Magill's and Rayburn's theorems on the homeomorphism of Stone-ˇ Cech remainders and some of their generalizations to the remainders of arbitrary Hausdorff compactifications of Tychonoff spaces are extended to some class of mappings.

In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff metric. This is the first part of our series work, which we try to establish some facts about the counterpart of Gromov-Hausdorff metric in the non-Archimedean spaces. One of the motivation of this work is to find some implied relations between this geometry and number theory via p-adic analysis, so that we can use the for...

The question of how complicated a critical set must be to have a nonnull image is answered by relating its Hausdorff dimension to the (Holder) differentiability of the map. This leads to a new extension of the Morse-Sard Theorem. The main tool is an extended version of Morse's Lemma.

We call a Gromov-Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. In this paper, we prove that any Ricci limit space has integral Hausdorff dimension provided that its Hausdorff dimension is not greater than two. We also classify one-dimensional Ricci limit spaces.

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in Rn. As an application, we find an upper bound close to n−k for the Hausdorff dimension of sets with large k-porosity. With k-porous sets we mean sets which have holes in k different directions on every small scale.

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R ′ . With Es(A, N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A, N) for each fixed s ≥ d. Moreover, if A has positi...

We consider, for G a simply connected domain and 0 < p < ∞, the Hardy space H(G) formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F |p over the curves τ({|z| = r}) be bounded for 0 < r < 1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is...

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (R, B, H) and (R, B, H) are not isomorphic if s 6= t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of R and H is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for ever...

For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on C\K are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) > 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but...