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     

准-Groebner基的计算

本文通过定义S－多项式，给出了系数环是整环的多项式中理想的准－Groebner基的一个算法，并据此给出了计算该理想极大无关变元组和维数的一种方法。     

变量重排优化Groebner基运算的研究

本文通过使用变量重排的方法,改变了多项式环中理想的Groebner基的计算过程,得到不同的过程的计算效率也不同,因此通过这种方法应该能够找出减少计算Groebner基时间的方法．     

零维代数簇的分解及其应用

本文利用Groebner基，给出了一种分解零维代数簇的方法，并且讨论了这种方法在理想的准素分解以及几何定理机器证明中的应用．     

多项式理想准素分解的算法

讨论了多项式理想准素分解的算法,对于理想坐标变换中关键的“一般位置”的计算,给出了一个确定算法,并建立了特别规范Groebner基的概念,利用这一概念及文中提供的相应算法,可以方便地计算理想的维数,进而使用降维的方法来得到高维理想的准素分解。     

Clifford Mbius变换与Hypergenic函数

首先,在实Clifford代数空间Cl_n+1,0(R)中给出了与Clifford Mbius变换相关的一些定理.其次,证明了hypergenic函数与Clifford Mobius变换的复合可以得到一个加权的hypergenic函数.     

Clifford M\"{o}bius变换与Hypergenic函数

首先,在实Clifford代数空间Cl_n+1,0(R)中给出了与Clifford Mbius变换相关的一些定理.其次,证明了hypergenic函数与Clifford Mobius变换的复合可以得到一个加权的hypergenic函数.     

求首尾和r-循环矩阵的极小多项式的算法

研究了域上首尾和r-循环矩阵，利用多项式环的理想的Groebner基的算法给出了任意域上首尾和r-循环矩阵的极小多项式和公共极小多项式的一种算法.同时给出了这类矩阵逆矩阵的一种求法。     

四维Clifford代数的相似与合相似

两四元数相似当且仅当它们有相同的实部和模,两四元数合相似当且仅当它们有相同的模.应用四维Clifford代数的矩阵表示,得到了四维Clifford代数中两元素相似或合相似的充分必要条件.     

Clifford 代数,几何计算和几何推理

Clifford代数是一种深深根植于几何学之中的代数系统,被它的创始人称为几何代数．历史上,E．Cartan,R．Brauer,H．Weyl,C．Chevalley等数学大师都曾研究和应用过Clifford代数,对它的发展起了重要作用．近年来,Clifford代数在微分几何、理论物理、经典分析等方面取得了辉煌的成就,是现代理论数学和物理的一个核心工具,并在现代科技的各个领域,如机器人学、信号处理、计算机视觉、计算生物学、量子计算等方面有广泛的应用．本文主要介绍Clifford代数在几何计算和几何推理中的应用．作为一种优秀的描述和计算几何问题的代数语言,Clifford代数对于几何体,几何关系和几何变换有不依赖于坐标的、易于计算的多种表示,因而应用它进行几何自动推理,不仅使困难定理的证明往往变得极为简单,而且能够解决一些著名的公开问题,目前在国际上,几何自动推理已经成为Clifford代数的一个重要应用领域。     

GROEBNER BASES FOR CONSTRUCTIBLE SETS

ABSTRACT

A constructible set can be defined in terms of a unique sequence of varieties. Given a monomial ordering, the reduced Groebner bases of the ideals of these varieties comprise a complete invariant for the constructible set. Morphic images of varieties, such as orbits under algebraic group actions and projections of varieties, can therefore be assigned complete invariants.     

Complexity of multiplication in commutative group algebras over fields of characteristic 0

Let rkA denote the bilinear complexity (also known as rank) of a finite-dimension associative algebra A. Algebras of minimal rank are widely studied from the point of view of bilinear complexity. These are the algebras A for which the Alder-Strassen inequality is satisfied as an equality, i.e., rkA = 2dimA ? t, where t is the number of maximum two-sided ideals in A. It is proved in this work that an arbitrary commutative group algebra over a field of characteristic 0 is an algebra of minimal rank. The structure and precise values of the bilinear complexity of commutative group algebras over a field of rational numbers are obtained.     

A generalization of the rees theorem to a class of regular semigroups

Let S be a regular semigroup for which Green's relations J and D coincide, and which is max-principal in the sense that every element of S is contained in maximal principal right, left and two-sided ideals of S. A construction is given of a max-principal regular semigroup W with J=D, which is also principally separated in the sense that distinct maximal principal right (or left) ideals of S are disjoint, and an epimorphism ψ: W→S that preserves maximality of principal left, right, and two sided ideals, and is in a sense locally one-to-one. If S is completely simple, this construction reduces to the Rees matrix representation of S. The main result of this paper has its origin in an incorrect result contained in the author's doctoral dissertation which was written at the University of California (Berkeley) under Professor John Rhodes. This theorem was first established for finite regular semigroups in [1] (Corollary 2.3), and the present generalization of this result to infinite semigroups was suggested by Professor A. H. Clifford, who the author would like to thank for this as well as his generous encouragement and many helpful editorial suggestions. The author would also like to thank Professor Rhodes for his encouragement.     

Grassmann Inequalities and Extremal Varieties in $${\mathbb {P}}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) $$ P ⋀ p R n

Journal of Optimization Theory and Applications - In continuation of the work in Leventides and Petroulakis (Adv Appl Clifford Algebras 27:1503–1515, 2016), Leventides et al. (J Optim Theory...     

Mixed ladder determinantal varieties from two-sided ladders

We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Gröbner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay, and we characterize the arithmetically Gorenstein ones. Our main result consists in proving that mixed ladder determinantal varieties belong to the same G-biliaison class of a linear variety.     

Reducing systems of linear differential equations to a passive form

In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger's algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented.     

Groups in the class semigroups of valuation domains

It is shown that the isomorphy classes of the ideals of a valuation domain form a Clifford semigroup, and the structure of this semigroup is investigated. The group constituents of this Clifford semigroup are exactly the quotients of totally ordered complete abelian groups, modulo dense subgroups. A characterization of these groups is obtained, and some realization results are proved when the skeleton of the totally ordered group is given. The authors are members of GNSAGA of CNR. This research was supported by Ministero dell'Università e della Ricerca Scientifica e Tecnologica, Italy.     

Structure spinorielle associée à un espace vectoriel quaternionien à droite e sur H, muni d’une forme sesquilinéaire b non dégénérée H-anti hermitienne

This paper, self-contained, deals with the study of Clifford Algebras associated with n-dimensional skew-hermitian spaces over the skew field H. The different structures associated with the spaces S of corresponding spinors are given and the natural imbeddings of associated spinor groups are revealed.     

Courbure et torsion dans les algebres de Clifford II

Clifford Algebras generalize easily the concepts of curvature and torsion inR(p,q) (p,q are positive integers andp+q=n dimension of the space).      

Hartogs Extension Theorem for Functions with Values in Complex Clifford Algebras

A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic function, i.e. a null solution of the firstorder, vector-valued Dirac operator in . The isotonic Clifford analysis is a refinement of the latter, which arises for even dimension. As such it also may be regarded as an elegant generalization to complex Clifford algebra-valued functions of both holomorphic functions of several complex variables and two-sided biregular function theories. The aim of this article is to present a Hartogs theorem on isotonic extendability of functions on a suitable domain of . As an application, the extension problem for holomorphic functions and so for the two-sided biregular ones is discussed.