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In this paper, we consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. We present the characterization of some known field extensions in terms of polynomial composites. This paper contains the open problem of characterization of ideals in polynomial composites with respect to various field extensions. We also present the full possible characterization of certain field extensions. Moreover, in this paper we show that any finite group is a Galois group of some field extensions and present the inverse Galois problem solved.

This paper is an introduction to a sequence of nonlinear Baskakov-Durrmeyer operators $(NBD_{n})$ of the form \[ (NBD_{n})(f;x) =\int_{0}^\infty K_{n}(x,t,f(t))\,dt \] with $x\in [0,\infty)$ and $n\in\mathbb{N}$. While $K_{n}(x,t,u)$ provide convenient assumptions, these operators work on bounded functions, which are defined on all finite subintervals of $[0,\infty)$. This paper comprise some pointwise convergence results for these operators in certain functional spaces. As well as this study can be seen as a continuation of studies about nonlinear operators, it is the first study on nonlinear Baskakov-Durrmeyer or modified Baskakov operators, while there were more papers on linear part of the operators.

In this paper, we study the qualitative behavior of the rational recursive equation \begin{equation*} x_{n+1}=\frac{x_{n-4}}{\pm1\pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, \quad n \in \mathbb{N}_{0}:=\{0\}\cup\mathbb N, \end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.

In the paper, we investigate an asymptotic behavior of the sharp upper bounds in the integral metric of deviations of the Abel-Poisson integrals from functions from the class $L^{\psi}_{\beta, 1}$. The Abel-Poisson integrals are solutions of the partial differential equations of elliptic type with corresponding boundary conditions, and they play an important role in applied problems. The approximative properties of the Abel-Poisson integrals on different classes of differentiable functions were studied in a number of papers. Nevertheless, a problem on the respective approximation on the classes $L^{\psi}_{\beta,1}$ in the metric of the space $L$ remained unsolved. We managed to obtain the estimates for the values of approximation of $(\psi, \beta)$-differentiable functions from the unit ball of the space $L$ by the Abel-Poisson integrals. In some cases, we also write down asymptotic equalities for these quantities, that is we solve the Kolmogorov-Nikol'skii problem for the the Abel-Poisson integrals on the classes $L^{\psi}_{\beta,1}$ in the integral metric.

The Editor-in-Chief has retracted this article. This paper was submitted to the journal Physics and Chemistry of Solid State on October 13, 2021. On November 17, 2021, the article was checked by the Unicheck system and was recommended for work with reviewers. But, the identical paper under the same title "Friction and wear of polystyrene crystal sliding against steel 100Cr6 in unlubricated contact" by the same authors was accepted in journal Tribology in Industry (http://www.tribology.rs) on September 29, 2021, and the signed Copyright Agreement. However, the authors did not report that they had previously signed a Copyright Agreement with the journal Tribology in Industry. One of the conditions of submission of a paper for publication is that authors declare explicitly that their work is original and has not appeared in a publication elsewhere. Re-use of any data should be appropriately cited. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and apologies are offered to readers of the journal that this was not detected during the submission process.

A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.

The article analyzes and systemizes current studies of leading world scientists on maritime economics and seaborne trade with the aim to reveal current trends and venues for future researches in this field. Special attention is paid to researches evaluating how the outbreak of coronavirus pandemic impacted shipping industry as a main global supplier of goods. All studies under review are conceptually grouped into two main branches. The first branch comprises papers focused on the world seaborne trade data dynamics, including official maritime reports. As opposed to Ukrainian and Russian maritime economics papers which predominantly describe and portray the statistical data available in official maritime reports issued by international organizations and shipping services providers, leading world scholars use this statistics as a baseline for individualized researches, mainly focused on investigation of correlation between various shipping indicators and prediction of same. The second branch comprises papers investigating trade of certain types of cargo, such as containers, crude oil, dry bulk. Several general peculiarities of both branches of researches are defined. Almost all of them attempt to provide an insight into the nature of a freight rate and to forecast the development of either general freight market or specific cargo related one. The utilized methodology is also identical. Depending on the aim of research and data availability, scholars employ various models of regression analysis, a standard tool of statistical modeling, which estimates the average relationship between two or more variables. No matter which freight market is under investigation, studies usually try to examine the connection of this market with others by evaluating the spillover effects between vessel types and vessel sizes. Distinguishing features of researches lie in the target stakeholders who could benefit from, either the industry in general or particular groups of market participants. In addition, nowcasting trade data is a real problem raised by the industry to modern science, which tries to tackle it by proposing innovative digitalized solutions.

