Том 89, № 1 (2025)

We show that the polymodal provability logic $\mathrm{GLP}$, in a language with at least two modalities and one variable, has nullary unification type. More specifically, we show that the formula $[1]p$ does not have maximal unifiers, and exhibit an infinite complete set of unifiers for it. Further, we discuss the algorithmic problem of whether a given formula is unifiable in $\mathrm{GLP}$ and remark that this problem has a positive solution. Finally, we state the arithmetical analogues of the unification and admissibility problems for $\mathrm{GLP}$ and formulate a number of open questions.

If $d$ is not a perfect square, we define $T(d)$ as the length of theminimal period of the simple continued fraction expansion for $\sqrt{d}$.Otherwise, we put $T(d)=0$. In the recent paper (2024), F. Battistoni,L. Grenie and G. Molteni established (in particular) an upper boundfor the second moment of $T(d)$ over the segment $x\alpha\sqrt{x}$. In this paper, we slightly improve thisresult of three authors.

In [1] it was shown that the degree (vertex) spanning tree enumerator polynomialof a connected graph $G$ is a real stable polynomial (that is, it does not vanish if all thevariables have positive imaginary parts) if and only if $G$ is a distance-hereditary graph.We prove a similar characterization for weighted graphs.With the help of this generalization, define the class of weighted distance-hereditary graphs.

It is proved that if$\mathcal M\to C$ is the Neron minimal model of a principally polarized $(d-1)$-dimensional Abelian variety$\mathcal M_\eta$ over the field $\kappa(\eta)$ of rational functions of a smooth projective curve $C$,$\operatorname{End}_{\overline{\kappa(\eta)}} (\mathcal M_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})=\mathbb Z,$ the complexification of the Lie algebra of the Hodge group$\operatorname{Hg}(M_\eta\otimes_{\kappa(\eta)}\mathbb {C})$ is a simple Lie algebra of type $C_{d-1}$, all bad reductions of the Abelian variety$\mathcal M_\eta$ are semi-stable,for any places $\delta,\delta'$ of bad reductionsthe $\mathbb Q$-space of Hodge cycles on the product$\operatorname{Alb}(\overline{\mathcal M_\delta^0}) \times   \operatorname{Alb}(\overline{\mathcal M_{\delta'}^0})$ of Albanese varietiesis generated by classes of algebraic cycles,thenthere exists a finite ramified covering $\widetilde{C}\to C$ such that, for any Künnemann compactification $\widetilde{X}$of the Neron minimal model of the Abelian variety $\mathcal M_\eta\otimes_{\kappa(\eta)}\kappa(\widetilde{\eta})$,the Grothendieck standard conjecture $B(\widetilde{X})$ of Lefschetz type is true.

In this paper, the inverse spectral problem method is used to integrate a Hirota type equation with additional terms in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proved.It is also shown that the Cauchy problem is solvable at all times for sufficiently smooth initial conditions.