[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 ...[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.展开更多
The impact of harmful algal blooms （HABs） on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a ...The impact of harmful algal blooms （HABs） on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a promising strategy for controlling HABs. Previous studies found that removal efficiency （RE） was influenced by many factors, including clay type and concentration, algal growth stage, and physiological aspects of HAB cells. To estimate the effect of morphological characteristics of the aggregates on HAB cell removal, fractal dimensions were measured and the RE of three species of HAB organism, Heterosigma akashiwo, Alexandrium tamarense, and Skeletonema eostatum, by original clay and modified clay, was determined. For all HAB species, the modified clay had a higher RE than original clay. For the original clay, the two-dimensional fractal dimension （D2） was 1.92 and three-dimensional ffactal dimension （D3） 2.81, while for the modified clay, D2 was 1.84 and D3 was 2.50. The addition of polyaluminum chloride （PAC1） lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D2 and D3. Due to the decrease of D3, and the increase of the effective sticking coefficient, the flocculation rate between modified clay particles and HAB organisms increased, and thus resulted in a high RE. The fractal dimensions of flocs differed in HAB species with different cell morphologies. For example, Alexandrium tamarense cells are ellipsoidal, and the D3 and D2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D3 and D2 of flocs were the lowest.展开更多
In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness o...In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.展开更多
The characteristics of the nonlinear dynamics in the Heavy Ion Collision (HIC) at intermediate energies have been studied by evaluating the productions of the Generalized Entropy (GE) and the Multifragmentation Entrop...The characteristics of the nonlinear dynamics in the Heavy Ion Collision (HIC) at intermediate energies have been studied by evaluating the productions of the Generalized Entropy (GE) and the Multifragmentation Entropy (ME) as well as the features of the information and fractal dimensions within the Isospin Quantum Molecular Dynamical Model compensated by the lattice methods. Results demonstrate from various views that the existence of deterministic chaos in the dynamical process of reaction.展开更多
MANDELBROT enunciated the uncertainty of the length of a coastline in his paper "How long is the coastline of Britain?" published in Science in 1967. The fractal concept was presented for the first time in t...MANDELBROT enunciated the uncertainty of the length of a coastline in his paper "How long is the coastline of Britain?" published in Science in 1967. The fractal concept was presented for the first time in that paper and has been applied to many fields ever since. Although fractal dimensions of lots of phenomena were calculated by the box-counting method,the quantitative influence of series of square grids on them is ignored. The issue is systematically discussed as a case study of the mountains of China′s Mainland in this paper. And some significant conclusions are drawn as follows: 1) Although the fractal character objectively exists in the mountains of China′s Mainland,and it does not vary with the changes of series of square grids,the fractal dimensions of the mountains of China′s Mainland are different with these changes. 2) The fractal dimensions of the mountains of China′s Mainland vary with the average lengths of sides of series of square grids. The fractal dimension of the mountains of China′s Mainland is the function of the average length of side of square grid. They conform to the formula D=f(r) (where D is the fractal dimension,and r is the average length of side of square grid). 3) Different dots of data collection can affect the fractal dimension of the mountains of China′s Mainland. 4) The same range of length of side of square grid and dots of data collection can ensure the comparison of fractal dimensions of the mountains of China′s Mainland. The research is helpful to get the more understanding of fractal and fractal dimension,and ensure that the fractal studies would be scientific.展开更多
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 ...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.展开更多
The assessment of emotions with fractal dimensions of EEG signals has been attempted before, but the quantification of the intensity and duration of sudden and short emotions remains a challenge. This paper suggests a...The assessment of emotions with fractal dimensions of EEG signals has been attempted before, but the quantification of the intensity and duration of sudden and short emotions remains a challenge. This paper suggests a method for this purpose, by using a new fractal dimension algorithm and by adjusting the amplitude of the EEG signal in order to obtain maximal separation of high and low fractal dimensions. The emotion was induced by embedding a scary image at 20 seconds in landscape videos of 60 seconds length. The new method did not only detect the onset of the emotion correctly, but also revealed its duration and intensity. The intensity is based on the magnitude and impulse of the fractal dimension signal. It is also shown that Higuchi’s method does not always detect emotion spikes correctly;on the contrary, the region of the expected emotional response can be represented by fractal dimensions smaller than the rest of the signal, whereas the new method directly reveals distinct spikes. The duration of these spikes was 10 - 11 seconds. The magnitude of these spikes varied across the EEG channels. The build-up and cool-down of the emotions can occur with steep and flat gradients.展开更多
