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Vol 211, No 2 (2020)
- Year: 2020
- Articles: 6
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7462
Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$
Abstract
Let $k$ be an uncountable algebraically closed field of characteristic $0$, and let $X$ be a smooth projective connected variety of dimension $2p$, embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplane section of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups of algebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulo rational equivalence on $Y$ and $X$, respectively. Assume that, whenever $Y$ is smooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$ and coincides with the subgroup of degree $0$ classes in the Chow group $\operatorname{CH}^p(Y)$. We prove that the kernel of the push-forward homomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countable collection of shifts of a certain abelian subvariety $A_0$ inside $A$. For a very general hyperplane section $Y$ either $A_0=0$ or $A_0$ coincides with an abelian subvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles $H^{2p-1}(Y)_\mathrm{van}$. Then we apply these general results to sections of a smooth cubic fourfold in $\mathbb P^5$. Bibliography: 33 titles.
Matematicheskii Sbornik. 2020;211(2):3-45
3-45
Integrable billiard systems realize toric foliations on lens spaces and the 3-torus
Abstract
An integrable billiard system on a book, a complex of several billiard sheets glued together along the common spine, is considered. Each sheet is a planar domain bounded by arcs of confocal quadrics; it is known that a billiard in such a domain is integrable. In a number of interesting special cases of such billiards the Fomenko-Zieschang invariants of Liouville equivalence (marked molecules $W^*$) turn out to describe nontrivial toric foliations on lens spaces and on the 3-torus, which are isoenergy manifolds for these billiards. Bibliography: 18 titles.
Matematicheskii Sbornik. 2020;211(2):46-73
46-73
Existence and uniqueness of a weak solution of an integro-differential aggregation equation on a Riemannian manifold
Abstract
A class of integro-differential aggregation equations with nonlinear parabolic term $b(x,u)_t$ is considered on a compact Riemannian manifold $\mathscr M$. The divergence term in the equations can degenerate with loss of coercivity and may contain nonlinearities of variable order. The impermeability boundary condition on the boundary $\partial\mathscr M\times[0,T]$ of the cylinder $Q^T=\mathscr M\times[0,T]$ is satisfied if there are no external sources of ‘mass’ conservation, $\int_\mathscr Mb(x,u(x,t)) d\nu=\mathrm{const}$. In a cylinder $Q^T$ for a sufficiently small $T$, the mixed problem for the aggregation equation is shown to have a bounded solution. The existence of a bounded solution of the problem in the cylinder $Q^\infty=\mathscr M\times[0,\infty)$ is proved under additional conditions. For equations of the form $b(x,u)_t=\Delta A(x,u)-\operatorname{div}(b(x,u)\mathscr G(u))+f(x,u)$ with the Laplace-Beltrami operator $\Delta$ and an integral operator $\mathscr G(u)$, the mixed problem is shown to have a unique bounded solution. Bibliography: 26 titles.
Matematicheskii Sbornik. 2020;211(2):74-105
74-105
Spectral representations of topological groups and near-openly generated groups
Abstract
Near-openly generated groups are introduced. They form a topological and multiplicative subclass of $\mathbb R$-factorizable groups. Dense and open subgroups, quotients and the Raikov completion of a near-openly generated group are near-openly generated. Almost connected pro-Lie groups, Lindelöf almost metrizable groups and the spaces $C_p(X)$ of all continuous real-valued functions on a Tychonoff space with pointwise convergence topology are near-openly generated. We provide characterizations of near-openly generated groups using methods of inverse spectra and topological game theory. Bibliography: 24 titles.
Matematicheskii Sbornik. 2020;211(2):106-124
106-124
Symmetries in left-invariant optimal control problems
Abstract
Left-invariant optimal control problems on Lie groups are considered. When studying the optimality of extreme trajectories, the crucial role is played by symmetries of the exponential map that are induced by symmetries of the conjugate subsystem of the Hamiltonian system of the Pontryagin maximum principle. A general construction is obtained for these symmetries of the exponential map for connected Lie groups with generic coadjoint orbits of codimension not exceeding one and with a connected stabilizer. Bibliography: 32 titles.
Matematicheskii Sbornik. 2020;211(2):125-140
125-140
Partially invertible strongly dependent $n$-ary operations
Abstract
We prove analogues of Malyshev's theorems on the structure of finite $n$-quasigroups with the weak invertibility condition and of Belousov's theorem with a description of $(i,j)$-associative $n$-quasigroups for the case of strongly dependent $n$-ary semigroup operations on a finite set. Bibliography: 8 titles.
Matematicheskii Sbornik. 2020;211(2):141-158
141-158