Том 98, № 1 (2018)

A simple method is given for a priori determination of the number of Jordan blocks and their orders in the regularizing decomposition of a square matrix. This decomposition was proposed by R. Horn and V. Sergeichuk. It is attained by congruence transformations and makes it possible to separate the regular and singular parts of the matrix.

A variant of the programmed iteration method is developed for solving game problems of realizing trajectories of a nonlinear conflict controlled system in a given set of functions. Solving these open-loop control problems yields an iterative process in a space of sets whose elements are functional positions, and the corresponding limit determines the set of successful solvability of the original problem in the class of set-valued quasi-strategies (nonanticipating responses to disturbances). For the conflict controlled system, the conditions of generalized uniqueness and uniform boundedness of programmed motions are assumed to hold. The use of an infinite-dimensional space of functional positions is essential, since the arising differential game generally does not satisfy the alternative solvability conditions in classes of feedback strategies.

The squared absolute value of the Dedekind zeta functions of imaginary quadratic fields of class number one on the critical line is expressed in terms of averaged values associated with the spectrum of the automorphic Laplacian with respect to the full modular group.

Number theory methods are used to construct quadrature formulas for exactly integrating trigonometric polynomials in several variables of degree as high as possible for a given number of integration nodes.

Stable rational cohomology groups of spaces of non-resultant homogeneous polynomial systems of growing degree in ℝⁿ are calculated.

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.

The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculated, in fact, using the same formulas (by integrating the Fejér kernels), so this ergodic theorem is, in fact, a statement about the asymptotics of the growth of the Fejér sums at zero for the spectral measure of the corresponding dynamical system. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates of the Fejér sums at the point for periodic measures. For example, natural criteria for the polynomial growth and polynomial decrease in these sums can be obtained. On the contrary, available in the literature, numerous estimates for the deviations of Fejér sums at a point can be used to obtain new estimates for the rate of convergence in this ergodic theorem.

The quotient space of an arbitrary Banach space B by any subspace B₁ ⊂ B equipped with the norm \(||b|B/{B_1}|| = \mathop {\inf }\limits_{f \in B/{B_1}} ||f|b||\) is considered. In the case where the infimum is a minimum, i.e., it is attained at some element, a formula for this element is presented. The proof is based on restating the original problem in dual spaces with the help of corresponding Legendre transforms. Although the original problem is nonlinear, it is found that its formulation in dual spaces is always linear and solvable. The results are applied to the general theory of boundary value problems for differential equations of mathematical physics.

A sub-Lorentzian area formula is proved for graph mappings constructed on the basis of intrinsically Lipschitz mappings defined on two-step Carnot groups.

Spectral asymptotics for a family of finite-dimensional perturbations of operators of trace class are found. The results are used to find exact asymptotics of small ball probabilities in the Hilbert norm for finitedimensional perturbations of Gaussian functions. As an example, Durbin processes appearing in the study of empirical processes with estimated parameters are considered.

A complete description of spaces associated with weighted Sobolev spaces of the first order on the real line with identical summation parameters for a function and its derivative is presented.

The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise C₂-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky plane. Namely, we obtain a complete classification of billiards for which the billiard geodesic flow has a nontrivial first integral depending polynomially on the velocity. According to this classification, every polynomially integrable convex bounded planar billiard with C²-smooth boundary is an ellipse. This is a joint result of M. Bialy, A.E. Mironov, and the author. The proof consists of two parts. The first part was given by Bialy and Mironov in their two joint papers, where the result was reduced to an algebraic-geometric problem, which was partially studied there. The second part is the complete solution of the algebraic-geometric problem presented below.

The influence exerted by the second time derivative with a small parameter added to the heat equation in the case of discontinuous periodic initial data is investigated. It is shown that, except for the initial instants of time, the error of hyperbolization vanishes as the square root of the addition.

Ultrasonic wave propagation in steel rails with explicit identification of flaws is numerically simulated. The problem is to detect a vertical crack in a railhead by applying ultrasonic nondestructive testing techniques. The propagation of elastic waves in the rail profile is simulated for various sizes and positions of the crack. It is shown that the finite-difference grid-characteristic method in the time domain and full-wave simulation can be used to analyze the effectiveness of rail flaw detection by applying ultrasonic nondestructive testing techniques. Full-wave simulation is also used to demonstrate the failure of the widely used echo-mirror method to detect flaws of certain types. It is shown that techniques for practical application of the ultrasonic delta method can be developed using full-wave supercomputer simulation. The study demonstrates a promising potential of geophysical methods as adapted to the analysis of ultrasonic nondestructive testing results.