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卷 80, 编号 5 (2025)

年: 2025
文章: 10
URL: https://journal-vniispk.ru/0042-1316/issue/view/22424

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Multidimensional Hamiltonian systems: non-integrability and diffusion

Kozlov V.

摘要

Hamiltonian systems of differential equations that are little different from completely integrable systems are under consideration. If such a system is integrable, then the action variables cannot change strongly, and there is no diffusion. Thus the non-integrable behaviour of a Hamiltonian system is closely linked with the diffusion of slow variables. This range of problems is discussed for a subclass of Hamiltonian systems. A new mechanism of diffusion, different from the ‘standard’ scheme of transition chains, is considered on these example. This mechanism is related to the breakdown of a large number of invariant tori of the non-perturbed problem which have almost resonance sets of frequencies. On the formal side, this phenomenon is based on the non-boundedness of integrals of conditionally-periodic functions of time with zero mean.

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Uspekhi Matematicheskikh Nauk. 2025;80(5):3-22

3-22

(Rus)

(JATS XML)

Brief introduction in greedy approximation

Temlyakov V.

摘要

Sparse approximation is important in many applications because of the concise form of an approximant and good accuracy guarantees. The theory of compressed sensing, which proved to be very useful in the image processing and data sciences, is based on the concept of sparsity. A fundamental issue of sparse approximation is the problem of the construction of efficient algorithms, which provide good approximation. It turns out that greedy algorithms with respect to dictionaries are very good from this point of view. They are simple in implementation, and there are well-developed theoretical guarantees of their efficiency. This survey/tutorial paper contains a brief description of different kinds of greedy algorithms and results on their convergence and rate of convergence. Also, in Sections 14 and 15 we give some typical proofs of convergence and rate of convergence results for important greedy algorithms and in Section 16 we list some open problems.

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Uspekhi Matematicheskikh Nauk. 2025;80(5):23-104

23-104

(Eng)

(JATS XML)

On the spectrum of random Gram matrices of large dimension in the case of partial dependence

Yaskov P.

摘要

A unified theory is proposed, which enables one to derive universal limiting spectral distributions for random Gram matrices of large dimension and related matrix models in the case of partial dependence.

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Uspekhi Matematicheskikh Nauk. 2025;80(5):105-174

105-174

(Rus)

(JATS XML)

Revaz Valer'yanovich Gamkrelidze (obituary)

Avakov E., Agrachev A., Aseev S., Giorgadze G., Davydov A., Zelikin M., Kozlov V., Lokutsievskiy L., Nikol'skii M., Ovchinnikov A., Osipov Y., Sarychev A., Sachkov Y., Treschev D.

Uspekhi Matematicheskikh Nauk. 2025;80(5):175-178

175-178