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Vol 216, No 2 (2025)
- Year: 2025
- Articles: 5
- URL: https://ogarev-online.ru/0368-8666/issue/view/20345
Lyapunov stability of an equilibrium of the nonlocal continuity equation
Abstract
3-31
The most symmetric smooth cubic surface
Abstract
32-80
John–Löwner ellipsoids and entropy of multiplier operators on rank $1$ compact homogeneous manifolds
Abstract
We present a new method of the evaluation of entropy, which is based on volume estimates for John–Löwner ellipsoids induced by the eigenfunctions of Laplace–Beltrami operator on compact homogeneous manifolds $\mathbb{M}^{d}$ of rank $1$. This approach gives the sharp orders of entropy in the situations where the known methods meet difficulties of fundamental nature. In particular, we calculate the sharp orders of the entropy of the Sobolev classes $W_{p}^{\gamma }(\mathbb{M}^{d})$, $\gamma>0$, in $L_{q}(\mathbb{M}^{d})$, $1 \leq q \leq p \leq \infty$. Bibliography: 35 titles.
81-109
Uniform rational approximation of the odd and even Cauchy transforms
Abstract
Best uniform rational approximations of the odd and even Cauchy transforms are considered. The results obtained form a basis for finding the weak asymptotics of best uniform rational approximations of the odd extension of the function $x^{\alpha}$, $x\in[0,1]$, to $[-1,1]$ for all $alpha\in(0,+\infty)\setminus(2\mathbb N-1)$, which complements some results due to Vyacheslavov. The strong asymptotics of the best rational approximations of this function on $[0,1]$ and its even extension to $[-1,1]$ were found by Stahl. It follows from these results that for $alpha\in(0,+\infty)\setminus\mathbb N$ the best rational approximations of the even and odd extensions of the above function show the same weak asymptotic behaviour. Bibliography: 29 titles.
110-127
Properties of at most countable unions of pairwise disjoint sets in asymmetric spaces
Abstract
We show that an at most countable nonsingleton union of pairwise disjoint proximinal sets is not a Chebyshev set. We also characterize the asymmetric linear spaces where each boundedly compact (approximatively compact) set is proximinal.Bibliography: 32 titles.
128-144