Singularity spectrum of fractal signals from wavelet analysis: Exact results

The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimensionD(h) of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at leastN>n vanishing moments. However, it is shown that aC behavior generally induces a phase transition in theD(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the numberN of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.     

On the generating function e xt+yϕ(t) and its fractional calculus

In this article, we derive the coefficient set {H _m (x,y)} _m=1 ^∞ using the generating function e ^xt+y?(t). When the complex function ?(t) is entire, using the inverse Mellin transform, and when ?(t) has singular points, using the inverse Laplace transform, the coefficient set is obtained. Also, bi-orthogonality of this set with its associated functions and its applications in the explicit solutions of partial fractional differential equations is discussed.     

Electrostatic and optical resonances of arrays of cylinders

The solutions of the electrostatic potential problem for the square and hexagonal arrays of circular cylinders with zero applied field (homogeneous or resonant solutions) are studied. We show that for non-touching cylinders, the set of resonances is discrete except in the neighbourhood of one point, at which the dielectric constant of the array has an essential singularity. For arrays of touching cylinders, the set is well represented by a continuous distribution. This representation enables the derivation of the asymptotic form of the expansion for the dielectric constant of the array when the dielectric constant of the cylinders is large. The known value of the first term in the expansion enables us to derive the second term. The physical characteristics of the resonant solutions are studied. Metals achieve values of dielectric constant which are close to the resonant values (real and negative) for certain wavelengths. Curves are given which enable the prediction of those wavelengths at which the optical resonances of both arrays occur, for any area fraction and composition of a columnar cerment film.     

Edge-detection algorithm based on DCT continuous extension technique

A new computational approach to the edge-detection problem, based on the continuous extension of discrete cosine transform (CEDCT) technique is proposed. This technique has some attractive properties, and other things being equal, it has more precise results than the usual discrete Fourier or discrete cosine transforms, especially at the intermediate points. That is why this technique allows one to estimate numerically a finite number of a derivatives of a discrete set of multidimensional points, using some specified properties of CEDCT. Because of using the spectrum of a given set of points, this approach is applicable to a wide area of signal-and image-processing problems. The results obtained by the proposed approach are compared with the well-known and widely used Canny algorithm. Some 1D and 2D numerical examples are given. The text was submitted by the authors in English.     

Exact solutions of discrete complex cubic-quintic Ginzburg-Landau equation with non-local quintic term

In this paper, via the extended tanh-function approach, the abundant exact solutions for discrete complex cubic-quintic Ginzburg-Landau equation, including chirpless bright soliton, chirpless dark soliton, constant magnitude solution (plane wave solution), triangular function solutions and some solutions with alternating phases, etc. are obtained. Meanwhile, the range of parameters where some exact solution exist are given. Among these solutions, solutions with alternating phases do not have continuous analogs. Moreover, in the lattice, the points of singularity of tan-type and sec-type solutions can be ‘between sites’ and thus the singularities can be avoided.     

Absence of Singular Continuous Diffraction for Discrete Multi-Component Particle Models

Particle models with finitely many types of particles are considered, both on ℤ ^d and on discrete point sets of finite local complexity. Such sets include many standard examples of aperiodic order such as model sets or certain substitution systems. The particle gas is defined by an interaction potential and a corresponding Gibbs measure. Under some reasonable conditions on the underlying point set and the potential, we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density. In particular, no singular continuous part is present.     

Equivalence of Euclidean and Wightman field theories

A new inversion formula for the Laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions implies that some of the limits needed for the inversion formula exist for any Euclidean Green's function with an even number of variables. We then prove that the initial Osterwalder-Schrader axioms [1] and the weak spectral condition are equivalent with the Wightman axioms.The research described in this publication was made possible in part by Grant No. 93-011-147 from the Russian Foundation for Basic Research     

Positive exponential sum method of inverting Laplace transform applied to photon correlation spectroscopy

A computer program for the positive exponential sum method of inverting Laplace transform of photon autocorrelation curves has been written and applied to both simulated and experimental data. The method recovers the decay time spectra from autocorrelation curves by replacing the decay time distribution by a set of delta peaks whose envelope roughly follows this distribution. Spacing of this set becomes slightly denser with decreasing noise and is about two peaks per decade under usual conditions (noise level about 10^–3). Occasional irregularities in peak spacing and an appearance of weak artifact peaks in regions where the decay time spectrum is of zero density may make the physical interpretation of results difficult.     

Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer

ABSTRACT

A new nonlocal theory of generalized thermoelastic materials with voids based on Eringen’s nonlocal elasticity and Caputo fractional derivative is established. The one-dimensional form of the above theory is then applied to study transient wave propagation in an infinite thermoelastic materials with voids due to a time-dependent continuous heat sources distributed in a plane area. Laplace transform and eigenvalue approach techniques are used to obtain the closed form solution in the transform domain. Numerical inversions of the studied physical variables are carried out by Zakian algorithm in the space-time domain. Numerical results are plotted graphically in some cases and the results obtained are analyzed. Some comparisons for different cases are also noted.     

The Δ- Method for the Boltzmann Equation

A new numerical method is proposed to solve the Boltzmann equation. A frame is set up by using a discrete velocity approximation in the infinite velocity space, but by considering only those distribution function points which are not too small. The distribution function points may occur anywhere in the infinite discrete velocity space and are not constrained to a pre-specified region. A fourth-order finite difference is used for the convection terms. A Monte Carlo-like method is applied to the discrete velocity model of the collision integral. The effort of the method is proportional to the number of discrete points. Numerical examples are given for the full Boltzmann equation and results for some benchmark problems are compared with analytical or prior solutions.     

