In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactars of a special class of iterated function systems. We show that such spaces are finite dimensional and give the...In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactars of a special class of iterated function systems. We show that such spaces are finite dimensional and give the bases of these spaces in an implicit way. Given such a space, we discuss how to obtain a set of knots for whah the Lagrange interpolation problem by the space is uniquely solvable.展开更多
In this paper,we furst construct a claas of fraetal funerions by means of b-adic fraction andinfinite series expressions.Then we investigate the fractal dimensions of the graphs of these funcrionsand Holder continuity...In this paper,we furst construct a claas of fraetal funerions by means of b-adic fraction andinfinite series expressions.Then we investigate the fractal dimensions of the graphs of these funcrionsand Holder continuity.Some of results of dimensions are obtained.展开更多
Bush type fractal functions were defined by means of the expression of Cantor series of real numbers. The upper and lower bound estimates for the K-dimension of such functions were given. In a typical case, the fracta...Bush type fractal functions were defined by means of the expression of Cantor series of real numbers. The upper and lower bound estimates for the K-dimension of such functions were given. In a typical case, the fractal dimensional relations in which the K-dimension equals the box dimension and packing dimension were presented; moreover, the exact Holder exponent were obtained for such Bush type functions.展开更多
Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional ca...Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional calculus and fractal functions,based only on fractal dimension considerations.Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves.Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist.After further discussion,fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Holder continuous functions.Investigation about other fractional calculus,such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary.This work is helpful to reveal the mechanism of fractional calculus on continuous functions.At the same time,it provides some theoretical basis for the rationality of the definition of fractional calculus.This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.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.
Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermit...Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].展开更多
In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their...In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.展开更多
Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions （FIFs） on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling fa...Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions （FIFs） on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket （SG）. As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5 ：△=0 on SG ／ {q1, q2, q3}, and u（qi）=ai, i = 1, 2, 3, where qi, i=1, 2, 3, are boundary points of SG.展开更多
Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the...Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.展开更多
The purpose of this paper is to prove a Holder property about the fractal interpolation function L(x), ω(L,δ)=O(δ~α), and an approximate estimate |f-L|≤2{α(h)+||f||/1-h^(2-D)·h^(2-D)}, where D is a fractal ...The purpose of this paper is to prove a Holder property about the fractal interpolation function L(x), ω(L,δ)=O(δ~α), and an approximate estimate |f-L|≤2{α(h)+||f||/1-h^(2-D)·h^(2-D)}, where D is a fractal dimension of L(x).展开更多
In this paper, we study on the application of radical B-spline wavelet scaling function in fractal function approximation system. The paper proposes a wavelet-based fractal function approximation algorithm in which th...In this paper, we study on the application of radical B-spline wavelet scaling function in fractal function approximation system. The paper proposes a wavelet-based fractal function approximation algorithm in which the coefficients can be determined by solving a convex quadraticprogramming problem. And the experiment result shows that the approximation error of this algorithm is smaller than that of the polynomial-based fractal function approximation. This newalgorithm exploits the consistency between fractal and scaling function in multi-scale and multiresolution, has a better approximation effect and high potential in data compression, especially inimage compression.展开更多
A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. T...A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. The variance of scattering intensity is presented to study the fractal characteristics through theoretical analysis and numerical calculations. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated as a slope of linear equation. This conclusion will be applicable for solving the inverse problem of reconstructing rough surface and remote sensing.展开更多
The EM scattering from rough surface has been investigated in the past years.Periodic and random models are often used in modeling the rough surface.Recently,the fractal geometry is rapidly improved.It provides a new ...The EM scattering from rough surface has been investigated in the past years.Periodic and random models are often used in modeling the rough surface.Recently,the fractal geometry is rapidly improved.It provides a new way to model the rough surface whose characteristics are long range order and short range disorder.In this paper,A fractal function is used to model the rough surface.A scattering coefficient for calculating the angular distribution and the amount of energy in the spectrally scattering field to the fractal characteristics of the surfaces by finding their analytical expressions is derived by using the Kirchhoff solution.In the end,we calculate some scattering patterns.展开更多
