Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimen...Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless;2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance;and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.展开更多
Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better underst...Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better understand many complex time series observed in nature [1-4]. The Hurst exponent H （0 〈 H 〈 1） is the most important parameter characterizing any given time series F（t）, where t represents the time steps, and the fractal dimension D is determined via the relation D = 2 - H.展开更多
原始精细的路面谱是研究车辆与路面耦合的重要基础,对车辆的通过性和噪声、振动及不平顺性NVH(noise,vibration and harshness)分析有重要的应用价值。为了构建与原始路面相同或相近的路面谱,利用非接触式激光路面不平度仪测量沥青路面...原始精细的路面谱是研究车辆与路面耦合的重要基础,对车辆的通过性和噪声、振动及不平顺性NVH(noise,vibration and harshness)分析有重要的应用价值。为了构建与原始路面相同或相近的路面谱,利用非接触式激光路面不平度仪测量沥青路面、水泥路面、比利时路面和砂石路面的三维路面不平度;基于分形理论,采用迭代函数法重构这四种路面谱;结合路面不平度的统计特性评价指标和分形维数对四种路面的原始谱和重构谱进行评价。研究结果表明:重构前、后路面谱的平均值、标准差、峰度系数以及分形维数变化范围在±5%以内,除水泥路面的偏态系数变化范围均在±9%以内,重构路面谱与原始谱具有一致性;重构路面谱保持了原始路面的结构特性,并具有路面的细微结构。展开更多
Polaron effects in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are studied by applying the fractal dimension method. In this paper, the polaron properties of GaAs/AlxGa1-xAs core-shell nanowires with different co...Polaron effects in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are studied by applying the fractal dimension method. In this paper, the polaron properties of GaAs/AlxGa1-xAs core-shell nanowires with different core radii and aluminum concentrations are discussed. The polaron binding energy, polaron mass shift, and fractal dimension parameter are numerically determined as functions of shell width. The calculation results reveal that the binding energy and mass shift of the polaron first increase and then decrease as the shell width increases. A maximum value appears at a certain shell width for different aluminum concentrations and a given core radius. By using the fractal dimension method, polaron problems in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are solved in a simple manner that avoids complex and lengthy calculations.展开更多
This paper presents the application of iterated function system (IFS) based three-dimensional (3D) fractal interpolation to elevation data compression. The parameters of contractive transformations are simplified by a...This paper presents the application of iterated function system (IFS) based three-dimensional (3D) fractal interpolation to elevation data compression. The parameters of contractive transformations are simplified by a concise fractal iteration form with geometric meaning. A local iteration algorithm is proposed, which can solve the non-separation problem when Collage theorem is applied to find the appropriate fractal parameters. The elevation data compression is proved experimentally to be effective in. reconstruction quality and time-saving.展开更多
文摘Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless;2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance;and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.
基金partially supported by the National Natural Science Foundation of China(Grant Nos.11173064,11233001,11233008,and U1531131)the Strategic Priority Research Program,the Emergence of Cosmological Structures of the Chinese Academy of Sciences(Grant No.XDB09000000)
文摘Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better understand many complex time series observed in nature [1-4]. The Hurst exponent H （0 〈 H 〈 1） is the most important parameter characterizing any given time series F（t）, where t represents the time steps, and the fractal dimension D is determined via the relation D = 2 - H.
文摘原始精细的路面谱是研究车辆与路面耦合的重要基础,对车辆的通过性和噪声、振动及不平顺性NVH(noise,vibration and harshness)分析有重要的应用价值。为了构建与原始路面相同或相近的路面谱,利用非接触式激光路面不平度仪测量沥青路面、水泥路面、比利时路面和砂石路面的三维路面不平度;基于分形理论,采用迭代函数法重构这四种路面谱;结合路面不平度的统计特性评价指标和分形维数对四种路面的原始谱和重构谱进行评价。研究结果表明:重构前、后路面谱的平均值、标准差、峰度系数以及分形维数变化范围在±5%以内,除水泥路面的偏态系数变化范围均在±9%以内,重构路面谱与原始谱具有一致性;重构路面谱保持了原始路面的结构特性,并具有路面的细微结构。
文摘Polaron effects in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are studied by applying the fractal dimension method. In this paper, the polaron properties of GaAs/AlxGa1-xAs core-shell nanowires with different core radii and aluminum concentrations are discussed. The polaron binding energy, polaron mass shift, and fractal dimension parameter are numerically determined as functions of shell width. The calculation results reveal that the binding energy and mass shift of the polaron first increase and then decrease as the shell width increases. A maximum value appears at a certain shell width for different aluminum concentrations and a given core radius. By using the fractal dimension method, polaron problems in cylindrical GaAs/AlxGa1-xAs core-shell nanowires are solved in a simple manner that avoids complex and lengthy calculations.
文摘This paper presents the application of iterated function system (IFS) based three-dimensional (3D) fractal interpolation to elevation data compression. The parameters of contractive transformations are simplified by a concise fractal iteration form with geometric meaning. A local iteration algorithm is proposed, which can solve the non-separation problem when Collage theorem is applied to find the appropriate fractal parameters. The elevation data compression is proved experimentally to be effective in. reconstruction quality and time-saving.
