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## The long-term oscillations in sunspots and related inter-sunspot sources in microwave emission.

This work presents the microwave long-term oscillations with periods of a few tens of minutes obtained from Nobeyama radioheliograph (NoRH) at frequency 17 GHz. In two active regions the fluctuations of radio emission of different types of intersunspot sources (ISS) (compact and extended) were compared with the fluctuations in magnetic fields of sunspots. Common periods in variations of microwave emission of different type of sources and magnetic field of sunspots were discovered. The delay of 17 minutes was revealed for oscillations of the extended ISS with respect to variations of magnetic field of its tail sunspot. The model of the sunspot magnetic structure based on the concept of three magnetic fluxes for explanation of this fact is discussed.

Radio-observations allow us to reveal the long-lived (2–5 days) intersunspot sources (ISS), whose centers are often located above the neutral line of the magnetic field separating leading and following parts of a whole active region (the first type of ISS (ISS-I)) or above the neutral line separating magnetic polarities into complex sunspots (the second type of ISS (ISS-II)). ISS-I and ISS-II demonstrate gyrocyclotron or gyrosynchrotron spectra, more dynamic pre-flare behavior than ISS-III with bremsstrahlung in the quiet active regions. The qualitative model of “three magnetic fluxes” explaining the origin of accelerated particles in ISS and their long-lasting existence and spectral features is proposed.

We have recently introduced an irregularity index *λ *for daily sunspot numbers *ISSN*, derived from the well-known Lyapunov exponent, that attempts to reflect irregularities in the chaotic process of solar activity. Like the Lyapunov exponent, the irregularity index is computed from the data for different embedding dimensions *m *(2-32). When *m* = 2, *λ* maxima match *ISSN* maxima of the Schwabe cycle, whereas when *m* = 3, *λ *maxima occur at *ISSN* minima. The patterns of *λ* as a function of time remain similar from *m* = 4 to 16: the dynamics of *λ *change between 1915 and 1935, separating two regimes, one from 1850 to 1915 and the other from 1935 to 2005, in which *λ* retains a similar structure. A sharp peak occurs at the time of the *ISSN* minimum between cycles 23 and 24, possibly a precursor of unusual cycle 24 and maybe a new regime change. *λ* is significantly smaller during the ascending and descending phases of solar cycles. Differences in values of the irregularity index observed for different cycles reflect differences in correlations in sunspot series at a scale much less than the 4-yr sliding window used in computing them; the lifetime of sunspots provides a source of correlation at that time scale. The burst of short-term irregularity evidenced by the strong *l*-peak at the minimum of cycle 23-24 would reflect a decrease in correlation at the time scale of several days rather than a change in the shape of the cycle.

We define, calculate and analyze irregularity indices λISSN of daily series of the International Sunspot Number ISSN as a function of increasing smoothing from *N* = 162 to 648 days. The irregularity indices λ are computed within 4-year sliding windows, with embedding dimensions *m* = 1 and 2. λISSN displays Schwabe cycles with ~5.5-year variations ("half Schwabe variations" HSV). The mean of λISSN undergoes a downward step and the amplitude of its variations strongly decreases around 1930. We observe changes in the ratio *R* of the mean amplitude of λ peaks at solar cycle minima with respect to peaks at solar maxima as a function of date, embedding dimension and, importantly, smoothing parameter *N*. We identify two distinct regimes, called Q1 and Q2, defined mainly by the evolution of *R* as a function of *N*: Q1, with increasing HSV behavior and *R* value as *N* is increased, occurs before 1915–1930; and Q2, with decreasing HSV behavior and *R* value as *N* is increased, occurs after ~1975. We attempt to account for these observations with an autoregressive (order 1) model with Poissonian noise and a mean modulated by two sine waves of periods *T*1 and *T*2 (*T*1 = 11 years, and intermediate *T*2 is tuned to mimic quasi-biennial oscillations QBO). The model can generate both Q1 and Q2 regimes. When *m* = 1, HSV appears in the absence of *T*2 variations. When *m* = 2, Q1 occurs when *T*2 variations are present, whereas Q2 occurs when *T*2 variations are suppressed. We propose that the HSV behavior of the irregularity index of ISSN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity.

Numerical simulation of the oscillation thick walled shell electromagnetoelasticity

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.