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.     

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.     

Stresses and strains in a deformable fractal medium and in its fractal continuum model

The model of fractal continuum accounting the topological, metric, and dynamic properties of deformable physical fractal medium is suggested. The kinematics of fractal continuum deformation is developed. The corresponding geometric interpretations are provided. The concept of stresses in the fractal continuum is defined. The conservation of linear and angular momentums is established. The mapping of mechanical problems for physical fractal media into the corresponding problems for fractal continuum is discussed.     

Dynamic conductivity of composites of fractal structure

We have studied the dynamic conductivity of a composite consisting of well- and weak-conducting components with random fractal structure. In order to calculate effective properties of composite medium, we used hierarchic structure model and innovative iterative averaging method based on renormalization group transformations idea. Our results show, that the behavior of a composite over a magnetic field become even more complicated.Unusual peaks and oscillations appear in frequency dependencies of effective conductivity, permittivity and other properties. We discuss the influence of fractal parameters of the composite structure on such unusual behavior of effective properties.     

Emergence of complex networks from diffusion on fractal lattices. A special case of the Sierpinski gasket and tetrahedron

A new approach to the assemblage of complex networks displaying the scale-free architecture is proposed. While the growth and the preferential attachment of incoming nodes assure an emergence of such networks according to the Barabási–Albert model, it is argued here that the preferential linking condition needs not to be a principal rule. To assert this statement a simple computer model based on random walks on fractal lattices is introduced. It is shown that the model successfully reproduces the degree distributions, the ultra-small-worldness and the high clustering arising from the topology of scale-free networks.     

Random distributed feedback fiber laser pumped by an ytterbium doped fiber laser

A random distributed feedback fiber laser operating at 1115 nm has been demonstrated experimentally in standard communication optical fibers by using a LD-pumped Yb-doped fiber laser as the pump source. We have studied the effect of different fiber spans on this new type of random fiber laser output power. It is shown that the generation power is the highest up to 198 mW in a 50 km fiber span. The slope efficiency is more than 28.7%. Stable, high-power continuous-wave (CW) lasing can be generated when the pump power is 3.6 W. The threshold power has also been calculated which well proves a random fiber laser operating via Rayleigh scattering, amplified through the Raman scattering.     

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.     

Noteworthy fractal features and transport properties of Cantor tartans

This Letter is focused on the impact of fractal topology on the transport processes governed by different kinds of random walks on Cantor tartans. We establish that the spectral dimension of the infinitely ramified Cantor tartan $d_s$ is equal to its fractal (self-similarity) dimension D. Consequently, the random walk on the Cantor tartan leads to a normal diffusion. On the other hand, the fractal geometry of Cantor tartans allows for a natural definition of power-law distributions of the waiting times and step lengths of random walkers. These distributions are Lévy stable if $D>1.5$. Accordingly, we found that the random walk with rests leads to sub-diffusion, whereas the Lévy walk leads to ballistic diffusion. The Lévy walk with rests leads to super-diffusion, if $D>\sqrt{3}$, or sub-diffusion, if $1.5<D<\sqrt{3}$.     

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.     

Organic glasses: cluster structure of the random energy landscape

An appropriate model for the random energy landscape in organic glasses is a spatially correlated Gaussian field. We calculated the distribution of the average value of a Gaussian random field in a finite domain. The results of the calculation demonstrate a strong dependence of the width of the distribution on the spatial correlations of the field. Comparison with the simulation results for the distribution of the size of the cluster indicates that the distribution of an average field could serve as a useful tool for the estimation of the asymptotic behavior of the distribution of the size of the clusters for “deep” clusters where value of the field on each site is much greater than the rms disorder. We also demonstrate significant modification of the properties of energetic disorder in organic glasses at the vicinity of the electrode.     

Dynamic structure factor in a random diffusion model

Let {X _t:0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew ^–1(X _t), where {w(x):x^d} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering function F(q,t)=E ₀ (q^d) is completely monotonic int (E ₀ denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factor S(q, w)=2 ₀ cos(t)F(q, t) exists for 0 and is strictly positive. Ind=1, 2 it diverges as 1/||^1/2, resp. –ln(||), in the limit 0; ind3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small such that ||D |q|², whereD is the diffusion constant.     

