Measurable envelopes, Hausdorff measures and Sierpiński sets

Hausdorff Dimension and Conformal Measures of Feigenbaum Julia Sets

1.1. Statement of the results. One of the first questions usually asked about a fractal subset of R is whether it has the maximal possible Hausdorff dimension, n. It certainly happens if the set has positive Lebesgue measure. On the other hand, it is easy to construct fractal sets of zero measure but of dimension n. Moreover, this phenomenon is often observable for fractal sets produced by conf...

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance

This paper presents a new method for similarity measures between intuitionistic fuzzy sets (IFSs). We will present a method to calculate the distance between IFSs on the basis of the Hausdorff distance. We will then use this distance to generate a new similarity measure to calculate the degree of similarity between IFSs. Finally we will prove some properties of the proposed similarity measure a...

Null-Control and Measurable Sets

We prove the interior and boundary null–controllability of some parabolic evolutions with controls acting over measurable sets.

Effectively approximating measurable sets by open sets

We answer a recent question of Bienvenu, Muchnik, Shen, and Vereshchagin. In particular, we prove an effective version of the standard fact from analysis which says that, for any ε > 0 and any Lebesgue-measurable subset of Cantor space, X ⊆ 2, there is an open set Uε ⊆ 2, Uε ⊇ X, such that μ(Uε) ≤ μ(X) + ε, where μ(Z) denotes the Lebesgue measure of Z ⊆ 2. More specifically, our main result sho...