V.V. Bilet, O.A. Dovgoshey, J. Prestin

BOUNDEDNESS OF LEBESGUE CONSTANTS AND INTERPOLATING FABER BASES

**Background.** We investigate the relationship between the boundedness of Lebesgue constants for the Lagrange polynomial interpolation on a compact subset of Υ and the existence of a Faber basis in the space of continuous functions on this compact set.

**Objective.** The aim of the paper is to describe the conditions on the matrix of interpolation nodes under which the interpolation of any continuous function coincides with the decomposition of this function in a series on the Faber basis.

**Methods.** The methods of general theory of Schauder bases and the results which describe the convergence of interpolating Lagrange processes are used.

**Results. **The structure of matrices of interpolation nodes which generate the interpolating Faber bases is described.

**Conclusions.** Every interpolating Faber basis is generated by the interpolating Lagrange process with the interpolating matrix of a special kind and bounded Lebesgue constants.

**Keywords:** Lebesgue constant; Lebesgue function; Lagrange polynomial interpolation; Faber basis.

O.M. Moklyachuk

ESTIMATION OF ACCURACY AND RELIABILITY OF MODELS OF **φ**-SUB-GAUSSIAN STOCHASTIC PROCESSES IN *С*(*T*) SPACES

**Background.** At present, in the theory of stochastic process modeling a problem of assessment of reliability and accuracy of stochastic process model in *C*(*T*) space wasn’t studied for the case of inexplicit decomposition of process in the form of a series with independent terms.

**Objective.** The goal is to study reliability and accuracy in *C*(*T*)** **of models of processes from Subj(W) that cannot be decomposed in a series with independent elements explicitly.

**Methods.** Using previous research in the field of modeling of stochastic processes, assumption is considered about possibility of decomposition of a stochastic process in the series with independent elements that can be found using approximations.

**Results.** Impact of approximation error of process decomposition in series with independent elements on reliability and accuracy of modeling of stochastic process in *C*(*T*) is studied.

**Conclusions.** Theorems are proved that allow estimation of reliability and accuracy of a model in *C*(*T*) of a stochastic process from Subj(W) in the case when decomposition of this process in a series with independent elements can be found only with some error, for example, using numerical approximations.

**Keywords:** stochastic processes; **φ**-sub-Gaussian processes; models of stochastic processes; reliability and accuracy of models of stochastic processes

I.V. Verygina

AN EFFICIENCY OF STRATEGIES IN MODELS OF CONFLICT REDISTRIBUTION OF RESOURCE SPACE

**Background. **The model of conflict redistribution of resource space (territory) between a pair of opponents in the case of infinite fractal division of space is investigated.

**Objective**.** **The aim of the paper is to consider the problem of conflict redistribution of resource space, if the opponent presence measures on the space are limited, but not necessarily probabilistic.

**Methods. **To specify the measures of the occupied territories and to find their limiting values a probabilistic approach was applied.** **

**Results. **The existence of limit values of measures of occupied territories with an infinite increase in the division step is established. The possible values are indicated. Using computer simulation graphs are obtained which show the behavior of the measures of the occupied territories with increasing of the division step.

**Conclusions. **It is shown that the limit values of measures of occupied territories depend only on the selected strategies (that is, the way of structured representation of measures that correspond to each of the opponents).

**Keywords:** dynamic conflict system; conflict redistribution of resource space; strategies of opponents; self-similar measures.

N.O. Virchenko, O.V. Ovcharenko

THE MAIN PROPERTIES OF** ***q*-FUNCTIONS

**Background.** The new generalization of the function of complex variable (*q*-function) is considered, its main properties are investigated. Such distributions have a special place among the special functions due to their widespread use in many areas of applied mathematics.

**Objective**. The aim of the paper is to study the new generalization of the function of complex variable for application in applied sciences.

**Methods.** To obtain scientific results the general methods of the mathematical analysis, and the theory of special functions have been used.

**Results.** The article deals with new generalization of the function of complex variable – *q*-functions, its main properties are investigated. The theorem on integral representation of *q* = *xk*-analytical functions is proved, its inverse formula is constructed.

**Conclusions.** Considered in the article new generalization of the function of complex variable opens up opportunities for the use of *q*-functions in the theory of special functions, and in the applications of mathematical and physical problems. In the future we plan to use the results to solve the boundary value problems of mathematical physics, in the theory of elasticity, for solving of, the theory of integral equations, etc.

