Search

Copyright of Bulletin of Ablai Khan KazUIRandWL: Series 'Pedagogical Sciences' is the property of Kazakh Ablai Khan University of International Relations & World Languages and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigate a system of linear ordinary differential equations containing point and integral loadings with nonlocal boundary conditions. Boundary conditions include integral and point values of the unknown function. An essential feature of the problem is that the kernels of the integral terms in the differential equations depend only on the integration variable. It is shown that similar problems arise during feedback control of objects with both lumped and distributed parameters during point and integral measurements of the current state of the controllable object. The problem statement considered in the paper generalizes a lot of previously studied problems regarding loaded differential equations with nonlocal boundary conditions. By introducing auxiliary parameters, we obtain necessary conditions for the existence and uniqueness of a solution to the problem under consideration. To solve the problem numerically, we propose to use a representation of the solution to the original problem, which includes four matrix functions that are solutions to four auxiliary Cauchy problems. Using solutions to the auxiliary problems in boundary conditions, we obtain the values of the unknown function at the loading points. This is enough to get the desired solution. The paper describes the application of the method using the example of solving a test model problem.

The purpose of this paper is to solve the problem of the sharp estimation of the modulus of the third Taylor coefficient on the class of holomorphic bounded nonvanishing in the unit disk functions (hereafter referred to as the class of bounded nonvanishing functions). The problem of sharp estimation of the moduli of all Taylor coefficients depending on the coefficient number on this class is usually called the Krzyz problem. Consider the class of normalized holomorphic bounded functions in the unit disk (hereafter referred to as the class of bounded functions). The coefficient problem on this class is posed as follows: find the necessary and sufficient conditions to impose on a sequence of complex numbers so that the power series with coefficients from this sequence is the Taylor series of some function from the class of bounded functions. In this paper, by means of the solution of the coefficient problem for the class of bounded functions, we solve the problem of obtaining the sharp estimates of moduli of the first three Taylor coefficients on the class of bounded functions. It is pointed out that it is possible to visualize the first three coefficient bodies of the subclass of the class of bounded functions, consisting of functions with real coefficients. Next, we solve the problem of obtaining the sharp upper estimation of the modulus of the third Taylor coefficient on the class of bounded nonvanishing functions, by reducing it to the functional over the class of bounded functions. On the basis of the above-mentioned estimates on the class of bounded functions, it was possible to obtain a functional that majorizes the original one. The problem is then reduced to the problem of finding the conditional maximum of the function of three real variables with inequality-type constraints, which made it possible to apply standard methods of differential calculus to obtain this main result.

The paper considers the problems of macroparameter control in a linear dynamic system. In the authors’ previous paper, such problems in controlling nonlinear systems were investigated. Variations of the considered functionals and necessary optimality conditions were obtained. This paper focuses on the trajectory beam control in the linear case. In this case, the problem of beam control can be reduced to the problem of the individual trajectory control. The developed approach may be of interest in the study of charged particle dynamics in accelerating and focusing structures, as well as in the construction of the velocity field in automatic image processing.

In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods rely on the recent results concerning regularization and asymptotic analysis for higher-order differential operators with distribution coefficients. The results of this paper have applications to the theory of inverse spectral problems as well as a separate significance.

Let $N\Phi(K)$ be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring $K$ with the identity associated with a root system $\Phi$. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings $N\Phi(K)$. Automorphisms of the Lie algebra $N\Phi(K)$ under restrictions $K=2K=3K$ on ring $K$ are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type $A_{n}$, a description of automorphisms of Lie rings $N\Phi(K)$ over $K$ was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring $N\Phi(K)$ are described by V.M. Levchuk (1990) for type $D_4$ over $K$, and for other types by A.V. Litavrin (2017), excluding types $G_2$ and $F_4$. In this paper we describe automorphisms of a nil-triangular Lie ring of type $G_2$ over an integrity domain $K$ under the following restrictions on $K$: either $K=2K=3K$, or $3K = 0$. To study automorphisms, the upper and lower central series described in this paper, are essentially used.

There is some evidence that in certain cases a contact of rough elastic solids is multiply connected, i.e. have regions in it where contact surfaces are apart from each other and the contact pressure is zero. The issue of the connectivity in rough elastic contacts has both theoretical and practical interest, especially for seals. In this paper, we extend the earlier conducted analysis of rough contacts without coatings in plane and axially symmetric formulations on the cases of plane and axially symmetric rough elastic contacts with special coatings and compare our findings. The main goal of the paper is to obtain the exact analytical solutions of plane and axially symmetric rough elastic contacts with a special coating and analyze their properties such as contact connectivity and contact pressure smoothness compared to the smoothness of the surface roughness profile. This goal is achieved by using solution expansions in Chebyshev and Legendre orthogonal polynomials. A range of contact parameters has been determined for which the contacts are connected individually.

