表面分形球、柱的光散射

分形和分维是近些年才兴起的一门处理复杂事物的新方法￣［１］。本文主要研究具有表面分形的球、柱的光散射特征。我们首先利用几何光学近似研究了一种表面满足自仿射原理的分形均匀球的散射，其散射特性与分维存在一定的关系。我们还利用微扰法讨论了表面具有沟槽的分形柱的光散射的分形特征。     

An Extended Correlation Dimension of Complex Networks

Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.     

Hausdorff Dimension of Non-Hyperbolic Repellers. I: Maps with Holes

This is the first paper in a two-part series devoted to studying the Hausdorff dimension of invariant sets of non-uniformly hyperbolic, non-conformal maps. Here we consider a general abstract model, that we call piecewise smooth maps with holes. We show that the Hausdorff dimension of the repeller is strictly less than the dimension of the ambient manifold. Our approach also provides information on escape rates and dynamical dimension of the repeller.     

Effects of Different Connectivity Topologies in Small World Networks on EEG-Like Activities

Based on our previously pulse-coupled integrate-and-fire neuron model in small world networks, we investigate the effects of different connectivity topologies on complex behavior of electroencephalographic-like signals produced by this model. We show that several times series analysis methods that are often used for analyzing complex behavior of electroencephalographic-like signals, such as reconstruction of the phase space, correlation dimension, fractal dimension, and the Hurst exponent within the rescaled range analysis (R/S). We find that the different connectivity topologies lead to different dynamical behaviors in models of integrate-and-fire neurons.     

Ising-type transitions in coupled map lattices

We study, by means of computer simulations, some models of coupled map lattices (CML) with symmetry, subject to diffusive nearest neighbor coupling, with the purpose of providing, a better understanding of the occurrence of Isingtype transitions of the type found by Miller and Huse. We argue, on the basis of numerical evidence, that such transitions are connected to the appearance of a minimum in the Lyapunov dimension of the system as a function of the coupling parameter. Two-dimensional CMLs similar to the one in Miller and Huse, but with no minimum in the Lyapunov dimension plot, have no Ising transition. The condition seems to be necessary, though by no means sufficient. We also argue, relying on the analysis of Bunimovich and Sinai, that coupled map lattices should behave differently, with respect to dimension, than Ising models.Dedicated to Yakov Grigorievich Sinai on his 60th birthday.     

Property of robustness to size and its realization on fractal dimension

Images in one class often have varied sizes due to different imaging system. Thus it will provide convenience to image classification if the indicator used in the classification is robust to the size of images. We regard the robustness to size of image as a property of image indicator. The property means that images from one class have small variance with the sizes, and is different from such traditional properties as the robustness to scale, rotation and illumination. Fractal dimension is an indicator which has the three traditional properties. We realize the property on fractal dimension in the statistical sense by modifying differential-box counting method. Tests on two classes of images demonstrate the effectiveness of the modifications. Tests on scaling process give a standard of FD’ robustness as 0.0611, and experiments on both the two class and four sets of images show the statistical validity of the standard and verify the realization. An indicator with this property can be a tool for the classification.     

Fractal structure of digital speckle patterns produced by rough surfaces

We report on the fractal analysis of digital speckle patterns experimentally generated using an optical setup to record the light scattered from metallic rough surfaces in the normal direction. Using the differential box counting technique, we have calculated the fractal dimension of digital speckle patterns for six samples with different roughness. Our results show a quadratic dependence between the surface roughness and the fractal dimension of the corresponding digital speckle pattern. As an application a method to determine the surface roughness of metallic surfaces is proposed.     

A Short Existence Proof for Correlation Dimension

The Grassberger–Hentschel–Procaccia correlation dimension has been put on a rigorous basis by Pesin and Tempelman. We simplify their proof that this dimension is given in terms of the measure of neighborhoods of the diagonal.     

单颗金刚石切削无氧铜的声发射关联维特征

声发射技术可以实现无氧铜切削加工特征的监测与评价。采用声发射技术监测单颗金刚石磨粒旋转切削无氧铜,利用G-P算法重构出声发射时域信号相空间,采用自相关函数法计算出相空间时间延迟参数,通过相空间双对数曲线的计算,得到不同切削工况下的关联维数。研究结果表明,进给速度和切削速度对声发射信号影响较不显著,切深与声发射信号振幅呈正效应关系;声发射信号双对数曲线呈现阶段性增加趋势,并逐渐收敛于饱和状态,关联维数随着嵌入维数的增加先快速下降后趋于平稳;金刚石切削无氧铜的声发射信号具有混沌运动变化特性,在较小嵌入维数时,关联维数与切深和切削速度呈现线性负效应关系,与进给速度呈现线性正效应关系。该研究为无氧铜的切削加工提供理论参考。     