The changes of fractal dimension ofPicea koraiensis seedlings under different light intensities in natural secondary forests was studied. The results showed that with the change of light environment, crown characters ...The changes of fractal dimension ofPicea koraiensis seedlings under different light intensities in natural secondary forests was studied. The results showed that with the change of light environment, crown characters ofPicea koraiensis seedlings exhibited a greater plastic in lateral number, lateral increment, lateral dry weight, and specific leaf area. The range of calculated fractal dimensions of seedling crowns was confined between 2.5728 and 2.1036, but maximum of fractal dimension achieved in term moderate shading and in extreme low light conditions fractal dimension was least.展开更多
This paper discusses application of fractal dimensions to speech processing. Generalized dimensions of arbitrary orders and associated fractal parameters are used in speaker identification. A characteristic vactor bas...This paper discusses application of fractal dimensions to speech processing. Generalized dimensions of arbitrary orders and associated fractal parameters are used in speaker identification. A characteristic vactor based on these parameters is formed, and a recognition criterion definded in order to identify individual speakers. Experimental results show the usefulness of fractal dimensions in characterizing speaker identity.展开更多
The tight tuff reservoir is an unusual type of unconventional reservoir with strong heterogeneity.However,there is a lack of research on the microscopic pore structure that causes the heterogeneity of tuff reservoirs....The tight tuff reservoir is an unusual type of unconventional reservoir with strong heterogeneity.However,there is a lack of research on the microscopic pore structure that causes the heterogeneity of tuff reservoirs.Using the Chang 7 Formation in Ordos Basin,China as a case study,carbon-dioxide gas adsorption,nitrogen gas adsorption and high-pressure mercury injection are integrated to investigate the multi-scale pore structure characteristics of tuff reservoirs.Meanwhile,the fractal dimension is introduced to characterize the complexity of pore structure in tuff reservoirs.By this multi-experimental method,the quantitative characterizations of the full-range pore size distribution of four tuff types were obtained and compared in the size ranges of micropores,mesopores and macropores.Fractal dimension curves derived from full-range pores are divided into six segments as D1,D2,D3,D4,D5 and D6 corresponding to fractal characteristics of micropores,smaller mesopores,larger mesopores,smaller macropores,medium macropores and larger macropores,respectively.The macropore volume,average macropore radius and fractal dimension D5 significantly control petrophysical properties.The larger macropore volume,average macropore radius and D5 correspond to favorable pore structure and good reservoir quality,which provides new indexes for the tuff reservoir evaluation.This study enriches the understanding of the heterogeneity of pore structures and contributes to unconventional oil and gas exploration and development.展开更多
After exposure to freeze-thaw cycles, scanning electron microscopy(SEM) and nuclear magnetic resonance(NMR) were used to test the four mixtures. The microstructure is qualitatively analyzed from the 2D SEM image and t...After exposure to freeze-thaw cycles, scanning electron microscopy(SEM) and nuclear magnetic resonance(NMR) were used to test the four mixtures. The microstructure is qualitatively analyzed from the 2D SEM image and the 3D pore distribution curve before and after freezing and thawing. The fractal dimension is utilized to characterize the two-dimensional topography image and the three-dimensional pore distribution, quantitatively. The results reveal that the surface porosity and volume porosity increase as the freeze-thaw action increases. Self-similarity characteristics exist in micro-damage inside the concrete. In the fractal dimension, it is possible to characterize pore evolution quantitatively. The fractal dimension correlates with pore damage evolution. The fractal dimension effectively quantitatively characterizes micro-damage features at various scales from the local to the global level.展开更多
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 withi...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.展开更多
To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for ...To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for numerical simulation.Numerical simulation results show that quartz in shale mainly affects the tensile and compressive strength of shale by increasing rock brittleness.The coupling of temperature and pressure will cause lateral and volume destruction of shale,which enables the shale body to be more easily broken.Fracture inclination is the key factor affecting shale damage patterns.The failure mode of shale with low-and high-angle fractures is mainly shear failure,and the compressive strength does not vary with crack inclination.The damage mode of obliquely intersecting fractured shale is slip damage along the fracture face,the compressive strength decreases and then increases with the fracture inclination,and a minimum value exists.The acoustic emission simulation results of the damage process effectively reflect the accumulated internal damage and macroscopic crack appearance until fracture instability when the prefabricated fractured shale is subjected to uniaxial compressive loading.The crack inclinations of 0°and 120℃ corresponds to the most complex"N"shape damage mode.The crack inclinations of 30°and 60°,and the damage mode is an inverted"λ"shape.展开更多
