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Acta Mathematica Scientia 2009年第3期29卷 697-704页

作者：李方段海豹 Department of Mathematics Jilin UniversityChangchun 130024 China Institute of Mathematics & Key Laboratory of Mathematics Mechanization Chinese Academy of SciencesBeijing 100190 China

Applying the theory of Grbner basis to the Schubert presentation for the cohomology of grassmannians [2], we extend the homology rigidity results known for the classical Grassmanians to the exceptional cases.

关键词： grassmannians cohomology

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Feature Patch Illumination Spaces and Karcher Compression for Face Recognition via grassmannians

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Advances in Pure Mathematics 2012年第4期2卷 226-242页

作者： Jen-Mei Chang Chris Peterson Michael Kirby Department of Mathematics Colorado State University Fort Collins USA Departmentof Mathematics and Statistics California State University Long Beach USA

Recent work has established that digital images of a human face, when collected with a fixed pose but under a variety of illumination conditions, possess discriminatory information that can be used in classification. In this paper we perform classification on grassmannians to demonstrate that sufficient discriminatory information persists in feature patch (e.g., nose or eye patch) illumination spaces. We further employ the use of Karcher mean on the grassmannians to demonstrate that this compressed representation can accelerate computations with relatively minor sacrifice on performance. The combination of these two ideas introduces a novel perspective in performing face recognition.

关键词： grassmannians Karcher Mean Face Recognition Illumination Spaces Compressions Feature Patches Principal Angles

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Real Hypersurfaces in Complex Two-Plane grassmannians Whose Jacobi Operators Corresponding to -Directions are of Codazzi Type

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Advances in Pure Mathematics 2011年第3期1卷 67-72页

作者： Carlos J. G. Machado Juan de Dios Pérez Young Jin Suh 不详

We prove the non-existence of Hopf real hypersurfaces in complex two-plane grassmannians whose Jacobi operators or the Jacobi corresponding to the directions in the distribution are of Codazzi type if they satisfy a further condition. We obtain that that they must be either of type (A) or of type (B) (see [2]), but no one of these satisfies our condition. As a consequence, we obtain the non-existence of Hopf real hypersurfaces in such ambient spaces whose Jacobi operators corresponding to -directions are parallel with the same further condition.

关键词： Real Hypersurfaces Complex Two-Plane grassmannians Jacobi Operators Codazzi Type

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Schubert Class and Cyclotomic NilHecke Algebras

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Algebra Colloquium 2021年第3期28卷 379-398页

作者： Kai Zhou Jun Hu School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081China

Let l and n be positive integers such that l≥n,and let Gn,l be the Grassmannian which consists of the set of n-dimensionsil subspaces of C^(l),There is a Z-graded algebra isomorphism between the cohomology H*(Gn,l,Z)of Gn,l and a natural Z-form B of the Z-graded basic algebra of the type A cyclotomic nilHecke algebraH^(0)l,n=.We show that the isomorphism can be chosen such that the image of each(geometrically defined)Schubert class(a1,...,an)coincides with the basis element bλconstructed by Hu and Liang by purely algebraic method,where 0≤q1≤q2≤…≤an≤l-n with ai∈Z for each i,andλis the l-€-multipartition of n associated to(l+1-(an+n),l+1-(an-1+n-),...,l+1-(a1+1)).A similar correspondence between the Schubert class basis of the cohomology of the Grassmanni-an Gl-n,l and the bλ's basis(λis anl-multipartition of n with each component being either(1)or empty)of the natural Z-form B of the Z-graded basic algebra of H^(0)_(l,n)is also *** an application,we obtain a second version of the Giambelli formula for Schubert classes.

关键词： cyclotomic nilHecke algebras grassmannians Schur polynomials

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