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Vol 88, No 3 (2024)
Articles
Geometric constructions in the theory of analytic complexity
Abstract
Two geometric constructions are considered in the context of analytic complexity.Using the first construction, on the set of analytic functions, we build a metric invariant under the actionof the gauge group. With the help of the second construction, we obtaina necessary differential algebraic condition for membership of a function in the tangent space to the class of bivariate functionsof analytic complexity $\le 2$ at the point $z_0=x^3 y^2 +xy$. From this result we show that thepolynomial $z=x^3y^2+xy + \pi x^2 y^3$ of degree 5 has analytic complexity 3.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(3):3-11
3-11
634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
Abstract
In 1987 Brehm and Kühnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for $d\in\{2,4,8,16\}$ only. There exist a unique $6$-vertex triangulation of $\mathbb{RP}^2$, a unique $9$-vertex triangulation of $\mathbb{CP}^2$, and at least three $15$-vertex triangulations of $\mathbb{HP}^2$. However, until now, the question of whether there exists a $27$-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct $634$ vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Four of them have symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$, and the other $630$ have symmetry group $\mathrm{C}_3^3$ of order $27$. Further, we construct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups $\mathrm{C}_3$, $\mathrm{C}_3^2$, and $\mathrm{C}_{13}$. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane $\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(3):12-60
12-60
On provability logics of Niebergall arithmetic
Abstract
K. G. Niebergall suggested a simple example of a non-gödelean arithmeticaltheory $\mathrm{NA}$, in which a natural formalization of its consistencyis derivable. In the present paper we consider the provability logicof $\mathrm{NA}$ with respect to Peano arithmetic. We describe the class of itsfinite Kripke frames and establish the corresponding completeness theorem.For a conservative extension of this logic in the language with an additionalpropositional constant, we obtain a finite axiomatization. We also considerthe truth provability logic of $\mathrm{NA}$ and the provability logic of $\mathrm{NA}$ with respect to $\mathrm{NA}$ itself. We describe the classes of Kripkemodels with respect to which these logics are complete. We establish$\mathrm{PSpace}$-completeness of the derivability problem in these logicsand describe their variable free fragments. We also prove thatthe provability logic of $\mathrm{NA}$ with respect to Peano arithmeticdoes not have the Craig interpolation property.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(3):61-100
61-100
Algebraic de Rham theorem and Baker–Akhiezer function
Abstract
For the case of algebraic curves (compact Riemann surfaces), it is shown thatde Rham cohomology group $H^1_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$of the Riemann surface $X$ has a natural structure of a symplectic vector space.Every choice of a non-special effective divisor $D$ of degree $g$ on $X$defines a symplectic basis of $H^1_{\mathrm{dR}}(X,\mathbb{C})$ consistingof holomorphic differentials and differentials of the second kind with poleson $D$. This result, which is the algebraic de Rham theorem, is used to describethe tangent space to Picard and Jacobian varieties of $X$in terms of differentials of the second kind, and to define a naturalvector fields on the Jacobian of the curve $X$ that move points of the divisor $D$.In terms of the Lax formalism on algebraic curves, these vector fieldscorrespond to the Dubrovin equations in the theory of integrable systems,and the Baker–Akhierzer function is naturally obtained by the integration alongthe integral curves.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(3):101-110
101-110
Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface
Abstract
The classical approach to the study of dynamical systems consistsin representing the dynamics of the system in the “source–sink” form,that is, by singling out a dual attractor–repeller pair, consisting ofthe attracting and repelling sets for all other trajectoriesof the system. A choice of an attractor-repeller dual pairso that the space of orbits in their complement (the characteristic spaceof orbits) is connected paves the way for finding completetopological invariants of the dynamical system. In this way, in particular,several classification results for Morse–Smale systems were obtained.Thus, a complete topological classification of Morse–Smale3-diffeomorphisms is essentially based on the existence of a connectedcharacteristic space of orbits associated with the choice ofa one-dimensional dual attractor–repeller pair.For Morse–Smale diffeomorphisms with heteroclinic points on surfaces,there are examples in which the characteristic spaces of orbits are disconnected in all cases.In this paper, we prove a criterion for the existence of a connectedcharacteristic space of orbits for gradient-like (without heteroclinicpoints) diffeomorphisms on surfaces. This result implies, in particular,that any orientation-preserving diffeomorphism admits a connectedcharacteristic space. For an orientable surface of any kind,we also construct an orientation-changing gradient-like diffeomorphism that does not havea connected characteristic space. On any non-orientable surface of any kind, we also constructa gradient-like diffeomorphism which does not admit a connected characteristic space.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(3):111-138
111-138
On adjacency operators of locally finite graphs
Abstract
A graph $\Gamma$ is called locally finite if,for each vertex $v\in \Gamma$, the set $\Gamma(v)$ of its adjacent vertices is finite.For an arbitrary locally finite graph $\Gamma$ withvertex set $V(\Gamma)$ and an arbitrary field $F$,let $F^{V(\Gamma)}$ be the vector space over $F$of all functions $V(\Gamma) \to F$ (with naturalcomponentwise operations) and let $A^{(\mathrm{alg})}_{\Gamma,F}$be the linear operator $F^{V(\Gamma)} \to F^{V(\Gamma)}$defined by$(A^{(\mathrm{alg})}_{\Gamma,F}(f))(v) = \sum_{u \in \Gamma(v)}f(u)$for all $f \in F^{V(\Gamma)}$, $v \in V(\Gamma)$.In the case of a finite graph $\Gamma$, themapping $A^{({\mathrm{alg}})}_{\Gamma,F}$ is the well-known operator defined by theadjacency matrix of the graph $\Gamma$ (over $F$), andthe theory of eigenvalues and eigenfunctions of such operatorsis a well developed part of the theory of finite graphs(at least, in the case $F = \mathbb{C}$). In the present paper, wedevelop the theory of eigenvalues and eigenfunctions of the operators$A^{({\mathrm{alg}})}_{\Gamma,F}$for infinite locally finite graphs $\Gamma$ (however, some results that followmay present certain interest for the theory of finite graphs) and arbitrary fields $F$,even though in the present paper special emphasis is placedon the case of a connected graph $\Gamma$ with uniformly bounded degrees of vertices and $F = \mathbb{C}$.The previous attempts in this direction were not, in the author's opinion,quite satisfactory in the sense that they have been concerned only with eigenfunctions (and corresponding eigenvalues)of rather special type.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(3):139-191
139-191
192-202