一类商代数上的双-Frobenius代数结构

设C[X]为复数域上的一元多项式代数,I为n+1次Dickson多项式E_(n+1)(X)生成的C[X]的理想,C[X]/I为商代数.证明了商代数C[X]/I既是Frobenius代数,又是Frobenius余代数.进一步,该商代数在恒等对极下还是双-Frobenius代数.     

Baer环和拟-Baer环的多项式扩张的一点注记

I1和I2分别是环R的一个左理想和右理想,T1=R[x]和T2=R[x,x-1]分别表示多项式环和洛朗多项式环.首先给出两个例子,分别说明了T1I1不一定是T1的左理想与T2L2不一定是T2的右理想.其次给出了环的多项式扩张及洛朗扩张的理想的性质.最后证明了,若R[X](R[x,x-1])是拟-Baer环,则R也是拟-...     

交换环上上三角矩阵代数在全矩阵代数中的扩代数及其若当导子

设R是任意含么交换环,2是R的可逆元.M(n,R)表示R上所有n×n级矩阵形成的代数,T(n,R)表示R上所有n×n级上三角矩阵形成的代数.决定了T(n,R)在M(n,R)中的扩代数,并具体刻画了这些扩代数的若当导子.     

可换环上一些线性李代数的扩代数

设R是任意含单位元的可换环,gl(n,R)是R上n级一般线性李代数.t表示gl(n,R)中所有上三角矩阵组成的子代数,d表示gl(n,R)中所有对角矩阵组成的子代数.本文将分别确定t在gl(n,R)中的扩代数和d在t中的扩代数.     

刘金旺刘晓奇周飞跃

1 引言 本文中R是指一个UFD,κ是R的商域,R[x]是以x为未定元的R上的多项式环.R上的半无限线性递归序列(lrs)与无限线性递归序列(Lrs)统记为LRS.LRS在代数编码、密码学、信号处理中是重要的研究对象,序列的综合问题主要是求出序列α的次数最小的特征多项式.     

环的中心幂等元与Boolean代数的表示

本文拟给出Boolean代数另一完全不同于Stone表示[1]的表示。文中所讨论的环均指结合环。 设A是一个有单位元1的半素环(即A不含非零幂零理想)。令E(A)是A的所有中心幂等元的集合。在E(A)中定义 则易知是E(A)上一个代数运算。又     

关于Gauss-Turán求积公式的注记

1.引言 设w(x)是区间[-1,1]上的权函数,N是自然数集,X1,…,Xn(n∈N)是对应于权函数w(x)的n次正交多项式的零点,则具有最高代数精度2n-1,其中Πn表示所有次数≤n的多项式空间. 1950年,Turan[1]将上述经典的Gauss求积公式予以推广,证明了,若     

上三角矩阵的线性交换映射的表示

设Trn(R)表示定义在实数域R上的n×n阶上三角矩阵的集合,φ是定义Trn(R)上线性映射.如果对任意X∈Trn(R)有Xφ(X)=φ(X)X成立,称φ是线性交换映射.本文利用初等的矩阵计算方法描述了当φ(I)=I时,线性交换映射φ的表示形式,而且给出了φ的Frobenius范数‖φ(X)‖F的估计.     

可换环上上三角矩阵代数的若当自同构分解(英文)

设R是含单位元1和可逆元2的可换环,Tn+1(R)表示R上(n+1)×(n+1)级上三角矩阵全体所形成的矩阵代数.本文证明了T(R)的每一个若当自同构都可唯一的分解为图自同构,内自同构和对角自同构的乘积.     

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.     

WELL-BEHAVED BASIS AND LR ARRAYS

ＷＥＬＬ－ＢＥＨＡＶＥＤＢＡＳＩＳＡＮＤＬＲＡＲＲＡＹＳＬＩＮＤＯＮＧＤＡＩ（林东岱）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅ，Ｂｅｉｊｉｎｇ１０００８０，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ...     

THE DEGENERACY PROBLEM OF TWO-DIMENSIONAL LINEAR RECURRING ARRAYS

The degeneracy degree and degeneracy position sets of a wo-dimensional linear recurrence relation set are characterized. The fact that a linear recurring array is essentially a doubly periodic array is shown. By using the Grbner base theory, a calculation formula for degeneracy degree is given and the existence of a special degeneracy position set is proved. In the present paper, the degeneracy problem of the two-dimensional linear recurring arrays is completely solved.     

