### CONSTRUCTION OF CONTINUOUS SOLUTIONS OF NONLINEAR FUNCTIONAL-DIFFERENCE EQUATIONS SYSTEMS /I.V. Betsko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background. **We study the structure of the set of solutions of functional-difference equations systems

*x*(*t*+ 1) = *Ax*(*t*) + *F*(*t*, *x*(*qt*)) (1)

under certain assumptions about the matrix ** A** and number

**.**

*q***Objective.** The aim is to build continuous limited solutions for t∈Ȳ^{+}(Ȳ^{-}) and study the structure of their set.

**Methods.** We use the classical methods of the theory of ordinary differential and difference equations.

**Results. **The existence of the family of continuous limited solutions for *t≥0* which depends on arbitrary one-periodic function dimension** k** is proved. A similar result was obtained for case *t≤0* (the theorem 2).

**Conclusions. **New sufficient conditions for the existence of continuous solutions of functional-difference equations systems (1) are established, we developed the method of constructing these solutions and investigated the structure of their set.

**Keywords: **functional-difference equations; continuous limited solutions.

### TWO-BODY PROBLEM BY THE SINGULAR RANK ONE NONSYMMETRIC PERTURBATION /T.I. Vdovenko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background. **We consider nonself-adjoint singular perturbation of rank one of a self-adjoint operator by nonsymmetrical potential, i.e., the formal expression Ã=A+α〈 ,ω_{1}〉ω_{2}, where * A* is semibounded self-adjoint operator in the separable Hilbert space

**H**, α∈J. In compare with many previous studies of self-adjoint perturbation, the vectors

**ω**

_{1},ω

_{2}∈H

_{-2}are different, i.e.,

**ω**

_{1}≠ω

_{2}, that is some general problem about nonlocal interactions. The additional difficulties are that the Hilbert space

*has a form of*

**H****tensor product of spaces H=K⊗H,**

**and the operator has a form A=B⊗I**

_{H}+I

_{K}⊗A,

**which together illustrate the problem of two bodies.**

**Objective. **The purpose of research in our work is the description of singular perturbation of operator of the form Ã=B⊗I_{H}+I_{K}⊗Ã, where * Ã* is a self-adjoint operator that is singularly perturbed by nonsymmetric potential of rank one.

**Methods.** We use the assertion of about the presentation in the resolvent form of a rank one singular perturbation of a self-adjoint operator by symmetric potential, which corresponds to the problem of two bodies, and the definition of the operator singularly rank one perturbed by nonsymmetric potential **Ã.**

**Results.** The representation of the operator * Ã*, given by formal expression in the form of resolvent is our main result.

**Conclusions.** We present the resolvent form for singularly perturbed self-adjoint rank one operator, which perturbed by nonsymmetric potential, appropriate the problem of two bodies. By representation we take into account the case when the perturbed operator requires additional parameterization. We look for the application of the general results to describe the problem with the use of two bodies Laplace operator perturbed by nonsymmetric potentials composed of d-functions Dirac in Ý^{n}; *n*=2,3.

**Keywords:** singular perturbations; nonsymmetrical perturbations; nonlocal interaction; problem of two bodies.

### GENERAL PROPERTIES OF GENERALIZED GAMMA-FUNCTIONS /N.O. Virchenko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background. **The article is dedicated to studies of the main properties of new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions for their best applications in applied sciences, for calculations of integrals which are absent in scientific literature.

**Objective.** Introduction and study of the basic properties of the new generalized gamma-functions, generalized incomplete gamma-functions, generalized digamma-functions and their applications.

**Methods.** We apply the following methods: the methods of the theory of functions of the real variable, the theory of the special functions, the theory of the mathematical physics, the methods of applied analysis.

**Results.** Some new forms of generalized gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of these generalized special functions are explored. Examples of application of new generalized gamma-functions are given.

**Conclusions.** With the help of the *r*-generalized confluent hypergeometric functions the new generalization of gamma-functions, incomplete gamma-functions, digamma-functions are introduced. The main properties of the new generalized special functions are explored, examples of application of these functions are given.