舰船辐射噪声超混沌现象研究

水下弱目标探测和识别一直是水声信号处理领域中研究的难点。从Lyapunov指数谱、吸引子相空间轨迹的演化、分形维数等方面，对船舶辐射噪声是否存在超混沌现象进行了研究。实验结果表明，船舶辐射噪声信号确实存在至少两个正的Lyapunov指数，即存在超混沌现象。辐射噪声吸引子在相空间中的轨迹具有多方向伸展的趋势，且不同类型目标的吸引子具有不同的分形维数。研究结果为建立精确描述辐射噪声信号的非线性模型、为水下弱目标信号探测和识别提供一定的理论依据。     

Macroscopic Dimension of 3-Manifolds

In this paper we give the answer to the Gromov's question about the macroscopic dimension of universal coverings of closed manifolds in dimension 3. We prove that the macroscopic dimension of a universal covering of a closed Riemannian 3-manifold cannot be equal to 2.     

基于分形脊波神经网络的遥感图像分类研究

在分形理论和脊波神经网络的基础上，综合利用彩色遥感图像的光谱、纹理和形状特征，提出了一种彩色遥感图像的分类新方法．该方法把彩色图像的蓝、绿、红波段作为3个光谱特征，由分形理论计算的DBC维和多重分形维数作为2个纹理特征，平均不变矩作为1个形状特征，并利用对曲线具有极强方向识别能力的脊波神经网络作为分类器．实验结果表明，提出的彩色遥感图像分类方法具有较高的分类准确率和较强的抗噪音能力．     

Scikit-Dimension: A Python Package for Intrinsic Dimension Estimation

Dealing with uncertainty in applications of machine learning to real-life data critically depends on the knowledge of intrinsic dimensionality (ID). A number of methods have been suggested for the purpose of estimating ID, but no standard package to easily apply them one by one or all at once has been implemented in Python. This technical note introduces scikit-dimension, an open-source Python package for intrinsic dimension estimation. The scikit-dimension package provides a uniform implementation of most of the known ID estimators based on the scikit-learn application programming interface to evaluate the global and local intrinsic dimension, as well as generators of synthetic toy and benchmark datasets widespread in the literature. The package is developed with tools assessing the code quality, coverage, unit testing and continuous integration. We briefly describe the package and demonstrate its use in a large-scale (more than 500 datasets) benchmarking of methods for ID estimation for real-life and synthetic data.     

Path integral over reparametrizations: Lévy flights versus random walks

We investigate the properties of the path integral over reparametrizations (or the boundary value of the Liouville field in string theory). Discretizing the path integral, we apply the Metropolis–Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Lévy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby previous results on QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.     

Fractal properties of critical invariant curves

We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.     

上海光源首批波荡器永磁块研制

介绍了上海光源2台混合型真空室内波荡器IVU25用636件Sm2Co17永磁块(规格为65 mm×25 mm×8.965 mm)和1台APPLE-Ⅱ型椭圆极化波荡器EPU100用668件NdFeB永磁块(规格为35 mm×35 mm×25 mm)的研制情况。指出了影响剩磁离散性、磁化偏角、机械尺寸精度等技术要求的基本因素,并对其原理进行了分析。研制的Sm2Co17和NdFeB永磁块的磁性能、磁场品质、机械尺寸精度等满足了上海光源首批波荡器插入件技术要求。     

基于径向基神经网络预测的混沌时间序列嵌入维数估计方法

根据Takens定理,研究了混沌时间序列相空间重构嵌入维数的选取问题.提出了基于径向基函数神经网络预测模型性能的嵌入维数估计方法,即根据嵌入维数与混沌时间序列预测模型性能的变化关系来确定嵌入维数.通过对几种典型混沌动力学系统的数值验证,结果表明该方法能够确定出合适的相空间重构嵌入维数. 关键词： 混沌 相空间重构 嵌入维数 预测     

A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions

In this paper we establish the complete multifractal formalism for equilibrium measures for Hölder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Hölder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.     

Some results on the behavior and estimation of the fractal dimensions of distributions on attractors

The strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation. In this paper we first provide a rigorous mathematical framework for the study of dimension, focusing on pointwise dimension(x) and the generalized Renyi dimensionsD(q), and give a rigorous proof of inequalities first derived by Grassberger and Procaccia and Hentschel and Procaccia. We then specialize to the problem of statistical estimation of the correlation dimension and information dimension. It has been recognized for some time that the error estimates accompanying the usual procedures (which generally involve least squares methods and nearest neighbor calculations) grossly underestimate the true statistical error involved. In least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates. We then develop an estimation procedure for which corrects for an important bias term (the local measure density) and provides confidence intervals for. The general applicability of this method is illustrated with various numerical examples.