Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.     

FRACTAL DIMENSION ESTIMATES FOR INVARIANT SETS OF NON-INJECTIVE MAPS

For a special class of non-injective maps on Riemannian manifolds an upper bound for the fractal dimension of invariant set in terms of singular values of the tangent map and degree of non-injectivity is given     

THE DIMENSION FOR RANDOM SUB-SELF-SIMILAR SET

In this article,the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions.Several examples are given at the end.     

The Hausdorff dimension of invariant measures for random dynamical systems

We consider random dynamical systems with jumps. The Hausdorff dimension of invariant measures for such systems is estimated.     

Computing the dimension of a semi-algebraic set

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of R ^k, where R is a real closed field. We prove that the dimension k′ of a semi-algebraic set described by s polynomials of degree d in k variables can be computed in time

Fractal dimension of brittle fracture

Summary Using a worldwide catalog of earthquakes we analyze the distribution of distances between pairs of earthquake hypocenters to determine the spatial fractal dimension of an earthquake fracture. As the time span of the catalog increases, asymptotically reaches the value 2.1–2.2 for shallow earthquakes. Approximately the same asymptotic value of dimension is obtained for a catalog of earthquakes with aftershocks removed. This value is compared with other known determinations of for brittle fractures. The fractal dimension declines to 1.8–1.9 for intermediate events (depth interval 71–280 km) and to about 1.5–1.6 for deeper events. Taking into account various possible errors and biases, we conclude that the fractal dimension of brittle shear fracture in rocks is 2.20 ± 0.05.     

THE RANDOM SHIFT SET AND RANDOM SUB-SELF-SIMILAR SET

First of all the authors introduce the concepts of random sub-self-similar set and random shift set and then construct the random sub-self-similar set by a random shift set and a collection of statistical contraction operators.     

On invariant random approximations

A result regarding invariant random approximation is proved. Supported by Kuwait University Research Grant No. SM-119.     

Amenable invariant random subgroups

We show that an amenable Invariant Random Subgroup of a locally compact second countable group lives in the amenable radical. This answers a question raised in the introduction of [2]. We also consider an opposite direction, property (T), and prove a similar statement for this property.     

Some remarks on the Hausdorff dimension of a multiple point set of random fields with continuous sample paths

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 32, No. 3, pp. 360–368, July–September, 1992.     

Resistance dimension,random walk dimension and fractal dimension

In this paper, we generalize a theorem due to Telcs concerning random walks on infinite graphs, which describes the relation of random walk dimension, fractal dimension and resistance dimension. Moreover, we obtain a reasonable upper bound and lower bound on the hitting time in terms of resistance for some nice graphs. In fact, the conditions given in this paper are weaker than those obtained by A. Telcs.Partly supported by National Natural Science Foundation and State Educational Committee of China.     

On matrices leaving invariant a nontrivial convex set

A real square matrix A leaves a nontrivial convex set invariant if there exists a convex set C, which is not a linear subspace, such that A(C) ? C. It is shown that this is equivalent to the statement that A has an eigenvalue λ with λ?0 or |λ|?1.     

The exact hausdorff dimension of a branching set

Summary We obtain a critical function for which the Hausdorff measure of a branching set generated by a simple Galton-Watson process is positive and finite.     

Fractal dimension and classification of music

The fractal aspect of different kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great number of different musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal dimension helps discriminate different categories of music. Then, we used an original statistical technique based on the Bootstrap assumption to find a time window in which fractal dimension reaches a high power of music discrimination. The best discrimination is obtained between 1/44100 and 16/44100 Hertz. We admit that to distinguish some different aspects of music well, the high information quantity is obtained in the high frequency domain. By calculating fractal dimension with the ANAM method, it was statistically proven that fractal dimension could distinguish different kinds of music very well: musics could be classified by their fractal dimensions.     

Asymptotics of random partitions of a set

This paper contains two results on the asymptotic behavior of uniform probability measure on partitions of a finite set as its cardinality tends to infinity. The first one states that there exists a normalization of the corresponding Young diagrams such that the induced measure has a weak limit. This limit is shown to be a δ-measure supported by the unit square (Theorem 1). It implies that the majority of partition blocks have approximately the same length. Theorem 2 clarifies the limit distribution of these blocks. The techniques used can also be useful for deriving a range of analogous results. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 227–250. The paper is supported by International Science Foundation, grant MQV-100. Translated by Yu. Yakubovich     

Fractal dimension and scale entropy applications in a spray

Multi-scale structure of spray images is investigated for varying ranges of pressure and temperature in quiescent air. For spray images a standard PIV set is used consisting basically on a CCD camera and a laser sheet. A deviation to fractality is evidenced: the scale analysis has a parabolic form. A scale-dependent fractal dimension is measured which displays a linear variation with scale-logarithm. The classical fractal dimension usually measured so far is reinterpreted as a mean slope for scales close to the outer cut-off scale. This multi-scale behaviour is described by a diffusion equation of a new geometrical quantity called scale entropy related to the wrinkling of a set over scales. This equation is based on the conservation of a scale entropy flux through scale-space which is interpreted as the evolutive potential of the system at a given scale. This gives access to the scale-dependency of fractal dimension and points to the importance of the variations through scale space of this evolutive potential and namely its gradient. It has been shown that for sprays, the evolution of the evolutive potential gradient is constant through scale space which corresponds to a parabolic behaviour for scale analysis.