MULTIFRACTAL ANALYSIS OF SOME WEIGHTED QUASI-SELF-SIMILAR FUNCTIONS

2011 ◽  
Vol 09 (06) ◽  
pp. 965-987 ◽  
Author(s):  
JAMIL AOUIDI ◽  
ANOUAR BEN MABROUK
Keyword(s):  
Finite Number ◽  
Multifractal Analysis ◽  
Similar Case ◽  
Gibbs Measures ◽  
Box Dimension ◽  
Multifractal Formalism ◽  
Spectrum Of Singularities ◽  
Self Similar

In this paper, a multifractal analysis of some non-self-similar functions based on the superposition of finite number of weighted quasi-self-similar ones ∑iωiFi is developed. In general, such superpositions do not yield neither a self-similar nor a quasi-self-similar outcome. Furthermore, there are two main problems that appear. Firstly, a phenomenon of regularity compensation may exist. Secondly, the computation of the spectrum of singularities and therefore the validity of the multifractal formalism based on the possibility of constructing Gibbs measures fail. In this paper, we propose to study such problems by conducting a multifractal analysis of such combinations and to check the validity of the multifractal formalism in the case where there is no compensation of regularity. Furthermore, we compute the box dimension of the associated graphs and provide some examples. The paper in its full subject re-considers the results of Ref. 3 in the quasi-self-similar case.

Conditional and Relative Multifractal Spectra

Fractals ◽  
1997 ◽  
Vol 05 (01) ◽  
pp. 153-168 ◽  
Author(s):  
Rudolf H. Riedi ◽  
Istvan Scheuring
Keyword(s):  
Multifractal Analysis ◽  
Mutual Influence ◽  
Multifractal Spectrum ◽  
Real Data ◽  
The Novel ◽  
Multifractal Formalism ◽  
Geometrical Properties ◽  
Single Mechanism ◽  
Novel Approach ◽  
Self Similar

In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relativemultifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.

scholarly journals WAVELET-BASED MULTIFRACTAL FORMALISM TO ASSIST IN DIAGNOSIS IN DIGITIZED MAMMOGRAMS

2011 ◽  
Vol 20 (3) ◽  
pp. 169 ◽  
Author(s):  
Pierre Kestener ◽  
Jean Marc Lina ◽  
Philippe Saint-Jean ◽  
Alain Arneodo
Keyword(s):  
Wavelet Transform ◽  
Hurst Exponent ◽  
Multifractal Analysis ◽  
Computer Assisted ◽  
Scaling Properties ◽  
Multifractal Formalism ◽  
Long Range Correlations ◽  
Multifractal Scaling ◽  
Self Similar ◽  
Assisted Diagnosis

We apply the 2D wavelet transform (WTMM) method to perform a multifractal analysis of digitized mammograms. We show that normal regions display monofractal scaling properties as characterized by the socalled Hurst exponent H =0.3±0.1 in fatty areas which look like antipersistent self-similar random surfaces, while H=0.65±0.1 in dense areas which exibit long-range correlations and possibly multifractal scaling properties. We further demonstrate that the 2D WTMM method provides a very efficient way to detect tumors as well as microcalcifications (MC) which correspond to much stronger singularities than those involved in the background tissue roughness fluctuations. These preliminary results indicate that the texture discriminatory power of the 2D WTMM method may lead to significant improvement in computer-assisted diagnosis in digitized mammograms.

A WAVELET MULTIFRACTAL FORMALISM FOR SIMULTANEOUS SINGULARITIES OF FUNCTIONS

2013 ◽  
Vol 12 (01) ◽  
pp. 1450009 ◽  
Author(s):  
JAMIL AOUIDI ◽  
ANOUAR BEN MABROUK
Keyword(s):  
Multifractal Analysis ◽  
Multifractal Formalism ◽  
Self Similar

In this paper, a wavelet multifractal analysis is developed which permits to characterize simultaneous singularities for a vector of functions. An associated multifractal formalism is introduced and checked for the case of functions involving self similar aspects.

scholarly journals A MIXED MULTIFRACTAL ANALYSIS FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

Fractals ◽  
2021 ◽  
pp. 2240001
Author(s):  
ANOUAR BEN MABROUK ◽  
ADEL FARHAT
Keyword(s):  
Large Class ◽  
Multifractal Analysis ◽  
Weak Condition ◽  
Probability Measures ◽  
Original Formulation ◽  
Multifractal Formalism ◽  
Self Similar ◽  
Joint Multifractal Analysis ◽  
Vector Valued ◽  
Special Classes

