In this work we consider generalized Kenmotsu manifolds, we introduce: the fourth and
the fifth fundamental identities of generalized Kenmotsu manifolds; the first and the second
structural tensors of generalized Kenmotsu manifolds (and we prove their properties); the
concept of adjoint Q-algebra for generalized Kenmotsu manifolds. We prove that generalized
Kenmotsu manifolds and the II kind special generalized Kenmotsu manifolds have
anticommutative adjoint Q-algebra. And the Kenmotsu manifolds and the I kind special
generalized Kenmotsu manifolds have Abelian adjoint Q-algebra. The type constancy contact
analog is introduced and the constant-type generalized Kenmotsu manifolds are thoroughly
examined. We have identified the type point constancy conditions of the generalized Kenmotsu
manifolds in the adjoint G-structure space. We prove that the zero constant type GKmanifold
class coincides with the Kenmotsu manifold class and the non-zero constant type
GK-manifold class can be concircularly transformed into the almost contact metric manifolds
locally equivalent to the product of the six dimensional NK-eigenmanifold and the real straight
line.

An upper bound and lower bound for the number of integer polynomials which have only two
close to each other roots, and small discriminant in terms of the Euclidean metric is obtained.

In the paper geometrical characteristics of metric spaces appearing in explicit formulas for
the Gromov–Hausdorff distance from this spaces to so-called simplexes, i.e., the metric spaces,
all whose non-zero distances are the same. For the calculation of those distances the geometry
of partitions of these spaces is important. In the case of finite metric spaces that leads to
some analogues of the edges lengths of minimal spanning trees. Earlier, a similar theory was
elaborated for compact metric spaces. These results are generalised to the case of an arbitrary
bounded metric space, explicit formulas are obtained, and some proofs are simplified.

In order to investigate arithmetic properties of the values of generalized hypergeometric functions with rational parameters one usually applies Siegel’s method. By means of this method have been achieved the most general results concerning the above mentioned properties.
The main deficiency of Siegel’s method consists in the impossibility of its application for the hypergeometric functions with irrational parameters. In this situation the investigation is usually based on the effective construction of the functional approximating form (in Siegel’s method the existence of that form is proved by means of pigeon-hole principle). The construction and investigation of such a form is the first step in the complicated reasoning which leads to the
achievement of arithmetic result. Applying effective method we encounter at least two problems which make extremely narrow the field of its employment. First, the more or less general effective construction of the approximating form for the products of hypergeometric functions is unknown. While using Siegel’s method one doesn’t deal with such a problem. Hence the investigator is compelled to consider only questions of linear independence of the values of hypergeometric functions over some algebraic field. Choosing this field is the second problem. The great majority of published results concerning corresponding questions deals with imaginary quadratic field (or the field of rational numbers). Only in exceptional situations it is possible to investigate the case of some other algebraic field.
We consider here the case of a real quadratic field. By means of a special technique we establish linear independence of the values of some hypergeometric function with irrational parameter over such a field.

It is a survey with respect to using trigonometric sums in the metric theory of Diophantine approximation on the manifolds in n-dimensional Euclidean space. We represent both classical results and contemporary theorems for Γ, dim Γ = m,
n/2 < m < n. We also discuss reduction of a problem about Diophantine approximation to trigonometric sum or trigonometric integral, and indicate measure-theoretic considerations.

If m ≤ n/2  then usually it is used the other methods. For example, the essential and inessential domains method or methods of Ergodic Theory.

Here we cite two fundamental theorems of this theory. One of them was obtained by V. G. Sprindzuk (1977). The other theorem was proved by D. Y. Kleinbock and G. A. Margulis (1998). The first result was obtained using method of trigonometric sums. The second theorem was proved using methods of Ergodic Theory. Here the authors applied new technique which linked Diophantine approximation and homogeneous dynamics.

In conclusion, we add a short comment concerning the tendencies of a development of the metric theory of Diophantine approximation of dependent quantities and its contemporary aspects.

