This is the first publication of I. M. Vinogravos’s senior thesis written under the scientific guidance of Ya. V. Uspensky at the mathematical department of the Faculty of physics and
mathematics of Petersbourg’s University in 1914.

One of the main problems in combinatorial group theory is the problem of equality and conjugacy of words. It is known that this problem is algorithmically unsolvable in the class of finitely defined groups. The problem arises of studying these problems in certain classes of groups, as well as whether subgroups of this class of groups inherit the algorithmic solvability of the word conjugacy problem.
D. Collins and K. Miller defined a group with a solvable word conjugacy problem containing a subgroup of finite index in which the word conjugacy problem is not solvable. We also construct a group with an unsolvable word conjugacy problem containing a subgroup of finite index with a solvable word conjugacy problem.
E. Artin defined braid groups and proved that the problem of word equality is algorithmically solvable in braid groups. A. A. Markov constructed an algebraic theory of braid groups and reproved, using the constructed theory, the problem of word equality.
F. Garside proved that the conjugacy problem of words in braid groups B_(n+1) is solvable.
Saito, using the ideas Of F. Garside, proved the solvability of the problem of equality and conjugacy of words in Artin groups of finite type. It is known that this class of groups belongs
to braid groups.
The interest is to study the solvability of this problem in subgroups of the class groups, in particular, in the normal divisor generated by the squares forming a group called painted
subgroup of this group.
In [1] it is proved that in a colored subgroup of Artin groups of finite type, the word conjugacy problem is solvable.
It is known that in Artin groups with a tree structure, the word conjugacy problem is also solvable. [2]. In this paper, we prove that colored subgroups of Artin groups with a tree structure
inherit the property of positive solvability of the word conjugacy problem.

In this paper, we study the sharp 𝐿𝑝-Nikol’skii constants for the case of Riemannian symmetric manifolds M𝑑 of rank 1. These spaces are fully classified and include the unit Euclidean sphere S𝑑, as well as the projective spaces P𝑑(R), P𝑑(C), P𝑑(H), P16(Ca). There is a common harmonic analysis on these manifolds, in particular, the subspaces of polynomials Π𝑛(M𝑑) of order at most 𝑛 are defined. In the general case, the sharp 𝐿𝑝-Nikol’skii constant for the subspace 𝑌 ⊂ 𝐿∞ is defined by the equality
𝒞(𝑌,𝐿𝑝) = sup
𝑓∈(𝑌 ∩𝐿𝑝)∖{0}
‖𝑓‖∞
‖𝑓‖𝑝
.
V.A. Ivanov (1983) gave the asymptotics
𝒞(Π𝑛(M𝑑),𝐿𝑝(M𝑑)) ≍ 𝑛𝑑/𝑝, 𝑛 → ∞, 𝑝 ∈ [1,∞).
For the case of a sphere, this result was significantly improved by the author together with F. Dai and S. Tikhonov (2020):
𝒞(Π𝑛(S𝑑),𝐿𝑝(S𝑑)) = 𝒞(ℰ𝑑
1 ,𝐿𝑝(R𝑑))𝑛𝑑/𝑝(1 + 𝑜(1)), 𝑛 → ∞, 𝑝 ∈ (0,∞),
where ℰ𝑑
1 is the set of entire functions of exponential spherical type at most 1 bounded on R𝑑.
M.I. Ganzburg (2020) transferred this equality to the case of the multidimensional torus T𝑑 and trigonometric polynomials. For 𝑑 = 1, these results follow from the fundamental work of
E. Levin and D. Lubinsky (2015).
In a joint work of the author and I.A. Martyanov (2020), the following explicit boundaries of the spherical Nikol’skii constant were proved, which refine the above results for 𝑝 ⩾ 1:
𝑛𝑑/𝑝 ⩽ 𝒞(Π𝑛(S𝑑),𝐿𝑝(S𝑑))
𝒞(ℰ𝑑
1 ,𝐿𝑝(R𝑑))
⩽
(︀
𝑛 + 2⌈𝑑+1
2𝑝 ⌉
)︀𝑑/𝑝
, 𝑛 ∈ Z+, 𝑝 ∈ [1,∞).
This result was proved using a one-dimensional version of the problem for the case of a periodic Gegenbauer weight.
The development of this method allows us to prove the following general result: for 𝑝 ⩾ 1
𝑛𝑑/𝑝 ⩽ 𝒞(Π𝑛(M𝑑),𝐿𝑝(M𝑑))
𝒞(ℰ𝑑
1 ,𝐿𝑝(R𝑑))
⩽
(︀
𝑛 + ⌈𝛼𝑑+3/2
𝑝 ⌉ + ⌈𝛽𝑑+1/2
𝑝 ⌉
)︀𝑑/𝑝
,
where 𝛼𝑑 = 𝑑/2 − 1, 𝛽𝑑 = 𝑑/2 − 1, −1/2, 0, 1, 3 respectively for S𝑑, P𝑑(R), P𝑑(C), P𝑑(H), P16(Ca). The proof of this result is based on the connection of harmonic analysis on M𝑑 with Jacobi analysis on [0, 𝜋] and T with periodic weight
⃒⃒
2 sin 𝑡
2
⃒⃒
2𝛼+1⃒⃒
cos 𝑡
2
⃒⃒
2𝛽+1. Also we give related results for the trigonometric Nikol’skii constants in 𝐿𝑝 on T with Jacobi weight and Nikol’skii
constants for entire functions of exponential type in 𝐿𝑝 on R with power weight.