This paper covers calculations of the activation energy of surface diffusion of ad-atoms on the substrate surface from the point of view of thermal oscillations of substrate atoms and ad-atoms. The main characteristic of oscillations of atoms and geometric mean frequency was calculated based on statistical approximation of the Debye model using the reference values of entropy and heat capacity of metals. The basic principle of the model of activation energy calculation presented in the paper is the formation of potential wells and barriers during oscillations of atoms localized in the sites of the lattice. Oscillations of atoms were considered in the framework of quasiclassical quantum approximation as the oscillations of harmonic oscillators in the potential parabolic wells. Dimensions of the negative part of values of the potential well energy were determined by the amplitude of thermal oscillations of atoms. Positive values constituted a significant part of the potential well energy values. Barriers were formed owing to interaction of positive values of the energy of parabolic wells of adjacent atoms. Therefore, in order to make the ad-atom jump, it is necessary to get out of the potential well having the negative values, and to overcome the potential barrier. The energy required for the ad-atom jump on the substrate surface was the activation energy of surface diffusion. The results obtained in this paper agree satisfactorily with the results of another method, which is based on determining the energy of ad-atom binding with the substrate atoms.

The paper calculates the electronic and lattice components of thermal conductivity coefficients for GeBiTe solid solutions. The calculation was carried out using two different models of the band structure of GeTe, which differ in the relative location of the zones of heavy and light holes. The first of the models is generally accepted for A4B6 compounds and assumes the location of the zone of light holes above the zone of heavy ones in the energy spectrum. Another model, obtained on the basis of DFT calculation, predicts the location of the zone of light holes below the zone of heavy ones. A significant difference was established in the numerical values of the electronic and lattice components of the thermal conductivity coefficients, depending on the adopted model. The influence of other calculation parameters on the investigated values was analyzed.

In this paper, we introduce the concept of orthogonal $m$-metric spaces and by using $F_m$-contraction in orthogonal $m$-metric spaces, we give the concept of orthogonal $F_m$-contraction (briefly, $\bot_{F_m}$-contraction) and investigate fixed point results for such mappings. Many existing results in the literature appear to be special case of results proved in this paper. An example to support our main results is also mentioned.

In this paper, based on generalized Herz-type function spaces $\dot{K}_{q}^{p}(\theta)$ were introduced by Y. Komori and K. Matsuoka in 2009, we define Herz-type Besov spaces $\dot{K}_{q}^{p}B_{\beta }^{s}(\theta)$ and Herz-type Triebel-Lizorkin spaces $\dot{K}_{q}^{p}F_{\beta }^{s}(\theta)$, which cover the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, where $\theta=\left\{\theta(k)\right\} _{k\in\mathbb{Z}}$ is a sequence of non-negative numbers $\theta(k)$ such that \begin{equation*} C^{-1}2^{\delta (k-j)}\leq \frac{\theta(k)}{\theta(j)} \leq C2^{\alpha (k-j)},\quad k>j, \end{equation*} for some $C\geq 1$ ($\alpha$ and $\delta $ are numbers in $\mathbb{R}$). Further, under the condition mentioned above on ${\theta }$, we prove that $\dot{K}_{q}^{p}\left({\theta }\right)$ and $\dot{K}_{q}^{p}B_{\beta }^{s}\left({\theta }\right)$ are localizable in the $\ell _{q}$-norm for $p=q$, and $\dot{K}_{q}^{p}F_{\beta }^{s}\left({\theta }\right)$ is localizable in the $\ell _{q}$-norm, i.e. there exists $\varphi \in \mathcal{D}({\mathbb{R}}^{n})$ satisfying $\sum_{k\in \mathbb{Z}^{n}}\varphi \left( x-k\right) =1$, for any $x\in \mathbb{R}^{n}$, such that \begin{equation*} \left\Vert f|E\right\Vert \approx \Big(\underset{k\in \mathbb{Z}^{n}}{\sum }\left\Vert \varphi (\cdot-k)\cdot f|E\right\Vert ^{q}\Big)^{1/q}. \end{equation*} Results presented in this paper improve and generalize some known corresponding results in some function spaces.