The system of matrix Lurie equations is considered. Such a system is of practical importance
in the study of the asymptotic stability of equilibrium states of a system of differential equations,
finding the regions of attraction of equilibrium states, determining the conditions for the
existence of limit cycles for systems of differential equations, investigating global stability,
hidden synchronization of phase and frequency-frequency self-tuning systems. It is known
that the conditions for the solvability of the matrix Lurie equations are determined by the
"Yakubovich-Kalman frequency theorem". To study nonlinear oscillations of phase systems, it
becomes necessary to find solutions of the matrix Lurie equations.
In this paper we consider the case when the matrix Lyapunov inequality, which is part of
the Lurie equation, has a matrix with real eigenvalues, some of which may be zero. For such a
case, necessary and sufficient conditions for the solvability of the Lurie equations are obtained
and the form of the solutions is determined, which makes it possible to carry out their spectral
analysis. The explicit form of the solutions of the matrix equations made it possible to make
their geometric interpretation depending on the spectrum, to show the relationship of the linear
connection equation to the quadratic forms of solutions of the Lurie equations. The method of
analyzing matrix equations is based on an approach based on the use of a direct product of
matrices and the application of generalized inverse matrices to find solutions to systems of linear
equations. The results of the work made it possible to investigate the system of three matrix
equations arising in the study of phase-frequency frequency-phase self-tuning systems.

Hypergraphic automata are automata, state sets and output symbol sets of which are hypergraphs, being invariant under actions of transition and output functions. Universally attracting objects in the category of such automata are called universal hypergraphic automata. The semigroups of input symbols of such automata are derivative algebras of mappings for such automata. Semigroup properties are interconnected with properties of the automaton. Therefore, we can study universal hypergraphic automata by investigation of their input symbol semigroups. In this paper, we solve a problem of abstract definability of such automata by their input symbol semigroups. This problem is to find the conditions of isomorphism of semigroups of input symbols of universal hypergraphic automata. The main result of the paper is the solving of this problem for universal hypergraphic automata over effective hypergraphs with p−definable edges. It is a wide and a very important class of automata because such algebraic systems contain automata whose state hypergraphs and output symbol hypergraphs are projective or affine planes. Also they include automata whose state hypergraphs and output symbol hypergraphs are divided into equivalence classes without singleton classes. In the current study, we proved that such automata were determined up to isomorphism by their input symbol semigroups and we described the structure of isomorphisms of such automata.

In paper proves an analogue of the theorem of F. Kubo [1] for almost locally solvable Lie algebras with zero Jacobson radical. The first section aims to clarify some aspects of the homological description of the Jacobson radical. We prove a theorem generalizing E. Marshall’s theorem to the case of almost locally solvable Lie algebras, the consequence of which is an analogue of Kubo’s theorem. In the second section, we investigate some properties of a locally nilpotent radical of a Lie algebra. Primitive Lie algebras are considered. Examples are given to show that infinite-dimensional commutative Lie algebras are primitive over any fields; finitedimensional Abelian algebra, dimensions greater than 1, over an algebraically closed field is not primitive; an example of a non-Artin noncommutative Lie algebra being primitive. It is shown that for special Lie algebras over the characteristic field, the zero PI-irreducibly presented radical coincides with the locally nilpotent one. An example of a Lie algebra whose locally nilpotent radical is neither locally nilpotent nor locally solvable is given. Sufficient conditions for the primitiveness of a Lie algebra are given, examples of primitive Lie algebras and a nonprimitive Lie algebra are given.

In this paper it is considered the definition of differentiability and regularity by Fueter [1, 2]
and examples of regular function by Fueter, and the definition of C-regularity and C-derivative
or Cullen derivative, on the basis of which a new theory of regular functions, which already
includes polynomials and converging series of hypercomplex variable as differentiable and regular
functions. Then a new definition of differentiability is proposed. It has a classical form, but
specific convergence, which allows to prove theorems about differentiability of the sum and
product of differentiable functions, differentiability of the “quotient” of differentiable functions.
Further, it is deduced the derivative of power and is proved differentiability of polynomials and
power series that allows to construct generalization of elementary functions for quaternionic
argument. An example is given to show that without specific convergence the given definition
of differentiability loses its meaning. With the help of power series functions are given, which
are solutions of differential equations with constant quaternion coefficients. It is considered the
problem of finding the roots of a square equation that arises in solving differential equations.

In paper the problem of the reflection and transmission of a plane sound wave through a
homogeneous elastic plate with an continuously inhomogeneous plate elastic coating adjoining
viscous liquids is considered. It is believed that heterogeneity laws of a coating are described by
differentiable functions.
The propagation of small perturbations in a viscous fluid in the case of steady state
oscillations is described by scalar and vector Helmholtz’s equations. The propagation of elastic
waves in a uniform isotropic elastic plate is described by two wave equations for longitudinal
and transverse waves. Oscillations of an inhomogeneous isotropic elastic coating are described
by general motion equations of the continuous medium.
The boundary-value problem for the system of ordinary second order differential equations
is constructed for determination of the displacement field in inhomogeneous coating.
An analytical description of the reflected and transmitted through the plate acoustic fields
is obtained.
The results of numerical calculations of dependences of coefficients of reflection and
transmission of longitudinal waves from the angle of incidence of plane wave are presented.

