Fractal dimension structure form of the Cosmos are explored, and the mathematical foundation, which include the expressions of fractal dimension differential and calculus, regular space integral solutions of fractal dimension differential equations, the fractal calculus definitions of fractal measure as well as the measure computational equation of self-similar fractal, of fractal dimension calculus and fractal measure are given.
Discuss the applications of some main embranchments of fractal theory: calculation of fractal dimension, fractal interpolation, fractal brownian motion, fractal measure, power exponent distribution, self-similarity and scaling invariance in the kinds of areas of practics, the mathematical models of these embranchments are discribed, some applications in some subjects related with it are introduced.
With a description of the features of image texture, a new method for edge detection based on the multiscale fractal measure is proposed by analyzing the difference between the variation of the local fractal measure of different texture features with the scale. Examples of the application of the algorithm for edge detection are given. Since the texture feature and gray feature are included in the edge detection operator, it is good in noise immunity and localized performance.
Based on the wavelet transform modulus maxima method, we performed multifractal measure analysis of solid hold-up fluctuation signal in CFB riser and got the multifractal singularity spectra of signals.
We study intermittency effects in high energy collisions introducing a fractal measure in rapidity space and formulationg the hadronization sector of the S-matrix within the Ginzburg-Landau approach.
We show that a fractal measure of crown can be used as the link between the mathematical models of crown growth and light propagation through the canopy.
It shows the dependence of fractal measure and fractal dimension on the amplitude of acoustic events along the axis of the tested pipe in their relation to the spatial distribution of defects in the material.
It is conjectured that a fractal J-integral should be the rate of potential energy release per unit of a fractal measure of crack growth.
Results fit the actual pattern of spread well, as measured by both visual inspection and a multiscale fractal measure of pattern.
Generalized thermodynamic ensembles for fractal measures
Fractal measures are characterized by means of suitable conservation laws which can be expressed either by introducing pointwise dimensions (local approach) or by evaluating global dynamical invariants like the dimension functionD(q).
The various thermodynamic ensembles are shown to be related to different covering procedures for fractal measures.
Non-parametric statistical estimation of the fractal measures hD max is finally applied.
Random recursive construction of self-similar fractal measures.
Let A be the Laplace-Beltrami operator on a Riemannian manifold M. The main aim of this paper is to study the Lp-boundedness of the map (I-t)-β,Lp(M,d)→Lp(M,dx). In particular, the author gives a new generalization of Wiener's theorem. Here,is a locally uniformly a-dimensional fractal measure on M,and dx is the volume element on M.
The obiect of this note is to establish two propositions on de-termining the Hausdorff dimension dim and the packing dimension Dim inR￣d(Theorem 1 and Theorem 2),and further seek the conditions that the Hausdorffdimension dim is equal to the pachng dimension Dim in R￣d(Theorem 3);thusthis enable introduction of a measure theory definition of fractal measures.