Showing papers in "Siberian Mathematical Journal in 1985"
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TL;DR: In this paper, the authors define a partial ordering of the set K of all sequences of natural numbers of the form (ml,..~,mr) with ml ~ ~ m r by putting (m,.... mr) ~ (m~..... m E) if and only if r ~ t and mri ~ mt_ i (i = 0, 1,..'.''r -I).
Abstract: In this paper we study subvarieties of the variety ~N~ of Lie algebras over a field K of characteristic 0 ( ~ denotes the variety of all Abelian Lie algebras, and Nc stands for the variety of all nilpotent Lie algebras of class
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TL;DR: In this paper, the possibility of describing families of recursively enumerable sets, whose index sets belong to a given class of the arithmetical hierarchy, is studied in terms of completely enumerable families.
Abstract: In [I] the problem about the possibility of "structural" discription of the families of recursively enumerable sets, whose index sets belong to a given class of the arithmetical hierarchy, is posed. In this article, we will study the possibility of this description in terms of completely enumerable families, and that too in the following more general situation: for abstract classes of numbered sets and for various hierarchies.
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TL;DR: In this article, it was shown that inf J = -o for F O > O for the case P > 0, and inf J > 0 for every (possibly non-definite) form of the problem with r > 0.
Abstract: We can show that inf J = -~o for F O. The above optimization problem has been extensively studied for F > O. The first results can be found in [I-3], they are related to the case P>0, ~(x, u)~0 (Vx, u). In [4] (see Introduction) the solution of this problem is given for every (possibly nondefinite) form ~(z, "u) with r > 0. This solution is based on the "frequency theorem" of [4]. In the same work the connection is established between this problem and the "Lur'e solving equations," which were introduced by Lur'e in [5] and which were used in fifties in numerous works on stability of control systems. (In seventies the Lur'e equations were named "algebraic Riceati equations" in the foreign literature. )
TL;DR: In this article, it was shown that the limit cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves.
Abstract: In this paper it is shown that cycles of planar dynamical systems with polynomial vector fields are in many ways similar to ovals of algebraic curves, and more generally to "separating solutions" of such systems, on general algebraic curves. For example, for separating solutions the analog of Bezout's theorem is valid. The proofs are straightforward. They are based on the version of Rolle's theorem and Bezout's theorem for planar algebraic curves offered below. We note that according to Hilbert's conjecture the total number of limit cycles of a dynamical system with a polynomial vector field is bounded above by the degree of the field. From the validity of the conjecture and the results of the present note it follows that the curve consisting of all the limit cycles of a polynomial dynamical system resembles an algebraic curve. At the present time, even the finiteness of the number of limit cycles has not been proved: Ii'yashenko proved that Dyudaktsproof [i] contains an unfillable gap (cf. [2]) . I. Rolle's Theorem for Dynamical Systems We consider a smooth dynamical system on the plane