卷 57, 编号 2 (2016)

We give criteria for the fundamental groups of compact Sol-3-manifolds to be residually nilpotent and residually finite p-groups.

We consider linear equations for analytic functions in the complex plane with cuts along a half of the boundary of a quadrangle. We propose a regularization method that reduces the equations to an equation with summary-difference kernels. Some applications are given to the moment problem for entire functions of exponential type.

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovskiĭ, and E. S. Zhukovskiĭ studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on n-fold coincidence points and n-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.

A formation F is superradical provided that: (1) F is a normally hereditary formation; (2) each group G = AB, where A and B are F-subnormal F-subgroups in G, belongs to F. We give an example of a hereditary superradical formation that is not soluble saturated. This gives a negative answer to Problem 14.99(b) in The Kourovka Notebook.

We obtain descriptions for the classes of maximal graph surfaces on four-dimensional two-step sub-Lorentzian structures. In particular, we deduce maximality conditions in terms of sub-Lorentzian mean curvature.

Under study are the metric, algebraic, spectral, and characteristic properties of measure compact, almost compact, and integral operators of the first, second, and third kind in spaces of integrable functions.

Some method is proposed for finding Ein components in moduli spaces of stable rank 2 vector bundles with first Chern class c₁ = 0 on the projective 3-space. We formulate and illustrate a conjecture on the growth rate of the number of Ein components in dependence on the numbers of the second Chern class. We present a method for calculating the spectra of the above bundles, a recurrent formula, and an explicit formula for computing the number of the spectra of these bundles.

We illustrate some well-known facts about the evolution of the 3-sphere (S³, g) generated by the Ricci flow. We define the Dirac flow and study the properties of the metric \(\bar g = dt^2 + g(t)\), where g(t) is a solution of the Dirac flow. In the case of a metric g conformally equivalent to the round metric on S³ the metric \(\bar g\) is of constant curvature. We study the properties of solutions in the case when g depends on two functional parameters. The flow on differential 1-forms whose solution generates the Eguchi–Hanson metric was written down. In particular cases we study the singularities developed by these flows.

We obtain sharp coefficient bounds for some p-valent starlike functions of order β, 0 ≤ β < p. Initially this problem was handled by Aouf in [1]. We pointed out that the proof by Aouf is incorrect and present a correct proof.