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...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.展开更多
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, althoug...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.展开更多
We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.
This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given bywhere 1 < s < 2, λk> tends to infinity as k→∞ and λk satisfies λk+1/λk≥λ...This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given bywhere 1 < s < 2, λk> tends to infinity as k→∞ and λk satisfies λk+1/λk≥λ>1. The results show thatis a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions. For the fractional Riemann-Liouvtlle differential operator Du and the fractional integral operator D-v, the results show that if A is sufficiently large, then a necessary and sufficient condition for box dimensionof Graph(D-v(B)), 0 < v < s - 1, to be s - v and box dimension of Graph(Du(B)), 0 < u < 2 - s, to bes + u is also lim.展开更多
Physics success is largely determined by using mathematics.Physics often themselves create the necessary mathematical apparatus.This article shows how you can construct a fractal calculus-mathematics of fractal geomet...Physics success is largely determined by using mathematics.Physics often themselves create the necessary mathematical apparatus.This article shows how you can construct a fractal calculus-mathematics of fractal geometry.In modem scientific literature often write from a firm that"there is no strict definition of fractals",to the more moderate that"objects in a certain sense,fractal and similar."We show that fractal geometry is a strict mathematical theory,defined by their axioms.This methodology allows the geometry of axiomatised naturally define fractal integrals and differentials.Consistent application on your input below the axiom gives the opportunity to develop effective methods of measurement of fractal dimension,geometri-cal interpretation of fractal derivative gain and open dual symmetry.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
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 contin...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.展开更多
The paper proposes a new multiple-factor clustering method(NMFCM)with consideration of both box fractal dimension(BFD)and orientation of joints.This method assumes that the BFDs of different clusters were uneven,and c...The paper proposes a new multiple-factor clustering method(NMFCM)with consideration of both box fractal dimension(BFD)and orientation of joints.This method assumes that the BFDs of different clusters were uneven,and clustering was performed by redistributing the joints near the boundaries of clusters on a polar map to maximize an index for estimating the difference of the BFD(DBFD).Three main aspects were studied to develop the NMFCM.First,procedures of the NMFCM and reasonableness of assumptions were illustrated.Second,main factors affecting the NMFCM were investigated by numerical simulations with disk joint models.Finally,two different sections of a rock slope were studied to verify the practicability of the NMFCM.The results demonstrated that:(1)The NMFCM was practical and could effectively alleviate the problem of hard boundary during clustering;(2)The DBFD tended to increase after the improvement of clustering accuracy;(3)The improvement degree of the NMFCM clustering accuracy was mainly influenced by three parameters,namely,the number of clusters,number of redistributed joints,and total number of joints;and(4)The accuracy rate of clustering could be effectively improved by the NMFCM.展开更多
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 integr...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.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
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.
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 ...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.展开更多
Complex nonlinear dynamic systems are ubiquitous in the landscapes and phenomena studied by earth sciences in general and by geomorphology in particular. Many natural landscape features have an aspect such as fractals...Complex nonlinear dynamic systems are ubiquitous in the landscapes and phenomena studied by earth sciences in general and by geomorphology in particular. Many natural landscape features have an aspect such as fractals. In the many geomorphologic phenomena such as river networks and coast lines this is visible. In recent years fractal geometry offers as an option for modeling river geometry and physical processes of rivers. The fractal dimension is a statistical quantity that indicates how a fractal scales with the shrink, the space occupied. This theory has the mathematical basis but also applied in geomorphology and also shown great success. Physical behavior of many natural processes as well as using fractal geometry is predictable relations. Behavior of complex natural phenomena as rivers has always been of interest to researchers. In this study using data basic maps, drainage networks map and Digital Elevation Model of the ground was prepared. Then applying the rules Horton-Strahler river network, fractal dimensions were calculated to examine the relationship between fractal dimension and some rivers geomorphic features were investigated. Results showed fractal dimension of watersheds have meaningful relations with factors such as shape form, area, bifurcation ratio and length ratio in the watersheds.展开更多
The insulating paper of the transformer is affected by many factors during the operation,meanwhile,the surface texture of the paper is easy to change.To explore the relationship between the aging state and surface tex...The insulating paper of the transformer is affected by many factors during the operation,meanwhile,the surface texture of the paper is easy to change.To explore the relationship between the aging state and surface texture change of insulating paper,firstly,the thermal aging experiment of insulating paper is carried out,and the insulating paper samples with different aging times are obtained.After then,the images of the aged insulating paper samples are collected and pre-processed.The pre-processing effect is verified by constructing and calculating the gray surface of the sample.Secondly,the texture features of the insulating paper image are extracted by box dimension and multifractal spectrum.Based on that,the extreme learning machine(ELM)is taken as the classification tool with texture features and aging time as the input and output,to train the algorithm and construct the corresponding relationship between the texture feature and the aging time.After then,the insulating paper with unknown aging time is predicted with a trained ELMalgorithm.The numerical test results show that the texture features extracted from the fractal dimension of the micro image can effectively characterize the aging state of insulating paper,the average accuracy can reach 91.6%.It proves that the fractal dimension theory can be utilized for assessing the aging state of insulating paper for onsite applications.展开更多
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, wh...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.展开更多
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.
In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.
文摘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.
