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Vol 210, No 9 (2019)
- Year: 2019
- Articles: 6
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7457
Convergence of formal Dulac series satisfying an algebraic ordinary differential equation
Abstract
A sufficient condition is proposed which ensures that a Dulac series that formally satisfies an algebraic ordinary differential equation (ODE) is convergent. Such formal solutions of algebraic ODEs are quite common: in particular, the Painleve III, V and VI equations have formal solutions given by Dulac series; they are convergent in view of the sufficient condition presented. Bibliography: 13 titles.
Matematicheskii Sbornik. 2019;210(9):3-18
3-18
Higher colimits, derived functors and homology
Abstract
We develop a theory of higher colimits over categories of free presentations. We show that different homology functors such as Hochschild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes' exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory, we show that the third reduced $K$-functor can be defined as the colimit of the second reduced $K$-functor applied to the fibre square of a free presentation of an algebra. We also prove a Hopf-type formula for odd-dimensional cyclic homology of an algebra over a field of characteristic zero. Bibliography: 17 titles.
Matematicheskii Sbornik. 2019;210(9):19-58
19-58
Hyperelliptic systems of sequences of rank 4
Abstract
Sequences of complex numbers satisfying functional relations of bilinear type are investigated. The results obtained are used in describing all 1-periodic entire functions $f\colon \mathbb C\to\mathbb C$ such that the expansion ${f(x+y)f(x-y)}=\varphi_1(x)\psi_1(y)+…+\varphi_4(x)\psi_4(y)$ holds for some $\varphi_j,\psi_j\colon\mathbb C\to\mathbb C$.Bibliography: 38 titles.
Matematicheskii Sbornik. 2019;210(9):59-88
59-88
Algebras of free holomorphic functions and localizations
Abstract
We consider the algebras of holomorphic functions on a free polydisc $\mathscr F^T(\mathbb D_R^n)$, $\mathscr F(\mathbb D_R^n)$ and the algebra of holomorphic functions on a free ball $\mathscr F(\mathbb B_r^n)$. We show that the algebra $\mathscr F(\mathbb D_R^n)$ is a localization of a free algebra and, moreover, is a free analytic algebra with $n$ generators (in the sense of J. Taylor), while the algebra $\mathscr F(\mathbb B_r^n)$ is not a localization of a free algebra. In addition we prove that the class of localizations of free algebras and the class of free analytic algebras are closed under the operation of the Arens-Michael free product. Bibliography: 21 titles.
Matematicheskii Sbornik. 2019;210(9):89-106
89-106
Local existence conditions for sweeping process solutions
Abstract
A sufficient condition for the existence of an absolutely continuous solution for a sweeping process is given by the absolute continuity, in a definite sense, of the multivalued mapping which generates the process. This property is described in terms of the Hausdorff distance between values of the multivalued mapping. However, there exist multivalued mappings for which the Hausdorff distance between those values is infinite; for instance, mappings which take hyperplanes as values. For such mappings absolute continuity cannot be described in terms of the Hausdorff distance. In this paper we study conditions which provide local absolute continuity of a multivalued mapping. By using these conditions we prove an existence theorem for an absolutely continuous solution of a sweeping process. We apply the results obtained to the study of sweeping processes with nonconvex and with convexified perturbations. For such sweeping processes we prove an existence theorem for solutions and a relaxation theorem. Bibliography: 13 titles.
Matematicheskii Sbornik. 2019;210(9):107-128
107-128
Weakly monotone sets and continuous selection in asymmetric spaces
Abstract
Sets admitting a continuous selection from the set of near best approximations are studied. Applications of the geometric theory of approximation to the existence of continuous selections for the sets of $n$-link piecewise linear functions, $n$-link piecewise polynomial functions and generalizations thereof are also discussed. Bibliography: 23 titles.
Matematicheskii Sbornik. 2019;210(9):129-152
129-152