The method of analysis of spatial fields of stresses and velocities in pro-cesses of plastic flow
is given, based on mapping of flow zones in deviator space of stresses. A generalized Mises flow
function corresponding to numer-ous experimental data is taken as the loading surface. It is
shown that the generalized Mises model is convenient for analysis of spatial deformation processes
with the power of a special depicting parametric space. The numer-ical implementation
of the method is illustrated by the example of plastic compression of a material under threedimensional
deformation conditions. It is shown that the distribution of stresses and flow rates
depends on the current ratio of layer sizes during settling.

This paper is devoted to the approximation of a quadratic algebraic lattice by an integer
lattice. It calculates the distances between a quadratic algebraic lattice and an integer lattice
when they are given by the numerator and denominator of a suitable fraction to the square root
of a Prime ???? of the form ???? = 2 or ???? = 4???? + 3.
The results of this work allow us to study questions about the best approximations of
quadratic algebraic lattices by integer lattices.

Outer billiards were introduced by B. Neumann in 1950s and became popular in 1970s due
to J. Moser; Moser considered outer, or dual, billiard as toy model of celestial mechanics. The
problem of stability of the Solar system has such a property that “it’s easy to write ???? equations
of particles motion down but hard to understand this motion intuitively”; according to this,
Moser suggested to consider Neumann’s outer billiard problem which has the same property.
One of classical examples of dynamical systems is an outer billiard outside regular ????-gon; in
particular, this billiard is connected with problems of existence of aperiodic trajectory and of
fullness of periodic points. These problems resolved only for a few number of a special cases.
In case ???? = 3, 4, 6 table is a lattice polygon; as a consequence, there are no aperiodic points,
and periodic points form a set of full measure. In 1993, S. Tabachnikov was managed to find an
aperiodic points in case of regual pentagon; it was done using renomalization scheme — method
which has a fundamental importance in research of self-similar dynamical systems.
According to R. Schwartz, cases which are next by complexity are ???? = 10, 8, 12; in these
cases, and also in case ???? = 5, it’s possible to build a renomalization scheme which, as R. Scwartz
writes, “allows one to give (at least in principle) a complete description of what is going on.”
Later, author was managed to discover self-similar sturctures and build renormalization
scheme for cases of regular octagon and dodecagon.
This article is devoted to outer billiard outside regular decagon. The existence of an aperiodic
orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of
such an outer billiard are proved to be periodic. All possible periods are explicitly listed. The
work is based on classical technology of search and research of renormalization scheme. Periodic
structures which occur in case ???? = 10 are similar to periodic structures in case ???? = 5, but has
their own features.

The study examines a model for creating a series of learning tasks, built on the basis of
ideas about the three-component structure of mental actions (orientation, execution, control).
With the help of problems of a dynamic nature, focused on the management of learning and
cognitive activity of students, the work reveals one of the possible approaches to the joint study
of the formulas for full probability and Bayes It is noted that for the effective management of
the educational and cognitive activity of students, the postoperative feedback is carried out,
turning into intermediate feedback and communication on the final result.
The article emphasizes that the optimality of the process of formation of elements of research
activity is carried out by increasing the efficiency of its mastering over time. The latter is
achieved through the methods of joint mastering by students of educational material related to
mastering, for example, the formulas of full probability and Bayesian.
In this perspective, it is noted that it is advisable to use the stages of information technology
problem solving, and the use of ICT as a process of forming tools in life and ICT-competencies
for the vocational guidance of a graduate school.

The article emphasizes the need to form ICT-competencies of the teacher as a requirement
of the federal state educational standard in the conditions of the information educational
environment, defines the principles for constructing a methodological system for the formation
of ICT-competencies within pedagogical universities through problem solving using computer
technologies.

The development of technology puts forward more complex problems, the effective solution
of which is associated with the refinement of mathematical models of the studied plastic forming
processes of metal systems (metals, steels, non-ferrous alloys) of various chemical compositions
and production technologies (traditional ingot processing, powder production, nanostructured
materials). To build a solution, the integration of differential equations to describe the physical
processes occurring during plastic flow or a variation approach based on the construction of
the original functional is used, but in either case the accuracy of the mathematical modelling
of the process will depend on the accepted mathematical model of the medium. The paper
discusses the development stages of different models of plastic media, from a simple rigidplastic
model, without taking into account the change in the material properties, to more
complex ones - a viscoplastic model that regards the appearance of viscosity with increasing
processing temperature and a dilating model that allows to take into consideration changes in
material density and thereby predict strain destruction. The correct use of mathematical models
of plastic media makes it possible to improve the accuracy of technological regimes calculation,
thereby reducing the time to develop the new product, and can be used for the development
of technological processes for obtaining products by means of the additive technology based
on laser sintering and fusing the powder alloys, thermoplastic processing technologies and
processes of thermochemical hardening and heat treatment of metallic systems of different
chemical compositions.

