主理想整环上保对合矩阵的线性映射

设R是特征不为2的交换主理想整环,Mn（R）表示R上n阶全矩阵模,本文基底生成元的方法刻划Mn（R）上保对合矩阵的R-线性映射的形式。     

交换主理想整环上立方幂等矩阵的线性保持

设Ｒ（≠Ｆ_３）是特征不为２的交换主理想整环,Ｍ_ｎ（Ｒ）定义Ｒ上的ｎ×ｎ矩阵模,本文刻划当ｎ≥ｍ时从Ｍ_ｎ（Ｒ）到Ｍ_ｎ（Ｒ）的保持立方幂等矩阵的线性映射的形式,由此推广了Ｃｈａｎ和Ｌｉｍ的一个结果（［１,定理３］）．     

保矩阵{1}逆的线性映射

设R是特征为2的主理想整环,Mn(R)表示R上n×n矩阵代数,在本文中我们给出了保Mn(R)中矩阵{1}逆的线性映射的一个刻划.     

主理想整环上的线性方程组的通解

将文[1]中整数环上的线性方程组问题推广到主理想整环上,利用主理想整环上的矩阵的初等变换及等价标准形导出了主理想整环上的线性方程组有解的一个充分必要条件和求解方法.最后,通过实例说明了算法.     

主理想整环上n阶矩阵环中的Goldbach问题

Abstract: In this paper, we consider the Goldbach's problem for matrix rings, namely, we decompose an n ×n (n > 1) matrix over a principal ideal domain R into a sum of two matrices in Mn(R) with given determinants. We prove the following result: Let n > 1 be a natural number and A = (αij) be a matrix in Mn(R). Define d(A) := g.c.d{αij}. Suppose that p and q are two elements in R. Then (1) If n > 1 is even, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p-q; (2) If n > 1 is odd, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p + q. We apply the result to the matrices in Mn(Z) and Mn(Q[x]) and prove that if R = Z or Q[x], then any nonzero matrix A in Mn(R) can be written as a sum of two matrices in Mn(R) with prime determinants.     

Lawson-Hoffmann对偶定理的推广

本文讨论了Z－连续偏序集的一系列性质,主要证明了Z－连续偏序集范畴对偶等价于完全分配格范畴的一个满子范畴．     

主理想整环上线性群中子群H_r的扩群

设R是一个主理想整环,GL(n,R)为R上n阶一般线性群,H_r为GL(n,R)的一个子群,在n≥3的情形下给出H_r在GL(n,R)中的所有扩群.     

关于环Z[m~（1/2）]的商环元素个数的一个新的证明

文献[1]热运用环论的方法证明了环Z[m~（1/2）]热的商环Z[m~（1/2）]/（a＋bm~（1/2））的元素个数是｜a2-b2m｜.我们将用主理想整环上的模的理论给出一种简洁的证明.     

分块带状矩阵的逆

1引言如果分块矩阵A=(A_(ij))_(n×n)满足A_(ij)=O(j-i＞p且i-j＞q),其中A_(ij)为m阶矩阵,则称A为(p,q)-分块带状矩阵．分块带状矩阵在一些实际问题中经常出现,例如在量子场论中用途很广的非线性Schr(?)dinger方程的差分离散问题,解热传导问题等,都会遇到分块带状矩阵．常见的分块三对角矩阵,分块五对角矩阵都是特殊的分块带状矩阵．采用通常的方法求解分块带状矩阵的逆矩阵时,需要进行O(n~3)次m阶矩阵的运算．本文首先将分块带状矩阵扩充成可逆的分块上(下)三角矩阵,利用其逆矩阵导出了分块带状矩阵的逆矩阵表达式;进而利用所得到的公式分别推导了分块三对角矩阵及分块五对角矩阵的逆矩阵的快速算法,所需运算量为O(n~2)次m阶矩阵的运算．本文的结果扩充了文[1]等关于分块三对角阵求逆的相关结果．     

一类整除性定理

本文给出复域上多元多项式环Ｃ［ｘ１，ｘ２，…，ｘｎ］的一类整除性定理，它是［１］中整除定理的直接推广     

REPRESENTATION OF POSETS

In this paper we give new criterions for left distributive posets to have neatest representations. We also illustrate a construction that would embed left distributive posets into representable semilattices.     