As a mathematical analysis method,fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties.Fractal analysis has been used to characterize the shapes of metal...As a mathematical analysis method,fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties.Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions.Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces.However,fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials.From the perspective of two-dimensional planes and three-dimensional curved surfaces,this paper reviews the current research status of the fractal analysis of metal precipitate interfaces,metal grain boundary interfaces,metal-deposited film surfaces,metal fracture surfaces,metal machined surfaces,and metal wear surfaces.The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized.Starting from three perspectives of fractal analysis,namely,research scope,image acquisition methods,and calculation methods,this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed.It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.展开更多
The microscopic characterization of isolated bubbles in gassy soil plays an important role in the macroscopic physical properties of sediments and is a key factor in the study of geological hazards in gas-bearing stra...The microscopic characterization of isolated bubbles in gassy soil plays an important role in the macroscopic physical properties of sediments and is a key factor in the study of geological hazards in gas-bearing strata.Based on the box-counting method and the pore fractal features in porous media,a fractal model of bubble microstructure parameters in gassy soil under different gas con-tents and vertical load conditions is established by using an industrial X-ray CT scanning system.The results show that the fractal di-mension of bubbles in the sample is correlated with the volume fraction of bubbles,and it is also restricted by the vertical load.The three-dimensional fractal dimension of the sample is about 1 larger than the average two-dimensional fractal dimension of all the slices from the same sample.The uniform porous media fractal model is used to test the equivalent diameter,and the results show that the variation of the measured pore diameter ratio is jointly restricted by the volume fraction and the vertical load.In addition,the measured self-similarity interval of the bubble area distribution is tested by the porous media fractal capillary bundle model,and the fitting curve of measured pore area ratio in a small loading range is obtained in this paper.展开更多
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...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.展开更多
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 w...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.展开更多
The study of the hydrate formation process in porous media is of great significance for hydrate application.In this work,the formation process of methane hydrate in porous media was monitored in situ by a low-field ma...The study of the hydrate formation process in porous media is of great significance for hydrate application.In this work,the formation process of methane hydrate in porous media was monitored in situ by a low-field magnetic resonance imaging(MRI)system.The formation characteristics of methane hydrate in porous media and the change of fractal dimension of pore space were studied through the change of residual water saturation and T2 relaxation time distribution.The experimental results show that the hydrate formation process is divided into two stages:fast and slow.During the formation process,the water in the pores is continuously consumed and transformed into hydrate,and the overall T2 distribution gradually shifts to the left.In the formation process of hydrate,the pore space becomes more complex,the change of fractal dimension from top to bottom of the reactor gradually increases,and the hydrate formation rate also gradually increases.展开更多
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 indicat...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.展开更多
In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f（x） of order v（v 〉 0） which is ...In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f（x） of order v（v 〉 0） which is written as D-Vf（x） has been proved to still be continuous and bounded. Furthermore, upper box dimension of D-v f（x） is no more than 2 and lower box dimension of D-v f（x） is no less than 1. If f（x） is a Lipshciz function, D-v f（x） also is a Lipshciz function. While f（x） is differentiable on [0, 1], D-v f（x） is differentiable on [0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f（x） satisfying HSlder condition, upper box dimension of Riemann-Liouville fractional integral of f（x） seems no more than upper box dimension of f（x）. Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying HSlder condition of order v（v 〉 0） is strictly less than 2 - v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.展开更多
基金Supported by National Natural Science Fund Program(40705038)~~
文摘[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.