In this article, within the framework of the general problem of the formation of modern science, the question of the transition from Aristotelian physics to classical mechanics in the 17th century is being considered. The distinction between these two forms of scientific approach to the study of nature is based on the opposition of ideas of holism, which guided the preclassical natural science, and reductionism, which became the foundation of classical mechanics. The reductionist nature of the postulates underlying the law of the parabolic trajectory of the projectile, the discovery of which inaugurated the scientific revolution of the 17th century, is disclosed and emphasized. Among them - the principle of inertial motion in the horizontal direction (in the absence of gravity), the law of free fall in the vertical direction under the influence of gravity, and the law of the parallelogram of forces and motions. It is shown that none of these postulates could be fulfilled in pre-classical mechanics due to its holistic orientation. Proceeding from the hypothesis that the theoretical constructions in the field of mechanics are secondary in relation to the forms of motion realized in practical mechanics, it is established that holism is associated with the use of muscular strength of man and animals as working engines. It is shown that it is the use of animate engines (characteristic of antiquity and the Middle Ages) that underlies the violation of the principle of the additivity of forces, upon which classical mechanics is based. In conclusion, proceeding from the aforementioned hypothesis about the secondary nature of theoretical constructions, the thesis is argued that the technological prerequisites for the formation of the idea of reduction are new types of technical movement realized in practical mechanics in the 15th-16th centuries. These are lifting devices using gravity as driving force and crank-and-rod mechanisms equipped with flywheels. Devices of the first type led to the idea of additivity of forces, while of the second to the possibility of a uniform motion.

As it develops, mathematical modeling finds more and more new areas of application,
remaining an effective tool, including engineering. Mathematical models go the way of
evolutionary development, increasing the adequacy in accordance with real physical processes.
One of the relevant areas of mathematical modeling is associated with the developing
technologies of additive prototyping. For example, in the manufacture of products from metal
powders by the methods of additive technologies, in particular, selective laser melting, one of
the practical issues is the selection of optimal parameters for the 3D printer. The solution
to the optimization problem x of the 3D printer operation parameters should be based on a
mathematical model of the process of heating and melting of metal particles. An approach
based on the formation and solution of the heat equation with boundary conditions that take
into account the spherical shape of the particle, the energy distribution in the cross section
of the laser beam, and the relative spatial position of the particle and the laser beam is used
as the basic concept of modeling. It is noted that to assess the structure of the formed parts,
this approach is redundant, and the algorithm for integrating the partial differential equation
has high computational complexity. To simplify the analysis task, the initial micromodel is
transformed into heating and melting macromodels in which the temperature distribution over
the volume of the particle is considered constant, and the external effect on the particle is
reduced to heat transfer through the surface of the ball, from the upper side from the laser beam
to the particle, and from the lower side - from particle to environment. For the macromodel,
we obtained time diagrams of the temperature increase and the accumulated internal particle
energy in time. It is concluded that it is possible to divide the space around the particle into
zones: complete and incomplete melting, as well as a heating zone insufficient for melting. It is
shown that the presence of such zones leads to the friability of the structure of the parts formed
on the 3D printer.Keywords: additive technology, laser heating, heat conduction equation, micromodel,
macro-model, heating-melting timing charts.

The FIFO (First In First Out) discipline of interrupt processing is widely used in Von Neumann type computers of informational and digital control systems. The goal of implementing such modes of operation - optimization time to data access - is achievable only when there is an adequate model, which describes data processing in the system. The analytical model is worked out with use the fundamental mathematical apparatus of Petri-Markov nets. The initial Petri-Markov model is divided into hierarchical levels in accordance with the number of interrupts in queue for processing. It is shown, that from the current level it is possible to switch both to the previous and to the next interrupt. Dependencies for determine the time of residence on the current level, and the probabilities of switching to conjugate levels are obtained. The method of Petri-Markov model transformation into the semi-Markov process is proposed. It is shown, that semi-Markov process obtained has the binary tree structure. Dependences for determining the time and probabilistic characteristics of wandering through a binary tree, are obtained.

