Ying Guang Shi(1995 & 1999) obtained some quadratures, which is based onthe zeros of the so-called s-orthogonal polynomials with respect to some JacobiB.Bojanov(1996) and our recent work, we give here a simple and...Ying Guang Shi(1995 & 1999) obtained some quadratures, which is based onthe zeros of the so-called s-orthogonal polynomials with respect to some JacobiB.Bojanov(1996) and our recent work, we give here a simple and unified approachto these questions of this type and obtain quadratures in terms of the divided differ-ences, which is based on an appropriate representation of the Hermite interpolatingpolynomial, of corresponding function at the zeros of the appropriate s-orthogonalpolynomial with multiplicities.展开更多
Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbne...Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a GrSbnerbasis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to GrSbner basis theory are reexamined and/or further developed for the parametric case: (i) Definition of a comprehensive Groebner basis, (ii) test for a comprehensive GrSbner basis, (iii) parameterized rewriting, (iv) S-polynomials among parametric polynomials, (v) completion algorithm for directly computing a comprehensive Groebner basis from a given basis of a parametric ideal. Elegant properties of Groebner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique GrSbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Groebner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.展开更多
In this paper, we study(n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a ...In this paper, we study(n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples.展开更多
文摘Ying Guang Shi(1995 & 1999) obtained some quadratures, which is based onthe zeros of the so-called s-orthogonal polynomials with respect to some JacobiB.Bojanov(1996) and our recent work, we give here a simple and unified approachto these questions of this type and obtain quadratures in terms of the divided differ-ences, which is based on an appropriate representation of the Hermite interpolatingpolynomial, of corresponding function at the zeros of the appropriate s-orthogonalpolynomial with multiplicities.
基金supported by the National Science Foundation under Grant No.DMS-1217054
文摘Groebner basis theory for parametric polynomial ideals is explored with the main objec- tive of nfinicking the Groebner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive GrSbner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a GrSbnerbasis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to GrSbner basis theory are reexamined and/or further developed for the parametric case: (i) Definition of a comprehensive Groebner basis, (ii) test for a comprehensive GrSbner basis, (iii) parameterized rewriting, (iv) S-polynomials among parametric polynomials, (v) completion algorithm for directly computing a comprehensive Groebner basis from a given basis of a parametric ideal. Elegant properties of Groebner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique GrSbner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Groebner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.
基金Supported by National Natural Science Foundation of China under Grant Nos.11471139,11271202,11221091,11425104Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20120031110022National Natural Science Foundation of Jilin Province under Grant No.20140520054JH
文摘In this paper, we study(n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples.