Vol 100, No 1-2 (2016)

Under natural conditions, we prove an existence theorem for stochastic differential equations with current velocities (mean derivatives) and with nonautonomous right-hand side.

The restriction of a monotone operator P to the cone Ω of nonnegative decreasing functions from a weighted Orlicz space L_φ,v without additional a priori assumptions on the properties of theOrlicz function φ and the weight function v is considered. An order-sharp two-sided estimate of the norm of this restriction is established by using a specially constructed discretization procedure. Similar estimates are also obtained for monotone operators over the corresponding Orlicz–Lorentz spaces Λ_φ,v. As applications, descriptions of associated spaces for the cone Ω and the Orlicz–Lorentz space are obtained. These new results are of current interest in the theory of such spaces.

The matrix Schrödinger operator with point interactions on the semiaxis is studied. Using the theory of boundary triplets and the corresponding Weyl functions, we establish a relationship between the spectral properties (deficiency indices, self-adjointness, semiboundedness, etc.) of the operators under study and block Jacobi matrices of certain class.

The averaging method is justified for normal systems of differential equations with rapidly oscillating summands proportional to the square root of the oscillation frequency in the case of the boundary-value problem on a finite interval and for the problem of bounded solutions on the positive semiaxis with boundary condition at its left endpoint.

A formula relating the Radon transform of functions of spherical symmetries to the Radon–Kipriyanov transform K_γ for a naturalmulti-index γ is given. For an arbitrary multi-index γ, formulas for the representation of the K_γ-transform of a radial function as fractional integrals of Erdelyi–Kober integral type and of Riemann–Liouville integral type are proved. The corresponding inversion formulas are obtained. These results are used to study the inversion of the Radon–Kipriyanov transform of the generalized (generated by a generalized shift) spherical mean values of functions that belong to a weighted Lebesgue space and are even with respect to each of the weight variables.

The quantum interpretation of quasisymmetric homeomorphisms of the circle, i.e., homeomorphisms that can be extended to quasiconformal homeomorphisms of the unit disk, and their relationship to basic constructions of quantum calculus are discussed.

In this paper, the following problem is studied. For what multipliers {λ_k,n} do the linear means of the Fourier series of functions f ∈ L₁[−π, π],

, converge as n→∞ at all points at which the derivative of the function ∫₀^xf exists? In the case λ_k,n = (1 − |k|/(n + 1)), a criterion of the convergence of the (C, 1)-means and, in the general case λ_k,n = ϕ(k/(n + 1)), a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether ϕ(x) and xϕ′(x) belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given.

In the paper, the additive complexity of matrices formed by positive integer powers of greatest common divisors and least common multiples of the indices of the rows and columns is considered. It is proved that the complexity of the n × n matrix formed by the numbers GCD^r(i, k) over the basis {x + y} is asymptotically equal to rn log²n as n→∞, and the complexity of the n × n matrix formed by the numbers LCM^r(i, k) over the basis {x + y,−x} is asymptotically equal to 2rn log₂n as n→∞.

The value of the deviation of a function f(x) from its de la Vallée-Poussin means V_mⁿ(f, x) with respect to the trigonometric system for classes of piecewise smooth 2π-periodic functions is estimated.

Spaces of differentiable functions constructed from the spaces L_p, 0 < p < ∞, and the interdependence of the differential properties of functions with their local approximations are considered.

In this paper, the Dirichlet-type problem for the system of elliptic equations of second order with the degeneracy at a line crossing the domain is considered. The Dirichlet-type problem with additionally given asymptotics of the solution at this line is discussed. The uniqueness and the existence of the solution of this problem in the class of Hölder functions is proved.

In this article we obtain the sharp asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operators generated by a system of the Sturm–Liouville equations with Dirichlet and Neumann boundary conditions. Using these asymptotic formulas, we find a condition on the potential for which the root functions of these operators form a Riesz basis.

The author constructs a new conception of thermodynamics which is based on new results in number theory. We consider a maximally wide range of gases, liquids, and fluids to which, in principle, the Carathéodory approach can be applied. The Carathéodory principle is studied using the Lennard-Jones potential as an example. On the basis of this example, we analyze the dispersive structure of a fluidwhose density exceeds the critical value. We introduce a new parameter, the “jamming factor,” which determines the jamming effect for such fluids. A comparison with experimental data for nonpolar molecules is carried out. The phase transition “liquid-amorphous solid” is studied in detail in the domain of negative pressures. We discuss the theoretical relationship between the obtained solutions and econophysics, some mysteries in biology, and other sciences.