A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

A Successive Shift Box-Counting Method for Calculating Fractal Dimensions and Its Application in Identification of Faults

Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This paper proposes a successive shift box-counting method,in which the studied object is divided into small sub-objects that are composed of a series of gridsaccording to its characteristic scaling. The terrain fractal dimensions in the grids are calculatedwith the successive shift box-counting method and the scattered points with values of fractaldimensions are obtained. The present research shows that the planar variation of fractal dimensionsis well consistent with fault traces and geological boundaries.

Modified Laguerre spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions

The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.

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.

Combining Fractals and Box-Counting Dimension

In this paper, the box-counting dimension is used to derive an explicit formula for the dimension of a fractal constructed using several contractions or by combining fractals. This dimension agrees with the Hausdorff dimension in the particular case when the scales factors considered are all the same. A more general sufficient condition for the box-counting dimension and the Hausdorff dimension to be the same is given. It is also shown that the dimension of the fractal obtained by combining two fractals is the weighted average of the dimensions of the two fractals.

DIMENSION AND DIFFERENTIABILIIY OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, a new iterated function system consisting of non-linear affinemaps is constructed. We investigate the fractal interpolation functions generated bysuch a system and get its differentiability, its box dimension, its packing dimension,and a lower bound of its Hansdorff dimension.

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control.We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians.Additionally,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of dimensionality.Using existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial costs.We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demonstrates the promising performance boosts FPGAs can achieve.Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

Research on the chaos recognition method based on differential entropy

Phase space reconstruction is the first step to recognizing the chaos from observed time series. On the basis of differential entropy, this paper introduces an efficient method to estimate the embedding dimension and the time delay simultaneously. The differential entropy is used to characterize the disorder degree of the reconstructed attractor. The minimum value of the differential entropy corresponds to the optimum set of the reconstructed parameters. Simulated experiments show that the original phase space can be effectively reconstructed from time series, and the accuracy of the invariants in phase space reconstruction is greatly improved. It provides a new method for the identification of chaotic signals from time series.

FINITE DIMENSION OF GLOBAL ATTRACTORS FOR DISSIPATIVE EQUATIONS GOVERNING MODULATED WAVE

The finite dimension of the global attractors for the systems of the perturbed and unperturbed dissipative Hamiltonian amplitude equations governing modulated wave are investigated.An interesting result is also obtained that the upper bound of the dimension of the global attractor for the perturbed equation is independent of ε.

PARAMETERS DETERMINATION METHOD OF PHASE-SPACE RECONSTRUCTION BASED ON DIFFERENTIAL ENTROPY RATIO AND RBF NEURAL NETWORK

Phase space reconstruction is the first step of recognizing the chaotic time series.On the basis of differential entropy ratio method,the embedding dimension opt m and time delay t are optimal for the state space reconstruction could be determined.But they are not the optimal parameters accepted for prediction.This study proposes an improved method based on the differential entropy ratio and Radial Basis Function(RBF)neural network to estimate the embedding dimension m and the time delay t,which have both optimal characteristics of the state space reconstruction and the prediction.Simulating experiments of Lorenz system and Doffing system show that the original phase space could be reconstructed from the time series effectively,and both the prediction accuracy and prediction length are improved greatly.

DIMENSIONS FOR RANDOM SELF-CONFORMAL SETS

A set is called regular if its Hausdorff dimension and upper box-counting dimension coincide. In this paper,we prove that the random self-conformal set is regular almost surely. Also we determine the dimensions for a class of random self-conformal sets.

Solvability of Non-homogeneous Mixed Type Multi-point BVPs for Second Order Differential Equations with p-Laplacian

A class of multi-point boundary value problems are studied.Easily verified suffcient conditions to guarantee the existence of at least one solutions of above mentioned BVPs are established.The examples are presented to illustrate the main results.

Fractal Dimension Based Shot Transition Detection in Sport Videos

Increase in application fields of video has boosted the demand to analyze and organize video libraries for efficient scene analysis and information retrieval. This paper addresses the detection of shot transitions, which plays a crucial role in scene analysis, using a novel method based on fractal dimension (FD) that carries information on roughness of image intensity surface and textural structure. The proposed method is tested on sport videos including soccer and tennis matches that contain considerable amount of abrupt and gradual shot transitions. Experimental results indicate that the FD based shot transition detection method offers promising performance with respect to pixel and histogram based methods available in the literature.

Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

We study a new algorithm for solvingparabolic partial differential equations(PDEs)and backward stochastic differential equations(BSDEs)in high dimension,which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function,and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE.The policy function is then approximated by a neural network,as is done in deep reinforcement learning.Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation,the Hamilton–Jacobi–Bellman equation,and a nonlinear pricing model for financial derivatives.