The paper proposes refined definitions for associated homogeneous functions (AHFs) of real variables, which are of great practical importance for a wide range of problems. I. M. Gelfand and Z. Ya. Shapiro were the first in 1955 to introduce AHFs into scientific use. However, the possibilities of using these functions in various applications have not been exhausted to this day. The proposed definitions inherit the basic idea of the original paper to define chains of new functions using the recurrent linear functional relations, where some homogeneous Euler function is the starting point. This makes it possible to apply the corresponding results not only for differentiable and continuous functions, but also for discontinuous functions, including discontinuous ones at all points. The possibility of constructing a detailed consistent theory of AHFs of real variables, defined by a chain of linear recurrent functional relations of a general form, is shown. The basic theorems are formulated and proven. Further ways of generalizing the functions under consideration are discussed.

In this paper, nonlinear mathematical models of functionally gradient porous nanobeams are constructed taking into account transverse shifts. Transverse shifts are described using kinematic models of the second (S. P. Timoshenko) and third approximations (Sheremetyev – Pelekh). From the Sheremetyev – Pelekh model, as a special case, the kinematic models of the second (S. P. Timoshenko) and first approximation (Bernoulli – Euler) follow. Geometric nonlinearity is accepted according to T. von Karman, nanoeffects are accepted according to the modified Yang moment theory of elasticity. The required equations are derived from the Ostrogradsky – Hamilton principle. An efficient algorithm has been developed that allows us to consider both static and chaotic dynamics problems. Numerical examples are given.

The paper is devoted to the study of one class of infinite systems of nonlinear two-dimensional equations with convex and monotone nonlinearity. The studied class of nonlinear systems of algebraic equations has both theoretical and practical significance, in particular, in the study of discrete analogs of problems in dynamic theory of $p$-adic open-closed strings, in the kinetic theory of gases, in mathematical biology in the study of space-time distribution of epidemics. Existence and uniqueness theorems for a positive solution in a certain class of non-negative and bounded matrices are proved. Some qualitative properties of the solution are revealed. The obtained results supplement and generalize some of the previously obtained ones. Illustrative examples of the corresponding matrices and nonlinearities (including those of an applied nature) that satisfy all the conditions of the formulated theorems are given.

Only finite groups are considered. $\frak F$-projectors and $\frak F$-covering subgroups, where $\frak F$ is a certain class of groups, were introduced into consideration by W.~Gaschutz as a natural generalization of Sylow and Hall subgroups in finite groups. Developing Gaschutz's idea, V. A. Vedernikov and M. M. Sorokina defined $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups, where $\omega$ is a non-empty set of primes, and established their main characteristics. The purpose of this work is to study the properties of $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups, establishing their relation with other subgroups in groups. The following tasks are solved: for a non-empty $\omega$-primitively closed homomorph $\frak F$ and a given set $\pi$ of primes, the conditions under which an $\frak F^{\omega}$-projector of a group coincides with its $\pi$-Hall subgroup are established; for a given formation $\frak F$, a relation between $\frak F^{\omega}$-covering subgroups of a group $G=A\rtimes B$ and $\frak F^{\omega}$-covering subgroups of the group $B$ is obtained. In the paper classical methods of the theory of finite groups, as well as methods of the theory of classes of groups are used.

The paper presents the engineering estimate of the dispersion level of the values of such fused quartz glass physical and mechanical properties as density and elasticity modulus. The value scatter of the considered parameters over material volume leads to negative effect appearance — working natural frequency splitting of gyroscope resonator. Evaluation is carried out on the basis of the consideration of the fused quartz glass micro-structure and variability of its parameters. As a result, resonator frequency splitting values from calculated scattering levels of the structural material properties are presented.

We study the minimum number of elements in the conjugacy class of an automorphism of a sporadic simple group that generate a subgroup containing all inner automorphisms. These results refine the estimates obtained earlier in the papers by Guralnick and Saxl and by Di Martino, Pellegrini, and Zalesski.

In this paper, it is proved that for any Boolean function of n variables, there are infinitely many functions, each of which is its concave continuation to the n-dimensional unit cube. For an arbitrary Boolean function of n variables, a concave function is constructed, which is the minimum among all its concave continuations to the n-dimensional unit cube. It is proven that this concave function on the n-dimensional unit cube is continuous and unique. Thanks to the results obtained, in particular, it has been constructively proved that the problem of solving a system of Boolean equations can be reduced to the problem of numerical maximization of a target function, any local maximum of which in the desired domain is a global maximum, and, thus, the problem of local maxima for such problems is completely solved.