Parity representations of posets

Parity representations, introduced in this paper, comprise a new method of representation of posets that yields insight into the combinatorics of the poset of all intervals of a poset. Results here generalize some results previously obtained for the face lattices of binary partition polytopes.     

交一致连续偏序集

In this paper, as a generalization of uniform continuous posets, the concept of meet uniform continuous posets via uniform Scott sets is introduced. Properties and characterizations of meet uniform continuous posets are presented. The main results are:(1) A uniform complete poset L is meet uniform continuous iff ↑(U ∩↓ x) is a uniform Scott set for each x ∈ L and each uniform Scott set U;(2) A uniform complete poset L is meet uniform continuous iff for each∨∨x∈ L and each uniform subset S, one has x ∧S ={x ∧ s | s ∈ S}. In particular, a complete lattice L is meet uniform continuous iff L is a complete Heyting algebra;(3) A uniform complete poset is meet uniform continuous iff every principal ideal is meet uniform continuous iff all closed intervals are meet uniform continuous iff all principal filters are meet uniform continuous;(4) A uniform complete poset L is meet uniform continuous if L1 obtained by adjoining a top element1 to L is a complete Heyting algebra;(5) Finite products and images of uniform continuous projections of meet uniform continuous posets are still meet uniform continuous.     

Vortex sheets with reflection symmetry in exterior domains

In this paper we prove the existence of a weak solution of the incompressible 2D Euler equations in the exterior of a reflection symmetric smooth bluff body with symmetric initial flow corresponding to vortex sheet type data whose vorticity is of distinguished sign on each side of the symmetry axis. This work extends the results proved for full plane flow by the authors in [M.C. Lopes Filho, H.J. Nussenzveig Lopes, Z. Xin, Existence of vortex sheets with reflection symmetry in two space dimensions, Arch. Ration. Mech. Anal. 158 (3) (2001) 235-257].     

多项式矩阵代数性质讨论

本文利用系统与控制论中有关多项式矩阵的结果,对多项式矩阵代数性质进行讨论,得到的主要结果有多项式方阵环是主理想环,也是主单侧理想环。     

Homotopy of Non-Modular Partitions and the Whitehouse Module

We present a class of subposets of the partition lattice _n with the following property: The order complex is homotopy equivalent to the order complex of _n – 1, and the S _n -module structure of the homology coincides with a recently discovered lifting of the S _n – 1-action on the homology of _n – 1. This is the Whitehouse representation on Robinson's space of fully-grown trees, and has also appeared in work of Getzler and Kapranov, Mathieu, Hanlon and Stanley, and Babson et al.One example is the subposet P _n ⁿ – 1 of the lattice of set partitions _n , obtained by removing all elements with a unique nontrivial block. More generally, for 2 k n – 1, let Q _n ^k denote the subposet of the partition lattice _n obtained by removing all elements with a unique nontrivial block of size equal to k, and let P _n ^k = _{i = 2} ^k Q _n ⁱ . We show that P _n ^k is Cohen-Macaulay, and that P _n ^k and Q _n ^k are both homotopy equivalent to a wedge of spheres of dimension (n – 4), with Betti number . The posets Q _n ^k are neither shellable nor Cohen-Macaulay. We show that the S _n -module structure of the homology generalises the Whitehouse module in a simple way.We also present a short proof of the well-known result that rank-selection in a poset preserves the Cohen-Macaulay property.     

Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.     

Basic Interval Orders

An interval order of length n has elements in correspondence with a collection of intervals in the linearly ordered set {1, 2, ..., n}. A basic interval order of length n has the property that removal of any element yields an order with length less than n. We construct and enumerate the set of basic length n interval orders.     

XST-环和Morita-Like等价

XST-环的概念于1999年由Garcia和Marin[1]引进.本文主要研究XST-环的Morita-Like等价.证明了在环R中,由两个右q-稠密的XST-环确定的两个完全可加的范畴是同构的.通过描述生成元AM的自同态环End(AM)的q-稠密右理想和稠密右理想的关系,得到了,对于有单位元的环A而言,位于FM (A)和FC (A)间的中间矩阵环上的所有完全可加范畴是同构的.如此,扩张了中间矩阵环的Morita-Like等价链[2].再则,改进了文[1]中的主要结果,刻画了一个右XST-环Morita-Like等价于一个有单位元的环的条件.