**Keywords:** generalized gamma-functions; incomplete gamma-functions; digamma-functions.

### GENERALIZATION OF EULER’ INTEGRAL OF THE FIRST KIND /N.O. Virchenko, O.V. Ovcharenko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** The new generalization of Euler’ integral of the I-kind (beta-functions) 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 generalization of the new r-generalized beta-function and its application to the calculation of the new integrals.

**Methods.** To obtain results the general methods of the theory of special functions have been used.

**Results.** The article deals with new generalization of Euler’ integral of the I-kind. For the corresponding r-generalized beta functions were obtained important functional relations and differentiation formulas. For a wide application in the theory of integral and differential equations are important theorems on the connection of new beta functions with classical hypergeometric functions, Macdonald’ and Whittaker’ functions.

**Conclusions.** Considered in the article new generalization of Euler’ integral of the I-kind opens up opportunities for the use of Euler’ integrals in the theory of special functions, in the application of mathematical and physical problems. In the future we plan to use r-generalized beta functions to solve the new problems of the theory of probability, mathematical statistics, the theory of integral equations, etc.

**Keywords:** generalization of Euler’ integral of the I-kind; *r*-generalized beta function; hypergeometric function; Macdonald’ function; Whittaker’ function.

* *

* *

### TERMS UNIQUENESS EXTENT APPROPRIATE POINTS IN THE TWO-DIMENSIONAL PROBLEM /N.E. Dudkin, V.I. Kozak - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** We continue to study the properties of block Jacobi matrices corresponding to the two-dimensional real problem. Repeating the reasoning applied to probabilistic measure with compact support, we get similar matrices associated with Borel measure without limitation. The difficulty in our research is that the probability measure on compact corresponds uniquely to the block Jacobi type matrices. If the measure is arbitrary, then the same set of matrices can fit infinite number of measures.

**Objective.** The objective of research is to find conditions under which some Borel measure without limitation corresponds to only one pair of block matrices.

**Methods.** Using previous publications established the form of block Jacobi matrix type, by coefficients of these matrices one can inferred about the above mentioned bijection.

**Results.** The result of research is a conditions on the coefficients in the form of divergent series in which the one-to-one correspondence holds true.

**Conclusions.** Using the solution of direct and inverse spectral problems for two-dimensional real moment problem of the previous work we found its condition to be determined (unique) by the coefficients of the block Jacobi type matrix. The result is a two-dimensional analogue of the well-known case in classical Hamburger moment problem.

**Keywords****:** two-dimensional moment problem; block Jacobi type matrix; determinism of a two-dimensional moment problem.

### TWO BIRTHDAY PROBLEM MODIFICATIONS: NON-UNIFORM CASE /P.A. Yendovitskij - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** The scheme of random allocation of particles in cells is studied both in probability theory and mathematical statistics. In probability theory usual study is concerning limit theorems, in mathematical statistics – construction statistical criteria’s. Birthday problem is one of main questions in this theory.

**Objective.** Two modifications of the birthday problem are considered in the paper. One was formulated in Fermi statistic scheme, another – in non-uniform and independent random allocation scheme. In both cases the objective was to solve a birthday problem.

**Methods.** Standard asymptotical methods were used. At first we needed to prove one limit theorem and to estimate rapidity of convergence in it. Using these results numerical calculation of probabilities from birthday problem was made. Also formulas for the group size from birthday problem were obtained.

**Results.** As a result numerical estimates for birthday problem probability and group size were obtained.

**Conclusion.** For both modifications main asymptotic values coincide, as in the formula for probability calculation, as in the formula for the group size. But second terms from their asymptotic series are already different.

**Keywords**: birthday problem; birthday paradox; random allocations; Fermi statistic; Uval attack.

### BOUNDED OPERATORS OF STOCHASTIC DIFFERENTIATION ON SPACES OF NONREGULAR GENERALIZED FUNCTIONS IN THE LÉVY WHITE NOISE ANALYSIS /N.A. Kachanovsky - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background**. Operators of stochastic differentiation play an important role in the Gaussian white noise analysis. In particular, they can be used in order to study properties of the extended stochastic integral and of solutions of normally ordered stochastic equations. Although the Gaussian analysis is a developed theory with numerous applications, in problems of mathematics not only Gaussian random processes arise. In particular, an important role in modern researches belongs to Lévy processes. So, it is necessary to develop a Lévy analysis, including the theory of operators of stochastic differentiation.