文摘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.
文摘We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.
基金Research supported by national Natural Science Foundation of China (10141001)Zhejiang Provincial Natural Science Foundation 9100042 and 1010009.
文摘This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given bywhere 1 < s < 2, λk> tends to infinity as k→∞ and λk satisfies λk+1/λk≥λ>1. The results show thatis a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions. For the fractional Riemann-Liouvtlle differential operator Du and the fractional integral operator D-v, the results show that if A is sufficiently large, then a necessary and sufficient condition for box dimensionof Graph(D-v(B)), 0 < v < s - 1, to be s - v and box dimension of Graph(Du(B)), 0 < u < 2 - s, to bes + u is also lim.
文摘Physics success is largely determined by using mathematics.Physics often themselves create the necessary mathematical apparatus.This article shows how you can construct a fractal calculus-mathematics of fractal geometry.In modem scientific literature often write from a firm that"there is no strict definition of fractals",to the more moderate that"objects in a certain sense,fractal and similar."We show that fractal geometry is a strict mathematical theory,defined by their axioms.This methodology allows the geometry of axiomatised naturally define fractal integrals and differentials.Consistent application on your input below the axiom gives the opportunity to develop effective methods of measurement of fractal dimension,geometri-cal interpretation of fractal derivative gain and open dual symmetry.
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.
文摘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.
基金funded by the National Natural Science Foundation of China(Grant Nos.41972264 and 52078093)Liaoning Revitalization Talents Program,China(Grant No.XLYC1905015)。
文摘The paper proposes a new multiple-factor clustering method(NMFCM)with consideration of both box fractal dimension(BFD)and orientation of joints.This method assumes that the BFDs of different clusters were uneven,and clustering was performed by redistributing the joints near the boundaries of clusters on a polar map to maximize an index for estimating the difference of the BFD(DBFD).Three main aspects were studied to develop the NMFCM.First,procedures of the NMFCM and reasonableness of assumptions were illustrated.Second,main factors affecting the NMFCM were investigated by numerical simulations with disk joint models.Finally,two different sections of a rock slope were studied to verify the practicability of the NMFCM.The results demonstrated that:(1)The NMFCM was practical and could effectively alleviate the problem of hard boundary during clustering;(2)The DBFD tended to increase after the improvement of clustering accuracy;(3)The improvement degree of the NMFCM clustering accuracy was mainly influenced by three parameters,namely,the number of clusters,number of redistributed joints,and total number of joints;and(4)The accuracy rate of clustering could be effectively improved by the NMFCM.
文摘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.
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.
基金Project supported by National Natural Science Foundation of China.
文摘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.
文摘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.
文摘Complex nonlinear dynamic systems are ubiquitous in the landscapes and phenomena studied by earth sciences in general and by geomorphology in particular. Many natural landscape features have an aspect such as fractals. In the many geomorphologic phenomena such as river networks and coast lines this is visible. In recent years fractal geometry offers as an option for modeling river geometry and physical processes of rivers. The fractal dimension is a statistical quantity that indicates how a fractal scales with the shrink, the space occupied. This theory has the mathematical basis but also applied in geomorphology and also shown great success. Physical behavior of many natural processes as well as using fractal geometry is predictable relations. Behavior of complex natural phenomena as rivers has always been of interest to researchers. In this study using data basic maps, drainage networks map and Digital Elevation Model of the ground was prepared. Then applying the rules Horton-Strahler river network, fractal dimensions were calculated to examine the relationship between fractal dimension and some rivers geomorphic features were investigated. Results showed fractal dimension of watersheds have meaningful relations with factors such as shape form, area, bifurcation ratio and length ratio in the watersheds.
基金This work was supported by the Tianyou Youth Talent Lift Program of Lanzhou Jiaotong University,the Youth Science Foundation of Lanzhou Jiaotong University(No.2019029)the University Innovation Fund Project of Gansu Provincial Department of Education(No.2020A-036)the Young Doctor Foundation of JYT.GANSU.GOV.CN(No.2021QB-060).
文摘The insulating paper of the transformer is affected by many factors during the operation,meanwhile,the surface texture of the paper is easy to change.To explore the relationship between the aging state and surface texture change of insulating paper,firstly,the thermal aging experiment of insulating paper is carried out,and the insulating paper samples with different aging times are obtained.After then,the images of the aged insulating paper samples are collected and pre-processed.The pre-processing effect is verified by constructing and calculating the gray surface of the sample.Secondly,the texture features of the insulating paper image are extracted by box dimension and multifractal spectrum.Based on that,the extreme learning machine(ELM)is taken as the classification tool with texture features and aging time as the input and output,to train the algorithm and construct the corresponding relationship between the texture feature and the aging time.After then,the insulating paper with unknown aging time is predicted with a trained ELMalgorithm.The numerical test results show that the texture features extracted from the fractal dimension of the micro image can effectively characterize the aging state of insulating paper,the average accuracy can reach 91.6%.It proves that the fractal dimension theory can be utilized for assessing the aging state of insulating paper for onsite applications.
文摘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.
基金National Natural Science Foundation of Zhejiang Province
文摘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.
文摘In this paper, the relationship between Riemann-Liouville fractional integral and the box-counting dimension of graphs of fractal functions is discussed.
