For compact and connected Lie group G with a maximal torus T the quotient space G/T is canonically a smooth projective manifold,known as the complete flag manifold of the group G.The cohomology ring map c^(∗):H^(∗)(B...For compact and connected Lie group G with a maximal torus T the quotient space G/T is canonically a smooth projective manifold,known as the complete flag manifold of the group G.The cohomology ring map c^(∗):H^(∗)(B T)→H∗(G/T)induced by the inclusion c:G/T→B_(T) is called the Borel’s characteristic map of the group G[7,8],where B T denotes the classifying space of T.Let G be simply-connected and simple.Based on the Schubert presentation of the cohomology H^(∗)(G/T)of the flag manifold G/T obtained in[10,11],we develop a method to find a basic set of explicit generators for the kernel ker c^(∗)⊂H^(∗)(B_(T))of the characteristic map c.展开更多
希尔伯特(Hilbert)第15问题要求,为舒伯特(Schubert)计数演算法建立严格基础.其中,长期悬而未解的部分是舒伯特特征数问题.在构建代数几何学的工作中,范·德·瓦尔登(van der Waerden)和韦伊(Weil)将经典的舒伯特演算归结于决...希尔伯特(Hilbert)第15问题要求,为舒伯特(Schubert)计数演算法建立严格基础.其中,长期悬而未解的部分是舒伯特特征数问题.在构建代数几何学的工作中,范·德·瓦尔登(van der Waerden)和韦伊(Weil)将经典的舒伯特演算归结于决定旗流形相交理论的问题.本文介绍第15问题的背景、内容以及解答历程.本文的重点是计算特征数的统一公式(定理5.1)以及解答韦伊问题的系统表述(定理5.2).本文通过应用示例5.1和5.2,说明该公式和算法的有效性.展开更多
We develop two parallel algorithms progressively based on C＋＋ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of ...We develop two parallel algorithms progressively based on C＋＋ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.展开更多
基金This work is Supported by Natural Science Foundation of China(Grant Nos.11771427,11961131004,12071309).
文摘For compact and connected Lie group G with a maximal torus T the quotient space G/T is canonically a smooth projective manifold,known as the complete flag manifold of the group G.The cohomology ring map c^(∗):H^(∗)(B T)→H∗(G/T)induced by the inclusion c:G/T→B_(T) is called the Borel’s characteristic map of the group G[7,8],where B T denotes the classifying space of T.Let G be simply-connected and simple.Based on the Schubert presentation of the cohomology H^(∗)(G/T)of the flag manifold G/T obtained in[10,11],we develop a method to find a basic set of explicit generators for the kernel ker c^(∗)⊂H^(∗)(B_(T))of the characteristic map c.
文摘希尔伯特(Hilbert)第15问题要求,为舒伯特(Schubert)计数演算法建立严格基础.其中,长期悬而未解的部分是舒伯特特征数问题.在构建代数几何学的工作中,范·德·瓦尔登(van der Waerden)和韦伊(Weil)将经典的舒伯特演算归结于决定旗流形相交理论的问题.本文介绍第15问题的背景、内容以及解答历程.本文的重点是计算特征数的统一公式(定理5.1)以及解答韦伊问题的系统表述(定理5.2).本文通过应用示例5.1和5.2,说明该公式和算法的有效性.
基金The authors sincerely appreciate the referees for acknowledging the manuscript and providing valuable comments and suggestions that benefit their manuscript. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11131008, 11271157, 11201453, 11471141）, the 973 Program （2011CB302400）, the Open Project Program of the State Key Lab of CAD＆CG （A1302） of Zhejiang University, and the Scientific Research Foundation for Returned Scholars, Ministry of Education of China. They also wish to thank the High Performance Computing Center of Jilin University and Computing Center of Jilin Province for essential computing support.
文摘We develop two parallel algorithms progressively based on C＋＋ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.
