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The distribution of the breaking strengths over the system is considered to be a fractal function.


As an example, a "Weierstrasslike" fractal function is considered, for which the wavelet transform is related to a Jacobi theta function.


In this paper, we use a fractal function to model the surface of sea, and fractal dimension is also an appropriate descriptor of roughness of sea surface.


A normalized two dimensional bandlimited Weierstrass fractal function is used for modeling the rough surface.


Signchanges of the ThueMorse fractal function and Dirichlet Lseries

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 In the paPer,we think that fractal dimension and nonscaling region are two important characteristic Parameters of seismic fractal selforganization. The characteristics of correlative dimension at two levels and the main limits on dimension calculations are discussed.The relations between the growing structures of the strong seismic sequences and fractal principles are studied Primarily.The new research methods and subject about the seismic fractal function and the law of the seismic temporal distribution... In the paPer,we think that fractal dimension and nonscaling region are two important characteristic Parameters of seismic fractal selforganization. The characteristics of correlative dimension at two levels and the main limits on dimension calculations are discussed.The relations between the growing structures of the strong seismic sequences and fractal principles are studied Primarily.The new research methods and subject about the seismic fractal function and the law of the seismic temporal distribution are suggested. The characteristics about the temporal changes of the fractal dimensions and their applications on earthquake Prediction are analyzed.  本文认为,分维及无标度区是地震分形自组织两个重要特征参数。讨论了两个层次关联维特征及关联维 D_2计算的基本限制。初步探索了强震序列生长结构与分形原理间关系。提出地震分维函数与地震时序分布律新的研究方法及课题。分析了分维随时间变化特征及其在地震预报上的应用。  In this paper we present a new method local accumulated deviation method for evaluating the fractal dimension of curves or onedimensional (1D) surfaces. Our method is tested on various types of curves for Weierstrass Mandelbrot fractal function and fractal Brownian motion with known fractal dimension. The results are good agreement with the theoritical values. Finally, using Monte Carlo method, we simulated the randam rough (1D) surfaces with Gauss spectrum, and the new method is applied... In this paper we present a new method local accumulated deviation method for evaluating the fractal dimension of curves or onedimensional (1D) surfaces. Our method is tested on various types of curves for Weierstrass Mandelbrot fractal function and fractal Brownian motion with known fractal dimension. The results are good agreement with the theoritical values. Finally, using Monte Carlo method, we simulated the randam rough (1D) surfaces with Gauss spectrum, and the new method is applied to data from simulating surfaces.  本文提出了一维分形的分维估计方法—局部方差累积法。通过对已知分维数的Weiers trass函数和分数布朗运动的检验,估计的分维数与理论值有很好的吻合。最后,利用蒙特卡罗方法模拟了高期分布随机粗糙面,并对它们的分维给予了估计。  The widely existing nonlinear phenomena revealed by highresolution geological records need to be explored With nonlinear scientific methods. As a comprehensive science which has farreaching influence, nonlinear science has become very popular with the international scientific circle. The development and application of nonlinear science in geosciences is the prelude of the birth of a series of nonlinear geosciences. Fractal theory, timeseries analysis and chaotic dynamics are hopefully the most promising... The widely existing nonlinear phenomena revealed by highresolution geological records need to be explored With nonlinear scientific methods. As a comprehensive science which has farreaching influence, nonlinear science has become very popular with the international scientific circle. The development and application of nonlinear science in geosciences is the prelude of the birth of a series of nonlinear geosciences. Fractal theory, timeseries analysis and chaotic dynamics are hopefully the most promising nonlinear methods in Quaternary studies. Both neotectonic vertical movement and ancient climate changes are oscillating movement which, as a complicated irregular oscillation between regular oscillation (e. g. BZ oscillation in chemistry) and chaos. The oscillation which is a socalled geophysical chaotic timeseries results from the internal stochastic nature of the deter mined Earth system. In the frequency change of oscillation, bifurcation space and width of frequency separation are