\sigma_{p3}(M_C)$和$\bigcap\limits_{C\in B(K,H)}\sigma_{p4}(M_C)$.     

（广义）四元数群的自同构群及其全形

该文讨论了一类2^n阶非交换群——（广义）四元数群Q_{2^n}=〈a,b|a^{2^n-1}=1,b^2=a^{2^{n-2}},{b^{-1}ab=a^{-1}〉(n≥3)的自同构群A（Q_{2^n}）与全形H（Q_{2^n}））的置换表示，给出了A（Q_{2^n}））与H（Q_{2^n}））的构造.     

系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调

研究了系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调, 其中~$\mathbb{F}$ 是一个素特征的代数闭域且~$\frak{gl}(2,\mathbb{F})$ 是系数在~$\mathbb{F}$ 上的~$2\times 2$ 阶矩阵李代数. 计算出所有~$\frak{gl}(2,\mathbb{F})$ 到模李超代数~$W(m,3,\underline{1})$ 的子模的导子和内导子. 从而一维上同调~$\textrm{H}^{1}(\frak{gl}(2,\mathbb{F}),W(m,3,\underline{1}))$ 可以完全用矩阵的形式表示.     

与群PSL2(\mathcal{R})相关的交叉积{\mathcal{R}(\mathcal{A}},α)的一点注记

在上半复平面$\mathbb{H}$上给定双曲测度$dxdy/y^{2}$, 群$G={\rm PSL}_{2}(\mathbb{R})$ 在$\mathbb{H}$上的分式线性作用导出了$G$在Hilbert空间$L^{2}(\mathbb{H}, dxdy/y^{2})$上的酉表示$\alpha$. 证明了交叉积 $\mathcal{R}(\mathcal{A}, \alpha)$是$\mathrm{I}$型von Neumann代数, 其中$\mathcal{A}= \{M_{f}:f\in L^{\infty}(\mathbb{H},dxdy/y^{2} )\}$. 具体地, 交叉积代数$\mathcal{R}(\mathcal{A}, \alpha)$与von Neumann代数$\mathcal{B}(L^{2}(P, \nu))\overline{\otimes}\mathcal{L}_{K}$是*-同构的, 其中$\mathcal{L}_{K}$是$G$中子群 $K$的左正则表示生成的群von Neumann代数.     

算子矩阵值域的闭性及其应用

令{H}和{K}均为无限复可分的Hilbert空间. 定义M_X=(A&C\\X&B\)为作用在{H}}\oplus{K}上的2x2算子矩阵, 其中X为从{H}到{K}上未知的有界线性算子.在本文中, 基于R(C)的闭性对某个(或任意的)X\in{B}}({H,K}}), 使得R(M_{X})为闭集的充要条件做了等价刻画.另外, 研究了算子矩阵M_{X的半Fredholm性与广义Weyl性并给出了一些相应的结论.     

Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

双曲空间中完备子流形的特征值估计

研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})相似文献   

SOME SIMPLE GROUPS OF LIE TYPE CONSTRUCTED BY THE INNER-AUTOMORPHISM

当\[L \cong {C_l}\]，l为偶数，且l≥4,域\[\mathcal{H}\]=\[{\mathcal{H}_0}(\sqrt { - 1} )\]，其中\[{\mathcal{H}_0}\]为一有序域（或\[{\mathcal{H}_0}\]满足： а)\[\sqrt { - 1} \notin {\mathcal{H}_0},{(\sqrt { - 1} )^2} = - 1\]; b)\[ch{\mathcal{H}_0} > 3\]; c)若\[a,b \in {\mathcal{H}_0}\]，则 \[{a^2} + {b^2} \ne - 1\]，设Ф和 \[\prod :\{ {\alpha _1},{\alpha _2},...,{\alpha _l}\} \]，\[{\alpha _l}\]为长根分别为L的一组根系和素根系.令\[\{ {h_r},r \in \prod ,{e_r}r \in \Phi \} \]为L的一组Chevalley基;\[G = L({\cal H})\]为对于这一组Chevalley基在域\[{\cal H}\]上的L型 Chevalley群,令\[{w_0} = {w_{{\alpha _1}}}{w_{{\alpha _2}}}...{w_{{\alpha _{l - 1}}}}\]，其中\[{\alpha _i} \in \prod \]且为对于垂直于\[{\alpha _i}\]的平面的反射，显然\[{w_0}\]为L的Weyl群中的元素.设N为G的单项子群，\[{n_0} \in N\]，\[{n_0}\]的自然同态 像为 \[{w_0}\]，且\[{n_0}^2{\rm{ = }}I\]，存在域\[{\cal H}\]的自同构f:f(a)=a,\[a \in {{\cal H}_0}\] , \[{\rm{f(}}\sqrt { - 1} {\rm{) = }} - \sqrt { - 1} \],f在G中的扩充为G的一个域自同构(仍记为f)，且令U(V)为G对于正(负)根生 成的么幂子群，令\[{U^1}\{ u \in U|{n_0}f(u){n_0}^{ - 1} = u\} \];\[{V^1}\{ v \in V|{n_0}f(v){n_0}^{ - 1} = v\} \], 本文证明了 \[{}^2{C_l}({\cal H}) = < {U^1},{V^1} > \]为一单群.     

弱斜配对双代数和弱相关Long双代数

该文在弱双代数$H$上给出了扭曲积$(H^\sigma,\cdot_\sigma)$成为弱双代数的充分必要条件.设$[B, H, \tau]$是一个弱斜配对, 并且$\tau$可逆,则在某个条件下弱双交叉积$B\bowtie_\tau H$是一个弱双代数. 如果$(B,H, \sigma)$是弱相关Long双代数, 并且$\sigma$可逆,则弱双交叉积$B^{OP}\bowtie_\sigma H$可以被构造. 它的乘法是:$(x\otimes h)(y\otimes g)=\Sigma\sigma(y_1, h_1)y_2x\otimes h_2g\sigma^{-1}(y_3, h_3),$ 特别地, 如果$(B, H,\sigma)$是相关Long双代数, 则$(B^{OP \bowtie_\sigma H,\beta)$是Long双代数当且仅当对任意$b, d\in B^{OP}; g, \ell\in H$,$\Sigma\sigma^{-1}(b, g_2\ell)\sigma(d, g_1)=\Sigma\sigma^{-1}(b,\ell g_1)\sigma(d, g_2),$ 其中$B$为$H$的子Hopf代数,$\beta$定义为$\beta(b\bowtie_\sigma h\otimes c\bowtie_\sigma g)=\varepsilon_H(h)\varepsilon_{B^{OP}}(c)\sigma^{-1}(b, g).$ 对于Sweedler 4维Hopf代数$H$, 作者给出一个例子说明:此弱双交叉积$(B^{OP}\bowtie_\sigma H, \beta)$不仅是一个Long双代数,而且是一个非可换和非余可换的8维Hopf代数. 最后, 设$B,H$都是弱双代数, $\sigma: B\otimes H\rightarrow k$是一个线性映射, 作者给出了$(B,\sigma,\leftharpoonup, \Delta_B)$是弱相关右$(H, B)$ -重模代数的充分必要条件.     

名校基础知识自测

★高一年级 北京第十二中学(100071)李有毅一、选择题1.卜列四个关系式中正确的是(). (A)g任{a}(B)a星{a} ((、){a}任{a,b}(D)a〔{a,b}2.满足{l}里A里{1,2,3}的集合A的个数为().(A)l(B)2(C)3(D)43.已知尸一{、}二2一3二+2一0},T一{y{yS一定之一一5}.则尸nTUS一().(A)2)(B){1,2}(C){一2,2}(D){1}设全集u一{2,3,5},A={}a一5{,2},CoA一{5},贝日u的值为().(A)2(B)8(C)2或8(D)一2或8已知集合{‘·{一2了.>2了,·>。)一{工}了<一5或二>4},则,丫+n的值为().(A)一8(11)l()(C)8(D)80若集合A一{二i“厂一a二+1<。}一②,则实数“的值的集合…     

边界可约的Heegaard分解

本文证明了任意边界可约流形的Heegaard分解都是n个不可约的、边界不可约的三维流形的Heegaard分解通过连通和、边界连通和及边界自连通和运算而得到.     