The paper investigates a hybrid system consisting of Lur’e subsystems with constant delays and time-dependent switching. It is assumed that nonlinearities from the right side of the systems have degrees less than unity. An analysis of such a property of the system as the uniform ultimate boundedness of all its solutions is conducted. The linear part of the system is supposed to be asymptotically stable. As is known, this means that there is a correspondent homogeneous Lyapunov function. Using this function, a common Lyapunov–Krasovskii functional is constructed which makes it possible to find sufficient conditions for the uniform ultimate boundedness with arbitrary choices of positive delays and switching laws. Moreover, delays can occur during switching, for example, when generating feedback. The derived conditions are found to be less conservative in the case of asynchronous switching compared to synchronous ones. The validity of the theoretical results is confirmed through numerical modeling.

In this paper we explore the linear logic of multi-agent knowledge using multivalued models. The logic of the language contains the unary operators $K_{j}$ --- $j$ --- the agent knows, $ULK_{G}$ --- unstable local knowledge, $E_{G}$ --- stable local knowledge in the group, and the binary logical operator $AP_{G}$ - the majority opinion. We will show some examples that demonstrate the diversity of this language and its capabilities. Technically we prove decidability of satisfiability problem in the resulting models for our multi-agent logic, develop verification technique and provide some examples.

In the quaternion formulation, the problem of mathematical modeling of the spacecraft motion in an elliptical orbit is considered. The control is a modulo-limited acceleration vector from the jet thrust, directed orthogonally to the plane of the spacecraft orbit. The motion of the spacecraft center of mass is described using the quaternion differential equation of the orientation of the orbital coordinate system. An approximate analytical solution of the quaternion differential equation of the orientation of the orbital coordinate system is constructed in the form of a uniformly suitable asymptotic expansion by degrees of eccentricity of the spacecraft orbit (small parameter). To eliminate the secular terms in this expansion, the renormalization method was applied. Taking into account the known solution of the equation of orientation of the orbital coordinate system for the case when the spacecraft orbit is circular, allowed to simplify the form of the above expansion. The nonlinear oscillation frequencies of each component of the desired quaternion were found. Analytical transformations were performed using the SymPy symbolic algebra package. To carry out numerical simulation of the spacecraft motion, a program was written in Python. Calculations based on analytical formulas obtained in the paper (in the absence of secular terms) and previously obtained results in the presence of secular terms are compared. An example of modeling the controlled motion of a spacecraft is given for the case when the initial orientation of the orbital coordinate system corresponds to the orientation of the orbit of one of the satellites of the GLONASS orbital grouping. Graphs were built to show error in the module (and components) of the quaternion describing the orientation of the orbital coordinate system. It is shown that the elimination of secular terms using the renormalization method made it possible to reduce the error in determining this module with an increase in the number of spacecraft revolutions around the Earth. The analysis of the obtained approximate analytical solution is carried out. The features and regularities of the spacecraft motion in an elliptical orbit are established.

A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. The theory of algebras of relations is an essential part of modern algebraic logic and has numerous applications in semigroup theory. The following problems naturally arise when classes of algebras of relation are considered: find a system of axioms for these classes, and find a basis of of identities (quasi-identities) for the varieties (quasi-varieties) generated by these classes. In the paper, these problems are solved for the class of semigroups of relation with the binary associative operation of the rectangular product, the result of which is the Cartesian product of the first projection of the first relation on the second projection of the second one.

There are perfect and ideal threshold secret sharing schemes, for example, Shamir’s secret sharing scheme. For the case of general secret sharing schemes with an arbitrary access structure, it is possible to construct a perfect scheme for any access structure (for example, the Ito – Saito – Nishizeki scheme, the Benaloh – Leichter scheme), but in general, such a scheme will not be an ideal secret sharing scheme. In the paper, for some classes of access structures, the construction of perfect and ideal secret sharing schemes based on linear codes is given. We also give a construction of perfect verifiable secret sharing schemes for any access structure for which there is a line code that implements this structure.

