Open Access
Access granted
Subscription Access
Vol 212, No 12 (2021)
- Year: 2021
- Articles: 6
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7484
Topological type of isoenergy surfaces of billiard books
Abstract
The homeomorphism class of the isoenergy surface of a billiard book, of low complexity and not necessarily integrable, is determined using methods of low-dimensional topology. In particular, a series of billiard books is constructed that realize isoenergy 3-surfaces homeomorphic to the connected sum of a number of lens spaces and direct products $S^1\times S^2$.The Fomenko-Zieschang invariants, which classify Liouville foliations on isoenergy surfaces up to fibrewise homeomorphisms – that is, up to Liouville equivalence of the corresponding integrable Hamiltonian systems – are calculated for several integrable billiards of this type.Bibliography: 14 titles.
Matematicheskii Sbornik. 2021;212(12):3-19
3-19
Uniqueness theorems for simple trigonometric series with application to multiple series
Abstract
For simple trigonometric series it is shown, in particular, that if the trigonometric series is Riemann summable in measure to an integrable function $f$ and if the Riemann majorant is finite everywhere except possibly on a countable set, then this series is the Fourier series of the function $f$. Uniqueness theorems for multiple trigonometric series are obtained on the basis of this result.Bibliography: 14 titles.
Matematicheskii Sbornik. 2021;212(12):20-39
20-39
The polynomial Hermite-Pade $m$-system for meromorphic functions on a compact Riemann surface
Abstract
Given a tuple of $m+1$ germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Pade $m$-system, which includes the Hermite-Pade polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Pade $m$-system constructed from the tuple of germs of functions $1, f_1,…,f_m$ that are meromorphic on an $(m+1)$-sheeted compact Riemann surface $\mathfrak R$. We show that if $f_j = f^j$ for some meromorphic function $f$ on $\mathfrak R$, then with the help of the ratios of polynomials of the Hermite-Pade $m$-system we recover the values of $f$ on all sheets of the Nuttall partition of $\mathfrak R$, apart from the last sheet. Bibliography: 18 titles.
Matematicheskii Sbornik. 2021;212(12):40-76
40-76
Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of $\mathbb R^2$
Abstract
Criteria for the uniform approximation of functions by solutions of second-order strongly elliptic equations on compact subsets of $\mathbb R^2$ are obtained using the method of reduction to similar problems in $\mathbb R^3$, which were previously investigated by Mazalov. A number of metric properties of the capacities used are established.Bibliography: 16 titles.
Matematicheskii Sbornik. 2021;212(12):77-94
77-94
Generalization of the Artin-Hasse logarithm for the Milnor $K$-groups of $\delta$-rings
Abstract
Let $R$ be a $p$-adically complete ring equipped with a $\delta$-structure. We construct a functorial group homomorphism from the Milnor $K$-group $K^{M}_{n}(R)$ to the quotient of the $p$-adic completion of the module of differential forms $\widehat{\Omega}^{n-1}_{R}/d\widehat{\Omega}^{n-2}_{R}$. This homomorphism is a $p$-adic analogue of the Bloch map defined for the relative Milnor $K$-groups of nilpotent extensions of rings of nilpotency degree $N$ for which the number $N!$ is invertible. Bibliography: 12 titles.
Matematicheskii Sbornik. 2021;212(12):95-114
95-114
Orbit spaces for torus actions on Hessenberg varieties
Abstract
In this paper we study effective actions of the compact torus $T^{n-1}$ on smooth compact manifolds $M^{2n}$ of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to $S^{n+1} \setminus (U_1 \sqcup …\sqcup U_l)$, the complement to the union of disjoint open subsets of the $(n + 1)$-sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.
Matematicheskii Sbornik. 2021;212(12):115-136
115-136