Oscillating singularities on cantor sets: A grand-canonical multifractal formalism

The singular behavior of functions is generally characterized by their Hölder exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then elaborate on a grand-canonical multifractal formalism that describes statistically the fluctuations of both the Hölder and the oscillation exponents. We prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities.     

Fractional Fourier transform of Cantor sets: further numerical study

This paper is a further work of the authors＇ paper published previously （Liao T H and Gao Q 2005 Chin. Phys. Lett. 22 2316）. The amplitudes of fractional Fourier transform of Cantor sets are analysed from the viewpoint of multifractal by wavelet transform maxima method （WTMM）. An integral operation is carried out before the application of WTMM, such that the function obtained can be considered as the perturbed devil staircase. Also, wavelets with large number of vanishing moments are used, which makes the complete singularity spectrum more accessible. The validity of multifractal formalism is guaranteed by restricting parameter q to a proper range, so that the phenomenon of multifractal phase transition can be explained reasonably. Particularly, the method of determining the range of parameter q in the above paper is developed to be more operational and rigorous.     

基于小波leaders的海杂波时变奇异谱分布分析

海杂波的奇异谱分析不仅能从理论上揭示海洋表面的动力学机理,同时也是对海探测雷达的关键技术之一.本文提出基于小波leaders的海杂波时变奇异谱分析方法,将时间信息引入海杂波的奇异谱分析之中,从而实现动态的解析描述海杂波随时间变化的奇异谱特性.在理论上,通过信号自身加窗,将时间信息引入传统的奇异谱(或称多重分形谱),实现了对海杂波时变奇异谱分布分析;在算法上,充分利用了小波leaders技术对于多种奇异性的提取能力(包括chirp奇异性和cusp奇异性),通过对时变奇异性指数和时变尺度函数的Legendre变换,实现对海杂波时变奇异谱分布的计算;在应用部分,采用经典的多重分形模型——随机小波序列(RWC)以及三级海态条件下连续波多普勒体制雷达海杂波进行仿真分析,实验结果表明:1)基于小波leaders的奇异谱分布能跟踪海杂波的时变尺度特性,有效展示其时变奇异性谱分布;2)算法具有较好的负矩特性和统计收敛性.该方法能为复杂非线性系统及随机多重分形信号分析提供参考.     

Dimension spectrum of axiom a diffeomorphisms. I. The Bowen-Margulis measure

We compute the dimension spectrumf() of the singularity sets of the Bowen-Margulis measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. It is proved thatf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we decompose these sets.     

Dimension spectrum of axiom a diffeomorphisms. II. Gibbs measures

We compute the dimension spectrumf() of the singularity sets of a Gibbs measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. This case is the generalization of the case where the measure studied is the Bowen-Margulis measure—the one that realizes the topological entropy. We obtain similar results; for example, the functionf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we finally decompose these sets.     

Wavelet-based multifractal analysis of DNA sequences by using chaos-game representation

Chaos game representation (CGR) is proposed as a scale-independent representation for DNA sequences and provides information about the statistical distribution of oligonucleotides in a DNA sequence. CGR images of DNA sequences represent some kinds of fractal patterns, but the common multifractal analysis based on the box counting method cannot deal with CGR images perfectly. Here, the wavelet transform modulus maxima (WTMM) method is applied to the multifractal analysis of CGR images. The results show that the scale-invariance range of CGR edge images can be extended to three orders of magnitude, and complete singularity spectra can be calculated. Spectrum parameters such as the singularity spectrum span are extracted to describe the statistical character of DNA sequences. Compared with the singularity spectrum span, exon sequences with a minimal spectrum span have the most uniform fractal structure. Also, the singularity spectrum parameters are related to oligonucleotide length, sequence component and species, thereby providing a method of studying the length polymorphism of repeat oligonucleotides.     

A new transition for projections of multifractal measures and random maps

Typical projections of simple multifractal measures with generalized dimensionsD _q onto subspaces of dimensionD are considered. It is known that forD _o > D almost all projections have Euclidean support. Here it is shown that if in additionD increases beyondD, a typical projection changes from a singular continuous distribution to an absolutely continuous measure with a squareintegrable, or even differentiable density, and thus from a multifractal to an ordinary distribution with trivial singularity spectrum. Since projections of strictly self-similar measures can be regarded as invariant distributions of iterated function systems, such a transition is found also there and is expected to occur in related systems.     

基于多尺度Radon变换的图像检索

根据Radon变换的统计特性构造了不变量,提出一种新的基于多尺度Radon变换的图像形状检索方法.对检索图像作小波变换,根据小波模极大原理得到边缘图像,对边缘图像构造Radon变换中心矩,在中心矩的基础上根据Radon的统计原理构造出尺度不变矩.由于矩阵的奇异值具有旋转不变性,因此针对不变矩向量矩阵求奇异值,该奇异值特征向量具有平移、尺度和旋转不变性.将该Radon变换的不变量作为形状特征,并进行高斯归一化,按照欧氏距离计算不同图像间的形状相似度.试验结果表明,该方法对高斯噪音具有较强的鲁棒性,与其它方法相比具有较好的检索效果.     

