Vol 80, No 3 (2025)

Year: 2025
Articles: 9
URL: https://journals.rcsi.science/0042-1316/issue/view/20354

Convergence of generalized power series satisfying functional equations

Gontsov R.R., Goryuchkina I.V.

Abstract

We consider questions relating to the convergence of generalized power series (with complex-valued exponents) that satisfy formally some analytic functional equations: a differential equation, a $q$-difference one, or Mahler's equation. We present new results, as well as generalizations of some of our earlier results, thus summing up our investigations of this subject. We also present a selection of results on the existence and uniqueness of local holomorphic solutions of such equations and review some classical results on the convergence of Taylor power series that solve them formally.

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Uspekhi Matematicheskikh Nauk. 2025;80(3):3-66

3-66

(Rus)

(JATS XML)

Asymptotics of convergence to a wave travelling from a saddle to a node

Kalyakin L.A.

Abstract

An asymptotic solution is constructed for semilinear partial differential equations (of parabolic and hyperbolic types) that converges as $t\to\infty$ to a wave travelling from a stable equilibrium to an unstable one. It is established that for a wave of this kind the velocity asymptotics contains $\ln t$ and cannot be represented as a series in powers of $1/t$. It is demonstrated how the matching method can be used for this problem. An efficient method is indicated for the calculation of the universal part of the asymptotic formula that is independent of the initial data.

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Uspekhi Matematicheskikh Nauk. 2025;80(3):67-112

67-112

(Rus)

(JATS XML)

Johnson graphs, their random subgraphs, and some of their extremal characteristics

Raigorodskii A.M., Sinelnikov-Murylev P.S.

Abstract

One of the most important objects of study in modern graph theory is the so-called Johnson graph $G(n,r,s)$. The vertices of this graph are all $C_n^r$ subsets of cardinality $r$ of the set $\{1,…,n\}$, and two vertices are joined by an edge if and only if the intersection of the corresponding subsets has cardinality $s$. Such graphs play a vital role in coding theory, extremal combinatorics and Ramsey theory, combinatorial geometry, and in other areas. In this survey we give an account of applications of these graphs and of some of their parameters including the independence number, the clique number, and the chromatic number. We also pay attention to a large part of this theory concerned with the investigation of random subgraphs of Johnson graphs, which was actively developed in recent years.

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Uspekhi Matematicheskikh Nauk. 2025;80(3):113-176

113-176