HAUSDORFF DIMENSION OF GENERALIZED STATISTICALLY SELF-AFFINE FRACTALS

The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.

The autocorrelation of speckles in deep Fresnel diffraction region and characterizations of random self-affine fractal surfaces

This paper studies the correlation properties of the speckles in the deep Fresnel diffraction region produced by the scattering of rough self-affine fractal surfaces. The autocorrelation function of the speckle intensities is formulated by the combination of the light scattering theory of Kirchhoff approximation and the principles of speckle statistics. We propose a method for extracting the three surface parameters, i.e. the roughness w, the lateral correlation length ξ and the roughness exponent α, from the autocorrelation functions of speckles. This method is verified by simulating the speckle intensities and calculating the speckle autocorrelation function. We also find the phenomenon that for rough surfaces with α= 1, the structure of the speckles resembles that of the surface heights, which results from the effect of the peak and the valley parts of the surface, acting as micro-lenses converging and diverging the light waves.

THE UPPER BOUND OF BOX DIMENSION OF THE WEYL-MARCHAUD DERIVATIVE OF SELF-AFFINE CURVES

A certain type of self-affine curves can be transferred into fractal functions. The upper bound of fractal dimensions of the Weyl-Marchaud derivative of these functions has been investigated, and further questions have been put forwarded.

Study on Properties of Intensity Profiles Scattered from the Self-Affine Fractal Random Surfaces: an Approximate Theory and Simulations

We study the properties of the intensity profiles scattered from the self-affine fractal random surfaces.We use the mathematical decay function to approximate the duple negative exponent function in the rigorous theory of scattering,by letting them have the same maximum value and half-width,and the expression for the half-widths of the intensity profiles in the whole range of the perpendicular wave vector component is obtained.The previous results in the two extreme cases are included in the results of this paper.In the simulational verification,we propose a method for the generation of self-affine fractal random surfaces,using the square-root of Fourier transform of the correlation function of the surface height.The simulated results conform well with the theory.

REFINABLE DISTRIBUTIONS SUPPORTED ON SELF-AFFINE TILES

In this paper,some conditions which assure the compactly supported refinable distributions supported on a self-affine tile to be Lebesgue-Stieltjes measures or absolutely continuous measures with respect to Lebesgue-Stieltjes measures are given.

Self-Affine Fractals and the Fractal Dimension of Fractured Rock Profiles

The measured profiles of laboratory fractured rocks should be self-affine fractal.The scaling properties of these profiles are described by two parameters-the fractal dimension D and the crossover length tc The D values of eight profiles are calculated by the ruler method and by the standard deviation method respectively.It is shown that if tc is far greater than the sampling step tc two methods yield the same results,although if it is far smaller than r,the D by the standard method will be about 1.20,while D by the ruler method will very close to 1.0,because two fractal dimensions,local and global,exist on two sides of tc In order to obtain the local fractal dimension which may be close to that of the standard deviation method,the ruler method must be modified.We propose a way to estimate the tc and to modify the ruler method.Finally,a profile having given D is generated in terms of the principle of non-integer order differential,through which the above two methods are verified and lead to the same

Non-Spectral Problem of Self-Affine Measures in R³

The problems of determining the spectrality or non-spectrality of a measure have been received much attention in recent years. One of the non-spectral problems on μM,D is to estimate the number of orthogonal exponentials in L2(μM,D). In the present paper, we establish some relations inside the zero set by the Fourier transform of the self-affine measure μM,D. Based on these facts, we show that μM,D is a non-spectral measure and there exist at most 4 mutually orthogonal exponential functions in L2(μM,D), where the number 4 is the best possible. This extends several known conclusions.

Spectrality of planar self-affine measures with two-element digit set

The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the self- affine measures have some surprising connections with a number of areas in mathematics, and have been received much attention in recent years. In the present paper, we shall determine the spectrality and non-spectrality of a class of self-aiffine measures with decomposable digit sets. We present a method to deal with such case, and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.

Spectral self-affine measures on the planar Sierpinski family

The present research will concentrate on the topic of Fourier analysis on fractals.It mainly deals with the problem of determining spectral self-affine measures on the typical fractals:the planar Sierpinski family.The previous researches on this subject have led to the problem within the possible fifteen cases.We shall show that among the fifteen cases,the nine cases correspond to the spectral measures,and reduce the remnant six cases to the three cases.Thus,for a large class of such measures,their spectrality and non-spectrality are clear.Moreover,an explicit formula for the existent spectrum of a spectral measure is obtained.We also give a concluding remark on the remnant three cases.

Experimental determination of Hurst exponent of the self-affine fractal patterns with optical fractional Fourier transform

By means of experimental technique of optical fractional Fourier transform, we have determined the Hurst exponent of a regular self-affine fractal pattern to demonstrate the feasibility of this approach. Then we extend this method to determine the Hurst exponents of some irregular self-affine fractal patterns. Experimental results show that optical fractional Fourier transform is a practical method for analyzing the self-affine fractal patterns.

