关于同伦元素 $\alpha_{1}\beta_{1}\beta_{2}\gamma_{s}$

设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s 相似文献   

${R}_{0}$代数中的滤子格

Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.     

vspace{-1cm}$W(2,\boldsymbol{n})$和$H(2,\boldsymbol{n})$的$p$-特征标高度为$2$的不可约模的分类$^{}$

研究$p$-\!\!特征标高度等于$2$的$W(2,\boldsymbol{n})$和$H(2,\boldsymbol{n})$ 的不可约表示, 给出了当 $p$-\!\!特征标$\chi $的 高度等于$2$时, $L=X(2,\boldsymbol{n})$, $X=W,H$ 的不可约$L$-\!\!模 同构类代表元集合.     

循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.     

Young表与$\textrm{U}_{q}(\mathrm{osp}(1|2n))$的晶体基

设 $U_q(\mathrm{osp}(1|2n))$是对应Lie超代数$\mathrm{osp}(1|2n)$的量子包络超代数. 利用满足一定条件的半标准Young表, 给出有限维既约$U_q(\mathrm{osp}(1|2n))$模晶体图的实现 . 建立晶体图张量积分解的广义Littlewood-Richardson法则.     

Hyperplanes of $DW(5,{\mathbb{K}})$ with ${\mathbb{K}}$ a perfect field of characteristic 2

Let \({\mathbb{K}}\) be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space \(DW(5,{\mathbb{K}})\) that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to 5+N, where N is the number of equivalence classes of the following equivalence relation R on the set \(\{\lambda\in {\mathbb{K}}\,|\,X^{2}+\lambda X+1\mbox{ isirreducible}\) \(\mbox{in }{\mathbb{K}}[X]\}\): (λ ₁,λ ₂)∈R whenever there exists an automorphism σ of \({\mathbb{K}}\) and an \(a\in {\mathbb{K}}\) such that (λ ₂ ^σ )^?1=λ ₁ ^?1 +a ²+a.     

Irreducible Modules Over Witt Algebras $\mathcal {W}_{n}$ and Over 𝖘𝖑n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$

In this paper, by using the “twisting technique” we obtain a class of new modules A _b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\). We give necessary and sufficient conditions for A _b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\)-modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\), all the above irreducible \(\mathcal {W}_{n}\)-modules A _b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules, denoted by Ω_1?a (λ ₁, λ ₂, ? ,λ _n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\), we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module Ω_1?a (λ ₁, λ ₂,? ,λ _n ) is reducible, we prove that it has a unique nontrivial submodule W _1?a (λ ₁, λ ₂,...λ _n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module with highest weight mΛ _n for some non-negative integer \(m\in \mathbb {Z}_{+}\). We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules of the form Ω_1?a (λ ₁, λ ₂,? ,λ _n ) or of the form W _1?a (λ ₁, λ ₂,...λ _n ) to be isomorphic.     

Orbit inequivalent actions for groups containing a copy of $\mathbb{F}_{2}$

We prove that if a countable group Γ contains a copy of \mathbbF₂\mathbb{F}_{2}, then it admits uncountably many non orbit equivalent actions.     

完全解决图$\overline{B_{n-8,1,4}}$的色等价类

Two graphs are defined to be adjointly equivalent if and only if their complements are chromatically equivalent.Using the properties of the adjoint polynomials and the fourth character R₄（G）,the adjoint equivalence class of graph B_n-8,1,4 is determined.According to the relations between adjoint polynomial and chromatic polynomial,we also simultaneously determine the chromatic equivalence class of B_n-8,1,4 that is the complement of B_n-8,1,4.     

$\mathbb{C}^{n}$中$\mu$-Bergman空间的刻画和微分复合算子

设$p>0$, $\mu$和$\mu_{1}$是$[0,1)$上的正规函数. 本文首先给出了$\mathbb{C}^{n}$中单位球上$\mu$-Bergman空间$A^{p}(\mu)$的几种等价刻画; 然后 分别刻画了$A^{p}(\mu)$到$A^{p}(\mu_{1})$的 微分复合算子$D_{\varphi}$为有界算子以及紧算子的充要条件, 同时给出了当$p>1$时$D_{\varphi}$为 $A^{p}(\mu)$到$A^{p}(\mu_{1})$上紧算子的一种简捷充分条件和必要条件.     

