Atsuji completions: Equivalent characterisations

A metric space (X,d) is called an Atsuji space if every real-valued continuous function on (X,d) is uniformly continuous. It is well known that an Atsuji space must be complete. A metric space (X,d) is said to have an Atsuji completion if its completion is an Atsuji space. In this paper, we study twenty-nine equivalent characterisations for a metric space to have an Atsuji completion.     

On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

We consider tensors T=fg on the pseudo-euclidean space Rn and on the hyperbolic space Hn, where n?3, g is the standard metric and f is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric , conformal to g, such that , and the existence of such a metric which satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability and we construct the solutions when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics , defined on Rn or Hn, are complete. As a consequence of these results, we get positive solutions for the equation , where g is the pseudo-euclidean metric.     

On continuous choice of retractions onto nonconvex subsets

For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate can be improved to and the constant can be replaced by the root of the equation α+α²+α³=1.     

Estimates of weighted Hardy-Littlewood averages on the p-adic vector space

In the p-adic vector space , we characterize those non-negative functions ψ defined on for which the weighted Hardy-Littlewood average is bounded on (1?r?∞), and on . Also, in each case, we find the corresponding operator norm ‖Uψ‖.     

Barycentric selectors and a Steiner-type point of a convex body in a Banach space

Let F be a mapping from a metric space into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping satisfying . The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector S_X^(m) defined on the family of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set.     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².     

The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

Approximation of the Hausdorff distance by the distance of continuous surjections

The aim of this paper is to answer the following question: let (X,?) and (Y,d) be metric spaces, let A,B⊂Y be continuous images of the space X and let be a fixed continuous surjection. When is the inequality

     

Two results on spaces with a sharp base

Example. There exists a space X with a sharp base and a perfect mapping onto a space Y which does not have a sharp base.     

Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces

In this paper, for a commuting pair consisting of a point-valued nonexpansive self-mapping t and a set-valued nonexpansive self-mapping T of a hyperconvex metric space (or a CAT(0) space) X, we look for a solution to the problem of existence of z∈E⊂X such that

     

Computing the minimal number of equations defining an affine curve ideal-theoretically

There is an algorithm which computes the minimal number of generators of the ideal of a reduced curve C in affine n-space over an algebraically closed field K, provided C is not a local complete intersection.The existence of such an algorithm follows from the fact that given , there exists , such that if is a height n−1 radical ideal in K[X₁,…,X_n], generated by polynomials of degree at most d, then admits a set of generators of minimal cardinality, with each generator having degree at most d′, except possibly when is an (unmixed) local complete intersection.     

Mappings and Prohorov spaces

We consider completely regular Hausdorff spaces. In this paper we investigate the space of probability Radon measures P(X) on a space X and the property to be a Prohorov space. We prove that the space P(X) is sieve-complete if and only if X is sieve-complete. Every mapping generates the mapping . Some properties of the mapping P(φ) are studied. In particular, we investigate under which conditions the open continuous image of a Prohorov space is Prohorov.     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety of ordered semigroups, let be the class of sofic subshifts whose syntactic semigroup lies in . It is proved that if contains then is closed under taking shift equivalent subshifts, and conversely, if is closed under taking conjugate subshifts then contains and . Almost finite type subshifts are characterized as the irreducible elements of , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.     

Inducing sensitivity on hyperspaces

Let (X,d) be a compact metric space and (K(X),dH) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric dH. The dynamical system (X,f) induces another dynamical system . We study the relations between the various forms of sensitivity of the systems (X,f) and . We prove that all forms of sensitivity of partly imply the same for (X,f), and the converse holds in some cases.