Fractal analysis of major faults and fractal dimension of lineaments in the Indo-Gangetic Plain on a regional scale

The Indo-Gangetic Plain(IGP)is one of the most seismically vulnerable areas due to its proximity to the Himalayas.Geographic information system(GIS)-based seismic characterization of the IGP was performed based on the degree of deformation and fractal dimension.The zone between the Main Boundary Thrust(MBT)and the Main Central Thrust(MCT)in the Himalayan Mountain Range(HMR)experienced large variations in earthquake magnitude,which were identified by Number-Size(NS)fractal modeling.The central IGP zone experienced only moderate to low mainshock levels.Fractal analysis of earthquake epicenters reveals a large scattering of earthquake epicenters in the HMR and central IGP zones.Similarly,the fault fractal analysis identifies the HMR,central IGP,and south-western IGP zones as having more faults.Overall,the seismicity of the study region is strong in the central IGP,south-western IGP,and HMR zones,moderate in the western and southern IGP,and low in the northern,eastern,and south-eastern IGP zones.

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.

Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

量化三维网格模型复杂度的Fractal Dimension^(+)方法

对三维网格模型复杂度的评估及应用进行研究,提出基于Fractal Dimension^(+)方法量化三维网格模型复杂度。该方法利用分形几何学的分形维度来确定三维网格模型的不规则性;为了全面量化复杂度并且提高量化的准确性和可信度,在分形维度算法中添加孔隙比和亏格数的概念,全面考虑介质内部空间分布和几何结构的连接性以及孔洞分布的信息。对大量三维网格模型进行实验,结果表明Fractal Dimension^(+)方法可以有效计算出准确的复杂度。

Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.

Scalable Temporal Dimension Preserved Tensor Completion for Missing Traffic Data Imputation With Orthogonal Initialization

Dear Editor,This letter puts forward a novel scalable temporal dimension preserved tensor completion model based on orthogonal initialization for missing traffic data(MTD)imputation.The MTD imputation acts directly on accessing the traffic state,and affects the traffic management.

An Exploration of the Three Dimensions and Epochal Strengths of Building a Human Community With a Shared Future

The idea of a human community with a shared future was proposed by the Communist Party of China(CPC)Central Committee with Comrade Xi Jinping at its core for the future development of human beings to face up to the most important question in today's world:“What is happening to the world and what should we do?”It profoundly answers the question of the world,history,and the times.The theory of a human community with a shared future is an innovative theory with a multidimensional formation logic that guides humanity toward continually seeking common interests and values.This paper dives into the profound motivations behind building a human community with a shared future from historical,cultural,and practical dimensions and analyzes its epochal value from both domestic and international perspectives.This not only helps exert China's role in the international community,contributing Chinese strength to the construction of a peaceful,stable,and prosperous human society,but also enhances the influence of the idea of a human community with a shared future in the international community,accelerating the building of a human community with a shared future that considers the legitimate concerns of all countries,and aiding in solving the crises facing the world.

A Dynamical System-Based Framework for Dimension Reduction

We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a nonlinear flow towards a lower-dimensional subspace;the projection onto the subspace gives the low-dimensional embedding.Training the model involves identifying the nonlinear flow and the subspace.Following the equation discovery method,we represent the vector field that defines the flow using a linear combination of dictionary elements,where each element is a pre-specified linear/nonlinear candidate function.A regularization term for the average total kinetic energy is also introduced and motivated by the optimal transport theory.We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method.We also show how the DDR method can be trained using a gradient-based optimization method,where the gradients are computed using the adjoint method from the optimal control theory.The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods,including the PCA,t-SNE,and Umap.

The Global Attractor and Its Dimension Estimation of Generalized Kolmogorov-Petrovlkii-Piskunov Equation

In this paper, the initial boundary value problem of a class of nonlinear generalized Kolmogorov-Petrovlkii-Piskunov equations is studied. The existence and uniqueness of the solution and the bounded absorption set are proved by the prior estimation and the Galerkin finite element method, thus the existence of the global attractor is proved and the upper bound estimate of the global attractor is obtained.

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.