完全图$K_{2m}$书本表示的唯一性

In the present paper we discuss some properties of book presentation of spatial graphs, and prove that the book presentation of minimum sheets of a complete graph K2m with even vertices is unique up to sheet translation and ambient isotopy. We also show this is true for K7.     

$T_{2}(T)$的几乎可裂序列及几乎$\mathcal {D}$-可裂序列

The AR-quiver and derived equivalence are two important subjects in the representation theory of finite dimensional algebras, and for them there are two important research tools-AR-sequences and D-split sequences. So in order to study the representations of triangular matrix algebra T2 (T ) = T0TT where T is a finite dimensional algebra over a field, it is important to determine its AR-sequences and D-split sequences. The aim of this paper is to construct the right(left) almost split morphisms, irreducible morphisms, almost split sequences and D-split sequences of T2 (T) through the corresponding morphisms and sequences of T. Some interesting results are obtained.     

$K_{5}\times S_{n}$ 的交叉数

把完全图$K_{5}$的五个顶点与另外$n$个顶点都联边得到一类特殊的图$H_{n}$.文中证明了$H_{n}$的交叉数为$Z(5,n)+2n+\lfloor \frac{n}{2}\rfloor+1$,并在此基础上证明了$K_{5}$与星$K_{1,n}$的笛卡尔积的交叉数为$Z(5,n)+5n+\lfloor\frac{n}{2} \rfloor+1$.     

Hilbert-Schmidtness of Submodules in $H^2 (\mathbb{D}^2 )$ Containing $θ(z)−\varphi (w)$

A closed subspace $M$ of the Hardy space $H^2(\mathbb{D}^2)$ over the bidisk is called submodule if it is invariant under multiplication by coordinate functions $z$ and $w.$ Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem. This paper proves that every finitely generated submodule $M$ containing $θ(z)−\varphi(w)$ is Hilbert-Schmidt, where $θ(z),$ $\varphi(w)$ are two finite Blaschke products.     

双圆盘加权Hardy空间上$T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$的约化子空间

本文中, 我们主要刻画了Toeplitz算子$T=M_{z^k}+M^*_{z^l}$的约化子空间, 其中 $k_i, l_i$ ($i=1,2$) 均是正整数, $k=(k_1,k_2), l=(l_1,l_2)$ 且 $k\neq l$, $M_{z^k}$, $M_{z^l}$ 是双圆盘加权Hardy空间$\mathcal{H}_\omega^2(\mathbb{D}^2)$上的乘法算子. 对权系数 $\omega$ 适当限制, 我们证明了由 $z^m$ 生成的 $T$ 的约化子空间均是极小的. 特别地, Bergman 空间和加权 Dirichlet 空间 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 均是满足该限制条件的加权Hardy空间. 作为应用, 我们刻画了 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 上 Toeplitz 算子 $T_{z^k+\bar{z}^l}$ 的约化子空间, 该结论是对双圆盘Bergman 空间上相关结论的推广.     

The Quasiapproximate $(\mathcal{U} + \mathcal{K})$ –Invariants of Essentially Normal Operators

For a class of essentially normal operators, we characterize their norm closures of –orbits. Moreover, we introduce a notion of the quasiapproximate – equivalence of essentially normal operators and determine completely the quasiapproximate –invariants. Finally, we give the canonical forms of essentially normal operators under this quasiapproximate –equivalence.     

Teichmüller curves in genus three and just likely intersections in $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).These constraints are akin to those that appear in Zilber and Pink’s conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\) (rather than the more standard \(\mathbf{G}_{m}^{n}\)). The ambient algebraic group lies outside the scope of Zilber’s Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum \(\Omega\mathcal{M}_{3}(1,1,1,1)\), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of \(\mathbf{G}_{m}^{9}\). These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.     

$M_{2}({\mathbb F})$上的$H$-模代数结构

设${\mathbb F}$是特征为零的代数闭域, $H$为非点化非幺模的8维非半单Hopf代数, $M_{2}({\mathbb F})$为${\mathbb F}$上二阶方阵组成的全矩阵代数. 本文的主要目的是讨论和分类$M_{2}({\mathbb F})$上所有的$H$-模代数结构.     

第二大特征值小于$\frac{\sqrt{5}-1}{2}$的平面图

若能将图$G$画在一个平面上,使得任何两条边仅在顶点处相交,则称$G$是平面图.本文刻画了第二大特征值小于$\frac{\sqrt{5}-1}{2}$的所有无孤立点的平面图.     

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.