Fractional Integro-Differential Equations for Electromagnetic Waves in Dielectric Media

We prove that the electromagnetic fields in dielectric media whose susceptibility follows a fractional power-law dependence in a wide frequency range can be described by differential equations with time derivatives of noninteger order. We obtain fractional integro-differential equations for electromagnetic waves in a dielectric. The electromagnetic fields in dielectrics demonstrate a fractional power-law relaxation. The fractional integro-differential equations for electromagnetic waves are common to a wide class of dielectric media regardless of the type of physical structure, the chemical composition, or the nature of the polarizing species (dipoles, electrons, or ions)

Extraction of reliable information from time-domain pressure and flow signals measured by means of forced oscillation techniques

This paper aims to give a proof-of-concept for the possible application of the forced oscillation lung function test to assess the viscoelastic properties of the airways and tissue. In particular, a novel signal processing algorithm is employed on non-stationary, noisy, (relatively) short time series of respiratory pressure and flow signals. This novel technique is employed to filter the useful information from the signals acquired under two measurement conditions: pseudo-functional residual capacity (PFRC) and pseudo-total lung capacity (PTLC). The PFRC is the measurement performed at lowest lung volume with maximum deflation, and the PTLC is measurement performed at the maximum lung volume under maximum inflation. The results suggest that the proposed technique is able to extract information on the viscoelastic properties of the lung tissue at a macroscopic level. The conclusion of this preliminary study is that the proposed combination of signal processing method and lung function test is suited to be employed on a large database in order to deliver reference values and perform further statistical analysis

Universal distribution function for the strongly-correlated fluctuations: General way for description of different random sequences

It has been proved that for the strongly-correlated fluctuations there is a universal distribution function for the relative fluctuations (UDFRF). The analytical form of this function follows from the solution of some types of the functional equations. For obtaining the UDFRF a procedure of the optimal linear smoothing (POLS) has been developed. This procedure based on criterion of the minimal relative error helps to separate correctly a possible trend (the "low-frequency" curve, defined as the generalized mean value curve or trend) from the "high-frequency" (HF) fluctuations, defined as a random sequence of relative fluctuations with zero trend. A universal treatment procedure outlined in this paper helps to find an optimal trend, separate it from the relative HF fluctuations and read them quantitatively. The statistics of the fractional moments outlined in this paper helps "to read" the found trends and express them in terms of the fitting parameters if the model for their description is absent. These new possibilities can be applied for description of different noises (quantum fluctuations, for example) that always present on the scale (10-6 ÷ 10-9 m). Quantitative reading of these noises with their subsequent classification is important for every developing nanotechnology that it has a possibility to be applied in this range of scales. © 2009 Elsevier B.V. All rights reserved

Quantitative Universal Label: How to Use It to Mark Any Randomness

It is possible to find a quantitative universal label (QUL) that can express quantitatively any random sequence in terms of a finite set of quantitative parameters. This label is associated with nine parameters of the generalized Gaussian distribution (GGD), describing all possible correlations (expressed in terms of symmetric products) for all amplitudes {yj} belonging to a random sequence considered. This noise label allows one to compare any randomness with another one and to calibrate these fitting parameters with respect to a possible external factor in the signature (character) space. It opens new resource in signal/noise data treatment and makes it possible to discover completely new relationships that might be hidden in random sequences. This label reflects the distribution of correlations between k stable points existing in the initial random sequence having N initial points (k ≤ N). This distribution is free of any model assumption and can be used as a universal quantitative measure characterizing some random sequence. Different examples considered in this paper confirm the effectiveness of the QUL expressed in terms of GGD fitting parameters. © 2009 Allerton Press, Inc

Dielectric relaxation phenomenon based on the fractional kinetics: Theory and its experimental confirmation