The purpose of this paper is to obtain some sufficient conditions to determine the relation between a meromorphic function and an $L$-function when certain differential polynomial generated by them sharing a one degree polynomial. The main theorem of the paper extends and improves all the results due to W.J. Hao, J.F. Chen [Discrete Dyn. Nat. Soc. 2018, 2018, article ID 4673165], F. Liu, X.M. Li, H.X. Yi [Proc. Japan Acad. Ser. A Math. Sci. 2017, 93 (5), 41-46], P. Sahoo, S. Halder [Tbilisi Math. J. 2018, 11 (4), 67-78].

The paper presents experimental values of lateral and extension transmission ultrasonic waves speed and their absorbance coefficient obtained with the help of discrete immersed method in metal filled polymer auxetic samples with polyurethane matrix. Poisson’s ratio, real and imaginary parts of complex dynamic elastic modulus (Young’s modulus, shear modulus, bulk modulus) and mechanical losses tangent of angle for extension, lateral and bulk deformation have been calculated regarding paper’s results. Viscoelastic properties of polyurethane auxetic have been examined regarding different theoretical approaches. To support negative values of Poisson’s ratio, a filled polymer model with critical filler amount and auxetic deformation models have been used. Models structural characteristics for filled polyurethane auxetics have been defined. Systems modeling with solid bulky inclusions, that are able to absorb and to disseminate ultrasonic waves, enables to consider suchlike systems as complex density materials. It is shown that in certain ratio of wave parameters and environment structural and mechanical properties, resonance effects take place, regarding filler’s particles vibration as inclusions in general.

The paper shows that the adaptation of energy methods to automated calculation of mobile machines frame constructions consists of developing a single algorithm applicable to different construction schemes. The calculation outset still remains the idea of getting a function of potential energy of deformation as a function with unknown inner power factors. Search for local function minimum of potential power of deformation has been based on the function’s discrete grid-surface. We managed to reach tactical flexibility of coordinate descent method in an attempt to continue approaching local minimum in cases of a dead end situation by changing the discrete course. The paper suggests extending the implemented algorithm from 3-D surface dealing only with two power factors, to n-D one with many unknown values.

The paper investigates the process of liquid-phase sintering of amorphous iron-based nanoparticles by the method of molecular dynamics simulations. The classical molecular dynamics package LAMMPS was used for modeling. Visual analysis of the atomic configurations of nanoparticles during their rapid cooling revealed the self-purification effect of the particles. Partial pair correlation functions and coordination number distribution functions were used to analyze the atomic structure of nanoparticles after sintering. As a result of the analysis of the main structural parameters, which were obtained using the specified functions, differences in the atomic composition and structure of the volume and surface of nanoparticles were established.

Solar cells rely on photogeneration of charge carriers in p-n junctions and their transport and subsequent recombination in the quasineutral regions. Several basic issues concerning the physics of the operation of solar cells remain obscure. This paper discusses some of those unsolved basic problems. In conventional solar cells, recombination of photogenerated charge carriers plays a major limiting role in the cell efficiency. High quality thin-film solar cells may overcome this limit if the minority diffusion lengths become large as compared to the cell dimensions, but, strikingly, the conventional model fails to describe the cell electric behavior under these conditions. A new formulation of the basic equations describing charge carrier transport in the cell along with a set of boundary conditions is presented. An analytical closed-form solution is obtained under the linear approximation. It is shown that the calculation of the open-circuit voltage of the solar cell diode does not lead to unphysical results in the new given framework.

