Mellin transforms and correlation dimensions

We consider the Mellin transform of the correlation integrals and show that the divergence abscissa is the correlation dimension. The analytic structure of the Mellin transform is explicitly described for some Julia and Cantor sets. The existence of oscillations in the correlation integral for the Cantor sets is proved. Extensions of the results to the order d correlation integrals are discussed.     

Strange objects in the complex plane

Julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems. Technically they are the closure of the set of unstable cycles of analytic maps. Physically, they are sets of points which lead to chaotic behavior. The mapf(z)=z²+p is analyzed for smallp where the Julia set is a closed curve, and for largep where the Julia set is completely disconnected. In both cases the Hausdorff dimension is calculated in perturbation theory and numerically. An expression for the rate at which points escape from the neighborhood of the Julia set is derived and tested in a numerical simulation of the escape.     

Meromorphic structure of the Mellin transforms and short-distance behavior of correlation integrals

The short-distance behavior of the measure of a sphere and of the correlation integral is determined, in the case of disconnected repellers, by scaling laws whose corrections are oscillating functions, periodic or aperiodic, depending on exact or approximate self-similarity of the measure. The Mellin transforms prove to be the correct analytic tool in order to investigate these corrections to scaling. It has been previously proved that they are meromorphic for linear Cantor sets and that the leading pole gives the correlation dimension in agreement with the results of the thermodynamic formalism. Here we show that the residues of these poles can also be computed to any desired accuracy with simple algorithms and that the knowledge of the singularity spectrum of the Mellin transforms provides the Fourier spectrum of the scaling correction for the self-similar measure and that it reproduces the damped oscillations in the generic case. The method applies to the nonlinear repellers such as the disconnected Julia sets by using an approximation theorem.     

Pattern classification using optical Mellin transform and circular photodiode array

Scale-invariant pattern classification using a hybrid system combining the optical Mellin transform and a digital signal processing technique is discussed. We accomplish the optical Mellin transform by a logarithmic coordinate transformation using a computer-generated hologram, followed by an optical Fourier transform. Mellin transform patterns are detected with a circular photodiode array, whose output signals are processed by a micro-computer. A new criterion is discussed, in which circular or periodic correlation is employed. Experimental examples are presented.     

Statistical theory of self-similarly distributed fields

A field theory is built for self-similar statistical systems with both generating functional being the Mellin transform of the Tsallis exponential and generator of the scale transformation that is reduced to the Jackson derivative. With such a choice, the role of a fluctuating order parameter is shown to play deformed logarithm of the amplitude of a hydrodynamic mode. Within the harmonic approach, deformed partition function and moments of the order parameter of lower powers are found. A set of equations for the generating functional is obtained to take into account constraints and symmetry of the statistical system.     

Four loop anomalous dimensions of gradient operators in {\phi^4} theory

We compute the anomalous dimensions of a set of composite operators which involve derivatives at four loops in in theory as a function of the operator moment . These operators are similar to the twist-2 operators which arise in QCD in the operator product expansion in deep inelastic scattering. By regarding their inverse Mellin transform as being equivalent to the DGLAP splitting functions we explore to what extent taking a restricted set of operator moments can give a good approximation to the exact four loop result. Received: 12 May 1997 / Published online: 20 February 1998     

Reproducing Kernels and Coherent States on Julia Sets

We construct classes of coherent states on domains arising from dynamical systems. An orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analogous results on domains arising from iterated function systems. The research of the first two authors was supported by Natural Sciences and Engineering Research Council of Canada.     

复系统Julia集的同步

非线性系统中的分形集——Julia集, 在工程技术中有着十分重要的应用,定义了不同系统间的Julia集同步的概念, 并引入一种非线性耦合的方法, 对同一系统不同参数的Julia集进行了有效的同步.并以多项式形式和三角函数形式的Julia集同步为例验证了该方法的有效性. 关键词： Julia集 同步 分形     

复系统Julia集的同步

非线性系统中的分形集——Julia集, 在工程技术中有着十分重要的应用,定义了不同系统间的Julia集同步的概念, 并引入一种非线性耦合的方法, 对同一系统不同参数的Julia集进行了有效的同步.并以多项式形式和三角函数形式的Julia集同步为例验证了该方法的有效性.     

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.     