基金Supported by the Fund for Creative Research Groups by National Natural Science Foundation of China (No. 40821004)the National Natural Science Foundation of China (No. 40906055)the National Basic Research Program of China (973 Program) (No. 2010CB428706)
文摘The impact of harmful algal blooms （HABs） on public health and related economics have been increasing in many coastal regions of the world. Sedimentation of algal cells through flocculation with clay particles is a promising strategy for controlling HABs. Previous studies found that removal efficiency （RE） was influenced by many factors, including clay type and concentration, algal growth stage, and physiological aspects of HAB cells. To estimate the effect of morphological characteristics of the aggregates on HAB cell removal, fractal dimensions were measured and the RE of three species of HAB organism, Heterosigma akashiwo, Alexandrium tamarense, and Skeletonema eostatum, by original clay and modified clay, was determined. For all HAB species, the modified clay had a higher RE than original clay. For the original clay, the two-dimensional fractal dimension （D2） was 1.92 and three-dimensional ffactal dimension （D3） 2.81, while for the modified clay, D2 was 1.84 and D3 was 2.50. The addition of polyaluminum chloride （PAC1） lead to a decrease of the repulsive barrier between clay particles, and resulted in lower D2 and D3. Due to the decrease of D3, and the increase of the effective sticking coefficient, the flocculation rate between modified clay particles and HAB organisms increased, and thus resulted in a high RE. The fractal dimensions of flocs differed in HAB species with different cell morphologies. For example, Alexandrium tamarense cells are ellipsoidal, and the D3 and D2 of flocs were the highest, while for Skeletonema costatum, which has filamentous cells, the D3 and D2 of flocs were the lowest.
文摘In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: . At first, we prove the existence and uniqueness of the solution by priori estimation and the Galerkin method. Then, we obtain to the existence of the global attractor. At last, we consider that the estimation of the upper bounds of Hausdorff and fractal dimensions for the global attractors are obtained.
文摘The characteristics of the nonlinear dynamics in the Heavy Ion Collision (HIC) at intermediate energies have been studied by evaluating the productions of the Generalized Entropy (GE) and the Multifragmentation Entropy (ME) as well as the features of the information and fractal dimensions within the Isospin Quantum Molecular Dynamical Model compensated by the lattice methods. Results demonstrate from various views that the existence of deterministic chaos in the dynamical process of reaction.
基金Under the auspices of the National Natural Science Foundation of China(No.40301002) and the Key Program of the National Natural Science Foundation of China(No.40335046)
文摘MANDELBROT enunciated the uncertainty of the length of a coastline in his paper "How long is the coastline of Britain?" published in Science in 1967. The fractal concept was presented for the first time in that paper and has been applied to many fields ever since. Although fractal dimensions of lots of phenomena were calculated by the box-counting method,the quantitative influence of series of square grids on them is ignored. The issue is systematically discussed as a case study of the mountains of China′s Mainland in this paper. And some significant conclusions are drawn as follows: 1) Although the fractal character objectively exists in the mountains of China′s Mainland,and it does not vary with the changes of series of square grids,the fractal dimensions of the mountains of China′s Mainland are different with these changes. 2) The fractal dimensions of the mountains of China′s Mainland vary with the average lengths of sides of series of square grids. The fractal dimension of the mountains of China′s Mainland is the function of the average length of side of square grid. They conform to the formula D=f(r) (where D is the fractal dimension,and r is the average length of side of square grid). 3) Different dots of data collection can affect the fractal dimension of the mountains of China′s Mainland. 4) The same range of length of side of square grid and dots of data collection can ensure the comparison of fractal dimensions of the mountains of China′s Mainland. The research is helpful to get the more understanding of fractal and fractal dimension,and ensure that the fractal studies would be scientific.
文摘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.
文摘The assessment of emotions with fractal dimensions of EEG signals has been attempted before, but the quantification of the intensity and duration of sudden and short emotions remains a challenge. This paper suggests a method for this purpose, by using a new fractal dimension algorithm and by adjusting the amplitude of the EEG signal in order to obtain maximal separation of high and low fractal dimensions. The emotion was induced by embedding a scary image at 20 seconds in landscape videos of 60 seconds length. The new method did not only detect the onset of the emotion correctly, but also revealed its duration and intensity. The intensity is based on the magnitude and impulse of the fractal dimension signal. It is also shown that Higuchi’s method does not always detect emotion spikes correctly;on the contrary, the region of the expected emotional response can be represented by fractal dimensions smaller than the rest of the signal, whereas the new method directly reveals distinct spikes. The duration of these spikes was 10 - 11 seconds. The magnitude of these spikes varied across the EEG channels. The build-up and cool-down of the emotions can occur with steep and flat gradients.
基金the National Natural Science Foundation of China!(No.39670152)Chinese Academy of Scietlces.
文摘The changes of fractal dimension ofPicea koraiensis seedlings under different light intensities in natural secondary forests was studied. The results showed that with the change of light environment, crown characters ofPicea koraiensis seedlings exhibited a greater plastic in lateral number, lateral increment, lateral dry weight, and specific leaf area. The range of calculated fractal dimensions of seedling crowns was confined between 2.5728 and 2.1036, but maximum of fractal dimension achieved in term moderate shading and in extreme low light conditions fractal dimension was least.
