Vol 94, No 2 (2016)

An operator equation with the second time derivative is represented in the form of a Hamiltonianadmissible equation. A relationship between solutions of Hamiltonian-admissible equations and the corresponding Hamiltonian equations is established.

We obtain broad sufficient conditions for the existence of proper conditional probabilities measurably depending on a parameter in the case of parametric families of measures and mappings.

A class of inhomogeneous Markovian queuing systems with possible catastrophic failures and group arrival of customers in the case of empty queue is considered; basic estimates of the rate of convergence and stability for this class are obtained.

Poincaré-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.

The method of lines is used to obtain semidiscrete equations for a bicompact scheme in operator form for the inhomogeneous linear transport equation in two and three dimensions. In each spatial direction, the scheme has a two-point stencil, on which the spatial derivatives are approximated to fourth-order accuracy due to expanding the list of unknown grid functions. This order of accuracy is preserved on an arbitrary nonuniform grid. The equations of the method of lines are integrated in time using diagonally implicit multistage Runge–Kutta methods of the third up fifth orders of accuracy. Test computations on refined meshes are presented. It is shown that the high-order accurate bicompact schemes can be efficiently parallelized on multicore and multiprocessor computers.

Given any nonzero entire function g: ℂ → ℂ, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+∙∙∙+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ℂ → ℂ. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.

Given a polynomial f of odd degree, the nontrivial S-units can be effectively related to the continued fraction expansions of the elements associated with \(\sqrt f \) only in the case where S contains an infinite valuation and a finite valuation determined by first-degree polynomial. A quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained. For key elements, a more accurate criterion is found. The criterion is used to show that, for S specified above, in the presence of a nontrivial S-unit, the expansion of \(\sqrt f \) can be both nonperiodic and periodic. Estimates relating the quasi-period to the degree of the fundamental S-unit are obtained. Examples in which the bounds of these estimates are attained are given.

On a model example of a linear hyperbolic equation with small parameter multiplying the highest time derivative it is shown that the closeness of individual trajectories to the dynamics of the limiting parabolic equation essentially depends on the Fourier spectra of the initial data. The trajectories stay close if the higher modes decay sufficiently fast. If the initial data are irregular and there are relatively high modes, then the convergence of the trajectories becomes non-uniform. Namely, the boundary layer is formed and there exist small moments of time such that the difference of the solutions reaches in the mean a finite value as the coefficient of the highest time derivative tends to zero. These results reflect the difficulties that may arise in the analysis of the systems of non-linear quasi-gasdynamic equations.

A new, generalized and strengthened, form of an assertion about an extremum of a linear-fractional integral functional given on a set of probability measures is presented. It is shown that the solution of the extremal problem for such a functional is completely determined by the extremal properties of the so-called test function, which is the ratio of the integrands of the numerator and the denominator. On the basis of this assertion, a theorem on an optimal strategy for controlling a semi-Markov process with a finite set of states is proved. In particular, it is established that if the test function of the objective functional of a control problem attains a global extremum, then an optimal control strategy exists, is deterministic, and is determined by the point of global extremum. The corresponding assertions are also obtained for the case where the test function does not attain the global extremum.

Generalized weighted Morrey spaces defined on spaces of homogeneous type are introduced by using weight functions in the Muckenhoupt class. Theorems on the boundedness of a large class of sublinear operators on these spaces are presented. The classes of sublinear operators under consideration contain a whole series of important operators of harmonic analysis, such as, e.g., maximal functions, singular and fractional integrals, Bochner–Riesz means, and so on.

There are some results concerning t-designs in which the number of points in the intersection of two blocks takes less than t values. For example, if t = 2, then the design is symmetric (in such a design, v = b or, equivalently, k = r). In 1974, B. Gross described t-(v, k, l) designs that, for some integer s, 0 < s < t, do not contain two blocks intersecting at exactly s points. Below, it is proved that potentially infinite series of designs from the claim of Gross’ theorem are finite. Gross’ theorem is substantially sharpened.

A special Harnack inequality is proved for solutions of nonlinear elliptic equations of the p(x)-Laplacian type with a variable exponent p(x) that takes different values on two sides of a hyperplane dividing the domain. Examples are given showing that the classical Harnack inequality does not hold in this case.

The behavior of solutions near an equilibrium of differential equations with two large different-order delays is studied. In critical cases, special equations (namely, quasi-normal forms) determining the dynamics of the original problem are constructed. As a rule, they have the form of nonlinear parabolic equations with two spatial variables. Asymptotic formulas are derived showing the relation between the solutions of a quasi-normal form and the original equation. The case of a small coefficient multiplying the term with a larger delay is studied separately.

A new problem of random choice for pills consumption process is formulated and considered. We find a kind of the Law of Large Numbers associated with any ordinary differential equation. This generalized LLN says that a stochastic analog of the Euler broken lines converges in probability to solution of the initial value problem for the ODE. This approach is applied to a stochastic process of pills consumption, and shows that after a suitable scaling the consumption process is almost deterministic, provided that the initial number of pills is large.

The control of a viscous incompressible flow within MHD is considered. The phenomenon of internal heat release is explored as applied to a layered plane flow in which the kinetic energy dissipates strictly into heat. The influence exerted by the electromagnetic field of the flowing fluid on the surrounding medium is examined, and methods for flow control in the context of digital deposit field technology are considered.

A fourth-order linear elliptic partial differential equation describing the displacements of a transversely isotropic linear elastic medium is considered. Its symmetries and the symmetries of an inhomogeneous equation with a delta function on the right-hand side are found. The latter symmetries are used to construct an invariant fundamental solution of the original equation in terms of elementary functions.

In physical and engineering applications, an important task is the processing of experimental curves measured with considerable errors. Such problems are solved by applying the regularization method, in which success relies heavily on the researcher’s intuition. We propose using an approximation based on the double period method designed for smooth aperiodic functions. Regularization makes use of a Tikhonov stabilizer with the second derivative squared. As a result, the spurious oscillations are suppressed and the shape of an experimental curve is well approximated. This approach makes it possible to solve broad classes of problems in a unified manner. The method is demonstrated as applied to the approximation of cross sections of nuclear reactions important for controlled thermonuclear fusion. Tables recommended as reference data are obtained.

The boundary control of vibrations of a plane membrane is considered. A constraint is imposed on the absolute value of the control function. The goal of the control is to drive the membrane to rest. The proof technique used in this paper can be applied to a membrane of any dimension, but the two-dimensional case is considered for simplicity and illustrative purposes.