This paper explores the use of deep learning techniques to improve the performance of web application firewalls (WAFs), describes a specific method for improving the performance of web application firewalls, and presents the results of its testing on publicly available CSIC 2010 data. Most web application firewalls work on the basis of rules that have been compiled by experts. When running, firewalls inspect HTTP requests exchanged between client and server to detect attacks and block potential threats. Manual drafting of rules requires experts' time, and distributed ready-made rule sets do not take into account the specifics of particular user applications, therefore they allow many false positives and miss many network attacks. In recent years, the use of pretrained language models has led to significant improvements in a diverse set of natural language processing tasks as they are able to perform knowledge transfer. The article describes the adaptation of these approaches to the field of information security, i.e. the use of a pretrained language model as a feature extractor to match an HTTP request with a feature vector. These vectors are then used to train the classifier. We offer a solution that consists of two stages. In the first step, we create a deep pre-trained language model based on normal HTTP requests to the web application. In the second step, we use this model as a feature extractor and train a one-class classifier. Both steps are performed for each application. The experimental results show that the proposed approach significantly outperforms the classical Mod-Security approaches based on rules configured using OWASP CRS and does not require the involvement of a security expert to define trigger rules.

In this paper, we examine the question about the approximation of the solution to a transport-diffusion equation in a half-space with the homogenous Neumann condition. Using heat kernel and translation corresponding to the transport in each step of time discretization, we construct a family of approximate solutions. By even extension the given functions and the approximate solutions are transformed into functions defined on the whole space, what makes it possible to establish estimates of approximate solutions and their derivatives and to prove their convergence. We show that the limit function satisfies the equation and the boundary condition.

One of the important characteristics of structures is degrees of semantic and syntactic rigidity, as well as indices of rigidity, showing how much the given structure differs from semantically rigid structures, i.e., structures with one-element automorphism groups, as well as syntactically rigid structures, i.e., structures covered by definable closure of the empty set. Issues of describing the degrees and indices of rigidity represents interest both in a general context and in relation to ordering theories and their models. In the given paper, we study possibilities for semantic and syntactic rigidity for ordered theories, i.e., the rigidity with respect to automorphism group and with respect to definable closure. We describe values for indices and degrees of semantic and syntactic rigidity for well-ordered sets, for discrete, dense, and mixed orders and for countable models of $\aleph_0$-categorical weakly o-minimal theories. All possibilities for degrees of rigidity for countable linear orderings are described.

In this paper, we investigate the problem on small motions of a compressible ideal stratified fluid in a bounded domain. The problem is studied on the base of approach connected with application of so-called operator matrices theory, as well as abstract differential operator equations. For this purpose, Hilbert spaces and some of their subspaces are introduced. The original initial-boundary value problem reduces to the Cauchy problem for a second-order differential operator equation in the orthogonal sum of some Hilbert spaces. Further, an equation with a closed operator is associated with the resulting equation. On this basis, sufficient conditions for the existence of a solution to the corresponding problem are found.

Multifunctions are discrete functions defined on a finite set and returning as their values all subsets of the considered set. The paper considers the classification of multifunctions defined on a two-element set with respect to the E-closure operator. E-closed sets of multifunctions are sets that are closed under superposition, the closure operator with branching by the equality predicate, the identification of variables, and the addition of dummy variables. The concept of separating sets was introduced using a greedy algorithm for the set covering problem, and 22 classes of separating sets were obtained. It is shown that the classification under consideration leads to a finite set of closed classes. The work describes all 359 E-closed classes of multifunctions, among which there are 138 pairs of dual classes and 83 self-dual classes. For each class consisting only of multifunctions, its generating system is indicated.

In 2001, Virchenko et al. published a paper on a new generalization of Gauss hypergeometric function, namely Wright-type generalized hypergeometric function. Present work aims to define the q-analogue generalized hypergeometric function, which reduces to generalized hypegeometric function by letting q tends to one, and study some new properties. Convergence of the series defining generalized q-hypergeometric function and properties including certain differentiation formulae and integral representations have been deduced.

In the paper, we consider nonlinear control systems that are linear with respect to controls with output; vector fields defining the system and the output are supposed to be real analytic. Following the algebraic approach, we consider series $S$ of iterated integrals corresponding to such systems. Iterated integrals form a free associative algebra, and all our constructions use its properties. First, we consider the set of all (formal) functions of such series $f(S)$ and define the set $N_S$ of terms of minimal order for all such functions. We introduce the definition of the maximal graded Lie generated left ideal ${\mathcal J}_S^{\rm max}$ which is orthogonal to the set $N_S$. We describe the relation between this maximal left ideal and the left ideal ${\mathcal J}_S$ generated by the core Lie subalgebra of the system which realizes the series. Namely, we show that ${\mathcal J}_S\subset {\mathcal J}_S^{\rm max}$. In particular, this implies that the graded Lie subalgebra that generates the left ideal ${\mathcal J}_S^{\rm max}$ has a finite codimension. Also, we give the algorithm which reduces the series $S$ to the triangular form and propose the definition of the homogeneous approximation for the series $S$. Namely, homogeneous approximation is a homogeneous series with components that are terms of minimal order in each component of this triangular form. We prove that the set $N_S$ coincides with the set of all shuffle polynomials of components of a homogeneous approximation. Unlike the case when the output is identical, the homogeneous approximation is not completely defined by the ideal ${\mathcal J}_S^{\rm max}$. In order to describe this property, we introduce two different concepts of equivalence of series: algebraic equivalence (when two series have the same homogeneous approximation) and weak algebraic equivalence (when two series have the same maximal left ideal and therefore have the same minimal realizing system). We prove that if two series are algebraically equivalent, then they are weakly algebraically equivalent. The examples show that in general the converse is not true.