Proposed paper presents the latest results in the framework of polarization-sensitive low-coherence interferometry related to new approaches for using the geometric phase to reproduce the polarization structure of a biological transparent anisotropic micro (nano) object. The polarization parameters of an anisotropic object are measured in real time on the basis of a modified Mach-Zehnder interferometer. The advantage of using the geometric phase is the diagnostic of polarization anisotropic surface (subsurface) nanosized layers in a non-contact, non-invasive manner.

The study was performed on alloys with a carbon content of 0,37-0,57 % (wt.), silicon 0,23-0,29 % (wt.), manganese 0,7-0,86 % (wt.), the rest– iron. To determine the phase composition of alloys used microstructural, microanalysis and X-ray analysis. In addition, the physical characteristics of the alloys studied in this paper were determined, such as alloy chemical dependence of extension and contraction ratio, impact toughness and hardness. The results obtained in this paper showed that the iron-based alloy with the content of carbon of 0.57 % (wt.), silicon of 0.28 % (wt.) and manganese of 0.86 % (wt.)) had the superior microstructure and physical properties. It was determined that after a number of crystallization and phase transformation the alloy phase structure includes two phases: a-iron and cement magnesium doping Fe2.7Mn0,3C.. For the first time using the method quasichemistry received an expression of the free energy of a γ-iron alloyed with silicon and magnesium, and determined the solubility limit of carbon, manganese and silicon. The maximum content in γ-iron can reach: carbon 6,8 % (at.), manganese – 67,5 % (at.), silicon – 2,3 % (at.).

The starting point of the present paper is a result by E.A. Boltarovych (1989) on convergence regions, dealing with branched continued fraction \[\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{1}{\atop+}\ldots{\atop+}\sum_{i_n=1}^N\frac{a_{i(n)}}{1}{\atop+}\ldots,\] where $|a_{i(2n-1)}|\le\alpha/N,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and $|a_{i(2n)}|\le r/(N-1),$ $i_{2n}\ne j_{2n},$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ where $N>1$ and $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions. In the present paper conditions for these regions are replaced by $\sum_{i_1=1}^N|a_{i(1)}|\le\alpha(1-\varepsilon),$ $\sum_{i_{2n+1}=1}^N|a_{i(2n+1)}|\le\alpha(1-\varepsilon),$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R$ and $\sum_{i_{2n}\in\{1,2,\ldots,N\}\backslash\{j_{2n}\}}|a_{i(2n)}|\le r,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ where $\varepsilon,$ $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions, and better convergence speed estimates are obtained.

In this paper, the short algorithmic formulas for computations of the crystals thermal and kinetic properties are given. These formulas are fulfilled the case of isotropic crystals for any one of dispersion laws of current carriers, where these carriers can be scattered by all kinds of the crystal lattice defects. In the paper, these algorithmic formulas were used to calculations the important properties of crystals with the nonparabolic Kane’s dispersion law of current carriers. Here, the passing from these non-parabolic dispersion law crystals to the parabolic dispersion law crystals was also described.

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

In this paper, we show that harmonic convex functions $f$ is strongly $(p, h)$-harmonic convex functions if and only if it can be decomposed as $g(x) = f(x) - c (\frac{1}{x^p})^2,$ where $g(x)$ is $(p, h)$-harmonic convex function. We obtain some new estimates class of strongly $(p, h)$-harmonic convex functions involving hypergeometric and beta functions. As applications of our results, several important special cases are discussed. We also introduce a new class of harmonic convex functions, which is called strongly $(p, h)$-harmonic $\log$-convex functions. Some new Hermite-Hadamard type inequalities for strongly $(p, h)$-harmonic $log$-convex functions are obtained. These results can be viewed as important refinement and significant improvements of the new and previous known results. The ideas and techniques of this paper may stimulate further research.

In the paper, a theoretical calculation of the coefficient of thermal conductivity of solid solutions of PbSnTe was carried out. The contribution of phonon scattering on substitution atoms to the effect of reducing thermal conductivity has been established. The composition of the PbSnTe solid solution, characterized by the lowest values of the lattice component of the thermal conductivity coefficient klat, was determined. The concentration of intrinsic charge carriers in solid solutions is calculated and their influence on the thermoelectric parameters of the material is shown.