The analysis of the RCS from the rough sea and ground surface is made. The two dimensionally band limited fractal function is used to model the sea and ground surface, the scattered electromagnetic field is calculat...The analysis of the RCS from the rough sea and ground surface is made. The two dimensionally band limited fractal function is used to model the sea and ground surface, the scattered electromagnetic field is calculated by using Kirchhoff approximation. The validity of this result is assured by some references, which indicates that the methods are reliable.展开更多
In this study,the fractal dimensions of velocity fluctuations and the Reynolds shear stresses propagation for flow around a circular bridge pier are presented.In the study reported herein,the fractal dimension of velo...In this study,the fractal dimensions of velocity fluctuations and the Reynolds shear stresses propagation for flow around a circular bridge pier are presented.In the study reported herein,the fractal dimension of velocity fluctuations（u′,v′,w′） and the Reynolds shear stresses（u′v′ and u′w′） of flow around a bridge pier were computed using a Fractal Interpolation Function（FIF） algorithm.The velocity fluctuations of flow along a horizontal plane above the bed were measured using Acoustic Doppler Velocity meter（ADV）and Particle Image Velocimetry（P1V）.The PIV is a powerful technique which enables us to attain high resolution spatial and temporal information of turbulent flow using instantaneous time snapshots.In this study,PIV was used for detection of high resolution fractal scaling around a bridge pier.The results showed that the fractal dimension of flow fluctuated significantly in the longitudinal and transverse directions in the vicinity of the pier.It was also found that the fractal dimension of velocity fluctuations and shear stresses increased rapidly at vicinity of pier at downstream whereas it remained approximately unchanged far downstream of the pier.The higher value of fractal dimension was found at a distance equal to one times of the pier diameter in the back of the pier.Furthermore,the average fractal dimension for the streamwise and transverse velocity fluctuations decreased from the centreline to the side wall of the flume.Finally,the results from ADV measurement were consistent with the result from PIV,therefore,the ADV enables to detect turbulent characteristics of flow around a circular bridge pier.展开更多
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
A one-dimensional continuous function of unbounded variation on [0, 1] has been con- structed. The length of its graph is infinite, while part＂ of this function displays fractal features. The Box dimension of its Rie...A one-dimensional continuous function of unbounded variation on [0, 1] has been con- structed. The length of its graph is infinite, while part＂ of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated.展开更多
The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on t...The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on the Euclidean space R^n and the local field K are compared,so that one may interpret the essential differences between the analyses on R^n and K.Finally,the Cantor type fractal functionθ(x)is showed in the Lipschitz class Lip(m,K),m＜(ln 2/ln 3).展开更多
文摘In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactars of a special class of iterated function systems. We show that such spaces are finite dimensional and give the bases of these spaces in an implicit way. Given such a space, we discuss how to obtain a set of knots for whah the Lagrange interpolation problem by the space is uniquely solvable.
基金Supported by the Scientific Research Foundation of Huaibei Coal Industry Teaccher's Colhge(00061).
文摘In this paper,we furst construct a claas of fraetal funerions by means of b-adic fraction andinfinite series expressions.Then we investigate the fractal dimensions of the graphs of these funcrionsand Holder continuity.Some of results of dimensions are obtained.
基金The National Natural Science Foundation of China (No.10171080)
文摘Bush type fractal functions were defined by means of the expression of Cantor series of real numbers. The upper and lower bound estimates for the K-dimension of such functions were given. In a typical case, the fractal dimensional relations in which the K-dimension equals the box dimension and packing dimension were presented; moreover, the exact Holder exponent were obtained for such Bush type functions.
基金Supported by National Natural Science Foundation of China(Grant No.12071218)the Fundamental Research Funds for the Central Universities(Grant No.30917011340)。
文摘Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional calculus and fractal functions,based only on fractal dimension considerations.Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves.Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist.After further discussion,fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Holder continuous functions.Investigation about other fractional calculus,such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary.This work is helpful to reveal the mechanism of fractional calculus on continuous functions.At the same time,it provides some theoretical basis for the rationality of the definition of fractional calculus.This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.
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.
基金partially supported by the CSIR India(Grant No.09/084(0531)/2010-EMR-I)the SERC,DST India(Project No.SR/S4/MS:694/10)
文摘Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].
文摘In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.
