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Yakuninagalina@yandex.ru UDC 517.9The modern theory of fully nonlinear second order partial differential equations is based on some algebraic facts and, in particular, on the theory of a-hyperbolic polynomials created by L. Gårding in 1959. The goal of this paper is to describe the Gårding cones in this context. Bibliography: 30 titles.

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The large (X-ray class > M1) and very large (X-ray class >X1) flares (according to the observations of GOES-15 and Preliminary data from Current Catalog of Flare Events) in solar activity cycle 24 were analyzed. The monthly average values of optical Flare Index for 2010 -2016 were calculated. The values of the total energy of the flare E (J · m −2 ) in the 0.1 -0.8 nm range at the level of the earth's atmosphere were estimated. The energy spectrum (the dependence of the number of flares with the full energy E from the value of this full energy) for 115 flares of M5 -X7 classes was built. The comparative study of monthly averaged values of several indices of solar activity in current cycle 24: the relative sunspot numbers (SSN), the 10.7 cm radio flux (F 10.7 ), the radiation flux in the Lyman-alpha line (F Lα ), the solar constant (TSI) and the Flare Index (FI) was made.

The solar activity in the current, that is, the 24-th, sunspot cycle is analyzed. Cyclic variations in the sunspot number (SSN) and radiation fluxes in various spectral ranges have been estimated in comparison with the general level of the solar radiation, which is traditionally determined by the radio emission flux F 10.7 at a wavelength of 10.7 cm (2.8 GHz). The comparative analysis of the variations in the solar constant and solar indices in the UV range, which are important for modeling the state of the Earth's atmosphere, in the weak 24th cycle and strong 22nd and 23rd cycles showed relative differences in the amplitudes of variations from the minimum to the maximum of the cycle. The influence of the hysteresis effect between the activity indices and F 10.7 in the 24-th cycle, which is considered here, makes it possible to refine the forecast of the UV indices and solar constant depending on the quadratic regression coefficients that associate the solar indices with F 10.7 depending on the phase of the cycle.

We prove that the Dirichlet problem is solvable in a generalized sense ]or a class of nonlinear elliptic equations and related equations of Monge-Ampdre type. Bibliography: 1~ titles. v u 1 [ Jm-l( )= n-m+i jv(u~);'~-l[u]dx' m=2,...,n, f~ where/~ = 1, and the operators #m-l," m = 1, .. . , n, are defined by recursion as the variational derivatives for the functionals J,~,_2(u); u U ~ u 1= It turns out [9, 11], that #V~[u] --~:m[A, u]with A(p) = vp-~ 1, where vpp is the Hessian matrix for v(p). In particular for the function v = X/1 + p2 of the matrix A = X/1 + p2(~hj. + pipj) we Obtain the curvature operator of order m with a vector k defining the principal curvatures of the graph of the function u(x). If v = p2/2, then A = E, the vector k is the vector of eigenvalues of the matrix u~, and we are dealing with operators of Monge-Amp&re type. It might be more correct to call them operators of Laplacian type.

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Abstract. The flare activity and the ultraviolet emission of the sun during its 24-th cycle are analysed. As compared to cycles 21-23, where the most powerful flares were observed during the decay phase, in cycle 24 the greatest number of powerful flares (>X2.7) occurred in the rising phase and at the maximum with the exception of the two largest flares of cycle 24 X9.3 and X8.2 in September 2017. We showed that regression fits of solar UV indices to the overall radiation level from the sun are substantially different for cycle 24 compared to cycles 21-23. It is found that for the flare of August 9, 2011 (SDO and GOES-15 observations), the flare propagates in a direction from the upper corona to the transition region and to the chromosphere. A study of the N-S asymmetry in the distribution of the flares in cycle 24 reveals a strong predominance of flares in the N-hemisphere in 2011 and in the S-hemisphere in 2014. It is also found that during cycles 23 and 24, the delays in the onset of proton events relative to the onset of the flares that cause them have a distribution with a distinct maximum corresponding to a delay of 2 hours for protons with energies >10 MeV, as well as for those with energies >100 MeV.