Vol 210, No 11 (2019)

We study the period mapping from the moduli space of real hyperelliptic curves to a Euclidean space. The mapping arises in the analysis of Chebyshev's construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of the moduli space into polyhedra labelled by planar graphs allows us to investigate the global topology of low-dimensional fibres of the period mapping. Bibliography: 23 titles.

We construct asymptotics with respect to a small parameter of a boundary layer solution of the boundary value problem for a system of two ordinary differential equations, one second order and the other first order, with a small parameter multiplying the derivatives in both equations. Systems of this type arise in chemical kinetics as stationary processes in models of fast reactions in the absence of diffusion for one of the reactants. An essential feature of the problem under study is a double root of one of the equations of the degenerate system. This leads to a qualitative change in the boundary layer component of the solution by comparison with the case when all the roots are simple. The boundary layer becomes multizoned, while the standard algorithm for constructing boundary layer series is no longer suitable and has to be replaced by a new one. Bibliography: 13 titles.

The spectrum of a two-dimensional problem in elasticity theory is investigated for a body $\Omega^h$ with a cuspidal sharpening with a short tip of length $h>0$ that is broken off. It is known that when the tip is in place, the spectrum of the problem for $\Omega^0$ has a continuous component $[\Lambda_\dagger,+\infty)$ with positive cut-off point $\Lambda_\dagger>0$. We show that each point $\Lambda>\Lambda_\dagger$ is a ‘blinking’ eigenvalue, that is, it is an actual eigenvalue of the problem in $\Omega^h$ ‘almost periodically’ in the scale of $|\ln h|$. Among families of eigenvalues $\Lambda^h_{m(h)}$, which continuously depend on $h$, we discover ‘gliding’ eigenvalues, which fall down along the real axis at a great rate, $O((\Lambda^h_{m(h)}-\Lambda_\dagger)h^{-1}|\ln h|^{-1})$, but then land softly on the threshold $\Lambda_\dagger$. This reveals a new way of forming the continuous spectrum of the problem for a cuspidal body $\Omega^0$ from the system of discrete spectra of the problems in the $\Omega^h$, $h>0$. In addition, there may be ‘hardly movable’ eigenvalues, which remain in a small neighbourhood of a fixed point for all small $h$, in contrast to ‘gliding’ eigenvalues. Bibliography: 30 titles.