DIMENSIONS OF SELF-AFFINE SETS WITH OVERLAPS

The authora develop an algorithm to show that a class of self-affine sets with overlaps can be viewed as sofic affine-invariant sets without overlaps,thus by using the results of [11] and [10],the hausdorff and minkowski dimensions are detemined

Asperity-based modification on theory of contact mechanics and rubber friction for self-affine fractal surfaces

Modeling the real contact area plays a key role in every tribological process,such as friction,adhesion,and wear.Contact between two solids does not necessarily occur everywhere within the apparent contact area.Considering the multiscale nature of roughness,Persson proposed a theory of contact mechanics for a soft and smooth solid in contact with a rigid rough surface.In this theory,he assumed that the vertical displacement on the soft surface could be approximated by the height profile of the substrate surface.Although this assumption gives an accurate pressure distribution at the interface for complete contact,when no gap exists between two surfaces,it results in an overestimation of elastic energy stored in the material for partial contact,which typically occurs in many practical applications.This issue was later addressed by Persson by including a correction factor obtained from the comparison of the theoretical results with molecular dynamics simulation.This paper proposes a different approach to correct the overestimation of vertical displacement in Persson’s contact theory for rough surfaces with self-affine fractal properties.The results are compared with the correction factor proposed by Persson.The main advantage of the proposed method is that it uses physical parameters such as the surface roughness characteristics,material properties,sliding velocity,and normal load to correct the model.This method is also implemented in the theory of rubber friction.The results of the corrected friction model are compared with experiments.The results confirm that the modified model predicts the friction coefficient as a function of sliding velocity more accurately than the original model.

Porosity of Self-affine Sets

In this paper, it is proved that any self-affine set satisfying the strong separation condition is uniformly porous. The author constructs a self-affine set which is not porous, although the open set condition holds. Besides, the author also gives a C^1 iterated function system such that its invariant set is not porous.

Multi-Dimensional Piece-Wise Self-Affine Fractal Interpolation Model

Iterated function system （IFS） models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 （R is the set of real numbers）. In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn （n is an integer greater than 3）, which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a ＂mapping partial derivative＂, and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in R^n. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.

Integral Self-affine Tiles of Bandt's Model

Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equals to the order of the associate symmetry group. We apply the result to the study of the Levy Dragon.

Double-porosity rock model and squirt flow in the laboratory frequency band

Biot theory research has been extended to the multi-scale heterogeneity in actual rocks. Focused on laboratory frequency bandwidth studies, we discuss the relationships between double-porosity and BISQ wave equations, analytically derive the degeneration method for double-porosity＇s return to BISQ, and give three necessary conditions which the degeneration must satisfy. By introducing dynamic permeability and tortuosity theory, a full set of dynamic double-porosity wave equations are derived. A narrow band approximation is made to simplify the numerical simulation for dynamic double-porosity wavefields. Finally, the pseudo-spectral method is used for wave simulation within the laboratory frequency band （50 kHz）. Numerical results have proved the feasibility for dynamic double-porosity＇s description of squirt flow and the validity of the quasi-static approximation method.

Pronounced Fluctuations of Target Fragments in Forward Hemisphere Only in Ultra Relativistic Nuclear Collision

Dynamical fluctuation of target evaporated black particles is investigated in both forward and backward hemispheres within the framework of multi-dimensional factorial moment methodology using the brilliant concept of the Hurst exponent. We analyse the black particles emitted in ^32S-AgBr interactions at 200AGeV and it is evident that the dynamical fluctuation in the backward hemisphere is self-affine. In the forward hemisphere, dynamical fluctuation is self-similar but not self-affine. However, study indicates that the fluctuation in the forward hemisphere is more pronounced than that in backward hemisphere.

Self－affine fractal features of earthquake time series before and after moderate earthquakes

In this paper we calculate the local fractal dimension values D of the self-affine feature of earthquake time series by RMS (root-mean-square) error method, and express the fractal dimensionality by the normalized correlation coefficient R. The fractal dimension values are given for earthquakes occurred in Tangshan, Haicheng, Songpan, Longling, Changshu, I.iyang in China and its vicinity by the moving scanning method with different magnitude thresholds and the fixed-window length (100 events). The results show the D values are characterized by decreasing, continued low level in values or by decreasing first and then increasing before moderate earthquakes.

Determination of water level design for an estuarine city

Based on the extreme value theory, self-affinity, and scale invariance, we studied the temporal and spatial relationship and the variation of water level and established a model of Gumbel-Pareto distribution for designed flood calculation. The model includes the previous extreme value models, the over-threshold data, and the fractal features shared by previous extreme value models. The model was simplified into a logarithmic normal distribution and a Pareto distribution for specific parameter values, and was used to calculate the designed flood values for the Shanghai Wusong Station in 100- and 1 000-year return periods. The calculated results show that the value of the designed flood height calculated in the Gumbel-Pareto distribution is between those in the Gumbel and Pearson-Ⅲ distributions. The designed flood values in the 100- and 1 000-year return periods of the model were 0.03% and 0.11% lower, respectively, than the Gumbel distribution and 0.06% and 1.54% higher, respectively, than the Pearson-Ⅲ distribution. Compared to the traditional model based solely on extreme probability, the Gumbel-Pareto distribution model could better describe the probabilistic characteristics of extreme marine elements and better use the data.