The subject of mathematical study and modelling in this paper is an inbound call center that receives calls initiated by customers. A closed exponential queueing network with customer retrials and impatient customers is used as a stochastic model of call processing. A brief review of published results on the application of queueing models in the mathematical modeling of customer service processes in call centers is discussed. The network model is described. The possible customer states, customer routing, parameters, and customer service features are given. The allocation of customers by network nodes at a fixed time fully describes the situation in the call center at that time. The state of the network model under study is represented by a continuous-time Markov chain on finite state space. The model is studied in the asymptotic case under the critical assumption of a large number of customers in the queueing network. The mathematical approach used makes it possible to use the passage to the limit from a Markov chain to a continuous-state Markov process. It is proved that the probability density function of the model state process satisfies the Fokker – Planck – Kolmogorov equation. Using the drift coefficients of the Fokker – Planck – Kolmogorov equation, a system of ordinary differential equations for calculating the expected number of customers in each network node over time can be written. The solution of this system allows for predicting the dynamics of the expected number of customers at the model nodes and regulating the parameters of the call center operation. The asymptotic technique used is applicable both in transient and steady states. The areas of implementation of research results are the design of call centers and the analysis of their workload.

The results of theoretical calculations of electronic states of the gallium oxide (Ga2O3) nanocrystals both doped with donor impurity and undoped have been presented in the paper. In the envelope function approximation, the structure, states and energy levels of size quantization in the nanocrystals were determined. According to our calculations, the electron-hole pair forms a bound state of the exciton type in the nanocrystal. The typical donor impurities in Ga2O3, such as silicon and tin, were shown to create bandgap states localized in a spatial domain being several times smaller than the nanocrystal’s volume. Forming a compact neutral pair, the electron and donor ions have no noticeable influence on the states of the optically excited electron-hole pairs. The effect of impurity implantation on recombination processes was also discussed.

In the paper, the effect of bombardment with Ar+ ions on the composition, electronic and crystal structure of the surface layers of bulk single crystal samples and CdF2(111) films has been studied using the methods of Auger electron and ultraviolet photoelectron spectroscopy, high-energy electron diffraction and recording the angular dependences of the reflectance factor of inelastically reflected electrons. The effect of this bombardment on the density of states of valence electrons and energy band parameters of CdF2 was investigated for the first time. The degree of disorder of CdF2 into components and evaporation of fluorine from the surface layers was established to depend on the energy and dose of Ar+ ions. The complete evaporation of F in the form of a diatomic gas was shown for the first time to be observed in the energy range of 1 – 2 keV at a saturation dose.

This paper presents the results of a study of polymer films based on a blend of low-density (high-pressure) polyethylene (LDPE, HPP) with a copolymer of ethylene and vinyl acetate (EVA, sevilen). The use of thermal activation, infrared and dielectric spectroscopy methods made it possible to describe the electric charge relaxation processes in the polymer blends investigated. The data obtained suggested the presence of an α-relaxation process in the samples in the temperature range 250–280 K. An increased value of the activation energy of this process was also found in the LDPE/EVA samples compared to that in the LDPE one. This effect has been interpreted as the appearance of deeper traps of charge carriers in the blends. The dependences obtained by dielectric spectroscopy indicated the presence of hopping conductivity in the subjects of research.

In the paper, the patterns of changes in the composition and structure of the surface layers of GaP(111) in bombardment by electrons with energies from 3 to 10 keV and doses in the range 1017 – 1020 cm–2 have been studied using the method of Auger electron spectroscopy and recording the angular dependence of the electron inelastic reflection coefficient. It was established that the surface layers of GaP were enriched with P atoms at E = 3 keV, and with Ga atoms at E = 10 keV. In both cases, the Ga atoms distribution profiles over the depth the sample were non-monotonic. The electron energy value at which an inversion of the surface composition took place was estimated. An analysis of the results obtained was given.

