CONNECTION BETWEEN THE ORDER OF FRACTIONAL CALCULUS AND FRACTIONAL DIMENSIONS OF A TYPE OF FRACTAL FUNCTIONS

The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.

FRACTIONAL INTEGRALS OF THE WEIERSTRASS FUNCTIONS: THE EXACT BOX DIMENSION

The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.

Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Continuous Functions

Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional integral of any continuous functions on a closed interval is no more than 2 - v.

On the Connection between the Order of Riemann-Liouvile Fractional Calculus and Hausdorff Dimension of a Fractal Function

This paper investigates the fractal dimension of the fractional integrals of a fractal function. It has been proved that there exists some linear connection between the order of Riemann-Liouvile fractional integrals and the Hausdorff dimension of a fractal function.

Box Dimension of Weyl Fractional Integral of Continuous Functions with Bounded Variation

We know that the Box dimension of f（x） ∈ C^1[0,1] is 1. In this paper, we prove that the Box dimension of continuous functions with bounded variation is still 1. Furthermore, Box dimension of Weyl fractional integral of above function is also 1.

AN UPBOUND OF HAUSDORFF’S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRODINGER OPERATOR ON H^(s)(R^(n))

Given n≥2 and α≥1/2,we obtained an improved upbound of Hausdorff's dimension of the fractional Schrodinger operator;that is,supf∈H^(s)(R^(n)) dim_(H){x∈R^(n):limt→0 e^(it)(-△)^(α) f(x)≠f(x)}≤n+1-2(n+1)s/n for n/2(n+1)<s≤n/2.

Application of the Algorithm of Fractional Dimension to Extraction Image Edge

The idea of fractional dimension was stated in brief firstly.Then,adopting the fractional statistical similar principle, the method of the least square minimum error was applied to evaluate the fractional dimension of per image pixel depending on the fractional property of image.And the image edge is extracted by magnitude of fractional dimension of image pixel.We presented the algorithm of the local fractional dimension,which made the rule of window size and sentencing the fractional dimension of edge.Although this algorithm was waste time,it is better than the classical ones in extraction edge and anti-jamming.

THE FRACTIONAL DIMENSION IDENTIFICATION METHOD OF CRITICAL BIFURCATED PARAMETERS OF BEARING-ROTOR SYSTEM

The stable problem of rotor system, seen in many fields, has been cared for more. Nowadays the reasons of most losing stability are caused by nonlinear behaviors This presents higher requirements to the designing of motor system : considering nonlinear elements, avoiding the unstable parameter points or regions where nonlinear phenomena will be presented. If a family of time series of the unknown nonlinear dynamical system cart only be got ( may be polluted by noise), how to identify the change of motive properties at different parameters? In this paper, through the study of Jeffcott rotor system, the result that using the figures between the fractional dimension of rime-serial and parameter can be gained, and the critical bifurcated parameters of bearing-rotor dynamical system can be identified.

Bounds on Fractional-Based Metric Dimension of Petersen Networks

The problem of investigating the minimum set of landmarks consisting of auto-machines(Robots)in a connected network is studied with the concept of location number ormetric dimension of this network.In this paper,we study the latest type of metric dimension called as local fractional metric dimension(LFMD)and find its upper bounds for generalized Petersen networks GP(n,3),where n≥7.For n≥9.The limiting values of LFMD for GP(n,3)are also obtained as 1(bounded)if n approaches to infinity.

Fractal Dimension of Graph of Weierstrass Function and Its Derivative of the Fractional Order

In this paper,we obtain the fractal dimension of the graph of the Weierstrass function, its derivative of the fractional order and the relation between the dimension and the order of the fractional derivative.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

A Technique for Estimation of Box Dimension about Fractional Integral

This paper discusses further the roughness of Riemann-Liouville fractional integral on an arbitrary fractal continuous functions that follows Rfs. [1]. A novel method is used to reach a similar result for an arbitrary fractal function , where is the Riemann-Liouville fractional integral. Furthermore, a general resultis arrived at for 1-dimensional fractal functions such as with unbounded variation and(or) infinite lengths, which can infer all previous studies such as [2] [3]. This paper’s estimation reveals that the fractional integral does not increase the fractal dimension of f(x), i.e. fractional integration does not increase at least the fractal roughness. And the result has partly answered the fractal calculus conjecture and completely answered this conjecture for all 1-dimensional fractal function (Xiao has not answered). It is significant with a comparison to the past researches that the box dimension connection between a fractal function and its Riemann-Liouville integral has been carried out only for Weierstrass type and Besicovitch type functions, and at most Hlder continuous. Here the proof technique for Riemann-Liouville fractional integral is possibly of methodology to other fractional integrals.

