Fractals and dynamical chaos in a random 2D Lorentz gas with sinks

Two-dimensional random Lorentz gases with absorbing traps are considered in which a moving point particle undergoes elastic collisions on hard disks and annihilates when reaching a trap. In systems of finite spatial extension, the asymptotic decay of the survival probability is exponential and characterized by an escape rate γ, which can be related to the average positive Lyapunov exponent and to the dimension of the fractal repeller of the system. For infinite systems, the survival probability obeys a stretched exponential law of the form P(c,t)∼exp(−Ct^1/2). The transition between the two regimes is studied and we show that, for a given trap density, the non-integer dimension of the fractal repeller increases with the system size to finally reach the integer dimension of the phase space. Nevertheless, the repeller remains fractal. We determine the special scaling properties of this fractal.     

Fractal Germanium Patterns: Annealing Strategies and Perspectives of Metal-Induced Crystallization

Semiconductor germanium (Ge) in contact with some metals, such as Al, Pd, and Au, etc., is a class of distinctive materials with non-integer dimensions (D) that differ from integer dimensional materials, such as nanoparticles (0D), nanowires/nanorods//nanotubes/nanoribbons (1D), and thin films (2D). In this article, we describe our efforts toward understanding the annealing strategies and perspectives of metal-induced crystallization for the amorphous Ge embedded in Al, Pd, and Au matrices prepared by high vacuum thermal evaporation techniques, highlighting contributions from our laboratory. First, we present the Al-induced crystallization of amorphous Ge and formation processes of fractal Ge patterns. In addition, the fractal Ge patterns induced by Pd nanoparticles with solid-state reactions will be summarized in detail. Temperature-dependent properties of resistance and fractal dimension in Pd/Ge bilayer films will be expounded. In particular, the nonlinear optical properties are discussed in detail. Finally, we will emphasize the in situ observations by transmission electron microscopy and multi-fractal analysis for the fractal Ge patterns induced by Au nanoparticles. Moreover, the polycondensation-type fractal Ge patterns with non-integer dimensions, thick branches and smooth edges, and metastable gamma-Au_0.6Ge_0.4 are further investigated. The computer simulation indicated that the experimental results are good agreement with the simulation patterns, which were carried out by a ripening mechanism of non uniform grains. This review may provide a novel insight to modulate their competent performance and promote rational design of micro/nanodevices.     

Calculation of Spectral Dimension on Inhomogeneous Fractal Lattices

We propose a scheme to derive the spectral dimension of inhomogeneous fractal lattice via renormalization procedure, in which the distribution of masses on the sites of fractals is introduced. The spectral dimension of diamond-type hierarchical lattice and Sierpinski gasket with b = 3 are re-investigated in this way. Moreover, the variants of Sierpinski gaskettype fractals are studied, the results show that the spectral dimension is independent of the details of internal structure of fractal, and thus implies the existence of universality. The source of the universality is also analyzed.     

Fractional Dynamics from Einstein Gravity,General Solutions,and Black Holes

We study the fractional gravity for spacetimes with non-integer fractional derivatives. Our constructions are based on a formalism with the fractional Caputo derivative and integral calculus adapted to nonholonomic distributions. This allows us to define a fractional spacetime geometry with fundamental geometric/physical objects and a generalized tensor calculus all being similar to respective integer dimension constructions. Such models of fractional gravity mimic the Einstein gravity theory and various Lagrange–Finsler and Hamilton–Cartan generalizations in nonholonomic variables. The approach suggests a number of new implications for gravity and matter field theories with singular, stochastic, kinetic, fractal, memory etc processes. We prove that the fractional gravitational field equations can be integrated in very general forms following the anholonomic deformation method for constructing exact solutions. Finally, we study some examples of fractional black hole solutions, ellipsoid gravitational configurations and imbedding of such objects in solitonic backgrounds.     

Subcooled pool boiling heat transfer in fractal nanofluids:A novel analytical model

A novel analytical model to determine the heat flux of subcooled pool boiling in fractal nanofluids is developed. The model considers the fractal character of nanofluids in terms of the fractal dimension of nanoparticles and the fractal dimen- sion of active cavities on the heated surfaces; it also takes into account the effect of the Brownian motion of nanoparticles, which has no empirical constant but has parameters with physical meanings. The proposed model is expressed as a function of the subcooling of fluids and the wall superheat. The fractal analytical model is verified by a reasonable agreement with the experimental data and the results obtained from existing models.     

