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## An excitation mechanism for discrete chorus elements in the magnetosphere

A beam pulsed amplifier mechanism responsible for effective amplification of short very low

frequency (VLF) electromagnetic pulses is proposed. Effective amplification near the magnetic

equator outside the plasmasphere is considered. A conditional growth rate of short electromagnetic

pulses is calculated. Obtained results can explain some important features of the oblique

electromagnetic chorus emissions without hiss-like radiation background.

The nonlinear response of the equatorial current jet to external actions is considered, as well as some geophysical effects produced by it. Irradiating the current jet by shortwave electromagnetic radiation with amplitude modulation in ranges of VLF emission and geomagnetic pulsations, one can provide for con ditions when a nonlinear antenna is realized in the current jet region. Estimations have shown that under the current jet on the ground surface electromagnetic signals on modulation frequencies can be two orders of magnitude larger than at middle latitudes due to modulated electron temperature and density. The effect of signals from the modulated equatorial current jet on the modes of operation of the plasma magnetospheric maser in VLF waveband is considered. The resonance modification of the spectra of natural electromagnetic waves of VLF range in the magnetosphere is shown to be possible.

We study the problem of energy exchange between waves and particles, which leads to energization of the latter, in an unstable plasma typical of the radiation belts. The ongoing Van Allen Probes space mission brought this problem among the most discussed in space physics. A free energy which is present in an unstable plasma provides the indispensable condition for energy transfer from lower energy particles to higher-energy particles via resonant wave-particle interaction. This process is studied in detail by the example of electron interactions with whistler mode wave packets originated from lightning-induced emission. We emphasize that in an unstable plasma, the energy source for electron energization is the energy of other particles, rather than the wave energy as is often assumed. The way by which the energy is transferred from lower energy to higher-energy particles includes two processes that operate concurrently, in the same space-time domain, or sequentially, in different space-time domains, in which a given wave packet is located. In the first process, one group of resonant particles gives the energy to the wave. The second process consists in wave absorption by another group of resonant particles, whose energy therefore increases. We argue that this mechanism represents an efficient means of electron energization in the radiation belts.

We propose a simple explanation of the prolonged existence of pancake-like electron velocity distributions inthe radiation belts. The pancake-like distribution function is characterized by a longitudinal particle velocity (along themagnetic field) of the order of the thermal velocity of the background plasma. The parameters of the tablet-like distribution function with a characteristic longitudinal particle velocity of the order of 20 Alfvèn velocities are refined. Such distribution functions can occur in the middle magnetosphere near the magnetic equator with appropriate sources of energetic particles. The stability of these distributions is examined. The results agree with known experimental data.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

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.