Vol 99, No 1 (2019)

We consider the problem of obtaining the restrictions on the zero set of an entire function of exponential type under which this function belongs to the Schwartz algebra and invertible in the sense of Ehrenpreis.

A.N. Krylov’s ideas concerning convergence acceleration for Fourier series are used to obtain explicit expressions for the classical solution of a mixed problem for a nonhomogeneous equation and explicit expressions for a weak solution in the case of arbitrary summable \(q(x)\), \(\varphi (x)\), \(\psi (x)\), and \(f(x,t)\).

A double asymptotic expansion (with respect to smoothness and low viscosity) of the resolving operator of the Cauchy problem for the linearized system of gas dynamics is obtained. Estimates for the summands and the remainder are derived in the Sobolev scale. Hydrodynamic and acoustic modes are explicitly described.

A scale of mappings that depends on two real parameters \(p,q\) (\(n - 1 \leqslant q \leqslant p < \infty \)) and a weight function \(\theta \) is defined. In the case \(q = p = n,\)\(\theta \equiv 1,\) well-known mappings with bounded distortion are obtained. The mappings of a two-index scale inherit many properties of mappings with bounded distortion. They are used to solve several problems in global analysis and applied problems.

For spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space, an embedding theorem into spaces of the same type is proved and related results are presented.

We consider symmetric random matrices \({{{\mathbf{X}}}_{n}} = [{{X}_{{jk}}}]_{{j,k = 1}}^{n},n \geqslant 1\), whose upper triangular entries are independent random variables with zero mean and unit variance. Under the assumption \(\mathbb{E}{\text{|}}{{X}_{{jk}}}{{{\text{|}}}^{4}} < C\), j, k = 1, 2, ..., n, it is shown that the fluctuations of the Stieltjes transform m_n(z), \(z = u + i{v},{v} > 0,\) of the empirical spectral distribution function of the matrix \({{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n \) about the Stieltjes transform \({{m}_{{{\text{sc}}}}}(z)\) of Wigner’s semicircle law are of order (n\({v}\))\(^{{ - 1}}\text{ln}n\). An application of the result obtained to the convergence rate in probability of the empirical spectral distribution function of \({{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n \) to Wigner’s semicircle law in the uniform metric is discussed.

Let ξ ₁, ξ₂, ξ₃ be independent random variables with values in a locally compact Abelian group X with nonvanishing characteristic functions, and a_j, b_j be continuous endomorphisms of X satisfying some restrictions. Let L₁ = a₁ξ ₁+ a₂ ξ₂ + a₃ξ ₃, L₂ = b₁ξ ₁ + b₂ξ ₂ + b₃ξ ₃. It was proved that the distribution of the random vector (L₁, L₂ ) determines the distributions of the random variables ξ_j up a shift. This result is a group analogue of the well-known C.R. Rao theorem. We also prove an analogue of another C.R. Rao's theorem for independent random variables with values in an a-adic solenoid.

The properties of a statistic called a self-consistent stationary level of nonstationary time series are examined. It is shown that a change in this statistic can be treated as a disorder in the nonstationarity properties of the series. The significance level of decision making is estimated, characteristic periods in a nonstationary stochastic process are detected, and an optimal sample size for constructing indicators in stochastic control problems is determined. A disorder indicator for electroencephalogram data from epilepsy patients is studied as a practical application.

A method for efficient comparison of a symbol sequence with all strings of a set is presented, which performs considerably faster than the naive enumeration of comparisons with all strings in succession. The procedure is accelerated by applying an original algorithm combining a prefix tree and a standard dynamic programming algorithm searching for the edit distance (Levenshtein distance) between strings. The efficiency of the method is confirmed by numerical experiments with arrays consisting of tens of millions of biological sequences of variable domains of monoclonal antibodies.

For some classes of Lévy processes, the notion of reflection from an interval boundary is introduced. It is shown that trajectories of a reflecting process define random operators that map functions defined on the interval boundaries into elements of \({{L}_{2}}\) on the whole interval.

The system of pressureless gas dynamics is a hydrodynamically justified generalization of the system consisting of the Burgers vector equation in the limit of vanishing viscosity and the mass conservation law. The latter system of equations was intensively used, in particular, in astrophysics to describe the large-scale structure of the Universe. The solutions of the vector Burgers equation involve interesting dynamics of singularities, which can describe concentration processes. However, this dynamics does not satisfy the law of momentum conservation, which prevents us from treating it as dynamics of material objects. In this paper, momentum-conserving dynamics of singularities is investigated on the basis of the pressureless gas dynamics system. Such dynamics turns out to be more diverse and complex, but it is also possible to formulate a variational approach, for which the basic principles and relations are obtained in the work.

The first mixed problem for the Vlasov–Poisson system in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a high-temperature two-component plasma under an external magnetic field. For an arbitrary electric field potential and a sufficiently strong external magnetic field, it is shown that the characteristics of the Vlasov equations do not reach the boundary of the cylinder. It is proved that the Vlasov–Poisson system with ion and electron distribution density functions supported at some distance from the cylinder boundary has a unique classical solution.

The local dynamics of a well-known model of an optoelectronic oscillator with delayed feedback is studied. In a neighborhood of zero equilibrium, normalized equations are constructed which are boundary value problems depending on a continual parameter. Asymptotic solutions of the original nonlinear system in the form of a combination of slow and fast oscillations are obtained by solving the boundary value problems. The frequencies and amplitudes of the components of these solutions are determined.

A new generalization of the Kravchenko–Kotelnikov theorem by spectra of compactly supported infinitely differentiable functions \(h_{{\mathbf{a}}}^{{(m)}}(x)\) is considered. These functions are solutions of linear integral equations of a special form. The spectrum of \(h_{{\mathbf{a}}}^{{(m)}}(x)\) is a multiple infinite product of the spectra of the atomic functions \({{h}_{a}}(x)\) dilated with respect to the argument. The resulting generalized series is characterized by fast convergence, which is confirmed by the truncation error bound presented in the study and by the results of a numerical experiment.

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and mean-square distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

The inversion of a linear time-invariant time-delay system is considered. For such systems, a canonical form with isolated zero dynamics is obtained, system invariant zeros are investigated, and their relation to the spectral observability of the zero-dynamics subsystem is described. Based on these results, an inversion algorithm for time-delay systems is suggested.