Vol 212, No 9 (2021)

In this paper, we construct complete sets of polynomials in bi-involution on nilpotent Lie algebras of dimension 7 in the list due to Gong. Thus we verify the generalized Mishchenko-Fomenko conjecture for all algebras in this list.Bibliography: 14 titles.

A nonlinear conflict control system in a finite-dimensional Euclidean space on a finite time interval is considered. Two interrelated game-theoretic problems of making a system approach a compact set at a fixed moment of time are studied. A method for constructing approximate solutions to game problems of approach is presented. Most attention is paid to problems related to constructing approximations of the solvability sets of game problems in the phase space. Bibliography: 35 titles.

We propose and justify an algorithm for producing Hermite-Pade polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,…,f_m]$, $m\geq1$, about the point $z=0$ ($f_j\in\mathbb{C}[[z]]$) under the assumption that the series have a certain (‘general position’) nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for constructing Pade polynomials (for $m=1$ our algorithm coincides with the Viskovatov algorithm).The algorithm is based on a recurrence relation and has the following feature: all the Hermite-Pade polynomials corresponding to the multi-indices $(k,k,k,…,k,k)$, $(k+1,k,k,…,k,k)$, $(k+1,k+1,k,…,k,k)$, …, $(k+1,k+1,k+1,…,k+1,k)$ are already known at the point when the algorithm produces the Hermite-Pade polynomials corresponding to the multi-index $(k+1,k+1,k+1,…,k+1,k+1)$.We show how the Hermite-Pade polynomials corresponding to different multi-indices can be found recursively via this algorithm by changing the initial conditions appropriately.At every step $n$, the algorithm can be parallelized in $m+1$ independent evaluations. Bibliography: 30 titles.

Galton-Watson random forests with a given number of root trees and a known number of nonroot vertices are investigated. The distribution of the number of direct offspring of each particle in the forest-generating process is assumed to have infinite variance. Branching processes of this kind are used successfully to study configuration graphs aimed at simulating the structure and development dynamics of complex communication networks, in particular the internet. The known relationship between configuration graphs and random forests reflects the local tree structure of simulated networks. Limit theorems are proved for the maximum size of a tree in a random forest in all basic zones where the number of trees and the number of vertices tend to infinity. Bibliography: 14 titles.