环上的Clifford代数

§1. 引言与记号 如众周知,域上的Clifford代数乃是概括域上的Grassmann代数(外代数)以及广义四元数代数的一个代数。它不但在数学的一些分支(如群表示论、二次型理论等)中有着重要的应用,而且也是近代理论物理中的有用工具之一(比如参看[1])。1954年,C.Chevalley在[2]中完美地给出了域上Clifford代数的基本理论。本文的主要目的是建立可换环上的Clifford代数,即给出它的定义、存在性与唯一性等。容易看出,这是域上的Clifford代     

模糊自动机的强连通性及群自动机

为了更好地研究模糊自动机的结构和性质,采用代数的方法,在传统的模糊有限状态自动机的基础上,通过定义状态集合为代数群的自动机,讨论了这一类自动机的连通性和正则性,这丰富了模糊自动机理论．     

探讨有限莫利秩的域的结构和性质

环具有加法群和乘法群的二元代数运算结构,利用有限莫利秩的群的性质具有降链条件,与在无限域上的代数扩张的伽罗瓦理论结合,来研究有限莫利秩的无限域的结构和性质,主要成果为:有限莫利秩的无限域K,任意a∈K,整数n>0,方程xⁿ=a在域K中有解;假设域K是有限莫利秩的无限域,那么域K一定是代数闭域.     

QF环上的Macaulay逆系问题

Macaulay的逆系理论研究多项式理想与域上线性递归阵列之间的对偶关系.本文致力于将逆系理论推广到Quasi-Frobenius环上.这项研究可以应用到多个领域中，例如高维卷积码、Galois环上的代数编码以及参数化系统论.首先以新的观点描述逆系问题.将其转换成阵列形式零点(Nullstellensatz)问题.然后建立代数理想与线性递归阵列之间的对应关系.对此相关的问题已有大量的研究，但几乎所有现有的有趣结果可以看成本文结果的特殊推论.     

粗糙除环、粗糙域和粗糙整环

对粗糙集上代数结构进行了研究,引入了粗糙除环、粗糙域和粗糙整环等概念.利用粗糙集理论方法和经典代数中的处理手段,讨论了粗糙除环和它们的结构特征,得到了它们的一些良好的性质.     

QF环上的Macaulay逆系问题

Ｍａｃａｕｌａｙ的逆系理论研究多项式理想与域上线性递归阵列之间的对偶关系．本文致力于将逆过系理论推广到 Ｑｕａｓｉ－Ｆｒｏｂｅｎｉｕｓ环上．这项研究可以应用到多个领域中,例如高维卷积码、 Ｇａｌｏｉｓ环上的代数编码以及参数化系统论．首先以新的观点描述逆系问题．将其转换成阵列形式零点（Ｎｕｌｌｓｔｅｌｌｅｎｓａｔｚ）问题．然后建立代数理想与线性递归阵列之间的对应关系．对此相关的问题已有大量的研究,但几乎所有现有的有趣结果可以看成本文结果的特殊推论．     

非零势能的耗散力学控制系统的位形能控性

在拉格朗日力学控制系统的仿射联络框架下,基于Sussmann对有限维流形上一般仿射非线性控制系统的能控性讨论,将简单力学控制系统短时间局部位形能控的一个可计算的充分条件推广到迷向耗散的系统上,并给出系统是平衡点能控的一个充分条件,其中,系统的拉格朗日函数为动能减势能A·D2在问题的讨论中,系统的能控李代数的向量场李括号运算,以及与系统位形流形的Levi-Civita联络相关的对称积起了重要作用．尽管势能项会使系统的位形能控性讨论复杂化,但Liouville向量场又简化了系统的能控李代数计算．     

线性系统若当不变量与标准形

1.引言1980年 Bosgra 和 Weiden 在满秩线性变换意义下,首先给出了能控能观线性系统的所谓输入输出不变量及其相应的标准形。这一结果在满秩线性变换群意义下将能控能观系统参数化,也即从输入输出方面来看,将系统参数化,给定一组参数,即可唯一确定一个输入输出关系;不同的参数组所确定的输入输出关系不同。它一方面揭示了线性系统的内在本质,同时在实现理论、部分实现理论、编码理论和有限自动机方面有着重要应用。本文依据矩阵的若当形,除了给出能控系统的一组完备独立不变量及其相应的标准形外,还给出了能控能观系统一组完备独立不变量及其相应的标准形,在某些方面这一结果比     

交换环上线性系统的多项式模理论

引言 在线性系统理论中,R.Kalman[1]首先应用模论研究域上线性系统。后来Fuhrmann在这一方向上做了重大发展,创造了域上线性系统的多项式模理论和方法。近年来,Fuhrmann的方法进一步被用于研究各种控制理论问题。 本文借助于[7]中引入的周期性,对交换环上线性系统建立类似于域上线性系统的多项式模理论。并在此基础上讨论传递函数阵的分解与实现问题。     

有限模糊树自动机的一些代数性质与相应的语言

将模糊自动机的同态、完全、容许关系等概念引入到模糊树自动机中,从代数的角度研究模糊树自动机的一些代数性质,并探讨了模糊树自动机的语言的相关问题.     

