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Vol 216, No 4 (2025)
- Year: 2025
- Articles: 8
- URL: https://ogarev-online.ru/0368-8666/issue/view/20347
On locally nilpotent derivations of polynomial algebra in three variables
Abstract
In this paper we investigate locally nilpotent derivations on the polynomial algebra in three variables over a field of characteristic zero. We introduce an iterating construction giving all locally nilpotent derivations of rank $2$. This construction allows us to obtain examples of non-triangularizable locally nilpotent derivations of rank $2$. We also show that the well-known example of a locally nilpotent derivation of rank $3$, given by Freudenburg, is a member of a large family of new examples of rank $3$ locally nilpotent derivations. Our approach is based on considering all locally nilpotent derivations commuting with a given derivation. We obtain a characterization of locally nilpotent derivations with a given rank in terms of sets of commuting locally nilpotent derivations. Bibliography: 32 titles.
3-34
On square of the Riemann zeta function modulus in the critical strip and estimates of means
Abstract
A new integral representation is derived for the square of the modulus of the Riemann zeta function. Estimates of the Laplace transform of the residual term $E_\sigma(T)$ and of the mean value with respect to a Gaussian function of the square of the modulus of the zeta function are obtained.Bibliography: 6 titles.
35-43
Sharp univalence and sharp univalent covering domains for the class of holomorphic self-maps of a disc with two fixed boundary points
Abstract
44-66
Optimal recovery of a solution of a system of linear differential equations from initial information given with a random error
Abstract
The problem of the optimal recovery of a solution of a system of linear differential equations from initial information containing a random error is considered. Optimal methods are searched for among all possible (not necessarily linear) recovery methods. Depending on the given variance of random errors, the optimal recovery methods constructed in the paper, which turn out to be linear, use only part of the available information. Bibliography: 17 titles.
67-89
Avkhadiev–Wirths conjecture on best Brezis–Marcus constants
Abstract
We study Hardy-type inequalities with additional terms. The constant $\lambda(\Omega)$ multiplying the additional term depends on the geometry of the multidimensional domain $\Omega$ and the numerical parameters of the problem. This constant (functional) is commonly called the Brezis–Marcus constant. Avkhadiev and Wirths [1] put forward the conjecture that, over all $n$-dimensional domains with fixed inner radius $\delta_0$, the maximum best Brezis–Marcus constant is $\lambda(B_n)$, where $B_n $ is the $n$-ball of radius $\delta_0$. We improve the previously available lower estimates for $\lambda(B_n)$, for $n=2$ and $n= 4,…,10$, which takes us closer to this conjecture. Bibliography: 18 titles.
90-112
Finiteness theorems for generalized Jacobians with nontrivial torsion
Abstract
Consider a curve $\mathcal C$ defined over an algebraic number field $k$. This work is concerned with the number of generalized Jacobians $J_{\mathfrak{m}}$ of $\mathcal C$ associated with moduli $\mathfrak{m}$ defined over $k$ such that a fixed class of finite order in the Jacobian $J$ of $\mathcal C$ is lifted to a torsion class in the generalized Jacobian $J_{\mathfrak{m}}$. On the one hand it is shown that there are infinitely many generalized Jacobians with the above property, and on the other hand, under some additional constraints on the support of $\mathfrak{m}$ or the structure of $J_{\mathfrak{m}}$, it is shown that the set of generalized Jacobians of this type is finite. In addition, it is proved that there are finitely many generalized Jacobians which have a lift of two given divisors to classes of finite orders in $J_{\mathfrak{m}}$. These results are applied to the problem of the periodicity of continued fractions in the field of formal power series $k((1/x))$ constructed for special elements of the function field $k(\widetilde{\mathcal{C}})$ of a hyperelliptic curve $\widetilde{\mathcal{C}}\colon y^2=f(x)$. In particular, it is shown that for each $n \in \mathbb N$ there is a finite number of monic polynomials $\omega(x) \in k[x]$ of degree at most $n$ such that the element $\omega(x) \sqrt{f(x)}$ has a periodic expansion in a continued fraction. Bibliography: 14 titles.
113-131
132-134
Letter to the editors
135-136