文摘This paper discusses application of fractal dimensions to speech processing. Generalized dimensions of arbitrary orders and associated fractal parameters are used in speaker identification. A characteristic vactor based on these parameters is formed, and a recognition criterion definded in order to identify individual speakers. Experimental results show the usefulness of fractal dimensions in characterizing speaker identity.
基金supported by the Strategic Cooperation Technology Projects of CNPC and CUPB(No.ZLZX2020-02)the National Science and Technology Special(No.2017ZX05049-006-001)+1 种基金the National Natural Science Foundation of China(No.41602137)Science Foundation of China University of Petroleum,Beijing(No.2462020YXZZ022).
文摘The tight tuff reservoir is an unusual type of unconventional reservoir with strong heterogeneity.However,there is a lack of research on the microscopic pore structure that causes the heterogeneity of tuff reservoirs.Using the Chang 7 Formation in Ordos Basin,China as a case study,carbon-dioxide gas adsorption,nitrogen gas adsorption and high-pressure mercury injection are integrated to investigate the multi-scale pore structure characteristics of tuff reservoirs.Meanwhile,the fractal dimension is introduced to characterize the complexity of pore structure in tuff reservoirs.By this multi-experimental method,the quantitative characterizations of the full-range pore size distribution of four tuff types were obtained and compared in the size ranges of micropores,mesopores and macropores.Fractal dimension curves derived from full-range pores are divided into six segments as D1,D2,D3,D4,D5 and D6 corresponding to fractal characteristics of micropores,smaller mesopores,larger mesopores,smaller macropores,medium macropores and larger macropores,respectively.The macropore volume,average macropore radius and fractal dimension D5 significantly control petrophysical properties.The larger macropore volume,average macropore radius and D5 correspond to favorable pore structure and good reservoir quality,which provides new indexes for the tuff reservoir evaluation.This study enriches the understanding of the heterogeneity of pore structures and contributes to unconventional oil and gas exploration and development.
基金Funded by the Key Project of Science and Technology Research in Higher Educational Institutions of Inner Mongolia Autonomous Region (No.NJZZ22518)Inner Mongolia Natural Science Foundation Project (No.2022MS05043)Inner Mongolia Autonomous Region Water Conservancy Research Special Project(No.NSK2016-S11)。
文摘After exposure to freeze-thaw cycles, scanning electron microscopy(SEM) and nuclear magnetic resonance(NMR) were used to test the four mixtures. The microstructure is qualitatively analyzed from the 2D SEM image and the 3D pore distribution curve before and after freezing and thawing. The fractal dimension is utilized to characterize the two-dimensional topography image and the three-dimensional pore distribution, quantitatively. The results reveal that the surface porosity and volume porosity increase as the freeze-thaw action increases. Self-similarity characteristics exist in micro-damage inside the concrete. In the fractal dimension, it is possible to characterize pore evolution quantitatively. The fractal dimension correlates with pore damage evolution. The fractal dimension effectively quantitatively characterizes micro-damage features at various scales from the local to the global level.
基金supported by the National Natural Science Foundation of China (Nos.52374078 and 52074043)the Fundamental Research Funds for the Central Universities (No.2023CDJKYJH021)。
文摘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.
基金Funded by the Guizhou Province Outstanding Young Scientifc and Technological Talents Training Plan(No.Qian Kehe Platform Talents-YQK[2023]012)National Natural Science Foundation of China(Nos.52104080,52264004)+4 种基金Guizhou Science and Technology Fund(No.[2021]401)Guizhou Science and Technology Fund(Qiankehe Support[2023]136)Guizhou Science and Technology Fund(Qiankehe Support[2022]227)Guizhou Science and Technology Fund(Qiankehe Strategic Search for Minerals[2022]ZD005)Natural Science Special(Special Post)Scientifc Research Fund Project of Guizhou University(No.[2021]51)。
文摘To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for numerical simulation.Numerical simulation results show that quartz in shale mainly affects the tensile and compressive strength of shale by increasing rock brittleness.The coupling of temperature and pressure will cause lateral and volume destruction of shale,which enables the shale body to be more easily broken.Fracture inclination is the key factor affecting shale damage patterns.The failure mode of shale with low-and high-angle fractures is mainly shear failure,and the compressive strength does not vary with crack inclination.The damage mode of obliquely intersecting fractured shale is slip damage along the fracture face,the compressive strength decreases and then increases with the fracture inclination,and a minimum value exists.The acoustic emission simulation results of the damage process effectively reflect the accumulated internal damage and macroscopic crack appearance until fracture instability when the prefabricated fractured shale is subjected to uniaxial compressive loading.The crack inclinations of 0°and 120℃ corresponds to the most complex"N"shape damage mode.The crack inclinations of 30°and 60°,and the damage mode is an inverted"λ"shape.
