Doklady Mathematics

Observation and control problems in Banach (B)-spaces are investigated. A criterion for controllability and optimal controllability is formulated on the basis of the BUME method and the monotone mapping method. An inverse controllability problem is introduced, and an abstract maximum principle is formulated in (B)-spaces. For PDE in Hilbert (H)-spaces and for ODE in \({{\mathbb{R}}^{n}}\), an integral maximum principle is proved and an optimality system is presented.

A combined scheme for the discontinuous Galerkin (DG) method is proposed. This scheme monotonically localizes the fronts of shock waves and simultaneously maintains increased accuracy in the regions of smoothness of the computed weak solutions. In this scheme, a nonmonotone version of the third-order DG method is used as a baseline scheme and a monotone version of this method is used as an internal scheme, in which a nonlinear correction of numerical fluxes is used. Tests demonstrate the advantages of the new scheme as compared to standard monotonized variants of the DG method.

A numerically implementable model of a three-dimensional homogeneous random field in a “horizontal” layer 0 < z < H is constructed assuming that the integral of the field with respect to the “vertical” coordinate z has a given infinitely divisible one-dimensional distribution and a given correlation function. An aggregate of n independent elementary horizontal layers of thickness h = H/n shifted vertically by a random variable uniformly distributed in the interval (0, h) is considered as a basic model.

According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in independent random variables given another. We prove an analogue of this theorem for linear forms in two independent random variables with values in an a-adic solenoind without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of the a-adic solenoid.

We consider the problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum (over both clusters) of the intracluster sums of the squared distances between the cluster elements and their centers. The center of one cluster is defined as a centroid (geometric center). The center of the other cluster is determined as an optimized point in the input set. We analyze the variant of the problem with given cluster sizes such that their sum is equal to the size of the input set. The strong NP-hardness of this problem is proved.

The Dirichlet problem for an elliptic functional differential equation involving a shifted and contracted argument under the Laplacian sign is studied. Sufficient conditions for the unique solvability of the problem are established. It is shown that the problem may have an infinite-dimensional solution manifold.

Polynomial identities and codimension growth of nonassociative algebras over a field of characteristic zero are considered. A new approach is proposed for constructing nonassociative algebras starting from a given infinite binary word. The sequence of codimensions of such an algebra is closely connected with the combinatorial complexity of the defining word. These constructions give new examples of algebras with abnormal codimension growth. The first important achievement of the given approach is that the algebras under study are finitely generated. The second one is that the asymptotic behavior of codimension sequences is widely different from all previous examples.

Fractional Brownian motion is studied. Statistical estimators of the Hurst exponent are proposed, and their properties are examined. This stochastic process is widely used in model development, trend forecasting, and, in particular, as a special case of long-memory processes. The first model involving the Hurst exponent appeared in the British hydrologist Harold Hurst’s research published in 1951, where he analyzed the flow of the Nile River. Later, an improved model of fractional Brownian motion was widely used in different financial market studies. Since such stochastic processes are of great interest, the extrapolation of fractional Brownian motion and the point estimation of the Hurst exponent H have become important problems. A new approach to the point estimation of the Hurst exponent is proposed in this article.

A model was constructed for reconstructing the initial stage of solidification of binary alloys treated as a nonequilibrium phase transition with a diffusion stratification mechanism. Numerical experiments concerning the self-excitation of a homogeneous state by applying a melt cooling boundary control condition were performed.

Today, the exploration of the Arctic region is one of the most important direction of research in our country, because large amounts of unexplored oil and gas deposits are located there. Large hydrocarbon deposits are situated within the Northern seas. Their development is complicated by gas explosions resulting from an accidental opening and further spread of the gas. Since the region with gas layers cannot be frequently monitored, an area with already detected gas deposits is numerically modeled instead. In this work, seismic waves propagating in multilayered geological models with gas-containing inclusions were numerically modeled over a four-year interval by applying the grid-characteristic method. Wave patterns of seismic reflections and seismograms for the described problem were obtained. The wave patterns and seismograms obtained in the two- and three-dimensional cases were compared. The results were found to be in good agreement.

New equations for Laplace transform inversion are obtained. The equations satisfy the causality principle. The impulse response of a channel is determined in order to analyze dispersion distortions in inhomogeneous media. The impulse response excludes the possibility that the signal exceeds the speed of light in the medium. The transmission bandwidth, the angular spectrum, and the Doppler shift in the ionosphere are computed.