A multifractal formalism in a probability space

In this paper, we mainly research a multifractal formalism in a probability space.The first objective of this paper is to define a multifractal generalisation of the Hausdorff measure and the packing measure. We explore the properties of the multifractal Hausdorff measure and the multifractal packing measure in a probability space, and invesigate the relative results of multifractal spectra functions in a probability space. We also describe a sufficient condion and a necessary one of validity for the multifractal formalism in a probability space.     

From Multifractal Measures to Multifractal Wavelet Series

Given a positive locally finite Borel measure μ on R, a natural way to construct multifractal wavelet series is to set , where . Indeed, under suitable conditions, it is shown that the function F_μ inherits the multifractal properties of μ. The transposition of multifractal properties works with many classes of statistically selfsimilar multifractal measures, enlarging the class of processes which have self-similarity properties and controlled multifractal behaviors. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of F_μ are studied. As an application, multifractal Gaussian processes associated with F_μ are created. We obtain results for the multifractal spectrum of the so-called W-cascades introduced by Arnéodo et al.     

具有有限记忆的随机分形的重分形分解

Dryakhlov和Tempelman对具有有限记忆的随机分形集的Hausdorff维数进行了研究,本文对具有有限记忆的随机分形集K(ω)的重分形分解集Kα(ω)进行研究,得到了在一定条件下,这种随机分形集重分形分解集Kα(ω)的Hausdorff维数表达式.     

Multifractal analysis of a measure of multifractal exact dimension

We focus on the multifractal generalization of the centered Hausdorff measure and dimension. We analyze the correlation among different approaches to the definition of the multifractal exact dimension of locally finite and Borel regular measures on the basis of fractal analysis of essential supports of these measures. Using characteristic multifractal measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.     

Multifractal variation for projections of measures

The aim of this work is to provide a relationship between the relative multifractal spectra of orthogonal projections of a measure μ in Euclidean space and those of μ. As an application we study the relative multifractal analysis of the projections of a measure.     

Analyse multifractale de fonctions continues

For the symbolic dynamic on a finite alphabet, the pressure of Walters allows the study of the multifractal analysis of continuous functions. The results obtained link up large deviation theory and apply to the classic multifractal analysis of a large class of ergodic measures not necessarilly homogeneous called g-measures and for which some phase transitions may occur.     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions φn:X→M on a metric space X, multifractal analysis refers to the study of the Hausdorff dimension of the level sets

     

Cosmology in one dimension: A two-component model

We investigate structure formation in a one dimensional model of a matter-dominated universe using a quasi-Newtonian formulation. In addition to dissipation-free dark matter, dissipative luminous matter is introduced to examine the potential bias in the distributions. We use multifractal analysis techniques to identify scale-dependent structures, including clusters and voids. Both dark matter and luminous matter exhibit multifractal geometry over a finite range as the universe evolves in time. We present the results for the generalized dimensions computed on various scales for each matter distribution which clearly supports the bottom-up structure formation scenario. We compare and contrast the multifractal dimensions of two types of matter for the first time and show how dynamical considerations cause them to differ.     

Multifractal scenarios for products of geometric Lévy-based stationary models

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We construct the multifractal, such as log-gamma, log-tempered stable, or log-normal tempered stable scenarios.     

Fixed Points for the Multifractal Spectrum Map

For all continuous function g having a specific form that we call with increasing visibility, we construct a function f whose multifractal spectrum is such that \( d_f =g\circ f\). The function f is obtained as an infinite superposition of piecewise \(C^1\) functions, is also with increasing visibility, and is homogeneously multifractal; i.e., its restriction on any subinterval of \([0,1]\) has the same multifractal spectrum as the function f itself. In particular, we prove the existence of a function f which is its own multifractal spectrum; i.e., \(f=d_f\).     

