基于分形编码的天体光谱降噪方法

为了降低噪声影响,恢复光谱的连续谱和谱线等主要特征,以便准确测量谱线的等值宽度,文章根据天体光谱自身具有局部自相似性,而随机白噪声不具有自相似性,首次将分形方法应用于天体光谱降噪.实验表明,分形降噪方法对于准确测量谱线的等值宽度、星系红移等参数是有效的,此外,还可以实现数据压缩.分形方法适用于海量光谱的降噪和数据存储.     

Surface fractal analysis of pore structure of high-volume fly-ash cement pastes

The surface fractal dimensions of high-volume fly-ash cement pastes are evaluated for their hardening processes on the basis of mercury intrusion porosimetry (MIP) data. Two surface fractal models are retained: Neimark's model with cylindrical pore hypothesis and Zhang's model without pore geometry assumption. From both models, the logarithm plots exhibit the scale-dependent fractal properties and three distinct fractal regions (I, II, III) are identified for the pore structures. For regions I and III, corresponding to the large (capillary) and small (C-S-H inter-granular) pore ranges respectively, the pore structure shows strong fractal property and the fractal dimensions are evaluated as 2.592-2.965 by Neimark's model and 2.487-2.695 by Zhang's model. The fractal dimension of region I increases with w/b ratio and hardening age but decreases with fly-ash content by its physical filling effect; the fractal dimension of region III does not evolve much with these factors. The region II of pore size range, corresponding to small capillary pores, turns out to be a transition region and show no clear fractal properties. The range of this region is much influenced by fly-ash content in the pastes. Finally, the correlation between the obtained fractal dimensions and pore structure evolution is discussed in depth.     

Lagrange formalism for particles moving in a space of fractal dimension

Analogs of the Lagrange equation for particles evolving in a space of fractal dimension are obtained. Two cases are considered: 1) when the space is formed by a set of material points (a so-called fractal continuum), and 2) when the space is a true fractal. In the latter case the fractional integrodifferential formalism is utilized, and a new principle for devising a fractal theory, viz., a generalized principle of least action, is proposed and used to obtain the corresponding Lagrange equation. The Lagrangians for a free particle and a closed system of interacting particles moving in a fractal continuum are derived. Zh. Tekh. Fiz. 68, 7–11 (February 1990)     

A physically based connection between fractional calculus and fractal geometry

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.     

Physical mechanisms of power fractal asymptotic forms of dispersion transport in disordered media

Particle drift in systems with anomalous diffusion is investigated. Physical mechanisms of power fractal asymptotic forms in dispersion transport are established and the physical meaning of the characteristic changeover time for asymptotic forms is clarified. It is shown that long-term power fractal asymptotic forms for particle mobility in subdiffusion problems corresponding to the behavior of transition currents in disordered systems (i.e., having different asymptotic forms for short and long time intervals) are associated with capture in traps (ribs in the comb structure).     

Hierarchical Monte Carlo methods for fractal random fields

Two hierarchical Monte Carlo methods for the generation of self-similar fractal random fields are compared and contrasted. The first technique, successive random addition (SRA), is currently popular in the physics community. Despite the intuitive appeal of SRA, rigorous mathematical reasoning reveals that SRA cannot be consistent with any stationary power-law Gaussian random field for any Hurst exponent; furthermore, there is an inherent ratio of largest to smallest putative scaling constant necessarily exceeding a factor of 2 for a wide range of Hurst exponentsH, with 0.30<H<0.85. Thus, SRA is inconsistent with a stationary power-law fractal random field and would not be useful for problems that do not utilize additional spatial averaging of the velocity field. The second hierarchical method for fractal random fields has recently been introduced by two of the authors and relies on a suitable explicit multiwavelet expansion (MWE) with high-moment cancellation. This method is described briefly, including a demonstration that, unlike SRA, MWE is consistent with a stationary power-law random field over many decades of scaling and has low variance.     

Mapping physical problems on fractals onto boundary value problems within continuum framework

In this Letter, we emphasize that methods of fractal homogenization should take into account a loop structure of the fractal, as well as its connectivity and geodesic metric. The fractal attributes can be quantified by a set of dimension numbers. Accordingly, physical problems on fractals can be mapped onto the boundary values problems in the fractional-dimensional space with metric induced by the fractal topology. The solutions of these problems represent analytical envelopes of non-analytical functions defined on the fractal. Some examples are briefly discussed. The interplay between effects of fractal connectivity, loop structure, and mass distributions on electromagnetic fields in fractal media is highlighted. The effects of fractal connectivity, geodesic metric, and loop structure are outlined.     

