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. ...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.展开更多
In this paper, we consider a new notion of generalized Tanaka Webster З-parallel shape operator for a real hypersurface in a complex two-plane Grassrnannian and prove a non-existence theorem of a real hypersurface.
Applying the theory of Grbner 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.
We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.
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 f...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.展开更多
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)...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=<ψ1,…ψn-1,y1,…,yn>.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 obtained.As an application,we obtain a second version of the Giambelli formula for Schubert classes.展开更多
In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coo...In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coordinates,and it is proved that such equations generate the defining ideal of variety of type C in those of type A.As applications of this result,the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed,and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections.Finally,the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed,filling gaps in the study of algebraic varieties of the same type.展开更多
We give a proof of an explicit formula for affine coodinates of points in the Sato’s infinite Grassmannian corresponding to tau-functions for the KdV hierarchy.
Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are o...Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition.In this work,we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes.These are symmetric matrices of metric trees satisfying compatibility conditions.We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections.As applications of the first,we find all 693,13612 and 346710 collections for(k,n)=(3,7),(3,8)and(3,9),respectively.As applications of the second kind,we find all90608 and 30659424 planar matrices that compute(k,n)=(4,8)and(4,9)biadjoint amplitudes,respectively.As an example of the evaluation of matrices of Feynman diagrams,we present the complete form of the(4,8)and(4,9)biadjoint amplitudes.We also start a study of higher-dimensional arrays of Feynman diagrams,including the combinatorial version of the duality between(k,n)and(n-k,n)objects.展开更多
In this article, a Grassmannian precoding multiuser multiple-input multiple-output （MU-MIMO） scheme for downlink transmission is proposed. The proposed MU-MIMO scheme will perform scheduling and precoding simultaneo...In this article, a Grassmannian precoding multiuser multiple-input multiple-output （MU-MIMO） scheme for downlink transmission is proposed. The proposed MU-MIMO scheme will perform scheduling and precoding simultaneously at the base station, to obtain both the multiuser diversity gain and the precoding gain, to maximize the system capacity. The precoding method is related to Grassmannian precoding, which extends the point-to-point single-user Grassmannian precoding to point-to-multipoint multiuser Grassmannian precoding. It provides further significant system capacity enhancement than the single user MIMO （SU-MIMO） system and also outperforms the block dia^onalization （BD） algorithm under the same simulation environment.展开更多
Given a harmonic map φfrom a Riemann surface M into the complex projectivespace CP^n, by using the -transform associated to the map φ, Chern et al. obtained the se-quence of harmonic maps. We call it a harmonic sequ...Given a harmonic map φfrom a Riemann surface M into the complex projectivespace CP^n, by using the -transform associated to the map φ, Chern et al. obtained the se-quence of harmonic maps. We call it a harmonic sequence. We will say that a harmonicmap φ: M→CP^n is k-orthogonal if k consecutive maps in the harmonic sequence of φ aremutually orthogonal. In particular, φis conformal if and only if φ is 3-orthogonal and, inall cases, φ is at most (n+1)-orthogonal. There are two possible ways in which the展开更多
We construct explicit Morse functions on Grassmannian manifolds, and use them to find explicit taut embeddings of the quadratic hypersurfaces in complex projective spaces into Euclidean spaces.
Further geometry and topology for pseudo-holomorphic curves in complex Grassmannians Gm(CN) are studied. Some curvature pinching theorems for pseudo- holomorphic curyes with constant Kahler angles in Gm(CN) are obtain...Further geometry and topology for pseudo-holomorphic curves in complex Grassmannians Gm(CN) are studied. Some curvature pinching theorems for pseudo- holomorphic curyes with constant Kahler angles in Gm(CN) are obtained, so that the corresponding results for pseudoholomorphic curves in complex projective spaces are generalized.展开更多
We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space Λ n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) ?Λ n (R n+p), and use the local...We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space Λ n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) ?Λ n (R n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG n (R n+p). As a consequence of this formula we get a rigidity theorem.展开更多
Let M be an oriented surface and G(2,k) be the Grassmannian.Smooth maps t1 M→G2（2，k) are studied to determine whether or not they are Gauss maps.Some new results have been obtained and some known results reproved.
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature,and in which rational homogeneous spaces play a prominent role.This selection i...This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature,and in which rational homogeneous spaces play a prominent role.This selection is largely arbitrary and mainly refiects the interests of the author.展开更多
文摘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.