In spaces with weight |𝑥|−1𝑣𝑘(𝑥), where 𝑣𝑘(𝑥) is the Dunkl weight, there is the (𝑘, 1)- generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, in
problems of quantum mechanics. We define the Riesz potential for the (𝑘, 1)-generalized Fourier transform and prove for it, a (𝐿𝑞,𝐿𝑝)-inequality with radial power weights, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential. For the Riesz potential we calculate the sharp value of the 𝐿𝑝-norm with radial power weights. The sharp value of the 𝐿𝑝-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.

In 2012, Salem Ben Saˇid, Kobayashi, and Orsted defined the two-parametric (𝑘, 𝑎)- generalized Fourier transform, acting in the space with weight |𝑥|𝑎−2𝑣𝑘(𝑥), 𝑎 > 0, where 𝑣𝑘(𝑥) is the Dunkl weight. The most interesting cases are 𝑎 = 2 and 𝑎 = 1. For 𝑎 = 2 the generalized Fourier transform coincides with the Dunkl transform and it is well studied. In case 𝑎 = 1 harmonic analysis, which is important, in particular, in problems of quantum mechanics, has not yet been sufficiently studied. One of the essential elements of harmonic analysis is the bounded translation operator, which allows one to determine the convolution and structural characteristics of functions. For 𝑎 = 1, there is a translation operator 𝜏 𝑦. Its 𝐿𝑝-boundedness
was recently established by Salem Ben Saˇid and Deleaval, but only on radial functions and for 1 ⩽ 𝑝 ⩽ 2. Earlier, we proposed for 𝑎 = 1 a new positive generalized translation operator and
proved that it is 𝐿𝑝 -bounded in 𝑥. In this paper, it is proved that it is 𝐿𝑝 -bounded in 𝑡. For the translation operator 𝜏 𝑦, 𝐿𝑝-boundedness on radial functions is established for 2 < 𝑝 < ∞. The operator 𝑇𝑡 is used to define a convolution and to prove Young’s inequality. For (𝑘, 1)-generalized means defined by convolution, sufficient conditions for 𝐿𝑝-convergence and convergence almost everywhere are established. The fulfillment of these conditions is verified for analogues of the classical summation methods of Gauss–Weierstrass, Poisson, Bochner–Riesz.

markets more and more often. For example, the taxi market in Moscow has transformed from a classical provision of material services — a passenger transportation service — into a market for the provision of information services, where the main players — digital platforms (Yandex taxi, Uber) provide information services to introduce the client-passenger to the supplier-driver. In this regard, it seems natural to consider the reasons why some firms win the market struggle in the new economic environment, while others lose it. The research is based on Frank Bass’s
informational approach, which assumes that the distribution of information about a new product among consumers has the main influence on the distribution of market shares (in our case, the consumer environment will be modeled using a network).
After setting out the basic principles of the Bass model, the model expands significantly, which allows us to describe the interaction of several firms competing within the same market
for an undifferentiated information product. It is necessary to find out exactly how the specific parameters of the models affect the final stable equilibrium position, how firms can and should
influence this parameter in order to win the struggle for market power. The study involves the construction of general theoretical conclusions, based on which, in the future, it will be possible
to move on to the practical part of the study.
The work uses the mathematical apparatus from the elementary theory of differential equations, the theory of catastrophes, the theory of populations. Before we begin, let us intrigue
the reader with the following thought: the market struggle of producers in a limited consumer market can be seen as a competition between two different types of predators for a limited population of prey.

