Vol 62, No 12 (2018)

In approximation theory, logarithmic derivatives of complex polynomials are called simple partial fractions (SPFs) as suggested by Dolzhenko. Many solved and unsolved extremal problems, related to SPFs, are traced back to works of Boole, Macintyre, Fuchs, Marstrand, Gorin, Gonchar, and Dolzhenko. Now many authors systematically develop methods for approximation and interpolation by SPFs and their modifications. Simultaneously, related problems, being of independent interest, arise for SPFs: obtaining inequalities of different metrics, estimation of derivatives, separation of singularities, etc.

In introduction to this survey, we systematize some of these problems. In themain part, we formulate principal results and outline methods to prove them whenever possible.

We discuss the problems of the connections of the modern theory of integrability and the corresponding overdetermined linear systems with works of geometers of the late nineteenth century. One of these questions is the generalization of the theory of Darboux–Laplace transforms for second-order equations with two independent variables to the case of three-dimensional linear hyperbolic equations of the third order. In this paper we construct examples of such transformations. We consider applications to the problem of orthogonal curvilinear coordinate systems in ℝ³.

If for any injective endomorphism α and surjective endomorphism β of an abelian group there exists its endomorphism γ such that βα = αγ (respectively, αβ = γα), then we say that the group possesses the R-property (respectively, the L-property). We show that if a reduced torsionfree group possesses the R-property or the L-property, then the endomorphism ring of the group is normal. We describe divisible groups and direct sums of cyclic groups possessing the R-property or the L-property.