The influence of lattice structures on the propagation of elastic waves in these structures is analyzed. The statement of problem is formulated within the framework of linear elasticity under small strains. To solve this problem, the finite element method and the Fidesys CAEsystem were used. Different variants of plane lattice structures are modeled: the classical lattice and the lattice with uniformly curved strips across the plane: elevator and star-shaped lattices. The relation between the structure of lattice structures and the propagation of perturbations in these structures is analyzed. The dependence of wave velocity in curved lattice structures on the frequency of the emitted waves is analyzed. The applied conceptual model of the device for measuring the frequency of waves is developed.

The development of additive technologies (3D printing) made it possible to manufacture
parts and products of a regular porous and cellular structure (in order to reduce the weight of
the structure). In this case, the characteristic cell size is much smaller than the scale of the whole
product. Numerical strength and related calculations of such structures require a preliminary
estimation of the effective properties of such a cellular structure. In this article, a method for the
numerical estimation of the effective elastic properties of regular cellular structures is presented,
which is based on the numerical solution of boundary value problems of the theory of elasticity
on a periodicity cell. Periodic boundary conditions in the form of restraints on the displacements
of opposite edges of the cell are successively applied to the cell. The boundary value problem of
the theory of elasticity is solved for each type of boundary conditions, and the resulting stress
field is averaged over the volume. The effective properties of the cellular material are estimated
as a generalized Hooke’s law.
Composite materials based on a rigid lattice skeleton filled with softer material are considered
in the paper. The calculations are carried out using the finite element method with the domestic
Fidesys CAE system. Beam finite elements are used in some calculations for the modeling of a
lattice skeleton. In some other calculations, a thin layer of a binder between the skeleton and
the matrix is taken into account. This layer is modeled using shell finite elements.
Graphs of comparing the results of calculations of composite materials with a lattice skeleton
modeled by beam elements and the results of similar calculations in which the skeleton is
modeled by three-dimensional finite elements are given in the article. In addition, graphs of
comparing the results of calculations in which the binder layer is modeled by shell elements and
the results of similar calculations in which the binder is modeled by three-dimensional elements
are given. The graphs show that with sufficiently thin framework elements (or with a sufficiently
thin layer of the binder), the results are quite close. It confirms the applicability of beam and
shell elements for the numerical solution of such problems.

Irreversible changes in the volume of material, called dilatancy, occur in many technological processes. It manifests itself in powder and porous metals, soils and rocks, concretes, metal alloys of different chemical composition and other materials. In addition, under the conditions of plastic deformation, an irreversible change in the volume of the deformable material occurs - its plastic dilatancy, which is the main physical mechanism of damage to various metal systems (metals, steels, non-ferrous alloys) with their large plastic deformations. In this regard, it is necessary to take into account the irreversible changes in the volume of the material in the calculations of many technological processes, for example, pressing powder metal materials, pressure treatment and cutting of porous metals and metal alloys. In compiling the basic mathematical relationships, a theoretical description of the change in volume uses various mathematical models of plastic dilatancy: discrete models, continual models, including stochastic models, which describe the behavior of dilating materials, both subject to preferential compaction and loosening. For the construction of the conditions of fluidity used in the calculation of dilating media, it is necessary to define material mathematical functions for specific processes and materials. The paper discusses the main conditions of fluidity and methods of their construction, which are used in the calculations of the processes of plastic processing of powder and ingot metallic materials in various conditions and states.

The biggest, but not the only, European mystification about the ancient origin of Chinese astronomy, occurred in the 17th century. According to Chinese sources, astronomy dates back to the middle of the third millennium B.C. The reliability of the sources is debatable. In 213–212 B.C., the emperor Qin Shi Huang ordered the destruction of all manuscripts except palace chronicles and practical books. At the turn of the 2 and 1 c. B.C. the historian Sima Qian wrote the "Records of the Grand Historian". This monumental work outlines the ancient legends and chronicles from the 8 century B.C. For a long time, China was closed to Europeans, the first popular European book about China was written by Marco Polo (13 c.). In the 17-18 centuries, in China the Jesuit mission resided. The mission included 36 astronomers who, with their reports, created among the Europeans an idea of the antiquity of Chinese astronomy. The subsequent discussion questioned this idea. From the 19th century, the Petersburg Academy of Sciences was represented in Pekin. Chinese studies (Sinology) arose due to the work of various missions, and many profound researchers emerged from the missionaries, the remarkable Russian Sinologists were among them. The works of the 19 and 20 centuries continue the debate about the authenticity of ancient sources. We consider the history of this discussion and the main arguments of the parties, including the works of P.-S. Laplace, K. A. Skachkov, G. N. Popov, A. V. Marakuev, N. J. Needham, N. Sivin, E. I. Beryozkina, Cheng-Yih Chen, V. Ye. Eremeyev.