This paper is devoted to the refinement of the results of V. A. Bykovsky on the estimation of the error of approximate integration on the Korobov class 𝐸𝛼 𝑠 for two-dimensional
parallelepipedal grids.
The necessary information from the theory of continued fractions and Euler brackets is given. With the help of the theory of best approximations of the second kind, the Bykovsky set consisting of local minima of the lattice of Dirichlet approximations for a rational number is described.
The Bykovsky set for a two-dimensional lattice of linear comparison solutions is explicitly described. A formula is obtained expressing the hyperbolic parameter of this lattice in terms of denominators of suitable fractions and Euler brackets and allowing it to be calculated in 𝑂(𝑁) arithmetic operations.
Estimates of the hyperbolic zeta function of a two-dimensional lattice of linear comparison solutions are obtained in terms of the Bykovsky sum, which is a partial sum of the zeta series
for the hyperbolic zeta function of the lattice. The partial sum is taken by the Bykovsky set.
For the Bykovsky sum, estimates are obtained from above and from below, from which it follows that the main term for these sums is the sum of the 𝛼-th degrees of the elements of the
continued fraction for 𝑎 𝑁 divided by 𝑁𝛼.
In conclusion, the current directions of research on this topic are noted.

Idempotents of the monoid play different roles in the formation of its properties. A set of idempotents is divided into three parts: incomparable with a unit, less and equal to a unit,
and more and equal to a unit. The idempotents of the first part are called primary and the idempotents comparable with a unit are called secondary. The properties of idempotents are
investigated in terms of partial orders and Green’s equivalences. In the article the main attention is given to finding connections among different classical and non-classical, stable and unstable
partial orders and roles which the idempotents play in that. In particular, as a result, the criterion of stability of Mitsch’s partial order is obtained. Different examples of ordered monoids
are shown in the context of the constructed theory of idempotents and partial orders.

For a number of additive problems with almost equal summands, in addition to the estimates for short exponential sums with primes of the form
𝑆𝑘(𝛼; 𝑥, 𝑦) =
Σ︁
𝑥−𝑦<𝑛⩽𝑥
Λ(𝑛)𝑒(𝛼𝑛𝑘),
in minor arcs, we need to have an estimate of these sums in major arcs, except for a small neighborhood of their centers. We also need to have an asymptotic formula on a small
neighborhood of the centers of major arcs.
In this paper, using the second moment of Dirichlet 𝐿-functions on the critical line, we obtained a nontrivial estimate of the form
𝑆𝑘(𝛼; 𝑥, 𝑦) ≪ 𝑦L−𝐴,
for 𝑆𝑘(𝛼; 𝑥, 𝑦) in major arcs 𝑀(L𝑏), 𝜏 = 𝑦5𝑥−2L−𝑏1 , L = ln 𝑥𝑞, except for a small neighborhood of their centers |𝛼 − 𝑎
𝑞 | >
(︀
2𝜋𝑘2𝑥𝑘−2𝑦2
)︀−1, when 𝑦 ⩾ 𝑥1− 1
2𝑘−1+𝜂𝑘 L𝑐𝑘 , where
𝜂𝑘 =
2
4𝑘 − 5 + 2
√︀
(2𝑘 − 2)(2𝑘 − 3)
, 𝑐𝑘 =
2𝐴 + 22 +
(︁
2
√
√ 2𝑘−3
2𝑘−2 − 1
)︁
𝑏1
2
√︀
(2𝑘 − 2)(2𝑘 − 3) − (2𝑘 − 3)
,
and 𝐴, 𝑏1, 𝑏 are arbitrary fixed positive numbers. Furthermore, and we also proved an asymptotic formula on a small neighborhood of the centers of major arcs.

