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- ДокументOn homotopy nilpotency of some suspended spaces(2021) Marek GolasinskiA homological criterium from [Golasiński, M., On homotopy nilpotency of loop spaces of Moore spaces, Canad. Math. Bull. (2021), 1–12] is applied to investigate the homotopy nilpotency of some suspended spaces. We investigate the homotopy nilpotency of the wedge sum and smash products of Moore spaces M (A, n) with n ≥ 1. The homotopy nilpotency of homological spheres are studied as well.
- ДокументOn symmetry reduction and some classes of invariant solutions of the (1+3)-dimensional homogeneous Monge-Ampère equation(2021) Vasyl Fedorchuk, Volodymyr FedorchukWe study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.
- ДокументOn tensor products of nuclear operators in Banach spaces(2021) Oleg ReinovThe following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.
- ДокументOn the Koebe Quarter Theorem for Polynomials(2022) Олександр Михайлович Стоколос, Jimmy Dillies, Dmitriy Dmitrishin, Andrey SmorodinThe Koebe One Quarter Theorem states that the range of any Schlicht function contains the centered disc of radius 1/4 which is sharp due to the value of the Koebe function at −1. A natural question is finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials. In particular, it was asked in [7] whether Suffridge polynomials [15] are optimal. For polynomials of degree 1 and 2 that is obviously true. It was demonstrated in [10] that Suffridge polynomials of degree 3 are not optimal and a promising alternative family of polynomials was introduced. These very polynomials were actually discovered earlier independently by M. Brandt [3] and D. Dimitrov [9]. In the current article we reintroduce these polynomials in a natural way and make a far-reaching conjecture that we verify for polynomials up to degree 6 and with computer aided proof up to degree 52. We then discuss the ensuing estimates for the value of the Koebe radius for polynomials of a specific degree.