Hopf Algebras of Linear Recurring Sequences over Rings and Modules

The module of linear recurring sequences over a commutative ring R can be considered as a Hopf algebra dual to the polynomial Hopf algebra over R. Under this approach, some notions and operations from the Hopf algebra theory have an interesting interpretation in terms of linear recurring sequences. Generalizations are also considered: linear recurring bisequences, sequences over modules, and k-sequences.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 113–148, 2003.     

结合一种新搜索的Broyden算法类的全局收敛性

文[1]给出了基本周期矩阵为对角形状的线性递归m-阵列的平移等价类的计数．本文在此基础上运用这表达式分别给出了：（1）具有任意一个可能的基本周期矩阵；（2）Grobner窗口为m×n；（3）Grobner窗口大小即级数为任意正整数w时的线性递归m-阵列不同平移等价类的个数．     

Gröbner bases of characteristic ideals of LRS over UFD

LetR be a unique factorization domain (UFD). A method of Gröbner bases and localization in commutative algebra is applied to compute and analyze the characteristic ideals of semi-infinite linear recurring sequences (lrs), infinite linear recurring sequences (LRS), and finite lrs over UFD. The canonical form of a minimal Gröbner basis of the homogeneous characteristic ideal is described for a finite segment of an lrs, from which a precise relation between every step in the classical Berlekamp-Massey algorithm and every member of the Gröbner basis is derived.     

Linear Recurring Arrays, Linear Systems and Multidimensional Cyclic Codes over Quasi-Frobenius Rings

This paper generalizes the duality between polynomial modules and their inverse systems (Macaulay), behaviors (Willems) or zero sets of arrays or multi-sequences from the known case of base fields to that of commutative quasi-Frobenius (QF) base rings or even to QF-modules over arbitrary commutative Artinian rings. The latter generalization was inspired by the work of Nechaev et al. who studied linear recurring arrays over QF-rings and modules. Such a duality can be and has been suggestively interpreted as a Nullstellensatz for polynomial ideals or modules. We also give an algorithmic characterization of principal systems. We use these results to define and characterize n-dimensional cyclic codes and their dual codes over QF rings for n>1. If the base ring is an Artinian principal ideal ring and hence QF, we give a sufficient condition on the codeword lengths so that each such code is generated by just one codeword. Our result is the n-dimensional extension of the results by Calderbank and Sloane, Kanwar and Lopez-Permouth, Z. X. Wan, and Norton and Salagean for n=1.     

Weak uniform distribution of un+1=aun+b in Dedekind domains

Let R be a Dedekind domain and I be an ideal of R such that the residue class ring R/I is finite. Necessary and sufficient conditions for the inital value u_o and the coefficients a,b are obtained such that the recurring sequence u_n+1=au_n+b is weakly uniformly distributed modulo I.     

Linear recurring sequences over modules

The aim of this paper is to extend some fundamental and applied results of the theory of linear recurring sequences over fields to the case of polylinear recurring sequences over rings and modules. Quasi-Frobenius modules and Galois rings play a very special role in this project.     

达到极大线性复杂度的前馈阵列

讨论了三类达到极大线性复杂度的前馈阵列的输入阵列的结构。一类是达到极大线性复杂度的乘积前馈阵列,完全确定了其输入阵列的结构;给出了两类输入阵列,证明了它们的任意前馈阵列达到极大线性复杂度。     

Sudoku-like arrays,codes and orthogonality

This paper is concerned with constructions and orthogonality of generalized Sudoku arrays of various forms. We characterize these arrays based on their constraints; for example Sudoku squares are characterized by having strip and sub-square constraints. First, we generalize Sudoku squares to be multi-dimensional arrays with strip and sub-cube constraints and construct mutually orthogonal sets of these arrays using linear polynomials. We add additional constraints motivated by elementary intervals for low discrepancy sequences and again give a construction of these arrays using linear polynomials in detail for 3 dimensional and a general construction method for arbitrary dimension. Then we give a different construction of these hypercubes due to MDS codes. We also analyze the orthogonality of all of the Sudoku-like hypercubes we consider in this paper.