基金Supported by the National Natural Science Foundation of China(11271327)Zhejiang Provincial National Science Foundation of China(LR14A010001)
文摘Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions （FIFs） on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket （SG）. As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5 ：△=0 on SG ／ {q1, q2, q3}, and u（qi）=ai, i = 1, 2, 3, where qi, i=1, 2, 3, are boundary points of SG.
文摘Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.
文摘The purpose of this paper is to prove a Holder property about the fractal interpolation function L(x), ω(L,δ)=O(δ~α), and an approximate estimate |f-L|≤2{α(h)+||f||/1-h^(2-D)·h^(2-D)}, where D is a fractal dimension of L(x).
文摘In this paper, we study on the application of radical B-spline wavelet scaling function in fractal function approximation system. The paper proposes a wavelet-based fractal function approximation algorithm in which the coefficients can be determined by solving a convex quadraticprogramming problem. And the experiment result shows that the approximation error of this algorithm is smaller than that of the polynomial-based fractal function approximation. This newalgorithm exploits the consistency between fractal and scaling function in multi-scale and multiresolution, has a better approximation effect and high potential in data compression, especially inimage compression.
基金supported by the National Natural Science Foundation of China (Grant No 60571058)Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant No 20070701010)
文摘A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. The variance of scattering intensity is presented to study the fractal characteristics through theoretical analysis and numerical calculations. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated as a slope of linear equation. This conclusion will be applicable for solving the inverse problem of reconstructing rough surface and remote sensing.
文摘The EM scattering from rough surface has been investigated in the past years.Periodic and random models are often used in modeling the rough surface.Recently,the fractal geometry is rapidly improved.It provides a new way to model the rough surface whose characteristics are long range order and short range disorder.In this paper,A fractal function is used to model the rough surface.A scattering coefficient for calculating the angular distribution and the amount of energy in the spectrally scattering field to the fractal characteristics of the surfaces by finding their analytical expressions is derived by using the Kirchhoff solution.In the end,we calculate some scattering patterns.
文摘The analysis of the RCS from the rough sea and ground surface is made. The two dimensionally band limited fractal function is used to model the sea and ground surface, the scattered electromagnetic field is calculated by using Kirchhoff approximation. The validity of this result is assured by some references, which indicates that the methods are reliable.
文摘In this study,the fractal dimensions of velocity fluctuations and the Reynolds shear stresses propagation for flow around a circular bridge pier are presented.In the study reported herein,the fractal dimension of velocity fluctuations（u′,v′,w′） and the Reynolds shear stresses（u′v′ and u′w′） of flow around a bridge pier were computed using a Fractal Interpolation Function（FIF） algorithm.The velocity fluctuations of flow along a horizontal plane above the bed were measured using Acoustic Doppler Velocity meter（ADV）and Particle Image Velocimetry（P1V）.The PIV is a powerful technique which enables us to attain high resolution spatial and temporal information of turbulent flow using instantaneous time snapshots.In this study,PIV was used for detection of high resolution fractal scaling around a bridge pier.The results showed that the fractal dimension of flow fluctuated significantly in the longitudinal and transverse directions in the vicinity of the pier.It was also found that the fractal dimension of velocity fluctuations and shear stresses increased rapidly at vicinity of pier at downstream whereas it remained approximately unchanged far downstream of the pier.The higher value of fractal dimension was found at a distance equal to one times of the pier diameter in the back of the pier.Furthermore,the average fractal dimension for the streamwise and transverse velocity fluctuations decreased from the centreline to the side wall of the flume.Finally,the results from ADV measurement were consistent with the result from PIV,therefore,the ADV enables to detect turbulent characteristics of flow around a circular bridge pier.
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.
基金Supported by National Natural Science Foundation of China(Grant No.11201230)Natural Science Foundation of Jiangsu Province(Grant No.BK2012398)
文摘A one-dimensional continuous function of unbounded variation on [0, 1] has been con- structed. The length of its graph is infinite, while part＂ of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated.
基金This work supported by the National Natural Science Foundation of China(Grant No.10571084)
文摘The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on the Euclidean space R^n and the local field K are compared,so that one may interpret the essential differences between the analyses on R^n and K.Finally,the Cantor type fractal functionθ(x)is showed in the Lipschitz class Lip(m,K),m＜(ln 2/ln 3).
