This work is the first, and main, of a series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results.

No abstract

We consider multicomponent local Poisson structures of the form P 3 + P 1 , under the assumption that the third order term P 3 is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. of linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular we show that each such Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with vertices labeled by natural numbers whose sum is the dimension of the manifold. These pencils are naturally related to certain (polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the Nijenhuis pencil.

The improvement of technologies for the collection and primary processing of secondary nonferrous metals is a prerequisite for the development of secondary non-ferrous metallurgy and the establishment of solid waste processing enterprises. One such technology is eddy-current (electrodynamic) separation in a traveling magnetic field. The main structures of electrodynamic separation installations are considered. Widely distributed separators based on linear inductors. Such separators easily fit into technological lines and are used in the extraction of non-ferrous metals from various types of metal-containing solid waste (automobile scrap, mixed production and municipal waste, cable and electronic scrap, etc.), as well as in the processing of complex non-ferrous scrap in preparation it to metallurgical redistribution. Еlectrodynamic separation problems can be conditionally divided into two groups. The first group includes the tasks of separating non-ferrous metals from non-metals, and the second group involves more complex tasks of sorting metals according to their physical properties. In both cases, the final results of separation depend on the combined action of electromagnetic and mechanical forces on metallic particles. This article describes some of the results of studies of electrodynamic separators based on linear inductors. The prospects for the use of linear inductors that create opposite direction traveling magnetic fields are shown. The use of such linear inductors in electrodynamic separators to remove non-ferrous metals from solid waste allows for an increase in the yield of metals. The use of opposite direction traveling magnetic fields in the induction sorting of metals and alloys provides increased plant productivity and improved sorting quality. It is shown that the distribution of the magnetic field and electromagnetic forces along the inductors depend on the three-phase winding circuit. The main attention is focused on the choice of the three-phase winding of the linear inductor, which provides these advantages. The article presents the results of experimental studies of prototypes of electrodynamic separators based on linear inductors, created in the laboratory of the Department of Electrical Engineering and Electrotechnological Systems of the Ural Federal University. The article presents the results of studies of of dual-purpose linear induction machines created in the laboratory of the Academic Department of Electrical Engineering and Electrotechnological Systems of the

No abstract