卷 98, 编号 2 (2018)

A subgroup H of a group G is called pronormal if, for any element g of G, the subgroups H and H^g are conjugate in the subgroup they generate. Some problems in the theory of permutation groups and combinatorics have been solved in terms of pronormality, and the characterization of pronormal subgroups in finite groups is a problem of importance for applications of group theory. A task of special interest is the study of pronormal subgroups in finite simple groups and direct products of such groups. In 2012 E.P. Vdovin and D.O. Revin conjectured that the subgroups of odd index in all finite simple groups are pronormal. We disproved this conjecture in 2016. Accordingly, a natural task is to classify finite simple groups in which the subgroups of odd index are pronormal. This paper completes the description of finite simple groups whose Sylow 2-subgroups contain their centralizers in the group and the subgroups of odd index in which are pronormal.

The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established.

The concept of an Ω-weakly regular problem is introduced. On the basis of the Zhuravlev operator approach combined with the neural network paradigm, it is shown that, for each such problem, a correct algorithm and a six-level spatial neural network reproducing the computations executed by this algorithm can be constructed. Moreover, the set of Ω-weakly regular problems includes the set of Ω-regular problems. It turns out that a three-level spatial network (μ-block) is a forward propagation network whose inner loop under estimation of the class membership for each test object consists of a single iteration. As a result, the amount of computations required for the six-level network is reduced noticeably.

We give a description of the cubic polynomials f(x) with coefficients in the quadratic number fields \(\mathbb{Q}(\sqrt{5})\) and \(\mathbb{Q}(\sqrt{-15})\) for which the continued fraction expansion of the irrationality \(\sqrt {f\left( x \right)} \) is periodic.

Random changes in the standard genetic code in which its structure is preserved are considered. The hypothesis that the best possibility of overlapping genes is provided by the standard genetic code of given structure is justified. This code is determined by the condition that the total number of admissible substitutions opening closed reading frames is maximal, while the number of closed reading frames for which there is no such substitution is minimal. In terms of this criterion, the estimated probability of a random choice of the standard genetic code is less than 10^–6.

Randomized Monte Carlo algorithms intended for statistical kernel estimation of the averaged solution to a problem with random baseline parameters are optimized. For this purpose, a criterion for the complexity of a functional Monte Carlo estimate is formulated. The algorithms involve a splitting method in which, for each realization of the parameters, a certain number of trajectories of the corresponding baseline process are constructed.

An explicit two-level in time and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L²-dissipativity of solutions to the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity and depending on the Mach number. It is shown that the condition on the Courant number can be made independent of the Mach number. The results provide a substantial development of the well-known stability analysis of the linearized Lax–Wendroff scheme.

We have found an infinite family of nonisomorphic hyperelliptic curves of genus two over the field of rational numbers whose Jacobian varieties contain rational points of order 28. Previously, only 10 such curves were known.

The problem of uniqueness of probability solutions to the two-dimensional stationary Fokker–Planck–Kolmogorov equation is considered. Under broad conditions, it is proved that the existence of two different probability solutions implies the existence of an infinite set of linearly independent probability solutions.

The eigenvalues and eigenfunctions of certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue squareintegrable functions on an interval are explicitly calculated, while the resolvents of these operators are integral operators with kernels for which the theorem on an eigenfunction expansion holds. In addition, each of these kernels is the Green’s function of a self-adjoint boundary value problem, and the procedure for its construction is well known. Thus, the Green’s functions of these problems can be expanded in series in terms of eigenfunctions. In this study, identities obtained by this method are used to calculate the sums of convergent number series and to represent the sums of certain power series in an intergral form.

For Monte Carlo estimators, a variance reduction method based on empirical variance minimization in the class of functions with zero mean (control functions) is described. An upper bound for the efficiency of the method is obtained in terms of the properties of the functional class.

Integro-differential equations with unbounded operator coefficients in a Hilbert space are considered. Such equations arise in viscoelasticity theory, thermal physics, and homogenization problems in multiphase media. Initial–boundary value problems for the indicated equations are proved to be well posed, and their spectral analysis is performed.

A sample X₁,...,X_n consisting of independent identically distributed vectors in ℝ^p with zero mean and a covariance matrix Σ is considered. The recovery of spectral projectors of high-dimensional covariance matrices from a sample of observations is a key problem in statistics arising in numerous applications. In their 2015 work, V. Koltchinskii and K. Lounici obtained nonasymptotic bounds for the Frobenius norm \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector P_r were constructed assuming that the moment characteristics of the observations are known. This paper describes a bootstrap procedure for constructing confidence sets for the spectral projector P_r of the covariance matrix Σ from given data. This approach does not use the asymptotical distribution of \(\parallel {P_r} - {\hat P_r}{\parallel _2}\) and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.

In the six-dimensional (6D) phase space, a new statement of the problem of calculating the electric current due to the pressure gradient in a toroidal plasma is considered. A semi-Lagrangian approach is applied. A new efficient parallel algorithm is developed for the numerical solution of a 6D kinetic equation with the Coulomb collision operator. By the DiFF-PK code developed by the authors, a 5D problem is solved under the conditions of ITER-scale facilities. The bootstrap current of electrons is calculated. A good agreement with the previously known limit cases is demonstrated. The method proposed can be applied to the high-precision computation of a large class of problems whose solution within existing approaches is complicated. For these purposes, the mini-supercomputer at the Research Institute for Systems Analysis, Russian Academy of Sciences, or pentaflop class supercomputers can be used. The problems include, for example, the calculation of the bootstrap current of alpha particles–a product of thermonuclear fusion, the calculation of the electric current produced by neutrals injected into a plasma, and simulation of radial electric fields and instabilities of plasma.

Mathematical models for practical calculations of technological progress (total productivity of production factors) and economic growth in the era of widespread digitalization and robotization of national economies, where the main factor of production is technological information, are developed and verified. For this purpose, models using different modes of information production are proposed for the first time. It is shown that the economic effect of the digitalization of an economy will not come immediately, but with a lag of about eight years. For the US economy, forecast calculations show that this will happen in 2022–2026 with total productivity increasing by 1.1 percentage points up to 2.5% per year.

A two-body system consisting of a rigid body with a linear oscillator attached to it is considered. The body moves along a horizontal line under the action of a control force and a small unknown disturbance. The phase state of the oscillator is assumed to be not available for measurement. A bounded feedback control is proposed which brings the body to a prescribed terminal state in a finite time.