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卷 215, 编号 12 (2024)
- 年: 2024
- 文章: 5
- URL: https://bakhtiniada.ru/0368-8666/issue/view/20343
Automorphisms of Du Val del Pezzo surfaces
摘要
We present a description for the automorphism groups of Du Val del Pezzo surfaces whose automorphism groups are infinite.Bibliography: 11 titles.
Matematicheskii Sbornik. 2024;215(12):3-29
3-29
Rate of convergence in the central limit theorem for the determinantal point process with Bessel kernel
摘要
We consider a family of linear operators diagonalized by the Hankel transform. We express explicitly the Fredholm determinants of these operators, as restricted to $L_2[0, R]$, so that the rate of their convergence as $R\to\infty$ can be found. We use the link between these determinants and the distribution of additive functionals in a determinantal point process with Bessel kernel and estimate the distance in the Kolmogorov–Smirnov metric between the distribution of these functionals and the Gaussian distribution. Bibliography: 27 titles.
Matematicheskii Sbornik. 2024;215(12):30-55
30-55
Zeros of discriminants constructed from Hermite–Pade polynomials of an algebraic function and their relation to branch points
摘要
Let $f_\infty$ be the germ at $\infty$ of some algebraic function $f$ of degree $m+1$. Let $Q_{n,j}$, $j=0,…,m$, be the Hermite–Pade polynomials of the first type of order $n\in\mathbb N$ constructed from the tuple of germs $[1, f_ \infty, f_\infty^2,…,f_\infty^m]$. We study the asymptotic properties of discriminants constructed from the Hermite–Pade polynomials in question, that is, the discriminants $D_n(z)$ of the polynomials $Q_{n,m}(z)w^m+Q_{n,m-1}(z)w^{m-1}+…+Q_{n,0}(z)$. We find their weak asymptotics, as well as the asymptotic behaviour of their ratio with the polynomial $Q_{n,m}^{2m-2}$. In addition, we refine the weak asymptotic formulae for $D_n$ at branch points of the original algebraic function $f$ and apply the results obtained to the problem of finding branch points of $f$ numerically on the basis of the prescribed germ $f_\infty$, which is used in applied problems. Bibliography: 49 titles.
Matematicheskii Sbornik. 2024;215(12):56-88
56-88
Strong asymptotics of the best rational approximation to the exponential function on a bounded interval
摘要
We apply recent findings of complex approximation theory to best rational approximation of degree $n$ to the function $\exp(-(n+\nu)x)$ on a finite interval $[0,c]$. We show that the error norm behaves like the $n$th power of the main approximation rate times the $\nu$th power of a secondary approximation rate. The computation of the first rate is a consequence of works of Gonchar, Rakhmanov and Stahl done in the 1980s; the complete asymptotic description was achieved by Aptekarev in the first years of the 21st century. The solution is given in terms of elliptic integrals of the third kind. Bibliography: 92 titles.
Matematicheskii Sbornik. 2024;215(12):89-147
89-147
Realization of permutations of even degree by products of three fixed-point-free involutions
摘要
We consider representations of a permutation $\pi$ of degree $2n$, $n\geqslant3$, by a product of three so-called pairwise-cycle permutations, all of whose cycles have length $2$. This is a valid question for even permutations if $n$ is even and for odd permutations if $n$ is odd. We prove constructively that for $n\geqslant4$, $n\neq8$, such a representation holds for all permutations $\pi$ of the same parity as $n$, apart from four exceptional conjugacy classes. For $n=8$ there are five exceptional conjugacy classes, and for $n=3$ there is one such class. Bibliography: 32 titles.
Matematicheskii Sbornik. 2024;215(12):148-182
148-182