# Vol 27, No 2 (2021)

**Year:**2021**Articles:**8**URL:**https://journals.ssau.ru/est/issue/view/533

## Articles

### ON DECODING ALGORITHMS FOR GENERALIZED REED — SOLOMON CODES WITH ERRORS AND ERASURES. II

#### Abstract

The article is a continuation of the authors’ work «On decoding algorithms for generalized Reed — Solomon codes with errors and erasures». In this work, another modification of the Gao algorithm and the Berlekamp — Massey algorithm is given. The first of these algorithms is a syndrome-free decoding algorithm, the second is a syndrome decoding algorithm. The relevance of these algorithms is that they are applicable for decoding Goppa codes, which are the basis of some promising post-quantum cryptosystems.

**Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya / Vestnik of Samara University. Natural Science Series**. 2021;27(2):7-15

### CRITICAL TRAVELLING WAVES IN ONE MODEL OF THE ”REACTION-DIFFUSION” TYPE

#### Abstract

The paper is devoted to the order reduction for critical traveling wave problems for a reaction – diffusion type systems. The mathematical apparatus is based on the geometric theory of singular perturbations and the canards technique. The use of the method of invariant manifolds of singularly perturbed systems allows us to replace the study of traveling waves of the original PDE system by analyzing their profiles in a ODE system of a lower order.

**Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya / Vestnik of Samara University. Natural Science Series**. 2021;27(2):16-24

### ON A CHARACTERISTIC OF STRONGLY EMBEDDED SUBSPACES IN SYMMETRIC SPACES

#### Abstract

It is shown that the presence of a lower p - estimate with constant 1 in the symmetric space E is sufficient for the condition of equivalence of convergence in norm and in measure on the subspace H of the space E to be satisfied if and only if the numerical characteristic ηE(H) < 1. The last criterion is also valid for symmetric spaces ”close ”to L1, more precisely, for which an analog of the Dunford - Pettis criterion of weak compactness is valid. In particular, it is shown that spaces ”close” to L1, have the binary property: the characteristic ηE(H) takes only two values, 0 and 1. This gives an example of binary Orlicz spaces different from the spaces Lp.

**Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya / Vestnik of Samara University. Natural Science Series**. 2021;27(2):25-32

### USE OF THE INDIRECT BOUNDARY ELEMENTS METHOD FOR ISOTROPIC PLATES ON AN ELASTIC WINKLER BASE AND PASTERNAK — VLASOV BASE

#### Abstract

The calculation tasks combining lightness, economy, high strength and reliability of thin-walled structures on an elastic base are relevant for modern mechanical engineering. In this regard, the use of isotropic materials on an elastic base seems justified, therefore their calculation is considered in this article. The problems of the theory of plates and shells belong to the class of boundary value problems, the analytical solution of which, due to various circumstances (the nonlinearity of differential equations, the complexity of geometry and boundary conditions, etc.), cannot be determined. Numerical methods help to solve this problem. Among numerical methods, undeservedly little attention is paid to the boundary element method. In this regard, the further development of indirect compensating loads method for solving problems of the theory of isotropic plates on an elastic base of Winkler and Pasternak — Vlasov, based on the application of exact fundamental solutions, is relevant.

### SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR ANISOTROPIC PLATES AND SHELLS BY BOUNDARY ELEMENTS METHOD

#### Abstract

Modern mechanical engineering sets the tasks of calculating thin-walled structures that combine lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of anisotropic materials and plastics seems justified. The problems of the theory of plates and shells belong to the class of boundary value problems, the analytical solution of which, due to various circumstances (nonlinearity of differential equations, complexity of geometry and boundary conditions, etc.), cannot be determined. Numerical methods help to solve this problem. Among numerical methods, undeservedly little attention is paid to the boundary element method. In this regard, the further development of indirect method of compensating loads for solving problems of the anisotropic plates and shells theory based on the application of exact fundamental solutions is relevant. The paper considers the application of the indirect boundary element method for solving of an anisotropic plates and shells nonlinear deformation problem. Since the kernels of the system of singular integral equations to which the solution of the problem is reduced are expressed in terms of the fundamental solution and its derivatives, first of all, the article provides a method for determining the fundamental solutions to the problem of bending and the plane stress state of an anisotropic plate. The displacement vector is determined from the solution of linear equations system describing the bending and plane stress state of an anisotropic plate. The solution of the system is performed by the method of compensating loads, according to which the area representing the plan of the shallow shell is supplemented to an infinite plane, and on the contour that limits the area, compensating loads are applied to the infinite plate. Integral equations of indirect BEM are given. In this paper, the study of nonlinear deformation of anisotropic plates and shallow shells is carried out using the “deflection – load” dependencies. The deflection at a given point on the median surface of the shell was taken as the leading parameter.

### INFLUENCE OF CURVATURE OF THE CRACK TIP RADIUS ON STRESSES

#### Abstract

In experimental mechanics, when conducting research on models, the question often arises of whether it is legitimate to replace a crack with a cut, whether the radius of curvature of the cut will have a large effect on the magnitude of stresses near its apex. In order to understand these questions and give answers to them, a number of experiments were carried out on samples made of piezo-optical material (Plexiglass of E2 grade). In the models, the crack was simulated using a cut, then a hole was made at the top of the cut with a drill. The models were investigated in pure bending by the photoelasticity method. Stress fields were obtained in two batches of samples at different loads. The intensity of stresses near cracks-cuts at different radius of curvature of their tops was determined by using the experimental data. An assessment of the influence of the crack-cut tip radius curvature on the magnitude of stresses near it has been carried out.

### INFLUENCE OF THE PARAMETERS OF THE BOTTOM BOREHOLE ZONE ON THE OWN VIBRATIONS OF THE LIQUID IN THE PUMP COMPRESSOR TUBING

#### Abstract

The problem of natural oscillations of a liquid column in a tubing string, arising after a sudden opening or closing of a vertical well (water hammer), is considered. For this, a mathematical model has been built that describes the dynamics of the fluid column in the well and the filtration flow in the bottomhole zone, analytical solutions of the system of equations have been obtained. To determine the frequency, period, coefficient and decrement of damping of oscillations, a characteristic equation is found. The impact of such parameters as the length of the open section, the perforation zones of the well, the length of the tubing string, the permeability coefficient on the dynamic nature of natural pressure fluctuations has been analyzed.

### SCENARIOS MODEL OF THE EFFECT OF A TEMPORARY SHARP REDUCTION OF POPULATION WITH A LARGE REPRODUCTIVE PARAMETER

#### Abstract

Our ongoing research is devoted to various aspects of predicting invasive processes in unstable biosystems. Extreme events are interesting for modeling. The purpose of this work is to describe in a computational experiment a scenario of active counteraction, which temporarily suppresses the development of an aggressive invasive process. The impact in a situation of slow regulation begins to affect not the small initial group N(0) ≈ L of individuals of the invading species, but only when the critical population threshold is reached. Relevance — let us consider in the model a scenario that can be interpreted as an artificially created resistance in case of delayed immune activation. In most cases, after invasion, the presence of the species remains, but below its biological optimum. Method — a modification of the equation with two delays is used. Novelty — a model has been obtained where it is possible to overcome the crisis or the death of the population, depending on the time of activation of the impact. The oscillatory scenario is not observed in the model. The equation with a threshold reaction assumes further expansion and use in the composition of multicomponent polymodel complexes.