We give an example of Cantor-type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in the Stahl–Totik sense. c ⃝ 2014 Elsevier Inc. All rights reserved. MSC: 30C85; 31A15; 28A78; 28A80

We develop several results on hitting probabilities of random fields which highlight the role of the dimension of the parameter space. This yields upper and lower bounds in terms of Hausdorff measure and BesselRiesz capacity, respectively. We apply these results to a system of stochastic wave equations in spatial dimension k ≥ 1 driven by a d-dimensional spatially homogeneous additive Gaussian ...

Abstract. With a map f : Ω → R,Ω ⊂ R, that belongs to the John Ball classAp,q(Ω) where n − 1 < p < n and q ≥ p/(p − 1) one can associate a set valued map F whose values F (x) ⊂ R are subsets ofR describing the topological character of the singularity of f at x ∈ Ω. Šverak conjectured that Hn−1(F (S)) = 0, where S is the set of points at which f is not continuous andHn−1 is the Hausdorff measure...

We prove that every separable metric space which admits an `1tree as a Lipschitz quotient, has a σ-porous subset which contains every Lipschitz curve up to a set of 1-dimensional Hausdorff measure zero. This applies to any Banach space containing `1. We also obtain an infinite-dimensional counterexample to the Fubini theorem for the σ-ideal of σ-porous sets.

We consider the problem of maximizing the Lebesgue measure of the convex hull of a connected compact set of prescribed onedimensional Hausdorff measure. In dimension two, we prove that the only solutions are semicircles. In higher dimension, we prove some isoperimetric inequalities for the convex hull of connected sets, we focus on a classical open problem and we discuss a new possible approach.

The aim of this paper is to show the global existence of weak solutions for a moving boundary problem arising in the non-isothermal crystallization of polymers. The main features of our works are (i) the moving interface is shown to be of co-dimension one; (ii)finite Hölder continuous propagation speed yields an intrinsic estimate of finite co-dimension one Hausdorff measure of the moving inter...

It has been shown (see [10]), that there are strongly MARTIN-LÖFε-random ω-words that behave in terms of complexity like random ωwords. That is, in particular, the a priori complexity of these ε-random ω-words is bounded from below and above by linear functions with the same slope ε. In this paper we will study the set of these ω-words in terms of HAUSDORFF measure and dimension. Additionally w...

The famous game Towers of Hanoi is related with a family of so–called Hanoi–graphs. We regard these non self–similar graphs as geometrical objects and obtain a sequence of fractals (HGα)α converging to the Sierpiński gasket which is one of the best studied fractals. It is shown that this convergence holds not only with respect to the Hausdorff distance, but that also Hausdorff dimension does co...

The paper investigates bounds on various notions of complexity for ω–languages. We understand the complexity of an ω–languages as the complexity of the most complex strings contained in it. There have been shown bounds on simple and prefix complexity using fractal Hausdorff dimension. Here these bounds are refined by using general Hausdorff measure originally introduced by Felix Hausdorff. Furt...