**Objective.** During recent years the operators of stochastic differentiation were introduced and studied, in particular, on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Lévy analysis. In this paper, we make the next step: introduce and study such operators on spaces of nonregular generalized functions.

**Methods.** We use, in particular, the theory of Hilbert equipments and Lytvynov’s generalization of the chaotic representation property.

**Results.** The main result is a theorem about properties of operators of stochastic differentiation.

**Conclusions.** The operators of stochastic differentiation are considered on the spaces of nonregular generalized functions of the Lévy white noise analysis. This can be interpreted as a contribution in a further development of the Lévy analysis. Applications of the introduced operators are quite analogous to the applications of the corresponding operators in the Gaussian analysis.

**Keywords:** operator of stochastic differentiation; extended stochastic integral; Hida stochastic derivative; Lévy process.

### EXISTENCE OF MOMENTS OF EMPIRICAL VERSIONS OF HSU—ROBBINS—BAUM—KATZ SERIES /O.I. Klesov, U. Stadtmüller - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background. **We study the so called empirical versions of Hsu–Robbins and Baum–Katz series that are the basic notion of the classical theory of complete convergence.

**Objective. **The aim of the** **paper is to find necessary and sufficient conditions for the almost sure convergence of empirical Baum–Katz series. These conditions are expressed in terms of the existence of certain moments of the underlying random variables.

**Methods. **For proving our results we develop some new technique based on truncation and studying the truncated random variables. A sufficient ingredient of our approach is to show that the behavior of the truncated versions and the original ones is the same. Despite some similarity between the original series and its empirical version, the methods for achieving the results are quite different.

**Results. **We find necessary and sufficient conditions for the existence of higher moments of empirical versions. A special attention is paid to the case of multi-indexed sums. The latter case differs essentially from the one-dimensional case, since the space of indices is not completely ordered and thus any approach based on the first hitting moment does not work here.

**Conclusions. **The results obtained in the paper may serve as a base for further studies of empirical versions that could be used in statistical procedures of estimating an unknown variance.

**Keywords: **complete convergence for sums of independent identically distributed random variables; empirical Hsu–Robbins and Baum–Katz series; multi-indexed sums; regularly varying weights.

### THE REGULARIZATION OF UNITARY MATRIX FUNCTIONS /V.V. Pavlenkov - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background****.**** **The limit behavior at infinity of unitary matrix functions of real argument and arbitrary finite dimension is considered. The question of regular variation in Karamata sense of these functions is studied.

**Objective****.**** **The main aim of this work is to find the conditions under which not regularly varying unitary matrix function of real argument can be regularized by variable substitution.

**Methods****.**** **It is shown in the paper that substitution *t→*log*t* converts two-dimensional unitary matrix function into a power function with known matrix degree. This property is the base of all main result’s proofs.

**Results****.**** **The conditions under which the unitary matrix function of arbitrary finite dimension can be regularized are obtained.

**Conclusions****.**** **A significant difference between the matrix functions of dimension lower than 4 and the matrix functions of higher dimension is established. The constructed example of unitary matrix function of 4-dimension that can’t be regularized by substitution *t→*log*t* shows this difference.

**Keywords:** regularly varying function; matrix function; linear operator.

### INFINITEDIMENSIONAL RIEMANNIAN MANIFOLDS WITH UNIFORM STRUCTURE /A.Yu. Potapenko** ** - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** Solving boundary value problems on infinitedimensional Riemmanian manifolds, in particular researching Dirichlet problem, seems to demand for metric completeness. It does not appear to be feasible to state metric completeness in the general case, hence stems the issue of giving sufficient conditions of it.

**Objective.** Giving sufficient conditions of metric completeness of infinitedimensional Riemmanian manifolds and essential examples that would satisfy them.

**Methods.** Basic results of functional analysis and contemporary differential geometry are used.

**Results.** Sufficient conditions of infinitedimensional Riemmanian manifolds completeness have been formulated and proved. It has been proved that given conditions are satisfied for by level surfaces of finite codimension with certain bounds on first and second derivatives of the respective functions.