A Study of the Teaching Effects of Case Analysis in Cross-Cultural Communication Class-Taking Individualism-Collectivism Cultural Dimension Class as an Example

This study introduces the individualism-collectivism dimension of the cultural dimension of cross-cultural communication initiated by Geert Hofstede.Different cultures must develop a way of correlating that strikes a balance between caring for themselves and showing concern for others.Individualist culture encourages uniqueness and independence while collectivist culture emphasizes conformity and mutual assistance.This article introduces how to use case analysis method to effectively carry out classroom teaching in this cultural dimension.

A Dimensional Reduction Approach Based on Essential Constraints in Linear Programming

This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted average and the cosine simplex algorithm. The first approach identifies binding constraints by using the weighted average of each constraint, whereas the second algorithm is based on the cosine similarity between the vector of the objective function and the constraints. These two approaches are complementary, and when used together, they locate the essential subset of initial constraints required for solving medium and large-scale linear programming problems. After reducing the dimension of the linear programming problem using the subset of the essential constraints, the solution method can be chosen from any suitable method for linear programming. The proposed approach was applied to a set of well-known benchmarks as well as more than 2000 random medium and large-scale linear programming problems. The results are promising, indicating that the new approach contributes to the reduction of both the size of the problems and the total number of iterations required. A tree-based classification model also confirmed the need for combining the two approaches. A detailed numerical example, the general numerical results, and the statistical analysis for the decision tree procedure are presented.

Measurement Uncertainty Analysis of the Rotary-scan Method for the Measurable Dimension of Cylindrical Workpieces

The measurement uncertainty analysis is carried out to investigate the measurable dimensions of cylindrical workpieces by the rotary-scan method in this paper.Due to the difficult alignment of the workpiece with a diameter of less than 3 mm by the rotary scan method,the measurement uncertainty of the cylindrical workpiece with a diameter of 3 mm and length of 50 mm which is measured by a roundness measuring machine,is evaluated according to GUM(Guide to the Expression of Uncertainty in Measurement)as an example.Since the uncertainty caused by the eccentricity of the measured workpiece is different with the dimension changing,the measurement uncertainty of cylindrical workpieces with other dimensions can be evaluated the same as the diameter of 3 mm but with different eccentricity.Measurement uncertainty caused by different eccentricities concerning the dimension of the measured cylindrical workpiece is set to simulate the evaluations.Compared to the target value of the measurement uncertainty of 0.1μm,the measurable dimensions of the cylindrical workpiece can be obtained.Experiments and analysis are presented to quantitatively evaluate the reliability of the rotary-scan method for the roundness measurement of cylindrical workpieces.

ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS

In this paper,we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions.More precisely,we show that,given a real number 1≤β≤2,any real-valued continuous function in C([0,1])can be decomposed into a product of two real-valued continuous functions,each having a graph of Hausdorff dimensionβ.In addition,a product decomposition result for the packing dimension is obtained.This work answers affirmatively two questions raised by Verma and Priyadarshi[14].

The Historical Position and Value Dimensions of Human Rights Civilization

Human beings are the mainstay and the ultimate goal of civilization.The history of human civilization is a continuous struggle to realize the respect,liberation,protection,and development of humanity.Human rights are an achievement of humanity and a symbol of progress,and the human rights civilization is an important component of human civilization.Understanding and interpreting human rights from the perspective of human rights civilization means that human rights are not only a concept or an idea but also a grand historical and long-term social practice.Up to now,the development of human rights civilization has roughly experienced four awakening eras:initialization,revolution,popularization,and globalization.In terms of its value dimensions,it has the characteristics of progressiveness,diversity,commonality,inclusiveness,indivisibility,openness,and so on.The historical position of human rights civilization and the development of its value dimensions have shown to the world that human rights are the common wealth of humanity,and human rights belong to all mankind;human rights are historical,concrete,and developmental;the concept of human rights is constantly evolving,and its connotations and categories are constantly expanding;achieving the free and well-rounded development of every person is the highest value realm of human rights civilization.The Chinese modernization endows Chinese civilization with modern strength and opens up new horizons for human rights civilization.The new pattern of human rights civilization to be created by Chinese modernization not only possesses the common characteristics of human rights civilization but also enjoys Chinese characteristics based on its own national conditions,enriching and developing the diversity of human rights civilization for all mankind.

