We observe that the compacta hyperspace K(X) of any separable, uniformly. The main difference between the adaptive landscape and the theoretical morphospace lies in their dimensions. Schori and West proved that K() is homeomorphic with the Hilbert cube, while Hohti showed that K() is not bi-Lipschitz equivalent with a variety of metric Hilbert cubes. We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. Theoretical morphospaces may be defined most explicitly, if a bit tersely, as ‘ n -dimensional geometric hyperspaces produced by systematically varying the parameter values of a geometric model of form’ (McGhee, 1991, p. By way of contrast, the hyperspace K() of the unit interval contains a bi-Lipschitz copy of a certain self-similar doubling series-parallel graph studied by Laakso, Lang-Plaut, and Lee-Mendel-Naor, and consequently admits no bi-Lipschitz embedding into any uniformly convex Banach space. How to cite abstract = $ in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. Schori and West proved that K() is homeomorphic with the Hilbert cube, while Hohti showed that K() is not bi-Lipschitz equivalent with a variety of metric Hilbert cubes. Some of these notions were introduced and considered in 9, 10 and 11, focussing 7 PDF View 2 excerpts, cites background Selection principles, -sets and i-properties in ech closure spaces M. If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in ℝ n + 1 in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. We study some topological properties of hyperspaces of Cech closure spaces endowed with Vietoris-like topologies. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ². The following class of mappings is central in modern geometric function theory 16. We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. The analogous results for self-conformal sets that satisfy the Open Set Condition are developed in Chapter 4.Bi-Lipschitz embeddings of hyperspaces of compact sets In Chapter 3, a one-parameter family of gauge functions is constructed which computes the dimensions of the hyperspaces of graph-self-similar sets that satisfy the Strong Separation Condition, after which the approximations of Chapter 2 are applied to extend the result to graph-self-similar sets which satisfy the Open Set Condition. In Chapter 2 it is shown that the dimensions of the underlying fractals may be approximated by the dimensions of sets invariant under particularly constructed subiterated function systems that satisfy the Strong Separation Condition. This dissertation further generalizes these results to include graph-self-similar and self-conformal fractals satisfying the Open Set Condition in Rd. Hyperspaces have been extensively studied by topologists since the 1970's, but the measure theoretical study of hyperspaces has lagged, Boardman and Goodey concurrently provided a characterization of a one-parameter family of Hausdorff gauge functions that determine the dimension of H(), and this result was extended by McClure to H(X) where X is a self-similar fractal satisfying the Open Set Condition. H(K) is itself a metric space under the Hausdorff metric dH. Given a metric space (K, d), the hyperspace of K is defined by H(K) =.
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