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Vol 84, No 2 (2020)
- Year: 2020
- Articles: 8
- URL: https://journal-vniispk.ru/1607-0046/issue/view/7543
Articles
A new approach to the question of the existence of bounded solutions of functional differential equations ofpoint type
Abstract
We develop an approach which we used to deduce conditions of a new type for the existence of periodic solutions ofordinary differential equations and functional differential equations of point type. These conditions are based onthe use of asymptotic properties of solutions of differential equations which can be observed on shifts of solutionsand stated in terms of averages over the period on a distinguished sphere in the phase space. The development ofthis approach enables us to obtain conditions for the existence of bounded solutions for the same classes offunctional differential equations.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):3-42
3-42
Greedy approximation by arbitrary set
Abstract
We define various algorithms for greedy approximations by elements of an arbitrary set $M$ in a Banach space. We study the convergence of these algorithms in a Hilbert space under various geometric conditions on $M$. As a consequence, we obtain sufficient conditions for the additive semigroup generated by $M$ to be dense.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):43-59
43-59
On singularly perturbed systems of ODE with a multiple root of the degenerate equation
Abstract
We consider a boundary-value problem for a system of two second-order ODE with distinct powers of asmall parameter at the second derivative in the first and second equations. Whenone of the two equations of the degenerate system has a double root, the asymptotic behaviour ofthe boundary-layer solution of the boundary-value problem turns out to be qualitatively different from the knownasymptotic behaviour in the case when those equations have simple roots. In particular,the scales of the boundary-layer variables and the very algorithm for constructing the boundary-layer seriesdepend on the type of the boundary conditions for the unknown functions. We construct and justify asymptoticexpansions of the boundary-layer solution for boundary conditions of a particular type. These expansions differfrom those for other boundary conditions.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):60-89
60-89
Conditions of modularity of the congruence lattice of an act over a rectangular band
Abstract
We describe polygons over rectangular bands that have modular, distributive or linearly ordered congruence lattice. It turns out that such polygons have at most 11 elements, and their congruence lattice has at most 300 elements. Furthermore, certain facts are established about the structure of polygons with modular congruence lattice over an arbitrary semigroup and about the structure of the congruence lattice of a polygon over a rectangular band. The work is based on the description of polygons over a completely (0-)simple semigroup obtained by Avdeev and Kozhukhov in 2000 and on the characterization of disconnected polygons with modular or distributive congruence lattice by Ptakhov and Stepanova in 2013.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):90-125
90-125
Vaught's conjecture for weakly $o$-minimal theories of finite convexity rank
Abstract
We prove that weakly $o$-minimal theories of finite convexity rank having lessthan $2^{\omega}$ countable models are binary. Our main result is theconfirmation of Vaught's conjecture for weakly $o$-minimal theories of finiteconvexity rank.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):126-151
126-151
$p$-adic monomial equations and their perturbations
Abstract
In this paper, we describe the set of solutionsof the monomial equation $x^k=a$ over $\mathbb Q_p$. Moreover, asan application, we study some perturbations of the equation under consideration over the $p$-adic field.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):152-165
152-165
Some trigonometric polynomials with extremely small uniform norm and their applications
Abstract
We construct orthogonal trigonometric polynomials satisfying a new spectral conditionand such that their $L^{1}$-norms are bounded below and the uniform norm of their partial sums has extremely small order of growth. We obtain new results that relate the uniform norm and $\mathrm{QC}$-norm on subspaces of the vector space of trigonometric polynomials.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):166-196
166-196
On $S$-units for valuations of the second degree in hyperelliptic fields
Abstract
In this paper we propose a new effective approach to the problem of finding and constructing non-trivial $S$-units of a hyperelliptic field $L$ for a set $S=S_h$ consisting of two conjugate valuations of the second degree. The results obtained are based on a deep connection between the problem of torsion in the Jacobians of hyperelliptic curves and the quasiperiodicity of continued $h$-fractions, that is, generalized functional continued fractions of special form constructed with respect to a valuation of the second degree. We find algorithms for searching for fundamental $S_h$-units which are comparable in effectiveness with known fast algorithms for two linear valuations.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2020;84(2):197-242
197-242