Vol 75, No 4 (2020)

Year: 2020
Articles: 8
URL: https://ogarev-online.ru/0042-1316/issue/view/7517

Spectral triangles of non-selfadjoint Hill and Dirac operators

Djakov P.B., Mityagin B.S.

Abstract

This is a survey of results from the last 10 to 12 years about the structure of the spectra of Hill–Schrödinger and Dirac operators. Let $L$ be a Hill operator or a one-dimensional Dirac operator on the interval $[0,\pi]$. If $L$ is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large $|n|$, close to $n^2$ in the Hill case or close to $n$ in the Dirac case ($n\in \mathbb{Z}$). There is one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda_n^-$ and $\lambda_n^+$ (counted with multiplicity). Asymptotic estimates are given for the spectral gaps $\gamma_n=\lambda_n^+-\lambda_n^-$ and the deviations $\delta_n=\mu_n-\lambda_n^+$ in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for $\gamma_n$ and $\delta_n$ are found for special potentials that are trigonometric polynomials.Bibliography: 45 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(4):3-44

3-44

(Rus)

(JATS XML)

Semantic limits of dense combinatorial objects

Coregliano L.N., Razborov A.A.

Abstract

The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as ‘flag algebras’, while the semantic, geometric approach is often associated with the name ‘graph limits’. The language of graph limits is generally more intuitive and expressible, but a price that one has to pay for it is that it is better suited for the case of ordinary graphs than for more general combinatorial objects. Accordingly, there have been several attempts in the literature, of varying degree of generality, to define limit objects for more complicated combinatorial structures. This paper is another attempt at a workable general theory of dense limit objects. Unlike previous efforts in this direction (with the notable exception of [5] by Aroskar and Cummings), our account is based on the same concepts from first-order logic and model theory as in the theory of flag algebras. It is shown how our definitions naturally encompass a host of previously considered cases (graphons, hypergraphons, digraphons, permutons, posetons, coloured graphs, and so on), and the fundamental properties of existence and uniqueness are extended to this more general case. Also given is an intuitive general proof of the continuous version of the Induced Removal Lemma based on the compactness theorem for propositional calculus. Use is made of the notion of open interpretation that often allows one to transfer methods and results from one situation to another. Again, it is shown that some previous arguments can be quite naturally framed using this language.Bibliography: 68 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(4):45-152

45-152

(Rus)

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Geometry of Banach limits and their applications

Semenov E.M., Sukochev F.A., Usachev A.S.

Abstract

A Banach limit is a positive shift-invariant functional on $\ell_\infty$ which extends the functional$$(x_1,x_2,…)\mapsto\lim_{n\to\infty}x_n$$from the set of convergent sequences to $\ell_\infty$. The history of Banach limits has its origins in classical papers by Banach and Mazur. The set of Banach limits has interesting properties which are useful in applications. This survey describes the current state of the theory of Banach limits and of the areas in analysis where they have found applications.Bibliography: 137 titles.

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Uspekhi Matematicheskikh Nauk. 2020;75(4):153-194

153-194