scholarly journals MULTIFRACTAL ANALYSIS OF FUNCTIONS ON HEISENBERG AND CARNOT GROUPS

2015 ◽  
Vol 16 (1) ◽  
pp. 1-38
Author(s):  
S. Seuret ◽  
F. Vigneron
Keyword(s):  
Heisenberg Group ◽  
Besov Space ◽  
Multifractal Analysis ◽  
A Priori ◽  
Multifractal Spectrum ◽  
Upper Bounds ◽  
Carnot Groups ◽  
Hölder Exponents ◽  
Global And Local

In this article, we investigate the pointwise behaviors of functions on the Heisenberg group. We find wavelet characterizations for the global and local Hölder exponents. Then we prove some a priori upper bounds for the multifractal spectrum of all functions in a given Hölder, Sobolev, or Besov space. These upper bounds turn out to be optimal, since in all cases they are reached by typical functions in the corresponding functional spaces. We also explain how to adapt our proof to extend our results to Carnot groups.

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%.

Numerical Methods of Multifractal Analysis in Information Communication Systems and Networks

2013 ◽  
pp. 16-46
Author(s):  
Oleg I. Sheluhin ◽  
Artem V. Garmashev
Keyword(s):  
Communication Systems ◽  
Multifractal Analysis ◽  
Fractal Dimensions ◽  
Multifractal Spectrum ◽  
Telecommunication Networks ◽  
Information Communication ◽  
Novel Approach ◽  
Wavelet Transform Modulus Maxima ◽  
Modulus Maxima ◽  
Development Methods

In this chapter, the main principles of the theory of fractals and multifractals are stated. A singularity spectrum is introduced for the random telecommunication traffic, concepts of fractal dimensions and scaling functions, and methods used in their determination by means of Wavelet Transform Modulus Maxima (WTMM) are proposed. Algorithm development methods for estimating multifractal spectrum are presented. A method based on multifractal data analysis at network layer level by means of WTMM is proposed for the detection of traffic anomalies in computer and telecommunication networks. The chapter also introduces WTMM as the informative indicator to exploit the distinction of fractal dimensions on various parts of a given dataset. A novel approach based on the use of multifractal spectrum parameters is proposed for estimating queuing performance for the generalized multifractal traffic on the input of a buffering device. It is shown that the multifractal character of traffic has significant impact on queuing performance characteristics.

scholarly journals High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengcheng Niu ◽  
Kelei Zhang
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Carnot Group ◽  
A Priori ◽  
Elliptic Operators ◽  
High Order ◽  
Carnot Groups ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Horizontal Vector

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

scholarly journals Multifractals Properties on the Near Infrared Spectroscopy of Human Brain Hemodynamic

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Truong Quang Dang Khoa ◽  
Vo Van Toi
Keyword(s):  
Time Series ◽  
Infrared Spectroscopy ◽  
Near Infrared Spectroscopy ◽  
Near Infrared ◽  
Real Life ◽  
Multifractal Spectrum ◽  
Singular Behavior ◽  
Maximum Expansion ◽  
Experimental Time Series ◽  
Hölder Exponents

Nonlinear physics presents us with a perplexing variety of complicated fractal objects and strange sets. Naturally one wishes to characterize the objects and describe the events occurring on them. Moreover, most time series found in “real-life” applications appear quite noisy. Therefore, at almost every point in time, they cannot be approximated either by the Taylor series or by the Fourier series of just a few terms. Many experimental time series have fractal features and display singular behavior, the so-called singularities. The multifractal spectrum quantifies the degree of fractals in the processes generating the time series. A novel definition is proposed called full-width Hölder exponents that indicate maximum expansion of multifractal spectrum. The obtained results have demonstrated the multifractal structure of near-infrared spectroscopy time series and the evidence for brain imagery activities.

scholarly journals ON THE INVERSE MULTIFRACTAL FORMALISM

2009 ◽  
Vol 52 (1) ◽  
pp. 179-194 ◽  
Author(s):  
L. OLSEN
Keyword(s):  
Multifractal Analysis ◽  
Regularity Condition ◽  
Upper Bounds ◽  
Legendre Transform ◽  
Self Similarity ◽  
Multifractal Formalism ◽  
Multifractal Spectra ◽  
Physics Literature ◽  
Image Position ◽  
Legendre Transforms

AbstractTwo of the main objects of study in multifractal analysis of measures are the coarse multifractal spectra and the Rényi dimensions. In the 1980s it was conjectured in the physics literature that for ‘good’ measures the following result, relating the coarse multifractal spectra to the Legendre transform of the Rényi dimensions, holds, namely This result is known as the multifractal formalism and has now been verified for many classes of measures exhibiting some degree of self-similarity. However, it is also well known that there is an abundance of measures not satisfying the multifractal formalism and that, in general, the Legendre transforms of the Rényi dimensions provide only upper bounds for the coarse multifractal spectra. The purpose of this paper is to prove that even though the multifractal formalism fails in general, it is nevertheless true that all measures (satisfying a mild regularity condition) satisfy the inverse of the multifractal formalism, namely

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 Multifractal Analysis of Infinite Products of Stationary Jump Processes

2010 ◽  
Vol 2010 ◽  
pp. 1-26
Author(s):  
Petteri Mannersalo ◽  
Ilkka Norros ◽  
Rudolf H. Riedi
Keyword(s):  
Stochastic Processes ◽  
Multifractal Analysis ◽  
Point Processes ◽  
Multifractal Spectrum ◽  
Jump Processes ◽  
Poisson Point Processes ◽  
Infinite Products ◽  
Stationary Measures ◽  
Basic Properties ◽  
Side Result

There has been a growing interest in constructing stationary measures with known multifractal properties. In an earlier paper, the authors introduced themultifractal products of stochastic processes(MPSP) and provided basic properties concerning convergence, nondegeneracy, and scaling of moments. This paper considers a subclass of MPSP which is determined by jump processes with i.i.d. exponentially distributed interjump times. Particularly, the information dimension and a multifractal spectrum of the MPSP are computed. As a side result it is shown that the random partitions imprinted naturally by a family of Poisson point processes are sufficient to determine the spectrum in this case.

scholarly journals Nonexistence Results for Semilinear Equations in Carnot Groups

2013 ◽  
Vol 1 ◽  
pp. 130-146 ◽  
Author(s):  
Fausto Ferrari ◽  
Andrea Pinamonti
Keyword(s):  
Heisenberg Group ◽  
Carnot Group ◽  
Carnot Groups ◽  
Semilinear Equations ◽  
Engel Group ◽  
Large Step

Abstract In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

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