Efficient algorithms and applications of symbolic computation

符号计算的高效算法及应用

• 批准号: RGPIN-2015-06197
• 负责人: Zhang, Yang
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669736
• 关键词: Efficient; algorithms; applications; symbolic; computation

The overarching theme of this proposed program is the design, analysis and implementation of efficient algorithms for a number of important non-commutative algebraic structures in symbolic computation (also called computer algebra), such as Ore (skew) polynomials, polynomials over some non-commutative rings, generalized differential-difference algebras and matrices over these non-commutative algebras. Their study is motivated by many applications in coding theory, control theory, cryptography and engineering, etc.******The challenge in this area is that these non-commutative algebras generally have a much more complex structure when compared with commutative case.  In particular, many of the algorithmic breakthroughs in symbolic computation over the past three decades do not obviously apply in non-commutative domains. This is due both to the non-existence of particular mathematical properties in these rings or their inherently more difficult formulations. My research under this program will be to consider some of the most important computational problems in these non-commutative algebras at both a theoretical and practical level, which include fast algorithms for factoring Ore polynomials (both simple variable and multivariate cases), computing normal forms and inverses of matrices over Ore polynomial rings, and fast algorithms for computing Groebner bases in generalized differential-difference algebra and hope to achieve a unified and efficient approach to many well-known non-commutative domains. We also wish to give some applications, e.g.,  computing Gelfand-Kirillov dimensions,  and classifying Post-Lie algebras. The algorithmic advances of this proposal will be implemented in computer algebra software such as Maple, SAGE and Singular.  ****The proposed research plans all involve extensive training of highly qualified personnel (HQP) for their future positions in academia and industry.********

该提出的程序的总体主题是在符号计算中的许多重要非指数代数结构（也称为计算机代数）的设计，分析和实施，例如矿石（skew）多项式，多项方面的多项式，在某些非指挥性的差异范围内，总体上的差异差异和分数。他们的研究是由编码理论，控制理论，密码学和工程等的许多应用所激发的。特别是，在过去三十年中，符号计算中的许多算法突破显然不适用于非交通域中。这既是由于这些环中特定的数学特性不存在，或者它们本质上更困难的公式。 My research under this program will be to consider some of the most important computational problems in these non-commutative algebras at both a theoretical and practical level, which include fast algorithms for factoring Ore polynomials (both simple variable and multivariate cases), computing normal forms and inverses of materies over Ore polynomial rings, and fast algorithms for computing Groebner bases in generalized differential-difference代数和希望为许多众所周知的非交流领域实现统一和有效的方法。我们还希望提供一些应用程序，例如计算gelfand-kirilov尺寸，并分类LIE后代数。该提案的算法进步将在计算机代数软件（例如Maple，Sage和Singular）中实施。 ****拟议的研究计划均涉及对高素质人员（HQP）在学术界和工业中的未来职位进行广泛培训。****