Vol 227 (2023)
Статьи
On orders of n-term approximations of functions of many variables in the Lorentz space
Abstract
In this paper, we consider the anisotropic Lorentz space of -periodic functions of many variables and the Nikolsky–Besov class in this space. We obtain estimates for the best approximations along the hyperbolic cross and the best -term approximations of functions of the Nikolsky—Besov class with respect to the norm of the anisotropic Lorentz space for various relations between the parameters of the class and the space.
3-19
Reconstruction of characteristic functions of quadratic functionals on trajectories of Gaussian stochastic processes
Abstract
In this paper, we examine the characteristic functions , , of stochastic variables determined by the values of the quadratic functionals on the space of trajectories of homogeneous Gaussian stochastic processes. We justify a method for calculating such characteristic functions, called reconstruction in the work, the application of which is not related to the use of the well-known Karhunen–Loeve–Pugachev method.
20-40
Exact solution of 3d Navier–Stokes equations for potential motions of an incompressible fluid
Abstract
A procedure for constructing an exact solution of the 3D Navier–Stokes equations for the case of potential motion of an incompressible fluid in a deep, large-volume reservoir is proposed. The solution is considered under asymptotic boundary conditions that correspond to a given value of the velocity vector at great depth. The procedure for constructing a solution is based on the integral of the 3D Navier–Stokes equations. By introducing functions of a complex variable, the problem is reduced to a system of Riccati equations, which can be solved analytically. The qualitative features of the solution are examined.
41-50
Problem of the equilibrium of a two-dimensional elastic body with two contacting thin rigid inclusions
Abstract
A new nonlinear mathematical model is proposed that describes the equilibrium of a two-dimensional elastic body with two thin rigid inclusions. The problem is formulated as a minimizing problem for the energy functional over a nonconvex set of possible displacements defined in a suitable Sobolev space. The existence of a variational solution to the problem is proved. Optimality conditions and differential relations are obtained that characterize the properties of the solution in the domain and on the inclusion; these conditions are satisfied for sufficiently smooth solutions.
51-60
On several models of population dynamics with distributed delay
Abstract
In this paper, we examine several models of population dynamics: the Hutchinson equation, the Mackey–Glass equation, the Lasota–Warzewski equation, and the Nicholson equation. The greatest attention is paid to models in which the aftereffect is considered distributed over a certain interval. The local stability of solutions to these equations is studied.
61-78
Completeness of exponential systems in functional spaces in terms of perimeter
Abstract
A new scale of completeness conditions for exponential systems is established for two types of functional spaces on subsets of the complex plane. The first type of spaces are Banach spaces of functions that are continuous on a compact set and holomorphic in the interior of this compact set (if it is nonempty) with the uniform norm. The second type consists of spaces of holomorphic functions on a bounded open set with the topology of uniform convergence on compact sets. These conditions are formulated in terms of majorizing the perimeter of the convex hull of the domain of functions from the space by new characteristics of the distribution of exponents of the exponential system.
79-91
Optimization problems in ordinary first-order autonomous systems
Abstract
In this paper, we examine mathematical control problems for first-order autonomous systems. Using Pontryagin’s maximum principle, we analyze the mathematical problem of optimizing the generation of income in the market for educational services, taking into account the deferment of investment.
92-99
Tensor invariants of geodesic, potential and dissipative systems. I. Systems on tangents bundles of two-dimensional manifolds
Abstract
In this paper, we present tensor invariants (first integrals and differential forms) for dynamical systems on the tangent bundles of smooth -dimensional manifolds separately for , , , , and for any finite . We demonstrate the connection between the existence of these invariants and the presence of a full set of first integrals that are necessary for integrating geodesic, potential, and dissipative systems. The force fields acting in systems considered make them dissipative (with alternating dissipation).
100-128