Fractal dimension and classification of music

The fractal aspect of different kinds of music was analyzed in keeping with the time domain. The fractal dimension of a great number of different musics (180 scores) is calculated by the Variation method. By using an analysis of variance, it is shown that fractal dimension helps discriminate different categories of music. Then, we used an original statistical technique based on the Bootstrap assumption to find a time window in which fractal dimension reaches a high power of music discrimination. The best discrimination is obtained between 1/44100 and 16/44100 Hertz. We admit that to distinguish some different aspects of music well, the high information quantity is obtained in the high frequency domain. By calculating fractal dimension with the ANAM method, it was statistically proven that fractal dimension could distinguish different kinds of music very well: musics could be classified by their fractal dimensions.     

On the fractional calculus of Besicovitch function

Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann–Liouville fractional calculus and Weyl–Marchaud fractional derivative of Besicovitch function have been discussed.     

Contour Integrals and Vector Calculus on Fractal Curves and Interfaces

This article develops the definition of contour integrals over fractal curves in the plane by introducing the notion of oriented Iterated Function Systems and directional pseudo-measures. An expression for the contour integral of continuous functions over fractal interfaces is obtained through renormalization. As a result, a vector calculus on fractal interfaces which are boundaries of regular two-dimensional domains is developed by extending Greens theorem in the plane, also to fractal curves.The use of moment analysis makes it possible to obtain recursive relations and closed-form expressions for contour integrals of algebraic functions. Several physical applications are analyzed, including the properties of double-layer potentials and connections with the solution of the Dirichlet problem on bounded two-dimensional domains possessing fractal boundaries.     

Extensions of certain classical integrals of Erdélyi for Gauss hypergeometric functions

It is shown how series manipulation technique and certain classical summation theorems for hypergeometric series can be used to prove Erdélyi's integral representations for ₂F₁(z), originally proved using fractional calculus. The method not only leads to generalizations but also leads to new integrals of Erdélyi type for certain _q+1F_q(z) and corresponding Pochhammer contour integrals. The technique outlined here, compared to the method of fractional calculus, seems to be more effective as it not only provides transparent elementary proofs of Erdélyi's integrals but even leads to various generalizations.     

Geometrical dimension versus smoothness

We study the relation between geometric dimension and smoothness, and give a precise characterization of the fractal dimension of the graph of a function in terms of smoothness classes of functions. We also express the fractal dimension in terms of different classical oscillation measures and in terms of wavelet expansions.     

A set of formulae on fractal dimension relations and its application to urban form

The area-perimeter scaling can be employed to evaluate the fractal dimension of urban boundaries. However, the formula in common use seems to be not correct. By means of mathematical method, a new formula of calculating the boundary dimension of cities is derived from the idea of box-counting measurement and the principle of dimensional consistency in this paper. Thus, several practical results are obtained as follows. First, I derive the hyperbolic relation between the boundary dimension and form dimension of cities. Using the relation, we can estimate the form dimension through the boundary dimension and vice versa. Second, I derive the proper scales of fractal dimension: the form dimension comes between 1.5 and 2, and the boundary dimension comes between 1 and 1.5. Third, I derive three form dimension values with special geometric meanings. The first is 4/3, the second is 3/2, and the third is 1 + 2^1/2/2 ≈ 1.7071. The fractal dimension relation formulae are applied to China’s cities and the cities of the United Kingdom, and the computations are consistent with the theoretical expectation. The formulae are useful in the fractal dimension estimation of urban form, and the findings about the fractal parameters are revealing for future city planning and the spatial optimization of cities.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

The concept of physical and fractal dimension II. The differential calculus in dimensional spaces

The projective dimensional analysis based on the projective extension of scaling group and projective dimensional function is studied. The differential calculus corresponding to geometry of dimensional spaces is constructed and examined. At the next step we explore the projective extension of dimensional derivatives. Simple fractal models of various processes with changing fractal dimension illustrate the proposed methods.     

Fractal dimension of zeolite surfaces by calculation

A theoretical method for the estimation of the fractal dimensions of the pore surfaces of zeolites is proposed. The method is an analogy to the commonly employed box-counting method and uses imaginary meshes of various sizes (s) to trace the pore surfaces determined by the frameworks of crystalline zeolites. The surfaces formed by the geometrical shapes of the secondary building units of zeolites are taken into account for the calculations performed. The characteristics of the framework structures of the zeolites 13X, 5A and silicalite are determined by the help of the solid models of these zeolites and the total numbers of grid boxes intersecting the surfaces are estimated by using equations proposed in this study. As a result, the fractal dimension values of the zeolites 13X, 5A and silicalite are generally observed to vary in significant amounts with the range of mesh size used, especially for the relatively larger mesh sizes that are close to the sizes of real adsorbates. For these relatively larger mesh sizes, the fractal dimension of silicalite falls below 1.60 while the fractal dimension values of zeolite 13X and 5A tend to rise above 2. The fractal dimension values obtained by the proposed method seem to be consistent with those determined by using experimental adsorption data in their relative magnitudes while the absolute magnitudes may differ due to the different size ranges employed. The results of this study show that fractal dimension values much different from 2 (both higher and lower than 2) may be obtained for crystalline adsorbents, such as zeolites, in ranges of size that are close to those of real adsorbates.     

