Vol 89, No 3 (2025)

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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.

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.

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).

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 $g\in V$.