Curvature of curvilinear 4-webs and pencils of one forms: Variation on a theorem of Poincaré, Mayrhofer and Reidemeister

A curvilinear d-web W = (F ₁ , . . . , F _d ) is a configuration of d curvilinear foliations F _i on a surface. When d = 3, Bott connections of the normal bundles of F _i extend naturally to equal affine connection, which is called Chern connection. For 3 < d, this is the case if and only if the modulus of tangents to the leaves of F _i at a point is constant. A d-web is associative if the modulus is constant and weakly associative if Chern connections of all 3-subwebs have equal curvature form. We give a geometric interpretation of the curvature form in terms of fake billiard in §2, and prove that a weakly associative d-web is associative if Chern connections of triples of the members are non flat, and then the foliations are defined by members of a pencil (projective linear family of dim 1) of 1-forms. This result completes the classification of weakly associative 4-webs initiated by Poincaré, Mayrhofer and Reidemeister for the flat case. In §4, we generalize the result for n + 2-webs of n-spaces. Received: September 23, 1996     

透镜空间L~1(p)上向量丛的示性类

本文讨论了透镜空间 L1( p)的上同调环中生成元的运算性质 ,进而利用 L1( p)的 KO -结构得到了L1( p)上任一向量丛的全 Stiefel-Whitney类     

A Joint Integral Test for the Locations of Extrema for Brownian Motion

Let μ⁺(t) and μ⁻(t) be the locations of the maximum and minimum, respectively, of a standard Brownian motion in the interval [0,t]. We establish a joint integral test for the lower functions of μ⁺(t) and μ⁻(t), in the sense of Paul Lévy. In particular, it yields the law of the iterated logarithm for max(μ⁺(t),μ⁻(t)) as a straightforward consequence. Our result is in agreement with well-known theorems of Chung and Erdős [(1952) Trans. Amer. Math. Soc. 72, 179–186.], and Csáki, F?ldes and Révész [(1987) Prob. Theory Relat. Fields 76, 477–497].      

Sobolev-Type Classes of Functions with Values in a Metric Space. II

We prove equivalence of the definitions by the author and by Korevaar and Schoen of the Sobolev classes of mappings of a domain of an arithmetic n-dimensional space to a metric space.     

S[a,b]类中边界Nevanlinna-Pick插值(Ⅱ)

继续作者最近的研究,用所谓的Hankel向量方法建立S[a,b]函数类中带边界插值数据的Nevanlinna-Pick插值问题与[a,b]上的某种带约束条件的Hausdorff矩量问题之间解集之间明确的一一对应关系.通过N[a,b]函数类与S[a,b]函数类之间的联系,从而由BNP(N[a,b])问题的可解性准则和解的参数化描述获得BNP(S[a,b)问题的可解性准则和解的参数化表示.     

Space-Time Topology (II)—Causality, the Fourth Stiefel–Whitney Class and Space-Time as a Boundary

We show that stable causality is related to the vanishing of the top Stiefel–Whitney class of a space-time manifold M, and that if M is a stably causal space-time manifold, then it is the boundary of a five-dimensional space-time. We then propose a scheme for making it both a necessary and sufficient condition.     

Sub-bundles of the complexified tangent bundle

We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.

     

Stone代数的Cayley表示定理

首先通过集代数得到了Stone代数的表示定理,然后证明了每一个Stone代数均嵌入到某个集合X上的一个Stone映射类S中.