Vol 27, No 1 (2021)

The aim of this paper is to study the solvability of solution of non-local problem with integral condition in spatial variables for high-order linear equation in the classe of regular solutions (which have all the squared derivatives generalized by S.L. Sobolev that are included in the corresponding equation). It is indicated that at first similar problems were studied for high-order equations either in the one-dimensional case, or under certain conditions of smallness by the value of T. A list of new works for the multidimensional case is also given. In this paper, we present new results on the solvability of non-local problem with integral spatial variables for high-order equation a) in the multidimensional case with respect to spatial variables; b) in the absence of smallness conditions by the value T; however, this condition exists for the kernel K(x; y; t). The research method is based on obtaining a priori estimates of the solution of the problem, which implies its existence and uniqueness in a given space.

The paper is devoted to the Berlekamp — Masssey algorithm and its equivalent version based on the extended Euclidean algorithm. An optimized Berlekamp — Massey algorithm is also given for the case of a field of characteristic 2. The Berlekamp — Massey algorithm has a quadratic complexity and is used, for example, to solve systems of linear equations in which the matrix of the system is the Toeplitz matrix. In particular, such systems of equations appear in algorithms for the syndrome decoding of BCH codes, Reed — Solomon codes, generalized Reed — Solomon codes, and Goppa codes. Algorithms for decoding the listed codes  based on the Berlekamp — Massey algorithm are given.

Proposed work is the sixth work of the cycle on differential and topological diagnostics. It is shown that the diagnostics in the case of trajectorial measurements corrupted by noise, which is a stochastic process of the normal white noise type with zero mean value and bounded spectrum, can be performed by using the diagnostic algorithms obtained in [5], i.e., the results of this section remain valid even in this rather general case; moreover, the diagnostic functional, which was introduced in the theorem of [5] a priori, is now obtained a posteriori.

The article discusses the production of prompt isolated photons with large transverse momenta at the LHC at energies √s = 8 and 13 TeV in the parton Reggeization approach, which is based on the factorization theorem for hard processes at high energies and the effective theory of Reggeized gluons and quarks by L.N. Lipatov. Unintegrated parton distributions in the parton Reggeization approach were obtained in the modified Kimber–Martin–Ryskin model proposed earlier by the authors of the article. In numerical calculations, only the contribution of the main parton process, R +Q(¯ Q) → γ + q(¯q), is taken into account, since the contribution of other processes does not exceed 5–10 %. The calculation results are compared with the predictions obtained in the collinear parton model. Good agreement of calculations in the parton Reggeization approach with experimental data obtained by the ATLAS collaboration is shown.