Fractal features of size distribution of Chinese intercity bus hubs

Size distribution characteristics of intercity bus hubs in China from 1997 to 2（104 were analyzed regarding highway passenger volume as a size index of intercity bus hubs. Yearly fractal dimensions of intercity bus hub sizes were exactly calculated by a novel model. Fractal dimensions of the 200 biggest intercity bus hubs from 2000 to 2004 were 1. 486 2 to 1. 511 8, and that is consistent with fractal dimensions of Chinese urban system sizes. It showed that the size distribution of intercity bus hubs had fractal structure. Fractal dimensions from 1997 to 2004 indicated that intercity bus hub size distribution grew from bi-fractal to single fractal. It is concluded that the intercity bus hub system is in evolutionary progress, and the Central Government should support large intercity bus hubs more to optimize system structure.

Fractal analysis on the spatial distribution of acoustic emission in the failure process of rock specimens

The spatial distribution of acoustic emission （AE） events in the failure process of several rock specimens was acquired using an advanced AE acquiring and analyzing system. The box counting method （BCM） was employed to calculate the fractal dimension （FD） of AE spatial distribution. There is a similar correlation between the fractal dimension and the load strength for different rock specimens. The fractal dimension presents a decreasing trend with the increase of load strength. For the same kind of specimens, their FD values will decrease to the level below a relatively same value when they reach failure. This value can be regarded as the critical value, which implies that the specimen will reach failure soon. The results reflect that it is possible to correlate the damage of rock with a macroscopic parameter, the FD value of AE signals. Furthermore, the FD value can be also used to forecast the final failure of rock. This conclusion allows identifying or predicting the damage in rock with a great advantage over the classic theory and is very crucial for forecasting rockburst or other dynamic disasters in mines.

Applicability of Fractal Models in Estimating Soil Water Retention Characteristics from Particle-Size Distribution Data

Soil water retention characteristics are the key information required in hydrological modeling. Frac-tal models provide a practical alternative for indirectly estimating soil water retention characteristics fromparticle-size distribution data. Predictive capabilities of three fractal models, i.e, Tyler-Wheatcraft model,Rieu-Sposito model, and Brooks-Corey model, were fully evaluated in this work using experimental datafrom an international database and literature. Particle-size distribution data were firstly interpolated into20 classes using a van Genuchten-type equation. Fractal dimensions of the tortuous pore wall and the poresurface were then calculated from the detailed particle-size distribution and incorporated as a parameter infractal water retention models. Comparisons between measured and model-estimated water retention cha-racteristics indicated that these three models were applicable to relatively different soil textures and pressurehead ranges. Tyler-Wheatcraft and Brooks-Corey models led to reasonable agreements for both coarse- andmedium-textured soils, while the latter showed applicability to a broader texture range. In contrast, Rieu-Sposito model was more suitable for fine-textured soils. Fractal models produced a better estimation of watercontents at low pressure heads than at high pressure heads.

Fractal Characteristic of Pits Distribution on 304 Stainless Steel Corroded Surface and Its Application in Corrosion Diagnosis

Electrochemical techniques and fractal theory were employed to study the corrosion behaviors and pits distribution characteristics on the corroded surfaces of 304 stainless steel exposed in FeCl3 solution. Fractal features of pits distribution over the corroded surfaces were observed and described by the fractal dimension. A 5-8-2 back-propagation （BP） artificial neural network model for the diagnoses of the pitting corrosion rate and pits deepness of 304 stainless steel under various conditions was developed by considering the fractal dimension as a key parameter for describing the pitting corrosion characteristics. The predicted results are well in agreement with the experimental data of pitting corrosion rate and pit deepness. The max relative errors between their experimental and simulation data are 6.69% and 4.62%, respectively.

Power-law Distribution of Normal Fault Displacement and Length and Estimation of Extensional Strain due to Normal Faults:A Case Study of the Sierra de San Miguelito, Mexico

The Sierra de San Miguelito is a relatively uplifted area and is constituted by a large amount of silicic volcanic rocks with ages from middle to late Cenozoic. The normal faults of the Sierra de San Miguelito are Domino-style and nearly parallel. The cumulative length and displacement of the faults obey power-law distribution. The fractal dimension of the fault traces is -1.49. Using the multi-line one-dimensional sampling, the calculated exponent of cumulative fault displacements is -0.66. A cumulative curve combining measurements of all four sections yielded a slope of -0.63. The displacement-length plot shows a non-linear relationship and large dispersion of data. The large dispersion in the plot is mainly due to the fault linkage during faulting. An estimation of extensional strain due to the normal faults is ca. 0.1830. The bed extension strain is always less than or equal to the horizontal extension strain. The deformation in the Sierra de San Miguelito occurred near the surface, producing pervasive faults and many faults are too small to appear in maps and sections at common scales. The stretching produced by small faults reach ca. 33% of the total horizontal elongation.