The paper proposes a model of an electromagnetic radiation sensor that uses the precession of the magnetization vector in a ferromagnet (ferromagnetic resonance) as a result of absorbing the energy of an incident electromagnetic wave, the generation of a spin current as a result of this precession, the generation of a spin-polarized current as a result of the passage of a spin current in a non-magnetic metal, and a change in the direction of magnetization of a ferromagnetic layer with a low coercive force (free layer) due to the passage of a spin-polarized current. Then the radiation will be detected by its effect on the electrical resistance of the entire structure, which depends on the mutual directions (parallel or antiparallel) of magnetization of the free and fixed (with a large coercive force) ferromagnetic layers (phenomenon of giant magnetic resistance). The dependence of the spin-polarized current in the device on the frequency and amplitude of the incident electromagnetic wave with linear polarization was calculated. A method of calculating the range of amplitude and frequency values of radiation that can be detected by the sensor has been developed. The parameters of this model are the detection time and the number of spin gates in one sensor. Calculations are given for a ferromagnetic layer made of permalloy and for spin valves with four different critical current values that determine the process of remagnetization of the free layer: 20, 50, 100, and 200 microamps.

This paper's goal is to investigate how petroleum's n-alkane isomerization reaction occurs on the surface of a solid nanocatalyst. In this experiment, the isomerization of aromatic-rich crude oil is being studied as a result of radiation exposure. It was discovered that sodium-rich bentonite –solid acid, from the Alpoid deposit works well as a catalyst for refining petroleum. Possible mechanisms of the radio-catalytic reaction that converts n-alkanes to branched hydrocarbons have been researched by methods of FT- IR spectroscopy and gas chromatography. The effect of acidity type of the nanostructured bentonite clay on Lewis acid cites is proposed to explain observed results. Additional radiation energy, which is mainly generated due to the nanobentonite volume, results in a more efficient aromatic- rich petroleum radiolysis process. Also, it was investigated the dynamics of dose –dependent changes in the amount of branched hydrocarbons. One of the primary objectives of this research work is increasing the efficiency of transformation of adsorbed beam energy in solid phase to the system and improvement of hydrocarbon isomerization reactions in crude oil. The results of hydrocarbon transformations are discussed in terms of the behavior of excited electrons in solids. Novel eco-friendly nanocatalyst creates new opportunities for hydrocarbons with multi- branched structures.

Thermal properties characterizations for carbon nanotubes (CNTs), carboxyl group-functionalized carbon nanotubes (FCNT), and graphite were presented in the paper. TGA/DSC and TEM techniques were used for characterization. The features of TGA characteristics transformation for synthesized carbon nanomaterials were investigated. The specific heat capacities of the samples at a constant pressure increased as the temperature increased.

The paper presents the results of computer modeling of the proposed functional-electrical circuit of integrated signal converters (ISC) from photosensitive elements based on CMOS operational amplifiers, which is intended for the construction of an element base of hybrid sensor microsystems for biomedical applications. A feature of this ISC is the regulation and filtering of the amplitude of the constant component in the amplified signal from the diode photosensitive element in the wave range of 400 - 1040 nm. Computer simulation of the functioning of the device was carried out, the constituent components were determined and their parametric optimization was carried out. The results of experimental studies and computer modeling agree well, which confirms the correct functioning of the proposed signal converter from photosensitive elements. The developed ISC is suitable for creating real devices, both on the basis of discrete components and in an integrated design, as an element of sensor microsystems-on-a-chip or intelligent sensors.

This paper deals with a study on classes of non linear operators. Let $SL(X,Y)$ be the set of all sublinear operators between two Riesz spaces $X$ and $Y$. It is a convex cone of the space $H(X,Y)$ of all positively homogeneous operators. In this paper we study some spaces generated by this cone, therefore we study several properties, which are well known in the theory of Riesz spaces, like order continuity, order boundedness etc. Finally, we try to generalise the concept of adjoint operator. First, by using the analytic form of Hahn-Banach theorem, we adapt the notion of adjoint operator to the category of positively homogeneous operators. Then we apply it to the class of operators generated by the sublinear operators.

Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based on a chaotic representation property (CRP): roughly speaking, any square integrable with respect to the Gaussian measure random variable can be decomposed in a series of Ito's stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP (except the Gaussian and Poissonian particular cases). Nevertheless, there are different generalizations of this property. Using these generalizations, one can construct different spaces of test and generalized functions. And in any case it is necessary to introduce a natural product on spaces of generalized functions, and to study related topics. This product is called a Wick product, as in the Gaussian analysis. The construction of the Wick product in the Levy analysis depends, in particular, on the selected generalization of the CRP. In this paper we deal with Lytvynov's generalization of the CRP and with the corresponding spaces of regular generalized functions. The goal of the paper is to introduce and to study the Wick product on these spaces, and to consider some related topics (Wick versions of holomorphic functions, interconnection of the Wick calculus with operators of stochastic differentiation). Main results of the paper consist in study of properties of the Wick product and of the Wick versions of holomorphic functions. In particular, we proved that an operator of stochastic differentiation is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication.

Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Levy white noise measure, the corresponding theory is called the Levy white noise analysis. In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itos stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP, but there are different generalizations of this property. In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Levy analysis, which is proposed by E.W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces. Our research is a contribution in a further development of the Levy white noise analysis.

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants to the same limit.

In this paper, the important thermal characteristics of matter (they describe thermodynamic systems in a state of thermodynamic equilibrium) were calculated. There are the following important thermodynamic functions: the system internal energy , the thermal function (or enthalpy) the free Helmholtz energy, the thermo-dynamic potential (or Gibbs free energy) , the Gibbs grand thermodynamic potential , the entropy , the specific heat capacity . These functions are explicit functions of system’s parameters, they fulfil some mathe-matical relationships and posses some total differentials. These functions are calculated in this paper and their physical sense is given in the cited works.Keywords: Gibbs potential, chemical potential, system’s heat capacity, equation of neutrality.

The main purpose of this paper is to study some geometric and topological properties of $c_0$-sum of the finite dimensional Banach lattice $\ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattice $c_0(\ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\infty}(\ell_2^n)$ is an almost limited set.

In this paper, a new accelerated extrapolation Krasnoselski-Mann type algorithm for finding common element in the solution set of the hierarchical fixed point and split monotone variational inclusion problems are introduced in the setting of a real Hilbert space. We then prove that a sequence generated by the algorithm converges strongly to such common element which also approximates solution of some fixed point problem of demimetric mapping in the space. Finally, some applications and numerical experiment are given to show effectiveness of the proposed algorithm over the recently known related results in the literature. The established results extend and generalize many recent ones announced in the literature.

The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{\omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.

The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. The main aim of this paper is to give the crossing numbers of the join products of eight graphs on six vertices with paths and cycles on $n$ vertices. The proofs are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs.

Dimonoids, generalized digroups and doppelsemigroups are algebras defined on a set with two binary associative operations. The notion of a dimonoid was introduced by J.-L. Loday during constructing the universal enveloping algebra for a Leibniz algebra. One of the important motivations for studying doppelsemigroups comes from their connections to interassociative semigroups. Generalized digroups are dimonoids with some additional conditions while commutative dimonoids provide the class of examples of doppelsemigroups. Let $V$ be a variety of universal algebras. One of the main problems is to describe free objects in $V$. The purpose of this paper is to construct new more convenient free objects in some varieties of dimonoids, generalized digroups and doppelsemigroups. We first construct a new class of abelian dimonoids, give a new model of the free abelian dimonoid of rank 2 and extend it to the case of an arbitrary rank. Then we show that the semigroups of the free generalized digroup are anti-isomorphic, present a new model of the free monogenic generalized digroup and characterize the least group congruence on it. We also prove that there do not exist commutative generalized digroups with different operations. Finally, we construct a new model of the free monogenic commutative doppelsemigroup, characterize the least semigroup congruence on it and establish that every monogenic abelian doppelsemigroup is the homomorphic image of the free monogenic commutative doppelsemigroup.