光纤陀螺随机漂移测试及分析

要实现光纤陀螺在高精度惯性导航领域的应用,抑制随机漂移是需解决的一个重要问题。光纤陀螺的随机漂移包含不同分量,分析不同的分量的特性及其产生的根源,是进行补偿的前提。设计了光纤陀螺数据采集系统,采集了光纤陀螺静态数据。应用Allan方差法,求出了光纤陀螺随机漂移中的噪声分量,并分析了各项噪声来源、特性,指出了抑制噪声方法。     

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.     

Effect of preparation conditions on fractal structure and phase transformations in the synthesis of nanoscale M-type barium hexaferrite

The conditions of the synthesis of carbonate-hydroxide precursors (pH of FeOOH precipitation and heat treatment regimes) were studied in terms of their effect on the fractal structure and physical-chemical properties of precursors. Phase transformations which occur during the synthesis of nanosize M-type barium hexaferrite (BHF) were studied as well.The first structural level of precursors' aggregation for mass fractals, the correlation between fractal dimension and precursors' activity during the synthesis of BHF were determined.Synthesis parameters for the precursors with the optimal fractal structure were determined. These data permit an enhancement of the filtration coefficient of the precipitates by a factor of 4-5, obtaining substantial decrease in the temperature required for synthesis of a single-phase BHF, and monodispersed plate-like nanoparticles (60 nm diameter) with the shape anisotropy and good magnetic characteristics (saturation magnetization (M_s)=68,7 emu/g and coercitivity (H_c)=5440 Oe).     

Solubility of fractal nanoparticles

A relationship between the fractal surface dimension of dissolving nanoparticles and their solubility has been derived showing that fractally rough nanoparticles are expected to exhibit significantly higher solubility than predicted from the classical Ostwald-Freundlich equation. The derived relationship estimates the latent solubility increase for a given material by manipulating the surface of the particles to be dissolved, and can thus be used for developing new strategies for improving the solubility of poorly soluble drug candidates.     

基于无芯-少模-无芯光纤结构的温度传感特性研究北大核心CSCD

提出了一种基于无芯-少模-无芯光纤结构的温度传感器,对传感器进行了理论分析和实验研究。该传感器将无芯光纤(coreless fiber,CLF)与少模光纤(few-mode fiber,FMF)同轴熔接,构建无芯-少模-无芯的光纤结构,结构两端熔接单模(single mode fiber,SMF)光纤作为输入输出光纤,第1段无芯光纤与单模光纤的模式失配起到激发高阶模的作用,少模光纤中的LP01与LP11两种模式沿少模光纤纤芯传输,在第2段无芯光纤的作用下LP01与LP11两种模式重新耦合回单模光纤,LP01与LP11两种模式发生干涉,形成干涉光谱。当外界温度变化时,两种模式的光程差发生变化,干涉光谱的干涉波谷发生漂移,选取2个不同的干涉波谷作为特征波长,进行实验分析。实验结果表明:波长在1550 nm和1534 nm附近的干涉谷均发生红移,相应的温度灵敏度分别为68 pm/℃和44.5 pm/℃。该传感结构制作简单、灵敏度高,有很好的应用前景。     

溶液蒸发冻结过程中冰体发展的分形模拟研究

根据溶液在蒸发冻结过程中的复杂性以及液面冰体分布的不规则,结合分形有限扩散聚集(DLA)理论,建立了湿度差作用下溶液蒸发冷冻过程中液面冰体生成、发展过程的数学模型,模拟分析了气流湿度差和气流流速对溶液表面冰体生成、发展的影响.结果表明,利用分形方法可以对溶液蒸发冻结过程中冰体的生成发展进行预测,同时也为研究溶液蒸发冻结和冰体生长提供了新的思路和方法.     

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.     

Scalar curvature of systems with fractal distribution functions

Starting with the relative entropy for two close statistical states we define the metric and calculate the scalar curvature R for systems with classical, boson and fermion fractal distribution functions with moment order parameter q  . In particular, we find that for q≠1$q\ne 1$ the scalar curvature is closer to zero implying that the fractal bosonic and fermionic systems are more stable than the standard ones.