Vol 237, No 1 (2019)

In this paper, we consider boundary-value problems with shift and show that such problems are equivalent to boundary-value problems for generalized analytic functions. We interpret the shift as a change of the complex structure on the complex plane with a given closed curve.

We discuss equilibrium configurations of the Coulomb potential of positive point charges with positions satisfying certain quadratic constraints in the plane and three-dimensional Euclidean space. The main attention is given to the case of three point charges satisfying a positive definite quadratic constraint in the form of equality or inequality. For a triple of points on the boundary of convex domain, we give a geometric criterion of the existence of positive point charges for which the given triple is an equilibrium configuration. Using this criterion, rather comprehensive results are obtained for three positive charges in the disc, ellipse, and three-dimensional ball. In the case of the circle, we strengthen these results by showing that any configuration consisting of an odd number of points on the circle can be realized as an equilibrium configuration of certain nonzero point charges and give a simple criterion for existence of positive charges with this property. Similar results are obtained for three point charges each of which belongs to one of the three concentric circles. Several related problems and possible generalizations are also discussed.

We present several results concerning the geometry and topology of quadratic mappings. The main attention is given to certain basic properties, namely, properness, surjectivity, stability, and topology of fibers. The structure of singular sets, discriminants, bifurcation diagrams, and Pareto sets is also discussed. After describing the setting and algebraic methods for computing the topological degree and Euler characteristic that are crucial for our approach, we concentrate on the study of quadratic endomorphisms and quadratic mappings into the plane. We begin by considering homogeneous quadratic endomorphisms of the plane and give criteria of properness and possible values of topological degree, as well as some geometric information about the structure of singular sets and discriminants. Next, we deal with analogs of the above results for proper quadratic endomorphisms in arbitrary dimension. In particular, we obtain an explicit estimate for the topological degree of quadratic endomorphism in terms of dimension and present examples showing that this estimate is exact. After this we discuss homogeneous quadratic mappings from ℝⁿ into the plane and obtain a number of results on the Euler characteristic and topology of fibers. Finally, we derive some corollaries in the case of numerical range mapping of a complex square matrix.