卷 75, 编号 1 (2020)

年: 2020
文章: 6
URL: https://journals.rcsi.science/0042-1316/issue/view/7514

Attractors of nonlinear Hamiltonian partial differential equations

Komech A., Kopylova E.

摘要

This is a survey of the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. Included are results on global attraction to stationary states, to solitons, and to stationary orbits, together with results on adiabatic effective dynamics of solitons and their asymptotic stability, and also results on numerical simulation. The results obtained are generalized in the formulation of a new general conjecture on attractors of $G$-invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr transitions between quantum stationary states, de Broglie's wave-particle duality, and Born's probabilistic interpretation.Bibliography: 212 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(1):3-94

3-94

(Rus)

(JATS XML)

Extremal problems in hypergraph colourings

Raigorodskii A., Cherkashin D.

摘要

Extremal problems in hypergraph colouring originate implicitly from Hilbert's theorem on monochromatic affine cubes (1892) and van der Waerden's theorem on monochromatic arithmetic progressions (1927). Later, with the advent and elaboration of Ramsey theory, the variety of problems related to colouring of explicitly specified hypergraphs widened rapidly. However, a systematic study of extremal problems on hypergraph colouring was initiated only in the works of Erdős and Hajnal in the 1960s. This paper is devoted to problems of finding edge-minimum hypergraphs belonging to particular classes of hypergraphs, variations of these problems, and their applications. The central problem of this kind is the Erdős–Hajnal problem of finding the minimum number of edges in an $n$-uniform hypergraph with chromatic number at least three. The main purpose of this survey is to spotlight the progress in this area over the last several years.Bibliography: 168 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(1):95-154

95-154

(Rus)

(JATS XML)

Etudes of the resolvent

Takhtadzhyan L.

摘要

Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number theory from a unified point of view. For instance, for the Sturm–Liouville operator there is a derivation of the Gelfand–Levitan trace formula, and for the one-dimensional Schrödinger operator a derivation of Faddeev's formula for the characteristic determinant and the Zakharov–Faddeev trace identities. Recent results on the spectral theory of a certain functional-difference operator arising in conformal field theory are then presented. The last section of the survey is devoted to the Laplace operator on a fundamental domain of a Fuchsian group of the first kind on the Lobachevsky plane. An algebraic scheme is given for proving analytic continuation of the integral kernel of the resolvent of the Laplace operator and the Eisenstein–Maass series. In conclusion there is a discussion of the relationship between the values of the Eisenstein–Maass series at Heegner points and the Dedekind zeta-functions of imaginary quadratic fields, and it is explained why pseudo-cusp forms for the case of the modular group do not provide any information about the zeros of the Riemann zeta-function.Bibliography: 50 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(1):155-194

155-194