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## Обменное взаимодействие электронных и позитронных сгустков

We analyze the role of local geometry in the spin and orbital interaction in transition metal compounds with orbital degeneracy. We stress that the tendency observed in the most studied case (transition metals in O6 octahedra with one common oxygen—common corner of neighboring octahedra—and with ~180° metal–oxygen–metal bonds), that ferro-orbital ordering renders antiferro-spin coupling and, vice versa, antiferro-orbitals give ferro-spin ordering, is not valid in the general case, in particular, for octahedra with a common edge and with ~90° M–O–M bonds. Special attention is paid to the “third case,” that of neighboring octahedra with a common face (three common oxygens), which has largely been disregarded until now, although there are many real systems with this geometry. Interestingly enough, the spin-orbit exchange in this case turns out to be simpler and more symmetric than in the first two cases. We also consider, which form the effective exchange takes for different geometries in the case of strong spin–orbit coupling.

The numerical simulation of nonlinear time-dependent processes of the resonant interaction condensations charges in electron-positron substance with the formation of quantum macroplasmoid in terms of classical large particles model and quantum macroscopic model of single-particle wave functions has been conducted. Unlike the point kinematic approach of quantum electrodynamics, the particles are treated as deformable clots charge.

The paper presents a framework for numerical simulation that allows you to ensure saving of resources due to the numerical selection of the optimal size and temperatures in the preparation of bimetallic castings. Modeling obtained boundary and initial conditions at which the metal parts submelting first layer in the contact area with the second layer and is saved in the unmelted state of the first layer with a thickness of 1.5-2 mm, which is in contact with the mold.

Electron-positron medium is considered as a promising implementation of powerful microwave devices on multibeam electron and positron streams. Simulation of the resonant interaction processes in electron-positron substance is carried out using macroscopic wave functions of electrons and positrons of macroscopic quantum theory. It is shown that at optimum space charge observed nonlinear resonance exchange process leading to the compensation of the Coulomb field.

In the present study, issues related to the hydrogeology of the basin of the Volga River from Rybinsk to Cheboksary Reservoir are reviewed and analyzed, evaluation of the current state of hydrogeology reservoirs on various parameters is performed. It is revealed that the erosion processes in the basin of the Gorky Reservoir has an average intensity in comparison with similar processes in the basins of the Rybinsk and Cheboksary reservoirs, but the activity is presented. Particular attention to the processes of erosion and shoreline erosion of the Gorky Reservoir is given. The mathematical and numerical model of the slope stability coefficient is presented.

A new mathematical model of heat transfer in silicon field emission pointed cathode of small dimensions is constructed which permits taking its partial melting into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type problems. The approach used by the authors is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the obtained mathematical model including its parallel implementation. The results of numerical simulation conclude the book.

The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.

In the present work the results of different scenario of the cliff of Cape Canaille hypothetic collapse (South of France) are presented. Three scenarios were considered: falling of one block, falling of several blocks in one time and debris flow avalanche. The analysis of the entire scenario was done.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.