The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct...The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.展开更多
Let R be a unique factorization domain (UFD). A method of Grbner bases and localization in commutative algebra is applied to compute an d analyze the characteristic ideals of semi-infinite linear recurring sequences (...Let R be a unique factorization domain (UFD). A method of Grbner bases and localization in commutative algebra is applied to compute an d analyze the characteristic ideals of semi-infinite linear recurring sequences (lrs), infinite linear recurring sequences (LRS), and finite lrs over UFD. The c an onical form of a minimal Grbner 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 membe r of the Grbner basis is derived.展开更多
In this paper we extend the theory of Grbner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module eq...In this paper we extend the theory of Grbner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for construct-ing these Grbner bases counterparts. To this aim we introduce the concept of "generalized term order" on Nm ×Zn and on difference-differential modules. Using Grbner bases on difference-differential mod-ules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations.展开更多
研究了NGP(nearly general stewart-gough platform)并联机构动平台位置与姿态变量之间的耦合关系,将9个变量中的6个用其余的3个表达出来,从而实现了位置变量和姿态变量的解耦.运用Gr bner基算法,得到了15个只含有其余3个变量的4次相容...研究了NGP(nearly general stewart-gough platform)并联机构动平台位置与姿态变量之间的耦合关系,将9个变量中的6个用其余的3个表达出来,从而实现了位置变量和姿态变量的解耦.运用Gr bner基算法,得到了15个只含有其余3个变量的4次相容方程.在此基础上,采用变量代换的方法消去其中的高次项,最终将NGP并联机构的运动学正解问题简化为求解一个一元20次的代数方程;这个方程是通过计算一个10阶行列式得出的,并且通过一个具体的算例验证了该方法的正确性.该方法适用于所有的NGP并联机构.展开更多
We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obn...We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.展开更多
基金supported by the grant LSS (Grant No. 344.2008.1)the SB RAS Integration Grant (GrantNo. 2006.1.9) (Russia)+1 种基金National Natural Science Foundation of China (Grant No. 10771077)Natural Science Foundation of Guangdong Province (Grant No. 06025062)
文摘The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.
文摘Let R be a unique factorization domain (UFD). A method of Grbner bases and localization in commutative algebra is applied to compute an d analyze the characteristic ideals of semi-infinite linear recurring sequences (lrs), infinite linear recurring sequences (LRS), and finite lrs over UFD. The c an onical form of a minimal Grbner 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 membe r of the Grbner basis is derived.
基金supported by the National Natural Science Foundation of China (Grant No. 60473019)the KLMM (Grant No. 0705)
文摘In this paper we extend the theory of Grbner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for construct-ing these Grbner bases counterparts. To this aim we introduce the concept of "generalized term order" on Nm ×Zn and on difference-differential modules. Using Grbner bases on difference-differential mod-ules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations.
文摘研究了NGP(nearly general stewart-gough platform)并联机构动平台位置与姿态变量之间的耦合关系,将9个变量中的6个用其余的3个表达出来,从而实现了位置变量和姿态变量的解耦.运用Gr bner基算法,得到了15个只含有其余3个变量的4次相容方程.在此基础上,采用变量代换的方法消去其中的高次项,最终将NGP并联机构的运动学正解问题简化为求解一个一元20次的代数方程;这个方程是通过计算一个10阶行列式得出的,并且通过一个具体的算例验证了该方法的正确性.该方法适用于所有的NGP并联机构.
基金supported by the Danish Council for Independent Research(Grant No.DFF–4002-00367),supported by the Danish Council for Independent Research(Grant No.DFF–6108-00362)supported by the Research Council of Norway(Project No.280731)supported by IRCC Award grant 12IRAWD009 from IIT Bombay
文摘We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.
基金partially supported by a National Science Foundation Grant(Grant No.ID:DMS-1601303)partially supported by a National Science Foundation Grant(Grant No.ID:DMS-1500966)+1 种基金Simons Foundation Collaboration Grant for MathematiciansSimons Fellowship
文摘This is a survey of some recent results on spherical tropical geometry.
