scholarly journals The multifractal nature of heterogeneous sums of Dirac masses

2008 ◽  
Vol 144 (3) ◽  
pp. 707-727 ◽  
Author(s):  
JULIEN BARRAL ◽  
STÉPHANE SEURET
Keyword(s):  
Multifractal Analysis ◽  
Multifractal Spectrum ◽  
Positive Sequence ◽  
Similar Measure ◽  
Multiscale Structure ◽  
Self Similar ◽  
Image Position ◽  
Multifractal Nature

AbstractThis paper investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses where (xn)n≥0 is a sequence of points in [0, 1]d and (wn)n≥0 is a positive sequence of weights such that Σn≥0wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1]d, and the weights wn depend on a random self-similar measure μ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following way The measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn,λn)n with respect to μ.

scholarly journals Multifractal spectra for random self-similar measures via branching processes

2011 ◽  
Vol 43 (01) ◽  
pp. 1-39
Author(s):  
J. D. Biggins ◽  
B. M. Hambly ◽  
O. D. Jones
Keyword(s):  
Random Number ◽  
Branching Processes ◽  
Multifractal Spectrum ◽  
Compact Set ◽  
Similar Measure ◽  
Random Mass ◽  
Random Fractal ◽  
Multifractal Spectra ◽  
Self Similar ◽  
Random Division

Start with a compact setK⊂Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy ofKand all of which are insideK. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal setFis the limit, asngoes to ∞, of the union of thenth generation sets. In addition,Khas a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure onF. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations inRdand drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

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.

SYMBOLIC AND GEOMETRIC LOCAL DIMENSIONS OF SELF-AFFINE MULTIFRACTAL SIERPINSKI SPONGES IN ℝd

2007 ◽  
Vol 07 (01) ◽  
pp. 37-51 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Multifractal Analysis ◽  
Multifractal Spectrum ◽  
Separation Condition ◽  
Open Set Condition ◽  
Local Scaling ◽  
Open Set ◽  
Strong Separation ◽  
Self Similar ◽  
Strong Separation Condition ◽  
Sierpinski Sponges

In this paper we study the multifractal structure of a certain class of self-affine measures known as self-affine multifractal Sierpinski sponges. Multifractal analysis studies the local scaling behaviour of measures. In particular, multifractal analysis studies the so-called local dimension and the multifractal spectrum of measures. The multifractal structure of self-similar measures satisfying the Open Set Condition is by now well understood. However, the multifractal structure of self-affine multifractal Sierpinski sponges is significantly less well understood. The local dimensions and the multifractal spectrum of self-affine multifractal Sierpinski sponges are only known provided a very restrictive separation condition, known as the Very Strong Separation Condition (VSSC), is satisfied. In this paper we investigate the multifractal structure of general self-affine multifractal Sierpinski sponges without assuming any additional conditions (and, in particular, without assuming the VSSC).

scholarly journals Multifractal spectra for random self-similar measures via branching processes

2011 ◽  
Vol 43 (1) ◽  
pp. 1-39 ◽  
Author(s):  
J. D. Biggins ◽  
B. M. Hambly ◽  
O. D. Jones
Keyword(s):  
Random Number ◽  
Branching Processes ◽  
Multifractal Spectrum ◽  
Compact Set ◽  
Similar Measure ◽  
Random Mass ◽  
Random Fractal ◽  
Multifractal Spectra ◽  
Self Similar ◽  
Random Division

Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

In-homogenous self-similar measures and their Fourier transforms

2008 ◽  
Vol 144 (2) ◽  
pp. 465-493 ◽  
Author(s):  
L. OLSEN ◽  
N. SNIGIREVA
Keyword(s):  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Compact Support ◽  
Fourier Transforms ◽  
Probability Vector ◽  
Similar Measure ◽  
Special Cases ◽  
Non Linear ◽  
Self Similar ◽  
Image Position

AbstractLetSj: ℝd→ ℝdforj= 1, . . .,Nbe contracting similarities. Also, let (p1,. . .,pN,p) be a probability vector and let ν be a probability measure on ℝdwith compact support. We show that there exists a unique probability measure μ such thatThe measure μ is called an in-homogenous self-similar measure. In this paper we study the asymptotic behaviour of the Fourier transforms of in-homogenous self-similar measures. Finally, we present a number of applications of our results. In particular, non-linear self-similar measures introduced and investigated by Glickenstein and Strichartz are special cases of in-homogenous self-similar measures, and as an application of our main results we obtain simple proofs of generalizations of Glickenstein and Strichartz's results on the asymptotic behaviour of the Fourier transforms of non-linear self-similar measures.

scholarly journals Equidistribution Results for Self-Similar Measures

2021 ◽  
Author(s):  
Simon Baker
Keyword(s):  
Sufficient Conditions ◽  
Cantor Set ◽  
Similar Measure ◽  
Natural Measure ◽  
Self Similar

Abstract A well-known theorem due to Koksma states that for Lebesgue almost every $x&gt;1$ the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one. In this paper, we give sufficient conditions for an analogue of this theorem to hold for a self-similar measure. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty }$ where $(f_n)_{n=1}^{\infty }$ is a sequence of sufficiently smooth real-valued functions satisfying some nonlinearity conditions. As a corollary of our main result, we show that if $C$ is equal to the middle 3rd Cantor set and $t\geq 1$, then with respect to the natural measure on $C+t,$ for almost every $x$, the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one.

WEIGHTED AVERAGE GEODESIC DISTANCE OF PENTADENDRITE NETWORKS

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050075
Author(s):  
YUANYUAN LI ◽  
XIAOMIN REN ◽  
KAN JIANG
Keyword(s):  
Complex Networks ◽  
Geodesic Distance ◽  
Average Distance ◽  
Weighted Average ◽  
Similar Measure ◽  
Important Index ◽  
Self Similar

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

Analog Circuits Fault Diagnosis Using Multifractal Analysis

2013 ◽  
Vol 721 ◽  
pp. 367-371
Author(s):  
Yong Kui Sun ◽  
Zhi Bin Yu
Keyword(s):  
Support Vector Machine ◽  
Fault Diagnosis ◽  
Analog Circuits ◽  
Multifractal Analysis ◽  
Multifractal Spectrum ◽  
Experimental Results ◽  
Support Vector ◽  
Diagnosis Accuracy ◽  
Circuit Under Test

Analog circuits fault diagnosis using multifractal analysis is presented in this paper. The faulty response of circuit under test is analyzed by multifratal formalism, and the fault feature consists of multifractal spectrum parameters. Support vector machine is used to identify the faults. Experimental results prove the proposed method is effective and the diagnosis accuracy reaches 98%.

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

Non-spectral Problem for Some Self-similar Measures

2019 ◽  
Vol 63 (2) ◽  
pp. 318-327
Author(s):  
Ye Wang ◽  
Xin-Han Dong ◽  
Yue-Ping Jiang
Keyword(s):  
Spectral Problem ◽  
The Self ◽  
Exponential Functions ◽  
Similar Measure ◽  
Self Similar

AbstractSuppose that $0<|\unicode[STIX]{x1D70C}|<1$ and $m\geqslant 2$ is an integer. Let $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$ be the self-similar measure defined by $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$. Assume that $\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$ for some $p,q,r\in \mathbb{N}^{+}$ with $(p,q)=1$ and $(p,m)=1$. We prove that if $(q,m)=1$, then there are at most $m$ mutually orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$ and $m$ is the best possible. If $(q,m)>1$, then there are any number of orthogonal exponential functions in $L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$.

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