attenuating respectively in aocordance with the Feigenbaum constants δ and α. The oscillating movement of the Earth system is a transition to chaos. The uprise of mountains, the rise of continents, the fall of ocean floors and the change of the global sea level and a series of such intense changes are indications of chaos. Oscillation movement appears to be nonperiodically, rhythmic, it together with erosion of external forces, creates the undulating geomorphological landforms which are similar hierarchically at some extent. The geomorphological phenomena which have similarity are therefore one of resources of fractal concepts. By a couple of popular simple fractal functions, it is possible to "create" Various geomorphological landforms which are similar fairly with natural landforms. The artificial landforms which we call fractal landforms are called Mandelbrot landscape by mathematicians. It is significant for Quaternary studies: no matter how complicated the modern and ancient landforms appear to be, their mechanisms are simple. It is in accordance with traditional concepts of geosciences: landforms are a result of interaction of both internal and external forces. Simple fractal functions are helpful for the study of the interaction and its evolution characteristics. Relief of landforms is a height field. The conformity between Mandelbrot landscape and real landforms indicates that the geomorphological height field is fractal. The complexity of landforms, i.e. the complexity of the height field (or roughness) can be estimated by fractal dimension of their surface. The difference of fractal dimensions of landforms shows the difference of environmental conditions and the difference of evolution stages. The timeseries of the Quaternary environmental change generally not random, i.e. not a Brownian condition, but a fractal Brownian movement(FBM), which has a longterm persistence. Whether a timeseries is FBM or not can be decided by Hurst exponent. This provides us a theoretical base for forcasting the mean behaviour of volution. he timeseries Hurst ex ponents of lots of natural phenomena such as the change of precipitation, temperature and the water surface of river, lake and sea are all larger than 1/2, which indicates that every successive value of the timeseries is not independent, i. e. they are not Gaussian variable or white Gaussian ncise. Although the traditional statistical methods have been used all the time, they cannot accurately describe the behaviour of timeseries. It is desirable to use R/S(range over standard deviation). In the eyes of statisticians, timeseries=trend +period + random Since the appearance of Chaotic theory, scientists have however found that the irregularity of timeseries results from the stochasticity inside the system, so timeserles=chaos + fluctuation The freedom of a timeseries is decided by its correlation dimension of the chaotic attractor of timeseries, which also decide whether the timeserits is a stochastic system or a determined system of a limited dimension. If it is the latter, a nonlinear dynamic model can be inferred. Besides fractal dimension, Lyapunov exponent and Kolmogonov entropy(Kentropy) are usually used to describle the chaotic characteristics of timeseries. When a system is chaotic the Lyapunov exponent is larger than 0(λ>0). In most cases Kentropy equals the sum of all plus Lyapunov exponents. For a order system K=0, for a stochastic system K=∞. When K=c (a limited value) the system is chaotic. The larger K is the stronger the chaos is. The characteristics of the Quaternary timeseries will help us to know the further changes of environment. In the field of Quaternary studies, as the resolving power of data is enhancing, we will have a new insight into the Quaternary process. As we are facing a stern challenge like the catastrophe process of climate change, the traditional linear methods are obviously not enough. The system of climate like many other geographical systems is a nonlinear system. At critical condition a sequence of reaction caused by small events will exert influences on lots of elements in the system and lead to happening of great events. To solve the problem, the theory SOC(selforganized criticality) is much helpful. Nonlinear science is very important for Quaternary studies and it also will develop in turn in the field of Quaternary studies.  第四纪研究中最常用的非线性科学方法可能是分形理论、时间序列分析及混沌动力学。振荡在新构造运动和古气候变化中都是普遍存在的,它是介于有规振荡和混沌之间的中间阶段和过渡。分形理论可以描述地貌的状态和演化特征,混沌吸引子的分维又是混沌的一个重要参量。在时间序列分析中计算混沌吸引子的关联维、李雅普诺夫指数、赫斯特指数、柯尔莫果洛夫熵等可以更深入地揭示时间序列的性质,加深对演化过程的认识。   << 更多相关文摘 
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