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Vol 77, No 1 (2022)
- Year: 2022
- Articles: 10
- URL: https://journal-vniispk.ru/0042-1316/issue/view/7526
What do Abelian categories form?
Abstract
Given two finitely presentable Abelian categories $A$ and $B$, we outline a construction of an Abelian category of functors from $A$ to $B$, which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of $A$ and $B$. The construction is absolute, so it makes it possible to recover not only Hochschild cohomology but also Mac Lane cohomology.Bibliography: 29 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(1):3-54
3-54
Structures of non-classical discontinuities in solutions of hyperbolic systems of equations
Abstract
Discontinuity structures in solutions of a hyperbolic system of equations are considered. The system of equations has a rather general form and, in particular, can describe the longitudinal and torsional non-linear waves in elastic rods in the simplest setting and also one-dimensional waves in unbounded elastic media. The properties of discontinuities in solutions of these equations have been investigated earlier under the assumption that only the relations following from the conservation laws for the longitudinal momentum and angular momentum about the axis of the rod and the displacement continuity condition hold on the discontinuities. The shock adiabat has been studied. This paper deals with stationary discontinuity structures under the assumption that viscosity is the main governing mechanism inside the structure. Some segments of the shock adiabat are shown to correspond to evolutionary discontinuities without structure. It is also shown that there are special discontinuities on which an additional relation must hold, which arises from the condition that a discontinuity structure exists. The additional relation depends on the processes in the structure. Special discontinuities satisfy evolutionary conditions that differ from the well-known Lax conditions. Conclusions are discussed, which can also be of interest in the case of other systems of hyperbolic equations.Bibliography: 58 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(1):55-90
55-90
Spectrum of the Laplace operator on closed surfaces
Abstract
A survey is given of classical and relatively recent results on the distribution of the eigenvalues of the Laplace operator on closed surfaces. For various classes of metrics the dependence of the behaviour of the second term in Weyl's formula on the geometry of the geodesic flow is considered. Various versions of trace formulae are presented, along with ensuing identities for the spectrum. The case of a compact Riemann surface with the Poincare metric is considered separately, with the use of Selberg's formula. A number of results on the stochastic properties of the spectrum in connection with the theory of quantum chaos and the universality conjecture are presented.Bibliography: 51 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(1):91-108
91-108
Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary 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 elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed.Bibliography: 91 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(1):109-176
109-176
Victor Nikolaevich Latyshev (obituary)
Uspekhi Matematicheskikh Nauk. 2022;77(1):177-182
177-182
An observation on the Gram matrices of systems of uniformly bounded functions and a problem of Olevskii
Uspekhi Matematicheskikh Nauk. 2022;77(1):183-184
183-184
On Voronoi's conjecture for four- and five-dimensional parallelohedra
Uspekhi Matematicheskikh Nauk. 2022;77(1):185-186
185-186
Inverse function theorem on the class of holomorphic self-maps of a disc with two fixed points
Uspekhi Matematicheskikh Nauk. 2022;77(1):187-188
187-188
On reduction and separation of projective sets in Tychonoff spaces
Uspekhi Matematicheskikh Nauk. 2022;77(1):189-190
189-190
Boris Abramovich Trakhtenbrot (on the centenary of his birth)
Uspekhi Matematicheskikh Nauk. 2022;77(1):191-195
191-195