A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications

We prove a splitting theorem for Alexandrov space of nonnegative curvature without properness assumption. As a corollary, we obtain a maximal radius theorem for Alexandrov spaces of curvature bounded from below by 1 without properness assumption. Also, we provide new examples of infinite dimensional Alexandrov spaces of nonnegative curvature.     

A volume convergence theorem for Alexandrov spaces with curvature bounded above

We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere. Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002     

A rigidity theorem in Alexandrov spaces with lower curvature bound

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.     

A fixed point theorem for affine mappings and its application to elasticity theory

In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.     

A glance at three-dimensional Alexandrov spaces

We discuss the topology and geometry of closed Alexandrov spaces of dimension three.     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

A ramsey theorem for trees,with an application to Banach spaces

LetS be the binary tree of all sequences of 0’s and 1’s. A chain ofS is any infinite linearly ordered subset. Letℋ be an analytic set of chains, we show that there exists a binary subtreeS’ ofS such that either all chains ofS’ lie inℋ or no chain ofS’ lies inℋ. As an application, we prove the following result on Banach spaces: If (x _s) _sɛs is a bounded sequence of elements in a Banach spaceE, there exists a subtreeS’ ofS such that for any chainβ ofS’ the sequence (x _s ) _{s ∈β} is either a weak Cauchy sequence or equivalent to the usuall ¹ basis.     

A common fixed point theorem for expansive mappings in 2-metric spaces and its application

In this paper, we establish a common fixed point theorem for expansive mappings by using the concept of compatibility of type (A) in 2-metric spaces of Cho [Cho Y. Fixed points for compatible mappings of type (A). Math Japonica 1993;38(3):497–508]. Our theorem generalizes a result of Kang et al. [Kang SM, Chang SS, Ryu JW. Common fixed points of expansion mappings. Math Japonica 1989;34(3):373–379]. Examples are given to support the generality of our result. Finally, we introduce an application of our main theorem to product spaces.     

A density theorem and its application

We compute the density of primes represented by a special quadratic form in a fixed square residue class. Using this result and a new method introduced by Thaine we prove the fact that for a prime p 3congruent to 3 modulo 4, the component e(p+1)/2of the p-Sylow subgroup of the ideal class group of Q(ζp) is trivial.     

A spectra comparison theorem and its applications

We give a sharp comparison between the spectra of two Riemannian manifolds (Y, g) and \((X,g_0)\) under the following assumptions: \((X,g_0)\) has bounded geometry, (Y, g) admits a continuous Gromov–Hausdorff \(\varepsilon \)-approximation onto \((X,g_0)\) of non zero absolute degree, and the volume of (Y, g) is almost smaller than the volume of \((X,g_0)\). These assumptions imply no restrictions on the local topology or geometry of (Y, g) in particular no curvature assumption is supposed or inferred.     

The comparison theorem for Hilbert spaces of entire functions

A generalization of the Sturm comparison theorem is obtained for formally self-adjoint ordinary differential operators of finite order given in canonical form. The result is stated within the vector theory of Hilbert spaces of entire functions when the coefficient space is a finite-dimensional vector space.     

A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems

In this article we present a fixed-point theorem of operator equations defined in cones of Banach spaces, where the spectral radius of the operator is involved. The operator is an S-type operator, a meaning introduced here. As an application of our main theorem, we get the existence of positive solutions to two boundary-value problems, extending several results from the literature.     

A Ramsey theorem for metric spaces

We show the Ramsey property of the metric spaces $(V,d)$ where $1\le d(x,y)\le 3$ ($x\ne y$).     

A decomposition theorem for convexity spaces

We utilize the unifying framework of families of convexity spaces for the treatment of various notions of planar convexity and the associated convex hulls. Our major goal is to prove the refinement and decomposition theorems for families of convexity spaces. These general theorems are then applied to two examples: restricted-oriented convex sets andNESW-convex sets. The applications demonstrate the usefulness of these general theorems, since they give rise to simple algorithms for the computation of the associated convex hulls of polygons.     

Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

We prove the compactness of the imbedding of the Sobolev space into for any relatively compact open subset of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally H?lder continuous heat kernel. Received: 27 December 1999 / in final form: 1 February 2000 / Published online: 4 May 2001