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Vol 81, No 1 (2026)
Modern theory of electrical networks: from matrix tree theorem to the theory of cluster varieties
Abstract
The theory of electrical networks, in its current state, covers several areas of contemporary mathematics and mathematical physics, such as the combinatorics of paths, forests, and woods on graphs, discrete harmonic analysis, problems of random walks, exactly solved models in statistical physics, cluster varieties related to spaces of totally positive matrices, discrete integrable systems, algebraic structures similar to Zamolodchikov's tetrahedron equation, and many others. The main aim of the survey is presenting some of these subjects, classical and recently discovered alike.
3-70
Annulus principle in the problem of the existence of an infinite-dimensional invariant torus
Abstract
An annulus-like set of the form $K=B\times\mathbb{T}^{\infty}$ is under consideration, where $B$ is a closed ball in a Banach space $V$ and $\mathbb{T}^{\infty}$ is the so-called standard infinite-dimensional torus, defined by $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional Banach space and $\mathbb{Z}^{\infty}$ is an abstract integer lattice in $E$. The main result is as follows: for a certain class of smooth maps $\Pi\colon K\to K$ we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form $A=\{(v,\varphi)\in K\colon v=h(\varphi)\in V,\break \varphi\in\mathbb{T}^{\infty}\}$, where $h(\varphi)$ is a continuous function of $\varphi\in\mathbb{T}^{\infty}$. We also answer a number of related questions. First, we consider the problem of the $C^m$-smoothness of the manifold $A$ for each positive integer $m$; second, we show that all trajectories of the map $\Pi$ with initial conditions in $K$ tend to $A$ while admitting an asymptotic phase; third, we extend our results to semiflows and then apply the theory developed to integral networks of nonlinear oscillators.
71-136
Algorithmic complexity of theories with Kleene iteration
Abstract
The Kleene iteration (star operator) is one of the most interesting algebraic operations arising in theoretical informatics. The studies of structures with this operation, Kleene algebras and their extensions, begin with the classical concept of regular expression describing formal languages. Subsequently, so-called action algebras (Pratt 1991, Kozen 1994), or Kleene algebras with division, were introduced. In these structures the Kleene star operator is combined with divisions compatible with a partial order (such operations had previously been introduced by Krull in 1924). A survey of results on algorithmic complexity for the logical theories of structures with Kleene iteration is given. Although the simplest of these theories, the theory of equality of regular expressions, is algorithmically solvable, some of its generalizations, such as Horn theories and fragments of these, as well as theories with division, almost immediately become unsolvable. Particularly interesting is the case of $*$-continuous Kleene algebras, where an iteration is defined as the limit of powers of an element (in the general case an iteration is defined as a fixed point). In the language of logic, $*$-continuity corresponds to the omega rule, and the complexity of such theories can attain the level of $\Pi^1_1$-completeness.
137-204
SHORT MESSAGES
Solution to Hart-van Mill's Problem 61
205-206
$n$ -valued monoids on $\mathbb{C}P^1$ and discriminants
207-208
Chattering extremals in Hamiltonian systems with control in a square
209-210
Mathematical Life
Anatolii Timofeevich Fomenko (on his 80th birthday)
211-222