**Keywords:** function of complex variable; generalized *q*-function; integral equation.

A.V. Ivanov, N.N. Karpova

SURFACE OF MAXIMUMS OF AR(2) PROCESS SPECTRAL DENSITIES AND ITS APPLICATION IN TIME SERIES STATISTICS

**Background.** In the problem on probabilities of large deviations of discrete time and sub-Gaussian AR(2) noise nonlinear regression model parameter least squares estimate a constant is determined that controls the rate of exponential convergence to zero of indicated probabilities.

**Objective.** The aim of the paper is to find the surface of maximums of AR(2) process spectral densities in the domain of its stationarity in explicit form.

**Methods.** The results were obtained on the use of methodology developed in the works by A. Sieders, K. Dzhaparidze (1987), A.V. Ivanov (1997, 2016) and standard Calculus methods.

**Results.** A complex formula that describes a continuous surface of maximums of AR(2) process spectral densities assigned on the stationary triangle of the time series of this type is obtained.

**Conclusions.** The obtained formula of surface of maximums of noise spectral densities gives an opportunity to realize for which values of AR(2) process characteristic polynomial coefficients it is possible to look for greater rate of convergence to zero of the probabilities of large deviations of the considered estimates.

**Keywords:** nonlinear regression model; least squares estimate; sub-Gaussian white noise; AR(2) process; probabilities of large deviations; surface of maximums of spectral densities.

A.V. Ivanov, O.V. Maliar

CONSISTENCY THE LEAST SQUARES ESTIMATE OF TEXTURED SURFACE PARAMETERS

**Background.** For sinusoidal observation model of textured surface, i.e. model where regression function is a two-parameter harmonic oscillation and noise is a homogeneous isotropic Gaussian random field on the plane with zero mean and covariance function of the special kind, the problem of statistical estimation of sinusoidal model unknown amplitudes and angular frequencies is considered.

**Objective.** The aim of the paper is to study asymptotic behavior of the least squares estimate (LSE) of unknown amplitudes and angular frequencies of textured surface sinusoidal model.

**Methods.** The results were obtained on the application of methodology developed in the papers by A.V. Ivanov et al. (2009, 2015) and monograph by A.V. Ivanov, N.N. Leonenko (1989).

**Results.** Sufficient conditions on random noise covariance function that ensure strong consistency of sinusoidal model parameters LSE are obtained.

**Conclusions.** The obtained results allow extending them on the models where regression function is a sum of several two-parameter harmonic oscillations. Besides LSE consistency property will allow proving asymptotic normality of these estimates in further works.

**Keywords:** textured surface; sinusoidal model; homogeneous and isotropic random field; covariance function; slowly varying function; least squares estimate; consistency.

A.B. Ilienko, V.V. Fatenko

A GENERALIZATION OF RÉNYI’S PARKING PROBLEM

**Background. **A generalization of the A. Rényi’s stochastic parking model is considered. In our model, the probability distribution of the left end of a parked car is a mixture of a uniform distribution and a degenerated one. This allows distinguishing drivers with different skills.

**Objective. **The aim of the paper is an asymptotic study of the mean number of parked cars m_{p}(x) when the length of parking space *x* increases unboundedly.

**Methods. **A functional-integral equation satisfied by the function m_{p} is derived. This equation admits an explicit solution in terms of the Laplace transform which allows for applying Tauberian theorems to study the required asymptotics.

**Results. **For the above model it is shown, that m_{p}(x)=λ_{p}x+0(x) as x→∞. The explicit form of the constant λ_{p} is given by . A similar asymptotic bound is obtained for a wider class of models in which the uniform component of the mixture of distributions is replaced by a more general one.

**Conclusions. **As a generalization of the classical A. Rényi’s random parking model, a new parking model is proposed. In this model the uniform distribution of the parking point is replaced by a mixture of a uniform distribution and a degenerated one. In the new model, a counterpart of the Rényi’s theorem on the asymptotic behavior of the mean number of parked cars is deduced.

**Keywords:** Rényi’s parking problem; mixture of distributions; asymptotic behavior; functional-integral equation; Laplace transform; Tauberian theorems.

O.I. Klesov, I.I. Sirenka, O.A. Tymoshenko

STRONG LAW OF LARGE NUMBERS FOR SOLUTIONS OF NON-AUTONOMUS STOCHASTIC DIFFERENTIAL EQUATIONS

**Background. **Asymptotic behavior at infinity of non-autonomous stochastic differential equation solutions is studied in the paper.