Generalizing the wavelet-based multifractal formalism to random vector fields: application to three-dimensional turbulence velocity and vorticity data

We use singular value decomposition techniques to generalize the wavelet transform modulus maxima method to the multifractal analysis of vector-valued random fields. The method is calibrated on synthetic multifractal 2D vector measures and monofractal 3D fractional Brownian vector fields. We report the results of some application to the velocity and vorticity fields issued from 3D isotropic turbulence simulations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.     

Renormalization group analysis of the global properties of a strange attractor

This paper considers the circle map at the special point: the one at which there is a trajectory with a golden mean winding number and at which the map just fails to be invertable at one point on the circle. The invariant density of this trajectory has fractal properties. Previous work has suggested that the global behavior of this fractal can be effectively analyzed using a kind of partition function formalism to generate anf versus curve. In this paper the partition function is obtained by using a renormalization group approach.     

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over , being algebraic varieties of the form , where R is a polynomial in two variables; that is, abelian deformations of the coordinate algebra ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr–Gerstenhaber–Schack decomposition. Homology is the same for all plane curves , but the cohomology depends on the local algebra of the singularity of R at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.      

Singularities and conjugate points in FLRW spacetimes

Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial curvature and scale factors that near the singularity have power law behavior or power law behavior times a logarithm. When the spatial curvature is negative, the theorem is applicable to a subset of these spacetimes.     

基于提升方案的CDF(1,5)整-整可逆双正交小波变换在渐进性无损图像压缩中的应用

讨论了基于双正交小波变换提升方案在图像压缩中的应用 ,研究了CDF( 1,3) ,CDF( 1,5 )双正交小波消失矩对图像压缩的影响。实验结果表明 ,在图像压缩中应尽可能选取具有较高消失矩的双正交小波 ,用CDF( 1,5 )双正交小波提升方案实现了基于整数运算的整 整可逆双正交小波变换 ,以及快速的、渐进性的直至无损图像压缩     

On the wavelet transformation of fractal objects

The wavelet transformation is briefly presented. It is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half-plane near the boundary of its domain of analyticity. As an example, a Weierstrass-like fractal function is considered, for which the wavelet transform is related to a Jacobi theta function. Some of the scalings of this theta function are analyzed, and give some information about the scaling behavior of this fractal.     

Wavelet Transformation and Wigner-Husimi Distribution Function

We find that the optical wavelet transformation can be used to study the Husimi distribution function in phase space theory of quantum optics. We prove that the Husimi distribution function of a quantum state |ψ〉 is just the modulus square of the wavelet transform of with ψ(x) being the mother wavelet up to a Gaussian function. Thus a convenient approach for calculating various Husimi distribution functions of miscellaneous quantum states is presented.     

Multi-Fractal Formalism for Quasi-Self-Similar Functions

The study of multi-fractal functions has proved important in several domains of physics. Some physical phenomena such as fully developed turbulence or diffusion limited aggregates seem to exhibit some sort of self-similarity. The validity of the multi-fractal formalism has been proved to be valid for self-similar functions. But, multi-fractals encountered in physics or image processing are not exactly self-similar. For this reason, we extend the validity of the multi-fractal formalism for a class of some non-self-similar functions. Our functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions. For the computation of their spectrum of singularities, it is unknown how to construct Gibbs measures. However, it suffices to use measures constructed according the Frostman's method. Besides, we compute the box dimension of the graphs.     

Structure of Malicious Singularities

In this paper, we investigate relativistic spacetimes, together with their singular boundaries (including the strongest singularities of the Big Bang type, called malicious singularities), as noncommutative spaces. Such a space is defined by a noncommutative algebra on the transformation groupoid = × G, where is the total space of the frame bundle over spacetime with its singular boundary, and G is its structural group. We show that there exists the bijective correspondence between unitary representations of the groupoid and the systems of imprimitivity of the group G. This allows us to apply the Mackey theorem to this case, and deduce from it some information concerning singular fibers of the groupoid . At regular points the group representation, which is a part of the corresponding system of imprimitivity, does not have discrete components, whereas at the malicious singularity such a group representation can be a single representation (in particular, an irreducible one) or a direct sum of such representations. A subgroup K G, from which—according to the Mackey theorem—the representation is induced to the whole of G, can be regarded as measuring the richness of the singularity structure. In this sense, the structure of malicious singularities is richer than those of milder ones.     

Variational evaluation of correlation functions for lattice electrons in high dimensions

We present a general formalism for the diagrammatic calculation of correlation functions for Hubbard-type models in terms of projected wave functions. It is shown that in the limit of high spatial dimensionsd only diagrams with bubble-structure remain. This causes correlation functions to have an overall RPA-type form ind. Exact evaluations are performed for the Gutzwiller wave function. Nearest neighbor correlations are shown to be proportional to their value in the non-interacting case, i.e. are renormalized. However, their absolute value is only of order 1/d. Hence this wave function does not describe spin correlations adequately in high dimensions. The asymptotic behavior of the spin-correlation function is extracted and is found to have a scaling form similar tod=1. Assuming this form to hold in all dimensions we show that the Brinkman-Rice transition only occurs ind=. Finite orders of perturbation theory in 1/d around this singular point are not sufficient to remove the transition.