The scattering of electromagnetic waves in fractal media

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

The study of fractal correlation method in the displacement measurement and its application

The classical digital speckle, or digital image, correlation method of deformation measurement is based on gray level correlation between unformed and deformed digital images. The pattern of artificial random speckles and the natural textures on some object's surfaces have fractal characteristics, and their fractal dimensions represent both gray and morph information. Furthermore, the fractal dimensions are stable feature parameters of the patterns. The digitized images of the patterns are confirmed to be also fractals. By this fact, a new method of displacement measurement is developed in the paper, based on the fractal dimensions correlation. The in-plane displacement fields of a body can be acquired. In order to verify the validity of the new method, an experiment has been designed and the results have been compared with those obtained from the classical digital image correlation method. The validity of the new method is not less than that of classical method. Further discussions about the traits and the developing vista of the method are given at the end.     

The scattering of electromagnetic waves in fractal media

Abstract

The scattering of electromagnetic waves in fractal media is studied. The fractal dimension is naturally involved in the formulation of two physical problems studied in this paper. The general theory of multiple scattering of electromagnetic wave in fractal media is developed by modifying Twersky's theory. Statistical quantities, such as the average field and average intensity of the multiple scattered wave, are studied for a wave propagating in a fractal medium. The scattering cross section of the medium is deduced. The backscattering of electromagnetic waves is also studied. The results showing the range of dependence of the backscattered signals are in agreement with numerical simulations by Rastogi and Scheucher (1990). It also suggests a method of measuring the fractal dimension of the fractal embedded media using radar sounding. The theory developed in this paper can also be used for problems related to multiple scattering of other kinds of waves, such as acoustic waves, elastic waves etc, in fractal media.     

Moisture diffusivity in structure of random fractal fiber bed

A theoretical expression related to effective moisture diffusivity to random fiber bed is derived by using fractal theory and considering both parallel and perpendicular channels to diffusion flow direction. In this Letter, macroporous structure of hydrophobic nonwoven material is investigated, and Knudsen diffusion and surface diffusion are neglected. The effective moisture diffusivity predicted by the present fractal model are compared with water vapor transfer rate (WVTR) experiment data and calculated values obtained from other theoretical models. This verifies the validity of the present fractal diffusivity of fibrous structural beds.     

The fractal energy measurement and the singularity energy spectrum analysis

The singularity exponent (SE) is the characteristic parameter of fractal and multifractal signals. Based on SE, the fractal dimension reflecting the global self-similar character, the instantaneous SE reflecting the local self-similar character, the multifractal spectrum (MFS) reflecting the distribution of SE, and the time-varying MFS reflecting pointwise multifractal spectrum were proposed. However, all the studies were based on the depiction of spatial or differentiability characters of fractal signals. Taking the SE as the independent dimension, this paper investigates the fractal energy measurement (FEM) and the singularity energy spectrum (SES) theory. Firstly, we study the energy measurement and the energy spectrum of a fractal signal in the singularity domain, propose the conception of FEM and SES of multifractal signals, and investigate the Hausdorff measure and the local direction angle of the fractal energy element. Then, we prove the compatibility between FEM and traditional energy, and point out that SES can be measured in the fractal space. Finally, we study the algorithm of SES under the condition of a continuous signal and a discrete signal, and give the approximation algorithm of the latter, and the estimations of FEM and SES of the Gaussian white noise, Fractal Brownian motion and the multifractal Brownian motion show the theoretical significance and application value of FEM and SES.     

Elastic properties of a disordered continuum of regular fractal structure: Self-consistent approach and finite-element method simulations

Macromolecular structures, as well as aggregation of filler in polymer-based composites, often may be described properly as fractals. Scaling behavior of the elastic moduli of a modeled fractal, the Sierpinski carpet, was the subject of this study. Sheng and Tao [1] and Patlazhan [2] found that, in the case of voids in on elastic host, axial and shear moduli exhibit distinct scaling dependencies on the size of the system. Nevertheless, it is widely accepted that moduli of random isotropic fractals (percolation clusters) scale with the same exponents. Explanation of the discrepancy is one of the main targets of the paper. The self-consistent approach and position space renormalization group technique (PSRG) have been applied for this goal. The mapping, corresponding to PSRG, was constructed numerically using the finite-element method (FEM) in the cases of voids and rigid inclusions. The self-consistent approach gives scaling behavior with exponents of values of about 0.11, independent of the modulus and type of inclusion, at developed stages of the fractal. It has been shown that mappings of PSRG on the plane, for two ratios of three independent moduli, have stable fixed points. This means that different elastic moduli exhibit scaling behavior with the same exponents (0.29 for voids and 0.17 for rigid squares) for developed fractal structure. The discrepancy in the exponent values obtained in the previous simulations is caused by the analysis of the initial stages of the structure. We believe that analogous results are valid for the wide class of self-similar fractals, and the dimension is the main parameter that governs the exponents and fixed point values.     

