On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions

We consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as Hausdorff dimension, box dimension, etc. We also introduce different ways to define and study the generalized spectrum for dimensions—a one-parameter family of characteristics of dimension type.     

A practical method to experimentally evaluate the Hausdorff dimension: an alternative phase-transition-based methodology

We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.     

拓扑光学体系中合成维度方法的进展

光子学中的合成维度是近年来微纳光学和拓扑光子学的研究热点。通常意义上,一个光学系统的物理维度受限于其空间几何维度,因此极大地制约了光学系统所支持研究的物理现象。而研究者通过引入合成维度,可以突破几何维度对物理系统维度的制约,研究高维空间的物理问题。同时,合成维度的高度可控性和选择的丰富多样性,为简化系统设计、观察高维物理现象提供了很大的便利。本文介绍了光子学中合成维度的基本概念,回顾了近年来实现合成维度的各种设计方案,并初步探讨了其在基础物理研究和应用上的未来前景。     

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     

Nonlinear dynamical analysis of turbulence in a stable cloud layer

An eight mode truncated spectral model based on Burgers' approximation to the one-dimensional Navier-Stokes equations is used to compute the Lyapunov dimension of the dynamical attractor for turbulence in a stable cloud layer. The model results are compared with the correlation dimension obtained earlier from a time series of radar Doppler and reflectivity signals from a turbulent layer in a marine stratus cloud. The analysis supports a weak coupling explanation for the lower correlation dimension found for the reflectivity time series compared with that for the Doppler time series. Turbulent Prandtl number emerges from the analysis as a flow parameter which can enlarge the dimension of the model's dynamical attractor, but the attractor dimension computed for the model remains lower than the radar Doppler correlation dimension. Linear stability analysis of the model's equilibrium states suggests that a nontruncated version of the model will possess an attractor which is also of lower dimension than the radar Doppler correlation dimension. (c) 1995 American Institute of Physics.     

混沌时序相空间重构参数确定的信息论方法

根据信息论基本原理，研究了混沌时间序列相空间重构参数延迟时间和嵌入维数的选取.提出了用符号分析的方法计算互信息函数，确定出延迟时间，在此基础上，提出了一种估计嵌入维数的信息论方法，即根据重构向量条件熵随向量维数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 互信息 条件熵 符号分析     

Dimensions of Attractors in Pinched Skew Products

We study dimensions of strange non-chaotic attractors and their associated physical measures in so-called pinched skew products, introduced by Grebogi and his coworkers in 1984. Our main results are that the Hausdorff dimension, the pointwise dimension and the information dimension are all equal to one, although the box-counting dimension is known to be two. The assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable. Our findings confirm a conjecture by Ding, Grebogi and Ott from 1989.     

纳米AT13陶瓷喷涂层结合强度的分形维数表征

应用分形维数对微弧等离子喷涂纳米AT13涂层的界面进行研究,结果表明：随喷涂电流的增加,纳米AT13涂层的界面分维不断增大,其结合强度也不断提高;随着氩气压力的增加,纳米AT13涂层的界面分维和结合强度都先增大后减小;界面分维D可用于表征涂层结合强度σ,σ随D的增大而增大,且ln(σ)与D之间呈近似线性关系：ln(σ)=17.6D-26.2. 关键词： 纳米涂层 分形数维 界面 结合强度     

Generalized thermal radiation from arbitrary fractional dimension

Properties of the thermal radiation from arbitrary fractional dimension are investigated. Generalized blackbody radiation for arbitrary dimension can be obtained and the energy density is shown as a function of arbitrary dimension as well as temperature. Maximum frequency factor representing the relation between most probable photon energy and thermal energy is shown as a function of arbitrary fractional dimension. It is also shown how to measure the arbitrary fractional dimension of the body with thermal radiation.     

Testing the dimension of Hilbert spaces

Given a set of correlations originating from measurements on a quantum state of unknown Hilbert space dimension, what is the minimal dimension d necessary to describe such correlations? We introduce the concept of dimension witness to put lower bounds on d. This work represents a first step in a broader research program aiming to characterize Hilbert space dimension in various contexts related to fundamental questions and quantum information applications.     