In the paper, the exact and asymptotic approximate solutions for symmetric and antisymmetric Lamb waves in the homogeneous isotropic elastic film have been analyzed. Using the numerical methods of the theory of continuation of solutions of nonlinear equations, the dispersion curves were calculated for waves with different variability across the layer thickness. Based on the results obtained, the nature of the displacement field and the variability of oscillation forms depending on the wave number were studied. The asymptotic correctness of the Timoshenko and Euler – Bernoulli beam models as long-wave asymptotics of Lamb waves was analyzed.

The paper presents asymptotic methods for solving the problem of small harmonic oscillations of a flat contour immersed in an incompressible viscous liquid. In the case of large values of the dimensionless viscosity parameter, asymptotic approximations up to the third order have been obtained. In the case of small values of this viscosity parameter, the main term of the asymptotic of the hydrodynamic force on an arbitrary smooth contour was constructed and its form was proved not to depend on the shape of the contour. The results obtained were confirmed by an example of the problem of oscillations of an elliptical cylinder.

In the paper, the coupled elastic-diffusion processes arising as a result of unsteady bending vibrations of an orthotropic plate that has a cantilever fastening on one side and hinged support on the sides adjacent to the cantilever have been analyzed. For a mathematical description of physical and mechanical processes, the Timoshenko plate model supplemented with mass transfer equations taking into account the finite speed of propagation of diffusion flows was used. The solution algorithm was based on the use of the equivalent boundary conditions method allowing to express the solution to the problem posed through a known solution to some auxiliary problem of a given class. The nature of the interaction between mechanical and diffusion fields was simulated using the example of a bendable three-component plate.

One of the main derived objects of a given structure is its automorphism group, which shows how freely elements of the structure can be related to each other by automorphisms. Two extremes are observed here: the automorphism group can be transitive and allow any two elements to be connected to each other, or can be one-element, when no two different elements are connected by automorphisms, i.e., the structure is rigid. The rigidity given by a one-element group of automorphisms is called semantic. It is of interest to study and describe structures that do not differ much from semantically rigid structures, i.e., become semantically rigid after selecting some finite set of elements in the form of constants. Another, syntactic form of rigidity is based on the possibility of getting all elements of the structure into a definable closure of the empty set. It is also of interest here to describe “almost” syntactically rigid structures, i.e., structures covered by the definable closure of some finite set. The paper explores the possibilities of semantic and syntactic rigidity. The concepts of the degrees of semantic and syntactic rigidity are defined, both with respect to existence and with respect to the universality of finite sets of elements of a given cardinality. The notion of a rigidity index is defined, which shows an upper bound for the cardinalities of algebraic types, and its possible values are described. Rigidity variations and their degrees are studied both in the general case, for special languages, including the one-place predicate signature, and for some natural operations with structures, including disjunctive unions and compositions of structures. The possible values of the degrees for a number of natural examples are shown, as well as the dynamics of the degrees when taking the considered operations.

This paper deals with multi-agent computation tree logic --- $\mathcal{CTLK}$ (Computation Tree Logic with Knowledge). Each agent represents its own computational route of the initial problem, and new branches of possible computational routes spawn new agents. The logic $\mathcal{CTLK}$ is a natural enrichment of $\mathcal{CTL}$ by new knowledge operators. We introduce alternative to automata Kripke's relational semantics, describes properties of $\mathcal{CTLK}^{Rel}$-frame and proves finite approximability.

This paper is devoted to solving the inverse problem (determining the parameters of a system of ordinary differential equations based on additional information determined at discrete points in time) and analyzing its solution for a mathematical model describing the dynamics of changes in the population and capital of two regions of the world. The inverse problem is reduced to the problem of minimizing the target functional and is solved by the method of differential evolution. A numerical method for solving direct and inverse problems is implemented. The developed method was tested on model and real data for countries such as Russia, China, India and the USA.

In this paper, we study the negative order modified Korteweg-de Vries (nmKdV) equation in the class of rapidly decreasing functions. In particular, we show that the inverse scattering transform technique can be applied to obtain the time dependence of scattering data of the operator Dirac with potential being the solution of the considered problem. We demonstrate the explicit representation of one soliton solution of nmKdV based on the obtained results.

