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Vol 213, No 3 (2022)
- Year: 2022
- Articles: 7
- URL: https://journals.rcsi.science/0368-8666/issue/view/7487
Lyapunov instability of stationary flows of a polymeric fluid in a channel with perforated walls
Abstract
The rheological Pokrovskii-Vinogradov model for flows of solutions or melts of an incompressible viscoelastic polymeric medium is studied in the case of flows in an infinite planar channel with perforated walls. The linear Lyapunov instability is proved for the base solution with constant flow rate in the class of perturbations periodic in the variable varying along the channel wall. Bibliography: 14 titles.
Matematicheskii Sbornik. 2022;213(3):3-20
3-20
On the cohomology rings of partially projective quaternionic Stiefel manifolds
Abstract
The quaternionic Stiefel manifold $V_{n,k}(\mathbb H)$ is the total space of a fibre bundle over the corresponding Grassmannian $G_{n,k}(\mathbb H)$. The group $\operatorname{Sp}(1)=S^3$ acts freely on the fibres of this bundle. The quotient space is called the quaternionic projective Stiefel manifold. Its real and complex analogues were actively studied earlier by a number of authors. A finite group acting freely on the three-dimensional sphere also acts freely and discretely on the fibres of the quaternionic Stiefel bundle. The corresponding quotient spaces are called partially projective Stiefel manifolds.The cohomology rings of partially projective quaternionic Stiefel manifolds with coefficients in $\mathbb Z_p$, where $p$ is prime, are calculated.Bibliography: 14 titles.
Matematicheskii Sbornik. 2022;213(3):21-40
21-40
41-63
64-80
Bifurcations changing the homotopy type of the closure of an invariant saddle manifold of a surface diffeomorphism
Abstract
It is well known from the homotopy theory of surfaces that an ambient isotopy does not change the homotopy type of a closed curve. Using the language of dynamical systems, this means that an arc in the space of diffeomorphisms that joins two isotopic diffeomorphisms with invariant closed curves in distinct homotopy classes must go through bifurcations. A scenario is described which changes the homotopy type of the closure of the invariant manifold of a saddle point of a polar diffeomorphism of a 2-torus to anyprescribed homotopically nontrivial type. The arc constructed in the process is stable and does not change the topological conjugacy class of the original diffeomorphism. The ideas that are proposed here for constructing such an arc for a 2-torus can naturally be generalized to surfaces of greater genus. Bibliography: 32 titles.
Matematicheskii Sbornik. 2022;213(3):81-110
81-110
Optimal recovery in weighted spaces with homogeneous weights
Abstract
The paper concerns problems of the recovery of operators from noisy information in weighted $L_q$-spaces with homogeneous weights. A number of general theorems are proved and applied to problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of the Laplace operator from a noisy Fourier transform in the $L_p$-metric.Bibliography: 30 titles.
Matematicheskii Sbornik. 2022;213(3):111-138
111-138
On the problem of periodicity of continued fraction expansions of $\sqrt{f}$ for cubic polynomials $f$ over algebraic number fields
Abstract
We obtain a complete description of the fields $\mathbb K$ that are extensions of $\mathbb Q$ of degree at most $3$ and the cubic polynomials $f \in\mathbb K[x]$ such that the expansion of $\sqrt{f}$ into a continued fraction in the field of formal power series $\mathbb K((x))$ is periodic. We prove a finiteness theorem for cubic polynomials $f \in\mathbb K[x]$ with a periodic expansion of $\sqrt{f}$ for extensions of $\mathbb Q$ of degree at most $6$. We obtain a description of the periodic elements $\sqrt{f}$ for the cubic polynomials $f(x)$ defining elliptic curves with points of order $3 \le N\le 42$, $N \ne 37, 41$.Bibliography: 19 titles.
Matematicheskii Sbornik. 2022;213(3):139-170
139-170