基于FrFT-FD和KPCA模拟电路故障特征提取方法

为获得有效的故障特征信息,提出一种基于分数阶傅里叶变换分形维的模拟电路故障特征提取方法。首先,把原始数据空间中的特征数据映射到不同的分数阶空间,分别计算不同分数阶次下故障响应信号的分形维数;然后,采用核主元分析进一步对候选特征实施降维;最后,将优化后的特征向量作为故障特征,利用神经网络进行分类诊断。仿真结果表明,本方法能很好地获取不同故障响应信号的细微差异,增强不同故障模式的可分性,提高故障诊断准确率。

A Class of Fractal Functions and Their Dimension Estimates

本文作者首先借助于b进制分数及无穷级数构造了一类分形函数，然后研究了这些函数图象的分形维数及其Hoelder性质，得到了一些维数结果。

基于FD特征的Snake模型及自动目标轮廓提取

用Snake模型分割自然背景下的人造目标时,Snake曲线往往被复杂背景所吸引,无法收敛到人造目标的边缘。针对该问题,文章从目标特征的角度,将分形维数特征引入Snake模型。利用自然背景和人造目标在分形维数特征上的显著区别,定义了基于目标分形维数特征的梯度加权函数,来自适应调整图像梯度幅值的大小,抑制自然背景的干扰。同时,该模型允许初始轮廓远离目标的真实边缘,降低了Snake模型对初始位置的依赖性。实验表明,该Snake模型能够克服复杂自然背景的干扰,提取出人造目标的边缘。

基于FDS的超细水雾抑制酒精火的研究

通过搭建小尺寸实验平台,研究超细水雾抑制酒精火的过程,分析受限空间中氧气体积分数以及烟气温度分布状况以及施加超细水雾对其的影响。基于FDS进行超细水雾与酒精火相互作用的数值模拟。研究结果表明,在超细水雾作用下,受限空间内的氧气体积分数与烟气温度下降速率高于自由燃烧状态;超细水雾抑制酒精火的后期,受限空间顶部会积聚大量高温烟气;数值模拟结果较好地预测了超细水雾施加前、后烟气温度的变化规律。

融合子带方差和FD变换的长记忆过程仿真

根据长记忆过程小波变换的频带分割及其去相关性,提出了“子带方差”的概念,并给出了在长记忆条件下的估算公式。在此基础上,构造了一种具有扩展记忆功能的FD变换,该变换可以将一个无记忆的、独立同分布的随机序列转化为具有指定长记忆特性的随机序列,由此实现对长记忆过程的仿真。对仿真序列进行模型比较和参数估计证实,基于子带方差的FD变换可以有效地扩展和控制随机序列的记忆性。

Polar Functions for Fractional Brownian Motion

Let X (t)(t∈R^N) be a d-dimensional fractional Brownian motion. A contiunous function f:R^N→R^d is called a polar function of X(t)(t∈R^N) if P{ t∈R^N\{0},X(t)=t(t)}=0. In this paper, the characteristies of the class of polar functions are studied. Our theorem 1 improves the previous results of Graversen and Legall. Theorem2 solves a problem of Legall (1987) on Brownian motion.

ON THE FRACTIONAL CALCULUS FUNCTIONS OF A FRACTAL FUNCTION

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

INTERSECTIONS AND POLAR FUNCTIONS OF FRACTIONAL BROWNIAN SHEETS

Let B^H={B^H（t）,t∈R^N＋}be a real-valued（N,d）fractional Brownian sheet with Hurst index H=（H1,…,HN）.The characteristics of the polar functions for B^H are discussed.The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained.The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented.Furthermore,it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions.A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Holder condition is also solved.