**Conclusions.** The Sufficient conditions of Riemmanian manifolds completeness – structure uniformity – look to be promising, since they are satisfied for at least by one relatively wide class of surfaces in Hilbert’s space. In terms of future researches, it now appears to be reasonable to devise approaches to considering boundary value problems on such infinitedimensional Riemmanian manifolds.

**Keywords:** infinitedimensional space; Riemmanian manifold; differential geometry.

### SMALL PERTURBATIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH POWER COEFFICIENTS /Y.E. Prykhodko** ** - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** Random perturbations of ordinary differential equations were considered by Bafico (1980), Bafico, Baldi (1982), Delarue, Flandoli (2014), Delarue, Flandoli, Vincenzi (2014), Krykun, Makhno (2013), Pilipenko, Proske (2015). Bafico, Baldi (1982) considered random perturbation of the differential equation that describes the Peano phenomenon. The coefficients of the initial differential equation are not Lipschitz continuous, so there may be no uniqueness of the solution. Then stochastic differential equation is considered instead of ordinary differential equation and the weak convergence of its solutions is proved.

**Objective.** The aim of this paper is to generalize the result of Bafico, Baldi (1982) to the case of stochastic differential equation dX(t)=α(X(t))dt+σ(X(t))dW(t) with power coefficients.

**Methods.** Small random perturbations of the initial equation dX(t)=α(X(t))dt+(ξ+σ(X(t)))dW(t) are considered and the limit behaviour of its solutions is studied.

The methods used to prove the weak convergence of the solutions are based on the methods developed in Pilipenko, Prykhodko (2015 and 2016).

**Results.** The limit behaviour of the solutions of stochastic differential equations with perturbations is considered and the weak convergence of such solutions is proved.

**Conclusions.** The result of Bafico, Baldi (1982) is thus generalized to the case of stochastic differential equation with power coefficients.

**Keywords:** stochastic differential equations; stochastic differential equations with power coefficients; stochastic differential equations with perturbations; asymptotic behavior; Peano phenomena.

### STOCHASTIC EQUIVALENCE OF GAUSSIAN PROCESS TO THE WIENER PROCESS, BROWNIAN BRIDGE, *ORNSTEIN*–*UHLENBECK PROCESS* /N.V. Prokhorenko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background. **We consider the Gaussian process with zero expectation and following covariance function: *R(s,t)=u(s)v(t),s≤t*. It was found the representation of the equivalent Wiener process for such process (Doob’s Transformation Theorem). We consider the representation of the Gaussian process via Wiener process, Brownian bridge and Ornstein–Uhlenbeck process in the case of monotonous function *u(t)/v(t)*.

**Objective. **The purpose of this paper is to find the criteria of equivalence between Gaussian process and Wiener process and to formulate similar criteria for Brownian bridge and Ornstein–Uhlenbeck process.

**Methods. **We constructed the system of the functional equations based on properties of Gaussian processes.

**Results. **Representation of Gaussian process with covariance function *R(s,t)* to equivalent Wiener process, Brownian bridge, Ornstein–Uhlenbeck process is discovered. Results are formulated in the form of criterion. Cases of decreasing and strictly increasing function *u(t)/v(t)* are considered.

**Conclusions. **The received outcomes can be used for research of functionals of the Gaussian processes. For example, to find the probability that Gaussian process crossing certain level. Representation of restriction of the Chentsov random field on polygonal line to equivalent Wiener process allowed finding the exact distribution of the maximum of the Chentsov random field on polygonal lines.

**Keywords:** Chentsov random field; the distribution of the maximum; Wiener process; Doob’s transformation theorem; the Brownian bridge.

### LIE SYMMETRIES AND FUNDAMENTAL SOLUTIONS OF THE LINEAR KRA-MERS EQUATION /V.I. Stogniy, I.M. Kopas, S.S. Kovalenko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** The group-theoretical analysis of fundamental solutions of the one-dimensional linear Kramers equation was carried out in the article.

**Objective.** The aim of the paper is to find the algebra of invariance of fundamental solutions of the equation under study using the Aksenov–Berest approach, and construct a fundamental solution of the one in the explicit form taking into account the algebra of Lie symmetries to be found.

**Methods.** The group-theoretical methods of analysis of partial differential equations are used. In particular, the Aksenov–Berest method of constructing in explicit form of fundamental solutions of linear partial differential equations is applied.