The problem of diffraction of a plane monochromatic sound wave by an elastic cylinder with a layered inhomogeneous transversely isotropic outer layer is considered. It is assumed that the cylinder is located near a plane with an ideal surface (absolutely hard or acoustically soft).
In order to get rid of the boundary conditions in the plane, in accordance with the so-called imaginary scatterer method, an additional obstacle is introduced in the form of a second elastic
cylinder, which is mirror-like with respect to the initial one on the other side of the plane. The plane itself is excluded from consideration, and the fulfillment of the boundary conditions on
it is ensured by introducing a second incident plane wave with the same amplitude as that of the first. The direction of propagation of the second wave is mirrored to the direction of the original wave relative to the plane. The phase shift in the second wave is equal to the phase shift in the first if the plane is absolutely rigid. If the plane is absolutely soft, the phase shift in
the second wave is shifted relative to the phase shift in the first one by 𝜋. Thus, the problem is reduced to the problem of scattering of two plane waves by two identical elastic cylinders with parallel axes. Assuming that the incident wave propagates along the normal to the cylinder axis, a two-dimensional problem is solved. The solution of the problem in a modified formulation is carried out using the finite element method. Numerical simulation of the solution in the near zone of a scattered acoustic field is carried out. The calculation results show that in a number
of cases of combinations of the parameters of the cylinder and the incident wave, the anisotropy and inhomogeneity of the material properties of the outer layer of the cylinder have a significant effect on the scattered field.

The article defines a class of 𝐷𝑛(𝑥) polynomials by specially designed nodes. Each of 𝐷𝑛(𝑥) is the factor of the Chebyshev polynomial of the first kind 𝑇2𝑛(𝑥). The research task for
polynomials 𝐷𝑛(𝑥) on the interval [0,1] is reduced to find values 𝐷𝑛(𝑥). The article contains exact expressions and estimates of values 𝐷𝑛(𝑥) in special nodes.

The work refers to the direction in the theory of polyhedra in 𝐸3, in which classes of convex polytopes are studied that extend the class of regular (Platonic) polyhedra: polyhedra of such
classes retain only some properties of regular polyhedra.
Earlier, the author found new classes of polyhedra united by such symmetry conditions under which the conditions for the regularity of the faces were not assumed in advance. At the
same time, the completeness of the lists of the considered classes was proved.
Further, the author considered the class of so-called 𝑅𝑅 -polyhedra. A 𝑅𝑅-polyhedron (from the words rombic and regular) is a convex polyhedron that has symmetric rhombic vertices and there are faces that do not belong to any star of these vertices; moreover, all faces that are not included in the star of the rhombic vertex are regular polygons.
If a faceted star 𝑆𝑡𝑎𝑟(𝑉 ) of a vertex 𝑉 of a polyhedron consists of 𝑛 equal and equally spaced rhombuses (not squares) with a common vertex 𝑉 , then 𝑉 is called rhombic. If the vertex 𝑉
belongs to the axis of rotation of the order 𝑛 of the star 𝑆𝑡𝑎𝑟(𝑉 ), then 𝑉 is called symmetric. A symmetric rhombic vertex 𝑉 is called obtuse if the rhombuses of the star 𝑆𝑡𝑎𝑟(𝑉 ) at the vertex
𝑉 converge at their obtuse angles.
An example of an 𝑅𝑅-polyhedron is an elongated rhombododecahedron.
Previously, the author found all 𝑅𝑅-polyhedra with two symmetric rhombic vertices.
In this paper, we consider the question of the existence of closed convex 𝑅𝑅-polyhedra in 𝐸3 with one symmetric obtuse rhombic vertex and regular faces of the same type. A theorem
is proved that there are only two such polyhedra, a 13-faced and a 19-faced. Both of these polyhedra are obtained from the regular — icosahedron. The proof of the existence of a 19-
hedron is based, in particular, on A.D. Aleksandrov’s theorem on the existence of a convex polyhedron with a given unfolding.