Multifractal Measure of Post Distribution in Post System

In order to investigate the true post distribution in the whole society, microelasticity （MIE） and macroelasticity （MAE） were defined by regarding all posts as a system. On this basis, the method for measuring post distribution was proposed. Using the Legendre dual transformation between MIE and MAE to highlight the probabilities of different levels, the post distribution were analyzed hierarchically. The two-scale Cantor model verified that the multifractal measure is applicable to the post distribution evolution process.

NEW METHOD TO ESTIMATE SCALING EXPONENTS OF POWER-LAW DEGREE DISTRIBUTION AND HIERARCHICAL CLUSTERING FUNCTION FOR COMPLEX NETWORKS

A new method and corresponding numerical procedure are introduced to estimate scaling exponents of power-law degree distribution and hierarchical clustering function for complex networks. This method can overcome the biased and inaccurate faults of graphical linear fitting methods commonly used in current network research. Furthermore, it is verified to have higher goodness-of-fit than graphical methods by comparing the KS （Kolmogorov-Smirnov） test statistics for 10 CNN （Connecting Nearest-Neighbor） networks.

DESCRIPTION OF G-S PARTICLE SIZE DISTRIBUTION OF ROCK COMMINUTION WITH FRACTAL GEOMETRY

The fractal model of rock comminution is presented with Mandelbrot's fractal geometry.The results show that it is difficult for those with only a linear similarity ratio to fit practical situations.The comminution prob ability of the central part should be considered so the geometric meaning of the constant in Gaudin-Schuhmann's distribution function can be explained more clearly.

On Origin of Power-Law Distributions in Self-Organized Criticality from Random Walk Treatment

The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is then regarded as a first-return random walk process in a one-dimensional lattice. We assume that the variation of the number of active sites has three possibilities in each update： to increase by 1 with probability f1, to decrease by 1 with probability f2, or remain unchanged with probability 1 - f1 - f2. This mimics the dynamics in the system. Power-law distributions of the lifetime are found when the random walk is unbiased with equal probability to move in opposite directions. This shows that power-law distributions in self-organized criticality may be caused by the balance of competitive interactions.

Heavy-Tailed Distributions Generated by Randomly Sampled Gaussian, Exponential and Power-Law Functions

A simple stochastic mechanism that produces exact and approximate power-law distributions is presented. The model considers radially symmetric Gaussian, exponential and power-law functions inn= 1, 2, 3 dimensions. Randomly sampling these functions with a radially uniform sampling scheme produces heavy-tailed distributions. For two-dimensional Gaussians and one-dimensional exponential functions, exact power-laws with exponent –1 are obtained. In other cases, densities with an approximate power-law behaviour close to the origin arise. These densities are analyzed using Padé approximants in order to show the approximate power-law behaviour. If the sampled function itself follows a power-law with exponent –α, random sampling leads to densities that also follow an exact power-law, with exponent -n/a – 1. The presented mechanism shows that power-laws can arise in generic situations different from previously considered specialized systems such as multi-particle systems close to phase transitions, dynamical systems at bifurcation points or systems displaying self-organized criticality. Thus, the presented mechanism may serve as an alternative hypothesis in system identification problems.

Variation of singularity of earthquake-size distribution with respect to tectonic regime

Frequency-size relation of earthquakes in a region can be approximated by the Gutenberg-Richter law(GR). This power-law model involves two parameters: a-value measuring seismic activity or earthquake productivity, and b-value describing the relation between frequencies of small and large earthquakes.The spatial and temporal variations of these two parameters, especially the b-value, have been substantially investigated. For example, it has been shown that b-value depends inversely on differential stress. The b-value has also been utilized as earthquake precursor in large earthquake prediction.However, the physical meaning and properties of b-value including its value range still remain as an open fundamental question. We explore the property of b-value from frequency-size GR model in a new form which relates average energy release and probability of large earthquakes. Based on this new form of GR relation the b-value can be related to the singularity index(1-2/3 b) of fractal energy-probability power-law model. This model as applied to the global database of earthquakes with size M ≥ 5 from 1964 to 2015 indicates a systematic increase of singularity from earthquakes occurring on mid-ocean ridges, to those in subduction zones and in collision zones.

