Том 100, № 1 (2019)

A sharp integral inequality is proved that is used to derive a Sobolev interpolation inequality. A generalization of the logarithmic Sobolev inequality is proposed based on the Sobolev interpolation inequality.

We consider the problem of partitioning a finite set of points in Euclidean space into clusters so as to minimize the sum, over all clusters, of the intracluster sums of the squared distances between cluster elements and their centers. The centers of some clusters are given as input, while the other centers are defined as centroids (geometric centers). It is well known that the general case of the problem is strongly NP-hard. In this paper, we have shown that there exists an exact polynomial-time algorithm for the one-dimensional case of the problem.

А high excursion probability for the modulus of a Gaussian vector process with independent identically distributed components is evaluated. It is assumed that the components have means zero and variances reaching its absolute maximum at a single point of the considered time interval. An important example of such processes is the Bessel process.

In this paper we propose a new method of constructing examples of nonuniqueness of probability solutions by reducing the stationary Fokker–Planck–Kolmogorov equation to a degenerate elliptic equation on a bounded domain.

New sufficient second-order optimality conditions for equality constrained optimization problems are proposed, which significantly strengthen and complement classical ones and are constructive. For example, they establish the equivalence between sufficient conditions for inequality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the former to equalities via introducing slack variables. Previously, in the case of classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete. Accordingly, the proposed optimality conditions complement the classical ones and solve the issue of equivalence between inequality and equality constrained problems when the former is reduced to the latter by introducing slack variables.

In 1932, E. Cartan described holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, but a similar study in the three-dimensional case remains incomplete. In a series of works performed by several international teams, the problem is reduced to describing homogeneous surfaces that are nondegenerate in the sense of Levi and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. Given in this paper, the complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space includes examples of new homogeneous hypersurfaces. These results bring us closer to the completion of a large-scale scientific study that is of interest in various branches of mathematics.

In the paper, a description of the Grothendieck–Serre duality on an arithmetic surface by means of fixing a horizontal divisor is given and this description is applied to the generalization of theta-invariants.

Generalized functions with a Laplace transform having a bounded argument in a tube domain over the positive orthant are considered. Necessary and sufficient conditions for the existence of quasi-asymptotics of such functions are found. A regularly varying function with respect to which such quasi-asymptotics exist is explicitly given. It turns out that the modulus of a holomorphic function in a tube domain over the positive orthant in the purely imaginary subspace on rays entering the origin behaves as a regularly varying function. The obtained results are used to find quasi-asymptotics of solutions to the generalized Cauchy problem for convolution equations with kernels being passive operators. Multidimensional passive operators and systems are often encountered in mathematical physics. Examples are operators that are hyperbolic with respect to a cone, transport equations, equations for complex electric circuits without energy pumping, the Maxwell and Dirac equations, etc.

An asynchronous threshold network is a network of threshold elements (agents) that operate in continuous time. The agents can be in one of two states: active or passive. An active agent generates a signal of certain sort (color) and power. This signal is received by all agents that have inputs of the same color. An agent has a potential that changes under exciting or inhibiting effects of signals; it is active only if its potential exceeds a threshold. Changes in agent activity are events that divide a continuous timeline into discrete time steps. The dependence of the behavior of an autonomous network on the values of its parameters is studied.

A new formalization of the accelerometer unit calibration problem is proposed within the framework of a guaranteeing approach to estimation. This problem reduces to an analysis of special variational problems. Based on the new formalization, the scalarization method is justified, which is widely used to calibrate accelerometer units. In particular, the limit of its applicability is determined.