A very important property of a deterministic self-similar set is that its Hausdorff dimension and upper box-counting dimension coincide. This paper considers the random case. We show that for a random self-similar set, its Hausdorff dimension and upper box-counting dimension are equal a.s.

Moran’s Theorem is one of the milestones in Fractal Geometry. It allows the calculation of the similarity dimension of any (strict) self-similar set lying under the open set condition. Throughout a new fractal dimension we provide in the context of fractal structures, we generalize such a classical result for attractors which are required to satisfy no separation properties.

In this short paper we introduce a proper method to perform Korcak-analysis and obtain the correct Korcak-exponent on a set of patches, embedded into two-dimensions. Both artificial and natural data sets are used for the demonstration. The independence of the Korcak-exponent from the classical (Hausdorff) fractal dimension is also demonstrated. 2012 Elsevier B.V. All rights reserved.

In the context of Kolmogorov’s algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, s...

We establish various relationships of the Hausdorff dimension, entropy dimension and L p-dimension of a measure without assuming that the local dimension of exists-a.e. These extend a well known result of Young.

The Hausdorff fractal dimension has been a fast-to-calculate method to estimate complexity of fractal shapes. In this work, a modified version of this fractal dimension is presented in order to make it more robust when applied in estimating complexity of non-fractal images. The modified Hausdorff fractal dimension stands on two features that weaken the requirement of presence of a shape and als...

We show that the Hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet. Using computable, feasible, and finite-state predictors, this equivalence also holds for the computable, feasible, and finite-state dimensions. Combining this with recent results of Fortnow and Lutz (2002), we have a tight relationship between prediction wi...

A proof that a concept is learnable provided the Vapnik-Chervonenkis dimension is finite is given. The proof is more explicit than previous proofs and introduces two new parameters which allow bounds on the sample size obtained to be improved by a factor of approximately 4log2(e).

An arrangement of oriented pseudohyperplanes in affine d-space defines on its set X of pseudohyperplanes a set system (or range space) (X,R), R ⊆ 2 of VCdimension d in a natural way: to every cell c in the arrangement assign the subset of pseudohyperplanes having c on their positive side, and let R be the collection of all these subsets. We investigate and characterize the range spaces correspo...

The classical Hausdorff dimension, denoted dimH , of finite or countable sets is zero. We define an analog for finite sets, called finite Hausdorff dimension and denoted dimfH , which is non-trivial. It turns out that a finite bound for dimfH (F ) guarantees that every point of F has ”nearby” neighbors. This property is important for many computer algorithms of great practical value, that obtai...