Vol 211, No 7 (2020)

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.

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.

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.

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.