Novel summability methods generalizing the Borel method

The paper deals with moment constant summability methods. A method is constructed which provides an analytic continuation of a function regular at the origin onto its Mittag-Leffler (principal) star. In this sense the method is optimal in contrast to the Borel one. In the next a one parameter family of such methods is introduced. The methods may be useful both in field theory and in statistical physics. Applications to the Nevanlinna theorem, the Rayleigh-Schrö-dinger perturbation theory and the dispersion-like integral are given. The proofs of theorems can be easily adapted to the study of the Mellin transform of some entire functions and a simpler proof of asymptotic properties of the gamma function can be obtained.I am indebted to J. Fuka for stimulating discussions and J. Fischer for continuous interest in my work and valuable comments. Financial support of the Czech Literatury Fund is also gratefully acknowledged.     

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.     

Continuous and discrete frames on Julia sets

A frame is an overcomplete family of vectors in a Hilbert space in which the orthogonality condition is relaxed. The Julia set is the chaotic regime of a rational function. In this note, we label frames of an abstract Hilbert space by elements of the Julia set of a rational function.     

Mellin Transform of the Limit Lognormal Distribution

The technique of intermittency expansions is applied to derive an exact formal power series representation for the Mellin transform of the probability distribution of the limit lognormal multifractal process. The negative integral moments are computed by a novel product formula of Selberg type. The power series is summed in general by means of its small intermittency asymptotic. The resulting integral formula for the Mellin transform is conjectured to be valid at all levels of intermittency. The conjecture is verified partially by proving that the integral formula reproduces known results for the positive and negative integral moments of the limit lognormal distribution and gives a valid characteristic function of the Lévy-Khinchine type for the logarithm of the distribution. The moment problem for the logarithm of the distribution is shown to be determinate, whereas the moment problems for the distribution and its reciprocal are shown to be indeterminate. The conjecture is used to represent the Mellin transform as an infinite product of gamma factors generalizing Selberg’s finite product. The conjectured probability density functions of the limit lognormal distribution and its logarithm are computed numerically by the inverse Fourier transform.     

Yang-Lee zeros of the Ising model on the closed Cayley tree

The Yang-Lee zeros of the partition function of the ferro-, antiferro- and of the partially antiferromagnetic anisotropic Ising models defined on the closed symmetric Cayley tree are studied. The applicability of the Yang-Lee theorem to the antiferromagnetic systems is shown to be a consequence of the invariance of the unit circle under the Bethe-Peierls map. The relationship as well as the distinction between the set of zeros and the Julia set is established. The fractal dimension of the Julia set is shown to be equal to one in the low temperature phase and to be a decreasing function of the temperature in the paramagnetic phase of the three systems.     

实现Mellin变换的一种新方法

本文讨论了用光学系统实现Mellin变换的一种新方法,叙述了用计算机进行全息透镜的设计和用计算机产生二元式全息图的制造原理。实验上实现了由一组16个抽样点组成的一维Mellin变换,并可直接推广到二维情况。 关键词：     

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.     

Universal properties of the transition from quasi-periodicity to chaos in dissipative systems

An exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi-periodic regime is held at a fixed irrational value. Our approach is a natural extension of K.A.M. theory to strong coupling. Most of our analysis is for analytic circle maps. We have found a strong coupling fixed point where invertibility is lost, which describes the universal features of the transition to chaos. We find numerically that any two such critical maps with the same winding number are C¹ conjugate. It follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency.When the winding number has a periodic continued fraction, our renormalization transform has a fixed point and spectra are self similar in addition. For a set of non-periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub-space of all critical maps. Physically one finds singular and universal spectra that do not scale.     

Image encryption algorithm based on the multi-order discrete fractional Mellin transform

A multi-order discrete fractional Mellin transform (MODFrMT) is constructed and directly used to encrypt the private images. The MODFrMT is a generalization of the fractional Mellin transform (FrMT) and is derived by transforming the image with multi-order discrete fractional Fourier transform (MODFrFT) in log-polar coordinates, where the MODFrFT is generalized from the closed-form expression of the discrete fractional Fourier transform (DFrFT) and can be calculated by fast Fourier transform (FFT) to reduce the computation burden. The fractional order vectors of the MODFrMT are sensitive enough to be the keys, and consequently key space of the encryption system is enlarged. The proposed image encryption algorithm has significant ability to resist some common attacks like known-plaintext attack, chosen-plaintext attack, etc. due to the nonlinear property of the MODFrMT. Additionally, Kaplan-Yorke map is employed in coordinate transformation process of the MODFrMT to further enhance the security of the encryption system. The computer simulation results show that the proposed encryption algorithm is feasible, secure and robust to noise attack and occlusion.