Cu-W薄膜表面形貌的分形表征与电阻率

磁控溅射制备铜钨薄膜，用原子力显微镜和功率谱密度法分析薄膜生长表面形貌的分形维数，发现频段的选择基本不影响分形维数与溅射时间的关系.随溅射时间延长，薄膜厚度增加，分形维数增大，电阻率随分形维数的增大而升高.分析分形维数与电阻率的关系，认为对同一物质的导电薄膜，其表面形貌与电阻率存在对应关系. 关键词： Cu-W薄膜 分形维数 功率谱密度 电阻率     

Dimension of similarity as a characteristic of the solid-surface relief

It is proposed that the dimension of similarity of the relief profile be used as an integral characteristic of solid-surface roughness. This parameter coincides with the fractal dimension of the line only in the case of the same length scales in the surface plane and in the direction that is normal to it. The interrelation between the dimension of similarity and the usually used fractal dimension of the relief profile is obtained and analyzed. The optical surface of single-crystal Ge is used as an example in order to reveal the correlation between the dimension of similarity of the relief surface and the transmission coefficient in the infrared range.     

An optical technique for measuring fractal dimensions of planar poincaré maps

An optical technique for measuring the fractal dimension of two-dimensional Poincaré maps has been developed. The parallel processing feature of this method provides a rapid way to calculate a correlation measure between distributed points in a plane. The method has been used to determine the correlation dimension of the Duffing-Holmes strange attractor for the two-well potential. The optically measured values of fractal dimension agree well with computer calculation of dimension for both numerical and experimental data.     

Zero Entropy Systems

This paper introduces the notion of entropy dimension to measure the complexity of zero entropy dynamical systems, including the probabilistic and the topological versions. These notions are isomorphism invariants for measure-preserving transformation and continuity. We discuss basic propositions for entropy dimension and construct some examples to show that the topological entropy dimension attains any value between 0 and 1. This paper also gives a symbolic subspace to achieve zero topological entropy, but with full entropy dimension.     

时间序列三维荧光光谱的异常值检测

异常值的存在往往干扰着时间序列三维荧光光谱的定性和定量分析。充分利用时间维和光谱维的内在特性, 提出了一种有效的异常值检测方法。在时间维结合方差提取异常值可能性最大的波长点;通过对异常值存在方式的分析,在任意两个三维荧光光谱的相似度基础上给出了光谱维上的累积相似度;最后利用时间维的校正矩阵对所有三维荧光光谱的每个波长点荧光强度进行修正并计算对应的累积相似度,从而根据累积相似度对异常值进行判断。时间维校正矩阵的采用不仅提高了算法的有效性而且其特征区域的选择大大减少了光谱维相似度的计算量。相关的数值试验表明光谱维选取50%的波长点仍然能有效对异常值进行检测。     

The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.Research in part supported by AFOSR and by the Department of Energy (Scientific Computing Staff Office of Energy Research)     

Determining the input dimension of a neural network for nonlinear time series prediction

Determining the input dimension of a feed-forward neural network for nonlinear time series prediction plays an important role in the modelling.The paper first summarizes the current methods for determining the input dimension of the neural network.Then inspired by the fact that the correlation dimension of a nonlinear dynamic system is the most important feature of it ,the paper pressents a new idea that the input dimension of the neural network for nonlinear time series prediction can be taken as an integer just greater than or equal to the correlation dimension.Fimally,some validation examples and results are given.     

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.     

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.     

Device-independent tests of classical and quantum dimensions

We address the problem of testing the dimensionality of classical and quantum systems in a "black-box" scenario. We develop a general formalism for tackling this problem. This allows us to derive lower bounds on the classical dimension necessary to reproduce given measurement data. Furthermore, we generalize the concept of quantum dimension witnesses to arbitrary quantum systems, allowing one to place a lower bound on the Hilbert space dimension necessary to reproduce certain data. Illustrating these ideas, we provide simple examples of classical and quantum dimension witnesses.