In this paper, we present some new integral inequalities of Hermite-Hadamard type. To obtain these results, general convex functions of various type are considered such as $(h,m)$-convex functions. The main results extend some previously known inequalities by taking fractional integral operators.

This paper deals with construction and studying wavelet type Bernstein operators by using the compactly supported Daubechies wavelets of the given function $f$. The basis used in this construction is the wavelet expansion of the function $f$ instead of its rational sampling values $f\big( \frac{k}{n}\big)$. After that, we investigate some properties of these operators in some function spaces.

This paper describes the physical model of the penetrating thermoelement with the developed surface of warmth exchange for the cooling of the air’s flow. It presents the theory of the calculation and the method of the computer modeling of the temperature’s and potential’s allocation, the definition of the effectiveness of the energy’s transformation – the cooling factor and the cooling efficiency. This work researches the 3D temperature’s and materials’ allocation for the material of the branches of the thermoelement based on Bi2Te3.

The paper provides a brief review of literature data on the synthesis, processing, structure, mechanical properties, and application of nitinol and alloys based on it, which are promising functional materials found application in a number of high technologies. In addition, machine learning methods were applied to predict the temperatures of phase transformations.

This work attempts to replace the sandwich core's traditional shape and material with a cellular pattern, where the cells have a regular shape, distribution, and size. The contribution of this paper is to design two structures, one open-celled and the other closed, and to evaluate the performance of sandwich plates with lattice cell core as it is used for many industrial applications, particularly in automobile engineering. The new theoretical formulations are constructed for two structures to find the free vibration characteristics. The results of the new design are compared with the traditional shape. Derivation of equations to predict mechanical properties based on relative density with the chosen shapes, specific vibration equation of three-layer sandwich plate, and substitution by equation using excel sheet. Results are promising, and the effectiveness of cellular pattern theoretical analysis estimation. Limitations and error rates for the mechanical properties come through the empirical equations, and their ratio to the relative density values are higher depending on the behavior of the core material. Findings reveal, with open cell decrease in modulus of elasticity by (PLA: -90.4%) and (TPU: -90.4%), increases natural frequency by (PLA: 44.5%) and (TPU: 46.4%), as for closed-cell decreases in the modulus of elasticity by (PLA: -66.9%) and (TPU: -64.4%), increases natural frequency by (PLA: 36%) and (TPU: 37.7%). Converting a solid substance or replacing a foam form with a cellular pattern is one way to better performance and save weight through the selected cell pattern in absorbing the energy of the vibration wave.

The paper deals with the study of the processes of obtaining polymer composites by chemo-mechanical activation technology (CMA-technology). The methods of research and testing of composite materials based on polytetrafluoroethylene (PTFE) and carbon fibers are analyzed and generalized. The influence of CMA technology on the structure and properties of PTFE composite was determined.

In this paper analyzes the electromagnetic radiation absorption by a composite based on bimetallic nanowires. Using the Drude-Lorentz theory it was obtained the relationship for the polarizability frequency dependences, as well as real and imaginary parts of the dielectric function of layered 1D-systems was obtained. It is shown that the structure in the form of a metal core covered with a layer of another metal leads to splitting and the appearance of two maxima in the frequency dependence of the absorption coefficient. The magnitude and position of the maxima are determined by the composition of bimetallic nanowires and the volume content of metals. The influence of the dimensionality of the systems has been established by comparing the frequency dependences of the composite absorption coefficient based on bimetallic nanowires and nanoparticles. Calculations were performed for Ag@Au and Au@Ag nanowires immersed in Teflon.

