可换环上矩阵代数的SZ-导子, PZ-导子 和 S-导子

Let R be a commutative ring with identity, Nn（R） the matrix algebra consisting of all n × n strictly upper triangular matrices over R with the usual product operation. An R-linear map φ ： Nn（R） → Nn（R） is said to be an SZ-derivation of Nn（R） if x2 = 0 implies that φ（x）x＋xφ（x） = 0. It is said to be an S-derivation of Nn（R） if φ（x2） = φ（x）x＋xφ（x） for any x ∈ Nn（R）. It is said to be a PZ-derivation of Nn（R） if xy = 0 implies that φ（x）y＋xφ（y） = 0. In this paper, by constructing several types of standard SZ-derivations of Nn（R）, we first characterize all SZ-derivations of Nn（R）. Then, as its application, we determine all S-derivations and PZ- derivations of Nn（R）, respectively.     

交换环上矩阵代数的局部化导子

Let R be a commutative ring with identity, Nn（R） the matrix algebra consisting of all n × n strictly upper triangular matrices over R. Several types of proper local derivations of Nn（R） （n ≤ 4） are constructed, based on which all local derivations of Nn（R） （n ≤ 4） are characterized when R is a domain.     

交换环上上三角矩阵的李三导子

Let T(n,R) be the Lie algebra consisting of all n × n upper triangular matrices over a commutative ring R with identity 1 and M be a 2-torsion free unital T(n,R)-bimodule.In this paper,we prove that every Lie triple derivation d : T(n,R) → M is the sum of a Jordan derivation and a central Lie triple derivation.     

中心对称本原矩阵的k点指数集

Let G = (V, E) be a primitive digraph. The vertex exponent of G at a vertex v ∈ V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. We choose to order the vertices of G in the k-point exponent of G and is denoted by expG(k), 1 ≤ k ≤ n. We define the k-point exponent set E(n, k) := {expG(k)| G = G(A) with A ∈ CSP(n)}, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n,k) for all n, k with 1 ≤ k ≤ n except n ≡ 1(mod 2) and 1 ≤ k ≤ n - 4. We also characterize the extremal graphs when k = 1.     

Topological and geometric property of matrix algebra

Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.     

Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.     

Additive preservers of idempotence and Jordan homorphisms between rings of square matrices

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by Mn（R） the ring of all n x n matrices over R. Let （Jn（R）） be the additive subgroup of Mn（R） generated additively by all idempotent matrices. Let JJ = （Jn（R）） or Mn（R）. We describe the additive preservers of idempotence from JJ to Mm（R） when 2 is a unit of R. Thereby, we also characterize the Jordan （respectively, ring and ring anti-） homomorphisms from Mn （R） to Mm （R） when 2 is a unit of R.     

交换环上上三角矩阵保对合与保立方幂等的模自同构

Suppose R is a commutative ring with 1, and 2 is a unit of R. Let Tn(R) be the n × n upper triangular matrix modular over R, and let (?)i(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that preserve involutory or tripotent. The main result in this paper is that f ∈ (?)i(R) if and only if there exists an invertible matrix U ∈ Tn(R) and orthogonal idempotent elements e1,e2,e3 ande4 in R with such that where     

可换环上上三角矩阵代数的局部若当导子和局部若当自同构

Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

双圈图的零度与其匹配数之间的关系

Let G be a graph with n(G) vertices and m(G) be its matching number.The nullity of G,denoted by η(G),is the multiplicity of the eigenvalue zero of adjacency matrix of G.It is well known that if G is a tree,then η(G) = n(G)-2m(G).Guo et al.[Jiming GUO,Weigen YAN,Yeongnan YEH.On the nullity and the matching number of unicyclic graphs.Linear Alg.Appl.,2009,431:1293 1301]proved that if G is a unicyclic graph,then η(G)equals n(G)-2m(G)-1,n(G)-2m(G),or n(G)-2m(G) +2.In this paper,we prove that if G is a bicyclic graph,then η(G) equals n(G)-2m(G),n(G)-2m(G)±1,n(G)-2m(G)±2or n(G)-2m(G) + 4.We also give a characterization of these six types of bicyclic graphs corresponding to each nullity.     

Asymptotics of Resolvent Iterates for Abstract Potentials

Let $A$ be an abstract potential, that is, an operator whose resolvent $R(\cdot)$ exists and satisfies the condition $||(\Re\lambda-a)R(\lambda)||\leq M$ in a halfplane $\Pi_a:=\{\lambda\in\Bbb C;\,\Re\lambda>a\}$. It is shown that the resolvent iterates satisfy in $\Pi_a$ the estimates $$||\Big[(\Re\lambda-a)R(\lambda)\Big]^n||< Men$$ for all $n\in\Bbb N$.     

Continuity of a Deformation in H<Superscript>1</Superscript> as a Function of Its Cauchy-Green Tensor in L<Superscript>1</Superscript>

Let $\Omega$ be a bounded Lipschitz domain in $\BBbR^n$. The Cauchy-Green, or metric, tensor field associated with a deformation of the set $\Omega$, i.e., a smooth-enough orientation-preserving mapping $\bTh\colon\Omega\to\BBbR^n$, is the $n\times n$ symmetric matrix field defined by $\bnabla\bTheta^T(x)\bnabla\bTheta(x)$ at each point $x\in\Omega$. We show that, under appropriate assumptions, the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the spaces $\bH^1(\Omega)$ for the deformations and $\bL^1(\Omega)$ for the Cauchy-Green tensors. When $n=3$ and $\Omega$ is viewed as a reference configuration of an elastic body, this result has potential applications to nonlinear three-dimensional elasticity, since the stored energy function of a hyperelastic material depends on the deformation gradient field $\bnabla\bTheta$ through the Cauchy-Green tensor.     

关于纤维锥的深度和Hilbert级数

Let (R,m) be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ? 1 and λ( JKKIIkk-1 ) = 1. We show that if depth G(I) ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK(I) assuming that depth G(I) ≥ d - 1.     

笛卡尔乘积图的圈点连通度

图G的圈点连通度,记为κ_c(G),是所有圈点割中最小的数目,其中每个圈点割S满足G-S不连通且至少它的两个分支含圈.这篇文章中给出了两个连通图的笛卡尔乘积的圈点连通度:(1)如果G_1≌K_m且G_2≌K_n,则κ_c(G_1×G_2)=min{3m+n-6,m+3n-6},其中m+n≥8,m≥n+2,或n≥m+2,且κ_c(G_1×G_2)=2m+2n-8,其中m+n≥8,m=n,或n=m+1,或m=n+11;(2)如果G_1≌K_m(m≥3)且G_2■K_n,则min{3m+κ(G_2)-4,m+3κ(G_2)-3,2m+2κ(G_2)-4}≤κ_c(G_1×G_2)≤mκ(G2);(3)如果G_1■K_m,K_(1,m-1)且G_2■K_n,K_(1,n-1),其中m≥4,n≥4,则min{3κ(G_1)+κ(G_2)-1,κ(G_1)+3κ(G_2)-1,2_κ(G_1)+2_κ(G_2)-2}≤κ_c(G_1×G_2)≤min{mκ(G_2),nκ(G_1),2m+2n-8}.