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Vol 214, No 7 (2023)
Geodesic flow on an intersection of several confocal quadrics in $\mathbb{R}^n$
Abstract
By the Jacobi-Chasles theorem, for each geodesic on an n-axial ellipsoid in n-dimensional Euclidean space, apart from it there exist also n−2 quadrics confocal with it that are tangent to all the tangent lines of this geodesic. It is shown that this result also holds for the geodesic flow on the intersection of several nondegenerate confocal quadrics. As in the case of the Jacobi-Chasles theorem, this fact ensures the integrability of the corresponding geodesic flow. For each compact intersection of several nondegenerate confocal quadrics its homeomorphism class is determined, and it turns out that such an intersection is always homeomorphic to a product of several spheres. Also, a sufficient condition for a potential is presented which ensures that the addition of this potential preserved the integrability of the corresponding dynamical system on the intersection of an arbitrary number of confocal quadrics.
3-26
27-41
Values of the $\mathfrak{sl}_2$ weight system on the chord diagrams whose intersection graphs are complete graphs.
Abstract
A weight system is a function on the chord diagrams that satisfies Vassiliev's
Although the definition of the
42-59
Metric description of flexible octahedra
Abstract
A new description of flexible Bricard octahedra is obtained using conditions in terms of edge lengths. It is suitable for the study of a number of problems in the metric geometry of octahedra and, in particular, for searching for a proof of the conjecture of Sabitov on the vanishing of all but the leading coefficients of the polynomial for the volume of a type 3 octahedron.
60-90
Spectral gaps in a thin-walled rectangular infinite box with a periodic family of cross-walls
Abstract
The Dirichlet spectral problem for the Laplace operator is considered in an infinite thin-walled rectangular box with a periodic family of cross walls whose thickness is proportional to that of the walls. Using asymptotic analysis it is shown that spectral gaps open up in the case of ‘thin’ or ‘sufficiently thick’ cross-walls whose relative thickness is bounded above or below by certain characteristics of model Dirichlet problems in L- and T-shaped domains in the plane and in a union of two pairwise orthogonal halves of space layers and a quarter of a space layer. A number of open questions are stated; in particular, because of the lack of information on threshold resonances in the three-dimensional model problem, the structure of the spectrum for cross walls of any intermediate thickness remains unknown.
91-133
Logarithmic character of large time asymptotics for solutions of Sobolev type nonlinear equations with cubic nonlinearity
Abstract
The Cauchy problem of the form
{i∂t(u−∂2xu)+∂2xu−a∂4xu=u3,t>0, x∈R,u(0,x)=u0(x),x∈R,
is considered for a Sobolev-type nonlinear equation with cubic nonlinearity, where a>1/5, a≠1. It is shown that the asymptotic behaviour of the solution is characterized by an additional logarithmic decay in comparison with the corresponding linear case. To find the asymptotics of solutions of the Cauchy problem for a nonlinear Sobolev-type equation, factorization technique is developed. To obtain estimates for derivatives of the defect operators, L2-estimates of pseudodifferential operators are used.
134-160