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

scholarly journals ACCRETION AND PLASMA OUTFLOW FROM DISSIPATIONLESS DISCS

2010 ◽  
Vol 19 (03) ◽  
pp. 339-365 ◽  
Author(s):  
S. V. BOGOVALOV ◽  
S. R. KELNER
Keyword(s):  
Angular Momentum ◽  
Similar Case ◽  
Gravitational Energy ◽  
Accretion Disc ◽  
The Self ◽  
Low Viscosity ◽  
Variable Polarity ◽  
Self Similar ◽  
Similar Solutions ◽  
High Electric Conductivity

We consider the specific case of disc accretion for negligibly low viscosity and infinitely high electric conductivity. The key component in this model is the outflowing magnetized wind from the accretion disc, since this wind effectively carries away angular momentum of the accreting matter. Assuming magnetic field has variable polarity in the disc (to avoid magnetic flux and energy accumulation at the gravitational center), this leads to radiatively inefficient accretion of the disc matter onto the gravitational center. In such a case, the wind forms an outflow, which carries away all the energy and angular momentum of the accreted matter. Interestingly, in this framework, the basic properties of the outflow (as well as angular momentum and energy flux per particle in the outflow) do not depend on the structure of accretion disc. The self-similar solutions obtained prove the existence of such an accreting regime. In the self-similar case, the disc accretion rate (Ṁ) depends on the distance to the gravitational center, r, as [Formula: see text], where λ is the dimensionless Alfvenic radius. Thus, the outflow predominantly occurs from the very central part of the disc provided that λ ≫ 1 (it follows from the conservation of matter). The accretion/outflow mechanism provides transformation of the gravitational energy from the accreted matter into the energy of the outflowing wind with efficiency close to 100%. The flow velocity can essentially exceed the Kepler velocity at the site of the wind launch.

scholarly journals Multifractal analysis for the ULF geomagnetic data during the 1993 Guam earthquake

2005 ◽  
Vol 12 (2) ◽  
pp. 157-162 ◽  
Author(s):  
Y. Ida ◽  
M. Hayakawa ◽  
A. Adalev ◽  
K. Gotoh
Keyword(s):  
Detrended Fluctuation Analysis ◽  
Multifractal Analysis ◽  
Fluctuation Analysis ◽  
Sensitive Parameter ◽  
Wide Range ◽  
Geomagnetic Data ◽  
Self Similar ◽  
Significant Change ◽  
Detrended Fluctuation

Abstract. In our previous papers we have shown that the fractal (monofractal) dimension (Do) showed a significant increase before the Guam earthquake occurred on 8 August, 1993. In order to have a further support to this precursory effect to the general rupture (earthquake) we have carried out the corresponding multifractal analysis (by means of detrended fluctuation analysis) for the same data to study the statistical self-similar properties in a wide range of scales. We have analyzed the ULF geomagnetic data (the most intense H component) observed at Guam observatory. As the result, we have found that we could observe significant changes in the multifractal parameters at Guam such that αmin showed a meaningful decrease about 25 days before the earthquake and correspondingly Δα increased because αmax exhibited no significant change at all. The most sensitive parameter seems to be non-uniformity factor Δ. Correspondingly, the generalized multifractal dimension Dq (q>1) showed a significant decrease (whereas Dq (q<0) showed no change) and D0 (=Dq (q=0) (as already found in our previous papers) is reconfirmed to increase before the earthquake. These multifractal characteristics seem to be a further support that these changes are closely associated with the earthquake as a precursor to the Guam earthquake, providing us with appreciable information on the pre-rupture evolution of the earthquake.

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

Multifractal analysis for disintegrations of Gibbs measures and conditional Birkhoff averages

2009 ◽  
Vol 29 (3) ◽  
pp. 885-918 ◽  
Author(s):  
DE-JUN FENG ◽  
LIN SHU
Keyword(s):  
Convex Analysis ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Gibbs Measures ◽  
Thermodynamic Formalism

AbstractThe paper is devoted to the study of the multifractal structure of disintegrations of Gibbs measures and conditional (random) Birkhoff averages. Our approach is based on the relativized thermodynamic formalism, convex analysis and, especially, the delicate constructions of Moran-like subsets of level sets.

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