Bilateral self-amalgamation of a Heegaard splitting and Hempel distance

A closed orientable Haken 3-manifold containing a non separating incompressible closed surface has two canonical Heegaard splittings, which are called self-amalgamation and bilateral self-amalgamation.Heegaard distance introduced by Hempel is a useful index in studying Heegaard splitting. This paper studies the stabilization problem for the bilateral self-amalgamation, and proves that if the distance of bilateral selfamalgamation of a Heegaard splitting is at least 9, then it is unstabilized, weakly reducible and irreducible.     

Unstabilized and uncritical self-amalgamation along essential subsurfaces

Suppose V ∪_S W is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F_1 and F_2 are pairwise disjoint homeomorphic essential subsurfaces in ?_V. In this paper,we give a sufficient condition such that the self-amalgamation of V ∪_S W along F_1 and F_2 is unstabilized and uncritical.     

Random methods in 3-manifold theory

The surface map arising from a random walk on the mapping class group may be used as the gluing map for a Heegaard splitting, and the resulting 3-manifold is known as a random Heegaard splitting. We show that the splitting distance of random Heegaard splittings grows linearly in the length of the random walk, with an exponential decay estimate for the proportion with slower growth. We use this to obtain the limiting distribution of Casson invariants of random Heegaard splittings.     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

Critical Heegaard surfaces

In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.     

A Note on Heegaard Splittings of Amalgamated 3-Manifolds

Let M be a compact orientable irreducible 3-manifold, and F be an essential connected closed surface in M which cuts M into two manifolds M1 and M2. If Mi has a minimal Heegaard splitting Mi = Vi ∪Hi Wi with d(H1) + d(H2) ≥ 2(g(M1) + g(M2) -g(F)) + 1, then g(M) = g(M1) + g(M2) - g(F).     

沿着不连通曲面的Heegaard分解的不可稳定化的融合积

Let M be a 3-manifold, F= {F1 , F2 , . . . , Fn } be a collection of essential closed surfaces in M (for any i, j ∈ {1, ..., n}, ifi≠j, Fi is not parallel to Fj and Fi ∩Fj = φ) and0 M be a collection of components of M. Suppose M-UFi ∈FFi×(-1, 1) contains k components M1 , M2 , . . . , Mk . If each M i has a Heegaard splitting ViUSiWi with d(Si) > 4(g(M1 ) + ··· + g(Mk )), then any minimal Heegaard splitting of M relative to 0M is obtained by doing amalgamations and self-amalgamations from minimal Heegaard splittings or -stabilization of minimal Heegaard splittings of M1 , M2 , . . . , Mk .     

Local detection of strongly irreducible Heegaard splittings

Let S be a Heegaard splitting surface of a compact orientable 3-manifold M. If S is strongly irreducible, the manner in which it can intersect a ball or a solid torus in M is very constrained and the allowable configurations are simple and useful. Splitting surfaces not conforming to these simple local pictures must be weakly reducible.     

平环和的Heegaard亏格的一个下界

Let Mi, i = 1,2, be a compact orientable 3-manifold, and Ai an incompressible annulus on a component Fi of OMi. Suppose A1 is separating on F1 and A2 is non-separating on F2. Let M be the annulus sum of M1 and M2 along A1 and A2. In the present paper, we give a lower bound for the genus of the annulus sum M in the condition of the Heegaard distances of the submanifolds M1 and M2