Realizing the Box-counting Method for Calculating Fractal Dimension of Urban Form Based on Remote Sensing Image

In the research of fractal cities, the fractal dimension is very important. It is used to describe the fractal character of the city. The authors have designed two approaches to calculate the fractal dimension by the box-counting method through an example of Beijing, which are called the vector method and the grid method, respectively. The former calculates the fractal dimension through an intersecting analysis in ArcView; and the latter is carried out by programming in Matlab. They are compared from three aspects： the calculating process, the limits in use, and the results. As a result, the conclusion is made that there are merits and faults on both methods, and they should be chosen to use properly in practical situation.

Grbner bases in difference-differential modules and difference-differential dimension polynomials

In this paper we extend the theory of Grbner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for construct-ing these Grbner bases counterparts. To this aim we introduce the concept of "generalized term order" on Nm ×Zn and on difference-differential modules. Using Grbner bases on difference-differential mod-ules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations.

中国市域新质生产力:时序演变、组群特征与发展策略

新质生产力代表了新一轮产业变革与科技革命的前进方向,是重塑全球竞争新优势的关键着力点。本文构建了新质生产力测量体系,运用熵权TOPSIS法测度中国270座城市的新质生产力发展水平,并借助Dagum基尼系数分解法、Kernel密度估计、自然断裂法、Moran’s I指数与障碍因子诊断模型对市域新质生产力的时序演进和空间分化进行组群分析。结果发现:(1)中国市域新质生产力发展水平总体呈现上升态势但水平有待提高,层级分布上呈现头部较少、腰部相当、尾部堆积的“金字塔”形态。(2)时序上,新质生产力发展水平的总体差距随时间演进呈现扩大化迹象,总体差距主要来源于地区间差异;空间上,新质生产力发展表现出正向空间集聚特征。(3)产教融合水平、高新技术企业数量、城市创新指数是制约市域新质生产力发展的主要障碍因子。研究结论丰富了新质生产力的特征维度和时空分异事实,为因地制宜发展新质生产力提供了主攻方向和经验证据。

中国与东南亚国家的文化距离测量及其对中华文化传播的影响

东南亚地区是中华文化全球传播格局的重要组成部分,具有鲜明的区域特征。根据吉尔特·霍夫斯泰德(Geert Hofstede)的国别文化维度数据,可以发现东南亚六国和中国在6个文化维度上既有共性,又有差异。采用Kogut和Singh的文化距离测量公式测得的数据显示,东南亚六国与中国的文化距离普遍较近,这对中华文化在东南亚地区的传播格局有明显的影响,与孔子学院在各国的分布也有一定的联系。考察国际间语言文化的传播与互动应注重差异化研究。参考文化距离指数,关注东南亚国家当地中华文化资源的不同状况,有助于科学研判中华文化输入需求,总结出适应当地文化特点的高效传播模式。

基于差分盒维数的井下电视图像裂缝分割识别方法研究

利用井下电视图像对岩心完整性进行识别、分割及裂缝宽度提取是解释岩性、裂隙的关键。目前,井下电视成像结果主要是通过人工方法进行识别解释,工作量大且易受主观因素影响。基于此,针对井下电视图像分割识别,提出从分形理论出发,采用差分盒维数(Differential Box-counting,DBC)算法来提取井下电视图像的岩性变化,以及裂缝、裂隙等岩层纹理特征,进一步通过基于最佳聚类数选择的K-均值聚类算法对井下电视图像进行纹理分割,实现对井下电视图像的岩层自动化分区。结果表明:结合Canny边缘检测算法提取裂缝宽度信息,裂缝识别准确率达到94.2%,实现了岩性的准确分层、裂缝的精细识别。与2D Log-Gabor算法相比,差分盒维数算法对井下电视图像分割效果好、速度快。

基于分形理论的闪络放电痕迹分析方法研究

绝缘子连续闪络会导致绝缘子表面出现碳化痕迹,导致绝缘劣化。对绝缘子表面碳迹的研究有助于解绝缘子长期运行时的闪络特性。文中通过对绝缘子进行直流闪络实验,获取连续放电后的碳化痕迹图像。基于分形理论,采用改进的差分盒计数法(improved differential box-counting method,IDBCM)对这些图像进行分析,对于边缘被切割尺寸框覆盖的矩形图像,在计数过程中提出加权项,所提出的方法可以估计绝缘子表面的放电传播特性和劣化特性。由于考虑了碳化痕迹图像的颜色深度,IDBCM在图像预处理和评估绝缘子劣化性能方面表现出了比传统盒计数方法更大的优势。该图像分析方法在处理碳化痕迹时具有通用性,可为闪络电压提供附加信息,包括闪络时间、电压大小、工况等。为今后研究放电机理及相应的绝缘子劣化问题提供了一种方法。