基金supported by National Research Foundation of Korea(NRF)(Grant Nos.2012-R1A1A3002031 and 2015-R1A2A1A-01002459)supported by KNU 2015(Bokhyun)Research Fund
文摘In this paper, we consider a new notion of generalized Tanaka Webster З-parallel shape operator for a real hypersurface in a complex two-plane Grassrnannian and prove a non-existence theorem of a real hypersurface.
文摘Applying the theory of Grbner 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.
基金Supported by National Research Foundation of Korea(Grant No.NRF-2011-220-1-C00002)partially supported by MCT(Grant No.MTM2010-18099)supported by NRF(Grant No.NRF-2012-R1A2A2A-01043023)
文摘We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.
文摘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.
基金The research was supported by the National Natural Science Foundation of China(No.11525102).
文摘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=<ψ1,…ψn-1,y1,…,yn>.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 obtained.As an application,we obtain a second version of the Giambelli formula for Schubert classes.
文摘In this paper,a description of the set-theoretical defining equations of symplectic(type C)Grassmannian/flag/Schubert varieties in corresponding(type A)algebraic varieties is given as linear polynomials in Plucker coordinates,and it is proved that such equations generate the defining ideal of variety of type C in those of type A.As applications of this result,the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed,and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections.Finally,the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed,filling gaps in the study of algebraic varieties of the same type.
文摘We give a proof of an explicit formula for affine coodinates of points in the Sato’s infinite Grassmannian corresponding to tau-functions for the KdV hierarchy.
基金supported in part by the Government of Canada through the Department of Innovation,Science and Economic Development Canadaby the Province of Ontario through the Ministry of Economic Development,Job Creation and Trade。
文摘Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition.In this work,we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes.These are symmetric matrices of metric trees satisfying compatibility conditions.We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections.As applications of the first,we find all 693,13612 and 346710 collections for(k,n)=(3,7),(3,8)and(3,9),respectively.As applications of the second kind,we find all90608 and 30659424 planar matrices that compute(k,n)=(4,8)and(4,9)biadjoint amplitudes,respectively.As an example of the evaluation of matrices of Feynman diagrams,we present the complete form of the(4,8)and(4,9)biadjoint amplitudes.We also start a study of higher-dimensional arrays of Feynman diagrams,including the combinatorial version of the duality between(k,n)and(n-k,n)objects.
基金the National Natural Science Foundation of China(60702051)the Hi-Tech Research and Development Program of China(2007AA01Z261)the Fujitsu‘the Research of Multiuser MIMO Precoding Technique’(K0703001)
文摘In this article, a Grassmannian precoding multiuser multiple-input multiple-output （MU-MIMO） scheme for downlink transmission is proposed. The proposed MU-MIMO scheme will perform scheduling and precoding simultaneously at the base station, to obtain both the multiuser diversity gain and the precoding gain, to maximize the system capacity. The precoding method is related to Grassmannian precoding, which extends the point-to-point single-user Grassmannian precoding to point-to-multipoint multiuser Grassmannian precoding. It provides further significant system capacity enhancement than the single user MIMO （SU-MIMO） system and also outperforms the block dia^onalization （BD） algorithm under the same simulation environment.
文摘Given a harmonic map φfrom a Riemann surface M into the complex projectivespace CP^n, by using the -transform associated to the map φ, Chern et al. obtained the se-quence of harmonic maps. We call it a harmonic sequence. We will say that a harmonicmap φ: M→CP^n is k-orthogonal if k consecutive maps in the harmonic sequence of φ aremutually orthogonal. In particular, φis conformal if and only if φ is 3-orthogonal and, inall cases, φ is at most (n+1)-orthogonal. There are two possible ways in which the
文摘We construct explicit Morse functions on Grassmannian manifolds, and use them to find explicit taut embeddings of the quadratic hypersurfaces in complex projective spaces into Euclidean spaces.
文摘Further geometry and topology for pseudo-holomorphic curves in complex Grassmannians Gm(CN) are studied. Some curvature pinching theorems for pseudo- holomorphic curyes with constant Kahler angles in Gm(CN) are obtained, so that the corresponding results for pseudoholomorphic curves in complex projective spaces are generalized.
文摘We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space Λ n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) ?Λ n (R n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG n (R n+p). As a consequence of this formula we get a rigidity theorem.
文摘Let M be an oriented surface and G(2,k) be the Grassmannian.Smooth maps t1 M→G2（2，k) are studied to determine whether or not they are Gauss maps.Some new results have been obtained and some known results reproved.
文摘This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature,and in which rational homogeneous spaces play a prominent role.This selection is largely arbitrary and mainly refiects the interests of the author.