Network analysis of the Íslendinga sögur – the Sagas of Icelanders

It is well-known that cooperative properties such as magnetic ordering can depend on the samples’ dimensions (Ds) in a qualitative way. However, there have been no samples with well-defined non-integer Ds. The dimension of a given sample has been always discussed on the anisotropy of the electronic/crystal/magnetic structures, which has no definition suitable for quantitative discussion on dimensions vs. properties. On the other hand a particular type of porous samples, i.e. fractal bodies, can have well-defined non-integer Ds dependent exclusively on the geometrical feature of structures, and physical properties of such materials remains unexplored. This paper reports on magnetic ordering in samples covering 2.5 ≤ D ≤ 3, in addition to a way of precise control of the fractal dimensions of given samples simply by wax (alkylketene dimer). The results show that the magnetic ordering temperatures, i.e. Néel temperatures (T _N s), of CoO depend on D, and rapidly enhance immediately below D = 3. This means that one can control or enhance the critical temperature simply by tuning D with keeping the remaining magnetic properties unchanged.     

Fractals in DNA sequence analysis

Fractal methods have been successfully used to study many problems in physics,mathematics,engineering,finance,and even in biology,There has been an increasing interest in unravelling the mysteries of DNA;for example,how can we distinguish coding and noncoding sequences,and the problems of classification and evolution relationship of organisms are key problems in bioinformatics,Although much research has been carried out by taking into consideration the long-range correlations in DNA sequences,and the global fractal dimension has been used in these works by other people,the models and methods are somewhat rough and the results are not satisfactory.In recent years,our group has introduced a time series model(statistical point of view)and a visual representation (geometrical point of view) to DNA sequence analysis.We have also used fractal dimension,correlation dimension,the Hurst exponent and the dimension spectrum (multifractal analysis)to discuss problems in this field.In this paper,we introduce these fractal models and methods and the results of DNA sequence analysis.     

Fractal dimensions of speech sounds: computation and application to automatic speech recognition

The dynamics of airflow during speech production may often result in some small or large degree of turbulence. In this paper, the geometry of speech turbulence as reflected in the fragmentation of the time signal is quantified by using fractal models. An efficient algorithm for estimating the short-time fractal dimension of speech signals based on multiscale morphological filtering is described, and its potential for speech segmentation and phonetic classification discussed. Also reported are experimental results on using the short-time fractal dimension of speech signals at multiple scales as additional features in an automatic speech-recognition system using hidden Markov models, which provide a modest improvement in speech-recognition performance.     

Fractal properties of aggregates of metal nanoclusters on solid surface

AFM images are used to determine and analyze fractal characteristics (cluster fraction dimension and lacunarity) of aggregates of Au and Ag nanoclusters on metal films of the same metal produced with the aid of thermal vacuum deposition on mica surface. A fractal dimension of 1.6 that corresponds to typical samples with relatively uniform distribution of nanoclusters on the film surface is in agreement with the mean value calculated from experimental data of Belko et al., who studied the fractal dimension of Au nanoclusters on a different dielectric (quartz) surface. When a compact single aggregate of Au nanoclusters is formed on a certain active center or defect, the fractal cluster dimension decreases to 1.4. The experimental data are compared with the results of existing theoretical models of association of nanoclusters in 2D systems.     

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.     

基于显微CT图像的岩芯孔隙分形特征

采用国内自主开发的高分辨率显微CT设备,对岩芯样本进行图像采集,运用数字图像分析方法获得图像中的岩芯孔隙目标,并以此为基础重建三维模型。然后基于分形理论,分别从二维图像和重建的三维模型角度计算孔隙的分形维数。最后,再采用基于多孔介质的分形模型对岩芯孔隙分形特征进行验证,得出岩芯的二维分形维数和三维分形维数之间的差值均值为1.000 3,最大偏差为0.004,很好地满足多孔介质分形模型中关于二维和三维分形维数的关系。     

基于显微CT图像的岩芯孔隙分形特征

采用国内自主开发的高分辨率显微CT设备,对岩芯样本进行图像采集,运用数字图像分析方法获得图像中的岩芯孔隙目标,并以此为基础重建三维模型。然后基于分形理论,分别从二维图像和重建的三维模型角度计算孔隙的分形维数。最后,再采用基于多孔介质的分形模型对岩芯孔隙分形特征进行验证,得出岩芯的二维分形维数和三维分形维数之间的差值均值为1.000 3,最大偏差为0.004,很好地满足多孔介质分形模型中关于二维和三维分形维数的关系。     

KDP晶体单点金刚石车削表面形貌分形分析

 分别使用2维和3维分形方法对单点金刚石车削加工的KDP晶体表面形貌进行了分析,并对表面的3维分形维数和3维粗糙度表征参数进行了比较,分析了二者对表面形貌表征的差异。使用2维轮廓分形方法计算了KDP晶体表面圆周各方向上的分形维数。通过分析得出：3维分形维数与表面粗糙度值成反比关系;使用单点金刚石车削方法加工KDP晶体会形成各向异性特征明显的已加工表面,在一定程度上容易形成小尺度波纹;已加工表面是否具有明显的小尺度波纹特征与表面粗糙度值并无直接关系,但与其表面轮廓分形状态分布密切相关;KDP晶体表面2维功率谱密度与其分形状态具有相近的方向性特征。     

