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卷 210, 编号 11 (2019)
- 年: 2019
- 文章: 6
- URL: https://bakhtiniada.ru/0368-8666/issue/view/7459
Sufficient conditions for the stability of linear periodic impulsive differential equations
摘要
Abstract linear periodic impulsive differential equations are considered. The impulse effect instants are assumed to satisfy the average dwell-time condition (the ADT condition). The stability problem is reduced to studying the stability of an auxiliary abstract impulsive differential equation. This is a perturbed periodic impulsive differential equation, which considerably simplifies the construction of a Lyapunov function. Sufficient conditions for the asymptotic stability of abstract linear periodic impulsive differential equations are obtained. It is shown that the ADT conditions lead to less conservative dwell-time estimates guaranteeing asymptotic stability. Bibliography: 24 titles.
Matematicheskii Sbornik. 2019;210(11):3-23
3-23
Combinatorial analysis of the period mapping: the topology of 2D fibres
摘要
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.
Matematicheskii Sbornik. 2019;210(11):24-57
24-57
Schur's criterion for formal power series
摘要
A criterion for when a formal power series can be represented by a formal Schur continued fraction is stated. The proof proposed is based on a relationship, revealed here, between Hankel two-point determinants of a series and its Schur determinants. Bibliography: 10 titles.
Matematicheskii Sbornik. 2019;210(11):58-75
58-75
Asymptotic behaviour of a boundary layer solution to a stationary partly dissipative system with a multiple root of the degenerate equation
摘要
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.
Matematicheskii Sbornik. 2019;210(11):76-102
76-102
Commuting homogeneous locally nilpotent derivations
摘要
Let $X$ be an affine algebraic variety endowed with an action of complexity one of an algebraic torus $\mathbb T$. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions $\mathbb K[X]$ can be described in terms of proper polyhedral divisors corresponding to the $\mathbb T$-variety $X$. We prove that homogeneous locally nilpotent derivations commute if and only if a certain combinatorial criterion holds. These results are used to describe actions of unipotent groups of dimension two on affine $\mathbb T$-varieties.Bibliography: 10 titles.
Matematicheskii Sbornik. 2019;210(11):103-128
103-128
‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings
摘要
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.
Matematicheskii Sbornik. 2019;210(11):129-158
129-158