The paper considers the need to create an innovative health and rehabilitation center due to the long-term military aggression of russia. The latter leads to the fact that many military and civilians receive physical and psychological injuries that need to be treated. At the first stage of rehabilitation, these problems are solved by special medical facilities, while the restoration of both physical and psychological condition should be carried out outside hospitals. One of these facilities could be health and rehabilitation centers on a national scale. Such centers can be formed on the basis of existing sports complexes, where there is a basis for carrying out appropriate actions. The article uses the example of the Olimp sports complex located on the Black Sea coast to illustrate the prospects of this area. The relevance of the work is due to the needs of the present and the principles of sustainable development of society, which combines social (human factor), economic (economic factor) and environmental (health improvement through a clean environment) components. The purpose of the study is to substantiate the creation of an innovative health and rehabilitation center on the basis of the Olimp sports complex for military and civilians suffering from physical and psychological injuries. Research methods: analysis and evaluation of economic activities; methods of generalization for the study of activities; observation method for the study of economic activities of FSC Olimp and the peculiarities of its services. The results of the work include: research on the needs of creating a health and rehabilitation center; identification of the main forms and types of work of the health and rehabilitation center; analysis of the activity of the physical culture and sports complex "Olympus"; assessment of the importance of introducing health and rehabilitation services; development of ways of innovative health and rehabilitation activities. The scientific novelty lies in the development of innovative approaches for the creation of a health and rehabilitation center that can operate on the basis of the Olimp sports complex. Practical significance of the results. The system for creating an innovative health and rehabilitation center is proposed for a specific existing enterprise and can be implemented in the near future. The developed recommendations can be used for a similar type of enterprise.

The purpose of the paper is investigation of the dissipative structures that occur in solids on the instance of rock mass during its damaging and irreversible deforming. Ground movement monitoring of surrounding underground roadway rock mass and landslides has been employed as the method of investigation. The author of this paper registered the dissipative structures that dispersed the ground pressure energy, periodically changing their patterns during irreversible deformation. It was found for the first time that short interaction of the rock fragments and far cooperation of the clusters minimize the entropy production, and are the form in which the dissipative structures appear. Space and time parameters of these structures evolution have been investigated. In order to achieve the practical relevance of the experimental findings a new principle of constructions stability enhancement were developed. Both the translational and rotational degree of freedom should be restricted for moving ground in three dimensional state. Key words: solids, strength, irreversible deformation, dissipative structure, self-organization, far interaction, cluster.

In the paper the physical properties and thermodynamic functions of borides Х2В (Х = W, Mo, Mn, Fe, Co, Ni та Cr) are studied with accounting for fluctuation processes. We use the microstructure analysis, the X-ray structural and the durometric analyses to determine the physical properties of alloys. In the paper it is determined the phase composition and physical properties of borides. In this paper for the first time it is determined the thermodynamic functions of borides using the Hillert and Staffansson model with accounting for the first degree approximation of high-temperature expansion for the free energy potential of binary alloys. We obtain the temperature dependences for such thermodynamic functions as Gibbs free energy, entropy, enthalpy and heat capacity Ср along with their values at the formation temperature for Х2В (Х = W, Mo, Mn, Fe, Co, Ni та Cr). The approach under consideration enables to give more thorough from the thermodynamic point of view description of borides formed from the liquid. The outcomes of the thermodynamic function calculation for borides are in good agreement with experimental data and results of other authors. Key words: borides, Gibbs energy, entropy, enthalpy, heat capacity, fluctuation process

In this paper, using Gibbs grand thermodynamic potential, kinetic tensors of electrical and thermal conductivity generalized equations known in non-equilibrium thermodynamics have been proven. These tensors determine calculation algorithms of the material tensors of conductor crystals and various galvanomagnetic and thermomagnetic effects coefficients. These algorithms are pragmatic formulas in calculation problems of crystals kinetic properties and in the problems for prediction of semiconductor crystals with preset properties. Their pragmatism is proven by the huge number of scientific papers dedicated to kinetic properties of semiconductor crystals study. Keywords: gibbs potential, electrical conductivity, thermal conductivity, algorithm, tensor.

This paper aims to modify the $(p,q)$-Lupaş Bernstein operators using King's technique and to establish convergence results of these operators by using of modulus of continuity and Lipschitz class functions. Some approximation results for this new sequence of operators are obtained. It has been shown that the convergence rate of King type modification is better than the $(p,q)$-Lupaş Bernstein operators. King type modification of operators also provide better error estimation within some subinterval of $[0,1]$ in comparison to $(p,q)$-Lupaş Bernstein operators. In the last section, some graphs and tables provided for simulation purposes using MATLAB (R2015a).