Vol 102, No 3-4 (2017)

Series in multiplicative systems χ with generalized monotone coefficients are studied. Necessary and sufficient Hardy–Littlewood type conditions for the sums of such series to belong to the Lorentz space are proved. As corollaries, we establish estimates of best approximation in the system χ and Konyushkov-type theorems on the equivalence of O- and ≍-relations for the weighted sums of the Fourier coefficients in the system χ and for the best approximations.

The consistency of the existence of a countable definable set of reals, containing no definable elements, is established. The model, where such a set exists, is obtained by means of a countable product of Jensen’s forcing with finite support.

It is proved that every uniquely divisible Abelian semigroup admits an injective subadditive embedding in a convex cone. As an application, the classical theory of generators of one-parameter operator semigroups is generalized to the case in which the parameter ranges over a uniquely divisible semigroup.

A problem with inhomogeneous boundary and initial conditions is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in a rectangular domain. The solution is constructed as the sum of an orthogonal series. A criterion for the uniqueness of the solution is established. It is shown that the uniqueness of the solution and the convergence of the series depend on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. On the basis of this problem, inverse problems for finding the factors of the time-dependent right-hand sides of the original equation of mixed type are stated and studied for the first time. The corresponding uniqueness theorems and the existence of solutions are proved using the theory of integral equations for inverse problems.

The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exactGreen function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.

In a Morrey space, the product of the convolution operator with summable kernel and the operator of multiplication by an essentially bounded function is considered. Sufficient conditions for such a product to be compact are obtained. In addition, it is shown that the commutator of the convolution operator and the operator of multiplication by a function of weakly oscillating type is compact in a Morrey space.

The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.

This note is devoted to the study of the smoothness of weak solutions to the Cauchy problem for three-dimensional magneto-hydrodynamic system in terms of the pressure. It is proved that if the pressure π belongs to L₂(0, T, Ḃ_∞,∞⁻¹(ℝ³)) or the gradient field of pressure ∇_π belongs to L_2/3(0, T, BMO(ℝ³)), then the corresponding weak solution (u, b) remains smooth on [0, T].

New lower bounds for the chromatic number of the rational space in the Minkowski metric for certain irrational values of the forbidden distance are obtained.

For a holomorphic function f, f′(0) ≠ 0, in the unit disk U, we establish a geometric constraint on the image f(U) for which the classical Kraus inequality |S_f (0)| ≤ 6 holds; earlier, it was known only in the case of the conformal mapping of f. Here S_f (0) is the Schwarzian derivative of the function f calculated at the point z = 0. The proof is based on the strengthened version of Lavrent’ev’s theorem on the extremal decomposition of the Riemann sphere into two disjoint domains.

The author attempts to change and supplement the standard scheme of partitions of integers in number theory to make it completely concur with the Bohr–Kalckar correspondence principle. In order to make the analogy between the the atomic nucleus and the theory of partitions of natural numbers more complete, to the notion of defect of mass author assigns the “defect” \(\overline {\left\{ a \right\}} \) = [a + 1] − a of any real number a (i.e., the fractional value that must be added to a in order to obtain the nearest larger integer). This allows to carry over the Einstein relation between mass and energy to a relation between the whole numbers M and N, where N is the number of summands in the partition ofM into positive summands, as well as to define the forbidding factor for the number M, and apply this to the Bohr–Kalckar model of heavy atomic nuclei and to the calculation of the maximal number of nucleons in the nucleus.

The paper is devoted to the study of the properties of a class of space mappings that is more general than that of bounded distortion mappings (aka quasiregular mappings). It is shown that the locally uniform limit of a sequence of mappings f: D → ℝⁿ of a domain D ⊂ ℝⁿ, n ≥ 2, satisfying one inequality for the p-modulus of families of curves is zero-dimensional. This statement generalizes a well-known theorem on the openness and discreteness of the uniform limit of a sequence of bounded distortion mappings.