In this short paper, we outline the basic results obtained for dielectric relaxation based on the fractional kinetics. One can prove that many self-similar dynamical processes taking place in microscopic scales are averaged on the mesoscale and from the mathematical point of view these averaged motions accept power-law dependence with real or complex-conjugated exponents. This means that in the region of the mesoscale the kinetic equation for the total polarization is described by equations containing non-integer operators with real or complex-conjugated exponents. This approach helps us to understand some empirical expressions (Cole-Cole and Cole-Davidson) suggested for the description of the complex permittivity in the frequency domain. Besides, this theory suggests new expressions for complex permittivity that follow from the stationary solution of equations containing non-integer differential operators. This theory is confirmed by experimental measurements; in particular, it helps us to understand the generalization of the empirical Vogel-Fulcher-Tamman (VFT) equation that relates the behavior of any extreme point of the dielectric spectra with temperature. For the first time, the kinetic processes that are described by real and complex-conjugated power-law exponents are confirmed experimentally in polymerization vitrification reactions. © 2009 The Royal Swedish Academy of Sciences

The statistics of the fractional moments: Is there any chance to "read quantitatively" any randomness?

The statistical meaning of higher ΔN (p) (p = 1, 2, ...) and fractional ΔN (p) (0 < p < ∞) moments for an arbitrary random sequence of the length N has been found. The higher moments help to reduce the sequence analyzed to a finite set of k statistically stable parameters, keeping invariant the values of the first kth moments Δk (p) (p = 1, 2, ..., k). The conditions of statistical stability and proximity expressed in terms of higher moments ΔN (p) = ΔN + k (p) (p = 1, 2, ..., k) help to find k unknown stable points and predict possible future behavior of the random sequence analyzed. The generalized mean value (GMV)-function defined as GN (p) = (ΔN (p))1 / p is turned to be very effective in analysis of statistically close random sequences or containing large numbers of measured points (N ≫ 1). The approximate analytical expression for an arbitrary p value from the range (- ∞ < p < ∞) entering into GN (p) has been found. It gives a possibility to transform any random sequence to the determined GMV curve and express quantitatively the reduced characteristics of any random sequence in terms of a 'universal' set of the fitting parameters defined by the determined GMV-function. Statistical proximity factor can be used for construction of calibration curves, when it is necessary to compare one random sequence with another one to respect of variations of some given external factor (small signal). The higher moments are easily generalized for the fractional and even complex moments. In turn, the GMV-function can be also generalized and then calculated for 2D and 3D random sequences. The approach developed in this paper is free from any model assumption and can be extremely helpful in comparison of different random sequences using for these purposes the 'unified' quantitative language based on introduction of the given set of fractional moments. The relationship between the value of the fractional moment and nonextensive parameter q entering into the definition of the nonextensive Tsallis entropy has been found. A possible model of statistical protection of plastic cards and other valuable documents demonstrating the effectiveness of the statistics of fractional moments has been considered. Some instructive examples in detection of superweak signals embedded into the basic random sequence (S / N = 10- 2, 10- 3) based on model and real data confirm the effectiveness of new approach and can serve a new basis for numerous practical applications. Analysis of dielectric spectroscopy data by means of fractional moments gives unique possibility to compare quantitatively each measurement with each other and express the influence of a neutral additive in terms of calibration curve without concrete knowledge of the corresponding fitting function. © 2006 Elsevier B.V. All rights reserved

Detection of weak signals based on a new class of transformations of random series

For detecting weak signals (when their amplitude is comparable with the amplitude of the noise track) a new class of transformations of random series to a straight line has been suggested. It has been shown that these transformations are quasi-linear and can be defined as a signal-to-staircase transformation (SST). The height of a step defines an amplitude of the detected signal and the step length its duration. The SST can be applicable for a wide class of random series having different statistical nature. The verification of this new method based on the analysis of the real signal/noise tracks containing registration of different earthquakes with small amplitudes has been realized. It has also been shown that different situations which have been found from the real-data analysis demonstrate the high sensitivity and efficiency of the new method suggested