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1.
《Chaos, solitons, and fractals》2007,31(1):5-13
Fractals in the large can be generated as the invariant set of an expansive, iterated function system. A number of dimensions have been introduced and studied for such fractals. In this note we show that these dimensions coincide for large fractals generated by functions with arithmetic expansion factors, and that this common dimension is equal to the dimension of the (small) fractal generated by the inverse functions. 相似文献
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Yanguang Chen 《Chaos, solitons, and fractals》2011,44(8):619-632
The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d = 2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D < d = 2 indicative of spatial intermittency. Thus the conventional central place models are converted into fractal central place models. If we further integrate the chance factors into the improved central place fractals, the theory will be able to explain the real patterns of urban places very well. As empirical analyses, the US cities and towns are employed to verify the fractal-based models of central places. 相似文献
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The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem. 相似文献
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Fractal properties of real world objects are commonly examined in digital images. Digital images are discrete representations of objects or scenes and are unavoidably contaminated with noise disturbing the representation of the captured objects. We evaluate the noise dependency of frequently applied algorithms for the calculation of the fractal dimension in digital images. Three mathematically defined fractals (Koch Curve, Sierpinski Gasket, Menger Carpet), representative for low, middle and high values of the fractal dimension, together with an experimentally obtained fractal structure were contaminated with well-defined levels of artificial noise. The Box-Counting Dimension, the Correlation Dimension and the rather unknown Tug-of-War Dimension were calculated for the data sets in order to estimate the fractal dimensionality under the presence of accumulated noise. We found that noise has a significant influence on the computed fractal dimensions (relative increases up to 20%) and that the influence is sensitive to the applied algorithm and the space filling characteristics of the investigated fractal structures. The similarities of the effect of noise on experimental and artificial fractals confirm the reliability of the obtained results. 相似文献
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Yi Wang 《Journal of Mathematical Analysis and Applications》2009,354(2):445-450
In this paper, we consider a class of fractals generated by the Cantor series expansions. By constructing some homogeneous Moran subsets, we prove that these sets have full dimension. 相似文献
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余旌胡 《数学物理学报(A辑)》2001,21(4):443-452
定义了一类广泛的随机自仿射集,得到了此类集合的Hausdorff维数估计.此前的随机自相似(包括Graf,Mauldin与Falconer等定义的随机自相似情形)和Falconer定义的(严格)自仿射以及作者定义的μ 统计自仿射情形均成为该文结果的特例. 相似文献
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In the paper, we try to classify Moran fractals by using the quasi-Lipschitz equivalence, and prove that two regular homogeneous Moran sets are quasi-Lipschitz equivalent if and only if they have the same Hausdorff dimension. 相似文献
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We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples. 相似文献
10.
G. Reza Rakhshandehroo M.R. Shaghaghian A.R. Keshavarzi N. Talebbeydokhti 《Applied Mathematical Modelling》2009
Fractals are objects which have similar appearances when viewed at different scales. Such objects have details at arbitrarily small scales, making them too complex to be represented by Euclidian space; hence, they are assigned a non-integer dimension. Some natural phenomena have been modeled as fractals with success; examples include geologic deposits, topographic surfaces and seismic activities. In particular, time series have been represented as a curve with fractal dimensions between one and two. There are different ways to define fractal dimension, most being equivalent in the continuous domain. However, when applied in practice to discrete data sets, different ways lead to different results. In this study, three methods for estimating fractal dimension are described and two standard algorithms, Hurst’s rescaled range analysis and box-counting method (BC), are compared with the recently introduced variation method (VM). It was confirmed that the last method offers a superior efficiency and accuracy, and hence may be recommended for fractal dimension calculations for time series data. All methods were applied to the measured temporal variation of velocity components in turbulent flows in an open channel in Shiraz University laboratory. The analyses were applied to 2500 measurements at different Reynold’s numbers and it was concluded that a certain degree of randomness may be associated with the velocity in all directions which is a unique character of the flow independent of the Reynold’s number. Results also suggest that the rigid lateral confinement of flow to the fixed channel width allows for designation of a more-or-less constant fractal dimension for the spanwise velocity component. On the contrary, in vertical and streamwise directions more freedom of movements for fluid particles sets more room for variation in fractal dimension at different Reynold’s numbers. 相似文献
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In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R~n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by "multirules" take the value in Marstrand's theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure L~m. When μV《 L~m, the connection of the local dimension ofμVand the box dimension of slices is given. 相似文献
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There exist several sets having similar structure on arbitrarily small scales. Mandelbrot called such sets fractals, and defined a dimension that assigns non-integer numbers to fractals. On the other hand, a dynamical system yielding a fractal set referred to as a strange attractor is a chaotic map. In this paper, a characterization of self-similarity for attractors is attempted by means of conditional entropy. 相似文献
13.
Xia Li Weiyi Su 《分析论及其应用》2007,23(3):283-300
In the book [1] H.Triebel introduces the distributional dimension of fractals in an analytical form and proves that: for Г as a non-empty set in R^n with Lebesgue measure |Г| = 0, one has dimH Г = dimD Г, where dimD Г and dimH Г are the Hausdorff dimension and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically. By
discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end. 相似文献
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We give a formula for the Hausdorff dimension of fractals which are the support of certain Riesz-product type measures. 相似文献
17.
Zeke Wang 《Annals of Operations Research》1990,24(1):261-271
PL homotopy methods are effective numerical methods for highly nonlinear problems. It is widely believed that the feasibility of a PL homotopy method depends on the nondegeneracy condition that the zero set (or the fixed point set in the case of finding fixed points instead of zeroes) of the PL approximation of the homotopy does not intersect the triangulation's skeletons of co-dimensions two and above. This paper shows that, although the sections of the PL approximation's zero set tracked by the PL homotopy method are of dimension one (while other sections may have higher dimensions), the paths generated by the pivoting method are potentially and essentially of dimension two. It makes pathcrossing a safe thing. Thus, this paper first sets up the without exception feasibility of PL homotopy methods geometrically.This work is supported in part by the Foundation of Zhongshan University Advanced Research Centre. 相似文献
18.
Linear dimension reduction plays an important role in classification problems. A variety of techniques have been developed for linear dimension reduction to be applied prior to classification. However, there is no single definitive method that works best under all circumstances. Rather a best method depends on various data characteristics. We develop a two-step adaptive procedure in which a best dimension reduction method is first selected based on the various data characteristics, which is then applied to the data at hand. It is shown using both simulated and real life data that such a procedure can significantly reduce the misclassification rate. 相似文献
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“Graph-directed” fractals are collections of metric spaces, each of which can be expressed as a union of several scaled copies of spaces from the collection. They give rise to weighted, directed graphs where the term comes from. We show in this note that any (finite) weighted, directed graph (with weights between 0 and 1) can be realized in a Euclidean space in the sense that, starting from the graph one can define a system of similitudes (with the similarity ratios being the given weights) on an appropriate Euclidean space. The point is that these maps satisfy a certain property (called the open set condition) so that the theory of Mauldin–Williams can be applied to compute the dimension of the emerging fractals. Additionally, we give a novel example of a system of graph-directed fractals. 相似文献