基金financially supported by the National Key R&D Program of China(No.2022YFE0121300)the National Natural Science Foundation of China(No.52374376)the Introduction Plan for High-end Foreign Experts(No.G2023105001L)。
文摘As a mathematical analysis method,fractal analysis can be used to quantitatively describe irregular shapes with self-similar or self-affine properties.Fractal analysis has been used to characterize the shapes of metal materials at various scales and dimensions.Conventional methods make it difficult to quantitatively describe the relationship between the regular characteristics and properties of metal material surfaces and interfaces.However,fractal analysis can be used to quantitatively describe the shape characteristics of metal materials and to establish the quantitative relationships between the shape characteristics and various properties of metal materials.From the perspective of two-dimensional planes and three-dimensional curved surfaces,this paper reviews the current research status of the fractal analysis of metal precipitate interfaces,metal grain boundary interfaces,metal-deposited film surfaces,metal fracture surfaces,metal machined surfaces,and metal wear surfaces.The relationship between the fractal dimensions and properties of metal material surfaces and interfaces is summarized.Starting from three perspectives of fractal analysis,namely,research scope,image acquisition methods,and calculation methods,this paper identifies the direction of research on fractal analysis of metal material surfaces and interfaces that need to be developed.It is believed that revealing the deep influence mechanism between the fractal dimensions and properties of metal material surfaces and interfaces will be the key research direction of the fractal analysis of metal materials in the future.
基金supported by the Open Research Fund Program of State Key Laboratory of Hydroscience and Engineering(No.sk lhse-2022-D-03)the National Natural Science Foundation of China(Nos.U2006213,42277139)the Taishan Scholars Program(No.tsqn202306297).
文摘The microscopic characterization of isolated bubbles in gassy soil plays an important role in the macroscopic physical properties of sediments and is a key factor in the study of geological hazards in gas-bearing strata.Based on the box-counting method and the pore fractal features in porous media,a fractal model of bubble microstructure parameters in gassy soil under different gas con-tents and vertical load conditions is established by using an industrial X-ray CT scanning system.The results show that the fractal di-mension of bubbles in the sample is correlated with the volume fraction of bubbles,and it is also restricted by the vertical load.The three-dimensional fractal dimension of the sample is about 1 larger than the average two-dimensional fractal dimension of all the slices from the same sample.The uniform porous media fractal model is used to test the equivalent diameter,and the results show that the variation of the measured pore diameter ratio is jointly restricted by the volume fraction and the vertical load.In addition,the measured self-similarity interval of the bubble area distribution is tested by the porous media fractal capillary bundle model,and the fitting curve of measured pore area ratio in a small loading range is obtained in this paper.
文摘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.
文摘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.
文摘The study of the hydrate formation process in porous media is of great significance for hydrate application.In this work,the formation process of methane hydrate in porous media was monitored in situ by a low-field magnetic resonance imaging(MRI)system.The formation characteristics of methane hydrate in porous media and the change of fractal dimension of pore space were studied through the change of residual water saturation and T2 relaxation time distribution.The experimental results show that the hydrate formation process is divided into two stages:fast and slow.During the formation process,the water in the pores is continuously consumed and transformed into hydrate,and the overall T2 distribution gradually shifts to the left.In the formation process of hydrate,the pore space becomes more complex,the change of fractal dimension from top to bottom of the reactor gradually increases,and the hydrate formation rate also gradually increases.
文摘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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11201230 and 11271182)
文摘In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f（x） of order v（v 〉 0） which is written as D-Vf（x） has been proved to still be continuous and bounded. Furthermore, upper box dimension of D-v f（x） is no more than 2 and lower box dimension of D-v f（x） is no less than 1. If f（x） is a Lipshciz function, D-v f（x） also is a Lipshciz function. While f（x） is differentiable on [0, 1], D-v f（x） is differentiable on [0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f（x） satisfying HSlder condition, upper box dimension of Riemann-Liouville fractional integral of f（x） seems no more than upper box dimension of f（x）. Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying HSlder condition of order v（v 〉 0） is strictly less than 2 - v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.