Non-differentiable variational principles

We develop a calculus of variations for functionals which are defined on a set of non-differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non-differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schrödinger equation can be obtained as extremals of a non-differentiable variational principle, leading to an extended Hamilton's principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time.     

Koch曲线及其分数阶微积分

给出了Koch曲线的一个复值表达式,并且估计了该表达式的分数阶微积分的分形维数,同时给出了此表达式的Weyl-Marchaud分数阶导数的图像.进一步讨论了Koch曲线的图像与某类自仿分形函数图像的联系.最后证明了这类自仿分形函数的分形维数与其分数阶微积分的分形维数成立着线性关系,一个特殊例子的图像和数值结果在文中给出.     

一类Hermite分形插值函数的构造算法及其运算性质

已知结点处的函数值和一阶导数值,给出了构造一类二次分形插值函数的方法.不同于仿射分形插值函数,得到的插值函数具有可微性,并讨论分形插值函数的微积分运算,最后给出一个构造例子.     

铜基复合材料组织形态分形特征的统计分析与研究

通过对铜基复合材料显微组织结构相图的分析和研究,根据分形理论,计算了不同实验条件下铜基复合材料横截面和平行压制力面的显微组织结构相图的分形维数,同时结合统计方法分析了铜基复合材料分形维数的一些统计特性,结果表明,分形维数反映了石墨在样品中的分布规律,分形维数越大,组织结构相图越复杂,石墨分布越不规则,故石墨分布的不规则性可用分形维数来刻画,分形维数可作为材料组织形态分析的一个表征参数,通过统计分析可知,铜基复合材料横截面和平行压制力面的组织结构相图的分形维数服从正态分布,且横截面和平行压制力面的分形维数随石墨含量变化的情况互不影响。     

一个分形函数的分数阶微积分函数

Based on the combination of fractional calculus with fractal functions, a new type of is introduced； the definition, graph, property and dimension of this function are discussed.     

A simple method for estimating the fractal dimension from digital images: The compression dimension

The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem.     

Surface investigation by electrochemical methods and application of chaos theory and fractal geometry

The present study has considered the application of the noise analysis and fractal geometry as a promising dynamic method for exploiting the corrosion mechanism of the stainless steel 304 that is immersed in different concentrations of FeCl₃. The fractal dimension calculated from the electrochemical noise technique has a good correlation with the surface fractal dimension obtained by electrochemical impedance spectroscopy and scanning electron microscopy results. The complexity of system increases by divergence of Electrochemical Potential noise fractal dimension from 1.5 value and also the roughness of surface increases by an increase in surface fractal dimension. As the concentration of FeCl₃ increases (0.001 M, 0.01 M and 0.1 M) the value of Electrochemical Potential noise fractal dimension diverges from 1.5 value (1.57, 1.33 and 1.01 respectively) and the value of surface fractal dimension increases (2.107, 2.425 and 2.756 for impedance results and 2.073, 2.425 and 2.672 for scanning electron microscopy images). These results show that the complexity of system and roughness of the surface increases by an increase in concentration of FeCl₃. The present study has shown that chaos and noise analysis are effective methods for the study of the mechanism of the corrosion process.     

Fractional Nottale’s Scale Relativity and emergence of complexified gravity

Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale’s Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.     

分形插值函数的分数阶微积分的分形维数

证明了线性分形插值函数的Riemann-Liouville分数阶微积分仍然是线性分形插值函数.在基于线性分形插值函数有关讨论的基础上,证明了线性分形插值函数的Box维数与Riemann-.Liouville分数阶微积分的阶之间成立着线性关系.文中给出的例子的图像和数值结果更进一步说明了这个结论.     

On the Beurling dimension of exponential frames

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff dimension of the fractal.     

On turbulence in fractal porous media

We examine three fundamental equations governing turbulence of an incompressible Newtonian fluid in a fractal porous medium: continuity, linear momentum balance and energy balance. We find that the Reynolds stress is modified when a local, rather than an integral, balance law is considered. The heat flux is modified from its classical form when either the integral or local form of the energy density balance law is studied, but the energy density is always unchanged. The modifications of Reynolds stress and heat flux are expressed directly in terms of the resolution length scale, the fractal dimension of mass distribution and the fractal dimension of a fractal’s surface. When both fractal dimensions become integer (respectively 3 and 2), classical equations are recovered.