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)

Fluctuation-noise spectroscopy and a "universal" fitting function of amplitudes of random sequences

A "universal" fitting function has been recognized that makes use of the eigen-coordinates method (Physica A 285 (2000) 547) to accurately describe the distribution of ordered amplitudes within random sequences (taken from a diversity of sources). It is shown that sequences with a discrete structure can be described in terms of specific distributions of relative frequencies with respect to the number of quantum levels. An investigation of these quantum distributions leads to an increase in both sensitivity and selectivity, when attempting the statistical detection of various predominant factors of the hidden signals. The physical meaning of this new function is discussed and a proposal is made as the basis of a new fluctuation-noise spectroscopy, in which the recognized function can be used for the detection of small signals and/or the forerunners of strong signals that are hidden within the random sequences analyzed. © 2002 Published by Elsevier Science B.V

The generalized mean value function approach: A new stastistical tool for the detection of weak signals in spectroscopy

A 'universal' set of fitting parameters have been recognized (with the help of the eigen-coordinates method) which fit well to any random sequence of points. The methodology is developed within the space of moments and is based upon the definition of the correct fit to the generalized mean value (GMV) function. By fitting G(p) N it is possible to express quantitatively the reduced characteristics of any random sequence, thereby providing a possible instrument for differentiating between statistically close random sequences. It is suggested that this new approach might find application in certain spectroscopic measurements, in cases where the signal to noise ratio is low, but the stability of the noise and the influence of other external factors can be maintained. Those fitting parameters from the approximate analytical expression, which depend on the concentration of the small additive, can then be used for the construction of the quasi-monotonic line, defined as the calibration curve. In certain well-defined cases, the new approach might allow significant improvements in the sensitivity of analytical instrumentation particularly when the available analysis methodology itself is non-optimal or even considered unsuitable. To test this possibility we examined the application of the GMV method to the near-infrared detection of model micro-particles (in our case yeast cells) in an aqueous suspension, and thereby demonstrated the possibility of increasing the sensitivity of a certain spectroscopy by at least one order of magnitude

Implementation of Fault-tolerant Quantum Logic Gates via Optimal Control

The implementation of fault-tolerant quantum gates on encoded logic qubits is considered. It is shown that transversal implementation of logic gates based on simple geometric control ideas is problematic for realistic physical systems suffering from imperfections such as qubit inhomogeneity or uncontrollable interactions between qubits. However, this problem can be overcome by formulating the task as an optimal control problem and designing efficient algorithms to solve it. In particular, we can find solutions that implement all of the elementary logic gates in a fixed amount of time with limited control resources for the five-qubit stabilizer code. Most importantly, logic gates that are extremely difficult to implement using conventional techniques even for ideal systems, such as the T-gate for the five-qubit stabilizer code, do not appear to pose a problem for optimal control.Comment: 18 pages, ioptex, many figure

The justified data-curve fitting approach: Recognition of the new type of kinetic equations in fractional derivatives from analysis of raw dielectric data

Usually, for the description of dielectric spectra one uses the empirical Cole-Davidson (CD) and Havriliak-Negami (HN) equations each of which contains one relaxation time. However, the parameters figuring in the CD and HN equations (or the linear combination of several CD or HN equations) do not have any clear physical meaning. For the description of such asymmetric dielectric spectra, we suggest complex permittivity functions containing two or more characteristic relaxation times. These complex susceptibility functions correspond, in the time-domain, to a new type of kinetic equation, which contains non-integer (fractional) integrals and derivatives. The physical meaning of these operators is discussed in [1]. We suppose that these kinetic equations describe a wide class of dielectric relaxation phenomena taking place in heterogeneous substances. To support and justify this statement, a special recognition procedure has been developed that helps to identify this new kinetic equation from real dielectric data. This recognition procedure can be considered as the justified data-curve fitting (JDCF) approach, in contrast to the conventional 'imposed' data-curve fitting (IDCF) treatment invariably used in modern dielectric spectroscopy. The JDCF approach incorporates the ratio presentation (RP) format and a separation procedure. It is shown how this separation procedure can be helpful in the detection of the many relaxation processes (each process is described by a characteristic relaxation time), which are taking place in the dielectric material under consideration

Application of the generalized mean value function to the statistical detection of water in decane by near-infrared spectroscopy

The generalized mean value (GMV) function, defined as GN(p)=(ΔN(p))1/ p(where ΔN(p) is the absolute value of the moment of the pth order), was used here to differentiate between statistically close random sequences or those sequences containing large numbers of measured points (N≧1). The approach taken was to find (with the help of the eigen-coordinates method) the approximate analytical function for any GN(p), and in so doing demonstrate that it is inherently possible to express quantitatively the reduced characteristics of any random sequence, in terms of the 'universal' set of fitting parameters defined by this function. The introduction of a 'universal' set of the reduced parameters in the moment space then provides the instrument for the comparison of different random sequences. Applications for this new method are evident for many branches of the analytical sciences, but especially in cases when visual 'labels' (e.g. resonance lines), which serve as an indication of the presence of an additive, are either absent or 'contaminated' strongly by noise. Those fitting parameters from the approximate analytical expression, which depend on the concentration of the small additive can then be used for the construction of the quasi-monotonic line, defined as the calibration curve. Real experiments based on the treatment of near-infrared (NIR) spectra obtained for decane (the initial matrix) with water (the additive) confirm the efficiency of this simple approach. In contrast, the more conventional statistical method, based on cluster analysis, failed to establish the desired calibration curve. This simple and universal approach, which is free from model assumptions, can be used for any set of random sequences (e.g. spectrograms) if it is necessary to compare them quantitatively with each other. © 2005 Elsevier B.V. All rights reserved

New approach in the description of dielectric relaxation phenomenon: Correct deduction and interpretation of the Vogel-Fulcher-Tamman equation

An empirical Vogel-Fulcher-Tamman (VFT) equation, connecting the maximum of the loss peak with temperature, was described. In order to establish the loss peak VFT dependence, a complex permittivity function should contain at least two relaxation times obeying the Arrhenius formula with two different set of parameters. It was shown that at a certain combination of initial parameters the parameter TVF can be negative or even accept complex value

Application of probability circles analysis to the construction of calibration curves for infra red spectroscopy

A new method, based on the presentation of initial data in the form of probability circles (PC), has been developed for constructing a calibration curve, which shows a monotonic dependence with respect to the given concentration. The centre of the reference probability circle is defined by its mean value and the radius of the circle is calculated as the value of the standard deviation of the sampling considered. The comparative probability circle is defined by the same corresponding parameters but rotated, relative to the initial reference circle, by an angle, which is related to the Pearson's correlation coefficient (PCC). The two parameters of the PCC and the statistical proximity factor (PCF), which defines the positions of the centres of the circles relative to each other, can be chosen as statistical parameters for the construction of the desired calibration curve. Experiments realized with the mixture of two liquids (chloroform serves as the basic matrix) and acetone (serves as an additive) confirm the efficiency of this new analytical method and demonstrate a possible increase sensitivity for the detection of lower concentration limit by approximately one order of magnitude. This new approach, which is free from model assumptions, and having very clear geometrical meaning, can be applied for different types of spectra and has many potential applications in the construction of calibration curves for different additives embedded within different matrices. © 2004 Elsevier B.V