In this paper, we consider the Fermat–Steiner problem in hyperspaces with the Hausdorff metric. If 𝑋 is a metric space, and a non-empty finite subset 𝒜 is fixed in the space of nonempty
closed and bounded subsets 𝐻(𝑋), then we will call the element 𝐾 ∈ 𝐻(𝑋), at which the minimum of the sum of the distances to the elements of 𝒜 is achieved, the Steiner astrovertex, the network connecting 𝒜 with 𝐾 — the minimal astronet, and 𝒜 itself — the border. In the case of proper 𝑋, all its elements are compact, and the set of Steiner astrovertices is nonempty.
In this article, we prove a criterion for when the Steiner astrovertex for one-point boundary compact sets in 𝐻(𝑋) is one-point. In addition, a lower estimate for the length of the minimal
parametric network is obtained in terms of the length of an astronet with one-point vertices contained in the boundary compact sets, and the properties of the boundaries for which an exact estimate is achieved are studied. Also bifurcations of Steiner astrovertices under 1-parameter deformation of three-element boundaries in 𝐻(R2), which illustrate geometric phenomena that are absent in the classical Steiner problem for points in R2, are studied.

This work is devoted to the study of one-parameter deformations of metrics. We assume that the lengths of curves are continuous when the parameter changes, and we study additional
conditions that will be sufficient for the continuity of the distances. We start from the presence of the continuity of the lengths of curves, since it is convenient in practice — the continuous dependence of the Riemannian or Finsler metric on the parameter obviously implies the continuity of the lengths of curves, and to obtain the continuity of the distance, it is enough
to check the fulfillment of certain conditions. It is shown in the paper that the compactness of space and the continuity of the lengths of curves when changing the parameter is not enough for the continuity of the distances, and an example is given. In addition, we give special conditions, which are sufficient for the continuity of the distances in combination with the boundedly
compactness of the space. As an application, we consider Finsler manifolds whose metrics continuously depend on a parameter. We show that sufficient conditions for the continuity
of the distance are satisfied on compact Finsler manifolds, from which it follows that the distance function on such manifolds also continuously depends on the parameter. The last result is generalized to complete Finsler manifolds. Since Finsler manifolds are a generalization of Riemannian manifolds, as a corollary we obtain similar results for Riemannian manifolds.

The paper describes the basic properties of polynomial comparisons modulo an ideal in the ring of integers of an algebraic number field, estimates of total rational trigonometric sums from a polynomial over an algebraic field are found, estimates of sums of Dirichlet characters modulo the degree of a prime ideal in an algebraic field are obtained, estimates of multiples of total rational trigonometric sums from polynomials over an algebraic field are given.

The authors of the article set themselves the tasks: to tell about the unexpected and long cooperation and interaction of teachers and scientists of the Tolstoy Tula State Pedagogical University and the Mikhail Iosifovich Kadets Kharkiv School, as well as about
some scientific works of the Kharkiv mathematicians of the Mikhail Iosifovich Kadets School and mathematicians of the city of Tula in the twenty-year period 1986-2006. he role of V. I. Rybakov is particularly noted. Under his leadership, a Tula student conducted scientific work, who later, after studying at the Mikhail Iosifovich Kadets Kharkiv School, became a candidate of physical and mathematical sciences. Vladislav Ivanovich Rybakov obtained deep, meaningful scientific results. For example, you can read about "the classical theory of Rybakov"in books and articles published in the international mathematical press. Mikhail Iosifovich Kadets was interested in Vladislav Ivanovich’s scientific activity. The Kharkiv school of Kadets at that time became
world famous. Not only scientific work, Mikhail Iosifovich paid a lot of attention and effort to pedagogical work. Nineteen of his students, including Rybakov’s student, defended their
PhD theses, seven of them became doctors of science. M. I. Kadets generously shared his mathematical ideas with his students. The article presents some results obtained by Kharkiv
mathematicians of the Kadets school and Tula mathematicians close to them in scientific interests in the period 1986-2006.

isotropic elastic sphere with a continuously inhomogeneous anisotropic elastic layer is considered.
It is believed that the body is placed in an infinite ideal fluid, the laws of heterogeneity of the coating material are described by continuous functions.
An analytical solution to the diffraction problem is obtained for the case when the material of the sphere layer is radially inhomogeneous and transversally isotropic.
Wave field in a containing medium and a homogeneous isotropic sphere are described by expansions in spherical wave functions. A boundary value problem is constructed for a system of ordinary differential equations of the second order for finding displacement fields in an inhomogeneous anisotropic layer of sphere.
The results of numerical calculations of directional patterns for scattered acoustic field in the far zone are presented. It is shown that anisotropy of continuously inhomogeneous elastic layer one can substantially change the scattering characteristics of spherical bodies.