Light-cone structure of vertex functions for composite fields

Vertex functions for composite fields are defined in a model field theory both on and off mass shell. Light-cone dominance at large momentum transfer is shown to hold, by the compositeness assumption, for the off-shell vertex function. On the other hand, it is in general untrue that the elastic form factor probes light-like distances between the constituents inside the nucleon. The relevant light-cone singularity (in the relative space-time separation x) is less important in this case than the large x₀ behaviour of the wave function at fixed x². It is found however that, under some conditions, the light-cone singularity determines the large x₀ behaviour of the wave function, and therefore the large q² behaviour of the form factor. For composite particles described by a Bethe-Salpeter equation, this result is equivalent to the known fact that at large q² the form factor depends on the binding interaction at small distances. A relation similar to that of Drell-Yan-West is finally established between the asymptotic behaviour of the elastic form factor and the threshold behaviour of the absorptive part of the vertex scaling function.     

The problem of the unambiguity of phase-shift analysis I

For the discrimination of phase-shift solutions, the new so-calledτ-criterion, based on the calculation of some functionτ of the experimental data, is used. This method can be applied, in general, to an arbitrary ambiguous phase-shift analysis. Theτ-criterion is more powerful than theχ ²-criterion, e.g. for rejecting a less probable set it is necessary to have a smaller quantity of experimental points (in statistics this means that theτ-criterion gives a smaller Type II error). For the calculation of the quantityτ, the standard programme for phase-shift analysis with a small additional part can be used. A computer time of about only 5–10 min. on a computer of M-20 type is needed. A Type I error is then obtained from the tables of the normal distribution function.     

Matrix continued fraction solutions of the Kramers equation and their inverse friction expansions

The distribution function in position and velocity space for the Brownian motion of particles in an external field is determined by the Kramers equation, i.e., by a two variable Fokker-Planck equation. By expanding the distribution function in Hermite functions (velocity part) and in another complete set satisfying boundary conditions (position part) the Laplace transform of the initial value problem is obtained in terms of matrix continued fractions. An inverse friction expansion of the matrix continued fractions is used to show that the first Hermite expansion coefficient may be determined by a generalized Smoluchowski equation. The first terms of the inverse friction expansion of this generalized Smoluchowski operator and of the memory kernel are given explicitly. The inverse friction expansion of the equation determining the eigenvalues and eigenfunctions is also given and the connection with the result of Titulaer is discussed.     

NMR of Liquid 3He in Pores of a Clay Sample

In the present work a new method for studying porous media by nuclear magnetic resonance of liquid ³He has been proposed. This method has been demonstrated by an example of a clay mineral sample. For the first time the integral porosity of the clay sample has been measured. For investigated samples the value of the integral porosity is in the range of 10–30%. The inverse Laplace transform of the ³He longitudinal magnetization recovery curve has been carried out and the distribution of the relaxation times T ₁ has been obtained.     

A Laplace transform on the Lorentz groups

As a first step in the generalisation of the Laplace transform to a non abelian group, we examine the representations of the groupsSO(n, 1) by means of transformations of (not necessarily integrable) functions defined over the hyperboloidsO(n, 1)/O(n). We define a regularised version of the Gel'fand-Graev transformation from then-dimensional hyperboloid to its associated cone, which is valid (under certain restrictions) for polynomially bounded functions. Upon the cone we then carry out a pair of classical Laplace transforms parallel to a generator. We give inversion formulas for both these procedures, and express the Laplace transform/inversion pair directly in terms of the function on the hyperboloid.For integrable functions our results reduce to those already known; in the nonintegrable case they are new. New features include the divergence of the transform for certain discrete asymptotic behaviours; the existence of a finite dimensional kernel subspace which is annihilated; good asymptotic behaviour of both Laplace projection and inversion formulas; and the existence of discrete terms contributing to the inversion formula for even dimension. Our results are valid for all dimensions and are completely independent of the usual Laplace transforms involving projection by means of second-kind representation functions; in a final section of the paper we examine briefly the significance of that approach in the light of our own.     

Determination of the S and P° Rydberg energy levels of rubidium-like ions by the method of interpolation of relativistic quantum defects

The method of interpolation of relativistic quantum defects is used for determining the energies of Rydberg levels of rubidium-like ions. For this purpose, the values of relativistic quantum defects calculated by the Dirac-Fock method at three points, two of which correspond to discrete levels and the third, to the ionization threshold, are approximated by a second-degree polynomial. By using the continuous function μ(E) thus obtained, one can readily determine the energy value for any discrete level. A formula for calculating the threshold value of the quantum defect μ(0) (the phase shift δ(0)) is given. The approximation coefficients corresponding to the nS _1/2, nP°_1/2, and nP°_3/2 levels are presented. For better agreement with the experimental results, an empirical correction to the quantum defect is introduced, which weakly depends on energy. The calculations were performed for 17 members of the rubidium isoelectronic sequence (from Rb to Fr⁵⁰⁺).     

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.     

Elastic scattering of acoustic phonons on point mass defects

The relaxation of homogeneous states of long-wave acoustic phonon gas scattered by point mass defects in transversely—isotropic media is studied. The spectrum of the suitable collision operator of the Boltzmann-Peierls equation is investigated. It consists of a continuous part and several discrete eigenvalues. Both continuous and discrete part of the spectrum depend on the values of components of the elastic constant tensor. For some values of elastic constants the continuous part splits up into two separate intervals and some of the discrete eigenvalues appear in the gap. The number of discrete eigenvalues and their arrangement are also affected by elastic properties of medium.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.