Multifractality in relativistic charged particles produced at SPS energies

The multifractal analysis of relativistic shower particles produced in ³²S–emulsion interactions at 200 AGeV has been investigated using the method of modified multifractal moments, G_q, in pseudo-rapidity space. The anomalous fractal dimension, d_q, and generalized fractal dimensions, D_q, are determined for the present data for different order of moment. The experimental data reflects multifractal geometry in a multipion production process. The downward concave shape of the multifractal spectral function, f(α_q), gives an evidence for self-similar cascade mechanism. The multifractal specific heat has also been evaluated for the present data using the generalized fractal dimensions, D_q. We compared our experimental results with those obtained from simulated events of the Lund Monte Carlo Code FRITIOF and uncorrelated Monte Carlo events, (MC-RAND) generated randomly in pseudorapidity space based on the assumption of independent emission of particles. The experimental data on multifractality has been found to exhibit a remarkable proximity to the analogous data obtained from the FRITIOF code and the uncorrelated Monte Carlo events.     

重分形Hausdorff测度的有限测度子集

设μ为 Ｒ^d上 Ｂｏｒｅｌ概率测度,ｑ,ｔ∈Ｒ,记 Ｈ_μ^q,t为 Ｏｌｓｅｎ定义的重分形 Ｈａｕｓｄｏｒｆｆ测度,证明了当μ为测度时,Ｈ_μ^q,t的有限测度子集存在．     

具有有限交性质的广义图定向递归集的重分形分解

本文作者利用马尔科夫型测度证明了对一类允许适当交叠的递归集,在满足有限交性质情况下,重分形分解仍成立。     

Approximation of the Fourier Transform and the Dual Gabor Window

Many results and problems in Fourier and Gabor analysis are formulated in the continuous variable case, i.e., for functions on . In contrast, a suitable setting for practical computations is the finite case, dealing with vectors of finite length. We establish fundamental results for the approximation of the continuous case by finite models, namely, the approximation of the Fourier transform and the approximation of the dual Gabor window of a Gabor frame. The appropriate function space for our approach is the Feichtinger space S₀. It is dense in L², much larger than the Schwartz space, and it is a Banach space.     

A new exact method for obtaining the multifractal spectrum of multiscaled multinomial measures and IFS invariant measures

In this paper we present a new exact method for obtaining the multifractal spectrum of multiscaled multinomial measures and invariant measures associated with iterated function systems (IFS). A multinomial measure is shown to be generated as the invariant measure of an associated IFS. Then, the multifractal spectrum of the measure is determined by a couple of parametric implicit equations. This analysis generalizes some results previously obtained for the case of single-scaled multinomial measures (e.g., the binomial measure). A geometric interpretation of this new framework working in the space of codes of the IFS gives new insight into the nature of the multifractal formalism. This paper extends the results presented in Gutiérrez et. al. Fractals 4, (1996) 17–27.     

有限义性质下一类广义MORAN集的重分形分解

本文在一类广义Moran集上定义了一种弱分离条件－有限交性质,并证明了重分形分解在广泛的一类分形上仍然成立。本文部分的推广了Cawley和Manlain的结果。     

Multifractal Components of Multiplicative Set Functions

We analyze the multifractal spectrumof multiplicative set functions on a self-similar set with open set condition. We show that the multifractal components carry self-similar measures which maximize the dimension. This gives the dimension of a multifractal component as the solution of a problem of maximization of a quasiconcave function satisfying a set of linear constraints. Our analysis covers the case of multifractal components of self-similar measures, the case of Besicovitch normal sets of points, the multifractal spectrum of the relative logarithmic density of a pair of self-similar measures, the multifractal spectrum of the Liapunov exponent of the shift mapping and the intersections of all these sets. We show that the dimension of an arbitrary union of multifractal components is the supremum of the dimensions of the multifractal components in the union. The multidimensional Legendre transform is introduced to obtain the dimension of the intersection of finitely many multifractal components.     

The p-variation of functions

A formula is derived for the p-variation of functions of the class Vq (1 q < p < ) and it is proved that this formula ceases to hold when 1 相似文献