The problem of obtaining estimates of the strength and resource characteristics of critical engineering infrastructure facilities under operational multiparametric nonstationary thermomechanical impacts is considered. The basic degradation mechanisms in structural materials (metals, alloys) under these influences are identified. The methodology of resource assessment of responsible engineering facilities based on end-to-end modeling of the entire life cycle of the object is substantiated. End-to-end modeling forms a set of computational experiments of different levels of complexity, each of which has its own characteristic features and semantics. From the perspective of the mechanics of the degraded continuum, a mathematical model of the damaged medium has been developed, in which the processes of thermoplasticity and damage accumulation are generated by thermal fatigue. The model describes the effects of cyclic thermoplastic deformation; kinetics of damage accumulation; conditions of macroscopic destruction of the material. The model postulates the representation of the yield surface and the principle of gradiency of the velocity vector of plastic deformations at the loading point. A variant of the thermoplasticity equations describes the main effects in proportional and disproportionate modes. The thermoplasticity model is constructed as a system of “nested” models and contains the forms of equations of the theory of plastic flow under small deformations: various variants of isotropic hardening (ideally plastic material with a constant flow surface, linear isotropic hardening, variant of isotropic nonlinear hardening), various cases of kinematic hardening (linear kinematic hardening, the case of purely nonlinear kinematic hardening) and the general case of translational isotropic hardening. The kinetics of fatigue damage accumulation is described by introducing a scalar damage parameter and based on the energy principles of taking into account the main effects of the damage accumulation process for arbitrary complex loading modes. The condition of reaching the critical damage value is used as a criterion for macroscopic destruction. The relationship between the components of the model is carried out by introducing effective stresses. The paper presents a numerical analysis of the thermal fatigue life of a compact sample with stress concentrators simulating the operation of parts in the nozzle box of a steam turbine of a nuclear power plant. During the analysis, the characteristic features of thermal fatigue in the details of power equipment were studied. It is shown that the end-to-end modeling technology can be effectively used to assess the resource characteristics of power equipment parts under operational loading conditions.

The paper examines the natural frequencies of prismatic thin shells, the cross-section of which is the regular polygon. Spectra of free vibration frequencies of such shells are analyzed as the number of cross-section sides increases, provided that the perimeter is preserved. The relation between fundamental frequencies of the prismatic shells with the regular polygonal cross-section and a circular cylindrical shell is discussed. For a small and large number of polygon sides analytical and asymptotic solutions are compared with numerical solutions obtained by the finite element method (COMSOL). The convergence of the numerical method is studied for the prismatic shell with a large number of facets.

The unsymmetrical buckling of annular plates with an elastically restrained edge which are subjected to normal pressure is studied in this paper. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. The effect of plate geometry (ratio of inner to outer radii) and boundary on the buckling load is examined. It is shown, that for an annulus the buckling pressure and the buckling mode number decreases as the inner radius increases. It is shown that as the internal radius increases, the plate loses stability as the buckling pressure decreases, which also leads to the buckling mode number decrease.

A differential equation is derived that describes free long-wave vibrations of a low-dimensional elastic isotropic strip-beam, taking into account effects on free surfaces. Boundary conditions on external surfaces are formulated within the framework of the Gurtin – Murdoch surface theory of elasticity, which takes into account surface inertia and shear stresses, including residual ones. Additional geometric dimensions are introduced, associated with the face surfaces, which are assumed to be small compared to the main geometric dimension — the wavelength. The ratio of the thickness of the ultrathin strip to the wavelength of bending vibrations is considered as the main small parameter. Using the method of asymptotic integration of two-dimensional equations of the theory of elasticity over the thickness of the strip-beam, relations for displacements and stresses in the volume of the strip were obtained in explicit form. The main result of the paper is a differential equation for low-frequency vibrations of a beam, which takes into account surface effects and generalizes the well-known equations of beam theory. It is shown that the presence of surface stresses leads to an increase in natural frequencies from the lower spectrum, while taking into account surface inertia, as well as transverse shears in volume, leads to a decrease in frequencies.

The paper is devoted to a wide range of problems related to the two-dimensional Nye figures for micropolar continua. The method of two-dimensional matrix representation of fourth-rank tensors is well known from monographs on crystallography. Such representations are used to simplify tensor notation of the equations of anisotropic solids. This method allows us to represent the asymmetric constitutive tensors and pseudotensors of the fourth, third and second ranks in the form of specific two-dimensional figures. The Nye figures for the constitutive hemitropic tensors of the fourth and second ranks are given. The matrix form of the constitutive equations of a hemitropic micropolar solid in the athermal case is obtained. The transformation of the pseudotensor governing equations of the micropolar theory to a formulation in terms of absolute tensors is carried out via the pseudoscalar units and their integer powers. The study is carried out in terms of absolute tensors in a Cartesian rectangular coordinate system.

The article is dedicated to the anniversary of the editor-in-chief of this journal Leonid Yu. Kossovich. The paper presents an overview of the scientific areas in which our anniversary celebrant worked and his publications over the past five years.

