Embeddings of

We show that there is only one embedding of in at the prime , up to self-maps of . We also describe the effect of the group of self-equivalences of at the prime on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of whose homotopy fixed point set coincide with

     

Discretely Compact Embeddings of Spaces of Cell-Centered Grid Functions

Compactness of embeddings in discrete counterparts of Sobolev spaces is considered. We study the embeddings in spaces of cell-centered grid functions, in one- and two-dimensional domains. No restrictions are made on the mesh-ratios of the underlying meshes.     

无穷维欧氏空间中球体及球面的Lebesgue测度

本分别给出了使在无穷维欧氏空间中球体和球面具有有限的,但又不是无穷小的测度的半径集合。     

Embeddings of self-similar ultrametric Cantor sets

We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R[Hdim(C)]+1, where [Hdim(C)] denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural sequence of refining coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of Rd is bi-Lipschitz embeddable in Rd+1.We also show that C is bi-Hölder embeddable in the real line. The image of C in R turns out to be the ω-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry.     

哈密顿系统的低维环面的保存性

本文给出了关于哈密顿系统低维环面的一个推广的KAM定理,它适用于同时存在法向频率和双曲法向分量的情况.其证明基于尤建功的一个定理的光滑性表述及法向双曲不变流形理论的应用.文中还给出了另外两种情况下的推广.     

关于伪欧氏空间

本文提出了一些新概念，ｐ—伪转置阵，ｐ—伪正交阵，ｐ—伪对称阵，从而能以这些矩阵为工具研究伪欧氏空间的性质以及空间中两个特殊线性变换，伪正交变换和伪对称变换．本文得到的主要结果是定理２，定理３，定理４及定理６．本文还指出了北大编《高等代数》第二版中的伪正交变换习题的一个错误．     

Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

Bi-Lipschitz Embeddings of Trees into Euclidean Buildings

Let be the affine Bruhat-Tits (or Euclidean) building associated to a semisimple, simply connected linear algebraic group defined over a non-Archimedean local field. We prove that there exist locally finite trees of degree 3 which are bi-Lipschitz embedded in . As a consequence such a Euclidean building cannot be bi-Lipschitz embedded into a Hilbert space.     

The Euclidean dimension of the join of two cycles

The Euclidean dimension of a graph G is the smallest integer p such that the vertices of G can be represented by points in the Euclidean space R^p with two points being 1 unit distance apart if and only if they represent adjacent vertices. We show that dim(C_m+C_n)=5 except that dim(C₄+C₄)=4, dim(C₅+C₅)=4, and dim(C₆+C₆)=6.     

Hamiltonian Stationary Tori in the Complex Projective Plane

Hamiltonian stationary Lagrangian surfaces are Lagrangian surfacesin a four-dimensional Kähler manifold which are criticalpoints of the area functional for Hamiltonian infinitesimaldeformations. In this paper we analyze these surfaces in thecomplex projective plane: in a previous work we showed thatthey correspond locally to solutions to an integrable system,formulated as a zero curvature on a (twisted) loop group. Herewe give an alternative formulation, using non-twisted loop groupsand, as an application, we show in detail why Hamiltonian stationaryLagrangian tori are finite type solutions, and eventually describethe simplest of them: the homogeneous ones. 2000 MathematicsSubject Classification 53C55 (primary), 53C42, 53C25, 58E12(secondary).     

On extremal point distributions in the Euclidean plane

We ask for the maximum σ _n ^γ of Σ _i,j=1 ⁿ ‖x _i-x _j‖^γ, where x ₁,χ,x _n are points in the Euclidean plane R ² with ‖x_i-x_j‖ ≦1 for all 1≦ i,j ≦ n and where ‖.‖^γ denotes the γ-th power of the Euclidean norm, γ ≧ 1. (For γ =1 this question was stated by L. Fejes Tóth in [1].) We calculate the exact value of σ _n ^γ for all γ γ 1,0758χ and give the distributions which attain the maximum σ _n ^γ . Moreover we prove upper bounds for σ _n ^γ for all γ ≧ 1 and calculate the exact value of σ ₄ ^γ for all γ ≧ 1. This revised version was published online in June 2006 with corrections to the Cover Date.     

论向量积在高维空间的推广

探讨了3维欧氏空间的向量积在高维空间推广的可能性.在三条假设下推导出高维空间中的向量积所应具备的若干性质,证明了在某些意义下向量积仅存在于R3,并举例说明R7与R3中向量积不同的根源.     

On Lower Dimensional Invariant Tori in C^d Reversible Systems

In this paper, a result on the persistence of lower dimensional invariant tori in Cd reversible systems is obtained under some conditions. The theorem is proved for any d which is larger than some constants.     

On Lower Dimensional Invariant Toriin C~d Reversible Systems

In this paper, a result on the persistence of lower dimensional invariant tori in Cd reversible systems is obtained under some conditions. The theorem is proved for any d which is larger than some constants.     

Carnot-Caratheodory空间中的变换论

Klein发表著名的埃尔兰根纲领,由群论角度研究了空间变换群的不变量,从而引进了各种不同的几何学．本文利用Felix Klein的观念,研究Carnot-Caratheodory空间{M,Q,g}(又称为次黎曼流形)上的类似问题,给出了次黎曼流形中的共形不变量和射影不变量．本文给出的共形和射影不变量可视为黎曼情形的一种自然推广．由于次黎曼流形与黎曼流形之间有着本质的差异,故此,本文通过次黎曼流形上存在的唯一非完整联络(Nonholonomic connections)来刻画所提的问题．     

On Vitali's Theorem for Groups of Motions of Euclidean Space

We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [1] holds true. This characterization is formulated in purely geometrical terms.     

紧致对称空间的热核、特征值和特征函数

本文利用群表示论研究李群以及紧致对称空间的热核，特征值与特征函数．特别讨论了复格拉斯曼流形以及流形Sp（n）／U（n）上特征值及特征函数。     

紧致黎曼4-对称空间的Cartan嵌入

本文首先给出齐性空间的Ｃａｒｔａｎ嵌入的概念,然后利用李群、李代数的自同构理论,决定了使其Ｃａｒｔａｎ嵌入是极小的那些紧致黎曼４-对称空间．     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .