# Vol 27, No 1 (2021)

**Year:**2021**Articles:**8**URL:**https://journals.ssau.ru/est/issue/view/514

## Full Issue

## Статьи

### A PROBLEM WITH NONLOCAL CONDITION FOR ONE-DIMENSIONAL HYPERBOLIC EQUATION

#### Abstract

In this paper, we study the problem with a dynamic nonlocal condition for the one-dimensional hyperbolic equation, which occurs in the study of rod vibrations. This problem may be used as a mathematical model of

longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. Conditions have been obtained for input data, providing unambiguous resolution of the task, proof of the existence and singularity of the problem in the space of Sobolev. The proof is based on the a priori estimates obtained in this paper, Galerkin’s procedure and the properties of the Sobolev spaces.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):7-14

### BOUNDARY VALUE PROBLEM WITH A NONLOCAL BOUNDARY CONDITION OF INTEGRAL FORM FOR A MULTIDIMENSIONAL EQUATION OF IV ORDER

#### Abstract

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.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):15-28

### FACTORIZATION OF ORDINARY AND HYPERBOLIC INTEGRO-DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS IN A BANACH SPACE

#### Abstract

The solvability condition and the unique exact solution by the universal factorization (decomposition) method for a class of the abstract operator equations of the type B1u = Au − SΦ(A0u) − GF(Au) = f, u ∈ D(B1),

where A,A0 are linear abstract operators, G, S are linear vectors and Φ, F are linear functional vectors is investigagted. This class is useful for solving Boundary Value Problems (BVPs) with Integro-Differential Equations (IDEs), where A,A0 are differential operators and F(Au), Φ(A0u) are Fredholm integrals. It was shown that the operators of the type B1 can be factorized in the some cases in the product of two more

simple operators BG, BG0 of special form, which are derived analytically. Further the solvability condition and the unique exact solution for B1u = f easily follow from the solvability condition and the unique exact solutions for the equations BGv = f and BG0u = v.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):29-43

### ON THE BERLEKAMP — MASSEY ALGORITHM AND ITS APPLICATION FOR DECODING ALGORITHMS

#### Abstract

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.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):44-61

### ON SOME CRYPTOSYSTEMS BASED ON ALGEBRAIC CODES

#### Abstract

In 1978 McEliece built the first public key cryptosystem based on error-correcting codes. At the same time, effective attacks on the secret keys of this cryptosystem have not yet been found. The work describes the classical and modernized cryptosystems of McEliece and Niederreiter, also examples of their practical application based on Goppa codes using the Patterson algorithm. Also the algorithms of two-step authentication protocols with zero disclosure based on error-correcting codes are given.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):62-73

### PROBLEMS OF DIFFERENTIAL AND TOPOLOGICAL DIAGNOSTICS. PART 6. STATISTICAL SOLVING OF THE PROBLEM OF DIFFERENTIAL DIAGNOSTICS

#### Abstract

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.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):74-80

### NONLINEAR DYNAMIC EQUATIONS FOR ELASTIC MICROMORPHIC SOLIDS AND SHELLS. PART I

#### Abstract

The present paper develops a general approach to deriving nonlinear equations of motion for solids whose material points possess additional degrees of freedom. The essential characteristic of this approach is the

account of incompatible deformations that may occur in the body due to distributed defects or in the result of the some kind of process like growth or remodelling. The mathematical formalism is based on least action principle and Noether symmetries. The peculiarity of such formalism is in formal description of reference shape of the body, which in the case of incompatible deformations has to be regarded either as a continual family of shapes or some shape embedded into non-Euclidean space. Although the general approach yields equations for Cosserat-type solids, micromorphic bodies and shells, the latter differ significantly in the formal description of enhanced geometric structures upon which the action integral has to be defined. Detailed discussion of this disparity is given.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):81-103

### PRODUCTION OF ISOLATED PHOTONS WITH LARGE TRANSVERSE MOMENTA AT LHC IN THE REGGE LIMIT OF QCD

#### Abstract

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.

**Vestnik of Samara University. Natural Science Series**. 2021;27(1):104-110