Novel Path Counting-Based Method for Fractal Dimension Estimation of the Ultra-Dense Networks

Next-generation networks,including the Internet of Things(IoT),fifth-generation cellular systems(5G),and sixth-generation cellular systems(6G),suf-fer from the dramatic increase of the number of deployed devices.This puts high constraints and challenges on the design of such networks.Structural changing of the network is one of such challenges that affect the network performance,includ-ing the required quality of service(QoS).The fractal dimension(FD)is consid-ered one of the main indicators used to represent the structure of the communication network.To this end,this work analyzes the FD of the network and its use for telecommunication networks investigation and planning.The clus-ter growing method for assessing the FD is introduced and analyzed.The article proposes a novel method for estimating the FD of a communication network,based on assessing the network’s connectivity,by searching for the shortest routes.Unlike the cluster growing method,the proposed method does not require multiple iterations,which reduces the number of calculations,and increases the stability of the results obtained.Thus,the proposed method requires less compu-tational cost than the cluster growing method and achieves higher stability.The method is quite simple to implement and can be used in the tasks of research and planning of modern and promising communication networks.The developed method is evaluated for two different network structures and compared with the cluster growing method.Results validate the developed method.

Cosmological Model in Four Time and Four Space Dimensions

We develop a cosmological model in a physical background scenario of four time and four space dimensions ((4+4)-dimensions or (4+4)-universe). We show that in this framework the (1+3)-universe is deeply connected with the (3+1)-universe. We argue that this means that in the (4+4)-universe there exists a duality relation between the (1+3)-universe and the (3+1)-universe.

Fractal dimension for porous metal materials of FeCrAl fiber

The FeCrA1 fiber was used to prepare porous metal materials with air-laid technology, and then followed by sintering at 1300 ℃ for a holding period of 2 h in the vacuum. In addition, a novel fractal soft, which was developed based on the fractal theory and the computer image processing technology, was explored to describe the pore structure of porous metal materials. Furthermore, the fractal dimension of pore structure was calculated by the soft and the effects of magnification and porosity on ffactal dimension were also discussed. The results show that the fractal dimension decreases with increase in the magnification, while it increases continuously with the porosity enhancing. The interrelationship between the fractal dimension and the magnification or porosity can be presented by the equation of D=α_0exp（-x/α_1）＋α_2和D=k_2-（k_1-k_2）/[1＋exp（（θ-k_0）/k_3）], respectively.

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。

Study on the Spatial Pattern Changes of Land Use Based on Fractal Dimensions in Tianjin New Coastal Area

[Objective] The aim was to discuss the spatial pattern changes of land use in Tianjin new coastal area based on fractal dimensions.[Method] By dint of remote and geographic information system technology to obtain the data of urban land use in new coastal area from 1993 to 2008,the boundary dimension,radius dimension and information dimension of each land use type were calculated based on fractal dimension.In addition,the revealed land use spatial dimension changes characteristics were analyzed.[Result] The spatial distribution of each land use type in new costal area had distinct fractal characteristics.And,the amount and changes of three types of dimension values effectively revealed the changes of complicatedness,centeredness and evenness of spatial pattern of land use in the study area.The boundary dimension of unused land and salty earth increased incessantly,which suggested its increasing complicatedness.The boundary of the port and wharf and shoal land was getting simpler.The radius dimension of the cultivated land was larger than 2,which suggested that its area spread from center to the surroundings;the one in salty land and waters distributed evenly within different radius space to the center of the city;the one in other land use types reduced gradually from center to the surroundings.The information dimension value in the woodland and orchard land,unused land and shoal land was small,and was in obvious concentrated distribution;the spatial distribution of cultivated and salty land concentrated in the outside area;the construction area in the port and wharf spread gradually on the basis of original state;the spatial distribution of waters and residents and mines were even.[Conclusion] Applying fractal dimensions to the study of spatial pattern changes of urban land use can make up for some disadvantages in classical urban spatial pattern quantitative research,which has favorable practical value.