Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Том 79, № 3 (2024)
Sequences of independent functions and structure of symmetric spaces
Аннотация
Основная цель обзора состоит в представлении результатов последнего десятилетия по описанию подпространств как $L_p$-пространств и пространств Орлича, так и общих симметричных пространств, порожденных независимыми функциями. Предлагается новый подход, основанный на использовании комбинации результатов теории симметричных пространств, методов теории интерполяции операторов и некоторых вероятностных идей. Изучается проблема единственности распределения функции, последовательность независимых копий которой порождает данное подпространство. Доказан общий принцип сравнения дополняемости подпространств, порожденных последовательностями независимых функций в симметричном пространстве на $[0,1]$ и их попарно дизъюнктных копий в некотором пространстве на полуоси $(0,\infty)$, одним из следствий которого является классическая теорема Дора–Стабеда о дополняемости подпространств $L_p$-пространств. Библиография: 103 названия.
Uspekhi Matematicheskikh Nauk. 2024;79(3):3-92
3-92
Boltzmann-type kinetic equations and discrete models
Аннотация
The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.
Uspekhi Matematicheskikh Nauk. 2024;79(3):93-148
93-148
One-sided discretization inequalities and recovery from samples
Аннотация
Recently, in a number of papers it was understood that results on sampling discretization and on the universal sampling discretization can be successfully used in the problem of sampling recovery. Moreover, it turns out that it is sufficient to only have a one-sided discretization inequality for some of those applications. This motivates us to write the present paper as a survey paper, which includes new results, with the focus on the one-sided discretization inequalities and their applications in the sampling recovery. In this sense the paper complements the two existing survey papers on sampling discretization.
Uspekhi Matematicheskikh Nauk. 2024;79(3):149-180
149-180
Maximum principle and asymptotic properties of Hermite–Pade polynomials
Uspekhi Matematicheskikh Nauk. 2024;79(3):181-182
181-182
On the adjacency of type $D$ singularities of a front
Uspekhi Matematicheskikh Nauk. 2024;79(3):183-184
183-184
Andrei Igorevich Shafarevich (on his sixtieth birthday)
Uspekhi Matematicheskikh Nauk. 2024;79(3):185-188
185-188
Steklov Institute — 90!
Uspekhi Matematicheskikh Nauk. 2024;79(3):189-193
189-193
194-194