Multifractal Geometry Research Articles

On comparing multifractal and classical features in minimum distance classification of AVHRR imagery

The ability to distinguish between different types of surfaces is the strength of texture descriptors in the analysis of satellite imagery. Although the most common analytical means are based on co‐occurrence analysis, considerable progress has been made in understanding the role of fractal and multifractal analysis in remote sensing. After indicating the limitations of using fractal dimensions as the only texture descriptor and introducing the concept of multifractal geometry, we consider the effectiveness of using multifractal and second‐order fractal features in image classification. In particular, we present the results of comparing two supervised classifications of an Advanced Very High Resolution Radiometer (AVHRR) image of Scotland using classical texture features and multifractal second‐order fractal ones. In terms of percentage correct and Khat statistics, this study provides evidence, with a confidence limit of 95%, that classifications using multifractal and second‐order fractal features are more accurate than those using classical features. The classification algorithm used for this study is a typical minimum distance classifier.

FRACTAL BEHAVIOR OF NUCLEAR FRAGMENTS IN HIGH ENERGY INTERACTIONS

The multifractal analysis of data on nuclear fragments obtained from 28 Si-AgBr interactions at 14.5 A GeV is performed using three different methods (the factorial moments, G-moments and Takagi moments). The generalized fractal dimensions Dq is determined from all these methods. Data reflects multifractal geometry for the nuclear fragments. From the knowledge of Dq, the multifractal specific heat is calculated for this data and also for 16 O-AgBr interactions at 60 A GeV and 32 S-AgBr interactions at 200 A GeV.

Determination of the morphology of surface groups formed and PVDF-binder materials dispersed on graphite composite electrodes in terms of fractal geometry

Morphological structures of surface groups formed and poly-vinylidene fluoride (PVDF)-binder materials dispersed on the PVDF-bonded graphite composite electrode were investigated in terms of fractal geometry using cyclic voltammetry combined with Kelvin probe force microscopy (KFM). When a fractal surface has single fractal geometry consisting of binder materials only, the overall fractal dimension was determined to be 1.82 from cyclic voltammetry based upon the peak current–scan rate relation, which is just the same in value as the individual fractal dimension of binder materials determined from KFM based upon the perimeter–area relation. By contrast, when a fractal surface has multifractal geometry composed of surface groups and binder materials, the overall fractal dimension was determined to be 1.77 from cyclic voltammetry. But the individual fractal dimensions were determined from KFM to distinguish the fractal dimension (1.70) of surface groups from that fractal dimension (1.82) of binder materials. The overall fractal dimension determined from cyclic voltammetry is just the average of the two individual fractal dimensions determined from KFM.

Multifractal geometry in stock market time series

It has been recently noticed that time series of returns in stock markets are of multifractal (multiscaling) character. In that context, multifractality has been always evidenced by its statistical signature (i.e., the scaling exponents associated to a related variable). However, a direct geometrical framework, much more revealing about the underlying dynamics, is possible. In this paper, we present the techniques allowing the multifractal decomposition. We will show that there exists a particular fractal component, the most singular manifold (MSM), which contains the relevant information about the dynamics of the series: it is possible to reconstruct the series (at a given precision) from the MSM. We analyze the dynamics of the MSM, which shows revealing features about the evolution of this type of series.

Gravitational relation of prolongable general relativity in the irreversible process of a dissipative system

The heterogeneity h（ρt）and coarse-grained entropy S（ρtε）of the density distribution ρ in phase space and the derived computational expression of multiscale factor η* are used in the expansion of Weyl curvature into multiscaled curvature R*; and the multifractal in relation to the singular distribution of physical quantities on geometry is used in the preliminary establishment of the multifractal geometry of growing structures set in Riemann space; on this basis, the equation of non-conservative attractive field is set up by the introduction of the general dimension Du in relation to the fractal and informational and incident dimensions. The analysis here shows that the new result could render the suitable basis for the settlement of the questions such as the singularity of cosmology the unidentified matter in the cluster of galaxies, the doubt of information from black hole and the unification between gravitation theory and quantum physics, and so on.

Nonextensive statistics: theoretical, experimental and computational evidences and connections

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various (possibly) relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.

Geometric Constructions in Multifractal Geometry

This is a survey on geometric constructions in multifractal analysis of measures. We show that the multifractal formalism introduced by Olsen [Ol11] and Peyriere [Pey] leads to a multifractal geometry for product measures, for slices of measures (i.e. intersections of measures with lower dimensional subspaces), and for general intersections of measures, which is analogous to the fractal geometry for product sets, for slices of sets (i.e. intersections of sets with lower dimensional subspaces), and for general intersections of sets, respectively.

Multifractal structure and intermittency of laser-generated turbulence in nematic liquid crystals.

We study the chaotic dynamics which occurs in a nematic liquid crystal film during the director reorientation induced by an intense optical field. We show the presence of intermittency in the fluctuations of the transmitted light intensity polarized perpendicularly to the incident beam. The intermittency can safely be described by the multifractal geometry, and, in this framework, we present a simple heuristic model for the phenomenon.