In the paper, we consider the 𝑞-integrability of functions 𝜆(𝑡)|ℱ𝑞,𝜈 (𝑓)(𝑡)|𝑟, where 𝜆(𝑡) is a Gogoladze-Meskhia-Moricz type weight and ℱ𝑞,𝜈(𝑓)(𝑡) is the 𝑞-Bessel Fourier transforms of a function 𝑓 from generalized integral Lipschitz classes. There are some corollaries for power type and constant weights, which are analogues of classical results of Titchmarsh et al. Also, a 𝑞-analogue of the famous Herz theorem is proved.

In this paper, we formulate two theorems on the disap­pearance of the kernel of the Hodge — de Rham Laplacian and refine the estimate for its smallest eigenvalue on closed Riemannian manifolds.

We continue to study the cocongruence of -dimensional planes us­ing the Cartan — Laptev method. In an -dimensional projective space , the cocongruence of -dimensional planes can be given by the following equations . Compositional clothing of a given cocongruence by fields of ()-planes : and points allows one to define connections of three types in the associated bundle. In the present paper, parallel transports of an analogue of Cartan plane are studied in the connections of three types. It is proved 4 theo­rems: 1. Parallel transport of the analogue of the Cartan plane in an arbitrary connection is freely degenerate, i. e., in general, there are no spe­cial transports of this clothing plane. 2. In the group connection of the first type, the parallel transport of an analog of the Cartan plane is connected degenerate, i. e., the plane will be fixed under parallel transport in this connection. 3. In the group connections of the second and third types, the parallel transport of the analogue of the Cartan plane is freely degenerate. 4. The analogue of the Cartan plane is transferred in parallel in a line­ar combination of the first type connection if and only if it is displaced in the plane .

The paper is devoted to affine connection in the frame bundle associ­ated with a manifold which structure equations and derivation formulas are constructed using deformations of the exterior and ordinary differen­tials. Curvature and torsion objects of this connection are not tensors. A cha­racteristic of a curvature which is a convolution of a deformation tensor and a torsion, is considered. Torsion-free connections are not distingui­shed on the introduced manifold, even in the case of symmetric deforma­tion, a class of semi-symmetric connections is distinguished, which is an analogue of symmetric connection on an ordinary smooth manifold. It is proved that if the connection deformation tensor is symmetric or zero, then the connection is semi-symmetric. Analogues of torsion-free and cur­vature-free connections are constructed. The torsion and curvature of this connection are expressed in terms of the symmetric deformation tensor for the connection. Canonical connection is a special case of this connec­tion, it is semi-symmetric and curvature-free.

The concept of a common path space was introduced by J. Duqlas. M. S. Knebelman was the first to consider affine and projective move­ments in these spaces. The general path space is a generalization of the space of affine connectivity. In this paper, we study spaces of paths that admit groups of affine motions with one-dimensional orbits. For each representation in the form of algebra of vector fields of the abelian Lie algebra and the Lr algebra containing the abelian ideal Lr-1, a system of equations of infinitesimal affine motions is compiled. The vector fields of each of these representations are operators of a group of transformations with one-dimensional orbits. Integrating this system, general spaces of paths are defined that admit a group of affine motions with one-dimensional orbits, the operators of which are the vector fields of these representations. The maximum order of these groups is set. It is shown that the spaces of paths admitting a group of affine motions with one-dimensional orbits of maximum order are projectively flat. The conditions that are necessary and sufficient for the space of paths to admit a group of affine motions with one-dimensional orbits of maximum order are given.

Among Thurston's famous list of eight three-dimensional geometries is the geometry of the manifold Sol. The variety Sol is a connected simply connected Lie group of real matrices of a special form. The manifold Sol has a left-invariant pseudo-Riemannian metric for which the group of left shifts is the maximal simply transitive isometry group. In this paper, we prove that on the manifold Sol there exists a left-invariant differential 1-form, which, together with the left-invariant pseudo-Riemannian metric, defines a paracontact metric structure on Sol. A three-parameter family of left-invariant paracontact metric connections is found, that is, linear con­nec­tions invariant under left shifts, in which the structure tensors of the par­acontact structure are covariantly constant. Among these connections, a flat connection is distinguished. It has been established that some geo­desics of a flat connection are geodesics of a truncated connection, which is an orthogonal projection of the original connection onto a 2n-dimen­sional con­tact distribution. This means that this connection is con­sistent with the con­tact distribution. Thus, the manifold Sol has a pseudo-sub-Riemannian structure determined by a completely non-holonomic contact distribution and the restriction of the original pseudo-Riemannian metric to it.