Eugene B. Postnikov

About calculation of the Hankel transform using preliminary wavelet transform

The purpose of this paper is to present an algorithm for evaluating Hankel transform of the null and the first kind. The result is the exact analytical representation as the series of the Bessel and Struve functions multiplied by the wavelet coefficients of the input function. Numerical evaluation of the test function with known analytical Hankel transform illustrates the proposed algorithm.

Tait Equation Revisited from the Entropic and Fluctuational Points of View

We consider the possibilities for prediction of liquid densities under pressure based on the inverse reduced fluctuations parameter, which is directly connected with the isothermal compressibility. This quantity can be determined based on the thermodynamic properties of the saturated liquid, and it consists of only two constant parameters within a relatively wide region close to the melting points. It is confirmed by the comparison with the experimental data on n-alkanes that the derived expression is a quite reasonable estimator without a necessity to fit data along some parts of isotherms for different temperatures. At the same time, the obtained formula (i) can be reduced to the form of the Tait equation and (ii) the resulting Tait parameters in this representation have a clear physical meaning as functions of the excess entropy, which determines the mentioned reduced fluctuations.

Limits of structure stability of simple liquids revealed by study of relative fluctuations

We analyse the inverse reduced fluctuations (inverse ratio of relative volume fluctuation to its value in the hypothetical case where the substance acts as an ideal gas for the same temperature-volume parameters) for simple liquids from experimental acoustic and thermophysical data along a coexistence line for both liquid and vapour phases. It has been determined that this quantity has a universal exponential character within the region close to the melting point. This behaviour satisfies the predictions of the mean-field (grand canonical ensemble) lattice fluid model and relates to the constant average structure of a fluid, i.e. redistribution of the free volume complementary to a number of vapour particles. The interconnection between experiment-based fluctuational parameters and self-diffusion characteristics is discussed. These results may suggest experimental methods for determination of self-diffusion and structural properties of real substances.

Transition in fluctuation behaviour of normal liquids under high pressures

We explore the behaviour of the inverse reduced density fluctuations and the isobaric expansion coefficient using α,ω-dibromoalkanes as an example. Two different states are revealed far from the critical point: the region of exponentially decaying fluctuations near the coexistence curve and the state with longer correlations under sufficiently high pressures. The crossing of the isotherms of the isobaric expansion coefficient occurs within the PVT range of the mentioned transition. We discuss the interplay of this crossing with the changes in molecular packing structure connected with the analysed function of the density, which represents inverse reduced volume fluctuations.

Nonspectral relaxation in one dimensional Ornstein-Uhlenbeck processes.

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized Lévy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to nonspectral relaxation can be considered exotic, for generalized OUPs driven by Lévy noise, such initial conditions are the rule.

Model of lateral diffusion in ultrathin layered films

We consider the diffusion of markers in a layered medium, with the lateral diffusion coefficient being the function of hight. We show that the probability density of the lateral displacements follows a one-dimensional Batchelor’s equation with time-dependent diffusion coefficient governed by the particles’ redistribution in height. For the film of a finite thickness the resulting mean squared displacement exhibits superdiffusion at short times and crosses over to normal diffusion at long times. The approach is used for a description of experimental results on inhomogeneous molecular diffusion in thin liquid films deposited on solid surfaces.

On precision of wavelet phase synchronization of chaotic systems

It is shown that time-scale synchronization of chaotic systems with ill-defined conventional phase is achieved by using wavelet transforms with center frequencies above a certain threshold value. It is found that the possibility of synchronization detection by introducing a wavelet phase is related to diffusion averaging of the analyzed signals.

Evaluation of a continuous wavelet transform by solving the Cauchy problem for a system of partial differential equations

It is shown that the problem of evaluating the continuous Morlet wavelet transform can be stated as the Cauchy problem for a system of two partial differential equations. The initial conditions for the desired functions, i.e., for the real and imaginary parts of the wavelet transform, are the analyzed function and a vanishing function, respectively. Numerical examples are given.

Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition

Recently, it has been proven [R. Soc. Open Sci. 1 (2014) 140124] that the continuous wavelet transform with non-admissible kernels (approximate wavelets) allows for an existence of the exact inverse transform. Here we consider the computational possibility for the realization of this approach. We provide modified simpler explanation of the reconstruction formula, restricted on the practical case of real valued finite (or periodic/periodized) samples and the standard (restricted) Morlet wavelet as a practically important example of an approximate wavelet. The provided examples of applications includes the test function and the non-stationary electro-physical signals arising in the problem of neuroscience.

On alternative wavelet reconstruction formula: a case study of approximate wavelets.

The application of the continuous wavelet transform to the study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies owing to the admissibility condition. We propose an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard reduced Morlet wavelet, which is an important example of such analysing functions, is discussed.