Introduction. Today, the development of the graphene theory to control its physical and mechanical properties is a relevant objective. The paper deals with the conducting properties of graphene. In particular, the paper investigates the linear law of electron dispersion and traces its corollaries. Materials and Methods. The development of the theory is based on the verified experimental data and on the foundamental principles of the solid body theory and quantum mechanics. The study follows the universal synergetic principle according to which, there have been developed two split-level mathematical models of the quasi-particle motion in graphene on exposure to the electric field. On the macroscopic level, we suggest that graphene should be analyzed as a crystal consisting of three parallel planes. Two of them are electron gas. The remaining one is the main body of the crystal. On the microscopic level, the quasi-particle motion of the electron wave is described through the Schroedinger equation. Results. The study has developed the alternative method for the explanation of the linear dispersion law in graphene on the macroscopic level. Basing on the analysis of the model, the paper provides a hypothesis of the cooperative motion of the electron pairs, which make up a boson particle. The given hypothesis is different from the traditional one. In accordance with the latter, quasi-particles in graphene are Dirac fermions. To prove the hypothesis consilience, the study examines Hall’s effect in grapheme. The linear dispersion law for a pair of electrons is also deduced from the Schroedinger equation. Both the macroscopic and microscopic models are in a reasonable agreement with the experimental data. Discussion and Conclusion. The main result of the research is the development of the multi-level mathematical model which properly features the conducting properties of graphene (linear dispersion law, anomalous Hall effect). The practical relevance consists in revealing the possibility to control the conducting properties of graphene through impacts on electron pairs.

Introduction. The deals with the multi/interdisciplinary approach to designing the unmanned aerial vehicle (quadcopter) based on the use of the integrated model environment. The designing process is implemented as creating different types of models: natural (physics) and virtual. Materials and Methods. The virtual model is understood to be a set of mathematical, algorithmic, program and 3D models maintaining its functioning in virtual environment. The design decision represents a set of the design-technology documentation including the integrated model of the designed project, whose components are connected with each other. The natural (physics) part of the integrated model environment includes the following components: a carrier system, shell details, electromechanical and electronic systems for controlling navigation, telemetry and sensory. For the carrier systems and shells there used polyamide bionic parts, which are purchased and printed on the 3D printer; the basic element of electronic system is the flight controller ArduPilot (ArduCopter). The virtual environment of modelling is formed on the basis of CAD/CAE/CAM/PDM/PLM SolidWorks (Motion, Simulation). The main tools, used for creating the communications between models of different types and levels, are the COM technology, API CAD/CAE/CAM/PDM/PLM system, MS Visual Studio C++, which allow developing the integrated interface for controlling the flight and planning the trajectory in the real and virtual environment. Results. The integrated (natural and virtual) model environment for the quadcopter is developed. On this basis, the design decision in the form of a real object and its virtual model is made. The state and behaviour of these objects is controlled and guided by the software having access both to a real object and to its 3D model. The received result can be considered as the tool of engineering for the solution of a wide range of scientific, technical and production tasks: performing defectoscopy, diagnosing emergencies, and 3D-scanning remote and hard-to-reach objects. Discussion and Conclusion. The research shows the efficiency of the approach to designing as to process of creating the multi/interdisciplinary models of different types and levels. At the same time, the problem of integrating these models into a coherent whole by forming bidirectional associative communications has assumed particular importance. The technological (program) means for synchronizing a state and behaviour of the natural and virtual models of design objects require further development.

Introduction. The article presents a simplest differential game with four participants. The players move on a plane and can do simple movements. The game under considering comes down to a cooperative differential game. The dynamic stability of such optimality principles as the S-kernel and Shapley vector is shown. Materials and Methods. The standard procedures of the cooperative game theory are applied to the analysis and decision of a cooperative differential game. The conditional and optimum trajectories, along which the players move, are found using the Pontryagin’s maximum principle. When constructing the characteristic function, the minimax approach is used. Results. The optimum strategy of the players, conditional and optimum trajectories of their movements at various ways of formation of coalitions are written out explicitly. The characteristic function is constructed according to the accepted max-min principle; the S-kernel and Shapley vector are considered as a decision. The components of the Shapley vector are written out explicitly; the fact that the Shapley vector is an element of the S-kernel and nonemptiness of the S-kernel, when the players are moving along an optimum trajectory, are shown. Using the results of the static cooperative game theory for researching differential games, we face the problems, which are connected with specifics of the differential equations of the movement. As a priority, the problem of the dynamic stability of the optimality principles under consideration is identified. In the work, the dynamic stability of the Shapley vector and S-kernel is shown. Discussion and Conclusion. The results of the research show that the analysis of the dynamic stability of the optimality principles considered is relevant.

Introduction. Modern manufacturing technologies of powder materials allow obtaining particles sized 0.1–0.5 μm. Powders with such dispersion have a very high surface energy and, consequently, unique properties. Powders with a given size can be used as an additive to motor oil to form the surface film with high tribological properties. However, the widespread use of ultra-nanoscale powder materials as additives to motor oil limits sedimentation and aggregation of the particles. As a result, the dimensions of the metal crystals increase to dozens of microns and their particles are then retained by oil filters of engines that leads to efficiency decrease and even to clogging oil-conducting channels and to setting the friction surfaces of the parts. The most technically competent solution for increasing sedimentation resistance of the lubricant composition is the use of chemical stabilizers forming the thinnest film at the surface of metal particles, which not only hinders particle aggregation, but also keeps them suspended for a long time. Materials and Methods.As stabilizers of dispersed metal powders in lubricant compositions, organic compounds of various classes are used. The analysis of several stabilizers of disperse metal powders is given in the paper and new compositions are proposed: apinezon MN, polyethylene glycol sebacate and commodity stabilizer apiezon L. The methods of stabilizers synthesis and examination of their activity are described. Results. The stabilization activity of the developed preparations was compared to the basic version: oleic acid. Addition of the ester of ethylene glycol sebacate had no stabilizing effect on the lubricant composition. The synthesized MN and Mg reagent apeizon L show stabilizing properties with respect to metallic powders of the lubricant composition, comparable with the properties of oleic acid. Discussion and Conclusion.Since apiezon greases are a mixture of hydrocarbons, their corrosivity is much lower than that of oleic acid, they can be recommended for practical use.