Vol 38, No 2 (2017)

In this present work, we continue studying the Nikol’skii and Bernstein–Walsh type estimations for complex algebraic polynomials in the bounded and unbounded regions bounded by asymptotically conformal curve.

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism M whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary ∂M. In addition to the standard conformal ergodic action of a uniformhyperbolic lattice on the round sphere Sⁿ⁻¹ and its quasiconformal deformations in Sⁿ, we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed n-ball into Sⁿ), on non-trivial (n − 1)-knots in Sⁿ⁺¹, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from “non-standard” components of its varieties of representations (faithful or with large kernels of defining homomorphisms).

We set forth solvability conditions and construction of the generalized Green’s operator for Noetherian boundary value problem for the matrix differential-algebraic equations and solvability conditions and the constructive scheme for constructing solutions of nonlinearNoetherian boundary value problem for matrix differential-algebraic equation. We show that the principal results in the theory of weakly nonlinear periodic oscillations remain valid for nonlinear Noetherian boundary value problem for matrix differential-algebraic equation. The study is illustrated by the periodic problem for a Duffing-type matrix differential equation.

The central part of the paper consists of a theorem on generalized nonions governing dynamical systems modelling of special ternary, quaternary, quinary, senary, etc. structures, due to the third named author. Let M_n(C), n ≥ 2, be the set of n × n-matrices with complex entries. The theorem states that in M_n(C) there exists a basis such that PQ− λ^sQP = 0, s = 0, 1, 2,.., n − 1, where {P,Q,}, u, v are specified in Section 1, formulae (1) and (2).The particular cases n = 2, 3, 4 with other choices of u, v were discussed by James Joseph Sylvester (1883, 1884) and by Charles Sanders Peirce (1882).In particular, λ = j, j³ = 1, j ≠ 1, generates nonions. Before the section on the above theorem and its visualization on a two-sheeted Riemann surface, we give three physical motivations for the topic: controlled noncommutativity: Sylvester–Peirce approach vs. Max Planck approach (1900), supersonic flow of a ternary alloy in gas, and changing hexagonal to pentagonal structure in pentacene.

In this paper, the Liénard system of differential equations is considered. The domain of location of the limit cycle of this system is estimated. The hypothesis under which the Liénard system has the unique stable limit cycle are also stated.

In terms of prime ends by Caratheodory, it is studied the boundary behavior of the socalled lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec as well as to finitely bi–Lipschitz mappings that a far–reaching extension of the known classes of isometric and quasiisometric mappings.

We discuss some new aspects in the theory of extremal problems for conformal mappings.

We obtain explicitly principal extensions of analytic functions of the complex variable into an infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We consider an extension of differentiable in the sense of Gâteaux functions with values in a topological vector space being an expansion of the mentioned algebra and its relations to spatial potentials.

The phase space of the closed string theory may be identified with the space of smooth loops. This reduces the problem of quantization of string theory to the quantization of the space of smooth loops. In this paper we describe the solution of the latter problem obtained in a series of papers. But the symplectic form of string theory is correctly defined not only on the space of smooth loops but also on its Hilbert completion coinciding with the Sobolev space of half-differentiable functions. So it is reasonable to consider this space as the phase manifold of non-smooth string theory. There is a natural group associated with this Sobolev space, namely the group of quasisymmetric homeomorphisms of the circle acting by change of variable. Unfortunately, this action is not smooth. However, we are able to quantize the Sobolev space of half-differentiable functions provided with the action of the group of quasisymmetric homeomorphisms using methods of noncommutative geometry.

This article is a survey of the earlier results of the author. We study space mappings with branching that satisfy modulus inequalities. For classes of these mappings, we obtain several sufficient conditions for the normality of families. Moreover, we prove that the normal families of the so-called Q-mappings have the logarithmic order of growth in a neighborhood of a point. We obtain a result on the normal families of open discrete mappings f: D → C {a, b} from the class W_loc^1,1 with finite distortion that do not take at least two fixed values a ≠ b in C whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions.

We consider the functions h_μ,ν introduced by Zastavnyi in 2002. The family of these functions is a subfamily of Buhmann’s functions and contains the families of functions introduced by Trigub in 1987 and Wendland in 1995. We investigate the problems of positive definiteness and smoothness at zero for the linear combinations β₂^εh_μ,ν (x/β₂) − β₁^ε_μ,ν (x/β₁).