Isometries of spaces of compact or compact convex subsets of metric manifolds

The isometries with respect to the Hausdorff metric of spaces of compact or compact convex subsets of certain compact metric spaces are precisely the mappings generated by isometries of the underlying spaces. In particular this holds when the underlying space is a finite dimensional torus or a sphere in a finite dimensional strictly convex smooth normed space.     

Strong consistency of density estimation by orthogonal series methods for dependent variables with applications

Among several widely use methods of nonparametric density estimation is the technique of orthogonal series advocated by several authors. For such estimate when the observations are assumed to have been taken from strong mixing sequence in the sense of Rosenblatt [7] we study strong consistency by developing probability inequality for bounded strongly mixing random variables. The results obtained are then applied to two estimates of the functional Δ(f)=∫f ² (x)dx were strong consistency is established. One of the suggested two estimates of Δ(f) was recently studied by Schuler and Wolff [8] in the case of independent and identically distributed observations where they established consistency in the second mean of the estimate. Research supported in part by the National Research Council of Canada and in part by McMaster University Research Board. Now at Memphis State University, Memphis, Tennessee 38152, U.S.A.     

Hermitian metric rigidity on compact foliated manifolds

We prove a Hermitian metric rigidity theorem for leafwise symmetric Kaehler metrics on compact manifolds with smooth foliations. This provides applications to the study of the geometry of foliations as well as Kaehler manifolds that contain some symmetric geometry.     

Expanding maps on compact metric spaces

The notions of locally expansive, positively expansive, expanding in the sense of Ruelle and expanding in the sense of Duvall and Husch are equivalent in a quite general setting.     

Dimension of compact metric spaces

We give a survey of old and new results in dimension theory of compact metric spaces. Most of the relatively new results presented in the survey are based on the cohomological dimension approach. We complement the survey by stating the basics of cohomological dimension theory and listing some of its applications beyond the dimension theory.     

Strong density results in trace spaces of maps between manifolds

We deal with strong density results of smooth maps between two manifolds and in the fractional spaces given by the traces of Sobolev maps in W ^1,p .     

Positively expansive homeomorphisms on compact metric spaces

We give a new and simple proof to show that no homeomorphism of infinite compact metric spaces is positively expansive.     

Generic properties of compact metric spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.     

Weak-open compact images of metric spaces

The main results of this paper are that (1) a space X is g-developable if and only if it is a weak-open π image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.     

Images of locally compact metric spaces

We establish relationships between locally compact-metric spaces and all kinds of spaces with mk-system (compact k-network) by means of compact-covering mappings, quotient mapping and closed mappings. This revised version was published online in June 2006 with corrections to the Cover Date.     

Curvature of a Riemannian metric on compact homogeneous manifolds

Translated from Matematicheskie Zametki, Vol. 43, No. 2, pp. 169–177, February, 1988.     

Inequalities in metric spaces with applications

Analogues of the parallelogram identity and the (CN) inequality of Bruhat and Tits in uniformly convex metric spaces are established. As an application of the new inequalities, we prove two fixed point results for single-valued and multi-valued Lipschitzian mappings.     

Isometric coactions of compact quantum groups on compact quantum metric spaces

We propose a notion of isometric coaction of a compact quantum group on a compact quantum metric space in the framework of Rieffel, where the metric structure is given by a Lipnorm. Within this setting we study the problem of the existence of a quantum isometry group.     

Strong uniform consistency of nonparametric regression function estimates

Let (X, Y) be a ^dx-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X _i, Y_i), i=1,..., n. We establish conditions ensuring that an estimate of the form

Structure of homogeneous locally compact spaces with intrinsic metric

Omsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 1, pp. 23–34, January–February, 1989.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Open subspaces of locally compact metric spaces

Although classically every open subspace of a locally compact space is also locally compact, constructively this is not generally true. This paper provides a locally compact remetrization for an open set in a compact metric space and constructs a one-point compactification. MSC: 54D45, 03F60, 03F65.     

On similarity homogeneous locally compact spaces with intrinsic metric

In this article, we generalize partially the theorem of V. N. Berestovskii on characterization of similarity homogeneous (nonhomogeneous) Riemannian manifolds, i.e., Riemannian manifolds admitting transitive group of metric similarities other than motions to the case of locally compact similarity homogeneous (nonhomogeneous) spaces with intrinsic metric satisfying the additional assumption that the canonically conformally equivalent homogeneous space is δ-homogeneous or a space of curvature bounded below in the sense of A. D. Aleksandrov. Under the same assumptions, we prove the conjecture of V. N. Berestovskii on topological structure of such spaces.