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Vol 78, No 1 (2023)
Automorphism groups of $\mathbb{P}^1$-bundles over a non-uniruled base
Abstract
In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p\colon X \to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We consider the groups $\operatorname{Aut}(X)$ and $\operatorname{Bim}(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively, and discuss when these groups are bounded, Jordan, strongly Jordan, or very Jordan.Bibliography: 88 titles.
Uspekhi Matematicheskikh Nauk. 2023;78(1):3-66
3-66
Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions
Abstract
Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing.The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elliptic functions. The focus is on describing extremal trajectories and their optimality, the cut time andcut locus, and optimal synthesis.Bibliography: 162 titles.
Uspekhi Matematicheskikh Nauk. 2023;78(1):67-166
67-166
Strong and weak associativity of weighted Sobolev spaces of the first order
Abstract
A brief overview of the recent results on the problem of characterization of associative and double associative spaces of function classes, including both ideal and non-ideal structures, is presented. The latter include two-weighted Sobolev spaces of the first order on the positive semi- axis. It is shown that, in contrast to the notion of duality, associativity can be ‘strong’ or ‘weak’. In addition, double associative spaces are further divided into three types. In this context it is established that a weighted Sobolev space of functions with compact support possesses weak associative reflexivity, while the strong associative space of a weak associative space is trivial. Weighted classes of Cesàro and Copson type have similar properties; for these classes the problem us fully investigated, and their connections with Sobolev spaces with power weights are established. As an application, the problem of boundedness of the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue space is considered.Bibliography: 49 titles.
Uspekhi Matematicheskikh Nauk. 2023;78(1):167-204
167-204
Igor Moiseevich Krichever (obituary)
Uspekhi Matematicheskikh Nauk. 2023;78(1):205-206
205-206
Cyclic Frobenius algebras
Uspekhi Matematicheskikh Nauk. 2023;78(1):207-208
207-208
The number of components of the Pell–Abel equations with primitive solutions of given degree
Uspekhi Matematicheskikh Nauk. 2023;78(1):209-210
209-210