Weighted Fractal Networks

In this paper we define a new class of weighted complex networks sharing several properties with fractal sets, and whose topology can be completely analytically characterized in terms of the involved parameters and of the fractal dimension. General networks with fractal or hierarchical structures can be set in the proposed framework that moreover could be used to provide some answers to the widespread emergence of fractal structures in nature.     

KDP晶体单点金刚石车削表面形貌分形分析

分别使用2维和3维分形方法对单点金刚石车削加工的KDP晶体表面形貌进行了分析,并对表面的3维分形维数和3维粗糙度表征参数进行了比较,分析了二者对表面形貌表征的差异。使用2维轮廓分形方法计算了KDP晶体表面圆周各方向上的分形维数。通过分析得出：3维分形维数与表面粗糙度值成反比关系;使用单点金刚石车削方法加工KDP晶体会形成各向异性特征明显的已加工表面,在一定程度上容易形成小尺度波纹;已加工表面是否具有明显的小尺度波纹特征与表面粗糙度值并无直接关系,但与其表面轮廓分形状态分布密切相关;KDP晶体表面2维功率谱密度与其分形状态具有相近的方向性特征。     

Electron propagators based on generalised density operators

ABSTRACT

Electron binding energies and Dyson orbitals may be obtained from the poles and residues of the electron propagator. The Dyson quasiparticle equation provides a convenient route to computing this information. Systematic approximations to the latter equation's self-energy, wherein electron correlation and final-state orbital relaxation are described, may be expressed in terms of the elements of the superoperator Hamiltonian matrix. Perturbative methods of electron propagator theory in wide use are based on a reference determinant constructed with canonical, Hartree–Fock orbitals. Generalised matrix elements of the superoperator Hamiltonian that accommodate non-integer occupation numbers associated with general, orthogonal spin orbitals are presented for the first time. Non-Hermitian terms may be systematically eliminated with perturbative corrections to generalised reference density operators. The structure of self-energy approximations that are complete through second, third, fourth or fifth order is presented in terms of superoperator Hamiltonian matrix elements. The present extensions pertain when generalised, zeroth-order density operators expressed in terms of orthonormal spin orbitals are employed.     

Topological Self-Similar Networks Introduced by Diffusion-Limited Aggregation Mechanism

We propose a model for growing fractal networks based on the mechanisms learned from the diffusion-limited aggregation （DLA） model in fractal geometries in the viewpoint of network. By studying the DLA network, our model introduces multiplicative growth, aging and geographical preferential attachment mechanisms, whereby featuring topological self-similar property and hierarchical modularity. According to the results of theoretical analysis and simulation, the degree distribution of the proposed model shows a mixed degree distribution （i.e., exponential and algebraic degree distribution） and the fractal dimension and clustering coefficient can be tuned by changing the values of parameters.     

高速铁路轮轨滚动噪声的分形描述及分形维估计

提出基于分形分析的轮轨滚动噪声研究方法,研究了不同尺度下轮轨滚动噪声增量的统计特性,证明轮轨滚动噪声具有分形特性,可由分形布朗运动描述。在此基础上,利用分形布朗运动的小波系数方差与分形维的幂律关系估计了轮轨滚动噪声的分形维,并估计了以裂纹扩展为代表的钢轨伤损声发射信号的分形维,对比研究发现:轮轨滚动噪声分形维本身具有随机性,不同车速下的轮轨滚动噪声分形维估计以1.5666为均值随机分布在小于2的小区间内;分形维是轮轨滚动噪声的固有特征,与车速无关,不同车速的轮轨滚动噪声可由统一的分形布朗运动模型描述;裂纹扩展信号与轮轨滚动噪声不同,其分形维估计大于2,不满足分形布朗运动模型的要求,但分形维可作为二者的区别特征。研究提取了轮轨滚动噪声的分形维作为固有特征,为高速轮轨滚动噪声描述及轮轨伤损检测提供了有效的分析方法。     

粗糙表面形貌对湿润性的影响

采用自仿射分形对粗糙表面形貌进行了定量描述。基于分形粗糙表面形貌,建立了粗糙表面润湿性的理论模型并进行了数值计算,分析讨论了均方根粗糙度和表面分形维数对接触角的影响。研究结果表明,均方根粗糙度对接触角的影响较大,而粗糙表面的分形维数对润湿性的影响则并不明显。