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The paper is a review of the latest results of I. R. Kayumov and S. Ponnusamy on the Bohr inequality. An exact estimate in the strong Bohr inequality is obtained and the Bohr-Rogosinski radius for a certain class of subordinations is examined. All results are exact., Author(s): A. A. Ismagilov [sup.1], A. V. Kayumova [sup.1], I. R. Kayumov [sup.1], S. Ponnusamy [sup.2] Author Affiliations: (1) grid.77268.3c, 0000 0004 0543 9688, Kazan (Volga region) Federal University, , [...]

Let [p.sub.nN] be the probability of successful allocation of n groups of particles in N cells with the following assumptions: (a) each group contains m particles and has allocation as a general allocation scheme; (b) each cell contains at most r particles from the same group; (c) events connected with different groups are independent. We obtain an asymptotically exact bound of [p.sub.nN] as n, N [right arrow] [infinity] such that n/N is bounded. Applications to problems in error-correcting coding are considered., UDC 519.212.2+517.537 1. Introduction and Main Result We will consider a general allocation scheme of n independent series of particles in N cells (for an account of such an allocation [...]

In this work, we explore the properties of antiferromagnetic cycloid and the phase transitions between commensurate and incommensurate magnetic states in epitaxial BiFeO3 film. Additional magnetic anisotropy induced by strain effects in the films allocates cycloids with the definite directions of spin rotation. Peculiar feature of the cycloids propagating in the films whose symmetry is different from the single crystals is the orientation of spin rotational plane that does not contain electric polarization in contrast with the bulk materials. We construct a diagram of phase transitions induced by magnetic field applied along normal to the surface and show considerable decrease of the strength of magnetic field destroying cycloid in films compared with the bulk. [ABSTRACT FROM AUTHOR]