MIXED MULTIFRACTAL DENSITIES FOR QUASI-AHLFORS VECTOR-VALUED MEASURES

In this work, some density estimations associated to vector-valued quasi-Ahlfors measures are developed within the mixed multifractal analysis framework. The principle idea reposes on the fact that being quasi-Ahlfors is sufficient to conduct a mixed multifractal analysis for vector-valued measures. In this work, we introduced a multifractal density for finitely many measures, and showed that such density may be estimated well by means of the mixed multifractal measures. Such estimation induces an exact computation of multifractal spectrum of the vector-valued quasi-Ahlfors measure.

Remarks on the mutual singularity of multifractal measures

In the present work, we study the mutual singularity of multifractal Hausdorff and packing measures which provide a positive answer to Olsen’s questions in a more general framework. Our main results apply to a family of measures supported by the full 5-adic grid of [0, 1], namely the quasi-Bernoulli measures.

The mutual singularity of the relative multifractal measures

M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

Another example of the mutual singularity of multifractal measures

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular.

Multifractal Nature of Diesel Engine Rattle Noise in Vehicle

The investigation results of the emission of noises accompanying the diesel engine powertrain running in unloaded condition at low or idle speed, are presented in the hereby paper. The multifractal nature of the tested signals was confirmed by means of two groups of parameters. The proposed local parameters are based on the distribution of the probability density of measures, segregated into subsets according to pointwise Holder exponents. The obtained spectra constitute a set of multifractal measures related to singularities representing the local scaling of measures in various points of the time series. Global parameters were defined on the basis of multifractal measures of the parametrised Renyi’s entropy. These are properties treating summarily the whole spectrum of the emitted noise and reflecting changes of the vibroacoustic energy levels generated by noise sources. The tests encompassing simultaneous application of both groups of parameters confirmed their efficiency in the comparative analysis as well as in the subjective assessment of the noise level generated in the drive system of vehicles with diesel engines. This opens new possibilities within the simulation range of vehicle noises, for the needs of constructing the passive and active reduction systems of the effects of the vibroacoustic energy propagation.