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Vol 220, No 1 (2017)

Article

Generalized Dunkl–Opdam Operator and its Properties in the Spaces of Functions Analytic in Domains

Linchuk Y.S.

Abstract

We define the generalized Dunkl–Opdam operator and study some of its properties in the space ℋ(G) of functions analytic in an arbitrary domain G . We also investigate the possibility of extension of a diagonal operator constructed by the Taylor coefficients of generalized hypergeometric function to the space ℋ(G).

Journal of Mathematical Sciences. 2017;220(1):1-14
pages 1-14 views

Gâteaux Differentiability of the Polynomial Test and Generalized Functions

Sharyn S.V.

Abstract

Let S+ and S+ be the Schwartz spaces of rapidly decreasing functions and tempered distributions on ℝ+ , respectively. Let P(S+) be the space of continuous polynomials over S+ and let P ′ (S+) be its strong dual. These spaces have representations in the form of Fock type spaces

\( \varGamma \left({S}_{+}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\oplus}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}\right)\kern2em \mathrm{and}\kern2em \varGamma \left({S}_{+}^{\prime}\right):=\underset{n\in {\mathbb{Z}}_{+}}{\times}\left({\oplus}_{s,\mathrm{p}}^n{S}_{+}^{\prime}\right), \)

respectively. In the present paper, the Gâteaux differentiability of elements of the spaces P(S+), P ′ (S+), Γ(S+), and Γ(S+) is investigated. The relationship between the Gâteaux derivative, the operators of creation and annihilation in the Fock type spaces, and the differentiations on Γ(S+) and Γ(S+) is established.

Journal of Mathematical Sciences. 2017;220(1):15-26
pages 15-26 views

On the Invariant Solutions of Some Five-Dimensional D’alembert Equations

Fedorchuk V.I.

Abstract

By using the invariants of nonconjugate subgroups of the Poincaré group P(1,4) [conjugation is considered with respect to the group P(1,4)], we propose ansatzes that reduce some linear and nonlinear five-dimensional d’Alembert equations to ordinary differential equations. On the basis of the solutions of the reduced equations, we construct the invariant solutions of these five-dimensional d’Alembert equations.

Journal of Mathematical Sciences. 2017;220(1):27-37
pages 27-37 views

Estimation of Decay of the Solutions of Initial-Boundary-Value Problem for the System of Semilinear Equations of Magnetoelasticity in Exterior Domains

Botsenyuk O.M.

Abstract

We study the behavior as t → ∞ of strong solutions of the initial-boundary-value problem for the system of semilinear equations of magnetoelasticity in exterior domains. For the vectors of magnetic induction and displacements, the estimates of decay are obtained in L and in the Sobolev spaces of fractional order, respectively.

Journal of Mathematical Sciences. 2017;220(1):38-49
pages 38-49 views

Interpolating Integral Continued Fraction of the Thiele Type

Makarov V.L., Demkiv I.I.

Abstract

We generalize the well-known results on the interpolation of a function of single variable by the Thiele–Hermite fraction with arbitrary multiplicity of each interpolation node to the case of a functional acting from the space of piecewise continuous functions with finitely many discontinuities of the first kind. We obtain an interpolating integral fraction of the Thiele type on the set of interpolation nodes one of which is continual. We also indicate an efficient approach to the construction of interpolating integral fraction of the Thiele type in the case where all interpolation nodes are continual. This case is important to balance the data used for the construction of the interpolant and its interpolating properties.

Journal of Mathematical Sciences. 2017;220(1):50-58
pages 50-58 views

Investigation of Different Versions of Formulation of the Problem of Soundproofing of Rectangular Plates Surrounded with Acoustic Media

Paimushin V.N., Tarlakovskii D.V., Gazizullin R.K., Lukashevich A.

Abstract

We consider five different formulations of the stationary problem of passage of plane acoustic waves through a rectangular plate. The first of these formulations corresponds to the application of the inertial mass model based on the hypothesis of the nondeformability of a nonfixed rigid plate in the course of its interaction with incident and plane acoustic waves formed in the surrounding half spaces. The other four statements correspond to taking into account (according to the model of the Winkler base) or neglecting the compliance of the support contour of a hingedly supported rectangular plate deformed according to the Kirchhoff model and to the application one- or three-dimensional wave equations for the description of motions of the acoustic media and the construction of the equation of motion of the plate with regard for its certain external damping. The use of these last four statements enables us to obtain smoothened graphic frequency dependences whose shapes agree with the experimental dependences obtained by testing specimens in the acoustic laboratory aimed at finding the soundproofing index of the plate.

Journal of Mathematical Sciences. 2017;220(1):59-81
pages 59-81 views

Equations of Motion of the Vortices in Bose–Einstein Condensates: Influence of Rotation and The Inhomogeneity of Density

Zueva T.I.

Abstract

We deduce equations of motion of quantized vortices in rotating Bose–Einstein condensates in two cases, namely, for a homogeneous condensate in a rigid rotating cylinder and for an inhomogeneous condensate in a rotating magnetic trap in the Thomas–Fermi approximation. The Schrödinger equations for both media are reduced to a convenient dimensionless form. By the method of expansion in a small parameter, we obtain two asymptotic solutions in different space scales. The comparison of the principal terms of these solutions yields the required equations. The equations of motion of the vortices in the homogeneous condensate are reduced to the well-known equations of vortices in the ideal liquid. The inhomogeneity of the medium results in the appearance of additional terms. We deduce the equations of motion of the vortices in the most general case: for any number of vortices, for vessels of different shapes, and both in the presence and in the absence of rotation. It is shown that, in the partial case of motion of a single vortex, the corresponding equations are reduced to the well-known equations of precession of the vortex. We present the plots of motion of several vortices for various initial data.

Journal of Mathematical Sciences. 2017;220(1):82-102
pages 82-102 views

Mass Sources and Modeling of Subsurface Heterogeneities in Deformable Solids

Nahirnyi T.S., Chervinka K.A.

Abstract

We propose an approach to the description of thermoelastic processes in deformable solids with regard for the structural heterogeneity of the material and the geometric heterogeneity of the surface of the body. In formulating the source relations, we use the methods of thermodynamics of nonequilibrium processes and nonlinear continuum mechanics. We take into account the structure of the material by introducing an irreversible component of the vector of mass flow. The geometric subsurface heterogeneity of the body is taken into account by introducing mass sources modeling its properties and by the dependence of the characteristics of materials, including the moduli of elasticity, on density. We study the equilibrium state of the half space. It is shown that two characteristic sizes correspond to the distributions of stresses and density. One of these sizes is connected with the structural heterogeneity of the material and the other is connected with the geometric heterogeneity of the actual surface of the body. We discuss the limits of applicability of the local gradient approach in the case of linearized approximation.

Journal of Mathematical Sciences. 2017;220(1):103-115
pages 103-115 views