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Volume 213, Nº 9 (2022)
- Ano: 2022
- Artigos: 5
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7493
Coincidence of set functions in quasiconformal analysis
Resumo
It is known that mappings occurring in quasiconformal analysis can be defined in several equivalent ways: 1) as homeomorphisms inducing bounded composition operators between Sobolev spaces; 2) as Sobolev-class homeomorphisms with bounded distortion whose operator distortion function is integrable; 3) as homeomorphism changing the capacity of the image of a condenser in a controllable way in terms of the weighted capacity of the condenser in the source space; 4) as homeomorphism changing the modulus of the image of a family of curves in a controllable way in terms of the weighted modulus of the family of curves in the source space. A certain set function, defined on open subsets, can be associated with each of these definitions. The main result consists in the fact that all these set functions coincide. Bibliography: 48 titles.
Matematicheskii Sbornik. 2022;213(9):3-33
3-33
Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology
Resumo
We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of these billiard systems in terms of Fomenko invariants. Then we provide periodicity conditions in terms of functional Pell equations and related extremal polynomials.Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases of billiards on the Minkowski and Euclidean planes.Dedicated to R. Baxter on the occasion of his 80th anniversary.Bibliography: 51 titles.
Matematicheskii Sbornik. 2022;213(9):34-69
34-69
Distribution of Korobov-Hlawka sequences
Resumo
Let $a_1, …, a_s$ be integers and $N$ be a positive integer. Korobov (1959) and Hlawka (1962) proposed to use the points$$x^{(k)}=(\{\frac{a_1 k}N\}, …, \{\frac{a_1 k}N\}),\qquad k=1,…, N,$$as nodes of multidimensional quadrature formulae. We obtain some new results related to the distribution of the sequence $K_N(a)=\{x^{(1)},…,x^{(N)}\}$. In particular, we prove that$$\frac{\ln^{s-1} N}{N \ln\ln N} \underset{s}\ll D(K_N(a))\underset{s}\ll \frac{\ln^{s-1} N}{N} \ln\ln N$$for ‘almost all’ $a\in (\mathbb Z_N^*)^s$, where $D(K_N(a))$ is the discrepancy of the sequence $K_N(a)$ from the uniform distribution and $\mathbb Z^*_N$ is the reduced system of residues modulo $N$.Bibliography: 18 titles.
Matematicheskii Sbornik. 2022;213(9):70-96
70-96
Solomyak-type eigenvalue estimates for the Birman-Schwinger operator
Resumo
We revise the Cwikel-type estimate for the singular values of the operator $(1-\Delta_{\mathbb{T}^d})^{-d/4}M_f(1-\Delta_{\mathbb{T}^d})^{-d/4}$ on the torus $\mathbb{T}^d$, for the ideal $\mathcal{L}_{1,\infty}$ and $f\in L\log L(\mathbb{T}^d)$ (the Orlicz space), which was established by Solomyak in even dimensions, and we extend it to odd dimensions. We show that this result does not literally extend to Laplacians on $\mathbb{R}^d$, neither for Orlicz spaces on $\mathbb{R}^d$, nor for any symmetric function space on $\mathbb{R}^d$. Nevertheless, we obtain a new positive result on (symmetrized) Solomyak-type estimates for Laplacians on $\mathbb{R}^d$ for an arbitrary positive integer $d$ and $f$ in $L\log L(\mathbb{R}^d)$. The last result reveals the conformal invariance of Solomyak-type estimates. Bibliography: 44 titles.
Matematicheskii Sbornik. 2022;213(9):97-137
97-137
Proper cyclic symmetries of multidimensional continued fractions
Resumo
We show that palindromic continued fractions exist in an arbitrary dimension. For dimension $n=4$ we also prove a criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry. Klein polyhedra are considered as multidimensional generalizations of continued fractions. Bibliography: 11 titles.
Matematicheskii Sbornik. 2022;213(9):138-166
138-166