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Volume 211, Nº 7 (2020)
- Ano: 2020
- Artigos: 7
- URL: https://bakhtiniada.ru/0368-8666/issue/view/7467
Stability of Poiseuille-type flows in an MHD model of an incompressible polymeric fluid
Resumo
A generalization of the Pokrovskii-Vinogradov model for flows of solutions and melts of incompressible viscoelastic polymeric media to the case of nonisothermic flows in an infinite plane channel under the effect of a magnetic field is considered. A formal asymptotic representation is derived for the eigenvalues of the linearized problem (the basic solution is an analogue of the Poiseuille flow of a viscous fluid in the Navier-Stokes model) as their absolute value increases. A necessary condition for the asymptotic stability of an analogue of the Poiseuille shear flow is deduced.Bibliography: 22 titles.
Matematicheskii Sbornik. 2020;211(7):3-23
3-23
On the derived category of $\mathrm{IGr}(3,8)$
Resumo
We construct a full exceptional collection of vector bundles in the bounded derived category of coherent sheaves on the Grassmannian $\mathrm{IGr}(3,8)$ of isotropic 3-dimensional subspaces in a symplectic vector space of dimension 8.Bibliography: 16 titles.
Matematicheskii Sbornik. 2020;211(7):24-59
24-59
First-order zero-one law for the uniform model of the random graph
Resumo
The paper considers the Erdős-Renyi random graph in the uniform model $G(n,m)$, where $m=m(n)$ is a sequence of nonnegative integers such that $m(n)\sim cn^{\alpha}<(2-\varepsilon)n^2$ for some $c>0$, $\alpha\in[0,2]$, and $\varepsilon>0$. It is shown that $G(n,m)$ obeys the zero-one law for the first-order language if and only if either $\alpha\in\{0,2\}$, or $\alpha$ is irrational, or $\alpha\in(0,1)$ and $\alpha$ is not a number of the form $1-1/\ell$, $\ell\in\mathbb{N}$. Bibliography: 15 titles.
Matematicheskii Sbornik. 2020;211(7):60-71
60-71
A new series of moduli components of rank-2 semistable sheaves on $\mathbb{P}^{3}$ with singularities of mixed dimension
Resumo
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathscr{M}(k)$, $k \geq 3$, of semistable rank-2 sheaves on $\mathbb{P}^3$ with Chern classes $c_1=0$, $c_2=k$ and $c_3=0$, whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $\mu$-semistable reflexive sheaves along disjoint unions of collections of points and smooth irreducible curves which are rational or complete intersection curves in $\mathbb{P}^{3}$. As a special member of this series, we obtain a new component of $\mathscr{M}(3)$. Bibliography: 12 titles.
Matematicheskii Sbornik. 2020;211(7):72-92
72-92
An elliptic billiard in a potential force field: classification of motions, topological analysis
Resumo
Given an ellipse ${\frac{x^2}{a}+\frac{y^2}{b}=1}$, $a>b>0$, we consider an absolutely elastic billiard in it with potential $\frac{k}{2}(x^2+y^2)+\frac{\alpha}{2x^2}+\frac{\beta}{2y^2}$, $a\geq0$, $\beta\geq0$. This dynamical system is integrable and has two degrees of freedom. We obtain the iso-energy invariants of rough and fine Liouville equivalence, and conduct a comparative analysis of other systems known in rigid body mechanics. To obtain the results we apply the method of separation of variables and construct a new method, which is equivalent to the bifurcation diagram but does not require it to be constructed. Bibliography: 17 titles.
Matematicheskii Sbornik. 2020;211(7):93-120
93-120
Analytic solutions of convolution equations on convex sets in the complex plane with an open obstacle on the boundary
Resumo
Conditions, including criteria, are established for the existence of a continuous linear right inverse to a surjective convolution operator in the space of germs of analytic functions on a convex subset of the complex plane which has a countable neighbourhood basis consisting of convex domains. These are stated in terms of the existence of special families of subharmonic functions and the boundary behaviour of convex conformal mappings related to the sets in question. Bibliography: 50 titles.
Matematicheskii Sbornik. 2020;211(7):121-150
121-150
Encodings of trajectories and invariant measures
Resumo
We consider a discrete dynamical system on a compact manifold $M$ generated by a homeomorphism $f$. Let $C=\{M(i)\}$ be a finite covering of $M$ by closed cells. The symbolic image of a dynamical system is a directed graph $G$ with vertices corresponding to cells in which vertices $i$ and $j$ are joined by an arc $i\to j$ if the image $f(M(i))$ intersects $M(j)$. We show that the set of paths of the symbolic image converges to the set of trajectories of the system in the Tychonoff topology as the diameter of the covering tends to zero. For a cycle on $G$ going through different vertices, a simple flow is by definition a uniform distribution on arcs of this cycle. We show that simple flows converge to ergodic measures in the weak topology as the diameter of the covering tends to zero. Bibliography: 28 titles.
Matematicheskii Sbornik. 2020;211(7):151-176
151-176