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Vol 86, No 2 (2022)
Articles
The optimal start control problem for two-dimensional Boussinesq equations
Abstract
We consider the problem of the optimal start control for two-dimensional Boussinesqequations describing non-isothermal flows of a viscous fluid in a boundeddomain. Using the study of the properties of admissible tuples and of the evolutionoperator, we prove the solubility of the optimization problem under naturalassumptions about the model data. We derive a variational inequality which issatisfied by the optimal control provided that the objective functional isdetermined by the final observation. We also obtain sufficient conditions for theuniqueness of an optimal control.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(2):3-24
3-24
On the number of epi-, mono- and homomorphisms of groups
Abstract
It is well known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We study the question of what can be said about the number of homomorphisms satisfying certain natural conditions like injectivity or surjectivity. A simple non-trivial consequence of our results is the fact that in any finite group the number of generating pairs $(x,y)$ such that $x^3=1=y^5$ is divisible by the greatest common divisor of fifteen and the order of the group $[G,G]\cdot\{g^{15}\mid g\in G\}$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(2):25-33
25-33
Foundations of Lie theory for $\mathcal E$-structures and some of its applications
Abstract
We construct an analogue of classical Lie theory in the case of Lie groups and Lie algebrasdefined over the algebra of dual numbers. As an application, we study approximate symmetriesof differential equations and construct analogues of Hjelmslev's natural geometry.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(2):34-61
34-61
The coloured Tverberg theorem, extensions and new results
Abstract
We prove a multiple coloured Tverberg theorem and a balanced coloured Tverbergtheorem, applying different methods, tools and ideas. The proof of the first theorem uses a multiplechessboard complex (as configuration space) and the Eilenberg–Krasnoselskii theory ofdegrees of equivariant maps for non-free group actions. The proof of the second result relies onthe high connectivity of the configuration space, established by using discrete Morse theory.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(2):62-79
62-79
The generalized Plücker–Klein map
Abstract
The intersection of two quadrics is called a biquadric. If we mark a non-singular quadricin the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surfacewas canonically associated with every three-dimensional marked biquadric (that is, witha quadratic line complex provided that the Plücker–Klein quadric is marked).In Reid's thesis, this correspondence was generalizedto odd-dimensional marked biquadrics of arbitrary dimension $\ge 3$. In this case,a Kummer variety of dimension $g$ corresponds to every biquadric of dimension $2g-1$.Reid only constructed the generalized Plücker–Klein correspondence. This map was notstudied later. The present paper is devoted to a partial solution of the problem of creatingthe corresponding theory.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(2):80-127
80-127
Birational geometry of varieties fibred into complete intersections of codimension two
Abstract
In this paper we prove the birational superrigidity of Fano–Mori fibre spaces$\pi\colon V\to S$ all of whose fibres are complete intersections of type$d_1\cdot d_2$ in the projective space ${\mathbb P}^{d_1+d_2}$ satisfying certainconditions of general position, under the assumption that the fibration $V/S$is sufficiently twisted over the base (in particular, under the assumption that the$K$-condition holds). The condition of general position for every fibre guaranteesthat the global log canonical threshold is equal to one. This condition also boundsthe dimension of the base $S$ by a constant depending only on the dimension $M$of the fibre (this constant grows like $M^2/2$ as $M\to\infty$). The fibres and the variety $V$may have quadratic and bi-quadratic singularities whose rank is bounded below.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2022;86(2):128-212
128-212