A point set 𝑀 in the Euclidean plane is said to be a planar integral point set if all the distances between the elements of 𝑀 are integers, and 𝑀 is not situated on a straight line.
A planar integral point set is said to be a set in semi-general position, if it does not contain collinear triples. The existing lower bound for mininal diameter of a planar integral point set is
linear with respect to its cardinality. There were no known special diameter bounds for planar integral point sets in semi-general position of given cardinality (the known upper bound for
planar integral point sets is constructive and employs planar integral point sets in semi-general position). We prove a new lower bound for minimal diameter of planar integral point sets in
semi-general position that is better than linear (polynomial of power 5/4). The proof is based on several lemmas and observations, including the ones established by Solymosi to prove the first linear lower bound for diameter of a planar integral point set.

Aking into account the microplasticity in the vicinity of the pores, an effective discharge equation is obtained. The influence of porosity on the residual deformation under uniaxial tension of a sample of a porous metal composite is taken into account. The nonlinearity in the deformations of the discharge curve, which is caused by microplasticity, is established. Numerical calculations of the residual strain as a function of the initial stress and porosity are given.

Many real dynamical systems are characterized by the presence of a set coexisting attractors.
This property of systems is called multistability. In multistable systems, a sudden transition to unwanted or unknown attractors can occur. Such a transition can lead to catastrophic events. It
turned out that multistability is also associated with the emergence of unpredictable attractors, which are called hidden attractors. An attractor is called hidden if its area of attraction does not intersect with small neighborhoods of an unstable fixed point. One of the defining reasons for studying multistable chaotic systems with different characteristics is a wide range of their potential engineering applications - in control theory, computer science, cryptology, artificial neural networks, image encryption, secure communication, and weak signal detection. In recent years, researchers have turned to developing methods for artificially designing systems with desired dynamics. In this case, the main efforts are focused on creating systems with an infinite
number of coexisting attractors - extremely multistable and megastable systems. It turned out that such systems open up new possibilities for solving some applied problems, for example, for realizing control of the signal amplitude and polarity in engineering systems or for creating new image encryption systems. In this paper, a new smooth three-dimensional dynamical system is constructed, reversible in time, containing an analytical solution and a strange multifractal hidden attractor. The basin of attraction of the attractor includes almost all three-dimensional space, and its dimension is "almost 3". By replacing one of the variables of the system with a periodic function of this variable, a system is constructed that has a 1-D strip of hidden chaotic attractors of dimension "almost 3"and, at the same time, an infinite number of analytical solutions. A special transformation of the latter system allows us to design a megastable system with a 2-D strip of hidden attractors.

The mechanical properties of composite and polymer materials widely used in engineering are analyzed. It is confirmed that the absolute majority of them have structural anisotropy of different classes. In addition, it is shown that these structural materials often exhibit a sensitivity of the deformation characteristics to the type of stress state. Due to the fact that classical mathematical models describing the states of such materials lead to gross errors in the calculation of structural elements, and the well-known, specially developed theories for them are quite contradictory and have significant drawbacks, the authors propose an energy model of the determining relations for media with structural and deformation anisotropies. This model is
based on the use of the normalized stress tensor space, which has an undoubted advantage over the singular functions and parameters having an infinite interval of change, which are used in the known versions of the theories of deformation of materials with double anisotropy. As a specific class of structural anisotropy, orthotropic materials are accepted, for which the strain potential defined in the main structural axes is postulated. By differentiating the formulated potential according to the recommendations of the Castigliano rules, the equations of connection of two tensors of the second rank - strains and stresses - are established. It is shown that these equations
have a nonlinear form, which aggravates the problem of uniqueness of solutions to boundary value problems. To identify the resulting model of the defining equations, we recommend an
experimental program that includes mechanical tests for uniaxial tension and compression along the main axes of the anisotropy of the material, as well as for a net shift in the three planes
of orthotropy. The main technical constants of a number of composite and polymer materials widely used in engineering are given. On the basis of the use of the postulate about the positive
certainty of the energy surface, the consistency of the proposed strain potential is verified. Using this test, we prove the uniqueness theorem for solving boundary value problems in the mechanics of a deformable solid. Taking into account the rules of transformation of the components of the second-rank tensors when the axes of the selected coordinate system are rotated, it is shown that the stresses calculated in the main axes of orthotropy are recalculated in the new system according to traditional formulas.