We study the ionization of helium Rydberg atoms in an electric field above the classical ionization threshold within the semiclassical theory.By introducing a fractal approach to describe the chaotic dynamical behavio...We study the ionization of helium Rydberg atoms in an electric field above the classical ionization threshold within the semiclassical theory.By introducing a fractal approach to describe the chaotic dynamical behavior of the ionization,we identify the fractal self-similarity structure of the escape time versus the distribution of the initial launch angles of electrons,and find that the self-similarity region shifts toward larger initial launch angles with a decrease in the scaled energy.We connect the fractal structure of the escape time plot to the escape dynamics of ionized electrons.Of particular note is that the fractal dimensions are sensitively controlled by the scaled energy and magnetic field,and exhibit excellent agreement with the chaotic extent of the ionization systems for both helium and hydrogen Rydberg atoms.It is shown that,besides the electric and magnetic fields,core scattering is a primary factor in the fractal dynamics.展开更多
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
Based on the fractal theory, this study establishes a Continuous Spatial Scaling Model （CSSM） of the Normalized Difference Vegetation Index （NDVI） to address issues arising with spatial up-scaling in quantitative ...Based on the fractal theory, this study establishes a Continuous Spatial Scaling Model （CSSM） of the Normalized Difference Vegetation Index （NDVI） to address issues arising with spatial up-scaling in quantitative remote sensing. This model is able to quantitatively describe transformation relationships of the NDVI on continuous scales. Then the following experiments are accomplished： （1） the validation of ETM＋ NDVI imagery is implemented based on the GEOEYE-1 image and its NDVI CSSM, and the following conclusion is obtained： because of bad stripes in the ETM＋ image and the limited effect of destriping, the ETM＋ NDVI image had a rather large error, and the error for the entire experimental imagery is about 25%, so the ETM＋ NDVI product is not suitable for direct practical application; （2） Shatian Byland （Beihai City, China） is taken as the experimental area, and four images （two ETM＋ images with wider and smaller coverage, respectively, a GEOEYE-1 image, and an HJ-1B CCD1 image） are studied. The most suitable scale levels are computed and compared for the four images, and a better understanding is obtained of the impact of various image characteristics （area of coverage, spatial resolution, and imaging quality） on determining the scale level for the NDVI CSSM.展开更多
Fractals are essentially characterized by their self-similarity at different scales and non-integer Hausdorff dimensions[1],while crystals always show certain symmetries and discrete diffraction diagrams[2].Thus,a fra...Fractals are essentially characterized by their self-similarity at different scales and non-integer Hausdorff dimensions[1],while crystals always show certain symmetries and discrete diffraction diagrams[2].Thus,a fractal crystal by definition must be identical at all scales with a compatible symmetry with crystals.Although fractals,e.g.snowflakes,trees,coastlines and blood-vascular systems,展开更多
基金Project supported by the Natural Science Foundation of Shandong Province,China(Grant No.ZR2014AM030)
文摘We study the ionization of helium Rydberg atoms in an electric field above the classical ionization threshold within the semiclassical theory.By introducing a fractal approach to describe the chaotic dynamical behavior of the ionization,we identify the fractal self-similarity structure of the escape time versus the distribution of the initial launch angles of electrons,and find that the self-similarity region shifts toward larger initial launch angles with a decrease in the scaled energy.We connect the fractal structure of the escape time plot to the escape dynamics of ionized electrons.Of particular note is that the fractal dimensions are sensitively controlled by the scaled energy and magnetic field,and exhibit excellent agreement with the chaotic extent of the ionization systems for both helium and hydrogen Rydberg atoms.It is shown that,besides the electric and magnetic fields,core scattering is a primary factor in the fractal dynamics.
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.
文摘Based on the fractal theory, this study establishes a Continuous Spatial Scaling Model （CSSM） of the Normalized Difference Vegetation Index （NDVI） to address issues arising with spatial up-scaling in quantitative remote sensing. This model is able to quantitatively describe transformation relationships of the NDVI on continuous scales. Then the following experiments are accomplished： （1） the validation of ETM＋ NDVI imagery is implemented based on the GEOEYE-1 image and its NDVI CSSM, and the following conclusion is obtained： because of bad stripes in the ETM＋ image and the limited effect of destriping, the ETM＋ NDVI image had a rather large error, and the error for the entire experimental imagery is about 25%, so the ETM＋ NDVI product is not suitable for direct practical application; （2） Shatian Byland （Beihai City, China） is taken as the experimental area, and four images （two ETM＋ images with wider and smaller coverage, respectively, a GEOEYE-1 image, and an HJ-1B CCD1 image） are studied. The most suitable scale levels are computed and compared for the four images, and a better understanding is obtained of the impact of various image characteristics （area of coverage, spatial resolution, and imaging quality） on determining the scale level for the NDVI CSSM.
文摘Fractals are essentially characterized by their self-similarity at different scales and non-integer Hausdorff dimensions[1],while crystals always show certain symmetries and discrete diffraction diagrams[2].Thus,a fractal crystal by definition must be identical at all scales with a compatible symmetry with crystals.Although fractals,e.g.snowflakes,trees,coastlines and blood-vascular systems,