**Objective.** The aim of the work is to find sufficient conditions for the strong law of large numbers for a random process which is a solution of non-autonomous stochastic differential equation.

**Methods.** Basic results of the theory of stochastic differential equations related to stochastic integrals estimation.

**Results.** Sufficient conditions for almost sure convergence to zero of normalized term related to diffusion of non-autonomous stochastic differential equation are obtained.

**Conclusions**. Results of the paper can be used for further research on the asymptotic behavior of stochastic differential equation solutions, finding the stability condition of stochastic differential equation solution and ergodic type problems also.

**Keywords:** strong law of large numbers; stochastic differential equation; Wiener process; asymptotic behavior.

K.V. Moravetska

AN ALTERNATIVE SURFACE MEASURES CONSTRUCTION IN FINITE-DIMENSIONAL SPACES AND ITS CONSISTENCY WITH THE CLASSICAL APPROACH

**Background. **The area formulae are well known for surfaces embedded into a finite-dimensional Euclidean space. However, in the case of an infinite-dimensional Banach manifold, such formulae cannot be used. Thus, a problem of finding an alternative approach to the surface measures construction appears, that, on the one hand, leads to classical results in finite-dimensional case, and on the other hand, can be used for infinite-dimensional Banach manifolds.

**Objective. **The aim of the paper is to get a construction of surface measure induced by the Lebesgue measure and the associated form for a parametrically defined surface embedded into finite-dimensional Euclidean space. Show the consistency of surface area calculation by this construction with an area calculated by using well-known classical formulae.

**Methods. **Basic results of mathematical analyses, measure theory and differential geometry are used.

**Results. **An alternative construction of surface measures induced by the Lebesgue measure on surfaces in finite-dimensional space R^{m} is obtained. It is shown that such approach is consistent with the classical definition of the surface area.

**Conclusions. **The construction of surface measures suggested for infinite-dimensional spaces is a generalization of the classical approach in finite-dimensional spaces. Therefore further investigation of the described approach seems to be reasonable.

**Keywords: **surface measure; surface area; vector field.

V.V. Kulish

MAGNETIC SPIN-WAVE PROPERTIES OF FERROMAGNETIC NANOSYSTEMS OF VARIOUS SHAPES. PECULIARITIES OF THE BORDER CONDITIONS ACCOUNTING IN THE PROCESS OF THE WAVENUMBER VALUES SPECTRUM FINDING

**Background.** The paper continues the investigation of linear dipole-exchange spin waves in ferromagnetic nanosystems started by the author in previous papers. The known papers investigating dipole-exchange spin waves obtain their spectral characteristics only for a narrow range of special cases. In the presented paper, an approach that allows expanding substantially this range of cases is described and applied.

**Objective.** The aim of the paper is to develop a method – based on the use of boundary conditions – for obtaining the spectral characteristics of dipole-exchange spin waves in a number of nanosystems’ configurations as well as an application of this method to specific configurations.

**Methods.** A method for obtaining a values’ spectrum of the wavenumbers for dipole-exchange spin waves in ferromagnetic nanosystems of a series of typical configurations is proposed. The method does not require specific assumptions, e.g. the absence of transverse spin excitations or the presence of a high-conductivity metal outside the ferromagnet. The method uses imposition of boundary conditions for the magnetic field and the magnetization on the boundary of the ferromagnetic medium for a linear spin wave in the magnetostatic approximation. This method allows obtaining the above-mentioned spectral characteristics for a wider range of cases compared to the known previous papers.

**Results.** The results of the paper are conditions for the magnetic potential – that imply from the above-mentioned boundary conditions – of a linear spin wave on the boundary of a ferromagnetic nanosystem as well as the spectrum of the wavenumbers’ values of such wave for the investigated nanosystems (in the implicit form). In particular, for a ferromagnetic nanosystem of an arbitrary cross section with a one-dimensional translational symmetry, the conditions specifying this spectrum are found. Such conditions are specified – and an implicit expression for this spectrum is obtained – for the case when such nanosystem is a nanotube with a circular cross section. The analysis of the obtained results is carried out.