Interactions between wind and water erosion change sediment yield and particle distribution under simulated conditions

Wind and water erosion are among the most important causes of soil loss, and understanding their interactions is important for estimating soil quality and environmental impacts in regions where both types of erosion occur. We used a wind tunnel and simulated rainfall to study sediment yield, particle-size distribution and the fractal dimension of the sediment particles under wind and water erosion. The experiment was conducted with wind ero- sion firstly and water erosion thereafter, under three wind speeds （0, 11 and 14 m/s） and three rainfall intensities （60, 80 and 100 ram/h）. The results showed that the sediment yield was positively correlated with wind speed and rain- fall intensity （P〈0.01）. Wind erosion exacerbated water erosion and increased sediment yield by 7.25%-38.97% relative to the absence of wind erosion. Wind erosion changed the sediment particle distribution by influencing the micro-topography of the sloping land surface. The clay, silt and sand contents of eroded sediment were also posi- tively correlated with wind speed and rainfall intensity （P〈0.01）. Wind erosion increased clay and silt contents by 0.35%-19.60% and 5.80%-21.10%, respectively, and decreased sand content by 2.40%-8.33%, relative to the absence of wind erosion. The effect of wind erosion on sediment particles became weaker with increasing rainfall intensities, which was consistent with the variation in sediment yield. However, particle-size distribution was not closely correlated with sediment yield （P〉0.05）. The fractal dimension of the sediment particles was significantly different under different intensities of water erosion （P〈0.05）, but no significant difference was found under wind and water erosion. The findings reported in this study implicated that both water and wind erosion should be controlled to reduce their intensifying effects, and the controlling of wind erosion could significantly reduce water erosion in this wind-water erosion crisscross region.

Generalized Extreme Value-Pareto Distribution Function and Its Applications in Ocean Engineering

In this paper, we establish a generalized extreme Value-Pareto distribution model and derive an analytical expression of Weibull–Pareto distribution model. Based on a data sample of 26-year wave height, we adopt the new model to estimate the design wave height for 500, 700 and 1000-year return periods. Results show that the design wave height from Weibull–Pareto distribution is between that of the Weibull distribution and that of the Pearson-Ⅲ distribution.For the 500-year return period design wave height, the results from the new model is 1.601% lower than those from the Weibull distribution and 1.319% higher than those from the Pearson-Ⅲ distribution. The Weibull–Pareto distribution innovatively considers the fractal features, extreme-value statistics and the truncated data in the derivation process. Therefore, it is a more holistic and practical model for estimating the design parameters in marine and coastal environments.

Variation in Grain Size Distribution in Debris Flow

Grain composition of debris flow varies considerably from fluid to deposit, making it uncertain to estimate flow properties （e.g., density, velocity and discharge） using deposit as done in practice. Tracing the variation of grain composition is thus more important than estimating some certain properties of flow because every debris flow event consists of a series of surges that are distinct in properties and flow regimes. We find that the materials of debris flows, both the fluid and the source soils, satisfy a universal grain size distribution （GSD） in a form of P （D） = CD-zexp（-D/Dc）, where the parameters C, p and De are determined by fitting the function to the grain size frequency. A small At implies a small porosity and possible high excess pore pressure in flow; and a large D~ means a wide range of grain composition and hence a high sediment concentration. Flow density increases as 11 decreases or Dc increases, in a power law form. A debris flow always achieves a state of certain mobility and density that can be well described by the coupling of p and Dc, which imposes a constraint on the fluctuations of flow surges. The GSD also describes the changes in grain composition in that it is always satisfied during the course of debris flow developing. Numerical simulation using the GSD can well illustrate the variation ofμ and Dc from source soils to deposits.

Influence of Processing Parameters on Granularity Distribution of Superalloy Powders during PREP

In order to investigate the influence of processing parameters on the granularity distribution of superalloy powders during the atomization of plasma rotating electrode processing (PREP), in this paper FGH95 superalloy powders is prepared under different processing conditions by PREP and the influence of PREP processing parameters on the granularity distribution of FGH95 superalloy powders is discussed based on fractal geometry theory. The results show that with the increase of rotating velocity of the self-consuming electrode, the fractal dimension of the granularity distribution increases linearly, which results in the increase of the proportion of smaller powders. The change of interval between plasma gun and the self-consuming electrode has a little effect on the granularity distribution, also the fractal dimension of the granularity distribution changed a little correspondingly.

