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Vol 212, No 2 (2021)

Year: 2021
Articles: 6
URL: https://journal-vniispk.ru/0368-8666/issue/view/8373

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Ramification filtration via deformations

Abrashkin V.A.

Abstract

Let $\mathscr K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$, $\mathscr G_{< p}$ the maximal quotient of the Galois group of $\mathscr K$ of period $p$ and nilpotency class $< p$ and {$\{\mathscr G_{< p}^{(v)}\}_{v\geqslant 1}$} the filtration by ramification subgroups in the upper numbering. Let $\mathscr G_{< p}=G(\mathscr L)$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathscr L)$ is the group obtained from a suitable profinite Lie $\mathbb{F}_p$-algebra $\mathscr L$ via the Campbell-Hausdorff composition law. We develop a new technique for describing the ideals $\mathscr L^{(v)}$ such that $G(\mathscr L^{(v)})=\mathscr G_{< p}^{(v)}$ and constructing their generators explicitly. Given $v_0\geqslant 1$, we construct an epimorphism of Lie algebras $\overline\eta^{\dagger}\colon \mathscr L\to \overline{\mathscr L}^{\dagger}$ and an action $\Omega_U$ of the formal group of order $p$, $\alpha_p=\operatorname{Spec}\mathbb{F}_p[U]$, $U^p=0$, on $\overline{\mathscr L}^{\dagger}$. Suppose $d\Omega_U=B^{\dagger}U$, where $B^{\dagger}\in\operatorname{Diff}\overline{\mathscr L}^{\dagger}$, and $\overline{\mathscr L}^{\dagger}[v_0]$ is the ideal of $\overline{\mathscr L}^{\dagger}$ generated by the elements of $B^{\dagger}(\overline{\mathscr L}^{\dagger})$. The main result in the paper states that $\mathscr L^{(v_0)}=(\overline\eta^{\dagger})^{-1}\overline{\mathscr L}^{\dagger}[v_0]$. In the last sections we relate this result to the explicit construction of generators of $\mathscr L^{(v_0)}$ obtained previously by the author, develop a more efficient version of it and apply it to recover the whole ramification filtration of $\mathscr G_{< p}$ from the set of its jumps.
Bibliography: 13 titles.

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Matematicheskii Sbornik. 2021;212(2):3-37

3-37

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On the phenomenon of the support shrinking of a solution with a time delay and on the extinction of the solution

Degtyarev S.P.

Abstract

The phenomenon of support shrinking with a time delay for the solution of a doubly nonlinear degenerate parabolic equation is studied in the case of slow diffusion and strong absorption. For a nonnegative solution, a sufficient condition for support shrinking beginning with some moment of time is deduced in terms of the local behaviour of the mass of the initial datum. It is also proved that the solution vanishes identically in finite time.Bibliography: 21 titles.

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Matematicheskii Sbornik. 2021;212(2):38-52

38-52

(Rus)

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Polyhomomorphisms of locally compact groups

Neretin Y.A.

Abstract

Let $G$ and $H$ be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism $G\rightarrowtail H$ is a closed subgroup $R\subset G\times H$ with fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the Haar measures on $G$ and $H$. A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with ‘uniform’ measures. For two polyhomomorphisms $G\rightarrowtail H$ and $H\rightarrowtail K$ there is a well-defined product $G\rightarrowtail K$. The set of polyhomomorphisms $G\rightarrowtail H$ is a metrizable compact space with respect to the Chabauty-Bourbaki topology and the product is separately continuous. A polyhomomorphism $G\rightarrowtail H$ determines a canonical operator $L^2(H)\to L^2(G)$, which is a partial isometry up to a scalar factor. For example, we consider locally compact linear spaces over finite fields and examine the closures of groups of linear operators in semigroups of polyhomomorphisms. Bibliography: 40 titles.

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Matematicheskii Sbornik. 2021;212(2):53-80

53-80

(Rus)

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Topological analysis of a billiard bounded by confocal quadrics in a potential field

Pustovoitov S.E.

Abstract

Consider a billiard in a plane domain bounded by confocal ellipses and hyperbolae. A Hooke potential acts on a point mass. This dynamical systems turns out to be completely Liouville integrable. A topological analysis of the Liouville foliation of isoenergy manifolds at all possible levels of the Hamiltonian is performed and the complete Fomenko-Zieschang invariants (marked molecules) of these manifolds are constructed. Bibliography: 15 titles.

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Matematicheskii Sbornik. 2021;212(2):81-105

81-105

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Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

Runovskii K.V.

Abstract

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric polynomials in the $L_p$-spaces, $1\le p\le+\infty$. As corollaries, estimates for the quality of approximation by Fourier means, an inverse theorem of approximation theory, comparison theorems, an analogue of the Marchaud inequality for generalized moduli of smoothness defined by a periodic generator, as well as some constructive sufficient conditions for generalized smoothness and Bernstein type inequalities for generalized derivatives of trigonometric polynomials are obtained. Bibliography: 49 titles.

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Matematicheskii Sbornik. 2021;212(2):106-137

106-137

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Maximal Lie subalgebras among locally nilpotent derivations

Skutin A.A.

Abstract

We study maximal Lie subalgebras among locally nilpotent derivations of the polynomial algebra. Freudenburg conjectured that the triangular Lie algebra of locally nilpotent derivations of the polynomial algebra is a maximal Lie algebra contained in the set of locally nilpotent derivations, and that every maximal Lie algebra contained in the set of locally nilpotent derivations is conjugate to the triangular Lie algebra. In this paper we prove the first part of the conjecture and present a counterexample to the second part. We also show that under a certain natural condition imposed on a maximal Lie algebra there is a conjugation taking this Lie algebra to the triangular Lie algebra. Bibliography: 2 titles.

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Matematicheskii Sbornik. 2021;212(2):138-146

138-146

(Rus)

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