**Conclusions.** The obtained expressions for the spectrum of the values of the investigated spin waves’ wavenumbers can be used for a wider range of cases than the ones obtained in the previous papers dedicated to the investigated configurations of nanosystems. For a nanotube of the circular cross-section with small (compared to the inverse characteristic size of the nanotube cross-section) values of the longitudinal wave number, the dependence of the latter on the transverse wave number is weak, as well as for the big longitudinal to transverse wavenumber component ratio. The obtained dependence is also represented graphically.

**Keywords:** spin waves; nanomagnetism; dipole-exchange theory; boundary conditions.

P.V. Lukianov, V.N. Turick

ANALYTICAL MODELS DEVELOPMENT OF COMPACT MONOPOLE VORTEX FLOWS

**Background. **Mathematical description of vortex flows as a part of basic fundamental concepts of continuum mechanics, hydro- and gas dynamics, thermal physics, theoretical basics of chemical engineering, geophysics, meteorology.

**Objective. **Replenishment of existing information in traditional scientific, technical and educational publications about vortices and their analytical models by new data for scientists, teachers, post-graduates and students with the purpose of using in further investigations of fluids and gas vortex motion.

**Methods. **Analytical review of existing vortex current models, including monopole compact vortices, on a basis of the latest data from scientific articles and specialized monographs and their comparison with traditional data containing in the known textbooks and educational materials.

**Results. **The lack of modern compact vortices models was revealed in traditional scientific, technical and educational literary sources. They include the compact analogs of the point vortex and Rankine vortex: quasi-point vortex and compact compensated vortex. On basis of these ones and similar to them solutions (circular vortex, vortex with triad constant vorticity zones) the models of compact helical flows and vortices with helical symmetry were elaborated. The solutions of such tasks were analyzed: diffusion of compact vortices (Taylor vortex, Kloosterziel solution), turbulent diffusion of compact vortex, solutions for quasi-compact (laminar and turbulent) vortex-source, vortex-sink and also solution of the problem about compact turbulent vortex generation by rotating cylinder. All considered models are consistent with the energy conservation law and have advantages in their use.

**Conclusions. **The article contains series of the latest analytical models that describe both laminar and turbulent dynamics of monopole vortex flows which have not been reflected in traditional publications up to the present. The further research must be directed to search of analytical models for the coherent vortical structures in flows of viscous fluids, particularly near curved surfaces, where known in hydromechanics “wall law” is disturbed and heat and mass transfer anomalies take place.

**Keywords:** compact vortex; analytical models; ideal fluid; viscous fluid; laminar flow; turbulent flow; vortex diffusion; vortex generation.

S.O. Reshetnyak, A.V. Lysak

FREQUENCY DEPENDENCIES OF THE EXCHANGE SPIN WAVE REFLECTION COEFFICIENT ON A ONE-DIMENSIONAL MAGNON CRYSTAL WITH COMPLEX INTERFACES

**Background.** This work is devoted to theoretical study of the behavior of spin waves passing through multilayer ferromagnetic with complex interfaces.

**Objective. The aim of the paper is to calculate **the **reflection coefficient of multilayer **ferromagnetic** with complex interfaces as function of spin wave frequency at variable material parameter and **constant** value of external magnetic field. **The formalism of geometric optics allows describing the spin wave refraction process and the reflection coefficient, as well as controlling this process by changing the frequency of the spin wave for given parameters of the medium.

**Methods. **To find the reflection coefficient from a **multilayer **ferromagnetic the mathematical apparatus of geometric optics was used. To describe the dynamics of the magnetization vector the formalism of the spin density order parameter was used allowing for the use of methods of quantum mechanics to calculate the reflection coefficient from a semi-infinite multilayer structure.

**Results.** The spin wave reflection coefficient of semi-infinite multilayer structure of ferromagnetic materials with complex interfaces has been found. The dependency graphs of the reflection coefficient from the frequency of the spin waves at different parameters of magnetic anisotropy inhomogeneity and constant value of the external magnetic field were obtained.

**Conclusions.** It is shown that the frequency dependencies are periodic, points of full transmission and areas, full of reflection. Decreasing exchange parameter value in interface causes the increase of reflectance coefficient. Changing the material parameters we get the necessary intensity value of the reflection coefficient depending on the frequency at a constant value of the external magnetic field.

**Keywords: **spin waves; multilayer ferromagnetic; reflection coefficient; complex interface; magnetization vector.

Оновлено: 02/11/2017 - 17:44

Версія для друку