Half-Bol quasi-fields

L. V. Sabinin and B. L. Sharma discussed a new type of quasi-field called a half-Bol quasi-field, which is a generalization of Bol quasi-field. This article improves their definition of half-Bol quasi-field and shows that such quasi-fields probably do not give any new and unusual translation planes. The main result is that a half-Bol quasi-field of odd dimension over its kernel is a generalized André system.Research for this article was supported in part by an NSF grant.     

一类非线性离散动态系统的分析

时序电路是一类重要的离散动态系统,它在数学控制、计算机、数字通信等工程系统中应用得很普遍。线性时序电路理论已比较成熟。然而,由于非线性带来的本质困难,非线性时序电路理论的某些方面还相当薄弱。即使对一类结构简单的非线性时序电路——非线性移位寄存器,其状态图中圈数与圈长的分析计算仍缺乏系统的理论。而这类寄存器在数字控制、计算机、编译码器、密码机等工程系统中有重要的应用。     

An efficient algorithm for factoring polynomials over algebraic extension field

An efficient algorithm is proposed for factoring polynomials over an algebraic extension field defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Gröbner basis, no extra Gröbner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a characteristic polynomial of a generic linear map. Then this polynomial is factorized over the ground field. From its factors, the factorization of the polynomial over the extension field is obtained. The algorithm has been implemented in Magma and computer experiments indicate that it is very efficient, particularly for complicated examples.     

On Extensions of Semilocal Prüfer Domains

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prüfer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prüfer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prüfer domain. Moreover, we show that any two extensions of a semilocal Prüfer domain are incomparable (by inclusion) in an algebraic extension of fields.     

The rational canonical form via the splitting field

It is clear that a given rational canonical form can be further resolved to a Jordan canonical form with entries from the splitting field of its minimal polynomial. Conversely, with an a priori knowledge of the existence and uniqueness of the rational canonical form of a matrix with entries from a general field, one can modify its Jordan canonical form in the splitting field of its minimal polynomial to construct its rational canonical form in the original field. No author has tried this converse with the a priori existence–uniqueness condition removed. It is feared that “in many occasions when, after a result has been established for a matrix with entries in a given field, considered as a matrix with entries in a finite extension of that field, we cannot go back from the extension field to get the desired information in the original field” [I.N. Herstein, Topics in Algebra, Ginn and Company, Waltham, MA, 1964 (pp. 262–263)]. The present paper removes this a priori condition and uses a “symmetrization” to “integrate” back the Jordan canonical form of a matrix to its rational canonical form in the original field.     

Dimension Polynomials of Intermediate Fields and Krull-type Dimension of Finitely Generated Differential Field Extensions

Let K be a differential field of zero characteristic and let L be a finitely generated differential field extension of K. We introduce the concept of a transcendence type and dimension of the extension L/K considering chains of intermediate differentials fields of this extension. Using the technique of differential dimension polynomials we obtain relationships between the transcendence type and dimension of L/K and differential birational invariants of this extension carried by its dimension polynomials.     

Hasse-Arf Property and Abelian Extensions

The paper contains discussions of relations between the property of a totally ramified p-extension of a local field to be abelian and the property of its Galois group to possess integer jumps with respect to the upper numbering (Hasse-Arf property). It is shown that such an extension L/F is abelian if and only if for any totally ramified abelian extension E/F the extension LE/F satisfies the Hasse-Arf property. An additional property to the Hasse-Arf property in terms of principal units which makes the extension abelian is established as well.     

A ruled residue theorem for function fields of elliptic curves

It is shown that a valuation of residue characteristic different from 2 and 3 on a field E has at most one extension to the function field of an elliptic curve over E, for which the residue field extension is transcendental but not ruled. The cases where such an extension is present are characterised.     

Picard-Vessiot theory for real fields

The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we prove the existence of a Picard-Vessiot extension for a homogeneous linear differential equation defined over a real differential field K with real closed field of constants. We give an adequate definition of the differential Galois group of a Picard-Vessiot extension of a real differential field with real closed field of constants and we prove a Galois correspondence theorem for such a Picard-Vessiot extension.