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Vol 210, No 8 (2019)
Isomorphisms and elementary equivalence of Chevalley groups over commutative rings
Abstract
It is proved that two Chevalley groups with indecomposable root systems of rank $>1$ over commutative rings (which contain in addition $1/2$ for the types $\mathbf A_2$, $\mathbf B_l$, $\mathbf C_l$, $\mathbf F_4$, and $\mathbf G_2$, and $1/3$ for the type $\mathbf G_2$) are isomorphic or elementarily equivalent if and only if the corresponding root systems coincide, the weight lattices of the representation of the Lie algebra coincide, and the rings are isomorphic or elementarily equivalent, respectively. The isomorphisms of adjoint (elementary) Chevalley groups over the rings of the above types are also described. Bibliography: 25 titles.
Matematicheskii Sbornik. 2019;210(8):3-28
3-28
An approach problem for a control system and a compact set in the phase space in the presence of phase constraints
Abstract
A control system with a phase constraint is considered in a finite-dimensional Euclidean space. The problem of making this system approach the target set at a fixed time instant is studied. A method for constructing an approximate solution to the approach problem is given, which involves the concept of the solvability set of an approach problem. Bibliography: 24 titles.
Matematicheskii Sbornik. 2019;210(8):29-66
29-66
On maximizers of a convolution operator in $L_p$-spaces
Abstract
The paper is concerned with convolution operators in $\mathbb R^d$, whose kernels are in $L_q$, which act from $L_p$ into $L_s$, where $1/p+1/q=1+1/s$. It is shown that for $1< q,p,s< \infty$ there exists a maximizer (a function with $L_p$-norm $1$) at which the supremum of the $s$-norm of the convolution is attained. A special analysis is carried out for the cases in which one of the exponents $q,p$, or $s$ is $1$ or $\infty$.
Bibliography: 12 titles.
Matematicheskii Sbornik. 2019;210(8):67-86
67-86
87-119
Convex trigonometry with applications to sub-Finsler geometry
Abstract
A new convenient method for describing flat convex compact sets and their polar sets is proposed. It generalizes the classical trigonometric functions $\sin$ and $\cos$. It is apparent that this method can be very useful for an explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Finsler problems with two-dimensional control lying in an arbitrary convex set $\Omega$ is investigated. Namely, problems on the Heisenberg, Engel, and Cartan groups and also Grushin's and Martinet's cases are considered. Particular attention is paid to the case when $\Omega$ is a convex polygon.Bibliography: 13 titles.
Matematicheskii Sbornik. 2019;210(8):120-148
120-148