S-curve networks and an approximate method for estimating degree distributions of complex networks

In the study of complex networks almost all theoretical models have the property of infinite growth, but the size of actual networks is finite. According to statistics from the China Internet IPv4 （Internet Protocol version 4） addresses, this paper proposes a forecasting model by using S curve （logistic curve）. The growing trend of IPv4 addresses in China is forecasted. There are some reference values for optimizing the distribution of IPv4 address resource and the development of IPv6. Based on the laws of IPv4 growth, that is, the bulk growth and the finitely growing limit, it proposes a finite network model with a bulk growth. The model is said to be an S-curve network. Analysis demonstrates that the analytic method based on uniform distributions （i.e., Barabasi-Albert method） is not suitable for the network. It develops an approximate method to predict the growth dynamics of the individual nodes, and uses this to calculate analytically the degree distribution and the scaling exponents. The analytical result agrees with the simulation well, obeying an approximately power-law form. This method can overcome a shortcoming of Barabasi-Albert method commonly used in current network research.

THE STUDY ON DEGREE DISTRIBUTION PROPERTY OF COMPLEX NETWORK BASED ON COOPERATIVE COMMUNICATION

Complex networks have been widely studied. Recently,many results show that the degree distributions of some large networks follow the form of power-law and these networks possess better robustness against random nodes failure. As an effective technology on combating the channel fading,wireless cooperative communication is becoming one of the most important methods to improve the wireless communication performances. In this paper,the complex network models based on cooperative communication and non-cooperative communication are established; and the degree distribution properties for them are studied. The simulation results show that the degree distributions of these networks also follow the form of power-law,which means that the addition of cooperative communi-cation links will not change the property of degree distribution and then these networks will possess better robustness against random nodes failure as well.

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

The steady oil production and pressure distribution formulae of vertically fractured well for power-law non-Newtonian fluid were derived on the basis of the elliptic flow model in fractal reservoirs. The corresponding transient flow in fractal reservoirs was studied by numerical differentiation method: the influence of fractal index to transient pressure of vertically fractured well was analyzed. Finally the approximate analytical solution of transient flow was given by average mass conservation law. The study shows that using elliptic flow method to analyze the flow of vertically fractured well is a simple method.

Spatial Density Distributions and Correlations in a Quasi-one-Dimensional Polydisperse Granular Gas

By Monte Carlo simulations, the effect of the dispersion of particle size distribution on the spatial density distributions and correlations of a quasi one-dimensional polydisperse granular gas with fractal size distribution is investigated in the same inelasticity. The dispersive degree of the particle size distribution can be measured by a fractal dimension dr, and the smooth particles are constrained to move along a circle of length L, colliding inelastically with each other and thermalized by a viscosity heat bath. When the typical relaxation time τ of the driving Brownian process is longer than the mean collision time To, the system can reach a nonequilibrium steady state. The average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation time τB to the steady state becomes shorter with increasing values of df. In the steady state, the spatial density distribution becomes more clusterized as df increases, which can be quantitatively characterized by statistical entropy of the system. Furthermore, the spatial correlation functions of density and velocities are found to be a power-law form for small separation distance of particles, and both of the correlations become stronger with the increase of df. Also, tile density clusterization is explained from the correlations.

Modeling porous structure of oil-pressboard interface and its effect on electric field distribution

The oil-pressboard insulation is a typical composite insulation system widely used in the design and manufactory of large power apparatus. The implement of oil-pressboard insulation may lead to surface electrification and discharge at the interface under certain condition. It is of significant importance to take an insight into the phenomenon occurring at the interface. Through experiment, the pressboard is found as a porous material. The interface changes abruptly from bulk pressboard to the bulk oil as a result of the porous structure. A new model is proposed which divides the interface into bulk oil region, transition region, and bulk pressboard region. The width of the transition region is decided according to the microtome figure. The effective permittivity of the transition region is calculated using a new model based on fractal theory. The model is validated and compared with previous calculation model. The effect of the existence of transition region on the electric field distribution is discussed.