Spiral waves in systems with fractal heterogeneity

The influence of fractal heterogeneity on a spiral wave in an excitable system is numerically studied based on the Barkley model. The heterogeneity is implemented by letting the diffusive coefficient in the heterogeneous area be different from the other area. The results show that fruitful transitions of the spiral tip trajectories are induced by the fractal heterogeneity. In particular, when the heterogeneity increases to a sufficiently high level the spiral tip trajectory always changes to a stable rotating trajectory (closed-circle tip trajectory), whatever transitions have been induced by a lower level of heterogeneity. We qualitatively ascribe the transitions to the attraction on the spiral tip exerted by the heterogeneous area.     

A highly efficient algorithm for the generation of random fractal aggregates

A technique to generate random fractal aggregates where the fractal dimension is fixed a priori is presented. The algorithm utilizes the box-counting measure of the fractal dimension to determine the number of hypercubes required to encompass the aggregate, on a set of length scales, over which the structure can be defined as fractal. At each length scale the hypercubes required to generate the structure are chosen using a simple random walk which ensures connectivity of the aggregate. The algorithm is highly efficient and overcomes the limitations on the magnitude of the fractal dimension encountered by previous techniques.     

A fractal streaming current model for charged microscale porous media

An analytical expression for the streaming current in fractal porous media is developed based on the capillary model and the fractal theory for porous media. The proposed fractal model is expressed as a function of the space charge density at the solid–liquid interface, the fluid flow rate, the Debye–Huckel parameter, the minimum and maximum pore/capillary radii and fractal dimensions for porous media. The results are compared with available experimental data and good agreement is found between them. In addition, factors influencing the streaming current in porous media are also analyzed.     

Real-time fractal signal processing in the time domain

Fractal analysis has proven useful for the quantitative characterization of complex time series by scale-free statistical measures in various applications. The analysis has commonly been done offline with the signal being resident in memory in full length, and the processing carried out in several distinct passes. However, in many relevant applications, such as monitoring or forecasting, algorithms are needed to capture changes in the fractal measure real-time. Here we introduce real-time variants of the Detrended Fluctuation Analysis (DFA) and the closely related Signal Summation Conversion (SSC) methods, which are suitable to estimate the fractal exponent in one pass. Compared to offline algorithms, the precision is the same, the memory requirement is significantly lower, and the execution time depends on the same factors but with different rates. Our tests show that dynamic changes in the fractal parameter can be efficiently detected. We demonstrate the applicability of our real-time methods on signals of cerebral hemodynamics acquired during open-heart surgery.     

Scaling reduction of the contrast of fractal speckles detected with a finite aperture

It is known that speckle patterns with fractal properties, called fractal speckles, are produced by illuminating a diffuser with the coherent light having the intensity distribution obeying a negative power law. One of key properties of fractal speckles is the spatial correlation function obeying a negative power law, which implies that such speckles have scaling properties. In detecting fractal speckles, the effect of the spatial integration is inevitable in most cases since they have speckle grains of various scales including very fine ones. To evaluate this effect, in this paper, the contrast of spatially integrated intensity distributions is investigated theoretically and experimentally for fractal speckles. The results show that the contrast reduction with the size of the detector aperture obeys a negative power function related with the exponent of the intensity correlation coefficient of fractal speckles.     

Propagation of a wavepacket on a model fractal lattice

Dynamics of a wavepacket on a model fractal surface has been studied by solving the pertinent time dependent Schrödinger equation. Spatial and temporal behaviour of charge and current densities and a local chemical potential for two different fractal lattices have been considered. Important insight into the dynamics has been obtained through time dependence of various quantities like macroscopic kinetic energy, global current, Shanon entropy, density correlation and global chemical potential. This study would be helpful in simulating adsorption and catalysis.