Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):3-4

3-4

On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings

Arzhantsev I.V., Gaifullin S.A., Lopatkin V.E.

Abstract

For a finite set of homogeneous locally nilpotent derivations of the algebraof polynomials in several variables, a finite dimensionality criterionfor the Lie algebra generated by these derivations is known.The structure of the corresponding finite-dimensional Lie algebraswas also described in previous works. In this paper, we obtaina finite dimensionality criterion for a Lie algebra generated by a finite setof homogeneous derivations, each of which is not locally nilpotent.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):5-22

5-22

Spectral asymptotics for Schrödinger operator perturbed by translation operator

Borisov D.I., Polyakov D.M.

Abstract

We consider the one-dimensional Schrödinger operator on the unitsegment with Dirichlet condition and perturb it by a translationoperator. The main result describes the asymptotics of theeigenvalues of this operator with respect to the index counting theeigenvalues, the resulting asymptotics is uniform in the translation. Inthe asymptotics, we explicitly find the terms generated by thetranslation operator. We establish that the system of eigenfunctionsand generalized eigenfunctions of the considered operator forms a Bari basis in the space of functions square integrable on the unitsegment.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):23-44

23-44

Existence of entropy solution to the Neumann problem for elliptic equation with measure-valued potential

Vildanova V.F., Mukminov F.K.

Abstract

In a bounded or unbounded domain in $\mathbb{R}^n$,the Neumann problem for a non-linear second order elliptic equation with measure-valued potential is considered.The assumptions on the structure of the equation are stated in terms of a generalized $N$-function.The existence of an entropy solution to the problem is proved.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):45-79

45-79

$1$-nodal Fano threefolds with Picard number $1$

Kuznetsov A.G., Prokhorov Y.G.

Abstract

We classify all $1$-nodal degenerations of smooth Fano threefolds with Picard number $1$ (both non-factorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds of higher Picard rank and with unprojections of complete intersection varieties.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):80-178

80-178

(Eng)

On completeness of the root function system of the $(2\times 2)$-Dirac operatorswith non-regular boundary conditions

Makin A.S.

Abstract

The paper is concerned with completeness of the system of root functions of the$(2\times2)$-Dirac operator with summable complex-valued potential and non-regular boundary conditions. Sufficient conditions for completeness of the root function system of this operator are found.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):179-192

179-192

On the decision problem for quantified probability logics

Speranski S.O.

Abstract

Let $\mathsf{QPL}^{\mathrm{e}}$ expand the quantifier-free “polynomial” probability logic of [4](R. Fagin et al., 1990)by adding quantifiers over arbitrary events; it can be viewed as a one-sorted elementary language for reasoning about probability spaces. We prove that the $\Sigma_2$-fragment of the $\mathsf{QPL}^{\mathrm{e}}$-theory of finite spaces is hereditarily undecidable. By earlier observations, this implies that $\Pi_2$ is the maximal decidable prefix fragment of $\mathsf{QPL}^{\mathrm{e}}$. Moreover, we obtain similar results for two natural one-sorted logics of probability that emerge from [1](M. Abadi and J. Y. Halpern, 1994).

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):193-211

193-211

(Eng)

On an analogue of the fundamental Voevodsky theorem

Tyurin D.N.

Abstract

Let $k$ be a field of zero characteristic, $X$ be a $k$-smooth scheme, and $F$be an $\mathbb{A}^1$-invariant quasi-stable presheave with framed transfers.Then the corresponding Gersten complex is exact.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):212-229

212-229

A criterion for the weak continuityof representations of topological groups in dual Frechet spaces

Shtern A.I.

Abstract

Sufficient conditions are obtained for the weak continuity of representations of topological groupsin Frechet spaces that are dual to some locally convex spaces by operators adjoint to continuous linear operators in a predual spaceIn particular, it is shownthat a representation $\pi$ of a topological group $G$ on a Frechet space $E$ dual to a locally convex space $E_*$ by adjoint operators is continuous inthe weak$^*$ operator topology if, for some number $q$, $0\le q<1$, there is a neighbourhood $V$ of the neutral element $e$ of $G$ such that, for anyneighbourhood $U$ of the zero element in $E$, for its polar $\mathring{U}$in $E^*$, and for any vector $\xi$ in $U$ and any element$\varphi\in\mathring{U}$ the inequality $|(\pi(g)\xi-\xi)(\varphi)|\le q$holds for each $g\in V$.

Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2025;89(3):230-240

230-240