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Vol 212, No 9 (2021)
- Year: 2021
- Articles: 8
- URL: https://bakhtiniada.ru/0368-8666/issue/view/7481
Complete sets of polynomials in bi-involution on nilpotent seven-dimensional Lie algebras
Abstract
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.
Matematicheskii Sbornik. 2021;212(9):3-17
3-17
An eigenfunction manifold generated by a family of periodic boundary value problems
Abstract
An analytic and topological description is given of the manifold of periodic eigenfunctions generated by the space of one-dimensional stationary Schrödinger equations with periodic real potentials. Connections with results due to Neuman, Ince and Uhlenbeck are discussed. Bibliography: 11 titles.
Matematicheskii Sbornik. 2021;212(9):18-39
18-39
Two game-theoretic problems of approach
Abstract
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.
Matematicheskii Sbornik. 2021;212(9):40-74
40-74
On irregular Sasaki-Einstein metrics in dimension $5$
Abstract
We show that there are no irregular Sasaki-Einstein structures on rational homology 5-spheres. On the other hand, using $\mathrm{K}$-stability we prove the existence of continuous families of nontoric irregular Sasaki-Einstein structures on odd connected sums of $S^2 \times S^3$.Bibliography: 30 titles.
Matematicheskii Sbornik. 2021;212(9):75-93
75-93
A Viskovatov algorithm for Hermite-Pade polynomials
Abstract
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.
Matematicheskii Sbornik. 2021;212(9):94-118
94-118
Rigid germs of finite morphisms of smooth surfaces and rational Belyi pairs
Abstract
In the paper “On rigid germs of finite morphisms of smooth surfaces” (Sb. Math., 211:10 (2020), 1354–1381), we defined a map $\beta\colon{\mathcal R\to\mathcal{B}el}$ from the set $\mathcal R$ of equivalence classes of rigid germs of finitemorphisms branched in germs of curves having $ADE$ singularity types onto the set $\mathcal{B}el$ of rational Belyi pairs $f\colon\mathbb P^1 {\to} \mathbb P^1$, considered up to the action of $\mathrm{PGL}(2,\mathbb C)$. In this article the inverse images of this map are investigated in terms of monodromies of Belyi pairs.Bibliography: 7 titles.
Matematicheskii Sbornik. 2021;212(9):119-145
119-145
The maximum tree of a random forest in the configuration graph
Abstract
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.
Matematicheskii Sbornik. 2021;212(9):146-163
146-163
Letter to the editors
Matematicheskii Sbornik. 2021;212(9):164-164
164-164