**Results.** The Lie algebra of non-trivial symmetries of the one-dimensional linear Kramers equation under consider was found. The fundamental solution in the explicit form of the equation was constructed. The effectiveness of using of symmetry methods in investigating of fundamental solutions of linear Kolmogorov–Fokker–Planck equations was shown.

**Conclusions.** Using the Aksenov–Berest approach, the algebra of invariance of fundamental solutions of one one-dimensional linear Kramers equation was found. The operators of the algebra were used in the process of constructing of invariant fundamental solutions of the equation. It was shown that the fundamental solution found early by S. Chandrasekhar without using the methods of symmetry analysis of differential equations is the weak invariant fundamental solution.

**Keywords:** linear Kramers equation; fundamental solution; Lie symmetries.

### GENERALIZATION OF ASYMPTOTIC BEHAVIOR OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATION /O.A. Tymoshenko - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena.

**Objective.** In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation.

**Methods.** We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions.

**Results.** We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process.

**Conclusions.** Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.

**Keywords****:** stochastic differential equation; Wiener process; asymptotic behavior.

### FREQUENCY CHARACTERISTICS OF REFLECTION AND REFRACTION COEFFICIENTS OF BULK SPIN WAVES IN SPIN LENS WITH NON-IDEAL INTERPHACES /S.O. Reshetnyak, S.V. Kovalchuk - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background****.** This work is devoted to the application of geometrical optics formalism to describe the behavior of spin waves, which is propagating in a ferromagnetic medium with non-uniform distribution of magnetic parameters. Use of this approach allows to describe the process of refraction of spin waves to determine the focal length of spin lenses or mirrors and to operate it by changing the frequency of the spin wave with a given magnetic parameters of a medium.

**Objective.** The objective is to calculate the index of refraction, reflection coefficient and focal length of spin lens as function of frequency of spin waves, the external magnetic field and magnetic parameters of the medium.

**Methods.** In this paper, to find the refractive index and the focal length was used the geometrical optics approach. To describe the dynamics of the magnetization vector the formalism was used of the parameter of order of spin density that also gives an opportunity to use the methods of quantum mechanics to calculate the spin wave reflection coefficient.

**Results.** In the paper the refractive index and focal length of a bulk spin lens have been found. By considering the generalized boundary conditions the expression has been found for the reflection coefficient of spin lens. In addition, results of investigation says that strong dependence exists of transparency of spin lens on the frequency of spin waves that is characterized by corresponding magnetic parameters of structure.

**Conclusions.** It is shown the possibility to change “optical” spin lens parameters in a wide range of values by changing only the frequency of spin waves and keeping constant the values of the external magnetic field and magnetic structure parameters. In addition, the results of studies prove that exists a strong dependence of the spin lens transparency on the quality of its borders, which is characterized by appropriate parameters of interface.

**Keywords:** spin lens; ferromagnet; anisotropy; exchange interaction; refraction.

### CORRELATION BETWEEN THERMODYNAMIC CHARACTERISTICS OF GLASS-FORMING SUBSTANCES /Ya.O. Shablovsky, V.V. Kiselevich - K. Naukovi visti NTUU “КPІ”. – 2016. – N 4

**Background.** Thermodynamics of the glass transition of the bulk samples of glass-forming substances.

**Objective.** The aim of the paper is analytical description of the correlations between thermodynamic characteristics of substance in the glass transition point and prediction of thermodynamic properties of organic and polymeric glass-forming substances.

**Methods.** Thermodynamic analysis of the glass transition in view of the formal consideration of the melt in the glass transition as the second-order phase transition.

**Results.** The applicability ranges of Ehrenfest-type relations for description of the correlation between thermodynamic characteristics of substance in the glass transition point are determined. The increments of the isobaric heat capacity and the thermal expansion coefficient at the glass transition point, as well as the pressure coefficient of the glass transition temperapture are predicted with the “entropic” correlation ratio for a number of organic and polymeric glass-forming substances.

**Conclusions.** “Entropic” correlation ratio is obtained and successfully tested. “Volumetric” correlation ratio is inapplicable to glass transition process.

**Keywords:** glass transition; thermodynamic properties of glasses; second-order phase transition; Ehrenfest-type relations.