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Том 237, № 1 (2019)
- Год: 2019
- Статей: 6
- URL: https://ogarev-online.ru/1072-3374/issue/view/14985
Article
Boundary-Value Problems with Shift and Beltrami Systems
Аннотация
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.
1-29
Singular Generalized Analytic Functions
Аннотация
In this paper, we consider solution spaces for some class of singular elliptic systems on Riemann surfaces and boundary-value problems for solution spaces of such systems. We also discuss some relations for the kernels of the Carleman–Vekua equation. In particular, representations of these kernels in the form of generalized power functions are completely analogous to the classical Cauchy kernel expansion. The obtained results are applied to some problems of the theory of generalized analytic functions.
30-109
Equilibria of Three Point Charges with Quadratic Constraints
Аннотация
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.
110-125
Average Discrete Energies of Spectra of Gaussian Random Matrices
Аннотация
We present exact formulas for the averages of various discrete energies of certain point processes in the plane and two-dimensional sphere. Specifically, we consider point processes defined by the spectra of gaussian random matrices and their inverse images under stereographic projection on the Riemann sphere. The main attention is given to the discrete Coulomb energy, discrete logarithmic energy, and their analogs involving geodesic distances on the Riemann sphere. It is shown that the average discrete energies are expressed by integrals of certain special type taken over the Riemann sphere. This enables us to estimate their values and asymptotics as the size of random matrix tends to infinity. Analogous results are obtained for the three-point energy function considered by M. Atiyah. We also present several related conjectures and possible generalizations.
126-134
Remarks on Quadratic Mappings
Аннотация
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.
135-146
Extremal Connectors for Disjoint Circles
Аннотация
We discuss a number of geometric extremal problems for configurations of points on several circles in the plane. Circles are assumed to be nested or have disjoint interior domains. For disjoint circles, we study the problem of minimal connecting cycle and Morse theory of perimeter function. It is shown that minimal connecting cycle is unique if the pairwise convex hulls of circles do not intersect any other circle. For concentric circles, the main attention is given to the critical points of perimeter considered as a function on the product of concentric circles. In this setting, we prove that aligned configurations are nondegenerate critical points of perimeter and give formulas for their Morse indices. If the number of circles does not exceed 4 we prove that the perimeter is a Morse function and describe the shape of maximal connecting cycles. Similar problems are studied for the oriented area of connecting cycle. In conclusion we briefly discuss some possible generalizations.
147-156