Under certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

This paper describes the Lempel-Ziv dimension (Hausdorff like dimension inspired in the LZ78 parsing), its fundamental properties and relation with Hausdorff dimension. It is shown that in the case of individual infinite sequences, the Lempel-Ziv dimension matches with the asymptotical Lempel-Ziv compression ratio. This fact is used to describe results on Lempel-Ziv compression in terms of dime...

A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B : B ≤T A} contains no 1-random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We...

Bennett and Gill (1981) proved that P 6= NP relative to a random oracle A, or in other words, that the set O[P=NP] = {A | P = NP} has Lebesgue measure 0. In contrast, we show that O[P=NP] has Hausdorff dimension 1. This follows from a much more general theorem: if there is a relativizable and paddable oracle construction for a complexity-theoretic statement Φ, then the set of oracles relative t...

The best possible lower bound for the Hausdorff dimension of the set of v-approximable points on a C extremal manifold is obtained.

The Sierpiński gasket in RN−1 has a representation as a quotient space of the shift space Σ = {1, ..., N}N. Its topology can be described by any Martin metric as introduced in Denker and Sato [Publ. Res. Inst. Math. Sci., Kyoto University 35, 1999, 769–794] and Imai, Kawasaki and Sato [Stochastics and Dynamics 3, 2003, 267–277]. These metrics are parametrized by all strictly positive 1-sequence...

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundary of the tree. The random subset 3 of the boundary consisti...

We report the degree of order of twenty two Jackson Pollock's paintings using Hausdorff-Besicovitch fractal dimension. Through the maximum value of each multi-fractal spectrum, the fractality of the artworks is presented as a function of the year in which they were painted. It has been reported that Pollock's paintings are fractal and it increased on his latest works. However our results show t...

This paper deals with three resolving parameters: the metric dimension, the upper dimension and the resolving number. We first answer a question raised by Chartrand and Zhang asking for a characterization of the graphswith equalmetric dimension and resolving number. We also solve in the affirmative a conjecture posed by Chartrand, Poisson and Zhang about the realization of the metric dimension ...

We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order √ t away from the origin [DKK1], there is an uncountable set of measure zero of points, which escape to infinity at the linear rate [CSS1]. In this paper we prove tha...