It is shown that every bi-Lipschitz bijection from Z to itself is at a bounded L1 distance from either the identity or the reflection.We then comment on the group-theoretic properties of the action of bi-Lipschitz bijections.
We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ². If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in ; in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace K([0,1]) of the...
When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small ("snowflake") deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, like rectifiability, differentiability, and curves of finite length. Here we discuss a (somewhat technical)...
We prove that every compact C1-submanifold of Rn, with or without boundary, has the bilipschitz extension property in Rn.
A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. The category is balanced if every bimorphism is an isomorphism. In the paper properties of bimorphisms of several categories are discussed (pro-homotopy, shape, proper homotopy) and the question of those categories being balanced is raised. Our most interesting result is that a bimorphism f:X → Y of is an isomorphism if Y is movable. Recall that is the full subcategory of consisting of...
Let X and Y be compacta and let f:X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map such that dim (f × g) = 0. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map such that dim (f × g) = 1. We improve this result of Sternfeld showing...
We study binormality, a separation property of spaces endowed with two topologies known in the real analysis as the Luzin-Menchoff property. The main object of our interest are Banach spaces with their norm and weak topologies. We show that every separable Banach space is binormal and the space is not binormal.
Let ϕ :(M,F)→ (N,h) be a harmonic map from a Finsler manifold to any Riemannian manifold. We establish Bochner's formula for the energy density of ϕ and maximum principle on Finsler manifolds, from which we deduce some properties of harmonic maps ϕ.
We prove the following theorem: Given a⊆ω and , if for some and all u ∈ WO of length η, a is , then a is .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: -Turing-determinacy implies the existence of .